SUPERGRAVITY
AMD SUPERSTRINGS
A Geometric Perspective.
Vol. 1: Mathematical Foundations
Leonardo Castellani
Istituto Nazionale di Fisica Nucleare
Sezione di Torino
Riccardo D* Auria
Dipartimento di Fisica
Universita di Padova
Pietro Fre
International School for Advanced Studies, Trieste

V»
World Scientific
Singapore • New Jersey • London • Hong Kong
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SUPERGRAVTTY AND SUPERSTRINGS —A Geometric Perspective
Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by any means, electronic or mechanical, including photocopying, recording or any
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ISBN 9971-50-037-X (set)
ISBN 9971-50-038-8 pbk (set)
Printed in Singapore by Loi Printing Pte. Ltd.
V
PREFACE
йЧС»6"
.уК» ‘ our hopes the present book is a self-contained account of the theory of supergravity
'Send of the theory of superstrings. It is meant to be both introductory and advanced, this
Sffiatiire explaining its considerable length.
W^The authors’ views on the relevance of the whole subject can be related in a few
’ words. On one hand we feel that reconciling quantum mechanics with general relativity
logical necessity one cannot overlook, while trying to explain the structure of the other
^fieractions. As far as we know the only serious candidate for a quantum theory of the
$ЬимЛоп is superstring theory, whose low-energy approximation is supergravity. On the
Higher hand, provided Higgs fields are necessary to explain spontaneous symmetry
Ж’йеаЮпд, the only satisfactory solution of the gauge hierarchy problem seems to be
5&en by spontaneously broken local supersy m metry. From the two opposite sides of the
^energy scale we come to the same suggestion: particle physics phenomenology should
Wbe described in terms of an effective supergravity model.
|	Whether such views are supported by experimental evidence is a question that might
M'be answered in the near future as soon as the LEP machine becomes operational.
В *!'i? In any case it should be stressed that supersymmetry is a profound symmetry
fyfeindple, with far reaching implications: it has the same standpoint as the principle of
SjOCneral covariance and, similarly, it provides an extremely elegant framework for the
f formulation of the laws of Nature.
Furthermore, the structure of these theories encompasses so many different aspects
gj «nd it is so ramified that they will continue to be interesting theoretical laboratories in
ft many respects.
> We have chosen to present the whole subject in a systematic way, aiming more at
•te.Ithe basic principles than at the specific applications: however the eventual use of the
|ytheory for the construction of a realistic model, describing particle phenomenology, has
b been our constant motivation directing our choices.
I We have tried to be exhaustive in the discussion of the different mechanisms, and
models, but not in covering the various formalisms that have been historically utilized to
Й derive the various results. Indeed we have presented everything in a unified language
% emphasizing the underlying geometrical structure. Furthermore, we have included
several mathematical chapters, explaining, at each stage of the theory development, the
| mathematics involved in the construction.
f The book is divided into three volumes and six parts. The first volume introduces the
J: flsometric and algebraic foundations of supergravity (PartsOne and Two, respectively).
? The second volume is devoted to the construction of supergravity theories limited to
g second order derivative interactions. Specifically, in Part Three we discuss supergravity
। models without elementary scalar fields. In Part Four we introduce also the coupling of
the scalars that leads to the most exciting properties of the theory: the scalar field
* ®*°metry and the superHiggs mechanism. In Part Five we discuss the Kaluza-Klein
Interpretation of D= 11 supergravity and we explain its limitations in providing a viable
i, W|fied theory. The techniques utilized in this analysis have a general applicability and
vi
can be utilized also in string theories. Finally, in the third volume (Part Six), we develop
superstring theory, with particular emphasis on the issue of modular invariance and the
construction of D - 4 models. Furthermore, we discuss higher curvature supergravities
with Lorentz-Chern Simons terms and their role as effective superstring Lagrangians.
We would like to express our gratitude to the many persons who gave us invaluable
help and advice during the completion of this work.
Firstly we want to thank our friends Sergio Cecotti, Sergio Ferrara, Luciano Girardello,
Ferdinando Gliozzi, Tullio Regge, Stefano Sciuto, and Peter van Nieuwenhuizen forthe
many enlightening discussions, comments and suggestions.
Secondly we are grateful to our students and collaborators, who read our manuscripts
correcting many errors: Anna Ceresole, Claudio Coriano’, Andrea Ruben Levi, Igor
Pesando, Annamaria Piras, and Marco Rego Monteiro.
Thirdly we are particularly indebted to Silvana Afrito for the professional care and
dedication in typing the manuscript of Parts One to Five.
Finally we want to thank our publisher for the friendly and efficient cooperation
including the typing of Part Six.
Leonardo Castellani
Riccardo D’Auria
Pietro Frd
vii
CONTENTS
?- -
Preface
Volume 1
PART ONE. GRAVITY AND DIFFERENTIAL GEOMETRY
Chapter 1.0.	Introduction	3
Chapter 1.1.	Exterior Calculus	9
	1.1.1.	Exterior forms on vector spaces	10
1.1.2.	Mappings and operations on forms	23
1.1.3.	Differentiable manifolds, vector fields and differential forms	28
1.1.4.	Functions, vector fields and differentia) forms	37
1.1.5.	Exterior differentiation and behaviour under mappings	47
- 1.1.6.	The vielbein basis	53
S 1.1.7.	Lie derivative, coordinate transformations and invariance	59
Appendix: The 8 Operator and the Hodge Decomposition		72
Chapter 1.2.	Riemannian Manifolds	75
1.2.1.	Introduction	75
c; I.2.2.	Geometry of the linear spaces	76
1.2.3.	The geometry of general Riemannian manifolds in the vielbein basis	80
1.2.4.	Relation with the standard world-tensor formalism	91
Chapter 1.3. Group Manifolds and Maurer-Cartan Equations		97
1.3.1.	Introduction	97
I.3.2.	Lie groups as manifolds: left and right invariant vector fields	98
1.3.3.	Maurer-Cartan equations	104
!	1.3.4.	Adjoint representation and Killing metric	107
1.3.5.	Killing metric	113
’ v 1.3.6.	Riemannian geometry of semisimple groups	116
1.3.7.	Soft group manifolds	119
Cl 1.3.8.	The example of Poincate and anti de Sitter soft group manifold	131
Chapter 1.4	. Poincard Gravity	141
1-4-1.  ГУ-	Poincate gravity	141
I.4.2.	Extension to the soft group manifold	152
л e 1.4.3.	Building rules for the gravity Lagrangians	157
* .. 1-4-4. t	Gravity in de Sitter and anti de Sitter space	166
Chapter 1.5	. Coupling of Gravity to Matter Fields	170
, f.-. 1-5.1.	Geometrical Lagrangian for scalar fields on a rigid background	170
152	Extension to the Poincate group manifold and interpretation	
	of the Lorentz transformation rules as variational equations	174
viii
1.5.3.	The interaction of the scalar fields with gravity and the effective
cosmological constant	176
1.5.4.	The field equation of a massless scalar field in anti de Sitter
space (in general in a curved space)	181
I.5.5.	Geometrical Lagrangian for spin 1 fields	185
I.5.6.	Geometrical Lagrangian for spin 1/2 fields	188
Chapter 1.6	. Differential Geometry of Coset Manifolds	190
1.6.1.	Introduction	190
I.6.2.	Classification of coset manifolds	195
1.6.3.	Coordinates on G/H and finite G-transformations	197
1.6.4.	Finite transformations on G/H	204
1.6.5.	Infinitesimal transformations and	Killing	vectors	210
1.6.6.	Vielbeins and metric on G/H	212
1.6.7.	Covariant Lie derivative	219
1.6.8.	Geodesics	222
1.6.9.	Invariant measure	225
1.6.10.	Connection and curvature	226
1.6.11.	Rescalings	231
1.6.12.	A Note on the isometries of G/H	235
1.6.13.	Some examples	240
1.6.14.	Elements of algebraic topology	254
1.6.15.	Homotopy and (co)homology of coset spaces	262
Chapter 1.7	. Applications of the Formalism and Miscellaneous
Examples	272
1.7.1.	The Brans-Dicke theory	272
1.7.2.	Minima! coupling of pseudoscalars through	a torsion mechanism	278
1.7.3.	The Schwarzschild solution	283
Bibliography	296
PART TWO. THE ALGEBRAIC BASIS OF SUPERSYMMETRY
Chapter 11.	1.	Introduction	301
Chapter 11.	2.	Super Lie algebras,	Supermanifolds and Supergroups	310
11.2.1	. The definition of superalgebras and the example of N-extended
super Poincard algebra	310
11.2.2	. Classification of the simple superalgebras whose Lie algebra
is reductive	323
11.2.3	.	Grassmann algebras	333
11.2.4	.	Supermanifolds	338
11.2.5	.	Supergroups and graded matrices	345
11.2.6	. Osp(4//V) as the Л/ extended supersymmetry algebra in anti de
Sitter space	352
Chapter 11.3	. Super Maurer-Cartan Equations and the Geometry
of Superspace	360
11.3.1.	Maurer-Cartan equations of supergroups on supergroup manifolds	360
11.3.2.	Maurer-Cartan equations of Osp(4/N)	and Osp(4/N)	364
ix
<. 11.3.3. Osp(4/W) Maurer-Cartan equations as the structural equations of	370 380
rigid superspace	
b' 11.3.4. Killing vectors on superspace, that is the generators of the	
supersymmetry algebra of superisometries	
Chapter И.4. Poincar6 Supermultiplets 11.4.1. How to construct the unitary irreducible representations of the	390
N-extended Poincard superalgebra	390
II.4.2.	Massive multiplets without central charges	395
II.4.3. Massive multiplets with central charges	411
II.4.4. Massless multiplets	416
Chapter 11.5. Supermultiplets in Anti de Sitter Space 11.5.1. Free field equations and the concept of mass in anti de	425
Sitter space	425
11.5.2. Unitary irreducible representations of SO(2,3)	435
11.5.3. Unitary irreducible representations of Osp(4//V)	448
11.5.4. Osp(4/1) supermultiplets 11.5.5. Remarks about the /V-extended case and the example of the	454
Osp(4/2) multiplets	464
Chapter 11.6. Supersymmetric Field Theories: The Example of
the Wess-Zu mine Multiplet	473
11.6.1.	Supersymmetric field-theories corresponding to an irreducible	
representation of the supersymmetry algebra	473
11.6.2. The Wess-Zumino model: the simplest example of a supersymmetric	
field theory	477
11.6.3. Superfield interpretation of the Wess-Zumino model and	
rheonomy	489
11.6.4. The integrability of the rheonomic conditions and the Bianchi	
identities	500
11.6.5. The rheonomic action principle	508
Chapter 11.7. Г-matrix Algebra and Spinors in 4 < D < 11	519
II.7.1. The construction of Г-matrices	519
11.7.2. The charge conjugation matrix	523
11.7.3. Majorana, Weyl and Majorana-Weyl spinors	526
11.7.4. Useful formulae in Г-matrix algebra	530
Chapter 11.8. Fierz Identities and Group Theory	535
11.8.1. Introduction	535
11.8.2. The structure of forms on A/-extended D = 4 superspace	537
H.8.3. Fierz decompositions in the N = 1, D = 4 superspace	545
11.8.4. The N = 2, D = 4 case	547
11.8.5, The N = 3, D = 4 case	552
11.8.6. The N = 2, D = 5 case	555
H.8.7. Systematics of Fierz identities in eleven dimensions	563
H.8.8. Irreducible representations of SO(1,9) and the irreducible basis	
of the D = 10 superspace	567
X
Chapter 11.9. Super Yang-Mills Theories	582
11.9.1.	Introduction	582
11.9.2.	Super Yang-Mills theories in D = 4	584
11.9.3.	The action principle tor N= 1, D= 10 super Yang-Mills theory	594
Historical Remarks and References	597
HI	xi
Volume 2
PART THREE. SUPERGRAVITY IN THE RHEONOMY FRAMEWORK
L
Chapter 111	.1. Introduction	607
Chapter III	.2. Supergravity in the Standard Component Approach	611
III.2.	1.	Local supersymmetry and gravity	611
III.2.	2.	Space-time Lagrangian of D = 4. N= 1 supergravity	615
111.2.	3.	The equations of motion of D = 4, N = 1 supergravity	618
111.2.	4.	Supersymmetry transformations and action invariance	624
111.2.	5.	On-shell supersymmetry invariance	629
<”	111.2.6.	The linearized theory of supergravity	633
Appendix III.2.A. Commutator of Two Supersymmetries on the
Gravitino Field	636
Chapter 111	.3. Supergravity in Superspace and the Rheonomy
Principle	641
111.3.	1.	From space-time to superspace	641
111.3.	2.	Geometry of superspace	643
111.3.	3.	The rheonomy principle	649
111.3.	4.	An extended action principle	661
111.3.	5.	D = 4, N = 1 supergravity and rheonomy	665
111.3.	6.	Rheonomic constraints and Bianchi identities	672
111.3.	7.	On-shell supersymmetry	677
111.3.	8.	Action invariance and off-shell supersymmetry	680
111.3.	9.	Building rules for supergravity Lagrangians	686
111.3.1	0.	Retrieving Л/= 1, D= 4 supergravity from the building rules	706
111.3.1	1.	Extension to anti de Sitter supergravity	709
111.3.1	2. Building rules for supergravity theories using rheonomy and
Bianchi identities	716
Chapter 111	.4. D = 4, N = 2 Simple Supergravity	726
111.4.	1.	Introduction	726
III.4.	2.	Rheonomic solution	of	the	N= 2. D = 4	Bianchi identities	728
III.4.	3.	The Lagrangian of N =	2.	D = 4 supergravity	737
Chapter III	.5. The D = 5, N = 2 Supergravity Theory	755
III.5.	1.	Introduction	755
111.5.	2.	Identification of the supergroup and construction of its curvatures	758
111.5.	3.	Construction of the Lagrangian	767
111.5.	4.	Superspace equations of motion and on-shell supersymmetry	777
Hl.5.5.	The second order formulation and the contracted version of the theory	783
Chapter 111	.6. The Theory of Free Differential Algebras and
Some Applications	794
111.6.	1.	Introduction	794
III.6.	2.	The concept of free differential algebra	795
Ш.6.З.	The structure of free differential algebras and some theorems	796
A
xii
II	1.6.4. Gauging of the free differential algebras and the building
rules revisited	806
111.6.	5. The Sohnius-West model (new minimal N = 1 supergravity):
the on-shell formulation	810
III.6.	6.	The Sohnius-West model:	off-shell extensions	817
111.6.	7. The building rules in their final form	826
Chapter 111.	7. Supergravity in 6 Dimensions	832
111.7.	1.	Introduction	832
III.7.	2. D = 6 Weyl spinors and selfdual tensors	834
II	1.7.3. The free differential algebra of D = 6 supergravity	841
III.7.	4. Construction of the model	844
III.7.	5. Non-invariance of the space-time action and how to cure it	855
Chapter III	.8. D « 11 Supergravity	861
111.8.	1. Introduction	861
111.8.	2. Free differential algebra of D = 11 supergravity	863
III.8.	3. Extended F.D.A. and the introduction of a 6-form	866
III.8.	4. The gauging of F.D.A. revisited	868
111.8.	5. Constructing the theory from Bianchi identities	873
III.8.	6. The action of D = 11 supergravity	881
II1.8.	7. The completion of the action and the equations of motion	897
Historical Remarks and References	911
PART FOUR. THE ROLE OF THE SCALAR HELDS: o-MODEL AND
SUPERHIGGS PHENOMENON IN SIMPLE AND
EXTENDED SUPERGRAVITY
Chapter IV	.1. Introduction	919
Chapter IV.2. Kahler manifolds	926
IV.2.1. o-models of supergravity and complex manifolds	926
IV.2.2. Almost complex and complex structures on a 2n-dimensional
manifold	928
IV.2.3.	Hermitean and Kahler metrics	934
IV.2.4.	The differential geometry of Kahler manifolds	937
Chapter IV.3. Coupling of N= 1 Supergravity to n Scalar Multiplets	943
IV.3.1.	Kahler geometry for the N = 1 coupling	943
IV.3.2.	Solution of the Bianchi identities and auxiliary fields	950
IV.3.3.	Construction of the action: generalities	967
IV.3.4.	Construction of У*”'1	969
IV.3.5.	Construction of Д У	976
Chapter IV.4. The Vector Multiplets and the Gauging of the Kahler
Manifold Isometries	983
IV.4.1.	Killing vectors and isometries of the scalar manifold	983
IV.4.2.	The vector multiplet	987
Chapter IV.5. The Super Higgs Phenomenon	997
IV.5.1. Introduction	997
IV.5.2. The mass relation in the minimal coupling case	1003
IV.5.3. Examples and flat potentials	1013
Chapter IV.6. Duality Transformations and the Coset Structure
of Extended Supergravities	1018
IV.6.1. How to extend the symmetries of the non linear o-model to the
vector fields	1016
IV.6.2. The coset structure of extended supergravities in D = 4	1030
Chapter IV.7. The Example of the N « 3 Theory	1033
IV.7.1. Introductory remarks	1033
IV.7.2. The N = 3 vector multiplet and the G/H structure of the
supergravity coupling	1034
IV.7.3. SU(3,n)/SU(3) ® SU(n) ® U(1) formalism and the solution
of Bianchi identities	1042
IV.7.4.	The Lagrangian	1051
IV.7.5.	The scalar field potential	1064
Appendix A:	The Scalar Potential	1067
Appendix B:	The A& Matrix and the Embedding of SU(3,W) into the
Symplectic Group	1069
Chapter IV.8. The Supersymmetry Breaking in the N = 3 Theory
and a Short Account of the N = 4 Theory	1077
IV.8.1.	Introduction to partial supersymmetry breaking	1077
IV.6.2.	Features of the N = 3 theory and of its potential	1085
IV.8.3.	A short discussion of simple N = 4 supergravity	1087
IV.8.4.	A short discussion of matter coupled N = 4 supergravity	1093
Chapter IV.9. The Directory of Supergravity Theories and the
A/= 8 Model	1106
IV	.9.1.	Introduction	1106
I	V.9.2. Classification of D = 4 supergravities and guide to the
related literature	1107
IV	.9.3.	The N= 8 Theory	1115
IV.	9.4.	Results for matter coupling in D = 4 supergravities	1131
References	1139
PART AVE. KALUZA-KLEIN SUPERGRAVITY
Chapter V.1. Introduction	1147
Chapter V.2. Spontaneous Compactification of D = 5 Pure Gravity	1157
V	.2.1. Spontaneous compactification of D =	5 pure gravity	1157
V	.2.2. Symmetries in D = 4	1162
V	.2.3. A preliminary example: the spectrum of Mt x S’ Maxwell theory	1166
V	.2.4 The Spectrum of Mt x S’ gravity	1169
xiv
Chapter V.3. Harmonic Expansions on Coset Manifolds	1175
V	.3.1. H -harmonics on G/H	1175
V	.3.2. Harmonic expansions in Kaluza-Klein theories	1162
V	.3.3. Yang-Mills fields from Mt x MK compactifications	1186
Chapter V.4. Compactifying Solutions of D « 11 Supergravity	1189
V	.4.1. The D = 4 vacuum: maximal symmetry	1189
V	.4.2.	AdS4 x M7 solutions (Freund-Rubin)	1192
V	.4.3. Properties of the internal space M7: Killing spinors and Weyl
holonomy	1194
V	.4.4.	Osp(4/W) formulation	1199
V	.4.5.	Differential operators on M7	1204
Appendix V.4.1. SO(7) Г-matrices	1217
Chapter V.5. The D = 4 Mass Spectrum in AdS4 x M7 Backgrounds	1221
V	.5.1. The linearized field equations of D = 11 supergravity	1221
V	.5.2.	Fermion masses	1224
V	.5.3.	Boson masses	1230
V	.5.4.	Supersymmetric mass relations	1232
V	.5.5.	Vacuum stability	1241
Chapter V.6. Classification of Compact Homogeneous D = 7
Einstein Spaces	1249
V	.6.1.	Homogeneous 7-manifolds	1249
V	.6.2.	The spaces Sll(3) xSU(2) x U(1)/SU(2) x U(1) x U(1)	1251
V	.6.3.	The other D = 7 Einstein spaces G/H	1254
Chapter V.	7. The Spectra of Specific Solutions: The Seven-Sphere	1259
V	.7.1.	How to compute on the G/H harmonics	1259
V	.7.2.	The spectrum of the round S7: harmonic analysis	1262
V	.7.3.	The spectrum of the round seven-sphere:	Osp(4/8) analysis	1277
Chapter V.8. The /И₽ч' spaces	1302
V	.8.1. The spaces	1302
V	.8.2.	Harmonics on the Mpqr spaces	1306
V	.8.3. The spectrum of the SU(3) x SU(2) x U(1) irreps in the
spinor expansion	1313
V	.8.4.	Conjugation in the longitudinal spectrum	1322
V	.8.5.	Calculation of the longitudinal mass eigenvalues	1329
Chapter V.9. Other Classical Solutions of D = 11 Supergravity	1343
V	.9.1.	Introduction	1343
V	.9.2.	Nonvanishing internal photon (Englert-type solutions)	1343
V	.9.3.	Symmetries of Englert-type solutions	1347
V	.9.4.	Stretched and warped solutions	1347
Chapter V.10. The Embedding of D = 4 S.G. into D=11 S.G.	1355
Chapter V.11. The Chirality Problem	1360
Bibliographical Note	1367
Volume 3
PART SIX. HETEROTIC SUPERSTRINGS AND SUPERGRAVITY
Chapter VI.	1. Introduction	1375
Chapter VI.2. Elements of Two-dimensional Differential Geometry
and of Riemann	Surface Theory	1391
V	I.2.1.	Introduction	1391
V	l.2.2. Definition of a Riemann surface; metrics, complex structures and
moduli space	1393
V	I.2.3. The simply connected Riemann surfaces and the uniformization
theorem	1400
V	I.2.4. Deformation of the metric, quadratic differentials and the complex
structure of Teichmiiller space	1416
V	I.2.5. Homology bases, abelian differentials and the period matrix	1426
V	I.2.6. Dehn twists, the mapping class group and its homomorphism
ontoSp(2g,Z)	1449
V	I.2.7. The group of divisors and the Riemann-Roch theorem	1458
V	l.2.6. The Jacobian variety: Riemann theta functions and spin structures	1472
Chapter VI.3. The Classical Action of the Heterotic Superstrings and
Their Canonical Quantization	1501
V	I.3.1.	Introduction	1501
V	l.3.2. Л/ = 1, D = 2 conformal supergravity and the heterotic
superspace geometry	1511
V	l.3.3.	Classical superconformal theories and the WZW-action	1518
V	l.3.4. Heterotic o-model on a general target space and the choice
of M leading to a classical superconformal theory	1531
V	l.3.5. Canonical quantization of the heterotic WZW-model and the
superconformal algebra	1541
Appendix: Rules for the Wick Rotation of Spinors	1556
Chapter VI.4. The BRST Charge and the Ghost Fields	1558
V	l.4.1.	Introduction	1558
V	i.4.2.	BRST quantization. Abstract properties of Q	1559
V	l.4.3.	Construction of Q	1561
V	I.4.4. The BRST invariant hamiltonian and the Fradkin-Vilkovski
theorem	1565
V	l.4.5.	BRST quantization of string theories	1566
Chapter VI.5. Quantum Determination of the Target Manifold and
Kac-Moody Algebras	1582
V	l.5.1.	Introduction	1582
V	l.5.2.	The BRST charge: cancellation of the conformal anomaly,
boundary conditions and intercepts	1585
V	l.5.3	Twisted Kac-Moody algebras and massless target fermions	1600
Chapter VI.6. The Polyakov Path Integral and the Partition Function
of String Models	1629
V	l.6.1.	Introduction	1629
xvi
V	I.6.2. The cosmological constant, the partition function and the
Polyakov path integral	1632
V	l.6.3. Operatorial evaluation of the bosonic string partition function	1646
V	L6.4. The Polyakov integration measure for the bosonic string	1654
V	l.6.5. Functional evaluation of the bosonic string partition function in the
case of the torus	1672
V	I.6.6. Functional determinants of the Laplacian and of the Dirac operator
on the torus	1677
V	l.6.7.	The gravitino ghost	1692
Appendix. A Detailed Treatment of Conformal Killing Vectors	1697
Chapter IV.7. Modular Invariance, Fermionization and the Particle
Spectrum of Heterotic Superstrings	1702
V	l.7.1.	Introduction	1702
V	I.7.2.	Modular invariance and GNO fermionization	1707
V	l.7.3.	Modular invariance and spin structures	1730
V	I.7.4.	An example in D= 10: the SO(32) superstring	1745
V	I.7.5. A second example in D = 10: the Et ® E't and SO(16) ® SO(16)
heterotic strings	1754
V	L7.6.	Examples in D = 4	1760
Chapter VI. 8. Quantum Conformal Field Theories, Vertex Operators
and String Tree Amplitudes	1766
V	I.8.1.	Introduction	1766
V	l.8.2.	Quantum conformal field theories and emission vertices	1769
V	L8.3.	Bosonization, vertex operators and spin fields in the matter sector	1787
V	l.8.4.	b-c systems, superghost bosonization and the background charge	1804
V	L8.5.	The covariant lattice for D= 10 superstrings	1819
V	I.8.6. Conjugacy classes and GSO projectors: the SO(32)
example in D = 10	1832
V	l.8.7. Massless emission vertices and the effective theory of D = 10
superstrings	1838
Chapter VI. 9. Effective Supergravity Theories and the Coupling of
the Lorentz Chern-Simons Term	1854
V	l.9.1.	Introduction	1854
V	l.9.2.	The algebraic basis of N = 1, D= 10 matter-coupled supergravity	1859
V	l.9.3.	The general solution of the D = 10 super Poincarri Bianchi	identities	1865
V	L9.4. The H-Bianchi identity in the (0,4)- and (1,3)-sectors: determination of
the H-parametrization	1891
V	l.9.5. The (2,2)- and (3,1)-sectors of the .^-Bianchi identity and the
equations of motion	1915
V	l.9.6. The Lagrangian of N= 1, D= 10 matter-coupled supergravity aty= 0	1942
V	l.9.7. Retrieving the superspace constraints from the к symmetry of
the Green-Schwarz string formulation	1959
V	l.9.8. Bianchi identities and off-shell formulations of N = 1. D = 4
supergravity revisited	1982
V	I.9.9. Chiral multiplets, the linear multiplet and the geometrical interpretation
of R-symmetiy	1997
V	l.9.10. D = 4 Chern-Simons cohomology and the linear multiplet	2012
xvii
К Chapter VI.10. (2,2) Superconformal Field Theories and the
Classification of N = 1, D » 4 Heterotic Superstring
Ж	Vacua	2028
Ж-	Vl.10.1.	Introduction	2028
ж-	VI. 10.2.	Type II superstrings on SU(2)3 groupfolds	2032
Ж.	VI. 10.3.	Construction of modular invariants and GSO projectors for the
C.	type II superstring	2037
У	VI. 10.4.	SU(2)3 groupfolds and superconformal field theories	2039
V	I. 10.5. The h-map	2050
V	l.10.6. Emission vertices of the massless multiplets in an N = 1 heterotic
model based on a (2,2),, internal theory	2056
V	I.10.7. Emission vertices of the massless multiplets in A/= 2 heterotic
models based on a (4,4)e, Ф (2,2)w internal theory	2068
V	I.10.8. Embedding of a (2,2),, into the direct sum (4,4)в,Ф (2,2)M	2076
V	I. 10.9. Classification of the SU(2)3 groupfold realizations of the internal
conformal field theory	2078
V	I.10.10. Details of the SU(2)3 groupfold construction with
emphasis on bosonization	2083
Appendix VI. 10.A. A bosonizable (2.2) vacuum of type A:
А1(1,2,3,4)е1Э	2091
Appenidx VI.10.B. A Bosonizable (2,2) Vacuum of Type B:
B5(1,2,3,4)fS3	2094
Appendix VI. 10 С.	An LRP (2,2) Vacuum of Type В: B26(1,2,3)	2097
Bibliographical Note	2102
Historical Remarks and References	2117
Index	2129


PART ONE
GRAVITY
and
DIFFERENTIAL GEOMETRY
... Olibri,
habile geometre, et grand physicien
fonda la secte des vorticoses.
Circino, habile physicien et grand
geometre fut le premier attractionnaire.
.... On entre sans preparation dans
Гёсо1е d’Olibri; tout le monde
en a la clef. Celle de Circino
n’est ouverte qu’aux premiers дёотё1ге$
Diderot, Les Bijoux Indiscrete, 1748

3
г
«•
CHAPTER I.О
INTRODUCTION TO PART ONE
In this part we are going to discuss mathematics and, in
particular, differential geometry. The chosen topics are classical
and can be found in many excellent textbooks, although it may be
difficult to find one where they are all collected together and treated
at the same elementary level as we treat them here. In spite of this,
the spirit underlying our presentation and the choice of the topics is
non conventional and it is physical in its nature.
In particular we want to stress, for the reader's benefit,
especially if he is a student approaching the subject of supergravity
for the first time, that Part One is not to be regarded as a summary
of mathematical literature to be used at the moment one needs to
refresh his memory, rather it is, in our intentions, the place where
he could begin the study of the whole subject. Indeed, as we already
stressed in the general introduction, ours is not a neutral book. It
is a biased one, and the first part is where the reader can develop a
feeling for our bias. Let us explain what we mean.
4
When the theories of supergravity and supersymmetry were first
developed, namely ten years ago, the familiarity of the physicists
community with the techniques and language of modern differential geo-
metry was considerably inferior to its 1986 level. Nowadays, probably
through the relevance gained by such topics as chiral anomalies, spon-
taneous compactifications and the like, the use of exterior forms,
Maurer Cartan equations, Lie derivatives and coset manifolds is topical
and fashionable in discussing both supergravity and superstring models.
These concepts have been accepted among those currently utilized in the
theoretical literature.
It was not so ten years ago. As a result the founders of super-
gravity theories developed their new models using mainly the formalism
of world-tensors of classical general relativity. In this formalism
the meaning of the new symmetry underlying the whole business, namely
supersymmetry, is much less clear than it is when one utilizes, right
from the start, the concepts of exterior algebra and differential
forms. Actually the interpretation of supersymmetry on which we base
our presentation of the whole subject of supergravity (that is "rheonomy")
is a natural yield of the use of exterior forms, almost impossible to
conceive in the world-tensor formalism. In our opinion it is the use of
a non well-adapted formalism what makes many aspects of supergravity
difficult to be grasped at the beginning, obscure in their meaning and
difficult from a computational point of view. Incidentally the use of
the world tensors makes even more remarkable the wealth of results,
obtained by the fathers of supergravity, relying only on the rather
cumbersome method popularly called "Noether" approach.
There is, therefore, in our opinion, an historical reason under-
lying what we consider a curious situation. The language of differential
geometry is used extensively in many developments of supergravity such as
the geometry of scalar manifolds appearing in matter coupled lagrangians
or the geometry of compact manifolds used in spontaneous compactifica-
tions, or such as the cohomological questions raised by anomalies; how-
ever this same language is not used very often in the formulation of the
gravity and supergravity models.
f	;
This notwithstanding, it is our opinion that the geometrical
dress suits supergravity magnificently and the whole subject can be
developed consistently in the language of exterior forms with many
distinct advantages.
It goes without saying that the models one obtains in this way
are the same one constructs via the Noether approach, the physics is
the same and even the explicit form of the Lagrangians, of the equations
of motion and of transformation rules is the same. What differs is the
geometric perspective, present in one case and lacking in the other,
which provides a powerful guide to the intuition.
Such a perspective is rooted in the Einstein-Cartan formulation
of gravity which we anti-historically regard as the most natural and
deepest. Taking it as primary, the development of supergravity as a
geometrical theory follows naturally.
In view of these considerations the following chapters pursue a
double goal. On one side we collect those mathematical results which
we need later not only in the development, but also in the applications
of supergravity theories; on the other hand we take the opportunity to
blend mathematics and physics together laying down, in our revisitation
of gravity, some of the concepts which form the starting point for the
rheonomy approach to supersymmetric field theories.
Chapter 1.	1 is an introduction to exterior differential forms on
manifolds. The essential operations we utilize in the sequel are
defined here: exterior derivatives, contractions, wedge products and
Lie derivatives.
Chapter 1.	2 is devoted to the reformulation of Riemannian geometry
in terms of vielbeins and spin connections. We meet here with the
concept of intrinsic components of the torsion and curvature 2-forms
which is going to play an essential role in the sequel.
The last part of the chapter provides the translation vocabulary
between the intrinsic vielbein formalism and the customary world-tensor
notation.
Chapter 1.	3 addresses the subject of group manifolds. In develop-
ing the differential geometry of these latter we typically pursue the
double goal mentioned before. On one side the Maurer-Cartan equations
6
which naturally arise in this context are utilized in specific appli-
cations. On the other hand they constitute the starting point for the
soft group manifold notion which underlies the rheonomy approach: on
this point we stand alone. Hence Chapter 1.3 is both preparatory for
Chapter 1.6 where the coset manifold geometries used in Kaluza-Klein
theories are discussed, and for Chapters II.3, II.6 and III.3 where
the rheonomy approach is developed. Furthermore Chapter 1.3 is prope-
deutic to Chapter 1.4 where we discuss the Einstein-Cartan formulation
of gravity. Here we proceed in a heuristic way going backward from the
usual formulation in terms of world-tensors to the formulation in terms
of vielbeins and connections. Once we have the geometric transcription
of the action principle we show how it can be reinterpreted as describ-
ing a theory on the soft group manifold of the Poincare group or on the
soft coset manifold ISO(1,3)/SO(1,3). This paves the way for the
construction of supergravity as a geometrical theory on a soft super-
group manifold.
Chapter 1.5 deals with the coupling to geometrical gravity of
matter fields such as spinors and spin 1 gauge fields. The problem of
incorporating these lower spin fields into the geometrical formalism is
very important for the subsequent development of supergravity. Indeed
supergravity multiplets involve, in general, fields of all spins and we
must be prepared to treat all of them on an equal footing. We show that
first order formalism is natural when treating both spin zero and spin 1
fields in Einstein Cartan gravitational theories. Namely the intrinsic
components Фа of the exterior differential d$, in the case of the
the scalar field ф, and F	of the field strength F = dA, in the
case of the spin one field A, must be regarded as independent varia-
bles. They are defined by decomposing dC and F in the vielbein
basis:
d<j> = <j>a Va
F = F , Va Vb
ab
(1.0.1a)
(I.0.1b)
7
This first order formalism will be seen to be essential in the rheonomy
approach to supergravity. Furthermore in Chapter 1.5 we discuss the
relation between the potential term of the scalar fields W(ф) and the
cosmological constant, which is of great momentum for the supersymmetry
breaking mechanism (super Higgs phenomenon) treated in Part Four.
Chapter 1.6 develops the differential geometry of coset manifolds
and contains much more advanced material than in previous chapters.
Coset manifolds find two main applications in supergravity theories:
on one side they appear in extended four dimensional supergravities as
the manifolds whose coordinates are the scalar fields of the theory.
On the other hand they can be utilized as Kaluza-Klein solutions of
D= 11 Supergravity by taking the eleven space	to be the direct
product:
M11 = M4 ® G/H	(1.0.2)
where dim(G/H)= 7 and where is anti de Sitter space.
Many local and global properties of the manifold G/H are
reflected into quite important physical properties such as existence
of fermions, residual supersymmetries, number of massless gauge fields
and the like.
In view of this a careful study of the coset manifolds is very
important although in a first reading the student may skip Chapter 1.6
and come back to it only at the moment of studying Part Four and Part
Five.
It must be said, anyhow, that coset manifolds provide very in-
structive explicit examples of all the differential geometric concepts
developed in previous chapters. Non trivial Riemann tensors, Lie
derivatives and Killing vectors can be looked into the eyes and this
considerably helps the general understanding.
Finally, Chapter 1.7 contains some miscellaneous examples. We
treat the non minimal coupling of scalar and pseudoscalars to gravity,
briefly discussing the Brans-Dicke theory. Furthermore we show how the
8
well known Schwarzschild solution is retrieved in the geometrical
formalism. This is, in our opinion, an excellent illustration of all
the advocated concepts.
9
CHAPTER 1.1
EXTERIOR CALCULUS
The content of this chapter is not exhaustive nor its presenta-
tion is rigorous.
We simply give a short account of the exterior algebra of k-forms
on a general manifold since this is the formalism we use throughout the
book for the construction of unified field theories. In this respect
it is useful to note that the Lagrangian itself is a d-form to be
integrated on a d-dimensional manifold. In choosing the material we
are going to discuss we have been guided by our need for the develop-
ment of subsequent chapters and not by the logic of a systematic math-
ematical presentation.
Furthermore not all the needed mathematics does appear in Chapter
1.1: more advanced developments will be presented in the following
chapters when motivated by an explicit physical application.
The first two sections of the present chapter are devoted to a
short account of exterior algebra of forms on a linear space; this
gives us the opportunity of stating the main definitions and properties
of tensors and к-forms in a pure algebraic way. This is a preparation
10
for their extension to a differentiable manifold. Such an extension,
together with a sketchy treatment of functions and vector fields on a
general manifold is given in Sections 3, 4 and 5. The reader will
notice that we do not discuss integration of forms since we do not
employ it explicitly anywhere in the book.
In Section 6 we discuss the relation between different local
frames on a general manifold; in particular the coordinate frame and
the vielbein frame. In Section 7 we present in some detail the notion
of Lie derivative which is important for our future discussion of symme-
tries on coset manifolds and in the Kaluza-Klein interpretation of
higher dimensional supergravities.
1.1.1 - Exterior forms on vector spaces
We introduce first the algebraic notion of 1-forms.
Let us consider a real vector space Vй whose elements we denote
by Vj,v2 ...
A 1-form (or covector) w is a linear functional on Vй:
w : Vn + R
(I.1.1a)
w(a1v1 + a2v2) = a1w(v1) + a2w(v2)
(I.1.1b)
The set of all 1-forms can be given the structure of a vector space,
V n, by defining:
(Wj + w2)(v) = Wjfv) + w2(v)
(Xw) (v) = Xto(v)
(I.1.2a)
(I.1.2b)
where X is a real number. V
fn
is called the dual vector space.
Given a basis of V11; e,,eo,....e . the dual basis o^,o^,...,on
*	1 4 n -----------------------
of V n is defined by
O1^) = <5*	(1.1.3)
It is straightforward to show that the G1 are linearly independent
vectors and that
w = w^o1	(1.1.4)
where
w.. = ш(ер	(1.1.5)
In the same way a vector v
izi	i
v = a (v)e. = v e.
i	i
can be expanded as:
(1.1.6)
O^v)
is then the oriented parallel projection of v along the i-th
coordinate axis. (see Fig. I.l.I).
Fig. I.l.I
12
According to (1.1.4) the two spaces Vn and V П have the same
dimensions; therefore they must be isomorphic. When a metric is intro-
duced this isomorphism can be made explicit (see Eq. (1.1.20) later).
• *n	*n
Since V is a vector space we may define the dual of V ,
denoted by (V ) n as the space of linear functionals on V П
* *n
u e (V )
*n
V
u(a1w1 + a2w2) = a1u(w1) + a2u(w2)
(1.1.7)
The space (V ) n is naturally isomorphic to V*1, the isomorphism
being given by
v*->u : u(w) = w(v)
(1.1.8)
and therefore (V ) n can be identified with V11, u=v.
The above identification implies:
v(w) = w(v)	(1.1.9)
so that the elements v 6 V11 can be thought as linear functionals from
*n	Г)	*n	~
V to F; v and V are thus dual of each other. In particular
from (1.1.6) one has
v = G1(v)ei = v(o1)ei	(1.1.10)
which parallels Eq. (1.1.4, 5).
Let us now introduce tensors.
Definition: A tensor ф^г’5^ on Vn of covariant order r and
contravariant order s is a multilinear map
13
$(r»s) . vxVx ... xv x V* x V* x ... x V* + R .
r-times	s-times
Thus <j/r,s) assigns to each r-tuple of vectors v^,...,v and
12 s
s-tuple of 1-forms ш , w ,...,w a real number and multilinearity is
defined as
ф(г»5) (v t #.. >au + a'u',... .и1,...) =
,(r,s),	1	,	,*(r,s),	,	1	.
= ссф (Vp . . . ,U, . . . ,Ш ,...) + СС'Ф4 (Vp ... ,u',... ,ш ,...)
(1.1.11)
for all a, a' e R and where u, u' can be either vectors or covectors.
Cr s 1
Let W ’ J be the collection of all tensors of order (r,s).
tr si
W4 ’ ' acquires the structure of a vector space if, for every Ф1,
fr si
Ф? € W , we introduce the natural definition:
1	2
(a^j + c^Hvp v2,...,w , w ,...) =
12	12
=	v2,...,w , co ,...) + “2®2(vp v2,...,w , co ,...)
(1.1.12)
1	2
where ctp a2 e JR, and v^, v2,...,co , co , . . . is a set of r vectors
and s 1-forms.
Let Ф and V be two covariant tensors of order 1 (1-forms or
(1,0) tensors); their product Ф ® T is a covariant tensor of order 2
((2,0) tensor) defined by:
(Ф ® T)(vrv2) = ®(vp • T(V2)
(1.1.13a)
Taking a basis {o\...,on} of Vй it is easy to show that
О 0 o
i1»i2 = {!.••>"}
(1.1.13b)
14
is a basis for the covariant tensors of order 2.
Indeed, since
if x2	X1 x2
о ® о (e. ,e. ) = о (e )cf (e. ) =
31 ’2	31	32
i i
6.6.
31 32
(I.1.14a)
given a generic (2,0)-tensor Ф and defining
1
Ф = Ф(е. ,e.	® о
X1 X2
(I. 1.14b)
we see that Ф and Ф agree on every set of basis elements e. , e. ;
4 x2
therefore ФЕФ. The product (1.1.13a) can be generalized to tensors
fr si	(к £1
of any order (r,s): the product of Ф1 ’ 1 and T1 ’ ' is a tensor
of order (r+k, s+£) defined by:
A(r,s) ш(к,£),
Ф ’ J ®	’ 1 (v
A(r,s),	1 S-x
= Фч '(v,,...,v , <j0	1 (v ,,...,v .
1 r’	v r+1	’ r+k
(1.1.15a)
Proceeding in the same way as before one shows that
jl	h
e. ® ... ® e. ®o ® ... ® a
X1	xk
(k £)
is a basis in IV ’ satisfying
3£ ^1	^"k
® О (a ....,0 , e ,. ..,e ) =
ml
Therefore any Ф e
can be written as follows:
(1.1.15b)
15
fk £1	k+£
Notice that the dimensionality of Wl ’ J is n . It is easy to
fk Я.-)
show that W = E ® W	is an associative algebra with respect to
the product defined in (I.1.15a).
Let us now consider a covariant tensor g of order 2:
gfavj + Bv2,w) = agfVj.w) + Bg(v2>w)
Using a basis of Vй we have:
(1.1. 16a)
(1.1.17)
If g is symmetric:
g(u.v) = g(v,u) «> g = g
(1.1.18)
and non degenerate:
Vu g(u,v) = 0 =» v = 0 «• det g.^ / 0	(1.1.19)
then g is called a metric on V*1.
A metric g induces an isomorphism between Vn and V n given
by:
o1 -> e. = g.. a3	(1.1.20)
i ij
A theorem in linear algebra guarantees that we can always find an
orthonormal frame o1 such that
-1	~1	-^k z~k ~k+l ~k+l -in ~n
g = a ® о + ...+O ® о - о «о - ... a ®o
(1.1.21)
16
Any other orthonormal frame is related to C* 1 il * * by a rotation of
SO(k.n-k).
The components of g in the orthonormal frame will be denoted
by in the following (6^ if к = n)
k-forms
Let us now introduce the к-forms: The space of к-forms is
spanned by the k-covariant and completely antisymmetric tensors.
Definition
fkl
A к-form co is a functional of к vectors v,,v ,...,v.
which is linear and completely antisymmetric, namely
(k)	Ck)
CU (v- , . . . ,CL V. + Ot„V.‘ ,...,¥.) = 0ЦШ	)
1 ’ll 21’	’ к' 1 v 1*	* i’ ’ n
СЮ Г	,	ч
2	1’	1’	’ Y
(1.1.22a)
w(k)(v, ,...,v ) = (-l)PwCk)(v . ,,v )	(1.1.22b)
il	ik	IK
{12	к 1
. * . ’ ’ ’ ‘ 9. У .
lpl2,...,lkJ
The set of к forms acquires the structure of a vector space via
the following definitions:
. (k) (k)w	,	(k),	,	(k),	.
Ц +ш2 Hvl..........Vk} = Ш1 J(v1,...,vk) +	'(Vp...,^)
(I.1.23a)
(Xw(k))(Vi.....vk) = XuCk)(Vp...,vk) .	(1.1.23b)
▼
17
Example:
The oriented volume element.
Let us recall that two frames e^
the same orientation iff detCa^) > 0.
equivalence class of all frames related
and e'.sa.^e. in Vй define
i i 1
An orientation is therefore the
by linear transformations with
positive determinant,
spanned by v^.. .vr
volume form й^:
Now the oriented volume of a parallelepided
in Vn is the value of an n-form, that is the

det
V11 v12"vln
V21 v22”v2n
vnl vn2’ ‘' vnn
(1.1.24)
where v.. is the j-th component of the i-th vector.
(k)
Let us denote by A' the vector space of the к-forms over the
n*dimensional vector space v”. Thus, in particular =V n and
we define A^ = R. Notice that any к-form w^k^ cA^ with k>n is
n
identically zero. This follows immediately from (I.1.23b).
We now introduce an operation on forms, the exterior (or wedge)
product. By means of it the vector space
Л = Л(0) ф AC1) ® ...®Atn)	(1.1.25)
n n n	n
(k)	(£)
acquires the structure of an associative algebra: if or and or
are respectively a к-form and an £-form, then their exterior product
denoted by
u(k+£) = ш(к) .
(1.1.26)
1* a (k+£)-form. In order to define o/k+^ we proceed by steps.
We first define the exterior product of two 1-forms
лШ2	2-form such that
18
Ш1 - “2(V1’V2) = |
Wj(Vj)
^(v2)
ш2(ур
W2(v2}
(1.1.27)
In other words ~ w2 is the antisymmetric part of the tensor product
(1.1.12):
Wj A w2(v1,v2) = j- [w1 ® w2 - w2 ® wj (VpV2) .
Consider Fig. I.l.II. Projecting the parallelogram spanned by the
vectors Vj and v2 on the (w^,w2)-plane we obtain another parallelo-
gram of which Wj л w2(VpV2) is the oriented area. This is the geo-
metrical interpretation of the exterior product л w?- It follows
from (1.1.27) that the exterior product is bilinear and antisymmetric:
Fig. I.l.II
Evaluating O1 ~ cP (e^.e^) where e^.e^ are basis vectors in Vй and
a1,cP are elements of the dual basis, we get:
19
i i,	,1
' - ° (ek’ep = 7
4
4
Every 2-form
(2)
or =
Indeed let
aij
4
4
= 6ij
k£
(I.
can be written as
a. . O1 zx oJ
(e.,e.)
x J
a
linear combination of the 0

