Nicole Berline Ezra Getzler
Michele Vergne
Heat Kernels
and Dirac Operators
Springer-Verlag
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Rndanest

Nicole Berline
Centre de Mathematique
Ecole Polytechnique
F-91128 Palaiseau Cedex, France
Ezra Getzler
MIT 2-267
Cambridge, MA 02139, USA
Michele Vergne
DMI, Ecole Normale Sup6rieure
45, rue d'Ulm
F-75005 Paris, France
Contents
Mathematics Subject Classification A991):
58G10, 58G11,58A10,53C05
ISBN 3-540-53340-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-53340-0 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. AH rights are reserved, whether the whole or part of the
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under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992
Printed in Germany
41/3140-543210 Printed on acid-free paper
Introduction 1
Acknowledgements 13
Chapter 1. Background on Differential Geometry 14
1.1. Fibre Bundles and Connections 14
1.2. Riemannian Manifolds 32
1.3. Superspaces 38
1.4. Superconnections 43
1.5. Characteristic Classes 46
1.6. The Euler and Thorn Classes 51
Bibliographic Notes 61
Chapter 2. Asymptotic Expansion of the Heat Kernel 63
2.1. Differential Operators 64
2.2. The Heat Kernel on Euclidean Space 72
2.3. Heat Kernels 74
2.4. Construction of the Heat Kernel 77
2.5. The Formal Solution 81
2.6. The Trace of the Heat Kernel 87
2.7. Heat Kernels Depending on a Parameter 98
Bibliographic Notes 101
Chapter 3. Clifford Modules and Dirac Operators 102
3.1. The Clifford Algebra 103
3.2. Spinors 109
3.3. Dirac Operators 113
3.4. Index of Dirac Operators 122
3.5. The Lichnerowicz Formula 126
3.6. Some Examples of Clifford Modules 127
The DeRham Operator 128
The Signature Operator 131
The Dirac Operator on a Spin-Manifold 134
The 9-Operator on a Kahler Manifold 135
Bibliographic Notes 141

VI
Contents
Chapter 4. Index Density of Dirac Operators 143
4.1. The Local Index Theorem 143
The Gauss-Bonnet-Chern Theorem — 148
The Hirzebruch Signature Theorem 150
The Index Theorem for the Dirac Operator 151
The Riemann-Roch-Hirzebruch Theorem 151
4.2. Mehler's Formula 153
4.3. Calculation of the Index Density 157
Bibliographic Notes 165
Chapter 5. The Exponential Map and the Index Density 167
5.1. Jacobian of the Exponential Map on Principal Bundles 168
5.2. The Heat Kernel of a Principal Bundle 172
5.3. Calculus with Grassmann and Clifford Variables 178
5.4. The Index of Dirac Operators 181
Bibliographic Notes 184
Chapter 6. The Equivariant Index Theorem 185
6.1. The Equivariant Index of Dirac Operators 186
6.2. The Atiyah-Bott Fixed Point Formula 187
6.3. Asymptotic Expansion of the Equivariant Heat Kernel 191
6.4. The Local Equivariant Index Theorem 194
6.5. Geodesic Distance on a Principal Bundle 198
6.6. The heat kernel of an equivariant vector bundle 200
6.7. Proof of Proposition 6.13 .204
Bibliographic Notes 206
Chapter 7. Equivariant Differential Forms 207
7.1. Equivariant Characteristic Classes 208
7.2. The Localization Formula ...:..... 215
7.3. Bott's Formulas for Characteristic Numbers 224
7.4. Exact Stationary Phase Approximation 226
7.5. The Fourier Transform of Coadjoint Orbits '. 228
7.6. Equivariant Cohomology and Families 234
7.7. The Bott Class 241
Bibliographic Notes , 246
Chapter 8. The Kirillov Formula for the Equivariant Index 248
8.1. The Kirillov Formula 249
8.2. The Weyl and Kirillov Character Formulas 253
8.3. The Heat Kernel Proof of the Kirillov Formula 258
Bibliographic Notes 267
Contents
VH
Chapter 9. The Index Bundle 268
9.1. The Index Bundle in Finite Dimensions 270
9.2. The Index Bundle of a Family of Dirac Operators 278
9.3. The Chern Character of the Index Bundle 282
9.4. The Equivariant Index and the Index Bundle 293
9.5. The Case of Varying Dimension 295
9.6. The Zeta-Function of a Laplacian 300
9.7. The Determinant Line Bundle 304
Appendix. More on Heat Kernels 310
Bibliographic Notes 315
Chapter 10. The Family Index Theorem 317
10.1. Riemannian Fibre Bundles 320
10.2. Clifford Modules on Fibre Bundles 325
10.3. The Bismut Superconnection 332
10.4. The Family Index Density 336
10.5. The Transgression Formula 345
10.6. The Curvature of the Determinant Line Bundle 348
10.7. The Kirillov Formula and Bismut's Index Theorem 351
Bibliographic Notes 355
Bibliography 356
List of Notations 361
Index 365

Introduction
. t>
ft)
Dirac operators on Riemannian manifolds, which were introduced in the ar-
articles of Atiyah and Singer [13] and Lichnerowicz [73], are of fundamental
importance in differential geometry: they occur in situations such as Hodge
theory, gauge theory, and geometric quantization, to name just a few ex-
examples. Most first-order linear differential operators of geometric origin are
Dirac operators. i
After Atiyah and Singer's fundamental work on the index for general el-
elliptic operators, methods based on the heat kernel were applied to prove the
Atiyah-Singer Index Theorem in the special case of Oirac operators, by Pa-
todi [82], Gilkey [58] and Atiyah-Bott-Patodi [8]. In recent years, new insights
into the local index theorem of Patodi and Gilkey have emerged, which have
simplified the proofs of their results, and permitted the extension of the lo-
local index theorem to other situations. Thus, we felt it worthwhile to write
a book in which the Atiyah-Singer Index Theorem for Dirac operators on
compact Riemannian manifolds and its more recent generalizations would
receive elementary proofs. Many of the theorems which we discuss are due
to J.M. Bismut, although we have replaced his use of probability theory by
classical asymptotic expansion methods.
Our book is based on a simple principle, which we learned from D. Quillen:
Dirac operators are a quantization of the theory of connections, and the su-
pertrace of the heat kernel of the square of a Dirac operator is the quantiza-
quantization of the Chern character of the corresponding connection. Prom this point
of view, the index theorem for Dirac operators is a statement about the re-
relationship between the heat kernel of the square of a Dirac operator and the
Chern character of the associated connection. This relationship holds at the
level of differential forms and not just in cohomology, and leads us to think
of index theory and heat kernels as a quantization of Chern-Weil theory.
Following the approach suggested by Atiyah-Bott and McKean-Singer,
and pursued by Patodi and Gilkey, the main technique used in the book is
an explicit geometric construction of the kernel of the heat operator e~tD
associated to the square of a Dirac operator D. The importance of the heat
kernel is that it interpolates between the identity operator, when t = 0,
and the projection onto the kernel of the Dirac operator D, when t = oo.
However, we will study the heat kernel, and more particularly its restriction
to the diagonal, in its own right, and not only as a tool in understanding the
kernel of D.

Introduction
Lastly, we attempt to express all of our constructions in such a way that
they generalize easily to the equivaxiant setting, in which a compact Lie group
G acts on the manifold and leaves the Dirac operator invariant.
We will consider the most general type of Dirac operators, associated to a
Clifford module over a manifold, to avoid restricting ourselves to manifolds
with spin structures. We will also work within Quillen's theory of supercon-
nections, since this is conceptually simple, and is needed for the formulation
of the local family index theorem of Bismut in Chapters 9 and 10.
We will now give a rapid account of some of the main results discussed
in our book. Dirac operators on a compact Riemannian manifold M are
closely related to the Clifford algebra bundle. The Clifford algebra CX(M) at
the point x 6 M is the associative complex algebra generated by cotangent
vectors a € T*XM with relations
ai-a2 +a2-ai = -2(alla2),
where @1,02) is the Riemannian metric on T*XM. If et is an orthonormal
basis of TXM with dual basis e1, then this amounts to saying that CX(M) is
generated by elements c* subject to the relations
(c*J = -1, and cV + cV = 0 for i ^ j.
The Clifford algebra CX{M) is a deformation of the exterior algebra AT*XM,
and there is a canonical bijection ax : CX(M) —> AT*XM, the symbol map,
defined by the formula
The inverse of this map is denoted by cx : AT*XM —» CX(M).
Let ? be a complex Z2-graded bundle on M, that is, ? = ?+ © ?~. We
say that ? is a bundle of Clifford modules, or just a Clifford module, if there
is a bundle map c : T*M —> End(?) such that
A) c(ai)c(a2) + c(a2)c(ai) = -2(ai,a2), and
B) c(q) swaps the bundles ?+ and ?~.
That is, ?x is a Z2-graded module for the algebra CX(M). If M is even-
dimensional, the Clifford algebra CX(M) is simple, and we obtain the decom-
decomposition
End(?) 2 C{M) ® EndC(M)(?)-
Prom now on, most of our considerations only apply to even-dimensional
oriented manifolds. If M is a spin manifold, that is, a Riemannian manifold
satisfying a certain topological condition, there is a Clifford module 5, known
as the spinor bundle, such that EndE) = C(M). On such a manifold, any
Clifford module may be written as a twisted spinor bundle W ® 5, with
Introduction 3
W = HomC(M)(S,?). Let TM e r(M,C(M)) be the chirality operator in
C(M), given by the formula
rM = idim(M)/2c1...cn,
so that rf^ = 1.
If ? is a vector bundle on M, let F(M, ?) be the space of smooth sections
of ?, and let A(M, ?) = T(M, KT*M <g> ?) be the space of differential forms
on M with values in ?. We make the obvious, but crucial, remark that if ?
is a Clifford module, then by the symbol map, the space of sections
r(M,End(?)) 3! r(M, C{M) ® Endc(M)(?))
is isomorphic to the space of bundle-valued differential forms
A(M,
, AT'M ® Endc(M)(? ))•
Thus, a section k 6 T(M, End(?)) corresponds to a differential form cr(k)
with values in Endc(M)(?)- If Af is a spin manifold and ? = W ® 5 is a
twisted spinor bundle, ff(fc) is a differential form with values in End(W).
A Clifford connection on a Clifford module ? is a connection V? on ?
satisfying the formula
where a is a one-form on M, X is a vector field, and V^a is the Levi-Civita
derivative of a.
The Dirac operator D associated to the Clifford connection V? is the
composition of arrows
, ?) -^ T(M, T"
With respect to a local frame e* of T*M, D may be written
A number of classical first-order elliptic differential operators are Dirac
operators associated to a Clifford connection. Let us list three examples:
A) The exterior bundle AT*M is a Clifford bundle with Clifford action by
the one-fom a € T(M, T*M) defined by the formula
c(a) =?(<*)-?«;
here e(a) : Y{M,K*T*M) -* T(M, A'+1T'M) is the operation of exterior
multiplication by a. The Levi-Civita connection on AT*M is a Clifford
connection. The associated Dirac operator is d + d*, where d is the
exterior differential operator. The kernel of this operator is just the
space of harmonic forms on M, which by Hodge's Theorem is isomorphic
to the deRham cohomology H'(M) of M.

Introduction
B) If M is a complex manifold and W is a Hermitian bundle over M, the
bundle A(T°'1M)* ® W is a Clifford module, with a = qx o + <*o i e
Alfl(M) ®A°'X{M) acting by
c(q) = y/2~(e(ao,i) - e{Wfi)*)-
The Levi-Civita connection on AT*M preserves A(T°'1M)* and defines
a Clifford connection if M is Kahler, and the Dirac operator associated
to this connection is y/2(B + 8*). If W is a holomorphic vector bundle
with its canonical connection, the tensor product of this connection with
the Levi-Civita connection on A(T°'1M)* is a Clifford connection with
associated Dirac operator V2(d + 8*). The kernel of this operator is the
space of harmonic forms on M lying in A°'*(M, W), which by Dolbeault's
theorem is isomorphic to the sheaf cohomology H*(M, W).
C) If M is a spin manifold, its spinor bundle 5 is a Clifford module, the
Levi-Civita connection is a Clifford connection, and the associated Dirac
operator is known simply as the Dirac operator. Its kernel is the space
of harmonic spinors.
Thus, we see from these examples that the kernel of a Dirac operator often
has a topological, or at least geometric, significance.
The heat kernel (x | e~tD | y) e Hom(?y,?x) of the square of the Dirac
operator D is the kernel of the heat semigroup e~ttJ, that is,
s)(x) = / (
Jm
\y)s(y)\dy\ forallSer(M,?),
where \dy\ is the Riemannian measure on M. The following properties of the
heat kernel are proved in Chapter 2:
A) it is smooth;
B) the fact that smooth kernels are trace-class, from which we see that the
kernel of D is finite-dimensional;
C) the uniform convergence of (x | e~tD | y) to the kernel of the projection
onto ker(D) as t —> oo;
D) the existence of an asymptotic expansion for {x | e~tD | y) at small t,
(where dim(M) = n = 2?),
(x
| y)
t=0
where fi is a sequence of smooth kernels for the bundle ? given by local
functions of the curvature of V? and the Riemannian curvature of M,
and d(x, y) is the geodesic distance between x and y.
Note that the restriction to the diagonal x h-» (x | e~tD | x) is a section
of End(?). The central object of our study will be the behaviour at small
Introduction
time of the differential form
e
~tD'
€ A(M,Eadollt)(e)),
obtained by taking the image under the symbol map a of (i | e *D | x).
Let us describe the differential forms which enter in the study of the asymp-
asymptotic expansion of {x | e~tD \x). If g is a unimodular Lie algebra, let js(X)
be the analytic function on g defined by the formula
it is the Jacobian of the exponential map exp : g —» G. We define jg' in a
neighbourhood of 0 ? 0 to be the square-root of j9 such that j9 @) = 1.
Let R 6 A2(M,bo(TM)) be the Riemannian curvature of M. Choose a
local orthonormal frame e< of TM, and consider the matrix R with two-form
coefficients,
Then f "'"p^2^ J is a matrix with even degree differential form coefficients,
and since the determinant is invariant under conjugation by invertible ma-
matrices,
'sinh(R/2)\
R/2 )
is an element of A{M) independent of the frame of TM used in its definition.
Note that the zero-form component ofj(R) equals 1. Thus, we can define the
A-genus A(M) of the manifold M by
it is a closed differential form whose cohomology class is independent of the
-. try
metric on M. It is a fascinating puzzle that the function jg occurs in a
basic formula of representation theory for Lie groups, the Kirillov character
formula, while its cousin, the A-genus, plays a similar role in a basic formula of
differential geometry, the index theorem for Dirac operators. Understanding
the relationship between these two objects is one of the aims of this book.
Note that our normalization of the A-genus, and of other characteristic
classes, is not the same as that preferred by topologists, who multiply the
2fc-degree component by (-27ri)fc, so that it lies in H2k(M,Q). We prefer to
leave out these powers of — 2ni, since it is in this form that they will arise in
the proof of the local index theorem.
Let ? be a Clifford module on M, with Clifford connection ? and curvature
FE. The twisting curvature F?ls of ? is defined by the formula
Fe/s = Fe_RSe A2(

6
where
Introduction
If M is a spin manifold with spinor bundle S and ? = W ® 5, F?/s is the
curvature of the bundle W.
For a e T(M, C(M)) ? T(M, AT*M), we denote the ifc-form component of
a(a) by ok{a).
The first four chapters lead up to the proof of the following theorem, which
calculates the leading order term, in a certain sense, of the heat kernel of a
Dirac operator restricted to the diagonal.
Theorem 0.1. Consider the asymptotic expansion of(x | e~tD | x) at small
times t,
(x
x)
i=0
with coefficients kt e T(M, C(M) ® Endc(M)(?)). Then
A) aj(ki) = Oforj > 1i
B) Let a(k) = ZLo°2i(ki) e A(M,EndC(M)(?)). Then
In Chapter 4, we give a proof of Theorem 0.1 which relies on an approx-
approximation of D by a harmonic oscillator, which is easily derived from Lich-
nerowicz's formula for the square of the Dirac operator, and properties of the
normal coordinate system.
Since the zero-form piece of the A-genus equals 1, we recover Weyl's for-
formula
lim D7rt)/(x | e-tD' | x) = rk(?);
t—*o
in this sense, Theorem 0.1 is a refinement of Weyl's formula for the square of
a Dirac operator.
Define the index of D to be the integer
ind(D) = dim(ker(D+)) - dim(ker(D~)),
where D* is the restriction of D to T(M,?±). For example, ind(d + d*) is
the Euler characteristic
n
Eul(M) = ^(-lJ'dimCff'fM))
i=0
of the manifold M, while ind(d + 5*) is the Euler characteristic of the sheaf
of holomorphic sections
Eul(M, W) =
, W)).
t=0
Introduction
These indexes are particular cases of the Atiyah-Singer Index Theorem, and
are given by well-known formulas, respectively the Gauss-Bonnet-Chern the-
theorem and the Riemann-Roch-Hirzebruch theorem; we will show in Section 4.1
how these formulas follow from Theorem 0.1.
In Chapter 3, we establish some well-known properties of the index, such
as its homotopy invariance, using the formula of McKean and Singer. If
E is a Z2-graded vector space and A is an endomorphism on E, we define
the supertrace Str(A) of A to bo the trace of the operator TA, where T G
End(E) is the chirality operator, which equals ±1 on E*. The McKean-
Singer formula says that for each t > 0, the index of D equals
ind(D) =
= / Str(x
) \dx\.
This formula shows that the index of D is given in terms of the restriction to
the diagonal of the heat kernel of D2 at arbitrarily small times, which we know
by the asymptotic expansion of the heat kernel to be given by local formulas
in the curvature of the connection V? and the Riemannian curvature of M.
Using the McKean-Singer formula, we see that
ind(D) = Dtt)-"/2 / Str(fcn/2(x)) dx.
Jm
From Theorem 0.1, we see that
ff(*»/2(*))H 6 An{M,Endc(M){?)) = r(M,Endc(M)(?))
equals the n-form component of
The differential form
ch(?/S) = 2~n'2 Str(rM • exp(-
is a generalization of the Chern character for Clifford modules, called the
relative Chern character; when M is a spin manifold and ? = W ® <S, the
form ch(?/S) equals the Chern character ch(VV) of the twisting bundle W-
In this way, we obtain the formula
ind(D) =
[
m
A(M) ch{?/S).
This is the Atiyah-Singer index theorem for the Dirac operator D. Thus, we
see that our main theorem is at the same time a generalization of Weyl's
formula for the leading term of the heat kernel, and of the Atiyah-Singer
index theorem. As conjectured by McKean-Singer, we establish that
lim Str (x | e
t—>o
"'0'
x)

Introduction
exists and is given by the above integrand. This result, due to Patodi [82]
and Gilkey [58], is known as the local index theorem. It is very important,
since it generalizes to certain situations inaccessible to the global index theo-
theorem of Atiyah and Singer, such as when the manifold M is no longer closed.
On important example is the signature theorem for manifolds with bound-
boundary of Atiyah-Patodi-Singer[ll]. As another example, it is possible in the
case of infinite covering spaces to prove a local index theorem which gives
the "index per unit volume" of the Dirac operator in question (Connes and
Moscovici [47]).
Note that the proofs of Patodi and Gilkey of the local index theorem re-
required the calculation of the index of certain Dirac operators on homogeneous
spaces. In the proof which we present, we are able to calculate the index den-
density by analysis of Lichnerowicz's formula for the square of the Dirac operator,
which was proved in [73].
In Chapter 5, we give another proof of Theorem 0.1. This proof, while
similar in some respects to the first, uses an integral expression for the heat
kernel of D2 in terms of the scalar heat kernel of the orthonormal frame bundle
SO(M); the Jacobian of the exponential map on SO(M) leads to the appear-
appearance of the .4-genus. Once more, Lichnerowicz's formula is fundamental to
the proof.
The second topic of this book is an equivariant version of the results of
Chapter 4. Suppose that a compact Lie group G acts on the manifold M
and on the bundle S, and that the operator D is invariant under this action
(for this to be possible, G must act on M by isometries). In this situation,
the action of the group G leaves the kernel of D invariant; in other words,
ker(D) is a representation of G. Thus, the index of D may be generalized to
a character-valued index, defined by the formula
indGG, D) = TrG, ker(D+)) - TrG, ker(D")).
The analogue of the McKean-Singer formula holds here:
indGG, D) =
/
m
x) \dx\,
where ~ye is the action of 7 e G on the space of sections of the vector bundle
?. In Chapter 6, we prove a formula, due to Gilkey [59], for the distribution
limStr(x|7?-e-'D2 \x).
It is a product of the delta-function along the fixed point set M7 of the action
of 7 on M and a function involving the curvature of M7, the curvature of
?, and the curvature of the normal bundle Af of M7. Integrating over M,
we obtain an expression for hide G, D) as an integral over M7, which is a
theorem due to Atiyah, Segal and Singer [12], [15]. The proof that we give is
modelled on that of Chapter 5, and is hardly more difficult.
Introduction
The Weyl character formula for representations of compact Lie groups
is a special case of the equivariant index theorem for Dirac operators. By
the Borel-Weil-Bott theorem, a finite-dimensional representation of G may
be realized as the kernel of a Dirac operator on the flag variety G/T of G,
where T is a maximal torus of G. If g is a regular element of G, then the
fixed-point set (G/TK is in one-to-one correspondence with the Weyl group,
and the fixed point formula for the index may be identified with the Weyl
character formula, as is explained by Atiyah-Bott [6].
Generalizing ideas of Bott [39] and Baum-Cheeger [18], we will show how
the fixed point formula for the equivariant index indo^e*, D) may be rewrit-
rewritten as an integral over the whole manifold. To do this, we introduce the
notion of equivariant differential forms in Chapter 7; this is of independent
interest. Let M be a manifold with an action of a compact Lie group G\ if
X is in the Lie algebra g of G, let Xm be the corresponding vector field on
M. Let C[fl] denote the algebra of complex valued polynomial functions on
g. Let Ag{M) be the algebra of G-invariant elements of C[g]® A(M), graded
by
deg(P ® a) = 2 deg(P) + deg(a)
for P e C[jj] and a 6 A(M). We define the equivariant exterior differential
d3 on Cfo] ® A(M) by
(dga)(X) = d(a(X))-L(X)(a(X)),
where i(X) denotes contraction by the vector field Xm- Thus, de increases
by one the total degree on C[g] ® A(M), and preserves Ag(M). Cartan's
homotopy formula implies that d2 vanishes on elements of Ag(M); thus
(Ac(M),d9) is a complex, called the complex of equivariant differential
forms; of course, if G = {1}, this is the deRham complex. Its cohomology
Hq(M) is called the equivariant cohomology of the manifold with coefficients
in R; it was proved by H. Cartan [42] that this is the real cohomology of the
homotopy quotient M xq EG. The Chern-Weil construction of differential
forms associated to a vector bundle with connection may be generalized to
this equivariant setting.
Integration over M defines a map
/„
: Ag(M)
by the formula (JMa)(X) = JMa(X); this map vanishes on equivariantly
exact forms dea, and hence defines a homomorphism from Hq{M) to C[g]G.
An important property of equivariant differential forms is the localization
formula, a generalization of a formula of Bott, which expresses the integral
of an equivariantly closed differential form evaluated at X € f( as an integral
over the zero set of the vector field Xm- This formula has many applications,
especially when the vector field Xm has discrete zeroes.

10
Introduction
Using the localization theorem, we can rewrite the formula for the equivari-
ant index as an integral over the entire manifold, at least in a neighbourhood
of the identity in G.
Theorem 0.2. The equivariant index is given by the formula, for X e fl
sufficiently small,
/ AB(X,M) ch(X,?/S),
m
where chB(X,?/S) E Ag(M) is a closed equivariant differential form which
specializes to the relative Chern character atX = 0, and A9(X, M) € Ag{M)
similarly specializes at X = 0 to the A-genus.
Thus, the integrand is a <28-closed equivariant differential form depending
analytically on X for X near 0 6 g, and which coincides with A(M) ch(?/S)
at X = 0.
Let G be a compact Lie group, and let T be a finite dimensional irre-
irreducible representation of G. As we will see in Chapter 8, Kirillov's formula
for the characters of the representation T of G is a special case of the above
equivariant index theorem: this formula says that there is an orbit M of the
coadjoint representation of G on g* such that
f
J
f
m
where d/3(f) is the Liouville measure on M.
This formula, which unlike the Weyl character formula has a generalization
to any Lie group, inspired our expression for the equivariant index in terms of
equivariant differential forms. Note the similarity with the equivariant index
theorem: the equivariant .4-genus corresponds to the factor j^l^2{X), while
the equivariant Chern character corresponds to the measure e'^'x^ df)(f).
The final major topic of the book is the local index theorem for families of
Dirac operators. This theorem, which is due to Bismut, is a relative version
of the local index theorem, in which we consider a fibre bundle n : M —* B
and a smooth family of Dirac operators D*, one for each fibre Mz = ic~l{z),
(z e B). The kernels of the Dirac operators D2 now fit together to form
a vector bundle, if they are of constant dimension. The index of a single
operator D, ind(D) = dim(ker(D+)) - dim(ker(D~)), is now replaced by the
notion of the index bundle, which is the difference bundle
ind(D) = [ker(D+)j - [ker(D~)];
following Atiyah-Singer, we define in Chapter 9 a smooth index bundle even if
the dimension of the kernel ker(Dz) is not constant, but it is no longer canon-
canonical. The index theorem of Atiyah-Singer [16] for a family gives a formula for
the Chern character of this index bundle.
It turns out that the formulation of a local version of this family index
theorem requires the introduction of generalization of connections due to
Introduction
11
Quillen, known as superconnections. Let H = H+ ®H be a Z2-graded
vector bundle. A superconnection on H is an operator acting on A(M, H) of
the form
A = A[oj + Am + A[2] + • ¦.,
where h[\\ is a connection preserving the Z2-grading on H, and A[<] for i j= 1
are i-form valued sections of End(Ti) which exchange H+ and H~ for i even
and preserves them for % odd; usually, A[0] e T(M, End(W)) is supposewd to
be self-adjoint as well. The curvature of a superconnection A2 is its square,
which is an element of A(M, End(W)) of the form A2 = Ah + .... Quillen
showed that as a deRham cohomology class, the Chern character of the
difference bundle [H+\ — [H~] is equal to the Chern character of A, defined
by the supertrace
ch(A) =
= Str(e~A'°i+-),
which is a closed even degree differential form whose cohomology class is in-
independent of A. Superconnections play a role in two situations: to define the
Chern character on non-compact manifolds, where A[o) grows at infinity, thus
allowing ch(A) to represent an integrable cohomology class, and for infinite
dimensional vector bundles, where A[0] is such that e~ [°i is trace-class. We
explain the extension of the Chern-Weil construction of characteristic forms
to the case of superconnections in Chapter 1.
Let D be an odd endomorphism of a finite-dimensional superbundle H
with components D* : W* —¦ 7iT, such that ker(D) has constant rank, and
let A be a superconnection on H with zero-degree term A[o] equal to D. At
the level of cohomology,
ch(A) = ch(ker(D+)) - ch(ker(D~)).
In Chapter 9, we prove a refinement of this equation which holds at the level
of differential forms.
If t > 0, we may define the rescaled superconnection on H, by the formula
with Chern character
We define a connection on the bundle ker(D) by the formula
Vo = Pq- A[!j -Po,
where Po is the projection from H onto its sub-bundle ker(D); thus,
ch(V0) = ch(ker(D+)) - ch(ker(D")).
Intuitively speaking, as t tends to infinity, the supertrace
ch(At) =

12
Introduction
is pushed onto the sub-bundle ker(D). In fact, we can prove the following
formula,
lim ch(At) = ch(Vo).
t—>oo
Let M —> B be a fibre bundle with compact Riemannian fibres of dimen-
dimension n, and let ? be a bundle of modules for the vertical Clifford algebras
CX(M/B) associated to the vertical tangent spaces of M —> B. Bismut
showed that the above picture can be extended to an infinite-dimensional
setting when D is a smooth family of Dirac operators on the fibres ?z —> Mz,
where ?z and Mz are the fibres over z e B of ? and M. He introduced the
infinite-dimensional bundle n,? whose fibre at z € B is the space of sections
of ? over the fibre Mz,
(*.?), =T(MZ,?Z).
Given a connection on the fibre bundle M —> B, that is, a choice of horizontal
tangent bundle in TM transverse to the vertical tangent bundle, we may
define a unitary connection on 7i\»?, whose formula involves the connection
on ? and the mean curvature of the fibres Mz. If we form the superconnection
At = t1/2D + V*'?, then the heat kernel of A? has a supertrace in A{B),
since
A? = tD2 + t1/2V'?D + (V"-eJ
is a perturbation of the operator tD2. In this way, we may define ch(At) 6
A{B).
Assume that ker(D) has constant rank. The projection Pq ¦ V'f • Po of
the connection Vir*e defines a connection Vo on the finite dimensional bundle
ker(D). We obtain the following transgression formula:
ch(At) - ch(V0) = d
In general, it is impossible to take the limit t —> 0 in this formula. However,
Bismut showed that this limit exists if a certain term Ap] is added to A,
proportional to Clifford multiplication by the curvature of the fibre bundle
M —* B; the curvature of this superconnection satisfies a formula very similar
to the Lichnerowicz formula for the square of a Dirac operator. By methods
almost identical to those of Chapter 4, we prove the following theorem in
Chapter 10.
Theorem 0.3. Let RMIB e A2(M,so{T(M/B})) be the curvature of the
vertical tangent bundle of M/B, with A-genus A(M/B). Then, if A is the
Bismut superconnection,
lim ch(At) =
f
Jm/b
A{M/B) ch{?/S) e A{B).
Acknowledgements
13
This result has turned out to be of interest to mathematicians working in
such varied areas as string theory, Arakelov theory, and the theory of moduli
spaces of Yang-Mills fields. Interestingly, when the fibre bundle M —* B has
a compact structure group G and all of the data are compatible with G, the
above theorem is equivalent to the Kirillov formula for the equivariant index.
The book is not necessarily meant to be read sequentially, and consists of
four groups of chapters:
A) Chapters 1, which gives various preliminary results in differential geom-
geometry, and Chapter 7, on equivariant differential forms, do not depend on
any other chapters.
B) Chapters 2, 3 and 4 introduce the main ideas of the book, and take the
reader through the main properties of Dirac operators, culminating in
the local index theorem.
C) Chapters 5, 6 and 8 are on the equivariant index theorem, and may be
read after the first four chapters, although Chapter 7 is needed in Chap-
Chapter 8.
D) Chapters 9 and 10 are on the family index theorem, and can be read
after the first four chapters, except for Sections 9.4 and 10.7, which have
Chapter 8 as a prerequisite.
Acknowledgements
This book began as a seminar in 1985 at MIT, and we would like to thank
the other participants in the seminar, especially Martin Andler and Varghese
Mathai, for their spirited participation. Discussions with many other people
have been important to us, among whom we would like to single out Jean-
Michel Bismut, Dan Freed and Dan Quillen. Finally, we are pleased to be
able to thank all of those people who read all or part of the book as it
developed and who made many comments which were crucial in improving the
book, both mathematically and stylistically, especially Jean-Fran§ois Burnol,
Michel Duflo, Christophe Soule, and Shlomo Sternberg. We also thank the
referee for suggestions which have improved the exposition.
We have not attempted to give a definitive bibliography of this very large
subject, but have only tried to draw attention to the articles that have influ-
influenced us.
To all of the following institutes and funds, we would like to express our
gratitude: the Centre for Mathematical Analysis of the ANU, ENS-Paris,
IHES, MIT, and Universite Paris-Sud. We also received some assistance
from the CNRS, the NSF, and the Sloan Foundation.

1.1. Fibre Bundles and Connections
15
Chapter 1. Background on Differential Geometry
In this chapter, we will survey some of the notions of differential geometry
that are used in this book: principal bundles and their associated vector bun-
bundles, connections and superconnections, and characteristic classes. For more
detail about these topics, we refer the reader to the bibliographic references
at the end of the chapter.
1.1. Fibre Bundles and Connections
Definition 1.1. Let % : ? —> M be a smooth map from a manifold ? to
a manifold M. We say that (?, n) is a fibre bundle with typical fibre E
over M if there is a covering of M by open sets Ui and diffeomorphisms
(pi : 7r~1(t/j) —U 1
with projection onto the first factor
total space of the fibre bundle, and M is called the base.
x E, such that it : 7r-1(i7i) —¦ Ui is the composition of 4>i
in Ui x E. The space ? is called the
It follows from this definition that Tr^) is diffeomorphic to E for all
x 6 M\ we will call E "the" fibre of ?. In this book, we will often make use
of the convenient convention that the bundles denoted ?, V, W, and so on,
have typical fibres E, V, W, etc.
Consider the diffeomorphism <\>j o <j>~1 of (f/< n Uj) x E which we obtain
from the definition of a fibre bundle. It is a map from Ui D Uj to the group
of diffeomorphisms Diff (E) of the fibre E.
Definition 1.2. A fibre bundle n : ? —> M is a vector bundle if its typical
fibre is a vector space E, and if the diffeomorphisms 4>i may be chosen in such
a way that the diffeomorphisms <j>j o 0 : {x} x E —> {x} x E are invertible
linear maps of E for all x e Ui D Uj.
A section a of a vector bundle ? over M is a map s : M —> ? such that
ns(x) = x for all x e M. In this book, we will use the word smooth to
mean infinitely differentiable. The space of all smooth sections is denoted by
T(M, ?), while the space of all compactly supported smooth sections (those
equal to 0 outside a compact subset of M) is denoted by FC(M,?).
If <j>: M\ —* Mi is a smooth map of manifolds and ? is a vector bundle on
M2, we denote by <f>*? the vector bundle over Mi obtained by pulling back ?:
it is the smooth vector bundle given by restricting the vector bundle Mi x ?
over Mi x M2 to the graph r(<j>) = {(xi,cj>(xi)) ? Mi x M2) of <j>, so that
4>'? = {(x,v) € Mi x ? I v 6 ?Hx)).
There is induced a pull-back map <f>* : T(M2,?) -> T(MU <f>*?).
Definition 1.3. A principal bundle P with structure group G is a fibre
bundle P with a right action of a Lie group G on the fibres, that is, (p • g) ¦ h =
p-(gh) for p e P, g,h e G such that
n(pg) = n(p) for all p e P and j€G,
and such that the action of G is free and transitive on the fibres. The fibres
of the bundle P are diffeomorphic to G itself, and so the base M may be
identified with the quotient manifold P/G.
A vertical vector on a fibre bundle ? with base M is a tangent vector on
? which is tangential to the fibres: that is, X(n"f) = 0 for any / € C°°(M),
where ir*f = f on. We denote the bundle of vertical tangent vectors by V?;
it is a subbundle of the tangent bundle T?. The space of vertical tangent
vectors to a point p in a principal bundle can be canonically identified with
0, the Lie algebra of G, by using the derivative of the right action of G on P.
If X 6 fl, we denote by XP the vector field such that if / 6 C°°(P),
Thus, we obtain a map from P x g to TP, whose image at p is just the vertical
subspace VPP. The vector field Xp will often be simply denoted by X.
If P is a principal bundle with structure group G and E is a left G-space,
with action of G on it given by the homomorphism
p:G-+ Diff(E),
we can form a fibre bundle over M with E as fibre, called the associated
bundle, by forming the fibred product PxqE, defined by taking the product
P x E and dividing it by the equivalence relation
(p-g,f)~(p,p(g)f),
for all p € P, g 6 G and / 6 E. In particular, if we take for E a vector space
which carries a linear representation of G, we obtain a vector bundle on M.
Every vector bundle with AT-dimensional fibres on M is an associated
bundle for a principal bundle on M with structure group GL(iV), called the
bundle of frames. In this book, we will use vector spaces over both R and
C, and when it is clear from the context, we will not explicitly mention
which field we take homomorphisms or tensor products over. Thus, when we
write here GL(JV), we mean either GL(N, R) or GL(AT,C), depending on the
context.

J.u 1. Background on Differential Geometry
Proposition 1.4. If? is a vector bundle over M with fibre RN, let GL(?)
be the bundle whose fibre over the point x e M is the space of all invertible
linear maps from RN to the fibre ?x. Then GL(?) is a principal bundle for
the group GL(N, R), under the action
(p-g)(v)=p(g-v),
wherep : RN —» ?x, g e GL(N, R) andv € RN, and? is naturally isomorphic
to GL(?) ><gl(jv,r) 1Ln- The same result holds when R is replaced by C. We
call GL(?) the frame bundle of ?.
Proof. Since locally, ? has the form U x RN, it follows that locally, GL(?)
has the form U x GL(AT,R), proving that P is a principal bundle.
The isomorphism from GL(?) Xql(jv,R) R^ to ? is given by mapping the
equivalence class of (p, v) 6 GL(?)x x R^ to p(v) 6 ?x; it is easily seen that
this map is equivariant under the action of GL(JV, R), that is, (p-g,v) maps
to the same point as (p, g • v). ?
The formation of an associated bundle is functorial with respect to mor-
phisms between G-spaces; in other words, we could consider it as a map from
the category of G-spaces to the category of fibre bundles on the base. Indeed,
if k : E\ —> Ei is a G-map from one G-space to another, then it induces a
map from P x a E\ to P x a E2 • This map is constructed by the commutative
diagram
pxEl
PxE2
PxGEy.
PxGE2
Definition 1.5. Let n : ? —» M be a fibre bundle and let G be a Lie group.
We say that ? is an G-equivariant bundle if G acts smoothly on the left
of both ? and M in a compatible fashion, that is,
7 • -it = n ¦ 7 for all 7 e G.
If ? is a vector bundle, we require in addition that the action y? : ?x —> ?7. x
is linear.
The group G acts on the space of sections T(M, ?) of an G-equivariant
bundle by the formula
A.1)
G •«)(*) =7'
= -v*.
-x).
Let 0 be the Lie algebra of G. We denote by C? (X), X e g, the corresponding
infinitesimal action on T(M, ?),
A.2)
C?(X)s =
e=0
exp(eX) ¦ s,
1.1. Fibre Bundles and Connections
17
which is called the Lie derivative. Clearly, C?(X) is a first-order differential
operator on F(M, ?).
Let TM be the tangent bundle of M. A section X ? T(M, TM) is called
a vector field on M. If <t> : Mj —» M? is a smooth map, it induces a map
<$>» : TM\ —* TM2, in such a way that
In particular, diffeomorphisms <j> of M induce a diffeomorphism 0» of TM.
Informally, TM is a Diff (M)-equivariant vector bundle.
Let GL(M) = GL(TM) be the frame bundle obtained by applying the
construction of Proposition 1.4 to the tangent bundle TM; it has structure
group GL(n) with n = dim(Af). From this principal bundle, we can con-
construct a vector bundle on M corresponding to any representation E of GL(n);
these vector bundles are called tensor bundles. We see from this that any
tensor bundle carries a natural action of Diff(M), making it into a Diff(M)-
equivariant vector bundle over M.
If V is a tensor bundle, the Lie derivative CV(X) by a vector field X on
M is the first-order differential operator defined on F(M, V) by the formula
4>t-s,
t=o
where <j>t is the family of diffeomorphisms generated by X. For example, if
V = TM, then we obtain the Lie bracket C(X)Y = [X, Y}.
One example of a tensor bundle is obtained by taking the exterior algebra
A(Rn)*; this leads to the bundle of exterior differentials AT*M. The space
of sections F(M, AT*M) of the bundle of exterior differentials is called the
space of differential forms A(M); it is an algebra graded by
Ai(M) = T(M,AiT*M).
Similarly, we denote by Ac{M) the algebra FC(M, AT*M) of compactly-
supported differential forms. If q is a differential form on M, we will de-
denote by Q[ij its component in A'(M). If a e Al(M), we will write |a| for
the degree i of a; such an a is called homogeneous. The space A(M) is
a super-commutative algebra (see Section3 for the definition of this term),
that is,
We denote by d the exterior differential; it is the unique operator on
A(M) such that
A) d : A'(M) -+ A'+1(M) satisfies d2 = 0;
B) if / € C°°(Af), then df e A*(M) is the one-form such that (df)(X) =
X(f) for a vector field X;
C) (Leibniz's rule) d is a derivation, that is, if a and /3 are homogeneous
differential forms on M, then
d(a A 0) = (da) A 0 + (-l)ltt|a A (d/3).

18 1. Background on Differential Geometry
If a e Ak{M\ and Xo,..., Xk e T(M, TM), then
«=o
+ ? (-l)i+ja([Xi,Xj},X0,...X,---,Xj,-.-,Xk).
A.3)
Definition 1.6. If V is a vector space, the contraction operator i(v) :
AV* —> AV*, (v E V), is the unique operator such that
A) i(v)a = a(v)itaeV*;
B) i(v)(aA0) = (i(v)a)A0 + (-l)MaA(i(v)P) if a and /3 are homogeneous
elements of AV*.
The exterior operator e(a) : AV* —> AV", is the operation of exterior
multiplication by a e K*.
We will denote the operation of contraction with a vector field X by
Similarly, on A(M, ?), l(X) is defined in such a way that for elements of the
form as, a e A(M) and s 6 T(M,?),
t(X){aa) = (L(X)a)s.
On differential forms, the Lie derivative satisfies the two properties,
C(X)d = dC{X),
C(X)(t(Y)a) = i([
from which follows easily E. Cartan's homotopy formula:
A.4)
Note that if a is an n-form, where n is the dimension of M, then this formula
implies that C(fX)a = C(X)(fa) for all / e C°°(M).
If <j> : Mi —» M2 is a smooth map of manifolds, we denote by <\>* :
A'{M2) —> A'(Mi) the pull-back of differential forms; it commutes with
exterior product and with d, and hence defines a homomorphism of differen-
differential graded algebras.
Since d2 = 0, we can form the cohomology of the complex (A(M),d),
which is
iT(.4(M),d) = ker(dU«(M))/im(dUi-I(M)).
We denote ^(^(MJ.d) by W{M), and call H'(M) the deRham coho-
cohomology groups of M. If da = 0, we say that a is closed, and we write its
class in H'(M) as [a]. If a = d/3 for some /3, we say that a is exact; exact
1.1. Fibre Bundles and Connections
19
forms represent zero in H'(M). The graded vector space 5D"=0
is a graded algebra, with product defined by
[ai] A [02] = [c*i A Q2] for closed forms ai and Q2.
This definition depends on the fact that if c*i and 02 are closed and one of
them is exact, then so is a\ A o^; for example, if a\ = d/3, then
Qi Aaj = d(/3 AQ2).
If M is a non-compact manifold, for example, a vector bundle, the defi-
definition of the de Rham cohomology may be repeated with A(M) replaced by
the differential forms of compact support AC(M). The cohomology of the
complex (Al(M),d) is denoted H*(M) and called the deRham cohomology
with compact supports.
Let P be a pricipal bundle with structure group G. There is a representa-
representation of the sections of an associated vector bundle P xq E as functions on the
corresponding principal bundle that is extremely useful in doing calculations
in a coordinate free way.
Proposition 1.7. LetC°°(P,E)G denote the space ofequivariant maps from
P to E, that is, those maps s : P —* E that satisfy s(p-g) = p(s~1)s(p).
There is a natural isomorphism between T(M,PxgE) and C°°(P, E)°, given
by sending s e C°°{P, E)G to sM defined by
here p is any element of ir~1(x) and \p,s{p)] is the element of P xq E cor-
corresponding to (p, s(p)) € P x E.
Proof. We first note that the definition of sm is unambiguous; if, for some
point x € M, we had chosen some lifting p-g instead of p 6 n~1(x), then
[p, s(p)] would have been replaced by
\p-9,s(p-g)] =
- \p,s(p)}.
To complete the proof, we only have to show that any section of P Xq E is
of the form 8m for some s e C°°(P,E)a. To do this, we merely define s{j>)
to equal the unique v 6 E such that (p, v) is a representative of 8m(x). ?
Observe the following infinitesimal version of equivariance: a function s in
C°°(P,E)G satisfies the formula
A.5)
(XP ¦ s)(x) + p(X)s(x) = 0, for X 6 g,
where we also denote by p the differential of the representation p.
If ? is a vector bundle on M, let A(M, ?) denote the space of differential
forms on M with values in ?, in other words,
Ak(M, ?) = T(M, AkT*M ® ?).

20
1. Background on Differential Geometry
The differential forms will be taken over the field of real numbers unless either
they take values in a complex bundle or unless explicitly stated otherwise.
If ? is the trivial bundle ? = M x E, we may write Ak(M, E) instead of
Ah(M, ?). When P Xq E is an associated bundle on M, we will describe
A(M,P xq E) as a subspace of the space of differential forms on P with
values in E.
The group G acts on A(P, E) = A(P) ® E by the formula
We denote by A{P, E)G C A(P) <8> E the space of invariant forms. Elements
of A(P, E)G satisfy the following infinitesimal form of invariance:
for all X e 0 and a 6 A{P, E)G.
Definition 1.8. A horizontal differential form on a fibre bundle ? is a
differential form a such that t(X)a — 0 for all vertical vector fields X.
A basic differential form on a principal bundle P with structure group
G, taking values in the representation (?, p) of G, is an invariant and hori-
horizontal differential form, that is, a form a € A(P, E) which satisfies
A) 5-a = a, geG;
B) i(X)a = 0 for any vertical vector field X on P.
The space of all basic forms with values in E is denoted A(P, J5)bas-
We will not prove the following proposition, since its proof is similar to
that of Proposition 1.7.
Proposition 1.9. If a 6 Aq{P,E)baa, define ctM € Aq(M,PxG E) by
«m(t.Xi,... ,w,Xq)(x) = \p,a(Xi,... ,
where p e P is any point such that ir(p) = x, and Xi 6 TPP. Then c*m is
well-defined, and the map a i-> qm from A(P, i?)bas to A(M, P xq E) is an
isomorphism.
Let ? be a fibre bundle with base M. The quotient of the tangent bundle
T? by its subbundle V? is isomorphic to the pull-back n*TM of the tangent
bundle of the base to ?; that is, we have a short exact sequence of vector
bundles
0 -> V? -> T? -» ttTM -+ 0.
There is no canonical choice of splitting of this short exact sequence, but
it is important to have such a splitting in many situations. Such a splitting
is called a connection.
1.1. Fibre Bundles and Connections
21
Definition 1.10. Let n : ? —> M be a fibre bundle. A connection one-
form is a one-form u) e Al(?, V?) such that i(X)u> = X for any vertical
vector field on ?. The kernel of the one-form iu is called the horizontal
bundle, and written H?; it is isomorphic to ir*TM.
If X is a vector field on the base M, we denote by Xg the section of H?
such that ntX? = X; we call X? the horizontal lift of X with respect to
the connection determined by u>.
Note that we can equally well specify a connection by the horizontal sub-
subbundle H? or by the connection form w. In many situations, however, the
connection form is more convenient.
An important object associated to a connection on a fibre bundle ? is its
curvature I), which is a section of the bundle A2tt*T*M ® V?: if X and
Y are vector fields on M and if Xe and Y? are their horizontal lifts, then
Q(X, Y) is the vertical vector field on ? defined by the formula
A.6) n(X,Y) = [X,Y]?-[Xe,Y?].
Note that this tensor is local on M, in the sense that for any / e C°°(M)
this follows immediately from the fact that X?(n*f) — n*(Xf).
If P is a principal bundle, we only consider connections on it which are sta-
stable under the action of the structure group G by multiplication on the right.
This condition is most conveniently characterized in terms of the connection
form. Observe that, by the identification of VP with the trivial bundle P x g,
a connection form on P will be an element of A1(P, g).
Definition 1.11. A connection one-form on a principal bundle P is an
invariant g-valued one-form u G A1{P, g)G such that i(Xp)u = X if X G g.
A differential form a G A(P, g)G satisfies the formula
A.7) C(X)a + [X,a] = 0 for all X G g.
Proposition 1.12. The space of connection one-forms on a principal bundle
P is an affine space modelled on the space of one-forms A1{M, P xcg).
Proof. Once a connection form uiq is chosen, all others are described uniquely
in the form uj0 + a for some a G A1(P,g)bas- The result now follows from
Proposition 1.9. ?
The curvature fi of the connection associated to w is the element of
-42(P,g)bas defined by
U(X,Y)P = PHP[X,Y] - [PhpX,PhpY},

22
1. Background on Differential Geometry
where Pup is the projection onto the horizontal sub-bundle of TP, and X
and Y are two vector fields on P. In stating the following proposition, we
use the Lie bracket on A(P, g) denned by
for a and /? in A(P), and Ai and A2 in g. This makes A(P,g) into a Lie
superalgebra, as we will see in the Section 1.3. If a is a g-valued one-form and
Xi and X2 are two tangent vectors, then [a,a](Xi,X2) = 2[a(Xi),a(X2)].
Proposition 1.13. The curvature form U satisfies the formula
Proof. If X € 0, then
t(X)(dui + i[w, w]) = ?(A> - d(t(A» + [A>] = 0
by A.7), since d(t(X)w) = dY = 0. Thus, the differential form dui + §[u>,w]
is seen to be horizontal; as it is clearly invariant, it is seen to be basic. To
complete the proof, we need only evaluate the two-form dui + \ \u>, u>] on the
horizontal vectors Y and Z; it is easy to see that we obtain — i([Y, Z])ui, which
is exactly equal to il(Y, Z). ?
The choice of a connection on a principal bundle P determines connections
on all associated bundles ? = P x<? E, by defining H? to be the image of
HP under the projection P x E —> ?. We will be especially interested in
this construction for associated vector bundles, since it provides an analogue
of the exterior differential on spaces of twisted differential forms A(M, ?),
called a covariant derivative.
Definition 1.14. If ? is a vector bundle over a manifold M, a covariant
derivative on ? is a differential operator
V : T(M, ?) -+ T(M, T*M ® ?)
which satisfies Leibniz's rule; that is, if a 6 T(M,?) and / G C°°(M), then
V(/s) = df ® s + /Vs.
Note that a covariant derivative extends in a unique way to a map
V:A'(M,?)-4A'+1(M,?)
that satisfies Leibniz's rule: if a 6 Ak(M) and 6 e A(M,?), then
V(a A 6) = da A 0 + (-l)fca A V0.
This formula is the starting point for the definition of a superconnection,
which generalizes the notion of a covariant derivative and forms the topic of
Section 3.
If X is a vector field on M, we will denote by Vx the differential operator
t(X)V, which is called the covariant derivative by the vector field X. It
1.1. Fibre Bundles and Connections
23
depends linearly on X, V/x = /Vx, while the commutator [Vx,/] of Vx
and the operator of multiplication by a smooth function / is multiplication
byX-f.
We see from the definition of a connection that if a 6 Ak(M, ?) and
X0,...,Xke T(M,TM), then
k
A.8) (Va)(X0,..., Xk) = ?(-1)^ (a(X0, ...,XU..., Xk))
«=o
Two covariant derivatives Vo and Vi on a vector bundle ? differ by a
one-form u 6 A1(M,End(?)):
= Vns
for s e T{M,?).
If ? = M x E is a trivial vector bundle, then the exterior differential d is
a covariant derivative on ?, so any covariant derivative on M x E may be
written
A.9)
Vs = ds + ui ¦ s,
for some w e >t1(M,End(?)) = A1(M) ® End(E). We say that u> is the
connection one-form of V in the given trivialization of ?.
The curvature of a covariant derivative V is the End(?)-valued two-form
on M defined by
for two vector fields X and Y. It is easy to check, using Leibniz's rule, that
F(X,Y) really is a local operator, in other words, that it commutes with
multiplication by a smooth function /:
[F(X, Y),f} = [Vx, 17] + [Xf, Vy] - [X, Y]f = 0.
Proposition 1.15. The operator V2 : A'(M,?) -* A*+2(M,?) is given by
the action of F 6 .42(M,End(?)) on A(M,?).
Proof. If ? is a trivial bundle and V = d + u>, then F is the End(?)-valued
two-form on M given by the formula
F = dw + w A w.
On the other hand, the square of the operator d + u> is also given by the
action of dui + u) A w. ?
We will study the curvature thoroughly later, in the more general context
of superconnections.

24
1. Background on Differential Geometry
If ? is an associated bundle ? = P xG E, then a covariant derivative V
may be formed from a connection one-form u on P, by means of the diagram
C°°(P,E)G
T(M,?)
To show this, we must check that if s is an equivariant map from P to E,
then ds + p(uj)s is basic. But choosing X G g, we see that
i(X)(ds + p(w)s) = X ¦ s + p(X)s = 0
since s e C°°(P,E)G. Similarly, the form ds + p(w)s € A1(P,E)G, since
s € C°°(P, E)G. It is easy to verify that V is indeed a covariant derivative.
There is another formula for the covariant derivative when written on the
principal bundle in terms of the projection onto the horizontal subspaces of
TP:
Vs = PHM ° ds for a € C°°{P, E)G.
Indeed, if Xh is a horizontal tangent vector, then
since Xh is horizontal, while if Xv is a vertical tangent vector,
since s e C°°(P,E)G. Thus, if sP € C°°(P,E)G corresponds to the section
s € T(M,?), and X € T{M,TM) is a vector field with horizontal lift XP,
then for any p 6 P above z,
Proposition 1.16. There is a one-to-one correspondence between connec-
connections on the frame bundle GL(?) and covariant derivatives on the vector
bundle ?.
Proof. If V is a covariant derivative on ?, we may construct a one-form
we.41(P,End(.E))G by
w(X>(p) = (V*.x')fr) - (X ¦ s)(p) for s G C°°(P, E)G, X e TPP.
This defines u> uniquely, since for any (p, v) e P x E, there exists an s €
C°°(P,E)G such that s(p) = u. The one-form w satisfies
for all X e g,
and hence is a connection form. D
As an example of this result, let ? = M x E be a trivial vector bundle
on M with connection V = d + w. The frame bundle GL(?) of ? is equal
1.1. Fibre Bundles and Connections
25
to 7T : M x GL(i?) —+ M. Let g~ldg be the Maurer-Cartan one-form in
-41(GL(?),End(?1)) defined using the tautological map
which sends an element of the group GL(E) to the corresponding linear map
in End(iJ). Then the connection one-form on GL(?) associated to V is
In this way, we see that the two different notions of connections, as a
connection on a principal bundle and as a covariant derivative on sections of
a vector bundle, are two faces of a single object.
If the covariant derivative has been defined in terms of a connection u on
a principal bundle P, then we obtain a representation of its curvature as a
basic two-form fi on P with values in g: F = p(il), with fl = du + ^[w,ui],
since for s€C°°(P,E)G,
F(X,Y)s = ([Xp,Yp] - [X,Y]p)s = -Sl(X,Y)s
The advantage of constructing a covariant derivative from a connection
on a principal bundle is that this provides a compatible family of covariant
derivatives on all vector bundles associated to P simultaneously, in a way
which is compatible with the natural linear maps between these bundles, and
with the tensor product of bundles. Thus, if we are given a G-equivariant
map k : V —+ W and s € T(M,P xqV), then we may form the bundle maps
k : P xG V -» P xG W and 1®k : T*M ® P xG V -* T*M®P xGW, and
then
Vk(s) = A ® k)(Vs) G Al{M,PxG W).
We now give some ways of contructing new covariant derivatives out of old.
If V is a covariant derivative on ?, we obtain a covariant derivative on the
dual bundle ?*, by the following formula, for s € T(M,?) and t € T(M,?*):
d(s,t) = (Vs,t) + (»,Vt>,
that is, if X e T(M, TM) is a vector field on M,
Here, we use the notation (s,t) to denote the pairing between ? and ?*.
If V?i is a covariant derivative on ?< for i = 1, 2, we obtain a covariant
derivative V?l®?j on the tensor product by
® s2) =
for Sj e r(M,?j). We refer
tive, and write
® s2
as the tensor product covariant deriva-

deriva26
1. Background on Differential Geometry
Let M be a manifold on which a compact Lie group G acts smoothly, and
let ? be an equivariant bundle over M. We say that a covariant derivative
V? is invariant if it commutes with the action of G,
for all g e G. If G is compact, there will exist such connections, since we
may average with respect to the Haar measure on G,
A.10)
g
is an invariant covariant derivative.
Let 0 : N —> M be a smooth map, and let ? be a vector bundle on M. If
Vs is a covariant derivative on ?, then the formula
V**?(/</.*s) = df ® 4>*s + f<t>*(V?s)
for / € C°°(N) and a e T{M,?), defines a covariant derivative V*"e on the
bundle 4>*?, which we call the pull-back of the covariant derivative Ve.
In particular, if *y(t): R —» M is a smooth curve in M with tangent vector
7@ e T-,(t)M, and if a : R —> ~/*? is smooth, then V?(t)s(t) is defined to
be the covariant derivative V7*?(s)(t). If ? is the trivial bundle M x E and
Vf = d + w, we may write s(t) = G(t), /(*)) with / : R -» ?, and
The parallel transport map along
is defined by solving the ordinary differential equation
A.11) V?(t)r7« = 0,
with initial condition T7@) = /.
Suppose U is an open ball in Rn and ? is a vector bundle over U with
connection V. If x* are the coordinates on U and let di are the corresponding
partial derivatives, let
be the radial vector field on U. The following result follows from the theory
of ordinary differential equations.
Proposition 1.17. Parallel tranaport along rays t i-> tv, v e U, gives a
smooth trivialization of ? over U, by identifying the fibres ?v with E = ?q.
1.1. Fibre Bundles and Connections
27
In terms of this trivialization, the connection one-form on ? is the End(i?)-
valued one-form u> = J^ Wjdx* such that V = d+w, and the curvature of ? is
the two-form F with values in End(E) defined by the equation F = dw+ioAu.
By the definition of this frame of ?, we have l(TV)lj = 0. Thus the following
formula for the radial derivative of the connection form u> holds:
A.12)
C(R.)u> = [i(,H), d]uj =
A useful property of this framing of ? is that all of the coefficients of the
Taylor expansion of w< at 0 are determined by the coefficients of the Taylor
expansion of F at 0.
Proposition 1.18. The Taylor expansion ofu)t atxo = O has the following
form:
Proof. Expanding the Taylor's series of both sides of A.12), we obtain
a a,k
By equating coefficients of xa on both sides, we see from this formula that
djUHixo) ^-\F{dudi)X0.
Furthermore, it follows that the Taylor coefficients of aij at xo to order m
only depend on those of F(di, dj) to order m — 1. ?
A metric on a vector bundle is a smooth family of non-degenerate inner
products on the fibres of ?. If ? is a real bundle of rank N, then this amounts
to requiring that ? is associated to a O(iV)-bundle P with fibre the standard
representation RN ¦ In fact, the principal bundle may be taken equal to
0E), the sub-bundle of GL(?) consisting of orthonormal frames. (There are
corresponding statements with R replaced by C and O(N) replaced by U(JV).)
A real vector bundle with metric will be called a Euclidean vector bundle,
while a complex vector bundle with metric will be called a Hermitian vector
bundle.
We say that a covariant derivative V preserves a metric if it satisfies the
formula
d{8Us2) = (Vsi,s2) + («i, Vs2) for 8i € T(M,?).
If V preserves a metric on ?, it is easy to see that its curvature FX(X, V) lies
in the Lie algebra so(?x) of skew-adjoint endomorphisms of ?x.
The covariant derivative V preserves a metric on ? if and only if the
parallel transport map also preserves this metric, since if s G ?7(o), we have
2 = 2 (r7(t)a, V7(t)r7(t)s) = 0.

28
1. Background on Differential Geometry
This shows that the trivialization of Proposition 1.17 is compatible with the
metrics on the fibres of ?. Thus, if ea is an orthonormal basis of E = ?o, we
obtain smooth sections, which we also denote by eo, which are orthonormal
at each point of U; these sections make up an orthonormal frame.
Let ir : ? —> M be a vector bundle on M. A covariant derivative V? on
? determines a connection on GL(?), and hence a connection on the total
space ?, that is, a splitting
T? = V? e HE.
The bundle V? is isomorphic to tt'?. We denote by 6 6 Al(?,V?) the
connection one-form of this connection.
Definition 1.10. If ? is a vector bundle, its tautological section x e
r(?,7r*?) is the smooth section which to a point v € ? assigns the point
(v,v) €7r*?.
Proposition 1.20. Let ? be a vector bundle with covariant derivative V?.
If V*'? denotes the pull-back of the covariant derivative on ? to n*?, then
0, thought of as an element of *A}{?,ir*?), equals V'^x.
Proof. Locally, ? may be trivialized; thus, we assume that ? = M x E, with
covariant derivative V = d + u>. Denote a point of ? by (x, v) e M x E; then
x(a;, v) = v, and V'?x = dv + u v. Thus
The frame bundle GL(?) has tangent vectors {X, A) e T^g) GL(?) =
TXM x 0[(E) at the point (x,g) € GL(?) = M x GL(E). The horizontal
tangent space at (x, g) is
H{X}9) GL(?) = {(X, -u>(X)) | X € TXM}.
Under the map ((x,g),v) t-» (x,gv) from GL(?) x E to ?, the image of
H{x,e) GL(?) is
H{x,v)? = {(X, -u(X)v) | X e TXM}.
Since V*'?x vanishes on this space and satisfies
we see that it equals 6. ?
We will now study connections on the tangent bundle TM: these are often
called afflne connections.
Definition 1.21. The fundamental one-form 6 on M is the element of
A1{M,TM) such that i(X)9 = X for X 6 TM. If V is a connection on
TM, then the torsion of V is the differential form T = V0 e A2(M,TM).
If T = 0, we say that the connection V is torsion-free.
1.1. Fibre Bundles and Connections
29
If X and Y are two vector fields on M, then T(X, Y) e T(Af, TM) is given
by the explicit formula
T(X,Y) =
-[X,Y\.
A connection on TM gives rise to a covariant derivative on any vector bun-
bundle which is associated to GL(M), in particular the exterior bundle AT*M;
by naturality, this connection satisfies
Vx(a A /?) = Vxa A 0 + a A VX0
for any homogeneous differential forms a and /3, and vector field X. If a is
a k-form and X, Yi, ... , Yjb are vector fields, we see that
X-a(Yu...,Yk) =
it... ,Yh).
t=i
In statement of the following proposition, we use the exterior product map
e : T*M ® AT'M — A#+1r*M.
Proposition 1.22. If' V is a torsion-free connection on TM, the exterior
differential is equal to the composition
T(M, AT'M) X T(M, TM ® AT'M) A T(M, A#+1r*M).
Proof. Let us denote the operator defined above by D. Using the fact that
V satisfies Leibniz's rule, we see that D does too:
D(a0) = DaA0 + (-I)|or|a A D/3.
(Note that this part of the proof is independent of whether V is torsion-free
or not.)
Given Leibniz's rule, it clearly suffices to show that D agrees with d on
functions (which is clear), and one-forms. The proof now rests upon the
following formula: for any / e C°°(M),
where T e A2(M,TM) is the torsion of the affine connection V. In partic-
particular, if T vanishes, then D agrees with d on one-forms. To prove this, we
choose a local coordinate system on M, and let dx' and Xi = d/dxl be the
corresponding frames of the cotangent and tangent bundles. If we denote the
covariant derivative in the direction X< by V<, D is given by the formula
Thus, D(df) equals

30
1. Background on Differentia) Geometry
But by Leibniz's rule,
0 = Vi{dxi,Xk) = (Vtdxi,Xk) +
so that
D(df) = -J2e(dxidxk)
. a
»<*
Let M be a manifold of dimension n. Another example of a vector bundle
associated to the principal bundle GL(M) is the bundle, denoted |Am|", of
^-densities on M, where s is any real number. (If the manifold M is clear
from the context, we will write simply |A|5.) This is the bundle associated
to the character A ¦-> |det(^4)|-" of GL(n,R). An element of the fibre |A|J
is a function ip : AnTxM - {0} -> R such that rp(XX) = \X\*ip(X) for A ^ 0.
If U is an open subset of Rn, we denote by \dx\' the s-density such that
where 9< is the vector field d/dx% corresponding to differentiation in the ith
coordinate direction. Using a partition of unity, we see that we can always
construct a nowhere-vanishing section of |A|*, so that |A|* is a trivializable
bundle.
The importance of the one-density bundle, or density bundle as it is
usually known, comes from the following result, which follows immediately
from the change of variables formula for integrals in several variables.
Proposition 1.23. Let M be a manifold of dimension n. There is a unique
linear form, the integral, denoted by
L
:rc(M,|A|)
which is invariant under diffeomorphisms, and agrees in local coordinates
with the Lebesgue integral:
f f(x)\dx\= f f(x)dx1...dxn.
Jm Jr»
For any vector field X on M and a any compactly supported C1 density,
we see that
A.13)
/ C(X)a = 0.
Jm
Since C(fX)a = fC(X)a + X{f)a if a G T(M, |A|), it is easy to see that
if /3 G T(M, IAI1'2) is a half-density on M,
A.14)
1.1. Fibre Bundles and Connections
31
The bundle of densities |A| is very closely related to the bundle of vol-
volume forms A"T*M; the first corresponds to the character |det(>l)|~1 of
GL(n) and the second to the character det(A)-1. Unlike |A|, the line-bundle
AnT*M is trivializable if and only if M is orientable. If M is an orientable
manifold, its frame bundle GL(M) has two components; an orientation of
M is the choice of one component, which is a principal bundle for the group
GL+(n) of matrices with positive determinant. We say that frames in the
chosen component GL+(M) are oriented. An orientation of a manifold M is
equivalent to the choice of an isomorphism between the bundles A"T*M and
jAjvr|. If v ? An(M) is a volume form on M, the corresponding density |t;| is
such that \v\(X) = v(X) if X e AnTxM is oriented.
If M is an n-dimensional oriented manifold, we define the integral of a
compactly supported differential form a 6 Ac(M) to be the integral of the
density corresponding to <*[„]:
I a= I |o|Bj|.
Jm Jm
There is a fibred version of this integral. Let it : M —> B be a fibre bundle
with n-dimensional fibre, such that both M and B are oriented. If a e A*{M)
is a compactly-supported differential form on M, its integral over the fibres
of M —> B is the differential form JM/B a g Ak~n(B) such that
A.15)
/(
b kJm/b
m
for all differential forms 0 on the base B. We sometimes write w,a instead
of JM/B a. It follows easily from A.15) that
A.16) 7r*(a A tt*/3) = tt.q: A/?
for all a G Ac{M) and 0 G A(B).
The integral over the fibres commutes with the exterior differential, in the
sense that
dB f a = (-1)" f dMa.
Jm/b Jm/b
A.17)
Im/b Jm/b
This formula shows that the integral over the fibres induces a map
Hkc~n{B)
M/B
on de Rham cohomology.
The following result is an example of a transgression formula: it says that
the cohomology class of the integral over the fibres does not change if the
map n is deformed smoothly.

32
1. Background on Differential Geometry
Proposition 1.24. Let n : M x R —+ B be a smooth, map such that re1 =
"¦|Mx{t} : M —* B is a fibre bundle for each t. Then there is a smooth
t-dependent map irj : Ak(M) -» AkcTn-'l(B) such that
Proof. Think of n as a fibre bundle map M x R —> B x R which preserves
the (-coordinate. Let q : M x R —> M be the projection which forgets the
t-coordinate. Given a e Ak{M), the differential form ir,q*a € Ak~n(B x R)
may be decomposed as follows:
nmq*a = it\a + jf*a A dt,
where this formula serves as the definition of 7r»a. Applying this formula to
da, we see that on the one hand,
n*q*da = ir*mda 4- Trjda A dt,
while on the other hand,
A dt.
Equating coefficients of dt, the proposition follows. ?
Corollary 1.25. The map induced by tt? on deRham cohomology with com-
compact support is independent of t.
1.2. Riemannian Manifolds
If M is an n-dimensional manifold, then a Riemannian structure on M
is a metric on the tangent bundle TM. Using the Riemannian structure, we
can construct the orthonormal frame bundle, which is denoted by O(M):
O(M) = { (x, (ei,... , en)) | ei form an orthonormal frame of TXM }
An orthonormal frame p € O(M) over x 6 M may be considered to be an
isometry u = («i,...,«„) 6 Rn >-» Y.uiei G TXM from Rn to TXM. We
will denote by it : O(M) —* M the projection map. The bundle O(M) is an
O(n)-principal bundle, and the tangent bundle of M is an associated bundle
of O(M), corresponding to the standard representation Rn of O(n):
TMS*O(M)xo(n1R".
If, in addition, M is oriented, then the bundle of oriented orthonormal frames
is a SO(n)-principal bundle which we denote by SO(M).
1.2. Riemannian Manifolds
33
The tangent bundle TM of a Riemannian manifold M has a canonical
covariant derivative V, called the Levi-Civita connection. This is the
unique covariant derivative which preserves the Riemannian metric,
for all vector fields X and Y on M, and which is torsion-free,
It is easily verified from the above two formulas that the Levi-Civita derivative
VxY is given by the formula
A.18)
+X(Y, Z) + Y{Z, X) - Z{X, Y).
Denote by bo(TM) the bundle O(M) xo(n)*o(n), whose fibre at x € M is
the Lie algebra consisting of antisymmetric endomorphisms of the inner prod-
product space TXM. The curvature of the Levi-Civita connection is an so(TM)-
valued two-form on At, called the Riemannian curvature, which we will
denote by R € A2{M,so(TM)), that is, if X, Y and Z are vector fields on
M,
R(X,Y)Z = VXVYZ - VyVxZ - V[X<Y]Z.
If X and Y are two vector fields on M, we will write R(X, Y) for the section
of so(TM) given by contracting the two-form R with X and Y, and {RX, Y)
for the two-form obtained by contracting the curvature in its so(TM) part.
For future reference, we list some well-known properties of the Riemannian
curvature.
Proposition 1.26. Let X, Y, Z andW be vector fields on M.
A) R(X,Y) = -R(Y,X)
B) (R(W, X)Y, Z) + (Y, R(W, X)Z)=0
C) R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0
D) (R(W, X)Y, Z) = (R(Y, Z)W, X)
Proof. A) says that R is a two-form, and B) that Rx{W, X) e so(TxM).
Let us prove C), which holds for the curvature of any torsion-free connec-
connection V on TM. If R is the curvature tensor of V,
VxVyZ -VXVZY = Vx[Y,Z],
and similarly if we cycle the three vectors X, Y and Z. In this way, we obtain
three equations which, when added together, give
VxVyZ + VyVZX +
- VXVZY
= VX[Y,Z] + Vy[Z,X] + VZ[X,Y).
Clearly, this is equivalent to C).

34
1. Background on Differential Geometry
The last property is a purely algebraic consequence of the first three, which
we leave as an exercise. ?
When we are given a frame Xi for the tangent bundle with dual frame
X1, we can write the components of the curvature matrix in the form Rjkl =
{R(Xk,Xi)Xj,X1); here, the i and j indices refer to the fact that R takes
values in End(TM), and the k and 2 indices are the two-form indices. Using
the metric, define
A.19)
u = (R{Xk,Xi)Xi,Xi).
The components Rijki satisfy RijM = Rkiij- If c» is an orthonormal frame of
TM with dual frame e*, the two-form (Rei,ej) is given by the formula
A.20) (Rei,ej) =
From the curvature tensor, we may form a symmetric tensor
Ricy =
€ T(M,S2(TM)),
called the Ricci curvature, and its trace the scalar tm = ?|m Rimim called
the scalar curvature of M.
We denote by \dx\ e T(M, |A|) the Riemannian density associated to
the Riemannian metric on M, which is such that
|dx|(eiA...Aen) = l
when e< is an orthonormal frame of TM. The existence of |dx| corresponds to
the fact that the character | det | of GL(n) used to define the line bundle |A| is
trivial on the structure group O(n), so that the density bundle is canonically
isomorphic to If x R. It follows that \dx\ is covariant constant under the
Levi-Civita derivative. We will often write dx instead of |dx|, although this
notation should be restricted to oriented manifolds.
If / is a smooth function on a Riemannian manifold M, the gradient of
M, denoted grad/, is the vector field on M dual to df, that is, for every
vector field X on M,
(grad/,*) = *•/•
By a smooth path xt : [0,1] —> M on a Riemannian manifold, we mean
the restriction to [0,1] of a smooth map x±: (—e, 1 +e) —¦ M for some e > 0.
If xt is such a path, we define its length by the formula
where by ±t we mean the tangent vector in TXt to the path xt at time t. Define
the Riemannian distance between two points d{xo,X\) to be the infimum of
L{xt) over all smooth paths between them.
1.2. Riemannian Manifolds
35
A smooth path xt is called a geodesic if it satisfies the ordinary differential
equation
Vitxt = 0.
Like any second-order ordinary differential equation, the equation for a geode-
geodesic has a unique solution for t small, given the starting point xo and the
derivative x = xo 6 TXoM at the starting point. The end point Xi of the
resulting curve is called the exponential of the vector x at xo and is written
expI0 x. It is defined if x is small enough. (This terminology arises because
on a compact Lie group with the left and right invariant metric, the geodesies
turn out to be the same as the one-parameter subgroups and we recover the
exponential map of the Lie group.)
The derivative dexpIo of the exponential map at xo itself is just the iden-
identity map of TXoM, so that by the inverse-function theorem, the exponential
map is a diffeomorphism from a small ball around zero in TX0M to a neigh-
neighbourhood of xo in M. (The radius of the largest ball in TX0M for which this
is true is called the injectivity radius at xo.) Thus the exponential map
defines a system of coordinates around xo, called the normal coordinate
system (this is also known as the canonical coordinate system). The impor-
important property of this coordinate system is that the rays emanating from the
origin in TXoM map onto geodesies in M. We will use letters of the form
x to denote coordinates in the normal coordinate system: thus, x 6 TXaM
represents the coordinates of the point exp^ x.
We will now collect some formulas for the pull-back of the metric and
covariant derivatives by the exponential map. In particular we will see that
the ray from 0 to x is the curve of shortest length between xo and expl0 x
when x is small.
If we apply the construction of Proposition 1.17 to the tangent bundle
with its Levi-Civita connection, we obtain a smooth trivialization of TM
over a normal coordinate chart centred at a point xo G M. Choose an
orthononnal frame of the tangent space TXoM, with respect to which the
coordinate functions are x\ and the corresponding partial derivatives are d,;
thus, the vectors di are orthonormal at x0, although not in general elsewhere.
In this coordinate system, the point xo has coordinates x = 0. The radial
vector field U is the vector field defined by the formula "R, = YH=\ x'#«- Le*
e< be the orthonormal frame of TM obtained from the smooth trivialization
of TM in this coordinate patch; thus, e< = di + 0(|x|) and V^e* = 0.
Proposition 1.27. In the normal coordinate system, the following formulas
hold:
(i) vnn = n
B) H = J^x'e,, and thus (n,K) = |x|2
C)(»,&)=xi i
D) d(x0)expIox) = |x|

36
1. Background on Differential Geometry
Proof. A) In the normal coordinate system, the curve xt = tx is a geodesic.
Since H{xt) — txt, the geodesic equation gives
B) If we differentiate the function G2., ej) with respect to the vector field
72, we obtain
72G2,6,) = (Vwfc,ej) + G2, V^) = G2, et).
In other words, G2, e<) is homogeneous of order one. Since
?^) = xj + O(|x|2),
the result follows from these two equations.
C) The function G2,0<) is equal to YLj
Oflxj2). On the other hand,
dj), so it must equal x< +
Since the Levi-Civita connection is torsion-free and [72,9<] = —di, it follows
that
G2, vKdi) = (n, vdin) + (n, \n,
If we assemble all of the terms that we have obtained, we see that
72G2,90 = |ai|7Z|2 = xi,
from which the result follows.
D) The equation C) shows that 72 is orthogonal for the Riemannian metric
to the vector fields x%dj — x'di. If xt, 0 < t < 1, is any curve from 0 to x,
decompose its tangent vector xt into the sum of a multiple of 72 and a vector
lying in the span of the vector fields x'dj — xPdi tangent to the sphere. It
follows that
dt
thus the length of the curve Xt is greater than or equal to |x|. Since the
length of the curve xt = tx is equal to |x|, the formula follows. ?
A property of the normal coordinate system is that near the origin, the
coefficients of the metric (dt,dj)x agree with the Kronecker delta 5y up to
second order. Although we will not use the full force of the next proposition
in the rest of this book, we give its proof because of its applications to the
theory of the heat kernel.
1.2. Riemannian Manifolds
37
Proposition 1.28. The Taylor expansion of the functions gy(x), defined by
(dj,dj)x, in normal coordinates at x0 has the following form:
SiS -
kl
|c|>3
Proof. To prove this result, we will make use of two distinct frames of the
tangent bundle TM in a normal coordinate chart around a point zo, the
first being given by the vectors di (which is not an orthonormal frame in
general), and the second by the orthonormal frame e< obtained by radial
parallel transport of the orthonormal frame (#«)z0 of TXoM. The matrix
relating these two frames is the matrix 9\ such that
With respect to the frame e,-, the fundamental one-form
equals ? &ejt where & = ?\ 0f dx*. Thus the Revalued one-form 6
satisfies the structure equation
where w is the End(Rn)-valued one-form of the Levi-Civita connection in the
frame ej. Observe that
since 72 = J^x'ej, as follows from Proposition 1.27B).
Let fi = dui + w Aw be the curvature of M in this coordinate system; as
in Proposition 1.18, we see that ?G2)o> = tG2)fi. Substituting the formulas
iTiu = 0 and (?G2) - l)x = 0, into the identity i-n{d6+w A0) = 0, we obtain
(?G2) - \)C{nH = (?G2)w)x = (i(H)n)x,
where we think of the curvature Q as a matrix of two-forms, so that fix is a
vector of two-forms. Using the formula 72 = 53 x'et once more gives
A.21)
i)
Now ej is expressed as a function of the di by the inverse of the matrix 0J,
and 0j(xo) = 0y. Thus this equation determines the Taylor expansion of
0? to order m in terms of the Taylor expansion of the curvature coefficients

38
1. Background on Differential Geometry
= (Q(dkidi)dj,di)x to order m - 2. In particular, the linear term is
zero, and the quadratic piece may be calculated from A.21)
6b = *¦» - I E *'*****» + O(|x|3).
Since the metric on M is expressed in terms of the one-form 0 by the formula
the Taylor coefficients of <?y may be expressed in terms of those of 0. ?
The differential dxexpl0 of the exponential map expx : TXoM —* M at
the point x e TIOM is a map from TXoM to TXM, where x = expl0 x. Since
TXoM and TXM are metric spaces, the Jacobian
j(x) = |det(dxexpIO)|
is well defined. By definition, if e< is an orthonormal frame of TXM and
di = ?_, #i ej, then j(x) = | det@j)|, so that
A.22)
= |det(fly(x))|1/a.
The frames dt and e< of the tangent bundle TM that were introduced at
the start of the above proof will be used constantly throughout this book,
and the reader must be very careful to distinguish them. The first frame is
frequently useful because it satisfies the formula [di,dj] = 0, while the frame
Ci possesses the advantage of being orthonormal at each point in the normal
coordinate chart. In fact, we will only use the following two consequences of
Propositions 1.18 and 1.28 in the rest of this book:
= 4 ?
i
A.23) w<i
A-24) flyl
1.3. Superspaces
Although this is not a book on supergeometry, we will constantly use the
terminology of super-objects. Recall that a superspace E is a Z2-graded
vector space
and that a superalgebra is an algebra whose underlying vector space is
a superspace, and whose product respects the Z2-grading; in other words,
Ax ¦ Aj C Al+i. (We will denote the two elements of Z2 by + and -, although
1.3. Superspaces
39
we should really denote them by 0 mod 2 and 1 mod 2.) An exterior algebra
is an example of a superalgebra, with the Z2-grading
An ungraded vector space E is implicitly Z2-graded with E+ = E and E~ =
0. The algebra of endomorphisms End(E) of a superspace is a superalgebra,
when graded in the usual way:
End+(?) = Hom(?+, E+) 0 Hom(?-, E~),
End~(?) = Eom(E+,E-)®Hom(E-,E+).
Definition 1.29. A superbundle on a manifold M is a bundle ? = ?+®?~,
where ?+ and ?~ are two vector bundles on M. Thus, the fibres of ? are
superspaces.
If ? is a vector bundle, we will identify it with the superbundle such that
?+ = ? and ?~ = 0.
There is a general principle in dealing with elements of a superspace, that
whenever a formula involves commuting an element a past another 6, one
must insert a sign (—1)'°'' '*', where \a\ is the parity of a, which equals 0 or 1
according to whether the degree of a is even or odd. Thus, the supercom-
mutator of a pair of odd parity elements of a superalgebra is actually their
anticommutator, due to the extra minus sign. We will use the same bracket
notation for this supercommutator as is usually used for the commutator:
This bracket satisfies the axioms of a Lie superalgebra:
A) [a,fc] + (-l)M-W[M=0,
B) [a,[6,c]] = [[a,61,c] + (-l)loH6l[6,[O)C]].
We say that a superalgebra is super-commutative if its superbracket
vanishes identically: an example is the exterior algebra kE. The following
definition gives the extension of the notion of a trace to the superalgebra
setting.
Definition 1.30. A supertrace on a superalgebra A is a linear form <j> on
A satisfying <t>([a, b\) = 0.
On the superalgebra End(E), there is a canonical linear form given by the
formula
f TTJ5+ (a) - TrB~ (a) if o is even,
10 if a is odd.
Str(o)=||
Proposition 1.31. The linear form Str defined above is a supertrace on
End(E).