(I-
(I.
Then the 2-form:
-(2)	к £
W = ak£ ° - °
(I-
takes the same value
(2)
as a:
is
к p
Since a ~о
£ к
о /х a
,	(2) _-(2)
on e.,e.: hence ы - w '
i J
the dimension of the vector space
Note that
®k£ " ' a£k
(I.
The previous definitions can be easily extended to the product
1-forms Wj л o>2
one sets:
of
wi
^(vl”"”vk)=^
0)1 (vp	0J2(v1)	... <VV1)	
ы1(у2)	w2(v2)	... <vv2)	
^i(vk)	0)2(vk)	•••	(I
1.29)
,s:
1.30)
1.31)
1.32)
1.33)
к
1.34)
20
Again (1.1.34) is just the completely antisymmetrized tensor product of
к 1-forms. Geometrically (1.1.34) represents the volume of the projec-
tion on the hyperplane	of the oriented hyperparallelepiped
spanned by the vectors v^,...,v^.
Eq. (1.1.34) implies multilinearity and antisymmetry under the
exchange of any two w's:
(I.1.35)
/1 ...k
where P is the parity of I
Vil-.-ik
In particular
11	xk
a 1 z. ... л о K (е. ,...,е. ) =
31	-"к
1 /2	?к
J1 h	J1
kJ • U. . • • kJ-
к! t J2 J2 J2
-1l’”1k
О .
Jr..Jk
(1.1.36)
In the same way as in the case
k = 2 we may easily prove that the
monomials:
X1	xk
О 1 л ... ~ О .	(1.1.37)
(к)	(к)
span M J, that is any к-form ы can be written as:
= a. .	0 1 z. ... z. 0 k	(1.1.38)
1l"1k
where
21
a. . = ш(к](е. ,...,e. )	(1.1.39)
хГ,хк	X1 xk
is a completely antisymmetric covariant tensor. It follows that
dim A^k) =	.	(1.1.40)
In particular if k= n, dim = 1 and
= v o1 . o2 ..... on	(1.1.41)
that is: any n-form is the oriented volume of a parallelepiped times
12	n
a unit volume а . о	a .
We may now define the exterior product of a general к-form with
il ik
a general £-form.using the expansion along the basis a ^ ...
and c?1 . ... .Л If
w(k) = a. .	a'1 ..... Л	(1.1.42)
XV ,Xk
= b.	..... Л	(1.1.43)
Jl-’Jp,
then
fk) И	к x?1 /к Jl хЛ
0) <s Cl) = a. . b. . G л ... лО лО л ... лО
хГ"хк
(1.1.44)
where we have defined
r/l	xk.	X1	xk jl
1° . .... a ) . (a . .... a ) = a . .... a .a . .... a
(1.1.45)
.	Ik) (£)
л more general way of defining the wedge product . w4 ‘ is the
following:
22
(к) U) <
---1---- Z (-1)Р </k)(v. ,...,v. ) wW(v.	)	(1.1.46)
(k + £)1	1 xk	xk+l k+£
where ip i2,— ,xk+£ xs a Permutati°n of 1, 2,...,k+£, P is its
parity, and the sum is over all the permutations. The equivalence
between (1.1.44) and (1.1.46) is easily proven; indeed setting
(к)	1	к (£)	k+1	k+£
ш= о z. ... л о ; or = о Z....Z.0
and recalling the definition (1.1.34), we see that the equality
1 2
° - ° —•—° Cvi,...,vk+Jt) =
v г r_l	sir k+1	k+£,	.i
= E (-1) [а л ... л o (v. ,... ,v. )J [a ^ ...	(v.	,.. . ,v.	)]
1 xk	Tk+1 xk+£
(1.1.47)
is just the Laplace rule for the development of a determinant. By
linearity the equality between (1.1.44) and (1.1.46) also holds for the
more general forms (1.1.42), (1.1.43). The basic properties of the
exterior product are associativity, distributivity and graded anti-
symmetry:

(1.1.48a)
...	(k)	(£)	. ..kJ. (£)	(k)
n) or 1 = (-1) <1Л ' „or 1
(I.1.48b)
Thus two forms of odd degree anticommute while the product is commuta-
tive if one of the degrees is even. The algebra	. - - ®
A^n^ is called the exterior algebra of forms on Vn.
One can establish a link between the exterior algebra and the
3
ordinary vector calculus in Euclidean space R endowed with the
3
metric g. = 6. . To each vector a e R we associate the 1-form
(1) 1J 1J
or J such that
23
о? J (v) = a • v = g.. a vJ - a
a
(2)
and the 2-form or J such that
u£2) (VpVp = a • (vj x v2)
a
12 3
Using an orthonormal basis О ,0 ,o we can write
(1.1.49a)
(I.1.49b)
w(1) = a.a1 + a a2 + a a3	(1.1.50)
a 1	2 d
and
<Д2) = bo2.o\bo\o2 + bo\o2	(1.1.51)
b 1	z	3
Then it is easy to show that
, о/1) = ш£2\	(1.1.52)
a b a x b
and
- ^2) = (a -W л а2 . о3	(1.1.53)
a b
1-1.2 - Mappings and operations on forms
In this section we introduce the notion of pull-back of an alge-
braic form induced by a linear mapping on Vй. We also discuss the
contraction of forms with vectors and the notion of Hodge duality.
24
Pull back
Let us consider a к-form	on Vй and a linear mapping
A : v"1 -> Vй
(1.1.54)
Ck)
We define the pull-back of w 7 with respect to A, denoted by
* (k)	in
Aid , as the к-form on V such that:
A*O)(k)(v1,.. . ,vk) = ШСк)(А vp ... ,A vk)	(1.1.55)
The following properties are easily verified:
(А* В)* = B* • A*	(1.1.56)
*, (k)	(£).	* (k)	* (£)	,T ,
A (or ' л or ') = A or 7 л А cr 1	(1.1.57)
Suppose n = m; let te^} and {e\} be basis of V™ and Vn respec-
tively and let {a1} and {o 1} be their duals. Representing A by
a matrix
e| = A.je.	(1.1.58)
*	’ i i	t
A : a + о is represented by the transposed matrix A :
ho'1 = o1 = o’^A? «• а'Э = (A-1)^1	(1.1.59)
*	(kl
Similarly, the action of A on a basis element of 7 is given by
the к-th antisymmetrized tensor power of A3:
*	'’’I	'^k	*1 Й
A (a ^ ... .o )Eo\...,o =
'jl 'jk 4	4
= 0	л...^0 A. \..A.	(1.1.60)
J1 Jk
25
.	,	. (n)
In particular on A
A*(o'1- О П) = det A о 1 ^ ... л О П	(1.1.61)
Contraction
Assume	and v e V11.
We define a mapping
A(r)Jl_A(r-l) n	n	(1.1.62)
called inner product or contraction of Ш with v by:
(ivw)(V1,...,vr l) = rw(v,v1,.--.,vr p	(1.1.63)
Another useful notation is
i = v] V —J	(1.1.64)
The contraction mapping is linear and satisfies the following properties
yj u] = - u| v]	(1.1.65)
au ♦ Bv| = a u| + В v|	(1.1.66)
vj (Л> .„») - (vju<k))	. CvjJ4)	(1.1.67)
Using a basis one has
26
i	1	к
v I gj = v e.lii). . а a ... a a K =
__iJ 1i--1k
i	?1 T2 xk /2 Х1 X3 гк
= V Ш. .(6.0 A ... A 0	-6.0 A O A ... A о +...
1l',1k 1	1
к *1	^k-1 *к
* (-1) O -------о K 6.K)	(1.1.68)
For 1-forms
vji/1-* = v1^	(1.1.69)
Hodge duality
We define a map from A^ to A^n-k\ called the Hodge
duality operator, (denoted by *) as follows:
*fXl ?k,	1	1l”’1k	xk+l 1k+n
(n - к)!	1k+l k+n
(1.1.70)
f kl
The square of this operator, when acting on a к-form co1 ’ gives
“/) = (-Dk("-kMk>
(1.1.71)
Notice that the contraction of the n-k indices in (1.1.70) implies the
use of the metric g. . = 6. in V .
bij ij	n
definite, g. • = q. - then, instead of
'ij
*	* (k)	, kfn k) * 2	(k)
ы -* = (-1)	ш 1
If the metric is not positive
(1.1.71) , one has
(1.1.72)
where t is the signature of the metric, namely the difference between
the number of + and - signs in q. . = (+ ... + - ... -) .
27
A consequence of the previous formulae is the following
n-t
a(k\ *B(k) = 6(k)/oW = (-1) 2 k!(a,B) c1 « ... лОП (1.1.73)
where (a,В) denotes the scalar product of the tensor a. . and
. , ,	-	•	.,*n	11‘ ’ ‘ 1k
n . with the metric p. . in V :
jj-'-^k	13
1r-1k
(а,В) = а .В
11”1к
*Р1 Vk
p •. .n
CL.
1
1*
xk J T ‘ ‘ jk
(1.1.74)
Indeed one has
L_r .x
1l'”1k	(n - k) ! 3r”Jk
„^k+1	^n
хе . .а л...ла =
Jk+T ' ’Jn
. e. . Д-Vk-r "4
(n - к) ! 1Г"1к Jr'3k
1l"1k	1	n
хе .	. о л ... л о =
Jk+1‘‘ ‘ ^n
-ЫСп-к)! (.ijV A x
(n-к)!	jr’-3k
-’1""^к 1	n
ха. . В ал...^о =
1Г"1к
Ч..Лк
= (-1) 1 к! а. - В	(1.1.75)
1Г"1к
28
1.1.3 - Differentiable manifolds, vector fields and differential forms
In the following sections we give a summary of the main proper-
ties of objects defined on manifolds rather than on Euclidean spaces.
First we introduce the concept of differentiable manifolds, that is of
spaces on which the usual notions of differentiation and integration
make sense. Then we extend the definition of к-forms from Euclidean
spaces to arbitrary manifolds. To this effect it is first necessary
to introduce the notion of vectors on a manifold: tensor and k-forms
are then defined as functionals of vectors in the same way as we defined
them in the case of Euclidean spaces. We are aware of the fact that
there are many global questions to be investigated, but since we touch
them only in a few cases we confine ourselves to local issues.
Differentiable manifolds
Before giving a formal definition of what is meant by a differ-
entiable manifold it is instructive to consider which difficulties arise
in a simple case, namely that of a circle.
One can define a circle in the x-y plane as the locus of points
satisfying
12	2
S : x + у = 1	(1.1.76)
This description, although very useful, is somewhat unsatisfactory;
first of all the circle is a 1-dimensional object and should be des-
cribed by a single coordinate; secondly the description (1.1.76)
depends on the position of the circle in the plane, and we would like
an intrinsic description.
One could try to avoid these difficulties by using as coordinate
the angle 0 formed by a fixed direction and the segment joining the
center of the circle with a point P. However 0(P), where P is a
point of the circle, is not continuous at the origin. One is easily
convinced that there is no continuous simple coordinate describing the
circle: the best we can do is to have different parametrizations for
different subsets of S1. The topology of the circle tells us when
functions on these open subsets are continuous, and this topology is
29
given by defining the circle S as the quotient space of the real line
by the abelian group of the integers:
S1 = JR/Z	(1.1.77)
A differentiable function on a subset of is a function which is
differentiable with respect to the coordinate describing this subset.
We can classify functions in terms of their differentiability proper-
ties and in any subset a given differentiable function can be used as a
coordinate if it is differentiable in term of the angular coordinate.
Thus the circle can be covered by intervals having each a coordinate
system. When two intervals overlap the change of coordinate is given
by a differentiable function. Notice that giving a coordinate in a
given subset of S means to map this subset onto a subset of К
(Fig. 1.1.3).
If this map is invertible and differentiable as often as we like, we
say that looks locally the same as the Euclidean space.
This procedure can be extended to manifolds of higher dimensions:
loosely speaking we shall call manifolds the spaces in which the neigh-
borhood of each point is just like a small piece of a Euclidean space.
For example, smooth surfaces like the sphere or the torus where
each point lies in a little curved disk which can be flattened into a
disk is a manifold. On the contrary a cone is not a manifold because
the neighborhood of the vertex does not look like a small piece of a
plane.
ll
30
After these preliminaries we give a formal definition of a
differentiable manifold.
Let us consider a topological space M which is Hausdorff and
which can be covered by a countable basis of open subsets; let us also
suppose that each point has a neighborhood U which is homeomorphic
to an open subset of IRn:
ф	n
U---->ф(и)ск"	(1.1.78)
Eq. (1.1.78) defines what is called a chart. The image ф(Р) of
Peis given by the value of n-coordinate functions:
ф(Р) : {xX(P), x2(P),...,xn(P)}	(1.1.79)
If the point P belongs also to a second neighborhood V and is
represented by a second chart ф(У) c ]Rn with
ф(Р) : {y^P), у2(Р),...,уП(Р))	(1.1.80)
then the homeomorphic mapping
ф О ф'1 : Ф(и Л V) * ф(и л V)
(1.1.80)
is a one-to-one mapping of open subsets of JRn .
Using the coordinates (1.1.79) and (1.1.80) it can be written as
i, 1 2 n.
= у (x ,x ,...,x )
(i = 1,... ,n)
(I.1.81a)
31
and its inverse is:
x1 = xi(y1,...,yn) (i = l,2,...,n)	(I.1.81b)
(1.1.81) define the transition functions between the overlapping neigh-
borhoods U and V; they are also called a change of coordinates on
U n V.
The Hausdorff space endowed with a set of charts stf cover-
ing Mn is called a topological manifold of dimension n.
If the transition functions (1.1.81) are Cn-differentiable the
two charts are said Cn-compatible and stf is called an atlas of class
cn.
C°°-differentiability is also called smoothness.
A differentiable manifold of class C°° (or smooth manifold) is a
00	г	/1е0
manifold with a maximal atlas of class C . Any atlas of class C is
contained in a unique maximal atlas: therefore to specify a differen-
tiable manifold it is sufficient to specify an atlas of class C .
If we substitute (I.1.81a) into (I.1.81b) we get the identity:
i i, 1, 1 n. 2, 1 n. n, 1 n.
x = x (y (x ,...,x ), у (x ,... ,x ) ,... ,y (x ,...,x )
(i = l,...,n)	(1.1.82)
Taking the Jacobian of both sides one obtains:
&L.	= gi	(1.1.83)
SyJ Эхк k
Thus
Эх1 3yJ	/ 0	;	Эу1 Эх3	/ 0
(1.1.84)
A
32
A smooth map f : U •* V of two subsets of a Euclidean space is a
diffeomorphism if it is one-to-one and onto and if the inverse map is
also smooth. We may then say that on a differentiable manifold the
mapping f= фо ф relating two different parametrizations on the over-
lapping of two neighborhoods of is a diffeomorphism between sub-
sets of the Euclidean space.
A manifold is called orientable if it is possible to choose local
coordinate patches so that each Jacobian of the corresponding transi-
tion functions is positive:
эу1
3xj
> 0
(1.1.85)
In the sequel manifolds will always be understood to be smooth manifolds.
Some simple examples of differentiable manifolds are:
2
a) The Euclidean plane IR . It is a differentiable manifold covered
by a single chart, e.g. a Cartesian system of coordinates. All the
charts described by different cartesian systems related by rotations
and/or translations are compatible.
Also the charts parametrized by polar coordinates
{r.e} : r > 0
0 < 6 < 2 it
(1.1.86)
are compatible with the cartesian systems since
1
IX = r cos 6
2	• o
X = r sin 6
(1.1.87)
are diffeomorphisms.
In the same way one shows that n-dimensional Euclidean spaces,
Kn, are smooth manifolds covered by a single chart.
As a non trivial example, let us show that S^:
33
s1 = {x, у e JR2 : x2 + y2= 1}	(1.1.88)
is a (smooth) manifold.
We show this using four charts: fu+, U~. U+. U”l
1 x’ x’ y’ yJ
 	'.--/«a	r ’ "w/'
Fig. I.l.V
where
и* = {у > o}	;	u* = {x > o}
/ A
UV = {y < °}	>	U" = {x < 0}	(1.1.89)
J
On U+:
У
ф(Р) S ф(х,у) = ф(х, /1-х2) = х	(1.1.90а)
ф-1(х) = (х, /1 - X2)	(1.1.90b)
On и*:
X
ф(Р) = ф(х,у) = ф(/1 - у2,у) = у
(1.1.91а)
34
ф-1(у) = (у, /1 - у2)
and on U n U we find
У X
,	л"1	Г 2
ф о ф : у = / 1 - х
А ,-1 Гл 2
ф о ф : х = ✓ 1 - у
(1.1.91b)
(1.1.92а)
(1.1.92b)
Since 0«	С X, у <	1, U+ У	and	U+ ; X	are compatible charts.	
In	this way on		u+ nu У	", u" x У	nU and u” HU one finds X	у X	
У =	/1 -	2 X	t	X =	- / 1 - у2	(I.1.93а)
У =	- /1	2 - X	»	X =	- /1-У2	(1.1.93b)
У =	- /1	2 - X	»	X =	/ 1 - у2	(1.1.93c)
respectively. The four charts Ux, Ux, U and U are compatible and
form an atlas.
Actually it is not hard to show that and, more generally,
Sn can be covered with two charts only. We show this explicitly in
2
the case of S :
S2 : {(5,4,5) : €2 + П2 + C2 = 1}	(1.1.94)
2	2
Using the stereographic projection from S - {0,0,1} or S - {0,0,-l}
to the plane we have:
35
On the intersection of the two charts one gets:
36
Ip о ф
(I.1.97а)
(I.1.97b)
Since (1.1.97) is a diffeomorphism the two charts are compatible and
2
S is a smooth manifold. Moreover
2
so that S is orientable.
As a final example we show that the real projective space JRPn
is a non-orientable smooth manifold.
Let us consider in Fn+^ - {0} the equivalence relation
x У ~ (Уг ’-Уп+Р =
(1.1.99)
for t / 0.
The set of equivalence classes is IRP^, namely the n-dimensional
projective space. It can be considered as the space of all the lines
through the origin. A set of n+1 compatible charts can be obtained
]RPn such that x^ / 0; for instance the set of all lines lying in the
shaded cone around the i-th axis (see Fig. I.l.VII).
37
Then we may set
(I.1.100a)
♦«)<»>
(I.1.100b)
к
л x
w = /
and therefore on IL n we have
к = xk . \
(Я) x£ x£
C(i)/?U)
(1.1.101)
Hence IL and are compatible charts.
Moreover one can easily verify that the Jacobian
3(4i),....cn(i))
does not maintain a constant sign in
smooth non-orientable manifold.
U. n U„; therefore JRP is a
i £	n
I•1.4 - Functions, vector fields and differential forms
Functions
Functions on a manifold are defined by giving their local para-
metrization: let P be a generic point on a manifold Mn: using a
chart with local parametrization ф, around P
P (x1,...^11)
one defines
f(P) = £(ФСР)) = f(x1,...,xn)
(1.1.102)
(1.1.103)
38
It is customary to omit the carets in this way identifying f(P) with
its local parametrization f(x ,...,xn).
Since the product of two functions f and g at P is a func-
tion at P, the set of (smooth) functions at P is a ring denoted by
Fp(M).
Mappings between manifolds are also defined by means of their
local parametrizations. The mapping
M N
(1.1.104)
from a manifold of dimension m to a manifold of dimension n is
written as follows:
r, f 1	in. r 1 f 1	m. 2 r 1	itk	n r 1	m,
f(x , ...,x ) = (y (x , ...,x ), у (x , ...,x ),...,у (x , ...,x )
(1.1.105)
where (x^,...,xm) and (y\...,yn) are local coordinates on M and
N respectively.
A smooth map f: X->Y between two open subsets of two manifolds
is a diffeomorphism if it is one-to-one and onto and if the inverse map
f-^ is also smooth. In particular, as we have already seen, the
coordinate transformations on the intersection of two neighborhoods of
a given manifold are diffeomorphisms: JRn •+ IRn.
Vector fields
On .an arbitrary manifold, vectors cannot be represented by merely
drawing arrows from a point as in Euclidean spaces.
However, one can define vectors in Rn without reference to
oriented line segments. In IRn a vector can be identified with a
directional differentiation; given
V f = grad f • v = v^ —f
v 6	a»1
(1.1.106)
39
the vector v can be identified with the first order differential
operator
Vp = V1 -Л-	(1.1.107)
Эх1 p
Vp maps the ring Fp(Fn) of the (smooth) functions at P on the real
numbers JR :
vp : Fp(Fn) * Ж	(1.1.108)
The definition (1.1.107) can be extended to a general smooth manifold.
Let Fp(M) be the ring of smooth functions on M in the neigh-
borhood of P; a tangent vector Vp at a point P of M is an
operator
vp:Fp(M)->R	(1.1.109)
satisfying linearity and the Leibnitz rule:
i)	vp(af+Bg) = avp(f) + Bvp(g)	(1.1.110а)
ii)	vp(f-g) = vp(f)g(P) + f(p)vp(g)	(1.1.110b)
Setting f=g=l in ii) yields:
v(l) = 2v(l) = 0	(1.1.111)
and, using i), we find that for any constant c:
vp(c) = c vp(l) = 0	(1.1.112)
40
Using local coordinates around a point
P one sees that the n opera-
tors :
(1.1.113)
fulfill the properties (1.1.110) defining a vector at P. Moreover
they are a basis for every tangent vector at P; indeed, let us expand
the function f around P= (c1,... ,cn) :
f(xk) = f(ck) + (x* - ci)gi(xk)	(1.1.114)
where
g.(c ) = —-
Эх1 к к	,т ,
Then using (1.1.112) and (I.1.110a), we get
(1.1.116)
Therefore
(1.1.117)
(1.1.118)
The set of all tangent vectors at P is called the tangent space at P
and is denoted by Tp(M). Note that Tp(M) and Rn are isomorphic
vector spaces.
41

of х'
If we let the coefficients a in (1.1.117) be smooth functions
on we get a vector field on
i/- i a
= a (x) —T-
Эх1
Induced
mappings on tangent spaces.
Let us suppose we have a mapping Ф
and N :
n
between two manifolds M
m
(1.1.120)
Using the chain rule for differentiation we have an induced mapping
between the tangent spaces Tp(M) and
let the map Ф be expressed in local coordinates by
Indeed
= y1(x1,...,xln)	(i=l,...,n)
(1.1.121)
eTp(M), the transformed vector Ф* (vp) e T$(p) is
then, if vp
defined by its action on a function f(y):
Ф*С^р)£(у) = vp(f°<j>)
= $p(f(y(x)))
(1.1.122)
Explicitly we
have:
vp - a
Эх
Эу£ Э
» i ~ ]
Эх 3yJ
- **tfp)
Ф(Р)
(1.1.123)
M :
n
v
(1.1.119)
Ф
M --------
m	n
Ф : у
a
P
a
The components of
О/Эу. I }
'p
on N
n
the transformed vector <j>*(vp) along the basis
are thus given by:
42
a'j(y) = ax(x)	(1.1.124)
Эх1
In particular the rank of the Jacobian Эу1/Эх1 equals the dimensions
of the image Ф*(Тр).
It is immediate to show that if we have a composition of mappings
Ф1 (j>2
M----» N-----> P and we call <j> = <J>^ ° <j>2 then it follows that
Ф*(?) = Фх* ° Ф2*(>;)
(1.1.125)
In components Eq. (1.1.125) implies the matrix multiplication of the
Jacobians.
The mapping Ф can be a diffeomorphism only if the dimensions of
the two spaces are equal and if the Jacobian is non zero:
dim M = dim N (m=n)	(1.1.125a)
det (^1) / 0	(1.1.125b)
Эх1
It follows easily that if Ф is a diffeomorphism then Ф4, namely the
induced mapping between tangent spaces, is an isomorphism. The con-
verse is also true. In particular if M and N are the same manifold
and Ф represents a change of coordinates relating two intersecting
charts then Eq. (1.1.124) represents the well known transformation law
of a contravariant vector field under a general coordinate transforma-
tion. Thus a vector field can be also represented by its contravariant
components along the natural frame {Э/Эх1} subject to the transforma-
tion law (1.1.124).
1-forms on a manifold
As previously shown, at each point P of a manifold M there
is a tangent space Tp(M) isomorphic to Rn .
43
1-forms (or covectors) at P can be defined as linear func-
tionals on vectors at the same point, in exactly the same way as we
did for IRn.
We begin by defining the differential of a function.
Let us consider a function on a manifold M; the differential of f
at P, df|p, is a 1-form:
dfl : df (v ) = v (f) = v1 -Д- fl	(1.1.126)
Ip	Эх1 'P
In particular, if we take as f the i-th coordinate function x1,
then (neglecting the suffix P) we have:
dx1 (vj -Л-) = vj dx1 (-Д-)	(1.1.127)
Эх3	Эх3
On the other hand
dx1(v) = v3 —x1 =	(1.1.128)
Эх3
It follows that:
dx1 (-Л-) = б!	(1.1.129)
Эх3 3
namely: the set of 1-form {dx1} is the basis of 1-forms dual to the
basis {Э/Эх1} of tangent vectors. Therefore a general 1-form at P
can be .written as
* In introducing vectors on a manifold we have denoted them with an
arrow (except in the case of the basis vectors ——). In the sequel
ЭхЦ
the arrow is frequently omitted for notational simplicity at least
when there is no possibility of confusion.
44
, i
ы = a. dx
i
(1.1.130)
In particular the value of a 1-fbrm co on a vector v corresponds to
the contraction of a covariant vector co. with a contravariant one v1:
i	’
indeed
co(v) = co.dx\vj —^-) = (0. V^dxi (-^-T-) = C0.vi
1	3xJ 1 3xJ 1
The space of all 1-forms at P is called the cotangent space at P
and is denoted as Tp (M). If we let the coefficient a£ in (1.1.130)
be smooth functions of the coordinates x1 then we obtain a covector
field or a field of 1-forms on M .
n
Tensors on manifolds are easily constructed following the method utilized
Ik £1
in the case of algebraic tensors. A tensor W ’ J of contravariant
order £ and covariant order к at a point P of a manifold M is a
multilinear map from
Tp(M) xTp(M) x ... xTp(M) xT*(M) xT*(M) x ... xT*(M)
k-times
£-times
to K.
Using the natural frames {Э/Эх1} and {dx1} for vectors and covectors
one may write a generic (k,£)-tensor A at P just mimicking Eq.
(1.1.14):
ir--ik
A 1 R
3Г”3£
3
“4
Эх
3
® ---—
a 1r
Эх
j 1 Jo
dx ® ... ® dx
(1.1.131)
i1...ik
A.1	/
lr..l£
(k,£) tensor field
If we let
be smooth functions
on M we get a
n a
generic
on the manifold.
A =
® ..
®
Let us consider in particular a (2,0)-tensor field g, that is,
a bilinear form on vector fields:
В gCVpttjV + ^w) = o^gfVpV) + a2g(vx,w)	(1.1.132)
Ел
IL is called a metric. In a natural basis we find:
f g = dxr ® dxS	(1.1.133)
« If a manifold is endowed with a metric which at each point P is non
^degenerate and symmetric (Eqs. (1.1.18) and (1.1.19)) then the manifold
11* Riemannian.
The importance of Riemannian manifolds derives from the fact that
ktheir tangent space is a vector space endowed with a scalar product
defined by:
g(v,w) = g dxr ® dxS(vX -Л- , w3 -X-)
rS	Эх1 Эх3
l	S grs vrwS	(1.1.134)
This enables us to define angles between curves and lengths of curves
j on the manifold. The "distance" between two points P^, P2e is
Fv usually defined by integrating (1.1.134) along a geodesic path.
f, k-forms on manifolds
Using the same constructions as in the case of algebraic forms we
ij may construct out of the basis dx1 a basis for forms of higher degree:
dx* л. dx3 ; dx* л dx3 л dx\. ..; dx ... Adx	(1.1.135)
X A generic differential к-form will be written as
-
| o/k) - a	dx , _____ л dx к	(1.1.136)
f	1l”1k
46
The space of all к-forms at P will be denoted by Tp k 7(M). In
*f0)	”
particular Tp k (M) is the vector space of all the О-forms, that
is the functions, at P previously denoted by Fp(M) and
*(1) * ”
Tp '(M)=Tp (M). Every definition given for the algebraic forms, in
particular the exterior product, can be extended to the forms on mani-
folds since the tangent space spanned by {Э/Эх1}р and the cotangent
space spanned by {dxx}p are isomorphic to the spaces Vй and V*n
of vectors and algebraic 1-forms, respectively.
A field of к-forms is obtained when the coefficients a. . in
1Г”1к
(1.1.136) are smooth functions of P = P(x^.. .xn) .
The set
T*(M) = T*(0)(M) ® T*(1)(M) ® ... ® T*(n)(M)	(1.1.137)
of all differential forms on M is then an (associative) algebra
called the exterior algebra on M^.
In particular the n-form
Ш(П) = V dx1 л ... , dx”	(1.1.138)
defines a volume element on M.
Using the volume element one may give a definition of the orien-
tation of a manifold alternative to that discussed at (1.1.85): M is
orientable if one can define a volume form on M which is never zero
at any point PeM.
The contraction of forms with tangent vectors is also immediately
extended from the algebraic case (see 1.1.63) to forms on a manifold
M. Let v be a vector field on M: the contraction of v with a
fk)
к-form ы J on M is defined by:
y] o/k-)(v1,...,vk p = kw(k\v,v1,. ..,vk l)	(1.1.139)
47
w and v| satisfies the properties given in (1.1.65-66-67). Notice that
'$	*fkl	*Гк-1")	।
•fa- vl is a linear mapping from T_ 1 J (M) to T 1 J (M) and that v|
—J	r	r
S' is not only linear but also f-linear; that is, if f is a smooth
’ function on M:
v|f • 0) = f v|te .	(1.1.140)
Moreover v| is f-linear also with respect to v:
fyjw = f v]w	(1.1.141)
J.1.5 - Exterior differentiation and behaviour under mappings
We shall now establish the existence and uniqueness of an operator
d : Т*(к\м) + т*(к+1) (M)	(1.1.142)
(k)	(£)
such that for any forms co , to :
i) d(0)(k) + wU)) = da>(k) ♦ da>U)	(1.1.143a)
ii) d(w(k) л wU)) = dw(k) л + (-l)Vk) л doj(£) (1.1.143b)
iii) d2 = 0	(1.1.144)
iv) for any function f (0-form)
df = 2Ldxi	(1.1.145)
Эх-*-
48
= а. . dx 1 „ ... „ dx к	(1.1.146)
1l'"1k
we set:
За.
du« = --------1 к dx£ л dx 1 л ... л dx к	(1.1.147)
Эх*
and it is easy to verify that (i)-(iv) are satisfied; in particular
dd = 0	(1.1.148)
is nothing else but the statement that partial derivatives commute.
Indeed we have:
32a.
d аЛ) = -.--1-:..1.k axm . dx* . dx11__________dxlk 5 0
й
d d	(1.1.149)
because of the antisymmetrization of m and Я. given by dx ~ dx .
Examples:
3
In 1R the exterior derivative is equivalent to the usual differ
ential operators of vector analysis: grad, rot, div. Indeed let us
define the differential forms analogous to (1.1.50) and (1.1.51), using
as a basis the differentials dx1:
ы£15-= dx1	(1.1.150)
A
о/2) = B. dx1 a. dx-1 = e. . B. dx-* л dxk	(1.1.151)
J ij	2 ijk i
D
where {x1} are cartesian coordinates; let f be a generic function
3
on К ; then
49
df = -^5- dx'*' = grad f • dx	(1.1.152)
Эх1
dw^l) = Э.А. dx'*' a. dx-1 = — (rot A). e. .. dx-1 л dx^ (1.1.153)
д 1 J	2	1 ijk
do/2) = 4 £- v Эо B. dx^ a. dx-1 л dx^ =
*	2 ijk £ i
D
1	r i D , 1	,2	, 3
= — E. E.Э„ B. dx a. dx a. dx =
2	ij к St,] к £ i
->	1	2	3
= div В dx a. dx л dx	(1.1.154)
Behaviour of k-forms under mappings.
We have seen before (Eqs. (1.1.122-124)) that the behaviour of vectors
under a mapping is defined through the chain rule. The same is true
* (0)
for k-forms. Let us consider first the ring Tp (M) of all 0-forms,
that is the ring of the functions, defined in a suitable neighborhood of
PeM.
Then a map between two manifolds M and N
Ф
M ---» N
(1.1.155)
which in local coordinates reads
У1 = yi(x1,...,xm)	(i=l,...,n)	(1.1.156)
can be used to pull-back a function f on N to a function Ф (f) on
M. It suffices to make a substitution of coordinates:
(®*f)(xi) = f(yi(x))
where
Ф*  Тф(р)М * T*C0)(M)
(1.1.157)
(1.1.158)
50
The same can be done for 1-forms: if
ы = a^(y) dy1
(1.1.159)
is a form on 1-form on M	N, then the pull back of such that:	Ш, Ф Ш,	is defined as a
Ф co(v)	= ш(ф*(у))		(1.1.160)
If v=vT Э/Эх1 then Ф*(у) is given by (1.1.123); hence (1.1.160)
implies
Ф ш = a. (y(x)) dx3 = b. (x) dx3
Эх3	3
(1.1.161)
where
b
J
a.
ay1
Эх3
(1.1.162)
is a diffeomorphism, then Ф
spaces T$(P)(N) and T p(M)
Considering now the case
change of lodal coordinates on
Therefore
Ф* : Tfjp/W ^‘(M)	(1.1.163)
*
pulls back 1-forms from N to M, and Ф ы is obtained from ы by
the chain rule of differentiation. The components of Ф ы are given
in terms of those of ы by means of the formula (1.1.161). In the
same way as for mapping of vector fields it is easy to show that if Ф
is an isomorphism between the vector
and vice versa,
of a diffeomorphism Ф representing a
M = N, we see that (1.1.161) represents
the effect of a change of coordinates on the components of a 1-form; it
51
coincides with the definition of a covariant vector in ordinary tensor
analysis. We may extend Eq. (1.1.160) to forms of higher degree in a
natural way. If
= a£j(y) dyi *	(1.1.164)
* (2)
Is a 2-form on N, then on M we have the pulled-back form Ф ы :
ф tu(2)(v,w) = a/2) (<f>*v,4>*w)	(1.1.165)
and one obtains explicitly
* (2)	,	, x , к , £	. , ..	Эу1 3yJ , к , £	,т .	. .
Ф w	b.	(х) dx л. dx	= а. . (у(х))	dx л dx	(1.1.166)
1J	ЭхК Эх*
(к)
and so on. In general any к-form or on N can be pulled back to a
к-form on M
Тф'(Р) (N)	Tpk) (M)	(1.1.167)
by merely substituting coordinates and by use of the chain rule of
di f ferent i at i on.
In particular if dim N= dim M= n then an n-form transforms as
follows:
Ф*ы(п) = det I — I w(n) .	(1.1.168)
। 3xi 1
We notice that the behaviour under mapping of the differential forms
is exactly the same as that already seen in the case of algebraic
forms, Eq. (1.1.55); indeed the Jacobian matrix at P is an nxn
52
matrix and therefore it can be identified with the matrix A of
Eq. (1.1.55).
Corresponding to Eqs. (1.1.56) and (1.1.57) we have:
i)	(ф о ф) = ф о )	(1.1.169)
ii)	ф (ых л ш2) = Ф*^) л ф*(ш2)	(1.1.170)
Moreover we can show that
iii)	d ф to = ф*dw .	(1.1.171)
Proof: the property is obvious for О-forms f(y) :
4>*(df) = Ф*(-Ц- dy1) = -Ц- dxk =
Эу1	Эу1 Эх
= df(y(x)) = d^*f)	.	(1.1.172)
Moreover if it is true for a (p-l)-form it is also true for a p-form;
it is sufficient to show this for a monomial p-form
= f(y) dy 1 л ... „ dy p	(1.1.173)
since any p-form can be written as a sum of monomials of type
(1.1.173).
We can write	as follows
= f d(y 1 dy 2 л . .. л dy p) ; f dr,	(1.1.174)
where r, is a p-1 form. Hence
53
dC<t> W) = d(<|> f • ф dr)) = d(ф f d ф n) =
= dф f ,, d ф г) = Ф df л ф dr) = ф d(fdq) = ф dw
(q.e.d.)	(1.1.175)
An important consequence of iii) is that diffeomorphisms between mani-
folds or general coordinate transformations commute with the exterior
derivative. This property will be important in building physical
theories invariant under diffeomorphisms.
We conclude this paragraph by observing that forms are objects
more naturally adapted to describe a field theory than vector fields;
indeed forms can be naturally differentiated by means of the d opera-
tor and they can be integrated (although we do not discuss the integra-
tions of forms). Moreover the pull-back of a form is always a form
while the mapping induced on vector fields by a mapping Ф does not
always give a vector field. (This happens for example if the map is
not surjective or if two vectors at different points are mapped into
two different vectors at the same point.)
1.1.6 - The Vielbein basis
We observe that we have always used the coordinate basis {dx1}
as the most natural setting for the explicit expansion of a general
к-form; however any other basis obtained from the {dxP} by a non
degenerate linear transformation could work equally well.
Therefore, if we denote by V1 such a new basis of 1-forms,
we can write:
V*(x) = V1p(x) dxP
dxP = VPi(x) V1
(1.1.176a)
(I.1.176b)
where by V1^ we have denoted the GL(n,JR) matrix connecting the two
coframes and we have defined
54
VP. = (У-1)\	(1.1.177)
A generic к-form can be written in two equivalent ways;
(k)	(k)	. P1 co	= co	dx л .		a - .. a dx	=	(k) co. ' ir.	. у11 . . xk	.. . v‘k (1.1.178)
and we have	ц1---цк the relations:				
co« ir.	P1	U2 . = V 1. v . ., ,xk	X1 X2	pk . . V . co Xk pr	..Pk		(I.1.179a)
Jk> Pr.	il	i „ = V	V	.. •pk	P1 p2	vTk . . V	to. рк xr	•Лк		(I.1.179b)
We have carefully distinguished between the coordinate indices.
labelled
by Greek letters, and the Latin indices labelling the new basis of 1-
forms {V1}; indeed they have quite different transformation properties.
In fact any n*n non singular matrix aV eGL(n,R)
{V1} basis into an equivalent one {V 1):
V*1 = A1. V3
J
transforms the
(1.1.180)
and therefore the Latin indices transform in the vector representa-
tion of GL(n,Jt) (cogrediently or contragrediently if they are upper
or lower indices respectively). Indices of this kind will be called
anholonomic or tangent space indices. Other denominations are flat or
intrinsic indices. Instead the Greek index of the coordinate basis
transforms with respect to coordinate changes with the Jacobian matrix,
as we have seen before (Eq. (1.1.161) with w= dxp)
~ *p
, 'p Эх , v
dx r = ------- dx
(1.1.181)
55
Coordinate indices are also called holonomic indices, world-indices,
or curved indices. It is true that also the Jacobian matrix is an
element of the GL(n,R) group at P, but these two GL(n,R) groups
acting on the tangent space at P are quite distinct in principle. In
particular it could be necessary for physical reasons to restrict the
GL(n,IR) group acting on the V1 basis to some subgroup G c GL(n,lR) .
This is the case if one wants to introduce spinor fields on a manifold
since, by definition, spinors are representations of the SO(n) group but
not of GL(n,K) . As such they can be defined at a point PeM of a
manifold only if the tangent group is SO(n) .
As an example let us take a Riemannian manifold so that we can
define a scalar product at each point P. Then we can construct a frame
of orthonormal vectors e^,...,e at P
e. = VP -3—	(I.1.182a)
1	1 3xP
-JL = v1 e.	(1.1.182b)
3xP p 1
where V1^jeGL(n,R) and VPi is the inverse matrix of V1^: VPi v\ =
= <5P, V1 VP. = 61 .
v’ p ) J
The orthonormal frame {e.} is called the moving frame. The
1 *
corresponding orthonormal dual frame of covectors in Tp(M) is called
the vielbein frame and is defined by:
V^e.) = б!
J J
that is
V1 = V1 dxP
U
(1.1.183)
(1.1.184)
56
From
gf^.ep = n ;	= (1,...,1 , -1,...,-1)	(1.1.185)
к	n-k
and
gCVV = gpv>	(1.1.186)
using (I.1.182b) one finds:
j
j
	= v1 V3 p V	nij	(1.1.187)
that is			
g =	v1 ® V3	%	(1.1.188)
In particular a canonical oriented volume element is given by the
n-form or volume form:
= V1 л V2 л ... „ Vn	(1.1.189)
Using (1.1.184) and (1.1.187) one finds
Q = V1 ... Vn dx 1 a. . .. „ dx n = det V dx1 л л dx"
P1 Pn
= / |gpJ dx1-----dx11	(1.1.190)
Q takes the value -Д- on the orthonormal frame e,__,e and on any
n!	In7
other frame related to it by an element of S0(k,n-k).
57
By expressing the volume form in two coordinate frames one finds:
V1--------Vй = V1' ... v”’ dx'^A... Adx'Pn =
4	4
1,4 j 1	, n
= V ... .V dx « ... Adx
V)
1 n
о» det V = det V (det J)	(1.1.190a)
where J = | Эх ^/3x^1. Therefore det V is a scalar density of weight
-1 and
j'l	j'n j т j 1	, П
dx л л dx = det J dx « ... « dx
(I.1.190b)
is a scalar density of weight +1. The two previous formulae simply
express the fact that the volume form л ~ Vй is obviously a
coordinate invariant.
Hodge-duality
As a further exercise we consider the extension to the case of a
arbitrary manifold M of the Hodge duality operator defined for forms
on a vector space (Eq. (1.1.70)).
Using the intrinsic vielbein basis we define the Hodge duality
mapping
T*(k) \ T*(n-k)
similarly to the algebraic case:
i-i	iv i ^l’"’^k	4	•’n-k
‘(V 1.... ,V k) = -i—e1	R. .	v4...„vnK
(n - k) !	-11’ ' '•’n-k
(1.1.191)
58
where anholomic indices are contracted by means of the metric
Let us now observe that using the determinant formula
(I.1.191a)
jP _ Эх—	one sees
v 3xv
coordinate basis is not a '
-1; analogously £
рГрп
i. -i
On the other hand e n,
in the anholomic basis, is a true
have the following relations:
where
in the
weight
i"
e
n
v 1
Hl
. . V n
yn
that the Levi-Civita antisymmetric symbol
tensor but rather a tensor density of
is a tensor density of weight +1.
the Levi-Civita antisymmetric tensor
Lorentz (invariant) tensor and we
pr..pn
£
(det V)"1
(I.1.191b)
e.
11"
. i
n
pl
V.
4
4
£
vr..pn
det V
(1.1.191c)
Therefore
(det
V)’1
Pr..Pn
£
and det V
coordinate tensors.
ец
Г"ип
are true
Using Eqs. (1.1.184), (1.1.191), and (I.1.191b,c) one
Hodge duality operation on the natural basis
finds the
^1
*(dx
pk 1 rr-v р1"-рк
dx ) = ------- / g €	x
(n-к)!	vVvn-k
V1	v V
X dx л ... л dx	(1.1.192)
where the Greek indices are saturated by means of the g^v Riemannian
metric.
59
Given any two forms оДк^ and B^ we have
n-t
a(k) A ‘gW = (-1) 2 k! и. В 1 k V1 л. ... л Vn =
1l"’1k
n-t
r 2 vi „ tJJr','Jk л—г ,1	, n
= (-1) к! a В /|g| dx л ... „dx
цГ"рк
(1.1.193)
which is the analogue of Eq. (1.1.73).
In Chapter II our local study of Riemannian geometry will be
based on the use of the moving frame (ej and of the corresponding
dual frame {V1}.
1.1.7 - Lie derivative, coordinate transformations and invariance
In this section we study the behaviour of functions, vector
fields and differential forms, and in general of any tensor field
under diffeomorphisms.
Of particular interest to us are the diffeomorphisms of a mani-
fold M on itself:
Ф : M -+ M	(1.1.194)
We already remarked that when Ф is expressed in local coordinates
Ф : x'1 = Ф1(х1,...,хп)	(i=l,...,n)	(1.1.195)
it can be also interpreted as a local general coordinate transformation
in the intersection of two charts of M.
Suppose we have a function f on M; under a general coordinate
transformation Ф we find:
f(x) -> f(<t>(x)) = 4>*f(x) = f'(x)
(1.1.196)
60
As a result of the transformation the function
which can be written as:
6f = f(x') - f(x) = -(f'(x) - f(x)) .
f undergoes a change
(1.1.197)
Thus 6f can be considered either as a change in the functional
form of f or the change in f due to the shift of the coordinate
point. On the other hand since f(x) is a coordinate scalar we also
have:
f(x) = fo<t>-1(x') = f(x')	(1.1.198)
the corresponding functional change 6f is:
6f = f(x) - f(x)	(1.1.199)
and is called the active change.
If x'p differs by an infinitesimal vector from xp
x'p = xP + ep	(1.1.200)
then 6£f is called the Lie derivative of f along the vector ep 3^
and is denoted f . One finds :
E
Uf = f(x+e) - f(x) = ep3 f .	(1.1.201)
e	U
Hence the Lie derivative along the tangent vector ep3^ is given by
the action of the vector e on f
JUf = e(f) .	(1.1.202)
e
From (1.1.198) and (1.1.199) one easily sees that the infinitesimal
active change 6 f is minus the Lie derivative 6^f:
e	£
Lf(x) = f(x) - f(x’) = -eP3 f = -eP3f	(1.1.202a)
E	OU
A more geometrical definition of the Lie derivative on functions, use-
ful for extension to vectors and forms, is given in terms of one-
61
parameter groups of transformations.
Let PeM^ and teB; the map:
(P,t) -* *t(P) e M	(1.1.203)
4>t+s = *£° *t	(1.1.204a)
фр s identity	(I.1.204b)
defines a one parameter group of diffeomorphisms or a flow on Mn-
Using coordinates as in Fig. I.1.VIII associated to the map
Fig. I.1.VIII
фг : xP -> yP(t,x)	(1.1.205)
there is a vector field e whose components at x are given by
Ep(x) = — yP(t,x)|	.	(1.1.206)
dt	't=0
The vector
E(x) = £P JL = ;P(0,x)	(1.1.207)
ЭхР	ЭхР
is called the infinitesimal generator of the diffeomorphism.
62
The Lie derivative of a function along the vector e is defined
by:
(A*f) (x) = lim i [f(y(t,x)) - f(x)] = A f(y(t,x))l
L	t + 0 г	dt	't=0
= ep(x) ap f(x) = e(f)
(1.1.208)
We see that the Lie derivative on a function, that is, the infinite-
simal change under a diffeomorphism, is given by the action of the
infinitesimal generator e on f in agreement with the definition
(1.1.202).
We will extend the notion of the Lie derivative to more general
objects like vector fields, к-forms or tensor fields. In order to do
that it is convenient to introduce first the notion of the Lie algebra
of vector fields. Let us first recall the definition of a Lie algebra:
Definition: a Lie algebra is a vector space on which an operation is
defined:
[A,B]	(1.1.209)
called "bracket operation", which is bilinear, antisymmetric and
verifies the Jacobi identity:
[[A,B],C] + [[B,C],Aj + [[C,a],b] = 0	.	(1.1.210)
Well-known examples of Lie algebras are the vector space of all n*n
matrices, where [A,b] is the commutator, or the vectors in where
[А.В] is the wedge product.
We show that the vector space of vector fields on M, named
E(M), becomes a Lie algebra if one defines
[a,b] = AB - BA
(A, Be E(M))
(1.1.211)
63
Indeed, using a basis:
A = AP —	(1.1.212a)
Эхр
В = BP —	(I.1.212b)
3xp
and evaluating the action of [А,В] on a generic function f we get:
[a,b] f =	(AP — BV	—	- BP —	AV —)Df	=
P	3xp	3xv 3xp 3xv P
= (AP — - BP —) —I f = CV — I f (1.1.213)
3xp 3x^ 3xV*P	3xV 'p
The important thing in (1.1.211) is that the second derivatives cancel.
Hence [a,b] is again a vector field C whose components are given by
CV = AP 3 BV - BP 3 AV
P	P
Moreover the Jacobi identity (1.1.208) is satisfied; indeed
[ [A, b] ,c] f = { (AP Эц BV - BP 3p Av) 3V Cp -
- Cv3 (AP 3 Bp - BP 3 Ap)} — f
v p	p 3xp
(1.1.214)
(1.1.215)
Adding the same expression with A,B,C circularly permuted one obtains
identically zero.
An important property of the bracket operation is that it is
invariant under diffeomorphisms, that is, if Ф: M*N is a diffeo-
morphism, then
64
Ф*[А,В] = [Ф*А.Ф*в] 
(1.1.216)
The proof of (1.1.216) is immediate using components.
We now compute the active change of a vector field A under an
infinitesimal diffeomorphism.
This is defined as the rate of change of vector field A in the
direction of e given in (1.1.206).
To obtain this change we consider a vector field A and its
associated flow xp-»yp(t,x):
Then we confront the vector field A(x) at x with Ф*[а(х)] =
A[y(t,x)] (we use the isomorphism between vector spaces at x and at
y(t,x)). The Lie derivative of A in the direction of e is defined
by the limit:
A = lira - [AP(y(t,x)) -5- - AP(x) -L-]	(1.1.217)
E t-*0 r	ЭуР	ЭхР
Setting
yp(t,x) = xp + yp(0,x)t + 0(t2) = xp + ept + 0(t2)	(1.1.218)
65
and therefore writing:
= ^ + t + oft^)	(1.1.219a)
3xv V 3yV
= 6v ' v 1 * 0(t ]	(1.1.219b)
Эу	3y
we obtain
Я+А = lim - [AP(xP + yp(O,x)t) - AV(x) -^-] =
t->0 t	Зук	Эхц
= lim - [(АЦ(х) + ep3 AWt)(^ - 3 evt) - АР(х) -M
t + 0 t	p p p Эх	3xp
= (eP3pAV - Ap 3p eV)	.	(1.1.220)
Эх
Hence using the definition (1.1.213) one gets
£+A = [e.A] = - Я*?	.	(1.1.221)
In tensor calculus one defines the Lie derivative of a contravariant
vector Ap as minus the active change induced by an infinitesimal
coordinate transformation; using equation (1.1.124) and putting
y^:xp + ep(x) with ep infinitesimal, one easily retrieves:
6£АР = a'P(x) - Ap(x) = AV3veP - cV3vAp
that is, using the definition (1.1.219):
<5+A = - JLgA
(1.1.222)
(1.1.223)
66
Therefore the Lie derivative £gA is minus the active change d+A.
From (1.1.221) it easily follows that is linear and, moreover, if
f is a smooth function one has:
S^(fB) = R,*f • В + f = A(f) • В + f • [A,B]
(1.1.224)
The Lie derivative of 1-forms can be defined in a way strictly analogous
to the definition given for vector fields.
Using again the flow (1.1.216) we define the Lie derivative of a
1-form w in the direction of e by means of the limit:
£->co = lim i [io (y(t,x))dyp
e t+ 0 t p
ш^(х)<1хр]
(1.1.225)
and using (1.1.218) and (1.1.219) one gets
£+io = (ш Э eP + ЕР Э w )dxV
e p v	p
(1.1.226)
It is very useful to rewrite this expression in terms of the operators
d and A| of the exterior algebra. One obtains:
= (ejd + d e]) G)
(1.1.227)
Indeed, using the properties of the inner product operator |, we
have:
e|io = ерЭ |ш dxv = epio 6V = e^io
—1	p| V	V p P
(1.1.228a)
e| dG) = £рэ I Э io dxP „ dxV
—1	 p| P V
2еРЭ г co -i dxV
[p v]
(I.1.228b)
67
Therefore:
= ( e| d+ de]) w = 2e^3j-^ iovjdxV + a^Ce^uypdx^ =
= (шр\еР + eyayG>v)dxV q.e.d.	(1.1.229)
Again one finds that the Lie derivative is minus the active change of
the 1-form. Property (1.1.224), true for vector fields, is also true
for 1-forms:
H-^(flo) = H-^f • to + f =
= A(f)<D + f( jjd + d_A])io .	(1.1.230)
The extension of the Lie derivative to arbitrary tensor fields is now
immediate. Indeed, on an arbitrary tensor field the Lie derivative is
defined in a way analogous to Eqs. (1.1.217) and (1.1.225). Given a
tensor X its Lie derivation along v is defined by:
£->X = — X(y(t,x)) I	(1.1.231)
v	dt	lt=0
where v is the generator of the flow x->y(t,x). To get the explicit
form of H+ on a general tensor field it is sufficient to observe that
from the definition (1.1.217) and (1.1.225) it follows easily that the
Lie derivative satisfies the Leibnitz rule when acting on tensor pro-
ducts of vectors and/or 1-forms U and V:
£+(U®V) = (fMJ) ® V + U ® («М0 .	(1.1.232)
Therefore, if we write the expansion of a general tensor X along a
basis:
68
1’ ’ к
X = X.1 .К
Зг • 'h
э ~ л э
—— ® . .. ®
Эх11
Эх к
jl	3P,
dx 1 ® ... ® dx (1.1.233)
and we use:
D , Э ,	г-> Э i	3v3 Э
v Эх1	Эх1	Эх1 Эх3
dx* = ( v| d + d|v)dx^ = dyi
(1.1.234a)
^tdx3
Эх3
(I.1.234b)
we easily obtain the explicit form of
the metric tensor which is a symmetric
Я+Х. In particular if we take
(0,2)-tensor one has
&->g = &-*(g dx^ ® dxv) = (vp3 g +3 vXg. +
v6 v6pv	1 p6pv p
+ 8pX3vvX)dxP ® d*V
(1.1.235)
which gives (minus) the "active change" of the metric tensor under a
general coordinate transformation.
Let us now restrict our attention to the Lie derivative acting
on the exterior algebra of forms on M^. In this case one finds that
on k-forms can be written in the same way as for 1-forms; that is
A
one has
SL-i = Al d + d A]
A —1	—1
(1.1.236)
on any k-form.
Moreover restricted to the exterior algebra satisfies the
following properties:
1+ d = d it
A	A
(1.1.237a)
69
л a>9 + ol л Jb*w9 .	(1.1.237b)
To prove the previous statements it is sufficient to observe that
(1.1.236) and (I.1.237a-b) are valid on О-forms and 1-forms: indeed
on 0-forms (1.1.236) reduces to (1.1.201), (I.1.237a) can be proved by
explicit computation, and (I.1.237b) is a consequence of (1.1.232).
On the other hand the whole exterior algebra on can be
generated by 0-forms and wedge products of 1-forms: therefore (1.1.236)
and (I.1.237a-b) are always true.
Another useful property of £д restricted to forms is the fol-
lowing:
[u], S,v] = [u,v]l .	(1.1.238)
I It can be easily shown to be true on 1-forms using components (on 0-
1 forms it is trivial); extension to к-forms is obtained by the same
I argument used before.
к	Coming back to the general case we show an important property of
 the Lie derivative: when acting on any tensor field the operator
| yields a representation of the Lie algebra of vector fields. This
I means the following:
I	 *[X,i) •	(I1-2!9)
To prove (1.1.239) we first observe that it is trivial on 0-forms. For
, a vector field C one has
♦ [ВД-* c] =
= [[a,b],C] + [B,[a,C]] .	(1.1.240)
70
Using Jacobi identities one finds
= [[A,B],C] = Jl[U]C
(1.1.242)
In the case of forms, using properties (1.1.237a) and (1.1.238), one
obtains:
= {[A,fl-]d ♦ d [А]Д|]}Ш =
= ([А,в] Id + d[A,Bj |)w = £ w .
[A,B]
(1.1.243)
The extension of (1.1.239) to any tensor field X is then
obtained using the formal properties of on tensor products (Eq.
(1.1.232)).
Finally we prove the following identity which gives the link
between the exterior derivative on forms and the bracket operation on
vector fields:
dw(X,Y) = | {X(io(Y)) - Y(io(X)) - W(|X,Y|)}
(1.1.244)
where w is a 1-form and X, Y are vector fields on M.
As in the proof of Eq. (1.1.171) it is not restrictive to consi-
der 1-forms of the type
w = f dr.
(1.1.245)
where f and n are 0-forms.
The left hand side of (1.1.244) then gives:
do>(X,Y) = df ^dn(X.Y) = | (df(X) ^dn(Y) - dn(X) ^df(Y)) =
= | {x(f)Y(n) - X(n)Y(f)}
(1.1.246)
71
The right hand side on the other hand gives:
| (Xto(Y) - Yw(X) - o>([x,y])) =
= | X(fY(n)) - | Y(fX(n)) - | f[x,Y] (n) =
= | {x(f)Y(n) - X(n)Y(f)} . q.e.d.	(1.1.247)
Consider now a general tensor field X and a general infinitesimal
diffeomorphism generated by the vector A: the active change of X
is given by:
X(x) •> X(x) + 6+ X(x)	(I. 1.248a)
- <5+ X(x) = X .	(1.1.248b)
Therefore the condition that a field X be invariant under a diffeo-
morphism generated by X is:
«.•> X = 0 .	(1.1.249)
A
This in particular applies if the diffeomorphism is a general coordi-
nate transformation generated by
A = еУ Э	(1.1.250)
P
where
ey = x'y - xy = 6xy .	(1.1.251)
Let us consider X = g, where g is the metric tensor on M. Then any
-»
A such that
72
«,-*g = 0
A * 6
(I.1.252)
is a Killing vector and the associated diffeomorphism is called an
isometry. We shall develop the study of the isometries of a manifold
in Chapter 1.6.
APPENDIX
Appendix: the 6 operator and the Hodge decomposition
Inner product
Consider the p-forms	An inner product can be defined
as the integral
(cJ₽), B(p)) s [ a(P> . *B(P) •	(A.l)
JM
Since o/P) л *g(P) = (З^Р) л *cJP) (see 1.1.193), the inner product (A.l)
is symmetric. Moreover, (io,o>) - 0, the equal sign holding only if
io= 0.
Adjoint of the exterior derivative
Consider (c/P\ dB^P ^). Integrating by parts, we find
(d(p), dB(p_1)) = (6a(p), e(p-1))	(A.2)
where 6, the adjoint of d, is given by
6 = (-l)nP+n+1 *d*	(A.3)
(for spaces with negative signature there are additional (-1) factors).
Note that 6 is nilpotent:
73
<56о/р:) = О	(А. 4)
and that бс/Р^ is a (p-1)-form.
With 6 we can define co-closure and co-exactness:
6i/p) = 0 =» to(P] is co-closed
(p) e (p+1)	(p)
w r = oa r =* (jj f is co-exact.
Laplacian
2
A = (d + 6) = d6 + 6d	(A.5)
Ao/P) is a p-form. A is a positive operator (since (i/p\ Ai/P^) >0
cfr. V.4.97). Harmonic p-forms are defined by
Ai/P) = 0 =* Ш(Р] is harmonic .	(A.6)
It is clear that if dio = 0, 6io = 0, then to is harmonic. The converse
is also true. Indeed
(Aio,to) = (d6io,to) + (6dto,to) = (6ш,6ш) + (dto,dto) .	(A.7)
If Аш = 0 (i.e. if io is harmonic), then
(6ш,6ш) + (dio.dto) = 0 .	(A. 8)
Each term is nonnegative, hence (6to,6io) = (dto,dw) = 0, and this implies
dto = 0,	= 0.
Hodge's theorem
If M is a compact manifold without boundary, any p-form a/P^
can be uniquely decomposed as a sum of exact, co-exact and harmonic
forms:
74
JP) = da(p-D + 6S(P*D + y(p)	(A .9)
with = harmonic p-form. The proof that a, 8, у exist is
difficult, whereas the uniqueness part is easy to settle, using the
semipositivity property of the inner product (see for ex. Ref. [б]).
75
CHAPTER 1.2
RIEMANNIAN MANIFOLDS
1.2.1 - Introduction
We already anticipated (Sect. 1.1.6) that the Riemannian geometry
of a manifold Mn will be developed using the moving frame {e^} and
the dual vielbein (co)frame {V1}.
We are aware of the fact that a rigorous treatment of differential
geometry should be based on the theory of fiber bundles, particularly
for what concerns the theory of connections and many global questions.
However the essential idea of the Cartan "moving frame" approach to
Riemannian geometry is to reduce, as far as possible, problems of
Riemannian geometry to problems of linear algebra in vector spaces.
In this way it is possible to give a simple intuitive interpreta-
tion to a number of properties which are usually hidden, in the usual
tensor approach, under a plethora of indices. There are some subtle
points in the derivations of some important formulae which we will
neglect; these defects in rigour are however greatly compensated in
our opinion by the gain in geometrical intuition. (For rigorous and
complete treatments see the books by E. Cartan and H. Flanders).
76
1.2.2 - Geometry of the linear spaces
To illustrate how the method works we begin with the case of a
linear space ]Rn and then we extend the procedure to a smooth Rieman-
nian manifold M .
n
Suppose we have curvilinear coordinates {x^} on ]Rn ; the
tangent vectors at P to the lines x^= const, span the natural basis.
It is convenient to use the symbol P to denote the position vector of
P referred to some origin in ]Rn . Then the vectors of the natural
frame are given by
Fig. 1.2.1
Each vector at P can be expressed in terms of its local components:
in particular the displacement vector dP is given by
dP = dxP
Эх^
(1.2.2)
P
Instead of the natural basis (1.2.1) any other frame could work equally
well; in particular it is obviously convenient to introduce a set of
77
vectors (ej which are orthonormal with respect to the n-dimensional
Minkowski metric n— = (1,-1,... ,-1) :
e. • e. = n-
i J
(1.2.3)
(The choice of the signature	which, from a rigorous point
of view corresponds to the choice of pseudo Riemannian rather than
Riemannian geometries, is motivated by our final goal which is the
theory of gravitation. We omit all the time the "pseudo".s and we use
Riemannian for pseudo Riemannian following a by now well established
tradition). The frame {ej is called the moving frame: it is related
to the natural frame (1.2.1) by a non singular matrix vV (see Eq.
(1.1.182))
Vp. V1	= 6P	;	Vp.	Vj =6?	.
i v v	’	i у i
(I.2.4b)
Introducting the 1-forms (Eq. (1.1.176-177))
V1 = V1 dxP	(1.2.4c)
V
Eq. (1.2.2) becomes:
dP = dx^V1 VV.) -5-P = V*e. (P) .	(1.2.5)
p 1 3xv	1
According to (1.2.4), the set of 1-forms {v1} is the vielbein frame
dual to the moving frame {e^}: indeed
yi(t) = Vv.dxp(3v) =	.
(1.2.6)
78
Notice that dP is a vectorial 1-form whose components along the basis
e^ ® VJ are 6*; in other words dP is that vectorial 1-form which
gives the identity map of Tp(M) onto itself:
dP(e^) = e..
(1.2.7)
The relation between two infinitesimally close frames {e^} and
(<L + de^} is given by
de.
i