40
1. Background on Differential Geometry
Proof. We must verify that Str[a, 6] =0. If o and b have opposite parity, this
is clear, since [a, b] is then odd in parity and hence Str[o, 6] = 0.
If a = (°n+ °_ ) and b = (b+ ° ) are both even, then
has vanishing supertrace, since
TrB+ [o+, 6+] = TrB- [a~, b~] = 0.
If a = (o°+ a0 ) and b = ( fc° b~ ) are both odd, then
a~b+ +&-a+
0
0
+ 6+a-
has supertrace
Str[a,b) = TrB+ {a~b+ + b~a+) - TrE- (a+b~ + b+a~) = 0. ?
If jE = E+(BE~ and F = F+ ©F~ are two superspaces, then their tensor
product E® F is the superspace with underlying vector space E®F and
grading
(E ® F)+ = F-+ ® F+ 0 E~
(F, ® F)~ = F-+ ® F~ © J5T
If A and B are superalgebras, then their tensor product A ® B is the
superalgebra whose underlying space is the Z2-graded tensor product of A
and B, and whose product is defined by the rule
(ai ® 6x) -(as ® h) = (-1I*1'' Mai<i2 ® 6i«>2.
(This product is imposed by the sign rule.) Note that this Z2-graded product
is not the same as the usual product on A ® B, so is this algebra is some-
sometimes denoted A®B; however, we will never make use of the ungraded tensor
product, so there is no ambiguity.
Definition 1.32. If A is a supercommutative algebra and ? is a superspace,
we extend the supertrace on End(i?) to a map
Str : A ® End(E) —> A,
by the formula Str(a ® M) = aStr(M) for a e A and M 6 End(E).
This extension of the supertrace still vanishes on supercommutators in
A ® End(E), since
A.25) [a ® M, b ® N] = (-1)IM|' l6|a6 ® [M, N]
when A is supercommutative.
1.3. Superspaces
41
If Ei and E^ are two superspaces, we may identify End(??i ® E2) with
End(?i) ® End(E2), the action being as follows:
® e2) = (-l)|aj| • |eil(aiei) ® (a2e2),
where o< € End(Ei) and ei € E^
If E is a superspace, such that dim(S±) = m±, we define its determinant
line to be the one-dimensional vector space
det(F-) =
-1 ® Am-F,-.
Here, we make use of the commonly-used notation L for L* when L is a
one-dimensional vector space. The vector space det(E) is one-dimensional,
and if E+ = E~, then det(U) is canonically isomorphic to R.
Definition 1.33. If ? = ?+ © E~ is a superbundle, then its determinant
line bundle det(?) is the bundle
det(f) = (Am+? +)-1 ® (Am-5-),
where tn± = rk(?±).
For example, if M is an n-dimensional manifold, det(TM) is the volume-
form bundle AnT*M.
Definition 1.34. A Hermitian superspace is a superspace E such that
both E+ and E~ are complex vector spaces with Hermitian structures.
If E is a Hermitian superspace, we say that « € End~(E) is odd self-
adjoint if it has the form
/0 «-\
U={u+ oj'
where u+ : E+ —> E , and u is the adjoint of u+.
Let V be a real vector space with basis e^. We will denote <* =
fc k k ^
e* =
where
fc) and
= 6'j,
p
e(efc). We will identify AkV* with (AkV)* by setting
/ = {1 < i1 < • • • < t* < dim(V)} and
J={l<3i<-<jk< dim(V)}
are multi-indices, that is, subsets of {1,..., dim V}, and e1 = e" ... e**, ej =
eh ¦ ¦ ¦ eik ¦ (Here, as occasionally in the rest of this book, we write v w instead
oivAw for the product in an exterior algebra.) Using this identification, it is
easy to see that e(a)* = t(a) € End(AF) and that i(t>)* = e{v) e End(AV).
If A e End(V), we denote by X(A) the unique derivation of the super-
algebra AV which coincides with A on V C AV; it is given by the explicit
formula
A.26)
\{A) =

42
1. Background on Differential Geometry
A non-zero linear map T : AF —¦ R which vanishes on A*V for k < dim(V)
is called a Berezin integral. Let us explain why such a linear map is called
an integral. If V is a real vector space, the superalgebra AV is the algebra of
polynomial functions on the purely odd superspace with E+ = 0, E~ = V.
If e< is a basis for V with dual basis e*, the elements e' 6 AF* play the
role of coordinate function on V. Prom this point of view, the contraction
l(v) : AV* —> AF*, (v 6 F), is the operation of differentiation in the direction
v. We now see that a Berezin integral is an analogue of the Lebesgue integral:
a Berezin integral T : AF* —+ R is a linear form on the function space AF*
which vanishes on "partial derivatives":
A.27)
T • i(v)a = 0 for all v g V and a e AF'.
If V is an oriented Euclidean vector space, there is a canonical Berezin
integral, defined by projecting a € AF onto the component of the monomial
ei A.. .Aen; here n is the dimension of V, and e< form an oriented orthonormal
basis of V. We will denote this Berezin integral by T:
A.28)
otherwise.
If a g AF, we will often denote T(a) by a[ni, although strictly speaking c*[n]
is an element of AnF and not of R. If A g ArV, its exponential in the algebra
AF will be denoted by expA A.
Definition 1.35. The Pfafflan of an element A 6 A2F is the number
The Pfaffian of an element A g so(V) is the number
Pf (A) = r(expA Y^iAei< ei)ei A
If V. = R2 with orthonormal basis {ei,e2}, and if
then the Pfaffian of A is 9.
The Pfaffian vanishes if the dimension of V is odd, while if it is even,
Pf (A) is a polynomial of homogeneous order n/2 in the components of A. If
the orientation of F is reversed, it changes sign.
Proposition 1.36. The Pfaffian of an antisymmetric linear map is a square
root of the determinant:
Pf(AJ = det(A).
1.4. Superconnections
43
Proof. By the spectral theorem, we can choose an oriented basis ej of F such
that there are real numbers Cj, 1 < j < n/2 for which
In this way, we reduce the proof to the case in which F is the vector space R2,
and Ae\ = 0e2, Ae-z = —Oei. In this case, Pf(^4) = 6, while the determinant
of>l=(«-o*)is02. D
We will frequently use the notation
A.29) det1/2(jl) = Pt(A).
However, the choice of sign in the square root det1//2(A) depends on the choice
of orientation of E.
1.4. Superconnections
In this section, we describe Quillen's concept of a superconnection. If M is a
manifold and ? = ?+ © ?~ is a superbundle on M, let A(M, ?) be the space
of ?-valued differential forms on M. This space has a Z-grading given by the
degree of a differential form, and we will denote the component of exterior
degree i of a g A(M,?) by apj. In addition, we have the total Z2-grading,
which we will denote by
A(M,?) = A+(M,?) ®A-(M,E),
and which is defined by
for example, the space of sections of f* is contained in
If 0 is a bundle of Lie superalgebras on M, then A(M, g) is a Lie superal-
superalgebra with respect to the Lie superbracket defined by
[ax ® Xua2 ® X2] = (-
Aq2) ® [X1,X2].
Likewise, if ? is a superbundle of modules for g with respect to an action p,
then A(M,?) is a supermodule fear A(M, g), with respect to the action
In particular, this construction may be applied to the bundle of Lie superal-
superalgebras End(?), where ? is a superbundle on M, since AT*M®? is a bundle of
modules for the superalgebra AT*M ® End(?); we see that A(M, End(?)) is
a Lie superalgebra, which has A(M,?) as a supermodule. We will frequently
use the fact that any differential operator on A(M, ?) which supercommutes

44
1. Background on Differential Geometry
with the action of A(M) is given by the action of an element of A(M, End(?));
such an operator will be called local. (This is consistent with the "super"
point of view, that A(M) is the algebra of functions on a supermanifold fibred
over M, and that elements of A(M, End(?)) are the zeroth order differential
operators on this supermanifold.)
In defining a superconnection, Quillen abstracted the main properties of
a covariant derivative, namely, that it is an odd operator satisfying Leibniz's
rule. It turns out that many of the results which hold for ordinary connections
continue to hold for superconnections.
Definition 1.37. A) If ? is a bundle of superspaces over a manifold M,
then a superconnection on ? is an odd-parity first-order differential
operator
A:A±(M,?)->A*(M,?)
which satisfies Leibniz's rule in the Z2-graded sense: if a 6 A(M) and
9 eA(M,?), then
A(a A 0) = da A 0 + (-l)^a A A0.
B) If A is a superconnection on ?, then A is extended to act on the space
A(M, End(?)) in a way consistent with Leibniz's rule:
Aa = [A, a] for a € A{M, End(?)).
In order to check that Aa, as defined by this formula, is an element of
.4(Af,End(?)), we need only observe that the operator [A, a] commutes
with exterior multiplication by any differential form 0 e A{M).
Define the curvature of a superconnection A to be the operator A2 on
A(M,?).
Proposition 1.38. The curvature F is a local operator, and hence is given
by the action of a differential form F e ^4.(M,End(?)), which has total degree
even, and satisfies the Bianchi identity KF — 0.
Proof. The operator A2 is local because it supercommutes with exterior mul-
multiplication by any a e A(M):
[A2, <¦(<*)] = [A, [A, ?(«)]] = e(d2a) = 0.
Thus, the curvature A2 is given by the action of a differential form F €
A(M, End(?)). The Bianchi identity is just another way of writing the obvi-
obvious identity [A, A2] = 0. ?
It is clear that in the case when the superconnection ? is a connection, the
above definition of curvature coincides with that of Section 1.
A superconnection is entirely determined by its restriction to T(M,?),
which may be any operator A : T{M,?±) -* ./^(M.f) that satisfies
A(/s) = df ¦ s + /As for all / € C°°(M), s € T(M, ?).
1.4. Superconnections
45
Indeed, if we define A(a ® s) = das + (-l)lalaAs for a € A(M) and s e
T(M, ?), this gives the extension of A to all of A(M, ?).
In order to better understand what a superconnection consists of, we can
break it up into its homogeneous components A[jj, which map T(M, ?) to
^(M,?):
A = A[o] + A[ij -I- A[2] + .. .
Proposition 1.39. A) The operator A[i] is a covariant derivative on the
bundle ? which preserves the sub-bundles ?+ and ?~.
B) The operators A[4] for i / 1 are given by the action of differential forms
ww € ^'(M,End(?)) on A(M,?), where ww lies in ^(M,End~(?)) if
i is even, and in Ai(M,End+(?)) ifi is odd.
Proof. Let us decompose Leibniz's rule for A acting on T(M, ?) according to
degree:
1=0 t=0
It follows immediately from this that the operator A[ij is a covariant deriva-
derivative on the bundle ?; it preserves the Z2-grading of ?, since A[i] has odd total
degree, so that it is the direct sum of covariant derivatives on the bundles
On the other hand, for i ^ 1, we see that A[j](/s) = /(A^js), in other
words, that A^ : T(M, ?) —» A*(M, ?) is given by the action of a differential
form W[i] in A^ (M, End(?)); from the fact that A is an odd operator it follows
that Wfl is of odd total degree.
The actions of the operators A and o;[0] + A^j + up] + ¦¦¦ on A(M,?)
must agree, since any two operators that agree when restricted to T(M, ?) =
A°(M,?) and satisfy Leibniz's rule are equal. ?
Corollary 1.40. The space of superconnections on ? is an affine space mod-
modelled on the vector space ^-(M,End(?)). Thus, if A, is a smooth one-
parameter family of superconnections, then dX3/ds lies in .A~(M,End(?))
for each s.
We will call Ajij the covariant derivative component of the supercon-
superconnection A.
As an example of a superconnection, let us take a trivial bundle on M
with fibre a complex superspace E. If L e C°°(M, End"(?)), we may form
a superconnection by adding L to the trivial connection d. The curvature of
the superconnection d + L is then F = dL + L2. Relative to the splitting
E = E+ © E~, we may decompose L as
L =

46
1. Background on Differential Geometry
where L+ 6 C°°(M,Hom(E+,E-)) and L~ e C°°(M,Hom(E-,E+)). In
terms of this decomposition, the curvature of d + L may be written
(L-L+ dL~ \
F-\dL+ L+L-y
If ? and T are superbundles, then ? ® T is the superbundle graded by
= ? +
If A and B are superconnections respectively on ? and T, there is a tensor
product superconnection A®l + l®Bon?®.F defined by the formula
(A ® 1 + 1 ® B)(q A /?) = ka A 0 + (-l)la|a.B/?,
for a e A(M, ?) and /? G A(M, T).
If <f> : M —> N is a smooth map and ? is a superbundle on N with
superconnnection A, there is a unique superconnection (j>*k on <j>*? such that
<t>*AD>*a) = 0*(Aa) for a e ,ABV, ?), called the pull-back of A by <f>.
1.5. Characteristic Classes
Let ? be a superbundle, either real or complex, with superconnection A, on
a manifold M. Following Quillen, we will associate to this superconnection
certain closed differential forms, called the characteristic forms of A. These
give invariants of the superbundle ? in the deRham cohomology H'(M)
of M; the cohomology classes thus obtained are called the characteristic
classes of ?. We will only dealt with the special case of characteristic forms
associated to invariant polynomials of the type Str(^4*). In Chapter 7, we will
give a generalization of the theory of characteristic classes to the equivariant
context, where a Lie group acts on the data.
In order to define the characteristic forms, we need the supertrace map
on the space of differential forms A(M, End(?)). This is defined by means of
the supertrace Strx : End(?)x —> C, defined in each fibre of End(?) by
= Tr(a)-Tr(d).
Applying the construction of Definition 1.32 with auxiliary supercommutative
algebra AT*XM, we obtain a bundle map
Str : AT*M ® End(?) -¦ AT'M
which, applied to the sections of these bundles, gives the desired supertrace:
Str : A(M, End(?)) -» A(M),
1.5. Characteristic Classes
47
characterized by the formula Str(a ® A) = a Str(-A) for a e A(M) and
A € F(M, End(?)). The algebra A(M) is supercommutative, so by A.25),
this extension behaves like a supertrace from a superalgebra to C, in that it
vanishes on supercommutators. It is clear that the supertrace map preserves
the Z2-gradings of these spaces, that is, it maps A±(M, End(?)) to ^4±(M).
Suppose f(z) is a polynomial in z. If A2 G A+(M, End(?)) is the curvature
of the superconnection A on ?, then /(A2) is the element of A+(M, End(?)),
defined by
If we apply the supertrace map
Str : A+(M,End(?)) -> A+(M)
to /(A2), we obtain a differential form on M denoted by Str/(A2). We will
call this differential form the characteristic form, or Chern-Weil form, of
A corresponding to the polynomial f(z).
Proposition 1.41. A) The characteristic form Str(/(A2)) is a closed diff-
differential form of even degree.
B) (Transgression formula) If At is a differentiable one-parameter family of
superconnections on ?, then
where dkt/dt G A(M, End(?)) is the derivative of At with respect to t.
C) // Ao and Ai are two superconnections on ?, then the differential forms
Str(/(Ajj)) and Str(/(A2)) lie in the same deRham cohomology class.
Proof. The proof makes use of the following lemma.
Lemma 1.42. For any a € -4(A/,End(?)), d(Stra) = Str([A,a]).
Proof. In local coordinates on a subset U C M and with respect to a triv-
ialization of ? over U, A = d + w, where u G A{U, End(?)) is the super-
connection form of A in this coordinate system, and hence Str([A, a]) =
Str([d,a]) + Str([w,a]). The second term Str([w,a]) vanishes, while the first
equals Str(da). ?
Using this lemma, we see that
dStr(/(A2))=Str([A,/(A2)]) = O.
This proves that Str(/(A2)) is closed; the fact that it has even degree is clear,
since /(A2) is even, and Str preserves degree.
Let At be a differentiable one-parameter family of superconnections on ?,
with curvatures A2. We see that
dkf

48
1. Background on Differential Geometry
(where the bracket is, of course, an anticommutator, since both At and dAt/dt
are odd operators). We now use the following formula: if at : [0,1] —¦
A+(M, End(?)) is a family of differential forms on M with values in End(?)
of even total degree, then
where at is the derivative of at with respect to t. To prove this, it suffices to
consider the case f(z) = zn:
jt Str«) =
=0
where the last step uses the fact that Str(-) is a supertrace in order to replace
Str(ajdta"~*~1) by Str(dtaJl~1). Applying this with at = A?, we obtain
= Str([Ae, ^/'(A?)]) since [At, A?] = 0
If Ao and Ai are two superconnections and ui = Ai — Ao, then we may
integrate the transgression formula applied to the family
= A - t)Ao + tAi = Ao + tui.
We see that
Str(/(A?)) - Str(/(A2)) = d C Str(a;/'(A2))dt
Jo
which shows explicitly that Str(/(Ag)) and Str(/(A2)) differ by an exact
form. ?
There is another way to obtain the transgression formula, which is similar
to Proposition 1.24. Introduce the space MxR, with projection q : MxR —»
M. Using the family At of superconnections, we may define a superconnection
A on the superbundle q*?, by the formula
ds
The curvature T of A is
¦r = T, — A ds,
1.5. Characteristic Classes
49
where T, = A2 is the curvature of A,; thus,
dA,
Since ds Ads = 0, we see as in the preceding proof that underneath the
supertrace,
^/'(JF.)) Ada.
Since Str(/(^)) is closed in A(M x R), we infer that
A special case of the above construction of characteristic forms is that
in which the bundle ? is ungraded and the superconnection A is a covari-
ant derivative V with curvature F E A2(M,End(?)); then Str(V2)* is just
the differential form Tr(Ffc) € A2k(M). In this case we can allow f(z)
to be a power series in z, since the curvature F is a nilpotent element in
<4(M,End(?)). The differential form Tr(/(F)) is usually called the Chern-
Weil form associated to the invariant power series A i-» Tr(/(j4)) on gl(N),
(where N is the rank of ?). If A is a superconnection, we can allow / to be
any power series with infinite radius of convergence.
It should be clear that any differential form which is a polynomial
P(Str(/1(A2)),...,Str(/fc(A2)))
is a closed even differential form. Since the ring of invariant polynomials
on g[(N) is generated by the polynomials Tr(afc), the ring of all forms thus
obtained coincides with the classical ring of Chern-Weil forms, in the case of
an ungraded bundle.
The reader is warned that topologists prefer to consider the characteristic
forms defined by the formula Str/(-F/27ri), where F is the curvature of a
covariant derivative V, because the Chern class, the cohomology class of
c(F) = det(l — F/2ni), defines an element of the integral cohomology of M
only if these negative powers of 2ni are included. However, from the point of
view of the local index theorem, this is not as natural, which is why we have
preferred to give our less conventional definition.
Three cases of the construction given above are of special importance for
us.
A) If ? = ?+ © ?~ is a complex Z2-graded vector bundle and f(z) —
exp(—z), the characteristic form obtained is called the Chern character
form of the superconnection,
A.30)

50
1. Background on Differential Geometry
The Chern character form is additive with respect to taking the direct sum
of superbundles with superconnections:
A.31)
ch(A © B) = ch(A) + ch(B).
Furthermore, it is multiplicative with respect to taking the tensor product of
superbundles with superconnections:
A.32)
ch(A ® 1 + 1 ® B) = ch(A) A ch(B).
These two formulas explain why the differential form ch(A) is called a char-
character.
The transgression formula for the Chern character of a family of super-
connections gives
which implies that
A.33) ch(Ao) - ch(Ai) = d f Str[^e-A?) dt.
This proves the following proposition.
Proposition 1.43. The cohomology class of ch(A) is independent of the
superconnection A on ?; we will denote it by ch(?).
In particular, we see that replacing A by its covariant derivative component
V = A[!j does not change the cohomology class of the Chern character:
[ch(A)] = [ch(V)].
However, the connection V preserves the bundles ?±, and if we denote by
V+ and V~ the restrictions of V to ?+ and ?~, we have
ch(A) = ch(V) +da = ch(V+) - ch(V~) + da
for some differential form a. In particular, if we take the corresponding
cohomology classes, we obtain
A.34) ch(?) = [ch(V+)] - [ch(V-)] = ch(?+) - &(?-).
B) If ? is a real vector bundle with covariant derivative V of curvature F,
we associate to it its A-genus form,
A.35)
Using the formula det(^4) = expTr(log A), we see that i(V) may be under-
understood as the exponential of the characteristic form associated to the power
series 5log((z/2)/sinh(z/2)), that is,
= expTrf Jl
1.6. The Euler and Thorn Classes
The .A-genus is multiplicative, in the sense that
51
The reason that A(V) only has non-vanishing terms in dimensions divisible
by 4 is that the function f(z) is an even function of z, so f(V2) only involves
even powers of the curvature V2. In fact, the coefficients of the .4-genus of ?
are polynomials in what are called the Pontryagin classes of M, but we will
not need to know how these are defined; the first few terms of the expansion
of A(V) in terms of the classes Tr(Fk) are as follows:
A.36)
, V) = 1 -
^ Tr(F<)
The cohomology class of A(V) will be denoted A(?). In particular, the A-
genus A(M) of a manifold M is defined to be the .A-genus of its tangent
bundle TM.
C) If ? is an ungraded complex vector bundle with a covariant derivative
V with curvature F, we define its Todd genus form by the formula
Thus
Like A(V), the Todd genus is multiplicative. The cohomology class of Td(V)
will be denoted TdE). In particular, the Todd genus Td(M) of a complex
manifold M is the Todd genus of its tangent bundle TM.
As we will see, the .A-genus and its cousin the Todd genus arise quite nat-
naturally in a number of situations in mathematics, for example, in the Atiyah-
Singer index theorem and in the Weyl and Kirillov character formulas.
1.6. The Euler and Thorn Classes
In this section, we will define a characteristic class, the Euler class, which
differs from the characteristic classes defined in the last section, in that it
is only defined for oriented bundles. However, we will see that it has a
definition quite analogous to that of the Chern character. This section is
only a prerequisite to reading Section 7.7.
We start by describing the Thom form of an oriented vector bundle. Let
M be a compact oriented manifold and let p : V —> M be a real oriented
vector bundle over M of rank N. Let j : M —* V be the inclusion of M into
V as the zero section.

52
1. Background on Differential Geometry
Definition 1.44. A Thorn form for the bundle V is a closed differential
form U 6 ^(V) such that
/ U = 1 € A°(M).
JV/M
fV/M
We will show that a Thorn form exists, following a method of Mathai-
Quillen.
The following lemma is a fibred version of the homotopy used to prove the
Poincare lemma.
Lemma 1.45. Let Tt be the vertical Eider vector field on V, and let ht(v) =
tv. With H : Am(V) -> A—1^) defined by the formula
h/3
= hi(
Jo
we have the homotopy formula
P-P* U*P) = (dH + HdH for all 0 6 A{V).
Proof. First, note that the integral converges in the definition of H because
¦R. vanishes at 0. If 0 € A(V), let 0t = h*t0, and observe that 0O = p*j*0 and
that 0i = 0. Differentiating by t and using integrating the Cartan homotopy
formula
we obtain the lemma.
Corollary 1.46. The maps
0y—+j*0:H*{V)->E*{M) and
are inverses to each other.
We need another lemma, which is a consequence of Proposition 1.24.
Lemma 1.47. Let p:V —» M be a real oriented vector bundle over a mani-
manifold M. There is a smooth bilinear map m(a,0) from Alc(V) x A3c(V) to
,4«+j-i(V) such that
{jfp.a) A 0 - a A (p>./3) = dm(a, 0) - (m(da, 0) + (-l)|o|m(a, d0)).
Proof. Consider the one-parameter family kl of submersions from the Whit-
Whitney sum V xp V to V given by the formula
(v, w)eVxxVxi—»tv + A - t)w € Vx.
1.6. The Euler and Thorn Classes
53
If a and 0 are differential forms, we may form their external product a t3p 0,
which is a differential form on V xp V. We easily see that
Thus, it suffices to take m(a, 0) equal to
m{a,0)= f ki(aBp0)dt. ?
Jo
The following result explains the importance of Thorn classes.
Proposition 1.48. Let M be a compact oriented manifold and letp:V—>
M be a real oriented vector bundle over M of rank N, with Thorn form
U 6 A?{V).
A) The maps
0<—>Bn)-N'2pt0:HZ(V)^H-N(M) and
a>—>Ut\p*a: H'(M) -» H^+N{V)
are inverses to each other.
B) For every closed differential form 0 in A{V),
/ U A 0 = / 3*0.
Jv Jm
Proof. By A.16), B7r)-JV/2p,(t7 Ap'a) = a for all a € A(M).
The other direction of A) follows from the homotopy formula
0 - BnyN^UA{p'p.0) = BnrN^dm(U,0) - (-l)NBn)-N/2m(U,d0),
which follows from Lemma 1.47 on setting a = Bir)~N/2U.
To prove B), we use the homotopy H of Lemma 1.45. It follows that if
0 G A(V),
+ Btt)-jv/2 / U A dH0 + Btt)-n/2 f U A Hd0.
Jv Jv
The first term on the right-hand side equals fM j*0, the second term vanishes
by Stokes's theorem, since U is closed, and the third term vanishes since 0 is
closed. ?
In this section, we will construct a Thom form using the Berezin integral.
Let us start with a model problem, the construction of a volume form U on
an oriented Euclidean vector space V with integral JVU = Bir)^2; this is a

54
1. Background on Differential Geometry
Thom class for V considered as a vector bundle over a point. If x1,..., xN
is an orthonormal coordinate system on V, we define U by
() A ... A dxN.
We will show later how to remedy the fact that U does not have compact
support.
Now, consider the identity map V —> V to be an element of A°(V, V), with
exterior differential dx 6 Al{V, V). The exponential e~idx lies in A{V, AV).
We extend the Berezin integral T : KV —> R to a map T : A(V, AV) —» A(V)
by
T(a ® 0 = aT(?) for a € A(V) and ? 6 AK.
Lemma 1.49. A) The form U equals
U(x) =
where e(N) — 1 if N is even and —i if N is odd.
B) The integral ofU over V equals Btx)n>2.
Proof. Let eu be an orthonormal basis of V dual to the basis x* of V*. We
have
N
A ... A dxw>
and A) follows, since (-t)JV(-l)JV(JV-1)/2 equals 1 if AT is even and -z if N
is odd. The integral of U is calculated using the Gaussian integral
F
We will define a Thom form on an oriented Euclidean vector bundle p :
V —> N of rank N by modification of the formula of Lemma 1.49. Choosing
a Euclidean connection Vv on V, we may replace dx by Vvx e ^41(V,p*V),
where x is the tautological section of p*V. However, because the connection
Vv need not be flat, we will see that some modification to the definition of
U is necessary.
Suppose ? is an oriented Euclidean vector bundle of rank N over B. The
Berezin integral T defines a map
T:A(B,A?)->A(B).
If V is a covariant derivative on ?, we obtain a canonical covariant derivative
on AV.
1.6. The Euler and Thom Classes
55
Proposition 1.50. If V is a covariant derivative on ? compatible with the
metric on ?, then for any a E A(B, A?), we have
dT(a) = T(Vq).
Proof. Let N be the dimension of the fibres of ?. The Berezin integral is
given by pairing a section of AN? with the section T € F(B,AN?*) given
with respect to a local orthonormal oriented frame e* of ?* by the formula
e1 A ... A eN. Since VT = 0, the result follows by Leibniz's rule. ?
We apply this proposition with B = V and ? — p*V, and with connection
Vp"v = p* Vv. The space of sections A = A(V, A(p*V)) is a bigraded algebra,
since it may be decomposed as a direct sum of subspaces
Ai-*=Ai(V,AiptV).
The covariant derivative V*v defines a map V : A*'* —» At+lti. If s e
r(V,p*V), the contraction a(s) is defined by the formula
i
a(s)(a ® (si A ... A s,)) = ]T)(-l)|ai|+*~1(s,
A ... A si A ... A Sj
*=i
for a e A(V) homogeneous and Sk e r(V,p*V), A < k < j). The contraction
a(s) defines a map a(s): AiJ -» A^'1. We will identify so(V) with A2V by
the map
A e so(V) i—-> J2(Aei<ei)ei A ei-
For any s € A0-1 = T(V,p*V) and a e A, T(a(s)a) = 0, since a(s)a has no
component in A'tN. It follows that the Berezin integral T : A(V, Ap*V) —>
A(V) also satisfies the formula
Let us list some elements of the algebra A:
A) the tautological section x e A0-1 = r(V,p*V);
B) the elements |x|2 € .40'0 and Vx G A1'1, formed from x;
C) the curvature F = (VvJ € A2(M,so(V)) gives an element of A2'2 =
A2(V,A2p*V) by identifying it with an element of .42(M,A2V), and
pulling it back to V by the projection p : V —> M; abusing notation,
we will write this pull-back as F also.
Lemma 1.51. A) Let Q, = |x|2/2 + iVx + F e A. Then
(V - za(x))fi = 0.

56 1. Background on Differential Geometry
B) Ifp G C°°(R), define p(fi) e A by the formula
fc=O
Then (V - *a(x))p(n) = 0.
Proof. By Leibniz's rule, we have V(|x|2) = -2a(x)Vx. By the definition of
curvature, we have V(Vx) = a(x)F, while by the Bianchi identity, we have
VF = 0. Combining all of this with the obvious facts that a(x)x = |x|2 and
a(x)|x|2 = 0, we see that (V - ia(x))ft = 0.
The formula (V — io(x))p(fl) = 0 now follows from the fact that V — ta(x)
is a derivation of the algebra A. D
Note that fi € Eo<fc<2^*'fc. ^ that P(Q) e Eo<*<at ¦**'*¦ We see that
T(p(Sl)) e AN(V), and the above lemma shows that
dT(p(il)) = T((V - to(x))p(fi)) = 0,
so that T(/j(fi)) is a closed JV-form on V.
We now define the form U on V by choosing p to be the function p(x) =
A.37) U = T(e-n) = r(e-<l
We will study in greater detail the analogies between this formula and that
of the Chern character in Section 7.7. Observe that the Thom form depends
on the orientation of V (through T), on its metric, and on its connection.
Proposition 1.52. The form Bn)~Nl2U has integral 1 over each fibre if N
is even, and i if N is odd.
Proof. To calculate JViM U, it suffices to consider the the case in which M
is a point, so that V is a vector space; this is just Lemma 1.49. ?
The Thom form U constructed above has rapid decay at infinity instead
of being compactly supported. However, using the diffeomorphism from the
interior of the unit ball bundle of V to V given by the formula
y (i-|y|2I/2'
we can pull back the Thom form U to obtain a Thom form with support in
the unit ball bundle of V.
Just as for the Chern character, there are transgression formulas for the
Thom form which show how it changes when the metric is rescaled and when
the connection changes. Let us first consider the effect of rescaling the metric:
this is equivalent to replacing fi by
Let Ut denote the Thom form corresponding to Qt-
1.6. The Euler and Thom Classes
Proposition 1.53. We have the transgression formula
57
dt
Proof. Observe that
^ = «|x|2 + tVx = i(V - tta(x))x.
at
Since (V - tto(x))nt = 0, we see that
Tte~°'= -^e"n' = "i(v - i
and hence that
(e-"'). ?
By a very similar proof, we obtain a transgression formula for the Thom
form under a change of connection. Let U, be the Thom form corresponding
to a smooth family of connections V, on V compatible with its inner product.
The derivative of this family dV,/ds is an element of A2(M, A2V); we also
denote by dV,/ds the element of A1'2 = >11(V,A2V) obtained by pulling
back dV,/ds to V, and let fi, = |x|2/2 + iV.x + F, be the element of A
corresponding to the connection Vs.
Proposition 1.54. We have the transgression formula
ds
Proof. It is easy to see that
ds
^i = (V -ia(x)) —
ds * ds
Using the formula (Va — ia(x))ilB = 0, we see that
and hence that
dU,
~ds~

58
1. Background on Differential Geometry
Let j : M —> V be the inclusion of M in V as the zero-section. The pull-
back j*U is a closed differential form on M called the Euler form of V, and
denoted x(Vv). Thus, if F is the curvature of Vv, we have
(L38) j*U = x(Vv) = Pf(-F).
Proposition 1.55. The Euler form x(Vv) is a closed differential form of
degree N on M whose cohomology class depends only on the bundle V and
its orientation and not on the metric and covariant derivative on V.
Proof. Choose a metric on V. If V, is a one-parameter family of Euclidean
connections on V with corresponding Thorn class Ua, the transgression for-
formula A.54) for U, implies that
X(VO) - x(Vi> = d^ T (^ A expA(-Ft) j dt,
which is the analogue of the trangression formula A.33) for the Chern char-
character. This shows that the cohomology class of x(V) is independent of the
connection used in its definition.
We can also show that the cohomology class of x(V) is independent of the
metric used in its definition. If (•, H and (•, )i are two metrics on V, then
we can find an invertible orientation preserving section of End(V) such that
If Vo is a Euclidean covariant derivative for the metric (•, H, then it is
easy to see that Vi = g~x ¦ Vo -g is a Euclidean covariant derivative with
respect to the metric (•, )i. If we now construct the Euler forms of the two
covariant derivatives V< with respect to the metrics (•, )j, we see that they
are equal. ?
A case of special importance is that in which the bundle V is the tangent
bundle TM of an oriented even-dimensional Riemannian manifold. In this
case, the differential form x(V), where V is the Levi-Civita connection, is
called the Euler form x(M) of the manifold, and lies in An(M), where n
is the dimension of M. For example, if M is a two-dimensional Riemannian
manifold, and if {ei,e2} is an orthonormal basis of TXM for which R(ei,e2) =
Let v 6 F(M, V) be a section of the vector bundle V. The pull-back v*U
is a closed differential form of degree TV on M, given by the formula
v*u = r(e-(|"|a/2+<vV"+F)).
The transgression formula Proposition 1.53 implies that v*U is cohomologous
to the Euler class for any v € F(M, V):
*(VV) - v*V = -id ^ T(v A e-
Jo
dt
1.6. The Euler and Thorn Classes
59
The above results imply a form of the Gauss-Bonnet-Chern theorem, which
is a formula for the Euler number of a manifold in terms of the integral of its
Euler class. This is the simplest example of an index theorem, which will is
the main topic of this book.
Let v e T(M, TM) be a vector field on M. At a zero p of v, the Lie bracket
induces an endomorphism Lp(v) of TPM; in a coordinate system in which
p = 0 and v = ?"=1 v*di, Lp(v) is given by the formula
A zero p of v is called non-degenerate if Lp{v) is invertible. We denote by
u(p, v) € {±1} the sign of the determinant of Lp(v).
If all of the zeros of v are non- degenerate, we say that t; is non-degenerate.
If M is compact, such a vector field has only a finite number of zeroes.
A non-degenerate vector field may be constructed with the help of a Morse
function /, that is, a real smooth function on M such that at each critical
point p of /, there is a coordinate system in which
If we choose a Riemannian metric on M which is equal to ?V dxi 0 dx*
in each of these coordinate neighbourhoods, we see that the vector field
grad / equals ?"=i A<x'c>j around the critical point p of /, and hence that
i/(p,grad/) = niLi^t- The following theorem is Chern's generalization of
the Gauss-Bonnet theorem to arbitrary even-dimensional orientable compact
manifolds. The form in which we derive it here is not the usual statement
of the Gauss-Bonnet-Theorem, because it is phrased in terms of a defini-
definition of the Euler number in terms of the zeroes of a non-degenerate vector
field on M; however, this agree!) with the definition of the Euler number as
alternating sum of the Betti numbers of M, as we will prove in Theorem 4.6.
Theorem 1.56 (Poincare-Hopf). If v is a non-degenerate vector field on
a compact manifold M, then
E
{p|«(p)=o}
v{p,v) = {2*)-n'2 f
Jm
In particular, the sum
Eul(M)= J2 "(P'")
{p|«(p)=0}
is independent of the non-degenerate vector field; it is called the Euler num-
number of the manifold.

60
1. Background on Differential Geometry
Proof. Choose a coordinate chart in a neighbourhood Up of each zero p of v.
In this coordinate chart, the tangent bundle is trivialized, TUP Slfx Rn.
The vector field v defines a smooth map from Up to R" such that v(p) = 0,
with invertible derivative Lp(y) at p. It follows that v maps an open subset
V of V diffeomorphically to a small ball BcR". The orientations on V
induced by M and by the diffeomorphism v to Rn differ by the sign i/(p, v).
We may assume the open sets V to be disjoint. Choose a Riemannian metric
on M which on V equals the metric induced by the diffeomorphism into Rn.
Since M is compact, we see that there is a positive constant e > 0 such that
\\v\\ > e on the set M\ Up V.
Let U be the Thom form of TM, with respect to the given Riemannian
metric and its associated Levi-Civita connection. For each t > 0, we have
the formula
I v*tU = f X(M).
M JM
Let i/>: R+ —+ [0,1] be any smooth function such that
ip(s) = 1 if a < e2/4
4>{s) = 0 if s > e2.
Then
A.39)
/
M
v*tU= f (l-rP(\\v\\2))v*U+ f rP(\\vf)v*U.
JM JM
Since v%U is of the form
i=0
where ai are differential forms on M, it is rapidly decreasing as a function
of t as t —> oo when ||u||2 > e2/4. We see that the first integral in A.39) is
a rapidly decreasing function of t. The second integral is a sum of integrals
over the neighbourhoods Vp.
On Vp, the metric and connection are trivial and v = J27=i x*^»> so tna*
v*tU\w = tni/(p,t;)e-'a||x||:'/2dx1 A ... A dxn.
Thus
/
vp
Making the change of variables x —»t-1x, we see that
lim tn f rPiWxW^e-t'W^dx1 A ... A dxn = B7r)n
t-+oo JR"
Taking the sum over the zeroes p of v, the result follows. ?
Bibliographic Notes
61
There is a generalization of this result, to the case where we only assume
that the zeroes of v are isolated.
Definition 1.57. The index i/(p, v) of a vector field v on a manifold M
at an isolated zero v(p) = 0 is defined by choosing a coordinate system
around p, in which case v(p, v) is the degree of the map from the sphere
Su = {x G R" | ||x|| = u} to the unit sphere defined by
The index v(p, v) is an integer which is independent of the coordinate chart
used in its definition, as well as the small parameter u.
We can again choose small balls V around the zeroes of v such that
||t>|| > e on the complement of UPVP. The map sending x € V to v(x) is
a covering map over the set of regular values, and using Sard's theorem, we
see that
f
This is the outline of the proof of the following extension of Theorem GBC.
Theorem 1.58 (Poincare-Hopf). Ifv is a vector field with isolated zeroes
on a compact oriented even dimensional manifold M, then
E •
{pKp)=o}
[
m
X{M).
Bibliographic Notes
The book of Kobayashi-Nomizu [67] is perhaps the best reference for the
theory of bundles and connections.
Section 2. For background on Riemannian manifolds, the reader may re-
refer to the books of Kobayashi-Nomizu, Milnor [76] and Berger-Gauduchon-
Mazet [19].
The proofs of Proposition 1.18 and 1.28 are taken from the second appendix
of Atiyah-Bott-Patodi [8].
Sections 3 and 4. It was Berezin[26], motivated by the problem of under-
understanding geometric quantization for fermions, who emphasized the strong
analogy between calculus on superspaces and ordinary vector spaces.
The notion of a superconnection was developed by Quillen in order to
provide a formalism for a local family index theorem for Dirac operators [86].
This hope was realised by Bismut, as we will see in Chapters 9 and 10. The
notion of superconnection has turned out to be fruitful in other contexts, too.

62
1. Background on Differential Geometry
Section 5. We have presented the theory of characteristic forms in the con-
context of superconnections, following Quillen. The construction of characteris-
characteristic forms for connections is due to Chern and Weil; see Kobayashi-Nomizu.
Our interest is more in the characteristic forms themselves than in their co-
homology classes. The reader interested in the topological point of view may
consult Bott-Tu[41] and Milnor[78]. This section is also an introduction to
Chapter 7, where we define equivariant characteristic forms; thus, the biblio-
bibliographic notes of that chapter are relevant here.
Section 6. The results on the Thorn form and the Euler form presented here
are due to Mathai and Quillen [74], although we give more direct proofs which
use the Berezin integral. The proof of Theorem 1.56 is close to Chern's proof
of the Gauss-Bonnet-Chern theorem [44]; we have presented it in such a way
as to show the analogy with the theorems of later chapters. For more on the
proof of the Poincare-Hopf Theorem 1.58, see Bott and Tu [41].
Chapter 2. Asymptotic Expansion of the Heat Kernel
Given a Riemannian structure on a manifold M, one may construct the scalar
Laplacian A = d*d, which is the operator on the space of functions on M
corresponding to the quadratic form
= / (df,df)x\dx\
J\f
This operator can be used to study the geometry of the manifold, and is also
important because it arises frequently in physics problems, such as quantum
mechanics and diffusion in curved spaces. All of the eigenfunctions of A are
smooth functions and, if the manifold M is compact, A has a unique self-
adjoint extension. The Laplacian has a heat kernel, which is the function
kt(x, y) on M+ x M x M which solves the equation
subject to the initial condition that lim kt(x, y) is the Dirac distribution
along the diagonal 6(x,y). It turns out that the heat kernel kt(x,y) is a
smooth function whose behaviour when t becomes small reflects the local
geometry of the manifold M. In particular, there is an approximation to
kt(x, y) near the diagonal, of the form
**(*, V)
, y),
«=o
where d(x, y) is the geodesic distance between x and y, the integer n is the
dimension of M, and the functions fi(x,y) are given by explicit formulas
when restricted to the diagonal.
In this chapter, we will construct the heat kernel of a more general class
of operators which generalizes the scalar Laplacian to vector bundles. The
construction is extremely intuitive geometrically, being based on the explicit
formula involving a Gaussian for the heat kernel of the Laplacian on Eu-
Euclidean space. Since the manifold on which we work is compact, we require
no estimates more sophisticated than the summation of geometric series.
These generalized Laplacians are introduced in Section 1, while the con-
construction of the heat kernel, when M is compact, occupies Sections 2-4. In
Section 5, we give some applications, such as the smoothness of eigenfunctions
of generalized Laplacians and the Weyl formula for the asymptotic number

64
2. Asymptotic Expansion of the Heat Kernel
of eigenfunctions in terms of the volume of the manifold and its dimension.
In Section 6, we discuss the dependence of the heat kernel on the Laplacian
if the coefficients of the Laplacian are varied, smoothly.
2.1. Differential Operators
Let M be a manifold, and let ? be a vector bundle over M. The alge-
algebra of differential operators on ?, denoted by V(M,S), is the subalgebra of
End(r(M,?)) generated by elements of r(M,End(?)) acting by multiplica-
multiplication on r(M,?), and the covariant derivatives Vx, where V is any covariant
derivative on ? and X ranges over all vector fields on M. The algebra V(M, ?)
has a natural filtration, defined by letting
V{(M, ?) = T(M, End(?)) • span{VXl... Vjr, | i < *}•
We call an element of I\(M, ?) an i-th differential order operator. Note that
if D is an i-th order operator and / is a smooth function, then (ad f)*D is a
zeroth order operator.
If A is any filtered algebra, that is, A = (JSo A* w'tn A* C Ai+i and
Ai ¦ Aj C Ai+j, we may define the associated graded algebra
gtA =
i=0
t=0
where the component in degree t is gr4 A. The projection from the graded
algebra 5^^0 A> *° &¦A 's c^l^ *ne symbol map, and written a; its com-
component in degree i is the projection Ai —* gcA with kernel Ai-i.
We will now apply this formalism to the algebra of differential operators
V(M,?). Let S(TM) be the infinite dimensional graded vector bundle over
M whose fibre at x is the symmetric algebra on TXM.
Proposition 2.1. The associated graded algebra
oo
gtV(M,?) = Y,Vk{M,?)/Vk^{M,?)
fc=0
to the filtered algebra of differential operators is isomorphic to the space of sec-
sections of the bundle S{TM) ® End(?). The isomorphism ak : grfc V{M, ?) —>
r(M, Sk(TM) ®End(?)) is given by the following formula: ifD 6 Vk(M,?),
then forxeM and ? e TXM,
B.1) <rfc(Z?)(x,0 = lim t-k(e-itf ¦ D ¦ eitf)(x) 6
where f is any smooth function on M such that df(x) = ?.
2.1. Differential Operators
65
Proof. Let D be the differential operator a Vx, • • ¦ Vxk > where X\,..., Xk are
vector fields on M and a 6 F(M, End(?)). By Leibniz's rule, the operator
e-'tf. J). e*tf is a differential operator depending polynomially on t, of the
form
(tt)*a(*)<*i(*),0 ... (Xh(x),t) + Oit"-1).
It follows that the leading order is a zeroth-order operator, and that the
limit lim t~k(e~uf • D ¦ e*'^)(x) is independent of which function / with
t—^oo
df(x) = ? we choose, and defines a linear isomorphism from grfc V(M, End(?))
to T(M, Sk(TM) ® End(?)). D
By differentiation of B.1), we see that
If D{eVki(M,?), i = l, 2, then
We will often write o-(D) instead of ak(D), if k is the degree of D.
We may identify T(M,S{TM) ® End(?)) with the subspace of sections
in T{T*M, it* End(?)) which are polynomial along the fibres of T'M. A
differential operator D of order k is called elliptic if the section crk(D) 6
T(T*M, n* End(?)) over the cotangent space is invertible over the open set
Definition 2.2. Let ? be a vector bundle over a Riemannian manifold M.
A generalized Laplacian on ? is a second-order differential operator H
such that
Clearly, such an operator is elliptic. An equivalent way of stating this
definition is that in any local coordinate system,
H - -^rgij(x)didj + first-order part,
ij
where p'-'(x) = (dx*,dx:')x is the metric on the cotangent bundle. By defini-
definition of the symbol, we can characterize a generalized Laplacian as follows:
Proposition 2.3. An operator H is a generalized Laplacian if and only for
any f 6 C°°(M),
Let ? be a vector bundle bundle on a Riemannian manifold M, with
connection V? : T(M,?) —^ T(M,T*M ® ?). Since M is Riemannian, it
possesses a canonical connection, the Levi-Civita connection V, by means of

66
2. Asymptotic Expansion of the Heat Kernel
which we define the tensor product connection on the bundle T*M ® ?. In
this way, we obtain an operator
as the composition of the two covariant derivatives.
Definition 2.4. The Laplacian A? on I\M,?) (which we denote by A if
there is no ambiguity) is the second-order differential operator
Here, we denote by TrE) G T{M, ?) the contraction of an element S
T(M, T'M <g> T*M 0 ?) with the metric g e T{M, TM ® TM).
If s 6 r(Af, ?) and X and y are two vector fields, then
If e* is a local orthonormal frame of TM, we have the following explicit
formula for A?:
On the other hand, with respect to the framing d/dxt defined by a coordinate
system around a point in M, the operator A? equals
where Tkj are the components of the Levi-Civita connection, defined by
Vj9,= ?fc Fijdk- Thus, the Laplacian A? is a generalized Laplacian.
In particular the scalar Laplacian is given for / e C°°(M) by the formula
a/ = -
If 5i and ?2 are two bundles on M with connections V?l and Vf3, let A?l,
A?i> and Afl®fa be the Laplacians defined with respect to the connections
V?>, V?a and V?l®fa = V?l ® 1 + 1 ® Ve3. The following formula is clear:
for Si er(M,?i),
Afl®?3(s1 81 s2) = (Aflsi) ® s2 - 2Tr(V?lsi ® V?as2) + si ® A?*s2.
As we will now show, any generalized Laplacian is of the form Ae + F,
where A? is the Laplacian associated to some connection V?, and F is a
section of the bundle End(?).
2.1. Differential Operators
67
Proposition 2.5. If H is a generalized Laplacian on a vector bundle ?,
there exists a connection V? on ? such that for any f 6 C°a(M), [H,f] is
the differential operator whose action on F(M, ?) is given by the formula
B.2)
,/] = -2<grad/,V?>
where A is the scalar Laplacian. Furthermore, F = H — A? is a zeroth-order
operator.
Proof. Let us define an operator Vs from F(M, ?) to A1(M, ?) by the formula
(/ograd/i, V?s) = \fo{~H(flS) + MH
for all /o and ft e C°°(M). The relation
f2 - 2(dfudf2) +
and the formula [[#,/i],/2] = -2(d/!,d/2) implies that this definition is
consistent, in the sense that
(/ogradtfc/aJ.V**) = </o/igrad/2 + /o/2grad/1, V?s),
and that V? is a connection on the bundle ?:
[(grad/, V?),/.] = [-i[F,/] + \Af,h\ = -Jp,/],fc] = (grad/.dfc).
It is now easy to show that F = i? — A? is a zeroth-order operator: if /
is any smooth function on M, then
since both [H, f] and [A?, f] equal -2(grad/, V?) + A/. D
To summarize, a generalized Laplacian is constructed from the following
three pieces of geometric data:
A) a metric g on M, which determines the second-order piece;
B) a connection Vs on the vector bundle ?, which determines the first-order
piece;
C) a section F of the bundle End(?), which determines the zeroth order
piece.
If ? is a vector bundle and ? * is its dual vector bundle, then the space
rc(M,?) of compactly supported sections of ? is naturally paired with the
space of sections of ?* ® |A|: if we take two sections si e TC(M,?) and
s2 e T(M, ? * ® |A|), then their pointwise scalar product (si, s2)x is a section
of the density bundle |A|, so can be integrated, giving a number fM(si,a2),
the pairing of the two sections. In particular, if ? is a Hermitian bundle, then
the space FC(M,? ® |A|1/2) has a natural inner-product.

68
2. Asymptotic Expansion of the Heat Kernel
Definition 2.6. A) If ?\ and ?2 are two vector bundles on M and D :
Y(M, ?1) -* T(M, ?2) is a differential operator, the formal adjoint D* of D
is the differential operator
such that
[ (Ds,t)= f {s,D't)
Jm Jm
for s e rc(M,f!) and t e r(M,?2' ® |A|).
B) If ? is a Hermitian vector bundle and D a differential operator acting
on T(M, ? ® |A|1/2), we say that D is symmetric if it is equal to its formal
adjoint, D = D*.
It is easy to see that the symbol of the formal adjoint D" is equal to
Although the bundle |A| has no canonical trivialization, in most of this
book, we will deal with Riemmanian manifolds, which have a canonical
nowhere vanishing density |dx| e T(M, |A|). Using the identification of
r(M,?) with T(M,? ® |A|) which results from multiplication by \dx\, the
formal adjoint of D : T(M,?i) —* T(M,?2) becomes an operator D* :
r(M,?3) —* F(M,?i). However, by taking into account the density bun-
bundles, we get the most geometrically invariant formulations of many theorems.
On a Riemannian manifold, the formal adjoint of the exterior differential
d : A*(M) -» A'+1{M) is an operator from the bundle A'+1TM ® |A| to
the bundle ATM ® |A|. Using the identification of the bundles AT*M and
A*TM® |A| which comes from the Riemannian metric on M, we may identify
it with a map d* : A'+1(M) -* A*(M). Thus, d* is defined by the formula
f (dfi,a)\dx\= f {I3,d'a)\dx\,
Jm Jm
where a and /3 are compactly supported differential forms on M, and \dx\ is
the Riemannian density.
If a is a one-form, then d*a is a function called the divergence of a.
Taking j3 = 1 in the preceding equation, we see that
B.3)
/ d'a\dx\ =
Jm
0
if a is a compactly supported one-form.
If a is a one-form, the function Tr(Va) is given by the formula
Tr(Va) = ^eia(e<) - a(VCiej),
where e< is a local orthonormal frame.
2.1. Differential Operators
69
L
Proposition 2.7. The divergence on one-forms is given by the formula
d*a = -Tr(Va),
and hence for any compactly supported C1 one-form a, we have
Tr(Va) |dx| = 0.
r
Proof. If X is a vector field on M, then
VX € T(M, T*M ® TM) ? T(M, End(TM))
is the endomorphism of TM which acts on Y G F(M, TM) by the formula
(VX) • Y = VyX. Since V is torsion free, we see that
C(X)Y = [X, Y] = VXY - (VX)Y,
hence, using the fact that \dx\ is preserved by the Levi-Civita connection,
that
C(X)\dx\ = Tr(VA-)|dx|.
If / € C°°(M), it follows from the formula fM C{X)(f \dx\) = 0 that
/ X(f)\dx\ = - f Tr(VX)f\dx\.
Jm Jm
If a is the one-form on M corresponding to X under the Riemannian metric
on M, so that X(f) = (a,df), we see that
(a,4f)|dx|= / X{f)\dx\ = - f Tr(Va)/|cfc|. D
Jm Jm
The following proposition extends the above formula for the divergence,
by giving a formula for d*a in any degree.
Proposition 2.8. Denote by 1 : T{M,T*M ® AT*M) -» T(M,AT*M) the
contraction operator on differential forms defined by means of the Rieman-
Riemannian metric on M. Then
Proof. Since the Levi-Civita covariant derivative is torsion free, we see from
Proposition 1.22 that d = e o V. Let /3P € A"(M), /3P+1 € .4p+1(Af), and
let a be the one-form given by the formula a(X) — (f3p,i(XHp+i). Using
Leibniz's rule and the fact that at each point x e M, e* = 1, we see that
(eV/3p,/?p+i)s = -{0p,iVl3p+1)x + Tr{Va)x.
Integration over M kills Tr(Va), and we obtain the proposition. ?
We will now calculate the formal adjoint of the Laplacian A?. Recall
that the bundle |A|8 of s-densities on a Riemannian manifold has a natural
connection, which preserves the canonical section \dx\'. If A?®'A'' is the

70
2. Asymptotic Expansion of the Heat Kernel
Laplacian associated to the connection Vf®'Al" on the bundle ? ® \h\', we
have
Because of the following result, it is often more natural to write formulas with
half-densities. Let JM(si,s2) denote the natural pairing between a compactly
supported section st of the bundle ? ® |A|1/2, and a section s2 of the bundle
?* ® IAI1/2. We consider on the bundle ?* the connection Ve' dual to the
connection V? on 5.
Proposition 2.9. If si is a compactly supported C2-section of ? ® |A|1/2
and s2 is a C2-section of ?* ® |A|1/2, then
f (A?^/2Sl,s2)= f Cl,
Jm Jm
m
where the last inner product is defined using the natural pairing between
T*M®? andT*M®?\
Proof. If we apply the Levi-Civita covariant derivative V to the density
(V?®|A|1/asi,s2)x, we obtain the continuous section of T*M <g> T'M
so that
By Proposition 2.7, the term TV(V(Vf®|Al1/3s1,s2)):c vanishes after integra-
integration over M. D
If ? has a Hermitian metric, then the bundle ? * is naturally identified
with ?, and there is a natural inner product on the space T(M,?® IAI1/2).
In this situation, the above proposition has the following consequence.
Corollary 2.10. // the connection V on ? is compatible with the Hermi-
Hermitian metric on ?, then the Laplacian A?®IAI1/S on the bundle ? ® |A|J/2 is
symmetric with respect to the inner product on T(M,? ® IAI1/2).
Applying Proposition 2.9 with ? = M x C the trivial line bundle, we see
in particular that the scalar Laplacian satisfies
f (Af,f)\dx\= I (df,df)\dx\,
Jm Jm
or, in other words, A/ = d'df.
2.1. Differential Operators
71
In the following lemma, we collect some formulas for the Laplacian on the
bundle of half-densities in normal coordinates x i-» expl0 x around a point
xq. These formulas involve the function j(x) on TxgM defined in Chapter 1
by the property that the pull-back of the volume form dx on M by the map
exp^ equals j(x) dx, or, in other words,
The reader should beware the distinction between the symbol x, denoting
any coordinate system on the manifold M, and x, denoting coordinates on
the tangent space TXoM, or normal coordinates on M.
Let H be the radial vector field 71 = X) x*^x' ¦ Denote the Laplacians on
functions and on half-densities by A.
Proposition 2.11. In normal coordinates x around xq, we have
A) V{f\dx\W) = (df-if- dilogMdxl1/2, for f 6 C°°(M).
B)
C)
D) If qt(x) is the time-dependent half-density on a normal coordinate chart
in M centred at xo given by the formula
qt(x) =
then
Proof. Parts A) and B) follow from the formula
= 0.
To prove C), we use integration by parts: if <f> ? C^°(TXoM), then
,, A\\xf) j(x) dx = /(#, d\\xf) j(x) dx = 2J{H<t>)j{x) dx;
here we have used that (d||x||2, dtj>)/2 = "R4>, which is a consequence of Propo-
Proposition 1.27 C). Since the adjoint of the vector field H with respect to the
density |rfx| is — n — 71, we see that
J (<j>, A||x||2) j(x) dx = -2 J(n + n(logj)) <t>{x) j(x) dx.
As for D), Leibniz's rule shows that
A(e-llxlla/«)|dx|i/2 +1- h-^
and that A(e-«xH3/4t) equals
equals
+ e~

72
2. Asymptotic Expansion of the Heat Kernel
FVom these formulas, it is easy to show that (dt + A — j'1/2(A- j
vanishes. ?
2.2. The Heat Kernel on Euclidean Space
Let V = R be the standard Euclidean vector space of dimension one, and let
be its Laplacian. If
it is easy to check that
gt(x,y) = 0,
where Ax denotes the Laplacian acting in the variable x. This equation is a
special case of Proposition2.il D), since j(x) = 1 on R.
We will need the following lemma.
Lemma 2.12. The Gaussian integrals
a(k) = D7T)-1/2 f e~v^4vkdv
Jr
are given by the formula
(
@, k odd.
Proof. Consider the generating function
A(t) = f) ?a(*) = D*)-V> f e-'
fc=ok] Jr
By the change of variables u = v — 2t, we see that this integral is equal to
A(t) = D7T)-1
from which the lemma easily follows. ?
Define the norm on C'-functions by
\\<j)\\e= sup sup
k<tx€R
dk
2.2. The Heat Kernel on Euclidean Space
Let Qt be the operator on functions on R defined by
73
Proposition 2.13. Let ? be an even number. If <j> is a function on R with
\\4>h+i finite, then
Jfc=O
Proof. The change of variables y *-* x + t^2v shows that
@«^)(x) = D7T)-1/2 f e-/V(x + <1/2f) dv.
Jr
If ||0||/+i < oo, Taylor's formula
** + v) = E ^«^(fc)(x) + V
shows that
V
+ to)dt
Thus,
*=0
fc=o
The proposition follows immediately from this estimate and the formula of
Lemma2.12 for a{k). D
In this chapter, we will show that there is an analogue of the function
<ft(x,y) on any compact Riemannian manifold, and also that the analogue of
the above proposition holds.
In Section 5.2, we will use the fact that on any Euclidean vector space V,
if <t> € C?°(V), there is an asymptotic expansion in powers of t1?2,
f
Jv
which we will denote by
Jv
This is proved by the same change of variables as was used in the proof of
the Proposition 2.13.

74
2. Asymptotic Expansion of the Heat Kernel
2.3. Heat Kernels
In this section, we will define the heat kernel of a generalized Laplacian on a
bundle ? ® |A|X/2 over a compact Riemannian manifold M. As background to
this definition, let us recall the kernel theorem of Schwartz. If ? is a bundle on
a compact manifold M, the space r~°°(M, ?) of generalized sections of a
vector bundle ? is defined as the topological dual space of the space of smooth
sections r(M,?*®|A|). There is a natural embeddingT(M,?) C r-°°(M,?).
We will denote by || ¦ j|/ a C'-norm on the space Tl(M, ?) of C'-sections of a
vector bundle ? on M or on M x M.
Let ?\ and ?2 be vector bundles on M, and let prl and pr2 be the pro-
projections from M x M onto the first and second factor M respectively. We
denote by ?1 El ?2 the vector bundle
prj ?1 ® prj ?2
over M x M.
We will call a section p e T(M x M, (F® IAI1/2) El (?* ® IAI1/2)) a kernel.
Using p, we may define an operator
p: r-°°(M,?:® |A|1/2) -> r(M, j-® iai1/2)
by
(Ps)(x)= f p(x,y)s(y).
Jy€M
By this, we mean that (Ps)(x) is the pairing of the distribution «(•) in
r-°°(M,? ® |Ap/2) with the smooth function p(x,-) e T(M,?* ® \A\l/2).
An operator with a smooth kernel will be called a smoothing operator.
Clearly, the composition of two operators Pt, P2 with smooth kernels pu P2
is the operator with smooth kernel
P(x,y)= I Pi(x,z)p2(z,y);
JzeM
Pi fa) *)pi{z, y) is a density in z, so can be integrated.
The following form of the Schwartz kernel theorem gives a characteri-
characterization of operators with smooth kernels.
Proposition 2.14. The map which assigns to the kernelp(x,y) the corre-
corresponding operator s i-» Ps is an isomorphism between the space of kernels
T(M x M, {? ® IAI/2) El (?* ® \A\^2)) and the space of bounded linear op-
operators from r-°°(M,f ® JAp/2) to r(M,T® I1/2
We will sometimes use Dirac's notation for the kernel of an operator P:
if x and y lie in M, then the kernel of an operator P at (x, y) 6 M x M is
written (x | P \ y), so that
B.4)
= l€M(x\p\v)*M-
2.3. Heat Kernels
75
There are two quite different ways to investigate the heat kernel of H, that
is, the kernel attached to the operator e~tH (t > 0) by the kernel theorem:
A) If H is symmetric, if we can show that the operator if has a self-
adjoint extension H that is boimded below, then by the spectral theorem,
the operator e~**, which we will simply denote by e~tH, is well defined
as a bounded operator on the Hilbert space of square integrable sections of
? ® |A|1/2. If, in addition, we can show that e~tS extends to an operator
which maps distributional sections to smooth sections, then it follows by the
kernel theorem that the kernel of e~ta is smooth.
B) The other method, which we will follow, starts by proving the existence
of a smooth kernel pt(x, y) which possesses the properties that the kernel of
the operator e~tH should possess, without first constructing the operator H.
The advantage of this method is that it gives more information about the
operator e~tH than the first method, is more closely related to the geom-
geometry of H, is technically more elementary, and does not require that H be
symmetric.
The following definition summarizes the properties that the kernel of an
operator e~tH must have.
Definition 2.15. Let H be a generalized Laplacian on ? ® |A|1/2. A heat
kernel for H is a continuous section Pt(x,y) of the bundle (? ® |Ap/2) El
(?* ® |A|1//2) over R+ x M x M, satisfying the following properties:
A) pt(x,y) is C1 with respect to t, that is, dpt(x,y)/dt is continuous in
(t,x,y);
B) pt{x,y) is C2 with respect to x, that is, the partial derivatives
dx'dx'
are continuous in (t, x, y) for any coordinate system x;
C) Pt(x,y) satisfies the heat equation
(dt+Hx)pt(x,y) = 0;
D) pt(x, y) satisfies the boundary condition at t = 0: if a is a smooth section
of ?® IAI1/2, then
lim Pts = s.
t—K)
Here we denote by Pt the operator defined by the kernel pt, and the limit
is meant in the uniform norm ||s||0 = supl6M ||s(x)||, for any metric on
?.
Our first task is to prove that the heat kernel is unique. We consider the
formal adjoint H* of H, which is a generalized Laplacian on ?* ® I1/2

76
2. Asymptotic Expansion of the Heat Kernel
Lemma 2.16. Assume that the operator H* has a heat kernel, which we
denote 6ypJ. l}s{t,x) is a map from R+ to the space of sections ofS ®|A|1/2
which is C1 int and C2 in x, such fAoMim st = 0 and which satisfies
the heat equation (dt + H)at = 0, then at = 0.
Proof. If si{t, •) and s2(t, ¦) are C2 sections of ? ® |A|1/2 for each t > 0, we
have
/ )()) /
For any « in T{M,?* ® IAI/2), consider the function
[ (()pls(,y){y))
(x,y)GMxM
where 0 < 9 < t. Differentiating with respect to 0 and using the heat
equation Definition2.15 C), we see that f(9) is constant. By considering the
limits when 0 tends to t and to 0, we obtain
for every w € T(M, ?* ® |A.|x/2>, so that st = 0 for all (> 0. D
Using this lemma, we obtain the following result.
Proposition 2.17. A) Suppose there exists a heat kernelp\ for the operator
H*. Then there is at most one heat kernel pt for H.
B) Suppose there exist heat kernels p\ for H*, and pt for H. Then pt (x, y) =
(pf(y,x))*.
C) Suppose there exist heat kernels pt for H*, and pt for H. Then the
operators Pts(x) = fu pt(x, y)s(y) dy form a semi-group.
Proof. Consider f(9) = fM{{P9s)(x), (Pt*_eu)(x)) for 0 < 0 < t, where s and
u are smooth sections of T(M, ? ® |A.|X/2); as before, this is independent of 0
and we obtain
by considering limits. This proves A) and B).
To show the semigroup property, we must show that if s e T(M, ?® |A|1/2)
and 0 > 0, then the unique solution st of the heat equation for H with
boundary condition limt_>0 st = Pgs is given by Pt+es; but this follows from
the lemma. ?
2.4. Construction of the Heat Kernel
2.4. Construction of the Heat Kernel
On Rn, the scalar heat kernel is given by the exact formula
77
For an arbitrary Riemannian manifold M and bundle ?, it is usually impos-
impossible to find an exact expression for the heat kernel pt(x,y). However, for
many problems, the use of an approximate solution is sufficient. The true
kernel pt is then calculated from the approximate solution by a convergent
iterative procedure.
In order to describe this technique, let us consider the following finite
dimensional analogue of the construction of pt. Let V be a finite dimensional
vector space with a linear endomorphism H (we have in mind here though
that V = T(M,? ®\A\V*) andH = Aee>Wl/2+F); we would like to construct
the family of operators Pt = e~tH.
Suppose that we are given a function Kt : R+ —» End(V) which is an
approximate solution of the heat equation for small t in the sense that:
at
_
and such that Ko = 1; such a function is also called a parametrix for the
heat equation. The function Rt is called the remainder.
Definition 2.18. The fc-simplex Ak is the following subset of Rk:
{(*!,... ,*fc) |0< ti < «a.-- <** < 1}-
Often, we will parametrize A/t by the coordinates
o~o = h i °~\ — U+i ~U, 0* = 1 — tk,
which satisfy <7o H h <Tfc = 1 and 0 < Oi < 1. For t > 0, we will write (A*
for the rescaled simplex
{(*i,...,tfc) I 0<ti<«2... <«»<*}•
Let Vk be the volume of A* for the Lebesgue measure dti... dt^. Since
vq = 1 and
= /
Jo
Jo Jo K
we see that Vk = l/k\. The rapid decay of Vk will be important in the
following proofs, when we must show that certain series converge.
Theorem 2.19. Let Qf : R+ -» End(K) be defined by the integral
Kt-tkRtk-tk_1...Rt:i-t1Rt1dt1...dtk;

78
2. Asymptotic Expansion of the Heat Kernel
in particular, Q°t = Kt. Then the sum of the convergent series E*>o(~1)*Q*
is equal to Pt = e~tH, and Pt - Kt + O(t1+a).
Proof. The fundamental theorem of calculus shows that the derivative of the
convolution f0 a(t - s)b(s) ds with respect to t equals
If we apply this with a{s) = K, and b(s) = #*>(«) defined by
R(k)(s) = / Ra_tk_t .. . Jfc,.,,*,, dtx . .. dtk-u
JtAit-i
then since Ko = 1, we obtain
Thus, the sum with respect to A; telescopes:
= o.
fc=O
To estimate Pt - Kt for small «, we use the uniform bounds over tAk,
\Kt-th\<Co and \Rti+l-ti\ < Cta,
which lead to the bound
10?I < CQCktka^ since vol(tAk) = ?.
let ic\
It is clear from this that the series defining Pt converges, and that Pt =
Kt + O(tl+a). ?
An important special case of this formula is the Volterra series for the
exponential of a perturbed operator: if H = Ho + Ht e End(F) and K. =
e tH°, we see that i?t = (d/dt + H)e~tH" = Jfie-«o, and we obtain
where
w --{
JlAk
This formula implies in particular that
c-«(H0+iri) = JT(-t)k I ,
k=o ^A»
B.5) =e-t«o-tf\
Jo
2.4. Construction of the Heat Kernel
79
If Hz is a one-parameter smooth family in End(V), by setting Ho — Hz and
Hi = edHz/dz we obtain the following formula for the derivative of e~tH':
B.6)
l(
dz
'ds.
We now turn to implementing this method in the case which interests us,
namely V = T(M,? ® lAI1^) and H = A?®lAll/2 + F. In formulas which
follow, we will omit the notation dti... dtk for the Lebesgue measure on
tAk, since this is the only measure which we will consider on this space.
Imitating the above method in an infinite dimensional setting, the steps in
the construction of the heat kernel are:
A) Construction of an approximate solution kt(x,y), and study of the
remainder rt(x,y) = (dt + Hx)kt{x,y);
B) Proof of convergence of the series
)* / / kt-
JtAh Jm>-
h-tl..1(zk,zk-i)...rtl(zuy)
to the heat kernel Pt(x,y).
The construction of kt will be performed in the next section, by solving
the heat equation in a formal power series in t. For the moment, we will
only use the properties of the approximate solution kt(x,y) summarized in
the following theorem, which will be proved in the next section.
Theorem 2.20. For every positive integer N, there exists a smooth one-
parameter family of smooth kernels k^(x,y) such that, for every integer ?
A) for every T > 0, the corresponding operators Kt form a uniformly bounded
family of operators on the space r*(M,? ® |A|1/2) for 0 < t < T;
B) for every s € r*(M,? ® |A|1/2), we have lim _^ KfS = s with respect to
the norm || • ||/;
C) the kernel rt(x,y) — (dt + Hx)k^(x,y) satisfies the estimate
We fix N and write kt for fcjf. For such a kernel kt, we would like to
consider the operator
Qt = I Kt-ttRtt-t,.-! ¦ ¦ ¦ Rt!
defined by the kernel '
9t(x,y)= / / kt-tk(z,Zk)rtk-tH-i(zk7>'k-i)---rti(LZi,y).
JtAk JMk
We will first prove that if TV is chosen large enough, this integral is convergent
and that the kernel qk is differentiable to some order depending on N. To do
this, we need some estimates.

2. Asymptotic Expansion of the Heat Kernel
Consider the kernel
-fc+1/
,y)= / rt-tk(x,zk)rth-.tk_l(zk,zk-1)...rtl(zuy)
Lemma 2.21. // TV satisfies N > (n +1)/2, then rk+1 is of class Cl with
respect to x and y, and
\\rk+% < Ck+1 *(*+D<"-»/2)-//2 ^(M)***/*!
Proof. If N > (n +1)/2, the section (x, y) --+ rt(x, y) and its derivatives up
to order I extend continuously to t — 0, so that the integral defining rk+1
is convergent. Now, the uniform estimates of the integrand over tAk lead to
the result, the factor tk/k\ being equal to the volume of tA*. ?
We now turn to the estimation of qk(x, y) and its derivatives.
Lemma 2.22. Assume that N>(n +1)/2, and thati>\.
A) The kernel qk(x, y) is of class Cl with respect to x and y, and there exists
C h
a constant C such that
\\l
for every k>l.
\\qk\\l
- 1)!
B) The kernel qk{x, y) is of class C1 with respect to t, and
(ft + Hx)qk(x, y) = rk+1(x, y) + rf (x, y).
Proof. We write the kernel qk in the following form:
{*,V) =
!-.(*,*)rf(*,y))ds.
Since composition with the kernel k, defines a uniformly bounded family of
operators on F'(? ® IAI1/2), the section
b(t,s,x,y)= / kt-,{x,z)rk{z,y)
depends continuously on s e [0, t], as do its derivatives up to order I with
respect to x and y. Furthermore, b(t,t,x,y) = rk(x,y); also
(ft + Hx)b(t, s, x,y)= f r«_.(*, z)rk(z, y) = rfc+1(x, y),
so that b(t, 3, x, y) is differentiable with respect to t, and dtb(t, s, x, y) is a con-
continuous function of s € [0, t). The proof of B) now follows by differentiating
qk{x, y) = f0 b(t, s, x, y)ds with respect to t.
To prove A), we use the uniform boundedness of Kt, which shows that for
0 < s < t,
\\b(t,s)\\t<C'\\rk\U.
2.5. The Formal Solution
81
Combining this with Lemma 2.21, we obtain
\\Ht,s)\\i < C'Cktkl'N-n/2)-t/2vo\{M)k-1tk-1/(k - 1)!
from which A) follows. ?
The following theorem shows that we can construct a heat kernel pt start-
starting from the approximate kernel kt.
Theorem 2.23. Assume that the kernel k^(x,y) satisfies the conditions of
Theorem2.20 with N > n/2 + 1.
A) For any t such that N>(n + l+ l)/2 the series
converges in the || • ||/+i-norm (over M x M) and defines a Cl-map from
R+ to Tl{M xM,(?® IAI1/2) E {?* ® IAI1/2)) which satisfies the heat
equation
(dt+Hx)Pt{x,y)=0.
B) The kernel k^1 approximates pt in the sense that
*f)ll« = O(t(jV-"/2)-fc-'/2+1) when t -» 0.
C) The kernel pt is a heat kernel for the operator H.
Proof. Our estimate on ||g*(x, j/)||/+i proves the absolute convergence of the
series defining pt(x,y) in the Banach space T/+1, uniformly with respect to
t. From the equation dtqk = rk+1 +rk — Hxqk, we obtain bounds for the
derivative with respect to t in the Cl norm. This shows the convergence of
the series ?)(—l)kdtqk in F', uniformly with respect to t, and the equation
(dt + Hx)pt(x,y) = 0 follows from the same telescoping as in Theorem 2.19.
To show C), we must show that pt satisfies the initial condition of a heat
kernel. However, we know that fcf satisfies this initial condition, so that C)
follows from B). ?
2.5. The Formal Solution
Let H = A?®lAlI/:> + JF be a generalized Laplacian on the bundle ? <g> |A|1/2
over a compact manifold M. In this section, we will denote the connection V?
on ? simply by V. We denote by Idxl1/2 the canonical half-density associated
to the Riemannian metric on M. To construct an approximate solution to
the heat equation for H, we start by finding a formal solution to the heat
equation as a series of the form
kt(x, y) = qt(x, y) f; t%(x, y, H) \dy\^2,
i=0

82
2. Asymptotic Expansion of the Heat Kernel
where the coefficients (x, y) >-> $<(«, y, H) are smooth sections of the bundle
? G3? * defined in a neighbourhood of the diagonal of M x M and qt is modelled
on the Euclidean heat kernel: in normal coordinates around y (x — exp x),
qt(x, y) =
(The precise definition of a formal solution is given below). We fix y 6 M
and write qt for the section x i-> qt{x,y) of the bundle |A|1/2. Abusing
notation once more, we denote by jxl2 the function on M around y given by
j1/2(x) = j(xI/2 if x = expyx.
To show the existence of a formal solution of the heat equation, we will
make use of the following formula. Here, we write H for the operator s i-*
\d\l/2\\1'2 on T(M,?).
Proposition 2.24. Let B be the operator acting on sections of? in a neigh-
neighbourhood of y defined by the formula
Then, for any time dependent section st of ?,
(ft + H)stqt = ((dt + r* VK + B)st)qt.
Proof. Let 11-» st be a smooth map from R+ to the space of smooth sections
of ? around y. By Leibniz's rule, we see that
(dt + H)(stqt) = ((dt + H)St)qt - 2(Vst, Vqt) + st((dt + A)qt),
where A is the scalar Laplacian. Proposition 2.11 gives explicit formulas for
the last two terms:
From this, we see that
(ft + H)(stqt) = {(dt +H + r'VR + Vdlog,-
By Leibniz's rule again, we obtain
qt-
Proposition 2.24 makes clear what we mean by a formal solution of the
heat equation.
2.5. The Formal Solution
83
Definition 2.25. Let $t(x, y) be a formal power series in t whose coefficients
are smooth sections of ? H ?* defined in a neighbourhood of the diagonal of
M x M. We will say that qt(x, y)$t(x, l/)|dj/|1/2 is a formal solution of the
heat equation around y if x t-t $t(a;> J/)> considered as a section of the bundle
?®?y over a neighbourhood of y in M, satisfies the equation
We can now prove the existence and uniqueness of the formal solution of
the heat equation. Let x and y be sufficiently close points in M, and write
x = expv x for x 6 TyM. Denote by r(x, y) : ?v -* ?x the parallel transport
along the geodesic curve x, = expv sx : [0,1] —* M.
Theorem 2.26. There exists a unique formal solution kt(x,y) of the heat
equation
of the form
h(x, y) = qt(x, y) f; *<*<(*, y, H) \dy\l'\
«=o
such that <bo(v,V,H) = Igy e End(?w). We have the followng explicit for-
formula for $<:
r(x,j/)-1$j(x,J/) = - / 8i-1T(xt,y)-1(Bm-*i-1)(x.,y)da.
Jo
In particular, $o(x,y) = t(x, y).
Proof. In the left side of the equation
+ r'Vn + Bx)*rt%(x,y,H) = 0,
«=o
we set the coefficients of each t* equal to zero. This gives the system of
equations:
W$o = 0
(VR + i)$j = -Bx$i-i if i > 0.
Moreover, we know that &0(y,y,H) = Iev e End(?w), and that $i(x,y,H)
should be continuous at x = y.
The parallel transport r(x, y): ?v —» ?x along the geodesic x, = expy sx :
[0,1] —» M satisfies the differential equation Vrt = 0 with boundary condi-
condition r(y, y) = 1, and we see that $o(z, y, H) = r(x, y).
To obtain a formula for $j in terms of $i_i, we consider the function
where x. = expysx. Then &@) = 0 and Vd/d,4>i = -»<(
and we obtain the required explicit formula for $<¦ D

84
2. Asymptotic Expansion of the Heat Kernel
Example 2.27. We can use the explicit iterative formula for $i(x,y) given
above to calculate the subleading term $i(z,x) in the asymptotic expansion
of kt{x, y) on the diagonal. We have
*i (*,*) = -(? •*,,)(*,*).
However, since $o{xiV) — 1"(a;>i/)> we see that
(x) —F(x),
Using Proposition 1.28, we see easily that this function equals
where tm (x) is the scalar curvature of M,
Consider the ?y-valued function on TyM defined by
Ui{x, y, H) = r(expy x,
x, y, H) x € TvM.
Let L be the differential operator which acts on the space of ?tf-valued func-
functions on M around y, defined by the formula
Then we see that
= - f
Jo
where x, = expy sx. It is clear from Propositions 1.18 and 1.28 that the
coefficients Ui<a(y, H) in the Taylor expansion
of the function x \-* Ui(x.,y,H) are polynomials in the covariant derivatives
at y of the Riemannian curvature, the curvature of the bundle ?, and the
potential F, since this is true of the Taylor coefficients of the coefficients of
the operator L at y.
Definition 2.28. If t/>: R+ —» [0,1] is a smooth function such that
ip(s) = 1, if s < ?2/4,
V»(s) = 0, if s > e2,
we will call ip a cut-off function.
We will now use the formal solution to the heat equation to construct
an approximate solution k^(x,y) satisfying the properties given in Theo-
Theorem 2.20. To do this, we truncate the formal series and extend it from the
diagonal to all of M x M by means of a cut-off function ip(d(x, yJ) along the
diagonal, where the small constant e is chosen smaller than the injectivity
2.5. The Formal Solution
85
radius of the manifold; thus, the exponential map is a diffeomorphism for
|x| < ?. We define the approxiamte solution by the formula
N
B.7) fcf (*, y) = W(x, yJ) qt(x, y) ? t%(x, y, H)\dy\y'2.
In the next theorem, we will prove that hf possesses the properties that
were used to construct the true heat kernel in Section 3. Indeed, we will
see that it satisfies somewhat stronger estimates than were stated in Theo-
Theorem 2.20.
Theorem 2.29. Let I be an even positive integer.
A) For anyT>0 the kernels kf, 0 < t < T, define a uniformly bounded
family of operators K? on T'(M, ? ® IAI1'2), and
lim \
t—»o
- s\\t = 0.
B) 7%ere exist differential operators ?>* of order less than or equal to 2k such
that Do is the identity and such that for any s € Tt+1(M,? ® \A\^2)
l/2-j
\\
\\2
C) The kernel r?(x, y) = (dt + Hx)k?(x, y) satisfies the estimates
for some constant C depending on I and k.
Proof. We fix N and write Kt for the operator corresponding to the kernel
fcf. In a neighbourhood of the diagonal, we write y = expx y, with y e TXM,
and identify ?x to ?y by parallel transport along the unique geodesic joining
x and y. If s is a section of ? ® lAp/2, we denote by s(x,y) the function of
y such that s(y) = s(a;,y)|dy|1/2. We see that
(Kts)(x)
= D7rf)-n/2 /
N
x,y)|dy| ® \dx\1'2,
with smooth compactly supported coefficients
(ar,y)«—>9i(x,y)eEnd(?x)
Furthermore, tfofoO) = I?x.
The statement of the theorem is local in x, so we can assume that the
vector spaces ?x and TXM are fixed vector spaces E and V. The change of
variables y = t1//2v on V shows that (Kts)(x) equals
(x, tl'2v)dv ® \dx\1'2,
from which A) follows.

86
2. Asymptotic Expansion of the Heat Kernel
2.6. The Ti&ce of the Heat Kernel
87
By Taylor's Theorem, there exist differential operators Dk of order less
than 2k such that
1/2
*=0
A similar argument may be used to estimate the C^-norm of
l/2-j
fc=O
The bound on r^(x, y) follows from the formula for k? by means of Propo-
Proposition 2.24, which shows that
r?(x,y) = [(dt
Bx)
N
»=o
The right-hand side in this formula is made up of two kinds of terms, those
which involve at least one differentiation of the factor t(>(d(x, j/J), and those
which only involve time derivatives or derivatives of the functions $j. The
first type vanishes for d(x, y) < e/2 and so is very easy to estimate, since for
d(x,y)>e/2
d2ip(d(x,yJ)e-dlx<tf'u = O(th),
for any k. Prom the system of differential equations defining the functions
$i, we see that the terms which do not involve a derivative of ip(d(x, j/J)
cancel, except for one remaining term equal to
tNqt(x,y)(Bx$N)(x,y),
which may be bounded by tN~n/2. The argument used to bound the deriva-
derivatives of rt(x, y) is similar, once one observes that
As a consequence of Theorems 2.29 and 2.23, we have now proved the
existence of a C'-heat kernel, for any operator H = Af®lAl' * + F and any
L from the unicity of the heat kernel (Proposition 2.17) it follows that the
heat kernel pt(x,y,H) is smooth as a function of (x,y). The heat equation
(dt + Hx)pt(x,y) = 0 implies of course that it is smooth as a function of
(t,x,y).
Let us summarize the properties of the heat kernel that will be used in the
proof of the local index theorem.
Theorem 2.30. Let pt(x,y,H) be the heat kernel of the generalized Lapla-
cian H = A? ®lAl1/a + F. There exist smooth sections $i € T(M xM.fBf)
such that, for every N > n = dim(M)/2, the kernel k^(x,y,H) defined by
the formula
N
i=Q
\dy\V2
is asymptotic to pt(x,y,H):
||a*(pt(x, y, H) - k?(x, y, H))\\t = O{tN-n'2-l'2-k).
The leading term $>o(x, y, H) is equal to the parallel transport t(x, y) : ?v —>
?x with respect to the connection associated to H along the unique geodesic
joining x and y.
If s e T(M,? ® \A\1/2) is a smooth section of ? <g> |A|1/2, we see from
Theorem 2.29 that Pts, defined by
(Pts)(x)= I pt(x,y,H)s(y)
has an asymptotic expansion in T{M, ?® \h\1^2) with respect to t of the form
Y, tkPkS- The heat equation {d/dt+H)Pts = 0 implies that Dk = (-H)k/k\.
Thus, we obtain the following estimate:
B.8)
t=0
This asymptotic expansion justifies writing e tH for the operator Pt.
2.6. The Trace of the Heat Kernel
In this section, we will derive some of the consequences of the existence of a
heat kernel for an operator of the form H = A?®'A' + F. In particular, we
show that if H is symmetric, then it has a unique self-adjoint extension H
in the space of square-integrable sections of the bundle ? ® IAI1/2, and the
kernel of the operator e~tH is the heat kernel pt(x, y, H) of H.
We first recall some elementary facts on trace class operators on a Hilbert
space. Let H be a Hilbert space with orthonormal basis ej. An operator A
is a Hilbert-Schmidt operator if
« a
is finite. The number \\A\\HS is called the Hilbert-Schmidt norm of A. If A is
a Hilbert-Schmidt operator, so is its adjoint A* and ||-A||HS = ||i4*||Ha; also,

88 2. Asymptotic Expansion of the Heat Kernel
if U is a bounded operator on H and A is an Hilbert-Schmidt operator, then
UA and AU are Hilbert-Schmidt operators and ||J7A||Hs < \\U\\ ||A||H3. If A
and B are Hilbert-Schmidt operators, then for any orthonormal basis e< of
n,
Let M be a compact manifold and let ? be a Hermitian vector bundle on
M, and denote by TLi(M,? ® IAI1/2) the Hilbert space of square-integrable
sections of ? ® |A|1/2. We will also frequently consider the Hilbert space
Fx,a (M, ?) where M is a compact manifold with canonical smooth positive
density (for example, a Riemannian manifold with the Riemannian density).
Lemma 2.31. The topological vector space underlying rLa(M,?<g> IAI1/2) is
independent of the metric on ?, although the metric on Ti?{M,? ® |A|X/2)
depends of course on the metric on ?. Similarly, the topological vector space
underlying YLi(M,?) is independent of the metric on ? and the density on
M.
Proof. If hi(-, •) and /i2(-, •) are two metrics on ?, then there is an invertible
section A € T(M, End(?)) such that fti(s, s) = h2(As, As) for all s G T(M, ?).
Since A and A~l induce bounded operators on T^(M,? ® IAI1/2), the first
part of the lemma follows.
If aii and W2 ale two smooth positive densities on M, then there is an
everywhere positive function <f> € C°°(M) such that u>i = (fx^. Since mul-
multiplication by 01/2 and 0/2 induce bounded operators on r^M, ?), the
independence of this space on the density on M follows. ?
An operator K with square-integrable kernel
k(x,y) 6 TL,(M x M, (? 9 W1/2) B (? 9 \M1/2))
is Hilbert-Schmidt, and
B.9)
= /
J(x
Tr(*(x,
/(i,VNMxM
as follows from the definition of the Hilbert-Schmidt norm,
In particular, we see that the heat kernel pt of a generalized Laplacian H is
the kernel of a Hilbert-Schmidt operator for all t > 0.
An operator K is said to be trace-class if it has the form AB, where
A and B are Hilbert-Schmidt. For such an operator, the sum ^{Ke^ei)
is absolutely summable, and the number X^(-^e»>e>) ls independent of the
Hilbert basis; it is called the trace of K:
2.6. The Trace of the Heat Kernel
89
Note that if A and B are Hilbert-Schmidt, then Tr(AB*) = (A,jB)hs; also
B.10) Tr(AB) = Tr(BA) = ^(Ae<) ej)(Beh e{).
y
Observe that the restriction of the bundle (? <g> IAI1/2) H (? <g>
M x Af to the diagonal is isomorphic to End(?) ® |A|. If
on
a(x,y) e T(M x M, (? ®
B (? ®
is a smooth kernel on M x M, denote by Tr(a(x, x)) the section of |A| obtained
by taking the pointwise trace of o(x,x) € r(M,End(?) ® |A|) acting on the
bundle ?.
Proposition 2.32. For all t > 0, the operator Pt with kernel pt the heat
kernel of a generalized Laplacian on a compact manifold M is trace class,
vrith trace given by the formula
Tt(Pt) = J Tr(p((x,x)).
Proo/. This follows from the fact that Pt = (Pt/iJ, and the fact that Pt/2 is
Hilbert-Schmidt. The trace of Pt is given by the formula
Pt/2(x,y)pt/2(y,x)
Pt(x,x). ?
= /
JxE
Let H = A?®|AI1/2 + F be a generalized Laplacian on ?, and consider the
operator H as an unbounded operator on rLi(M,? <g> |A|1/2) with domain
equal to the space of smooth sections of ?<g> IAI1/2; its formal adjoint H* will
also be considered on the same domain.
Proposition 2.33. The closure of H* is equal to the adjoint of the operator
H with domain T{M,? 9 \h\1/2). In particular, if H is symmetric, then it
is essentially self-adjoint; in other words, H is a self-adjoint extension of H,
and is the only one.
Proof. Consider the action of the formal adjoint H* on the space of distri-
distributional sections of ? <g> IAI1/2. It is clear that the domain of the adjoint of
H is
V = {s € TL,(M,?® |A|V2) | H's e rL,(M,?® \\\1'2)},
where H*s is defined in the distributional sense. Thus, we need to prove that
for s € V, there is a sequence sn of smooth sections of ? ® IAI1/2 such that
sn converges to s and H*sn converges to H*s in TL*(M,? ® |A|X/2). We
will do this using the operators Pt with kernel pt(x, y, H); first, we need two
lemmas.

90
2. Asymptotic Expansion of the Heat Kernel
Lemma 2.34. If s € TLi(M,? ®\\\1/2), then limj_>oPts = a in the L2-
norm.
Proof. If 0 is a continuous section of ? ® IAI1/2, we know that Pt<t> converges
to 4> in the uniform norm, and hence in the L2-norm. Writing a — (a — <j>) + <j>,
we see that
\\Pts - «||L. < \\Pt{s -
\\Pt<t> -
- 4>\\»-
Thus, it suffices to establish that the operators Pt are uniformly bounded on
rL2(M,? ® |A|V2) for small t.
Theorem 2.30 shows that it is enough to prove the lemma for the operator
Qt associated to the kernel qt(x, y)\dx\1/2 ® \dy\1/2 with
qt(x,y) = {^tyn'2e-<x'^'AtrP(d{x,yJ)\dx\1'2 ® \dy\1'2.
There exists a constant C such that fM \qt(x,y)\dy < C for t > 0. By
Schwarz's inequality, we see that if / € Tii(M,? ® lA^/2),
J\fqt(x,y)f(y)dy\2dx < J\Jqt(x,y)dy) [Jqt{x,y)\f{y)\2dy) dx
< C J jqt(x,y)\f(y)\2dxdy < C2\\f\\2. D
Lemma 2.35. If a 6 T> and Pt* is the operator with kernel pt(x,y,H") =
pt(y,x,H)', then H*Pt*s = Pt'H*s.
Proof. Use T(M,? ® IAI1/2), we have HPts = PtHs, since both sides
satisfy the heat equation and Pts — a converges uniformly to 0 as well as all
its derivatives, when t-»0. Taking adjoints, we see that P?H* = H*P? on
Y~°°{M,? ® IAI1/2), and in particular on 27. Q
Given a, we may take the sections an to be given by Pf/na. Asn-K»,
the first lemma shows that sn -> s in T^a, while the second shows that
H*sn = Pf/nH*s, which again converges to H*s by the first lemma.
Applying this result with H replaced by its formal adjoint H*, we see
that the closure of H with domain T{M,?® IAI1/2) is the adjoint of H*. In
particular, if if is symmetric, that is H = H*, then H is self-adjoint, and we
obtain Proposition 2.33. D
Prom now on, we consider only the case in which H is symmetric. If we
simultaneously diagonalize the trace-class self-adjoint operators Pt, we obtain
a Hilbert basis of eigensections <j>t; the semigroup property for Pt now implies
that for some Aj 6 R,
Since Ptfa is a smooth section of ? <g> |A|X/2, we see that the eigensections
4>i are smooth. The eigenvalues A< are bounded below, since the operators
Pt are bounded. The fact that the function Pt<t>i satisfies the heat equation
implies that Hfc — Xi4>%- Furthermore, since the operators Pt are trace-class,
2.6. The Trace of the Heat Kernel
91
we see that Yli e~tXi is finite for each t > 0. This shows that the spectrum
of H is discrete with finite multiplicity. Thus, we have proved the following
result:
Proposition 2.36. If H is a symmetric generalized Laplacian, then the op-
operator Pt is equal to e~iH. The operator H has discrete spectrum, bounded
below, and each eigenspace is finite dimensional, with eigenvectors given by
smooth sections of ? ® IAI1/2
We will not be very careful about distinguishing between the operator H
and its closure H, and will write e~tH for the operator Pt = e~tH. From the
spectral decomposition of H it follows that
Pt(x,y) = ]Te-«A^-(z) ® 4>j{y).
j
Proposition 2.37. If H is a generalized Laplacian, and if P(o,oo) is the pro-
projection onto the eigenfunctions of H with positive eigenvalue, then for each
? 6 N, there exists a constant C(?) > 0 such that for t sufficiently large,
IK* I JWo)*-'"JWo) I v)\U < c(e)e-^2,
where Aj is the smallest non-zero eigenvalue of H.
Proof. For t large, we may write
(x | P@,ao)e-fHF(o,oo) | V)
= f
oo) | z2)k1/2(z2,y).
The kernel fc1/2(z, y) is smooth, so we only have to show that
L
|<|(.)(o1c
/(i,y)€Af»
However, the left-hand side is nothing but the square of the Hilbert-Schmidt
norm of P(o,oo)e~(t~1)KP(o,«>)- Thus, if 0 < Ai < A2 < ... are the positive
eigenvalues of H, repeated according to their multiplicities, we must show
that
Thus, we see for t large that
= {J <* I
P(o.oo)e
-*
P(o.oo)
*>)
D

-~ z. Asymptotic Expansion of the Heat Kernel
In particular, if H is positive, that is, if it is non-negative and has vanishing
kernel, then P(o,oo) is the identity, and each ? € N, there exists a constant
C(?) > 0 such that for t sufficiently large,
where Ai is the smallest eigenvalue of H.
The Green operator G of a positive generalized Laplacian H is by def-
definition the inverse of H on rLi(M,? ® |A|1/2). Clearly G is bounded on
Fia(M,?® (Al1/2). In the following theorem, we discuss the properties of G.
Theorem 2.38. Assume that H is a positive generalized Laplacian. Let
G be the inverse of H on TLi(M,? ® |A|1/2). Then we have the integral
representation
^ 1
for each natural number k. The operator Gk has a Cl kernel for k > 1 +
(dim(M)-M)/2.
Proof. If k > 1, define the operator Gk by the integral
If <f> is an eigenfunction of H with eigenvalue A, it is easy to see that the
operator Gk applied to <j> equals
We will show that for k sufficiently large, the integral
converges and defines a kernel which is in fact the kernel of the operator G*.
We start by splitting the integral into two pieces,
We bound the first integral by means of the following lemma.
Lemma 2.39. The family of kernels tk~lpt(x,y), (t < 1), is uniformly
bounded in the Cl-norm when k> 1 + (? + n)/2.
2.6. The Trace of the Heat Kernel
93
Proof. Using the asymptotic expansion for the heat kernel Pt(x,y), we see
that it suffices to prove.the lemma for the kernel
where rj) is a smooth function on R+ which equals 1 near zero and 0 in
[e2,oo). Using the fact that \dae-d<-'<tf'4t\ < Ct~W2, it is easily seen
that for k > 1 + (? + n)/2, the family of kernels tfc~1aagt(x, j/) is uniformly
continuous for 0 < t < 1, a < t. ?
We will complete the proof by showing that the second integral converges
to a smooth kernel, using Proposition 2.37. Indeed, for any C'-norm,
which is finite. ?
In Section 9.6, we will extend this theorem to fractional powers of the
Green operator.
It is straightforward to extend this theorem to an arbitrary self-adjoint
generalized Laplacian H. We define the Green operator of H to be the inverse
of H on ker(.ff)-1-, and to vanish on ker(/f). The integral representation of
G is similar to the above one, except that we must handle the non-positive
eigenvalues separately: if A_m < • • • < A_i < 0 are the strictly negative
eigenvalues of H, and if P\t is the projection onto the eigenspace of H with
eigenvalue A<, then we see that
B.11) Gk =
dt
Since the operators P-\t are smoothing operators, the estimates on the kernel
of G* may be carried out in exactly the same fashion as when H is positive.
The above theorem has the following important consequence, known as
elliptic regularity.
Corollary 2.40. Let H be a symmetric generalized Laplacian on ?® IAI1/2.
A) If s € FLi(M,? ® IAI1/2) is such that Hks, in the sense of generalized
sections, is square-integrable for all k, then s is smooth.
B) If A is a bounded operator on rLi(M,? ® |A|X/2) which commutes with
H, and if s 6 T{M,S® \K\1'2) is smooth, then As € T{M,? ® \A\^2) is
smooth.
Proof. We may assume that H is positive by adding a large positive constant
to H. It follows that 1 = G* o Hk = Hk o G*.
Since Gk commutes with H, we see that s = GkHks. By hypothesis,
Hks is square-integrable, hence by the above estimates, GkHks lies in Cl for
I < Ik — n — 1. Letting k tend to oo, we obtain the first part.

94 2. Asymptotic Expansion of the Heat Kernel
The second part is proved in an analogous way, using the formula
As = (<?* o Hk o A)a = (GkoAo Hk)s.
If a € r(M,?(8)|A|1/'2) is smooth, then so is Hks; in particular, .ff*s is square-
integrable. Since A is bounded on the Hilbert space Fii(M, ? ® lAI1*1'2), we
see that AHks e Tv(M,S® \A\1/2). It follows that As = Gk(AHks) is Cl
for ? < 2k — dim(M) — 1. Taking k large, we obtain the result. ?
It follows from Theorem 2.30 and the asymptotic expansion of p((x, y) that
the trace of the heat operator e~tH possesses an asymptotic expansion in a
Laurent series in t1?2 for small t:
/
m
In particular, using the fact that
Theorem.
»s equal to Is, we obtain WeyPs
Theorem 2.41 (Weyl). Let A< be the eigenvalues of H, each counted with
multiplicities. Then as t —> 0, we have the asymptotic formula for the Laplace
transform of the spectral measure of H:
?VtA< = D7rt)-"/2rk(?)vol(M) + O(rn/2+1).
t
Note that, at least in principle, it is possible to compute all the coefficients
$i(x, x, H) in the asymptotic expansion of the heat kernel Pt(x, y) explicitly,
so that we can, if we desire, obtain a full asymptotic expansion for the Laplace
transform of the spectral measure of H.
It is possible to derive from this theorem information about the asymp-
asymptotic distribution of the eigenvalues of H at infinity, by means of Karamata's
Theorem.
Theorem 2.42 (Karamata). Let dfi(X) be a positive measure on R+ such
that the integral
F
Jo
converges for t> 0, and such that
lim ta / e~tx
Jo
t-+o
for some positive constants a and C. If f(x) is a continuous function on the
unit interval [0,1], then
limt"
2.6. The IVace of the Heat Kernel
95
Proof. By Weierstrass's Theorem, we can approximate the function f(x) ar-
arbitrarily closely by a polynomial p:
l/(x)-P(x)|<? for x 6 [0,1].
It follows that it suffices to prove the limit formula for polynomials on the
unit interval, and, in fact, for the monomial f(x) = xk. But this is a straight-
straightforward calculation: on the one hand,
lim ta [°° e-(k+1)tx dn(X) = C(k
t—>o Jo
while on the other,
T(a),
also equals C(fc + l)~a. D
Using Karamata's Theorem we obtain the following restatement of Weyl's
theorem.
Corollary 2.43. // N(X) is the number of eigenvalues of H that are less
than X, then
rk(g)vol(M) /a
Proof. Taking in Karamata's Theorem a decreasing sequence of continuous
functions converging to the function equal to x~l on the interval [1/e, 1] and
zero on the interval [0,1/e), we see that
t—*o
f efc(A) = fo f
-'o I (a) Jo
The corollary follows by considering a constant a such that H + a is positive,
and applying the above result to the spectral measure /i = ]?V Sy, of H + a,
for which a = n/2 and C = D7r)-n/2 rk(?) vol(M). O
It is interesting to see how this works in the simplest possible case, where
M is equal to the circle of circumference 2ir, and H is the scalar Laplacian
A. The eigenvalues of the Laplacian are A = 0, with multiplicity 1, and the
squares A = n2 (n > 1), with multiplicity 2. It follows that JV(A) = [2>/A+l].
On the other hand, Weyl's Theorem tells us that
V '
v/47FrC/2)
and it is easily checked that the coefficient of </X equals 2 (since vo^S1) = 2w
and TC/2) = y/v/2).
In the next proposition, we will extend the formula for the trace of the
heat kernel to any operator with smooth kernel. The proof is an application
of the Green operator.

96
2. Asymptotic Expansion of the Heat Kernel
Proposition 2.44. If A is an operator with smooth kernel
a(x,y) 6 T(M x M, {? ® \A\^2) f3(?®
then A is trace class, with trace given by the formula
Tr(A)= Tr(a(x,x)).
JxZM
Proof. Let H be a positive generalized Laplacian on the bundle ?, with Green
operator G. By Theorem 2.38, the operator Gk has continuous kernel for
k > 1 + (dim(M)/2), and hence is Hilbert-Schmidt.
The operator Ak associated to the smooth kernel Hka(x,y) is Hilbert-
Schmidt, as is any operator with smooth kernel, and A = GkAk is the product
of two Hilbert-Schmidt operators and hence is trace class. The trace of A is
now easy to calculate:
= Tr(GkAk)
gk(x,y)Hka(y,x)
a(x,x).
Let D be a differential operator acting on T{M,? <g> |A|1/2). If if is an
operator with smooth kernel, then so are DK and KD, and hence they are
trace class. More precisely, the operator DK has smooth kernel Dxk(x, y)
while integration by parts shows that KD has smooth kernel D*k(x,y).
Proposition 2.45. Let D be a differential operator acting on the bundle ?.
If K is an operator with smooth kernel, then Tr(DK) - Tr(KD).
Proof. Choose a positive generalized Laplacian on the bundle ?, and let G
be its Green operator. For sufficiently large N, the operator DGN has a
continuous kernel, and hence is a Hilbert-Schmidt operator. Thus,
Tt{DK) = Tr((DGN)(HNK)) = Tr((HNK)(DGN)) by B.10)
= Tr({HNKD)GN) = Tr(GN(HNKD)) = Tt(KD). ?
Let us construct an asymptotic expansion for the kernel of the operator
DPt, where D is a differential operator.
Proposition 2.46. // Pt is the heat kernel of a generalized Laplacian H and
D is a differential operator of degree m on the bundle ?, then there exist
smooth kernels $i(x,y,H,D) such that the kernel (x | DPt | y) satisfies
\\(x\DPt\y)-ht(x,y)
N
2.6. The Trace of the Heat Kernel
97
where ftt(x,y) = (A*t)-nl2e-d(x<tflutl){d{x,yJ)\dx\xl* ® \dy\xl2, and V> is a
cut- off function.
Proof. If we apply the differential operator d/dxi to the Euclidean heat kernel
9t(x,y) = i
we obtain
By induction, we see that d^qt has the form
,y)=ft(x,y) 22
The proof now follows by combining this with Theorem 2.30. D
From the proof of Proposition 2.46, we see that the restriction of the kernel
of the operator DPt to the diagonal has the form
D7rt)-"/2 f; *9t{x,x,H,D)\dx\,
i=-[m/2)
since the restriction of the coefficients $i(x, y, H, D) to the diagonal vanishes
for i < —m/2. It follows that the trace Tr(DP4) has an asymptotic expansion
of the form
Tr(Z?P()
where
Jm
i=-[m/2]
.= / 9t(x,x,H,D)\dx\.
Jm
It is also possible to give an asymptotic expansion of the trace of the
operator KPt, if K is a smoothing operator on T(M, ? ® IAI1/2).
Proposition 2.47. There is an asymptotic expansion of the form
oo
Tt(KPt) ¦ —
Proof. Let k(x,y) |dx|x/2 ® \dy\V2 be the kernel of K, and let kv € T(M,?)
be the section x >-* k(x,y). Since Tr(KPt) = Tr(PtK), we see that
TT(KPt) = / (Ptkx)(x) \dx\.
JM
Applying the asymptotic expansion B.8), we see that
•=o
from which the proposition follows. ?

98
2. Asymptotic Expansion of the Heat Kernel
2.7. Heat Kernels Depending on a Parameter
Let M be a compact manifold and let ? be a vector bundle over M. Let
(Hz | z e R), be a smooth family of generalized Laplacians; in other words,
we are given the following data, from which we construct the generalized
Laplacians Hz in the canonical way:
A) a family gz of Riemannian metrics on M depending smoothly on z 6 R;
B) a family of connections V1 = V? + u>z on the bundle ?, where u>z is a
family of one-forms in A1(M, End(?)) depending smoothly on z 6 R;
C) a smooth family of potentials Fz 6 T(M,End(?)).
Our goal in this section is to prove the following result.
Theorem 2.48. If Hz is a smooth family of generalized Laplacians, then
for each t > 0, the corresponding family of heat kernels pt{x,y,z) defines
a smooth family of smoothing operators on the bundle ?. Furthermore, the
derivative ofpt(x,y,z) with respect to z is given by Duhamel's formula
d
d~z'
-Pt(x,y,z) = - ( Pt—(x,yi,z)(dzHx)Vlp,(yi,y,z))d8;
2 Jo yJVleM '
alternatively, writing this in operator form, we have
As a first step in the proof, we will prove a result about the dependence
of the formal heat kernel of the family Hz on the parameter z. Since we are
only interested in results local in z, we may assume without loss of generality
that the injectivity radius is everywhere bounded below by e > 0. Let xp :
R+ —+ [0,1] be a cut-off function.
Lemma 2.49. For each t > 0, the formal solution k?(x, y, z) = k?(x, y, Hz)
depends smoothly on z. Given T > 0 and z lying in a compact subset of Hi,
the family of kernels dik?{x,y,z), 0 < t < T, form a uniformly bounded
collection of operators on Tl(M,? <g> |A|1/2) for each t>Q.
Proof. Consider the explicit formula for k^(x,y,zy.
•=o
,v) \dzx\1'2 \dzy
In this formula, dz(x,yJ is the distance between x and y 6 M, jz(x,y) is
the Jacobian of the exponential map, and \dzx\ is the Riemannian volume,
all with respect to the Riemannian metric gz. All of these objects depend
2.7. Heat Kernels Depending on a Parameter
99
smoothly on z, as does the parallel transport rz(x,y) in ? with respect to
the connection Vz.
We can show, by induction on i, that the terms $j(z,y, z) in the asymp-
asymptotic expansion depend smoothly on z near the diagonal in M x M. Indeed,
we have the following formula for $j(z,y, z) in terms of $j_i(z, y, z):
ri(zlJ/)-1$i(z,y,Z) = - / si-1Tz(x.,V)-1(B*x-$i-1)(x.,y,z)ds.
Jo
It remains to observe that the operator Bz depends smoothly on z, as is clear
from the formula
The proof of uniform boundedness of the kernels k^(x,y,z) for z in a
compact subset of R is the same as in the proof of Theorem 2.29. ?
We now turn to the proof of Theorem 2.48.
Proof. Consider the formula for Pt(x, y, z),
where r^(x,y,z) = (dt + H^k^x, y,z). By Lemma2.49, we know that
r^(x, y, z) depends smoothly on z. Prom now on, restrict z to lie in a bounded
interval. It is easy to see that rf satisfies the uniform estimate
dz
< C(t)t
N-n/2-t/2-m
where the C'-norm is that of Tl(M xM,{?® \h\1'2) B (? ® \k\1'2)).
Consider the space of all families of C'-sections ot?® | Al1/2 which are C*
in the parameter z, with norm
\ers\
m<k
Application of the operator K^(z) with kernel fcf (z, y, z), 0 < t < 1, gives a
uniformly bounded family of linear operators for this norm; this follows from
the fact that the operator d^K^(z) equals
combined with the fact that the family diKt'(z) is uniformly bounded on
T((M,? ® |A|1/2) for t e [0,1]. Thus, we see that the series for dl"pt(x,y,z)
converges uniformly in the C'-norm as long as N > m + (? + n)/2. Since N
is arbitrary, this proves the smoothness of pt(x,y,z) in z.

100
2. Asymptotic Expansion of the Heat Kernel
Let us now prove Duhamel's formula for dpt(x, y, z)/dz. Let <j> be a smooth
section of ? ®\\\ll2. If we apply the operator dt + Hz to the section dzPf4> €
T(M,?®\S\1'2), we obtain
(ft + Hz)dzP?<t> = 0,(ft
From this, it is easy to see that
Ai(x) = ?(/?
satisfies the heat equation (9t + Hz)Af = 0. We will show that A'(x) con-
converges to zero as t —> 0; by the uniqueness of solutions to the heat equation,
it follows that it vanishes. Since <j> is an arbitrary element of T(M, ?® [AI/2),
this proves Duhamel's formula.
We have the following asymptotic expansion for Pf<j) in T(M,? ® |A|1/f2):
Furthermore, we may take derivatives of both sides of this asymptotic expan-
expansion for d™P*<j>. In particular,
d \
On the other hand, the section
/ ^Pt->(;yi,z){dzH''%1p,(y1,y2,zL>{y2)
is a smooth function of (s, t,x, z), and hence
limy (/ 3Pt-s(;yuz)(dzHz)yips(yuy2,z)(i)(y2}jds = 0. D
The following corollary will be used repeatedly in the rest of the book.
Corollary 2.50. // Hz is a one-parameter smooth family of generalized La-
placians on a vector bundle ? over a compact manifold, then
Proof. Upt(x, y, z) is the kernel of the operator e~iH", then Duhamel's for-
formula shows that
-z-Pt(x,y,z) = - / Pt-.(x,yi,z)(dzHz)yip,(y1,y,z)ds.
oz J0 Jyi€M
Bibliographic Notes
Taking the supertrace of both sides gives
101
from which the result is clear, since the integrand is independent of s. ?
Bibliographic Notes
The pioneers of the approach to the analysis of the heat kernel that we
present in this chapter were Minakshisundaram and Pleijel[79]. A more
recent work which presents some applications to Riemannian geometry is the
book by Berger-Gauduchon-Mazet [19]; we have followed this book closely in
Sections 2-4.

Chapter 3. Clifford Modules and Dirac Operators
In this chapter, we begin the study of Dirac operators, which are a gener-
generalization of the operator discovered by Dirac in his study of the quantum
mechanics of the electron.
The problem that Dirac asked was the following (he was actually working
in four-dimensional Minkowski space and not in a Riemannian manifold):
what are the first-order differential operators whose square is the Laplacian?
Generalizing the question, one is motivated to ask: if M is a manifold and ?
is a vector bundle over M, what are the first-order differential operators D
whose square is a generalized Laplacian on F(M, ?)?
At this level of generality, the answer is quite easy to give. The operator
D, if written in local coordinates, takes the form
where ak(x) and b(x) are sections of End(?), and it is easily seen that
X^a^zjdfc transforms as a section of the bundle Hom(T*M, End(?)); this
section is —i times the symbol of D introduced in Chapter 2,
<r(P)(z,
The square of D is
D2 = A ^(o'(x)o-'(x) + ai(x)ai(x))didi + first order operator,
y
so that D2 is a generalized Laplacian if and only if for any f and n e T*XM,
we have
where (•, -)x is the metric on T*XM. This is the defining relation for the
Clifford algebra of T*XM, which we introduce in Section 1, and study the
representations of in Section 2.
In Sections 3-5, we will define and study the notion of a generalized Dirac
operator, and in particular, introduce the index, which is our main object of
interest in this book. Our general application of the term "Dirac operator"
should be carefully distinguished from its more usual use, which reserves
this name for the operators introduced by Lichnerowicz and Atiyah-Singer,
3.1. The Clifford Algebra
103
which act on a twisted spinor bundle. We will discuss this special case in
Section 6. Also in Section 6, we show how some of other the classical operators
of differential geometry are generalized Dirac operators, such as d + d* on a
Riemannian manifold, and 5 + 8* on a Kahler manifold.
3.1. The Clifford Algebra
The first step in defining the Dirac operator is to understand the linear alge-
algebra underlying it: the Clifford algebra.
Definition 3.1. Let V be & real vector space with quadratic form Q (which
need not be non-degenerate). The Clifford algebra of (V,Q), denoted by
C(V, Q), is the algebra over R generated by V with the relations
v • w + w ¦ v = — '2Q(v, w) for v and w in V,
in other words, v2 = — Q(v) for all v E V.
When Q is fixed, we may write C(V) for C(V,Q), and abbreviate Q(a,b)
to (a,b).
The Clifford algebra of (V, Q) is the solution to the following universal
problem.
Proposition 3.2. If A is an algebra and c : V —> A is a linear map satisfying
c(w)c(v) + c(v)c(w) = — 2Q(v,w),
for all v, w e V, then there is a unique algebra homomorphism from C(V, Q)
to A extending the given map from V to A.
The Clifford algebra may be realized as the quotient of the tensor algebra
T(V) by the ideal generated by the set
Jq = {v ® w + w ® v + 2Q(v,w) | v,w € V}.
Observe that the algebra T(V) is a Z2-graded algebra, or superalgebra,
with Z2-grading obtained from the natural N-grading after reduction mod 2.
Since the generating set of the above ideal is contained in the evenly graded
subalgebra of T(V), it follows that C(V) itself is a superalgebra, C(V) =
C+(V) 0 C~(V), with V C C"(V). We will mainly consider modules over
R or C for the Clifford algebra which are Z2-graded, by which we mean that
the module ? is a superspace E = E+ © E~, and the Clifford action is even
with respect to this grading:
C+(V)-E± cJS*,

104
3. Clifford Modules and Dirac Operators
Since the algebra T(V) carries a natural action of the group O( V, Q) of
linear maps on V preserving the quadratic form Q and the above ideal is in-
invariant under this action, it follows that the Clifford algebra carries a natural
action of O( V, Q) as well.
Let o i-» a* be the anti-automorphism of T(V) such that v € V is sent
to —v. Since the ideal Jq is stable under this map, we obtain an anti-
automorphism a i-+ a* of C(V).
Definition 3.3. If Q is a positive-definite quadratic form, we say that a
Clifford module E of C(V) with an inner-product is self-adjoint if c(a*) =
c(a)*. This is equivalent to the operators c(v), v € V, being skew-adjoint.
We will denote by c(v) the action of an element of V on a Clifford module
of C{V), and in particular, on the Clifford algebra C(V) itself. If E is a
Z2-graded Clifford module, we denote by Endc{v)(E) the algebra of endo-
morphisms of E supercommuting with the action of C(V).
Our first example of a Clifford module is the exterior algebra of V. To
define the Clifford module action of C(V) on AV, by the above lemma, we
need only specify how V acts on AV. To do this, we introduce the notations
e(v)a for the exterior product of v with a, and i(y) for the contraction with
the covector Q(v, •) e V*. We now define the Clifford action by the formula
C.1) c(v)a = e(y)a — i(v)a.
To check that this defines a Clifford module action on AV, we use the formula
C.2)
t,(w)e{v) = Q(v, w).
If Q is positive-definite, the operator t(v) is the adjoint of e(v), so that the
Clifford module AV is self-adjoint.
Definition 3.4. The symbol map a : C(V) -» AV is defined in terms of
the Clifford module structure on AV by
a(a) = c(a)l € AV,
where 1 € A0 V is the identity in the exterior algebra AV. Note that <r(l) = 1.
Let e-i be an orthogonal basis of V, and denote by c< the element of C(V)
corresponding to ej.
Proposition 3.5. The symbol map a has an inverse, denoted c : AV —>
C(V), which is given by the formula
c(eh A...Aejj) = Cj1...cij.
We call c the quantization map. It follows that C(V) has the same
dimension as AV, namely 2dlm(v). In fact, a is an isomorphism of Z2-graded
O(V)-modules.
3.1. The Clifford Algebra
105
The Clifford algebra has a natural increasing filtration C(V) = IJ{ Cj(V),
defined as the smallest filtration such that
O>(V) = R,
It follows that a G C(V) lies in Ct(V) if and only if it lies in the span of
elements of the form vx... vk, where Vj € V and k < i. The following result is
analogous to the identification of gr V(M, ?) with sections of S(TM)®End(?)
in the theory of differential operators; in fact, these two results are unified in
the theory of supermanifolds.
Proposition 3.6. The associated graded algebra grC(V) is naturally iso-
morphic to the exterior algebra AV, the isomorphism being given by sending
vi A ... A Vi € A'V to a(vi ¦ ¦ ¦ vt) e gr4 C(V).
The symbol map a extends the symbol map at: Cj(V) —> grrf C(V) = A*V,
in the sense that if a e Cj(V), then ct(o)w = a<(a). The filtration Cj(V) may
be written
fc=0
Using a, the Clifford algebra C(V) may be identified with the exterior
algebra AV with a twisted, or quantized, multiplication a -q /3. We will need
the following formula later: if v € V and a ? C(V), then
C.3)
a([v,a\) =-2t{v)a{a)
We will now show that if V is a real vector space with positive-definite
scalar product, then there is a natural embedding of the Lie algebra so(V)
into C(V).
Proposition 3.7. The space C2(V) = c(A2V) is a Lie subalgebra ofC(V),
with bracket the commutator in C(V). It is isomorphic to the Lie algebra
S0(V), under the map r : C2(V) -+ so(V) obtained by letting a G C2(V) act
on Cl{V) = V by the adjoint action:
r(a)-v = [a,v].
Proof. It is easy to check that the map r(a) really does preserve CJ(V), so
defines a Lie algebra homomorphism from C2(V) to Ql{V). To see that it
maps into «o(V), observe that
Since [[i;,iu],a] = 0, Jacobi's identity shows that this vanishes.
The map r must be an isomorphism, since it is injective and since the
dimensions of C2(V) and eo(V) are the same, namely n(n — l)/2. ?

106
3. Clifford Modules and Dirac Operators
We will make constant use in this book of the fact that a matrix A e eo(V)
corresponds to the Clifford element
C.4) t~\A) =
Note the following confusing fact: if we identify A with an element of A2V
by the standard isomorphism
A € eo(V)
A ei>
then c(A) equals
which differs from r~1(A) by a factor of two.
If a e C(V), then we may form its exponential in C(V), which we will
denote by expc(o); on the other hand, if q 6 AV, we can exponentiate it in
AV, to form expA(a).
Let a € A2V and consider the exponential expA a in the algebra AV. If T
is the Berezin integral on AV introduced in A.28), then by Proposition 1.36,
we have the following result.
Lemma 3.8. Assume that V is even-dimensional and oriented. If a 6 C2(V),
so that a{a) 6 A2V, then
T(expAa(a)) = 2~dimW2 det1/2(r(a)).
We will often use the following formula for the exponential of an element of
C2(V) inside C(V). If v and w are vectors in V such that (v, v) = (w, w) = 1
and (v, w) = 0, then
expc t(v ¦ w) = 2_) -g (y ¦ w)k
C.5)
If we are given a Lie subalgebra of the Lie algebra underlying a finite
dimensional algebra, then we can exponentiate it inside this algebra to obtain
an associated Lie group.
Definition 3.9. The group Spin( V) is the group obtained by exponentiating
the Lie algebra C2(V) inside the Clifford algebra C(V).
The adjoint action r of C2(V) on V exponentiates to an orthogonal action
still denoted by r of Spin(V) on V. For g € Spin(V) and v eV,we have the
fundamental relation
C.6)
gvg 1 = T(g) ¦ v.
3.1. The Clifford Algebra
107
Indeed, writing g = expc(o) with a 6 A2V, we see from the fact that [a, v] =
r(a)v that expc(a)vexpc(a)~1 = exp(r(o)) -v.
Proposition 3.10. //dim(V) > 1, the homomorphism
t : Spin(F) -> SO(F)
is a double covering.
Proof. The map r is clearly surjective, since the exponential map is surjective
on SO(V). Let g e Spin(V) be such that r(g) = 1. Then [g,v] = 0 for all
v e V. Formula C.3) implies that t(v)a(g) = 0 for all v 6 V, so that g is a
scalar.
If dim(V) > 1, then -1 € Spin(V), as follows from C.5) with t = tt.
Consider the canonical anti-automorphism a i-+ a* of C(V). On C2(V),
this anti-automorphism equals multiplication by — 1, since if v and w are
orthogonal vectors,
(v ¦ w)* = (-w) •(-«) = -v ¦ w.
Since
expc(o)* = expc(a") = expc(-a)
for o e C2(V), we see that g ¦ g* ~ 1 for all g 6 Spin(V). For g in the kernel
of r, and hence scalar, the relation g-g* = 1 implies that g = ±1. Since
—16 Spin(F), we see that r is a double cover. ?
Any Clifford module restricts to a representation of the group Spin(F) C
' C+(V); this gives a nice way to construct those representations of Spin(F)
which do not descend to representations of SO(V). These are known as spinor
representations, and will be discussed in the next section.
Proposition 3.11. // the Clifford module E is self-adjoint, then the repre-
representation of Spin(V) on E is unitary.
Proof. The representation of C2(V) on E is skew-adjoint, since c(a)* =
c(o*) = -c(a) for o 6 C2(F). It follows that the Clifford action of C2(V)
exponentiates to a unitary representation of Spin(V). ?
Definition 3.12. Let V be a vector space. If X € End(V), let
sinh(A-/2) ex'2 - e~xl2
JV{X) =
X/2
and let
jv(X) = det{Jv(X)) = det I—? ' ' \.

108
3. Clifford Modules and Dirac Operators
Let us compare the two functions <7(expc(o)) and expA<r(o) from C(V)
to AV. IfXeEnd(V), let
=
be the analytic map from End(F) to itself which is defined on the set of
X whose spectral radius is less than 2n. Since Hy@) = 1, there is a well
defined square root HV/2(X) e GL(V) if X lies in a neighbourhood of 0. For
o e C2{V), we will denote Hv(r(a)) by Hv{a), and jv(r(o)) by jv(a). We
denote by a y-+ g ¦ a the automorphism of the algebra AV corresponding to
an element g € GL(K).
Proposition 3.13. If a € (^{V) = c(A2F) is sufficiently small, we have
<r(expc(o)) = jl/2(a) det(^/2(a)) (ffv1/2(a)-expA(<r(a))).
Proof. We can choose an orthonormal basis fa)^ of V such that
+ 92e3e4 + ... .
In this way we reduce the proof to the case in which V is two-dimensional and
a = 0eie2. In this case, expc a = cosfl + sin0eie2, while jy2{a) = d~l sin0,
and the matrix Hv (a) equals 0 cot 6. U
The preceeding proof shows that on the Cartan subalgebra of so(V) of
matrices of the form
< ? < dim(V)/2,
we have the formula
It follows that on this subalgebra, jv(X) has an analytic square root
We may now apply the following theorem of Chevalley, which is proved in
Chapter 7 (see Theorem 7.28).
Proposition 3.14. Let G be a connected compact Lie group with Lie algebra
0,o nd let T be a maximal torus of G, with Lie algebra t and Weyl group
W(G,T). If(j> is an analytic function on t which is invariant under W(G,T),
then it extends to an analytic function on all of g.
Corollary 3.15. The function jv/2(X) has an analytic extension to eo(V).
3.2. Spinors
109
Let T be the Berezin integral on AV. Since T(g-a) = det(g)T(a), we
obtain the following corollary of Proposition 3.13; however, we do not use
this result in the rest of this book.
Proposition 3.16. If a 6 C2(V), then
T(<7(expc(a))) = jl/2{a) T(expA a(o)).
3.2. Spinors
In this section, we will construct the spinor representation of the Clifford
algebra of a Euclidean vector space V. We will suppose that V has even
dimension n, and, in addition, that an orientation on V has been chosen, to
remove an ambiguity which otherwise exists in defining the Z2-grading of the
spinor space.
In the following lemma, we do not assume that V is even-dimensional.
Lemma 3.17. Let eit 1 < j < n, be an oriented, orthonormal basis ofV,
and define the chirality operator
wherep = n/2 ifn is even, andp = (n+l)/2 ifn is odd. Then T e C(V)®C
does not depend on the basis of V used in its definition. It satisfies F2 = 1,
and super-anticommutes with v for v € V, in other words, Tv = — vF ifnis
even, while Tv = vrifnis odd.
It follows from this lemma that T belongs to the centre of C(V) ® C if n
is odd, while if n is even and E is a complex Clifford module, we can define
a Z2-grading on E by
E* = {v G E | Tv = ±v}.
Furthermore, when the dimension of V is divisible by four, T belongs to the
real Clifford algebra C(V), so that in that case, real Clifford modules are also
Z2-graded.
Definition 3.18. A polarization of the complex space V® C is a subspace
P which is isotropic, that is, Q(w, w) = 0 for all w € P, and such that V®C =
P © P; here, we extend the bilinear form Q on V ® C by complex linearity.
The polarization is called oriented if there is an oriented orthonormal basis
et of V, such that P is spanned by the vectors {e2j_i - ie2j | 1 < j < n/2}.
Proposition 3.19. // V is an even-dimensional oriented Euclidean vector
space, then there is a unique 7^-graded Clifford module S = S+ © S~, called
the spinor module, such that
In particular, dim(S) =
C(V) <g> C =* End(S).
and dim(S+) = dimE~) = 2<"/2>-1.

110
3. Clifford Modules and Dirac Operators
Proof. Given a. polarization of V, finding a realization of 5 is very simple.
The bilinear form Q on V ® C places P and P in duality. The spinor space
5 is now defined to be equal to the exterior algebra AP of P. An element
v of V is represented on S by splitting it into its components in P and P;
the first of these acts on AP by exterior product, while the second, which
may be considered to be an element of P* by the above duality, acts by
contraction (there are some constants inserted to make the relations of the
Clifford algebra come out precisely): if s ? S — AP, we have
c(w) ¦ s = 21/2e(w)s, if w ? P,
c(w) • s = -21/2i(w)s,
To see that C(V) ® C is in fact isomorphic to End(S), we only need to
observe that dim(C(V)) = dim(SJ, and that no element of C(V) acts as
the zero map on 5. The uniqueness of the spinor representation is now a
consequence of the fact that the algebra of matrices is simple, that is, it has
a unique irreducible module.
We will now show that if the polarization P of V is oriented, the operator
c(r) on S = AP is equal to (-1)* on A*P, so that
C.7)
= A±P.
In terms of the vectors Wj = 2~1/2(e2J-i — ie2j), we may rewrite F as
r = 2~n/2{u>iW1 - ivivii)... (wn/2wn/2 - wn/2wn/2).
Thus, T acts on AP by
(-I)n/2(e(wi)t(tui) - t(wi)e(«;i))... (e(wn/2)t.(wn/2) - i(wn/2)e(wn/2));
this is easily seen to equal (—l)fc on A*P. ?
Since Spin(F) C C+(V), it follows that both S+ and S~ are representa-
representations of Spin(F). They are called the half-spinor representations, and have a
great importance in differential geometry.
Definition 3.20. We denote by p the spinor representation of Spin(V)
in
There is a natural supertrace on C(V), defined by
We will derive an explicit formula for this supertrace; before doing so, we will
prove a result about abstract supertraces on C(V).
3.2. Spinors
111
Proposition 3.21. // the quadratic form Q is non-degenerate, then there is,
up to a constant factor, a unique supertrace on C(V), equal to T oa. The
supertrace Str(a) defined in C.8) equals
a) = (-2i)n/2To«7(a).
Proof. We will show that (C(V),C(V)] = Cn_i(V). It follows that any
supertrace on C(V) must vanish on Cn_i(V), and hence be proportional to
To a.
Let (ej)"=1 be an orthonormal basis of V. For any multi-index / C
{1,... ,n}, we form the Clifford product cj = Yli€I q; the set
is a basis for C(V). If |/| < n, them there is at least one j such that j ^ I,
and we obtain
On the other hand, the space of supertraces on C(V) is at least one-di-
one-dimensional, since we have constructed a non-vanishing supertrace in C.8).
To calculate the normalization of this particular supertrace, it suffices to
calculate its value on the single element F, which by C.7) equals dim(AP) =
2n^2. In this way, we obtain the formula Str(o) = (-2i)"/2T o o-(a). D
In particular, for a € C2(V), we see by Corollary 3.16 that
C.9) Str(expc a) = (-2i)n'2jl/a(a) T(expA a(a)).
Lemma 3.22. If g e Spin(V), then T{a(g)J = det((l - r(g))/4).
Proof. The calculation is easily reduced to the case in which dim(V) = 2 and
g = cosO + sin0eie2, so that
det((l - T(p))/4) = A - <"fl)(l - e-2i«)/16 = \ sin2 $. ?
It follows from Lemma3.22 that if g € Spin(V) is such that det(l — r(g))
is invertible, then T(a(g)) € AnV is non-zero. Define e(g) to equal +1
(respectively —1) if T(a(g)) is a positive (negative) element of AnV. With
this notation, we obtain the following formula for the supertrace of the Spin-
representation. This result expresses in a quantitative way the notion that
the spinors are a "square-root" representation.
Proposition 3.23. If g € Spin(V), then
Str(p(g)) = i-n<2e{g)\ det(l -
When g ? SO(V), the eigenvalues of g occur either in conjugate complex
pairs, or are ±1, and hence det(l — g) > 0. The following result does not
mention the spinor representation, but we state it here because the proof
makes use of this representation. ;

112
3. Clifford Modules and Dirac Operators
Proposition 3.24. If g G SO(V), then the function det(l -gexpA) ofA€
so(V) has an analytic square root.
Proof. The function W2 Str(p(jrexp A)), where g is any element of Spin(V)
which covers g e SO(F), is an analytic square root of det(l — gexpA). O
Definition 3.25. If g e SO(V), then Dg(A), A € so(V), is the analytic
square root of det(l - gexpA) such that Dg{0) = | det(l - gI/2\.
We will use the more suggestive notation det1//2(l — gexpA) for Dg(A).
Next, we show that the spinor representation is self-adjoint.
Proposition 3.26. There is, up to a constant, a unique scalar product on
the spinor space S for which it becomes a self-adjoint Clifford module. With
respect to this metric, the spaces S+ and S~ are orthogonal to each other.
Proof. Uniqueness is clear, since it is implied by the irreducibility of 5 as
a C(V)-module. To show the existence of a scalar product, we construct
one explicitly on the realization AP that we obtained above. Indeed, it is
sufficient to have a scalar product on the space P, which can be extended
in the usual way to all of AP. We choose the scalar product furnished by
{w,w), where w is the complex conjugate of w and is thus an element of P.
In particular, it is immediate that S+ = A+P is orthogonal under this scalar
product to S~ = A~P.
To show that this metric makes the Clifford module S self-adjoint, it suf-
suffices to show that the operator c(v) is skew-adjoint for v 6 V. But an element
of V, being real, has the form w + w; thus it acts on AP by v^^w) — t(u))),
which is clearly skew-adjoint. ?
The next result shows that the spinor space S is the building block for
complex representations of C(V).
Proposition 3.27. IfV is an even-dimensional real Euclidean vector space,
then every finite-dimensional Z2 -graded complex module E of the Clifford
algebra C(V) is isomorphic to W<8>S, for the Z2-graded complex vector space
W = Homc(v)E, E) which carries a trivial C(V)-action, and End(W) is
isomorphic to Endc(v)(E)-
Proof. This follows from the fact that any finite dimensional module for the
matrix algebra End(S) is of the form W ® S. The isomorphism between
HomC(V)(S, E) ® S and E is given byw®«i-» w(s). ?
The space W will be called the twisting space for the Clifford module
E.
Since Strs(r) = 2"/2, the supertrace over W of an element F e End(W) ^
is given by the formula
C.10) Str,y(F) = 2-"/2 StrE(I\F).
Motivated by this, we make the following definition.
3.3. Dirac Operators
113
Definition 3.28. The relative supertrace StrB/s : EndC(v){E) -* C
equals
If W is ungraded, the relative supertrace equals the W-trace, and is pro-
proportional to the 2?-trace:
Trw(F) = StiE/s(F) = 2-"/2TrB(F).
We end this section with some formulas for the special case in which V
is a complex Hermitian vector space; these formulas will be of use in study-
studying the 5-operator on a Kahler manifold. In this case, the group U(V) of
unitary transformations of V is a subgroup of SO(V). Let us consider the
decomposition
where V1'0 and V0'1 are the spaces on which the complex structure J of V
acts by +i and —* respectively. Then V1>0 is a polarization of V ® C and
we can take 5 = AV'0 as the spinor space. On the other hand a complex
transformation of V leaves V1'0 stable, so there is a natural representation
A of U(V) on AV1'0. The representation A and the restriction of the spinor
representation p of Spin(V) to r U(V) differ by the character det^/2 of the
double cover t~1(U(V)) of U(V). More precisely, the infinitesimal represen-
representations of the Lie algebras on S, which we denote also by A and p, are related
by the following result.
Lemma 3.29. Let a e C2(V). Ifr(a) is complex linear on V, then
p(a) = A(r(o))-iTrv.,oT(a).
Proof. Choose a complex basis w, of V1-0 such that t(g)wj =
j, and (vjj,Wi) = Sij. Then a = | Ei<«<«/2 M«W - uw),
A(r(a)) = ]T Ai?(wi)i(i(>i),
Prom these formulas, the result is evident. ?
for every
, so that
3.3. Dirac Operators
Definition 3.30. If M is a Riemannian manifold, the Clifford bundle
C{M) is the bundle of Clifford algebras over M whose fibre at x e M is
the Clifford algebra C(T*XM) of the Euclidean spaces T*XM.

114 3. Clifford Modules and Dirac Operators
The Clifford bundle C(M) is an associated bundle to the orthonormal
frame bundle,
C{M) = O(M) xo(n) C(Rn).
FYom this representation of C(M), we see that it inherits a connection, the
Levi-Civita connection, which is compatible with the Clifford product, in the
sense that
V(ab) = (VaN + o(V6) for all a and 6 e T{M, C{M)).
Definition 3.31. The symbol map a
a : C{M) -» AT*M,
which identifies the Clifford bundle C{M) with the exterior bundle AT*M,
is defined by means of the symbol map ax : C(T*XM) —* AT*XM.
Definition 3.32. A) A Clifford module ? on an even-dimensional Rie-
mannian manifold M is a Z2-graded bundle ? on M with a graded action
of the bundle of algebras C(M) on it, which we write as follows:
(a, s) i—> c(a)s, where a e T(M, C(M)) and s e T(M, ?).
B) If ? is a Clifford module with metric h, for which ?+ and ?~ are orthog-
orthogonal, we say that the Clifford module ? is self-adjoint if the Clifford
action is self-adjoint at each point of M, in other words, if the operators
c(o) with a ? T*M are skew-adjoint.
C) If W is a vector bundle, then the twisted Clifford module obtained from
? by twisting with W is the bundle VV<8>?, with Clifford action 1 ®c{a).
The most basic example of a Clifford module is the bundle of spinors. In
order to define this, we need a spin-structure on the manifold.
Definition 3.33. A spin-structure on a manifold M is a Spin(n)-principal
bundle Spin(iW) on M such that the cotangent bundle of M is isomorphic to
the associated bundle Spin(M) Xgpin(n) R".
Note that a manifold with a spin-structure is, in particular, an oriented
Riemannian manifold, with frame bundle
SO(M) = Spin(M) xSpin(n) SO(n).
Thus the principal bundle Spin(M) is a double cover of SO(M) and inherits
from SO(M) the Levi-Civita connection.
There may be topological obstructions to the existence of a spin-structure
on M, while on the other hand, there may be more than one. This question
is understood using the classification of isomorphism classes of principal G-
bundles on a manifold G by the cohomology group HX(M, G), combined with
the Bockstein long exact sequence
) -> H2(M,Z2)
3.3. Dirac Operators 115
associated to the short exact sequence of groups
1 -+ Z2 -» Spin(n) -¦ SO(n) -+ 1.
The frame bundle SO(M) defines an element a e H1 (M, SO(n)) whose image
in H2(M, Z2) may be identified with the second Stieffel-Whitney class of M,
written w2(M). This at least motivates the following result.
Proposition 3.34. An orientable manifold M has a spin-structure if and
only if its second Stieffel-Whitney class u>2(M) G H2(M, Z2) vanishes. If this
is the case, then the different spin-structures are parametrized by elements of
/f1(M,Z2) S Hom(jri(M),Z2).
If M is an even-dimensional spin-manifold, the spinor bundle S is defined
to be the associated bundle
5 = Spin(M) xSpin(n) 5.
It is clear that S is a Clifford module, since the action of C(R") on S leads
to an action of the associated bundle C(M) = Spin(M) xSpin(n) C(Rn) on S.
The following is the analogue for manifolds of Proposition 3.27.
Proposition 3.35. ifM is an even-dimensional oriented manifold with spin-
structure, every Clifford module ? is a twisted bundle ? = W<g>S.
Proof. We may define a complex vector bundle W such that ? = W ® S
as Clifford modules by the formula W = Endc(M)($>?)• The isomorphism
between Homc(M)(S,?)®S and ? is given by w <g> a i-» w(s). D
Locally, we may trivialize the bundle C(M) by choosing an orthonormal
frame of the cotangent bundle. Over such an open set U, the Clifford bun-
bundle C(M) ^ U x C(Rn). Hence, we see that a spinor bundle 5 always
exists locally, and we can always locally decompose a Clifford module ? as
Homcm(S,?)®S.
If o € A(M, C(M)) is a Clifford algebra-valued differential form on M,
we can define an operator c(a) which acts on the space A(M, ?) by means of
the following formula: if a and E are differential forms on M, a is a Clifford
algebra section, and a is a section of ?, all homogeneous with respect to the
Z2-grading, then
C.11)
(c(a ® o))(/? ® s) = (-1)IQI ¦ W(a A 0) ® (c{a)s)
The interest of Clifford modules is that they are naturally associated to a
class of first-order differential operators, known as Dirac operators. We will
use a more general definition of Dirac operators than is usual, since it fits in
better with the notion of a superconnection, and does not make any of the
proofs more difficult.

Definition 3.36. A Dirac operator D on a Z2-graded vector bundle ? is
a first-order differential operator of odd parity on ?,
such that D2 is a generalized Laplacian.
Applying the results of Chapter 2, we obtain the following result. From
now on, when discussing the heat kernel of a generalized Laplacian H on a
Riemannian manifold M and vector bundle ?, we will use the Riemannian
density to identity the heat kernel (x \ e~tH \ y) of H with a section of ?B?*
over M x M.
Proposition 3.37. A) If D is a Dirac operator on a compact manifold M,
then D2, acting on F(M,?), has a smooth heat kernel
(x\e-tD* \y)er(MxM,?R?').
B) If the manifold M is compact, the Dirac operator D has finite dimensional
kernel.
In particular, a symmetric Dirac operator D is essentially self-adjoint.
Proof. The operator D2 has a finite dimensional kernel, and hence so does
the Dirac operator D. The proof that D is essentially self-adjoint uses the
heat kernel (a; | e"*0' | y) in the same way as the proof of the corresponding
result for D2 given in Proposition 2.33 did. D
The purpose of this section is to work out what are the geometric data
needed to specify a Dirac operator: this is analogous to the one-to-one cor-
correspondence between generalized Laplacians and the three pieces of data con-
consisting of a metric on M, a connection on ? and a potential F G F(M, End(?)).
If we are given a Dirac operator on a ,vector-bundle ?, then ? inherits a
natural Clifford module structure.
Proposition 3.38. The action of T*M on ? defined by
[D, /] = c(df), where f is a smooth function on M,
is a Clifford action, which is self-adjoint with respect to a metric on ? if the
operator D is symmetric. Conversely, any differential operator D such that
[D, /] = c(df) for all f 6 C°°{M) is a Dirac operator.
Proof. First of all, this definition is unambiguous, since if / and h are both
smooth functions on M, we have
c(d(fh)) = [D, fh] = fc(dh) + hc(df).
To check the relations for c(df) to be a Clifford action, we need only observe
that
c(dff = i[[D,/],[D,/]] = i[[D2,/],/] = -\df\\
o.o. uirac operators
117
since D2 is by assumption a generalized Laplacian; we have used the fact
that [[D,/],/j = 0 and that [D,[D,4]] = [D2,A], which is true for any odd
operator D and even operator A. This equation also shows that any operator
satisfying [D,/] = c(df) for all / € C°°{M) is in fact a Dirac operator.
Finally, if D is self-adjoint with respect to some metric on ?, then for all real
functions / on M,
It is clear that there is always at least one Dirac operator on a Clifford
module; we can construct one explicitly by taking the operators J^ c(dx')di
on local coordinate patches and combining them by means of a partition of
unity.
If D is a Dirac operator on a bundle ? and A is an odd section of the
bundle End(?), the operator D + A will be another Dirac operator on ?
corresponding to the same Clifford action on ?. Conversely, any two Dirac
operators on ? corresponding to the same Clifford action will differ by such
a section. Indeed, if Do and Di are two Dirac operators on ?,
for all / € C°°(M), and hence Di - Do may be represented by the action of a
section of End~(?). Thus, the collection of all Dirac operators on a Clifford
module is an affine space modelled on F(M,End~(?)). In order to sharpen
this identification, we will introduce the notion of a Clifford superconnection
on a Clifford module.
Definition 3.39. A) If VE is a connection on a Clifford module ?, we say
that V? is a Clifford connection if for any a e F(M, C{M)) and X €
r(M,TM),
In this formula, Vx is the Levi-Civita covariant derivative extended to
the bundle C(M).
B) If A is a superconnection on a Clifford module ?, we say that A is a
Clifford superconnection if for any o € F(M, C(M)),
C.12) [A,c(a)]=c(Va).
In this formula, Va is the Levi-Civita covariant derivative of a, which is
an element of Al(M,C{M)).
It follows from this definition that if A is a Clifford superconnection, then
the formula C.12) holds for all a € A(M,C(M)).
The collection of all Clifford superconnections on ? is an affine space based
on A~ (M, Endc(M) {?)), the space of sections of AT*M (g> Endc(M) (?) of odd
total degree; this follows from the fact that if A is a Clifford superconnection
on ? and a e ^4~(M,EndC(M)(^)), then A + a is again a Clifford supercon-
superconnection.

118
3. Clifford Modules and Dirac Operators
The Levi-Civita connection V5 on the spinor bundle S of a spin manifold is
a Clifford connection. If u>? are the coefficients of the Levi-Civita connection
with respect to an orthonormal frame e< of the tangent bundle, defined by
Va(ej = uj^ek, then C.4) shows that V5 is given in local coordinates by the
formula
C.13)
7*
Corollary 3.41. There exists a Clifford superconnection on any Clifford
module ?.
Proposition 3.40. Let M be a spin manifold and let ? = W®S be a twisted
spinor bundle.
A) If Aw is a superconnection on the bundle W, then the tensor product
superconnection A = Aw ® 1 + 1 ® V5 on the twisted Clifford module
W ® «S is a Clifford superconnection. We call Aw the twisting super-
connection associated to the Clifford superconnection A, and denote it
by A?'s.
B) There iso one-to-one correspondence between Clifford superconnections
on the twisted Clifford module W®S and superconnections on the twisting
bundle W, induced by the correspondence Aw >-> (Aw ® 1 + 1 ® Vs).
Proof. PartA) follows from the formula for the Clifford action onW®5,
which sends a 6 C(M) to 1 ® c(a) e T(M, End(W ® S)). Thus, we see that
[Aw ® 1 + 1 ® V5,1 ® c(a)} = 1 ® [V5, c(a)] = 1 ® c(Va).
To see that the assignment of a twisted Clifford superconnection on W® <S
to a superconnection on W is a bijection, we simply observe that the space of
Clifford superconnections on W® 5 is an affine space modelled on the space
of sections A~(M, Endc(M)(W ® ?)) = A~(M, End(W)), as is the space of
superconnections on W.
If A is a Clifford superconnection on the twisted Clifford module W ® S,
we define an operator Aw acting on sections of W by the formula
(Awu) ® a = A(u ® s) - u ® (Vss),
where s is a non-vanishing section of S. It is easy to see that this formula is
independent of the section s used, since any two sections Si and s2 of S are
related locally by the formula s2 = c(o)«i for some section a e F(M,C(M))
of the Clifford bundle C(M). We then see that
A(u ® s2) - u ® (V5s2) = A(u ® (cia^S!) - u ® (Vsc{a)s1)
3.3. Dirac Operators
119
Proof. To construct a Clifford connection on the Clifford module ?, it is
sufficient to do this locally and then patch the local Clifford connections
together by means of a partition of unity. But locally, we may decompose
? as W ® 5. It suffices to take any connection Vw and form the Clifford
connection Vw® 1 + 1® V5. D
We wil now study the relationship between Clifford superconnections and
Dirac operators. Let us start with the case of a Clifford connection V?. We
may define a Dirac operator on ? by means of the following composition:
T(M, ?) -^ r(M, T*M ® ?) A T(M, ?)
In local coordinates, this operator may be written
D = ?>(<*
from which it is clear that it is a Dirac operator, that is, that [D, /] = c(df).
Often in the results of this book, we will have to restrict attention to operators
in the subclass of Dirac operators constructed from a Clifford connection. In
terms of an orthonormal frame ej of the tangent bundle TM over an open
set U with dual frame e* of T*M, we may also write
If M is a spin manifold, the Dirac operator on S associated to the Levi-
Civita connection V5 is the operator often referred to as the Dirac operator
associated to the spin-structure, with no further qualification. Other authors
reserve the term Dirac operator for twisted versions of this operator; we have
chosen a far more liberal definition of Dirac operators because many results
of index theory deserve to be stated in greater generality.
Let A be a Clifford superconnection on the Clifford module ?. We can
define a first-order differential operator on T(M, ?) , which we will write as
DA, by composing the superconnection with the Clifford multiplication map:
T(M,?) -^ A(M,?) A T(M, C{M) ® ?) A T(M,?)
Here, we have made use of the isomorphism
A(M,?)?*,r{M,C(M)®?)
given by the quantization map c : KT*M —» C(M) of Proposition 3.5. With
respect to a local coordinate system and orthonormal frame ej of the tangent
bundle in which

Aj being sections of End(?), we have
Proposition 3.42. The map sending a Clifford superconnection A to the
operator Da is a one-to-one correspondence between Clifford superconnections
and Dirac operators <x>mpatible with the given Clifford action on ?.
Proof. First, we must check that the operator Da is in fact a Dirac operator
compatible with the given Clifford action. This follows from Leibniz's formula
A(/s) = df ® s + f • As 6 A(M, ?), where / € C°°(M) and s 6 T(M, ?).
If we apply the Clifford map
A(M, ?) — T(M, C(M) ®?)-+ r(Af, ?)
to both sides, we obtain the desired equation, namely
It remains to show that any Dirac operator on ? may be realised in a unique
way in the form DA for some Clifford superconnection A. The uniqueness is
immediately deduced from the fact that any two Clifford superconnections
differ by an element of A~{M,EndC(M){?))- K Do and Di are two Dirac
operators on ?, Di — Do may be represented by the action of a section of the
bundle End~(?). However, applying the symbol map
r(M,End~(?)) 2 r(Af,C(M)
A~(M,End(?))
to this section, we obtain a differential form w 6 A~(M,Endc(M)(?))- I*
follows from this that Di = Do + c(u>). If Do is the Dirac operator associ-
associated to the Clifford superconnection A, we see that Di is the Dirac operator
associated to the Clifford superconnection A + u. ?
Proposition 3.43. If A is a Clifford superconnection on ?, then the curva-
curvature A2 € A(M, End(?)) of A decomposes under the isomorphism End(?) =
C(M) ®Endc(M)(?) as follows:
In this formula, R? 6 A2{M,C(M)) C ,42(M,End(?)) is the action of the
Riemannian curvature R of M, on the bundle ?, given by the formula
kl
and Fe/S € ,4(M,EndC(M)(?)) ** an invariant of A, called the twisting
curvature of the Clifford module ?.
Proof. Define F?/s to equal A2 — R?. If we can show that the operation
e(F?/s) of exterior multiplication by F?/s, acting on A(M,?), commutes
with the operators c(a), where a € r(M,T"M) C F(M,C(M)), then it will
follow immediately that F?ts is actually a differential form with values in
Endc(M)(?)- But this commutation follows immediately from the the fact
that A is compatible with the Clifford action, as expressed in C.12):
[A2,c(o)] = [A, [A,c(a)]] = [A,c(Va)] = c(V2a).
To finish the proof, we use the fact that V2a = flo,where R is the Riemannian
curvature of M and that \R?, c(a)) = c(Ra), so that
[A2,c(a)] = c(Ra) = [R?,c(a)]. Q
Let M be a spin manifold and ? — W®S be a twisted spinor bundle. If Aw
is a superconnection on W, with curvature Fw, and if A = AW<8>1+1®VS is
the corresponding twisted superconnection on ? = W®S, then the curvature
A2 equals
Thus, the twisting curvature F?/s of A equals the curvature Fw of the twist-
twisting superconnection A^; this explains why we call it the twisting curvature.
In the next proposition we compute the formal adjoint of a Dirac operator.
Recall that the adjoint V* of an ordinary connection V on a Hermitian vector
bundle is defined by the equality of one-forms
d(su s2) = (Vsi, s2) + («i, V*s2) for all s< e T(M, ?).
Likewise, if A = V + u is a superconnection, where u> ? A~(M, End(?")),
then A* is defined to equal V* + u>*, where by u* we mean the differential
form ??=o(-1)i(<+1)/2w[<l; the reason for the sign (-l)««+i)/2 is that the
differential form dfi...dfi = {-I)i^-1^2dfi...dfx and taking adjoints will
introduce an additional (—1)* sign.
Proposition 3.44. The adjoint of the Dirac operator Da is the Dirac oper-
operator Da* .
Proof. It is easy to see that it suffices to prove this when A is a connection V.
Let si and 82 be two sections of ? and let X be the vector field on M given by
a(X) = (si,c(a)s2) for a e A1{M). Thus VX belongs to T(M, TM®T*M).
It is easy to see that
(DAs1,s2)I = (si,DvS2)x-Tr(VX)x.
The integral of the last term vanishes by Proposition 2.7, proving the propo-
proposition. D
Dirac operators are characterized by the formula [D, /] = c(df) for all
/ e C°°(M). In the next proposition, we calculate [D,c@)] when 9 is a one-
form on M and D is a Dirac operator. Let V? be the connection associated
to the generalized Laplacian D2 on the bundle ?.

122 3. Clifford Modules and Dirac Operators
Proposition 3.45. 7/0 € AX{M) is dual to the vector field X, then
where we recall that
Proof. It suffices to prove the theorem for differential forms of the type fdg.
Let X = gradgi be the vector field dual to dg. Observe that
[D, c(/ dg)] = /[D, c(dg)] + c{df) c{dg).
Since c(df) c(dg) = c(df A dg) - (df, dg), we see that
[O,c(fdg)\ = f[D,c(dg)}+c(d(fdg)) - (df,dg).
Now observe that
since D is a generalized Laplacian associated to the connection Vs. We
that
see
[D,c(fdg)} = -2/v? + c(d(fdg)) + f(cTdg) - (df,dg).
Using the fact that d'(fdg) = fAg - (df, dg), the result follows. ?
3.4. Index of Dirac Operators
Let ? be a Z2-graded vector bundle on a compact Riemannian manifold M,
and let D : T(M,?) -> T(M,?) be a self-adjoint Dirac operator. We denote
by D* the restrictions of D to T(M, ?±), so that
0
where D~ = (D+)*.
An important invariant that can be associated to the operator D is its
index. If E = E+®E~ is a finite-dimensional superspace, define its dimension
to be
- dim(?T).
The superspace ker(D) is finite dimensional.
3.4. Index of Dirac Operators
123
Definition 3.46. The index space of the self-adjoint Dirac operator D is
its kernel
ker(D) = ker(D+)eker(D-).
The index of D is the dimension of the superspace ker(D),
ind(D) = dim(ker(D+)) - dim(ker(D")).
Lemma 3.47. If u = ( *+ u0 ) is an odd self-adjoint endomorphism of a
superspace E = E+ © E~, then coker(w+) S ker(u~)*, where coker(u+) =
E-/im(u+).
Proof. Since u~ is the adjoint of u+, we have for all e 6 E+ and / 6 E~ that
so that / e ker(u~) if and only if (u+e, f) = 0 for all e e E+. D
We will now prove an analogous result for self-adjoint Dirac operators.
Thus let S be a Z2-graded Clifford module on a compact Riemannian manifold
M of even dimension, and let D be a self-adjoint Dirac operator on F(M, ?).
As above, we denote by D the restrictions of D to T(M, *
Proposition 3.48. If D is a self-adjoint Dirac operator on a Clifford module
? over a compact manifold M, then
IXM.f*) = ker(D±) © imCD*),
and hence
ind(D) = dim(ker(D+)) - dim(coker(D+)).
Proof. Since D is self-adjoint, (D2*,s) = (Ds, Da) and ker(D) = ker(D2).
Let G be the Green operator for the square of D; this is the operator
which inverts D2 on the orthogonal complement of ker(D). Since D2 is a
generalized Laplacian, G preserves F(M, ?) by Proposition 2.38. Let Po be
the projection operator onto ker(D). If s G F(M, ?), then s = D(DGs) + Po^
is a decomposition of s into a section D(DGs) in the image of the D and a
section in the kernel of D. This shows that
T(M, ?±) = im(DT) © ker(D±). D
From this proposition, we see that the index ind(D) may be identified with
the dimension of the superspace ker(D+)©coker(D+) if D is self-adjoint. This
is the usual definition of the index of D+. However, we prefer to phrase the
definition of the index in a more symmetric form, involving both D+ and D~,
so we adopt our first definition of the index. It is also for this reason that we
write ind(D) and not the more usual notation ind(D+).
The index of D is a topological invariant of the manifold M and the Clifford
bundle ?: that is, if Dz is a one-parameter family of operators on T(M, ?)
which are Dirac operators with respect to a family of metrics gz on M and
Clifford actions c* of C(M,gz) on ?, then the index of Dz is independent of

1/4 3. Clifford Modules and Dirac Operators
z. We will prove this by means of a beautiful formula for ind(D) in terms of
the heat kernel e~tD called the McKean-Singer formula.
If A is a trace-class operator on a Z2-graded Hilbert space H^, then its
supertrace is defined as in the finite dimensional case by the formula
Sti(A) =
TrM+ (A) - Tr^- (A) if A is even,
0 if A is odd.
Consider the Hilbert spaces H± of L2-sections of E^. The following lemma
is the analogue for the supertrace of Proposition 2.45.
Lemma 3.49. If D is a differential operator on ? and if K has a smooth
kernel, then Str[?>, K] = 0.
Proof. If K and D are both even, this follows immediately from Lemma 2.45.
On the other hand, if D and K are both odd, that is, D = (?+ D~) and
K=(k+ Ko )'then Sti{DK + KD) = Tr[ii:-,D+] + Tr[?>-)A:+] = 0, again
by Proposition 2.45. D
The supertrace of the projection onto the kernel of D is clearly equal to
the index of D:
Str(Po) = dim(ker(D+)) - dim(ker(D-)) = ind(D).
The McKean-Singer formula generalizes this formula, and will allow us to use
the kernel of e~tD to calculate the index of D.
Theorem 3.50 (McKean-Singer). Let (x | e'10' \ y) be the heat kernel
of the operator D2. Then for any t>0
ind(D) = Str(e
-tD') = / Str«x I e-tD'
Jm
x))dx.
Proof. We give two proofs of this formula. The first one uses the spectral
theorem for the essentially self-adjoint operator D2. If A is a real number,
let n* be the dimension of the A-eigenspace H f of the generalized Laplacian
D2, acting on T(M,€±). We have the following formula for the supertrace of
A>0
Since the Dirac operator D commutes with D2, it interchanges H* and WJ,
and moreover induces an isomorphism between them if A / 0, since in that
case, the composition
is equal to the identity. Thus, n* — nx = 0 for A > 0, and we are left with
rig —n^, which is of course nothing but the index of D.
3.4. Index of Dirac Operators
125
The second proof is more algebraic; we give it because it is a model for the
proof of more general versions of the McKean-Singer formula. Let a(t) be
the function Str(e~'D ). If Pi = 1 — Pq is the projection onto the orthogonal
complement of ker(D), then by Lemma2.37, for t large,
|Str(e-tD2 - Po)| = 11 Str«x
J M
x» dx
where Ai is the smallest non-zero eigenvalue of D2. This shows that
o(oo) = lim Str(e-tD3) = ind(D).
t—>oo
The proof is completed by showing that the function a(t) is independent of
t, so that a(t) = a(oo) = ind(D). We do this by differentiating with respect
to t, which is legitimate because the heat equation tells us that the operator
d(e~to2)/dt has a smooth kernel equal to —D2e~tD , and is thus trace-class
for t > 0. We see that
= -IStr([D,De-'°3]),
since D is odd, and this vanishes by Lemma 3.49. Q
With the McKean-Singer formula in hand, it is easy to see that the index of
a smooth one-parameter family of Dirac operators (Dz | z G R) is a smooth
function of z, since the heat kernel of (D*J is. However, the index is an
integer, so it follows that it is independent of s. Even without using the fact
that the index is an integer, we can show explicitly that the variation of the
supertrace of e~tD* vanishes, by means of the following result.
Theorem 3.51. The index is an invariant of the manifold M and Clifford
module S.
Proof. Given a one-parameter family of Dirac operators D2 on ?, we see that
(DzJ is a one-parameter family of Laplacians. Applying Lemma2.50 to the
McKean-Singer formula, it follows that
dz
dz
Once again, this last quantity vanishes by Lemma 3.49.
?

126
3.5. The Lichnerowicz Formula
3. Clifford Modules and Oirac Operators
If A is a Clifford connection, there is an explicit formula for Dj which is of
great importance in differential geometry, as we will see when we describe
examples of Clifford modules in the next section.
Theorem 3.52 (Lichnerowicz formula). Let A be a Clifford connection
on the Clifford module ?. Denote the Laplacian with respect to the connection
A by AA. Ifr\f is the scalar curvature of M, then
where F?/s € A2(M,Endc(M)(?)) " the twisting curvature of the Clifford
connection A defined in Proposition 3.43, and
Proof. Choose a covering of the manifold by coordinate patches. In a coordi-
coordinate system, we may write the Dirac operator as Da = Yl c(dx')Ai, where Aj
is the covariant differentiation in the direction d/dxi. Squaring this formula
gives
ij
=i '?\c(dxi),c(dx')]AiAi
In this calculation, V is the Levi-Civita connection on M, and Fy are its
coefficients with respect to the frame di, which satisfy Vjda:-7 = — T'ikdxk.
Here, we have used the compatibility of the connection A with the Clifford
action, so that
[Aitc(dxi)} = cWidx*) = -Tikc(dxk),
and the symmetry r3ik = Fj^ which follows from Vidk = Vjt^t, so that
ik
ik
3.6. Some Examples of Clifford Modules
127
Let A2 be the curvature tensor of the connection A, and let ej, A < i < n),
be an orthonormal frame of TM. The above calculation gives the formula
C.14) = AA +
By Proposition 3.43, we see that
c{ei)c(ei)c(ek)c(el) + c(Fe's),
ijkl
where R+jki are the components of the Riemannian curvature of M.
Observe that c{et)c(ei)c(ek) equals
- 6ijc{ek) - 6ikc{e{) +6kic(ei).
(T€S3
Using the fact that the antisymmetrization of Rkiij over ijk vanishes, we
obtain
ijkl
We see that
ijkl
ijl
ijl
But J2ij c(eJ)c(ei)Bifcjfc = - ?\ Rikik, since FUkjk = Rjkik- Hence, the right-
hand side equals — 2r&f, and the Lichnerowicz formula follows. Q
3.6. Some Examples of Clifford Modules
In this section, we will show that the classical linear first-order differential
operators of differential geometry, are Dirac operators with respect to very
natural Clifford modules and Clifford connections. By classical, we mean an
operator considered by Atiyah and Singer [13]. Our exposition follows closely
that of Atiyah and Bott [6]. .
The operators that we describe here, rather briefly, are those for which
the index theorem of the next chapter might be considered to be classical;
both the Gauss-Bonnet-Chern theorem and the Riemann-Roch-Hirzebruch
theorem turn out to be special cases, in which we take as our Clifford mod-
module respectively the operator d+d* on A(M), and the operator d + 8* on
A°''(M, ?), where M is a Kahler manifold and ? is a holomorphic Hermitian
bundle.

128
3. Clifford Modules and Dirac Operators
The DeRham Operator. Our first example is related to the deRham
operator
0 -»A°(M) -^ A\M)^ A\M) X
On a Riemannian manifold, the bundle AT*M is a Clifford module by the
action of C.1): if a e T(M,T*M) and ,9 e A(M), then c(aH = e(aH-t(aH.
Clearly, the Levi-Civita connection on the bundle AT*M is compatible with
this Clifford action.
Proposition 3.53. The Dirac operator associated to the Clifford module
AT*M and its Levi-Civita connection is the operator d + d*, where
is the adjoint of the exterior differential d.
Proof. Since V is torsion-free, Proposition 1.22 shows that d equals e o V,
where V is the Levi-Civita connection. On the other hand, Proposition 2.8
shows that d* = -t o V. It follows that d + d* = (e - l) o V = c o V. ?
The square dd* + d*d of the operator d + d* is called the Laplace-Beltrami
operator. Recall the statement of de Rham's theorem: the cohomology
H*(A(M),d) = kBr(*)/im(*-i)
of the deRham complex is isomorphic to the singular cohomology H'(M, R).
A differential form in the kernel of dd* + d'd is called a harmonic form.
It is useful to consider a more general situation than the above, in which
?' —> M is a Z-graded vector bundle (with ?x non-zero for only a finite
number of values of i) with first-order differential operator d : T(M, ?') —¦
r(M,?'+1), such that d2 = 0 and A = dd* + d*d is a generalized Laplacian.
We will write H'(d) for the cohomology of this complex,
Fi(d)=ker(di)/im(di_1).
In this situation, the operator d + d* is a Dirac operator, since (d + d*J =
dd* + d*d is a generalized Laplacian; the ^-grading on the bundle ?V ? * is
obtained by taking the Z-degree modulo two. It is clear from the following
equation that ker(A) = ker(d + d*J = ker(d) D ker(d*):
/ (a, (dd* + d*d)a) = / (da,da) + (d*a,d*a).
m Jm
Theorem 3.54 (Hodge). // (?,d) is as above, then the kernel ker(Aj) of
the operator Aj = d'+1dj + dj_id* is finite-dimensional in each dimension,
and naturally isomorphic to H'(d) by the decomposition ker(di) = im(dj_i)©
ker(Ai).
The Dirac operator d+d* on the bundle Y,i ?' has index equal to the Euler
number of the complex (?,d), defined by J2i(—iydim(Hl(d)).
3.6. Some Examples of Clifford Modules
129
Proof. Since the Laplacian A has a smooth heat kernel, we see that it is
essentially self-adjoint, has a discrete spectrum with finite multiplicities, and
smooth eigenfunctions. Thus, the space ker(Aj) coincides with the zero-
eigenspace of the closure of d*+1a\ + dj_id* in the Hilbert space F^a (M, ?*)
of square-integrable sections of ?*.
On the orthogonal complement to ker(A), we can invert the generalized
Laplacian A to obtain its Green operator G, which we extend by zero on
ker(A); moreover, if a e r(M,?*) is smooth, then Get e T(M,?') is also
smooth (Corollary 2.38). Observe that A commutes with d and d* as does
G. The operator d*G on T(M,?) satisfies
[d, d*G] = (dd* + d*d)G = 1 - Po,
where Pq is the orthogonal projection onto ker(A). Thus, d*G is what is
known as a homotopy between the identity operator on YLi(M,?*) and Po,
and gives us an explicit decomposition ker(dj) = im(dj_i) © ker(Aj).
Thus, we may identify the kernel of the Dirac operator d + d* with H'(d).
The dimension of the kernel of d + d* on the space of even-degree sections
J2i^(M,?2i) is equal to ?<dim(iJ2i(d)), while the dimension of its kernel
on ~?i T(M, ?2i+1) is equal to ?, dim(H2<+1(d)). Thus, the index of d + d*
equals
which equals the Euler number of (?, d) by definition. ?
Corollary 3.55. The kernel of the Laplace-Beltrami operator dd* + d*d on
A'(M) is naturally isomorphic to the deRham cohomology space H'(M,R).
The index of the Dirac operator d+d* on A(M) is equal to the Euler number
Eul(M) =
of the manifold M.
The Euler number defined using the de Rham cohomology is equal to the
Euler number defined in Theorem 1.56 by means of a vector field with non-
degenerate zeroes; this may be shown using Morse theory, but would take us
too far afield to discuss here.
Corollary 3.56 (Kiinneth). If M and N are two compact manifolds,
H'(M xN)* H*(M) ® H*(N).
Proof. Choose Riemannian metrics on M and N, with Laplace-Beltrami op-
operators Am and An. The Laplace-Beltrami AMxN on M x N with the
product metric is AM ® 1 +1 ® AN. Using the spectral decomposition of the
positive operators Am and Ajv, we see that the zero eigenspace of A
the tensor product of the zero eigenspaces of Am and An- ?

130
3. Clifford Modules and Dirac Operators
The Lichnerowicz formula for the square of a Dirac operator, when applied
to the operator d+d', gives a formula for the Laplace-Beltrami operator which
is known as Weitzenbock's formula. Let us denote by AAT M the Laplacian
of the exterior bundle AT'M. The bundle AT'M is the bundle associated
to the principal bundle GL(M) by the representation A : GL(n) —> A(Rn)*
of A.26), so that the curvature of the Levi-Civita connection on the bundle
AT'M is given by the formula
C.15)
Hence by C.14),
(d + d-J = AAT"M + i53^w(?fc
ijkl
Using the fact that the antisymmetrization of
ishes, we see that
over three indices van-
van= 0.
Retaining only the non-zero terms, we obtain Weitzenbock's formula: the
decomposition of the Laplace-Beltami operator as a generalized Laplacian is
C.16)
ijkl
Observe that on the space of one-forms, the Weitzenbock formula becomes
where the Ricci curvature is identified with a section of the bundle End (T'M)
by means of the Riemannian metric on M. This follows from the following
formula,
53 JloMeVeV = - ? flyweW*' - 53 Ricy eV,
where the first term on the right-hand side vanishes on one-forms. In partic-
particular, if the manifold M has Ricci curvature uniformly bounded below by a
positive constant c, then the Laplace-Beltrami operator is bounded below by
c on one-forms, since the operator AAT"M itself positive. By Hodge's The-
Theorem, this implies that the first de Rham cohomology space of the manifold
vanishes; the Hurewicz theorem then tells us that the manifold has finite fun-
fundamental group. This result is a simple example of the so-called vanishing
theorems that follow from lower bounds for the square of the Dirac opera-
operator, and which are proved by combining assumptions on curvature and the
Lichnerowicz theorem.
3.6. Some Examples of Clifford Modules 131
\
JPhere is a generalization of the de Rham complex, in which we twist the
bundle of differential forms by a flat bundle W, with covariant derivative V:
0 -» A°(M,W)
A2(M,W)
This sequence is a complex precisely when V2 = 0, that is, when the bundle
W is flat^There is a simple way to characterize flat bundles, as an associated
bundle M xTi(m) W, where W is representation of Tri(Af). Here, M is the
covering space of M, which is a principal bundle with structure group tti(M)
the fundamental group of M, acting on M by means of the so-called deck
transformations.
The Signature Operator. There is a variant of the first example, in which
the Dirac operator is the same but the definition of the Z2-grading on the
Clifford module AT'M is changed. In order to define this new grading, we
use the Hodge star operator, which is defined on an oriented Riemannian
manifold.
Definition 3.57. If V is an oriented Euclidean vector space with complex-
ification Vc, the Hodge star operator • on AVc equals the action of the
chirality element T 6 C(V) ® C, defined in Proposition 3.17, on the Clifford
module AVfc.
This definition differs from the usual one by a power of i in order that
*2 = 1.
Applying this operator to each fibre of the complexified exterior bundle
AqT'M = AT'M ®r C of an oriented n-dimensional Riemannian manifold
M, we obtain the Hodge star operator
• : ? j
Proposition 3.58. A) If a e Al{M,C), then *-1e(a)* = (-l)ni(a).
B) // a and E are. k-forms on M, then
f
JM
f / (a,0)dx,
JM JM
where dx is the Riemannian volume form on M corresponding to the
chosen orientation, and p = n./2 for even and (n + l)/2 for n odd.
C) The operator d* equals (—l)n+1 *-1 d* on the space of k-forms.
In these formulas, we can replace *-1 by *, since *2 = 1.
Proof. A) If ej is an orthonormal frame of V, we may assume that a = en,
and then we see that
e(en)c(ei)... c(en) = (-lJ-Mei) • • • c(en_1)e(en)c(en).
But for any v e V, we have e(v)c(v) = e(v)(e(v) — l(v)) = — (e(v) — i(v))t.(v),
from which the formula follows.