(1.2.8)
and since de^ is a vectorial 1-form we find:
d<L = - e(1.2.9)
where o1. is an infinitesimal matrix of 1-forms:
J
wji = wji|p dxP .	(1.2.10)
Differentiating the orthonormality relation <L« e. = and using
(1.2.9) one obtains:
d(e. • e.) = de. • e. + e. • de =
1 J	i j 1 j
к ->	-» к
= - (e,	• e.	cj .	+ e.	• e,	co .)	=
v к j i i к j'
= - (о. . + ш..) = 0 .
1J Ji
(1.2.11)
79
Therefore a?\ is an infinitesimal "rotation" matrix of SO(l,n-l); it
is called the spin connection.
The Lorentzian ErouP SO(l.n-l) emerges because of our choice of the
signature	In an arbitrary signature
Ш j turns out to be an SO(k,£) Lie algebra element and in particular
for the strictly Riemannian case (k=n, R,= 0) it belongs to S0(n).
We apply the d-operator to both sides of (1.2.5) and (1.2.9); the
2 _
integrability condition d =0 gives the following 2-form equations:
. UC JL .
R1 = dV1 - W1. AV] = 0
J
def .	.	.
R1. = dw1. - co1. . = 0
J J к J
(1.2.12a)
(1.2.12b)
The left-hand sides of these equations, R1 and R1^, are called the
torsion 2-form and the curvature 2-form respectively. In the ]Rn case
they are identically zero. This is so because in Euclidean spaces,
such as ]Rn , it is always possible at each point P to introduce an
orthogonal matrix В such that each moving frame {e.} can be aligned
c. , c	f+(0)i	1
a given fixed frame te? )
: = :(0)B
(1.2.13)
Then from (1.2.9) we get
w = - в 1dB
(1.2.14)
Eq. (1.2.8) says that the spin connection associated to the gauge group
SO(l.n-l), acting locally on the moving frame, is a pure gauge. Accord-
ingly, Eq. (1.2.12b) expresses the fact that the associated field
strength is identically zero.
80
Let us now consider a vector field v defined over a region of
Kn; referring it to the moving frame we have
i
v
v
e
(1.2.IS)
Using (1.2.9) we evaluate the change dv due to an infinitesimal
displacement:
dv = dv^ e.
J
v^aP. e. = (dv^
i J
i 11-*
co . vJ)e.
J i
(1.2.16)
where
.	-	. UCL
dv1 - u)1. v^ = © v1
J
(1.2.17)
is called the covariant derivative of v1.
1.2.3 - The geometry of general Riemannian manifolds in the vielbein
basis
We want now to extend the formalism developed for the almost
trivial case of ]Rn to general manifolds. Suppose we consider an
n-dimensional manifold M on which a metric g. has been defined;
n
according to the general definition given in Chapter I (see considera-
tions following Eq. (1.1.132)) is by definition a Riemannian
manifold.
In the same way as we did for lRn at each point P we set up an
orthonormal local reference frame {e^} spanning a basis of Tp(Mn):
(1.2.18)
81
where n.j is the Minkowskian metric on the tangent space. We insist
on taking only orthonormal frames since we are going to introduce spinor
fields on Mn and since they are SO(l,n-l) representations. Therefore
we are forced to restrict the set of affine frames at P, related to
each other by elements of GL(n,IR)
e! = e. Aj. A e GL(n,R)	(1.2.19)
to the subset of orthonormal frames related to each other by elements
of SO(l,n-l). In particular spinors cannot be described in the natural
frame {Э }. Indeed under a coordinate transformation the vectors Э
U	U
transform as (see Eq. (1.1.123)):
Э _ 9xv Э
9x'p ’ 9x'p 9xv
(1.2.20)
where the Jacobian matrix (9xV/9x,,J)p is, in general, an element of
GL(n,JR).
The relation between the moving (orthonormal) frame and the
natural one is obviously the same as in the Euclidean case i.e.:
e. = V? —	(1.2.21)
1	1 9xp
With and vV satisfying (1.2.4b).
From now on we use only moving frames. The relation with the
usual tensor formulation which utilizes the natural frame will be given
afterwards.
82
Proceeding now in analogy to the Rn case, we express an
infinitesimal displacement dP in terras of the moving frame at Tp(M):
dP = V1 1	(1.2.23)
where the V1 are the vielbein fields dual to the moving frame:
V1 = V* dxp .	(1.2.24)
They are a basis for the 1-forms on the cotangent plane at P.
—>-
The notation dP for the infinitesimal displacement is due to
the fact that (1.2.3) is not in general an exact differential since P,
contrary to what happens in the Euclidean case, is not a function of
the coordinates. The same remark applies to the evaluation of the
change of the moving frame under an infinitesimal translation P-’-P+dP:
de. = -e.oP. ,
i J 1
(1.2.25)
where as in the Euclidean case
(1.2.26)
Since dP and de. are not exact differentials we do not expect the
torsion 2-form R1 and the curvature 2-form R1-1 defined by Eqs.
(1.2.12) to be zero. Therefore on any manifold we introduce the
concept of torsion and the curvature 2-forms by means of the following
definitions:
. def
R1 E dV1 - л Vj	(1.2.27a)
. . def . .	.	. -
R1J E dw1J - </k л w J .	(1.2.27b)
where E. In general R1 and R1^ have non vanishing values.
83
The metric tensor on is given in terms of the vielbeins by
Eqs. (1.1.187-188), or equivalently it is defined by taking the square
of (1.2.23):
(dP)2 = ds^ = V*e. ®	V^e.	= 1Л	dx^ ® 0	dxVe. • e.	=
i J у	v i j
= n. .V1 Vj dxP ® dxV .	(1.2.28)
1J у V
Therefore:
g = V1 Vj n..	.	(1.2.29)
6yv у v ij
We notice that the structure equations (1.2.27) could also be retrieved
in the following heuristic manner. Let us take the exterior derivative
of both sides of Eqs. (1.1.23) and (1.1.25). We get:
d(dP) = dV1^ - V1 d^ = (dV1 -	~ Vj)e.	(1.2.30a)
d(de^) = - de. • ar\ ~ ®j " j =
= - ^.(dlA - Jk ~ Шк.) .	(1.2.30b)
where the 2-form components along the {e.} frame define again the
torsion and (minus) the curvature.
The reason we call this derivation heuristic is that in the
differentiation we have substituted the d-operator by d when acting
on the moving frame. Using the same rule for the differentiation of
* _
e. • e. = n • • we get:
1 J 4
d(e. • e.) = de. • e. + e. • de. =
i J 1 J 1	3
= - e, • шк.е. - e. • e,wk. .	(1.2.31)
к i j i к j
84
that is:
wij = - Wji •	(1.2.32)
Therefore our heuristic arguments tell us that the connection co1,
belongs to the Lie algebra of SO(l,n-l) as in the JRn case. In the
sequel we assume the validity of (1.2.32). In this case co1.. is called
a spin connection. Differentiating both sides of Eqs. (1.2.27), and
2
using d =0, we get the following integrability conditions:
dR1 = - dco1 . „ Vj + co1. л dV-i =
3	3
=	- (R1. + w1 л <Л) л Vj + co1. л (Rj + coJ. л Vk) (1.2.33a)
J K J	J	K
=	- r\ л + W1 л R^	(1.2.33b)
,Di , i к i , к ,„i i S, . к
dR . = dco k л co j - л dco j = (R k * и л co p „ co . -
i ._k к Я. .
-co. я (R . + co . A co .) =
к J Я. j
= R\ л Л - co\ „ Rk. .	(1.2.33c)
Equations (1.1.33) are referred to as the Bianchi identities obeyed by
the curvatures R1 and R1..
Let us observe explicitly that all the equations introduced so
far are exterior equations and as such they are scalars under diffeo-
morphisms, according to Eq. (1.1.160). Latin indices are inert under
diffeomorphisms being indices of the local gauge group S0(l,n-l).
The same is true if we expand co1., R1 and R1. in a local
4	3	J
cotangent basis {V }:
co\ = (Л|к Vk	(I. 2.34a)
R1 = R\l ук ~ V*	(I. 2. 34b)
R\ = R1j|k£ Vk » Vя .	(1.2.34c)
85
Indeed the component
Latin type and hence
fields w1.]., R1,	R1.).»
j|k*	k£’ j|k£
inert under diffeomorphisms.
have indices of the
Let us collect our results: we started with a Riemannian manifold
M endowed with a local (orthonormal) moving frame and its dual {V1}
n	i
in the cotangent plane. The frame {V } is acted on by the local gauge
group
SO(l,n-l). We also introduced a local connection 1-form co1..
The local geometry is described by:
i)
Structure equations
d? = V1 e.
= - ®/i
R1 = dV1 -co1. л
J
R^ = dcoij - (A .
к
(I.2.35a)
(I.2.35b)
(1.2.35c)
(I.2.35d)
ii)	Bianchi identities: i.e. the integrability conditions of the
structure equations:
dR1 + со1. „ R^ + R1. -	= 0	(1.2.36a)
J	J
dR1. - R\ л cok. + co1. л Rk.	(1.2.36b)
J к j к J
iii)	The metric postulate
(0ij = - c?1	.	(1.2.37)
If one further assumes:
iv)	R1 = 0	.	(1.2.38)
then M is a (Riemannian) manifold with a Riemannian connection.
86
In this case one can express the spin connection
vielbein field. Indeed let us expand the 1-forms
dV^ along the V1-basis:
in terms of the
and the 2-form
dV1 = c1.. Vj л Vk
(I.2.39a)
(I.2.39b)
Inserting in (1.2.38) we get:
;ijk = I 4k|j -“W	t1-2-40)
where we have lowered the upper index with the metric •
Adding and subtracting the two equations analogous to Eq. (1.2.40),
but with ijk indices circularly permuted one obtains:
co.. i, = c. .. + c.. . - c. . .
ij|k ijk jki kij
(1.2.41)
where we have used Eq. (1.2.37).
If one wants to express the spin connection in terms of the space-
time derivatives of V1, it is better to expand Eq. (1.2.38) in the
coordinate basis {dx'1}:
3 r v\	= 1	(co1. । Vj	- to1 .	i Vj)	.
lu vj 2 J |u v j P
(1.2.42)
Converting the tangent index into a world index by multiplication with
к
n.,V = V. , we obtain:
ik p ip’
,,k i 1 . i ,,i,,k i ,,j,,k.
lbvV Эг V 1 = — (w .1 V V - co . i V-’V )r).,
ik p [ц у] 2	V P iNu P ik
(1.2.43)
This equation can be solved as Eq. (1.2.40) by permuting the yup
indices. We get
where
f. ।	= vt3r V-’in- •	-	(1.2.45)
x|yv x Ip vj 'ij
Let us now explore the local gauge invariance under the local Lorentz
group SO(l,n-l).
Suppose we perform an SO(l,n-l) gauge transformation on the local
frames
e! = е.л\ A e SO(l,n-l) .	(1.2.46)
i J i
From
dP = e.V1 = e.'V11	(1.2.47)
1 1
we obtain:
V4 = (A"1)1^	.	(1.2.48)
Then from
de' = - e’w’	(1.2.49)
(where we use a matrix notation) using (1.2.25) and	(1.2.46), we have:
- ewA + edA = - е/ш'	(1.2.50)
and therefore we can write
88
w' = - Л'1 (dfli- w)A
=» w j = (Л )кы£Л.-(Л ) k(dA) .	fl.2.51)
The result is that the spin connection w undergoes an SO(l,n-l) gauge
trans formation.
One easily finds that the torsion 2-form R1 and the curvature
2-form r\ transform in the vector and in the adjoint representation
of SO(l,n-l) respectively:
R 1 = (A-1)1 Rj	(1.2.52a)
R1j =	R\ ‘	(1.2.52b)
Next we compute the change of a vector
v = v1 1	(1.2.53)
under an infinitesimal displacement. Differentiating both sides of
(1.2.53) and using (1.2.25), we find:
dv = e^ (dv1

(1.2.54)
Hence we define the SO(l,n-l) covariant exterior derivative of v1 by:
. def
@ V1 H dv1
1 1
w .VJ
J
(1.2.55)
(In the following it will be referred to as the Lorentz covariant
derivative).
Indeed taking into account Eq. (1.2.48,51) ©v1 transforms in the
same way as v1.
We can extend the notion of covariant derivative to any tensor-
il^2'’’
valued p-form Ф . . as follows:
J1J2---
89
’p2‘”	^p2'“ k Зр2’“
к 1р2" ’ ’
Ж	- Ш ф .	- ...	(1.2.56)
*J2
and verify that this is indeed a SO(l,n-l) covariant derivative.
Ж	As we have already pointed out one can also introduce p-form
,^K. fields which are in spinor representations of the tangent group
SO(l.n-l). Let a be one such field in the lowest spinor representa-
tion and let
f rij = | K’rjJ	(1.2.57)
be the Lorentz generators in the spinor representation, with Г1 Dirac
у-matrices for SO(l,n-l). Then
S Sa = da - -w.. л Гх’а	(1.2.58)
Ж	4 13
S is the covariant derivative of the spinorial p-form a.
"	This can be easily checked using (1.2.51) and
_ 1	cd
Ж	L Г . L = Г , Л A .	(1.2.59)
Ж	ab	cd a b
3F where L and Л are elements of SO(l,n-l) in the spinor and vector
Ж representation respectively. Quite generally for a p-form A with
Ц indices in any representation D of SO(l,n-l) the covariant deriva-
sj; tive is defined by:
4*	def
ЭА = dA + wXJ л D(T )A	(1.2.60)
where D(T^) is the representation of the generators T„ of
S0(l,n-l). We shall come back to this general formula in Chapter 1.5.
Using the Lorentz covariant derivative the torsion 2-form is
rewritten as follows:
R1 = SV1
(1.2.61)
{	90
j	and the Bianchi identities (1.2.36) become:
I''
= 0	(1.2.62a)
&R1 + R1 A = 0	(1.2.62b)
f	Let us make the symmetries of the intrinsic curvature tensor R1.!,.
j|k«.
explicit; from Eq. (1.2.34c) one immediately gets
R1j|n = - R1j|zk	П-2-63)
and from the metric postulate (1.2.37):
Rij |k£ = - Rji|k£ ‘	t1-2-64)
Furthermore, when w1\ is a Riemannian connection, that is when Eq.
(1.2.38) holds, from (1.2.62b), we get:
R1j a Vj = 0	(1.2.65)
Expanding along the vielbein basis we find
|UVj л Vk л V* = 0	(1.2.66)
which gives the cyclic identity:
Rij|k£ * Rik|£j + R\|jk = °	C1-2’67)
By repeated use of Eqs. (1.2.63), (1.2.64) and (1.2.67) one also
derives:
91
Rij |kJ, " Rk£|ij Rij|kf. + Rki|j£ + Rkj |£i
Rij |kJ. + Rik|£j " Rjк|Jti ~ RiSl|jk Rjк|Я1
= RS.i | jk * (Rjj|ik * Rji|kP =
= (RJi | jk + R«.j |ki] + Rji|k£ °
R£k | i j + Rj i | k£
= - Rij|k£+ RU|ij ’	(I-2’68)
Therefore:
RijM = Rk£|ij •	(I2-69)
From R\ one may construct the Ricci tensor
def
R1.,.,	= R..	(1.2.70)
j|ik jk
which turns out to be symmetric in j,k, and the curvature scalar
def
n1J = R •	(1.2.71)
Because of the aforementioned symmetry properties any other contraction
possibility gives at most a change of sign with respect to definitions
(1.2.70) and (1.2.71).
1,2.4 - Relation with the standard world-tensor formalism
Up to now all the fields defined on Mp have been expressed in
terms of their components along the vielbein basis so that all the
92
indices transform linearly with respect to SO(l,n-l). Of course it is
also possible to use the natural frame {Эр} in the tangent plane and
its dual {dx*1}. In that case we get the description of the classical
tensor calculus where all the indices transform covariantly with respect
to a change of local charts.
We give briefly the expression of all the geometrical tensors in
the natural frame and their relation to the same objects in the intrinsic
frame. In the natural frame Op = ep) the relation between two infinite-
simally close frames is given by
d? = e Гр	(1.2.72)
where*
ГЦ = ГЦ । dxP	(1.2.73)
v v|p
is called the affine connection since it makes the transition between
two frames {Э } and O'} related by an element of GL(n,R).
Proceeding as before we can define the torsion 2-form R and the
curvature 2-form RPy by the replacement
V1 > dxp	(1.2.74)
w1. > - ГЦ	(1.2.75)
jv
Accordingly in the natural frame the torsion is:
Ry = d(dxy) + rUv „ dxV = - ryv|p dxV л dxP .	(1.2.76)
* Notice the change of sign in (1.2.72) with respect to (1.2.25) in
order to adhere to the usual conventions in the world tensor
formalism (see e.g. Ref. 17).
93
The	antisymmetry condition (1.2.37) becomes: dg , = d(e • e ) = de • e + e -de = 6UV	U \>	p v	p	v = + e • ГР e + e • e Гр	(1.2.77) P P v U P v
that is:
	- '"Av - r₽Ap ’ °
which is the metric postulate in the coordinate basis.
	On the other hand the condition Rp=0 defining the Riemannian
connection, upon use of Eq. (1.2.76), yields the symmetry properties of
the	Christoffel symbol
	f p л гц = ГР = 4	(1.2.79) Vp pV I vp J
where
	{"} - J «““(’’Ар *	* Map’ '
Eq.	(1.2.80) is obtained from (1.2.78) in the same way as we obtained
(1.2.41) from (1.2.39).
Finally the curvature 2-form becomes
	RP = drp + ГР л ГР .	(1.2.81) V	\)	p	V
Expanding along the natural basis we retrieve the definition of the
Riemann curvature tensor
	ru । = - {a rp - a ru * rp rx - rp rA } .	(1.2.82) \>|po	2 1 P vo о vp Ap vo Ao vpJ
94
Let us now perform a change of local chart; in local coordinates
•i i<- 1 n.
x = x (x ,... ,x )
(1.2.83)
Recalling our discussion of vector fields and forms, the natural frames
and coframes transform as follows (Eqs. (1.1.123) and (1.1.161)):
Э' = (J-1)V Э
p k 1 p \)
,. p Tp . \)
d'x = J dx
where
= -^ £ GL(n,K) .
(1.2.84a)
(1.2.84b)
(1.2.85)
Eq. (1.2.84) are analogous to Eqs. (1.2.46) and (1.2.48). Therefore a
coordinate transformation induces a local gauge transformation of
GL(n,R) on the basis frames. With the same procedure used in deriving
Eq. (1.2.51) from Eq. (1.2.49) one finds the transformation of
under a change of local chart:
(Г')Уу = + J(dD ♦ DJ-1 .	(1.2.86)
Correspondingly we have:
R,y = RV	(1.2.87a)
R,p = Jp RP (J"1)0	(1.2.87b)
v p О V
Proceeding in the same way as from Eqs. (1.2.53) to (1.2.55) one finds
for the GL(n,R) covariant derivative of a vector field:
95
(Vv)u = dvu + Fpv Vх1
(1.2.88)
The extension to a general (к,X.)-tensor field is:
U
VA
1" ’uk
Vr--v«.
• V
Pr--Uk
A 1 K
vr.
P1 W2---1Jk
+ Г A
p
vr-'vx
A

(1.2.89)
We stress that we cannot introduce spinor fields in the GL(n,K)-
covariant basis {Э }.
Let us now observe that the formula (1.2.24) giving the change of
frame from the natural to the intrinsic basis can be thought of as
induced by a coordinate transformation from a general basis dxP to
an orthonormal basis V1, V1 being an element of GL(n,]R). Conso-
le	i
quently the relation between the spin connection co . and the affine
u
connection ГH	is given by the law (1.2.51):
Гуу = - V^tAv^ * V^.dV^	(I.2.90a)
(Л = - ViyryvVVj - VivdVVj .	(1.2.90b)
Multiplying (1.2.90a) by we find:
dV1 - lAv* - V1 ГЦ = 0 .	(1.2.91)
v j V p v
Taking into account Eqs. (1.2.55) and (1.2.88), Eq. (1.2.91) can be
interpreted as the vanishing of the combined Lorentz and GL(n,F)
covariant derivative on V1 .
An analogous equation follows from (I.2.90b) for the inverse
vielbein.
Finally we observe that the relation between objects evaluated in
the coordinate basis and in the intrinsic basis is given by the inter-
96
twining vielbein matrix V1^: every coordinate index can be transformed
into an intrinsic one by V1 :
U
A1’" = V1^ Ap"‘	(1.2.92)
and vice versa
Au--- = VP£ A1’"	(1.2.93)
In particular
A\ = nijAiAj =	= gyvAPAV = APAp .	(1.2.94)
where we have used Eq. (1.2.29).
Therefore coordinate scalars are also Lorentz scalars and vice
versa.
Other useful relations are the following ones:
0 A1 = VAP
VAU = VU. SA1
i
(I.2.95a)
(I.2.95b)
where Q and V are the covariant derivatives in the tangent or
natural frames respectively.
Eqs. (1.2.95) can be proved by direct computation using Eqs.
(1.2.90). Notice that the affine connection entering (1.2.95) is
symmetric in its lower indices which implies that the torsion Ra is
zero.
Therefore (1.2.95) is not true in presence of a nonvanishing
torsion.
97
CHAPTER 1.3
GROUP MANIFOLDS AND MAURER-CARTAN EQUATIONS
1.3.1 - Introduction
In this chapter we discuss Lie groups from a differential geometric
point of view. As in previous chapters we just give those main defini-
tions and properties which are relevant for the subsequent developments;
previous knowledge of group theory is required.
The chapter is divided in two parts; in the first (Section 1 to 6)
we concentrate on the study of those properties which are peculiar to
group manifolds, like the existence of left- and right-invariant vector
fields or 1-forms. This leads to the discussion of the Lie algebra
associated to Lie groups and to the dual concept of Maurer-Cartan equa-
tions. Within the same framework we shortly discuss the adjoint and co-
adjoint representations of groups and algebras and the Killing metric;
finally a short account is given of the Riemannian geometry of semi-
simple group manifolds.
The second part of this chapter is devoted to the study of mani-
folds which are locally diffeomorphic to group manifolds. They are
98
obtained by softening the rigid metric structure of the group manifolds
in the same way as the manifold of the translation group Rn can be
locally softened to a general Riemannian manifold.
Manifolds obtained in this way are named soft-group manifolds.
In particular we discuss the process of factorization of the curvatures
which gives rise to the fiber bundle structure of the manifold.
The last section of the chapter is devoted to a detailed study of
the important cases of (anti)-de Sitter and Poincare soft group mani-
folds.
1.3.2 - Lie groups as manifolds: left and right invariant vector fields
A group G is a Lie group if it is a smooth manifold and if the
map
G x g + G	(1.3.1)
defined by
(x,y) = xy x,y e G	(1.3.2)
and the inverse mapping
G * G	(1.3.3a)
defined by
x-/'1	(1.3.3b)
are both smooth.
In particular if a is a fixed element of G, then the left
translation:
L : G * G	(1.3.4a)
2.
99
kjg) = ag	(I.3.4b)
and the right translation
R : G * G a	(I.3.5a)
Ra(g) = ga	(I.3.5b)
are diffeomorphisms.
From the associativity of the G multiplication
(ax)b = a(xb)	(1.3.6)
it follows that the left and right translations commute
[La,Rb] = 0	(1.3.7)
Let us consider the tangent space Te£G) di ffeomorphi sm e + ge : g	,	g £ G	at the identity e. The (1.3.8)
induces a map between Te(G) and T^(G) v = (L ) v g	g * e	according to (1.1.123); (1.3.9)
The vector field obtained in this way is (1.1.125) and (1.3.9):	left-invariant. Indeed using
(L ,) v = (L ,) (L ) v = (L , g’ * g	g' * g * e	g'	V» = V„.	-	(1.3.10) g * e g g
This shows that the functional form of as that of v at g. g	(Lt)* vg at g'g is the same
100
In an analogous way one can show that the vector field
vg = CRg}* ve	(1.3.11)
is right invariant.
Since the left- (right-) translation is a diffeomorphism, by
taking into account property (1.1.216), one sees that the subset of
left- (right-) invariant vector fields is closed under the Lie bracket
operation.
Hence the left- (right-) invariant vector fields form a Lie
algebra.
Definition: The Lie algebra, E, of the left- (right-) invariant vector
fields on G is called the Lie algebra of the group G. As a matter of
convention in the following we shall mainly refer to left-invariant
vector fields.
Since any left-invariant vector field is uniquely determined by its
value at e, the identity element of G, G can be identified with
Te(G).
Let us introduce a basis, Тд, A=l,... dim(G) on Te(G): then
tTA’TBJ = САВСТС	(I’3’12)
c
where the CAD are constants. Indeed, since the left hand side of
(1.3.12) is left-invariant, the same must be true for the right hand
C
side; this inplies that the C._ are left-invariant, that is constant.
C
The Сдв are called the structure constants of the Lie algebra of G.
Actually the presence in Eq. (1.3.12) of structure constants instead of
structure functions is what characterizes the Lie algebra of left-
invariant vector fields on G.
From the Jacobi identity of vector fields:
+ [TB>[TC’TJJ + МТА’ТВП = 0
(1.3.13a)
101
one finds:
A	R
c b[c clm] =°	(1.3.13b)
which is the Jacobi identity for structure constants.
We now show that the left- (right-) invariant vector fields are
the generators of right- (left-) translations and that each generator
is in a one-to-one correspondence with the one-parameter subgroups of
G. Let us consider a one-parameter subgroup H of G, that is the
homomorphic map
R * H c G .	(1.3.14)
Let IR be parametrized by t and g by the n coordinates x1. Then
the right translation
RH(t): g * gH(t) = g’	(1.3.15)
is a flow (1.1.203)
x4 = ^(t.x)	(1.3.16)
where x' are the coordinates of g'.
According to the definition (1.1.207), there is an infinitesimal
associated to the flow whose components are given by
^(t.x) = ^(O.x)	.	(1.3.17)
t=0
generator t1 '
tCR)i = A
dt
It is easy to see that this vector is left-invariant: indeed if aeG,
associativity of the group composition law in^lies:
La[g-H(t)] = (ag)H(t)
(1.3.18)
102
Непсе, if we parametrize the left translation
g * ag = g'
by
(1.3.19)
z = z(x,y)
(1.3.20)
where x,y and z are the coordinates of a,g and g' respectively, the
value of the infinitesimal generator at g’ is:
(L ) t (R) = H1(0,z(x,y)) — e t ,(R)	.
a * g	Sz1 8
(1.3.21)
The last equality expresses the fact that the functional form of the
components of
(1.3.17). In
with the left
(L ) t
a'* g
other words
is the same
„ (R) •
t is
as the one of t given in Eq.
left-invariant. If we had started
action of a one-parameter subgroup HcG.
g
g * H(t)g = g'
(1.3.22)
the corresponding generator t^ would have been right-invariant.
Hence we have shown that to each 1-parameter subgroup H of G
there corresponds a generator of right- (left-) translations and the
associated vector field is left- (right-) invariant.
It follows that the Lie derivative along the generator of right
translations, t^R\ of the right-invariant vector field t^^ is
zero
t, fD.t^ = [t^,t^] = 0
11RJ
(1.3.23)
(Actually (1.3.23) is the infinitesimal form of (1.3.7)).
As an example let us consider the group
G = GL(n, R)
(1.3.24)
103
The coordinates of a given element geGL(n.JR) can be taken to be the
entries x1J of the matrix g. The tangent space at the identity,
Te(G), is spanned by the basis vectors
I-	X. . [ 13	= -Лт	(1.3.25) Эх13 e
I so that a	generic vector T at e can be written as
f	T =	T1J —iL.	.	(1.3.26) Эх13 e
The Lie algebra of the tangent vectors at the identity (1.3,25) is thus
isomorphic to the Lie algebra Mn(R) of the nxn matrices T1J.
In particular, the mapping:
t * exp tT E H(t) , T e Mn(R)	(1.3.27)
yields the one-parameter subgroup of GL(n,R) whose infinitesimal
generator is given by Eq. (1.3.26).
If x*j(t) denote the entries of the matrix etT then
xij(t) = 6ij + tTij + 0(t2)	.	(1.3.28)
f Hence the	components of the infinitesimal generator T1J are given by
f	T!j	= — xi3	.	(1.3.29) dt lt=0
Subalgebras of Mn(R) give rise to subgroups of GL(n,R). For instance
if
T = - T1	(1.3.31)
104
(i.e. T is antisymmetric) then the corresponding 1-parameter subgroup
lies in S0(n).
In the following we shall often use the same notation T for both
the tangent vector (1.3.26) and for the matrix of its components:
{T1-*}=T, especially when we deal with a basis Тд, (A=l,...,dim G).
When confusion can arise, the vector field T will be written with an
arrow: T.
I.3.3 - Maurer-Cartan equations
Now we introduce left-invariant 1-forms on a group-manifold G
and we give a formulation of its Lie algebra in terms of 1-forms.
Let us consider again the diffeomorphic map L (1.3.8). On
* S
-forms co we can define the pull-
L .co = co
g-i e g
where
, -1	-1
L ге = g e = g
g
Writing the mapping
h -»• g"1 • h
in coordinates
z = z(x,y)
where x are the fixed coordinates
of h, by setting
coe = (^(x.Ojdy1
map L i (recall Eq. 1.1.
(1.3.32)
(1.3.33)
(1.3.34)
(1.3.35)
-1 .
g and у are the coordinates
(1.3.36)
105
we obtain
w = co. dz^ .
8	1 3zJ
(1.3.37)
The 1-form	w is left invariant, g	Indeed
w ag	L	= L	w = ,	x-1 e	-1 -1 e (ag)	g a	L* L* ,co = L*	(1.3.38) -1 -1 e	-1 £ a g	а Б
where we have used property (1.1.169). Thus a field of left-invariant
1-forms is completely determined by its value at e.
Let us take a basis of left-invariant 1-forms at Te*(G):
{oA}: (A= l,...,dim G) .	(1.3.39)
Taking into account property (1.1.171) we have
* A * A A
L do = dL о = (do )	.	(1.3.40)
a
д
Since do is also left-invariant we may expand it in the complete
basis of 2-forms at e:
, A _	1 „А	В С	ст z ли
def _ — C Dл о	(1.3.41J
2 be
д
where the С functions being left invariant are actually constants.
Equations (1.3.41) are called the Maurer-Cartan equations for the left-
д
invariant 1-forms о . The content of the Maurer-Cartan equations
(1.3.41) is completely equivalent to that of equations (1.3.12).
Indeed, Eqs. (1.3.41) provide the dual formulation of the Lie algebra.
To show this, let us introduce the basis of left-invariant vectors
T dual to the cotangent basis oA of the left-invariant 1-forms.
oA(T (R)) = 6A
D	D
(1.3.42)
106
fR)
The label R is a reminder that ’ generate right translations on
G; for notational simplicity in the sequel we omit the label R.
Evaluating both sides of (1.3.41) on the vectors Т^,Т№ we get:
d<ATM,TN) = - | CABCOB . 0C(TM.TN) -	(1.3.43)
Using Eq. (1.1.244) one obtains
Лл = \ tTMaA(TrP - TN°A(TM^ -	=
= - I cABC°B - °C(W •	C1-3-44)
A	A
Since T^o (Tjj) -	(T^) = 0 because of Eq. (1.3.42), we have:
°A([TM’TN^ = cAmn	C1-3-45)
and therefore
Г	T A
TU,TM] = С“т.	.	(1.3.46)
L M NJ MN A
In this way we also see that the constants entering the Maurer-Cartan
equations are the structure constants defined by the Lie algebra.
In particular we notice that the Jacobi identity for the structure
constants, Eq. (1.3.13), can be retrieved from the integrability condi-
2
tion d = 0 of the Maurer-Cartan equations. Indeed, taking the exterior
derivative of both sides of Eq. (1.3.41), one obtains
d(daA) = 0 = - | 2 CA doB л oC =
2 dl
A rB L M C
C BCC LM° л ° ~ °	'	(1.3.47)
107
Taking into account the antisymmetry of о л о „ a we get
C*B(C c"LMj  0	(1.3.48)
that is Eq. (1.3.41).
1,3,4 - Adjoint representation and Killing metric
From the commutativity of the left and right translations on G,
(Eq. (1.3.7)) one easily deduces the commutativity of the corresponding
mapping between tangent and cotangent vectors:*
[L ,R *1=0	(I.3.49a)
L a* a J
= 0	.	(1.3.49b)
In particular if w is a left invariant 1-form then
L (R^w) =	(1.3.50)
*
that is, R^u is also left-invariant. The same is true for left
invariant vector fields X
La*Rb*X = Rb*X	(I-3’51)
and in general for any left-invariant (k,£) tensor on G.
Let us consider in particular the inner automorphism
I: G ->• G	(1.3.52)
given by
* (L ) is now written as L .
a *	a*
108
г	.	V-1
I: х -> bxb	(1.3.53)
where b is some fixed element of G.
On the Lie algebra generators Тд e ffi, I induces the mapping
W = 4* *1 TA = *1 TA •	(I-3-54)
D * Ь *
This is a representation of G on the vector space (E and is called
the adjoint representation of G. It will be denoted by Adj(b).
In coordinates

is the Jacobian of the map (1.3.53) evalu-
ated at e.
As an example we take G=GL(n,R). Using as coordinates the
entries x1-’ of g e GL(n,F), the map (1.3.53) becomes:
Adj(b): x'1’ = blk xk£(b
whose Jacobian is
.	Эх'1-’!	. ik,.-l.fj
J = VkT = b (b >
Эх 'e
Therefore if
(1.3.55)
(1.3.56)
T = T1^
3
3xlj
(1.3.57)
e
is a generator of the Lie algebra, the components of Adj(b)T are
given by
T,lj = blk(b’1)5yTM	(1.3.58)
that is, in matrix notation
t' = bTb-1	.	(1.3.59)
109
If b is given by a one-parameter subgroup
b = etB	(1.3.60)
then the corresponding representation for the Lie algebra ffi:
Adj(B): E + (E	(1.3.61)
can be obtained using the analogue of Eq. (1.3.17) applied to the
automorphism (1.3.59); one finds:
Adj RA = — Adj (etB) AI	= — (etBAe'tB) I =[b,A] . (1.3.62)
B dt	lt=0 dt	lt=o
If A and В are two basis generators of E:
AST. ; В = T_	(1.3.63)
A	D
then
“Va = fTB’TJ = CLBATL ’	(I-3‘64)
Hence
.	O.3.6S)
In an analogous way one can represent the automorphism (1.3.53) on the
left invariant 1-forms oB. In that case one deals with the coadj pint
representation of G in <E. The coadjoint representation of the Lie
algebra acts on forms rather than on vectors and can be defined by
[coadj (T )]oB(T ) = OB(Adj(TA)Tc) = oB([TA,Tc]) =
def
=	= CBAC '	(I-3’66)
110
Therefore
coadj(Тд)оВ = СВдсоС .
(1.3.67)
We also notice that the adjoint and the coadjoint representations give
the change of the left-invariant vector fields and 1-forms under the
Lie derivative along the left-invariant generators T,Indeed,
(R)	Л
recalling that Тд generate right translations and using definitions
(1.1.223) and (1.1.227), we obtain
\TB = MiJ = cLabtl = AdWTB	C1-3-68)
\°B = <2A|d + d2Ap°B =	+ d<) =
= - I ^LM^V' = V = - coadj (TA)oB .
(1.3.69)
A set of independent 1-forms, i.e. a cotangent basis on G, can be
obtained in terms of the group element g. Consider the 1-form:
-1,
O = g dg •	(1.3.70)
This 1-form is left-invariant. Indeed, under translation through a
fixed element beG we have
* -1 “1 “1 -1
4,° = (bg) d(bg) = g b bdg = g dg = о .	(1.3.71)
111
Differentiating both sides of Eq. (1.3.70) one obtains
da = dg"1 л dg	(1.3.72)
and using
dg-1*g = " g-1dg = - <7	(1.3.73)
one obtains
do + О л O = 0 .	(1.3.74)
We notice that
<7 = g-1dg	(1.3.75)
is a Lie algebra valued matrix of 1-forms and therefore can be expanded
along the set of generators Тд (in their matrix realization):
a = aAT A .	(1.3.76)
This can be proved by evaluating g ^dg at the origin e. The 1-forms
д
о span a cotangent basis.
Introducing the expansion (1.3.76) in (1.3.74) and using (1.3.12)
one obtains again the Maurer-Cartan Eqs. (1.3.41). In a matrix repre-
sentation of G, Eq. (1.3.74) is a matrix equation for a set of dim G
linearly independent 1-forms, and can be used to explicitly compute the
structure constants of G.
As an example let us derive the Maurer-Cartan equations for the
Poincare group in D dimensions (the group of rigid motions in D
dimensions).
As in the four dimensional case it is defined to be the semi-
direct product of the Lorentz group, SO(1,D-1), with the D-dimensional
translations and will be denoted by ISO(1,D-1). It can be realized as
the group of (D+l) x (D+l) matrices g of the form:
112
/ Л	С \
g =	|
\ 0	1	(1.3.77)
where Л is a matrix of SO(1,D-1) in the vector representation and
C eIrD, the group of translations in D dimensions (with the vector
addition as composition law).
The inverse of (1.3.77) is given by:
-1
g
(1.3.78)
so that the matrix
g dg of left-invariant 1-forms is:
-1,
g dg
V '
0	(1.3.79)
where we have defined
w = - A-1dA = dA-1A
V = A-1d?
(1.3.80)
(1.3.81)
By differentiation we immediately see that
dw = w л w
dV = - A-1dA „ A-1d£ = ш л V
(1.3.82)
(1.3.83)
that is we obtain the Maurer-Cartan equations of ISO(l.D-l).
To arrive at the usual formulation of the Lie algebra of
ISO(1,D-1) in terms of commutation relations among generators, we
introduce the dual tangent vectors Тд:
113
Т.: {Р ;J , }
А 1 a abJ
(1.3.84)
Р and being respectively the generators of the translations and
of the Lorentz transformations; by definition of dual algebra:
Va(Pb) = 6b ; Va(Jc(J) = 0	(1.3.85)
wab(J ,) = 6ab ; wab(P ) = 0	.	(1.3.86)
cd cd	c
Evaluating Eqs. (1.3.83-4) on the couples of vectors
{Pa,Jbc) and ^ab’^cd^ respectively and using Eq. (1.1.244) we
obtain the commutation relations of the Poincare group in D-dimensions:
[P ,P.l = 0	(1.3.87a)
L a bJ
|j l.J j] = - — (J Tk, + dvjP - J jHv - J. П j) (1.3.87b)
L ab’ cdJ 2 ac bd bd 'ac ad be be ad'
[j . ,P ] = - - (Г) P, - m PJ -	(1.3.87c)
L ab cJ 2 ac b be a
1.3.5 - Killing metric
Now we introduce a metric on G which is biinvariant, namely it
is both left- and right-invariant.
Suppose we take a metric 4>e at Te(G):
Ф: X ,Y -> ф (X ,Y ) £ R .	(1.3.88)
e’ e ev e’ e
By left translation we determine a metric field on G

(1.3.89)
114
As in the case of 1-forms one easily shows that Ф^ is left-invariant:
LAg = * -1	= 4>g  •	(1.3.90)
6 a ag e
Analogously R _^Фе is a right-invariant metric,
g
In order to get a biinvariant metric we need:
ЕХ-1Фе = Adj(gHe = *e •	(1.3.91)
To fulfill this requirement one defines the metric on Te(G) by the
so-called Killing form:
g(xe>\J = TrCAdj (Xe) Adj(Ye)) .	(1.3.92)
This form is obviously bilinear and symmetric and therefore defines a
metric on Te(G). Moreover it is biinvariant; indeed, using the cyclic
property of the trace and recalling Eq. (1.3.89), one has
Adj(a)g(X Y ) = Tr(a Adj (X )a4 a Adj (Y Ja'1)
c c	c	e
= g(Xe>V •	(1.3.93)
If we take X = T.. Y = Tn then
e A’ e В
Adj (TA) -Adj (TB) (Tc) = Adj(TA)CLB(.TL =
= Valtm •	f1-3’94)
Hence
®AB = «W = Tr(Adj(TA) Adj (TB)) = CLbmCMal .	(1.3.95)
115
If the Killing form is non-degenerate the group is said to be semi-
simple. For compact groups one can prove that gAB is negative
definite. To see what is the implication of biinvariance of g let
tA	A
us rewrite Eq. (1.3.93) using a = e : one has
Tr(etAAdj(Xe)e'tAetAAdj(Ye)e'tA) = Tr(Adj (Xe)Adj (Ye))	(1.3.96)
Differentiating with respect to t at t = 0 one obtains:
Tr([A, Adj (X )Adj (Y )] ) = Tr([A, Adj (X )]Adj Y + Adj (X ) x
X [A, Adj (Ye)]) = 0	(1.3.97)
that is
+ S^e’^el) = °	•	(1.3.98)
Taking A=T , X =T Y =T we obtain
A e о e и
C ABeLC + C ACgBL = °	(1.3.99)
Therefore defining
r = n cL	(1.3.100
CABC gAL BC
one obtains
С + C = 0	.	(1.3.101
ABC ACB
Taking into account the antisymmetry of C AB in A, В Eq. (1.3.101)
implies complete antisymmetry of the lowered structure constants
(1.3.100).
116
Finally we note that for semisimple groups the Killing metric
can be used to lower or raise the indices of the Lie algebra; in parti-
cular the adjoint and coadjoint representations of the algebra are
equivalent.
1.3.6 - Riemannian geometry of semisimple groups
On semisimple groups the Killing metric gA^ is non degenerate
and, after some linear transformation on the generators, it can always
be reduced to the diagonal form:
gAB = nAB = dia^+  +	(1.3.102)
p-times q-times
In this basis the Killing metric coincides with the tangent metric used
in Chapter 1.2 to define Riemannian geometry on an arbitrary manifold.
For this reason in this section we confine ourselves to the study of
the Riemannian geometry of semisimple Lie groups.
A Riemannian connection is introduced in the following way: let
us consider again the Maurer-Cartan equations for the left-invariant
. _ A
1-forms о :
daA + | C^o6 - oC = 0	.	(1.3.103)
A
Since the о are a set of n independent 1-forms on G they can be
used as a set of vielbeins on G. The associated dual moving frame is
given by the Lie algebra vector fields T .
A
As connection we take the left-invariant 1-form co defined by:
by:
Л = J 4°C = |	°C •	d.3.104)
117
E Then the Maurer-Cartan equations merely express the fact the G is
I'1 torsionless:
' R=do-co_^a=0 .	(1.3.105)
D
' Moreover using the Killing metric:
C
шлв = ®ac“ в = ~ “ba 	(1.3.Ю6)
A
 We conclude that to g is a Riemannian connection (see Eqs. (1.2.37-38)).
Let us compute the corresponding curvature: using the definition
(1.2.27b) we find:
dA _ , А A С 1 ._A rC „А „С . D E _
R В - dw в u> c ~ W в "	4 (CBCCDE CCDCBE)o л
= - 1 c£cEnoD л oE .	(1.3.107)
4 CC DU
where in the last step we have used the Jacobi identity for the struc-
ture constants (1.3.13). Hence the intrinsic components of the curva-
ture are constants.
RARlnF = - 7 <C?ECRD - CCDCRF? ’	(1.3.108)
D {De g Ct DU CD DC
Using the Killing metric Вдц to contract indices, one computes the
Ricci tensor and the scalar curvature:
rbe = rAb|ae = - I ccVba = 1 6be	"-3-109>
R = gBE Rpn = - dim G .	(1.3.110)
6 BE 8
118
In particular from Eq. (1.3.109) we see that any semisimple group
manifold is an Einstein space with respect to the Riemannian connection
(1.3.104).
It is also possible to introduce a (non Riemannian) left-invariant
A
connection on G such that R is identically zero. It is sufficient
to set
“AB = CBC°C - (Adj Tc)AB°C •	(1.3.111)
Again we find:
but now the corresponding torsion is different from zero:
RA(w) = doA - wAB л oB = j- CAcoB л oC .	(1.3.113)
Д
Hence gj D is non Riemannian.
D
The intrinsic components of the torsion are given by the structure
constants
RA(w) =lcA .	(1.3.114)
2
The curvature tensor is:
R BtG))  CBCd  CCD° л CBF° “ ’ 2 (CBCCDF 2CCDCBF)0 - ° =
_ r 1 rA „С 1 „А „С	1 „А C D F _ _	,T -
2 CBCCDF ' 2 CCDCBF	2 CCFCBD^° " ° ~ ° •	(1.3.115)
In deriving (1.3.115) we have used (1.3.103) and the Jacobi identities
(1.3.13).
119
A manifold M is said to be parallelizable if one can find a
connection 1-form such that the curvature r\ (w) defined by
(1.2.27b) is identically zero. is called the parallelizing con-
nection. Thus Eq. (1.3.115) expresses the fact that every semisimple
group manifold is parallelizable.
Any other left invariant connection not given by (1.3.104) or
(1.3.111) gives rise to a non Riemannian manifold with non vanishing
curvature.
1.3.7 - Soft group manifolds
Group manifolds G have a rigid structure: the left- or right-
invariant vector fields and 1-forms have (in a given chart) a fixed
coordinate dependence and, moreover, the Riemannian geometry of G is
(locally) fixed in terms of its structure constants. As such they can-
not be used as domains of definition of fields which should dynamically
describe the space-time structure.
Nevertheless, as we show in a moment, the group manifolds G can
be identified with the vacuum configurations of gravitational theories.
So we are led to consider manifolds G in which the rigid topo-
logical and metric structure of G has been "softened" in order to
describe non trivial physical configurations. G manifolds are locally
diffeomorphic to G and will be called soft group manifolds.
A well known example is space-time itself which, being diffeo-
morphic to IR^ , can be thought of as the soft group manifold of the
four-dimensional translations. As a further example let us consider
the soft Poincare group manifold, naturally appearing in the vielbein
formulation of gravity.
We first consider a flat Minkowskian space-time: its geometry
is described by the vielbein Va and a spin connection io3’3 fulfilling
Eqs. (1.2.12). In a particular Lorentz gauge the solution is
Va(x) = dxa
(1.3.116a)
wab(x) = 0
(I.3.116b)
120
while in a general Lorentz gauge it reads:
Va(x,n) = (A-1(n)dx)a	(1.3.117a)
wab(x,n) = (A-1(n)dA(n))аЬ	(I.3.117b)
ab
(ri are the Lorentz parameters).
The solution (1.3.117) corresponds to the left-invariant 1-forms
(1.3.81,82) of the Poincare group (in four dimensions). Indeed we can
identify the xa and the ri with the parameters associated to the
translations and the Lorentz rotations respectively. Therefore (1.3.117)
satisfy the Maurer-Cartan equations (1.3.83-84). Moreover, since the
Poincare group, ISO(1,3), is locally isomorphic to JR4®SO(1,3), it
can also be considered as a (trivial) principal bundle, P (H4, SO(1,3)),
with base space given by
IR4 E ISO(1,3)/SO(1,3)	(1.3.118)
and SO(1,3) as fiber.
It follows that the rigid Poincare group manifold describes the
trivial configuration corresponding to flat Minkowski space.
Now suppose that the space-time M. is not flat: the fields Va
ab	4
and w , subject to the gauge transformation laws (1.2.48) and
(1.2.51) are now defined on a fiber bundle P(M^, S0(l,3)).
P(M^, SO(1,3)) is not isomorphic, but just locally diffeomorphic to
G= IS0(l,3) due to the diffeomorphism M^n-JR4. in other words we have
"softened" the rigid structure of the base space, p4->-M4, maintaining
the structural group S0(l,3), which guarantees Lorentz covariance.
Notice that the curvatures Ra and Ra associated to Va and
ab
w are defined on the bundle through the gauge transformations
(1.2.52) and these in turn imply "horizontality": the 2-forms Ra, Ra^
ab
do not contain the differential dr, ; we express this by the equations:
^ablRab = MRa=°
121
where J is the left invariant vector field associated to the fiber
S0(l,3).
If we now soften the Poincare group manifold also in the direction
of the fiber, we obtain the soft Poincare group manifold, a manifold
diffeomorphic to IS0(l,3) with no fiber bundle structure.
On this manifold the configurations of the Va and co3’3 1-forms
are more general since their dependence on the parameters which were
previously associated to Lorentz transformations is no longer factorized
by a Lorentz transformation, and the curvatures Ra, Ra are no longer
horizontal.
In the particular case we are now discussing, this extension of
the field configurations to the soft group manifold is not very signi-
ficant from the physical point of view. Indeed a gravitational theory
must be locally Lorentz invariant: hence we must end up with fields
Va, gj having the correct gauge dependence on the Lorentz parameters.
The usual way to obtain this is to start directly with Va and
w defined on the principal bundle P(M^, S0(l,3)). (In Chapter 1.4
we will show that the fiber bundle structure can also be obtained from
the variational principle starting with an action defined on the soft
group manifold. This is certainly an interesting possibility from the
point of view of the economy of concepts, but it is not of any funda-
mental importance).
In more general theories like supergravity theories, we will see
that it is neither required nor desirable to factorize all the coordi-
nates which are not associated to the translations. Indeed, starting
from the super Poincare group, only the Lorentz gauge transformations
will be factorized: the gauge transformations of supersymmetry will
not. The resulting theory will be described on a principal fiber
4/4	4/4
bundle P(M ' , SO(1,3)) whose base space is the superspace M
This is discussed in detail in Chapters II.6 and III.3.
With this motivation in mind we now turn to the formal definitions
and the important formulae concerning soft group manifolds.
122
Soft forms and curvatures
Let us start with the rigid group G and the set of left-
Д
invariant 1-forms о satisfying the Maurer-Cartan equations (1.3.41):
doA + | C^coB л oC = 0	.	(1.3.120)
We soften G to the locally diffeomorphic soft group manifold G by
introducing new Lie algebra valued 1-forms
A
У = У Тд	(1.3.121)
which are also soft, that is, non left-invariant. We note that the Lie
algebra generators are considered in their matrix realization as in
A
Eq. (1.3.76). The p do not satisfy the Maurer-Cartan equations; the
A
shift from zero of the l.h.s. of (1.3.120) defines the curvature of p
RA
d!f , A 1 „А В C
" dp 7 CBCU л u
(1.3.122)
or, using R =
R = dp + p z. p
(1.3.123)
We note that the definition of the soft 1-forms and of the associated
curvature is the same as in Yang-Mills theory except that in our case
A
p is defined on a manifold G which does not have an "a priori"
fiber bundle structure.
A
When a fiber bundle structure is imposed on G then p becomes
a Yang-Mills potential on G = G(G/H, H) if H is the fiber.
A
The p , A = l,...,dim G, span a basis on the cotangent plane of
G; therefore, they have to be interpreted as vielbein and not as Yang-
Mills connections on G (there is no structural group acting on Tp(G)).
123
Fig. I.3.1
Let us take the exterior derivative of both sides of Eq. (1.3.122):
2
from d =0 we get the Bianchi identity:
dRA + cSB л RC = 0	(1.3.124)
dL
which can also be rewritten as:
VRA = 0	.	(1.3.125)
In (I.3.125) we have introduced the covariant derivative operator V.
On a tensor	with indices [•] in a representation D of ffi,
V is defined as follows
VA^ = (d + UB л D(Tb))aH .	(1.3.126)
In the case of A^ s RA:
[D(TB)]AC = Cjc	(1.3.127)
and (1.3.125) follows.
124
Applying the covariant exterior derivative to both sides of
(1.3.126) one obtains
V2A = d(VA) + uB л D(T_)VA =
D
= dpB л D(TB)A - pB л D(TB)dA +
+ UB ~ D(TB)dA + pB л D(TB)uC л D(TC)A =
= (dpA + | cJcpB л uC)D(Ta)A =
A
= R л О(ТД)А .	(1.3.128)
We have introduced the soft forms starting from the dual formulation of
the Lie algebra, namely the Maurer-Cartan equations. The same can be
done in the language of vector fields. It suffices to consider a basis
A
of soft vectors dual to the 1-forms p :
Д	Д
U (TB) = 6g .	(1.3.129)
*	~	A
The new vector fields TB> being the dual of the soft 1-forms p are
also soft (non left-invariant). Therefore they close a Lie algebra with
structure functions rather than structure constants, according to the
general formula (1.1.213).
Indeed the structure functions can be immediately computed
evaluating Eq. (1.3.122) on two tangent vectors T., T . Expanding
A	A В	L M
R along the intrinsic basis p p :
RA = RA pB л pC	(1.3.130)
dL
We omit the arrow on TA for notational clarity. However one must
keep in mind the distinction between T the Lie algebra generator
in its matrix realization, and T the vector field dual to pA.
A	A
Notice that T , thought as a vector field, is dual to о .
125
Eq. (1.3.122) can be rewritten as
dvA < I (CA - 2RA )pB л pC = o .	(1.3.131)
2 DC DC
Therefore, in the same way as we derived Eq. (1.3.46) from (1.3.41),
taking the value of both sides on T,, T,„ we obtain:
tW = (cab - 2rab^c •	»-3-132>
We note that the structure functions are given in terms of the curvature
intrinsic components.
Lie derivative on soft group manifolds
д
We now study the Lie derivative of the soft forms p along the
generic vector field T,.
A	A
Let us consider a generic infinitesimal diffeomorphism on p
generated by
t = eAT.	(1.3.133)
A
where eA= 6xA is the infinitesimal parameter associated to the shift
xA •+ xA + 6xA (A= 1,2,...,dim G) .	(1.3.134)
A .	...
We want to compute the Lie derivative p ; using the definition
(1.1.227), we obtain
VA = ( tjd + d t])pA = t]dpA * d(eB TgjpA) =
= t]dpA + deA .	(I.3.135)
Adding and subtracting 1/2 CA pBлpC to dpA and using the definition
ВС	д
(1.3.122) and (1.3.126), we reconstruct the curvatures R :
126
0 , А . I ,, А 1 „А В С. L„A С .А
= t|(dp + - CBCu л u ) - e CLCu + de =
= (Ve)A + jtjRA .	(1.3.136)
A
The first term (Ve) corresponds to an infinitesimal gauge transforma-
tion of the group G.
Hence an infinitesimal diffeomorphism on the soft manifold G is
a G-gauge transformation plus curvature correction terms.
In particular if the curvature RA has vanishing projection along
a tangent vector t:
jt]RA E 2eBR^cuC = 0	(1.3.137)
then the action of the Lie derivative coincides with a gauge
transformation.
Horizontality and factorization
Because of property (1.1.239), the set of vectors {t} satisfying
(1.3.137) must span a subalgebra H of the general algebra of diffeo-
morphisms (1.3.132). Let us denote by T^, (H=l,...,dim H) a basis
of vector fields in H. Then the condition
.A , . A
T^jR = о RHB = 0
(1.3.138)
reduces (1.3.132) to:
= C- T1
(1.3.139)
That is, H is the Lie algebra spanned by the left invariant vector
fields of HcG. Condition (1.3.138) will be referred to as the H-
Д
horizontality condition for the curvatures R . We see that gauge
д
transformations of p can be considered as diffeomorphisms along the
directions of the Lie algebra vector fields T^.
127
This is strictly related to the fact that H-horizontality of the
curvature is the condition under which the manifold G assumes the
structure of a principal fiber bundle with base space Mp=G/H and
fiber H:
G = G(G/H, H) .	(1.3.140)
Indeed when (1.3.138) holds the gauge transformation generated by
t =
г e H
6(gauge) =	= fVe)A	(1.3.141)
can be explicitly integrated.
To obtain the explicit expression of the finite transformation
д
associated to (1.3.141) we split the coordinates x of the soft group
~	к
manifold G into coordinates x relative to the base space and co-
JJ
ordinates p relative to the fiber, which is the rigid group-manifold
of the subgroup H.
If the horizontality condition (1.3.138) holds then the dependence
A	H
of Ц (x,p) on p is factorized.
д
By factorization we mean that every p (x»n) is determined by its
boundary value on the base space:
pA(x) = pA(x, r> = 0)	.	(1.3.142)
Indeed taking any set of 1-forms
Ax) = p£(x)dxv	(1.3.143)
on the base manifold we can lift them to forms defined on the whole G
via a finite gauge transformation of the subgroup H.
Let
h(n) = exp(nH T^)	(1.3.144)
128
be an element of H and let be the H-generators in the coadjoint
representation of G (see Eq. (1.3.66)): h(n) is a dim G x dim G
matrix with the block form:
h(n) = coadj h(n) =	/ coadj H h(n)	° \	} dim H (1.3.145)
	°	M(h(n)) у	
			} dim G/H
coadj h(q) is the coadjoint representation of H and M(h(n)) is
the representation of H on the К subspace of the G-Lie algebra, К
being the coset generators (see Chapter 1.6).
Let us define the lifted 1-form
'	A’
U (x.n) = у (х,п)Тд	(1.3.146)
as the h(n)-gauge transform of u(x) = рА(х)Тд:
U (x.n) = h-1(n)p(x)h(r)) + h-1(n)dh(n)	.	(1.3.147)
Expanding along the generators we have:
UH(x,n) = [coadjH(h(n))]HH, pH (x) + oH(n)	(I.3.148a)
UK(x,n) = M(h(n))K pK (x)	(I. 3.148b)
which gives the general expression of the 1-forms pA where the n
dependence is factorized through a gauge transformation.
It is instructive to see the reason behind the integrability of
the gauge transformation (1.3.137).
A complete set of initial data for the integration of the flow
(1.3.137) is given by the boundary value of yA(x,0) and its "normal
derivatives", that is
Jh| d^A
(1.3.149)
129
since the T„ span the orthogonal complement to G/H. Now since
dyA(TH,TL) = (RA ♦ |cA.pR л pC)(TH,fL)	(1.3.ISO)
A
the horizontality condition R (T ,T ) = 0 determines the "normal deri-
n L
vatives" making the flow integrable in terms of its boundary value
Fig. 1.3. II
A
The conclusion is that we can interpret the original vielbein p (x,ri)
~	A
on G as a Yang-Mills G-Lie algebra valued connection p (x) defined
on a general base space = G/H and subj ect to the transformation
law (1.3.147) or (1.3.148).
Let us observe that the definitions of curvatures and Bianchi
~	A	A
identities on G keep the same form when we restrict p and R to
the base space; this is a consequence of the fact that under the mapping
ф: G + G/H
the wedge-product and the exterior derivative commute with the pull
back ф* (see Eq. (1.1.170-171)).
130
Before concluding this section we give the explicit expression
of the curvatures associated to the l.h.s. of (1.3.148) in the special
case where the Lie algebra of G:
(E = JH + Ж
Li
is (weakly) reductive and symmetric, namely (see Chapter 1.6) Си1=
К	П К
Cn = 0 or equivalently:
К . к
	[h.h] c JH (H is	a subalgebra)	(I.3.150a)
	[u,JK] £ JK (JK is	weakly reductive)	(1.3.150b)
	[К, К] c p (JK is	symmetric)	(1.3.150c)
In	this case the curvatures	of (1.3.148) read:	
RH -	t 1 CH UK R - « + 2 CKK' р л RK=^H> pK where		K' V	(I.3.151a) (I.3.151b)
	ЛН , H 1 „Н	H' « = dp + - CH,H„ p	H" P	(1.3.152)
is	the curvature of the subgroup H and		
	9® = < ♦ pH . К [°(ТН)] к. = Снк,	[DCrH)]KK.	(I. 3.153a) (I. 3.153b)
defines the H-covariant derivative in the coadjoint representation.
131
The same decomposition on the Lie derivative (see Eq. (1.3.136))
gives:
о	JrH K' |„H _
— de - e	~ e	+ £ IR ~~
<?(H) H K_H K' |_H
= & e - e С у + elR	(1.3.154a)
я/ = deK - eK’c«,HpH - eH4,uK’ ♦ e]RK =
= AK-eV' ^|RK	C*-3.1S4b)
nix
where
e =	+ e*?.,	.	(1.3.155)
41	lx
If e = e and £]R =e]R =0 then the Lie derivative coincides with
the H-gauge transformation:
£eUH = <5^gauge) цН = 0(H)eH	(1.3.156a)
Z UK = <5Cgauge) цК = - et/' .	(1.3.156b)
E	£	пК
1.3,8 - The example of Poincare and anti de Sitter soft group manifold
These groups are of particular relevance to the formulation of
gravity and supergravity. We discuss first the de Sitter or anti-de
Sitter group in D dimensions:
de Sitter: G = SO(1,D)
Anti de-Sitter: G = SO(2,D-1)
(1.3.157a)
(1.3.157b)
and their Lorentz subgroup H=SO(1,D-1).
132
The Poincare group
G = ISO(l.D-l)	(1.3.158)
will be discussed afterwards as an Inonii-Wigner contraction of G. We
begin by computing the curvatures of the soft-manifold G.
De Sitter curvatures
Let A be an SO(2,D-1) or an SO(1,D) matrix, for convenience in
the defining representation, and let oag be the corresponding matrix
of left-invariant 1-fbrms:
о = сЛг = A-1dA
A
satisfying the Maurer-Cartan equations
(I.3.159)
do + о л c = 0
(1.3.160)
Being SO(2,D-1) or SO(1,D) Lie algebra valued о is antisymmetric with
respect to the metric Ogg:
ab
о
ba
о
( ab be
(a = n
a >
° a)
(1.3.160a)
= (1, -1,...,-1,±1)	; a, b 0,...,D
ab
(I.3.160b)
D-l times
signs in the D-D component
Sitter and de Sitter case respectively.
= U Jgg has the same symmetry proper-
associated curvature is
where the 2
the anti-de
The soft potential =
ties; the
of distinguish between
Rab
, ab ac b
= dp - p ~ Pg
(1.3.161)
133
Decomposing the indices with respect to the Lorentz subgroup			
SO(1,D-1) :			
ab	ab aD			(1.3.162a)
SO(1,D-1)			
JSb SO(l,D-l)>Jab’			(1.3.162b)
Rab	>Rab, RaD SO(1,D-1)			(1.3.162c)
aD	Da , DaD _ gj = - lj	; R = -	RDa > J n = - aD	Jn = P Da	a	(I.3.162d)
{a,b} = {0,1,...,D)	;	{a,b) = {0,..	,,D-1)	(I.3.162e)
(1.3.161) split as follows:			
„ab , ab	ac	b R = da) - ш л Шс	aD Db - ш	ш riDD		(I. 3.163a)
„aD . aD a	bD R = du - w . л ш D			(1.3.163b)
We set			
/ = 2 ё Va			(1.3.164a)
RaD = 2 e Ra			(I.3.164b)
JaD = ₽a			(1.3.164c)
134
where e is an arbitrary scaling factor (introduced in order to per-
form the contraction to the Poincare group). Eqs. (I.3.163a-b) become
_ab , ab a cb . , _2 ,.a ,rb	,
R = du - co £ л ш ± 4 e V V	(1.3.165a)
Ra = dva - wab л vb = @va	(1.3.165b)
where Sa is the Lorentz covariant derivative
•а 1л
9 = d + ш D(J )	(1.3.166)
according to (1.3.126).
The plus and minus signs (in 1.3.165a) refer to the SO(2,D-1) or
50(1,0) cases respectively. In the following we restrict our attention
to the SO(2,D-1) (anti de Sitter) case only, bearing in mind that the
SO(1,D) case can be obtained from SO(2,D-1) by the replacement
2	2
4e -> - 4e .	(1.3.167)
Comparing (1.3.165) with (1.3.151-153) and using the identification
pH = C0ab  R^R^ ; UKHVa ; RKERa
we find that the Lie algebra valued curvatures of SO(2,0-1) are those
of a weakly reductive and symmetric algebra.
With the same decomposition of indices the SO(2,D-1) Bianchi
identities
v(S0(2,D-l))Rab E dRSb _ 2шДа Rb] c = 0	(1.3.168)
split as follows:
135
2^(SO(l,D-l))Rab + gg2 у[а ~ Rb] = 0	(1.3.169a)
^(SO(l,D-l))Ra + Rab = 0	(1.3.169b)
The explicit form of the curvature (I.3.165) completely specify the
Lie algebra of SO(2,D-1); indeed one may extract the structure constants
by comparing (1.3.165) with the general definition (1.3.122).
If we are interested in the commutation relations among the
generators we set Rab=Ra=0 so that wab and Va become left-
invariant; then setting
Oab(J-j) = бЙ	(1.3.170)
v cd'	cd
that is
uab(J .) = 6ab	;	Va(J .) = 0	(1.3.171a)
cd	cd	co
wab(Pc) = 0	;	Va(Pb) = 6a	(1.3.171b)
one finds (see Eqs. (1.3.43-46) and (1.1.244)):
[Ja6’Jeal = -1	- Jge nga)	(1.3.172)
^Jab’Jcdl = ' I (Jac nbd * Jbd nac ’ Jad r)bc ' Jbc nad5	(1.3.173a)
tJab’Pc i= 4 (ПасРЬ " Va>	(1.3.173b)
2
|p ,P. ] = - 2e J ,
L a bJ	ab
(I.3.173c)
136
Poincare group curvatures
The Poincare group in D dimensions, ISO(1,D-1) can be retrieved
as the Inonii-Wigner contraction of SO(2,D-1). Indeed, performing the
contraction limit e->0 one obtains from (1.3.165) the Poincare
curvatures:
ab _ ab _ , ab	a	cb R “ frt ~ QU) “to л U) c	(1.3.174)
Ra = 9Va E dVa - Ша л Vb D	(1.3.175)
and setting Ra = Ra= 0 one recovers the Maurer-Cartan equations of
the Poincare group given in Eqs. (1.3.83-84) (for 0=4).
The same limit applied to Eqs. (1.3.169) and to the Lie algebra
(1.3.173) gives the Poincare-Bianchi identities
3Rab = 0	(1.3.176a)
0Ra + Rab „ V. = 0
b
and the Poincare group algebra:
[j	J = - (J Пкд + J, л - J jTk - J. rj ,) L ab’ cdJ	2 ac bd bd ac ad be be adJ	(I.3.177a)
[j . ,P 1 = - - (n P. - П. P ) L ab’ cJ	2 ac b be a.	(I.3.177b)
[P ,P 1 = 0 L a’ cJ	(1.3.177c)
Fiber bundle structure
Let us now suppose that the H-horizontality conditions are satisfied by H=SO(1,D-1):	(1.3.138)
J. |RA = 0	A = {ab;a} £171	(1.3.178)
137
Eq. (1.3.178) amounts to saying that the general expansion of R -
(Rab,Ra) on the cotangent basis:
ra =	_A В С _ „А a RBC^ ~ У = RabV	,,b _A £m ,,a ~ V + R„ uj л V + £m,a
reduces to	♦ RA	л соРЧ m,pq	(1.3.179)
RA =	ra, va „ vb ab	(1.3. 180)
As we previously discussed, Eq. (1.3.178) implies that G acquires a
fiber bundle structure, where SO(1,D-1) is the gauge group and
MD = SO(2,D-1)/SO(1,D-1)
MD = ISO(1,D-1)/SO(1,D-1)
(1.3.181)
(1.3.182)
are the base manifolds, for the de Sitter or Poincare case respectively.
In this case Eqs. (I.3.174-17S) and (1.3.176) become the curva-
tures and the Bianchi identities of the connections and Va
gauging the Poincare (or de Sitter) group.
Restricting now our attention to the Poincare group we study the
explicit form of the Lie derivative of шаЬ, Va when condition
(1.3.178) holds. Considering the effect of an infinitesimal diffeo-
morphism on G generated by t we have that (1.3.136) split as follows:
£tuab = ®Eab + _tjRab
£ Va = ^ea + eabV + tjRa
t	b —
(I.3.183a)
(I. 3.183b)
138
.	_ ab т a~
where t = e J , + £ P .
ab a
Taking ea=0, i.e.
ab-j
ab
t
e
(1.3.184)
we have that an infinitesimal coordinate transformation in the J ,
ab
directions coincides with a Lorentz gauge transformation; indeed from
(1.3.178)
tjR31’ = RabUcd.TA) = 0	(1.3.185a)
tjRa = Ra0bc.TA)	= 0	(I. 3.185b)
and (1.3.183) become
lt<Dab = aeab	(1.3.186a)
LtVa = eabVb	(I. 3.186b)
that is an infinitesimal Lorentz gauge transformation of the fields.
According to the general discussion given in Section 1.3.6 the vector
fields Jab = Jab are left-invariant and close the Lie algebra of
SO(1,D-1).
Eqs. (1.3.186) can be integrated to a finite gauge transformation
according to (1.3.148a); we obtain
>	-1	-1
w (x,q) = A (n)w(x,O)A(n) - A (n)dA(n)	(1.3.187a)
V (x,n) = A 1(n)V(x,0)
(I.3.187b)
139
Thus we recover the gauge transformation law of the vielbein and of
the connection fields describing the Riemannian geometry of a D-
dimensional space:
MD = G/H = ISO(1,D-1)/SO(1,D-1)	(1.3.188)
given in (1.2.48) and (1.2.51).
On the other hand the Lie derivatives on along the tangent
vectors
t = ea Pa (eab = 0)	(I..3.189)
generate infinitesimal coordinate transformations on Mp:
Я.шаЬ = t|Rab = 2eCRab Vm	(1.3.190a)
t	—1	cm
Я Va = @ea + t|Ra = @ea + 2ecRa v"1	(1.3.190b)
t	—	cm
Notice that the ₽a vector fields are not left-invariant since they
are related to the generator of translations = 3^ by
a = va У У	й	ab Pa + %	J к ab	(1.3.191a)
=> p a	= vw(a  a Ц	be T , - co J, ) p be'	(1.3.191b)
It is worth to see in more detail the relation between the coordinate
transformation (1.3.190) and the Poincare transformations (1.3.183).
Writing a generic tangent vector e on G in the intrinsic
basis T. or in the coordinate basis Tv one has
A	L
eA = pA J .	(1.3.192)
140
Hj, being the components of the vielbein. Decomposing the indices A
and Г in (1.3.192) one finds:
ab ab u . ab
e = w e + h
U