132 3. Clifford Modules and Dirac Operators
B) Let a, 0 € AkV. We will prove by induction on k that
a A *0 = (-l)fcn+fe(fc-W2ip(a, 0)et A • • • A en.
For k = 0, 0 6 A°V = C, we have by definition
*0 = ip0ei A • • • A en.
If a e AkV, 0 e Afc+1 V and e € V, we have
Q A e A (•/?) = (~l)na A w(e)/3
= (_1)n+*n+fc(fc-i)/aip^a> L(e)p)ei A ... A c
)'2ip(e(e)a, /3)ei A • ¦ • A en
which is the desired formula for k + 1.
C) To prove this, we simply insert B) in the formula which defines d*,
and apply Stokes' formula: if a € «4fc(M)'and 0 € Ak+l(M), then
/ a,d*0)dx = f (da,
m Jm
/
Jm
f
Jm
r») / daA*/3
Jm
= -(-!)" / (a,*d*/3)dx. D
Jm
Corollary 3.59 (Poincare duality). LetM be an oriented compact n-dim-
ensional manifold. The bilinear form fM a A C induces a non-degenerate
pairing
Hk(M) x Hn~k{M) -> R.
Proof. Let a € Ak(M) be a harmonic form representing an element of
Hk(M); thus, *q is closed. If JMa A C = 0 for all j3 € Hn~k(M), then
in particular fMot/\ (*a) = 0. It follows that JM ]a\2 dx = 0, and hence that
a = 0. ?
Suppose that n = dim(M) is even. Then the operator • anticommutes with
the Dirac operator d+d* = d—kd-k, hence with the Laplace-Beltrami operator.
Since *2 = 1 and * anticommutes with the action of c(v) — e{v) — t(v), we
can define another Z2-grading on the exterior bundle AT*M ® C using the
operator *; the differential forms satisfying *a = +e* (respectively *a = —a)
are called self-dual (respectively anti-self-dual). If the dimension of M is
divisible by four, F is actually an element of the real Clifford algebra, so *
operates on the space of real valued forms.
3.6. Some Examples of Clifford Modules
Note as a special case of Proposition 3.58 that if dim(M) = n is even and
a and /? e Anl2(M), then
m
where ep = 1 if n = 4Z and e
/
m
{a,*C)dx,
p p = i if n = 4/ + 2
Let us recall the definition of the signature of a quadratic form.
Definition 3.60. Let Q be a quadratic form on a real vector space V.
Choose a basis in which
Q(x) = x2 + ... + x2p-x2p+1-...-x2p+q.
The signature a(Q) is the number p—q, which depends only on the quadratic
form Q.
If M is a manifold of even dimension n, the bilinear form on ifn/2(M,R)
defined by (a, 0) *-+ fM a A j3 satisfies
Jm
/
m
/3Aa
Thus, if the dimension of M is divisible by four, this quadratic form is sym-
symmetric. The signature of M is by definition the signature of the restriction
of this quadratic form to the space Hn/2{M,R), and we denote it by a{M).
If M is an oriented compact Riemannian manifold of dimension divisible
by four, the signature operator of M is the Dirac operator d + d* for the
Clifford module AT*M with Z2-grading coming from the Hodge star *.
Proposition 3.61. The index of the signature operator d + d* equals the
signature a(M), so by the McKean-Singer formula,
a{M) = Tr(*e-tA),
where A is the Laplace-Beltrami operator.
Proof. Let ctj G Ak(M) be a basis for the harmonic forms Hk, where k < n/2.
Note that the differential forms ctf = at ± *cti form a basis for the direct
sum of the spaces of harmonic forms Hk © Hn~k since * maps the space of
harmonic forms into itself. Now, af is self-dual and a,~ is anti-self-dual, thus
the index of the Dirac operator d + d* restricted to Ak{M) © An~k(M) is
zero.
Now consider the harmonic forms of degree n/2. We may find a basis a,
such that *an — a< for i < m and *aj = — a< for i > m; it is clear that
the index of d + d* with respect to the ^-grading defined by * is 2m — n.
Since n is divisible by four, the signature form may be rewritten as follows:
if a € H"/2 satisfies *a = ±a, then
/ «Aa= / (a,*a)dx = ±f |a|2da;;
Jm Jm Jm

134
3. Clifford Modules and Dirac Operators
from this we.see that the signature of the manifold is equal to 2m — n as
well. ?
If W is a Z2-graded vector bundle on M with connection, we may twist
the Dirac operator d + d* by W to obtain a twisted signature operator on
the tensor product AT*M ® W, whose index is denoted <r(VV). It is clear
from the homotopy invariance of the index that the signature ct(VV) is an
invariant which depends solely on the manifold M, its orientation, and the
vector bundle W.
The Dirac Operator on a Spin-Manifold. Let S be the spinor bundle
over an even-dimensional spin manifold M. The most basic example of a
Dirac operator is the Dirac operator on S associated to the Levi-Civita con-
connection V5 on 5; this operator is often referred to as the Dirac operator on
M. A section of the spinor bundle S lying in the kernel of D is known as
a harmonic spinor. More generally, we may consider the Dirac operator
Dvv®5 on a twisted spinor bundle W®«S with respect to a Clifford connection
of the form
Vw®5 = Vw ® 1 + 1 ® V5. If Fw is the curvature of Vw, then the
twisting curvature F?^s of Vw®s equals Fw, and the Lichnerowicz formula
becomes
T-f.
In particular, for the spinor bundle S itself, we have
Lichnerowicz used this formula to prove the following result.
Proposition 3.62. If M is a compact spin manifold whose scalar curvature
is non-negative, and strictly positive at at least one point, then the kernel of
the Dirac operator on the spinor bundle S vanishes; in particular, its index
isO.
Proof. If s e T(M,S), then
(D2s,s)dx=
] f rM\\sfdx.
4 Jm
/ [
Thus, if Da = 0, we obtain Vs = 0 and rMs — 0. Then (s, s) is constant on
M, hence s = 0. ?
On a spin-manifold M, the signature operator of the last section may be
rewritten as a twisted Dirac operator in an explicit way. Indeed, from the
isomorphism of Clifford modules
AT'M ®CS C{M) ® C ?* End(<S) * S* ® <S,
we see that the Dirac operator d+d* on AT*M is the Dirac operator obtained
by twisting the Dirac operator on the spinor bundle S by the bundle S*. The
3.6. Some Examples of Clifford Modules
135
two different Z2-gradings on AT'M correspond to the two different ways of
grading the twisting bundle S*: if we treat it as a Z2-graded bundle, we
recover the/grading of AT*M according to the parity of the degree, while if
we treat S* as an ungraded bundle, we recover the grading coming from the
Hodge star *.
The 8-Operator on a Kahler Manifold. Our last example of a Dirac
operator is important in algebraic geometry.
Definition 3.63. An almost-complex manifold M is a manifold of real
dimension In with a reduction of its structure group to the group GL(n, C);
that is, there is given a GL(n, C)-principal bundle P and an isomorphism
between the vector bundles P XcL(n,c) ^ an<^ TM.
An equivalent way of defining an almost-complex structure is to give a
section J of End(TM) satisfying J2 = — 1.
On an almost-complex manifold, the complexification of the tangent bun-
bundle splits into two pieces, called the holomorphic and anti-holomorphic tan-
tangent spaces,
TM ®R C = Tl'°M 0 T°<XM,
on which J acts by i and — i, respectively. In terms of a local frame Xi, Yi,
A < i < n), such that JX{ = Yt and JY{ = -Xu we see that TlfiM is
spanned by Wj = (Xj - iYj), and T^^M is spanned by Wj = (Xj + iYj).
It follows from this splitting that if W is a complex vector bundle on M,
the space of differential forms is bigraded,
A(M, W) =
Ap>q{M, W) = T(M, Ap(T1'°M)*
1 M)* ® W).
A Hermitian structure on an almost-complex manifold M is a Hermi-
tian metric on the tangent space TM. Underlying the Hermitian structure is
a Riemannian structure, and its associated Levi-Civita connection. However,
in general the Levi-Civita connection does not preserve T^^M and T1>0M.
If M is a Hermitian almost-complex manifold, and W a vector bundle on M,
then the vector bundle of anti-holomorphic differential forms A(X°'1M)* ® W
is a Clifford module, with the following action: if / e F(M, T*M) decomposes
as / = /o.1 + /!.° with Z1-0 6 T(M, (T0-^)*) and Z0'1 6 T(M, (T^M)*),
then its Clifford action on v 6 T(M, A^XM)* ® W) equals
Note that if W is Hermitian, eCf0'1)-^/1'0) is skew-adjoint, since f0'1 =
and the Clifford action is self-adjoint.
From now on, we will assume that M is a complex manifold, in other
words, there is a collection of local charts for M parametrized by an open
ball in C" for which the transition functions are holomorphic. Let z* be a
set of local holomorphic coordinates. Then Tlt0M is spanned by the vectors

136
3. Clifford Modules and Dirac Operators
dz<, while 7®<lM is spanned by the vectors dp. The elements in AP'"{M)
have the form
Y, /<!.... ,wi udzil A ¦ • •A rfzil> A dih A • • •A ^
On a complex manifold, the exterior differential d splits into two pieces,
d = 9 + 5, where 9 increases the degree of a form (p,g) by A,0) and B
increases it by @,1), and 92 = dd + dd = B2 = 0. Both of these operators
satisfies Leibniz's rule:
d{a A 0) = da A 0 + (-l)|a|Q A 9/3,
B(a A 0) = 9a A 0 + (-l)|a|a A 90.
Now let W be a complex vector bundle over M with covariant derivative
V. With respect to the decomposition T*M ®R C = {Tx<0My © (T0-^)*,
V splits into two pieces V = V1'0 + V0'1, which satisfy the following Leibniz
formulas: if a G A(M) and v e A{M, W), then
V1'°(q A v) = dot A 1/ + (-l)wa A V1>oi/,
V°'l{a A i/) = 9a A v + (-l)la'a A V0'1^
Definition 3.64. A holomorphic vector bundle on a complex manifold M
is a complex vector bundle W which has a chart, that is, a covering of M
by open sets Ui and framings fa : Wi\Ui —» U{ x CN, in which the transition
functions fa o 0T1 : \Ji n [/,• —> GL(n, C) are complex analytic.
If W is a holomorphic vector bundle over M, there is a unique operator
9 : .APi«(M, W) -+ ^M+1(M, W) which in each coordinate patch of a chart
with complex analytic transition functions equals
t=l
The sheaf O(W) of holomorphic sections of a holomorphic bundle W is the
kernel of 8 acting on the sheaf T(M, W) = A°'°(M, W) of smooth sections of
W. We say that a covariant derivative V on a holomorphic vector bundle is
holomorphic if V01 = 9.
Proposition 3.65. If W is a holomorphic vector bundle with a Hermitian
metric h(s,t), there exists a unique holomorphic covariant derivative on W
preserving the metric.
Proof. If V is holomorphic and preserves tie Hermitian metric h, then taking
the component of dh(s,t) lying in ,A0>1(M), we see that
s,t) + h{s,Vlfit).
Bh(s,t) =
If moreover V0'1 = 9, this equation determines V1'0. It is then easy to check
that V = V1'0 + V0-1 is a covariant derivative. ?
3.6. Some Examples of Clifford Modules
13Y
Let V be a holomorphic covariant derivative on W which preserves a
Hermitian metric h. Since the curvature is skew-adjoint as a section of
A2T*M 8) End(W), we see that (V1-0J = 0, so that the curvature of V
is the section of ^-'(M.End^)) equal to [V1'0, V0'1].
It is interesting to see how the Hermitian metric and holomorphic covariant
derivative corresponding to it are related in a chart. If the Hermitian metric
on W is given at a point z e M by a Hermitian matrix h(z), then the
corresponding covariant derivative Vw equals
Its curvature is the A, l)-form
Fh = h~lBdh - (h^Bh) A {h-ldh).
If W is a line bundle, h is a positive real function and the curvature is thus
Fh = 8dlog(h).
If s is a non-vanishing holomorphic section of W over the chart, we see that
If M is a complex manifold with a Hermitian metric h, the real part of h
restricted to TM is a Riemannian metric on M, while the imaginary part il
of h restricted to TM is a two-form on M: if X and Y are vector fields on
M, then
Q{X, Y) = Im h(X, Y) = Im h(Y, X) = - Im h{Y, X) = -Q(Y, X).
If X and Y are vector fields on M, we see that g{X,Y) = Q(JX,Y), where
J is the almost-complex structure on M.
We call a complex manifold M Kahler if SI is a closed two-form. The
following result gives another characterization of Kahler manifolds.
Proposition 3.66. A complex Hermitian manifold M is Kahler if and only
if the bundles Tl'°M and T^^M are preserved by the Levi-Civita connection
V of the underlying Riemannian structure, or in other words, if V J = 0,
where J is the almost-complex structure of M.
Proof. Using the fact that (•, •) = fi(J •, •), we see that
({VXJ)Y, Z) = (VxiJY), Z) + Q(VXY, Z)
= -XQ(Y, Z) + il{VxY, Z) + il{Y, VXZ)
Since the Levi-Civita connection is torsion free, the right hand side of this
equation when antisymmetrized over {X, Y, Z} is nothing but dil(X, Y,Z).
Thus, if V J vanishes, we see that dil = 0.

138
3. Clifford Modules and Dirac Operators
Using the explicit formula for the Levi-Civita connection A.18), we com-
compute that in a holomorphic coordinate system z'
2(Vari a,*, a,-,) = a,4 (
i , a*) - a,,
», a*) - a,,
a,*, a,*) = t
a,», a*
Thus we see that if dfl = 0, the Levi-Civita covariant derivative preserves
sections of TlfiM and induces on the complex vector bundle TlfiM the
canonical holomorphic covariant derivative. ?
This result has the following beautiful consequence.
Proposition 3.67. Let W be a holomorphic vector bundle with Hermitian
metric on a Kahler manifold. The tensor product of the Levi-Civita con-
connection with the canonical connection of W is a Clifford connection on the
Clifford module AfT0'1 Af )* ® W, with associated Dirac operator s/2(8 + 8*).
Proof. It is clear that the Levi-Civita connection is a Clifford connection; a
quick look at the formula for the Clifford action of T*M on AfT^M)* shows
that what is needed is that the splitting TM ®R C = TlfiM © T°-lM is
preserved by the Levi-Civita connection, in other words, that V J — 0.
If Zt is a local frame of Tl<°M, with dual frame Z{ e (TlfiM)*, then
from which we see that
If we choose a holomorphic trivialisation of W over an open set U C M, so
that W\u — U x CN, the operator dyy may be identified over U with the
operator 8 on A(U,CN). On the other hand, if V is the tensor product of
the Levi-Civita connection of A(Tll0M)* with the holomorphic connection
of W, then the operator
also coincide with 8 in the holomorphic trivialisation of W over U, and thus
must be equal to 9yv-
It remains to show that
Let a be the one-form on M such that for any /?, € AP'9(M,W) and /3,+i G
A™+1(MW)
3.6. Some Examples of Clifford Modules
139
Since V preserves the splitting TM®RC = r1'°M®T°'1M, the contraction
Tr(Va) of the two-form Va with the Hermitian structure on TM ®rC equals
where Zj is a local orthonormal frame of Tll0M with respect to the Hermitian
structure on Tl'°M. Thus
) () +Tr(Va)x.
The last term vanishes after integration over M and we obtain the desired
formula for 8*, from which it is clear that \/2(8 + 8*) = c o V. ?
The cohomology of the complex
0
, W)
, W)
, W)
is called the Dolbeault cohomology of the holomorphic bundle W. Dol-
beault's theorem shows that this Ls isomorphic to the sheaf cohomology space
H'(M, O(W)). Hodge's theorem gives an explicit realization of this cohomol-
cohomology as the kernel of the Laplace-Beltrami operator on A°''(M, W).
Theorem 3.68 (Hodge). The kernel of the Dirac operator y/2(8 + 8*) on
the Clifford module A(T°'1M)* ®W is naturally isomorphic to the sheaf co-
cohomology space H* (M, O(W)).
Corollary 3.69. The index of 8+8' on the Clifford module A(T°'1M)*®W
is equal to the Euler number of the holomorphic vector bundle W:
ind(d + 8*) = Eul(VV) = ?(-1)' dim(#*(M, O(W))).
The Lichnerowicz formula for the square of the operator 8 + 8* on a
Kahler manifold is called the Bochner-Kodaira formula. Define the general-
generalized Laplacian A0-* on A°-'(M,W) by the formula
/ (
m
/ (
m
Lemma 3.70. IfZi is a local orthonormal frame ofTlt0M for the Hermitian
metric on M, we have the formula
Proof. Introduce the (l,0)-form a(X) = (Vxi,oa,s). Then
Tr(Va) = ^

140 3. Clifford Modules and Dirac Operators
since T1>0M is preserved by the Levi-Civita connection. Thus
= (V-VV0-1*), -Tr(Va)x.
i
The lemma follows by integrating over M, using Proposition 2.7. D
The canonical line bundle of a Kahler manifold is the holomorphic line
bundle K = A"(T^°M)* on M. The curvature of K* = ATT^M with
()
respect to the Levi-Civita connection is the (l,l)-form ^(RZi, Zi), where R
is the Riemannian curvature of M.
Proposition 3.71 (Bochner-Kodaira). LetW be an Hermitian holomor-
holomorphic vector bundle over the Kahler manifold M. In a local holomorphic co-
coordinate system,
88* + 8*8 = A°'# + J2 e{dzi)i{dz^)Fw®K' (dzi ,0*).
a
Proof. The proof is similar to the proof of Lichnerowicz's formula. Writing
8 = Y, <dzl)V0/8f< and 8* = - ? i(dz')V8/s,«, We see that
88* +8*8 = A°-»
where R+ is the curvature of A(T0>1M)*. Thus it only remains to show that
The left side is equal to
ijfcl
Using the fact that the Levi-Civita connection is torsion-free, we see that
R(Zi,Zi)Zk + R(Zk, Zj)Zt + R{ZU Zk)Z, = 0,
and the fact that R(Zk, Zj) = 0, we see that
R(Zj,Zi)Zk = R(Zk,Zi)Zj.
It follows that
ijk ik
completing the proof of the theorem. ?
This formula implies some important vanishing results. If ? is a Hermitian
holomorphic line bundle with curvature F = J^i^-dz* A dzJ, we say that C
Bibliographic Notes
141
is positive if the Hermitian form v >-> F(v,v) on Tl'°M is positive. This
positivity condition is related to the existence of global holomorphic sections
of C. For example, if s is a global holomorphic section of W, and x is a
point in M at which \s\2 attains a strict maximum, then it is easy to see that
Fx = 9dlog|s|2 is positive on T*'°M.
Proposition 3.72 (Kodaira). A) If C is a Hermitian holomorphic line
bundle on a compact Kahler manifold such that C® K* is positive, then
Hi{M,O(C)) = 0 fori>0.
B) If C is a positive Hermitian holomorphic line bundle and W is a Hermi-
Hermitian holomorphic vector bundle on M, then for m sufficiently large
W{M, O(Cm ® W)) = 0 for i > 0.
Proof. A) Denote by FC®K* = ? Ffjm"dzi A dP the curvature of C ® K*
and by \{FC®K') the endomorphism ?y eid^iidz^F^' of A
Then the Bochner-Kodaira formula shows that
I ((88*+d*8)a,a)= I (V°-1a,V°'1a)+ / (A(F?®K')a,a)
Jm Jm Jm
J
m
The right-hand side is strictly positive if i > 0 and a ^ 0; thus the conditions
da = 0 and d*a = 0 imply that a = 0.
B) For the bundle Cm ® W, the Bochner-Kodaira formula gives
88* + 8*8 = A^# + m\(Fc) + other curvature terms independent of m.
For m sufficiently large the term m\(Fc) dominates the other curvature
terms, and we obtain B). D
The reason for the importance of this result is that it shows that for m
large, dim{H°{M,O(Cm<2>W))) = Eul(?m<g>W). Riemann-Roch-Hirzebruch
theorem that we prove in the next chapter gives an explicit formula for
Eul(?m ® W) and hence for dim(J/0(M,P(?m ® W))). This combination
of an index theorem with a vanishing theorem is a very powerful technique
in differential geometry.
Bibliographic Notes
The book of Lawson and Michelson [72] is a valuable reference for many of
the topics in this chapter.
Section 1. The theory of Clifford algebras and their representations is clas-
classical; we mention as references Chevalley [46], Atiyah-Bott-Shapiro[9], and
Karoubi[64].

142
3. Clifford Modules and Dirac Operators
Sections 2 and 3. The characterization of spin manifolds in terms of u>2(M) of
Proposition 3.34 is proved by Borel and Hirzebruch [35] (see also Milnor[77]
and Lawson and Michelson [72]). The Dirac operator on spin manifolds ap-
appears in Atiyah-Singer [13]; they introduced the idea of twisting a Dirac oper-
operator. Lichnerowicz also studied this operator in [73], proving the fundamen-
fundamental formula for its square; he also proves the vanishing theorem for harmonic
spinors on manifolds of positive scalar curvature.
Sections 4 and 5. The supertrace formula for the index of a Dirac operator
is formulated by Atiyah-Bott [5] in terms of the zeta-function of D2; the
introduction of the heat kernel in this context is due to McKean-Singer [75].
Section 6. We do not attempt here to give references to the original arti-
articles on the geometric Dirac operators discussed in this section. However,
there are some good textbooks on this subject, among which we mention
Gilkey [60], Lawson and Michelson [72] for Dirac operators, and Wells [96] for
the 0-operator. Many examples are contained in the article of Atiyah-Bott [6].
Chapter 4. Index Density of Dirac Operators
Let M be a compact oriented Riemannian manifold of even dimension n.
Let D be a Dirac operator on a Clifford module ? on M associated to a
Clifford connection Vf. Since D is an elliptic operator, the Atiyah-Singer
Index Theorem gives a formula for the index of D in terms of the Chern
character of ? and the .A-genus of M. In this chapter, we will give a proof
of this formula which uses the asymptotic expansion of the heat kernel of D2
for t small. This approach was suggested by Atiyah-Bott [5] and McKean-
Singer [75], and first carried out by Patodi [82] and Gilkey [58]. However, our
method differs from theirs, in that the j4-genus and Chern character emerge
in a natural way from purely local calculations, while previous proofs all
required that the index density be determined by calculating the index of a
range of examples.
We will prove a refined form of the local index theorem mentioned above.
We define a filtration on sections of the bundle End(?) over M x R+. We take
the heat kernel of the operator D2 restricted to the diagonal, and calculate
it explicitly modulo lower order terms with respect to this filtration. In this
way, we obtain a generalization of the local index theorem which extracts a
differential form on M from the heat kernel for small times; this differential
form is nothing other than the product of the .A-genus A(M) with respect to
the Riemannian curvature and e~F e A(M, Endc(M)?)i the exponential
of the twisting curvature of ?.
4.1. The Local Index Theorem
Let %(x, x) = (a; | e~*D | x) be the restriction of the heat kernel of D2 to the
diagonal. In this chapter, we will calculate the leading term in the asymptotic
expansion of Str(fct(x,x)) as t —¦ 0. In particular, when combined with the
McKean-Singer formula, this will imply a formula for the index of a Dirac
operator.
The heat kernel kt{x,x) is a section of the bundle of filtered algebras
End(?) S C{M) ® EndC(M)(?)> where the filtration is induced by the fil-
filtration of C(T*M) and elements of Endc(M)(?) are given degree zero. We
denote by Ct(M) the sub-bundle of C(M) of Clifford elements of degree less
than or equal to i. The associated bundle of graded algebras is the bundle

142
3. CliiFord Modules and Dirac Operators
Sections 2 and 3. The characterization of spin manifolds in terms of w%(M) of
Proposition 3.34 is proved by Borel and Hirzebruch [35] (see also Milnor [77]
and Lawson and Michelson[72]). The Dirac operator on spin manifolds ap-
appears in Atiyah-Singer [13]; they introduced the idea of twisting a Dirac oper-
operator. Lichnerowicz also studied this operator in [73], proving the fundamen-
fundamental formula for its square; he also proves the vanishing theorem for harmonic
spinors on manifolds of positive scalar curvature.
Sections 4 and 5. The supertrace formula for the index of a Dirac operator
is formulated by Atiyah-Bott [5] in terms of the zeta-function of D2; the
introduction of the heat kernel in this context is due to McKean-Singer [75].
Section 6. We do not attempt here to give references to the original arti-
articles on the geometric Dirac operators discussed in this section. However,
there are some good textbooks on this subject, among which we mention
Gilkey [60], Lawson and Michelson [72] for Dirac operators, and Wells [96] for
the 9-operator. Many examples are contained in the article of Atiyah-Bott [6].
Chapter 4. Index Density of Dirac Operators
Let M be a compact oriented Riemannian manifold of even dimension n.
Let D be a Dirac operator on a Clifford module ? on M associated to a
Clifford connection V?. Since D is an elliptic operator, the Atiyah-Singer
Index Theorem gives a formula for the index of D in terms of the Chern
character of ? and the A-genus of M. In this chapter, we will give a proof
of this formula which uses the asymptotic expansion of the heat kernel of D2
for t small. This approach was suggested by Atiyah-Bott [5] and McKean-
Singer [75], and first carried out by Patodi [82] and Gilkey [58]. However, our
method differs from theirs, in that the A-genus and Chern character emerge
in a natural way from purely local calculations, while previous proofs all
required that the index density be determined by calculating the index of a
range of examples.
We will prove a refined form of the local index theorem mentioned above.
We define a filtration on sections of the bundle End(?) over M x R+. We take
the heat kernel of the operator D2 restricted to the diagonal, and calculate
it explicitly modulo lower order terms with respect to this filtration. In this
way, we obtain a generalization of the local index theorem which extracts a
differential form on M from the heat kernel for small times; this differential
form is nothing other than the product of the .A-genus A(M) with respect to
the Riemannian curvature and e~F e ,A(Af,Endcr(M)?)>tne exponential
of the twisting curvature of ?.
4.1. The Local Index Theorem
Let kt(x, x) = (x | e"*0* | x) be the restriction of the heat kernel of D2 to the
diagonal. In this chapter, we will calculate the leading term in the asymptotic
expansion of Str(fct(a;,x)) as t —> 0. In particular, when combined with the
McKean-Singer formula, this will imply a formula for the index of a Dirac
operator.
The heat kernel fct(x,a;) is a section of the bundle of filtered algebras
End(?) = C(M) ® EndC(M)(?),' where the filtration is induced by the fil-
filtration of C(T*M) and elements of EndC(M)(?) are given degree zero. We
denote by Cj(M) the sub-bundle of C(M) of Clifford elements of degree less
than or equal to i. The associated bundle of graded algebras is the bundle

4. Index Density of Dirac Operators
AT*M®EndC(M)(f)- Let
smh(R/2)
be the .A-genus form of the manifold M with respect to the Riemannian
curvature R.
Let F?/s be the twisting curvature of the connection V?, which was de-
defined in C.43). The aim of this chapter is to prove the following theorem.
Theorem 4.1. Consider the asymptotic expansion ofkt(x,x),
»=o
with coefficients kt € T(M, C{M) ® EndC(M)(?))-
A) The coefficient hi is a section of the bundle C2i(M) ® EndC(M)(?)•
t=n/2
B) Let<r(k)= ? <rM(*j) e A(M,Endc{M)(?))- Then
t=0
-*-'"•>•
This result need not hold for a Dirac operator associated to a Clifford
superconnection which is not a connection; the theorem must be modified in
order to hold for the Dirac operator associated to a general Clifford super-
connection.
The i = 0 piece of Theorem 4.1 is precisely Weyl's formula
t—+o
Thus, Theorem 4.1 is an extension of Weyl's theorem in the case of a gen-
generalized Laplacian which is the square o:f a Dirac operator associated to a
Clifford connection.
The idea of the proof is to work in normal coordinates x € U around a
point xo 6 M, where U is an open neighbourhood of zero in V = TXoM.
Using the parallel transport map for the connection V?, we may trivialize
the bundle ? over the set U, obtaining an identification of Y(U, End(?))
with C°°(U, End(-E)), where E = ?Xo. There is an isomorphism a of the
vector space End(?) = C(V*)®EndC(v)(E) with AF*®Endc(K.)(.E). We
introduce a rescaling on the space of functions on R+ x U with values in
AV* ® Endc(v){E) by the formula
i=0
4.1. The Local Index Theorem
145
Using this rescaling, we may restate Theorem 4.1 as follows: if k(t,x) =
r((expI
-tD'
x | e
lim (
u—><)
x0)), then
The rescaling operator 8U induces a filtration on the algebra of differential
operators acting on C°°(R+ x U, AV* ® EBdc(v)(E))\ an operator D has
filtration degree m if
lim um/26uD6Z1
u—»0
exists. Thus, a polynomial P(x) has degree — deg(P), a polynomial P(t)
has degree —2deg(P), a derivative d/dx' has degree one, a derivative d/dt
has degree two, an exterior multiplication operator e* has degree one, and
an interior multiplication operator t* has degree — 1. It will be shown, using
Lichnerowicz's formula, that the operator D2 has degree two; this is why
Theorem 4.1 holds. Indeed, up to operators of lower order, we will see that
the operator D2 may be identified with a harmonic oscillator with differential
form coefficients.
Given r and /6R, let H be the harmonic oscillator on the real line
In Section 3, we will prove Mehler's formula, which states that the heat kernel
Pt(x, y; H) of the harmonic oscillator H is equal to
( tr/2
\l/2
\4;rtsinh(tr/2)
j exp (- ~ (coth{tr/2) (x2+y2) - 2 cosech(tr/2)x2/) - <
The above filtration argument reduces the calculation of {6uk){t,s) to the
calculation of the heat kernel of a harmonic oscillator, with differential form
coefficients, in which r is replaced by the Riemannian curvature R of M and
/ by the twisting curvature F?/s of ?. The v4-genus of the manifold M and
the twisting curvature F?/s appear as consequences of the above formula for
Pt@,0;H).
In the rest of this section, we will discuss some of the consequences of this
theorem. The main consequence of Theorem 4.1 is that it provides the local
index theorem for Dirac operators, which is a strengthening of Atiyah and
Singer's result. By the McKean-Singer formula, for every t > 0,
ind(D)= / StT(kt(x,x))dx.
J
/
M
Theorem 4.1 implies that the integrand itself has a limit when t converges
to zero, if D is associated to a connection. Indeed, as the supertrace of an

146
4. Index Density of Dirac Operators
element a e C(M) vanishes on all elements of Clifford filtration strictly less
than n = dim(M), the first part of Theorem 4.1 implies that
Str(fct(x,x))~D7rt)
-"/2
«>n/2
thus there are no poles in the asymptotic expansion of Str(fct(x, x)). Further-
Furthermore, as the left-hand side ind(D) of the McKean-Singer formula is indepen-
independent of t, we necessarily have
ind(D) = Dtt)-"/2 I Str(fcn/2(x))dx,
Jm
while the integrals of all other terms fM Sti(kj(x)) dx, j ^ n/2, vanish.
To identify the term Str(fcn/2(x)) as a characteristic form on M, we must
introduce some notation. If we write a Clifford module ? locally as W ® S,
where S is a spinor bundle and W is an auxiliary bundle, then by C.28), for
a 6 r(M,End(W)) «* T(M,Endc{M)(?)),
where T 6 T(M,C(M)) is the chirality operator. We will avoid making
use of this decomposition, which is often only possible locally. Instead, we
use the relative supertrace of Definition 3.28; the relative supertrace of o 6
r(M,Endc(M)(?))is
We extend the relative supertrace to a linear map
Str?/S : A(M,EndC(M)(?)) - A(M),
and define, for a Clifford superconnection Ve on ?, the relative Chern
character form of the bundle ? by the formula
The relative Chern character ch(?/S) is a closed differential form on M.
The cohomology class of this differential form is independent of the choice
of Clifford superconnection on ?, and will also be denoted by ch(?/S). In
the special case where the Clifford module ? = W®S and V? is the tensor
product of the Levi-Civita connection on S and a connection on W with cur-
curvature Fw, we see that ch(?/S) = Strvv(exp(-.FVV')) is the Chern character
form of the twisting bundle W.
If a e T(M,C(M)) and 6 6 r(M,Endc(M)(?)), then the point-wise su-
supertrace of the section a ® b e T(M, C{M) ® EndC(jw) (?)) = T{M, End(?))
was shown in C.21) to equal the Berezin integral
Stre(a(x) ® b(x)) = (-2i)n/2an(a(x))Stre/s(b(x)),
4.1. The Local Index Theorem
SO that
147
Str?(fcn/2)(x) = (-2i)n/2Str?/s(<7n(fcn/2(x)).
Thus Theorem 4.1 implies the following theorem for the index of a Dirac
operator associated to a Clifford connection.
Theorem 4.2 (Patodi, Gilkey). Let M be a compact oriented Rieman-
nian manifold of even dimension n, with Clifford module ? and Clifford con-
connection V?, and let D be the associated Dirac operator. If kt(x,x)\dx\ 6
T(M, End(?) ® |A|) is the restriction of the heat kernel of the operator D2 to
the diagonal, then lim _^ Str(kt(x,x)) \dx\ exists and is the volume-form on
M obtained by taking the n-form piece of
It is important to note that there is no a priori reason to suppose that
the limit lim Strjfct(x,x) exists at each point, and indeed, this is not
necessarily true for Dirac operators which are not associated to a connection.
However, since the index of a Dirac operator is independent of the Clifford
superconnection used to define it, we obtain the Atiyah-Singer formula for
the index of an arbitrary Dirac operator.
Theorem 4.3 (Atiyah-Singer). The index of a Dirac operator on a Clif-
Clifford module ? over a compact oriented even-dimensional manifold is given
by the cohomological formula
ind(D) = B7ri)-n/2 f A{M)ch(?/S).
Jm
The factor B7ri)~n/2 is absent from the usual statement of this theorem,
since it is part of the topologist's definition of the characteristic classes, the
purpose of which is to ensure that the characteristic Chern classes are inte-
integral. However, in this book, we prefer the more natural (to the geometer)
definitions of the A-genus form
and of the Chern character form ch(?) = Str(exp(-V2)).
In the rest of this section, we will derive the usual formulas for the index
density Bni)~n^ A{M) ch(?/S) for each of the four classical Dirac operators
that were described in the last chapter. The resulting formulas are special
cases of the Atiyah-Singer Index Theorem for elliptic operators.

148
4. Index Density of Dirac Operators
The Gauss-Bonnet-Chern Theorem. We consider the Clifford bundle
? = AT*M with its Levi-Civita connection Ve. In order to analyse what
the above index theorem says for the case of the de Rham operator, we must
calculate F?ls in this case. We will do this in a local orthonormal frame e*.
We start with some general results about the Clifford module E = AV
of the Clifford algebra C(V), for V an even-dimensional space. Prom the
isomorphisms
<* End(S) S C(V)
AV<8>RC,
we see that in this case, we may identify the auxiliary bundle W with S*.
Thus, there are two natural Z2-gradings on S*: the grading 5* = (S*)+ ffi
(S*)~ induced by the grading on AF, and the grading defined by the chirality
operator, for which S* is purely even. We denote the first corresponding
supertrace on Endc(v)(AF) by Str^v/si the second one, the relative trace,
we denote by
Let e* be a positive orthonormal basis of V. Consider the following elements
of End(E):
e = e(e') - i(e'), b\ = e(e') + t(e*).
We have the relations [c',<J] = -25^, [b\&>] = 26", and [c'.fr'] = 0, so that
the algebra Endc(v)(-^) ls generated by the elements 6*. We freely identify
C2{V) and A2F by the map a.
Lemma 4.4. If a = "Ei<j <Hj^ e A2V, define b(a) = YLi<j oy&W €
Endc(v)(E)- We have the following formulas:
^) = (-2i)"/2jV(<zI/2 PfA(a),
TrAv/s(e6(o)) = 2"/
Proof. By definition, StrAVys(efc<o)) = (-2i)"n/2 Strife1 • • -cncfcW). It
suffices to prove the lemma when dim(F) = 2 and o = Be e2, so that b(a) =
ObW and exp^b2) = cos« + (sin^b^2.
It is easily checked that StrAV(c1c2) = 0 and that Str^cVM2) = -4,
so that
StrAV/s(e6(a)) = (-2i)-x Str AV{clc2 cos 9 + {sin ey
= -1i sin 0.
On the other hand, jvFe1e2I^ = ^-Ksinfl, while Pf^fleV) = 0. Putting
this all together proves the first formula.
On the other hand, det(coshv(r(fle1e2)/2I/2 = Cos0, while
TrAV(e9blb') = TrAV(cos6>) +rrrAV((sin»N162) = 4cos0,
since TrAvF1b2) = 0. This proves the second formula. ?
4.1. The Local Index Theorem
149
The curvature tensor (Vf) is equal to ?)y fc<i ilijfcieVe* Ae', which when
rewritten in terms of the operators c* and b* equals
? ^ + bi)(c> -b>)ek Ael = -\ ^ ^V*'
where we have used the antisymmetry of Rijki in i and j to show that
? W - 6V) = ?flyH(cW + c*t') = 0.
Prom this formula and A.20), it is easy to see that F?/s is given by the
formula
y
Let us now calculate the term A(M) Sti?/S(exp(-F?/S)) which enters in the
formula for the index of d + d*. Let x(TM) € An(M) be the Euler form of
the manifold M associated to the Levi-Civita connection A.38).
Proposition 4.5. We have A(M) Str?/S(exp(-F?/S)) = inl2X{TM).
Proof. When a = Yli<j aijei A ej 6 A2F, we have seen that
^p^ayW) = (-2i)"/2PfA(-a).
By analytic continuation of this identity, we see that the same formula re-
remains true when a = | l]i<J-(-Rej, e^)e' A e-3 is an element of A2F with two-
form coefficients, and we obtain the proposition. ?
As a result, we see that the index formula for
n
Eul(M) = ind(d + d*) = J2(-iy dim(JT(M)),
«=o
the Euler number of the manifold, simplifies greatly, and we obtain the fol-
following consequence of Theorem 4.1.
Theorem 4.6 (Gauss-Bonnet-Chern). The Euler number of an even-di-
even-dimensional oriented manifold M is given by the formula
Eul(M) = B7r)-"/2 / Pf(-.R).
Jm
For example, if the manifold is two-dimensional, we recover the classical
Gauss-Bonnet theorem
Eul(M) = — / rtA dx.
27r Jm
Of course, this is possibly the most difficult proof of this basic theorem
of differential geometry, with the exception of proofs using quantum field

150
4. Index Density of Dirac Operators
theory. Indeed, the usual proof of the Gauss-Bonnet-Chern Theorem was
given in Section 1.6, although we did not give there the topological argument,
involving Morse theory, which is needed to show that the Euler number cal-
calculated from a Morse function equals the Euler number calculated from the
de Rham cohomology. However, the above proof does provide a test for the
index theorem.
The Hirzebruch Signature Theorem. To derive Hirzebruch's formula
for the index of the signature operator, we have only to combine the formula
that we have already obtained for F?ls with the general index theorem.
Recall that the grading of AT*M for which the supertrace is computed for
the signature operator is the grading induced by the Hodge star-operator *.
Define the characteristic form L(M) e AU(M) by
it is known as the L-genus. By Lemma4.4, A(M) Tr?/S(exp(-F?/'s)) =
2"/2L(M), and we obtain the following result.
Theorem 4.7 (Hirzebruch). Let M be an oriented Riemannian manifold
of dimension divisible by four; then the signature a(M) is given by the formula
a(M) = {m)-n'2 [ L(M).
[
m
It follows by the same calculation that the twisted signature with respect
to an auxiliary vector bundle W is given by the formula
t(W) = (jri)-"/2 / L(M) ch(VV).
Jm
This formula is due to Atiyah-Singer [13], and is one of the main steps in
their calculation of the index of an arbitrary elliptic operator on a compact
manifold.
The first non-trivial term in the Taylor expansion of the L-genus may be
easily calculated:
24
Prom this, we see that for a four-manifold,
Tr(fl2)
24tt2 '
4.1. The Local Index Theorem
151
The Index Theorem for the Dirac Operator. In the case of the Dirac
operator on a spinor bundle over a spin-manifold, the curvature F?'s = 0,
so that
ind(D) =
/
m
A(M).
This formula has a number of immediate corollaries:
A) The index of the Dirac operator is independent of the spin-structure.
B) If the right-hand side (which is always a rational number) is not an
integer, then the manifold M does not have a spin-structure.
C) If the manifold M has a metric with respect to which the scalar curvature
tm is positive and non-zero, then the integral of the A-genus is zero, since
by Lichnerowicz's vanishing theorem, the index of the Dirac operator
vanishes.
Using the formula for the first few terms in the expansion of A(R) that we
gave in A.36), we see that if dim(M) = 4, then a(M) = -8ind(D).
Consider the twisted Clifford module ? = W ®S, where W is an auxil-
auxiliary Hermitian vector bundle with a Hermitian connection of curvature Fw.
In the index formula for the twisted Dirac operator D^, associated to the
Clifford connection obtained by twisting the Levi-Civita connection on S by
the connection of W, the term F?/s is nothing but the curvature Fw of the
twisting bundle, so we obtain the following case of the index theorem, which
is in some ways the most fundamental.
Theorem 4.8 (Atiyah-Singer). Let M be an oriented spin manifold of
even dimension. Let S be the spin bundle and let W be a vector bundle
over M. If Dyy is the twisted Dirac operator on T(M, W <8> <S), then
ind(Dw) = Birt)—/2 / i(M) ch(W).
Jm
M
The Riemann-Roch-Hirzebruch Theorem. Let M be a Kahler mani-
manifold, and consider the Clifford module ? = A(T°-1M)*. Let V? be the Clifford
connection on this Clifford module induced from the Levi-Civita connection.
Let us calculate the curvature F?Is of this Clifford module.
Considering the Riemannian curvature R to be a matrix with two-form
coefficients, the curvature operator (V?) is
kl
where tu* is a basis of Tlfi(M) and wk is the dual basis, while the End(?)-
valued two-form Re of Proposition 3.43 equals
kl
kl

4. Index Density of Dirac Operators
Thus, if the basis w* is orthonormal, we have
If W is a holomorphic Hermitian vector bundle and V? is the Clifford
connection on A°'#T*M®W obtained by twisting the Levi-Civita connection
with the connection of W compatible with the metric, we obtain
where R+ is the curvature of the bundle TlfiM and Fw the curvature of the
bundle W.
Since TM ®R C = T^M © T*>'lM, we see that
so that
A(M) Tr?/S(exp(-F?'s)) = Td(M) Tr(exp(-Fw)),
where Td(Af) is the Todd genus of the complex manifold M:
R+ \ . ./ R+
Td(M)=det(^T-T)=det(-
)exp(-Tr(i?+/2)).
We obtain the following theorem, which is due to Hirzebruch in the case of
projective manifolds, and to Atiyah and Singer in general.
Theorem 4.9 (Riemann-Roch-Hirzebruch). The Euler number of the
holomorphic bundle W is given by the formula
Eul(W) = B«)-n/2 / Td(M)ch(tt>).
Jm
For example, if M is a Riemann surface (has complex dimension one)
and curvature R+ € A2(M, C), and if ? is a line-bundle with curvature
F e A2{M,C), then Td(M) = 1 - R+/2 and ch(?) = 1 - F, and we obtain
the classical Riemann-Roch Theorem,
ind@?) = dim H°(M,C) - dim H\M,C) = t^ [ (R++ 2F)-
4« Jm
If we take C = C to be the trivial line-bundle in this formula, we see that
m
where g = dim^1'°(M) = dim if0'1 (M) = idimF^M) is the genus of M.
If we define the degree deg(?) e Z of the line bundle C by the formula
deg(?) =
4.2. Mehler's Formula
we may restate the Riemann-Roch Theorem in its classical form,
153
If M is a compact complex manifold which is not Kahler, then for any holo-
holomorphic vector bundle W we may still form the Clifford module A(T*'1M)* <g>
W, and the operator y/2(d + 8 ) is a generalized Dirac operator. The index
of the operator 8 + 8* is equal to the Euler number of the bundle W by
Hodge's Theorem, as for a Kahler manifold,
n/2
*) = Eul(W) =
M, W).
t=0
However, although the operator -^2(8+8*) is a Dirac operator associated to a
superconnection, it is no longer the Dirac operator associated to a connection,
so we cannot expect the local index theorem to hold. Nevertheless, there is
a natural Dirac operator D on this Clifford module which does satisfy the
local index theorem, which is formed simply by taking the Dirac operator
associated to the Levi-Civita connection of the bundle A(T°'1M)* ® W. It
is not hard to extend our above treatment to this operator, and it is easy
to see that its index is given by the same formula as in the Riemann-Roch-
Hirzebruch Theorem:
ind(D) = B«)-n/2 f Td(M)ch(W).
Jm
Since the Dirac operator D and \/2(d + d*) are both Dirac operators for
the same Clifford module, they must have the same index. In this way, we
see that the Riemann-Roch-Hirzebruch Theorem is valid on any compact
complex manifold, but not the local formula.
4.2. Mehler's Formula
In this section, we derive Mehler's formula for the heat kernel of the harmonic
oscillator, in the form in which it is needed for the proof of Theorem 4.1. In
order to give an idea of the result in its most basic form, let us consider the
harmonic oscillator on the real line:
We would like to find a function pt(x, y) satisfying the following requirements:
for each t > 0, the map
f°°
= / pt{x,y)<l>{y)dy
J-oo

154 4. Index Density of Dirac Operators
is bounded on the space of test functions S(R), satisfies the heat equation
and the initial condition
lim
pt(x,
To find a solution to this problem, one is guided by the fact that the operator
H is quadratic in differentiation and multiplication to seek a solution which
is a Gaussian function of x and y; in addition, the solution must clearly be
symmetric in a; and y, since the harmonic oscillator is self-adjoint:
Pt{x,y) = exp(atx2/2 + btxy + aty2/2
Applying the heat operator to this ansatz, and dividing by Pt(x, y), we find
pt(x,y) + Hpt(x,y) =
(o,x2/2 + kxy + aty2/2 + tt- (otx + btyf - at + x2)pt(x,y).
In particular, we obtain the following ordinary differential equations for the
coefficients:
dt/2 = a2 — 1 = b2
and
= ot.
These equations have the solutions at = — cothBt + C), bt = cosechBt + C)
and Ct = —l/21ogsinhBi + C) + D, and in order to determine the values of
the integration constants C and D, we simply have to substitute in the initial
conditions for pt(x,y), which show that C = 0 and D = \ogB/jr)~1^2. Thus,
we have obtained the following solution to the heat equation for H:
Pt(x,y) = B7rsinh2t)~1/2exp(-l((coth2t)(x2 + y2) - 2(cosech2t)xy)Y
This formula is known as Mehler's formula.
We will mainly use this formula when y = 0. By change of variables (or
by direct calculation), for any real numbers r,f, it follows that the function
Pt{x,r,f) defined by
D.1) D7rt)-1/2
is a solution of the equation
) coth(tr/2)x2/4t - tf)
s? + is+
We will now generalize the above argument so that it applies to the formal
heat equation that we use to prove Theorem 4.1, as well as in the proof of
the index theorem for families in Chapter 10.
4.2. Mehler's Formula
155
Let V be an w-dimensional Euclidean vector space with orthonormal basis
e'. Let A be a finite dimensional commutative algebra over C with identity
(in the applications A will be the even part of an exterior algebra).
If R is an n x n antisymmetric matrix with coefficients in A, we associate
to it the .A-valued one-form wonV given by
+ ±
acts on the space of
The operator V{ = 9< + u)(di) =
A-valued functions on V.
Definition 4.10. Let R be an n x n antisymmetric matrix and let F be an
N x N matrix, both with coefficients in the commutative algebra A. The
generalised harmonic operator H associated to R and F is the differential
operator acting on .4®End(C^)-valued functions on V defined by the formula
Consider the following .A-valued function defined on the space ofnxn
antisymmetric matrices with values in A-
R/2
Since jy@) = 1, jy1/2(tR) is defined for t sufficiently near 0, and is an
analytic function of t with a Taylor series of the form
D.2)
fc=i
where fk(R) »s a homogeneous polynomial function of degree k > 1 with
respect to the coefficients fly of the matrix R. Similarly, (tfl/2) coth(t#/2)
is an n x n-symmetric matrix with coefficients in A, defined for t small, and
with Taylor expansion l+Y,f=i i2kCkR2k- Thus the .A-valued quadratic form
is defined for t small, and has a Taylor series of the form
D.3) ||x||2
fe=i
Proposition 4.11. The kernel pt{x,R,F) taking values in A® End(CJV),
defined for t small by the formula
Pt(x,R,F) = D^)-«/2^1

t» a solution of the heat equation
Proof. We have to prove that dpt/dt = —Hpt. Both sides of this formula
are analytic with respect to the coefficients Rij of the matrix R. Thus it
is sufficient to prove the result when R has real coefficients. Choosing an
appropriate orthononnal basis of V, it suffices to verify the above formula
when V is two dimensional and Re1 = re2, Re2 = —re1, so that
H = -
and
l + x2)
exp(-(tr/2)cot(tr/2)||x||V4<)exp(-tF).
Since the function ||x||2 is annihilated by the infinitesimal rotation X2#i —
Zic*2, the fact that p< satisfies the heat aquation follows from D.1) which
gives the heat kernel for the harmonic oscillator on the real line, except that
r must be replaced by ir. ?
In the applications we will make of the above formula, the coefficients of
the matrices R and F will be nilpotent, so that the kernel will be defined
for all t. Let qt(x) = D7rt)~n/2e~1*" ^4i be the solution of the heat equation
for the Euclidean Laplacian on V. The kernel pt(x, R, F) has an asymptotic
expansion of the form
for small t, with $o = 1-
We will prove again the unicity result for formal solutions of the heat equa-
equation for the harmonic oscillator, even though a theorem of greater generality
has already been proved in Theorem 2.26. Observe that H is the Laplacian
on the Euclidean vector space V associated to the non-trivial connection
V = d + u>. Let H = Yji x$i be the radial vector field on V. If st is a
A ® End(CJV)-valued smooth function on V, then
(dt + H)qtst = qt{dt + t-lTl + H)st;
this follows from Proposition 2.24, if we bear in mind that the antisymmetry
of R implies that u)(H) = 0, so that Vr == U.
Let $t(ar) be a formal power series in t, whose coefficients are smooth
A ®End(CJV)-valued functions defined in a neighbourhood of 0 in V. We will
say that qt(x)$t(z) is a formal solution of the heat equation (dt + H)pt = 0,
if $((x) satisfies the equation
4.3. Calculation of the Index Density
157
Theorem 4.12. For any oq e A ® End(CN), there exists a unique formal
solution pt(x, R, F, Oo) of the heat equation
(ft + Hx)pt(x) = 0
of the form
pt{x) =
fc=0
and such that $o(O) = a®. The function pt(x, R,F,ao) is given by the formula
^ coth(^) |z)) exp(-*F)a0.
Proof. Although this follows from the results of Chapter 2, it is worth giving
the direct proof in this special case, since it is made so easy by our formula
for the formal solution. On the left side of the equation
(ft
fc=0
we set the coefficients of each tk to zero. This gives the system of equations
= -.ff*$fc-i ifJfe>0.
Thus we see that $o is the constant function equal to oq; uniqueness is then
clear, since if oo = 0, then $t = 0 for all k.
The function Pt(x, R, F) that we found above is analytic in t for small t and
is a solution of the heat equation, thus the corresponding formal expansion
pt(x, R, F) = qt(x) YlkLo tk$k(x), is a formal solution. ?
4.3. Calculation of the Index Density
Let ? be a graded Clifford module on an even-dimensional Riemannian mani-
manifold M, with Clifford connection V? and associated Dirac operator D. Recall
Lichnerowicz's formula, Theorem 3.52, for D2
where A is the Laplacian of the bundle ? with respect to the connection V?,
ei a local orthonormal frame of TM with dual frame e' of T*M, c' equals c(e*),
and F?/S(ei,ej) € Endc{\i)(?) are the coefficients of the twisting curvature
of the Clifford connection V?.
Fix xo 6 M and trivialize the vector bundle ? in a neighbourhood of xo by
parallel transport along geodesies. More precisely, let V = TXoM, E = ?Xo
and U = {x e V | ||x|| < e}, where e is smaller than the injectivity radius

158
4. Index Density of Dirac Operators
of the manifold M at xq. We identify U by means of the exponential map
x *-* expXo x with a neighbourhood of xo in M. For x = expl0 x, the fibre ?x
and E are identified by the parallel transport map t(xo,x) : ?x —*-E along
the geodesic xa = expl0 sx. Thus the space C°°(U,S) of sections of ? over
U may be identified with the space of ^-valued C°°-functions on V, defined
in the neighbourhood U. A differential operator D is written in this local
trivialization as
where aa(x) e End(E).
Let us compute D in the above trivialization. For this, we need to compute
the Clifford action and the connection. Choose an orthonormal basis di of
V = TX0M, with dual basis rfx* of T*XoM, and let c* = c(dx') e End(?). Let
5 be the spinor space of V* and let W = Ylomc(v')(S,E) be the auxiliary-
vector space such that E = 5 ® W, so that End(E) = End(S) ® End(W) =
C(V*)®End(W). Let et be the local orthonormal frame obtained by parallel
transport along geodesies from the orthonormal basis di of TXoM, and let e*
be the dual frame of T*M.
Lemma 4.13. In the trivialization of ? by parallel transport along geodesies,
the End(E)-valued function c(e')x is the constant endomorphism c*.
Proof. Let H be the radial vector field on V. In the trivialization of ? given
by parallel transport along geodesies, the covariant derivative Vfj is, by def-
definition, differentiation by 72. By construction, V^e* = 0, and since V? is
a Clifford connection, [Ve,c(e*)] = c(Ve'). Thus H-c(e% = 0, so that the
Clifford action of the cotangent vector ex is constant and equal to c*. ?
It follows from the above lemma that in this trivialization the bundle
Endc(M)(?), restricted to U, is the trivial bundle with fibre Endc(v)(.E) =
End(W).
Lemma 4.14. The action of the covariant derivative V|( on C°°(U,E) is
given by the formula
j\k<l
where RkUj = (R{di,dj)di,dk)Xo is the Riemannian curvature at xq, and
= W,End(W)),
fikl(x) = O(|x|2) 6
9i(x) = O(|x|) e
are error terms.
4.3. Calculation of the Index Density
159
Proof. If F? is the curvature of the connection V?, and R is the Riemannian
curvature of M, Proposition 3.43 shows that
= 5
'dj)ek,ei)ckcldx{ AdxP + F?'s{dud^dx* Adx*,
F?'s{di,dj) e T{U,Endc(v.)(?)) ? T(U,End(HO), and
where
Observe that (R(di,dj)ek,ei)Xo = —RkUj, since at the point xo, e{ = 8,. If
we apply the formula C(TV)uj = i{1l)FE (see 1.12), we obtain the lemma. ?
Let Pt(x, xo) be the heat kernel of the operator D . We transfer this kernel
to the neighbourhood U of 0 e V, thinking of it as taking values in End(i?),
by writing
k(t,x) = T(xo,x)pt(x,xo),
where x = expl0 x. Since we may identify End(i?) = C(V*) ® End(W) with
AV* ® End(W) by means of the full symbol map a, k(t, x) is identified with a
AV* ® End(W0-valued function on U. Consider the space AV ® End(W) as
a C(V*) ® End(W) module, where the action of C(V) on AV* is the usual
one c(a) — e(a) — i(a). The next lemma shows that k(t, x) is a solution to a
heat equation on the open manifold U.
Lemma 4.15. Let L be the differential operator on U C V, with coefficients
in C{V) ® End(W), defined by the formula
C i,e,)cV.
The AV* ® End(W)-valued function k(t,x) satisfies the differential equation
Proof. Since the kernelPt(x,xo) ^?x®?Xo satisfies the differential equation
it follows that r(xo,x)pt(x,xo) € C°°(U,End(E)) satisfies the differential
equation (9t + L)r(xo,x)pt(x,xo) = 0, where C(V*) ® End(W) s End(?)
acts on the left on E. But under the symbol isomorphism End(?) = C(V*)®
Ead(W) a AV* ® End(W0,the ^ion of C(V') ® End(W) on E intertwines
with the usual action of C(V) ® End(W) on AV* ® End(W). D
To explain the method which we will use to prove the local index theorem,
let us explain it in a simplified setting. Let H be a generalized Laplacian on a
vector bundle ?. If 0 < u < 1 is & small parameter, let us rescale space-time
by sending t to ut and x to u^X; t rescales by the same power of « as ||x||2
because the heat equation satisfied by k(t, x) has two space derivatives but

4. Index Density of Dirac Operators
only one time derivative. Consider the rescaled heat kernel k(u, t, x), defined
by
D.4) k(u,t,x) = ;
The factor of unl2 in this definition is included because the heat kernel is a
density in the variable x; to put this in a. more elementary way, it is needed
in order that the rescaled kernel continues to satisfy the initial condition
lim _^ k(u,t,x) = 6(x) for all 0 < u < 1. Note that the Euclidean heat
kernel qt(x) = Dnt)~n/2e~^ /4t is invariant under this rescaling.
If ^(t.x) = S^:o***<(x) m a formal power series in t with polynomial
coefficients $i(x), we define the rescaled series by
i=0
This is a formal series in u1/2 with coefficients polynomial functions of (t, x).
Proposition 4.16. There exist End(E)-valued polynomials *i(x) on V with
$0@) = 1, such that the function u t-> k(u, t,x) has an asymptotic expansion
in m1/2 when « —> 0 of the form
k(u,t,x)
«=0
This expansion is uniform for (t, x) lying in compact subsets of @,1) x U
and asymptotic expansions for the derivatives u i-> djfd%(k(u,t, x)) may be
obtained by differentiation.
Proof. By Theorem2.30, there exist functions fa G C°°(U,Ead(E)), with
<?o @) = 1, such that for any x€U,
\k(t,x) -
N
t=0
< C{N)tN~n'2.
Using the bounds \xke~x2/it\ < Cktk/2, we can replace <j>i{x) by its Taylor
expansion ^i(x) of order 2(N — i):
\k(t,x) - *(
Thus we obtain
(«,*, x) -
t=0
N
t=0
x)| < C'(N)uN,
4.3. Calculation of the Index Density
161
for (t, x) lying in compact subsets of @,1) x U and 0 < u < 1, which gives the
desired asymptotic expansion of the function u ^ k(u,t,x). The argument
for the derivatives of k(u, t, x) is similar. D
The main idea of our proof of the index theorem is to modify the rescaling
of space-time of D.4) by inclusion of a "rescaling" of AV. Given 0 < u < 1
and a e C°°(R+ x U, AV* ® End(WO), let Sua equal
Fua)(t,x) = ? u-^a(ut, ull2x)w
«=o
This rescaling has the following effect on operators on C°°(R+ x U, AV*
6u<t>(xN-1 = ^u^x) for <t> € C°°{U),
6ue(aNZl = «~1/2e(a) for a 6 V,
Definition 4.17. The rescaled heat kernel r(u,t,x) is defined by
The above choice is motivated by the fact that
lim r(u,t = l,x = 0) = lim ?u(nW
u—>0 u—>0 j=o
The right-hand side of this equation is precisely what we would like to prove
to be equal to
Dn)-n/2A(M) exp(~F?/s).
It is clear that r(u, t, x) satisfies the differential equation
The most important step in the proof of Theorem 4.1 is the calculation of the
leading term of the asymptotic expansion of L(u) = uS^LS'1 as a function
of u; we show that
L(u) = K + O(u1/2),
where K is a harmonic oscillator of the type that we discussed in the last
section. To show that this implies that r(u, t, x) has a limit as u —t 0, we use
the following lemma, which is a consequence of Proposition 4.16.

162
4. Index Density of Dirac Operators
Lemma 4.18. There exist AV*®End(W)-valued polynomials ~ti[t,x.) onRx
V, such that for every integer N, the Junction rN(u, t, x) defined by
approximates r(u,t,x) in the following sense: for N > j + \a\/2, there is a
constant C(N,j,a) such that
\\didZ(r(u,t,x)-rN(u,t,x))\\ < C(N,j,a)uN,
for 0 < u < 1 and (t,x) e @,1) x U. Furthermore, 7i@,0) = 0 if i ^ 0,
(
Proof. By Proposition 4.16, there exist AV* ® End(W)-valued polynomial
functions *<(x) with *0@) = 1 such that
i=0
t=0
||fc(*.x)-<
If we project out the p-form component of this equation, we see that
w
|fc(*.x)W-ft(
Rescaling by u 6 @,1), we obtain
r(u,t,x)w - w~p/2ft(x) V(
«=0
It is clear that we must define 7j(t,x)[p] to be the coefficient of v?l2 in the
sum
i=0
which is a polynomial on R+ x V with values in APV* ® End(W). It is clear
that the sum 7j(t,x) = 5Zp=07j(<.x)[p] satisfies 6u-yj = v?I2^j. Furthermore,
7j(t,x)[p] = 0 for j < -p, and hence 7j(t,x) = 0 for j < -n.
We see that for (t,x) G @,1) x U and 0 < u < 1,
r(ult,x)-ft(x) Y, "'^(t.xjI^Cu*.
t=-n/2
The argument for the derivatives is similar. The values of 7j@,0) are deter-
determined by
t=-n/2
4.3. Calculation of the Index Density
163
Using the fact that L(u) = K + Ofa1/2), which will be proved later, we
can now show that there are no poles in the Laurent series expansion in u1/2
ofr(u,t,x),
Expanding the equation (dt + L(u))r(u,t,x) = 0 in a Laurent series in m1/2,
we see that the leading term u~'/2gt(xO_<(t,x) of the asymptotic expan-
expansion of r(u, t, x) satisfies the heat equation (dt + i(Tx)((jt(xO_<(i,x)) = 0.
Since formal solutions of the heat equation for the harmonic oscillator are
uniquely determined by 7-*@,0), by the results of the last section, and since
7-<@,0) = 0 for I > 0, we see that 7_* is identically equal to zero unless
? = 0. In particular, we see that there are no poles in the Laurent expansion
of r(u, t, x) in powers of u1/2.
The other thing that we learn from this argument is that the leading term
of the expansion of r(u, t,x), namely r@, t,x) = <jt(xOo(t,x), satisfies the
heat equation for the operator L@) = K, with initial condition
7o@,0) = l.
Thus, to calculate r@,t,x), we have only to obtain a formula for K, and to
apply the results of the last section.
In order to state the formula for K, let us denote by R the matrix with
nilpotent entries equal to
where Rx0 is the Riemannian curvature of Af at io, operating on the algebra
AF* by exterior multiplication. Similarly, let F be the element of A2V* ®
End(W0 obtained by evaluating the twisting curvature Fe^s at the point aso>
again, it acts on AV* ® End(W) by exterior multiplication.
Proposition 4.19. The family of differential operators L(u) = vSuLS'1 act-
acting on C°°(U, AV* <8>End(W)) has a limit K when u tends to 0, given by the
formula
Proof. By Lemma 4.14, the differential operator vf:u = u1/25uVf.5~1 is
equal to
fc<!

164 4. Index Density of Dirac Operators
Since /titi(x) = O(|x|2), we see that V|j™ has a limit as u —» 0, equal to
9< + I E (R*oUij*Jekel-
Using the fundamental symmetry of the curvature Rijki = RkUj and the
definition D.5) of Ry = Y!lk<i(R*o)jU'i?kEli the above limit is seen to equal
The operator L(u) equals ?i(u) + ^(u), where
Clearly, lim _^ 1*2A1) = 0, since its leading term is the sum of ur\f(xo)/4
_^
and w1^2 times lim _^ ]TV Vy" ej, which we have just shown to be nonsin-
gular. The first term of Li (u) has limit
while its second term converges to
lim ? F?(e,, e^Xu^xXe1 - n**)^ - 1^) = ]T F(ft, d^eV = F,
since the vector fields ej coincide with 04 when x = 0. ?
The operator ?@) = K is a generalized harmonic oscillator, in the sense
of Definition 4.10, associated to the n x n antisymmetric curvature matrix
—Rij and the N x iV-matrix F (where N = dim(W)), with coefficients in the
commutative algebra A+V. Applying Theorem4.12, we obtain the following
result.
Theorem 4.20. The limit lim _^ r(u, t, x) exists, and is given by the for-
formula
Setting t = 1 and x = 0, we obtain Theorem 4.1.
Bibliographic Notes
Bibliographic Notes
165
Theorem4.1, is proved in Getzler[55], using a symbol calculus for pseudo-
differential operators which combines the usual symbol for pseudodifferential
operators with the Clifford symbol a. It is also in this article that the ap-
approximation of D2 by a harmonic oscillator is introduced. Another exposition
of this proof may be found in Roe's book [89].
Earlier proofs which were special cases of the proof of [55] were those
of the Gauss-Bonnet-Chern theorem by Patodi[81], and the Riemann-Roch
theorem for Riemann surfaces by Kotake[70]. The previous proofs of the
local index theorem, by Patodi[82], Gilkey[58], and Atiyah-Bott-Patodi [8],
use power counting and symmetries of the Riemannian curvature to show
that the top-order piece <rn(fct(x,x)) must be a characteristic form given by
a universal formula all n-dimensional manifolds, and then calculate sufficient
special cases of the global index theorem to characterize this characteristic
form completely. Thus, this route to calculating the local index density is
less direct. This approach to the local index theorem is explained in Gilkey's
book [60].
There have been a number of other proofs of the local index theorem
similar to ours, among which we mention Bismut[27], Priedan-Windey [54]
and Yu [99]. Also, the work of the physicist Alvarez-Gaume [1] was influential
in the development of this point of view.
The Gauss-Bonnet-Chern theorem is due to Chern [44]; we gave an adap-
adaptation of his original proof in Section 1.6. The local Gauss-Bonnet-Chern
theorem is due to Patodi [81].
The signature theorem is proved by Hirzebruch in [62], using cobordism
theory; in [63], Hirzebruch tells the story of his discovery of this theorem.
The twisted signatures were first calculated by Atiyah-Singer[13], also using
cobordism theory. In the same article, they introduce the twisted Dirac
operator and calculate its index. Their work was motivated by one of the
central problems of analysis: to calculate the index of an arbitrary elliptic
differential operator on a compact manifold. Atiyah and Singer used Bott
periodicity to show that the index of any elliptic pseudodifferential operator
may be written in terms of the numbers <t(W). References for their general
index theorem are two seminars directed by Cartan [43] and Palais [80], and
of course Atiyah's Collected Works [2]; an excellent account of Atiyah and
Singer's discovery of their theorem may be found in the introduction to [2].
The local index theorem is proved for the 9-operator on Kahler manifolds
by Patodi [81], and for the twisted signature operator by Gilkey [58]. Atiyah-
Bott-Patodi [8] give a unified treatment of the local index theorem for the
classical Dirac operators by Gilkey's method.
The index of the 9-operator was first calculated by Hirzebruch [62], in
the special case of smooth projective varieties; another proof, which proves
a much more general result, and led to the introduction of if-theory into
the study of index problems, is due to Grothendieck [36]. Atiyah and Singer

166
4. Index Density of Dirac Operators
gave the first proof valid for an arbitrary compact complex manifold, as a
corollary of their general index theorem. The local index theorem for these
operators was proved by Kotake [70] for Riemann surfaces, and by Patodi [81]
and Gilkey [58] in general.
Chapter 5. The Exponential Map
and the Index Density
In this chapter, we will give another proof of Theorem 4.1 for a Dirac operator
D, independent of the one in Chapter 4. This proof generalizes easily to obtain
the fixed point formula for the equivariant index of a group of isometries, as
we will see in the next chapter. A feature of the proof is that it gives an
explanation for the striking similarity between the .A-genus and the Jacobian
of the exponential map on a Lie group, both of which involve the j-function
J(X) =
sinh(X/2)
X/2 "
Indeed, the .A-genus will appear naturally through the Jacobian of the expo-
exponential map on the principal bundle S0(M) over M: this Jacobian can be
computed along the fibres in terms of the curvature of M, and this is where
the j-function makes its appearance, as we will see in Section 1.
Let us explain the idea of this proof. Let P —+ M be a principal bundle
endowed with a connection, with compact structure group G, and let ? be
a vector bundle associated to P, with Laplacian A?. It is easy to see that
the kernel fej of the heat semi-group e~tA is obtained from the scalar heat
kernel ht on the manifold P by averaging over the fibres Px = G. This gives
integral formulas for the coefficients of the asymptotic expansion of kt(x, x),
as we will see in Section 2.
Let M be a spin-manifold of dimension n, and let Spin(M) be the cor-
corresponding principal bundle with structure group Spin(n); Spin(M) is a
double cover of S0(M). Let D be the Dirac operator on the spin bun-
bundle S of M. Since by Lichnerowicz's formula, the square of the Dirac op-
operator is a generalized Laplacian, we obtain a formula for the restriction
kt(x, x) = (x | e~to2 | x) of the heat kernel to the diagonal in terms of the
scalar heat kernel ht of the Riemannian manifold Spin(M): apart from minor
factors, kt is given by the integral
fct(z,x)= / ht(p,pg)p(g) *dg,
JSpin(n)
where p : Spin(n) —> End(S) is the spin representation and p is an element of
Spin(M) lying above x 6 M. In exponential coordinates, this is a Gaussian
integral over A2Rn, which we identify with the Lie algebra of Spin(n). Using

166
4. Index Density of Dirac Operators
gave the first proof valid for an arbitrary compact complex manifold, as a
corollary of their general index theorem. The local index theorem for these
operators was proved by Kotake [70] for Riemann surfaces, and by Patodi [81]
and Gilkey [58] in general.
Chapter 5. The Exponential Map
and the Index Density
In this chapter, we will give another proof of Theorem 4.1 for a Dirac operator
D, independent of the one in Chapter 4. This proof generalizes easily to obtain
the fixed point formula for the equivariant index of a group of isometries, as
we will see in the next chapter. A feature of the proof is that it gives an
explanation for the striking similarity between the A-genus and the Jacobian
of the exponential map on a Lie group, both of which involve the j-function
J(X) -
sinh(X/2)
JC/2 "
Indeed, the .4-genus will appear naturally through the Jacobian of the expo-
exponential map on the principal bundle S0(M) over M: this Jacobian can be
computed along the fibres in terms of the curvature of M, and this is where
the j-function makes its appearance, as we will see in Section 1.
Let us explain the idea of this proof. Let P —+ M be a principal bundle
endowed with a connection, with compact structure group G, and let ? be
a vector bundle associated to P, with Laplacian A?. It is easy to see that
the kernel kt of the heat semi-group e~tA is obtained from the scalar heat
kernel ht on the manifold P by averaging over the fibres Px = G. This gives
integral formulas for the coefficients of the asymptotic expansion of kt(x, x),
as we will see in Section 2.
Let M be a spin-manifold of dimension n, and let Spin(M) be the cor-
corresponding principal bundle with structure group Spin(n); Spin(M) is a
double cover of S0(M). Let D be the Dirac operator on the spin bun-
bundle S of M. Since by Lichnerowicz's formula, the square of the Dirac op-
operator is a generalized Laplacian, we obtain a formula for the restriction
kt(x,x) = (a; | e~tD | x) of the heat kernel to the diagonal in terms of the
scalar heat kernel ht of the Riemannian manifold Spin(M): apart from minor
factors, kt is given by the integral
kt(x,x)= / ht(p,pg)p(g) * dg,
•/Spinfn)
JSpin(n)
where p : Spin(n) —* End(S) is the spin representation and p is an element of
Spin(M) lying above x € M. In exponential coordinates, this is a Gaussian
integral over A2Rn, which we identify with the Lie algebra of Spin(n). Using

100
5. The Exponential Map and the Index Density
this formula, it is easy to compute a(k), and the answer only involves the first
term of the asymptotic expansion of /ij. The elimination of the singular part
of Str(fct(z,z)) is reduced to the following simple lemma: if ? is a nilpotent
element of an algebra A (which for us will be an exterior algebra), and if /
is a polynomial in one variable, then
lim D7rt
t—>o
•L
As we saw in Chapter 2, the first term in the asymptotic expansion of
ht can be computed explicitly, for any Riemannian manifold, in terms of
the Jacobian of the Riemannian exponential map, so we obtain the equal-
equality a(k) = A(M). This argument is extended to the case of a twisted
Dirac operator, in which case ht must be replaced by the heat kernel of
a generalized Laplacian associated to the twisting bundle W, and we obtain
a{k) — A(M)exp(-F?'s); the twisting curvature F?/s is brought into this
formula by the parallel transport factor in the first term of the asymptotic
expansion of ht-
5.1. Jacobian of the Exponential Map on Principal Bundles
Let G be a compact Lie group with Lie algebra g, let M be a compact
Riemannian manifold and let n : P —> M be a principal bundle over M
with structure group G and connection form ui. Let us choose on g a G-
invariant positive scalar product; we will denote by fG f(g) dg the integral
with respect to the corresponding Riemannian volume form; because the
Riemannian metric is invariant, this is a Haar measure. For p e P above
x e M, the tangent space TPP is the direct sum of the horizontal space HPP,
which is identified with TXM by the projection nt : TP —> w*TM, and the
vertical space identified with g. Taking HVP and g orthogonal to each other
turns P into a Riemannian manifold, with metric depending on u).
For p e P, we denote by
J(p,a):TpP
¦TqP
the derivative of the map expp at the tangent vector a € TPP. The purpose
of this section is to compute J(p, a) when a € g. This computation is very
similar to the computation of the derivative of the exponential map in a Lie
group.
If a e g, let je(a) equal
E.D
Note that j,@) = 1.
5.1. Jacobian of the Exponential Map on Principal Bundles
109
Proposition 5.1. If dg is a Haar measure on G, then d(expa) = jB(a)da
in exponential coordinates.
Proof. If G is a compact Lie group with Lie algebra g, then
d 1 —
— exp(-a) exp(a + e6)|?=0 =
6,
E.2)
for a and 6 € g. In a matrix group, this follows from Duhamel's formula B.6);
since G is compact, it has a faithful matrix representation. ?
Let V be an Euclidean vector space with orthonormal basis e<. The Eu-
Euclidean scalar product on A2V has orthonormal basis {ejAe,- | i < j}. Identify
A2V with so(V) by the map r : A2V -+ so(V) of Proposition 3.7, defined by
the formula (r(a)ei, e^) = 2(a, e< A e,).
We denote by il the curvature of the connection w on P. It is a horizontal
fl-valued 2-form on P so, for p e P, ilp is an element of A2HpP ® g. If a e g,
the contraction of J2P with o is an element of A2H*P which we will denote
by ilp a.
If X is a vector field on M, we denote by Xp its horizontal lift on P. If
o ? g, we denote by o the corresponding vertical vector field on P. Since
Xp is invariant by G, we see that [a, Xp] = 0. From the definition of the
curvature of a connection A.6), we see that
[XP,YP] = [X,Y}P-Q(XP,YP)P,
where n(Xp,Yp) e g induces a vertical vector field U(XP,YP)P on P. We
will usually write U(XP, Yp) instead of il(Xp, YP)P. The formula
follows from the G-invariance of ui.
The skew-symmetric endomorphism r(fip • a) e End(TxM) is given by
• a)X = 2
,ep),a)ei,
and it varies along the fibre containing p by the formula
E.3)
Denote by VM and Vp the Levi-Civita covariant derivatives on the tan-
tangent bundles TM and TP.
Lemma 5.2. Let X and Y be vector fields on M, and let a and b be elements
ofg. Then
A)
B)
C) VPPYP = {V%Y)P - \Sl{Xp,Yp)

170
5. The Exponential Map and the Index Density
Proof. By the definition of Vp, if X, Y and Z are vector fields on P,
2(V?Y, Z) equals
X(Y, Z) + Y{Z, X) - Z(X, Y) + ([X, Y], Z) - ([Y, Z},X) + ([Z, X], Y).
To prove A), we observe that (Vf b,Xp) = 0, while
(Vf&.c) = i([a,6],c) - i([6,c],a) + i([c,a],6) = A([a,6],c),
since the scalar product on g is G—invariant.
To prove B), we observe that
2(VPXP,YP) = -([Xp,Yp],a) = ^r(Q.a)Xp,Yp)
by the definition of t, while (VPXP, b) = 0. A similar argument leads to the
proof of C). D
As a consequence of Lemma 5.2, we obtain a description of the horizontal
and vertical geodesies in P.
Lemma 5.3. A) Fora 6 0 andp G P, the curve s >-* pexp(sa) is a geodesic,
so that expp a = pexp a.
B) // x(s) is a geodesic in M, its horizontal lift is a geodesic in P.
Proof. By definition we have d,(pexpsa) — a, so A) follows from the relation
Vpa = 0. Similarly B) follows from Lemma5.2 C). ?
Fix x e M and let V = TXM. For p e P such that 7r(p) = x, identify HPP
with V and TPP with V © g. Then the map
becomes a linear map from V © g to itself.
Theorem 5.4. The map J(p, a) preserves the subspaces V and g of TPP,
and we have
l_e-T(npo)/2
r(fip-a)/2 '
J(p,o)|9 =
ad a
Proof. The formula for J(p,a)\B follows from E.2) and Lemma5.3(l). To
calculate J(p,a)\v, we need to compute dtexpp(a + tX)\t=o, for X € V.
Introduce p(s, t) = expp s(a + tX) and p(s) = expp(sa). For every t 6 K,
the curve s ¦-» p(s,t) is a geodesic. Put Y(s) = dtp(s,t)\t=o, so that Y(s) is
a vector field along the curve p(s), corresponding to the deformation of the
geodesic p(s) into the geodesic p(s,e); such a vector field is called a Jacobi
vector field. We have J(p,a)X = y{s)\.=1. Let Rp e A2(P,End(TP)) be
5.1. Jacobian of the Exponential Map on Principal Bundles
171
the curvature of the manifold P. The Jacobi vector field Y(s) is determined
by the differential equation
E.4)
? VpY(s) = Rp(d,p, Y(s)) ¦ 3sp
with initial conditions Y(Q) = 0 and V?y(s)|s=o = X. Let us recall how
these equations are obtained: since V is torsion free, taking the covariant
derivative of Va# dBp = 0 gives
0 = Va, Va.d.P = Va. Vdtdsp + R(dtp, dap)dap
= Va, Va. dtP + R(dtP, dsP)dsP,
Let A" be a vector field on M. Since Vfo = 0 and [a,Xp] = 0, we see
that
Rp(a,Xp)a = VpVpPa.
Since exp(sa)o = a, it follows from E.3) that T(fipexp(sa) -a) = T(ilp-a),
hence is independent of s and
It follows that
If V(s) is an horizontal vector field on P along the curve p(s), the vectors
VpVpV(s) and Rp(a, V(s))a are horizontal. Thus the differential equation
E.4) implies that Y(s) remains horizontal, and identifies with an element
y(s) of V. In this context, the differential equation E.4) says that
(d. + r(Qp ¦ a)/4Jj,(s) = (T(Op
so that we obtain the theorem. Q
For X € End(F), recall Definition3.12 of jv,
a)/4)\(s),
3v{X) = det I — I
X/2 )¦
The Hirzebruch A-genus will appear in the index formula because of the
following corollary to the above theorem.
Corollary 5.5. det(J(p,a)) = je(a)jv(T{Slp-a)/2))
Proof. Let A be the matrix r(f2p-o)/2. Since A is antisymmetric, we see
that det(e^/2) = 1, so that
det
There is no conflict of notation between je(a) and Jv(X) since the compact
group G is unimodular, so that j9(a) and j,(ada) are equal.

5.2. The Heat Kernel of a Principal Bundle
Let P-tMbea principal bundle with compact structure group G and
connection form a>. Let (p, E) be a representation of G in a vector space E
and let ? = P xa E be the associated vector bundle on M, with covariant
derivative V associated to the connection u on P. Let A? be the Laplacian
of ?; with respect to a local orthonormal frame e< of TM,
We can identify the space F(M, ?) with the subspace of C°°(P)®E invariant
under the group G, denoted C°°(P,E)a, as was explained in Proposition 1.7.
In this section, we will relate the Laplacian A? and its heat kernel to the
scalar Laplacian Ap on P, which acts on C°°(P).
If <j> e C°°(P) is a function on P, we have the integration formula
j <t>(p)dp = j (?<t>(pg)dg)dx.
In this formula, note that fG </>{pg)dg depends only on x = 7r(p).
Let Ea be an orthonormal basis of the Lie algebra g of G and let Cas =
"%2a p(EaJ 6 EndB?) be the Casimir operator; it is independent of the
choice of the orthonormal basis Ea of g and commutes with the action of G
on E. In particular, if the representation p is irreducible, Cas is a scalar.
Proposition 5.6. The Laplacian A? coincides with the restriction of the
operator Ap ® 1 + 1 ® C to (C°°(P) <g> E)G.
Proof. Let ef be the horizontal lift of the vector field e< on M to the principal
bundle P. The vector fields ef and Ea form a local orthonormal basis of TP.
By E.2) we see that
In the identification of F(M,?) with (C°°(P) <S> E)G, the operator Vej cor-
corresponds to ef, while, using the G-invariance of elements of (C°°{P) <g> E)G,
the vertical vector field Ea may be replaced by the multiplication operator
—p(Ea); the proposition follows. ?
Let us compare the semigroups of operators e~*A on T(M, ?) and e~tAP
on C°°(P). The Schwartz kernel of e~tA with respect to the Riemannian
density \dx\ of M is denoted by (x | e~*A | y). On pulling back to P, it
becomes an element (pi,p2)
which satisfies
(pi | e~'A? | P2) of C°°(P x P) ® End(?)
5.2. The Heat Kernel of a Principal Bundle
Prom Proposition 5.6, we see that
173
Thus we obtain an integral representation of the heat kernel of the Laplacian
of an associated vector bundle ? in terms of the scalar heat kernel on the
manifold P.
Proposition 5.7. lfp\ andpi are points in the principal bundle P, then
(Pi | e~tA | Pi) — e~tc
Proof. We have
I pjp) pig)'1 dg) <6(p2) <fc,
from which the theorem follows. ?
This theorem leads to an integral representation for the asymptotic ex-
expansion of (x I e~tA I a;) in terms of the asymptotic expansion of the heat
kernel of P.
If 4> € C%°(R) is a smooth function with compact support, then by Propo-
Proposition 2.13, the integral
has an asymptotic expansion in powers of t1/2 near t = 0 which depends only
on the Taylor series of </> at 0.
Let (p e C?°(g) be a smooth function with compact support on the vector
space {j. We will denote by
E.5)
the asymptotic expansion in t1/2 of the integral
This power series has the form tdim(o)/2 ?f~0 ^A^.
If <j> is a smooth function defined only in a neighbourhood U of 0 € fl, and
is a cut-off function on g with support in U and equal to 1 near 0, then
|| /
j«
is independent of the choice of ifi, so we will write

174
5. The Exponential Map and the Index Density
it simply as f^mpe-M*/*tip(a)da. If #(t>a) = ?°ZQt%(a) is a formal
power series in t with coefficients in C°°(U), we will write
for the formal series
t=0
To simplify the statement of the next theorem, we will suppose that the
Casimir Cas of the representation p of G on E is a multiple of the identity.
Theorem 5.8. There exist smooth functions $j on P x g such that
D7rt)-(dim(M)+dim(B))/2
•'» t=o
Furthermore, denoting Tw(p)M by V, we have
Proof. Let
we have
Let i/>(t) be a cut-oflF function which vanishes for t greater than half the
square of the injectivity radius of M.
In Section 2.5, we showed how to obtain a formal series ?^ot'*i, whose
coefficients $4 are smooth functions in the neighbourhood
{(puP2)\d(Pl,p2)<e}
of the diagonal of P x P such that if
pup*) be the heat kernel of Ap on C°°(P). By Theorem5.8,
<P I e-tA? I p) = e-tc
«=o
we have
By Theorem 2.30 and Corollary 5.5, we have, for a small,
N
ht(p,Pexpa) =
«=0
5.2. The Heat Kernel of a Principal Bundle
175
where $0(p,a) = 39 1/2(o)Jv1/2(T(np'a)/2)- The Proof is completed by mul-
multiplying through by the formal power series e"tCas = X)?L0(-<Cas)Vfc!> and
inserting the formula dg = J8(o) da. ?
Let us extend this theorem to the case of a generalized Laplacian on a
twisted bundle W ® ?. Let W be a bundle over M, which we will call the
twisting bundle. We can make the following identifications between space
of sections:
T(M, W®?) = (T(P, n'W) <g> E)G,
r(M,End(W® ?)) = (r(P,7r*End(W)) ® End(?))G.
End(E), we
If F € (r(P,7r* End(VV)) ® g)G, then applying the map p : fl
obtain a section
, End(W ® ?)),
p(F) e (r(P, 7T* End(W))
which we will simply denote by F. We will also abbreviate our notation
for the bundles n*W and Tr'End(W) on P to W and End(W). If F €
(T(P, End(W)) ® b)g and a e fl, then for all p e P, the contraction Fp ¦ a is a
well-defined element of End(Wz), where x = ir(p), and we have the invariance
for geG,
E.6)
We will consider generalized Laplacians on W ® ? of the form
where
Here, W is a twisting bundle with connection Vw, and Avv®e is the Lapla-
Laplacian on F(M, W®f) with respect to the tensor product connection on W®?.
The main result of this section will be an integral formula for the asymp-
asymptotic expansion of the heat kernel of H. As before, we may view the kernel
of e~tH as a section j
(PLP2) 1—* (Pi \e~tH |p2>
of the bundle WBW8 End(?) over PxP. Restricted to the diagonal, the
kernel (p | e~tH | p) gives a section of {T(P, End(W)) ® End(E))G. We will
give an asymptotic expansion of (p \ e~tH \ p) compatible with this tensor
product decomposition.
F° e r(M,End(W)) C r(M,End(W ® ?)),
F1e(r(P,End(W))®g)G.