a
e
(I.3.193a)
(1.3.193b)
have set rif'C;d>a\ n130 being the Lorentz parameters of
infinitesimal Lorentz transformation hab on the fiber, and
where we
a generic
V(po)0 by a coordinate choice.
Let us now substitute (1.3.193b) into the r.h.s. of (I.3.190b);
recalling that Ja^lRa= 0 because of SO(1,3) factorization one finds:
«, Vй = 0(vV) * еРЭ |Ra dxp л dx° =
e 1 у	li | po
= #Vaep + Vadep + 2epRa dx°
У У	pa
(£ Va) = (@ Va - 9 Va)ep + 9 Vaep * Va9 ep +
e Jp p у у p yp у p
* 2s4P  VpeU •	-
 <V>“ •	• “?vb|p'“ 
We see that the final result differs from the genuine general coordinate
transformation (1.1.220) of the coordinate vector Va by the term
u ab	.	u
ь ^bp’ which can be interpreted as a field-dependent Lorentz
transformation of parameter eab - hab since from Eq. (1.3.193a):
у ab,, , ab , ab,,,
V, = (e - h )V.
p bp	'bp
(1.3.195)
In other words a diffeomorphism on the soft group manifold gives rise,
on a Lorentz vector, to a diffeomorphism on the base space plus a field
dependent Lorentz transformation.
141
CHAPTER 1.4
POINCARE GRAVITY
1.4.1 - Poincare Gravity
In this chapter we utilize the vielbein Va and the spin
ab
connection w to describe the Einstein theory of gravitation.
On one hand this formalism reveals that gravity is a gauge theory,
precisely the gauge theory of the Poincare group ISO(1,3) (ISO(1,D-1)
in a D-dimensional space-time); on the other hand, however, the action
from which we deduce the gravitational field equations is essentially
different from the Yang-Mills action utilized in ordinary gauge
theories.
To understand this difference and to clarify the formal proper-
ties of "gravity" is essential for the formulation of its supersymmetric
extension, namely "supergravity".
We begin by writing the Einstein-Cartan action:
A = [ Rab(u) л Vе л Vde .	(1.4.1)
v	abed
JUPP»'-'»*-
142
The notations are those utilized in Chapter 1.2 for the study of an n-
ditnensional Riemannian manifold M . In our case n=4. In particular
according to (1.2.35)
„ab . ab a cb	,, .
R = du - ii) c л ш	(1.4.2)
is the curvature 2-form and Va is the vielbein.
We take units such that the gravitational coupling constant к is
equal to 1. Let us show the equivalence of Eq. (1.4.1) with the action
of gravity written in tensor formalism. Expanding R on the complete
2-form basis V1 а (see (1.2.34c) we get:
Now
Rab a VC a Vde . . =
abed
Rab
. .V
R31’. .?
Vе
P
Rab vi vj vc yd !
ij p v> p о
Vj
e , , =
abed
Vd e . . dxP л dxV a dxp
о abed
E^poe b . d‘
abed
)4x
,ab it cd . . ,, .4
: . . e J e . ,det V d x = - 4
ij abed
r’"-’ . . det V d4x
ij
J о
dx
(1.4.3)
Rij..
1J
R'JV = R
pv
(1.4.4)
„ v-*
p v
R
is the scalar curvature and det V = /- g is the square root of the
metric determinant (g = det g- ). Hence we get:
(1.4.5)
Let us examine the group-theoretical significance of Eq. (1.4.1).
ab
There are two gauge fields, the spin connection ш and the
vielbein Va:
143
I	шаЬ = dx^	(1.4.6)
I	Va = V* dxW	(1-4.7)
[ Working in first order formalism both gauge fields are treated as
:	independent. The equation Ra=0 which allows to express uab in
	terms of Va is not taken as an "a priori constraint", rather, as we
К are going to see, it follows as a variational equation from (1.4.1).
-	ab
» This means that "off the mass-shell" the connection ш is not
f necessarily Riemannian.
i	The key observation is that {Va,uab}, considered as a single
K- entity, constitute a multiplet in the adjoint representation of the
К Poincare group. That is we can write:
j	рА(х)Тд 5 wab(x)Jab * Va(x)Pa	(1.4.8)
i where
‘	pA(x) = yA(x)dx	(1.4.9)
is the gauge field of the Poincare group, Jab and ₽a being the
generators of the Lorentz transformations and of the four dimensional
translations, respectively. Hence gravity, as we claimed, is the "gauge
theory" of the Poincare group.
The field strength associated to p is defined as the Poincare
Lie algebra-valued curvature 2-form
RA=dpA + |cApB.pC .	(1.4.10)
Splitting the index A= (ab,a), we get:
Rab = du)ab - Ша лШсЬ	(1.4. Ila)
c
144
Ra = dVa - wa, Vb = S) Va
D
(1.4.11b)
which coincide with Eqs. (1.3.174-175). The associated Bianchi identi-
ties are given by Eqs. (1.2.62) or (1.3.176) which we rewrite here for
completeness:
®Rab = 0
SRa + Rab л V, = 0
b
(1.4.12a)
(1.4.12b)
Therefore the Lorentz-algebra valued curvature is the field strength of
the spin connection while the vector valued curvature (or torsion) is
the field strength of the vielbein field.
Let us emphasize that, although pA S (toa ,Va) is a Yang-Mills
potential and RA= (Ra ,Ra) the corresponding field strength the
action (1.4.1) is not of the Yang-Mills type; a Yang-Mills action for
Д
V would have the following form:
[ RA . *R = [ RA R . n epGaed^ . dxV л dxa . dx6 =
A 4 Alpa
Rpv Ra| per
M,
4
pp va ,— ,4
g g /-g d x
(1.4.13)
The main differences between an action of the form (1.4.13) and the
Einstein-Cartan action (1.4.1) are the following:
a) A Yang-Mills action is invariant under the whole gauge group
G of which the pA.s are the Lie algebra valued potentials.
The action (1.4.1) instead is not invariant under the whole gauge
group IS0(l,3), but only under the Lorentz subgroup S0(l,3).
The invariance under Lorentz gauge transformations is manifest
since Ra , Va and £ C are good Lorentz tensors.
145
To show the non invariance of (1.4.1) under a gauge translation
we recall that under any Poincare gauge transformation we have (see
Eq. (1.3.141))
6 pA = VeA	(1.4.14
gauge
„	A
where V is the Poincare covariant derivative, and e is the gauge
A - r ab a.
parameter: e = (e ,e ).
The Lorentz content of (1.4.14) is easily obtained by setting
tj Rab = _tj Ra = 0	(1.4.15)
in Eqs. (1.3.183); we obtain
6gauge wab = ^ab = @ £ab	(1.4.16a)
6gauge ya = (V£)a = фЕа + £ab	(1.4.16b)
clb
and setting e = 0 we get the infinitesimal action of the gauge
translation on the fields
6wab = 0	(1.4.17a)
6Va = Sea .	(1.4.17b)
Since (1.4.17a) implies (1.4.17) is:	ab 6R = 0, the variation of the	action under
6A =	Rab л ®eC	л Vd e , , = г[раЬ л RdeCE , , abed J	abed	/ 0
146
where we have used (1.4.12a) and (1.4.11b). Notice that we cannot use
the constraint R = 0 since it is not invariant under the gauge
translation:
6Ra = 6 0Va = S6Va = g>0ea = - Rabe, / 0
D
The non invariance of the Cartan-Einstein action under a gauge transla-
tion seems at first sight strange since one usually thinks of a trans-
lation as a coordinate transformation. This however is not right since
the generator of a coordinate transformations is not a gauge translation
rather a Lie derivative. Indeed, the Lie-derivative along the tangent
vector
e = cV3 = ea P + wpq J	(1.4.18)
V a p pq
where
ea = vaeu	(1.4.19a)
and
P = Vv(3 - wpq J )	(1.4.19b)
a av v V pq	v
yields the transformation laws (1.3.190):
£ewab = ejRab	(1.4.20a)
£eVa = ®ea +_fijRa .	(1.4.20b)
The Einstein-Cartan action is obviously invariant under general coordi-
nate transformations generated by Lie derivatives of the type (1.4.20).
Indeed since the integrand of (1.4.1) is written using only exterior
147
products of forms and exterior derivatives d thereof, invariance
under diffeomorphisms is guaranteed by the general law of transforma-
tion of forms under diffeomorphisms (see Eqs. (1.1.170-171)).
Furthermore invariance under diffeomorphisms can be directly
checked using the explicit form of the Lie derivative (1.1.236). We
obtain
M4
(de] * e]d)jZ
(1.4.21)
Now the second term is zero since the 5-form dJZ vanishes identically
on the 4-dimensional space-time : hence 6A= 0 since the first
term is a total derivative.
Sometimes the gauge translations generated by eaPa=eaV^
(P being left-invariant) are confused with the general coordinate
transformations generated by the (non left-invariant) tangent vector
ea P , the relation between the two generators being given in (1.3.191).
As we have just seen, however, the associated transformations (1.4.20)
and (1.4.17) are quite distinct; actually the former leaves the action
(1.4.1) invariant, while the latter does not.
What really people do when speaking of "equivalence between the
two kinds of transformations" is to observe that the gauge transforma-
tion (1.4.17) can be traded with the diffeomorphism (1.4.20) if one
keeps Ra= 0 (second order formalism) and amend the transformation law
ab
of w , which is a dependent field,
with (1.4.20a).
Since the transformation law of
order formalism one finds that on the
transformations are the same.
in such a way that it coincides
ab
co is uninteresting in second
vielbein field Va the two
U
It is evident however from our discussion, that what one is really
performing is in any case a general coordinate transformation, since the
as calculated from (1.4.20b) at Ra=0 exactly reproduces
(I.4.20a).
As a final remark we notice that the algebra of the diffeomor-
phisns being given by (1.1.239), it closes with structure functions
148
rather than with structure constants as it would be the case for the
group of translations. Indeed using (1.1.239) and (1.3.132) and
a ~	b ~
setting £. = P . £_ = P, one has:
1	1 a’ 2	2 b
К Л J = ЯГе p i 4	(1.4.22a)
bl 2 Lbi>e2J 3
where is given by:
e_ = [e® P сЬ P ] = ea e^j?. - 2RA )T. =
5 L 1 a 2 bJ 1 2 ab ab A
= - 2 e® E^(RC, P + Rc5* J ,)	(1.4.22b)
1 2 ab c	ab cd	1
and we have used the fact that the structure constants of two transla-
tions are zero for IS0(l,3).
b) A second difference we want to discuss between the Einstein-
Cartan and the Yang-Mills action is the following: the action (1.4.1)
A
is linear in the curvature forms R , while the Yang-Mills action
(1.4.13) is quadratic. A quadratic action is necessary in ordinary
Yang-Mills theories to produce second order propagation equations for
the potential A^. How is it, then, that the Cartan-Einstein action,
which is linear, gives second order propagation equations for the
graviton?
The answer is that we are using first order formalism. As anti-
ab
cipated, the variation of (1.4.1) in Sto yields the torsion equation
Ra = 0
ab
which can be algebraically solved for the spin connection w in terms
of the vielbein first order derivatives. Substituting this result into
the other field equation, obtained by varying (1.4.1) with respect to
<5Va, we get a second order differential equation for the vielbein Va.
149
Let us study how this works in more detail. Varying (1.4.1) with
respect to the vielbein field we get:
2Rab „ VCe , , = 0	.	(1.4.23)
abed	1	7
In order to retrieve from (1.4.23) the corresponding equation for the
components B8** we proceed as follows: we expand the 2-form Rab
along a complete basis of vielbeins as in (1.2.34c) and we obtain:
2Rab V® „v” „VC£, , = 0	.	(I.4.24a)
mn	abed
Setting
v"1 А v” л vc =	(1.4.24b)
where is a non zero 3-form, one deduces:
nab mnc£ „	xmn£ Dab _
R e £ , , = - 3! 6 , , R =0
mn	abed	abd mn
that is
Ra£ l6aR=0	(1.4.25)
b£ 2b
which is the usual Einstein field equation of pure gravity (the only
difference being that we are using intrinsic components of the curva-
ture instead of the world-components). Equation (1.4.25) is a 1st-
ab
order equation for the field 0) .
Another equation is obtained if we vary the independent field
coab* this variation is easily derived using the following formula:
6RA = V(6pA)
(1.4.26)
150
which is an immediate consequence of the definitions (1.3.122) and
(1.3.126). In our case (1.4.26) becomes:
6Rab = V6wab = 9 <Swab
(1.4.27a)
where the last equality follows from the definition of the Poincare
covariant derivatives of an adjoint multiplet (see Eq. (1.4.16a)).
ab
Therefore the	variation of (1.4.1) yields
!‘К‘Ь - V' - Vd %bcd - I - V' . г , K .
J	abed
= г[бюаЬ ®vc л vd E . ,
J	abed
(1.4.27b)
Notice that there is no minus sign in the partial integration since we
are partially integrating a 1-form. It follows that:
c d
RxVe ,	= 0	(1.4.28)
abed	1 J
where we have used the definition (I.4.11b).
It is easy to verify that (1.4.28) implies
RC = 0	.	(1.4.29)
c
Indeed, let us expand the torsion R along the vielbein basis
Rc = RC V™ . Vn
(1.4.30)
(1.4.28) becomes:
Rc V1" л Vn
mn
л vd
Eabcd
(1.4.31)
151
Setting as before:
Vm
yn yd = emnd£

(1.4.32)
we get:
RC
inn
mnd£
£
e , , =
abed
Rc 3! 6mn£ = 0
mn abc
(1.4.33)
that is
r£ h
ab
2
Rm Г
id a b
0
(1.4.34)
Contracting
£
with a
we obtain:
r£ -
R №
(1.4.35)
Hence we find:
R£ , = °
ab
(1.4.36)
Therefore formula (1.4.29) holds.
From Ra = 0 (and wa'D = -w^a) we deduce that the Riemannian
manifold is endowed with a Riemannian spin connection, w3'3 is
given in terms of the vielbein through formula (1.2.44) and (1.2.45).
ab
Inserting (1.2.44) into Eq. (1.4.25) which is lst-order in the ш
we get a second-order equation for the vielbein field (since (1.2.44)
is first order in 9 Va).
v p
The conclusion is that starting from the Cartan-Einstein action
(1.4.1), which is linear in the curvature, the propagation of the viel-
bein field Va is obtained via the torsion mechanism Ra=0, which
P	a
allows the elimination of the spin connection in terms of V . There-
fore only the degrees of freedom of Va are physical since they corres-
pond to a propagating field.
152
1.4.2 - Extension to the soft group manifold
The Cartan-Einstein Lagrangian has still another striking dif-
ference as compared with the Yang-Mills one (1.4.13). It is built
using only forms, wedge products and exterior derivative with exclusion
of the Hodge duality operator* (see Eqs. (1.1.191-192)):
appearing instead in (1.4.13). As a consequence, the equations of
motion, stating that certain 3-forms are zero, can be naturally extended
to a larger manifold by an inclusion mapping:
M. G э M.
4	4
(1.4.38)
In presence of the Hodge duality operator this would be forbidden since
in the definition (1.1.191) of the operator * the dimension of the mani-
fold enters in an essential way.
In our case the forms {w3'3, V3}, being Yang-Mills potentials
subject to the gauge transformations (1.4.16), are already defined
on a larger manifold G э M^ which is the principal fiber bundle
G = G[G/H, H]
(1.4.39)
where G is defined by the structure Eqs. (1.3.174-175) and
G/H => M4
(I.4.40a)
H = S0(l,3)
(I.4.40b)
As discussed in the previous chapter, this means that the inclusion
mapping extending the fields from M^ to G is given by the Lorentz
transformations (1.2.48) and (1.2.51-52) and that the curvatures
153

(1.4.11)	are horizontal. It is therefore possible to extend the field
equations (1.4.23) and (1.4.28) to G э M^.
We will now show that it is not necessary to start with the fiber
bundle structure (1.4.39) in constructing the action (1.4.1).
Indeed the fiber bundle structure can be obtained as a result of
the (suitable extended) variational principle, if we start with a field
A	„	--------
p defined on the soft Poincare group manifold IS0(l,3). In other
words, we will show that S0(l,3) horizontality of the curvatures can be
obtained as a variational equation from the same Cartan-Einstein action.
According to the discussion of the previous chapter we start with
A	-
a set of fields p which are Poincare Lie algebra valued soft 1-forms
spanning a basis of the cotangent plane to the 10-dimensional soft
Poincare manifold IS0(l,3).
The group curvatures are given by
„ A	j A 1 r>A В С	ЛТЛЛ1Л
R = dp + - CB(.p л p	(1.4.41)
or, in terms of S0(l,3) representations by (1.3.174-175). A priori
A ab
these forms are not horizontal. The Lagrangian for the fields p ,
Va is formally taken to be the same as before
У= Rab . Vе . Vd e , ,	(1.4.42)
abed
but, being a 4-form, it must be integrated on a 4-dimensional submani-
fold of G.
Therefore we write
Rab . Vе . Vd eabcd
M4c G
(1.4.43)
where is any 4-dimensional submanifold of G.
In principle when we vary (1.4.43) we should consider not only
arbitrary variations of the fields wab, Va but also arbitrary varia-
tions of M^. Observe, however, that a variation of Мд can always
154
be compensated by a diffeomorphism on the fields ioa , Va under which
the Lagrangian is invariant. It suffices, therefore, to restrict our
attention to the field variations.
The equations of motion
Rab
£abcd
Ra . Vb
£abcd
(1.4.44)
(1.4.45)
, Vе
0
0
are identical in form to the previous ones, (1.4.23) and (1.4.28),
except for the fact that they hold now on the whole 10-dimensional
manifold G.
д
To examine their content we must expand the curvatures R on a
cotangent basis of G which is given by the set of 2-forms:
A В _ r,,a ,,b ab ,,c ab cd.
(1.4.46)
Hence the expansion of the curvature in this local frame is
_A А В С _A ,,a	,,b	__A	,,a be
R = RD_p л p = R , V л V * 2R V л it +
ВС r ab	a,be
„А ab cd	zT ,
+ R , ,ш .ч io	.	(1.4.47)
ab,cd	J
It is now easy to verify that on the larger manifold G the equations
д
of motion (1.4.44-45) imply S0(l,3) horizontality of R , besides the
usual implications for Ra and Rab , obtained on M..
r	mn	cd	4
Using (1.4.47) with (A) = (ab) we have:
Rab Vp л Vq zs Vе e , , + 2Rab Vp zs io£m л Vе
pq	abed	p, хлп
£abcd +
Rab
Л л
£m,rs
£m rs
tO Z4 io
Vе
£abcd
(1.4.48)
0
155
Since WV, VwV and wcow are independent 3-forms each term of
(1.4.48) must be separately zero:
	Rab VP л V4 л Vе £ . , = 0	(I.4.49a) pq	abed Rab . VP л	л VC E , . = 0	(1.4.49b) рДт	abed Rab„	л wrs л Vе e . , = 0	(1.4.49c) £m,rs	abed
Equation (I.4.49a) is formally the same as Eq. (1.4.22) and we deduce
again (1.4.25). Moreover from (1.4.49b) and (1.4.49c) it easily
follows that:
	R*	= 0	(1.4.50a) Rab„ c = 0	(I.4.50b) Jcm.rs
In an analogous way the torsion equation gives:
Eqs.	Ra	= 0	(I.4.51a) be Ra	= 0	(I.4.51b) рДп) Ra„	= 0	.	(I.4.51c) £m,rs (I.4.50a,b) and (I.4.51b-c) are encompassed by the single equation: RA(J„. TR} = RL R = °	(1.4.52) XJD	D	XJH,D
which is equivalent to:
156
(1.4.53)
J |rA = 0
мп I
Since the S0(l,3) horizontality condition is satisfied, we may restrict
the equations (1.4.44) and (1.4.45) to the base space M4EG/SO(1,3)
and they coincide with (1.4.23) and (1.4.28).
Eq. (1.4.53) enforcing the S0(l,3) fiber bundle structure of the
theory is due to its Lorentz gauge invariance, that is to the absence
a.b
of the bare field w in the Lagrangian. Hence the Lorentz gauge
invariance of the extended action (1.4.43) is responsible for the fac-
torization of the Lorentz coordinates: effectively the theory lives
only on the base space M4 = ISO(1,3)/SO(1,3). This is so because the
fields depend on the "Lorentz coordinates" only via the finite Lorentz
transformations (1.3.148). Since the Lagrangian is invariant (by
construction!) under such transformations, the dependence on the
Lorentz coordinates disappears.
In supergravity theories we will always confine ourselves, for
obvious physical reasons, to Lorentz invariant Lagrangians, so that,
starting from soft super-group manifolds, factorization of the Lorentz
coordinates will always be guaranteed and the fields will effectively
depend only on the base space coordinates.
The use of the entire (soft)group manifold G instead of
G/SO(1,3) has therefore a rather formal value. Furthermore we have
pursued a pedagogical goal since, in the future, we will compare the
"almost factorization" of the supersymmetry parameters of supergravity
theories, due to rheonomy, with the complete factorization of the
Lorentz parameters due to Lorentz invariance.
It is in this spirit that in the next section we will insist on
giving the building principles for a geometrical theory on a soft group-
manifold G, rather than on a coset manifold G/H. The problems con-
nected with the extension from G/H to G, which is in itself trivial
in the present case without supersymmetry, are however similar to the
problems connected with the extension from space to superspace which is
non trivial and crucial for the geometric formulation of supergravities.
157
1.4.3 - Building rules for the gravity Lagrangians
Let us summarize our discussion. We started with the potential
д
U and its corresponding curvature (1.4.11) defined on the whole soft
group manifold GSISO(1,3). Variation of the action (1.4.43) gave
the 3-form equations of motion (1.4.44) and (1.4.45).
д
These imply the vanishing of the curvature R along the Lorentz
directions (1.4.53) and the consequent factorization of the Lorentz
parameters through gauge transformations.
Projection of the equations of motion along the directions of the
base space
G/H = ISO(1,3)/SO(1,3) = M4
(1.4.54)
identified with the physical space-time, gave the equations of motion
on M4 (1.4.23) and (1.4.28) for the factorized curvatures and
potentials.
As we have seen, they are the usual Einstein equations of gravity
in first order formalism.
One may wonder how one could have invented the Lagrangian (1.4.43)
possessing all the aforementioned good properties without previous know-
ledge of gravitational theory.
It is worthwhile to note that (1.4.1), or its extended form
(1.4.43), can be uniquely determined using a small set of building
rules which appear to be very different from the usual ones used in the
derivation of the Einstein action in the theory of gravitation. The
formal nature of these principles will prove useful in finding generali-
zations of gravity Lagrangians to supergravity Lagrangians, one of the
main goals of this book.
Before giving and discussing the aforementioned building princi-
ple for the construction of the action let us discuss the general philo-
sophy behind them.
We observe the following: if we want to identify the space-time
components of the 1-forms {ыа^, Va) with the physical fields Va and
158
Юр , without destroying their geometrical meaning, we should construct
the action in a way consistent with the equations (I.4.8,9), (1.4.11,12)
,	, ab	,,a,	, ,	, ab	a,
defining	(to ,	V )	and their	curvatures (R ,	R ).
Now Eqs. (1.4.11-12) have a number of properties and a symmetries
that we want to be conserved by the action describing the physical
theory. They are:
i)	Coordinate invariance: this is an obvious consequence of
the fact that (1.4.11-12) are equations among forms, where only the
coordinate invariant operations of exterior product and derivative are
used; in other words the equations defining the curvatures and their
Bianchi identities have an intrinsic geometrical meaning.
ii)	SO(1,3) gauge invariance: in fact all the equations
(1.4.11-12) are covariantly defined in terms of good Lorentz tensors.
We notice that, in contrast, (1.4.11-12) are not invariant under
(1.4.17), the gauge translation.
Д
iii)	R = 0 is a solution of (1.4.11-12): indeed in this case
(1.4.11) reduce to the Maurer-Cartan equations for the IS0(l,3) left-
invariant 1-forms Va, wab and (1.4.12) to the Jacobi identities for
the structure constants.
iv)	Rigid scale invariance: (1.4.11-12) are invariant under
the rigid transformation
wab -> wab	;	Va -> eVa	(1.4.55a)
Rab Rab	.	Ra _ eRa	(I.4.55b)
where e is a constant non zero parameter.
Accordingly we shall require that the action constructed in
terms of wab, Va and Rab, Ra will respect all the symmetries and
properties expressed by i)-iv). This leads us to formulate the follow-
ing building rules:
159
i)	The Lagrangian must be geometrical: by that we mean that
A ~
it must be a 4-form constructed using the potential 1-form p on G
and the diffeomorphic invariant operations among them, the wedge product
and the exterior differential "d".
Actually the requirement that the only physical fields of the
theory should be given by the Lie algebra valued 1-forms pA turns out
to be too restrictive for more general theories. In particular, when
coupling matter multiplets to gravity or supergravity, or considering
extended supergravities, one must also allow new fields which are 0-
forms, i.e. functions on G. For the moment we restrict ourselves to
these "strong geometricity" allowing; the presence of 0-forms will be
discussed in the next chapters.
Notice that we have excluded the duality operator p->*p since
it depends on the dimensions of the embedding space. As our Lagrangian
is a 4-form it must be integrated on a 4-dimensional surface embedded in
the ten-dimensional manifold G. The duality mapping would bring poten-
tials and their curvatures out of the 4-dimensional integration domain.
ii)	The Lagrangian must be invariant under the subgroup
H=SO(1,3) of G. To this we also add the obvious physical require-
ment that it must be a scalar density of definite parity.
iii)	The Lagrangian must be such that the eqs. of motion should
admit as a particular solution the zero-curvature solution:
RA = 0 A= {ab; a}	(1.4.56)
A
so that the corresponding potential p are given by the left invariant
1-forms a .
The solution (1.4.56) will be referred to as the "vacuum" of the
theory. In our case G=ISO(1,3) and on G we have:
wab = (A-1(n)dA)ab	(1.4.57a)
A _
О -
va = (A-1(n))a 6b dxP	(1.4.57b)
160
4
or in a particular cross section (Л = 1) G/H = R
Va = 6a dx13
V
(1.4.58a)
(I.4.58b)
which correspond to the vielbein and spin connection of flat Minkowski
space.
iv)	Finally we impose that the Lagrangian should scale homo-
geneously with respect to the transformation (1.4.55): if it were not
so the equations of motion derived from it would give relations among
the curvatures RaB, Ra depending on the parameter e; this would be
inconsistent with the Bianchi identities (1.4.12) which scale homo-
geneously in e.
Let us see how one can retrieve the action (1.4.43) from these
principles.
Condition i) implies that the Lagrangian is a 4-form expressible
as a polynomial (in the exterior calculus sense) in the pA.s and the
A	A
curvature R . Indeed the exterior differential dp can be written
A	A
in terms of the curvature R : moreover the exterior differential dR
д
is linear in R due to the Bianchi identities. Therefore the most
general Lagrangian is given by:
У =	+ RA л + - RA RB + total differential
A 2	AB
(1.4.59)
since the Lagrangian is defined modulo a total divergence.
Here Л^4\	are polynomials of degree four, two and
А	АВ Д
zero, respectively, in the p 's and their coefficients are constant
tensors.
Л(4) = CabcdvA . pB л pC л UD	(I.4.60a)
161
VA = САР(/ - uQ
VAB = CAB
(I.4.60b)
(I,4.60c)
is a scalar,	is in the coadjoint representation and vajP
in the coadjoint ® coadjoint representation.
Moreover requirement ii) implies that the constant tensors
CAPQ’ CAB be Lorentz (S0(l,3)) invariant tensors.
Now we show that the quadratic terms can always be dropped since
they are equivalent to a total differential.
Indeed the only constant tensors Сдв which are invariant under
S0(l,3) are the following
CAB
C(ab),(cd) "
r, = xab
ab,cd cd
Eabcd
(1.4.61)
therefore we can write:
RA . RBVab = C1Rab
Rcd
nab
Eabcd + c2R
R , + c,Ra л R
ab 3 a
(1.4.62)
where c^ c9, c^ are constants.
The first two terms are closed forms. Indeed
d(Rab л Rcd eabcd) = fz (Rab . Rcd eabcd) = 0	(1.4.63)
where ® is the Lorentz covariant derivative.
162
In this proof we have used the fact the Rab л Rcd c is
abed
Lorentz invariant and the Bianchi identity (1.4.12a).
In the same way we find:
яЪ
d(R л R , ) = 0
ab'
(1.4.64)
From these results we conclude that Rab л Rc e , , and
abed
are locally exact. Explicitly we can write:
Rab „ R k
ab
„ab _cd	,, ab „cd	a£. b cd,
R л R e , . = d(E , ,w „ R - e , ,w „	)
abed abed	abed	H
(1.4.65)
Rab /, R , = d(wab „ R
ab	ab
1 Л
— w
3 a
(1.4.66)
Since our manifold Мд is without boundary the integral is either zero
or a topological number; indeed
R к
ab
(1.4.67)
1
32tt2
e Rab
м abed
M4
Rcd
(1.4.68)
Af Rab
SlT Jm.
4
= Pl
E
where the integer p^ is the first Pontriagyn number and the integer
E is the Euler characteristic of the manifold .
4
In any case the two terms (1.4.65) and (1.4.66) give no contribu-
tion to the variation of the action and we can drop them. (Let us
stress, however, that this conclusion holds because c^ and c? are
constant numbers; if we allow c^ and C£ to be functions on G which
is the case when we couple gravity to matter fields, then the contribu-
tion of these two terms to the variation of the action is not zero since
in the partial integration the derivative hits c^ and c?).
163
We may arrive at the same conclusion in a quicker way by using
the requirement iv), namely the homogeneous scaling of all the terms of
the Lagrangian under (1.4.55). Since as we shall see in a moment the
linear terms of the Lagrangian contain of course the Einstein term
Rab „Vе e , , which scales as [e2l under (1.4.55), the same must
be true for all the other terms; we see that the first two terms in
(1.4.62) have the wrong scale [e^] = 1 so that they must not be
included in the Lagrangian in order to have a consistent theory. (Again
the argument fails if those terms appear multiplied by functions on G
which scale as [e2] due to the presence of some dimensional constant).
The last term in (1.4.62)
Ra a R	(1.4.69)
a
(which scales as [e2]) can be reduced to a linear term in the curva-
ture RA through partial integration. Indeed we have:
Ra a R = © Va л © V = © (Va S' V ) + Va a © R =
a	a	a	a
= d(Va л & V ) + Va „ (-R , л Vb)	(1.4.70)
1	a	ab
where we have used the Bianchi identity (I.4.12b). Therefore
Ra R = - Rab a. Va л + total divergence	(1.4.71)
so that (1.4.60) just redefines the coefficient of R ~ Vg л
already present in the general term RA л vA of Eq. (1.4.59).
Therefore the most general Lagrangian can be rewritten as follows
y=A^+RA.v
(1.4.72)
164
ab
Now we observe that requirement ii) allows the appearance of co , the
bare gauge field of S0(l,3), only through the SO(1,3)-covariant
ab
curvature R ; therefore (1.4.72) becomes
= a e , ,Va ,Vb ,VC A Vd + В e , ,Rab . Vе „ Vd +
abed	abed
+ у Rab . va . Vb
and the constant G-tensor
ABCD
(1.4.73)
have been identified with
and CAPQ
the Lorentz invariant tensors as follows:
CABCD *	Cabcd “ 01 eabcd	(1.4.74a) /C(ab)cd = 6 eabcd
CAPQ\\	4C(ab)cd^62	t1'4-741»)
with а, В, у	constant numbers.
Moreover requirement iii) implies 0=0; indeed if we vary the
action with respect to the vielbein field Vd, we find:
2yRad .	, V + 4ae , ,Va , Vb . VC + 2B e , ,Rab л VC = 0 a	abed	abed (1.4.75)
Requiring the	"vacuum" (or flat group-manifold)
Rab = Ra = 0	(1.4.76)
to be a solution of (1.4.75) implies:
4ae , ,Va „ Vb л Vе = 0	(1.4.77)
abed	7
165
and this can be true only if a = 0 since Va л Vb „ Vе is an indepen-
dent 3-form on G.
Finally since ii) requires a definite parity for У we must
discard either
В РгЬ , VC , Vd e , ,	(1.4.78)
abed	J
or
ski
Y R - va " vb ’	(1.4.79)
The equations of motion resulting from the second choice are
Rab - Va = 0	(1.4.80)
Ra V'h - Rb , Va = 0	(1.4.81)
which are identically satisfied by the choice
Ra = 0
(1.4.82)
since Bianchi identity (1.4.12b) implies (1.4.80) when (1.4.82) holds,
ab
The curvature R remains therefore completely free. We conclude
that (1.4.79) is not a physical Lagrangian.
We are thus left with the Einstein-Cartan action
. Rab
M.c G
4
Vе ~ Vd e ,,
abed
(1.4.83)
extended to the soft group manifold G, or, if horizontality has been
assumed, to its restriction to = G/SO(1,3).
166
1.4.4 - Gravity in de Sitter and anti de Sitter space
In the previous section we have discussed the formal properties
of the Cartan-Einstein formulation of pure gravity. This study will
prove to be extremely useful when we try to construct more sophisticated
theories generalizing gravity: that is supergravity, matter coupling in
gravity and supergravity, higher dimensional theories.
In this section we present the very simple extension of the
A
Cartan-Einstein Lagrangian to the case where the potentials p are
defined on a de Sitter or anti de Sitter soft group manifold G. The
two cases are respectively:
Gd s = SOCM)	(1.4.84a)
or
6A.d.S. = SO<2’3)
(1.4.84b)
In the following we restrict ourselves to the anti-de Sitter case; the
modification needed for the de Sitter case were discussed in Section
1.3.7.
As we are going to see the new Lagrangian corresponds in tensor
calculus formalism to ordinary gravity plus a cosmological term. To
construct the action of anti de Sitter gravity we apply the building
rules discussed in the previous section.
д
We start from the soft 1-form Ц of the SO(2,3) Lie algebra:
p = / Тд = pab	a, b = (0,1,...,4)	.	(1.4.85)
We use the formalism developed in Section 1.3.8 with D=4; here we
just rewrite the anti de Sitter curvatures and Bianchi identities:
167
Rab = dwab - coac л Wcb + 4e2Va Л? =flab * 4e2Va л Vb (1.4.86a]
Ra = dVa - шаь л Vb = &Va	(1.4.86b)
29Rab + 8e2 V^a л Rbl = 0	(1.4.87a)
9Ra + Rab Vb = 0	.	(1.4.87b)
From Eqs. (1.4.86a) we see that, for e/0, the (anti) de Sitter
ab
curvature R differs from the Lorentz curvature
«ab = dwab - шас л cocb	(1.4.88)
_? a h
by the term 4e V л V .
In particular zero anti-de Sitter curvature corresponds to a
constant Riemann tensor: indeed we have
Rab = 0 - «ab = -	~ Vb	(1.4.89)
which implies:
6?ab . = - 4e2 <5ab .	(1.4.90)
cd	cd
Hence the anti de Sitter "vacuum" (Ra= Ra =0) is a 4-dimensional
_2
manifold characterized by a constant negative curvature (- 4e ).
Since we are going to require Lorentz invariance we assume that
the anti de Sitter curvature is already horizontal; hence all the
fields live on the base space HS0(2,3)/S0(l,3). The extension to
the soft group manifold can be done exactly in the same way as in the
Poincare case.
168
The Lagrangian in the anti de Sitter case can be written down
following closely the procedure and the notations of the Poincare case.
Indeed if one decomposes the adjoint SO(2,3) indices of the general
Lagrangian (I.4.59) with respect to S0(l,3) and uses the Lorentz
invariant tensors <5^ and ea^c<j» as required by Lorentz gauge inva-
riance, then one gets exactly the same terms as in (1.4.62) and (1.4.73)
sb
the only difference being that the curvature R is given by (1.4.86a)
instead of (1.4.Ila). Using
Rab = fiab + 4e2Va . Vb
(1.4.91)
where Я^Ь is given by (1.4.88), we see that the argument which permits
to drop the quadratic terms is still valid. Indeed we have:
nab cd	_ab cd	__2,,a	,.b „cd
R ~ R e . . = Я ~ Я e , , t 8e V л V л e , , +
abed	abed	abed
+ 16e4Va ~ Vb л Vе л Vd e ..	(1.4.92)
abed	4	’
and
Rab ~ R . = <Rab л Я . + 8e2Va л Vb л Я .	.	(1.4.93)
ab	ab	ab	v '
Hence the new quadratic terms differ from those occurring in the Poincare
case by terms of lower order in the curvatures and these just redefine
the constant coefficients of the linear terms. The third quadratic term
Ra л Ra> (Eq. (1.4.70)), is eliminated exactly as before. Therefore we
are left with the Lagrangian linear in the curvatures given in Eq.
(1.4.73). By the same argument which leads from (1.4.73) to (1.4.75)
we get
У...	= a £ah ~ Vb ~ Vе - Vd + В Rab ~ Vе ~ Vd e . ,
(AdSJ abed	abed
(1.4.94)
ab
where R is given by Eq. (I.4.86a) and where we have already dis-
ab
carded the parity non conserving term R л va я (see the discus-
sion following (1.4.79)).
169
A non trivial difference with respect to the Poincare case comes
into play when we require the existence of the "vacuum" solution; indeed
recalling (1.4.86a) the variation of the vielbein field gives:
(4a + 8e2B)e . ,Va , Vе t 26 e ,Rab л Vе = 0
abed	abed
(1.4.95)
In order for the vacuum Rab = Ra= 0 to be a solution we must set:
э
a = - 2ё В .	(1.4.96)
Choosing B=1 the Lagrangian (1.4.94) becomes
O’	_ г>аЬ 4е r-	,,b	,,c	~
^(AdS) = R - V л V eabcd - 2e V . V . V . V
(1.4.97)
ab
or in terms of the Lorentz curvature 6?
,,ab ,,c	,,d _	п-2,,a	,Jb ,,c	,,d „
^(AdS) = л V л V eabcd ♦ 2e V о V" о V . V	.
'	(1.4.98)
In the tensor formalism (1.4.98) reads:
^AdS) " - “«Ab * 12Л Л 
This is the Einstein Lagrangian with the addition of the cosmological
1o-2
term 12e .
2
Finally we observe that in the contraction limit ё *0 from
(1.4.98) we recover the Cartan-Einstein action (1.4.1) (or its extended
form (1.4.43)).
170
CHAPTER 1.5
COUPLING OF GRAVITY TO MATTER FIELDS
1-5.1 - Geometrical Lagrangian for scalar fields on a rigid background
In the previous section we studied the formal properties of the
Einstein-Cartan action for gravity and we elucidated its group theoret-
ical and geometrical structure.
If it were only for the theory of gravity these formal develop-
ments would be of little interest since they do not involve any new
physics. However, as we already emphasized, the knowledge of this
formal structure proves extremely useful for the generalization to
supergravity, and for the interpretation of its group theoretical
structure.
For the same reason we make now a similar (although somewhat
shorter) study of matter Lagrangians. Indeed a geometrical formula-
tion of matter Lagrangians will be very useful not only for the study
of globally supersymmetric theories but also for their coupling to
geometrical supergravity. Let us begin by recalling that the essential
171
point in the derivation of the formal properties of gravity was the use
of first order formalism.
In this formalism one recognizes that:
A
i)	the 1-forms p are the Yang-Mills potentials of the Poincare
(or anti de Sitter) group;
ii)	the equations of motion allow the elimination of the non
ab
propagating spin connection co in terms of the physical field,
namely the vielbein Vя;
iii)	one can avoid the use of the Hodge duality operator while
writing the kinetic term of Vя.
This is important for the extension of the theory to the whole
soft group G and for the derivation of the Lorentz transformation
properties as variational equations. Keeping this in mind, we look for
matter Lagrangians which are first order in the fields and which are
geometrical. Recalling our previous discussion, by a geometrical
Lagrangian we mean a 4-form constructed out of 1-forms using only
invariant operations. (Wedge product and exterior differentiation,
with the exclusion of the Hodge duality operator).
Let us consider a particularly simple, but very important example;
a multiplet of n scalar fields ф1, (I = l,...,n) on a gravitational
background described by the vielbein Vя. In the ordinary formalism
the kinetic part of the action is:
dtp1 „ dфI =
A =
abn <1	,I j . тг .4 _
П Эф Э, ф det V d x -
a d
M4
(1.5.1)
where Э
a
represents the "anholonomized derivative", namely
(1.5.2)
172
The action (1.5.1) gives the so called minimal coupling of the scalar
field to gravity: this means that, in the flat background
V = о о И - П
Ц Ц ЬЦ-и pv
(1.5.За)
/ - g = det V = 1
(I.5.3b)
the Lagrangian (1.5.1) reduces to the ordinary one for a set of n
massless scalar fields in Minkowsky space.
To avoid the Hodge operator we use first order formalism. For
each field <jJ (I = l,...,n), we introduce a О-form ф\х), in the
a	I
vector representation of S0(l,3), to be later identified with Э ф
a
through its own equation of motion.
The first order action we shall use is the following:
Scalar = fM ^4 - W(*»vP ' v4 Л'Г ' Vs£pqrs +
M4
1 AaI J лЛ ,,b wC ,,d_	1
+ — ф dф л V ~ V л V e , .1
3	abcdJ
(1.5.4)
It contains 2n independent fields (ф1, ф^), it is first order in the
derivative d and it does not involve the Hodge operator.
Furthermore, we have introduced a selfinteraction term given by
the scalar potential И(ф), which is some function of the fields ф1.
Without supersymmetry ДО(ф) is a completely arbitrary object, while in
supersymmetric theories its structure is completely fixed for each
model (see Part Four, in particular Chapters IV.3 and IV.7). The parti-
cular normalizations chosen in (1.5.4) are such as to agree with those
used in the supersymmetric case (see, e.g., Table IV.3.II).
Performing the variation of (1.5.4) with respect to ф^ we
obtain:
.Л ,,r	,,s	1 ,,I .b ,,c	,,d	_
2 а Ф „ V1 л V л V e + — dф л v , V „ V e = 0
a	pqrs з	abed
(1.5.5)
173
Expanding дф1 along the vielbein basis
аф1 = э^ф1/
and setting
vP . Vq . Vr . Vs = ep4rsn
(1.5.6)
(1.5.7)
12	3	4
where R=V «V „V is the four dimensional volume element, we
obtain:
- 4'Заф1
a
31 I
-—Эф
3 a
0
(1.5.8)
Choosing a= -1/4! we find
.1 n Л1
Ф = Э ф
a av
(1.5.9)
Varying now with respect to ф one has:
- i d($IaVb л Vе „ Vde . ,) - fi = 0
3	abcd 6ф1
(1.5.10)
that is
1 x.. 11 a . ,b
- — Ф л V
3
(1.5.11)
where we have used the fact the d = ii on a Lorentz scalar and moreover
we have utilized equation (1.4.29): indeed the torsion equation R = 0
holds true since the matter Lagrangian does not contain the spin connec-
tion o>ab. Expanding @фЬ along the vielbein basis and using (1.5.7)
we find:
174
9 ф1а + 12 — = О
а	бф1
(1,5.12)
Now Ra=О implies that ша^ is a Riemannian connection; therefore,
using (1.2.95), one has
de f
9 (ш)Эаф1 = V ЭРфТ 5 □ ф1	(1.5.13)
a '	p	cov
where V, is the covariant derivative in the {3 }-frame and □
M	p	cov
is the covariant d'Alembertian.
Hence one finally obtains the equation of motion of ф1 in its
usual form:
□ ф1 + 12 — = 0
C0V	бф1
(1.5.14)
1.5.2 - Extension to the Poincare group manifold and interpretation of
the Lorentz transformation rules as variational equations
We shall now extend the theory on to the soft Poincare group.
We may argue that, since the equations of motion derived from the action
(1.5.4) are equations among forms, they can be extended to the soft
Poincare group via a Lorentz transformation on the fields ф and фа: *
ф'^х.П) = Л%) ф*(х)	(1.5.15а)
d	dD
ф'^х.п) = ф\х) .	(1.5.15b)
Viceversa we may obtain the factorization property (1.5.15) by extending
our matter action principle in exactly the same way as we did for
gravity.
According to the discussion given in Sect. 1.4.2 the present section
should be viewed as a pedagogical example in view of the non trivial
extension to superspace of the corresponding supersymmetric theories
175
Indeed let us start with fields*
Ф„ = Ф_(х,п)	(1.5.16a)
a a.
ф = ф(х,п)	(1.5.16b)
defined on G and let us consider the extended action for (ф, ф ):
3.
A .	= [ £abcd ad yb vc yd _
scalar M.cG	3
4
- (| фСфс + W»))Va л Vb . Vc „ Vd] .	(1.5.17)
The difference between (1.5.4) and (1.5.17) is that the fields entering
in (1.5.17) are defined on G instead of M. = G/H and that the inte-
gration is to be made on an arbitrary submanifold of G. As we
have already observed in the case of gravity, the variation of is
immaterial: the equations of motion derived from (1.5.17) have the
same form as (1.5.5) and (1.5.11). The only difference is that now
they hold on G.
From (1.5.5), expanding dф on the complete basis (ioab, Va)
of the cotangent plane to G, besides (1.5.12), which corresponds to
the 4-vielbein projection, one has a further equation associated to
the independent 4-form л Vb л VC л V eabcd> namely:
—фаР4 л Vb л Vе л Vd e . , = 0	.	(1.5.18)
snpq	abcd
This inplies:
3
-----ф=0	(1.5.19)
ЭпРЧ
For notational simplicity we omit the index I labelling the n scalar
fields.
176
which is equivalent to Eq. (1.5.15b).
Proceeding in an analogous way for Eq. (1.5.10), besides Eq.
(1.5.11) one gets the following equation associated to the „Va л
b c
V z.V sector:
~Vb. Vc.Vdeabcd = 0	(1.5.20)
that is
d^GJ..) = ojab(Ji;j)<^b
(1.5.21)
Since the left-hand side is the Lie derivative of the 0-form ф^ along
the Lorentz generator J^_., we find:
*(ij}Фа(х,п) = 26аЬфь = (6аф_. - 6аф.)	(1.5.22)
which tells us that фа(х,о) transforms as a vector with respect to
Lorentz transformations.
Eq. (1.5.22) is the infinitesimal version of Eq. (1.5.15a).
1.5.3 - The interaction of the scalar fields with gravity and the
effective cosmological constant
Let us now consider the vielbein Va as a dynamical field by
adding the Einstein-Cartan action to (1.5.4):
A + A	=
scalar gravity
Rab . Vе . Vd - i ф1а d ф1 „ Vb . Vе . Vd)e , . -
3	‘ abed
(— ф1афГ + и(ф1))уР , V4 . vr , VSe
4! v a '•*	pqrs
(1.5.23)
177
where R is the Lorentz curvature defined in (1.4.Ila).
In the gravitational sector the only change comes from the varia-
tion of the vielbein. The other equation Ra=0 is maintained.
From the 6V -variation one finds the modified Einstein equation:
_ ab c. r .al , .1	,,b с 1 HI a ,,b	,,c
2R ~ V eabcd L* d* ~ V ~ V ’If* V ~ V ~ V ~
- 4 W(<i>I)Va ~ Vb л VC]eabcd = 0 .	(1.5.24)
3-b	I
Expanding R	and d<j> along the vielbein basis one obtains:
R\b - I 6b R = - I pa* Эьф - I 6b - 3w6b •
(1.5.25)
The r.h.s. is the energy momentum tensor of the scalar field. In the
second order formalism Eq. (1.5.25) reduces to the usual Einstein
equation with a scalar field source term.
Let us now recall that in the case of pure gravity a vacuum con-
figuration was defined a solution of the field equations corresponding
to vanishing curvatures
Rab = Ra = 0	.	(1.5.26)
ab
In terms of the Riemann tensor Й . this has a different meaning for
cd
the Poincare and anti de Sitter case: we respectively obtain (see Eq.
(1.4.91))
fiab . = 0 (Poincare)	(1.5.27)
cd
<Rab = _ 4 e2 6ab (A.d.S)	.	(1.5.28)
cd	cd
178
The Poincare case corresponds to a vacuum which is flat Minkowski space,
while in the A.d.S. case the vacuum is a manifold with constant negative
curvature.
As we are going to show, the value of the cosmological constant
for the coupled system A + A . is determined by the scalar
scalar gravity	J
field potential И(ф).
Let us first extend the definition of the "vacuum configuration"
in presence of the scalar matter by supplementing Eqs. (1.5.26) with the
further requirement:
dij)1 = 0	(1.5.29)
which can also be interpreted as the vanishing of the "curvature", d<)>,
of the О-form Ф. Since
d^1 = Э фМ = 4>*va	(1.5.30)
a	a
Eq. (1.5.29) is equivalent to
ф1 = <)>q = constant	(1.5.31a)
Ф* = 0	.	(1.5.31b)
a
Inserting (1.5.31) into Eq. (1.5.14) or (1.5.12), we obtain
6W
бф1
= 0	(1.5.32)
Ф1=Фо
that is, the vacuum expectation value ф^ must correspond to an extre-
mum of the scalar field potential И(ф).
Considering the field equation (1.5.24) we see that (1.5.26) and
(1.5.31) imply
179

2 fiab л VCe , . - 4 W(<)>bva л Vb л Vе e . , = 0	.	(1.5.33)
abed	0	abed	k '
Expanding on the vielbein basis we get
«abcd = 2 WCV 6cd •	C1-5-34)
Confronting (1.5.34) with (1.5.28) wee that for the coupled system
A ,	+ A . we have three cases:
scalar gravity
i)	И(Фд) = 0; the vacuum is Minkowski space.
ii)	W(<j>Q) > 0; the vacuum is de Sitter space.
In this case we may identify the parameter 2e introduced in the de
Sitter algebra (Eq. 1.3.164) with the extremum value of W:
2
4e = 2 W(<)>0) * ё =
*(ФО)1Ъ
2
(I.5.35)
iii) У1(ф0) < 0;
the vacuum is anti de Sitter space, and
ё = [- W(<)>0)/2]^
(1.5.36)
We now address the question of stability of this type of solutions. To
this effect we compute the scalar field mass matrix in the background.
We expand W(ф) around the extremum ф„ and we obtain the following
U	II
linearized equation (holding for small fields ф - ф^):
2
□ (ф1 - ф£) + 6	(фЛ - Фд) = 0	.	(1.5.37)
Эф Эф
In Minkowski space (И(ф())= 0) the eigenvalues of the mass matrix
180
2
2	,	3 W
mT т = 6 -------
ЭфЛЭфЛ
(1.5.38)
define the physical squared masses appearing in the diagonalized
Klein Gordon equations:
I 2	1
□	ф + ш^ф = 0
(1.5.39)
A stable vacuum corresponds to the absence of tachions, namely to the
lower bound:
2
m(I^ >0 V I = 1,... ,n
(1.5.40)
Condition (1.5.40) implies that ф = фр is a local minimum of И(ф).
In anti-de Sitter space the discussion of stability is more
subtle.
First of all in AdS the propagation equation of a scalar field of
2
squared mass m^ is not (1.5.39), rather it is
□	Ф1 + (| + m^H1 = 0
(1.5.41)
as we shall explain in the next section.
Therefore the mass matrix becomes
2
2	а	9 W
miJ	6
dtp dtp
’ 4 6IJW
(1.5.42)
Naively one could associate stability in AdS space with non negativity
of the eigenvalues of (1.5.42): however a peculiarity of AdS space, to
be discussed in Chapter II.5, is the following: energy is positive for
2
a range of the m^ parameters including negative values
2	2
m^n > -eZ = И(Ф0)/2
(1.5.43)
181
Яг This inqslies that saddle points of the potential may still correspond
K, to stable vacua.
We omit the discussion of stability in the de Sitter case (ii)
К since de Sitter space is uninteresting in supersymmetric theories,
У' corresponding in general to completely broken supersymmetry (see Chapter
В	IV.5).
I
1.5.4 - The field equation of a massless scalar in anti de Sitter
<5	space (in general in a curved space)
Let us now come back to the derivation of Eq. (1.5.41). We recall
that the mass of a particle is not a good quantum number for the anti
\ de Sitter group, but only for its contraction, the Poincare group.
Therefore we have to define a massless state in a way independent from
SO(2,3) representation properties and such that in the contraction
limit e->0 one recovers the massless equations for a Poincare field
theory.
By definition we call massless a field which obeys a propagation
equation invariant under the conformal group in four dimensions.
To find out such equation we compute the change in the curvature
induced by a conformal transformation on the vielbein
Va = e’A Va - gyv = e’2A gyv	(1.5.44)
where A is an arbitrary function.
The corresponding change in the spin connection can be found by
solving the new torsion equation; indeed Eq. (1.5.44) implies:
Ra = dVa - £ab „ Vb = e’A(Ra - Awab л Vfa - dA „ Vй) (1.5.45)
where
~ab ab . ab
w = w + Aw
(1.5.46)
182
If we impose that both wab and wab are Riemannian connections then
Ra = Ra = 0 and we obtain
Awab л Vb = - dX л Va .	(1.5.47)
ab
Expanding Aw and dX on the vielbein basis we find:
Awab - Aw£C = - ЭсХ6а + ЭьХ6а	.	(1.5.48)
The equations has the same structure as (1.2.40) and can be solved in
the same way. One finds
Awab = - Э Хб5 + Э X6C .	(1.5.49)
c a b b a	1
Therefore the new Riemannian connection is given by
'ab ab . ab ab „a, ,,b _b , ,,a	r
w = w + Aw = w - 3 XV + 3 XV .	(1.5.50)
Computation of the transformed Riemann tensor gives:
gab .'-ab ^a ~cb ,,ab ab a .cd b „ ca
R = dw - w л w = R + dAw - w л Aw + w л Aw
c	c	c
- Awac л Awcb = Rab + S(w)Awab - Awac л Awcb (1.5.51)
ab
and substituting Aw as given in (1.5.49) one obtains
Rab = Rab + (g| эал + эаХ9 >)уЬ yc +
c	c
+ (© эьх + эьхэ x)vc л va + эсхэ xva л vb
c	c	c
(1.5.52)
183
If we expand both R and R in the intrinsic basis
Rab = Rab VC л Vd = e'2XRab Vе „ Vd	(1.5.53a)
cd	cd	J
Rab = Rab Vе л Vd	(1.5.53b)
cd	v
we find:
2Xf^ab —.ab 1	ч j?b
e R j = R i------------Э X + Э ХЭ Jo, +
cd cd 2 c	c d
+ -,эал + ЭаХЭ.А)6Ь + abx + эьаэ X)6a -
2 d	d c 2 c	c d
- |(®аЭЬА * ЭЬАЭаЛ)6а + эгАЭ£Л<5аЬ .	(1.5,54)
From this equation we obtain the transformation law of the Ricci tensor
Ra’ and of the curvature scalar Ra аь ” R‘ F°r our purposes we are
interested in the latter; contracting the indices (a,c) and (b,d)
in (1.5.54) we finally obtain:
R = e2A[R + 3(ЭаАЭаА - DA)J
(1.5.55)
Let us now consider the behaviour of the zero mass Klein-Gordon equation
®аЭаф E □ ф = 0	(1.5.56)
under a conformal mapping, with the transformation law
ф = e <p
(1.5.57)
assigned to the scalar field.
184
Since
v\ = va3	(1.5.58)
it follows that:
9S = еЛЭ .	(1.5.59)
Therefore
□ ф = ®§й)Эаф = eA ®a(w) [еАЭа(еЛф)] =
= eA^(w + Aw) [е2АЭаЛф + Эаф] =
= eA®a(w)[e2A(9 Хф + Э ф)] - Awabe2A(9, Аф + Э.ф) =
a a	a d d
= еЗА(Сф + фПА - фЭаЛЭаХ) .	(1.5.60)
Thus Оф is not invariant under a conformal mapping. However from
(1.5.60) and (1.5.55) one finds:
| R$ = e3A(£ ф - фПА + фЭаЛЭаЛ) .	(1.5.61)
Hence equation (1.5.61) combined with (1.5.60) gives:
(□ + |)ф = e3A(O + |)ф	(1.5.62)
so that the modified Klein-Gordon equation
(□ + |)ф = 0	(1.5.63)
185
is a conformally invariant equation, provided the conformal transforma-
tion law of Ф is ф=е^ф.
1.5.5 - Geometrical Lagrangians for spin 1 fields
Let us now discuss the geometrical coupling of a massless spin one
field to the Cartan-Einstein action.
We introduce the Yang-Mills connection 1-form A defined over
the base space M^:
л .An - »AJ Pn - лА1гаг>
A = A P, = A dx P. - A V P,
A p A a A
CI.5.64)
P^ are the generators
strength (or curvature
of the Yang-Mills group 'S. The associated field
2-form) is defined as in (1.3.123):
F = dA + А л A
(I.5.65a)
or
cA jaA
F = dA
1 CA AB . AC
2
(I.5.65b)
F satisfies the Bianchi identity
F = dF * А л F = 0
or, using the E-Lie algebra indices:
dFA + CA AB л FC = 0
dC
(1.5.66)
(1.5.67)
Equations (1.5.65-67) are analogous to those defining the curvatures
and the Bianchi identities of the gravitational sector (see Eqs.
(1.4.10) and (1.4.12)).
186
As we have already observed several times the usual second order
Yang-Mills action is not suitable because of the presence of the duality
operator:
CA _pv /—— ,4 CA ab ,	. ,4
FpvFA d * = FabFA det V d x =
= _ 1 FA va vb
2 ab
~ e Vr
21 rscd
vsFcd
A
1.
2
FA
(1.5.68)
which forbids the extension of the equations of motion to the whole G.
To avoid this we proceed as in the scalar field case using a first order
formalism: we introduce a new field fab which is a О-form and an anti-
symmetric tensor
₽ab
A
J>a
A
(1.5.69)
The equations of motion will identify fa with the intrinsic compo-
nent of the curvature F along the vielbeins.
The geometrical first order action of the Yang-Mills field A
coupled to gravity is
ay.m. = f ~[Rab Vе . vd +
Jm4cg
+ Tr(- — frsf Va л Vb + fabF) л Vе z. Vdle , .
12 rs	J abed
(1.5.70)
where Tr means the trace over the adjoint indices of У.
TS
Varying the 0-form fft one obtains:
- / Va л Vb л Vе л Vde , , + FA л Vе л Vde = 0
6 rs	abed	rscd
(1.5.71)
187
Setting
FA = FAKVa . Vb + F* , ^4^va + FA w Ao?4.o:rS CI.5.72)
ab	CpqJ.a	Cpq)(rs)
one obtains from the wV and co-w sectors the factorization of the
Lorentz coordinates, as in CI-5-19)» while from the V-V sector one
has:
which provides the transition from the first to the second order
formalism.
To vary A Cin 1.5.70) we utilize the formula
6F = VW6A	CI-5.74)
which follows immediately from CI-5.65a) and Cl-5.66), and we get
V(S° (fabvc d£ = о .	CI-5.75)
k A	abed
Since the expression between the brackets is a Lorentz scalar we may
extend V to a '0 ® SOC1,3) covariant derivative:
+V('#®SOC1,3))	(I 5 76)
Then using
VC» ® S0C1.3)) ya _ aSO(l,3)va _ Ra = 0	CI-5.77)
where the last equality is the equation of motion of the spin connec-
tion which remains unchanged in the presence of spin 1 matter, we find
C* ® SOC1,3)) pib c d	0	CI.5.78)
A	abed
188
Expanding as usual
Vffb = V ffV + Vf^b(J )о?4
А с А	A pq7
(1.5.79)
and using the identification (1.5.73) we finally get obtain from the
Va-sector and (J^-sector respectively:
^«80(1,3))^^	(I58()a)
a	a
fab = - 2(Ja fb^ - <sja f^b .	(1.5.80b)
Eqs. (1.5.80) are obtained following the same kind of computations as
in (1.5.20-22).	(e1^ is the parameter of the Lorentz transformation
generated by ex^J^.). Variation of the Poincar^ potentials шаЬ and
Va, besides Ra=0, which defines the transition from first to second
order formalism, give the modified Einstein equation:
R£a - 1 6aR = - 1 (FAaF^ - 1 <5aFA F™1) .	(1.5.81)
ЯЬ	2 b в сАЬдЬтпА	4	1
and the horizontality of the Poincare curvatures, as shown in Sect.
1.4.2.
1.5,6 - Geometrical Lagrangian for spin 1/2 fields
Finally we treat the minimal coupling of a spin 1/2 field to the
Cartan-Einstein gravity action.
In this case our task is much easier since the usual Lagrangian
of a spinor field does not contain the Hodge duality operator, being
already first order in the derivatives.
Let us consider for instance a Majorana massless spinor A:
X = X+y° = XbC
(1.5.82)
189
where C is the charge conjugation matrix (for conventions on spinors
and у-matrices see Chapter II.7). A minimally coupled geometrical
Lagrangian for A is given by
A .	,,, = [ Ауа@А л Vb л VC л Vde , .	(1.5.83)
sPin 1/2 к cG	abcd
4
where ya are the Dirac у-matrices and Q is the Lorentz covariant
derivative acting on a spinor (see (1.2.58)):
@A = dA - | yabw .A	(1.5.84)
4 ab
with
Yab = | [Ya.Yb] •	(1.5.85)
The equation of motion of A is given by the variation in <5A:
ya@A л Vb л Vе л Vdeabcd = 0	.	(1.5.86)
Projection along wrsVbVCVd gives the factorization property:
ya(<iA - |Yr£/sA)(J£m) - °	(I-5-87a)
• <IS'7b)
On space-time (projection VVW) we get
уаФ A = 0	.	(1.5.88)
' a
This is the usual Dirac equation of a massless spin 1/2 field.
190
CHAPTER 1.6
DIFFERENTIAL GEOMETRY OF COSET MANIFOLDS
1.6,1 - Introduction
Coset manifolds are a natural generalization of group manifolds
(see 1.3), and play an important role in Kaluza-Klein (super) gravity
theories, to be discussed later (Part V).
We begin by defining homogeneous spaces.
Def. A metric space is said to be homogeneous if it admits as an
isometry the transitive action of a group G. A group acts transi-
tively if any point of the space can be reached from any other by the
group action.
3
Example. The unit sphere in JR is isometric under the transitive
action of S0(3): any point (x,y,z) on the sphere can be carried into
any other point (x',y', z') by a three dimensional rotation R
191
У'
, z'
R с S0(3)	(1.6.1)
Def. The subgroup H of G which leaves a point X fixed is called
the isotropy subgroup. Because of the transitive action of G, any
other point X' = gX (geG, g С H) is invariant under a subgroup gHg'1
of G, isomorphic to H.
In our example, the North pole (0,0,1) is invariant under that S0(2)
subgroup of S0(3) which rotates the sphere around the z-axis.
It is natural to label the point X of a homogeneous space by the
parameters describing the G-group element which carries a conventional
Xq (origin) into X. However these parameters are redundant: because
of H-isotropy
HXo^o
(1.6.2)
there are infinitely many ways to reach X from X^. Indeed, if g
carries Xp into X, any other G-element of the form gH does the
same and one is led to characterize the points of a homogeneous space
by the coset gH.
A homogeneous space is therefore a coset space G/H, i.e. the
set of equivalence classes of elements of G, where the equivalence is
defined by right H multiplication (g~g' if g=g'h, with g,
g' e G and h e H).
The two-sphere can be written as the coset space SO(3)/SO(2).
In general, for a n-sphere
sn = SO(n ♦ 1)
SO(n)
(1.6.3)
Taking G to be a Lie group we obtain coset manifolds (endowed with a
Riemannian structure, see Chapter 1.2), parametrized by D coordinates
, 1	-.u
(y •»У ). with
192
D = dim G - dim H
(1.6.4)
(Cfr. CI.6.17) below)
In each coset, corresponding to a D-plet of coordinates y= (y\...,y^)
we can choose a representative group element L(y) e G. Under left
multiplication by geG, L(y) is in general carried into another
coset, with representative element LCy’)- Thus
gLCy) = LCy')h
h e H
(1.6.5)
where y' and h are functions of у and g, and depend on the way
of choosing coset representatives. Pictorially:
(1.6.6)
In the case of S = SO(3)/SO(2), we assign to each point y, i.e.
each coset, an element of S0(3) (the coset representative L(y)) which
maps the North pole y^, chosen as origin, into y.
(1.6.7)
193
Denoting by L(y)^ (A, 1= 1,...,3) the SO(3) matrix element, we must
have
L(y)Az * L(y)nZ = 6n (orthogonality)
Цу)Л£ = УЛ (L(y) maps the North pole into y) (1.6.8)
Using the stereographic coordinates z ,z , it is easy to prove
that*- )
(1.6.9)
(1.6.10)
satisfies (1.6.8).
Notice that any
/ Л2х2^
0
L'(z) = L(z)
(1.6.11)
7A VA	.	3 z2 - 4	, A 4zA
1 1 From — =	- one derives у = —---------- , and у------x---
2	1 - y3	z2 + 4	z + 4
(z = z zp.
194
with A2 x 2 E S0(2) still satisfies (1.6.8). The coset representative
is chosen to have ^2x2 = ^2x2‘ t^der left multiplication by a general
S0(3) matrix S we have:
where H2x9eSO(2). Eq. (1.6.12) is an illustration of the general
formula (1.6.5). We leave as an exercise to compute	and
z'. The general method to obtain y1 and h of Eq. (1.6.5) is
discussed in Sect. (1.6.4).
The Lie algebra of G can be split as
I
€ = К ф «	(1.6.13)
where H is the Lie algebra of H. IK contains the remaining genera-
tors, henceforth referred to as "coset generators".	,
The structure constants of G are defined by
[Hi,Hj] = ck..Hk	H. e JH ,
[H. ,K ] = C< H. + cb. К, К ЕЖ ,
Li aJ 1a j iao a
[‘a-Sl =cjabHj +cCabKc	(1.6.14)
and we use the index conventions
a, b, c ...	flat coset indices
a, 6, у -..	curved coset indices
i. j, k ...	H-indices
А, В, C ...	G-indices	.	(1.6.15)
195
Any g is expressible in the form
у К x H.
g = e ae * 1	e JH, Ka e IK	(1.6.16)
which is just a particular way to exponentiate К to obtain finite
elements of G. Eq. (1.6.16) suggests a natural parametrization of
coset spaces by the representative choice
av
У Ka
L(y) = e	(1.6.17)
corresponding to x1 = 0.
An explicit matrix representation of L(y) is given in Section
1.6.3.
1.6.2 - Classification of coset manifolds
The simple Lie algebras (L.A.) are classified in the А, В, C, D
series, corresponding to the classical matrix groups, and in the five
exceptional G£, F^, Eg, E?, Eg algebras. Any semisimple L.A. is
the direct sum of simple L.A., and any L.A. is the semidirect sum of a
semisimple and a solvable L.A.
In what follows we shall consider G/H spaces with E semi-
simple, or at most semisimple ® abelian algebras (a particular case of
the general decomposition semisimple$ solvable). These cover most of
the G/H spaces used in the course of this book. There are, however,
some physically interesting G/H spaces with nonsemisimple E that do
not admit the semisimple abelian decomposition, e.g. E =Poincare and
super Poincare algebras.
To completely specify a coset space G/H, both topologically and
metrically, two informations are necessary:
i) The particular embedding of H in G. This determines the
topology.
f **)
1 J This holds true for g e compact G. For noncompact G, g must be
"not too far" from the origin.
196
ii) The particular invariant metric on G/H. In general there
are infinitely many, labelled by a finite number of rescaling parameters
(see Section 1.6.10).
The G isometry is realized on G/H in the most economical way, i.e.
G/H has the lowest dimension, when H is a maximal subgroup of G.
There exist tables listing all the maximal subgroups of a given
(simple) G (see for example ref. [?]); these provide, therefore, also a
classification of ^glMPLE^^VlAXIMAL coset spaces. The same tables can
be used to find maximal subgroups of semisimple G, and also, more
generally, to find any subgroup of a semisimple G (since one can find
the maximal subgroups of ^maximal an<^ so on) • Thus, for a given G,
or for a given dim G/H, all G/H are in principle known. An example
of such a classification is provided by Table V.6.1, which lists all
the coset spaces of dimension 7.
Reductive G/H
We can in general perform a tensor transformation
A AR
T = S D TD	(1.6.18)
D
В
on the generators T of a semisimple group so that its Cartan-Killing
metric [see (1.3.95)]
CD	C
gAB = C AD C BC C AD = structure const, of G (1.6.19)
becomes diagonal. On that basis, G/H (for any subgroup H) is
reductive, i.e. the decomposition (1.6.13) satisfies
[1Н,Ж] с К .	(1.6.20)
* gAB is a real symmetric matrix, transforming as g'=S gS under
(1.6.18). Then g can always be diagonalized with a particular
(orthogonal) S.
197
Proof: CJ. is proportional to C., = C ..=0 (because [и, М] с ]H).
-----la	jia aj i
Indices are lowered with the Killing metric, and we recall that
СдВс is totally antisymmetric because of Jacobi identities.
It is straightforward to prove the existence of a reductive decom-
position (1.6.13) also for semisimple ® abelian L.A. In the following
we shall always consider diagonal gAB (unless otherwise specified) and
hence reductive G/H spaces.
Symmetric G/H: when
[Ж, Ж] = H	(1.6.21)
G/H is said to be symmetric. This typically happens when G is simple
and H is maximal (for a proof see Ref. [7], Chapter 9).
For a discussion on the freedom in choosing bases of generators
in M and JK (preserving reductivity) see P. van Nieuwenhuizen in
Ref. [16].
1.6.3 - Coordinates on G/H and finite G-transformation
Compact G/H
When the decomposition
.B
' AC
В 1
representation (T^)	= C
E = IH+K is
has a simple
reductive, the regular
block diagonal structure:
regular
Ф repres..
| dim №
} dim К
(1.6.22)
198
since only Ca.^» ik 311 d C1^, ^ai are nonvan^s^^n8- They corres-
pond to the real matrices Ap B, -B^ respectively ^* (**)).
If ffi is a simple classical Lie algebra, the defining matrix
representation reduces in all but two cases to the block diagonal
structure
defining / h 0 \	/0 B\
repres.	\	\
E ---------»	®	j
10 A2 /	\ -B+	0 /	(1.6.23)
similar to (1.6.22)1	. Note that here the submatrices Ар A2, В
are in general complex.
The coset representatives are obtained by exponentiating the
coset generators:
L(B) = exp
(1.6.24)
With the substitution
(1.6.25)
(*) G is compact by assumption, implying gAB negative definite.
Then ^ab = "^ai	are antisymnietric-
(**) The two cosets G/H whose defining matrix representatives do not
have the form (1.6.23) are S£(n)/S0(n) and SU*(2n)/Usp(2n).
199
Eq. (1.6.24) becomes
L(X) =	dim H (I - XX+)’3	dim Ж X	\	dim H
	-X+	(I - X+X)^ /	dim К
			
(1.6.26)
The range of the parameters describing the submatrix
X is limited by
the requirement
0 < X+X < I,
к
k = dim Ж ,
= kxk identity matrix
(1.6.27)
where the inequality refers
kxk hermitian matrix X^X.
to the k positive real eigenvalues of the
Since X^X and XX^ have the same non-
zero set of eigenvalues,
0 < X+X < I, » 0 < XX+ < I, (h = dim IH) .
k	h
(1.6.28)
Conditions (1.6.28) are necessary for L(X) to be a group element.
The coset coordinates X are bounded by (1.6.27), and therefore describe
a compact coset space.
Example: S11 = s°Cn+1) where SO(n) leaves the (n+1)-direction fixed.
---—	SO(n)
generators of SO(n+l) : T^B	A,В = l,...,n+l
generators of SO(n) : T	a,b = l,...,n
coset generators :	s n+i
CD CD
In the vector (defining) representation (T )	=	' Hence the n
CD
coset generators (T&) take the off-diagonal form:
200
0
О
1
*- а-th row
T
a
О
0. .-1. .0
О
t
a-th
column
(1.6.29)
The general
element of the "coset algebra"
ba T
a
in the vector representation is therefore
bl
0
b
0
Ж :
b
n
-br..
-b
n
0
-bT
0
(1.6.30)
Coset representatives:
with x = b
a a
exp IK :
T
-x
- xx J
n
9 V
sin(Zb.) '
(2 b?P
The range of the parameters
x
a
is defined by
x
(1 - xTx)
(1.6.31)
n 7
I x2 < 1
a
a=l
(1.6.32)
X
201
Setting x , = ±/l- X
6	n+1	a a
the xa satisfy:
2	2	2	2	,
X, + X- + ... X +X ,=1
12	n n+1
and the coset S0(n+l)/S0(n)
sphere embedded in Rn+^.
can be identified with the n-dimensional
S0(n+m)
Example:
S0(n) xSO(m)
generators of S0(n+m)
generators of S0(n)
generators of S0(m)
coset generators
T._ A,В = 1,...,n+m
AB ’
T..	i,j = l,...,n
T _	a.e = l,...,m
aS
T. с К
ra
In the vector representation
n + a-th column
Ж : (T. )CD = 4 ia	Л n X n	о	:	o\ 	i	' о	:	о	i-th row
	о	:	о \....-i.... \o	:	о	о	/ m x m /	
(1.6.33)
and for a general element of JK
b10tT.
ia
(1.6.34)
b is now an nxm matrix, and one can use formulas (1.6.24-26) with
b= B.
202
Noncompact G/H
Given a compact G/H, the "Weyl unitary trick"
К----->i К
(1.6.35)
yields the noncompact coset space G*/H, provided it is consistent
with the commutation relations
[зн, ж] = ж
[ж,ж] = К + И .	(1.6.36)
While the first trivially becomes
[И, iЖ] = i К	(1.6.37)
in the second one we must have Ca, =0:
be
[ж,ж] = и -> [i К, i к] = - И	(1.6.38)
i.e. G/H must be symmetric.
The structure constants Са^ are unchanged, whereas C1^ do
change sign, and the metric gAB becomes:
gAB =
(1.6.39)
203
The regular and defining representations						have	the same block diagonal
structure as in (1.6.22)			, with	-B ,	-BT	respectively replaced by	
1	T +B , +B	in the coset generators (since					C1 ab	= C .) ai
	regular	A1	0		0	В'	
E .	rePr- ,	0	A2	®	T в	0	
	defining	[A1	0 '		 0	В 	
E	repr.	0	A2-	®	B+	0	
			H			к	(1.6.40)
The coset representative exp(iK),
with hermitian generators iJK, is
now "unbounded". Indeed
В
В
0
BTB
exp (iK) : exp
B+
0
Hb+
B+B
(1.6.41)
and
after
substitution:
(1.6.42)
we find
L(X) =
[l ♦ XX+]^
X
[i + x+x]^
(1.6.43)
without bounds on XX or XX .
204
In the previous examples, the "Weyl unitary trick" brings G/H
into G*/H as follows:
SO(n+1)  S0(n,l)
S0(n) K+-iK S0(n)
§0(n+m)S0(n,m)	(1.6.44)
S0(n)xS0(m) Ж + iIK S0(n) x S0(m)
1.6.4 - Finite transformations on G/H
We now derive an explicit expression for y' in the transforma-
tion law
gL(y) = L(y')h	(1.6.45)
giving the mapping of G/H into itself under the (left) action of G:
g : у -+ y'
An arbitrary element of G has the structure
A В m
+
1
C D	n
+
+-Ш-+	+-n-+
(1.6.46)
(1.6.47)