176
5. The Exponential Map and the Index Density
5.2. The Heat Kernel of a Principal Bundle
177
Theorem 5.9. Let H = AW8? + F, where F = F° + F1 is a potential
such that F° € r(M,End(>V)) and F1 e(r(P,End(W)) ® fl)G. Assume that
the Casimir Cas of the representation p is scalar. Then there exist smooth
sections $j € Y{P x g, End(W)) (where we denote by End(W) the pull-back
o/End(W) onM toPxg) such that
Furthermore, letting V = Tv(p)m , ">e have
*o(p, a) = j;1/2(a)jy1/2(r(np • o
• a/2).
Pnoo/. The proof is very similar to the proof of Theorem 5.8. The first step
is to write H as the restriction to T(M, W ® S) = (T(P, W) <g> E)G of a
carefully chosen generalized Laplacian Hp on P. We will thus obtain an
integral formula similar to Theorem 5.7 for the kernel (p | e~tH | p).
The map a <-y \F* a 6 End(Wx) defines a vertical one-form on P. We
associate to the potential F a connection V on n*W by adding to the pull-
back 7r'Vw of the connection Vw on W the vertical one-form 5F1:
Let A be the Laplacian of the bundle tt'W associated to the Riemannian
structure of P and the connection V.
Lemma 5.10. There exists a potential F e r(P,End(W))G such that H is
the restriction of the operator (A + F) ® 1 + 1 ® Cas to T(M, W ® ?) =
{Y{P,W)®E)G.
Proof. Let F = i
We have
whereas
Observe that
• EaJ + F°, where Ea is an orthonormal basis of g.
Prom the relation E.6), we see that F^exp^eE y Eais independent of s, so that
the action of the vector field Ea commutes with the endomorphism F1 • Ea,
and we obtain
As before, we can replace the second order operator J2a E% on (F(P, W) ®
E)G, by Cas, while we can replace Ea by —p(Ea); the lemma follows eas-
easily. ?
For pi and p2 G P, let us denote by ht (pi, P2) € Hom( Wp,, WPl) the kernel
of e~t^+F^. As in Theorem 5.7, we have
(Pi | e~tH | pa) = e-'Cas / kfa.pwjpfor1*.
Jg
There exists smooth sections (pi.pa) •-+ *j(Pi,P2) G Hom(WPJ, VVPl), defined
in a neighbourhood of the diagonal of P x P, such that if
N
i=0
1 ¦ Ea)
we have
Furthermore, the first term $o(Pi)P2) does not depend on F, and is given
by the formula
*o(Pi,P2) = det~1/2(J(pi,expp11p2)) r(pi,p2),
where f(pi,P2) e Hom(Hp2,WPl) is the geodesic parallel transport with
respect to the connection V. We will compute f in the next lemma.
Lemma 5.11. For p G P and a € g, we have
f(p,pexpa) = exp(iFp1 a).
Proof. The function 11-» f (t) = f (p, p exp ta) is the solution of the differential
equation
^T(t) - i (F^,,, • a) f(*) = 0, f@) = 1,
and, by E.6), we have F}expta-a = Fp1-a. D
The rest of the proof of Theorem 5.9 is the same as that of Theorem 5.8,
except that the $0 term is multiplied by f(p,pexpa), calculated above. D
When we apply the above theorem to the heat kernel of D2 in Section 4,
the factor jy1/2(T(a ¦ ftp)/2) in
*o(p, a) = j,T1/2(a) Jv1'2 (t(Vp ¦ o)/2) exp(Fp1 ¦ a/2)

178
5. The Exponential Map and the Index Density
will gives rise, to the A-genus of M, while the factor exp(Fp • a/2) will lead
to the relative Chern character.
5.3. Calculus with Grassmann and Clifford Variables
Let <p be a continuous function on R slowly increasing at infinity. For a 6 R,
we have:
lim (
t—>o
/ e-{x-af'**4>{x)dx = <j>{a),
as follows from the change of variable x i-> tl/2(a + x). We will prove a
generalization of this formula, which allows a to lie in a more general algebra.
Let A be a finite dimensional supercommutative algebra with unit. (In
practice, .4 will be an exterior algebra). Let ? = (?i,..., ?n) be a n-tuple of
even nilpotent elements of A. Let <j> be an A-valued smooth function on R".
We define:
E.7)
(* + 0 =
for x € R".
This sum is finite because the elements ? are nilpotent. We will write \\x—?||2
for the function ]Cj=i(xj ~ fjJ-
Lemma 5.12. Let <j> be an A-valued smooth function on R™ slowly increasing
at infinity (as well as all its derivatives). Let f = (?1,... ,?„) be an n-tuple
of even nilpotent elements of A. Then
lim D7rt)-"/2
t—»o
Proof. If $ and its derivatives are rapidly decreasing functions on Rn, we
have
Thus
f e-"x-W4t<t>(x)dx= [ c-'-'V^x
Jr« Jr»
In the limit t —» 0, we obtain the lemma. ?
Let V be a Euclidean vector space. We denote by expA(a) the exponential
of a € AV, and by g = C2(V) the Lie algebra of Spin(V), which we can
identify as a vector space with A2V by the symbol map. By the universal
5.3. Calculus with Grassmann and Clifford Variables
179
property of the symmetric algebra C[g] of polynomial functions on g, the
injection of g into the even part of AV extends to an algebra homomorphism
A : Cfo] -» AV.
We can extend the map A to the algebra to C°°(b) by composing with the
Taylor series expansion at 0. Let us describe this homomorphism in the case
in which V = R", with orthonormal basis e\ An element / of C°°(g) is a
smooth function /(ay) of the coordinates {ay | 1 < i < j < n} on A2Rn,
and A(f) is the element /(el A e>) e AV defined in E.7), which is obtained
by replacing the variables ay by the elements e* A eJ e A2R". Denote by
e Ae* the antisymmetric matrix whose (i, j)-coefficient equals e* Ae3 G A2V.
We will use the suggestive notation f(e A e*) for A(f), and will call it the
evaluation of / at e A e*.
Since e* A e* = 0, many polynomial expressions in the variables e< A e,- will
vanish, and consequently some functions on $j will have a vanishing evaluation
at e A e*. In the following lemma, we collect a number of such functions.
Recall that r is the canonical representation g —> eo(V).
Lemma 5.13. A) Let v, w e V and let /*iU,(a) = (T(a)kv, w) be the cor-
corresponding coefficient of the matrix r(a)k. Then fkw{e A e*) = 0 for
k>2.
B) Let /it(a) = Tr((ada)fc), where ad is the adjoint representation of g.
Thenfk(eAe*) = Qfork> 1.
C)
Proof. The coefficients of the matrix t(oJ are linear combinations of mono-
monomials aikakj. Thus A) follows from the relation e* A e/t = 0.
The adjoint representation can be written as ad(a) = r(a) A1 +1A r(a) for
a e g. Hence it follows from A) that the evaluation at e Ae* of any coefficient
of (ad(a))* vanishes, when k > 2. Since r(a) is antisymmetric, we also have
Tr(r(a) A r{a)) = -Tr(-r(aJ) and Tr(ada) = 0. Thus, for any k > 1, the
evaluation at e A e* of Tr((ada)*) vanishes.
Using the relation det(>l) = exp(Tr(log A)), we can write jg(a) in the form
jB(a)=exP(Tr(^cfc(ada)fc)),
and C) follows from B). (Alternatively, B) and C) can be seen as a conse-
consequence of the fact that the constants are the only elements in AV which are
invariants under the full orthogonal group O(V).) ?
The following proposition is the crucial technical tool leading to the elim-
elimination of the singular part of the heat kernel index density.

n. t lie exponential Map and the Index Density
Proposition 5.14. If $ is a smooth slowly increasing function on g, then
lim D7rt)-dim(fl)/2 /Vw3/
t—>0 •'8
Proof. Since (e' A e-7J = 0, we obtain by completion of squares the formula
e-NlV« expA(a/2*) = expA(-i ?(ay - e< A
Applying Lemma 5.12 gives the desired formula. ?
For the proof of Theorem 4.1, we need to replace Grassmann variables by
Clifford variables in Proposition 5.14. The next proposition shows that this
is indeed possible when taking the limit t —> 0. Let 6t be the automorphism
of AV such that
^G) = rj/27 for 7 g AjV.
As usual, we will identify the spaces C(V) and AV by means of the symbol
map.
Proposition 5.15. If $ is a smooth slowly increasing function on g, and
7 e AkV, then
lim (lirt)-Aim(9)/2tk/2 f
= 7 A $(e A e*).
Proof. Recall the relation C.13) between the exponentials in the exterior
algebra and the Clifford algebra:
expc a = jv/2{a) tet{Hv/2{a)) (Hy1/2{a) ¦ expA a),
where Hv(a) : g —> GL(V) is given by the formula
r(a)/2
Hv(a) =
tanh.(T(o)/2)'
In particular, the matrix entries of Hv1'2(a) are analytic function of the
coefficients of the matrix r(oJ.
Let ej be the orthonormal basis of A V obtained from an orthonormal basis
ej of V. We have
<5t(expc(a/2))
= jv/2
/2(
jv/2(a) det(Hv/2(a))
(a)eue3) ej.
u
By Lemma5.13, the evaluation of jv/2(a) det(Hv/2(a)) (Hy1/2(a) ej,ej)
at e A e* is equal to Sij. Thus, applying Proposition 5.14, we obtain the
result for 7 = 1.
s.4. The Index of Dirac Operators 181
If 7 = ei A ... A efc, we have
7 • expc(a/2) = (?l - n) • • ¦ (ek - ik) expc(a/2).
Applying 6t, we obtain
t^Xi-y ¦ expc(a/2)) = (?l - in) ¦••(?*- tckNt(expc(a/2)).
and taking the limit, we deduce the result from the case 7 = 1. ?
We restate the above result in a slightly different way.
Proposition 5.16. Let $ be a smooth function on g defined in a neighbour-
neighbourhood ofO. Let 7 6 Cm{V), with symbol am(y) € AmV. Consider the formal
power series
{Ant)
a)G • expc a)
where Ct € C(V). Then the element d is of Clifford degree less than 2i + m,
and we have
n/2
J2 <T2i+m(Ci) = 7 A $Be A e*).
t=0
Proof. Proposition 5.15 asserts that the Laurent series
no poles, and computes the constant term. ?
>=o
j) has
5.4. The Index of Dirac Operators
We now give our second proof of Theorem 4.1. Thus M is an oriented compact
Riemannian manifold of even dimension n. For simplicity, we assume that
M admits a spin structure; this is always true locally, and since the theorem
is local, it is sufficient to prove the theorem in this case.
Let V = Rn, with its standard orthonormal basis ej. Let G = Spin(n),
with Lie algebra g = C2(V), and let r : g —> so(V) be the map de-
defined in Proposition 3.7. We denote by it : P —* M the principal bundle
Spin(M) which defines the spin structure, and provide it with its Levi-Civita
connection w. Let Q e .A2(Spin(M),(j) be the curvature of the princi-
principal bundle Spin(M). Then the Riemannian curvature R of M is equal to
t(Q) e A2(M,eo(TM)). At a point p € P, which corresponds under the
two-fold covering Spin(M) —> SO(M) to an orthonormal frame at the point
x — ir(p) 6 M, we may identify both TXM and T*XM with the vector space
V.

182
5. The Exponential Map and the Index Density
Let p : G —* End E) be the spinor representation, and let S be the asso-
associated superbundle of spinors, which carries the Levi-Civita connection Vs.
The Casimir Cas of the representation p is given by
Cas =
= - dim(fl) = -n(n -
so is a scalar.
We suppose given a Hermitian complex vector bundle W —» M and a
Hermitian connection Vw on W with curvature Fw, and we let D be the
corresponding twisted Dirac operator of the twisted spinor bundle W ® S.
Let fct(x, y) be the heat kernel of the operator D , along the diagonal, it is a
section of the bundle End(W ® $) S C(M) ® End(W).
Since the operator D2 satisfies the Lichnerowicz formula
we may apply to it the analysis of Section 2. Indeed,
where
Fp =
' ef
e< A ei
By Theorem 5.9, the asymptotic expansion of (x | e~*°a | x) has the form
j (a) ® p(exp o) j9(a) da,
j=o
where $j form a series of smooth functions on g with values in End(Wx),
and
*o(a) = Jfl/2(a) Jv1/2(r(np-a)/2) exp(-Fpl ¦ a/2).
In the identification End(S) S C(V) S AV by the symbol map a, the matrix
/j(expa) corresponds to the element expc(a) of AV. The first assertion of
Theorem 4.1 follows immediately from Proposition 5.15 with 7 = 1.
The formula in Proposition 5.15 for the leading order of the above asymp-
asymptotic expansion shows that
By Lemma5.13, the function j9Ba) evaluated at e A e* equals 1. Thus, it
remains to calculate $oBe A e*).
Identifying V and V*, we consider the curvature F^ as an element of
A2V ® End(VVj;) and the Riemannian curvature ftp as an element of A2V <8>
End(Ta;Af). The function a i-+ Fp • a is a End(WI)-valued linear function on
0, thus its evaluation at e A e* is an element of A2V ® End(Wx)- Similarly
5.4. The Index of Dirac Operators
183
the function a t-+ t(Hp • o) is an End(TxM)-valued linear function on g, thus
its evaluation at e A e* is an element of A2V ® End(TxM).
Lemma 5.17. We have Fpx -(e A e*) = F™ and r(f)p -(e A e*)) = Rp.
Proof. The curvature F™ is the element
of A2V ® End(Wx). If Fpx G EndCVV,,) ® g is given by E.4) and a =
Ei<ja<jei Aej e g, we see that F$-a = Ei<j aiJ^PW(e.?.ef )• Replacing
the scalar variables aij by the Grassmann variables e< A e^, we obtain the
first formula.
The second formula is more subtle. Let us write Up € A2V ® g as
Let Eij - r(ej A e^), so that {Eij | i < 3} is a basis of so(V). Since the
Riemannian curvature Rp ? A2V ® so(V) is the image of the curvature flp
by the representation r of g in V, we see that
The fundamental symmetry of the Riemannian curvature shows that
Rp= 5Z e* AeJ® (fip(e*.ei).ei Aei)Efcj.
•<j;fc<i ;
For a = J]i<:j OyCi A e7 6 g, the element flp • a of A2V equals
so that
r(np • o) =
by the Grassmann variable e< A ej, we obtain
Replacing the variable
B). D
The function $0Ba) is the product of the real function jgl/2Ba), the real
function jy1/2(T(fy> •a)) ^dthe End(Wx)-valued function exp(-F^ • a). We
know that j^~1/2Be Ae') = l. By the preceding lemma, the evaluation of
the function Jv1//2(T(np-a)) at ej A e* equals the A-genus of M, while the
evaluation of the End(WI)-valued function exp(-F^ • a) at e A e* equals the
element exp(-Fw) of ^(M,End(W)). This completes our second proof of
Theorem 4.1.

5. The Exponential Map and the Index Density
Bibliographic Notes
The method used in this chapter first appeared in Berline-Vergne[24]. This
work was influenced by the articles of Getzler [55] and Bismut [27].
Chapter 6. The Equivariant Index Theorem
Let M be a compact oriented Riemannian manifold of even dimension n,
and let H be a compact group of orientation preserving isometries acting on
M. Let ? —» M be a Clifford module with Clifford connection; if H acts on
? compatibly with the Clifford action and Clifford connection, we call ? an
equivariant Clifford module. If D is the Dirac operator on ? associated to
the given data, then D commutes with the action of H; hence, the kernel of
D is a finite-dimensional representation of H. The equivariant index is the
virtual character of H given for 7 e H by the formula
indH G, D) = TrG,ker D+) - TrG, ker D").
In this chapter, we will prove a generalization of the local index theorem
for D which enables us to calculate indnG, D) f°r arbitrary 7 e H. Let
(z I -ye~tD I y) be the equivariant heat kernel of D, that is, the kernel of the
operator ye~tD . The restriction of (x | 'ye0 | j/) to the diagonal of M is
a section of End(?) which we denote by kt(f,x).
Fix 7 G if, and denote its fixed point set by M7. The main result of
this chapter, Theorem6.11, is that kt(-y, x) has an asymptotic expansion, as
t —> 0, of the form
:=0
where the coefficients $4G, x) are generalized sections of End(?) which are
supported on M7, and such that $4G, z) has Clifford filtration degree 2i +
dim(M) -dim(M7). We will calculate 02i+dim(Af)-dim(M->)$»G,x) explicitly,
obtaining a formula of a similar type to that of Chapter 4, which is the case
where H = {1}; this formula involves the curvature of the normal bundle A/"
of M7 in M and the action of H on it.
We begin the chapter with some generalities on the equivariant index. In
Section 2, we give a proof of the Atiyah-Bott fixed point formula for the equi-
equivariant index of the d-operator when the fixed point set is discrete; this serves
as a simple introduction to the equivariant index theorem. In Section 3 we
state Theorem 6.11, while in Section 4, we show that it implies the equivari-
equivariant index formula of Atiyah-Segal-Singer for indyj G, D) as an integral over
M7 and the local formula of Gilkey. The rest of the chapter is devoted to
the proof of Theorem 6.11, by a method generalizing that of Chapter 5.

186 6. The Equivariant Index Theorem
6.1. The Equivariant Index of Dirac Operators
Let M be a compact oriented Riemannian manifold of even dimension n,
on which a compact Lie group H acts by orientation-preserving isometries.
There is induced an action of H on the Clifford bundle C(M), which preserves
the product.
Definition 6.1. An equivariant Clifford module ? over M is an H-
equivariant bundle, with a Clifford module structure and Hermitian inner
product preserved by the group action. We suppose that the action of H
preserves the ^-grading of ?.
Let D be a Dirac operator associated to a Clifford connection Vf on ?.
The following lemma is evident.
Lemma 6.2. The action of the group H commutes with the Dirac operator
if and only if the Clifford connection V? is H-invariant.
Prom now on, we only consider invariant Dirac operators D on ?. Since an
element 7 € H preserves the Dirac operator D, it must map ker(D) to itself,
so that ker(D) becomes a Z2-graded representation of H. The character of
this representation, that is, the supertrace
StrG,ker(D)) = TrG,ker(D+)) - TrG,ker(D-)),
is called the equivariant index of the Dirac operator D, and we will denote
it by indHG, D); it is an element of the representation ring R(H) of the
compact group H. The purpose of this chapter is to calculate ind,HG, D) in
terms of local data at the fixed point set of 7 acting on M; for 7 = 1 e H, this
is just the local index theorem of the Chapter 4. The other extreme occurs
when the fixed point set of 7 consists of isolated points; if D is associated to
an elliptic complex, the formula for indf/('),D) in this case, due to Atiyah
and Bott, is particularly simple to derive, and we will present it in Section 2.
The foundation of the calculation of the equivariant index of a Dirac op-
operator is the following generalization of the McKean-Singer formula.
Proposition 6.3. Let fetG,a;) \dx\ — (x | 7e~tDa | x) be the restriction of
the kernel of the operator -ye~iD to the diagonal. Then for any t > 0,
indHG)D) = StrGe-tD2)= / Str(fctG,x)) \dx\.
Proof. We will give two proof of this result, which are transcriptions of the
two proofs of the McKean-Singer formula Theorem 3.50 in its simplest ver-
ver6.2. The Atiyah-Bott Fixed Point Formula
187
If A is a real number, the A-eigenspace Tif of the generalized Laplacian
D2, acting on T(M, ?±) is a finite dimensional representation of H, with
character x*- We have the following formula for the supertrace of 7e~tD
tD
A>0
The Dirac operator D induces an isomorphism between H\ and H^ when
A/0. Since D commutes with H, this isomorphism respects the action of
H, so that
xJG)-XaG)=0 forA>0,
and we are left with xt(l) ~~ Xoil)' which is nothing but the equivariant
index of D.
The second proof of Theorem 3.50 generalizes as follows. If aG, t) is the
function StrGe~*D ), we again see by Lemma 2.37 that for t large,
|StrGe-tDa -y*U(D))| < Ce"^2 vol(M),
where Ai is the smallest non-zero eigenvalue of D2. This shows that
0G,00)= lim Str^e"'02) = indHG,D).
t—+oo
The proof is completed by showing that the function o(t,7) is independent
of t, so that 0G, t) = 0G,00) = iiidHG>D). Differentiating with respect to
t, we obtain
since D is odd and [0,7] = 0. This last line vanishes by Lemma 3.49. ?
Thus, the calculation of the equivariant index is reduced to the calculation
of the limiting generalized section lim kt("f,x) of ?.
o
6.2. The Atiyah-Bott Fixed Point Formula
The Atiyah-Bott fixed point formula is a generalization of the Lefschetz fixed
point formula, which gives the supertrace of the action of a diffeomorphism
on the deRham cohomology of a manifold. In some respects, it is more
general than the equivariant index theorem which we will prove later, since
the group H is allowed to be non-compact. However, it is more restrictive, in
that it only gives a formula for elements 7 € H whose fixed point set consists
of non-degenerate isolated points, and D must be associated to an elliptic
complex.

188
6. The Equivariant Index Theorem
Let D : T(M,?') —> r(M,?#+1) be a differential operator acting on the
space of sections F(M, ?) of a Z-graded finite dimensional vector bundle ?
over a compact manifold M. We say that (T(M,?),D) is a complex if
D2 = 0. Suppose that the cohomology space H'(D) = ker(D)/im(D) is
finite dimensional.
Consider a group H acting on ? —* M and assume that the action of H
on T(M, ?) commutes with D. The group H acts on the graded vector space
/**(?>) and we define, for 7 € H,
= ?(-!)'TrG,
If 7 acts on M with isolated nondegenerate fixed points, Atiyah and Bott
gave a fixed point formula for the index: of the transformation 7, even if H is
a non-compact group.
In this section, we will give a proof of Atiyah-Bott formula for those com-
complexes D for which there exist a Hermit ian structure on ? and a Riemannian
metric on M such that the operator D + D* is a Dirac operator. Let ? —* M
be a If-equivariant vector bundle over a Riemannian manifold M. However,
we do not need to assume that H acts by isometries on M; in particular, H
need not be compact.
If x is an isolated fixed point of the action of an element 7 € H on M,
denote by yx the tangent action at the fixed point x, and by -yx the endomor-
phism of the fibre ?x induced by 7. The point x is called a non-degenerate
fixed point if the endomorphism A — 7^) of TXM is invertible; for example,
this condition is satisfied if x is an isolated fixed point and H is a compact
group. Let I be a generalized Laplacian on ? and let kt(-y, x) \dx\ be the
restriction of the kernel of -ye~tL to the diagonal; it is a section of ? (g> |A|,
where |A| is the bundle of smooth densities on M.
Recall that a generalized section s € r~°°(M, ?) of a bundle ? is a contin-
continuous linear form on the space of smooth sections of ?*® |A|; thus, the space of
smooth sections T(M,?) of ? may be embedded in r-°°(M,?). We use the
notation JM{s(x),(j>(x)) \dx\ for (s,<j>\dx\). If x € M, the delta distribution
6X at x is the generalized section (Sx, <j>) = <j>(x) of the bundle of densities |A|.
Lemma 6.4. Let xq 6 M be a non-degenerate isolated fixed point 0/ 7 €
H. If x{x) ? C°°(M) is a function equal to 1 near xo and vanishing in
a neighbourhood of all other fixed points of the action of 7 on M, then as
t->0, the section x(x)M7,x) o/End(?) ® |A| has a limit in the space of
generalized sections of End(?) ® |A|, gi.ven by
t—>o
|det(l-7J01)|
€r-oo(M,End(?)®|A|).
6.2. The Atiyah-Bott Fixed Point Formula
189
Proof. We must compute
lim / x(x) kt(j,x) <f>(x) \dx\ for <j> € T{M, ?).
Since fc<G,a;) = 7f G~1x | e~iL | x), we see that the function x(x)kt("f, x) is
rapidly decreasing as a function of t, except in a small neighbourhood U of
x0. Let V = TXBM, and let F : U —> V be a diffeomorphism of U with a
small neighbourhood of 0 in V such that dXoF = /. We transport the integral
over M to an integral on a small neighbourhood of 0 in V, and trivialize the
bundle ? on this neighbourhood.
The approximation of (x | e~tL \ y) shows that
Urn f X{x)kt{i,x)<t>{x)\dx\
-"/2
= limD7rf)
where ip(x) is a smooth compactly supported section of End(?) equal to t|0
for x = 0. Consider the change of variables x —> t^2x on V. We see by
Proposition 1.28 that
lim r1dG
t—>0
where || • || is the Euclidean norm on V, and the lemma follows. ?
If D is a complex on a Z-graded Hermitian vector bundle ? —> M over
a Riemannian manifold M such that D + D* is a Dirac operator, then L =
DD* + D*D = (D + D*J is a generalized Laplacian which commutes with
D:
(DD* + D*D)D = DD'D = D(DD* + D*D).
It follows from Hodge's Theorem 3.54 that D + D* is essentially self-adjoint,
that the cohomology H(D) is a finite-dimensional vector space and that
ker(D + D') = ker(?>)nker(Z>*) is a subspace of ker(D) isomorphic to H{D).
Lemma 6.5. Let L be a generalized Laplacian of the form DD* + D*D with
D2 = 0. Assume that the action of H on T(M,?) commutes with D. Then,
for every t > 0,
indifG,I?)=StrGe-ti).
Proof. The proof is very similar to that of the equivariant McKean-Singer
formula of Section 1. First, note that the right-hand side of the equation is
independent of t; indeed, its derivative with respect to
is
dt
= - StrG(Z??>* + D*D)e-tL)

Fix a component Mo of M7 and let 71 be the transformation induced
on M\m0- Since the only possible real eigenvalue of 71 is —1, we see that
det(l —71) > 0. Furthermore, since detGi) = 1, we see that dim(A40) =
dim(M)— dim(Mo) must be even. We denote by to and ?1 the locally constant
functions on M7 such that dim(M0) = 2?0 and dim(AfXo) = 2^i for all x0 €
Mo; it is clear that n = 2f0 + 2^i- Denote by det(A/) the line bundle over M7
which over the component Mo of M7 is equal to A2'1 .A/"*; this line bundle is
contained in AT*M\m-i.
Lemma 6.9. The function det(l—71) is constant on each component ofM.
Proof. If Mo is a component of M, each fibre Afx, x € Mo, of the bundle M
carries a finite-dimensional representation of the closure G(j) of the group
generated by 7 in the group of isometries of M. Since M is compact, this
is a compact group. Since the dual of the compact group G(y) is discrete,
the representation must be independent of x G Mo, from which the lemma
follows. D
Let 7 e SO(V) and A € so(V). In Definition 3.25, we have defined the
analytic square root det1/'2(l — 7exp.A). Applying this with 71 6 SO(A40)
and A = -Rt 6 so(A40) ® A2T*X0W, we obtain an element of hT*XoM~<; as
the point x0 varies, we obtain a differential form detI/2(l - 71 exp(--Ri)) €
«4(M7). It follows from the theory of characteristic classes that this form
is closed. Its zero-degree component is equal to |det(l — 7i)|1/'2 so that
det1/2(l - 71 exp(-R1)) is a well-defined invertible element of .4(M7).
We may trivialize the density bundles |Am| and |AAf| by means of the
canonical sections \dx\ and \dxo\ derived from the Riemannian metrics on M
and M7 respectively. Thus, the delta-function Sw of M7 is the generalized
function on M defined by
m
6M,(x)<f>(x)\dx\ = f <t>(xo)\dxo\,
where \dx\ and |dxo| are the Riemannian densities of M and M7. If T is a
bundle over M and v e T{M~<,F), then vSmi is a generalized section of T
over M.
From now on, ? will be a /7-equivariant Clifford module on M, with
//-invariant Clifford action and Clifford connection. As usual, we write
End(?) = C(M) ® EndC(M)(?)- If M has a spin structure, then ? is iso-
morphic to W ® S for some bundle W, and Endc(M)(?) — End(W).
Lemma 6.10. Along M7, the action of 7 on ? may be identified with a
section -y? of C{M*) ® Endc(M)(f).
Proof. We have 7c(qO-1 = c{-ya) for a e r(M,T*M). Thus the action
on ?xo of 7 at a point xo e M7 commutes with the Clifford action of
C(T*xaMi). It is clear that the commutant of C(T*X0MJ) in End(?l0) is
- ?
6.3. Asymptotic Expansion of the Equivariant Heat Kernel
193
By this lemma, we may form the highest degree symbol adim(^)G),
which is a section of Endc(M)(?)®Aet(M) C Endc(M){?)® AT*M\My, since
det(Af) = Adim^^*.
Let Fe?s be the twisting curvature of the Clifford module ?. Recall that
if M has a spin-structure with spinor bundle S, we may write ? = W ® S,
and then F?/s = Fw is the curvature of the twisting bundle W. We denote
the restriction of FE'S to M7 by Fof /s € T(M7, AT*M7 ® Endc(M) (?)).
Let D be the Dirac operator of the Clifford module ?; it is an //-invariant
operator on F(M, ?). Let
AtG,x) \dx\ € T(M,C{M) ® EndC(M){?) ® |A|)
be the restriction of the kernel of the operator je'10' to the diagonal. The
main result of this chapter is to show that A* G, a;) has an asymptotic ex-
expansion when t —> 0 in the space of generalized sections T~°°(M, C(M) ®
Endc(M) (?) ® jA|), and to calculate the leading order of the expansion. This
theorem is a generalization of a result of Gilkey.
Theorem 6.11. The section kt(-y) of the bundle C{M)®En&c(M)(?) ® |A|
has an asymptotic expansion ast —> 0 in the space of generalized sections of
C(M) ® Endc(M) (?) ® IA| of the form
where $^G) is a generalized section of C2i+dim{U)(M) ® Endc(M)E) ® |A|
supported on M7. The symbol of$i(y) is given by the formula
< G)) = Hi) ¦
i=Q
where I(y) € T(M7, AT'M ® Endcw(f)) equals
7G) =
det1/2(l - 7l)det1/2(l - 7l exp(-7?»))"
Observe that when 7 is the identity, this theorem reduces to the local
index theorem of the last chapter, Theorem 4.1.
From the formula
kth,x) =
x),
we see that if -yx ^ x, the function t -» kt(-y,x) is rapidly decreasing as
t -* 0; thus only generalized sections with support in M7 will appear in the
asymptotics of the generalized section ^G), and we need only study ^G)
in a neighbourhood of M7.
The map (x0, v) e M h+ exp^ v defines a diffeomorphism between a neigh-
neighbourhood of the zero section of M and a neighbourhood U of M7. We may

194
6. The Equivariant Index Theorem
identify ?eXpx v with ?XQ by parallel transport along the geodesic s € [0,1] ^
expXosv; this respects the filtration on End(?) S C(M) ® Endc(M)(?).
Thus A;tG,expIo v) may be thought of as lying in the fixed filtered algebra
End(?l0). If cj> is a smooth function on M with support in U, the integral
I(t,7, <t>, x0) = / kt(-y,expXo v) cj>(expXo v) \dv\
defines an element of End(?l0), where |<fo| is the Euclidean density of NXo.
Using the rapid decrease of kt{i,x) as t —> 0 for x ^ M7 and a partition of
unity argument, we see that Theorem 6.11 is implied by the following result.
Proposition 6.12. Let Mq be the component of the point x0 in M7, and
let <j> be a smooth function on M supported in a small neighbourhood
Then
A) I(t, 7, (j>, xo) has an asymptotic expansion as t —> 0 of the form
t, 7, <t>, x0)
t=0
where x0 i-+ fyty,x0) e r(MQ,C2(i+ll)(M) ® Endc(M)(?)).
B) Ifo-(k(j,4>,x0)) is given by the formula S
, then
t>, x0)) = 7G, x0) 4>{x0).
We will prove this proposition later in this chapter. However, if x0 is an
isolated point of M7, the proof is much the same as that of Lemma 6.4, which
we proved in the course of proving the Atiyah-Bott fixed point theorem in
the last section. Indeed we see, using the change of variables v —»t1/2v, that
lim
6.4. The Local Equivariant Index Theorem
The Atiyah-Segal-Singer fixed point formula for ind/f G, D) is a formula for
the equivariant index ind#G, D) of an equivariant Dirac operator D as an
integral over the fixed point manifold M7. In this section, we will show how
combining Theorem 6.11 with the equivariant McKean-Singer formula leads
to a local version of this theorem.
Let W be an //-equivariant superbundle on M. Fix 7 e H. If A is an
iZ-invariant superconnection on W, denote by Fq the restriction to M7 of
the curvature F = A2 of A. Define the differential form chwG, A) 6 A(M~<),
by the formula
chHG, A) = StrwG • exp(-F0)).
6.4. The Local Equivariant Index Theorem
195
This is a closed differential form on Af7 whose cohomology class is indepen-
independent of the if-invariant superconnection A, by a proof which is identical to
that for the ordinary Chern character. In this section, we denote ch;/G, A)
bychHG,>V).
If ? is an H-equivariant Clifford module over M with H-invariant super-
connection A, we will define a differential form
chtfG,?/S) € A(M^,det(tf)) = r(M7,AT*M7 <8>det(JV")),
which agrees with chn G, W) up to a sign when ? = W® S is an equivariant
twisted spinor bundle. Recall thai; we have defined a map
¦ r(M7,Endc(M)(?)) - <?°°(M7).
Iff = H'igi5andae r(M7,Ertd(W)), Str?/S(o) = Strw(a). We extend
Str?/s to a map
Str?/S : r(M7,Endc(M)(?) ® det(jV))
Let Fq' be the restriction to M7 of the twisting curvature of A. Then
Stre/s((Tdim(AoG?) exp(-Fof/S))
is a differential form on M7 with values in the line bundle det(A/").
Definition 6.13. Define the localized relative Chern character form
ch«G,?/S) = ^1/2^' _ Str?/s(<Tdiro(A0G?)exp(-F05/s))
G^(M7,det(//))-
Obviously, this rather peculiar' definition requires some justification; we
will rewrite it in terms of more familiar objects when M has an H-equivariant
spin-structure; this is by definition a spin-structure determined by a princi-
principal bundle Spin(Af) which carries an action of the group H, such that the
projection map Spin(M) —* SO(M) is equivariant.
Proposition 6.14. If M has an H-equivariant spin-structure, then for any
7 g H, the manifold M1 is naturally oriented.
Proof. The action of 7 on the fibre of the spinor bundle Sxo at the point xq e
M7 gives an element 7 e End(<SI0) = C(T*XoM) such that -yc(a) = 0G^O
for all a € A?o. With respect to the decomposition TXaM = V0®V1, where
Vo = TXoM~t and V\ = MXa, we see that 7 is an element of C(V{).
Let e*, 1 < i < 2?q, be an orthonormal basis of V^*. If
To = exPc(? ? e2-1 A e2<) = c{e' A ... A e2t") € Spin(^),
then To -7 e Spin(V) maps to (-1) x 71 e SO(F0*) x SO(V,*).

6. The Equivariant Index Theorem
Let T : AT*XoM —» R be the Berezin integral. By Lemma3.28,
Thus <rdijn(jV) G) is a non-vanishing section of det(A/") such that
Thus (Tdim(AO (^) defines a trivialization of det(A/")> which is the same thing
as an orientation of Af. But given an orientation of M and .A/"*, there is a
unique orientation on AP compatible with these data. ?
Fix an orientation of M'. Sometimes, there is a natural orientation on M7
(for example, if M7 is discrete), which may not agree with the orientation
which we obtained above. We will write e(-y)(xo) = 1 if the two orientations
agree, and eG)(zo) = — 1 if they do not; thus, for fixed 7, ?G) is a function
from M7 to {±1} which is constant on each component.
This discussion implies the following result.
Proposition 6.15. Let M be an H-equivariant even-dimensional spin mani-
manifold, and let W be an H-equivariant vector bundle with H-equivariant super-
connection A. Consider the H-equivariant Clifford connection Vw <8> 1 +1 ®
V5 on the Clifford module ? = W ® S. Then
Proof. In this case, 7^ = 7^® 7, and
@ - 7lOW- ?
We now return to the general case, in which M does not necessarily have
a spin-structure, and state the equivariant local index theorem. Consider the
section of AT*M over M7 given by
i(M7)chflG,?/S)
Since M is orientable, we may take the Berezin integral of this section, to
obtain a function on M7, which we denote by
'
We can now state the equivariant index theorem.
Theorem 6.16 (Atiyah-Segal-Singer). The equivariant index ind#G, D)
of an equivariant Dime operator D associated to an ordinary connection is
given by the formula
indHG,D)=
det1/2(l-71exp(-.R1))
\dxo\.
6.4. The Local Equivariant Index Theorem
197
Proof. We will show that this theorem is a consequence of Theorem 6.11.
Let U be a tubular neighbourhood of M7 and let ip be a smooth function
with support in U identically equal to 1 in a neighbourhood of A/7. Using a
partition of unity, we see that
/
m
f
Ju
/ Str(fctG,exp v))ip(exp v)b(xo,v)dxo\dv\.
JM
/
JM
Here, we have rewritten the Riemannian volume \dx\ in terms of the Rieman-
nian volume on J\f and a smooth function b(xo, v) satisfying b(xo, 0) = 1, by
the formula
[ <t>{x)\dx\= I f ^ejqpSot»N(xo.t»)|A'||«fco|.
Jm Jm-t Jmxo
Applying Proposition6.12 with <j>(v) = ip(expxo v) b(xo, v), we see that
I(t, 7, <t>, x0) = / fc*G, exp v) 4>{v) \dv\
has an asymptotic expansion of the form
t=0
where the Clifford degree of $i(io) is less than or equal to 2(i + ?i). We
now use the fact that the supertrace vanishes on elements of Clifford degree
strictly less than n. It follows from Theorem6.11 that StrG(<,7,x0)) has an
asymptotic expansion without singular part, so that
indwG,D)= lim / Str(fctG,:r)) \dx\
from which the theorem follows immediately on substituting the formula for
O~n (*dim(M-')/2)- ?
There are two special cases of this formula in which the geometry is easier
to understand. The first case is that in which M has an //-equivariant spin-
structure, with spinor bundle S. Thus, the Clifford module ? is a twisted
spinor bundle W®5, where W = HomC(M)(S>?)> and D is the twisted Dirac

198
6. The Equivariant Index Theorem
operator associated to an invariant connection Vw on W with curvature Fw.
Then Proposition 6.15 shows that
indHG, D) =
2(l -
-R1))
A further simplification of the formula occurs if in addition the fixed point
set M7 has a spin-structure. In this case, the normal bundle M has associated
to it a spinor bundle 5(A/"), defined by the formula
The actions of 7 on the spinor bundles S^f and Sm induces an automorphism
71 of the vector bundle «S(A0- Choosing compatible orientations on M7 and
A/", we have
chHG,5(A0) =
= r'l?G,M7)det1/2(l -
and the equivariant index formula may be rewritten as follows:
ind«G,D)=/ B^(-^
6.5. Geodesic Distance on a Principal Bundle
In this section, we prove a technical result which is basic to the proof of
Proposition 6.12. Let 7r : P —¦ M be a principal bundle over a Riemannian
manifold, with compact structure group G, connection form w € A1(P) ® g,
and curvature il.
We denote by cP(pi,p2) the square of the geodesic distance between two
points p\ and P2 of P. If p\ and p% belong to the same fibre ir~1(x), the
horizontal tangent spaces at pi and P2 may be identified with TXM. Given
Hi, v2 in TXM, we are going to compute the Taylor expansion up to second
order in t e R of cP(exppi ti^exp^ tv2). We have P2 = Piexpa for some
a e g. Recall from Theorem 5.4 that
where
F.2)
exp (a + tv) = exp (Ufa ,a)v + o(t)) for v e TXM,
Pl ¦ a)/2
Proposition 6.17. For small a, d2(exppi tui,exppatt>2) equals
6.5. Geodesic Distance on a Principal Bundle
199
Proof. Let Pi(U) = exp,,.^^), i = 1, 2, and define X(t1,t2) e TPl{tl)P by
P2(t2) = exppi(tl)X(ti,t2). Thus, we must compute ||X(ti,t2)l|2 to second
order with respect to (ti,t2)-
We identify a neighbourhood of pi in P with a neighbourhood of the origin
in V x g by means of the exponential map; here, V — TXM. Thus, for t small,
Tpi(t)P may be identified with V © g. Under this identification, the metric
on Tp^P coincides with the orthogonal sum metric on V ffi g S TPIP up to
second order.
Let us start by computing X(ti,*2) up to first order. We have
X{t) = X(t,t) = o + tXx + O(t2).
Differentiating both sides of the equation exppi(t) X(t) = p2{t) with respect
to t, we obtain
vi + J(pi,a)Xi =v2;
here, we have used the fact that exppi(t) a = expp2(tvi), which follows because
both sides of this equation are geodesic curves starting at p2 with tangent
vector vi. Thus, we obtain
X(tut2) = o +
\t2\).
In particular, since the horizontal and vertical tangent spaces are orthogonal
for ti = t2 = 0, this implies that ||X(ti,t2)||2 - ||a||2 vanishes to at least
second order.
Denote by Vp the Riemannian covariant differentiation on P, and by Rp
its curvature. Consider the map p : R3 —» P given by
p(s,tllt2) = exppi(tl)(sX(ti,t2))-
This map satisfies the equations
Vpd.p = 0, \\d.P\\2 = HXC^.ta)!!2, and asp|8=0 = X(tut2).
Thus, for t = l,2, ]
^ tne vanishing of the torsion,
= {Rp(d.p,dtip)d.p,d.p),
= 0, since Rp is antisymmetric.
Prom this, we see that {dt(p, dap) is an affine function of s. Furthermore, the
vector 9t,p|«=i is tangent to the curve
tx 1—* exppi(tl) X(ti,t2) =
Thus 9tiP|,=i = 0, from which it follows that

6. The Equivariant Index Theorem
Writing
= 2d,{dtlp,dBp)
=-2(dhP,d,p)l=0,
we obtain the differential equation
= 2(t>i, J{pi,ay\txVl - t2v2)) + o(\h\ + \t2\).
Let \\X(tut2)\\2 = \\a\\2+t21xn + 2t1t2z12+t2ix22+o(tl+t?,) be the Taylor
expansion of H-X^i,^)!!2- We see that
a>n = («i,.7(pi,a)~V)> and
X12 = -{vi,J(pi,a)~1V2).
Now d2(pi(<i),p2(«2)) = rf2(P2(t2),exppa,,xp(_o)tii;i). Thus, exchanging the
roles of (pi,ti) and (P2,<2) we obtain
x22 = (v2,
Since J(pi,o) = J(p2,a), this completes the proof. ?
6.6. The heat kernel of an equivariant vector bundle
As in Chapter 5, let n : P —> M be a principal bundle with compact structure
group G, and let ? = P Xq E be an associated vector bundle corresponding
to the representation p of G on the vector space E. Let u be a connection
one-form on P, and let V be the corresponding covariant derivative on ?;
we will always consider P with the Riemannian structure associated to the
connection w. In this section, we will generalize the results of Section 5.2 to
the situation in which a compact group H acts on the left on P by orientation-
preserving isometries, and leaves the connection one-form u> invariant.
Also, suppose we are given an auxiliary bundle W with an action of if,
endowed with an H-invariant Hermitian metric and an if-invariant Hermitian
connection. Let F be a potential of the form F = F° + F1, where F° €
r(M,End(W)) and F1 e r(P,End(W)®fl)G, as in Section 5.2. Assume that
F commutes with the action of 7.
The group H acts on S, hence on the space of sections F(M, ?) by the
formula (¦y-s)(x) = 7sG~1x), where 7 e if and x 6 M. If Aw®? is the
Laplacian on the bundle W ® ? associated to the connection Vw ® 1 +1 ® V?
on W <8> ?, we will interested in the smooth kernel
(*o|7e-«AVV®E+F)|zi).
6.6. The heat kernel of an equivariant vector bundle 201
We can lift this kernel to the principal bundle P, obtaining there the kernel
(po I -re-tiAW9e+F) I pi) € T(P x P, WEl W) ® End(E).
Fix 7 € if, and let A/" be the normal bundle to Af7 in M. The exponential
map (xq,v) h-> expa.ov, for xo € Af7 and t; € MXo, gives an isomorphism
between a neighbourhood of the zero section in M and a neighborhood of
M7 in M, and if p 6 tt~1(xo), the point expp v projects to expxo v under it.
We identify Wexp, v and Wl0 by parallel transport along the geodesic. Let
<f> be a smooth function on AfXB with small support. Our aim is to study the
asymptotic behaviour of the time-dependent element of End(Wx0) ® End(?')
I(t,y,4>,P) =
F)) | exPpt>>
Generalizing Theorem4.1, we will prove that I(t,y, <j>,p) has an asymptotic
expansion as a Laurent power series in t1/2 when t —* 0, and give an integral
formula for this expansion.
Consider the quadratic form <5O on TXoM ffi TXoM defined for «i, v2 6
TIOM by the formula
Qa{vi,v2) = («i, J(p,a) vi) -2{v1,J(p,a)~1v2) + {v2,J(p,a)~lv2).
By Proposition 6.17, Qa is the Hessian of the function
(«i, «2)'—> d2 (expp vi, expp exp o v2)
at the critical point i>i = v2 = 0, and for o = 0, we have that Qa(vi, v2)\a=o =
||r;i — v2\\2. Let Qi(a,7) be the quadratic form on A/^o given by
Qi(a,7)(w) = Qa(t>,7") for ue A/"l0;
for a = 0, we have det(<2i@,7)) = det(l - 7XJ ^ 0. Finally, let 7 be the
element of G such that 7P = P7.
Theorem 6.18. Assume that the Casimir Cas of the representation p is a
scalar. There exist smooth sections $i e T(P x g, End(W)) such that as
t —> 0, I(t,j,<j>,p) is asymptotic to the Laurent series
Djrt)
p, a
js(a) da.
»=o
Furthermore, $o(Pii) is given by the formula

202
6. The Equivariant Index Theorem
Proof. The proof is a generalization of that of Theorem 5.9. However, we
will only give the details in the scalar case, in which W is the trivial bundle,
since the extension to non-zero potential F is similar to the corresponding
extension in the non-equivariant case, apart from the occurrence of 7*^ in the
formula for $o- Thus, from now on, we study the heat kernel of the operator
Af.
Let ht(po,Pi) be the scalar heat kernel of P and let Cas be the Casimir
of the representation (p, E) of G. The following lemma is the equivariant
version of Proposition 5.7.
Lemma 6.19.
'? I pi) = e~tCas [ httpo,'mg)plg)-1dg
JG
Proof. In the identification of T{M,S) with C°°(P, E)G, the action of H on
F(M, ?) corresponds to the action on C°°(P) given by (^s)(x) = s('y~1x).
Now the operator ye~tA on C°°{P) has kernel ht("f~1po,pi) = /it(Po,7Pi);
the rest of the proof is as in that of Proposition 5.7. ?
Identifying v € TXoM with a horizontal tangent vector at p e P, the
geodesic expp tv is a horizontal curve which projects to the curve expI0 tv in
M. It follows from the above lemma that
(exppv I 7e~tA* | exppi;) = e~tC!ul / ht(exppv,-yexpp(v) ¦ g)p(g)-1 dg.
J G
The curve -yexp^tv) is a geodesic in P starting at 7p = P7 with a horizontal
tangent vector which projects onto 71 v. Thus 7expp v = exppGii>) -7 and,
replacing g by j~1g, we obtain
(expp v I 7e~tA? | exp v) = / ht(exp v,expJ^v) -g)e~tCaap(g~1;y)dg.
JG
When g is far from the identity and v is small enough, the function
11—» ht(expp v, exppG!v) • g)
decreases rapidly as t —+ 0. Thus, since only a small neighbourhood of the
identity contributes asymptotically, we can work in exponential coordinates
on G. Let tp 6 C?°(g) be a smooth cut-off function equal to 1 on a small
neighbourhood of 0. Since |d(expa)| = jB(o) \da\, we see that I(t, 7, </>, x0) is
asymptotic to
-tCas
/oaymp
Mexpp v,exp
ea)p(e aj) (j>(v) ip(a)jB(a) \da\ \dv\.
6.6. The heat kernel of an equivariant vector bundle
203
Applying Theorem2.30, we may replace /it(exppi;,exppG1t;) -expo) by its
approximation
(Ant)- <««
i=0
where f(a,v) = d2(exppt;,exppG1v) exp a) and
$j(a, v) = Ui(expp v, exppG!t;) ¦ exp a).
Furthermore, Theorem 5.8 shows that
*o(a,O) = 3B(a)-l'%^M(T(ilp
By Proposition 6.17, the function v 1-* f(a,v) on MXo has its only critical
point at v = 0, where its Hessian is the quadratic form v h-> Q1(a,7)(f).
When 0 = 0, we have Qi@,7)(i>) = ||A - 7iH|2- Thus, for small a, the
quadratic form <2i(a,7) is not degenerate on NXo. By Morse's Lemma, there
exists a local diffeomorphism v t-» w = Fa(v) of MXo such that .Fo@) = 0 and
/(a,i;) = ||a||2-|-H|2. Note that
dv_
dw
w=0
= |det@1(o,7))|
-1/2
Thus, we see that I(t, 7, <j>, x0) is asymptotic to
t=0
a)-1^) j9(a)
flfo
\da\ \dw\.
The change of variable w —> t~1^'-lw gives us the asymptotic expansion of the
right-hand side that we wished to prove, with
#0)
In the case of a twisting bundle W, we consider the heat kernel ht of the
operator A + F of Theorem 5.9, and we use the formula
(po
Pi) = / Mpo,7Pi
?

6. The Equivariant Index Theorem
6.7. Proof of Proposition 6.13
In this section, we will prove Proposition 6.12, and hence Theorem 6.11. The
proof is a generalization of the method of Chapter 5. As in that chapter, we
will take advantage of the fact that the theorem is local to require that the
manifolds M has an equivariant spin structure, with spinor bundle S. Thus,
the Clifford module ? is isomorphic to W®<S, where W is a Hermitian vector
bundle with connection Vw.
Let P = Spin(M) be the double-cover of the orthonormal frame bundle
SO(M) corresponding to the spin-structure on M. A point p e Spin(M)
projects to an orthonormal frame at the point xo = n(p) 6 M, which we will
denote by p : V —» TXoM, where V is the vector space Rn. The theorem will
follow from an application of Theorem6.18, with group G = Spin(n).
Let Mo be a component of the fixed point set 7. Decompose V = Rn into
the direct sum of Vo = R2/° and V\ = R2*1. If x0 is a point in Mo, we will
denote by p a point in the fibre of the bundle Spin(M) at x0 such p restricted
to Vo is a frame of TXoMq C TXoM, while p restricted to Vi is a frame of AfXo.
Let 7 be the element of Spin(V) such that *yu = uj. The element 7 induces
an endomorphism of SXa. Since 7 lies above a transformation 71 € SO(A40)
such that det(l -71) / 0, we see that 7 belongs to C(Vi), and that its symbol
<72*jG) is non-zero.
We may identify the curvature F^ of the connection Vw with an element
of A2V®End(WXo). Lichnerowicz's formula (Theorem3.52) shows that D2 =
Aw®s + f°+f\ where F% = r^/4 and F* = F™. Thus, by Theorem6.18,
we obtain the following result.
Lemma 6.20. The section I(t, 7, <f>, xo) has the asymptotic expansion
/s
t%(a) p(exp(a)i) j9{a)da,
where $o(«) equals
Proposition 5.16 implies the first assertion of Theorem 6.12, as well as giv-
giving us the formula
a(fcG, <t>, xo)) = joBe A e*) *0Be A e*) A a2tl G).
Let us consider the decomposition V — Vo ffi Vi = R2*0 © R2'1. Denote by
eo A eg the antisymmetric n x n-matrix whose (i, j)-coefficient equals e* A e?
if i < 2@ and j < 21$, and such that all other coefficients vanish. If $ is
a function on g, we denote by $(eo A ej) its evaluation at this element of
0 ® A2^*. It is clear that $(eo A e^) depends only on the restriction of $ to
the subspace go = A2Vo of g = A2V.
6.7. Proof of Proposition 6.13
205
Since <T2*,G) e AMlVi, it is clear that
$Be A e*) A o2tl G) = $Be0 A ej
We now have the following results:
A) jBBe0Ae5) = l
B) Fw -(e0 A eS) equals the restriction to AP of the curvature Fw\
C) if R = il0©/*1 is the decomposition of the curvature of the bundle TM\Mo
of F.1), corresponding to the decomposition TM\m0 = TM0 © M, it
follows that for o e g0 = A2Vb,
jv(t(SIp-a)
It only remains to calculate QiBe0 A eg,71).
Lemma 6.21. For v G Vi and a e flo> we have
(<?i(a,7)«,«) = 2(»,Jv1(T(nJ.a)/2)-1(l --n)«).
Proo/. By Proposition 6.17, we have
«, Jv(p, a^-
For a e 00) the endomorphism Jv(Pi«) preserves the decomposition V =
Vq © Vi, and
Jv(p,a)|v,= Jv,(t(OJ-o)/2).
The lemma follows from the fact that Jv, (r(ft? ¦ o)/2) commutes with 71. ?
Let A = Jv,(r(nJ -a)l2)~l. The matrix .4 A - 71) is not symmetric, so
we must rewrite (Qi(a,i)v,v) in the form (Bv,v), where B is a symmetric
matrix. Thus, we write
We have A* = eT(n*"o)/2^, and hence
Thus for a G 0o, we find that
We now replace a by 2eo A ej. Using the fundamental symmetry of the
Riemannian curvature as in Lemma 5.17 we see that
which completes the proof of Proposition 6.12.

206
Bibliographic Notes
6. The Equivariant Index Theorem
The fixed point formula Theorem 6.16 for elliptic operators invariant un-
under a compact Lie group G was proved by Atiyah-Segal [12]; the proof uses
the Atiyah-Singer index theorem [14] and the localization result for the ring
KG(M). The Atiyah-Bott fixed point formula was proved in [5].
The generalization of the local index theorem to a fixed point formula
m the presence of a compact group action, was announced by Patodi [83]
m the case of the d-operator, and was proved by Donnelly-Patodi [491 for
the G-signature. It was first proved for the twisted G-signature operator
by Gilkey [59]. Our proof of Theorem 6.11 is taken from Berline-Vergne [241
which was inspired by Bismut [28], For a proof more in the style of Chapter 4
see the article of Lafferty-Yu-Zhang [71]. '
Chapter 7. Equivariant Differential Forms
Although it is rarely possible to calculate the integral of a differential form
exactly, it was a beautiful discovery of Bott that it is sometimes possible
to localize the calculation of such integrals to the zero set of a vector field
on the manifold. In this chapter, we will describe a generalization of this,
a localization formula for equivariant differential forms. Only the results of
Chapter 1 are a prerequisite to reading this chapter.
Let M be an n-dimensional manifold acted on by a Lie group G with
Lie algebra g. A G-equivariant differential form on M is defined to be a
polynomial map a : g —> A(M) such that a(gX) = g ¦ a(X) for jgG. The
equivariant exterior differential dg is the operator defined by the formula
where Xm is the vector field on M which generates the action of the one-
parameter group exptX. The space of equivariant differential forms with
this differential is a complex, With cohomology the equivariant cohomology
of M, denoted H^{M).
The condition that X i-+ a(X) is equivariantly closed means that for each
X e g, the homogeneous components of the differential form
satisfy the series of relations
These relations imply that a(X)[n\ is exact outside the set Mq of zeroes of
the vector field Xm, if the one-parameter group of diffeomorphism generated
by X is a circle. The localization formula then reduces the integral JM a(X)
to a certain integral over Mq.
Equivariantly closed differential forms arise naturally in a variety of sit-
situations. Bott's formulas for characteristic numbers, the exact stationary
phase formula of Duistermaat-Heckmann for a Hamiltonian action, Harish-
Chandra's formula for the Fourier transform of orbits of the coadjoint repre-
representation of a compact Lie group, and Chevalley's theorem on the structure
of the algebra of invariant functions on a semi-simple Lie algebra are all con-
consequences of the localization formula. These results will not be needed in the
rest of the book, but are included for their intrinsic interest.

208
7. Equivariant Differential Forms
We will also use equivariant differential forms to give a generalization of
the Chern-Weil map, thereby relating the equivariant cohomology of a G-
manifold M to the cohomology of fibre bundles with M as fibre. As an
illustration, we consider in Section 7 the simple case of a Euclidean vector
bundle with an action of the orthogonal group, which leads to the Mathai-
Quillen universal Thom form of a vector bundle.
7.1. Equivariant Characteristic Classes
Let M be a G°°-manifold with an action of a Lie group G, and let g be the
Lie algebra of G. The group G acts on C°°(M) by the formula (g ¦ <j>)(x) =
<f>(g~1x). For X 6 g, we denote by Xm (or sometimes simply X) the vector
field on M given by
e=0
The minus sign is there so that X >-+ Xm is a Lie algebra homomorphism.
On the other hand, if G acts on the right on a manifold P then we must write
de
e=0
Let C[jj] denote the algebra of complex valued polynomial functions on g.
We may view the tensor product C[{j] ® A(M) as the algebra of polynomial
maps from g to A(M). The group G acts on an element a e C[g] ® A(M)
by the formula
(g ¦ a)(X) = g (a(g-x ¦ X)) for all g 6 G and X 6 g.
Let Ag(M) = (C[g] ® A(M))G be the subalgebra of G-invariant elements;
an element a of Ag{M) satisfies the relation a(g ¦ X) = g ¦ a(X), and will be
called an equivariant differential form.
The algebra C[fl] ® A{M) has a Z-grading, defined by the formula
deg(P ® a) = 2 deg(P) + deg(a)
for P 6 C[fl] and a € .4(M). We define the equivariant exterior differential
d8 on C[fl] ® .4(M) by the formula
(dsa){X) = d(o(A")) - t(X)(a(X)),
where t(X) denotes contraction by the vector field Xm; from now on, we will
frequently write X instead of XM- Thus, dg increases by one the total degree
on C[g] ® A(M), and preserves Ag(M). The homotopy formula i(X)d +
dt{X) = C(X) (see A.4)) implies that
(d2.a)(X) = -L(X)a(X),
7.1. Equivariant Characteristic Classes
209
for any a € C[g] ® .4(M), hence {Aa{M),d3) is a complex. The elements
of Ag(M) such that dea = 0 are called equivariantly closed forms; those of
the form a = de0 are called equivariantly exact forms. This complex was
introduced by H. Cartan.
Definition 7.1. The equivariant cohomology HG(M) of M is the coho-
cohomology of the complex (Ag(M), dg).
We will also consider equivariant differential forms X t-+ a(X) which de-
depend smoothly on A" in a neighbourhood of 0 € fl, and not polynomially; the
algebra of all of these is written A'q (M). Although A"§ (M) is not a Z-graded
algebra, it has a differential rf9, defined by the same formula as before, which
is odd with respect to the Z2-grading.
If H —> G is a homomorphism of Lie groups, there is a pull-back map
Ag(M) —> Ah(M) on equivariant forms, defined using the restriction map
C[fl] —> C[l)]. This is a homomorphism of differential graded algebras; that
is, it sends exterior product to exterior product and equivariant exterior
differential dB to exterior differential dj,, and hence induces a map HG(M) —>
H'H(M).
It is clear that when H is the trivial group {1}, the equivariant cohomology
of M is the ordinary deRham cohomology. Thus, letting H = {1} and
pulling back by the inclusion of the identity {1} —> G, we obtain a map
Ag(M) —> A(M), given explicitly by evaluation a i-+ a@) at X = 0.
If <f>: N —> M is a map of G-manifolds which intertwines the actions of G,
then pull-back by <j> induces a homomorphism of differential graded algebras
(f>* : Ag(M) —+ Ag{N). In particular, if N is a G-invariant submanifold of
M, the restriction map maps Ag(M) to Ag(N).
If M is a compact oriented manifold, we can integrate equivariant differ-
differential forms over M, obtaining a map
L
: Ac(M) -»C[g]G,
by the formula (jMa)(X) = fMa(X); here, if a is a non-homogeneous
differential form, the integral fM a is understood to mean the integral of the
homogeneous component of top degree. If a is equivariantly exact, that is,
a — dsv for some v 6 Ag(M), and dim(M) = n, then
a(X)[n] = d(u(X)[n^),
so that fM a(X) — 0. Thus, if a is an equivariantly closed form, fM a is an
element of C[jj]g which only depends on the equivariant cohomology class of
a.
We will now associate to an equivariant superbundle certain equivariantly
closed forms whose classes in equivariantly cohomology will be called equivari-
equivariant characteristic classes. This construction follows closely the construction
of non-equivariant characteristic forms in Chapter 1.
Recall the definition of a G-equivariant vector bundle Definition 1.5.

210
7. Equivariant Differential Forms
Definition 7.2. If ? is a G-equivariant vector bundle, the space of equivari-
equivariant differential forms with values in ? is the space
Aa(M,?) = (C[g] ®A(M,?)f,
with Z-grading defined by a formula analogous to that on Ag(M).
A G-equivariant superbundle ? = ?+ ®?~ is a superbundle such that ?+
and ?~ are G-equivariant vector bundles. If A is a superconnection on a G-
equivariant superbundle ? which commutes with the action of G on A(M, ?),
we see that
for all X € g. We say that A is a G-invariant superconnection. If G is
compact, then every G-equivariant bundle admits an invariant connection,
constructed as in A.10).
Definition 7.3. The equivariant superconnection Ag corresponding to
a G-invariant superconnection A is the operator on C[g] ® A(M, ?) defined
by the formula
(Ata)(X) = (A - i(X))(a(X)), X 6 g,
where i(X) denotes the contraction operator o(Xm) on A(M, ?).
The justification for this definition is that
for all a 6 C[g]®A(M) and 0 e C{g]®A(M, ?). The operator Ag preserves the
subspace Ag{M, ?) C C[jj] <8> A(M, ?), and we will also denote its restriction
to this space by AB.
Bearing in mind the formula
we are motivated to define the equivariant curvature Fe of the equivariant
superconnection Ag by the formula
If 0 e Aa(M,End(?)), let A90 be the element of Ag(M,End(?)) such that
e(A,0) = [A,, ?(*)].
Proposition 7.4. The equivariant curvature Fe is in Aa(M, End(?)),
and satisfies the equivariant Bianchi formula
AaFa = 0.
7.1. Equivariant Characteristic Classes
211
Proof. To prove that Ft e Ag(M, End(?)), we must show that A,(XJ +
C?(X) commutes with multiplication by any a € C[{j] ® A{M):
[A8(XJ + C?(X),e(a(X))\ = \A9(X), [Ae(X),e(a(X))}} + {C?(X),e(a(X))\
= [AB(X),e(d8a(X))]+e(C(X)a(X))
= e{d\a{X) + C(X)a(X)) = 0.
The equivariant Bianchi identity is just another way of writing the obvious
identity [A,(X), A8(XJ + C?(X)\ = 0. ?
If we expand the definition of Fg, we see (identifying Fg with the operator
e(Fg) on ^(M,EndE))) that
where [t(Jf),A] = i(X)A + Al{X) is the supercommutator, and F = A2 is
the curvature of A. In particular, Fe@) = F. Motivated by this formula for
F9(X), we make the following definition.
Definition 7.5. The moment of X e g (relative to a superconnection A)
is the element of A+(M, End(?)) given by the formula
Observe that /x = Fe — F is an element of Ag(M,End(?)), so that
F,(X) = F + n(X).
The following formula is a restatement of the equivariant Bianchi identity,
G.1)
An(X) = t(X)F.
The use of the word moment is justified by the similarity between this equa-
equation and the definition of the moment of a Hamiltonian vector field in sym-
plectic geometry, as we will see later.
In the special case in which the invariant superconnection A on the equi-
equivariant bundle ? is a connection V, the above formulas take a simplified form.
Since [V,t(X)] = Vx, we see that the moment fi(X), given by the formula
n(x)-=c?(x)-vx,
lies in r(M,End(?)), and hence
H e (g* ® r(M,End(?)))G C ^(M,End(?)).
If the vector field Xm vanishes at a point xo 6 M, the endomorphism
fi(X)(xo) coincides with the infinitesimal action C(X)(xo) ofexptX in ?I0.
We can give a geometric interpretation of the moment n{X) in this case.
Denote by 7r : ? —¦ M the projection onto the base. The action of G on ?

212
7. Equivariant Differential Forms
determines vector fields Xg on the total space ? corresponding with X e fl.
The connection V on ? determines a splitting of the tangent space T?,
into a vertical bundle isomorphic to n*? and a horizontal bundle isomorphic
to n*TM.
Proposition 7.6. Let x be the tautological section of the bundle ir*? over
?. The vertical component of Xe may be identified with —fi(X)x.
Proof. This follows from Proposition 1.20, once we observe that C'e(X)x =
0. ?
We now construct the equivariant generalization of characteristic forms.
If f(z) is a polynomial in the indeterminate z, then f{F9) is an element of
Aq(M, End(?)). When we apply the supertrace map
Str : AUM,EM(?)) - A+(M),
we obtain an element of Aq(M), which we call an equivariant character-
characteristic form.
Theorem 7.7. The equivariant differential form Str(/(F8)) is equivariantly
closed, and its equivariant cohomology class is independent of the choice of
the G-invariant superconnection A.
Proof. If a € A?(M,End(f)), it follows from Lemma 1.42 that
dg Str(a) = Str(ABa).
The equation dB(Str(/(FB))) = 0 now follows from the equivariant Bianchi
identity ABF8 = 0. Similarly, if A' is a one-parameter family of G-invariant
superconnections, with equivariant curvature Ft, we have
so that the difference of the two equivariant characteristic forms
Str(/(FB1))-Str(/(F80))
is the equivariant coboundary of
As in the non-equivariant case discussed in Section 1.4, we can also allow
f(z) to be a power series with infinite radius of convergence. In particular, the
equivariant Chern character form chB(A) of an equivariant superbundle
7.1. Equivariant Characteristic Classes
213
? with superconnection A is the element of the analytic completion A@(M)
of Ag{M) defined by
G.2)
ch8(A) = Str(exp(-FB)),
in other words, chB(A)(X) = Str(exp(-Fe(X))).
In the case of a G-invariant connection V, the component of exterior degree
zero of the equivariant curvature Fe(X) = n(X) + F is fi(X), which satisfies
H(X) = O(\X\). Thus we can define f(Fe(X)) = f{fi{X) + F) even if /
is a germ of an analytic function at 0 € g. As an example, we have the
equivariant A-genus Ae(V) of a vector bundle ? over a compact manifold
M,
Finally, we define the equivariant Euler form. Let ? be a G-equivariant
oriented vector bundle with G-invariant metric and G-invariant connection
V compatible with the metric. The curvature F and the moment fi are both
elements of Ag{M,bo{?)). We define the equivariant Euler form of ? by
Xs(V)(X) = Pf(-F8p0) = det^-F.W),
where the notations are those of Definition 1.35. An argument similar to that
of Theorem 7.7 shows that Xb(^) is &11 equivariantly closed form, and that its
class in cohomology depends neither on the connection nor on the Euclidean
structure of ?, but only on the orientation of ?.
We close this section with two important examples of the equivariant mo-
moment map.
Example 7.8. Let us compute the moment nM{X) € r(M,so(TM)) of an
infinitesimal isometry X acting on the tangent bundle of a Riemannian mani-
manifold; this is called the Riemannian moment of M. The group G of isome-
tries of M preserves the Levi-Civita connection V on TM. By the definition
of the moment, we see that
and from the vanishing of the torsion we obtain
G.3)
nM{X)Y = -VyX,
in other words, fiM{X) = — VX. Since C(X) and Vx preserve the metric
on M, /J.M(X) is skew-symmetric. In this example, G.1) states that the
covariant derivative of the moment fiM is given by the formula
G.4) [VY,HM(X)] = R(X,Y),
where R is the Riemannian curvature of M.
The equivariant curvature of a Riemannian manifold M acted on isometri-
cally by a compact group G is defined by the formula Re(X) = R+/iM(X) ?
A2G(M,so(TM)).

214
7. Equivariant Differential Forms
Example 7.9. The next example of a moment map is due to Kostant and
Souriau; it is fundamental to the theory of geometric quantization. Let M
be a symplectic manifold with symplectic two-form ft, that is, ft ? A2(M)
is a closed two-form, dft = 0, such that the bilinear form QX(X, Y) on TXM
is non-degenerate for each x e M. If / 6 CCO(M), then the Hamiltonian
vector field generated by / is defined as the unique vector field Hj such
that
df = t(Hf)n.
Assume that a Lie group G acts on M and that the action is Hamiltonian:
this means that, for every X € g, there is given a function (i{X) on M such
that
A) fi(X) depends linearly on X;
B) the vector field Xm is the Hamiltonian vector field generated by n(X),
that is
C) n(X) is equivariant, that is, g ¦ p(X) = n(g • X) for jeC.
The symplectic moment map of the action is the C°° map y.: M
defined by (X,fi(m)) = n(X)(m). One sees easily that
is an equivariantly closed form on M such that ftfl@) = ft.
This equivariant differential form may in some cases be identified with i
times the equivariant curvature of an equivariant line bundle. Let C be a
complex line bundle on M. Suppose that ? carries a connection V? whose
curvature
(V?J = ift g A2(M)
equals i times the symplectic form ft on M. Furthermore, suppose that the
Lie group G acts on the manifold M and on the line bundle ?, in such a way
as to preserve the connection V?. It follows immediately that the symplectic
form ft is preserved by the action of G.
We may define the moment of this action, nc(X) = CC(X) — Vx, and the
equivariant curvature
In fact, nc equals •/—I times the moment /j defined above, since by G.1),
u(xu)n = dfic{X).
It follows that FB = ifto.
7.2. The Localization Formula
215
7.2. The Localization Formula
If M is a manifold acted on by a compact group G, there is a Riemannian
metric (¦, •) on M which is invariant under the action of G. For example,
such a metric may be formed by averaging any metric on M with respect to
the Haar measure of G. In this section, we will show how such a metric may
be used to study equivariant differential forms.
Proposition 7.10. Let G be a compact He group and X —+ a(X) be an
equivariantly closed differential form on M. If X eg, let Mo(X) be the set
of zeroes of the vector field Xm ¦ Then for each X € g, the differential form
a(X)[n\ is exact outside Mq(X).
Proof. Fix X 6 g and write dx = d — i(X). Let 0 be a differential form
on M such that C(X)8 = 0 and such that dx& is invertible outside Mq(X),
Such a differential form may be constructed using a O-invariant Riemannian
metric (•, •) on M. We define 0{?) = (XM,€) for ? 6 T(M,TM), then 9 is a
one-form on M invariant under the action of X such that dxO = |X|2 + d^,
which is clearly invertible outside the set Mo{X) = {|X|2 = 0}.
On the set M — Mq(X), we have
for every equivariantly closed differential form a on M. Indeed, since d\9 =
0, we have
and the result follows by taking the highest degree piece of each side. Q
Note that it is essential to assume that G is compact. For example, con-
consider M = S1 x S1 with coordinates x, y e R/2ttZ. Let X be the nowhere-
vanishing vector field (l + | sinx)9v, and let
a(X) = -^Gcosx + ain2x) + (l-4smx)dx A dy.
It is easy to verify that dxa(X) = 0. However, JMa(X) = BnJ, so that
a(X)[2j is not exact.
Proposition 7.10 strongly suggests that when G is a compact Lie group
and M is compact, the integral of an equivariantly closed form a(X) depends
only on the restriction of a(X) to Mq(X). In the rest of this section, we will
prove the localization formula, which expresses the integral JMa(X) of an
equivariantly closed differential form a as an integral over the set of zeroes
of the vector field Xm-
We will first state and prove the localization formula in the important
special case where Xm has isolated zeroes. Here, at each point p € Mo(X),
the Lie action C(X)? = [Xm,?{ on T(M,TM) gives rise to an invertible
transformation Lp of TPM. This1 can be proved using the exponential map

i. nxjuivariant uitterential Forms
with respect to a G-invariant Riemannian metric: if ? G TPM was annihilated
by Lp, all the points of the curve expp(s?) would be fixed by exp(tX). The
transformation Lp being the Lie derivative of an action of a compact Lie
group, it has only imaginary eigenvalues. Thus the dimension of M is even
and there exists an oriented basis e<, 1 < i < n, of TPM such that for
1 < i < I = n/2,
We have det(Lp) = A2Af... A|, and it is natural to take the following square
root (dependent only on the orientation of the manifold):
Theorem 7.11. Let G be a compact Lie group with Lie algebra g acting on a
compact oriented manifold M, and let a be an equivariantly closed differential
form on M. Let X 6 0 be such that Xm has only isolated zeroes. Then
<*(X)(p)
where ? — dim(M)/2, and by a(X)(p), we mean the value of the function
a(X)[Oj at the point p e M.
Proof. Let p € Mq(X). Using a G-invariant metric and the exponential
map, the vector field X can be linearized around p. Thus there exist local
coordinates X\,. ¦. ,xn around p such that Xm is the vector field
Let 0P be the one-form in a neighbourhood Up of p given by
0P = Af 1(x2dxi -
+ ... +
Xn-i - xn-idxn).
Then 0" is such that ?(XHP = 0 and 0P(XM) = Ei A = INI2- Using
a G-invariant partition of unity subordinate to the covering of M by the
G-invariant open sets Up and M — Mo{X) (which may be constructed by
averaging any partition of unity with respect to the action of G), we can
construct a one-form 9 such that C(X)9 = 0, dx9 is invertible outside M0(X),
and such that 9 coincides with 9P in a neighbourhood of p.
217
7.2. The Localization Formula
Consider the neighbourhood BP of p in M given by Bp = {x \ \\x\\2 < e}.
If Sp = {x | ||a;||2 = e}, then
/ a{X) = lim / a(X)
= lim / dl ,\')
(The sign change comes from exchanging the interior for the exterior orienta-
orientation of SP). Let us fix a point p. Near p, 9 = 9P. Rescaling the variable x by
a factor of e1^2, the sphere Se becomes the unit sphere Si, while 9{dx9)~l
being homogeneous of degree zero is invariant. Then
0 A a(X) _
0Aa?(X)
dx9 '
where aE(X)(x,dx) = Q(X)(e1/2a.j?i/2da.) When e -> 0, ae(X) tends to
the constant a(X)(p). To prove the proposition, it remains to compute
- / 0(dx9)-* = /
•/st Jsl
* = jB {deY
But
. . ~, —ji " "-*n'
Since the volume of the 2^-dimensional unit ball equals ne/?\, we obtain the
theorem. ?
We now turn to the localization formula in the general case. Fix an element
X e 0, and let Mo denote the set of zeroes of X.
Proposition 7.12. The set Mo of zeroes of the vector field Xm is a sub-
manifold of M, which may have several components of different dimension.
The normal bundle M of Mo in M is an even-dimensional orientable vector
bundle.
Proof. Fix a G-invariant Riemannian structure on M. If xq is a zero of Xm ,
the one parameter group exptX acts on TXoM. If a tangent vector at Xo is
fixed by exptX, then the geodesic tangent to it is contained in Mo, which
proves that Mg is a submanifold of M.
The Lie derivative Y *-* C(X)Y = [Xm, Y] is an endomorphism of AfXo =
TXoM/TXoMo which is invertible and antisymmetric. This implies that the
real vector bundle Af can be given a complex structure and is therefore even-
dimensional and orientable. Indeed, observe that the eigenvalues of C(X) in

218
7. Equivariant Differential Forms
A/Jo = Mxo ®r C come in pairs ±i\j, with A7 > 0. Therefore, we have a
decomposition
where A/!* is the sum of the eigenspaces such that Xj > 0. Q
Let rk(Af) : M) —* Z be the locally constant function on Mo which gives
the codimension of each component. Let go = {Y e g | [Y, X] = 0} be
the centralizer of X 6 g, and let Go be the connected Lie subgroup of G
with Lie algebra go- Then Go preserves the submanifold Mo and acts on
the normal bundle Af. We choose a Go-invariant Euclidean structure and a
Go-invariant metric connection V"^ on A/". The normal moment of Y € go is
the endomorphism f/^(Y) of AT such that
In particular, since the vector field Xm vanishes on Mo, we have
^[XM,C] for ?e AT,
and ft (X) is an invertible transformation of AT. Choose an orientation on N
and consider the equivariant Euler form X&>W) e A+{Mq). Its degree zero
piece equals det(-/ijVI/2, and therefore XeoW){Y) is invertible in A+{M0)
when Y is sufficiently close to X. We can now state the general localization
formula.
Theorem 7.13. Let G be a compact Lie group acting on a compact manifold
M. Let a be an equivariantly closed form on M. Let X € g, let Mo be the
zero set of the vector field XM and let AT be the normal bundle of Mo in
M. Choose an orientation on Af and impose the corresponding orientation
on Mo. Then for Y in the centralizer g0 of X in g and sufficiently close to
X, we have
f a(Y)=[ (a*)
JM JMo
where Xao{AP) i» the equivariant Euler form of the normal bundle. In partic-
particular,
[
m
a(X) =
The proof follows from several lemmas. As before, let 0 be the Go-invariant
one-form on M dual to the vector field Xm with respect to the G-invariant
Riemannian metric on M,
0{Z) = {Xu, Z) for Z e T{M, TM).
Then we have
{da0)(Y) = <W- (XM,YM), for Y € go.
7.2. The Localization Formula
Lemma 7.14. For allteR and Y 6 g0,
219
[ a(Y) = /
m Jm
Proof. We give two arguments, the second of which extends better to non-
compact manifolds. The first proof follows from the fact that etd'9 - 1 is
equivariantly exact: _ /fl(e^'-l)\
The other argument is analogous to the proof of the McKean-Singer for-
formula Theorem 3.50: we observe that
J* f eWH0a(r) = [ (
dt Jm Jm
= / d8(<?etd«8a)(r) = 0,
Jm
since a is equivariantly closed. Thus, we may set t = 0 without changing the
integral, whence the result. D
Thus, we must compute the limit when t —> 0 of
/ e(V)(n/'a(y)= /
Jm Jm
/ ^
Jm Jm t^o
To do this, we study the form 0 in the neighbourhood of Mo-
By orthogonal projection, the Levi-Civita connection V gives a connection
V^ on the normal bundle M which is compatible with the induced metric.
Identifying Mq with the zero section of A/", we obtain a canonical isomorphism
Consider the moment map fj.M{Y)Z = ~VZY of TM. Since nM{Y) com-
commutes with the operator C(X) on Mo, it preserves the decomposition
TM\Mo=TM0®Af;
thus, the restriction of nM(X) to AT coincides with the moment endomor-
endomorphism {^(Y). The endomorphism f*M{X) of M coincides with the infinites-
infinitesimal action C{X). The action of Go on the manifold M determines vec-
vector fields on Kf. The vector field X// is vertical and is given at the point
(z,y) e Mo x A4 by the vector -(iM(X)y € A4-
The connection V^ determines a splitting of the tangent space to the total
space Af into horizontal and vertical tangent spaces. We denote by (t>i,i>2)o
the Euclidean structure of the vector bundle A/". Consider the one-form 6q
on the total space Af given by
G.5) 0q(Z) = {Xjs,Zv)o
where Z\ is the vertical part of a vector field Z on AT.

220
7. Equivariant Differential Forms
To prove Theorem 7.13, we could proceed as in the proof of Theorem 7.11,
multiplying the one-form <?o by a cut-off function to obtain a one-form 0 which
coincides with 9q in a neighbourhood of Mo. However, we will see in the next
lemma that this is not necessary, since #o arises naturally as a scaling limit
of the one-form 9.
Let ip be a diffeomorphism of a neighbourhood U of the zero section in J\f
with a neighbourhood of Mo in M, such that
V»(x, 0) = x for x € Mo,
<ty|Mo=/ on TM\Mo = TM\Mo.
An example of such a map is the exponential map if>(x, y) = expx y, but the
exponential map is difficult to compute explicitly in most cases, so that it
is interesting to permit a different choice of rp. Transport the metric (•, •)
and the one-form 9 to U, by means of the diffeomorphism rft. Consider the
dilation 6t, t > 0, of M given by the formula
(>t(x,y) = (x, t1/2y), where x € Mo and y e Mx.
We assume that U is preserved by 6t for t < 1.
Lemma 7.15. A) d9(ZuZ2) = -2(fj,M(X)Z1,Z2) for Zu Z2 e T(M,TM)
B) lim f-^SlB = 0O
C) For Ye so, Bm t
t—>o
= (nM(X)y,^(Y)y)Q.
Proof. It is easy to calculate d9, using the explicit formula for the exterior
differential of a one-form: if Z\ and Z2 are vector fields on M, then
Z2) = Zx(X, Z2) - Z2{X, Zi) - (X, [Zi,Z,])
= -2(fiM(X)Z1,Z2).
Since /iM(X)Zi = 0 if Z\ is tangent to Mo, we see that d9(Zi, Z2) vanishes
if Z\ or Z2 is tangent to Mo. Consider local coordinates (x,y) on U such
that Mo is the set y = 0. Since 9 vanishes on Mo, as does i(Zi)d0 if Z\ is
tangential to Mo, we see that
ijk ij
for some smooth coefficients a,ijk(x,y) and bij(x,y). This shows that
ijk
7.2. The Localization Formula 221
has a limit when t —> 0:
J2Y,
t—>0 ijk ij
In order to obtain B), it remains to show that
G.6) dyi8(dyj)\y=0 = <V. W,)ly-o
and
G.7) fly,flyi«W.JIy=O = dyidyj90(dXk)\y=0.
Denote by 0 the one-form on Mo which represents the connection V in the
local basis dyj of the normal bundle M. Thus
By Proposition 1.20, we have
Since X vanishes on Mo, we have on Mo the formula
dyt (X, dyj) |y=0 = (Va,,. X, dyj) | Mo
which proves G.6).
Let us prove 7.7. We have
k)dyj)o - (nM(X)dyj,B(dXk)dyiH.
By G.4), we have
VYlVY,X = -VYl(,xM(X)Y2) = -nM(X)VYlY2 - R(X,Y!)Y3.
If Z is tangential to Md, so that fiM(X)Z and R(X, Yi) vanish, on Mo, we
see that
{VYiVy,X,Z)\Mo = (VYiY2,nM(X)Z)\Mo - (RiX^Y^Z)^ = 0.
It follows that
G.8) yiKaCY.ZJlM,, = (VYlX,VYlZ)\Mo + (VY2X,VYlZ)\Mo
We apply this with Z = dXk, Y\ = dyn and Y2 = dyy Using the relation
^dy.9Xk = Vgx dyi and the fact that nM(X) preserves the decomposition
TM*|M0 = TMq®N, we obtain
(v9yix,vdyjdXk)\Mo = -(ti
=-(nM{X)dyi,B(dXk)dyjH,

222
and hence
7. Equivariant Differential Forms
proving the second paxt of the lemma.
The vector field Xft is vertical and is given at the point (a:,y), x e M,
y € Afx, by the vector ~^(X)y € Mx. If Y € g0, the vector field YM
commutes with Xjy, and hence is tangent to Mo, so that the function (X, Y)
vanishes to second order on Mo. Equation G.8) shows that
YiY2(X,Y)\Ma =
which proves the third part of the lemma. D
Using the diffeomorphism ij), we can identify the neighbourhood U of the
zero section in jV with a neighbourhood U of Mo in M. Outside the neigh-
neighbourhood U, the function (X, X) is strictly positive, so we can find e > 0
such that if F € flo »s sufficiently close to X, (X, Y)y > e for all y ^ U. Since
d9 is nilpotent, a partition of unity argument now shows that
Urn /
t-taJM
= lim / i
~(X,Y)ltpdB/t
a{YL>,
if (j> is a compactly supported function on V equal to one on a neighbourhood
of Mo.
Let us consider a{YL> as a differential form on A/*. If we denote the one-
form a(Y) on Mo and its pull-back to M by the same notation, we see that
lim^ St(a(Y)(f>) — a(Y)\Ms>. Thus, it follows from Lemma7.15 that
lim «;(
t—+o
Furthermore, for Y near X and y in a neighbourhood of Mo, we have a
bound (X, F)(*,y) > c||y||2 with c > 0, so that by dominated convergence
and change of variables (x,y) t-» fat^y), we see that
lim
= / e
Mo-
Consider the differential form defined by integration over the fibre,
/ e (t* )y.M )y)oe o_
We have
Recall that V^ is invariant under C{X), so that
that d90 equals
= 0. We see
7.2. The Localization Formula
It follows that
223
The proof of the theorem is completed by the following lemma. Here, the
element xBo(A0(^) = detl/2(-(fiu(Y) + RU)) is an invertible element of
A(Mo), for Y sufficiently near X.
Lemma 7.16.
Proof. Choose a point xo € Mo and let V — MXo. Given a local oriented
orthonormal frame of A/" around xq, we may write V*^ = d + 0; choose the
frame in such a way that the connection one-form 0 vanishes at xo. We must
show that
f
Jv
where fj. = ^ e g* ® End(K) and R = R.% e A2T*IOM0 ® End(V).
It is a simple matter to evaluate this integral. First we apply the definition
of the Pfaffian, which shows that
I
/
Jv
A ... A dyn.
Recall the formula for a Gaussian integral: if A is a positive-definite en-
domorphism, then
Since ft(X) and fi(Y) + R are antisymmetric and commute, the matrix
is symmetric. Its zero-degree component — n(X)n(Y) is positive definite when
X = V, and hence for all Y sufficiently close to X. Hence, we obtain
f
Jv
A . L
n = w"/2 det(-»(X)(fi(Y) + R))-1'2
1/2MX))det-1/2MK) + R). D

224 : 7. Equivariant Differential Forms
7.3. Bott's Formulas for Characteristic Numbers
In the next three sections, we will give some applications of the localization
formula. The first of these predates this formula, and is due to Bott.
If $ e C[so(n)]°(n) is an invariant polynomial function on the Lie algebra
Bo(n), where n = 2? is even, then $ is uniquely determined by its restriction
to the Cartan subalgebra t C so{n) of matrices X of the form
Xe2i = -Xie2i-\.
Let pk be the A;-th elementary symmetric polynomials in x?, given by the
formula
¦••*
?*•
<ik<t
(In particular, pt = det(X).) If * is O(n)-invariant, the restriction of $ to
t is a symmetric function of the variables x?, so belongs to the polynomial
ring generated by pu 1 < i < (., that is, C[ao(n)]°<n) = C[pi,... ,pt}.
The function detl^2(X) = xi...xt = pt is only invariant under the
group SO(n), and is the additional generator of the ring of SO(n)-invariant
polynomials over the ring of O(n)-invariant polynomials; it is nothing but
the classical Pfaffian function P{(X).
Let M be a compact oriented Riemannian manifold of dimension n =
2?, and let R be the curvature of a connection on the tangent bundle TM
compatible with the metric. The Chern-Weil map defines a homomorphism
d>: C[So(n)]so<") = C[pi,... ,pi,p\12} - A(M),
by the formula $($) = $(R), and $(i?) is a closed form on M whose coho-
mology class is independent of the choice of connection. Define
Jm
The number $(M) is called a characteristic number of M; it is a classical
theorem of algebraic topology that if $ € Z[pi,p/ ], $(Af) is an integer.
Suppose that M admits a circular symmetry with isolated fixed points.
At each fixed point p, the action of the infinitesimal generator X of the circle
group G = {e27™*} on the tangent space TPM determines an endomorphism
Lp, given in an appropriate oriented basis by
where Aj are integers called the exponents of the action at p. Recall that by
definition det1' 2(LP) = Ai... A/, and depends on the orientation of TPM but
not on its metric.
7.3. Bott's Formulas for Characteristic Numbers
225
Theorem 7.17 (Bott). Let M be a manifold on which the circle acts with
isolated fixed points. If $ is a homogeneous polynomial of degree k<t,
_ ($(M)
k<?.
Proof. Identify the Lie algebra of G with {uX | u e R}. Choose a G-invariant
metric, and let ufiM(X) + R be the equivariant curvature of the associated
Levi-Civita connection. If $ is a polynomial, u —> $(u(iM (X) + R) is an
equivariantly closed form on M, and we see from Theorem 7.11 that
=u-1 y
(-27T)-
The left hand side of this equation is a polynomial in u with value at zero
the characteristic number $(Af), while the right hand side is a Laurent poly-
polynomial in u. The equality of the two sides implies remarkable cancellations
properties over fixed points of the values of $(?p) if deg($) < (, while the
equality of the constant term of this Laurent polynomial gives the formula
when deg($) = L ?
When applied to the Pfaffian function
= Vi{X) on so(n), the char-
char() {) (),
acteristic number (—2n)~l fM Pi(R) is (—1)' times the Euler characteristic
of M, by the Gauss-Bonnet-Chern theorem, and we see that the number of
fixed points of the circle action is the Euler characteristic of M. Since the
vector field X generating the circle action is an infinitesimal isometry, it al-
always has index u(p, X) = 1 at an isolated zero X(p) = 0. Thus, this result
is a very particular case of the Poincare-Hopf theorem, Theorem 1.58.
The general localization theorem allows us to generalize the above formula
to situations in which Mq{X) is not zero-dimensional; this extension was
made by Baum and Cheeger. Let us merely mention the case of the Pffafian,
in which we obtain the following result.
Proposition 7.18. Let M be a compact oriented even-dimensional Rieman-
Riemannian manifold, and let X be an isometry of M with fixed point set Mo- Then
the Euler characteristic of M equals the Euler characteristic of Mo.
Proof. Observe that
where R° is the Riemannian curvature of Mo- By the localization theorem,
we see that
/ det1'*{-Ra(X))= f Bnrdi
Setting X = 0 and applying the Gauss-Bonnet-Chern theorem to both sides,
the result follows. ?

226
7. Equivariant Differential Forms
7.4. Exact Stationary Phase Approximation
One important application of the localization formula is the "exact stationary
phase approximation" of Duistermaat-Heckmann.
Let M be a compact manifold of dimension n = 2?, f a smooth function
on M and let dx be a smooth density on M. Let teR and consider the
function
F(t) = / eitfdx.
Jm
The major contribution to the value of this integral when t tends to infinity
arises from the neighbourhood of stationary points, that is, points where
the differential df of the phase function / vanishes. Assume that the phase
function / is non degenerate, which means the set Mo where the differential
of / vanishes consists in a finite number of points and at these points the
Hessian Hp = Vpdf of / is a non-degenerate quadratic form on TPM. Let
ff(Hp) be the signature of the quadratic form Hp. If TpM = T+ © T~ is
an orthogonal splitting for Hp such that Hp is positive definite on T+ and
negative definite on T~ then a(Hp) - dim(T+) - dim(r~).
If ei is a basis of TPM such that (|dar|p, ei A ... A en) = 1, let
It is not hard to show that when t tends to infinity,
Duistermaat and Heckman discovered a class of examples of "exact stationary
phase approximation", where the error term in the above formula vanishes.
Let (M, Q) be a compact symplectic manifold of dimension n, and let G be
a compact group of Hamiltonian transformations of M. For X 6 g, let ft(X)
be the symplectic moment of X, defined in G.9). Since dft(X) = t(Xjv/)ft,
the set of points where the one-form dfi(X) vanishes coincides with the zero
set M0(X) of the vector field XM. For p € M0(X), we denote by CP(X) the
infinitesimal action of X on TPM. The Liouville form of the symplectic
manifold M is the form
this is a volume form which defines a canonical orientation on M.
Theorem 7.19 (Duistermaat-Heckman). Let (M,fl) be a compact sym-
symplectic manifold, with compact Hamiltonian symmetry group G. If X is an
7.4. Exact Stationary Phase Approximation
227
element of the Lie algebra ofG such that Mq(X) consists of a finite number
of points, then
Jm
:r(p)
(The square root of det(?p(X)) is computed vrith respect to the canonical
orientation ofTpM.)
Proof. The equivariant symplectic form ilB (X) = fi(X)+Q is an equivariantly
closed form on M, as is
e»n,(X) =ein(x)em
The theorem follows from the application of the localization theorem, Theo-
Theorem 7.11 to the integral
/ fWdp == Bjt*)-< / ein.W. ?
Jm Jm
Corollary 7.20. Let X be a Hamiltonian vector field with Hamiltonian f
on a compact symplectic manifold M, such that the flow generated by X is
periodic, and X has discrete zeroes. Then the error term in the stationary-
phase approximation to the integral fM eltfd0 vanishes:
f eufd/3
Proof. We need the following lemma.
Lemma 7.21. The Hessian HP{Y, Z) of the function f at the point p ?
Mq(X) is given by the formula
Proof. Since ?(Z)f = t(Z)i(X)n, we have
t(Z)i(C(Y)X)Q + l{Z
Since X vanishes at p, we obtain HP(Y,Z) — — il([X, Y],Z), proving the
lemma. ?
By the lemma, we see that
Thus, we must prove that for each critical point p of /,
G.9) sgn(det1/2(?p(Jf))) = j-^V"^/4
Let (V, Q) be a symplectic vector space, and let A be an invertible semisim-
ple endomorphism of V with purely imaginary eigenvalues. The quadratic

228
7. Equivariant Differential Forms
form H(v, w) = —il(Av, w) is non-degenerate, and A is a skew-adjoint endo-
morphism of V with respect to H. Choose a basis for V of vectors pt and
9», 1 < * < ^i such that Sl(pi,qj) = % O(,Pi,Pj) — ?l(qi,qj) - 0, Apt = Ajft,
and Aqt = —AjPj. We see that
while
It is now easy to check G.9). ?
Of course, using the general localization formula, it is possible to remove
the condition in the above theorem that Mo(X) is zero-dimensional, at the
cost of introducing the curvature of the normal bundle to Mq(X).
7.5. The Fourier Transform of Coadjoint Orbits
Let G be a Lie group with Lie algebra g, and let g* be the dual vector space
to the vector space g. The group G acts on g by the adjoint action, and
the dual action of G on g* is called the coadjoint action. One of the most
important examples of a Hamiltonian action arises in this situation.
Let M be an orbit of the coadjoint representation; thus, for some / G
g*, M = G-f. The vector fields Xm, X G g, are sections of TM and
span the tangent space at each point, and we have (Xm)/ = —X ¦ f, where
-(X-f)[Y) = (f,[X,Y}). Let ii(X) be the restriction to M of the linear
form / i-+ f(X) on g*.
Lemma 7.22. The form il(XM,YM)f = -(/, [X,Y]) defines a G-invariant
symplectic form on M. Furthermore the action of G on M is Hamiltonian
and the symplectic moment of X is the function
Proof. The form fi is clearly non-degenerate and G-invariant. The function
fi(X) being the restriction of a linear function obviously satisfies
dn{X)(YM)} = (YM)f(X) = -(/, [X,Y]),
and hence dfi(X) = l(Xm)^, which shows that ^i(X) is the Hamiltonian for
the vector field Xm. Furthermore, it follows that
0 = d2n(X) = di(XM)Q. = -i(XM)dil,
where in the last equality, we used the invariance of 0, C{Xm)Q — 0. Since
the vector fields Xm span the tangent space, we see that dii = 0. D
7.5. The Fourier Transform of Coadjoint Orbits
229
Let rf/8 be the Liouville measure on M. By the Fourier transform of an
orbit, we mean the integral
FM(X) = f e'
Jm
this is a generalized function on g if the orbit M is sufficiently well-behaved.
Such integrals are very important in representation theory. The above lemma
shows that such a Fourier transform is the integral of an equivariantly closed
differential form, since
FM(X) =
/
m
When G is a compact Lie group, the localization theorem will lead to a
formula for the Fourier transform of the Liouville measure of M, due to
Harish-Chandra. This formula in turn implies Chevalley's theorem on the
structure of C[g]G. To state Harish-Chandra's formula precisely, we first
recall a few results on the structure of compact Lie groups.
Let gc be a reductive Lie algebra over C; in other words,
0c=3©(flc,flc],
where 3 is the centre of gc and [gc, gc] is semisimple. Consider the collection
of all abelian subalgebras 0 of gc such that the transformations
{ad A" I X e a} C End(gc)
are simultaneously diagonalizable; a maximal subalgebra among this set is
called a Cartan subalgebra, and as is well known, any two Cartan subalgebras
are conjugate under the action of the adjoint group Ad(gc).
Choose a Cartan subalgebra h of gc- If a € h*, define
(flc)a = {X € gc I [H, X] = (a, H)X}, for all H € f).
If q 7^ 0 and (gc)a 7^ 0, a is called a root of f) in gc- The set of roots of fj in
gc is denoted by A(gc, f)), or A when h is fixed. If a 6 A, then dim(gc)a = 1>
-a G A and dim[(gc)a,(gc)-a] = 1- Furthermore, [(gc)Q, (gc)-a] C fj, and
there exists a unique element Ha G [(gc)a, (gc)-a] such that (a,Ha) = 2.
For q € A, we may consider the reflection of f) about the plane orthogonal
to Ha, sa(H) = H — {a,H)Ha, which satisfies s\ = 1. The subgroup
W = VF(gc,h) of transformations of f) generated by the reflections sa is a
finite group, called the Weyl group. We denote by e(w) = ±1 the determinant
of the transformation w G W. *
The Killing form B(X,Y) = Tr(adXadF) is positive definite on the
real span ?aR#Q. The vector Ha = 2B(Ha,Ha)-1Ha G h is the unique
element of I) such that for all H in h, B(Ha,H) = (a, H).

230
7. Equivariant Differential Forms
A positive System P of roots is a subset of A such that for a € A, either
a or —a, but not both, belongs to P, and such that if a ? P, 0 ? P and
a + 0eA then a + 0eP. We define
A simple root with respect to a fixed positive system P is a root which cannot
be written as the sum of two elements of P. The set {sa | a simple} is a set
of generators for W. If w € W, we denote by |w| the length of w with respect
to this set of generators.
A compact connected abelian Lie group T is called a torus; it is the quo-
quotient of its Lie algebra t by a lattice. If T acts on a finite dimensional real
vector space V, then V = Vo © /C*et*\{o} ^[*1> w^ere Vo is the subspace of
V fixed by T and Vjtj is an even-dimensional real vector space on which the
spectrum of the action of t is ±ik.
If G is a compact connected Lie group, every element of G belongs to
a torus and any two maximal tori of G are conjugate under the adjoint
action of G. The complexification gc of the Lie algebra g of G is a complex
reductive Lie algebra. If T is a maximal torus in G with Lie algebra t, we
have g = t © r, where t = [t, g], and thus g* = t* © t*, and tc is a Cartan
subalgebra of gc- Every element of g is conjugate to an element of t by the
action of an element of G, and every element of g* is conjugate to an element
A ? t*. Let A = A(gc,tc). Roots a € A take imaginary values on t, and
iHa ? t for all a € A. The group T acts on r and we write t[<Qj for the two-
dimensional subspace of t on which X € t acts by the infinitesimal rotation
of angle ia(X).
If T is a maximal torus in a compact Lie group G, let
be the normalizer of T in G. The group N(T) acts on t by the restriction of
the adjoint action. Since T acts trivially on t, this defines an action of the
quotient group W(G, T) = N(T)/T, which can be identified with the Weyl
group W(gc, tc)- We will denote it by W.
For a compact Lie group G, every coadjoint orbit is of the form M\ = G • X,
with A € t*. We will give an explicit formula for the Liouville measure of an
orbit M\ and for its Fourier transform Fx(X). For A 6 t*, let Px be the set
of roots
Px = {aeA\(X,iHa)>0},
and let t> = YlaePx r[»aJ- The OI"bit M\ may be identified with the homo-
homogeneous space G/G(A), where G(X) is the stabiliser of A. The space tA is
isomorphic to the tangent space Te(G/G(X)) at the base point e = G(X).
In particular, if (A, iHa) =? 0 for every a 6 A, then G(X) = T and tA = t.
Such a point A is called regular and its orbit has maximal dimension. The
restriction of minus the Killing form to tA is positive definite and determines
a G(A)-invariant inner product on Te(G/G(X)). Thus the homogeneous space
7.5. The Fourier Transform of Coadjoint Orbits
231
G/G(A) is a Riemannian manifold, with a G-invariant metric which coincides
with —5|tA at the point e. Denote by dg the corresponding G-invariant mea-
measure on G/G(A). This measure depends only on the subgroup G(A), which
varies over a finite number of subgroups of G.
Lemma 7.23. The Liouville measure d/3x of Mx is given by the formula
4>(gX) dg.
G/G(X)
Proof. We can choose X±a such that {ea, /„} is an orthonormal basis of t[iaj,
where
Xa — X_<> , , iXa + iX-a
Since ?l\(fa • X,ea ¦ A) = iX(Ha), we easily obtain the lemma. ?
The Fourier transform of the measure d0\ is a G-invariant analytic func-
function F\(X) on g, which is clearly determined by its restriction to t.
Theorem 7.24 (Harish-Chandra). Given X 6 t*, let Wx = {w e W \
wX = A} be the stabilizer of A in the Weyl group W. For X e t, X regular,
the Fourier transform F\(X) = jM e'^x^d/3\(f) is given by the formula
Proof. It is sufficient to prove this formula for a regular element X of t. In
this case the zero set of the vector field generated by the action of exp tX on
g* is the subspace t* of g* which is fixed by the whole of T. Thus the zero
set of the vector field Xmx is the finite set
Ma n f = {wX | w e W/Wx}.
Applying Theorem 7.11, we see that it suffices to compute det1'2(?u,A(^))-
The element IIaepA fa A ea determines the orientation of tA = TeMx- Since
Cx(X)fa = ia(X)ea, we find that det^CxiX)) = Ua€Px (MX)). D
It X is regular, Pa is "half of; A, and up to a sign, the denominator
detlJ^(C(X))= H(wa,iX)
a€P\
is the same at each fixed point. Thus we obtain the following corollary.
Corollary 7.25 (Harish-Chandra). If Mx is a regular orbit of the coad-
coadjoint representation, then for X 6 t, X regular,

232
7. Equivariant Differential Forms
When A is regular, we can compute the volume vol(M\) of the symplectic
manifold (M,(l\) as the limit when X tends to 0 of Fmx(X).
Proposition 7.26. Let A be regular and A = iX. Then
L>o (
Proof. We will also write p for the element of tc dual to p e t*, which is half
the sum of the roots in the positive system Pa C A. The volume of M\ is
the limit when t tends to 0 of FM,,(tp). Since
w€W
a>0
tends to ria>o(a> A)i wnen t —> 0, we obtain the proposition. ?
In particular, for iX = p the symplectic volume is 1. Using Lemma 7.23,
which relates the symplectic volume and. the Riemannian volume, we obtain
the following result.
Corollary 7.27. The Riemannian volume of the flag manifold G/T equals
II 1
Using the above results, we will now analyse the structure of the algebra
of G-invariant functions on g. If j : t —» g is the inclusion map, then we
obtain a restriction map
j' : C°°(g)G - C°°(l)w.
There are similar maps where the space of functions C°°(g) is replaced by
analytic functions Cu(g) or polynomials C[fl]. We will prove that the restric-
restriction map is an isomorphism in all of these cases, by constructing an explicit
inverse for it.
Choose a system P C A of positive roots, and define
Let us denote by du the constant coefficient differential operator on t, given
by
p€P
where dtfy is differentiation in the direction of the vector iHa.
Let pr be the projection from g = t©t to t. Let 7rB/t be the real polynomial
on t
7.5. The Fourier Transform of Coadjoint Orbits
233
All integrals over G are with respect to the Haar measure of volume one.
Theorem 7.28 (Chevalley). If <j> e C°°(i), let c{<j>) be the function on g
defined by the formula
Then c is an isomorphism of C°°(t)w, Cu{i)w and ?{i*]w to C°°(g)G,
Cu'(g)G and C[fl]G, and is the inverse of the restriction map j*.
Proof. The theorem follows from the following formula: if X is a regular
element of t, then
To see that j*c((f>) = <j> if <j> is invariant under the Weyl group, we apply this
formula to the function wa/t^>
It suffices to prove G.10) for <j>(X) = e^x>x\ A € t* regular. Indeed, this
proves it by continuity for all A, hence for all <j> € C?°(t), and finally, since G
is compact, for all <j> G C°°(t).
For the function <j>(X) = e'<A-*>, (du<t>)(X) = w(iA)e<<A'X> and the left-
hand side of G.10) becomes
Pi^,gX)
dg.
If A is regular, this is equal to Fmx (X) up to a constant independent of X. To
evaluate this constant, we compare both sides with X = 0; Proposition 7.26
shows that
L
Applying Corollary 7.25, we obtain G.10). ?
When G is a non-compact Lie group and M is a coadjoint orbit of G,
the Fourier transform Fm(X) = fMel^x^d0 may often still be defined as
a generalized function on g. If K is a compact subgroup of G, the action
of K on M is the Hamiltonian action of a compact Lie group, and so, by
Proposition 7.10, for X 6 t the form ex^x^dj3 is exact outside the zeroes of
Xm . Thus we can hope that when Fm admits a restriction as a generalized
function on the Lie algebra t of K, the localization formula will allow us to
compute this restriction Fm\i- We state here a result of this type, where
G is a real semisimple Lie group with maximal compact subgroup K. This
result can be proved using Stokes's theorem and estimates at infinity on the
noncompact manifold M.
Let G be a connected real semisimple Lie groug with Lie algebra g, and
let A' be a maximal compact subgroup of G with Lie algebra t. Let g = t ffi p
be the Cartan decomposition of g, and let T be a maximal torus of K with

234
7. Bquivariant Differential Forms
Lie algebra t: If the Cartan subalgebra tc of tc is also a Cartan subalgebra
of gc, we say that G and K have equal rank. We have then g = t© t, where
t = [t, g] and we identify t* with a subset of g*.
Let A = A(gc,tc), and Ap = {a | (flc)a C pc}; a root belonging to Ap is
called noncompact. For A e t", let P\ = {a 6 A | (A,i/?a) > 0}, and let n\
be the number of non-compact roots contained in Pa- K
lr = {if ? 11 {a,H} ^ 0 for all a € A}
is the set of regular elements in t, then G-tr is an open set of g. The
function F\f (X) is a generalized function on g and is analytic on G-tr. The
restriction of Fm to G-tr is thus determined by its restriction to V. When
M is of maximal dimension, the following formula is due to Rossmann.
Theorem 7.29 (Rossman). Let M be a closed orbit of the coadjoint rep-
representation of a real semisimple Lie group G such that G and K have equal
rank. Let W = W(tc, tc) be the compact Weyl group. Then for X e t,., we
have the following results:
A) IfM D f = 0, then FM(X) = 0.
B) If M = G\ with A 6 t*, and W\ is the subgroup of W stabilizing A,
wew/Wx
7.6. Equivariant Cohomology and Families
In this section, we will define the equivariant Chern-Weil homomorphism,
which maps the equivariant deRham complex of a G-manifold M to the
ordinary deRham complex of a fibre bundle with fibre M and structure
group G, which we do not need to assume is compact. This is an extension
of the theory of characteristic classes in Section 1.5, although here we only
consider the case of connections, and not of superconnections.
Let P —» B be a principal bundle with structure group G, with connection
one-form ui € A1(P,g)a and curvature Q = dui+\\uj,ui\. The decomposition
TP = HP © VP of the tangent bundle TP into a horizontal subbundle
HP = ker(u;) and a vertical subbundle VP determines a projection operator
h from A(P) onto the subalgebra of horizontal forms
A(P)bol = {a | i(X)a = 0 for all X 6 g}.
With respect to a basis Xit 1 < i < m, of g, we may write
m m
w = y u>'Xj and li = j il'Xj,
where lj1 are one-forms on P, and ft' are horizontal two-forms.
7.6. Equivariant Cohomology and Families
235
Lemma 7.30. The projection h onto the algebra -4(P)hor of horizontal forms
is given by the explicit formula
Proof If Pi = I — iv'i
), the relations
XiUiXj)] = 0 and
imply that t(Xi)pi = \pi,Pj] = 0 and p? = pt. It is also clear that Pi is
an algebra homomorphism and that (YliLiPi)<*>k = 0. Thus ITZLiP> 's *ne
projection h on the space of horizontal forms. Q
Let D denote the operator h-d-h on A(P)\ under the identification of
A{P)\,a* with A(B), the restriction of D to A{P)baa corresponds to d, since
h acts as the identity on -4(P)hor and d preserves .4(P)bas-
Lemma 7.31. A) D = h(d- ??i ni>-(xi))' and
B) ?>2 + /l-(EfiiA^))=0
Proof. Since d is a derivation and hw* = 0, hdw' = fi', we have for a € -4(P),
hd-h{a) =
To calculate D2, we observe that
since
= 0- ?
The following result shows how the covariant derivative Vv on an associ-
associated vector bundle V = P x<j V is related to the horizontal projection.

236
7. Equivariant Differential Forms
Proposition 7.32. The covariant derivative Vv on V coincides with the
restriction of the operator D®I on A(P, V) to A(B, V) = A(P, V)bSLB.
Proof. If a e A(P, V)^, we have u{X)da = {C(X) ® I)a = -p(X)a. Hence
t(Xi)i(Xj)da = 0 and
h{dha) = h(da) =
= Vva.
If M is a G-manifold, let M = P xG M be the associated fibre bundle,
with base B and typical fibre M. Since M is the quotient of P x M by a free
action of G, we may identify A(M) with the space A(P x M)bM of forms a
on P x Af which are basic with respect to the action of G. The form a; is a
connection form for the action of G oaPx M. Thus, if we write simply l(X)
instead of t{XPxM) = l{Xp) + l{Xm), the projection h onto the algebra of
G-horizontal forms A(P x M)hor is given by the formula
and the operator D = hdh restricts to dM on the space A(M) = A(P x
M)bas.
Consider the differential graded algebra (C[g] ®A{M), d3) of A(M)-valued
polynomial functions on 0. For a = / <g> 0 e C[g] ® .4(M), we define a(ft) G
.4(P) ® .4(M) by a(«) = (
Definition 7.33. The map </>u : C[g] ® .4(M) -> ,4(P x M)hor defined by
is called the Chern-Weil homomorphism.
Observe that the restriction of <t>w to Ac(M) - (C[g] ® A{M))° sends
Aa(M) into >t(X) =* A(P x M)bas.
Theorem 7.34. Let G be a Lie group and let P —* B be a principal bun-
bundle with structure group G and connection form u>. Let M be a manifold
with smooth action of G. Then the Chern- Weil homomorphism induces a
homomorphism of differential graded algebras
+a : (AG(M),d9) = ((Cfe] ® A(M)f,dB) - (A(M),d)
Proof. Since 4>w clearly preserves products, we have only to show that it
intertwines the differential d9 on Ag(M) and the differential d on A(M). In
fact, we will prove that the map <?w : C[flj ®A{M) —» (A(P x M))hor satisfies
the formula
D-(f>u = 4>w 'de.
7.6. Equivariant Cohomology and Families
If / ® a e C[fl] ® A(M), we see by Lemma7.31 that
237
<g> a))
since
= 0 and
= 0,
Note that when M is a point, the above Chern-Weil homomorphism be-
becomes a map
4>u : C[0]G - .4(B),
which is just the ordinary Chern-Weil homomorphism. In the general case,
the map <j>u still has a geometric interpretation. The connection on P deter-
determines a horizontal subbundle HM of the tangent bundle TM, where HM
is defined as the image of HP under the projection P x M —> M. We have
TM = HM © VM, where VM is the vertical tangent bundle. Thus, if
(p,m) g P x M projects to y = \p,m) 6 M, we have the isomorphism
which induces an isomorphism between AH*P <8> AT^M and AT* M.
If / ® /3 G C[g] ® A(M), then substituting Q for X € 0 in /, we obtain
a horizontal form f{il)p € AH'P, while /3m € AT^M may be considered
as a vertical form on P x M. If we define (/ ® 0)(il) to be f(il) ® /3, then
by linearity we can extend this to define a(fi) for any a € C[g] ® ,4(M). It
is easy to see that if a is G-invariant, then j(p,m)(a(^)(p,m)) depends only
on y € M- Thus, we obtain a differential form on M which corresponds to
h(a(fl)), since h(a(il)) is a basic form on P x M whose restriction to the
subspace AHP ® ATM coincides with that of a(ft).
The Chern-Weil homomorphism has the following functorial properties.
A morphism / : AT —> M of G-manifolds gives rise to a G-invariant map of
fibre bundles /p : M = P Xq N —> M = P xq M, and to the commutative
diagram
Ae(M)
A{M)

238
7. Equivariant Differential Forms
Proposition 7.35. Let G be a Lie group, and let M be a compact oriented
manifold with smooth action of G. We obtain the commutative diagram
At(M)
i
C[8]G
A(M)
I
A(B)
where the left vertical arrow is the integration over M and the right vertical
arrow is the integration over the fibres of M —» B.
We will now relate the equivariant curvature with the ordinary curvature.
Let E = E+ © E~ be a G-equivariant superbundle on M, with invariant
superconnection A. Let FE = A2 e A(M, End(-E)) be the curvature of A
and let nE be its moment. Consider the associated family of vector bundles
? = P xG E -> M = P xG M.
Then A(M, ?) may be identified with the space of basic forms A(PxM,
where we consider E as a vector bundle on P x M. The operator
is a projection from A(PxM, E) onto A(PxM, E)hor. Consider the operator
D? = hE -{dp® 1 + 1® A)- hE
on A(P x M,E), where dp <g> 1 + 1 ® A is the pull-back of the supercon-
superconnection A on P x M. Clearly D? preserves A(M,?) and induces on it a
superconnection.
Definition 7.36. The superconnection Ae on A(M, ?) is the restriction of
the operator D? to A(P x M,?)ba8.
Let us compute the curvature F? = (Af J of the superconnection A?. The
operator D? on A(P x M, ?) satisfies
D?(as) = (Da)s + (-l)^h(a)D?s
for all a e A{P x M) and s ? A(P x M, ?), where D = h ¦ dPxM- h. By the
formula D2 + hC^Li ^'A-X*)) = 0 of Lemma 7.31, we see that the operator
(D?f + hE -JTWCiXi)
commutes with exterior multiplication by any horizontal differential form,
and hence equals hE ¦ F?. We will now compute F? as a function of the
curvature FE of A and the moment nE.
Lemma 7.37. A) D? = hB ¦ (dP + A - YT=i ft*4*»))
7.6. Equivariant Cohomology and Families
B) The curvature F? equals FE + YZLi n>B(xi)> and
239
Proof. The proof of this lemma is almost the same as that of Lemma 7.31,
except that the operator d = dp + dM on A(P x M) is replaced by dP + A
on A(P x M, E). Since A is a superconnection and h(u>i) — 0,
D? = hEidP + A)hE
FVom this, it follows that
{D?f = hE{dP + h-
Using the relation
[dp + A, L(Xi)} = CE(X{) -
we see that
[dp + a, ah(Xi)} = drt ¦ i(Xi)
It follows that
+ C{(Xi)P),
and hence that
hE -
)) = hE-(
-(FE
since h(d{l{) = 0. D
Recall that the equivariant superconnection Ag is the operator on C[g] ®
A{M,E) defined by
(A9a)(X) = A(a(X)) - i(X)(a(X))
and that the operator A8(XJ + ?(X) is given by exterior multiplication by
the equivariant curvature Fg{X) = FE + fiE(X).
Theorem 7.38. Let G be a Lie group and let P -+ B be a principal bundle
with structure group G and connection form u>. Let M be a manifold with
smooth action of G, and let E —» M be a G-equivariant bundle on M with
G-invariant superconnection A.
A) The map
<t>u : C[0] ® A(M, E) - A{P x M, E)hor
given by <j>u(a) = hE(a(il)) satisfies the formula <j>u ¦ Ag = D? • <j>u.

240
7. Equivariant Differential Forms
B) // we also denote by <t>u the map
> A(P x M, End(S))hor,
4>u : C[g] ® A(M, End(E))
then 4>U(F9) = F?.
C) Let chB(A) be the equivariant Chern character of the G -equivariant super-
bundle E —> M, with G-invariant superconnection A, and let ch(Af) =
Str(e~'A ' ) be the Chern character form of the associated family of su-
perbundles ? —* M with superconnection A?. Then the image o/chg(A)
by the Chern- Weil homomorphism is the Chern character form of A?
<k,(ch9(A)) = ch(Af).
Proof. The proof of A) follows the proof in Theorem 7.34 that V • <j>u = <j>u-d,
except that the operator d = dp + d\i on A(P x M) is replaced by dp + A.
To prove B), we must show that
Let /' € 8* be the dual basis to the basis Xt of g. If / e C[g] and a e A(M),
we have
m
4>u{C{X)(f 9 a)) = <t>u (? Pf ® C(Xi)a)
1
(
1 = 1
since Z?=i tfC{Xi)f{n) = 0.
Part C) now follows immediately from B). ?
Let us see what these formulas become in the case of a trivial principal
bundle P = B x G. A connection one-form on P is just a g-valued one-form
on B,
Let M be a manifold with G action, and E be a G-equivariant vector bundle
over M with invariant connection V and moment fiB € g* ® F(M, End(E)).
The manifold M =PxgMis the direct product B x M, and the bundle
? = P Xq E is the pullback of E by the projection of B x M onto M.
Using the connection form w, we obtain a connection Vf on ?. The following
formula is easily shown.
7.7. The Bott Class
241
Proposition 7.39. Letu>ixE = YliLi^HE(Xi) be the contraction ofu with
pp. Then V? = d + V+ u-nE.
7.7. The Bott Class
Let V —* M be an oriented Euclidean vector bundle of even rank with spin
structure, let S —* M be the corresponding spin superbundle, and S\> —> V be
the pull-back of S to V. In this section, we will describe the "Riemann-Roch"
formula of Mathai and Quillen relating the Chern character of a supercon-
superconnection on S\> and of the Thom class U(V) of the bundle V. This result
is an application of the functorial properties of the equivariant Chern-Weil
differential forms proved in the last section.
Consider the case where M is a point. Let V be an oriented Euclidean
vector space of dimension 1L Let G = Spin(V), with Lie algebra j = A2Vc
C(V), and let r : fl -> .80A0 be the action of g on V defined in C.4). Let
5 be the spinor space of V and let p be the representation of G in S. The
trivial bundle Sv = V x 5 is a G-equivariant vector bundle with action
g(x,s) = (g-x,p(g)s).
Let c : C(V) —* End(S') be the spin representation of the Clifford algebra
of V, and consider the odd endomorphism c(x) e T(V, End~(Sv)), where x e
F(V, V x V) is the tautological section of the trivial bundle with fibre V. Prom
the section c(x) = X^fcxfccfc> we can construct a G-invariant superconnection
A = d + ic(x) on Sy —* V with curvature
, End(S)) et A(V, C(V)).
Fv = ||x||2 + «fc(x) = ||x||2 + i
The vector bundle Sv —* V, with superconnection d + ic(x), is called the
Bott class; by Bott periodicity, it represents an element of the .K-theory of
V which generates K(V) as a free module, but we will not discuss this point
of view.
If X = Yli<j Xijei A ej is an element of g, the vector field on V defined
by the action of G on V is
If X e g, the moment p.(X) = C(X) — Vx of X is equal to c(X), hence the
equivariant curvature of the superconnection is the EndE)-valued form on
V
FV(X) =
ufc(x) + c(X) =
dxkck
In this context, the Bianchi identity [A — i(X),Fg(X)] = 0 becomes
G.11) (d - i(Xv))Fv(X) + i[c{x),Fv(X)) = 0,
which is also easy to check directly.

242
7. Equivariant Differential Forms
The equivariant Chern character Ay of the trivial bundle Sv —> V with
superconnection A = d + ic(x) is the equivariantly closed form on V given
by the formula
AV(X) = Str(exp(-Fv(X)))
= Str(e~<l|x||2+<dc(x)+c(;f)))
_ gtr/e-(||x||2-»C(dx)+c(X))\
This differential form decays rapidly at infinity. Such a behaviour would
not have been possible if we had taken instead the Chern character of a
connection. This illustrates one of the applications of superconnections: the
construction of Chern characters which decrease rapidly at infinity on non-
compacts manifolds.
Consider now an oriented Euclidean vector bundle V —> M of even rank
n — 2? over a compact manifold M, and assume that V has a spin structure.
Thus V is associated to a principal bundle P —» M with structure group G =
Spin(V), in other words, V = PxaV. Let a; be a connection form on P with
curvature form Cl. The connection w defines a Chern-Weil homomorphism
<t>« : Ac(V) -»A{V).
Let S = P y-G S be the corresponding spinor bundle over M, and let Sv =
P x<3 Sy be a bundle over V defined using the trivial bundle Sv = V x S over
V; Sy is the pull-back of the bundle S —> M to V by the projection V —» M.
The bundle map c(x) : Sv —> Sv defines a bundle map c(x) : Sv —* S\>-
Consider the G-invariant superconnection A = d + ic(x) on Sv —+ V. As in
the preceding section, the connection w allows us to lift the superconnection
A to a superconnection A5v on Sv —* V. (Beware that the base of our
principal bundle P is M, and the fibre of the associated bundle is V, whereas
in the last section, the base was B and the fibre was M.) If we denote by
Vw the connection induced on the pull-back Sv —¦ V by the connection on
S —» M with connection form w, then
If dx G Al{V, V) is the canonical one-form on V, Lemma 7.30 shows that
4>u(dx) = dx + [w,x] e Al(P x V, V).
Thus the Chern-Weil homomorphism <j>w : C[g] 0 A(V) -> A(P x F)hor is
obtained by substituting Q for X 6 g, and the one-form dx + [u, x] on P x V
for the one-form dx on V. It follows that
4>U(AV) = Str(e-<M3-ic<*'+[w'xl>+c<n»).
The following result follows from the functorial properties of the Chern-Weil
homomorphism.
7.7. The Bott Class
243
Proposition 7.40. The closed form (f>u(Av) € A(V) is the Chern character
form ch(ASv) of the bundle Sv —> V with superconnection ASv.
The differential form <j>u{Av) has the important property of decaying
rapidly in the fibre directions.
In Chapter 1, we defined the Thom form of an oriented Euclidean vector
bundle V —* M. This may be obtained by a construction analogous to that
used above. We define the equivariant Thom form Uv of the vector space
V by a formula analogous to that of Ay, except that we work within the
Grassman algebra of V instead of its Clifford algebra. The analogy between
the Clifford algebra and the Grassmann algebra leads us replace the C(V)-
valued equivariant curvature Fy of Sy by the AV-valued form fy on V, given
by the formula
fv{X) = ||x||2 + fdx + X = Y,*l + J^dxtejfc + ]Txfcie/fc Ae,.
fc k k<i
This is the analogue of the differential form which we denoted Cl in Section 1.6.
If e e V, we denote by i\(e) the derivation of A(V, AV) defined by
iv{a 0 ?) = (-l)lala 0 tA(C) for a e Ak{V) and ? e A^,
and by i\(x) the operator
fc=i
It is easy to verify that for every X 6 fl,
(d - i(Xv))fv(X) - 2uA(x)fv(X) = 0.
This formula is the analogue of Part A) of Proposition 1.51.
Let exp(—fy(X)) be the exponential of fv(X) in the algebra A(V, AV).
Since the operators d, t(-XV) and ia(x) are derivations, we see that
G.12)
(d - iA(Xv) - 2uA(x
= 0.
The equivariant Thom form Uy on V is defined using the Berezin integral
T : A(V, AV) — A(V) in place of the supertrace Str : A(V, C(V)) — A(V):
UV(X) = T(exp(-/y(X))) = T(e-(|lxllJ+<dx+x)).
More explicitly, Uv(X) is given by a sum over multi-indices / C {1,... ,n}
of the form
where Pi(X) are homogeneous polynomials of degree (n — |/|/2) in X, I' is
the complement of I, and Pi>(X) coincides up to a sign with the Pfaffian of

244
In particular,
Uy(x) = Pf(-;
7. Equivariant Differential Forms
n/2-1
where q< G .42l(V) ® C[g]<_j, and e(n) = 1 for n even and — i for n odd.
Theorem 7.41. A) The form Uy is an equivariantly closed form on V, and
for all X e g, we have
I1' »/dim(*0 is even,
_.f ifdim(v)isodd_
B) // i: {0} —> V is the injection of the origin into V,
?{Uv{X))=Pt(-X).
Proof. It is clear that T(iA(x)a) = 0 for every a e A(V, AV). Applying the
Berezin integral to exp(—fv(X)), we see from G.12) that
(d - i{Xv))T(exp(-MX))) = 0.
The other properties of Uy(X) are obvious. ?
Definition 7.42. Define W(V) € A(V) by the formulas
u(v) = 2<<U?V) = r(e-(iixii3+<(*c+u-x)+n').
The differential form U{V) is the Thom form of Section 1.6, for the metric
||x||2/2. We obtain another proof that U(V) is closed, from the properties of
the Chern-Weil map <f>u. We also see that
v/m
is even,
is odd.
Recall the function jy(X), which we have already met a number of times
in this book:
/smh(rpO/2)\
I / y\ /O I *
By definition,
3v(
Proposition 7.43. The equivariant differential forms Ay and Uy are re-
related by the formula
(-2iyeAy(X)=jv/2(X)Uv(X).
7.7. The Bott Class
245
Proof. Choose an oriented orthonormal basis e< of V such that
i
X = ^Aje2i-i Ae2».
From this, we see that it suffices to consider the case in which dim(V) = 2
and X = Aei A e<i. Then
-Fy{X) = -||x||2 - idxici - idx2C2 - Acic2
= -||x||2 - A(Cl + tA
If we let ?i = C\ + i\~1dx.2 and ^2 = c2 — iA~1dxi, then Ci = ^2 = —1
?162 + ^2^1 = 0, so that e"^1^3 = cos A — fi^2 sin A. Thus, we see that
e-Fv(X) = e-||xf ^cosA _ sinAciC2 _ i
sin A — A cos A
sin A
Since Str(cic2) = —2i, we obtain
Str(e-Fv<x)) = 2isinAe-llx'|2(l - A-1dx1dx2).
On the other hand,
-fv(X) = -||x||2 - idxiei - idx2e2 - Aei A e2
and hence
e-fv(X) _ e-||x||2 /j __ j^Xl6l _ idx2e2 - Aei A e2 + dxidx2ei A e2).
It follows that T^-^W) = -Ae-llx"a(l - A-^xidxz).
The matrix t(X) has eigenvalues ±2iA, so that jy2{X) = A sin A. Thus,
it follows that
. ?
Thus, we obtain the following result, which is a refinement at the level of
differential forms of the well known "Riemann-Roch" relation between the
Thom classes in Jf-theory and in cohomology. In cohomology, this formula
is due to Atiyah-Hirzebruch [10].
Theorem 7.44. Let V —+ M be an oriented Euclidean vector bundle of even
dimension with spin structure over a manifold M. The Chern character
ch(A'Sv) of the Thom bundle Sv —* V with superconnection A5v and the
Thom form U(V) are related by the formula

246
Bibliographic Notes
7. Equivariant Differential Forms
Section 1. The complex of equivariant differential forms was introduced in
H. Cartan [42]. He shows that if G is compact, the cohomology Hq(M) of
this complex agrees with the topological definition of equivariant cohomology
H'(M XqBG, R). The map between the two theories may be described using
the Chern-Weil map of Section 6, with base the universal classifying space BG
(or a finite-dimensional approximation).
The definitions of the moment of an equivariant connection, the equivari-
equivariant curvature and equivariant characteristic classes are taken from Berline
and Vergne[22], except that here we allow superconnections instead of just
connections.
Sections 2, 3 and 4. Theorem 7.13 is due to Berline-Vergne [20] when X = Y,
and to Bismut [30] as stated. The original proof used an argument due to
Bott, of approximating Mo by a small tubular neighbourhood, similar to
that which we give here for the proof of Theorem 7.11. Here, we use an
argument by Gaussian approximation; it is interesting to observe its close
analogy to the proof of the McKean-Singer formula. Special cases which were
proved earlier include Bott's formulas for characteristic numbers, when the
zero-set of the vector field is discrete [39], corresponding formulas when the
zero-set may have arbitrary codimension proved in the Riemannian case by
Baum and Cheeger [18] and in the holomorphic case by Bott [40], Duistermaat
and Heckman's theorem on the exactness of stationary phase for functions
whose Hamiltonian vector field generates a circle action Corollary [52], and
the localization theorem for equivariant differential forms, proved by Berline
and Vergne [20], and by Atiyah and Bott [7]. A generalization of Theorem 7.11
to algebraic manifolds with singularities has been given by Rossman [91].
A localization theorem for equivariant if-theory was proved by Segal [94],
and by Quillen for equivariant generalized cohomology theories [84]. The
similarity between the localization properties of the complex of equivariant
differential forms and the localization theorem in K-theory were observed by
Berline and Vergne [20] and Witten [97].
Witten has proposed an interesting analogy between the local index the-
theorem for the Dirac operator on a manifold M and the localization theorem
in equivariant cohomology for the action of the circle group on on the loop
space LM (see Atiyah [4]). Bismut generalizes their results to twisted Dirac
operators in [30].
Section 5. The homogeneous Hamiltonian actions of Section 5 were intro-
introduced by Kostant [68]. A general relationship between the Fourier transform
of a codjoint orbit and the character of the corresponding representation was
conjectured by Kirillov [65], [66], based on his work on his formulas for the
characters of compact and niolpotent groups. Further conjectures on this
formula are stated in Vergne [95].
Bibliographic Notes
247
Further results on the relationship between equivariant cohomology and
Fourier transforms of orbits may be found in Berline and Vergne [21], Duflo-
Heckman-Vergne [50], Duflo-Vergne [51]. The proof we give of Chevalley's
Theorem 7.28 is in the last of these references. Theorem 7.29 is due to Ross-
Rossman [90].
Section 7. The results of this section are due to Mathai and Quillen [74],
although we have made greater use of the Berezin integral in the exposition
than they did. As explained in Atiyah-Singer [15], it allows one to pass from
the Atiyah-Singer Theorem for the index of an elliptic operator, expressed in
terms of if-theory, to formulas in terms of characteristic classes.

Chapter 8. The Kirillov Formula
for the Equivariant Index
The character of a finite-dimensional irreducible representation of a compact
Lie group G can be described by the Weyl character formula, which is a spe-
special case of the fixed point formula for the equivariant index of Chapter 6.
However, there is another formula, the universal character formula of Kiril-
Kirillov [65], which presents the character not as a sum over fixed points but as
an integral over a certain orbit of G in its coadjoint representation on g*;
this second formula is in principle much more general than the first, since it
applies to many cases other than that of compact groups.
Let M be a compact oriented even-dimensional Riemannian G-manifold,
where G is a compact Lie group, and let ? be a G-equivariant Clifford module
over M. In this chapter we will give an analogue of Kirillov's formula which
computes the equivariant index of a Dirac operator on the bundle ? over
M as the integral over M of an equivariantly closed differential form. Let g
be the Lie algebra of G, let AB(X,M), X e g, be the equivariant A-genus
of the tangent bundle of M, and let che(X,?/S) be the equivariant relative
Chern character of ? (see (8.2)). If D is a Dirac operator on ? associated to
a G-invariant connection on ?, we call the Kirillov formula the formula for
the equivariant index of D, which holds for X e g sufficiently small,
/ Ae(X,M) ch9(X,?/S).
Jm
In Section 1, we will prove this formula by combining the fixed point formula
for the index with the localization formula of the last chapter. We discuss
the special case in which D is an invariant Dirac operator on a homogeneous
space of a compact Lie group in Section 2.
In Section3, we introduce Bismut's "quantized equivariant differential",
namely, we replace the Dirac operator D by the operator D+ \c(Xm), where
Xm is the vector field on M corresponding to the Lie algebra element l€g.
Following Bismut, we rewrite the general Kirillov formula as a local index
theorem for this operator; the proof of this theorem follows closely the proof
of the local index theorem of Chapter 4.
8.1. The Kirillov Formula
8.1. The Kirillov Formula
249
In Chapter 6, we proved for the equivariant index of the Dirac operator
indcG, D) the fixed point formula (where n = dim(M))
(8.1)
f
which expresses the equivariant index as an integral over the fixed point set
M7 of 7 in M. In this section, we will rewrite this for 7 near the identity in
G, in terms of the equivariant cohomology of M; we call the resulting formula
the Kirillov formula, by analogy with Kirillov's formulas for characters of
Lie groups.
Let M be a compact oriented Riemannian manifold of even dimension n,
and let G be a Lie group, with Lie algebra g, acting on M by positively
oriented isometries. Let R be the Riemannian curvature of M, and let
/iM € (S* ® r(M,so(M)))G C ^(M,so(M))
be the Riemannian moment of X defined in G.3), that is, the skew-endo-
morphism of TM given by
HM(X)Y = -VYX.
Let R9 = nM + R e Aa(M,so{M)) be the equivariant Riemannian curvature
of M. Let X i-» A9(X, M) be the equivariant A-genus of the tangent bundle
of M,
Let ? be a G-equivariant Clifford module over M with G-invariant Her-
mitian metric, and invariant Clifford connection Vc. Let C? (X) be the Lie
derivative of the action of X e g on F(M,?), and let
ii?{X) = C?(X) - Vi 6 r(M,End(?))
be the moment of X with respect to the connection V?; thus, ne is an element
of (g* ® r(M,End(?)))G C A2G(M, End(?)).
Recall from C.7) the canonical map
t : A2T*M — ao(M);
if a e A2(M) and ? G A1(M), then as operators on ?,

250
8. The KiriUov Formula for the Equivariant Index
If /xM is the Riemannian moment of M, then c(r~l(nM)) is the element of
,AG(M,End(?)) given by the formula
where e', A < i < n), is an orthonormal frame of T*M. Thus
\c(r-\nM{X))),c{a)} = c{f"M(X)a)
for all a € A1(M), where nT'M is the moment of the bundle T*M, equal to
minus the adjoint of /xM.
Define the twisting moment /i?/s(X) for the action of X on ? by the
formula
Lemma 8.1. 77ie twisting moment p,?ls commutes with c(a) for all a €
A^M), and hence lies in (g*®r(M,Endc(M)(?)))G C .AG(M, Endc(M)(?))-
Proof. Since //(X) = ??(X) - Vf^ and /xT*M(X) = ?T*M(X) - V?*M, we
see, since V? is a G-invariant Clifford connection, that
[//(X), c(a)] = c(?T'M(X)a) - c(V™a)
We now define the equivariant twisting curvature of ? to be
F?/S(X) = /'*(*) + F?'s g ,4G(M,Endc(M)(?)).
Note that if Af has an equivariant spin-structure with spinor bundle S, so that
? is isomorphic to the equivariant twisted spinor bundle W<8M, then we may
identify F?/s e Aq{Mt'Eudc{M){S)) with Fgw 6 Aa{M,End{W)) under the
isomorphism of bundles EndC(M)(?) = End(W). Define the equivariant
relative Chern character form of ? to be
(8.2)
chB(X,?/S) = Str?/s(exp(-F?/s(X))).
In this chapter, we will study the following reformulation of the equivariant
index theorem near the identity in G. We will explain in the next section
why we call this the Kirillov formula.
Theorem 8.2 (Kirillov formula). For X 6 g sufficiently close to zero,
/ AB(X,M)che(X,?:/S).
8.1. The Kirillov Formula
251
Let us first comment on some differences between the fixed point formula
(8.1) and the Kirillov formula for indoG,D). In the first formula, it is not a
priori clear, and in fact quite remarkable that the right-hand side of the fixed
point formula depends analytically on 7 near 7 = 1, and in particular that it
has a limit when 7 —> 1 equal to
ind(D) = {2m)'n/2 [ A(M)ch(?/S).
Jm
In contrast the analytic behaviour of indG(e~x, D) near X = 0 is exhibited
in the Kirillov formula. However, the fact that this analytic function of X
is the restriction to a neighbourhood of zero of an analytic function of e~x
is not apparent. This fact is an analogue of the integrality property of the
A-genus of a spin manifold.
The Kirillov formula can be deduced from the localization formula, since
the integrals over M of an equivariantly closed form a(X) localize on the set
of zeroes of Xm-
We now start the proof of Theorem 8.2. Choose X 6 9 sufficiently close
to zero that the zero set of Xm coincides with the fixed point set Mo of
7 = exp(—X). By Proposition7.12, the bundle M is orientable, and hence
so is Mo. Fix compatible orientations on Mo and its normal bundle A/\ Let
TV be the Berezin integral
Since 7? e r(M0,C(AH ®EndC{M)(?)) ^ r(M0,AJv"* ® Endc(M)(?)), we
see that
The fixed point formula for the equivariant index states that
Jm0 det '*A-7^exp(-.RA/))
where R^ is the curvature of the normal bundle A/" —¦ Mo, 7^ is the induced
action of 7 on A/", and cha{i,?/S) is the differential form on Mo given by
the formula
Str?/sGVG?)-exp(-F?/s)).
By the localization theorem, we have
/In I y*. ) ,
Thus, we need only prove that, for 7 = e~x with X sufficiently small,
i(Mp) chGG, SIS) _ (A,(X, M) cha(X, S/S)) |M.

8. The KiriUov Formula for the Equivariant Index
252
Observe that the differential fonns det]^(-R^(X)) and chGG, S/S) both
depend on the orientation of AT. We will choose an orientation such that
tf > 0. Under the splitting
the Riemannian curvature restricted to Ado splits as R = Rq © R1*, where
Ro is the Riemannian curvature of Mo. The Riemannian moment nM(X) of
X restricted to Mo is the infinitesimal action of X on TM, and equals the
moment ^(X). Thus,
It follows that
detV2 f
\si
The exponential of R*{X) equals (V^)~:l exp(flA/') where 7 = e~x, since
H^iX) may be identified with the action of the vector field X on the bundle
N, and nM{X) and fl^ commute. Since Rf € ^g(Mo,bo(A0) is antisym-
antisymmetric and A/" is even-dimensional, we see that
AB(X,M)
i(M0)
and hence
It remains to show that
c\(X,e/S)\Mo
Consider the decomposition
where t~1(/xm(X)) G A2V*. Since the two terms on the right-hand side
commute, we see on taking the exponential of both sides that
7? = exp(-c(r-1(MM(*)))) -exp(-//s(X))
Let us write o for exp(c(r(/xM(X)))) € C(N*). With our choice of orien-
orientation of M, TV(a) > 0, and
8.2. The Weyl and KiriUov Character Formulas
so that
253
im(AO/2
det ' A — y™)
e ...
Strf/s(TvG?)exp(-F0?/s))
chGG,?/S) =
This completes the proof of Theorem 8.2.
8.2. The Weyl and Kirillov Character Formulas
Let G be a connected compact Lie group with maximal torus T. The irre-
irreducible finite dimensional representations of G were classified by E. Cartan,
and their characters were calculated by H. Weyl. In the notation of Sec-
Section?^, let LT be the lattice {X € t | ex = 1}, so that T = i/LT, and
denote by LJ. the dual lattice
VT = {I 6 I* 11{X) 6 2ttZ for every X e M C t*.
Choose a system P C A of positive roots. The subset XG = ipp + L^ of
t* is independent of the choice of P and ^-invariant. In the notation of
Section7.5, we write Weyl's character formula as follows.
Theorem 8.3 (Weyl). IfX is a regular element ofXa, there exists a unique
finite-dimensional irreducible representation T\ ofG, such that for X e t,
Tr(TA(ex)) = _ ^
Let us now describe Kirillov's formula for the character of the representa-
representation 1\. As usual, for X e g, let
sinh(adX/2)\
We can define an analytic square root of jg(X) on the whole of {j. Indeed, if
xet,
=n
e(a,X)/2 _ e-(a,X)/2
e(c,X)/2_e-(a,X>/2\2
(a,X>
)¦
Thus, by Chevalley's Theorem 7.28, there exists a unique analytic G-invariant
function jB on g, which on t is equal to

254
8. The Kirillov Formula for the Equivariant Index
Theorem 8.4 (Kirillov). Let X be a regular element of Xg and let M\ be
the symplectic manifold G ¦ A xoith its canonical symplectic structure fix- For
X € g, we have
jB1/2(X)Tr(Tx(e*))= f e"<*> dm(/)
where dm is the Liouville measure of Mx-
Proof. Both sides of the equation are G-invariant functions on g. Thus it
is sufficient to check this equality when X 6 t- The equality then follows
immediately from Corollary 7.25. D
In this section, we will explain how these formulas are special cases of two
forms of the equivariant index theorem, respectively the fixed point formula
Theorem 6.16 and the Kirillov formula Theorem 8.2. To do this, we must
find an equivariant Dirac operator whose equivariant index is the character
of Tx- As underlying space, we take the flag manifold G/T. Consider G/T
as a Riemannian manifold, with the metric induced by the restriction of the
opposite of the Killing form to t = Te(G/T). Choosing a system P C A of
positive roots, let
/« =
iXa + iX_Q
Choose the orientation on G/T such that IlaeP e" ^ /<* 's tne orientation of
t = Te(G/T). The manifold G/T is an even-dimensional oriented Riemannian
manifold with spin-structure.
Lemma 8.5. Let A8(X, G/T) be the equivariant A-genus of the Riemannian
manifold G/T. Then A9{X,G/T) andj91/2(X) represent the same class in
the equivariant deRham cohomology H*(G/T).
Proof. Consider the direct sum decomposition g = t © t of g. Prom this, we
obtain a direct sum decomposition of the bundle Gx^g,
GxTg = (GxTt)©T(G/T).
Since T acts trivially on t, the bundle G x r t is isomorphic to the trivial bundle
G/T x t. The trivial connection on this bundle has vanishing equivariant
curvature, so the equivariant A-genus equals 1. Thus, the bundle G Xt 9,
with connection the direct sum of the trivial connection on G/T x t and
the Levi-Civita connection on T(G/T), has equivariant A-genus equal to
A9{X,G/T).
However, since g is a G-module, the bundle G Xj g may be trivialized to
G/T x g by the map
l(g,X)] e G xT g h— ([g],adp • X) € G/T x g.
8.2. The Weyl and Kirillov Character Formulas
255
The trivial connection on G/T x g has equivariant curvature FS(X) = ad X,
so that for the trivial connection,
A8(X,GxTg)=j~1/2(X)-
Since the equivariant cohomology class of the equivariant A-genus is inde-
independent of the connection used in its definition, we obtain the lemma. ?
Of course, A,(X, G/T) and j7V2(X) are not in general equal as differential
forms, since j91/2{X) is a function whereas the differential form A{G/T) =
A8@, G/T) has non-vanishing higher degree terms in general.
If A 6 Lj., there exists a unique character of T with differential i\ on t.
Denote this character by eiA and by Cx the homogeneous line bundle G xTCx,
where Cx is the one-dimensional vector space C with the representation e'x
of T. Let us denote by nx{X) tne restriction to Ma of the linear function
/ i-+ (/, X). If A is regular, the map g >-* g • A is an isomorphism of G/T with
M\ = G\. Thus, Cx is a line bundle over M\.
Proposition 8.6 (Kostant). The homogeneous line bundle C\ has a G-
invariant canonical Hermitian connection. The equivariant curvature of L\
is the equivariantly closed form i(/xx(X) + ft*).
Proof. For X e g, define an operator Vx on sections of C\ by the equation
Since the vector fields XMx span the tangent bundle of Mx, it is only neces-
necessary to verify the following condition in order for Vx to be a connection: if
XMx vanishes at a point / G M\, in other words, if X e g(/), and if </> is a
section of C\, then Vx<^>(/) = 0. But if X e g(/),
(Cx(X)<t>)(f) = i{f,X)<t>(f) = i,/(X)(/)<K/)
by definition of the homogeneous vector bundle ?v Thus V is a connec-
connection with moment map i/i*(X), from which it follows that its equivariant
curvature equals i(/x*(X) + ft*). ?
If G is simply connected, the element ipp € LJ, so that if Xg = L^.
Consider the G-invariant twisted Dirac operator D* associated to the twist-
twisting bundle C\:
Dx : r(G/T,5 ®'?>) - T(G/T,5 ® Cx).
Its equivariant index
indG((j, DA) = <fttar(D+)(fl) - Ttker(D-)(9)
is the character of a virtual representation of G. Denote by e(P, P\) — ±1
the quotient of the two orientations on t determined by P and P\. The
following theorem identifies the Weyl and Kirillov character formulas with
the equivariant index theorem for the Dirac operator Dx-

256
8. The Kirillov Formula for the Equivariant Index
8.2. The Weyl and Kirillov Character Formulas
257
Theorem 8.7. The fixed point formula for the equivariant index of D\ is
given for X el by the formula
Thus, the index indG(g,DA) equals the character of the representation T\ up
to a sign.
The Kirillov formula Theorem&.2 for the equivariant index o/D* at X € g
small is
indG(ex,DX) = e(P,Px) f j
JMx
Mx
where dm is the Liouvtile measure of M\.
Proof. Let us start with the fixed point formula. The action of T on the flag
manifold leaves stable the points
{wT | to S W{G,T) = N{G,T)/T} C G/T,
and for generic t e T, the action of t on G/T has a finite number of fixed
points indexed by the Weyl group W. Thus it is easy to write the fixed point
formula for the character of the virtual representation ind D\: it equals
where the sign e(w, t, P) is the quotient of the orientations of t determined
respectively by P and by the element p(wtw~1) in the Clifford algebra C(t)
of the Euclidean space r. We have, for t = ex, X € t,
= J](cosh(a,X/2) -»sinh(a,X/2)ea/o).
It follows that
?{w,P,ex) = (-l)lp|e(w) n sgn(isinh(a,X/2».
a€P
Since
dett(l-ex)=
a€P
the fixed point formula follows.
Since the Clifford module S®C\ that we are considering is a twisted spinor
bundle, the Kirillov formula may be rewritten as
tad(DA)(e"*) =
"*) =
/
g/t
Ag(X,G/T)ch9(X,Cx).
By Lemma 8.6, the equivariant Chern character of C\ equals
The integral on G/T of an equivariantly closed differential form depends only
on its equivariant cohomology class. By Lemma 8.5, we can choose as a repre-
representative for AB(X, G/T) the function i^1/2(X). It only remains to compute
the highest degree term of ch,(A", Cx), which is (-27ri)dim(G/T)/2e-<<>'*>dm.
The replacement of X by —X and a careful comparison of orientations proves
the second formula. Q
Bott has given another realization of the representation T\, as the 8-
cohomology space of a line bundle on the flag manifold G/T. Define t+ =
X]aep(oc)a- Then t+ is a subalgebra of flc end t has a complex structure
J such that t+ = {X € tc | JX = iX}. There exists a unique G-invariant
complex structure on G/T such that T*'°(G/T) = t+. Furthermore, for any
(i? LJ., the line bundle ?,, = G xT C^ has a structure of an holomorphic
line bundle over G/T. It is easy to write down the fixed point formula for
the character of the virtual representation Y,(-^)'H°'i(G/T^n) on the 9-
cohomology spaces of the line bundle C^.
Theorem 8.8. If ft € L^, then for X et,
Q p
(e<a,X)/2_e-<a,X)/2)-
This theorem shows that the virtual representation ?(-l)p.ff0'p(G/T, ?,,)
is zero if A = \i — ipp is not a regular element of t*. If A is regular, let wx
be the element of W such that w(P\) = P. Then the virtual representation
^(-l)P/z'P(G/T,?fl) is the representation e(wx)Tx. In fact a more refined
result, the Borel-Weil-Bott theorem, is true:
where |w| is the length of the Weyl group element w e W(G, T).

258 8. The Kirillov Formula for the Equivariant Index
8.3. The Heat Kernel Proof of the Kirillov Formula
Let M be a compact oriented Riemannian manifold of even dimension n, and
let G be a compact Lie group with Lie algebra g acting on M by orientation-
preserving isometries. Let 6 be a G-equivariant Clifford module over M with
G-equivariant Hermitian structure and G-invariant Clifford connection V?.
Definition 8.9. The Bismut Laplacian is the second-order differential op-
operator on F(Af, S) given by the formula
H(X) = {D+\c(X)J+C(X),
where we write c(X) for the Clifford action of the dual of the vector field Xm
on ?.
The operator H(X) is a generalized Laplacian depending polynomially
on X e g. It may be thought of as a quantum analogue of the equivariant
Riemannian curvature
The restriction kt{x, x, X) \dx\ of the heat kernel of the semigroup
to the diagonal is a section of the bundle End(?) = C(M)®Endc(M)(?)®|A|.
The asymptotic expansion of kt(x, x, X) as a function of t for t small has the
form
kt(x,x,X)
t=0
where ki(x, X) depends polynomially on X (as we will show later). Denoting
by C[g] the algebra of formal power series on g, we may identify kt with
a section of the equivariant Clifford algebra C[g] ® C(M) ® Endc(jw)(?)-
The associated graded algebra of this filtered algebra is C[g] ® AT*M ®
Endc(M)(?).
If V is a Euclidean vector space with Clifford algebra C(V'), consider the
algebra C[g] ® C(V*) with filtration
where C[g]fc is the space of homogeneous polynomials on g of degree fc. This
filtration is analogous to the grading of the equivariant cohomology complex,
and will be called the equivariant Clifford filtration. The associated graded
algebra is isomorphic to C[g] ® AV*, with equivariant grading
deg(P ® o) = 2 deg(P) + deg(a) for P 6 C[g] and o € AV.
The next theorem is the generalization of Theorem 4.1 to the equivariant
context. It generalizes Bismut's infinitesimal Lefschetz formula [30].
8.3. The Heat Kernel Proof of the Kirillov Formula
259
Theorem 8.10. Let kt(x,x,X) = (x | e~tW(x) | x) be the restriction of the
heat kernel of the semigroup e~tH(x*> to the diagonal. Consider the asymp-
asymptotic expansion
kt(x,x,X)
t=0
A) The equivariant Clifford degree of ki is less or equal than 2i.
B) Let a(k) = ??o *[*](*;*) e C[gl ® ^(M,Endc(M)(?)). Then
Before proving this theorem, let us show how it implies the Kirillov formula
for the equivariant index. Recall that the heat equation computation of the
index is based on the McKean-Singer formula Proposition 6.3: for every t > 0,
For ti 6 C, we consider the operator
Since the operators D and c(X) commute with the action of etX on T(M,?),
so does Du.
Proposition 8.11. For all t > 0 and all u € C, we have
indG(expX, D) = Str(exe-'D«).
Proof. The result is true for u =¦ 0; we will show that the supertrace on the
right-hand side does not depend on u. By Proposition 2.50, and the fact ex
and the heat operator e~tD« commute, we have
~ Str(exe-tD«) = -tStr
du v '
du
du
Since ex commutes with c(X) and D, we obtain
4- Str(exe-tD') = -tStr f ^exe~t
du v ' \ du
= -tStr((Duc(X) + c(X)Du)exe-tD»)
= -tStr([Du,c(X)exe-tD-]).
This vanishes by Lemma 3.49 D
Let us now set u = \. It follows from this proposition that for every t > 0,
indG(e-tx,D) = Strte-^We-^+'W^') = Str(e-'H<X>).

260 8. The Kirillov Formula for the Equivariant Index
Thus we obtain the formula
indG(e-tx,D)= f Str(kt(x,x,X))\dx\.
Jm
Consider the asymptotic expansion in t of both sides of this formula. We
have
Str(fcf(*,x, X))
fo X)).
3=0
It follows from the first assertion of Theorem 8.10 that if j < n/2, the total
degree of kj is strictly less than n. In particular, its exterior degree is strictly
less than the top degree and the supertrace of kj vanishes. Thus there is no
singular part in the asymptotic development of Str(fct(s, x, X)). Using the
formula Str(o) = (—2i)n/2 Strg/s(a)[n], we obtain
/ Stre/s{kn/2+j(x,X)ln]) \dx\
3=0
Jm
The first assertion of Theorem8.10 also implies that kn/2+j(X,x))[n] is a
polynomial in X of degree less or equal to j. The asymptotic expansion of
the left-hand side indG(e~tx, D) of the above equation is a convergent series:
indG(e
-tx
y) - Trker(D
Comparing the coefficient of V in these two power series in t, we see that
only the component of a(kn/2+j(X, x)) lying in C[g,]j ® AnT*xM contributes
to the equivariant index. Using the explicit formula of the theorem for <r(k),
we obtain
/
Jm
AB{X,M) che(X,?/S).
which is the Kirillov formula.
Let us now prove Theorem 8.10. Following a plan which should be familiar
by now, we start by proving a generalization of Lichnerowicz's formula. Let
6X G A1(M) be the one-form associated to X,
Let Vf 'x be the Clifford connection V? - \QX on the bundle ?; since $x
is scalar-valued, it is clear that Vf 'x is again a Clifford connection. Let us
show how Bismut's Laplacian H(X) can be written as a generalized Laplacian
associated to the connection V?'x on ?. Let Ax be the Laplacian on ?
associated to V?'x ¦
8.3. The Heat Kernel Proof of the Kirillov Formula
261
Proposition 8.12. IfF?/s and n?ls{X) are, respectively, the twisting cur-
curvature and the twisting moment of X for the connection V? and rM is the
scalar curvature of M, then
H(X) =
\rM
Proof. Recall from Lemma 7.15 that (nM(X)Y,Z) = -\d9x{Y,Z) for Y,Z e
F(M,TM). Since 0x is a scalar-valued one-form, the twisting curvature
F?/s,x of the connection V?<x is equal to F?'s - \d9x- Let Dx = D +
jc(.X'); since Dx is the Dirac operator associated to the connection V?'~x,
Lichnerowicz's formula shows that
= A_x + \rM + c(F?'s) + i c(d3x).
By this formula and Proposition 3.45, we see that
A_x - Ax = Dx - D2_x - \<d6x)
= -vl + \<rex.
But d*0x = Tr(/iM(X)) € C°°{M) vanishes, since \iM is antisymmetric; this
implies that
Ax - A_x = Vx.
Since C?{X) = V?x + fi?{X), we see that
= D2X+C?(X)
= A_x + \
\rM
\ c{dBx).
Since Ax = A_x + Vx and fi?/s(X) = n?(X) + \ c(d6x) (by definition of
the twisting moment), the result is proved. ?
We could model the proof of Theorem 8.10 on the proofs of the simple index
theorem of either Chapter 4 or Chapter 5. We will follow the former. Thus,
as in Section4.3, let V = TXoM, E = ?Xo and U = {x 6 V | ||x|| < e}, where
e is smaller than the injectivity radius of the manifold M at xo- We identify
U by means of the exponential map x h-» expXo x with a neighbourhood of
zo in M.
For x = expl0 x, the fibre ?x and E are identified by the parallel transport
map t(xo,x) : ?x —» E along the geodesic xa = expj.o sx. Thus the space
T(U, ?) of sections of ? over U may be identified with the space C°°{U, E).
Choose an orthonormal basis di of V = TX0M, with dual basis dx* of
T'XoM, and let c' = c(dx.1) G End(E). Let et be the local orthonormal frame
obtained by parallel transport along geodesies from the orthonormal basis di
of TXoM, and let el be the dual frame of T*M.

262 8. The Kirillov Formula for the Equivariant Index
Let S be the spinor space of V and let W = Homc(v)(S, E) be the
auxiliary vector space such that E = S ® W, so that End(E) = End(S) ®
End(W) S C(V) ® End(W), and Endc(v.)(E) = End(W).
We must study the operator H(X) in the above coordinate system, but
with respect to the frame of the bundle ? obtained by parallel transport along
geodesies emerging from Xo with respect to the connection V?>x. Define
a : U x g —> Cx by the formula
so that flax = -\v{R.)9x- Let p(X,x) = ea*W. Then p(X,x) is the
parallel transport map on the trivial line bundle over M with respect to the
connection d — \6x, along the geodesic leading from x and xq.
Lemma 8.13. The conjugate
of the covariant derivative Vg!X = Vf. — \B\{di) is given on elements of
C°°(U,E) by the formula
Here, HyW = (fl(&,$)*,«,,fl.),. and^(X) = (jiM (¦*)>„*, »*)«. are
the Riemannian curvature and moment at xo, and /ijfc(x) e C°°(U), 9t(x) e
Coo(l/,EndC(v.)(?)) and hi(x) e C°°(U) ® g* are error terms lo/iic/i satisji/
2
/yfc(x) = O(|x|2), ft(x) =
), and hf(x) = O(|x|2).
Proof. Let us write
p(X,x)(d- J
where
i
satisfies wx^) = 0 and depends linearly on X G g. As a special case of
A.12), we have
The one-form -\i.(R.)d8x is dual to \nM{X)H. Taking the Taylor expansion
of both sides of this formula, we see that
j
where hi(x) = O(|x|2). Since p(X,x)(Va;x)p(X,x)-1 = Vee.+wu the lemma
follows from Lemma4.14. D
8.3. The Heat Kernel Proof of the Kirillov Formula 263
Let (x | e~tH(x) I xo) be the heat kernel of the operator H{X); by The-
Theorem 2.48, (x | e~tif(x' | Xo) depends smoothly on X e g. We transfer this
kernel to the neighbourhood U of 0 e V, thinking of it as taking values in
End(E), by writing
g
fc(t,x,X) = p(X,x)t(xo,x)(x I e-tH(x) | x0),
where x = expl0 x. Since we may identify End(JE) = C(V*) ® End(W) with
AV*®End(W) by means of the full symbol map a, fc(t,x, X) is identified with
a AV* ®End(lV)-vamed function on U x g. Consider the space AV* ®End(W)
as a C(V*)®End(W) module, where the action of C(V*) on AV* is the usual
one c(q) = e(a) — i(a). The following lemma shows that fc(t,x, X) may be
characterized as a solution to a heat equation on the open manifold U; the
proof is analogous to that of Lemma 4.15.
Lemma 8.14. Let L(X) be the differential operator on U, with coefficients
in C(V) ® End(W), defined by the formula
p(X,x)-1L(X)p(X,x) = E1 ^
The function fe(t,x,X) e C°°(R+ x U, AV* ®End(W)) satisfies the differen-
differential equation
(dt + L(X))fc(t,x,X) = O.
Using the explicit recurrence formula of Theorem 2.26 and the fact that
the coefficients of L(X) are polynomial in X, we see that the coefficients of
the asymptotic expansion of fc(t,|x,X) are polynomial in X.
To prove the Kirillov formula, we will rescale the kernel in a way similar to
that used in Section 4.3, except that we will perform an additional rescaling
on the Lie algebra variable X 6 g. If a e C°°(R+ x U x g, AV ® End(W)),
let 6ua, @ < u < 1), equal
Fua)(t,x,X) = f^u-^aiuW^u
i=0
Define the reseated heat kernel r(u, t, x, X) by
• \—7 -1—7
We define an operator L(u, X) by the formula u8uL(XN~l; it is clear that
r(u,t,x, X) satisfies the heat equation
(at + L(«,X))r(u,t,x,X)=0.
Using Lemma8.13, we can give, an asymptotic expansion for L(u, X). We
will show that

264
8. The Kirillov Formula for the Equivariant Index
where K(X) is a harmonic oscillator similar to that which entered in Sec-
Section 4.3.
In order to state the formula for K(X), recall the matrix R with nilpotent
entries equal to Ry = (.Rxodi,dj) G A2V* where RXo is the Riemannian
curvature of M at xo, and the element F of A2V* ® End(W) obtained by
evaluating the twisting curvature F?ls at the point xo- Let fiM(X) e End(V)
be the evaluation of the moment of the manifold M at the point xo and Lie
algebra element X e g, and let ///S(X) € End(VT) be the evaluation of
the relative moment of the Clifford module ? at xo and Lie algebra element
Xeg.
Proposition 8.15. The family of differential operators
L(u,X)=uSllL(XNZ1
acting on C°°(U x g, AV ® End(WO) has a limit K(X) when u tends to 0,
given by the formula
K(X) = - f
Proof. Using the fact that
differential operator
where
Z1 = e(e*) - tu(e'), we see that the
A2(u) =u-
The functions
that lim _^ A
equal to
3<k
uik)
x), 9i(x) and hi(x) vanish to sufficiently high order in x
= 0. Thus, we see that V^x>u has a limit as u —> 0,
Since Ry = 53fc<1 Rjiki?kel, the above limit is seen to equal
of
Let c'(u) = u1^26uct5u x = e' — tit*. The operator L(u, X) equals the sum
8.3. The Heat Kernel Proof of the Kirillov Formula
and
265
i
Clearly, these last two terms vanish in the limit u —> 0. The first term has
the limit
while the second and third terms converge to
since the vector fields ei coincide with di when x = 0. D
The operator 1,@, X) = K(X) is a generalized harmonic oscillator, in
the sense of Definition 4.10, associated to the n x n antisymmetric curvature
matrix Ry+/$(X) and the N x iV-matrix F+ne's(X) (where N = dimVT),
with coefficients in the commutative algebra A+V*. It follows that there is
a unique solution of the heat equation
such that lim __^ r(t,-x.,X) = 6(x), given by the formula
If J is an integer, let C[g](j) be the quotient of the algebra of polynomials
on the vector space g by the ideal of polynomials of order J+1. If o e C[g] (j),
it has a unique polynomial representative of the form a(X) = 52|a|<J a«-^ai
we define the norm of a to be supQ |aa|.
We can think of r(u, t, x, X) as defining an element
rj(u,t,x,X) e CX(R+ x 17, AV ® End(W)) <8>C[g](J)
by taking the Taylor series expansion of r(u, t, x, X) at X = 0 up to order J,
in other words,
Lemma 8.16. There exist polynomials 7<ij(t,x) onRxV, with values in
AV* ® End(W) ® C[g](j), such that for every integer N, the function
2N
i=-n-2J

266 8. The Kirillov Formula for the Equivariant Index
approximates rj(u,t,x,X) in the following sense:
||d/da(r,/(u, <,x, X) — rj(u,t,x,X))\\ . <C(J,N,j,a)uN,
for 0 < w < 1 and (t, x) lying in a compact subset of @,1) x U. Furthermore,
7m@,0) = 0ifi^0, while 70,J @,0) = 1.
Proof. By Theorem 2.48, there exist AV* ® End(W)-valued smooth functions
ipi on U x g such that
for t e @,1), x e U, and X in a bounded neighbourhood of 0 6 0. Fur-
Furthermore, we can diiFerentiate this asymptotic expansion with respect to the
parameter X. It follows that there exist AF* ® End(W) ®
smooth functions xpi}j on U such that
\\kj(t,x,X) -
t% j(x,X)
< C{N,J)tN-n'2
for t e @,1) and xeU, where kj(t, x, X) is the Taylor expansion of k(t, x, X)
around X — 0 to order J. As in the proof of Proposition 4.16, we can suppose
that ipi:j is polynomial in x.
Rescaling by u e @,1), we obtain
i=0
Thus, we see that the function u i-» r j(m, t, x, JQ has an asymptotic expansion
of the desired form, where ii,j(t,x, X)^\ is the coefficient of u*/2 in
j=0
Since Vv,./ is a polynomial in X of order at most J, we see that Gt, j)[p] = 0
if i < —p/2 - J. The argument for the derivatives is similar. Since
we see that 7i,j@,0,X)[p] = 0 unless i - p = 0, and that 7o,j@,0,X) =
1. D
It is clear that rj(u, t,x,X) satisfies the differential equation
(8.3) (dt + Lj(u>X))rJ(u,t,x,X) = Q,
Bibliographic Notes
267
where Lj(u,X) is the action induced by the differential operator L(u, X)
on C°°(U) ® AV"* ® End(W) ® C[g](J). Let us show how this implies that
rj(u,t,x,X) has a limit as u —> 0, using the fact that L(u,X) = K(X) +
O^1/2). Let ? be the largest integer for which 7_^j(?,x,.X") is non-zero.
Expanding (8.3) in a Laurent series in u1/2, we see that the leading term
u~l^2qt(x)')^itj(t, x, X) of the asymptotic expansion of rj(u, t, x, X) satisfies
the heat equation (dt + K(X))(qt(x)f-itj(t,x,X)) — 0. Since formal solu-
solutions of the heat equation (dt + K(X))(qt(x)'y(t,x, X)) = 0 for the harmonic
oscillator are uniquely determined by 7@,0, X), and since 7i,j@,0, X) = 0
for i < 0, we see that 7_t,j must equal zero unless ? < 0. In particular,
we see that there are no poles in the Laurent expansion of rj(u, t,x, X) in
powers of u1/2.
We also learn from this argument that the leading term of the expansion of
rj(u, t,x), namely rj@,t,x, X) = qt(x)-yotj(t,x, X), satisfies the heat equa-
equation for the operator L@, X) = K(X), with initial condition
Applying Theorem 4.12 to the operator K(X), we obtain the following result.
Proposition 8.17. The limit lim rj(u, t,x,X) exists, and is given by re-
u—>0
duction modulo XJ+l of the formula
D
7rt)-"/2det^
\
1/2
x expl —
x) — 1
Since this is true for arbitrary J, we have only to set t = 1 and x = 0 to
obtain Theorem 8.10.
Bibliographic Notes
The Kirillov formula Theorem 8.2 was proved in Berline-Vergne [23], using
the localization theorem for equivariant differential forms and the equivari-
equivariant index theorem of Atiyah-Segal-Singer. The local version of the theorem
proved in Section 3 is a generalization of the results of Bismut [31]; in partic-
particular, the definition of the "quantized equivariant curvature" H(X), and the
proofs of Propositions 8.11 and 8.12 are due to him.
In Sections 9.4 and 10.7, we will explain the relationship between the Kiril-
Kirillov formula and the index theorem for families with compact structure group;
the relationship comes about by considering the family P Xq M, where P is
an approximation of the universal bundle EG for G. We will see that the
Kirillov formula is equivalent to Bismut's local family index theorem, in the
case of families with compact structure group.