If g is in the adjoint representation, m=dim H, n = dim K.
Depending on which classical G we consider, there are various
relations between the submatrices А, В, C, D. Using the parametriza-
tion (1.6.26) of coset representatives, the abstract formula (1.6.45)
becomes
205
A	В			X		(Vx’x’V	X'		H m*m	0
C	D		-x+	(I -X+X)^ n		-x' +	(I -Х^Х'Г* n		0	Hnxn
(1.6.48)
i.e. :
AX +	B(In - X+X)^ =	: X1 H n	x n	(I.6.49a)
CX +	D(In - X+X)^ =	: <Xn -		(I.6.49b)
A(I m	+ p	+ - XX ) 2 - BX	= ^m	- Х’Х'^Н.»г,	(1.6.49c)
C(I m	+ i,	+ - XX ) 2 - DX	= -x,+	Hm xm	(1.6. 49d)
From these and A, B,	equations one C, D.	finds	X', H	and H n xn	m xm	in terms of X
Notice that multiplying (1.6.49a) by the inverse of (1.6.49b)
yields:
[AX + B(In - X+X)^][CX + D(In - X+X)^] 1 = x'(In - X*+x')
or
[AX(In - X+X) + B][CX(In - X+X) + D]*1 = X'(In - X,+X’) 4 .
(I.6.49e)
This last equation suggests the use of new coordinates Z:
Z = X(In - X+X)-!s =» X = Z(In + Z+Z)_!2	(1.6.50)
Z and X are 1-1 related (protectively). For compact cosets G/H,
X is bounded and Z is unbounded:
0 < X+X < 1 * 0 < Z+Z < °°	.	(1.6.51)
206
The Z's are called projective coordinates on G/H, and have simple
transformation properties under left action of G (cfr. (I.6.49e)):
Z* = (AZ + B)(CZ + D)
(1.6.52)
Thus, the group action is realized on the projective coset repre-
sentatives by a fractional linear transformation.
Exercise: prove (1.6.52).
Example - SO(3)/SO(2) = Sz
2	3
The coset representatives of the sphere S c R are
2	2	2
with x1 + x2 + x3=l, x3 > 0 (see (1.6.31)).
The projective coordinates of the upper hemisphere are
Z.
i
(1.6.54)
3
207
The figure illustrates three different parametrizations of the upper
hemisphere of SO(3)/SO(2), A fourth one was given in Eq. (1.6.7)
(stereographic coordinates).
2
Example - SO(2,1)/SO(2) = H (hyperboloid)
HZ is obtained from SO(3)/SO(2) via JK + ilK.
The coset representatives are
(1.6.56)
2	2	2
with x3 - *2 ~ xi = 1» -°0 <	*2 - + °° ’
Projective coordinates:
-1 < z.
i
(1.6.57)
Example - SU(n+l)/SU(n) xu(l) = (CP^
The generators of SU(n+l) are antihermitian traceless complex
matrices A1-1,	(i, j = 1,... ,n+l) .
Aij = -(Aji)*	, AU = 0	.	(1.6.58)
There are (n+l)Z-l independent matrices satisfying (1.6.58). These can
be decomposed in a real and an imaginary part:
A1} =	+ i Clj
B1J = -B^1 e R
C1^ = Cjl e R
(1.6.59)
Note: the antisymmetric B1’’ generate the maximal S0(n+l) subgroup of
SU(n+l).
208
A convenient basis for the SU(n+l) Lie algebra is provided by the
(n+l)n/2 antisymmetric matrices
i.....j
(1.6.60)
and by the (n'l'2)
2
1 symmetric traceless matrices:
[f. ,]кЯ = i[(6k<5fc + 6Я6к)/2 - <5..6k ,6я J
L ijJ	i J i J ij n+1 n+lJ
i and j
not both
equal to
n+1
j
*Fij Ci/j) =
-1
Fij (i = j) =
n+1
j
j
j
n+1
(1.6.61)
The SU(n) x U(l) = U(n) maximal subgroup of SU(n+l) is generated by
the E. . and F. . matrices with i,j = l,...,n. (Note: the extra
ij	i j
U(l) can be thought of as generated by F^. It is easy to check that
F commutes with all the E.. and F.., i.i = 1....,n).
nn	ij	ij’	J
SU(n+l)
The generators of the coset JK = ------------- are therefore given
SU(n) xu(l)
by the 2n matrices
209
Е. .	,	F.
i n+1	’ i n+1
(1.6.62)
i = 1»... ,n
and an arbitrary element of IK takes the form
Ж :	0	bl’ :	Г о	b
	b	+ n	-b+	0
-b* ..	-b* 1	n	0 .	
(1.6.63)
Coset representatives are obtained by exponentiating:
exp JK =
(1.6.64)
+ 1-
with r = (1 - x x)2.
n+1
Ф
The representatives (1.6.64) are bounded by 0<x x<l whereas the
projective coordinates
(1.6.65)
are unbounded.
The projective coset representatives are points in the complex
projective space <CPn.
210
1,6,5 - Infinitesimal transformations and Killing vectors
We consider the transformation law
gL(y) = L(y')h
(1.6.66)
for infinitesimal g:		
8 = 1 + ЛА	’	ТЛ E E	(1.6.67)
h = 1 - eAwiA(y)T.	,	T. E И 1	(1.6.68)
'a a Av a. . У = У + e Кд (у)		(1.6.69)
The induced h transformation depends in general on the infinite-
.	A
simal G-parameters e and on y, as shown in Eq. (1.6.68). The y-
dependent matrix WA(y), defined by (1.6.68), is sometimes called the
H-compensator. The shift in the coordinates {y} is also proportional
д
to e , and the у-dependent differential operator
КА(У) E KAa(y)	(1.6.70)
Эу
is the Killing vector on G/H associated to the G-generator Тд.
The variation of L(y) is then expressed as
L(y’) - L(y) = €AKa L(y)	(1.6.71)
and Eq. (1.6.66), after insertion of (1.6.67-69), becomes
ТдЕ(у) = Кд(у)Е(у) - L(y)T.WA1(y) .	(1.6.72)
211
Consider now the commutator g” g, g_ g acting on L(y). If
е1-1,еЛл-
«z'eAz'i"’'1 '* (1 ’ чЧ'ЬлДвЙьОТ •	(1.6.73)
Let us compute [Тд,Тв]ь(у):
[tA-Tb1LW = TA[TBL(y)] - TB[TAL(y)] =
= TA[KBL - LT.WB4 - (A ++ B) =
= Kb(TaL) -	- (A В) =
= К (К L - LT W/) - (LL - LT W?)T W J - (A B) =
D г\	X r\	M	A r\ J D
= [KB’KJL - LTi^KBWA^ - <KAWB^ + 2cijkWAjWBk] •
(1.6.74)
On the other hand:
Lta>tb]l = cCabtcl = cCab[kcl - LTiwc4 	(1.6.75)
Equating the r.h.s. of (1.6.74) and (1.6.75) yields
1Vb1 - - cVc	(I-6-76’
кв< - Vb‘ * 2с‘)Л‘"вк ‘ ’ с\в"с‘	(1'6'77)
where we have separately compared terms with and terms without Ws,
since the decomposition of a group element into L(y)h is unique, i.e.
L(y)h = L'(y)h’
(1.6.78)
212
implies L(y) = L'(y), h = h'.
Eq. (1.6.76) shows that the Killing vectors -Кд satisfy the
G-Lie algebra. Eq. (1.6.77) is the integrability condition for the
H-covariant Lie derivatives (see later).
1.6.6 - Vielbeins and metric on G/H
Consider the 1-form
V(y) = L'1(y) dL(y)	(1.6.79)
generalizing the left-invariant 1-form g ^dg defined on group mani-
folds (see Chapter 1.3). V(y) is Lie algebra-valued and may be expanded
on the <Б generators:
V(y) = Va(y)T + ^(yjT	(1.6.80)
«*	A
Va(y) =Vaa(y)dya is a covariant frame (vielbein) on G/H and fi1 (y) =
fi^fyidy01 is called the H-connection.
Under left multiplication by a constant geG, L bdL is not
invariant, but transforms as
V(y’) = hL(y)-1g	d(gL(y)h-1) =
= hV(y)h 1 + hdh 1	.	(1.6.81)
Projecting on the coset generators:
Va(y') = (hV(y)h-1)a= Vb(y)Dba(h_1)	(1.6.82)
213
where 1)д (g) is the adjoint representation defined by
g’XTAg = DAB(g)TB .	(1.6.83)
The infinitesimal form of (1.6.82) reads
Ау+бу) - Ay) = - e\1(y)Ca.bVb(y)
6ya = e\a(y)	(1.6.84)
g
easily derived by observing that (Уд are the generators of the adjoint
representation of H, and Ca^ =0. Eq. (1.6.84) implies that the left
action of G on Va(y) is equivalent to an SO(N) rotation on Va(y)
(N=dim G/H), since Ca^b for semisimple G is antisymmetric in a,b.
Projecting (1.6.81) on the H generators yields:
Ay') = (hV(y)h’1)1 + (hdh-1)1 =
= (y)D.i (h-1) + (hdh'1)1 (D ^h'1) = 0)	(1.6.85)
J	a
whose infinitesimal version is
Ay + «У) - Ay) = - C1kjWAkeAfij - е^Ид1 .	(1.6.86)
From Eq. (1.6.72), and using the definition in (1.6.79), we derive an
ejqilicit expression for the Killing vector Кда(у) and the H compensa-
tor Ид1 (y). Multiplying (1.6.72) by L-1(y) from the left yields:
D B(L(y))T = L-1(y)	“(у) - T W Чу) =
A	D	гч 1л	A
dy
= V aK,aT + n 1K.OtT. - w/t. .	(1.6.87)
a A a a A i Ai
214
Projecting on the IK and H generators gives respectively
Ка“(у) = DAa(L(y))Vaa(y)	(1.6.88)
W?(y) = V(y)KAa(y) - VfLCy))	•	(1.6.89)
M	Cl	2Л	2Л
A G-left invariant metric on G/H is given by
я h
Wy) = Yabva (y)VB (y)	(I-6-90:i
where у , is the Cartan-Killing group metric (1.6.19) restricted to
•	CL В
G/H. The invariance of g D(y)dy dyp under the infinitesimal trans-
ap
formations (1.6.84) is easy to prove:
6g = у , Ca- VCVb + у KVaCb. Vе =
6 'ab ic	'ab ic
= C .. VbVc + C, . VbVC = 0
cib	bic
since C, . = - C .	(1.6.91)
hi t'	h	\	j
To show that g o is invariant under the finite transformations
Clp
(1.6.82), it is sufficient to prove the following identity:
C d
Yab = Da (h)Db (h)Ycd h £ H	(1.6.92)
which can be obtained by squaring the definition
h'^h = DaB(h)TB = Dab(h)Tb .	(1.6.93)
The last equality is due to D X(h) =0.
a
215
Indeed
-xXT. xXT.
h'1 T h = e 1 T e 1 = T + /[t ,T.] +
a	a	a L a iJ
* xjsl [[т T ],T ] + ... (1.6.94)
2! a 1 J
produces only Ж generators (t?	= 0).
Squaring (1.6.93) gives
h-1T h h-1T,h = D C(h)D, d(h)T T, .	(1.6.95)
a b a b c a
The trace of (1.6.95), with the Ж generators in the adjoint represen-
tation, yields the identity in (1.6.92). G-invariance of gag easily
follows:
gaB(y'-)dy,ady'6 = gaB(y *dyOdyf	•	(1.6.96)
Exercise: show that gag(y) is insensitive to the particular choice
of coset representative L(y).
Exercise: prove that
y»K 6у\ (уа.ву‘)К
е е = e	e	(.i.o.y/J
i.e.
L(y)L(6y) = L(y + 6y)h
with
-бу^ a(y)w X(y)K.
h = e a a 1	.	(1.6.98)
216
Vaa(y) is defined as the inverse of Va3(y) :
V “(y)V Ь(у) = 6Ь , V = 6^	(1.6.99)
CL	U,	d	LI d	LI
(1.6.97) is a particular case of (1.6.66), and expresses the transforma-
tion of an infinitesimal displacement dx (at the origin) under left
multiplication by a coset representative.
Curved and flat coset indices are connected by V^a(y):
(vector)3 = Vaa (vector)01	(1.6.100)
Example: vielbein, H-connection and metric on Sn
We use the stereographic coordinates (1.6.10).
-2z°dz - 2z°z 4z°:
P P
z2 + 4
z*dz
z :
P
, 2 ..2
(z +4)
4dz
Z2 + 4
dL(z)
4dz
L (z)dL(z)
- 2pdzA)
-4dz
_____P
z2 + 4
8z z*dz
P
(z2 + 4)2
z“dz
____P
2 .
z +4
16z-dz
(z2 + 4)2
o ° J
8z z*dz
(z2 + 4)‘
= V a(z)T dz01 + 1(y)T.dza
a v J a a i
0
(1.6.101)
217
Vielbein:
V a(z) =
a v J
4<5a
a
z2 + 4
(1.6.102)
H-connection:
(z) =
a v
%ab(z) =
2(6a2 - V j
(z2 + 4)
(1.6.103)
since
,T sAB	,AB ЛАХВ ХВЛА
СТа] = (Та п+Р = багп+1 ' 6абп+1
(Т.)АВ = (Т , )АВ = 6^А6В^
1 v v ab' a b
(1.6.104)
Metric:
16 6„a
M(Z) = YabVaa(2)VB CZ) = - 777^
(Y . = ТгГс c, ] = - <5 , )
41ab	L a bJ ab'
(1.6.105)
As a check, we compute the length of half a meridian on Sn:
/•South pole
J equator
ds = |/-ga6dzadzB = jo
I 1 я
= 2 arc tg = —
'O 2
4	Г1 1
—Z— dz.	=	----- 2dt
z2 + 4	(z=2t) Jo t2+l
(0 < гг < 2, z2 = 0) .
(1.6.106)
The metric tensor at the origin of G/H is just the Cartan-Killing
metric of G restricted to G/H:
я h
= YabVa (°)VB (0) = YaB
(1.6.107)
218
since
V a(0) = 6a
a	a
(L’1(y)dL(y)l	= dyaT )
ly=O a
(1.6.108)
Invariance of the metric means
ds2 ° gaBWdAY)dye(Y) = gaB(0)dya(0)dye(0)	(1.6.109)
or
эуа(о) ayS(o)
M(Y) = YaB -----------~		(1.6.110)
3y'(Y) 3y°(Y)
Comparing (1.6.110) with the definition:
= YabVya(Y)V6b(Y)	U-6-11*)
we arrive at the following expression for the vielbein V^a(Y):
Va(Y)=-^l	(1.6.112)
“ 8ya(Y)
and we can interpret the vielbein at a point Y as the matrix connect-
ing the two infinitesimal displacements dya(0) and dya(Y):
dya(0) = Vaa(Y)dya(Y)
(1.6.113)
219
1.6.7 - Covariant Lie derivative
For an arbitrary tensor	, the (ordinary) Lie derivative
along a vector v is defined by (cfr. Section 1.1.7)
JU T Q(x) = va — T о + (Э vY)T „+...+
v a... В	gxY a...В a y---B
+ Ocvr)T
В a. ..у
(1.6.114)
so that SL_^ generates a general coordinate transformation with (in-
v
Y
finitesimal) parameter v . For example, the Lie derivatives along the
Killing vectors of the vielbein and the H-connection are:
- Va ’ ‘Va4*
‘кА1  V»1 ’ '’«“лЧ1 
А
Note that Я only acts on curved indices,
v
The Lie derivative on p-forms is defined by
= v|dw + d v|w
v
(1.6.117)
(cfr. Section 1.1.7).
Writing (1.6.117) in components one retrieves the definition
(1.6.114) in the case of antisymmetric tensors.
Exercise: prove this.
Since generates coordinate transformations ^->y + ev, we have
v
. , ,	,. w(y + ev) - ш(у)
Я^ш(у) = lim —“---------------
v £-> 0 e
(1.6.118)
220
and the transformation laws of the vielbein and H-connection (Eqs.
(1.6.84) and (1.6.86)) can be written as
£к/а(У) = <(у)са1Ь?(у)
Якд^(У) = cljkwAk(y)nJ - dw/fy) •
(1.6.119)
(1.6.120)
The H-connection transforms as a gauge-field, but note that the func-
tions Ид1(у) are not arbitrary, but are fixed by (1.6.89). Eqs.
(1.6.119) and (1.6.120) can be combined into a single formula for the
infinitesimal variation of V(y) = L”^(y) dL(y) :
ЯКдУ(У) = dWA(y) - [wA(y], V(y)]	.	(1.6.121)
We recall some properties of Lie derivatives:
i) [Я , d] = 0 i.e. the Lie derivative commutes with the
v
exterior derivative.
ii) , Я ] = Я . Hence
v u |v , uj
[£K ’	= ' C AB fcK
Ad	C
with K,, KD, К • Killing vectors	(1.6.122)
A В L
since [k.,K_] =- CC,DK_ and const. Я+= Я	.
1 A’	AB С	const, v
The integrability condition for (1.6.119) yields
якЛ - Va - Vwb1 = - cCabwc	t1-6-123)
ah i
with (WA) =WA C£ . This formula was already derived in (1.6.77).
221
The transformation laws (1.6.119), (1.6.120) suggest the defini-
tion of an H-covariant Lie derivative L :
-------------------------------------- KA
\ 5 \ (1-6Л24)
A A
where T. acts as (C.)	on Ж and as (С.)Л=С^.. on И.
1	' i a ia	i j ij
Then
L„ Va(y) = 0	(1.6.125)
KA
LK Я\у) = dWAT(y)	.	(1.6.126)
A
For later use we define the action of L on the coset representative
rA
L(y) as
LK L(y) = KA(y)L(y) - L(y)T.WA1(y)	(1.6.127)
A
with T. in the same G-representation as L(y). Eq. (1.6.72) implies
LK L(y) = TA L(y) A	(1.6.128)
so that [Ч’Ч)Цу)   cCA»Vly)  Lrc ,LW  A B	c	-c ABKC = L[KA.KB]L(y)	(1.6.129)
222
Exercise: prove that
Ч’Ч1 "Чл!
(1.6.130)
also on the vielbein Va and on the H-connection fi1.
1.6.8 - Geodesics
Geodesics through the origin of
G/H are obtained by exponentiat-
ing straight lines through the origin of К
(the "coset algebra"):
exp (tA) A e IK
(1.6.131)
In the off-diagonal representation of Ж
(cfr. (1.6.23))
cos t/в^В
exp (tA) = exp t
-B+ 0
В
sin t
„ sint/в^В
D —~" i ---
/в+в
cos t/B^B
(1.6.132)
0 В
for compact G/H.
The geodesic coordinates
are then given by:
x(t) = В -1П B
B+B
(1.6.133)
Theorem:
The length of the geodesic connecting the origin t = 0 and the point
д
t = 1 (i.e. the element e ) is equal to the length of the vector
A e K, i.e.
223
dn,eA) = ||А|| = /- YabAaAb
(1.6.134)
with А = AaX& , {Х^}=basis for Ж and у =Cartan Killing metric
restricted on G/H.
д
Proof: d( ‘n , e ) =
1 V-g „(t)dXa(t)dX₽(t) =
t=0 D
= j* X /-yabVa(t)Vb(t)dXa(t)dXe(t)
= Г	/-Ynh [L-1 (t) dL (t) ] a [L-1 (t) dL (t) ]b
Jt=0
= [	/-y . (e'tAAdtetA)a	(e’tAAdtetA)b
Jt=O	ab
= [	J-y , AaAb	dt	=	J-y , AaAb	=	|| A	|| .
Jt_Q	ab	ab
(1.6.135)
A
Any other line connecting the origin and e has greater length, since
in the IK linear vector space it would correspond to a curved line
connecting 0 with A
Ж-vector space
(1.6.136)
224
Непсе е (0 < t< 1) is really the coset representative of a geodesic
between 11 and the point represented by on G/H,
Example: Sn
A = b3!
b1 ’
bn
0 .
exp (tA) =
„1	,1 sin tb
x = о —-----
cos tb
(1.6.137)
д
A vector AeK such that e
f*)
must have
represents a point on the equator
||A|| = / -Yabbabb = b = n/2
(1.6.138)
so that x^ = x^ = .. .xn = 0, xn+1 = cos b=l (coordinates of the equator).
Thus: rr/2 = length of a geodesic connecting the pole and equator of an
n-sphere of unit radius.
Def: the distance of a point у £ G/H and the origin 0 is the length
of the shortest geodesic connecting 0 and y:
d(O,y) (=d(H,L(y))	.	(1.6.139)
As entries of the distance function d, we use indifferently coordinates
у of points in G/H or their coset representatives L(y).
(*) We take here the "natural" metric у , =-6 , rather than the
ab ab
Cartan-Killing metric restricted to Sn (proportional to -6 , ,
see the example at the end of Section 1.6.10).	atl
225
Theorem:
The length of a curve on G/H is group invariant. This is intuitively
obvious since the infinitesimal lengths are G-invariant
ds(y') = ds(y)
(1.6.140)
A corollary of (1.6.140) is that geodesics through the origin are mapped
into geodesics through у via (left) multiplication by L(y):
geodesics through 0
geodesics through у
(1.6.141)
We can therefore compute the length of a geodesic connecting any two
points x, у on G/H:
d(L(x), L(y)) = d(fl, L-1(x)L(y))
(1.6.142)
and define the distance between x and у as the length d(L(x), L(y))
of the shortest geodesic x-»y.
1,6.9 - Invariant measure
An element of volume at the origin
dV(0) = dxX(0) ~ dx2(0)--------dxn(0)
(1.6.143)
can be moved from 0 to у by the group operation L(y), and becomes:
dV(y) = dx\y) ~ dx2(y) ^ ... .dxn(y) =
= detlV^y)!’1 dx^O).... ,dxn(0)
(1.6.144)
226
An invariant measure can therefore be defined:
dp(y) = det|Vaa(y) |dV(y) dp(y) = dp(O) .	(1.6.145)
Theorem:
dp(g) = dp(y)dp(h) if g e G
у e G/H
h e H .	(1.6.146)
Proof: det|VAB(g)| = det|Vaa(y)| det|V.j(h)|
since h 1dh has no components along the К generators.
Integration of (1.6.146) yields
vol G = dp(g) =	dp(y) dp(h) =
' g e G	J у e G/H ' h e H
= vol (G/H) • vol (H)
(1.6.147)
so that the volume of coset spaces G/H is just vol G/vol H.
Exercise: find the volume of Sn.
1.6,10 - Connection and curvature
The differential properties of V = L \1L are expressed by the
Maurer-Cartan equation:
dV + V л V = 0	,	(1.6.150)
an immediate consequence of the definition of V:
dV = dL’1 л dL = -L-1dL L'1 л dL = -V л V .	(1.6.151)
In components (see (1.6.80)) we have:
dVa + - Ca Vb л Vе * Ca Vb л Q1 = 0	(1.6.152)
2 be	bi	v	J
dfi1 +1C1 , Va л Vb + i cV.iP л Qk = 0	(1.6.153)
2 ab	2 jk
The torsion 2-form is defined by
Ta = dVa - Bab Vb	(1.6.154)
where the 1-form Bab is the spin connection (see Chapter 1.2).
The spin connection defines parallel transport on the manifold.
The simplest choice for Bab corresponds to vanishing torsion on G/H
and	dVa - Ba, л Vb = 0 D Ba. is then called a Riemannian connection, b		 Combining Eqs. (1.6.152) and (1.6.155) yields	(1.6.155)
For	vC * ca. .fi1 b	2 be	bi symmetric G/H spaces, Bab takes the simple form	(1.6.156)
The	Ba. = Ca. .Q1 b	bi curvature 2-form is defined in terms of Ba, as 	 b	(1.6.157)
	r5 = dBa„ - ва л Be. = Ra ,vc , vd . Ъ	b	e	b	bed	(1.6.158)
Substituting (1.6.156) into (1.6.158), using the Maurer-Cartan Eqs.
(1.6.152-153) for dVa and dfi1, and using Jacobi identities for
228
products of structure constants, one derives the following formula for
the curvature tensor:
r>a — —	— P^ pi	1 p^l p® . i p^ pC
bed 4 be cd 2 bi cd ' J ec bd 8 ed be ’
(1.6.159)
Exercise: derive (1.6.159).
Exercise: prove the symmetry of (1.6.159) under (ab) +-* (cd) inter-
change.
The Ricci tensor
%d “ R bad
(1.6.160)
is easily obtained from (1.6.159) by contracting a and c.
Notice that for symmetric algebras
R . =------у .
ab 4 'ab
(1.6.161)
because
у , = - cA ncD = - c1 ,cd 'ab	aD bA	ad bi	- cd .c1 = 2Cd .c1.. ai bd	ai db	(1.6.162)
Exercise: prove that for symmetric	G/H	
D Ra , = 0 e bed		(1.6.163)
where the covariant derivative D is constructed via the Riemannian
e
connection Ba in (1.6.156):
229
О = a * П?(у)С.	.	(1.6.164)
сии	л
Example: the round Sn:
SO(n+l) algebra:
[TAB’TCd] = 7 6ADTBC + 7 6BCTAD " 2 6ACTBD " 7 6BDTAC
A, B... = 1,.. .,n+l
(1.6.165)
Structure constants
c[AB)[cdJ[ef] Фс/пЧ1 * I W? sf ’
- J sce{dA ‘f1 - J {dfscA $ •
SO(n) generators: T^, a,b = l,...,n
coset generators: T^ n+i =
SO(n+l)/SO(n) is a symmetric coset, so that structure constants with
all three indices in К directions vanish. This greatly simplifies the
expressions for the connection and the curvature.
metric:
Y =Yr ir i = ctEFl
ab ' 4a, n+1] [b, n+1]	[a> n+1] [cd] [b> n+1] [EF]
= _ 4Je’ n+1J Jcdl	=	(1.6.167)
[a, n+1] [cd] [e, n+l][b, n+1]
230
- - 6e6, [бсб?* - 6d6f] = - - [пб, - 6 , ]
2 с daL е b е bJ 2 L а^ abJ
= - 1 (П - 1)6 .
2	ab
Riemann connection:
a, b run on К = Sn
i runs on H = S0(n)
В a. =
a b
Ca. .П 1 = Ca, r
bi a b [cd] a
гЗ	г а
- 0 z, + о , z
a b ab
z2 + 4
(1.6.168)
We have used the stereographic coordinates za (cfr. (1.6.7)) and the
explicit expression (1.6.103) for	Notice that for Sn the H-
connection coincides with the Riemann connection.
Curvature:
na	1 _a „i 1 xa с x ie if]
bed 2 bi cd 4 [e f]b c d
= 4 [6|c 6d|b - 6b|c 6d|]	C1-6-169)
and
Dab bb’Da	2 xbb'Da	1 Jab]
R j = Y R i_f j = - ----------<S R ii j = ---------6r ;	(1.6.170)
cd y b'cd (n-!)	b'cd 4(n_x) [cd] 1	J
231
1.6.11 - Rescalings
In general the metric:
Wy) = YabVa(y)VB(y)
(1.6.172)
is not the only G-invariant metric on G/H. Let us study the extent
of this non-uniqueness.
First, consider the tensor y^,» i.e. minus the Killing metric
restricted to G/H. By an appropriate choice of basis in у ,
ab
can always be brought to the form:
(1.6.173)
The tangent group, i.e. the group of local rotations on the vielbeins
leaving g^g unchanged, is SO(p,n). If either n or p vanish, G/H
is compact.
We can choose the basis of (E generators once for all: as far
as the metric (1.6.172) is concerned, tensor transformations on y^
are equivalent to the same transformations on the vielbeins. We there-
fore assume у , as in (1.6.173), and consider the invertible linear
ab
mappings
Va = Mabv'b det M / 0
Mab e R .	(1.6.174)
Now we ask ourselves under which conditions the new metric
gaB
= у KMaCv'C Mbdvid
'ab a В
(1.6.175)
is still a G-invariant one.
232
г*)
A real nonsingular matrix M always admits the decomposition '
0 : (pseudo) orthogonal matrix
D : real diagonal matrix	(1.6.176)
The (pseudo) orthogonal part of M has a trivial action on Eag: by
construction the metric is insensitive to SO(p,n) rotations of the
vielbeins. The interesting part of (1.6.176) is D, and essentially
different metrics are obtained by rescaling the vielbeins with D:
V = DV' => Va = г%'а
(no sum on a)
(1.6.177)
An arbitrary rescaling (1.6.177) will in general destroy the G-
isometry of G/H. For example, on Sn only the uniform dilatation of
all directions maintains the SO(n*l) isometry. If some directions
expand differently from others, the resulting "squashed" Sn has a lower
symmetry (see next section).
The rescaled vielbeins Va' transform under left multiplication
by G:
v'a(y + 6y) - v'a(y) = -е\ X(y) ^Ca.bv'b(y) .	(1.6.178)
a
The new metric
= W«a(yJVBb(y)	(I-6-179)
(*)
A particular case of the Iwasawa decomposition
M = ODN
with N = nilpotent matrix. When det M/0, N is absent.
233
is invariant under (1.6.178) only if r,/r =1, for a,b such that
v,	ba'
cb. /0.
la b
If C ia is block-diagonal in some subspaces Sj, S2-.. of JK,
we can satisfy г^/га=1 by choosing a common rescaling tj. for all
the vielbeins within the same block S*. Then 6g' = 0 under the
variation (1.6.178), and we have a G-symmetric rescaling.
We summarize this result in the following
Theorem: a rescaling
“1
va2 -> r va2
a2
a^ labels the subspace
&2 labels the subspace
(1.6.180)
is a G-symmetric rescaling if and only if
(C.) b (= Cb. )
1 j?a	ra
al a2
is block-diagonal in the spaces spanned by V , V ,... .
The number N of rescaling parameters, i.e. the number of para-
meters necessary to specify the particular G-invariant metric, is equal
to the number of irreducible blocks of (C.) b. This matrix describes
i a
how H acts on the subspace K. If H acts irreducibly, the coset is
called isotropy irreducible, and only the trivial rescaling Va-*rVa
(same r for all a) is G-symmetric. If G/H is isotropy reducible,
we have an independent rescaling parameter for each irreducible sub-
space.
The rescalings must be non singular (r=0, r = °° are excluded),
but are otherwise unconstrained. We now derive the rescaled expressions
for the connection and the curvature.
The Cartan Maurer eqs. become(*)
dropping primes on vielbeins.
234
dVa + 1 -A_£ ca. vb л Vе + — ca, ,vb л fi1 = 0
2	be	bi
1	17&	1 ?Ь 1	Z~\J Z~l^ ГХ
dQ + — г г. С , V л V + - C .,QJ л Q = 0 .
2	a b ab	2 J*
The zero-torsion condition (1.6.155) determines B3^ up
Ka^c,	symmetric in	b,c:
Ba = 1 AS ca vc + — Ca Q1 + ка, Vе
b 2 r	be	r	bi be
a	xa
Ka^c is determined by the requirement Ba^ + B^a=0 (i.<
Riemann connection. Indices are raised and lowered with
a _ ra a r jc %
- 2 be	•
and
ва = + A c\ vc(ab) * ca,.
b 7 be 1 c7 bi r
a
with
, r r r, r r r,
^ab) _ a e + b c a b
The Riemann curvature, defined by (1.6.158), is now:
Ra. , =
bed
r r
1 a re ,ab. c d £ „a i
' 4 be cd1 eJ re 2 bi cd
r r, -
c d
.ae. .be. 1 „a .ae. .be.
c I d 8ed bc^ d ^ c
(1.6.181a)
(1.6.181b)
to a tensor
(1.6.182)
„a
s. В , is a
b
у , ). Then:
'ab7
(1.6.183)
(1.6.184)
(1.6.185)
(1.6.186)
_ 1 ca ce
8 ec bd
235
The Ricci tensor for symmetric G/H is
R . = - - Cc .C1 . r r, = - 1 у , (r )2 = - 1 у Ur,.)2 .
ab 2 ai cb c b 4 'ab1 a 4 'ab1 bJ
(1.6.187)
(cfr. (1.6.161). Note that Са^ is block diagonal).
1.6.12 - A note on the isometries of G/H
The "natural" isometry group of the coset space G/H is G, and
we have seen in Section 1.6.6 how to construct a G-invariant metric.
Our analysis, however, has been restricted to the left action of
G on G/H. For example, the metric (1.6.172) is a left invariant
metric, and the transformation law considered in (1.6.5) expresses how
L(y) changes under left multiplication by geG.
One can also examine what happens to L(y) under right action of
G. The left action of G induces SO(N) rotations on the vielbein, thus
leaving Rag(y) invariant. What will be the vielbein transformation
law under right action of G? Is there a subgroup of G such that its
right action on G/H only rotates the vielbein? This subgroup would be
an additional isometry of G/H.
We start by studying the behaviour of the coset representative
L(y):
L(y)g = L(y')h : right action of geG on L(y) .	(1.6.188)
For the expression L(y)g to make sense, it should not depend on the
choice of coset representatives. This happens if and only if g belongs
to the normalizer of H in G, denoted N(H), and defined by
gHg’1 = H «• g e N(H) .	(1.6.189)
Indeed one can easily verify that L(y)'vL(y)'	(i.e. L(y) and L(y)'
belong to the same coset) implies L(y)g,v L(y) 'g if and only if
geN(H).
236
This discussion was not necessary for left multiplication since
gL(y) is well defined for every geG.
It is clear that if geH, its right action on L(y) is trivial:
it does not move the point у on the coset space. Thus we need to
consider only elements of N(H)/H, which has a natural group structure.
We proceed to prove that N(H)/H is the right isometry group of
G/H.
Consider the transformation law of the 1-form V(y) under right
multiplication by geN(H)/H:
V(y') = L(y')dL(y') = hg_1(L-1(y)dL(y))gh-1 + hdh-1 .	(1.6.190)
Projecting on the coset generators T& we find:
Va(y') = (hg-1V(y)gh"1)a = Vb(y)Dba(gh-1) =
= Vb(y)DbE(g)DEa(h-1) = AyJO^gJO^OT1) .	(1.6.191)
where we have used (h ) = 0.
Infinitesimally, taking g and h as in (1.6.67-68):
Va(y + 6y) - Va(y) = - eAVB(y)CABa + eAWA1(y)Vb(y)Ca.b =
= ^Ab + И>Ка.ь)УЬ(у) - ebCabV(y) .
(1.6.192)
Thus, the right action of g on the vielbein induces an SO(N) rotation
of Vb(y) if and only if ebCab^ = 0 for every a,i. This happens if
the generators I^c N(H)/H commute with H, since this implies
b -» N(H)/H
Cabi = 0 if i + H	.	(1.6.193)
a + G/H
237
Now, the generators of N(H)/H are defined to act on U as
И] c JH «ъ e N(H)/H	(1.6.194)
which is just the infinitesimal form of gHg-1 = H. Reductivity of G,
however, requires
[l^, H] с Ж	.	(1.6.195)
Eqs. (1.6.194) and (1.6.195) together imply
[l^, H] = 0	.	(1.6.196)
Therefore the generators K^ of N(H)/H commute with H, and conse-
quently N(H)/H is an isometry of G/H.
The Killing vectors Кд(у) of N(H)/H and the corresponding
H-compensators Ид1(у) can be derived as in Section 1.6.6, Eqs.
(1.6.88-89). Consider first the infinitesimal form of L(y)g = L(y')h,
i.e.:
L(y)T = К (y)L(y) - L(y)W 1(y)T.	.	(1.6.197)
A A	Al
Multiplying by L l(y) on the left:
TA = L'1(y) -Ц?	‘ TiWA1(y) =
Эу
= V aK.aT + n 1K.aT. - w/т. .	(1.6.198)
aAa a A i Ai
Projecting on the Ж generators T^ yields:
к“(у) = v“(y)	(1.6.199)
cl	a.
238
к.“(у) = о
(1.6.200)
Eq. (1.6.200) is consistent with the fact that the right action of H
is trivial on G/H. Eq. (1.6.199) gives the Killing vectors correspond-
ing to the right action of N(H)/H: they are just the inverse vielbeins
vaa(y).
Projecting Eq. (1.6.198) on the F generators yields
w/(y) = 6A ’ fia1(y)KAa(y) ’	(1.6.201)
It is evident from Eqs. (1.6.66) and (1.6.188) that left- and right-
isometries on G/H commute.
Exercise: check that left and right Killing vectors commute.
From the preceding discussion, the reader could infer that the
isometries of a coset space G/H are at least
N(H)
H
G x
(1.6.202)
In most cases this is indeed correct. However, there are two instances
in which (1.6.202) fails to give the actual isometry group;
1) Some of the right Killing vectors coincide with left Killing
vectors. As each right isometry commutes with each left isometry,
these common Killing vectors can only correspond to explicit U(l)
factors occurring in G and N(H)/H. The isometry of G/H is there-
fore reduced to
G' x N(H)/H	(1.6.203)
where G=G* x (common U(l) factors). This happens whenever G contains
explicit U(l) factors: their right and left actions clearly coincide,
as they commute with all of G. An example if provided by the coset
spaces N^r discussed in Chapter V.6.
239
2) The symmetry may be larger than (1.6.202). This happens when the
coset manifold can be described by more than one quotient G/H. If
G/H~G/H, with GdG the maximal group for which this is possible,
the true isometry group of the coset manifold will be
G x N(H)/H	(1.6.204)
modulo the considerations in 1). A classic example is given by the 7-
7	7
sphere S : as a coset space, S can be written in many ways:
S0(5) ~ SU(4) .. SO(7) ~ S0(8)	(1.6.205)
S0(3)	SU(3) G2 S0(7)
In the first two cases, the isometry group is in general GxN(H)/H,
but is increased to S0(8) by a particular rescaling of the vielbeins
(see later the example of S0(5)/S0(3)).
On S0(7)/G?, the unique SO(7)-invariant metric is also S0(8)
*	7
invariant, so that SO(7)/G2 is the round S .
Symmetric rescalings
We now discuss rescalings preserving the full GxN(H)/H iso-
metry. Recall the transformation laws of the coset vielbeins:
left action of G:
Va(y+6y) - Va(y) =
- еА[о/(Ь(у)) - KA6(y)VBi(y)] Ca.bVb(y) .	(1.6.206)
(A runs on (E)
240
right action of N(H)/H:
Va(y+6y) - Va(y) =
' ^^bc + Vb6(y4i(y)CaiJvC(y)	(1.6.207)
(b runs on N(H)/H)
By the same argument used in Section 1.6.11, it is clear that if
(Cb)ca and (cpca are block diagonal in the same subspaces S^,S2...
of K, the vielbeins of these subspaces can be independently rescaled
without loss of G*N(H)/H symmetry.
We have therefore the following extension of Theorem (1.6.180):
Theorem: A rescaling
a2
• V
a2
(1.6.208)
is a G*N(H)/H symmetric rescaling if and only if
(CD)ba D runs on N(H)
fli a?
is block-diagonal in the spaces spanned by V , V ,...
1.6.15 Some examples
SO (5)
SO(S)* 1
The root diagram1 J of SO(5) is:
(*) We recall that in the Cartan basis (H ,E.), the commutators are
AX	ал
[н ,E J = г E. r = а-component of the root r corresponding
а л ал a ,	°
to E^
[E. ,E ] = 2 rXH
L X -XJ £ a a
241

(1.6.209)
is
Fp ?2> Hi» H2’ More precisely, it
El’ E2’
generated by the combinations
S0(3)I;	(E1 + E2), E., = (Ej - Ep,
E3 = - (Hj + Hp	(1.6.210)
S0(3)J:	= i/A (F1 + F2), F2 = (F^Fp,
F =i-^(H. -H,)	(1.6.211)
a 2 i z
Note that we are forced to introduce complex combinations of E, F and
H generators in order to have a compact S0(3) *S0(3). If the i fac-
tors were omitted in E^, E^, JF^, F^, the resulting algebra would be
footnote cont'd...
7
[e. ,E 1 = N, E, with N7 = n,(l + m,) ------------—
L X’ pJ Xp X+p	Xp Xk X' 2
where n,, m, are defined by the following conditions:
Л Л
r^+nr^ is a root and r^ + (n+l)r^ is not a root
r^ - nr^ is a root and r^ - (m+l)r^ is not a root
Furthermore note that r = -r^.
242
S0(2,l) xS0(2,l). In a compact coset space, the coset generators must
4-	Л.
be antihermitian (see Section (1.6.3)). Since Q' = Q o' = Qo, we
b 7	о
consider the four antihermitian combinations:
% = +
"51 = % - Q8
0?2 = - HQ6 * Q8)
(1.6.212)
The S0(5) structure constants in the E, IF, ф basis are
c\ = jk	Eijk		eij£
c° =	- 6 .	• C^n =	- — 6 .	= — 6
ia	2 ai	10	2 ai ’ aO 2 ai
c2 =	- - 6	• c?n =	- 6 <•	, c\ = - - 6 «
ia	2 ai	’ 10	2 ai	aO	з ai
	£	„i	£
C. la	2 Eiab	’ Cab "	3 Eiab
	1	„i	1
Co = ia	— , 2 iab	, C ,= ab	"з eiab	•	(1.6.213)
with the index conventions:
i,	j,	к	run on S0(3)X	
i,	j.	к	run on S0(3)J	
a>	b,	с	correspond to фр Щ2, Q3	coset indices
	0		corresponds to	
243
The Killing metric
n г
УАВ = C ACC BD A‘ B-‘- run on SO(5)
is given by
Y-. = - 36.., ym=- Збос, у , = - 26 , ,	= - 2
ij ij 'ij ij 'ab ab’ ’00
with vanishing off-diagonal parts.
We examine next the possible SO(5)-symmetric rescalings of the
i а П
coset vielbeins V , V , V .
From the root diagram, it appears that the 7-dimensional coset
space S0(5)/S0(3) splits into 5 irreducible subspaces under the
action of S0(3)\ namely the three singlets F^, F2, F3 and the two
doublets (Q^,Qy) and (Qg,Qg). However, because of the change of
basis (1.6.212) necessary to obtain a compact coset, the matrix (C\)
is not reducible any more in the dj subspace via a real tensor trans-
formation on the Q's, and mixes the 0,a directions	Са^ are
nonvanishing). According to Theorem (1.6.180), the rescalings that
preserve the S0(5) isometry involve four independent parameters
r1 (i = 1,2,3), r
V1' = r¥ Va' = rVa V°' = rV°	(1.6.214)
Before proceeding to compute the rescaled curvature, we observe
that the symmetry of S0(5)/S0(3)^ is actually greater than S0(5).
Indeed the normalizer N of S0(3)^ in SO(5) is
БО(З)1 x S0(3)J	(1.6.215)
and N/H is therefore S0(3)^. According to Theorem (1.6.208), the
full isometry of S0(5)/S0(3)^ is
244
G x N/H = S0(5) x S0(3)J
(1.6.216)
and the rescaling preserving this isometry involves only two parameters
since
DB
a,6
a, i, 0 G/H
1, i -* N(H)
(1.6.217)
D
is block diagonal in the spaces {i} and {a,0).
We therefore consider the rescalings
V1' •* aV1
Va' + bVa
V0' -> bV°
(1.6.218)
Applying formula (1.6.186), the rescaled Riemann curvature reads:
2
’’bed ‘ (5ае6Ы - Wbe>‘8 ‘
a
2
R оьо - - W8  —)b
a
Ri = _ л
°j°	48 ij „2
c*
4	2
R‘lbj = - iSlj6»b J* 12 l6a)sib ’ Vl,?» ’
Di _	1 rij 2
R jk£ ’ 4 6kfc a
R1. , = - — (6. 6., - <5..6 .)(— - 2b2 * *)
jab 12 la jb ib aj 2
a
2
1	b „2
R jOa = + 24 Eija(2 ’ ~)b	‘	(1.6.219)
a
245
The corresponding Ricci tensor is
R к
ab
R00
2
6rl_lL_)b2
ab 2	8 2
a
2	1 R4
+ IL.)
ч 4	8 2
a
2
i^)b2
8 a2
ij
(1.6.220)
Let us now look for rescalings a,b such that the resulting space
becomes an Einstein space, i.e. a space for which the Ricci tensor
R^p is proportional to the metric	As discussed in Part V,
Einstein spaces are of special relevance in Kaluza-Klein supergravity.
It is an easy exercises to check that if
b2
(1.6.221)
the Riemann curvature becomes that of the round 7-sphere
аЗ 1 <-aB
Y<S 24 °Y<5
(1.6.222)
This is an interesting illustration of how the symmetry G * N/H of a
coset G/H can be increased by a rescaling that brings G/H to be
equivalent to G/H, with GcG. Here the SO(5)*SO(3) symmetry of
SO(5)/SO(3)I becomes the full S0(8) of the round S2.
Another Einstein space can be reached continuously from S2 at the
value
b2 2
a2’ 5
(1.6.223)
of the rescalings.
Cfr. the example of Sn at the end of Section 1.6.10, with n-7.
R. .
2
a
R
2
1
2
--------------------------------------------------------------------1
246
This space has only the "canonical" symmetry of SO(5)/SO(3) s
i.e.
S0(5) x S0(3)J	(1.6.224)
and is called the "squashed" seven-sphere; it is the only other Einstein
space with the topology of S . Its use in 11-dimensional supergravity
is discussed in Chapter V.6.
The MP4r spaces
Topology and symmetries
Consider the 7-dimensional coset manifolds
I
G = SU(3) x SU(2) x U(l)
H SU(2) xu(l) xu(l)
(1.6.225)
where SU(2) is embedded as an "isospin” subgroup of SU(3), i.e. the
triplet .3 of SU(3) decomposes as 2+1^ under SU(2) . SU(3)xSU(2)x
U(l) has three commuting U(l) generators:
Ag : hypercharge of SU(3), commutes with the "isospin" subgroup
SU(2) c SU(3)
Tj : the isospin of SU(2)
Y : the U(l) charge
The surviving U(l) generator Z in the "coset algebra" is in general
a linear combination	of Xo, T_ and o -5	Y:
Z=ip/3X8 +	| 4T5 * irV	(1.6.226)
I
V	247
The embedding of the two U(l) factors of H is thus characterized by
K the three integers p,q,r, and the corresponding spaces are denoted by
» MPqr.
For Z to be a compact generator, p,q and r must be rational
numbers. Since an overall rescaling is inessential, we can always
® choose p,q,r to be coprime integers.
The topology of the MPqr spaces can be
the quotient in separate pieces: SU(3)/SU(2)
3
SU(2) is S , and thus
5	3
j^qr = s * s0 * u(i)
u(i) x U(l)
For simplicity, consider first	In this
one of the U(l) factors of H is mapped into
j^qO = S5 x S5
U(l)
Next observe that S^n+1 may be considered as
since
s2n+l = SU(n+l) x U(l)
SU(n) x Ufl)
(remember that (CPn = ——	—г see Section
SU(n) x u(l)
c	2	3
Then S is a Ufl) bundle over (CP , and S is a U(l) bundle
1	2
over (CP = S . The factoring by U(l) in (1.6.228) causes the identifi-
cation of the two fibers, and therefore	can be considered as a
U(l) bundle over (CP x S . The identification of the two fibers is done
in such a way that going q times around the U(l) fiber of S$ is
3
equivalent to going -3/2p times around the fiber in S (see Eq.
(1.6.231)); this implies that the topology of	only depends on
the ratio p/q.
understood by considering
is topologically S^,
(1.6.227)
case the generator of
Y. Then
(1.6.228)
a U(l) bundle over (CPn,
(1.6.229)
1.6.3).
I’
248	i
The isometry group of	is at least SU(3) xsU(2) xU(l).
Indeed М₽Ч° may also be written as
SU(3) x SU(2)
SU(2) x Ufl)
CI.6.230)
with the SU(3) xsu(2) symmetry generated by the left action of the
group on the coset space. However, the metric is also invariant under
right multiplication by the U(l) generated by Z, i.e. the extra
generator (outside SU(2)xu(l) itself) in the normalizer of SU(2) x
U(l) in SU(3)xsU(2).
In certain cases the symmetry of М₽Ч° may be even larger. For
example, in ~ CP^ x s$ or = S$ xthe symmetries are
SU(3) x SU(2) x SU(2) and S0(6) x S0(3) respectively.
Returning to the general case of	we may choose one of the
U(l) factors in H to be generated by
z" = - i q /Ъ X + — p т	(1.6.231)
2 й 2
and consequently
rfl'-A'J----------- . xu(l.) .	(I 6 232)
U(l)	U(l)
The last quotient by U(l) is almost trivial: it has the effect of
cancelling the U(l) in the numerator, and factoring the space
by a finite cyclic group. If p and q are not both zero, the
embedding of U(l) x u(l) may be defined by
iC> $Z'/k
e e
(1.6.233)
where
249
Z' = 2r[i p /3 X + 1 q т ] - i(3p2 + p2)Y	(1.6.234)
2	°	2	°
2	2
and к is the highest common factor of 2rp, rq and (3p +q ).
Dividing by к ensures that (1.6.233) is a one-to-one embedding. If
2	2
Ф changes by 2ттк/(3р +q ), then one has returned to the identity in
the U(l) generated by Y: it follows that points in	which differ
by an integer power of
exp [—~~ (|p/3 X + | q т )]	(1.6.235)
L(3p2 + q2) 2	2
must be identified with each other. Thus
^pqr = м₽Ч~	(1.6.236)
Z£
with Я = (3p2 + q2)/k and Z ={1), and the spaces М^г have funda-
mental group (see Section 1.6.14):
IT1(MPqr) = Zfc .	(1.6.237)
Their universal covering space is mP4°.
Rescaled curvatures
We now turn to study the geometry of M^qr spaces. As for any
G/H space, this involves three steps:
i)	to determine the G structure constants in a suitable generator
basis for в> = H ® X
ii)	to determine how the coset vector space X decomposes under
ad (H),
(CHJK ‘
of X.
the adjoint representation of H whose matrix elements are
A rescaling parameter is assigned to each irreducible subspace
250
iii)	to compute the rescaled curvature of G/H.
Of particular relevance for d=ll supergravity are the Einstein
metrics, for which
R „ = const, p _
aB
(1.6.238)
Condition (1.6.238) translates into algebraic equations for the rescal-
ing parameters. If these can be solved, G/H can be rescaled to an
Einstein space.
A suitable basis for G= SU^xSU2 xgenerators is given by*
Z, Z', z" .	(1.6.239)
where Z, Z' and Z" are the generators of the three commuting 11(1) 's
(see Eqs. (1.6.226-231-234)) in G. The subgroup H = Sll(2) x (j(l) x u(i)
is generated by
i [Xr X2, X3] : SU(2)
z', z" : U(l) X U(l)	(1.6.240)
so that the coset directions correspond to
~ Xj., X^, X? ; Tj, T?]	; Z .	(1.6.241)
* In the following X.,...,XO are the standard Gell-Mann matrices
satisfying [X^.X^] = 2i \ where the non vanishing components
of the SU(3) structure constants f. .. are f,_= 1: f, ., = f =
ijk	123	147	246
f257 ° f345 = f367 =‘f156 = 1^2; f458 = f678 = ^2' T1 3116 T2 are the
first two Pauli matrices.
251
On the basis (1.6.239), the non vanishing structure constants read:^
rm nZ	q — E 2 mn	CA = L iB	fiAB
c»-	q — E 2 mn	CA c Z'B	= 2 /3 rp f8AB
CM-	/7 . 2 P f8AB	C Z"B	= * 73 4 f8AB
CA = b BZ	/Т . 2 P f8AB	cm_, Z'n	= 2rq Emn
		c”1 c Z"n	= 3p e r mn
(1.6.242)
By inspection of the C t structure constants, we see that the
coset linear space К splits into three irreducible subspaces spanned
respectively by [Хд, Xp X6, X-,] , [tp т2] and Z. We can therefore
introduce 3 rescaling parameters a, b, c:
Vй = a VA'
Vй = b v"1'
VZ = c VZ'
(1.6.243)
and the SU^ xSU2 xUj - invariant metrics on мРЧг spaces are charac-
terized by the values of a, b, c.
The rescaled Mpqr curvatures are easily derived from the general
formula (1.6.186):
R1”1	= b2(l
rs
2
2
3 bZ 2, /
4 c2
ZYWH