9. The Index Bundle
269
Chapter 9. The Index Bundle
Consider a manifold B and a finite dimensional Hermitian superbundle H =
H+ ®H~ —+ B. Let D be an odd endomorphism of H with components
D* : W* —> WT, such that ker(D) has constant rank, so that the family of
superspaces (ker(D*) \ z e B) forms a superbundle over B, called the index
bundle of D. Let A be a superconnection on H with zero-degree term equal
to the odd endomorphism D of H, and let T = A2 G A{B,End{H)) be the
curvature of A; all notations are as in Section 1.4. If we assume that D is
self-adjoint and has kernel of constant rank, then at the level of cohomology,
ch(A) = ch(ker(D)).
In fact, this equation has a refinement at the level of differential forms.
If t > 0, we may define a new superconnection on H, by the formula
A« = t1/2D + AA] + r1/2A[2] + ...,
where Ajij is the connection associated to A. This superconnection has Chern
character form
ch(At) = Str(e-A?).
Let Po be the projection from H onto the subbundle ker(D). We may also
define a connection on the superbundle ker(D) by projection,
In Section 1, we will prove the following result:
lim ch(At) = ch(V0).
(—>oo
Intuitively speaking, as t tends to infinity, the supertrace
ch(At) =
is pushed onto the sub-bundle ker(D). The proof, which is given in Section 1,
is substantially more complicated than this simple idea.
Our real interest lies in an infinite-dimensional version of this, in which
the family D of operators on the finite dimensional bundle H is replaced
by a family of Dirac operators. Our technical tools will be the estimates
of Chapter 2 on the heat kernels of generalized Laplacians, and the spectral
theorem.
Consider a fibre bundle n : M —> B with compact fibres. We denote
by T(M/B) the vertical tangent bundle, which is the sub-bundle of TM
consisting of vertical vectors, and by \An\ the vertical density bundle, which
is isomorphic to |Am| ® tt*\Ab\~1. If ? is a superbundle on M, we denote by
7T»? the infinite-dimensional superbundle on B whose fibre at z G B is the
Frechet space
here, Mz — n~1(z) is the fibre over z € B, and ?z is the restriction of the
bundle ? to Mz. The space of sections of %t? is defined by setting
and the space of differential forms on B with values in 7r«? by
A(B, n,?) = T(M, ir*(AT"B) ® ? 0 \A*\1/2).
Let D = (Dz | 2 G B) be a smooth family of Dirac operators. By a super-
connection on tt»? adapted to D, we mean a differential operator A on the
bundle E = n* (AT*B) ® ? ® |A* | ^2 over M, of odd parity, such that
A) (Leibniz's rule) if v € A{B) and s G r(M,E), then
B) A = D + ??i™(B) AW, where AM : A'(B,*.?) - A'+i(B,*.?).
In the appendix to this chapter, we show that the curvature of the super-
connection A has a smooth heat kernel (x | e~A | j/) varying smoothly as a
function of z e B and x, y € Mz. Because of this we can extend the defini-
definition of the Chern character of a superconnection to the infinite dimensional
bundle irt? —> B.
Suppose that the dimension of ker(D*) is independent of z G B. In this
case, the vector spaces ker(Dz) combine to form a vector bundle ind(D), which
is the index bundle of the family D; the index of a single Dirac operator is
a special case of this construction. In Section 2, we show that, as in the
finite-dimensional case, we can define a connection on the index bundle by
the formula Vo = Po • A^j • Po, and the result of this chapter is a formula
due to Bismut for the Chern form ch(Vo), which is a generalization of the
McKean-Singer formula. Just as in the finite-dimensional case, and by the
same proof, we will show in Section 3 that
lim ch(A«) = ch(V0).
t—>oo
In Section 4, we consider the special case where the family D is associated
to a principal bundle P —> B with compact structure group G. In this case, it
is possible to calculate directly the Chern character form ch(ker(D)) = ch(Vo)
using the Kirillov formula of the last chapter, and we obtain the formula
ch(ker(D)) =
/
JM/B
A(M/B) ch(?/S),

270
9. The Index Bundle
where A(M/B) is the A-genus of the vertical tangent bundle T(M/B), and
dim(M/B) = dim(M) — dim(B) (we assume that M is connected). This
formula is true at the level of cohomology for any family of Dirac operators,
as we will show in the next chapter.
Section 5 is devoted to studying the case when the vector spaces ker(D2)
vary in dimension. The index bundle ind(D) is now only locally a vector
bundle, but it turns out that it has a well-defined Chern character in de Rham
cohomology, and that ch(At) lies in the same cohomology class for all t > 0
if A is a superconnection on n,? such that A[o] = D. In Chapter 10, in the
case where the family D is associated to a Clifford connection, we will present
Bismut's construction of a very particular superconnection for which ch(At)
possesses a limit lim ch(At); combined with the results of this chapter,
this enables us to calculate the Chern character of the index bundle ind(D).
In Section 6, we define the zeta-function and zeta-function determinant of
a generalized Laplacian. Using this, we define in Section 7 the determinant
line bundle detGr»?, D) associated to a smooth family of Dirac operators, and,
in the case in which the index of Dz is zero, a section det(D+) of detGr»?, D).
The determinant line bundle has a canonical metric, known as the Quillen
metric, and we define a connection on detGr,?, D) preserving this metric given
the data of a connection on the bundle M —> B and a Hermitian connection
on the bundle ? over M.
9.1. The Index Bundle in Finite Dimensions
In this section, we will discuss in detail the finite-dimensional analogue of the
index bundle, since it is easier to visualize what is happening here than in
the infinite-dimensional case. Notation will be as in Sections 1.4 and 1.5.
Consider a manifold B and a superbundle H = H+ © H~ —» B. Let D
be an odd endomorphism of H with components D± : H^ —» WT, such that
ker(D) has constant rank, so that the family of superspaces (ker(D*) | z e B)
forms a superbundle over B, called the index bundle of D. (Prom now
on, points of B will always be denoted by z.) Suppose that H+ and H~
have Hermitian structures for which D~ is the adjoint of D+ (so that D is
self-adjoint). If x e H and y e ker(D), then by the assumption that D is
self-adjoint,
(Di,») = (x,Dy)=0,
from which we see that H\ is the orthogonal complement of Ho,
H~ =Hoi,
and hence that the bundle ker(D~) is isomorphic to the bundle coker(D+) =
H~/im(D+), and the bundles H* and H^ are isomorphic.
9.1. The Index Bundle in Finite Dimensions
271
Let A = A[0] + A[X] + Apj +... be a superconnection on H, with curvature
T = A2 e A{B,End(H)). By A.34), the Chern character form of A,
ch(A) = St^e"^),
is equal in de Rham cohomology to the difference of the Chern characters of
the bundles H+ and H~.
Let A be a superconnection whose zero-degree term A(o] equals the odd
endomorphism D of H. If we assume that D is self-adjoint and has kernel of
constant rank, then by the above discussion and A.34), we have the equality
in cohomology
ch(A) =
= ch(ker(D)).
We will refine this equation to the level of differential forms.
Let us denote by Ho C H the superbundle ker(D), graded by H% =
ker(D±), and by Pq the orthogonal projection of H on Ho- Likewise, let
Hi = im(D) C H be the image of the operator D, and let Pi = 1 - Po
be the orthogonal projection of H onHi. The endomorphism D+ gives an
isomorphism between Hf and H^ ¦
Let A be the superconnection
which preserves the spaces A(B,Ho) and A{B,Hi) C A(B,H).
We will make constant use of the following notation: if K e A(B, End(W)),
we write
a
K =
which means simply that
P0KP0
a p
7 6
with a G T(B, End(H0)) etc.
Since A commutes with Po, we see that its curvature has the form
R 0
0 S
Denote by Vo the connection on the bundle Ho given by projecting the
connection A[i] onto the bundle Ho-

272 9. The Index Bundle
We filter the algebra M = A(B, End(ft)) by the subspaces
Lemma 9.1. The differential form R lies in M2, and the curvature of the
connection Vo equals Rp] ¦
Proof. The superconnection Ao = Po • A • Pq on the bundle Ho has curvature
Ag = R. Since Po • A@] • Po = Po • D ¦ Po = 0, we see that
from which the lemma is clear. ?
For t > 0, let 6t be the automorphism of A(B, H) which acts on A{{B, H)
by multiplication by t~i/2. Then At = t1/25t • A • S^1 is again a superconnec-
superconnection on H, and the decomposition of At into homogeneous components with
respect to the exterior degree is given by the formula
The curvature Tt = A2 of the superconnection At is the operator
and the cohomology class of ch(At) = Str(e~-F() is independent of t, and
is equal to the difference of the Chern characters chGio") — chGijj~) for all
t > 0. We will study the limit of ch(At) as t —* 00; it is remarkable that the
following stronger result holds.
Theorem 9.2. Let H = H+ © H~ be a Hermitian super-vector bundle and
let D be a Hermitian odd endomorphism ofH whose kernel has constant rank.
Let A be a superconnection on H with zero-degree term equal to D. For t > 0,
let
A* = tl'a6t ¦ A • Sr1 = t1/2D + Ap, + r1/2A[2] + ...
be the reseated superconnection, with curvature Tt ¦ Then for t large,
uniformly on compact subsets of B and for all Cl -norms.
Proof. We start by proving the following lemma.
Lemma 9.3. A) Under the decomposition H = Ho © Hi, the curvature T
can be written as
X Y
Z T
M2 Mi
Mi Mo
B) The endomorphism T[0] G T(B,End(Hi)) is equal to PfD2-P1 and is
positive definite.
9.1. The Index Bundle in Finite Dimensions
273
C) Denote the inverse of T[0] on Hi by G. The curvature R[2] of the con-
connection Vo on Ho equals
R[2)=Xm-Yli]GZ[i].
Proof. We write A = A + u, where u = Po • A • Px + Pi • A • Po e Mi:
0 fi
v 0
Expanding the right-hand side of the equation
we obtain
*r
P0[A,4Pi
Pi[k,u]P0
If we write
we see that
D2
X Y
Z T
mod
M3 M2
M2 Mi
= (R[2\
= R[2). D
The following lemma is the key step in the proof. We will use the fact that
the space of matrices g of the form
g = l + K, where K € Mi,
form a group; to obtain the inverse of such a matrix, we use the formula
(which is a finite sum, since B has finite dimension),
fc=i
Lemma 9.4. There exists a matrix g with g — 1 € Mi, such that
= 9
X Y
Z T
U 0
0 V
Furthermore
U = X- YGZ (mod M3),
V = T (mod Mi).

274
9. The Index Bundle
Proof. To construct a matrix g which puts T in diagonal form, we argue by
induction on dim(B) — i. Thus, assume that we have found <fr such that
=
Xi
\M2 Mi
Mi Mo
with T< = D2 (mod Mi); in particular,
0 0
0 1 - GTi
We write
1 -Yt
HZi 1
1 -YiG
IZi 1
ij Yi
Zi Ti
Since
0 -YiG
IZi 0
€ Mi, we see that
1 -YiG
IZi 1
-l
€M2i.
\-GZi 1
We have the following explicit formulas for Xi, Yi, Zit and fi mod M2i\
i) 6
= Xj (mod M2i);
y4 = Yi(l - G21) + (Xi - (YiGWiYiG) e Mi+1;
Zi = (\- TiG)Zi + (GZi)Xi - i
fi = Ti + (GZi)Xi(YiG) + Zi(YiG)
= % (mod Mi).
Thus, we can continue the induction.
Now, suppose that we have a matrix g of the required form which dia-
gonalizes T. This implies that
l + K M
N 1 + L
X Y
Z T
U 0
0 V
l + K M
N 1 +
for some
K M
\N L
.Mi, from which we obtain the equation
'.+KX + MZ Y + KY + MT\_\UA + K) UM
- . _ .... m.Tm|-| VN v(l + L)
\NX + Z + LZ NY + T + LT\ ,
Since X is in M2 and K, L, M, N, Y and Z are all in Mi, we see that
A) V = (T + LT + NY)A + L)-1 = T (mod Mi),
B) hence GV = 1 (mod Mi),
C) U = (X + KX + MZ)A + K)-1 =X + MZ (mod M3),
D) Y + MT = UM-KY e M2,
9.1. The Index Bundle in Finite Dimensions 275
E) hence, multiplying on the right by G, M = —YG (mod M2).
Combining the last two formulas, we see that
U = X-YGZ (modM3). D
By this lemma, we may write
e-.««=«arfT e.,?,(
Lemma 9.5. We have the estimate |e~t6'(y)| < Ce~et for some positive e,
as well as for all of its derivatives.
Proof. Essentially, this is true because Vj0] = T[0] = D2 is positive definite on
Hi. The proof uses the Volterra series for e't&t^ (see 2.5): if V = D2 + A,
then
fc>0
where
Ik= [ e-00*0*6t(A)e-aitD3 ...e~'"-^6t(A)e-'"'iDidal...dak.
JAfc
On the simplex
Afc = {(a0,... ,ak) 6 Kfc+1 I Eto "i = 1, ^ > 0},
one of the o^ must be greater than (k + I). Since the operator e~CTitD
decays exponentially when t —> oo, while the operators e~°itXi , j / i, are
bounded, we see that Ik decays exponentially. The sum is finite, since Ik is a
sum of terms of degree at least k with respect to the grading of the differential
forms on B. D
Using this lemma and the fact that t6t(U) — R[2j + O(t-1/2), which is a
consequence of Lemma 9.4, it follows that
'I2) 0
0
Both 6t[g) and 6t(g)~l = HJa~l\ have the form
ll
It follows that
(9.1)
-t«,CF) _
The argument is similar for the derivatives with respect to the base, using
the formula of Theorem 2.48. ?

276
9. The Index Bundle
Corollary 9.6. The limit lim ch(At) exists, and equals the Chern char-
character of the connection Vn on the superbundle Hq = ker(D).
We will now give an explicit homotopy between ch(At) and lim _^ ch(At).
Let us write
The transgression formula A.33) states that
dch(As)
ds
= -*»(«)¦
This may also be seen by the following construction, which is valid for jiny
smooth family A, of superconnections on a superbundle ? —» B. Let B =
B x R+, and let ? be the superbundle ? x R+ over B, which is the pull-back
to jB of 5. Define a superconnection A on ? by the formula
{A0){x,a) = (A,/J(-,«))(*) +da A ^j^.
The curvature T of A is
where T, = A2, is the curvature of As. Since (dsJ = 0, the Volterra series
expansion of eT* is
A ds.
Let
'"¦)¦
Taking the supertrace of the above formula, we see that
ch(A) = ch(As) + a(s) A ds,
with respect to the decomposition B = B x R+. Since ch(A) is closed in
A(B) and ch(As) is closed in A(B), we infer that
dch(As)
ds
---- -da{s).
Integrating this formula, we see that
rT d
ch(At) - ch(AT) = - / ^ ch(A,) ds
= d a(s)ds.
Jt
9.1. The Index Bundle in Finite Dimensions
277
Our task is to show that this integral converges as T —* oo. We will prove
this by a small modification of the proof of Theorem 9.2.
Theorem 9.7. The differential form
a(t) = Str (^±e~^eA(B)
satisfies the estimates, for t large,
\Ht)\\t < c{t)t-v2
on compact subsets of B, for all Cl-norms.
For all t>0, the integral
a(s)ds€A(B)
is convergent, and defines a differential form which satisfies the formula
ch(At) - ch(V0) = d / a(s) ds.
Jt
Proof. We have the following formula for a(t):
Since
dAt D
dim(B)
we have
Inserting
dkt
~dTe
this into
=
(9
1
2*1
O(
0
dt
dkt
~d7'~
1), we
(\°
/2VI°
t-3'2)
(t-1)
' 2fV2
1
" 2tV2
0
0
see that
0
D
O(t~2)
o{t-3'2)
i=2
0
D
-u
)
O(r3/2).
Since only the diagonal blocks contribute to the supertrace, this proves the
theorem. ?
In the special case in which A has the form V + D, so that A[jj = 0 for
i > 2, we see that At = V + <1/2D equals V at t = 0. Furthermore,
fort —oo.

278
9. The Index Bundle
Thus, the integral /0°° a(t) dt defines a differential form on B, and we have
(9.2) ch(V) - ch(Vo) = d f a(t)dt.
Jo
This is a refinement of the fact that
[ch(H+)] - \ch(H-)] =
to the level of differential forms.
H'{B)
9.2. The Index Bundle of a Family of Dirac Operators
The goal of this chapter is to give an infinite-dimensional version of the last
section, in which the family D of operators on the finite dimensional bundle
H is replaced by a family of Dirac operators. The details are slightly more
complicated for two reasons: firstly, because we work in infinite dimensions,
and secondly because we will be interested in the more general case in which
the dimension of ker(Dz) varies as we move about the family.
By a family of manifolds (Mz | z e B), we will mean the family of
fibres Mz = n~1(z) of a smooth fibre bundle tt : M —> B. We assume that
the fibres Mz are compact. However, we will not assume that the base is
compact. Let T(M/B) C TM be the bundle of vertical tangent vectors,
with dual T*(M/B) = T*M/n*T*B. To the family M —> B, we associate the
vertical density bundle
|A,| = \AT\M/B)\ st |AM| ® (t'IAbI)-1.
When restricted to a fibre Mz of M, \An\ may be identified with the bundle
|Am, | of densities along the fibre. We will call a section of |AW| a vertical
density.
By a family of vector bundles (?z | z € B) we mean a (smooth) vector
bundle ? —> M, so that ?z is the restriction of the bundle ? to Mz. For us, ?
will always be a complex superbundle with Hermitian metric. To the family
? —> M, we associate the infinite-dimensional superbundle tt,? = tt»?+ ©
7T.?- over B, whose fibre at z e B is the Frechet space T(MZ,?Z <S>\AM, |1/2)-
A smooth section of n,? over B is defined to be a smooth section of ?® IA* I1/2
over M:
(9.3)
= T(M,? ®\AV\1/2).
The bundle n*? carries a metric, defined by means of the canonical metric on
each fibre r(Mz,?z<g>|AMJ1/2). Indeed, if 4>z € ir.?z = r(Mz,?z®|AMJ1/2),
we see that |<?z(a;)|2 ? ^(Mz, |Am,|), and hence may be integrated:
= f
Jmz
\4>z(x)\2.
9.2. The Index Bundle of a Family of Dirac Operators 279
The reason for inserting the line bundle |AW|X^2 is precisely in order to have
such a canonical metric along the fibres 7r»?. The fibre bundle M will have
a metric along the fibres, which trivializes |AW|. However, unlike in the case
where B is a point, it is important to distinguish between the bundles 7T»?
and z y-» T(MZ,?Z).
Instead of working with the bundle EndGr»?) of all endomorphisms of ir.f,
we will restrict our attention to only those endomorphisms of the fibres which
are the sum of a differential operator and a smoothing operator. Thus, we will
denote by T>(?) the bundle of algebras over B whose fibre at z is the algebra
T>(MZ,?Z <8> |AM,|l/2) of differential operators on ?z ® |AM,|1/2, and whose
smooth sections are families of differential operators Dz whose coefficients in
a local trivialization of M and ? depend smoothly on the coordinates in B.
We denote by K.(?) the bundle of algebras whose fibre at z is the algebra of
smoothing operators on the bundle ?z <g> |Am, l1^2, and whose smooth sections
are smooth families of smoothing operators Kz. (We refer to reader to the
appendix for more details on this bundle.) Since K.(?) is a bundle of modules
for T>(?), we may form an algebra from the sums of operators in T>(?) and
Definition 9.8. The P-endomorphisms of the infinite-dimensional bun-
bundle tt,? are smooth sections of the bundle T>(?) + K(?), that is, families of
operators which may be written in the form
Dz+Kz,
where (Dz \ z 6 B) is a smooth family of differential operators acting on
the family of bundles ?z ® lAjvf.j1'2, and (Kz \ z 6 B) is a smooth family
of smoothing operators on the same family pf bundles. We write End-p(ir,,?)
for the space of smooth sections of this bundle.
We will consider a smooth family D = (D* | z € B) e T(B,V(?)) of Dirac
operators on ?. Thus, D is an odd operator with respect to the Z2-grading,
with components D* : r(B,7r,?±) -» T(B,k*?*), and D2 6 T(B,T>{?)) is
a family of generalized Laplacians. Note that the Dirac operator Dz defines
a Riemannian structure on the fibre Mz (that is, an inner product on the
vector bundle T(M/B) —> M of vertical tangent vectors), and an action of
the Clifford algebra bundle C(T*{M/B)) on ?. In particular, we obtain a
canonical trivialization of the bundle |AV|.
Now assume that for each z ,€ B, the operator Dz is self-adjoint with
respect to the Hermitian structure on the vector bundle ?z. The vector space
ker(D*) has finite dimension. The aim of this section is to define the index
bundle of the family D as a superbundle on B whose fibre at z € B is equal
to ker(D2) when the dimension of this vector space is independent of z; in
Section 5, we will show that the difference bundle [ker(D+)] — [ker(D~)] makes
sense even without this condition.
If A is not an eigenvalue for any operator of the family D*, then the su-
superbundle "H[o,x) whose fibre at the point z is the spectral subspace of (DzJ

282
9. The Index Bundle
If X < n, let (P?x ^ | z 6 U\ n Up) be the family of orthogonal projectors
onto the spectral subspace (A,n) of (DzJ, so that
P[o,n) = P[o,\) + P(\,»)-
Since P(a,/j) is a smooth family of projections, the family of vector spaces
7t?A , = PA , forms a smooth superbundle over UxOUp. It is clear that
9.3. The Chern Character of the Index Bundle
In this section, following Bismut, we will extend Quillen's theory of super-
connections to the infinite dimensional bundle n,? —¦ B, thereby obtaining
a formula for the Chern character of ind(D) in terms of heat kernels. Our
treatment is modelled on the finite-dimensional case, which we discussed in
Section 1.
The space of sections of ir,? is defined by T(B, nm?) = T(M, ? ® |AT|^2).
It is natural to define the space of differential forms on B with values in 7r»?
by
A(B,n,?) = T(M,n*(AT*B) <g>?® \
A differential operator on the space A(B, n,?) is by definition a differential
operator on T(M, w*(AT*B) ® ? ® lA^1/2). Let
A(B, V(?)) = T(B, AT*B ® V{?))
be the space of vertical differential operators with differential form coeffi-
coefficients. If a differential operator D on A(B, 7r«?) supercommutes with the
action of A(B), then this operator is given by the action of an element of
A{B,V{?)). Similarly, we write
A(B, K{?)) = T(B, AT'B ® ?(?))
for the space of smooth families of smoothing operators K" with differential
form coefficients. Denote by da the exterior differential on B.
Definition 9.12. Let D be a smooth family of Dirac operators on ?. A
superconnection adapted to the family D is a differential operator A on
A(B, 7r,?) of odd parity such that
A) (Leibniz's rule) for all v e A(B) and <f> e A{B,nt?),
B) A = D +
A(vct>) =
fB) Aw, where Aw : Am(B,*.e) — A-"{B,*.?).
9.3. The Chetn Character of the Index Bundle 283
It is easy to see that A[j] supercommutes with A(B) if i ^ 1, and hence
Let us show how to construct a superconnection adapted to a family of
Dirac operators D. It suffices to define a connection V**? on the bundle 7r»?,
that is, a differential operator from r(B,7r,?±) to A1(B,tt*?^:) such that
V'-(/</>) = d/A4 /
for all / e C°°(B) and cj> 6 r(J3,7r,?). Let us assume that the bundle M/B
possesses the additional structure of a connection, that is, a choice of a split-
splitting TM = THM®T{M/B), so that the subbundle THM is isomorphic to the
vector bundle tt'TB. From this, we can define a canonical linear connection
on the vertical tangent space T{M/B) using the projection operator
P : TM -» T(M/B)
with kernel the chosen horizontal tangent space TuM. If X is a vector field
on the base B, denote by Xm its horizontal lift on M, that is, the vector
field on M which is a section of TuM and which projects to X under the
pushforward tt, : (THM)X -> T^x)B.
Furthermore, let us suppose that the bundle ? over M is provided with a
connection Ve compatible with its Hermitian structure.
Proposition 9.13. Let a 6 r(M,|AM|l/2), & 6 r(B,|AB|/2) and a €
T(M,?). Define the action o/V^'?, where X is a vector field on B, on
e the
s ® a ® tt*j3 € T(M,? ® IA*!1'2) = T(B,ir.fi)
by the formula
,=^^-^s?m
and that if h€C-(B), then
These
follow easily from Leibniz's rule.

284
9. The Index Bundle
To show that V*'e is a connection, we must show that
/ 6 C°°(B). Using Lemma 1.14, we see that
= /V?f for
s) ® a ® ir*/3
+ s® (nmf?{XM)a + ±n*
+ a ® a ® (n*fn*C(X)/3 - ±n*
+ s
*/3 + s ® a ® tt*
We must now show that V'e preserves the inner-product on irt?• This
follows from the assumption that Vs is compatible with the Hermitian metric
on ?, so that
C(X)\s ® a ® n*0\2 = C{X) ( f \s\2 a2 ® tt*/?2)
\Im/b '
-L
IM/B
= 2(s ® a ® n*0, V?f (s ® a ® *•*/?)).
Thus, associated to a connection on the fibre bundle M/B and a connec-
connection on the bundle ?, there is a natural superconnection D + V*'? for the
family of Dirac operators (D2 | z G B). In Chapter 10, we will also have to
consider superconnections A for which Ap] does not vanish.
The curvature T = A2 € A(B,V(?)) of a superconnection is a vertical
differential operator with differential form coefficients. We have
where D2 is a smooth family of generalized Laplacians and
is a smooth family of differential operators with differential form coefficients
which raises exterior degree in
The results of Appendix 1 apply in this situation, and we obtain the exis-
existence of a smooth family of heat kernels for !F, which we denote by e~ij: €
A(B,K(?)):
C — i
fc>0
where
Ik=
Since Ik vanishes for A; > dim(B), the above sum is finite.
9.3. The Chern Character of the Index Bundle
285
We will show that the Chern character of the index bundle ind(D) can be
computed from a superconnection with A[0] = D, by the same formula as in
the case of a finite dimensional superbundle, namely ch(ind(D)) = Str(e~A ).
Let K = (K* | z € B) e A(B, K{?)) be a smooth family of smoothing
operators with coefficients in A(B), given by a kernel
(x | K | y)
M,iv*A.T'B ® (? ® \K\1/2) ®* (?* ® \K\1/2)).
Here M x,, M = {(x, y) G M x M \ n{x) = tt(j/)} with projections prx(x, y) =
x and pr2(x, y) = y to M, and if ?\ and ?2 are two vector bundles on M, the
vector bundle ?\ HT ?2 over M xT M is given by the formula
?1 B* ?2= prjf!®pr; fa-
There is a supertrace on K(?z) over each fibre Mz of M/B, which gives a
supertrace
Str :T(B,K{?))-*C'X{B).
Suppose that K € A(B, ?(?)); when restricted to the diagonal, its kernel
(x I K' I a;) is a smooth section of the bundle 7r*AT*J3®End(?)®|A,r| over M,
and its pointwise supertrace Strf (x | K" | x) is a section in T(M, ir*AT*B ®
|AW|). This section can be integrated over the fibres to obtain a differential
form on B.
Lemma 9.14. The A{B)-valued supertrace Str : A(B,K(?))
family of operators K is the differential form on B
A(B) of the
f Str?«
Jmz
x\K'\ x)).
If A is a superconnection on 7r«?, then [A, K] is again a family of smoothing
operators with differential form coefficients. The following lemma is one of
the main steps in the extension of Quillen's construction of Chern characters
to the infinite dimensional setting.
Lemma 9.15. dB Six(K) = Str([A, if]) e A(B)
Proof. As in the proof of Proposition 9.9, we may assume that the base B is
an open neighbourhood U C Rp of 0 e Rp, that the family M is the product
Mo x U, and that the bundle ? is the pull-back to M of the bundle ?q —> Mo-
With respect to the identification A(B,irt?) = A(U)®T(M0,?0® |AMo|1/2),
we may write
where D\ aredifferential operators on F(Mo, ?o®|Am0 l1^2) depending smooth-
smoothly on z € U; we also have

286
9. The Index Bundle
where Kf are smoothing operators on T(M0, ?0®| AMo I1/2) depending smooth-
smoothly on z 6 U. By Lemma3.49, we have Str([Df,X5]) = 0, so that we may
assume that A = dv in this trivialization. We see that
a
On the other hand,
dv SU(K) = Y, ^T Str(Kf) dz{
Str(x
= ? j^
A dzt
). ?
Definition 9.16. The Chern character form of a superconnection A on
the bundle rr»?, adapted to a family of Dirac operators D, is the differential
form on B given by the formula
this is well-defined, since e^ = e'* 6 A(B, ?(?)).
Theorem 9.17. Let A be a superconnection on the bundle it*? for the family
of Dirac operators D.
A) The differential form ch(A) is closed.
B) If h.a is a one-parameter family of superconnections on the bundle ir»?
for the family of Dirac operators Dff, then
Thus, the class o/ch(A<T) in deRham cohomology is a homotopy invariant
of the superconnection A.
Proof. The fact that ch(A) is closed follows from Lemma 9.15, since
dB Str(c-') = - Str([A, e~F\) = 0.
9.3. The Chern Character of the Index Bundle
287
The proof of the homotopy invariance is similar. As in Proposition 1.41, we
argue that
where all of the steps may be justified by using the fact that dka/da is a
family of vertical differential operators with differential form coefficients.
The fact that the cohomology class of ch(ACT) depends only on the homo-
homotopy class of DCT follows as in the finite-dimensional case from the homotopy
invariance of ch(A): if Ao and Ai are both superconnections for D, then
a e [0,1] i-» Ao- = crAi + A — <r)Ao is a one-parameter family of supercon-
superconnections for D. Similarly, when Da varies smoothly, we can form the smooth
family of superconnections ACT = Da + V7r-?, and the cohomology class of
ch(Aff) will be independent of c. O
Our goal in the remainder of this section is to prove the following funda-.
mental theorem, which generalizes the results of Section 1 to the case of a
family of Dirac operators. Assume that D is a family of Dirac operators such
that ker(D*) has constant dimension, so that ker(D) is a superbundle over B.
If Pq is the orthogonal projection from nt?z to ker(Dz), then Po G r(B, K(?))
is a smooth family of smoothing operators. The following lemma is clear.
Lemma 9.18. The operator Vo defined by the formula Vo = PoA[i]Po is a
connection on the superbundle ker(D).
For t > 0, let fit be the automorphism of A(B,n,?) which multiplies
Ai{B,tr.?) by t~'/2. Then At =[t1/26t • A-6^1 is a superconnection for the
family of Dirac operators t1^2D.
Theorem 9.19. For t > 0, let
At = t1/26t • A • fir1 = *1/2D + AW + r1'2 A[21 + ...
be the rescaled superconnection, with curvature 7\ = tfit(F). Then for t large,
uniformly on compact subsets of M xw M, for all Cl-norms.
Proof. The proof is formally almost exactly the same as the proof in the finite-
dimensional case given in Section 1, and we give only a few steps. Consider the
algebra M = A(B,EndP(?)) = V(B, ir*\T*B®Endv{?)), where Endv{w,?)
is as in Definition 9.8; we filter M by the subspaces

We will also need the algebra Af = A(B, K.{?)), filtered in the same way.
Let G = (Gz | z e B) be the family of Green operators Gz of (D*J.
Proposition 9.20. The action of left and right multiplication by G preserves
Af.
Proof. Since this is a local question, we may suppose that the bundles M and
S over B are trivial, by replacing B by an open subset U. We may rewrite
the integral representation of G given in B.11) as follows, where Po is the
projection onto the kernel of D:
= f
Jo
Thus, we may decompose da((G + Po)K), where K eAf, into terms propor-
proportional to
r
Jo
where ai + a-i = a. Repeated application of Theorem 2.48 shows that this
integral has the general form
where k = |ai |+1 and Dt € T(B, Endp (nm?)). This integral may be bounded
using the exponential decay of (x | e~'(D +p°^ | y) proved in Proposition 2.37.
Using the formula
^(GK) =da({G+ P0)K) - d°(P0K),
we see that GK is in Af. The case where G multiplies on the right is simi-
similar. D
Let Pi = 1 - Po be the projection onto im(D). If K € M, we will write
Let
be the curvature of the superconnection A; then X, Y and Z are in Af.
Let i?|2j be the curvature of the connection Vo on the bundle ker(D). By
the same proof as in Section 1, we have the formula Rp] = Xp] — ^[lj^^fij-
The space of endomorphisms g € M of the form g = 1+K, where K € Mi,
form a group, with
a
7
P
6
PoKF
>o
PiKPo
X
Z
Po}
(P\
PiKK
Y
T
Af
Af
Af
M
V.U. A IIC \ji
uern onaracter of the Index Bundle
289
Lemma 9.21. There exists g € M with g - 1 e Af\, such that
Furthermore
U = X- YGZ (mod Af3),
V = T (mod M).
The proof is the same as in the finite-dimensional case of Section 1, once
we observe that if G is the Green operator of D2, and if Y and Z € .A/J, then
the operators YG and GZ are also in Aft, and Y{1 - GT) and A - TG)Z are
in A/j+i. Thus, we may construct g as in Section 1 as a product of matrices
of the form
1
GZ
-YG
1
6
1 -f-
0
M
M
0
Thus, as before, we may put T into block diagonal form; there is a family
of operators K 6 A/1 such that, with g = 1+K,
U 0
0 V
where U = R[2) (mod Af3) and K = D2 (mod Afi). Note that the family of
operators 6t(V) is for each t > 0 the sum of a family of generalized Lapla-
cians D2 and an element of Pi -Mi Pi. It follows from Appendix2 that
the family of heat kernels e~tStf-v'1 is a section in A(B,K(?)) for each t > 0.
Furthermore, the uniqueness of solutions of the heat equation implies that
tiv)
{9).
If A(t) : @, oo) -+ T(B, K{S)), we will write A(t) = O(f{t)) if for all e > 0
and / e N and each function (j> e C?°(B) of compact support, there is a
constant C(?, e, <j>) such that
\\«'D>){x){x
y)\\t < C(t,e, <t>)f(t) for all * > e.
Let U be a relatively compact open subset of B. If A is the infimum over
U of the lowest non-zero eigenvalue of the operators D2, then it follows from
Proposition 9.49 that over U,
Thus, we have
St(g)
-i

290
9. The Index Bundle
Since St(g) - 1 = OH'1'2), we see that
e-Rm
0
() (
OH'1'2) O{t~l)
from which the theorem follows. ?
Corollary 9.22. The limit
lim ch(A«) = ch(P0A[i]P0) e A{B)
t—>oo
holds with respect to each C'-norm on compact subsets of B.
The above corollary is equivalent to the McKean-Singer theorem when B
is a point. In that case, a family of Dirac operators is just a single Dirac
operator D on a Clifford bundle ?-»M and the superconnection A equals
D. The Chern character of the index bundle ind(D) is just a sophisticated
name for its dimension, which is the index of D in the usual sense. On the
other hand, we see that Str(e~A<) is nothing but Str(e~tD2), which by the
McKean-Singer theorem equals the index of D.
We will now prove an infinite-dimensional version of the transgression for-
formula, which gives an explicit formula for a differential form whose differential
is the difference ch(Aj) — lim ch(At).
Theorem 9.23. Let D be a family of Dirac operators such that ker(D*) has
constant dimension. The differential form
satisfies the estimates
\\a(t)\\t <
on compact subsets of B, for all Cl -norms, and
ch(At) = lim ch(At)+d [°° a{s)ds
J
= ch(P0A[i]P0) + d / a(a) ds.
Jt
Proof. It is clear that the formula for ch(A() — lim ch(Aj), known as the
transgression formula, is an immediate consequence of the estimate on a(t)
as t —¦ oo.
The proof of the estimate on a(t) is again much the same as that of
Theorem 9.7, except that the estimates are all rewritten in terms of the kernels
of the operators in question. Since
dt It1'2
9.3. The Chern Character of the Index Bundle
291
and Aw e M2, we see that if K(t) : @,oo) -» T(B,fC(?)) is a family of
kernels such that K(t) = O(t~°), then
Furthermore, the proof of Theorem 9.19 showed that
It follows that a(t) is the supertrace of
O{t'1'2)
0 0
0 D
0{t~x'2)
Oit~3'2)
Oit~3'2)
Since only the diagonal blocks contribute to the supertrace, this proves the
theorem. ?
In the next chapter, following Bismut, we will construct a superconnection
A for the family D of Dirac operators associated to a Clifford connection, such
that as t —» 0,
ait) = Oir1'2).
Prom this will follow the existence of a limit lim _^ ch(At), which can in fact
be calculated explicitly, and the following analogue of (9.2):
lim ch(A4) - ch(Vo) = d f a(t) dt.
t—K) JO
Let R € T(.B,/C(?)) be a smooth family of self-adjoint smoothing opera-
operators, such that each operator Rz is odd. Consider the perturbed family of
Dirac operators
(9.4) D + fl
If we square D + R, we obtain
(D + RJ = D2
RD + R2).
Since D2 is a smooth family of generalized Laplacians and DR + RD + R2 is
a smooth family of smoothing operators, we obtain the following lemma.
Lemma 9.24. The square of the operator D + R has the form H = Ho + K,
where Ho is a smooth family of generalized Laplacians and K is a smooth
family of smoothing operators.

292
9. The Index Bundle
There is no difficulty in extending the results in Sections 2 and 3 to the
more general case where the family D of Dirac operators is everywhere re-
replaced by a family D + R of the form discussed above. In particular, the
vector space ker((D + R)*) is finite-dimensional for each z e B, the index of
(D + R)z
ind((D + R)*) = dim(ker+((D + R)z)) - dimmer"((D + R)z))
is independent of z e B (and equals ind(D*), by the homotopy invariance of
the index). As in Proposition9.20, we can also prove that if the dimension
of ker((D + R)z) is constant for all z & B, then the Green operator Gr =
(GZR | 2 € B) of D + R depends smoothly on z, in the sense that if X is a
family of smoothing operators, then XGr and GrX are families of smoothing
operators.
Note that a superconnection A adapted to D + R will have Ap] = D + R,
and hence cannot be a differential operator when considered as an operator
on r(M,7r*AT*B®5® lA^I1/2). Thus, we will allow a superconnection A for
D+R to be the sum of a differential operator on A{B, 7r»?) = T(M, it* AT*B<g>
? ® |A* |) and an element of A(B, Endp(?)). This does not cause any change
in the wording of the analogues of Theorem 9.17 and 9.19 for D + R, nor in
the estimates required in their proof.
First we have the analogue of Theorem 9.17.
Theorem 9.25. Let D + R be the sum of family of Dirac operators D and an
odd self-adjoint family of smoothing operators R. Let A be a superconnection
for the family D + R.
A) The differential form ch(A) is closed.
B) 1/ Au is a one-parameter family of superconnections on the bundle ir*?,
then
Thus, the class o/ch(Aff) in de Rham cohomology is a homotopy invariant
of the superconnection A.
C) In particular, the class of ch(A) in de Rham cohomology depends only on
the homotopy class of the Dirac operator D.
Next, we have the analogue of Theorem 9.19.
Theorem 9.26. Assume that ker((D + R)z) has constant dimension for z e
B.
A) The vector spaces ker((D + R)z) form a smooth vector bundle ker(D + R)
over B.
B) The limit
lim ch(At) = ch(ker(D + R)) € A{B)
holds with respect to each Ct-norm on compact subsets of B. ?
9.4. The Equivariant Index and the Index Bundle
293
We could also state the analogue of the transgression formula, but since
we will not make use of it, we leave its formulation to the reader.
9.4. The Equivariant Index and the Index Bundle
In this section, we will consider the special case of a family of Dirac operators
on a bundle M —» B with compact structure group G. We will see that the
Chern character of the index bundle may be calculated using the Kirillov
formula (Theorem 8.2). This section is not needed to understand the rest of
this chapter or the next.
Let N be a compact Riemannian manifold with an action of a compact Lie
group G. Let E be an equivariant Hermitian Clifford bundle over N, with
equivariant Clifford connection VB and associated Dirac operator Djv acting
on T(N,E). Then ker(D^) and ker(D^) are finite-dimensional G-modules;
we will denote the actions of G and of its Lie algebra g on ker(Djv) by pn ¦
Let P —* B be a principal bundle with structure group G, and let M =
Py.cN be the associated fibre bundle, with base B and typical fibre N.
We also form the associated bundle ? = P xG E, which we consider to be a
bundle over M; thus, ? defines a family of bundles (?z —> Mz \ z € B). A
section of ? is a G-invariant section of the pull-back of E to P x N, that is
a section <t> € T{P x N, E) such that
4>(P9, 9~1x) =
,x) for p € P and x € N.
The operator Djv acts fibrewise on T(P x N, E) and preserves the subspace
of G-invariant sections. Thus Djv induces a family of Dirac operators D =
(Dz | z € B). It is clear that in this special case the vector space ker(Dz) has
constant dimension, and that the superbundle ker(D) over B is induced by
the Z2-graded G-module ker(Djv):
ker(D) = P xG ker(Djv).
Using the Riemannian structure on N, we identify |Ajvlx^2 with the trivial
line bundle. Let 9 = 5Zi^'-^« e -41(-f>) ® g be a connection one-form on the
principal bundle P —+ B, with curvature two-form Up € ^42(P,0)bas- Since
tv,? is an associated vector bundle P xG T(N,E) with infinite dimensional
fibre, we deduce from the connection 9 on P a. natural connection on the
bundle n*? —* B. If CE(Xi) is the action of the Lie algebra element Xi e 9
on T(N, E), we see that
on G-invariant sections.
The following lemma follows from the obvious fact that V"'? preserves
the bundle ker(D), which itself follows from the fact that CE(Xi) preserves
ker(DN).

294
9. The Index Bundle
Lemma 9.27. The projected connection Vo on ker(D) may be identified with
the connection on P xq ker(Djv) associated to the connection one-form 9.
Denote by ch(ker(D)) the Chern character of ker(D) with respect to the
connection Vo. It follows from this lemma that the Chern character form
ch(Vo) is given in terms of the representation pn by the formula
0) = Str(e-'"v("p)).
Denote by I9 the G-invariant analytic function on g given by the formula
Ig(X) = indG(e-x,DN).
The equivariant Chern-Weil homomorphism (j>g maps the equivariant de Rham
complex (Aa(N),de) to the deRham complex (A{M),d). On C[g]G, it co-
coincides with the ordinary Chern-Weil homomorphism, that is, a >-* a(QP),
and maps into the space of basic forms on P, which may be identified with
forms on B. It follows that
Mh) = Str(e~'"'(n'->) = ch(ker(D))
The vertical tangent bundle T(M/B) of M may be identified with the
associated bundle P xGTN. It has a connection VN'e, obtained by combin-
combining the Levi-Civita connection on TAT with the connection on the principal
bundle P as in Section 7.6. Let RMIB be the curvature of the connection
VN'e. Denote by A(M/B) the i-genus of the connection VN'9,
To simplify the discussion, we will assume that TV has a spin structure
with spinor bundle S(N), and that E = W ® S(N) for some G-equivariant
Hermitian twisting superbundle W with connection Vw. Let W = P Xq W
be the corresponding Hermitian superbundle over M. Given a connection on
W, we obtain a connection Vw on W; denote by ch(W) the Chern character
form of the connection Vw:
ch(VV) = Str(exp(-Fvv)).
The differential form A(M/B) ch(VW) is a closed differential form on M,
whose integral over the fibres of M/B is a closed differential form on B.
Proposition 9.28. The Chern character of the connection Vo on the index
bundle ker(D) is given by the formula
ch(Vo) =
m/b
A(M/B) ch(W).
9.5. The Case of Varying Dimension
Proof. The proof is based on the Kirillov formula of Chapter 8:
295
IB(X) =
Ae(X,N) chB(X, W),
where AB(X,N) and chg(X,W) are equivariant characteristic classes given
by the formulas
Ae(X, N) = det1/2
By Proposition7.38 B), we see that the Chern-Weil map (j>e : Ag(N) —>
A(M) sends these equivariant characteristic forms to the corresponding char-
characteristic forms of the associated bundles:
<t>e(che(W)) = ch(VV).
The result follows from the formula fM,B o<j>g = 4>e ° JN- D
We will see in Chapter 10 that the above formula for ch(ker(D)) holds in
cohomology for any family of Dirac operators. We will also show that the
local version of the family index theorem implies the local Kirillov formula.
9.5. The Case of Varying Dimension
In this section, following Atiyah and Singer, we will show that the index bun-
bundle of a family of Dirac operators over a compact base B may be represented
as the formal difference of two vector bundles, even without the assumption
that ker(D) has constant dimension.
Let us start by recalling the definition of a difference bundle. This is
an equivalence class of superbundles on a manifold B, with respect to the
equivalence relation that ? ~ T if there are vector bundles Q and H such
that
^r~ en.
The class of ? is denoted by {?+\ — [?~~] and is called the difference bundle
of ?. If Q is a vector bundle, we will associate to it the difference bundle
[Q\ - [0].
If B is compact, the difference bundles forms an abelian ring K{B) called
the K-theory of B. The sum in this ring is induced by taking the direct
sum, and the product by taking the tensor product, of superbundle represen-
representatives. It is easily verified that the additive identity is the zero bundle, the

296
9. The Index Bundle
multiplicative identity is the class of the trivial line bundle, which we will
denote by [C], and the negative of [?+] - [?~\ is [?~] - [?+]. We will not give
any further results about the ring K{B): as explained in the introduction,
our emphasis in this book is on explicit representatives, by superbundles or
differential forms, rather than the classes that they represent in K-theory
K(B) or deRham cohomology H'(B). We do note, however, that the Chern
character defines a homomorphism of rings from K{B) to the deRham co-
cohomology H*(B) of B; this follows from the fact that the Chern character
vanishes on superbundles of the form Q ffi Q. In fact, this map may be shown
to induce an isomorphism of K(B) ® C with the even deRham cohomology,
by the Atiyah-Hirzebruch spectral sequence.
If ker(D) has constant dimension, we have defined the index bundle to
be the difference bundle represented by ker(D). Since M is compact, there
is a positive e such that U\ = M for 0 < A < e. It follows that ind(D)
is also represented by H[o,a) for 0 < A < e. Motivated by this, we would
like to define ind(D) in such a way that when restricted to U\, it has the
superbundle W^.a) as representative. The following lemma shows that this
approach is consistent on U\ l~l Up, for 0 < A < /x.
Lemma 9.29. // 0 < A < n, then the difference bundles represented by
are equal.
Proof. By the isomorphism
we see that we must prove that on U\ D Up,
On the bundle H*x ^, the operator (D+)*D+ = D~D+ has spectrum lying in
the interval (A, /j), and hence is invertible (an explicit inverse is given by the
Green operator G). It follows that the operator
induces the desired isomorphism of bundles. D
We will show that if B is compact, there is a difference bundle of the
form [E] — [CN] which when restricted to U\ is equivalent to the difference
bundle W[o,a)- This will show that ind(D) may be considered as an element of
K(B). We will also generalize Theorem 9.19 to give a formula for the Chern
character of ind(D) in the general case.
We make use of the simple observation that if D is a family of Dirac
operators such that ker(D~) vanishes, then
dim(ker(D*)) = ind(Dz),
and is thus independent of z. Thus, one way to perturb a family so that
its kernel ker(D) becomes a vector bundle is to modify it in such a way as
9.5. The Case of Varying Dimension
297
to ensure that D+ is surjective for each z, and hence that D has vanishing
kernel. This is not difficult if we permit ourselves to perturb Dz by a family
of smoothing operators.
We introduce the trick of replacing M by an enlarged version M' = MU B,
obtained by adding one point pz to each fibre Mz of M —> B. Similarly, we
replace ? by the bundle ?', which coincides with ? over M and whose fibre
over the point pz is a constant vector space V with V+ = CN and V" = 0.
Note that n*?' is isomorphic to ir,? 9(Bx C^), so that
r(B,7r,f) 2 r(B,ir.f) <BC°°{B,CN).
The family D of Dirac operators corresponds in a natural way to a family of
Dirac operators on the family of vector bundles ?' —> M'.
Let ip : CN -v T{B,it*?-) be a linear map from CN to T{B,nt?-).
Let ij)i be the sections of tt,? ~ corresponding to the basis vector e< € CN.
We define a family of smoothing operators Rj, 6 T(B,K{?')) on the vector
bundles ?' —» M' by the formulas
Y,e*W"*« *)' for *
It is easy to see that R^ is odd and symmetric. We define D,/, to equal
given explicitly by the formula
D *
N
= D+<t>
N
N
for 4> e r(B,ir,?+) and r(B,7r,?-) respectively, and u{ ? C°°(B). We thus
have a family (Dj)z of the type described in (9.4).
In order for ker(D^) to vanish, we must demand that Dj is surjective for
all z 6 B. The following lemma shows how to choose suitable ipi.
Lemma 9.30. There exists an integer N and a map tfi : CN —> T{B,ir*?~)
such that the map (Dj)z is surjective at each point in z e B.
Proof. By Proposition 3.48, we know that
ker(DJ) © D+[r(ilf,,?+ 9 \K\1/2)} = T(MZ,?- ® \K\^2)
at each point z & B. We need to construct a collection ipi of sections of irt?~
such that at each point zgB, the projection onto the vector space ker(D;5)
of the sections ipf span it. We proceed as follows.
Given z0 € B, we can find a A > 0 such that z0 ? U\, a ball U{zo) C B
around z0 and sections <j>i e T{U{za),ir*?) such that for every z G U(zq)
the elements <$>\ form a basis of {'H[0<x)) z- If X € Q°(^(zo)) >s a cut-off

298
9. The Index Bundle
function on U(zq) equal to 1 around z0, the sections \(z) <f>f can be extended
to smooth sections of the bundle n,? on B. Thus, for each zq 6 B, there
exists a A > 0, an open set V(zo) C U(zq) and sections fa 6 T(B, 7r,?) such
that V* span (W^,J, and hence their projections onto the vector space
ker((D~)z) span this space, for z € V(zo). Since B is compact, it is covered
by a finite number of sets V(zo), and we may take for (fa) the union of the
sections ip* for just these sets. D
If A is a positive number, let Ux^ be the open subset of U\ consisting of
those points z e U\ such that the composition of
C -t M-), ^ (***>).
is surjective. If Ai is the smallest non-zero eigenvalue of (Dz°J, then zq is in
{7a,v> if A < Ai. Thus, B is covered by the open sets {U\t^ | A € R}.
Proposition 9.31. A) Ifi> satisfies the conditions o/Lemma9.30, we have
the short exact sequence of vector bundles on U\^,
0 - ker(D,)
. 0.
T/ius, over [7a, the difference bundles [ker(D^,)] — [CN] and 7i[o,x) are
equivalent.
B) The difference bundle [ker(D^)] - [CN] o/ if (B) is independent of the
choice of map ip : CN —> F(B, 7r»?~) satisfying Lemma 9.30.
Pnoo/. Let us show that the bundle map
P[0,x) 0 1: ker(Dv,) C 7r,?+ © (B x CN)
) © (B x CN)
[)
is injective. Indeed, if (</>, u) e ker(D^,) lies in the kernel of this map, then we
see that u = 0, and hence that 4> 6 ker(D). However, the projection P[o,a) is
the identity on ker(D) if A > 0, and hence 0 = 0.
The bundle map
is surjective by the definition of U\^.
The composition of the two maps
ker^) C 7T.5+ 8 (B x CN) H?^L> W+x) © (B x CN)
is the bundle map from ker(D^) to HZ, ^ which sends (<t>,u) e ker(D^,) to
P[o,\){?>P[o,\)<t>) + P[o,A)iK«) = -P[o,A)(D0 + V(u)) = P[o,a)D^(^, u),
which vanishes, since (<j>, u) € ker(D^,). Thus, we see that
(B x CN)
9.5. The Case of Varying Dimension 299
is a complex. Let us show that it is a short exact sequence. If 4> € [Ht x^)z
and u e Cw are such that P.qX.(Dz<? + ipz{u)) = 0, then there exists an
a € {n,?+)z such that P*OtX)a = 0 and (D+)z0 + ^(u) = (D+)za. Thus
D> — a,u) 6 ker(D^), and its image by P[o,a) is our element D>,u).
It remains to prove the independence of this construction from the map
i> : CN -> r(B,ir.f-). Consider another map $ : C" -* r(B,ir,?~). The
bundle ker(D^,) may be identified with a subbundle of ker(D^,e^) by sending
((/>,u) e (n*?+)z © CN to (^,ii,0) e {**?+)z © CN © C*. Consider the
sequence of bundles
0 -» ker(D^) - ker(D^^) -»(B x C*) -» 0,
where the last arrow maps (</>, u, u) 6 ker(D^,^) to u 6 C'1'. This sequence
is exact at each point z 6 B. To see this, choose u € CN. The surjectivity
of D^ implies that there exists u € CN and 4> 6 7T,?Z such that i/5(u) =
?
^ + VCu) ?
Using this proposition, we may define the index bundle of any family of
Dirac operators on a compact base.
Definition 9.32. The index bundle of a smooth family of Dirac operators
over a compact base B is the virtual bundle
ind(D) = |ker(D*)] - ?"],
where i/> is chosen as in Proposition 9.31.
We can now prove a generalization of the McKean-Singer formula to su-
perconnections associated to families of Dirac operators.
Theorem 9.33. Let A. be a superconnection on the bundle n,? for the family
D. Then the cohomology class of ch(A) is equal to the Chern character of the
index bundle ind(D).
Unlike in the case where ker(D) had constant dimension, the differen-
differenrm Str(e~**'(A 1) may have no limit as t —> oo when the dimension of
tt However we may replace the family D over M —>B
Proof.
tial form Str(e~**'(A 1) may have no limit as t —> oo when the dime
ker(D) is not constant. However, we may replace the family D over M
by a family D^ over M' —¦ B such that ker(D^,) is a vector bundle.
Let A be a superconnection for the family D, and transfer the operators
A[i] to operators on A{B,n*?'). Let Ae denote the superconnection A + eJt,/,
for the family D + eR$. Applying Theorem 9.25 to the one-parameter family
AE, we see that the cohomology class of the differential form

300
9. The Index Bundle
does not depend on e. For e = 1, we know by Theorem 9.26 that ch(Ae)
represents the cohomology class of ch(ker(D^,)), the Chern character of the
vector bundle ker(D^). Taking e = 0, we see that
ch(Ao)=ch(A)+JV 6.4E).
It follows that ch(A) and ch(ker(D^,)) — N represent the same class in coho-
cohomology, from which the theorem follows. ?
9.6. The Zeta-Function of a Laplacian
In this section, we give an application of the existence of the asymptotic
expansion for the heat kernel: the construction of the zeta-function of a gen-
generalized Laplacian and the definition of its renormalized determinant. This
construction, applied to a family of Dirac operators, is used in the theory of
determinant line bundles, which we will describe in the next section. The
reader only interested in the family index theorem may omit the rest of this
chapter.
If / € C°° @, oo) is a function on the positive real line (well-behaved at 0
and oo in a sense which we will describe shortly) the Mellin transform of
/ is the function
r
Lemma 9.34. Let f e C°°@,oo) be a function with asymptotic expansion
for small t of the form
k>-n
and which decays exponentially at infinity, that is, for some A > 0, and t
sufficiently large,
\
Then
A) the Mellin transform M[f] is a meromorphic function with poles contained
in the set n/2 - N/2;
B) the Laurent series of M[f] around s = 0 is —gos'1 + /o + O(s).
Let h € C°°@, oo) be a function with asymptotic expansion for small t of
the form
fc>-n
and such that for some A > 0, and t large,
\h(t)\ < Ce~tx.
Then f(t) = J(°° h(s) ds satisfies the above assumptions.
9.6. The Zeta-Function of a Laplacian
301
Proof. Choose Re s large, and split the integration over [0, oo) into the inter-
intervals [0,1] and A, oo):
r(»)M[/](«)= f1 f(t)t'~1dt+ r
jo Ji
The second term, the contribution of the integral at infinity, is clearly entire,
by the exponential decay of f{i) ast-*oo. The first term may be expanded
in an asymptotic series, and then integrated explicitly, term by term:
k<K
+ g [\logt)t'-1dt+ f
Jo Jo
where r(s) is holomorphic in the right-half plane Res > -K/2. Since the
inverse of the Gamma-function T(s) is entire, A) follows.
Because T(s) has a simple pole at s = 0 with residue 1, it follows that
around s = 0, sM[f] has the Taylor series
Prom this, B) is clear.
If h is as in the statement of the lemma, then to show that J"t°° h(s) ds has
an asymptotic expansion of the required form, we write, for t small,
/ w = r D ihksk/2)ds - f (h^ - Dhksk/i)ds
Jt fc<0 J° k<0
fc<o
from which we obtain
where c is the constant
_ + f (h{s) _ Y, hksh'2) ds + ^ h(s) ds. D
1 Jo ^ t^n '1
afc/2"
If we define LIMt^0 /(*) to be the coefficient /0 in the asymptotic expan-
expansion of /, we see that part B) of the above lemma implies the formula
ds
«=o
= L.lM/(t),

302
We will call LIMt_0 f(t) the renormalized limit
If H is a self-adjoint generalized Laplacian on a
A is a positive real number, the zeta-function of
9. The Index Bundle
of f(t) as t -> 0.
compact manifold M and
H is the Mellin transform
C(s,H,X) =
1 (s)
f
Jo
where P(a,oo) is the spectral projection associated
lying in (A, oo). This function equals Tr(P(\tOO)H
the dimension of the manifold M.
As an example, consider the scalar Laplacian
eigenvalues {0,1,1,4,4,..., n2, n2,... }. If 0 < A
that
-1 dt,
to H onto the eigenvalues
*) for s > n/2, where n is
d?/dt2 on the circle, with
< 1 and s > 1/2, we see
1W
n~2' = 2<Bs),
n=l
where C(s) is the Riemann zeta function.
We can understand the behaviour of the zeta-function as a function on
the complex plane by means of the Minakshisundaram-Pleijel asymptotic
expansion of the heat kernel.
Proposition 9.35. The function ((a, H, A) possesses o meromorphic exten-
extension to the whole complex plane, and is holomorphic at s = 0.
Proof. As t —+ 0, we have
Tr(P(A,oo)e-t")=Tr(e-'H)-X;e
,-tAk
fc=0
k=-n/2
where Ao < • • ¦ < Am < A are the eigenvalues of H lying in (—oo, A], enumer-
enumerated according to multiplicity. The fact that Tt(P(\jOO)e~tH) decays expo-
exponentially fast as ( —> oo has been proved in Lemma 2.37. ?
Denote by C'(s, H, A) the derivative of ?(s, H, A) with respect to s. Fol-
Following Ray and Singer, we define the zeta-function determinant of a
generalized Laplacian H for which 0 is not an eigenvalue to be
det(ff) = e-<'<0'H'°\
This definition is motivated by the fact that if H is a positive endomorphism
of a finite dimensional Hermitian vector space V, and
C(«, H) = M [Tr(e-tH)] = Tr(iT'),
then C'@> H) = — log det(H). The zeta-function determinant of a generalized
Laplacian H is a renormalized determinant adapted to the class of generalized
Laplacians.
303
9.6. The Zeta-Punction of a Laplacian
If H is a self-adjoint Laplacian, let 0 < A < /x, and let A < Ai < • • • <
Am < M be an enumeration of the eigenvalues of H lying in the interval (A, n].
The following result is clear.
Proposition 9.36. The zeta-function of H satisfies the formula
m
<(«,H,A) = <(«.¦»,/i)
i=l
and hence its derivative at s = 0 satisfies the formula
m
C'(O,ff,A) = C@,H,n) -5>gAi.
t=i
In the following proposition, we calculate the variation of the zeta-function
?(s, H',X) of a family H" of positive generalized Laplacians on M with res-
respect to the parameter z. We work on the set U\ where A is not an eigenvalue
oiHz.
Proposition 9.37. Over the set V\, the derivative with respect to the pa-
parameter z of the zeta-function of a family of generalized Laplacians is given
by the formula
Proof. We argue at large Re s, where both sides of the formula are given by
convergent integrals. If we differentiate the expression
c(s> h' 'x)=wi r tk^w^k1 dt
for the zeta-function with respect to z and apply Corollary 2.50, we obtain
that
(9.5)
Since P = P*x ^ is a projection, Leibniz's rule shows that
dzP = PdzP + dzPP;
multiplying on the left and right by P, we see that PdzPP = 0, so that
dzP = F(8,P)A - P) + A - P){3ZP)P.
Thus the second term on the right-hand side of (9.5) vanishes. On the other
hand, integration by parts shows that in general,
M[t/'] = -sM[f),
so that the first term can be rewritten in the desired form. D

304
9. The Index Bundle
The following theorem is the analogue for the zeta-function determinant
of the formula
^ logdet(H') = Tt{dzH*(H *)~l)
for the variation of the determinant of a family of operators Hz on a finite-
dimensional vector space.
Proposition 9.38; IfH* is a family of generalized Laplacians on a compact
manifold M, then over the set U\,
^'{s = 0,H',X) = -^)
Proof. To apply Lemma 9.34 B), we must show that
has an asymptotic expansion. Consider the asymptotic expansion
-siH ~ f) ak(z)sk.
k=-n/2
Recalling from the proof of the previous proposition that
we see that Tr(PA ^dzH'e''11') has an asymptotic expansion of the form
k=-n/2
Thus, we can apply Lemma 9.34 to conclude that
/oo
has an asymptotic expansion. ?
9.7. The Determinant Line Bundle
Using the results of the last section, we will now define the determinant line
bundle associated to a family of Dirac operators, its Quillen metric, and its
natural connection. It is helpful, however, to see first how the theory looks
in finite dimensions.
Let Ti. = 7i+ © 7i~ be a finite-dimensional superbundle over a manifold
B, such that that dim(W+) = Aim(U~) = m. The map
AmD+ : kmU+ -> AmH~
9.7. The Determinant Line Bundle
305
is a smooth section of the determinant line bundle
det(W) = (A171?*-1-)-1 ® AmH~,
called the determinant of D+, which we denote by det(D+). The section
det(D+) vanishes at precisely those points where D+ is not invertible. In
particular, if D+ is everywhere invertible, the section det(D+) defines a triv-
ialization of the line bundle det('H), since it is nowhere vanishing.
Let Vw be a connection on the bundle H which preserves the decompo-
decomposition H = H+®H~; it induces connections Vdet(T<) and VEnd(Ti) on the
bundles det(W) and End(H).
Proposition 9.39. The section det(D+) satisfies the differential equation
Vdet<w>det(D+) = -det(D+) TrH+((D+)-1VEnd(w)D+).
Proof. We may assume that B is an open set on which H has been trivialized,
W* = U x if*, so that V = d + w, where u = u+ © u~. In this frame, the
connection Vdet'w' is given by the formula d — Str(w). If we replace D+ by
a small perturbation D+ + e^D+, where 5D e T(B,Hom(H+ ,H~)), we see
that
det(D+ + e«5D+) = det(D+(l
= det(D+) det(l
= det(D+) + edet(D+) Tr((D+)-16D+) + O(e2),
where we use the fact that
d
de
?=0
From this, applying the sign rule for graded tensor products, we see that
VdetKdet(D+) = -det(D+OYw+((D+)-1dD+) - Str(w) det(D+).
On the other hand,
Trw+((D+)-1VEnd(w»D+)=TrTi+((D+)-1(dD++u;-D+ + D+w+))
= Trw+ ((D+)-1dD+) + Str(w). D
We will now show how to define a determinant line bundle using a family
of Dirac operators D parametrized by a base B. Recall that if ? and T are
two vector bundles, then
det(? © T) = det(?) ® det(J').
Lot 0 < A < fi be a pair of positive real numbers, and consider the decompo-
decomposition
over the open set U\ C\ U^. Prom this, we see that
det(ttlOl/1)) =

9. The Index Bundle
Denote by P(x,fi) the Dirac operator restricted to W(a,/j). The line bundle
det(W(A,^)) over U\ fl U^ is trivial, since it has a nowhere-vanishing section
det(D^ ^j). Thus, the line bundles det(W[0,A)) piece together to form a line
bundle over all of B, called the determinant line bundle associated to
the family of Dirac operators D and denoted by detGr»?, D); to a section s €
r(U\PU,t, detGi[o,A))), we associate the section s®det(D*x ^) of detGi[0|A,)).
We have assumed that the vector bundle ? has a Hermitian metric, and
hence that irt? has a metric too, given on sections sz and tz € n,?z =
/.
'xeM,
We would like to define a natural metric on the line bundle detGr,?, D). From
the metric on tt,?, we obtain a natural metric on each of the vector bundles
W[o,A)i which is known as the ii'-metric. It follows that each of the line
bundles det(W[n,A)) has a natural metric, which it inherits from the metric
on H[o,x) in the natural way. However, the metrics on the bundles det(W[OiA))
and det(W(o,^)) over Ux n U^ are not equal, and they differ by a factor equal
to
where the metric is the ?2-metric on the trivial line bundle det(H(\llt)).
The family of generalized Laplacians D~D+ on nt?+ consists of self-adjoint
operators, since (D+)* = D~. The following lemma is clear.
L
Lemma 9.40. If X < Aj < • • • < A
between A and ft, then
< fx are the eigenvalues of D~D+ lying
By choosing local trivializations of the bundles M and ? over B, we see
that the zeta-function of the family D"~D+ is a meromorphic function of s
with smooth dependence on z e B. By Proposition 9.36, its derivative at
s = 0 satisfies the formula
From this, we easily see that the metric
i i
on the line bundle det(W[o,A)) agrees with the corresponding metric on the
line bundle det(W[0)M)), and hence all of these metrics patch together to form
a metric on the line bundle detGr,,?, D), which is the Quillen metric.
9.7. The Determinant Line Bundle 307
If the family D has index zero, then dim(WiAj) = dim{M70y.) for all
A > 0, and so that it makes sense to speak of the section det(Di A^) of the
determinant line bundle detGi[oiA)) over the set Ux- Since
det(DjU) =det(D+ A)) ®det(D(+i/0) 6 T{UX D t^det(«[„,„))),
we see that these sections patch together to give a section det(D+) of the line
bundle detGr,f,D).
Proposition 9.41. Let D be a family of Dirac operators of index zero on a
vector bundle ? over the family of manifolds M —* B. There is a canonical
smooth section det(D+) of the determinant line bundle det(yr»f, D), which
vanishes precisely where D is not invertible. The section det(D~*~) satisfies
the formula
|det(D+)|a = det(D-D+I/2,
where by det(D~D+) we mean the zeta-function determinant e~^ @>D D |0'
of the symmetric Laplacian D~D+. (This is the analogue for Dirac operators
of the formula det(ylM) = | det(^4)|2 in finite dimensions.)
We will now define a natural connection on the determinant line bundle
detGT«?, D), compatible with the Quillen metric. To do this, we need a
unitary connection V*** on the bundle ir,?, for example, the one defined in
Section 2 from connections on the bundle M/B and on the Hermitian vector
bundle ?.
From the connection V'0|A) = ^o,A)^v'?P[o,X) on W[o,a)> we obtain a con-
connection on detGi|o,A)) compatible with the ?2-metric, which we denote by
The operator D(aiOO) = P(x,oo)& °ver the open set U\ has the property
that the bundle ker(D(A|0o)) equals W[o,a)- We may define a superconnection
associated to the operator D(a?0o) by the formula
¦ VT-?,
and its rescaled version
Define two differential forms a±(«, A) 6 A(B), given by traces over nt?+ and
¦k*?~ respectively:
)
Lemma 9.42. The one-form components of the differential forms a+(s, A)
anda~(s, A) satisfy

308 9- The Index Bundle
and have asymptotic expansions for small s of the form
Proof. To see that q+(s,A)[1] = a~(s, A)|i], we use the facts that D* = D
and that V'e respects the metric on n,?.
The one-form component of a±(s, A) is equal to
= -1TW (A - P[0,
since P(A,0o)(V'r>?P(x,oo))f>(x,oo) = 0. The result now follows from Propo-
Propositions 2.46 and 2.47, which give asymptotic expansions for Tr(DPt) and
Tr(XPt) at small t respectively, where Pt is a heat kernel, D is a differential
operator, and K is a smoothing operator. ?
By Lemma9.34, we conclude that the functions Jt°°a±(s,A)[i]ds have
asymptotic expansions for t small. Lert
The sum 0i + 0^ is real, while the difference 0% — 0^ is imaginary.
Lemma 9.43. dC'@,D-D+, A) = -@+ + 0^)
Proof. Using the formula /t°° e~sW ds = H-xe~tH', we see that
ft = - LIM r Tr..?+ (P(AiOo)D-[V-e, D+]e-sD*) ds
t-*o jt
t-.o
and
=-LIM rTr..?
*-*0 Jt
= LIM f°° Tr..?+ (P(Ai0o)[V"-?, D-]D+e-»°2) ds
*~*° Jt
Adding the one-forms 0i and 0^ together, we obtain
The lemma follows immediately from Proposition 9.38. ?
9.7. The Determinant Line Bundle
3U9
We can now define the connection on detGr,?, D). On detGi[0,A))) we take
the connection
ydet(Kl0>i)) +;0+
Lemma 9.44. The section det(DJ^ ^) is parallel with respect to the connec-
tion Vdet(*<*.M>) + (/3+ - Pi) on det(HM).
Proof. Observe that
By Proposition9.39 applied to the bundle H = Ti(\tli), we see that
det(D+>M)) = -TrH, ^(det(D+ifl))(D+,/i))~1
+i(j))O3+-/3+). D
The connection Vdet^Hi°"'' on detGi[o,^)) is the tensor product of the
connections Vdet(Kicx)) an dGi)
ydet(H[0%tl)) _
Lemma 9.44 now shows that the connections
on the line bundles det(W[o,A)) and det(W[o,M)) agree over the set V\ PI U^,
when we identify these two bundles by multiplying by the section det(D^ >)
of det (W(a,h)) • Denote the resulting connection on detGr,?, D) by Vdet('r'?'D>.
{tii the Quillen
Proposition 9.45. The connection ydet(jr,?,D) ^s
metric \-\q on detGr,?, D).
Proof. If | • | and || • || are two metrics on a line bundle C such that || • || = e^| • |
and V is a connection compatible with the metric | • |, then it is easy to see
that V + df + u> will be compatible with the metric || • || for any imaginary
one-form w.
The connection Vdet(Mi°x^ is compatible with the L2-metric | ¦ |. Thus,
we see that any connection of the form
Vdet<«io.*)) - id?'(o,D~D+,A) + u = Vdett-n^^ + l{0i+0j;)+u>
where w is an imaginary one-form, will be compatible with the Quillen metric
| • |Q = e-<'(o,o-o*,\)/2\. |. Choosing w = (/3+ -/3A")/2, we obtain the desired
result. ?

310
9. The Index Bundle
Appendix. More on Heat Kernels
In this appendix, we will prove a generalization of the construction of heat
kernels of Chapter 2 which we need in the course of this chapter.
Let M be a compact manifold, let ? be a vector bundle on M, and let
A = Y17=o ' 'De an finite-dimensional algebra with identity graded by the
natural numbers. (In practice, .4 will be an exterior algebra.) We will denote
by T> the algebra of differential operators T>(M, ? ® \h\1/2), by K. the algebra
of smoothing operators on ? ® IAI1/2, and by V the algebra of operators
acting on Y{M,? ® IAI1/2) generated by V and by K. We call an element of
V a ¦P-endomorphism.
Let M be the algebra V ® A. We define a decreasing filtration of the
algebra M by setting Mt = J2j>i V ® A'. The space T(M, ? ® |A|1/2) ® A
is be made into a module for M, by letting V act on T{M,? <g> IAI1/2), and
letting A act on itself by left multiplication.
We will prove that an operator of the form
where Ho is & generalized Laplacian, K e K and ,F[+] ? Mi, has a heat
kernel satisfying many of the properties of the heat kernel of a generalized
Laplacian. We define a heat kernel for T to be a continuous map (t, x, y) t->
pt(x,y) e(?® \A\1/2)X ® (f ® \A\1/2)y ® A which is C1 in t, C2 in x, and
satisfies the equation
(
with the boundary condition that for every s 6 T(M, ? ® lA^I1/2) ® A,
lim / Pt(x, y) a(y) = s(x) uniformly in x ? M.
t—>o->y€M
t—>o-
Let us start with the case where A is the algebra of complex numbers C,
so that there is no component ^[+j. Thus, we wish to construct the heat
kernel of an operator H of the form H = Hq + K, which differs from a
generalized Laplacian Ho by a smoothing operator K. Since we know that
Ho has a smooth heat kernel satisfying certain strong estimates, it is not very
surprising that we can prove the same things for H. To construct the heat
kernel of if, we use a generalization of the Volterra series of Proposition 2.48.
Proposition 9.46. The series
Qt =
da
Appendix. More on Heat Kernels
311
converges in the sense that the corresponding series of kernels converges for
t > 0, with respect to any Cl-norm, ?>0, to a kernel
qt e T(M x M, {? ® IAI1/2) IS {?* ® \K\112)).
The sum is C°° with respect to t and is a solution of the heat equation
with the following boundary condition at t — 0: if <j> is a smooth section of
?®\A\1l2,then
lim Qts = s in the uniform norm.
t—>o
Thus qt(x, y) is a heat kernel for the operator H = Hq + K. Furthermore the
kernel of the difference
e-tH
-CTJktHn
= ][](-*)* / e-aotH°Ke-aitHo ¦ ¦ ¦ Ke-"k
do-
tends to 0 when t —> 0.
Proof. Since K is a, smoothing operator, the operator e~tH°K has a smooth
kernel for all t > 0, and
\\e-tH°K\\t<C(e)\\K\\t,
for some constant C(?) depending on 1. It follows that, for A; > 1,
k\
Thus the series ?]*>i converges with respect to the C'-norm, uniformly for
t > 0, with similar estimates for the derivatives with respect to t. It is easy
to verify, as in theorem 2.23, that the complete sum ?)fc>o ^ a sol11*'011 of
the heat equation. The estimate \\e~tH — e~tH° \\e = O(t) is clear, and implies
the boundary condition for t = 0. D
If the operator H = Hq + K is symmetric, then we can extend a number
of properties which hold for a generalized Laplacian Ho to the case where K
is non-zero.
A) As in Proposition 2.33, H is essentially self-adjoint on the Hilbert space
of L2-sections of ? ® |A|1/2.
B) The unique self-adjoint extension of H has a discrete spectrum bounded
below; each eigenspace is finite-dimensional and is contained in the space
of smooth sections.

312
9. The Index Bundle
C) The analogue of Proposition 2.37 holds, namely, if H - Ho + K, and if Pi
is the projection onto the eigenfunctions of H with positive eigenvalue,
then for each ? € N, there exists a constant C(?) > 0 such that for t
sufficiently large,
||( | | y)\\t <
where Ai is the smallest non-zero eigenvalue of H.
D) The Green operator G, which is H on ker(ff)-1- and 0 on ker(i?),
operates on the space of smooth sections of ? ® IAI1/2, and is given by
the formula ^
G= / Pie~mPldt.
Jo
We now turn to the case of an operator T of the form
which differs from a generalized Laplacian by the sum of a smoothing operator
K e K. and an operator J^j lying in the ideal M\. The operators T and H
differ by an operator J[+] of positive degree in the finite-dimensional graded
algebra A. Since we have already constructed the heat kernel of H = Ho + K,
the Volterra series once more gives a candidate for the heat kernel of T: for
fixed t > 0, define the operator e~<jr to equal
where
h =
i...e
hWe
_-<7l,tH
da.
The sum is finite, since Ik € Mk, and for k large, we know that Ak = 0,
and hence S\k is zero. Thus, we only have to make sense of each term in the
sum. We need the following lemma.
Lemma 9.47. Let D be a differential operator of order k. There exists a
constant C > 0 such that if K is a smoothing operator, one has the following
bounds: for t € [0, T], where T is a positive real number, we have
\\De-tHK\\t < C\\K\\k+t
\\Ke-tHD\\t < C\\K\\k+t.
Proof. There exists a constant C(l) such that for 4> € Tl(M, ?) one has for
t e [0,31, \\e-tH4>h < C{t)W\\t. The bound WDe'^KWt < C(?)\\K\\k+t
follows easily. Then we write
Ke~tHD = {D*e-tH'K'Y.
The above bound applies to the kernel of D*e~tH'K* , yielding \\Ke-tHD\\t <
c(e)\\K\\k+l. a
Appendix. More on Heat Kernels 313
We can now complete the proof of the following theorem.
Theorem 9.48. There exists a unique heat kernelPt(x,y) for T, which is a
smooth map from t G @, oo) to T(M x M, {? ® |A|V2) B (?• ® \A\V2)).
Proof. We must show that each term Ik has a smooth kernel. On the simplex
A*, one of the at must be greater than (k + l)~l. Since for (fc + 1) < a < 1
and fixed t, the operator e~atH has uniformly smooth kernel, it follows by
iterated application of Lemma 9.47 that the operator
has a smooth kernel which depends continuously on (cro, •. ¦, <?*) € Afc. Thus
the integral makes sense as an operator with smooth kernel. As in Chapter 2,
we see easily that e~tH + X]fc>o(~*)*^t satisfies the heat equation for T. The
boundary condition as t —> 0 holds, since for </> smooth, each term Ikcf> has a
limit as t —* 0; when multiplied by (—t)k, each term contributes zero except
the term Iq = e~tH. Uniqueness is proved as in Chapter 2 as a consequence
of the existence of the heat kernel for the adjoint operator T*. ?
There is an analogue of Proposition 2.37 for operators of the type T.
Proposition 9.49. Suppose T = H + J^+j where H = Ho + K is the sum of
a symmetric generalized Laplacian Ho and a symmetric smoothing operator
K, and J|+] € Mi- If Pi is the projection onto the positive eigenspace of H
and P\ commutes with ^[+j, then there exists an e > 0 such that the kernel
of the operator e~t:F satisfies the estimates
\\(x | Ae-'^P! 12,)||,
on AI x M as t —> oo.
Proof. Using the Volterra series, we may write
fc=0
where
4= / e
It follows that Pi/fcPi equals
-""tH
L
... {P1T[+]Pl){Pxe-'">tHPl).
By the analogue of Proposition 2.37 to the operator H, we see that for t
sufficiently large, and for each I,
\\(x | Ple~t"Pl | y)\U <

314
9. The Index Bundle
On the simplex Akt one of the at must be greater than (it + I). The
estimate now follows easily by induction using Lemma 9.47 combined with
the above decay for e~"itH, ?
Let tt: M —> B be a family of manifolds over a base B. Denote by M x „. M
the fibre-product
MxnM = {(x, y) e M x M | tt(x) = 7r(y)},
which is a fibre bundle over B with fibre at z e B equal to Mz x Mz. Let
prx and pr2 be the two projections from M x, M to M, and let \AV\ be the
vertical density bundle of the fibre bundle n : M —* B. If ?x and ?2 are
vector bundles on M, let ?i El* ?2 be the vector bundle over M xvM given
by the formula
?i B* ?2 = prj ?i ® prj ?2.
Let ? —* M be a family of vector bundles ?z —> Mz. We define a smooth
family of smoothing operators acting on the bundles ?z -+ Mz along the
fibres of it: M —> B to be a family of operators with a kernel
k e T(M x% M, (? ® l
(?*
When restricted to the fibre Mz x Mz, the kernel k may be viewed as a
kernel kz in T{MZ xMz,(?® \AMA1/2) a, (?• ® |AM, l1^)), which defines a
smoothing operator Kz on ¦Kt?z = r(Mz,?z) by the formula
(Kz<t>)(x) = J kz(x,y)<t>(y)
for <f> e r(Mz,?z ® IAmJ1/2). We define the bundle ?(?) over B to be the
bundle whose smooth sections are given by
r(?,/C(?)) = T(M xv M, (? ® |A^|1/2) H (?* ® \A*\l/2));
as explained in Definition9.8, ?(?) is a sub-bundle of Endp(?).
Suppose we are given a smooth family of generalized Laplacians Hz along
the fibres of M —> B; in other words, we are given the following data, from
which we construct the Laplacians Hz in the canonical way:
A) a smooth family gM/B € T(M, S2(T(M/B))) of metrics along the fibres;
B) a smooth family of connections acting on ? along the fibres, which we
may suppose to be the restriction of a smooth connection on the bundle
C) a smooth family of potentials Fz e T(Mz,End(?z)), in other words, an
element of T(M, End(?)).
We have the following result, which generalizes Theorem 2.48.
Theorem 9.50. If Hz is a smooth family of generalized Laplacians, then
the corresponding heat kernel pt(x,y,z) defines a smooth family of smoothing
operators, that is, a section in T(B,K.(?)).
Bibliographic Notes
315
Proof. As usual, around any point zo ? B, we can find a neighbourhood on
which the families M and ? are trivialized. Thus, we may replace B by a ball
U CW centred at zero, M by the trivial bundle Mo x U, and ? by the bundle
?o x U, where ?0 is a bundle over Mo. Since the changes of coordinates used
to obtain this trivialization are smooth, as are their inverses, we see that the
data used to define the family of generalized Laplacians Hz give a smooth
family of data for defining generalized Laplacians on the bundle ?o® |Am0 I1''2!
parametrized by the ball U. By Theorem 2.48, we know that for any t > 0,
the derivative d%pt(x, y, Hz) is a smooth family of smooth kernels on ?0, from
which the theorem follows. ?
The manifold M embeds inside M x* M as the diagonal M C M xw M.
If we restrict a kernel k € T(M x* M, (? ® \A*\1/2) &„ (?* ® \K\1/2)) to the
diagonal, it becomes a section of End(?) ® |A* |, and its pointwise trace Tr(fc)
becomes a section of the bundle of vertical densities |A^|. Such a section
may be integrated over the fibres to give a function on B; we will denote
this integral by JM/B Tr(fc(x, x)). In this language, the formula for the trace
along the fibres of the family of operators Kz becomes
= f Tr((x | Kz | x))
Jm/b
f
Jm/b
We may also state the family version of this theorem. Let A be a bundle
of finite-dimensional graded algebras with identity over B (in practice, A =
AT'B) and let M be the bundle of filtered algebras M = A ® Endp(?). Let
T e T(B, A ® Endp(?)) be a family of P-endomorphism with coefficients in
A, of the form
where HQ ? T(B, !>(?)) is a smooth family of generalized Laplacians, K €
T(B, /C(?)) is a smooth family of smoothing operators, and F[+] is an element
of T{B,Mi) = ET-i r(BM( « EndP(?)).
Theorem 9.51. For each t > 0, the kernel of the operator e~t:F is a smooth
family of smoothing operators with coefficients in A, that is, a smooth section
in T(B, A® ?(?))¦
Bibliographic Notes
The main theorem of this chapter, Theorem 9.33, is proved in the first part of
Bismut's article on the family index theorem [29]. This theorem is a synthesis
of the heat kernel approach of McKean and Singer towards the local index
theorem with Quillen's theory of superconnections. Our other main reference
for this chapter is Berline and Vergne [25], where Theorem 9.2 is proved. The
proof of our refinement Theorem 9.19 of Theorem 9.33 is new.

316
9. The Index Bundle
Sections 1, 2 and 3. Atiyah and Singer first defined the family index of a
continuous family of elliptic pseudodifferential operators, as an element of the
K-theory of the base [16]. They then derived a formula for this family index,
generalizing the Riemann-Roch theorem of Grothendieck [36]. For expositions
of topological K-theory, see the books of Atiyah [3] and Karoubi [64].
Bismut, by contrast, obtained a formula for the Chern character of the
family index in the special case of a family of Dirac operators; in this, he
was inspired by Quillen's definition of the Chern character of a superconnec-
superconnection [86].
The results on the convergence of the transgressed Chern character (Theo-
(Theorems 9.7 and 9.23) are new, but were inspired by an analogue of Theorem 9.23
for complex manifolds due to Gillet and Soule1 [61].
Section 4. The example presented in this section comes from Section 5 of
Atiyah and Singer [16]. However, since we have the Kirillov formula of Chap-
Chapter 8 at our disposal, we are able to give a formula for the Chern character
form of the index bundle.
Section5. The techniques of this section are those of Atiyah and Singer [16].
Section 6. The zeta-function of the Laplace-Beltrami operator was introduced
by Minakshisundaram and Pleijel[79]. This work was extended to elliptic
pseudodifferential operators by Seeley [92], and applied in the original proof of
the Atiyah-Bott fixed point formula [5] (see Section 6.2). The definition of the
zeta-function determinant of a Laplacian was given by Ray and Singer [87].
Section 7. Our definition of the determinant line bundle is taken from an
article by Bismut and Freed [32]; the construction is based on the ideas of
Quillen[85]. This theory began with the work of physicists, in particular
Alvarez-Gaume and Witten on anomalies of quantum field theories (which is
in fact precisely the theory of the Chern class of the determinant line bundle
of a family of Dirac operators); see for example [98].
Chapter 10. The Family Index Theorem
Let f : M -» B be a family of oriented Riemannian manifolds (Mz \ z e B),
and let ? be a bundle on M such that ?z — ?\m* is a Clifford module for
each z; suppose in addition that there is a connection V? given on ? whose
restriction to each bundle ?z is a Clifford connection. Let n,? be the infinite-
dimensional bundle over B whose fibre at z e B is the space T(MZ,?Z); let
D = (D* | z 6 B) be the family of Dirac operators acting on the fibres of ir,?,
constructed from the Clifford module structure and Clifford connection on
?. The aim of this chapter is to calculate the Chern character of the index
bundle ind(D) 6 K(B), by introducing a superconnection for the family
of Dirac operators D whose Chern character is explicitly calculable. (Note
that because the fibres of the bundle M/B have Riemannian metrics, the
line bundle |Aw|1/f2 has a canonical trivialization; this reconciles the above
definition of 7r,? with that in Chapter 9.) This theorem is a generalization of
Theorem 4.1 of Chapter 4; as in that chapter, we must assume that the Dirac
operators Dz are associated to Clifford connections on ?z.
We introduce the bundle E = ¦w*kT*B <g> ?, which has the property that
Recall that a superconnection for the family D is a differential operator A on
the bundle E —> M, which is odd with respect to the total Z2-grading of the
bundle E, such that
A) (Leibniz's rule) if v e A{B) and s e T(M,E), then
B) A = D + ??™(B) A[i], where Aw : A*(B, n.?) — A*+i{B, n,?).
Let At = t^26fA-6t = ES(B) tA-i)/2Aw be the rescaled supercon-
superconnection corresponding to A. In this chapter, we will describe a particular
superconnection A for the family of Dirac operators D, for which the limit
lim ch(At) = lim Str(e
t—>o t—»o
~A<
exists in the space A(B). Note that there is no reason a priori why the limit
lim _^ cli(A() should exist; this is analogous to the fact that the local index
theorem is not true for arbitrary Dirac operators, but only those associated to

318
10. The Family Index Theorem
an ordinary Clifford connection. Since the superconnection A was constructed
by Bismut, we will call it the Bismut superconnection.
Recall that in the last chapter, we proved that ch(Aj) is a differential form
which lies in the deRham cohomology class of the index bundle ind(D) e
K(B) of the family D. It follows that the limit lim _> ch(At) lies in the
same cohomology class. The formula for this limit, explained below, gives a
way of calculating the Chern character of the index bundle. Thus, Bismut's
theorem implies a local version of the family index theorem of Atiyah and
Singer for the family D.
To define the Bismut superconnection, we need a connection for the fibra-
tion MjB. In other words, if T(M/B) is the bundle of vertical vectors, we
need a decomposition of TM into a direct sum of T(M/B) and a horizontal
tangent bundle ThM, isomorphic to n'TB. This allows us to construct a
canonical connection VM/B on T(MjB), with curvature RM/B; we do this in
Section 1. The bundle ? is assumed to be a Clifford module for the Clifford
bundle C(M/B) = C{T*(M/B)). We assume the connection Vf is compat-
compatible with the connection VM/B. Denote by F?/s the twisting curvature of
Vf: if we are given a spin-structure on the bundle T(M/B) with spinors
Sm/b, so that ? = SM/b ® W for the bundle W = Homc(M/B)EM/B,5),
the twisting curvature F?ls is the curvature Fw of the twisting bundle W.
Let
be the ,4-genus of the connection
, and let
ch(?/5) = Stre/S(e-F*/5)
be the relative Chern character of the Clifford module ?. We will Bismut's
theorem,
lim ch(At) = B7ri)-"/2 / A(M/B) ch(?/S),
t—K) Jm/B
where n is the dimension of the fibre.
Now make the stronger assumption that the dimension of the kernel ker(D)
is constant, so that ker(D) is a finite-dimensional subbundle of n,?. If Po
is the projection from 7r»? onto the finite dimensional bundle ker(D), we
may form the projected connection Vo = PoA.[i]Po on ind(D). The Chern
character ch(At) has a limit as t —» co as well, equal to the Chern character
ofV0:
lim ch(At) = ch(Vo).
t—>oo
We will prove that the integral
10. The Family Index Theorem
319
converges in A(B) under the hypothesis that ker(D) is constant in dimension,
and thus derive the transgression formula for the Chern character associated
to the Bismut connection,
ch(Vo) =
[ A{M/B) ch(?/S) + d [
m/b Jo
dt.
In Sections 1-3, we study the geometry of a family MjB with a given
horizontal tangent bundle. Thinking of M as a Riemannian manifold with
degenerate metric g = gM/B on T*M, defined to vanish on the horizontal
cotangent vectors, then we can construct a natural Clifford module structure
and Clifford connection on E using one more piece of data, a metric on the
base B.
Having constructed the degenerate Clifford module structure on E, we
define the Bismut superconnection A to be the associated Dirac operator; we
then prove the important result that A does not depend on the horizontal
metric gs used in the definition of the Clifford module structure and Clifford
connection on E. It is interesting to observe that A has in general terms up
to degree two:
Let T = A2 be the curvature of the Bismut superconnection. In Section,3,
we will prove Bismut's explicit formula for F, and explain how it is a gener-
generalization of the Lichnerowicz formula.
By the results of Chapter 9, for each t > 0, the heat operator e~tjr acting
on T(M, E) has a kernel
kt € T(M x* M, tt'AT'B ® {? B* ?')).
If we restrict the kernel kt to the diagonal, we obtain a section kt(x,x) of
the bundle of algebras ir*AT*B ® End(?) over M. The bundle End(?) is
isomorphic to C(M/B) ® EndC(M/B)(?), and applying the symbol map to
C(M/B), we obtain a section of n*\T*B ® h.T*(M/B) <8> EndC(M/B)(?)•
Using the connection on the fibre bundle M —» B, we obtain an isomorphism
of 7r*AT*fl ® h.T\M/B) with AT'M, so that we obtain a differential form
kt(x,x) on M, with values in the bundle EndC(M/B)(?). We will calculate
lim _^ ch(A<) by showing that kt(x,x) has an asymptotic expansion
kt(x,x)
«=0
with coefficients fcj e Y^jk-h-^(M,EndC(M/B)(?)), such that
dim(M)/2
ki)m = A(M/B) exp(-
t=0
From this result, we immediately deduce that lim _^ ch(At) exists.

320
10. The Family Index Theorem
In Section 5, we prove the transgression formula, and use this in Section 6
to calculate the "anomaly" formula of Bismut-Freed for the curvature of the
connection on the determinant line bundle introduced in Section 9.7. Finally,
in Section 7, we return to the case studied in Section 9.5 in which the bundle
M/B is associated to a principal bundle P —* B with compact structure
group G, and show that in this case the local family index theorem is equiv-
equivalent to the local Kirillov formula of Chapter 8.
10.1. Riemannian Fibre Bundles
Let n : M —+ B be a fibre bundle with Riemannian metrics on the fibres. In
this section, we will describe two different families of connections that may be
constructed on the tangent bundle TM of the total space M. This material
is used in the rest of this chapter in the proof of the family index theorem.
Let 7T: M —> B be a fibre bundle. We will denote by T(M/B) the bundle
of vertical tangent vectors. Let us assume that the bundle M/B possesses
the following additional structures:
A) a connection, that is, a choice of a splitting TM = THM e T(M/B), so
that the subbundle ThM is isomorphic to the vector bundle n'TB;
B) a connection VM/B on T(M/B).
We denote by P the projection operator
P:TM -> T(M/B)
with kernel the chosen horizontal tangent space ThM. If X is a vector field
on the base B, denote by Xm its horizontal lift on M, that is, the vector
field on M which is a section of TjjM and which projects to X under the
pushforward tt» : (ThM)x —> T^X)B. We will often make use of a local frame
ej of the vertical tangent bundle T(M/B), and of a local frame fa of TB,
with dual frames el and fa; using the connection, we obtain a local frame of
the tangent bundle TM.
There are three tensors, S, k and ft, canonically associated to a family of
Riemannian manifolds with connection M/B. These are defined as follows:
A) The tensor S (the second fundamental form) is the section of the
bundle
End(T(Af/B)) ® T*HM <* T\M/B) ® T{M/B) ® T'HM
defined by
(S(x,e),z) = (v*f/Bx - P[z,x},0)
for Z € T{M,THM), X <E T{M,T{M/B)) and 6 e T{M,T*{M/B)). It
is clear that this is a tensor, since P vanishes on ThM.
10.1. Riemannian Fibre Bundles 321
B) The one-form k € A1(M) (the mean curvature) is the trace of S:
C) The tensor ft is the section of the bundle Rom(A2THM,T(M/B)) over
M defined by the formula
ft(X, Y) = -P[X, Y] for X and Y in T(M, THM).
This is clearly antisymmetric in X and Y, and is a tensor since if we
replace Y by fY, for / e C°°{M), then
tl(XJY) = -fP[X,Y] - X(f)PY = ftt{X,Y).
By comparison with A.6), we see that ft may be identified with the
curvature of the fibre bundle M/B.
The total exterior differential djvf may be expressed in terms of VM^B, 5,
n and the vertical exterior differential d.M/B of the fibre bundle M —> B. We
extend dM/B to an operator on T(M, AT*HM ® AT*(M/B)) by the formula
dM/B{**v ® P) = (-1)HtV ® dM/B(S
for v € A{B) and /? e T(M, AT\M/B)). We cannot define a horizontal
differential on T(M, AT*HM ® AT*(M/B)), but we may define an operator
&B by means of the connection VM/B on T(M, AT*(M/B)):
(wV ® 0) = n*{
A p + (-l
f*
If 6 G r(M,T*(M/B)), we define the contraction (S,0) € T(M,T*HM ®
T\M/B)) C A2{M) by the formula
and the contraction {Q,6) € T{M,A2T*HM) C A2{M).
Proposition 10.1. dM = dM/B + 6B-
+
Proof. It is easy to check that both sides are derivations of A(M), and that
they agree on horizontal forms. Thus, we only need to check that they agree
on the vertical one forms e*. We start by checking this when both sides are
evaluated on e* A fa: on the one hand,
while by the definition of S, the right-hand side equals

322
10. The Family Index Theorem
We must also evaluate both sides on fa A f@: but the equality comes down
to
(dMei)(fonf0) = (n(fa,f0),ei). ?
Let us now suppose that M —> B is a family of Riemannian manifolds,
thus, on each fibre Mz = n~1(z), z 6 B, there is given a Riemannian metric,
which we will denote by gM/B', m other words, we are given an inner product
on the bundle of vertical tangent vectors T(M/B). There is then a canonical
connection VM/B on T(M/B), constructed as follows.
If we choose a Riemannian metric gB on the base B and pull it up to
ThM by means of the identification TjjM = n*TB given by the connection
on the bundle M/B, we obtain an inner product on the bundle ThM, which
we will call a horizontal metric. We may now form the total metric g =
9b ffi 9m/b on the tangent bundle TM of M, by means of the identification
TM ? THM © T(M/B). Let V9 be the Levi-Civita connection on TM with
respect to this metric, and define a connection VM/B on the bundle T(M/B)
by projecting this connection
•yM/B — p .y9 .p
The following proposition shows that the choice of metric gB on the base is
irrelevant in constructing VM/B. In what follows, we will denote by g(X, Y)
the inner product of two vector fields X and Y with respect to the metric g,
or simply (X, Y) if the metric g is evident from the context.
Proposition 10.2. The connection VM/B on T(M/B) is independent of the
metric gB on TB used in its definition.
Proof. This follows easily from the formula A.18) for the Levi-Civita connec-
connection V9, which we will use constantly in this chapter:
2(v?vy, z) =([x, y),z)- ([y, z), x) + ([z, x\, y)
+ X{Y, Z) + Y(Z, X) - Z{X, Y).
If the vectors X, Y and Z are all vertical, that is, sections of T(M/B), then
the right-hand side reduces to the Levi-Civita connection on the fibres for
the vertical metric gM/B- On the other hand, if X is horizontal but Y and
Z are vertical, then [F, Z] is vertical, so that ([Y, Z], X) vanishes, and we see
that
2{vfBY, Z) = (P[X, Y},Z) + (P[Z, X},Y) + X(Y, Z).
Prom this formula, it is clear that only the vertical metric Sat/b and the
vertical projection P are used to define V^/B for X horizontal. Q
The vertical metric allows us to think of the tensor 5 as a section of
T*{M/B) ® T*(M/B) ® T*HM. Let us give a more explicit formula for the
tensor 5.
10.1. Riemannian Fibre Bundles
323
Lemma 10.3. // VM/B is the connection on T(M/B) associated to a verti-
vertical Riemannian metric gM/B, then S is given by the formula
2(S(X,Y),Z) = Z(X,Y) - (P[Z,X],Y) - (P\Z,Y],X).
Proof. This follows easily from the explicit formula for the Levi-Civita con-
connection. D
It is evident from this lemma that (S(X, Y), Z) is symmetric in X and Y.
In fact, it represents the horizontal covariant derivative of the metric.
Let n be the dimension of the fibres of M/B, and let vm/b ? An(M) be
the Riemannian volume form along the fibres, where we identify a section of
the bundle KnT\M/B) with a section of the bundle A"T*M by means of the
connection on the fibre bundle M/B.
Lemma 10.4. dvM/B = k A vM/B +
Proof. We use the formula of Proposition 10.1. Since VM/B preserves the
metric gM/B on T(M/B), and hence Vm/bvM/b = 0> we see that
dvM/B = ~
Furthermore,
= -k(fa)vM/B. a
If we choose a metric gs on B, we obtain a Levi-Civita connection on TB,
which may be pulled back to give a connection on the bundle ThM = n*TB.
We can form a new connection on TM taking the direct sum of the connection
VM/fl on T(M/B) and tt*Vb on THM (which we will write simply as Vs).
We will denote this connection by V®:
v® = vB©vM/B.
Note that if we replace g by the rescaled metric ugB + gM/B> where u > 0,
then VB does not change, and so neither does V®.
The connection V® preserves the metric g, but has non-vanishing torsion
in general. Our next task is to compare the connections V9 and V® on TM.
In order to do this, we introduce a three-tensor w on M which only depends
on the data that we are given for the fibre bundle M —¦ B, namely the
vertical metrics gM/B axi^ the choice of connection on M/B.

324
10. The Family Index Theorem
Definition 10.5. Let w G A1{M,A2T*M) be the A2T'M-valued one-form
on M defined by the formula
u(X){Y, Z) = S{X, Z)(Y) - S{X,Y)(Z)
+ \{fl{XtZ),Y) - \(fl(X,Y),Z) + ±(<l(Y,Z),X).
The above formula is antisymmetric in Y and Z, so that w takes its values
in A2T*M, as claimed.
The following result, due to Bismut, is the main result of this section.
Proposition 10.6. The Levi-Civita connection V9 is related to the connec-
connection V® by the following formula:
g(VxY, Z) = g(V%Y, Z) + U{X){Y, Z).
Proof. We will make use of the following lemma
Lemma 10.7. IfYis the horizontal lift of a vector field on B, and if X is
vertical, then [X, Y] is vertical.
Proof. If 0 G C°°{B), then Y(n*<j>) = n*{ntY<j>), while, since X is vertical,
X{k*4>) = 0 and (X ¦ Y)n*<t> = 0. This shows that [X,Y]{n*<l>) = 0, and
hence that [X, Y] is vertical. ?
Now, observe that (V^Y, Z) - (V|Y, Z) is antisymmetric in Y and Z,
because each of the connections V9 and V® preserves the total metric g on
M. The proof of the theorem now consists on a case-by-case examination of
the different situations in which X, Y and Z are horizontal or vertical.
A) If X, Y and Z are all horizontal lifts of vector fields on the base or all ver-
vertical, it is easy to see that (V9XY,Z) equals (VXY,Z) while u(X)(Y,Z)
is zero.
B) If both Y and Z are vertical, then {V9XY, Z) and (Vf Y, Z) are both
equal to (Vx/BY, Z), and once more w(X)(Y, Z) is zero.
In each of the remaining cases, (V® Y, Z) is equal to zero.
C) If X and Y are vertical and Z is horizontal, then (V^Y, Z) equals
-{S(X, Y),Z), which equals u(X)(Y, Z).
D) If X and Y are horizontal lifts of vector fields on B and Z is vertical,
then {VgxY, Z) equals
\([X,Y],Z) + \{[Z,Y),X) + \{[Z,X],Y) - \Z(X,Y) = \([X,Y\,Z),
as follows from the above lemma. But this equals — \ (fl(X, Y), Z) by the
definition of Q.
E) If X is vertical and Y and Z are the horizontal lifts of vector fields on
B, then {VgxY, Z) is equal to
i([X,Y],Z) - \{[X,Z\,Y) - i([Y,Z],X)
which equals ^(fi(Y, Z),X) by the same argument as for D). D
10.2. Clifford Modules on Fibre Bundles
10.2. Clifford Modules on Fibre Bundles
325
A fibre bundle M with the data that we considered in the last section may
be thought of as a Riemannian manifold with a degenerate metric 30 on its
cotangent bundle T*M, which vanishes in the horizontal cotangent directions
T(M/B)L C T*M. This metric explodes in the horizontal directions of the
tangent bundle TM of M, so that in contrast to Riemannian geometry, we
distinguish between T*M and TM. Since the Clifford algebra of a vector
space V with vanishing metric is isomorphic to the exterior algebra AV, we
see that the Clifford bundle C(T*M) of M with respect to the degenerate
metric g°, which we will denote by C0(M), may be identified with the bundle
of algebras tt'AT'B ® C(M/B), where C(M/B) denotes the Clifford bundle
C(T*{M/B)).
In this section, we will generalize the discussion of Clifford modules and
Clifford connections of Chapter 3 to the setting of the Clifford bundle Cq{M).
Fix a metric gB on the base. If u € @,1] is a small positive number, let
g" be the metric on T*M equal to the vertical metric gM/B °n the vertical
cotangent vectors, and w/b on the horizontal ones:
fl" =
In particular, g° - lim gu is the degenerate metric that we introduced
above. The family of metrics gu is only a tool in investigating the geometry
of g°. Let gu be the dual metric on TM,
which explodes in the horizontal directions as u —> 0.
Let CU{M) = C(T*M,gu) be the Clifford algebra bundle of the bundle of
inner-product spaces (T*M,gu), and denote the canonical quantization map
of Proposition 3.5 from AT*M to CU{M) by cu. Thus, the Clifford bundle
C0(M) is the limit of the one-parameter family of algebras bundles CU(M).
Let ru denote the bundle map from A2T*M to End(T*M) which is defined
implicitly by
where a e \2T*XM and ? € T\M; then
>i
- gu(v2,Z)vi) ¦
In order to understand this definition better, we will write it out explicitly
in terms of an orthogonal frame of T*XM of the form {e'} U {/"}, where e'
is an orthonormal frame for T'(M/B) and f* is an orthonormal frame for

326 10. The Family Index Theorem
T*B, with respect to the metric gB. In terms of this frame,
±ru(eV)e* = 6ike> - S^e*
A0.1)
* = 0
r"(eV)/° = 0.
It is clear from these formulas that the family of actions t" has a well-defined
limit as u —+ 0, which we denote by r°:
A0.2)
r/V = o
T°(a)fa = 0 for all a e A2T*M.
It is important to note that r°(a) vanishes on T*B.
We will denote the negative of the adjoint of tu(o) 6 End(T"M), where
a € A2T*M, by Tu(a) e End(TM), and the negative of the adjoint of r°(a)
by To (a). Note the formula
for X, Y 6 T(M,TM) and o € A2T*M.
It is clear that A2T*M has a family of Lie brackets induced on it by the
maps r", defined by the formula
^[01,02], = [r"(a1),T-(oa)J.
We denote the limit of [ai,a2]u as u —> 0 by [ai,a2]o, which satisfies
Let VMi", u > 0, be the Levi-Civita connection on TM corresponding to
the metric gu, and let V® be the direct-sum connection introduced in the
last section. In the notation of this section, we may restate Proposition 10.6
as
where a> is the element of AX{M, A2T*M) of Definition 10.5. It follows that
the family VM'" of connections has a well-defined limit as u —> 0, which we
will denote by VT'M'°, and which is given by the formula
= V® + I7b(u>).
10.2. Clifford Modules on Fibre Bundles 327
We denote by VT"M'U and VT*M'° the dual connections on TM, that is,
A0.3)
VT'M,O =
Proposition 10.8. A) The connection VM'U is torsion-free.
B) The projection ofVM'u to the sub-bundle T(M/B), equals VMIB.
C) Restricted to each fibre, the connection VM'° depends only on the vertical
metric gM/B an^ the connection on the fibre bundle M/B.
Proof. For u > 0, the connection VM'" is a Levi-Civita connection, so is
torsion-free. Since lim VM>U = VM'°, it follows that VM'° is torsion-
u—»0
free. The connection VM/S on the bundle T(M/B) is by definition equal to
the projection of VM|U onto T(M/B), and taking the limit u —> 0, we see
that this is true for VM'°. Finally, the fact that the restriction of VM'° to
a fibre of the bundle M/B depends only on the vertical metric gM/B and
the connection on the bundle M/B follows from A0.3), since tt*Vb = 0 on
vertical vectors. D
Consider the section of the bundle A2T*M ® A2T*(M/B),
RM'B(W,X)(Y,Z) = (RM'B{W,X)Y,Z)M/B,
where RM/B e A2(M,End(T(M/B))) is the curvature of the connection
yM/B on j*(M/B). Using the connection on the bundle M —* B, we may
extend RM'B to a four-tensor on M by setting RM/B(W, X){Y, Z) = 0 if Y or
Z is horizontal. Let R" 6 A2(M, End(TAf)) be the Riemannian curvature of
the metric gu, and let RB e A2(B, EndG!B)) be the Riemannian curvature
of the metric gs on the base B.
Proposition 10.9. Let R 6 T(M, A2T*M ® A2T*M) be the four-tensor de-
defined by the formula
R(W,X,Y,Z) = lim (gu{R»(W,X)Y,Z)~u-l{RB{W,X)Y,Z)B).
A) The above limit exists and equals RM>B + V®w + j[w,w]0.
B) When Y and Z are vertical, R(W, X, Y, Z) equals RM/B{W, X, Y, Z).
C) IfW, X, Y and Z are vector fields on M, then
R{W, X, Y, Z) + R(X, Y, W, Z) + R(Y, W, X, Z) = 0.
D) R(W, X, Y, Z) = R(Y, Z, W, X)

muex ineorem
10.2. Clifford Modules on Fibre Bundles
329
Proof. Written as a four-tensor gu(Ru(W, X)Y, Z), the curvature of the met-
metric gu is a polynomial in u and u, and we must show that R is the constant
term in this expansion, by proving that
lim ugu(Ru(W,X)Y,Z) = (RB{W,X)Y,Z)B.
u—»0
From the formula VM'U = V® -I- |ru(w), we see that
(VM-<f = (V®J + ±[VV«M] + ifr.M, tb(W)]
= (VeJ + ir.(V*w) + \tu{\[u>Mu\).
Since (V®J = (n*VBJ + (VM/BJ, it follows that
gu(IV(W, X)Y, Z) =u-l(n*RB{W, X)Y, Z)B + {RM'B{W, X)Y, Z)
+ (V*u)(W, X)(Y, Z) + \{w, u]u(W, X)(Y, Z),
where we recall that [u>,w]u = (r")-1([r"(w),ru(ai)]). The existence of the
limit as u —* 0 now follows from the fact that [u>, u>]« = [w,w]o + O(u), and
we have
+ (V9w)(W,X)(Y,Z) + i\u,
The other properties of R are clear from its definition: for example, C)
follows from the fact that both Ru and RB, being Riemannian curvatures,
satisfy the desired symmetry. ?
Let ? be a Clifford module along the fibres of the bundle M/B, that is, a
Hermitian vector bundle over M with a skew-adjoint action
c : C(T*(M/B)) -> End(f)
of the vertical Clifford bundle of M/B, and a Hermitian connection V? com-
compatible with this action,
for X e T{M, TM) and a e T(Af, T\M/B)).
Denote by E the vector bundle over M defined by the formula
This bundle carries a natural action tuq of the degenerate Clifford module
Cq(M). In order to define this action, it suffices to define the actions of
T*HM C Co(M) and T\M/B) c C0(M). The Clifford action of a horizontal
cotangent vector a e T(M, T*uM) is given by exterior multiplication
mo{a) = e(a)
acting on the first factor AT'nM in E, while the Clifford action of a vertical
cotangent vector simply equals its Clifford action on 6. This Clifford module
will be the one of the main tools in calculating the family index of the family
D. In order to study E, we will write it as the limit of a family of Clifford
modules for the bundles of Clifford algebras CU(M), all constructed on the
same underlying vector bundle as E.
In order to define the Clifford action
mu : CU(M) -> End(E) S End(Ar^M ® ?),
it suffices to define the actions of T*HM C CU(M) and T*{M/B) C CU{M).
The Clifford action of a horizontal cotangent vector a € T(M, T*hM) is given
by the formula
' acting on the first factor AT*nM in E, while the Clifford action of a vertical
i cotangent vector simply equals its Clifford action on ?. It is a straightforward
i task to check that
j tnu(aJ = -gu{a,a) for a 6 T{M,T*M).
• In particular, we see that the limiting Clifford module action lim mu is
just the degenerate action mo introduced above.
There are connections VE'® and VE'" on the Clifford module E analogous
to the connections V® and V7""*^" that we have constructed on the bundle
T'M. As before, to construct these connections, we must choose a horizontal
metric gB on B. The connection VE'® on E = Air"T*B <g> ? is defined by
taking the sum of the Levi-Civita connection n*VB on the bundle An*T*B
with the connection V? on ?,
the connection VE'U is defined by the formula, inspired by Proposition 10.6,
where u> is the one-form defined in Section 1.
Proposition 10.10. The connection VE'U is a Clifford connection for the
Clifford action mu of CU{M) on E, in other words,
for X e T(M, TM) and a € T(M, T'M). In particular, the connection
VE'° = lim VE'U = VE'® +
u—>0
is a Clifford connection for the Clifford action rriQ of the Clifford algebra
bundle Co(M) onE,

330
10. The Family Index Theorem
The restriction ofVE'u, and in particular of VE>0, to each fibre of the bundle
M/B is independent of the horizontal metric gs used in its definition.
Proof. The first step of the proof is to show that VE'e is a Clifford connec-
connection with respect to any of the Clifford actions mu, if CU(M) is given the
connection V®:
This formula follows from considering the two cases, in which X respectively
horizontal and vertical. The proof that VE>U is a Clifford connection for
the action mu and the connection VT*M'U on CU{M) follows by observing
that, almost by definition, [mu(w(X)),mu(a)] = mu(Tu(w(X))a). Finally,
the independence of the horizontal metric gs of the restriction of VElU to a
fibre of the bundle M/B follows as it did for VM'U from the fact that ir* VB
vanishes on vertical vectors. D
In the rest of this chapter, we will make use of the frame {e'} U {/"} of
T*M introduced in the last section, as well as the dual frame {ej} U {/a}
of TM. We adopt the convention for indices that i, j, ... , label vertical
vectors, a, /?,..., label horizontal vectors, while a, 6, ... , label all vectors,
horizontal or vertical. With this convention, we denote by ea any one element
of the cotangent frame {e*} U {/"}, and the Clifford action mu(ea) by m?.
In the next section, we will need a formula for the curvature of the Clif-
Clifford connection VEl°. First, observe that an easy generalization of Proposi-
Proposition 3.43 shows that the curvature (V?) of the connection V? on ? equals
2 Z-. »J<"
a<b
where R^B = (RM/B(ea,eb)ej,ei) and F?ls 6 A2 {M,Endc{M/B)(?)) is
by definition the twisting curvature of the Clifford connection V? on the
Clifford module ? over C(M/B). In the special case in which the fibres
M/B have a spin-structure with spinor bundle S(M/B), we may write ? as
? = S(M/B)®W, where the twisting bundle W is a Hermitian vector bundle
with connection Vw, and the twisting curvature F?ls is then nothing but
the curvature of the connection Vw.
Let A be the natural action of End(TB) on ATB defined in A.26). Then
X(RB) is given by the formula C.15).
Proposition 10.11. The curvature of the connection VE'° on E equals
\{RB) + imo(R) + Fe's.
Proof. We have
The proposition follows from the formulas
A) (VE-®J = A((VBJ) + (V?J = \{RB) + ^
10.2. Clifford Modules on Fibre Bundles 331
B) [VE'®,mn(w)] = mo(V<M, and
C) [mo(a;),mo(w)] =mo([w,w]o). ?
If the base B has a spin-structure, with associated spinor bundle S{B),
there is another Clifford module on M, this time considered with the non-
degenerate metric g = qb + g\t/B, namely n*S(B) ® ?. Indeed, there is a
natural decomposition
C{M)*iTT*C{B)®C{M/B)
of Clifford algebra bundles over M, corresponding to the decomposition
T*M = n*T*B © T*(M/B). We may introduce two natural connections in
this situation, which are respectively Clifford connections on S(B) ® ? with
respect to the connections V® and V9 on C(M). We will not give the details
of the proof, since it is very similar to that of Proposition 10.10.
Proposition 10.12. A) The connection
VB ® 1 + 1 <g) V?
on S(B) ® ? is a Clifford connection with respect to the connection V®
onC(M).
B) Denote by c(w) € A1{M,C{M)) the one-form defined by the formula
c{u) = i
® c(e6)c(ec).
abc
The connection
on the bundle S(B) ®? is a Clifford connection with respect to the Levi-
Civita connection V9 on C(M).
C) If M is a spin manifold with spinor bundle SM, and ? is the relative
spinor bundle S(M/B) ? HomC(B)(ir*5(J5),5M), then S(B)®? is nat-
naturally isomorphic to SM, and V5(B)®? may be identified with the Levi-
Civita connection on S(M).
To finish this section, we give a lemma that we will need later.
Lemma 10.13. // E is any Clifford module for the Clifford algebra bundle
CU(M), with action mu, then we have the equality
abc

332
10. The Family Index Theorem
Proof. Using the fact that u>(eo)(ei,,ec) vanishes when the three indices a, 6
and c are all horizontal and also when both b and c are vertical, the left-hand
side is seen to equal
ija
ta/3
Writing this ought explicitly, we get
ia0
Since S(ej,ej)(/Q) is symmetric in i and j, only the trace of the tensor S
contributes to the first sum, and we obtain our formula. Q
10.3. The Bismut Superconnection
Let n : M —+ B be a fibre bundle with data as in the first section, namely
vertical Riemannian metrics and a connection. Recall that given a vector
bundle over M, ir,? is the infinite-dimensional bundle over B whose fibre
at z € B is the space T(MZ,?). If ? is a Clifford module for the vertical
Clifford algebra bundle C(M/B), the Dirac operators D* along the fibres
(Mz | z e B), associated to the Clifford module ?z over Mz, combine to
give a smooth family of Dirac operators D = (Dz | z e B), which is a
smooth section of EndpGr,?) C EndGr*?) over B. In this section, we will
describe a superconnection, which we call the Bismut superconnection, on
the bundle Kt?, with zero-form component D, whose Chern character form
is particularly well-behaved. In view of its fundamental character, we will
present the Bismut superconnection from two different viewpoints which are
not obviously equivalent.
Prom now on, we will only consider the bundle E = n*AT*B ® ? over M
with its Clifford module structure for the Clifford algebra bundle Co(M), so
we will write E in place of the notation E of the last section. Recall the
following definition from Chapter 9.
Definition 10.14. The space A(B, n,?) of differential forms on B with coef-
coefficients in 7r,? is the space of sections of the Clifford module E = ir*(kT*B)®?
over M.
We introduce the Bismut superconnection
A: A{B, tt,?)-> A(B,n,?)
as a Dirac operator for the Clifford module E —» M. This is the motivation
for introducing the whole apparatus of the last section.
10.3. The Bismut Superconnection 333
Proposition 10.15. The Dirac operator on E, given by the formula
is a superconnection when thought of as an operator on the space A(B, n,?);
we call A the Bismut superconnection. More explicitly, the restriction of
A to a map T(M, ?) —» T(M, E) is given by the formula
t a a<0 i
The superconnection A is independent of the horizontal metric ps used in its
definition.
Proof. It is clear that A satisfies the first condition for it to be a supercon-
superconnection on the bundle 7r»?, namely it is a differential operator on the bundle
E = v'KT'B <g> ? over M which is odd with respect to the total Z2-grading.
Next, we check that A satisfies the formula A(us) = {dBv)s + (-l)l"I^As,
for v e A(B) and s € r(M,E), where ds is the exterior differential on B. It
is easily seen that A(i/s) = Y,a eQVfi/ + (-\)^vKa. Since the Levi-Civita
connection is torsion free, we see that $3Q e'Vf v = dgf, as required.
In order to check the explicit formula for A on T(M,?), observe that
A =
3 E w(
abc
The proof of the formula is completed by Lemma 10.13. In particular, we see
that A|0] = D. ?
The decomposition of the Bismut superconnection with respect to degree
A = A[0] + A[!j + A[2] has in general terms up to degree two. Indeed, this
supcrconnection is a natural example of a superconnection having non-zero
terms of degree higher than one.
Proposition 10.16. The connection part A[i] of the Bismut superconnection
Li the natural unitary connection V7^ on the bundle nt? of Proposition 9.13,
bearing in mind the identification of ? ® lA^I1/2 with ? coming from the
vertical metric gM/B on M/B.
Proof. Choose a metric ps on B, and let \vb\ and \vm\ be the Riemannian
densities on B and M with respect to the metrics gs and gM = 9b ® 9m/b\
clearly, \vM/B\ = \vm\<8>x*\vb\~1- It will suffice to check that if X is a vector
field on B, then
C{Xm)\vm\
since this will imply that
M
\k(X))s ®

334
10. The Family Index Theorem
But this is. a local result, so we may replace the vertical density \vm\ ®
"¦"I^bI" by the vertical volume form vm/b m the calculation. By A.4), we
see that
C(Xm)vM/b = t-{XM)dvM/B+d{i(XM)vM/B)-
The second term vanishes, since vm/b is vertical, while the second equals
k(X)vM/B by Lemma 10.4. D
In order to compute the Chern character of the superconnection A, we need
the formula for its curvature A2 which we referred to above. Let us introduce
the smooth family AM/B = (A* | z e B) € r(B,EndGr,?)) of generalized
Laplacians along the fibres Mz of the bundle M/B\ each Laplacian A* is just
the generalized Laplacian acting on the bundle ?z —> Mz corresponding to
the connection V6'0^ and with zero potential.
Theorem 10.17. Let F?/s € A{M,EndC(M/B){?)) be the twisting curva-
curvature of the Clifford module ?, and let tm/b te the scalar curvature of the
fibres ofMjB. Then
A2 =
\rM/B
a<b
Proof. The proof of this theorem is similar to the proof of Lichnerowicz's
formula, Theorem 3.52. To begin with, denoting the covariant derivative Veo
in the direction eo by Vo for any connection V on a bundle over M, we have
2 -
ad
ab
ab
ab
Since our frame e° is the union of an orthonormal basis e' of T*(M/B) and a
basis fa of T*B, we see that the first term of this sum is equal to - ?j(Vf'0J-
Using the fact that VE'° is a Clifford connection, proved in Proposition 10.10,
the second term is equal to
ab
abc
Here we have used the facts that VM'° agrees with VM//B when restricted to a
fibre Mz, that the connection VM'° on TM is torsion-free, and the adjunction
formula
10.3. The Bismut Superconnection
It follows that
A2 = .
335
ab
By Propositionl0.il, Ea6momo^E'°(e<"ei>) e(luals
ab
abed
where Rabcd = R(ec,ed)(e(,,eo). The first term of this sum is equal to
— \YlaB s^^^^RySaB* wmch vanishes, since the antisymmetrization of
RB over any three indices is zero. Furthermore, we have
^ mlmbQmlmi Rabed = ^
abed <^>d
= 2
+ Y, <rnbombom$Rabbd
abd
adi ij
where we have used for the first equality Proposition 10.9 and antisymmetriza-
tion over the first three indices, and for the second equality the symmetry
Raidi = Rdtot- But the four-tensor R coincides on vertical tangent vectors
with the Riemannian curvature tensor of the fibre. ?
We will finish this section by giving another definition of the Bismut super-
connection. Locally, we may assume that the base B is a spin-manifold, with
spinor bundle S(B). Let Dm be the Dirac operator on T(M,? ® tt*S(B))
corresponding to the connection V?®5(B) = V5(B)®?'® + \c{u>) introduced
in the last section.
If VV is a finite-dimensional bundle over B with superconnection A, we
can construct a Clifford superconnection B = A®1 + 1® V5^B^ on the
twisted Clifford module >V ® S(B) over B. Let Db be the Dirac operator on
r(B,W® S{B)) associated to this superconnection. We proved in Proposi-
Proposition 3.42 that the map A ¦-» Db defined a one-to-one correspondence between
superconnections on VV and Dirac operators on T(B, VV®S(B)), that is, odd
differential operators D satisfying the identity [D,/] = c(df), for / 6 C°°(J3).
We will say that Db is the Dirac operator associated to A.
We may identify the spaces of sections T(B,S(B) ® 7r»?) with F(M,? ®
n*S(B)). We define a Dirac operator on the Clifford module ir.? ® S(B) to
be an odd differential operator D on the space of sections F(B, irm?<2>S(B)) =
T{M,? ® ir*S(B)) such that [D,tt*/] = c(ir*(df)) for all / E C°°(B). The
following result is an infinite-dimensional extension of Proposition 3.42.
Proposition 10.18. There is a one-to-one correspondence between super-
connections A on the bundle ir,? and Dirac operators on the Clifford module

336
10. The Family Index Theorem
The total Dirac operator Dm on r(M,?<8)ir*S(B)), when considered as an
operator on the isomorphic space r(B,S(B) <g> n,?), is a Dirac operator for
the Clifford module n.?®S(B) over B. Indeed, the fact that [Dm,**/] =
c{n*(df)) for / e C°°(B) is just a special case of the formula [DM, /] = c(df)
for / e C°°{M). Thus, there is a superconnection A on the bundle it,?
whose associated Dirac operator may be identified with Dm- In fact, this
superconnection on irt? is just the Bismut superconnection.
Theorem 10.19. The Dirac operator on
T(B, n.? ® S(B)) <* T(M, ? ® ir*S(B))
associated to the Bismut superconnection A is the total Dirac operator Dm .
Proof. Let B = V5<B) ® 1 + 1 ® A. If s € Y{B,S{B)) and t e T{M,?), we
see that
Db(s ® t) = s ® ^ c* Vf t +
a, f0),
a<0 i
Thus, we see that DB = ?a c° vf (S)®?'® + \h, where h is the section of
C(M) given by the formula
h = 2?k(fa)ca
afii
By Lemma 10.13, h equals ?oI(cw(ea)(et,ec)cocl>cc.
Since Vf®5(B> = V5<B>®?>® + ic(w), the operator DM equals
DM =
which we see is the same as D^. ?
abc
10.4. The Family Index Density
In this section, we use the method of Chapter 4 to prove the local family index
theorem of Bismut, that is, to calculate lim ch(At). Let us summarize
the data that we are given:
A) A fibre bundle n : M —> B with a vertical metric <7m/b and connection,
that is, a decomposition of the tangent bundle of M into the direct sum of
the vertical tangent bundle T(M/B) and the horizontal tangent bundle
n*TB; from this, we obtain a connection VMIB on the vertical tangent
bundle T(M/B).
10.4. The Family Index Density
337
B) a Clifford module ? for the vertical Clifford bundle C(M/B), with Clif-
Clifford connection Vf compatible with VMIB.
Using just this data, we can construct a family of twisted Dirac operators
D = (D* | z e B), and the Bismut superconnection A, which satisfies A[0] =
D. Let n denote the dimension of the fibres Mz.
Let Az be the finite dimensional algebra AT*ZB. The curvature T = A2
of A acts on the space Az <8> {irm?)z of sections of the bundle E = An*T*B <g> ?
along the fibre Mz.
The following lemma is a restatement of Theorem 9.48.
Lemma 10.20. For each t > 0, the heat operator e~tJr acting on T(M, E)
has a kernel
(x
e~t:F
y) e T{M xw M,n*
?*),
in the sense that if <j> e r(M,E), we have
(x
e-^
yL>(y)dy,
where dy is the Riemannian volume form of the fibre Mz and z = n(x).
The manifold M embeds inside Mx,Mas the diagonal M C M x* M.
If we restrict the heat kernel of 3- to the diagonal, it becomes a section of
the bundle of algebras n*AT*B <g> End(?). Since ? is a Clifford module over
the Clifford algebra C(M/B), we see that
n'AT'B ® End(?) =* tt'AT'5 ® C(M/B) ® Endc(M/B) {?)
s w*AT*B ® AT*[M/B) ® Endc(M/B)(^)
= AT*M ® Endc{M/B)(? )•
Thus, the restriction to the diagonal h(x, x) of the heat kernel of T is iden-
identified with a differential form on M with values in the bundle Endc(M/B)(?)-
Let A(M/B) be the i-genus of the bundle T(M/B) for the connection
and let F?/s e T(M, Endc(M/B)(?)) De tne twisting curvature of the bun-
bundle ?. The aim of this section is to prove the following generalization of
Theorem 4.1.
Theorem 10.21. Consider the asymptotic expansion ofkt(x,x)
oo
kt(x,x) ~ D7r<)-n/2E<ifci(x).
t=0
A) The coefficient fc4 lies in ?\<2i A2j(M, Endc(M/B){?))¦

338
B) The full symbol ofkt(x, x), defined by <r{k) = ?t™(M)/2
by the formula
10. The Family Index Theorem
is given
a(k) =
G A(M, Endc(M/B)(?))
Before proving the theorem, let us show how it allows us to compute the
Chern character of the index bundle ind(D) of the family of vertical Dirac
operators D. Recall the operator 6B on A(B) which multiplies differential
forms of degree i by t~1/2. Let
¦ A -F?r1 = t1/2A
@l
r^A
P]
be the rescaled Bismut superconnection, with curvature Tt = t6B • T ¦F^)~1 ¦
By Theorem9.19, the Chern character of the bundle ind(D) is represented,
for every t > 0, by the closed differential form ch(At) = Str(e-Jr') on B.
Lemma 10.22. Extend the map Str : r(M2,End(?*)) -+ C°°{MZ) to a map
Str : T(MZ, Az ® End(?z)) -+ Az ® C°°(MZ). Then
ch(At)=/ Sf(stT?l(kt(x,x))dx.
JM, V J
Proof. Using the formula Tt = t6f ¦ T \6f )~x, we see that
ch(At) = Str(e-^') = Str(e"< • rtfr1)
= Str(«f(e-"r))
Now, the kernel of the operator e~tF is by definition equal to kt(x,y), and
the lemma follows now from the fact that
Str(e~1:F*)= / Str?l(fet(x,x))dxe ST*ZB. D
JM,
Let us show that the local family index theorem of Bismut is a corollary
of Theorem 10.21. Let ch(?/S) = Str?/s(e~F?/S) be the relative Chern
character of the Clifford module ?. Let
TM/b
be the map given by decomposing the bundle AT*M as a tensor product
AT*(M/B) ® n'AT'B and applying the Berezin integral map to the first
factor, that is, projecting onto AnT*(M/B) ® n*AT*B and then dividing by
the vertical Riemannian volume form v^/b in T(M,AnT*(M/B)).
Theorem 10.23 (Bismut). The section
6?(Stre.(fc,(x,x))) € T(M,n'AT'B)
10.4. The Family Index Density
339
has a limit when t —> 0 equal to
Bm)-n'2TM/B{A(M/B) ch(?/S)) € r(M,7r'Ar'B).
Consequently, the differential form ch(At) ? A{B) has a limit when t —> 0
given by the formula
lim ch(At) = Bm)~n/2 f A{RM/B) ch(S/S).
t>o Jm/b
t—>o
Proof. Consider the bigrading on AT*XM given by
If we decompose the symbol of a 6 Az ® End(f.) according to this bigrading,
where
trip,,,(a) 6 Apz ® A"T*X{M/B) ® Endc(ji
we have the formula
Str?.(a) = (-2i)n/
p
Applying this formula to the supertrace of
we see that
Since o\p,n\ (fcj) = 0 if 2j < n + p, we see that there is no singular term in the
asymptotic expansion of 6B (Str?!r(fct(x,x))) as t —* 0. The explicit formula
of Theorem 10.21 for ^ crijikj) implies the formula for the leading order in
the asymptotic expansion. ?
Bearing in mind the family McKean-Singer theorem of Chapter 9, we ob-
obtain the cohomological form of the Atiyah-Singer family index theorem.
Corollary 10.24 (Atiyah-Siuger).
ch(ind(D)) = Biri)"/2 f A{M/B) ch(?/S) 6 H2'{B).
Jm/b

340
10. The Family Index Theorem
We now turn to the proof of Theorem 10.21, which is a fairly straight-
straightforward generalization of Theorem 4.1, the case in which B is a point. We
start by choosing a point z € B, but instead of working with the generalized
Laplacian D2 on the manifold M as in Chapter 4, we will work with the cur-
curvature Fz of the Bismut superconnection A restricted to the fibre Mz, which
we may think of as a generalized Laplacian with coefficients in the exterior
algebra Az.
Fix a point x0 e Mz, and let V = TXo{M/B) and H = TZB be the vertical
and horizontal tangent spaces at Xq\ thus
T = TXoM = V®H.
Let U = {x 6 V | ||x|| < e}, where e is smaller than the injectivity radius
of the fibre Mz. We identify U by the exponential map x i-» expl0 x with a
neighborhood of xq in Mz. Let rM/B(xo, x) be the parallel transport map in
the bundle T(M/B) along the geodesic from x to xq, defined with respect to
the connection VM/B defined in Section 1; since we are working on a single
fibre Mz, this connection is nothing but the Levi-Civita connection of Mt.
Using this map, we can identify the fibre TX(M/B) with the space V, so that
the space of differential forms .4A0 i" identified with C°°(U,AV*).
Choose an orthonormal basis dx4 of V* = T*Xo(M/B), and let e' e
T(U,T*{M/B)) = F{U,V*) be the orthonormal frame of T*(M/B) over U
obtained by parallel transport of dxj along geodesies by the Levi-Civita con-
connection on Mz. We denote by e° a local frame of T*M on U consisting of the
union of the cotangent frame e* and of a fixed basis f of T*ZB.
Let E = ?Xo be the fibre of the Clifford module ? at x0, let Sv be the
spinor space of V*, and let W = Horac{v){Sv,E), so that E is naturally
isomorphic to Sv®W. Recall the Clifford action m0 of T*M on Aw*T*B®?,
for which vertical cotangent vectors act by Clifford multiplication on ?, and
horizontal tangent vectors act by exterior multiplication on A = An'T'B.
Let te(xq, x) be the parallel transport map in the bundle Az ® ? along the
geodesic from x to x<y, defined with respect to the Clifford connection VEl°
of Section 2. Using this map, we can identify the fibre Az ® ?x of E at x
with the space AH * ® Sv ® W, and the space of sections r([/, Az ® ?) with
Although the bundle E over Mz is the tensor product of the bundle ? with
the trivial bundle Az, the term |mo(w) in the definition of the connection
VE'° means that parallel transport with respect to VE'° is not equal simply
to the tensor product of the identity on Az with parallel transport in ? by the
connection V?. Using the parallel transport map te(xq,x), we can analyse
the Clifford action mo of T*XM on Ea: by transporting it back to x0, where
it becomes an x-dependent action of T* = H* © V* on Az <8> E.
10.4. The Family Index Density
341
Lemma 10.25. //c' is the Clifford action of the cotangent vector dxj on E,
and ea is multiplication by fa in the exterior algebra AH*, we have
mo(D = ea,
where u'a are smooth functions on U satisfying u%a(x) = O(|x|).
Proof. Let TZ be the radial vector field on U. The frame e* satisfies Vn e* =
0. It follows that
by A0.3), and the fact that u{X,Y) = 0 if both X and Y are vertical.
Now, we use the fact, proved in Proposition 10.10, that VE'° is a Clifford
connection: [V^°,mo(a)] = mo(V^'°a). This shows that
Integrating this ordinary differential equation with the initial condition that
at x — 0, mo(e*) = c*, we obtain the first formula.
The second formula is clear: since T°(a)fa = 0 for all a € A2T*XM, the
connection VM'° coincides with tt*Vb on the sub-bundle n'TB of horizontal
vectors, so that V^'°/a = Vf fa = 0. D
We need another simple lemma.
Lemma 10.26. In the trivialization of E over U induced by the parallel
transport map rE(xo,x), the connection VE>0 equals d + 0, where
j;a<b a<b
here fiab(x) = O(|x|2) e C°°(U) and <fc(x) = O(|x|) € C°°([/) ® End(W).
Proof. Let V. be the radial vector field on U. Since 6G?.) = 0, the one-form
9 is determined (see A.12)) by the equation
l{U) (Ve'°J
= t(Tl)(\{RB) + imo(R) +
by Proposition 10.11,
the last line holds because <,G?.)A(.RB) = 0, Ti. being a vertical vector field
and RB being horizontal. D

342
10. The Family Index Theorem
Let Az be the Laplacian on Mz associated to the connection VEl°, so that
the curvature of the Bismut superconnection equals
F* =
a, eb).
a<b
We can transfer this operator to C°°(U, AT*®End(W)) using the quantization
map, that is, by replacing the Clifford action mo(ea) at x = 0 by the action
m* = ei - i{ , ma = ea.
The resulting operator L is given by the explicit formula
o<6
Next, we introduce the rescaling operator
6U : C°°(U, AT* ® End(W)) -> C°°([/, AT* ® End(W)),
which is basic to the proof; if a € C°°(f/, A*T' ® End(W)), then
Furthermore, if o is an element of C°°(U x R, A{T* ® End(W)), we define
By Lemma 10.25,
lim u1/2Su ¦ mo@) ¦ SZ1 = e@) for 0 € C°°([/, T*).
u—»0
In the next lemma, we calculate the leading term of the asymptotic expansion
of tfuV6'0^1. The following lemma follows immediately from Lemma 10.26.
Lemma 10.27. In the trivialization of Az®?z over U induced by the parallel
transport map te(xq,x),
R(di,dj,ea,eb)xomambtf
ambtf
j;a<b
We can now show that the family of differential operators u5uL6~J acting
on C°°(l/,Ar* ® End(H')) has a limit as u —» 0; this is the most impor-
important step in the proof. Since the twisting curvature F = Fxl of the Clif-
Clifford module ? at xo is an element of A2T* ® End(W), it acts on the space
C°°(U,AT* ® End(W^)) by multiplication. Similarly, the matrix-coefficients
ay =
a<b
of the curvature of the bundle T(M/B) belong to A2T* and act on the space
C°°([/,AT*®End(W0).
10.4. The Family Index Density
343
Proposition 10.28. The differential operator u6u-L-6~1 on C°°(U, AT* <
End(W)) has a limit K when u tends to 0, equal to
Proof. By Lemma 10.27, the differential operator u1(/26uVg!06u J equals
The fundamental symmetry of the four-tensor R (Proposition 10.9 C)) shows
that
R(8fe,a,,eeie6) = R{ea,eh,dudj)
from which it follows that
The operator u5u ¦ L ¦ 5 is equal to
• m'm' • tf
a<b
It is clear that both the third and fourth terms behave as O(u1^2) as u —+ 0,
while
a<b
a<b
Putting all of these equations together, and using the fact that e, = 9, at
x = 0, we obtain the theorem. D
Consider the time-dependent function on U given by transferring the heat
kernel (x | e~tF" | xo) to the open set U C TXOMZ by means of the parallel
transport map ^()
fc(t,x) = rE(x0,x)(x | e~tF* \ x0), where a; = expI0 x.
Then fc(t,x) lies in C°°{U,AH* .® End(Sv) ® End(W)); using the symbol
map End(Sv) —¦ C(V*) —» AV*, we may think of fc(t,x) as a map from U
to AV ® AH* ® End(W) s AT* ® End(W), and is a solution to the heat
equation
with initial condition lim _^ k(t,x) = 6(x). As in Chapter 4, we rescale the
heat kernel as follows.

344
10. The Family Index Theorem
Definition 10.29. The rescaled heat kernel r(u, t, x) equals
r(u,i,x) = 5>
t=0
The choice of the particular rescaling operator 6U is motivated by the fact
that
lim r(u,t,0) = lim 6u(fet(xo)) € AT*XaM,
u—»0 «—»0
and the right-hand side of this equation is precisely what we must calculate
to prove Theorem 10.21.
It is clear that r(u, t, x) satisfies the differential equation
{dt+u6u-L-6Z1)r{u,t,x) = 0.
The rest of the proof proceeds in exactly the same way as the proof of Theo-
rem4.1. Lemma4.18 applies to the operator A2 on Mz just as well as it does
when the base B is a point, and we obtain AT* ® End(Wr)-valued polynomi-
polynomials 7t(t,x) onRxV such that for every integer JV, the function rN{u,t,x)
defined by
2JV
«(x) ? «i/27i(t,x)
t=-n
approximates r(u, t,x) in the following sense: for N > j + |a|/2,
u,t,x) -rw(u,t,x))|| < C(N,j,a)uN,
for 0 < u < 1 and (t,x) 6 @,1) x U. Just as in Chapter4, 7i@,0) = 0 if
i ^ 0, while 7o@,0) = 1. Using the fact that L(u) - K + O(w1/2), we can
show in the same way as when the base B is a point that there are no poles
in the Laurent series expansion of r(u, t, x), that is, that 7i(t, x) = 0 for t < 0
so that
r(ti,t,x)~ft(x)fy'a7i(t,x).
i=0
i=0
The limit lim r(u,t,x) = qt(xOo(t,x) is a formal solution to the heat
equation
(9
(9t + )qt(Oo(
with initial condition 7o@,0) = 1. The following theorem follows from The-
Theorem 4.12.
Theorem 10.30. The limit lim r(u, t, x) exists and is given by the for-
formula
xexp I--
'»¦
10.5. The Transgression Formula 345
Theorem 10.21 follows from this lemma when we set t = 1 and x = 0.
10.5. The Transgression Formula
In this section, we will show that the transgression formula
ch(As) - ch(AT) = d JTStr[^e-X') dt
has a limit as s —» 0, when A is the Bismut superconnection; of course, for
an arbitrary superconnection for the family D, this is not true.
Let us start with an analogue in finite dimensions of the technique which
we will use. Recall the construction of Section 9.1, in which we associated to
a smooth family of superconnections A, on a superbundle ? over a manifold
B parametrized by R+ a superconnection A on the superbundle ? = ? x R+
over the manifold B = B x R+. Consider the special case in which As is the
rescaled superconnection
and let Aj be the dilation of the corresponding superconnection A on the
superbundle ?:
with curvature Tt — A2.
Lemma 10.31. We have
where a(t) equals
ds A —,
os
'M - ta(t) A ds,
Proof. Since At = A«, + dR+, it follows that at s = 1,
at
The lemma now follows from the Volterra expansion and the definition of
a(t). D
The following theorem is proved by applying the family index theorem to
a superconnection over B = B x R+ constructed from the Bismut supercon-
superconnection in the same way as above.

346
Theorem 10.32. A) The differential form
fdAt _
10. The Family Index Theorem
has an asymptotic expansion as t —> 0 of the form
where a,/2 € A(B).
B) The Chern form ch(At) is given by the transgression formula
ch(A() = B«rn/2 / A(M/B) ch(?/S) -d f a(s) ds.
Jm/b Jo
C) jydim(ker(D*)) is independent of z e B, so that ind(D) is a vector bundle
with connection Vo = PoA[i]Po> then
ch(Vo) = lim ch(A,)
= B7ri)-"/2 f A(M/B) ch(?/S) - d f°° a(s) ds.
Jm/b Jo
f
Jm/b Jo
Proof. It is clear that Parts B) and C) are an immediate consequence of
Part A) and the explicit formula for lim ch(At) which comes from the
local family index theorem.
Let R+ be the positive real line @, oo), and consider the family of manifolds
M = M x R+ over the base B = Bx R+. To prove that the function s>-* a(s)
is integrable as s —» 0, we will apply the family index theorem to this family,
with respect to the following additional data:
A) the vertical metric g^m on ^ equals 8~19m/b\
B) the connection on the family M —* B equals the natural extension of
the given connection on the family M/B in the directions tangential to
B, and the trivial one in the direction tangential to R+, for which the
horizontal tangent space is spanned at each point by d/ds;
C) the Clifford bundle over M is ? x R+, with Clifford connection Vs+dn+.
We denote bj; S, k and T, the metric and tensors corresponding to the
above data on M, and by A and T the corresponding Bismut superconnection
and its curvature. Let n = dim(M) — dimE) be the dimension of the fibres
of the family M -+ B.
Lemma 10.33. A) If X and Y are vertical vector fields on M, then the
tensor S(X, Y) equals
±(
10.5. The Transgression Formula
347
B) The one-form k equals k = k — —ds.
C) The tensor ft equals Q.
D) The Bismut superconnection for the above data on the family M —» B
equals
A = A, + dR - — ds.
+ 4s
Proof. Given vector fields on M, we may extend them to vector fields on all
of M by taking the extension which is independent of s € R+. Denote by ds
the vector field on M which generates translation along R+. If X and Y are
vertical vector fields on M and Z is horizontal, S(X, Y) may be calculated
as follows:
(S(X,Y),Z) = \(Z-(X,Y)S - (P[Z,X],Y)~ - {P[Z,Y],X)S)
= s'1{S(X,Y),Z).
The formula for the one-form k follows by taking the trace of 5.
It is easy to see that fi = fi, since [ds, X] = 0 if X is any vector field on
M.
To compare the Bismut superconnections A and A, we use the above for-
formulas, the explicit formula for A given in Proposition 10.15, and the fact that
if a 6 A1(M), the Clifford action c(a) on ? equals s^2c(a). ?
If we take the square of the above formula for the total Bismut supercon-
superconnection, we see that
which has Chern character
= ch(A,)-a(s)ds.
The theorem will follow from the local family index theorem applied to the
Bismut superconnection A. By this theorem, we know that 6f Strf(e~*^) G
A(B) has an asymptotic expansion in t as t —» 0 of the form
Sf
ds
3=0

348
10. The Family Index Theorem
where 4>i/i *= <MB) and otj/2 G A(B) are differential forms on B which do
not involve ds. Using the fact that
0A.
da
= t-
we may set s = 1, and we obtain the asymptotic expansion
i=o
To complete the proof, we must show that ao = 0. We use the explicit
formula for <j>o — otods furnished by the family index theorem:
4>o -<
„ _A{M/B)ch(?/S).
M/B
But the differential form A(M/B) ch(?/S) e A(M) is the pull-back to M
of the differential form A(M/B) ch(?/.S) € A(M); hence, it does not involve
ds, and we see that ao vanishes. ?
10.6. The Curvature of the Determinant Line Bundle
In Section 9.7, we denned the determinant line bundle detGr»?, D), its Quillen
metric | ¦ \q, whose definition involves the zeta-function determinant of the
operators D~D+ restricted to the eigenspace (A, co), and its natural con-
connection. In this section, following Bismut and Freed, we will calculate the
curvature of the natural connection on detGi\,?, D), when D is a family of
Dirac operators associated to a connection on ?.
If ? is a finite-dimensional superbundle, there is a relationship between the
Chern character of ? with respect to a connection V? and the curvature of its
determinant line bundle det(?) with respect to the corresponding connection
ydet(f).
2 = _str((VfJ) = ch(V?)[2l.
We will see a similar relationship in infinite dimensions; the curvature of the
connection that we described in Section 9.7 on the line bundle detGr«?, D)
will be shown to equal
lira ch(At)[2] = B7rt)""n/2( /" A{M/B) ch(?/S)) .
t—»o
Let us first consider a special case, in which ker(D) is a vector bundle, so
detGr»?, D) and det(ker(D)) may be identified. In this case, the connec-
10.6. The Curvature of the Determinant Line Bundle
349
tion Vo = PoA[i]Po on ker(D) is compatible with the L2-metric, and Theo-
Theorem 10.32 shows that
ch(V0) = Bni)-n/2 f A(M/B) ch(?/5) - d f°° a(s) ds.
Jm/b Jo
If we take the two-form component of this formula, we see that
(Vdet(ker<D))J = B«)-"/2( / A(M/B) ch(?/5)) -d[°° a{s)m ds,
KJM/B ' [2] Jo
from which we see immediately that
vdet(ker(D))+ r°QE) .
./O
is a connection on the bundle det(ker(D)) with curvature
B™)-"/2(/ A(M/B) ch(?/5))
VM/B ' PI
This connection is compatible with the Quillen metric. It would be very
interesting to have similar geometric interpretations of the other components
of the transgression formula.
Recall that the Quillen metric is defined on the representative det(W[o,A))
of the line bundle detGr»?, D) over U\ as
where | • | is the L2-metric on ([,))
The operator D(Aoo) = P(a,oo)D over the open set U\ has the property
that the bundle ker(D(,\,oo)) equals W[o,a)- We may define a superconnection
A a by the formula
Similarly, we define the rescaled version of this superconnection
Define two one-forms q±(s, A)[ij e A1(B), given by traces over 7r»?+ and
*.?- respectively:
[1]'
By Lemma9.42, we see that a+(s, A)[X] = a~(s, A)[!j.
As in Section9.7, we denote the one-forms 2LIMt_,o/t°°c*:':(s, A)[ij ds by
Jf; these satisfy /3a + = Px- Thus, the sum /?J + 0^ is real, while the
difference C^ — 0^ is imaginary. Also, we proved that
-dC'@,D-D+,A)=/3+ +&-.

°°V 10. The Family Index Theorem
Thus, we see that the sum of differential forms 0+ + 07 is an exact real
one-form.
We defined the connection V<"*<*.?) on **(*.?, D) as follows: on the
bundle det(«@ A)), we took the connection
g
Our goal is to calculate the curvature of this connection. In the following
temma, we calculate the exterior differential of the imaginary one-form 0+ -
Lemma 10 34. The difference a+(s, A)A) -a~(s, A)(lj is bounded by a mul-
multiple ofs V2 in A(Ux) for small 3> d
ft ~ K = 2 jf(a+(a, A) - a-(s, A))AJ da.
The exterior differential of this differential form is given by the formula
HPt ~ 0D + ch(Vl°.*))[2] = Brt)-/»( / A(M/B)
JM/Ux
Proof. We have seen that
[2]
Since the Bismut superconnection A coincides with D + V*? up to terms of
degree 2, we see that
[i]
Prom this, it follows that
The operator Pjo,A) is a smoothing operator, so that by Lemma 9.47 for
small,
The result now follows from Theorem 10.32. D
We now arrive at the main result of this section (this is what is known to
physicists as the "anomaly formula").
Theorem 10.35. The curvature of the connection Vdet(*«?>D> equals
A(M/B) ch(?/S)) .
10.7. The Kirillov Formula and Bismut's Index Theorem
Proof. We have already done all of the work. Indeed,
351
since the differential form 0% + 0X is closed. The theorem follows from
Lemma 10.34. ?
10.7. The Kirillov Formula and Bismut's Index Theorem
In this section, we will show that the local family index theorem, in the case
in which the family M —¦ B is associated to a principal bundle P —> B with
compact structure group G, is equivalent to the Kirillov formula of Chapter 8.
All of our notations are as in Section 9.4.
We start by calculating the Bismut superconnection A. The vertical tan-
tangent bundle T(M/B) = P Xq TN has a connection VN>e, obtained by com-
combining the Levi-Civita connection VN on TN with the connection 0 on the
principal bundle P —> B, as explained in Section 7.6. On the other hand,
the connection 0 determines a connection on the fibre bundle M —» B, and
hence, by the construction of Section 1, a connection VM/B on T(M/B).
Lemma 10.36. The two connections VN>e and VM/B on T(M/B) are equal.
Proof. Since this formula is local, we assume that P is the trivial bundle P =
B x G, with connection form 0 = J^T=i &kxk € -4(B. 0). so tnat M = BxN.
If V is a vector field on B, its horizontal lift is V + 0{V)n- K Y and Z are
vector fields on N, then by the definition of VM/B, we see that
'z) =
+ 0(V)N(Y, Z)
where we have used the fact that the vertical metric on M is independent of
the point z in the base B, and that 6(V)n is a Killing vector field. On the
other hand, Lemma 7.37 shows that
= {0(v)n,y\. a
In the same way, the bundle ? — PxqE has a connection VE'e, obtained
by combining the Clifford connection VB on E —> N with the connection 0.
Using the fact that for any Y e g and o e T(N, T*N),

352
10. The Family Index Theorem
we see that VB'e is a Clifford connection on ? compatible with VN'e = VM/B.
We denote this connection by V?.
The equivariant Chem-Weil homomorphism <j>$ maps the equivariant de
Rham complex (Aa(N),dt) to the de Rham complex (A{M),d). Using the
fact that Vw and VE'8 equal VM/B and V? respectively, we see that
= A(M/B),
where As and cht{E/S) are the equivariant A-genus of N and the equivariant
relative Chern character of E.
In the fibre bundle M —* B, the tensor 5 is equal to zero, and hence
its trace k is also zero. To calculate the Bismut superconnection, we must
describe the curvature tensor fi of M.
Lemma 10.37. IfV and W are two vector fields on B, then
where VP and WP are the horizontal lifts ofV and W to P, QP € A2{P,$)
is the curvature of P, and for any element X ?. 0, Xn is the corresponding
vector field onN, as well as the induced vertical vector field onM = Px.gN.
Since the bundle 7r*? is the associated bundle to the representation of G
on T{N,E), it has a canonical connection given on the space
A(B,nt?)<*{A(P)®T(N,E))
haa
by the formula
This connection has curvature
There is a linear map c : 9 -> T{N,End(E)), defind by sending X ? g
to c(Xjv), the operator of Clifford multiplication by the vector field XN ?
T(N,TN) associated to X. By means of the formula
we obtain an action of the curvature ilP of P on A(P) ® T(N, E).
Theorem 10.38. A) The Bismut superconnection of the bundle P xGN is
the restriction to (A{P) ® r(JV,E))bM of the operator
10.7. The Kirillov Formula and Bismut's Index Theorem
353
B) The Chern character ch(At) = Str(e~A?) € A{B) is independent oft.
Proof, fr-om the formula for A given in Proposition 10.15, we see that A[Oj =
Djv and A[2] = jc(nP). To check that
we may assume that P is trivial, as in the proof of Lemma 10.36. We see that
Eeoy? _ r.«y
? Vo~Z^? V
The connection V*? commutes with D^, since Djv is G-invariant. Fur-
Furthermore, V*<? and c(Op) commute, by the Bianchi identity Vftp = 0. If
we let Qt be the operator
Qt = At - V? = tl'2DN + ^
we see that
€ T(B,Endv(w,?)),
dk2t \dAt
dkt
and hence that
since the supertrace vanishes by Lemma 3.49. D
We have seen in Section9.4 that Kirillov's formula for indc(e"x, Djv) im-
implies that
lim
t—>oo
?) = {2m)-n'2 f A(M/B) ch(?/S).
JM/B
We see from the above theorem that the formula
lim Str(e-A«) = {2ni)-n/2 f A{M/B) ch(?/S),
t—*0 JM/B
is, in the case of a compact structure group, a consequence of the Kirillov
formula for the equivariant index of DN. We will strengthen this result by
showing that the the local family index theorem of Bismut is implied in this

354
10. The EWily Index Theorem
context by the local Kirillov formula for the operator K(X) = (D+jc(X)) +
CE{X) of Section 8.3.
The operator K(X) is a differential operator depending polynomially on
X, and hence acts on the space C[g] ® T(N, E), by the formula
{Ks)(X) = K(X)s(X).
If p is a point in the principal bundle P lying in the fibre of z € B, we may
specialize the operator K(X) to an operator K(ilp) acting on A2T%B ®
T{N, E), by replacing X = ?™=1 akXk € g, ak & C, by
«p = f; ak(UkP)pXk e A2TV
t=i
On the other hand, the point p € P gives an identification of the fibres Mz
with N, so that we may also consider the curvature A2 of the Bismut super-
connection to be a differential operator A2 acting on h?T\(p)B ® T(N, E).
Proposition 10.39. The two operators K(UP) and A2 are equal.
Proof. This follows from Theorem 10.38 and the fact that V*'fi supercom-
mutes with Dat and c(Q):
D
Prom this lemma, we see that the coefficients of the asymptotic expansion
of (p | e~tA | p) depend polynomially on Qk, and are obtained from the
coefficients of the asymptotic expansion of (p | e~iK^x^ | p) by replacing X
byfip.
The converse result is also true: the local family index theorem implies
the local Kirillov formula. To show this, we use the following lemma from
the theory of classifying spaces.
Lemma 10.40. There exist fibre bundles Pf/ —> B^ for which the Chern-
Weil map C[fl]G —» A(Bf/) is an injection up to polynomials of degree N, for
any N > 0.
Proof. Let V be a faithful unitary representation of G. Prom the embedding
\ny ® cN) x u(v ® cN) c u(v ® c2N),
we see that U(V ® CN) acts freely on PN = U(V ® C2N)/V(V ® CN).
There is an embedding of G in U(V ® CN), in which we map g e G to
g ® /. Let B;v = Pn/G. The result follows from the fact that if m > n,
7Ti(U(m)/ U(n)) = 0 for i < 2n. ?
The local index theorem for families for the bundle Pn —* Bjv implies
the local Kirillov formula for K(X) up to order N. Since N is arbitrary, we
obtain the local Kirillov formula to all orders.
Bibliographic Notes
Bibliographic Notes
355
Sections 1-4. The results of the first four sections of this chapter are due to
Bismut [29], although the proof of the main result (Theorem 10.21), the local
family index theorem, follows our proof of the local index theorem in Chap-
Chapter 4 of this book. In his work, Bismut uses the term Levi-Civita superconnec-
tion where, for obvious reasons, we prefer the term Bismut superconnection.
Theorem 10.19 is new, and gives a natural interpretation of the Bismut super-
connection. For closely related approaches to the family index theorem, see
the articles of Berline-Vergne [25], Donnelly [48] and Zhang [100].
Sections 5 and 6. The "anomaly" formula for the curvature of the natural
connection on the determinant line bundle was proved by Witten in the
special case where there are no zero modes (see for example [98]), based
on the work of many thertical physicists. Further special cases were proved
by Atiyah-Singer [17] and Quillen [85], and the theorem is stated and proved
in full generality in the article of Bismut and Freed [33]. In this chapter, it
appears in a generalized form inspired by the article of Gillet and Soule [61].
Section 7. The relationship between the family index theorem for a family
with compact structure group and the equivariant index theorem is in Atiyah
and Singer [16]. The remarkable observation that such a relationship holds
at the level of differential operators, as in Proposition 10.39, is due to Bis-
Bismut [29].
For an extension of the results of this chapter in the direction of algebraic
geometry, see Bismut, Gillet and Soule [34].

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List of Notations
T[M,?)
TC(M,?)
GL(?)
C(X)
C?(X)
A(M)
MM)
°IO
M
i(v)
?(q)
H*(M)
H'C(M)
A{M,S)
MO
n
m/b
O(Af)
»o(TM)
R{X,Y)
Ric
r.\t
l«tr|
grad/
\a.b)
Sir
det(f)
T(a)
Pl(A)
space of smooth sections of ? 14
space of compactly supported smooth sections of ? 14
frame bundle of ? 16
Lie derivative acting on vector fields 17
Lie derivative acting on sections of ? 17
space of differential forms on M 17
space of compactly supported differential forms on M 17
i-form component of a 17
degree of a 17
contraction 18, 69
exterior multiplication by a 18
de Rham cohomology of M 18
compactly supported de Rham cohomology of M 19
differential forms on manifold M with values in ? 19, 43
tensor product of covariant derivatives 25
parallel translation along smooth path 7 26
Euler (radial) vector field 26, 52
bundle of half-densities 30
integral of a along the fibres of a fibre bundle 31
another notation for 7r»a 31
orthonormal frame bundle of Riemannian manifold M 32
O(M) xo(n) so(n) 33
Riemannian curvature 33
Ricci curvature 34
scalar curvature 34
Riemannian density 34
gradient of the function / 34
point in the normal coordinate system around Xo 35
Jacobian of exponential map around xq 38
supercommutator 39
supertrace 39
determinant line-bundle of ? 41
Berezin integral 42
PfafRan of A 42

362
A
ch(A)
A(M)
Td(M)
X(M)
Eul(M)
A?
D*
d*a
9t(x,y)
/•asyrap
C
Afc
r(x, y)
U\\»s
rL,(M,
Tr(K)
Gk
C(V)
c(v)
List of Notations
smother notation for Pf(i4) 43
superconnection 44
degree i component of A 45
Chern character form of A 49, 285
Chern character of ? 50
.A-genus of M 50
Todd genus of the almost-complex manifold M 51
Euler class of the oriented manifold M 58
Euler number of M 59
Laplacian associated to ? 66
formal adjoint of D 68
divergence of a 68, 127
heat kernel of Euclidean space 72
asymptotic expansion of a Gaussian integral 73, 173
space of generalized sections of ? 74
space of C'-sections of ? 74
external tensor product of ?i and ?2 74
kernel of the operator P at (x, y) 74
heat kernel of Laplacian H 75, 87
fc-simplex 77
parallel translation along geodesic joining x and y 83
Hilbert-Schmidt norm of A 87
space of square-summable sections of ? ® IAI1/2 88
trace of K 88
k-th Green kernel of H 93
Clifford algebra of V 103
action of v 6 V on a Clifford module 104
symbol map from C(V) to AV 104
degree i subspace of Clifford algebra 104
canonical homomorphism from C2(V) to so(V) 105
double cover of SO(V) 106
detx/2 (*J!Mm.) 108
T
Spin(V)
chirality element of C(V) 109
spinor representation of C(V) 109
EndC(y)(.E) endorphisms of Clifford module E over C(V) 112
Str?/s relative supertrace for Clifford module E 113
X(X) action of U(V) on AT^V 113
C(M) Clifford bundle of Riemannian manifold M 113
Spin(M) principal Spin(n)-bundle of spin-manifold M 114
Endc(M){?) endorphisms of Clifford module E over C(M) 114
5 spinor bundle 115
D Dirac operator 116
F?>s twisting curvature of Clifford module ? 120
R? Clifford action of Riemannian curvature on ? 120
List of Notations
363
ind(D)
<x(M)
kt(x, x)
/
ch(?/S)
Cas
C[g]
M
, A)
, ?/•!>)
Ag{M)
d9
Hq(M)
Ag(M, ?)
A8
F9(X)
n(X)
ch,(A)
HM{X)
fi
fi{X)
Mo
Xg(AT)
dp
AJX, M)
H*/S(X)
Fe'S
che(X,?/S)
H(X)
Vo
At
Tt
a(i)
{Mx I z e B)
T(M/B)
I A* I
{?z I z 6 B)
n,?
(D* I 2 6 B)
index of D 123
Hodge star 131
signature of M 136
heat kernel of D2 restricted to the diagonal 143
rescaling of a 144
relative supertrace for Clifford module ? 146
relative Chern character of Clifford module ? 146
Casimir of a representation 172
polynomials on the Lie algebra fl 179
fixed point set of 7 185
equivariant index of D 186
normal bundle of M7 C M 191
localized Chern character of A 194
localized twisting Chern character of ? 195
space of equivariant differential forms 208
equivariant exterior differential 208
equivariant deRham cohomology 209
equivariant differential forms with values in ? 210
equivariant superconnection 210
equivariant curvature 210
moment map 211
equivariant Chern character 212
Riemannian moment 213
symplectic form 214
symplectic moment 214
zero set of vector field X 217
normal bundle of Mo C M 217
equivariant Euler class of Af 218
Liouville form 226
equivariant A-genus 249
twisting moment 250
equivariant twisting curvature 250
equivariant relative Chern character of ? 250
Bismut Laplacian 258
projection of a superconnection A onto ker(D) 271, 287
rescaling of the superconnection A 272, 287
curvature of At 272, 287
transgression form of A 277, 290
family of manifolds 278
vertical tangent bundle 278
vertical density bundle 278
family of vector bundles 278
vector bundle of sections along fibres of ? 278
family of Dirac operators 279

364
List of Notations
ind(D)
UM/(t)
det(tf)
detGr.f, D)
det(D+)
d(?D)
A(M/B)
subbundle of ntS on which D2 < A 279
spectral projection onto D < A 280
subset of B on which A is not an eigenvalue of D2 280
family index of D 281, 298
subbundle of n»€ on which A < D2 < n 282
spectral projection onto A < D2 < fi 282
natural connection on nt? 283
Mellin transform of / 300
renonnalized limit of / 301
zeta function of H 302
zeta function determinant of H 302
determinant line bundle associated to the family D 306
determinant of D+ (section of detGr,?, D)) 307
Quillen metric 307
natural connection on detGr,?, D) 309
i-genus of T(M/B) 294, 337
Index
adjoint, formal 68, 89
affine connection 28
i4-genus 50, 144
equivariant 213
almost-complex manifold 135
anomaly formula 350
associated bundle 15
Atiyah, M.F. 147, 150, 187, 196,
245, 295, 339
Atiyah-Bott fixed point formula 187
Atiyah-Segal-Singer fixed point
formula 196
Baum, P. 225
Berezin integral 42, 54, 106, 109,
146, 196, 243
Bianchi identity 44
Bismut, J.-M. 258, 282, 332, 338,
348
Bismut
Laplacian 258
superconnection 332, 347
Bochner-Kodaira formula 140
Borel-Weil-Bott Theorem 257
Bott class 241
Bott, R. 187, 224, 257
bundle
associated 15
Clifford 113
equivariant 16, 209
frame 15, 32
holomorphic 136
horizontal 21
index 281, 299
principal 15
Cartan subalgebra 229
canonical line bundle 140
Casimir operator 172
character formula
Kirillov 254
Weyl 253
characteristic class, 47
i-genus 50, 151, 337
Chern character 49, 286
equivariant 212
Euler class 58, 148
Z-genus 150
Todd genus 51, 152
characteristic number
Bott's formula 225
Cheeger, J. 225
Chern, S. S. 59
Chern character 49, 286
equivariant 212
relative 146
equivariant 250
localized 195
Chern-Weil homomorphism 47
equivariant 236, 294, 352
Chevalley, C. 108, 233
chirality operator 109
Clifford algebra 103
quantization map 104
supertrace 110
symbol map 104
Clifford bundle 113
Clifford connection 117, 329
Clifford superconnection 117
Clifford module 103, 114, 127, 325
equivariant 186
twisted 114

366
coadjoint orbit
Fourier transform 228
cohomology
de Rham 18, 128, 296
Dolbeault 139
equivariant 209
complex manifold 135
connection 20
affine28
Clifford 117, 329
invariant 26, 210
Levi-Civita 33
on a principal bundle 21
torsion-free 28
connection form 21, 26
contraction 18, 69
covariant derivative 22
holomorphic 136
tensor product 25
curvature 21, 23, 44, 321
equivariant 210
Ricci 34, 130
Riemannian 33
scalar 34, 126, 134
twisting 120
equivariant 250
cut-off function 84
5-operator 136
deRham cohomology 18, 128, 296
compact supports 19
deRham's theorem 128
delta distribution 192
density 30
half 68
integral of 30
Riemannian 34
vertical 278
determinant, zeta-function 302
determinant line bundle 41, 306, 348
connection 309, 350
curvature 348
L2-metric 306
Quillen metric 306
difference bundle 295
Index
differential form 17
basic 20
equivariant 208
harmonic 128
horizontal 20
integral along the fibre 31
self-dual 132
volume 31
differential operator 64
elliptic 65
Dirac, P. A. M. 102
Dirac operator 116
classical 127, 134
index of 123
divergence 68
Dolbeault cohomology 139
Dolbeault's theorem 139
Duhamel's formula 98, 169
Duistermaat-Heckman theorem 226
Duistermaat, J. J. 226
equivariant
4-genus 213, 249
bundle 16, 209
characteristic class 212
Chern character 212
relative 192, 250
Chern-Weil homomorphism 236,
294, 352
Clifford module 186
cohomology 209
curvature 210
differential form 208
exterior differential 208
Euler class 213, 218
index 186, 250, 293
relative Chern character 250
superconnection 210
twisting curvature 250
Euler class 58
equivariant 213, 218
Euler number 59, 129, 149
holomorphic vector bundle 139,
152
exponential map 35
Index
367
Jacobian of 38, 168
exterior algebra 103
exterior differential 17, 29
equivariant 208
family 278
of smoothing operators 314
family index theorem 338
flag manifold 232, 254
Fourier transform of orbit 224
frame bundle 16
orthonormal 32
Freed, D. A. 348
fundamental one-form 28
Gauss-Bonnet-Chern theorem 59, 148
Gaussian integral 54, 72, 173, 178,
223
generalized section 74
geodesic 35
Gilkey, P. B. 147, 193
gradient 34
Green operator 92, 288
Hamiltonian vector field 214, 227
Harish-Chandra 231
harmonic form 128
harmonic oscillator 145, 153
harmonic spinor 134
heat equation 75
formal solution 83
heat kernel 72, 75, 310
asymptotic expansion 87
Heckman, G. 226
Hilbert-Schmidt operator 87
Hirzebruch, F. 150, 152, 245
Hodge star 131
Hodge's theorem 128, 139
holomorphic vector bundle 136
horizontal bundle 21
horizontal lift 21
index 123, 145
equivariant 186, 250, 293
family (see index bundle)
index bundle 270, 281, 299
index density 146
equivariant 196
family 338
index theorem
local 146, 181
equivariant 196, 250
injectivity radius 35
Jacobian of exponential map 38,168
tf-theory 295
Kahler manifold 137, 150, 257
Karamata, J. 94
kernel 74
Kirillov, A. A. 253
Kirillov formula 249, 295, 351
Kirillov character formula 253
Kodaira vanishing theorem 141
Kostant, B. 255
Kiinneth theorem 129
L-genus 149
L2-metric 302
Laplace-Beltrami operator 128,133
Laplacian, generalized 65
Leibniz's rule 17, 22, 44, 282
Levi-Civita connection 33
Lichnerowicz, A. 134
Lichnerowicz formula 126,135,156,
334
Lie bracket 17
Lie derivative 16
E. Cartan's homotopy formula
18
Lie superalgebra 39
Liouville form 226, 231
local index theorem 146, 181
equivariant 196, 250
family 338
localization theorem 215, 251
manifold
almost-complex 135
complex 135

368
Index
Hermitian 135
Kahler 137, 150
Riemannian 32
spin 114, 144
symplectic 214
Mathai, V. 52, 241
McKean, H. 124
McKean-Singer formula 124, 145
equivariant 186
family 299
mean curvature 321
Mehler's formula 145,154, 265, 344
Mellin transform 300
Minakshisundaram, S. 101
moment 211
Riemannian 213, 249
symplectic 214
twisting 250
normal coordinates 35, 71
orientation 31
P-endomorphism 279, 310
parallel transport 26
parametrix 77
Patodi, V. K. 147
Pfaffian 42, 224
Pleijel, A. 101
Poincar^ duality 132
Poincare lemma 52
Poincar6-Hopf theorem 59, 61, 225
polarization 109
principal bundle 15
geodesic distance 198
heat kernel 175, 200
quantization map 104
Quillen, D. 43, 46, 52, 241, 306
Quillen metric 306, 348
radial vector field 35
relative Chern character 146
equivariant 250
localized 195
relative supertrace 113, 147
renormalized limit 302
Ricci curvature 34, 130
Riemann-Roch theorem 152, 245
Riemann-Roch-Hirzebruch theorem
152
Riemannian
curvature 33
manifold 32
metric 32
moment 213, 249
Rossman, W. 234
scalar curvature 34, 126, 134
Schwartz kernel theorem 74
second fundamental form 320
section 14
generalized 74
Segal, G. B. 196
signature 133, 150
signature operator 133
signature theorem 150
simplex 77
Singer, I. M. 124,147,150,196,295.
339
smoothing operator 74
family 314
Spin group 106
spin structure 114
spinor bundle 115
spinor module 109
supertrace 110
stationary phase 226
superalgebra 38
Lie 39
supercommutative 39
superbundle 39
supercommutator 39
superconnection 44, 282
Bismut 318
Clifford 117
curvature 44
equivariant 210
tensor product 46
superspace 38
Index
369
Hermitian 41
tensor product 40
supertrace 39, 46
relative 113, 147
spinor 110
symbol
Clifford algebra 104
differential operator 64
symplectic manifold 214
tangent bundle 16
vertical 279, 320
tautological section 28, 55, 212
tensor product 40
Thorn form 52, 243
Todd genus 51, 152
torsion 28
trace-class 88
transgression formula 31, 47,57, 277,
290, 345
twisting
bundle 114, 131, 134, 175
curvature 120, 145
equivariant 250
moment 250
vanishing theorem
Bochner 130
Kodaira 141
Lichnerowicz 134
vertical density 278
Volterra series 78, 275, 310
volume form 31
Weitzenbock formula 130
Weyl, H. 94, 144
Weyl character formula 253
Weyl group 230
zeta function 302
zeta-function determinant 302
P'inting: Merceclesdruck, Berlin
Binu'inp: Buchbindcrei Liiderilz & Bauer, Berlin