2U2
mn _	/-= a b mn
R AB " ‘ 73 pq . 2	f 8AB
4c
„Zin 2 1Л I
R _ = q —=- —
П 4c2
(*)
The indices have the following ranges: A,B = 4,5,6,7; m,n-l,2.
252
9	2 A
ZB = 16 р с2 6В
mA
R nB
РЧ 4с2 8АВ
е1™1 1
£
AR 1	2	t 2 2
R CD = 2 а (fiABfiCD + C1 ' '^~7’)f8ABf8CD) '
ZC
4 Р 2 SAC 8BD
с
rab
mn
2.2
/V а Ь mn
_ /3 pq е	f gAB
4с
(1.6.244)
and the Ricci
tensor is block diagonal in the A,m,Z indices:
4	2
Dm 1 ,m c. 2	1 b 2i A 3 2,A	9 a 2^
R „ = — о |b----------□	, R = - a 6. 2------------p)
n 2 n 1	2 J B 8 В 2 c2
Einstein metrics on MPqr
In view of the later applications to Kaluza-Klein supergravity,
we investigate here the possibility of having Einstein metrics on
mPV
The question is whether there exists a triplet a,b,c such that
the components of the diagonal Ricci tensor (1.6.245) are all equal.
We distinguish 4 cases:
i) q = p = 0. The topology is that of IP2 *	* S1, which
obviously cannot be rescaled to an Einstein space, since has
vanishing curvature.
3
ii) P=0, q/0. In this case the topology is that of x S ,
and the rescalings
253
a = 4е , b = 4 /Ъ e , с = 2 qe
bring the Ricci tensor Ra to be
p
RaR = 12 e2 6“	.
p	p
5	2
iii) p/0, q=0. The topology is S *S and
a = 2 /6 e ,	b=2 /6 e , c = 3 /6 pe
(1.6.246)
(1.6.247)
(1.6.248)
are the rescalings for condition (1.6.247) to hold.
iv) p/ 0, q/ 0. The topology is no longer that of a direct
product of spaces, and is different for different ratios q/p. For each
q/p there exists an Einstein metric, corresponding to the rescalings
b = у /2f	, c = qy
(1.6.249)
where В is a real positive root of the following cubic equation:
4B3 - 6B2 + (- + 3y)g - 1 3=- = 0
4 p2 2 p2
(1.6.250)
a and у are linked to В by the relations:
n	2
a =	(3₽ - 4B )
q
/ 12e2
•V B(l-B)
(1.6.251)
Equation (1.6.250) has always (for all values of q/p) one and only
one positive real root В whose range is
0 < 6 < |
(1.6.252)
254
We can conclude that for all values q,p (except p=q=0) there is
always one (and only one) SU(3) x SU(2) x(j(l) Einstein metric on
1.6.14 - Elements of algebraic topology
This Section contains a micro-review of homotopy and homology,
and is quite non-rigorous. Its purpose is to recall some definitions
and theorems that will be useful in Section 1.6.15, and in later parts
of this book.
A path in a topological space X is a continuous map of some
closed interval I into X. Two paths, with the same end points, are
said to be equivalent if they can be continuously deformed into one
another.
The two paths fp f2 on the two-dimensional
cylinder are inequivalent: f^
(1.6.253)
The product of two paths AB and BC, defined when the terminal
point of the first path coincides with the initial point of the second
path, is just the path ABC.
If two paths fp f£ are respectively equivalent to gp g2,
their product fj’f2 is equivalent to gj’g2 (the simple proof is
left to the reader), and we can consider the multiplication of equiva-
lence classes. This multiplication is associative.
A path, or path class, is a loop if the initial and terminal
points are the same. The loop is said to be based at the common end
point.
255
The set of all loop classes based at any point x of X is a
group, with the above-defined multiplication. The identity is the
trivial loop, and the inverse of a loop is just the same loop traversed
in the opposite direction.
This group is called the fundamental group of X at the base
point x, denoted by w(X,x). If x and у are two points of X
connected by a path y, we can define an isomorphism u: tt(X,x)тг(Х,у)
induced by Ф-»-у Фу (we use Greek letters to denote path classes).
Exercise: prove that u really is an isomorphism.
The group structure of tt(X,x) is therefore independent of the particu-
lar point x € X.
Example: the fundamental group of a circle is Z (infinite cyclic).
Class representatives are paths wrapping around the circle 0,1,2... 00
times both in clockwise and anticlockwise directions.
0,1,2... are called the winding numbers of the loop classes.
Opposite winding numbers conventionally refer to the same loop traversed
in opposite directions.
Example: the fundamental group of a n-torus is ZxZx ... xz (n-times).
This is immediately proved by using the
Theorem: the fundamental group of a product space is isomorphic to the
product of the fundamental groups, i.e.
?r(X x Y, (x,y)) » tt(X,x) x 7T(Y,y)	(1.6.254)
The fundamental group of a 2-torus
is generated by the paths a and f3.
(1.6.255)
256
The isomorphism between n(X x Y, (x,y)) and ir(X,x) x n(Y,y) is defined
by assigning to each element aen(Xxy,(x,y)) the ordered pair
(pa,qa), where p: XxY + X and q; X><Y + Y denote the projections of
the product space into its factors.
Higher homotopy groups can be introduced via a generalization of
the mapping defining a loop, and are essentially sets of equivalence
classes of closed hypersurfaces. Consider the continuous map
f : In •* X	(1.6.256)
where In is the n-dimensional unit hypercube, satisfying:
f(9In = boundary of In) = x^ e X
(1.6.257)
If the images of two maps fp can be continuously deformed into
each other, fj is equivalent to fp The set of equivalence classes
of mappings (1.6.256) is easily seen to form a group, the n-th homotopy
group of X about the point x^:
лп(Х, x0)
As for the fundamental group, the particular point x^ is inessential
up to isomorphisms. Examples of higher homotopy groups are provided
in the next section.
From their definition, it is clear that homotopy groups are
topological invariants: if two spaces are homeomorphic, their homo-
topy groups are isomorphic.
2	3
The converse is not true: see the example of (EP x S and
5	2
S xS of next section. These spaces have isomorphic for all n,
but are topologically inequivalent.
For a more complete information on topological differences between
two spaces, it is necessary to study their homology or cohomology groups.
These also provide a powerful link between the topological aspects
of manifolds and their differentiable structure.
257
Let M be a smooth connected manifold. A p-chain a is
P
defined by the formal sum:
a = У c. N.	(1.6.258)
p	. i i
where the lb are smooth p-dimensional oriented submanifolds of M.
The coefficients c. can be taken complex, real, integers, Z^...;
for present purposes c^ e R-
Let us denote by Э the operation of taking the oriented boundary.
Then
Эа = У c. SN.
p	i i
(1.6.259)
Cycles
are defined to
p-chains without boundary,
written
be
is a (p-l)-chain.
Boundaries are those chains which can be
Since the boundary of a boundary is always empty (99an=0)
the set of boundaries is contained in the set of
as a =3a for
P P+1
some a
cycles.
P
The set of equivalence classes of cycles of M differing only by
boundaries is called the simplicial homology of M:
set of p-cycles
set of p-boundaries
(1.6.260)
and two p-cycles z , z’ are equivalent if
z* = z + 9a ,	for some a , .	(1.6.261)
P P P+1	P+1
The curves a and b belong to the same
homology class, since they bound the
two-dimensional strip a (a + b = 9a).
The curves a and c are not in the
same homology class.
(1.6.262)
258
Example: the homology groups of the 2-torus are
H0 = R
= » ® 1R
H2 = R	(1.6.263)
De Rahm cohomology
We define the De Rahm cohomology groups as the set of equivalence
classes of closed forms which differ only by exact forms
	hPr(M) =	set of closed p-forms		(1.6.264)
		set of	exact p-forms	
Two	forms w P	and w' P	are equivalent if	
	Ш’ = CJ p p	+ da , P-l		(1.6.265)
for	some a 5			
	p-l			
Note: since the exterior derivative of a constant is zero
H^r(M) = {space of constant functions}	(1.6.266)
and
dim (M) = number of connected pieces of M.	(1.6.267)
UK
Poincare's lemma: The De Rahm cohomology of Rn is trivial, since any
closed p-form can be expressed as the exterior derivative of a p-1 form
in Rn (for p > 0) .
259
Непсе for any manifold M, the De Rahm cohomology is locally trivial,
in the sense that it is trivial in any local Rn coordinate patch.
Only when local coordinate neighbourhoods are patched together in a
globally non-trivial way, the resulting manifold has non-trivial De Rahm
cohomology.
Inner product:
is defined as
the inner product of a cycle c
and a closed form ш
P
(c ,W ) = W £ Ж
p p Jc p
p
(1.6.268)
By Stokes' theorem:
(3cp = 0)	(1.6.269)
(dwp = 0)	(1.6.270)
The inner product is therefore independent of the choice of representa-
tives in the equivalence classes.
When M is a compact manifold without boundary, the following
theorem (De Rahm) holds:
Let {c.} be a set of independent p-cycles forming a basis for
H (M): let {w.} be a basis for HnD(M).
Pl	pk
Then the matrix (c^, w.) is invertible:
det (c^, ок) / 0
(1.6.271)
Hence HPR(M)
is dual to H (M) with respect to the inner product:
simplicial homology and De Rahm cohomology are naturally isomorphic.
260
The p-th Betti number (b ) of M is defined to be the dimension
of the p-th homology (or cohomology) group:
b = dim H (M) = dim HP (M)	,	(1.6,272)
p	p	UK
The alternating sum of the Betti numbers is the Euler characteris-
tic:
X(M) = £ (-1)P b .	(1.6.273)
p=0	p
Poincare duality: нР(M) is dual to Hn"p(M) (n= dim M) with
respect to the inner product
%’ Ln-p’ = L ШР л Шп-р •	d.6.274)
цР(М) and Hn’P(M) are therefore isomorphic vector spaces, and
dim IjP(M) = dim Hn P(M)	(1.6.275)
As a consequence the Betti numbers are related by
bp = bn_p .	(1.6.276)
Product formula (Kunneth)
Нк(Мг x м2) = ф Hp(Mx) x нЧСм2) .	(1.6.277)
p+q=k
For Betti numbers:
Ьк(М1 * M2} =	1 hp^P bq(M2} •	(1-6.278)
p+q=k
261
Непсе the Euler characteristic satisfies:
x(Mj x мр = х(мр х(м2)
(1.6.279)
Using Hodge's decomposition theorem on compact manifolds without
boundary:
to = da . + 6g , ♦ Y
p p-1 Pp+! P
(Y harmonic)
(1.6.280)
de Rahm cohomology classes are seen to be isomorphic to the set of
harmonic forms. Indeed dto=0 implies d66 = 0 so that 6g = 0, and
to=da + Y is in the same cohomology class of the harmonic form y.
Conversely, if to is harmonic, w = Y. and w is closed but not exact
(also: to is co-closed but not coexact).
Thus { set of harmonic p-forms on M } ® H^(M)	.	(1.6.281)
Examples: Rn :
All closed forms are exact except 0-forms eH°. Hence
dim H°(IRn) = 1 (space of constant functions) and dim H°(]Rn) = 0 for
к f 0.
dim H°(Sn) = 1 : space of constant functions
dim Hn(Sn) = 1 : constant multiples of volume element
all other н (Sn) vanish.
262
1.6.15 - Homotopy and (co)homology of coset spaces
Homotopy
It is surprisingly easy to obtain detailed information concerning
the homotopy group of a coset space G/H. While knowledge of the funda-
mental group of a coset space is particularly valuable, one finds that
the higher homotopy groups yield only very basic details of the topo-
logy. More precisely, the higher homotopy groups tell one about how the
topology is changed under various embeddings of the non-abelian factors
of H, such as SU(2) and SU(3). Since there are only very few ways
in which such factors may be embedded, and because such embeddings can
usually be understood quite directly, the information gleaned from
higher homotopy is not great. The more subtle variations in topology
occur through the embeddings of the U(l) factors, and essentially since
TTn[u(l)]=O for n > 2, the higher homotopy groups do not measure these
differences in topology.
For these reasons we will concentrate principally on the funda-
mental group, and return to the question of the other homotopy groups
later.
There are two reasons why one can easily calculate the homotopy
of the coset space G/H. First, the homotopy groups of Lie groups are
known, at least up to (See for ex. Encyclopedic dictionary of
mathematics, ref. [10]). Furthermore, for products of spaces one has
7^ (A x B) =iTn(A) xtt^(B). The second reason is the homotopy exact
sequence for fiber spaces. This says that if (E,p,B) is a fiber space
with fiber F, then there is an exact sequence
A i* p*
•••+irn+l(B)+1,n(F) ” VE) ” VB) ” •••	(1.6.282)
where the maps i* and p* are induced by the inclusion i: F + E and
the projection p: E^B. The definition of the map Д is somewhat
complicated, and will not concern us here. Knowledge that such a map
exists is usually sufficient. It should be recalled that the notation
263
(E,p,B) means that E is the total space, В is the base manifold
and p is the projection map of E onto B. The fiber F is iso-
morphic to p X(x) where x is any point in the base space B.
Furthermore an exact sequence
В -* C
CI.6.283)
by definition means that Image f = Kernel g at each point in the
sequence.
The relevance to coset spaces is that (G,p,G/H) is a fiber space
with fiber H, where p is the projection which takes geG to the
coset gH. Thus we have an exact sequence:
A
-> nn+1(G/H) - тгпСН)
i* P*
\(G) Ч- TTn(G/H)
(1.6.284)
where i* is induced by the embedding i: H-*G. It is important to
stress that the map i must be one to one for the sequence (1.6.284)
to be exact. When considering U(l) factors, i may be written in many
ways, but for (1.6.284) to be applicable, i must be written in its
unique one to one form.
In order to calculate the fundamental group of G/H, consider the
last part of the sequence. For any Lie group, one has tt^IG) =0, and
so one obtains
i* P* A i*
0 -> tt2(G/H) тг^Н) ir^G) * TTjCG/H) •* tt0(H) TrQ(G)
(1.6.285)
For any connected space M, тт^(М) = Z (the
measures the number of components), and the
dimension of 71q over Z
fact that the only compo-
nent of Н maps into the only component of G,
means that i„: iTq(H) +
7Tq (G) is an isomoiphism.
connected). Consequently
(We are assuming that
G and H are both
Im Al .... = Ker i. I .... = 0.
%(H) ^qCh)
Hence we may
replace the sequence with the exact sequence
264
A 	i* P*	A
0 -> -Пг(G/H)	-> 7Гг(Н)	+	tt^G) -	7^ (G/H) -	0
(1.6.286)
Since
Ker \(G/H)=
one has
Dorn p* 7Г (G)
tt^G/H) = Im p, = -------= -2-----
Ker p* Im i*
Furthermore Ker Al ...... =0
ТГ2 (G/H)
and hence
7T2(G/H) = Im A = Ker i* in тг^(Н)
(1.6.287)
(1.6.288)
Therefore the properties of the map i* completely determine tt^(G/H)
and 7^ (G/H). The map i* is readily understood: it simply takes a
noncontractible loop with a particular set of winding numbers in H,
and gives its winding numbers as a loop in G.
Consider the example of M^cir (see the previous section). The
fundamental groups of G and H are given by
tt1(H) = Z ® Z , tt1(G) = Z
(1.6.289)
These groups measure the winding numbers of a closed loop around the
U(l) factors of the group. Since one of the U(l) factors of H is
mapped directly into SU(3)xSU(2), and because Tr^(SU(3) xSU(2)) = 0,
one sees that i* maps one of the Z's in тг^(Н) to zero. If p = q=O
then both U(l) factors of H are mapped into SU(3) *SU(2) and con-
sequently
Ker i* = 7r1(H) = Z ® Z
Im i* = 0
(1.6.290)
265
Hence, from (1.6.287) and (1.6.288):
7^ (G/H) = tt^G) = Z
^(G/H) = Kor i* = Z ® Z
(1.6.291)
(1.6.292)
This is consistent with the earlier observation that M^r - CP^ x x s ,
2	2	7	7
when one recalls that тг^(СР )=tt^(S ) = 0 and ттг (IP ) = rr^ (S ) = Z.
If p/0 or q/0 then the generator Z' in (1.6.234) defines
the embedding, i. In particular this shows that a loop which goes once
around the second U(l) factor of H, goes around the U(l) factor of G
2	2
precisely i = (3p +q )/k times. Therefore, on the fundamental groups,
the map
i* : n1(H) = Z ® Z + tt1(G) = Z	(1.6.293)
takes (a,b) into bi. and so
7Г2 (G/H) = Ker i* = Z
тг/G) _
7T. (G/H) = —--- = — = Z .	(1.6.294)
1	Im i* £Z Z
One now sees the importance of using the one to one map, i, which
defines the embedding of H into G. If one had used an "m to one"
embedding, one would have obtained ^(G/H) = Z
Having classified тг^ and of	let us now consider the
higher homotopy groups. To do this, we have to refine our techniques
a little. Recall that	may be considered as the quotient space
5	3	1
S * S____* S .	(1.6.295)
U(l) x u(l)
266
5	3	1
Clearly this is also a fiber space. That is, S *S xS is an
S x s fibration over wPqr. Applying the homotopy exact sequence
one obtains
* •’Tn+1(MP4r) * ^(S^S1) * Trn[S5xS3xS1) -+
* Trn(MPqr) - ^ ^S^S1) -	.	(1.6.296)
However for n>2, iTn(S1xs1)=O and so one has
0 + irn(S5xS3xs1) -- тгп(МРЧГ) + 0	.	(1.6.297)
Hence
V1^41) = %(S5xS3xS1)	, n> 3	.	(1.6.298)
In other words, the higher homotopy groups give no further information,
in particular they do not tell anything about the role of the ratio p/q
in determining the topology of M₽qr.
A direct consequence of the foregoing is that if p and q are
not both zero, then	is the same for all p and q. That is,
homotopy cannot tell the difference between any of the spaces MPq0. In
particular
2	3	5	2
7Гп((ЕР x s ) = ?rn(S x S )	.	(1.6.299)
Thus one sees both the power, and the limitations of the homotopy groups.
The calculation of w^(M^qr) and ^(M^1") can be generalized to
arbitrary G/H.
Suppose that G=G'xU(l) and H=H' *U(1) where G' is simply con-
nected, and H' is mapped by the embedding, i, into G'. Then
267
i) If i maps all of H into G' ,
tt^G/H) = TTjfG) = Z
tt2(G/H) = тг^(Н) = Z Ф TTjJH') .	(1.6.300)
ii) If i maps the explicit 11(1) factor of H’xU(l) s0 that a
simple loop in this U(l) winds £ times around the U(l) of G' *0(1),
then
TTjfG/H) = Z£
"2(G/H) = TTjJH') .	(1.6.301)
This theorem enables one to calculate the m^(G/H) and tt2(G/H)
for every coset space in the TABLE V.6.1 of Chapter V.6.
A word of caution is advisable concerning cosets with nontrivial
fundamental groups. Let M be a manifold with (M) /0, and let M
be its universal covering space (for example M = М^г and M = M^^^).
If we solve Killing's equation in M, or solve the Killing spinor
equation to determine the surviving supersymmetry (see Section V.4.3),
then these calculations are performed locally, that is, in some coordi-
nate patch. If one is not careful one might conclude that solutions to
these equations exist in M if and only if they exist in M, since
these equations are only expressed in local terms. However, for the
existence of a supersymmetry, or a global Lie group symmetry, these
Killing spinors and Killing vectors must exist globally. When the mani-
fold is simply connected, the solutions which exist locally may be
consistently patched together to produce a global solution, and so there
is no problem on M. However, the manifold M is isomorphic to M/X,
where X is some discrete group, and the condition that Killing vectors
or Killing spinors exist on M is that they exist on M, and are con-
sistent with the factoring by the discrete group X. As a simple example
268
of this, consider the harmonicscos (шФ) and sin (тФ) on the unit
circle. If one divides the circle by the finite group Z^, then the
only well defined harmonics are those where m=k£ for some integer k.
The conclusion is that a coset space G/H will always have a Lie
group symmetry G, however it may appear locally to have more Killing
vectors than those obtained from G. These extra Killing vectors may
not be globally well defined on G/H, and thus one may not have a larger
Lie group symmetry. If the space G/H is simply connected, then the
extra local Killing symmetries can be made into global ones; but if the
space is not simply connected, the possibility of extending to global
symmetries depends on the details of the particular situation, and it
will usually be impossible.
Homology
It is interesting to note at this juncture that there are also
exact sequences on the homology and cohomology of fiber spaces, and
that the complete homology and cohomology of Lie groups is known (cfr.
Encyclopedic Dictionary of Mathematics, Ref. [10]). Thus one can, in
principle, determine the homology and cohomology modules of G/H. It
turns out that these modules are considerably more informative about
the structure of G/H. The difficulty is that it is no longer quite
so straightforward to determine the behavior of the induced maps i*
and p* on the homology. Furthermore, the homology of a product space
is not the product of the homologies. Instead one has to apply the
Kunneth formula (1.6.277) to find the homology of a product, and to
trace the action of i* and p* through these formulae is somewhat
complicated.
However, for a large class of cosets G/H there exists a
straightforward way to obtain the Betti numbers, based on the Poincare
polynomials. These are defined as follows:
( harmonic analysis on G/H is discussed in Chapter V.3.
269
РМ<”  Ьо • V * Ь2'2 •	• V”
М = n-dim. manifold
b. = Betti numbers
i
(1.6.302)
For the classical groups, the Poincare polynomials are given by (see
for example, Ref. [в]):
PU(1) - 1 + t
p	= A(£)	(l+t3)(l+t5) ..	. (l + t2£+1)	+- SU(£ + 1)
PB(£) =	3	7 (1 +td)(l + t')..	,. (l + t4^1)	+ S0(2R. + 1)
PC(£) =	3	7 (1 + td)(l + tz)..	..(1+t4*-1)	Sp(2£)
PD(£) =	(l + t3)(l + t7)..	.. (1+ t4K'5)(l+ t2vl)	<- S0(2£)
(1.6.303)
Examples:
T	3
pcnrn = pcnrn=1 + t • The Betti numbers of S0(3) ~ S are therefore
bA =1, b. = 0, b = 0, b_ = 1 and the Euler characteristic x = I (-) b.
0	!	2	3	k=l
is zero. This generalizes to all odd spheres, while all even spheres
have x=2» also, for all Lie group manifolds X=o-
P„lrnl = (l + t3)(l + t5) = l + t3 + t5 + t8 , x=o
3	7	3	7	10
PSO(5) = PSp(4)=<1 + t ^1 + t ) = 1 + t +t +t ’ X=°
PS0(4) = PSU(2) xSU(2) = fl + 1 5 (1 + 1 J = 1 + 2t +t ’ X“°
270
Some useful theorems are:
PG1*G2 = PG1 X PG2 fCfr‘ the examPle of PsU(2)xSU(2p
ii) Xg = 0 for G = Lie group manifold.
This can be easily inferred from the structure
(even power of t + odd power) (even power + odd power) ...
of the G-Poincare polynomials in (1.6.303).
The beauty of (1.6.303) is that it can also be applied to coset
manifolds G/H when G and H are of equal rank. The theorem is:
p
G/H
= PG/PH
(1.6.304)
where P' is the polynomial obtained from P in (1.6.303) by changing
all + signs into - signs and raising all powers of t by 1 unit.
Examples:

_ pS0(3)
SO(3) 
S0(2)	S0(2)
1-t4
1 -12
1 + t2
2
so that the Betti numbers of S are = 1, b^= 0, b£= 1, yielding
X = 2
= PS0(2n+l) =	(1 г Hl t ) ... (! t ) 
PS0(2n)	(1 - t4) (1 - t8) .. . (1 - t4n~4) (1 - t2n)
= 1
2n
t
iii) in general P = 1 + tn for any n.
sn
271
iv)
p	= p X p
? S 2	3
CP x S CP S
P
SU(3)
SU(2) x(J(l)
Ps3
pt
SU(3)
pt
SU(2) xU(l)
(1 - t4) (1 - t6) . (1 + t3}
(l-t4)(l-t2)
= 1
t2+t3+t4+t5+t7
x=o
2	5	2	5	7
V) P, ,. = P,XP =(lttZ)(l + t5) = bt%t\t'
S x s S S
The last two examples show that homology indeed distinguishes between
EP2xS3 and S2xs5 (whereas homotopy does not, cfr. (1.6.299)).
Tables of Poincare polynomials for more general G/H coset spaces
can be found in Ref. [8] , Vol. Ill, pp. 492-497.
272
CHAPTER 1.7
APPLICATIONS OF THE FORMALISM AND MISCELLANEOUS EXAMPLES
1.7.1 - The Brans-Dicke theory
In the classical treatment of gravity the Brans-Dicke theory can
be thought as a non-standard way of coupling a scalar field to the
gravitational field in such a way that its average value is related to
the gravitational constant G: in other words G is regarded as a
scalar field coupled to the mass density of the universe (see Ref. 17).
In this section we present a very elegant and simple way to
obtain the equations of motion of the gravitational and scalar fields
using the first order formalism developed in Chapter 1.4. Let us con-
sider once again the Einstein-Cartan Lagrangian introduced in Eq.
(1.4.1):
JC = —— Rab л V*
grav‘ 32ttG
Vе1 e
v abed
(1.7.1)
273
We have written explicitly the Newton coupling constant G which was
previously set equal to 1/32тг, since, according to the interpreta-
tion of the Brans-Dicke theory, we are going to regard it as a scalar
field. Setting, for notational simplicity
32irG = е2ф	(1.7.2)
we have the following first order action;
ABRANS-DICKE = | e *R ~V "V Eabcd ’	(1.7.3)
M4
Let us vary Vd and ф; we find respectively;
2e"2*Rab л Vе e . , = 0	(I.7.4a)
abed
-2e"2*Rab .VC„Vde,=0 .	(I.7.4b)
abed
Notice that, through exterior multiplication of (I.7.4a) by Vd, the
бф-equation is a consequence of the vielbein equation. Next we vary
toab. Using (1.4.27a) by partial integration one obtains:
0 = [ е'2ф ®(<5wab) „ Vе = f (de"2* л Vе л Vd н 4 Therefore we get: Rc л Vd e . . = dф л Vе , abed	л Vd e , , = abed „ -2ф c ,,d4	x ab H 2e ^R л V ) Eabcd . Vd e , , . abed
Here the new feature is the appearance of a non zero torsion; indeed
from (1.7.6) it follows immediately that:
Ra = d<f> л Va .	(1.7.7)
ab
Hence the spin connection w is non-Riemannian (that is, it is not
ab
given by Eqs. (1.2.44-45)). To calculate w we proceed as follows.
Let us set
ab	°ab , ab
w = w + h	(1.7.8)
where £>ab is the Riemannian part of wab, that is it satisfies
Ra(w) = 0. Then (1.7.7) becomes
- hab л Vfe = d<f> л Va .	(1.7.9)
ab	,
Expanding h and d<J> along the vielbein basis we obtain:
hab = 6a ЭЪф - 6Ь Эаф	(1.7.10)
or as a 1-form
hab = 2 V^a ЭЬ^ф .	(1.7.11)
In particular from (1.7.10) we have:
,aC	1	ГТ ГТ 104
h^ = - 3 Э ф	(I.7.12)
which will be useful later on.
ab
Having computed w we may now substitute in (I.7.4a,b) in
order to find the Einstein equation and the ф-equation in the second
order formalism.
275
As in Eqs. (1.4.23-25) equation (I.7.4a) implies:
r'?(w) --6?r"(<o) =0	.	(1.7.13)
•d 2 b
Now using (1.7.8) we find:
„ab,o .. ,,oab ,ab. ,oa ,a . ,ocb ,cb.
R (w + h)=d(w +h )-(а>с+Ьс)л(ш +h ) =
.oab oa ocb ,, ab рас ,b , ac pb
= dco - ш aO) +dh -ii) z. n - h л (j =
c	c	c
- hac л hcb = Rab (8) + ® (8)hab - hac л hcb s
S^ab+Hab	(1.7.14)
where RabSRab(8) is the curvature 2-form in terms of the Riemannian
connection 8ab (Ra(8) = 0) and where we have put
Hab = ®(8)hab - hac л hcb .	(1.7.15)
Therefore Eq. (1.7.13) can be rewritten as follows:
R,a* - 1 6a R" + Ha' - 1 6a H” = 0	(1.7.16)
D •	2 D	b •	2 b
where the tensor 0-form ll^=Ha. is given by:
H£ = | {- 3®ь(Эаф) + 2&с(6^с Эа1ф) ♦
+ 6 6^a ЭС1ф + 4 6^a Э^ф 3е! ф} =
= -5ьЭаф - | 6ь^сЭСф + 6^ЭСФЭСФ - ЭафЭьФ .	(1.7.17)
276
Contracting a,b we obtain (H’’ = H):
°
H = - 3(	- 3m4>am4>)
(1.7.18)
Substituting (1.7.17-18) in (1.7.16) we find the Einstein equation for
the Brans-Dicke theory in the second order formalism:
R,® - 1 6? R = (к Эаф - 6a* Этф +
O2b b b m
+ 3(ЭафЭьФ - | 6аЭтфЭтф) .	(I.7.19a)
If one sets f = e and converts (1.7.19a) to the world-tensor
formalism using the relation
в 0
V di. = V
P b p
(which is valid only for the Riemannian connection &, see Eq.
(1.2.95)) one finds the alternative form:
R -lpR=J_{&V9f-V3ff +
y\)	2	2f	v 1
4 7{W'7 W“f9afI f1-719’’)
which is the same as that appearing in standard books.
ab
Finally we compute the ф equation (I.7.4b). Expanding R in
the vielbein basis we obtain successively:
Rab V™. Vn . Vе . Vd e = 0
mn	abed
_ab mned	. ,,ab rmn „
=» R E E . , = - 4 R 6 , = 0
mn	abed	mn ab
=» r”(w) = 0
(1.7.20)
277
that is, using the decomposition (1.7.14):
R + H = 0	.	(1.7.21)
Substituting (1.7.18) one finds:
У 3m<t> - ЭтфЭ ф - R/3 = 0	.	(1.7.21)
in	m
The change of variable:
F = е’ф	(1.7.22)
transforms (1.7.21) into:
(^ 3m + 8/3)F = 0	(1.7.23)
m
that is the standard (conformal invariant) equation of motion of a
massless scalar field F (see Eq. (1.5.63)). As a final observation
we note that the Eq. (1.7.21) or (1.7.23) can be also obtained from
Eq. (1.7.19) as a consequence of the contracted Bianchi identities
V»ai. - f 6ь <>  °	(I-7-24’
which can be immediately derived from (1.4.12a) after double contrac-
tion of indices:
©Rab = 0=>® Rab + ® Rab + ®.Rab = 0
r st s tr t rs
=>5» r” + 9 RS" + R'b = 0
r •• ST t r-
=»© (RS* - 6s r”) = o .	(1.7.25)
s r- 2 r
278
X7.2 - Minimal coupling of pseudoscalars through a torsion mechanism
In the previous section we obtained a first order version of the
Brans-Dicke theory by the non-standard coupling of a scalar field e2*.
The peculiarity of this coupling is that although (1.7.3) does not
involve derivatives of ф, yet it gives rise, in second order
formalism, to the propagation equation (1.7.23).
Actually this was possible because the torsion Ra turned out
ab
to be non zero so that the spin connection w could be expressed as
a function of Va, 9^Va and of the ф first-derivatives (namely Eq.
(1.7.11)).
One may wonder whether the same kind of mechanism could work for
the ordinary minimal coupling of spinless field; we obtained such
coupling in Sect. 1.5.1 using auxiliary О-forms Фа, in order to avoid
Hodge duality. Here, instead, we pursue a different approach closely
related to that of the previous section; the appearance of a non-zero
torsion will be the mechanism allowing for the propagation of the spin-
less field.
Let us consider the following Lagrangian:
£ = const x [Rab л vC л Vd eabcd + 2F(x)Rab л л vj (1.7.26)
where we have mimicked the Brans-Dicke case by using the field F(x)
as a coupling function. Actually the latter is a scalar while the
former is a pseudoscalar: therefore we are forced to choose F(x) to
transform as a pseudoscalar under parity. Varying the vielbein and the
F-field we obtain, respectively
R3b - vC eabcd * 2РМК% - Va = °	(I-7‘27a)
аЪ
2R л Va л Vb = 0	(I.7.27b)
279
while from the variation of the spin connection we get:
RC . Vd e , , + 2 F Rr л V, i + dF л V - V. = 0	.	(1.7.28)
abed La bJ	a b
Let us solve (1.7.28); expanding Ra and dF in the vielbein basis
one obtains:
(RC eab . + 2 F Ra + Э F 6a6b)Vm л Vn л Vd = 0 .	(1.7.29)
v mn cd	mn d m nd
Putting as usual Vm - Vn л Vd =	after some algebra one arrives
at the following formula for Ra:
„ Э F
Ra = - 1-I—dF л Va ♦ 1 eabcr V, . V .	(1.7.30)
2 F2 - 1	2 F2 - 1	Ь C
To calculate wab we use again (1.7.8); since R (to) = h ~ Eq.
(I.-7.30) gives
hab =___L_Vfa ЭЬЬ + - eabcr V .	(1.7.31)
F2 - 1	2 F2 - 1	c
We now use (1.7.8) and (1.7.30) to find the propagation equations of
the graviton and of F in the second order formalism.
The quickest way to compute (I.7.27b) is to use the Bianchi
identity (1.4.12b): using the explicit formula (1.7.30) one has
wa = -Rab.vb= (.1 ^Ч*ь;г;)х
x ea Vго „ Vn . Vb	(I.7.32a)
mnr
and (I.7.27b) then gives:
280
- I —г—2 V3^ = 0	•	(1.7.32b)
F - 1	2 (F - 1) a
Considering now Eq. (I.7.27a) we may rewrite it using again the Bianchi
identity (1.4.12b) for the second term:
Rab
~ VCe = 2F0R, = 2F(- - —-— 3. F3rF +
abed	d	2 F2 . t b
+	-А1_)е Vm л Vn Vb .	(1.7.33)
2 b 2	' dmnr	1 J
z Fz - 1
The l.h.s. of (1.7.33) is evaluated via use of the decompositions
(1.7.14), (1.7.15) and of the l.h.s. of (1.7.16). One obtains
Ra'(&) - i 6a R”(S) + Ha’ - A 6a H** =
t-	2 1	2 b "
= ' |	| ~T~	+ | &Ъ -T^-)6bt	(I-7-34>
4 F - 1 r	F - 1 Dt
where now Ht1* is computed by means of the formula
Ht= *[tharr] -hak[thkrr]	t1-7-35)
ab
with ha given by (1.7.31).
Г	° _ л
After some algebra one finds (5"= 5* (co)) :
Ht = 7 ^t(-2~ зар) + 7	3rp)6t +
2 r F - 1	2 r F - 1 г
1 f2
+ ----y-y (6a3CF3 F - 3tF3aF)}	(1.7.36)
(1-F) tec
and the final form for the Einstein equation is (Rab_Rab(S)):
281
D *	1 X П * * _	1	( 1 X П	rjn г?	*A r'l
R.------o, R = —	--- (— 6, Э F9 F - 9 F9,F)
°	2 b “	4 F2. j 2 b m	b
(1.7.37)
To see that (1.7.32b) and (1.7.37) are the usual second order equations
for the minimal coupling we first observe that (1.7.32) can be rewritten
9aF -	— Э F9CF = 0
a F2 - 1 C
(1.7.38)
where we have used the relation
= 0(fi)9aF - har9rF =>
© (w)9aF = 9 (S)9aF - har 9 F
a	a	ar
(1.7.39)
which follows from (1.7.8) and the definition of covariant derivative;
and
(1.7.40)
which follows from (1.7.31).
Moreover setting:
F = cos Ф
(1.7.41)
Eqs. (1.7.37) and (1.7.38)	become respectively
Raf 1 6ag-’ = I b* 2b”	4	(Эаф9ьф - | 6аЭСфЭсф)
0 a <4 Эаф = 0 a	
(I.7.42a)
(I.7.42b)
which coincide with the equations describing the minimal coupling of
the pseudoscalar massless field to gravity as derived from the second
order Lagrangian:
282
£ = const x (R + Э^фЭ^ф) /^g
(1.7.43)
We conclude by adding a few considerations on the general construction
of the Lagrangians presented in the last two sections.
The important observation is that this kind of couplings can also
be obtained from the general building rules presented in Sect. 1.4.3
provided we allow the coefficients of the general polynomials (1.4.60)
to be functions on Мд (or G). Let us see what changes with respect
to the general derivation of pure gravity theory given in Sect. 1.4.3.
We first notice that if we allow the coefficients c^ and c2
appearing in (1.4.62) to be functions on Мд, then we cannot discard
any more the corresponding terms since they are not total derivatives.
After partial integration, however, they can be rewritten as
follows
dcl - (eabcdwab - rCJ + 3 Sbcd^ л ~	(1.7.44a)
dc2 л «/Ь л Rab - | о,/ л „ co/) .	(1.7.44b)
•	•	ab
Varying them in the action we obtain equations of motion for w which
are differential rather than algebraic in the spin connection. This
would lead to higher order derivative terms in the field equation for
Va. Now these kind of theories are very interesting in their own right
since they are related to the field-theoretic limit of string theories.
However we do not pursue this subject here and therefore we may
discard again the corresponding terms. See, however. Part VI.
Secondly if c3 = c3(x) the argument given in (1.4.69-70-71) for
the reduction of the term c, Ra лR to linear ones now gives
э a
c3(x)Ra л Ra = -Rab л Va л Vb + Ra л Vadc3(x) .	(1.7.44c)
283
The first term in the r.h.s. is the coupling studied in the present
section; also the second term could have been added to our Lagrangian
(1.7.26): we leave however to the reader to verify that the only effect
of this further term is to perform a rigid rescaling of the value of the
spin connection and of the torsion field. Therefore it does not add any
essentially new feature.
Finally if in Eq. (1.4.73) we allow the coefficients а, В, у to
be functions of x, then we find again a = 0. Furthermore B = B(x)
is the Brans-Dicke case theory already studied in the previous section
and y = y(x) is the case of the present section.
Therefore we conclude that the non standard coupling are the most
general ones one may consider by allowing the presence of 0-forms on
(or more generally on the soft group manifold G).
1.7.3 - The Schwarzschild solution
In this section we rederive the well-known Schwarzschild solution
of pure gravity; the exercise is done in order to convince the reader
that one may actually use the intrinsic formalism to perform every com-
putation; in general the resulting formulae are much simpler and more
transparent than in the usual world-tensor approach; in particular the
coordinate dependent diseases of the usual approach like the singularity
at the Schwarzschild radius disappear in the intrinsic formalism.
To begin with we derive the explicit formula for the vielbein (and
the metric) in a spherically symmetric and static space-time. In terms
of the concepts developed in Chapter 1.6 this means that the 4-D space-
time metric
g = Va Vb n .
p \> ab
(1.7.45)
must be invariant against the action of Killing vectors Kj, K^ which
generate the S0(3) ® R group of three-dimensional spatial rotations
times time-translations (t-» t + a) :
284
	S0(3): [KT, Kt] = eTT„ K„	(I,J,К = 1,2,3)	(1.7.46a) J.	J	LJ Jx lx [Ko, Ko] = 0	;	[Ko, Kj] = 0	.	(I.7.46b)
Indeed G = S0(3) ® 1R is the isometry group of the 3-dimensional coset
space	G/H = SO(3)/SO(2) ® JR = S2 x JR	.	(1.7.47)
If we	2 introduce polar coordinates (6, Ф) on S and we call t the
(time) coordinate of 1R the explicit expression of K^ and Kg is
given by the usual angular momentum operators and energy operator
	Э	Э K. = sin Ф—+ cotg0 cos Ф—	(I. 7.48a) 1	30	ЭФ Э	Э K_ = - cos Ф—+ cotg 0 sin Ф —	(1.7.48b) *	30	ЭФ 3 K, =	 (1.7.48c) 3	ЭФ 3 Kn = —	•	(1.7.49) u 3t
We know from Chapter 1.6 that the invariance of the metric of a coset
space under an isometry of G corresponds to the fact that under the
same isometry the vielbein undergoes just an H-gauge transformation.
According to Eq. (1.6.119) in our case we have:
	SLV e° = SLV e° = 0	(1.7.50a) K0	KI eA = W* e_AR eB	(I.7.50b) Kj	1 JiXD
285
д
e = 0	(1.7.50c)
0
where e° denotes the "einbein" of the t-space and e& (A= 1,2) the
2
"zweibein" of S .
Hence the invariance of (1.7.47) under G is guaranteed if we
choose as 4-D vielbein the following ones:
A A
V	= r e	(1.7.51)
V	° = s e°	(1.7.52)
V	3 :	V3 = V3 = 0	(1.7.53)
K0 KI
where r and s are the rescaling factors of Sect. 1.6.11 and V3
satisfies the condition of being invariant under G. The presence of
the two independent scaling factors r,s is due to the fact that G/H
is a product of coset manifolds; in other words, using the notation of
(1.6.11), the matrix C.^ is obviously block diagronal, being the
direct sum of two matrices:
(1.7.54)
We further note that r and s are constant with respect to the
action of the Killing vectors and therefore they will in general depend
on the fourth coordinate, say u, labelling the direction of the ortho-
2	3
gonal complement to S x R, namely the V -direction:
V3 = du
(I.7.54a)
286
It is convenient, however, to assume as fourth coordinate the function
r itself because of its geometrical meaning (r is the radius of the
2
rescaled S as we shall see in a moment). Therefore we may rewrite
the ansatz (1.7.51-52-53) as follows:
И = r eA
„0	0
V = /B(r) e
= du = / A(r) dr
(1.7.55)
(1.7.56)
(1.7.57)
where we have denoted the
scaling factors as square roots for later
convenience.
3
Notice that V satisfies automatically Eq. (1.7.53) since it is
a function only of r.
and
At this point we must simply determine the explicit form of e®
eA to get the most general form of the V& and consequently of
As far as e^ is concerned it is clear that we may always choose
coordinate on ]R such that
0
e =
dt
(1.7.58)
The general expression of the Sn-vielbein and
connection has been given
in (1.6.102-103) using a stereographic parametrization. Since we are
going to use the standard polar coordinates (6, Ф) we rederive expli-
citly the S£-zweibein in this coordinate system using Eq. (1.6.8). Let
J. .
1J
be the generators of SO(3) and let
according to the decomposition (1.6.13).
{J31’ J23} c K
Then we set
and Jj2 с H
.	.,	0J12	6J23
L(0, Ф) = e • e
(I 7.59)
Ф and 6 being parameters.
287
One easily verifies that the action of (1.7.59) on the north pole
vector gives a unit vector of polar angles (6, Ф). Indeed using
, , ,k£ <k£	... ,
(J..)	=o.. one finds:
ij ij
/ 0
°
\ 1 ,
ФЛ12 6J23
e e
sin Ф sin 6
cos Ф sin 6
cos 6
(1.7.60)
The zweibein and connection are computed using Eqs. (1.6.79-80); we
have:
,-1,,	~6J23 "ФЛ12	$J12 6J23.
LdL = e e d(e e J
-0J23	6J23
= d<₽ e Jj2 e + du “
= dФ(cos6J.э + sin6J,.) + dO J_,	.	(1.7.61)
1Z	<51
Hence setting
L^dL = e3J19 ♦ еЬ„ ♦ e2J,,	(1.7.62)
1 z	z о	ox
we have:
e3	=	cos	6dO	(1.7.63a)
e1	=	d6	(I.7.63b)
e2	=	sin	6dФ	(1.7.63c)
Let us observe that the S0(3)-Maurer Cartan equations obeyed by L dL,
namely:
288
, 1	2	3 n de + e	e =0	(I. 7.64a)
de2 + e3 л e1 = 0	(I.7.64b)
de3 + e1 л e2 = 0	(1.7.64c)
are now reinterpreted as the structural equations of the coset manifold
S2 = SO(3)/SO(2) with e& (A =1,2) as zweibein and as spin connec-
tion. Indeed setting e3 e^B=w^B Eqs. (1.7.64) can be rewritten as
nA , A AB В	, _
R = de - ш л e = 0	(1.7.65)
dAB .AB А В	,T ,
R = dw = e л e	(1.7.66)
and they tell us that SO(3)/SO(2) is a Riemannian manifold (R^ = 0,
AB BA	12
co = -w ) with constant curvature R 17= 1. If we had used the
A
rescaled zweibein V we would have found curvature equal to 1/r.
As far as the group of time translations is concerned we have that the
single vielbein e^ obeys the trivial Maurer-Cartan equation
de° = 0
(1.7.67)
Using the expressions (1.7.63) and (1.7.55-58) the general metric of a
spherically symmetric and time independent 4D-manifold takes the follow-
ing form
. 0	0	.. . , 2	2 A ~ A
g = B(r) e ® e + A(r)dr + r e ® e
2	2	2	2	2	2
=» gpV = B(r)dt + A(r)dr + r (d6 + sin 6d4 )	.	(1.7.68)
Let us now compute the Einstein equations
Ra’ - R = 0
b- 2 b
(1.7.69)
289
or, after contraction (a,b) (which implies R=0),
Ra * = 0 D •	(1.7.70)
To compute R, we must first compute the curvature 2-form R which	
°"	ab in turn implies the knowledge of the spin connection io . The latter	
is obtained from the	first structural equation of the 4-D Riemannian
manifold namely (see	(1.2.35c) and (1.2.38)):
,,a _ ,,,a a R = dV - io . D	„ Vb = 0 .	(1.7.71)
Decomposing the index a into {0,3,A} we obtain the following set of
equations (remember the signature of П ,) :
dV°	03 + co	Л V3	0A + co	л / =	0	(1.7.72)
dV3	03 + co	л v°	ЗА + co	л vA =	0	(1.7.73)
dVA	03 + co	. v°	ЗА - co	л v3 +	AB .,B	„ w л V = 0	(1.7.74)
Let us consider Eq. (1.7.72); using (1.7.56) we have:
dV° = d/B(r)	0	1 B* e = — 	 dr л 2 /Т	0	1 B’ „3 e = — 	V л 2 B/A	V°	(1.7.75)
(The prime indicates	differentiation	with respect to	r).	
Hence Eq. (1.7.72) is solved by setting
0A ,A
ш = a V
03	1	B* „0	0 „3
co = — ------- V + p V
2 B/A
(1.7.76)
(1.7.77)
290
dv3	+ W3A л / + 3 v3 ~ V° = 0	.	(1.7.78)
3 Since V	is given by (1.7.57), dV =0 and the solution of (1.7.78)
is simply
ЗА U)	= f(r) Vй	(1.7.79)
where f(r	•) is a so far arbitrary function of r.
Fina	lly Eq. (1.7.74), after use of (1.7.76) and (1.7.79) becomes:
d/	= w3A л V3 - шАВ л VB - a / л V°	(1.7.80)
that is, u	sing (1.7.55)
d(r	Ач	x г > A	AB	В	..A	. .0	е т	0 14 e J = f(r)r e V + r co л e - a V л V	.	(1.7.81)
Eq. (1.7.8	l) implies
. A de	AR - co . eB = 0	(I.7.82)
r f(	г) /А = -l	;	a = 0	(1.7.83)
that is co	AB is exactly the spin connection of the 2-sphere, and f(r)
is given b	
f(r)	=			 .	(1.7.84) r /A(r)
The final :	ab solution for ш	is therefore:
03 0)	= -	V°=l	dt	(I. 7.85) 2 В /Д'	2 /вд
291
Ш01 = 0	(1.7.86)
Л =. _2_ vi = - JL ei г >Ла	/а~	(1.7.87)
12	Q w = cos 9 d1!»	(1.7.88)
We are now ready to compute the curvature 2-form from the second struc-
tural equation of namely: (see (I.2.35d)
Rab	j ab	a = dco -co /s c	cb co	(1.7.89)			
Splitting	the indices as	before we		obtain:		
R03	, 03	1 = d<i) = - —— 2AB	(B" -	B’A’ 2A	- - ^)V° 2B	л V3	(1.7.90)
roa	03 ЗА = со л co = -	B1 2rAB	V°.	vA		(1.7.91)
r3a	, ЗА ЗВ = dco + co /s	BA co =	A2 ,	A 3/2 dr л e 2AZ		= А' v3 э й2 2 гА	л VA (1.7.92)
rab	j AB	A3 = dco + co л	3B co =	, AB dco	1	1 A - — e л A	В е	
	,, к A = (1 - -)e л < A	Л _ J r	2 f1	- -)И ~ 1 A	.	(1.7.93)
We have e:	, „ab .	. xpressed R in terms of			; the VC.	V basis	so that we may
read out immediately the		intrinsic		components of R =		Rab vc vd cd
We may now calculate the		Ricci	tensor:			
R0' Ro-	- R03 ♦ R°A - 03 0A	1 4AB	(B" -	'2 B'A' _ В	j _ В’ 2A	2B	2rAB		(1.7.94)
292
	= R30	+ R3A = -	(B" -	- 12) + -AL.	(1.7.95) ЛА	4AB	2A	2B	2rA2
	= rA°ao + rA3a3 + rABab = - — + ~~2 + 4 (1 -	•	C1-7-96) A0	A3	AB	2rAB 2rA2 r2	A
Setting (1.7.94-95-96) equal to zero we get the explicit form of the
Einstein equation (1.7.70). For completeness we also rederive here the
well known solution of (1.7.94-95-96). We first observe that from
(1.7.94) and (1.7.95) it follows successively:
	R°' - R3’ = - — - (—+—) = о	(1.7.97a) 0-	3-	2r A В A J R' A' — + — = 0	(1.7.97b) В A A(r) = —-— + const.	(I. 7.97c) B(r)
The value of the integration constant is fixed by the requirement that
the vielbein and metric should reduce to those of a Minkowskian space
for	r->°°. This implies lim A(r) = lim B(r) = 1	(1.7.98) p -> ОЭ	p -> ОЭ
since in this case (1.7.55-56-57) and (1.7.68) reduce to the flat
Minkowski expression in polar coordinates. Equation (I.7.97c) then
gives
	A = -	.	(1.7.99) В
(1.7.95) and (1.7.96) take the following form when expressed in term
of the single function B(r):
293
R3‘ R 3-	B" B' = -T-^=0	(1.7.100)
ra‘ R A-	, = - 2 — + -y (1 - B) = 0	.	(1.7.101) r r
It is easy to see that these two equations are not independent; indeed
(-1 + В + rB')' = r(B" + I B')	.	(1.7.102)
This functional dependence is actually a consequence of the (contracted)
Bianchi identity (1.7.24).
Solving (1.7.100) we get immediately
(rB(r)) = 1
dr
	B(r) = 1 +	;	A(r) = 			 .	(1.7.103) r	1 + a/r
Introducing these results into (1.7.55-56-57) and (1.7.68) we find
V1 =	= r de	(I. 7.104a)
V2 =	= r sin 6d$	(I.7.104b)
V3 =	=	1„. dr	(1.7.104c) / 1 + a/r
v° =	= / 1 + a/r dt	(I.7.104d)
dt2	0	j 2 j u, v	dr a dxHdx - (1 + )dt -	+ r	1 + a/r + r2(d02 + sin2<|> d®2)	.	(1.7.105)
294
We now use the fact that in the Newtonian limit (see Ref. 16)
g00 = = +1 + 2ф	(1.7.106)
where ф is the Newtonian potential, that is in our case
ф = - GM/r	(1.7.107)
where G is the Newton coupling constant and M is the mass of the
object generating the gravitational field. In this ways one finally
obtains the well known Schwarzschild solution.
Let us conclude with two observations.
a)	The solution (1.7.104) or (1.7.105) displays a singularity at
rg = 2MG known as the Schwarzschild radius. This singularity is actually
a coordinate dependent phenomenon and therefore deprived of any physical
meaning; indeed if we substitute (1.7.103) into the expression of the
curvatures (1.7.90-91-92-93) we find:
r03=2MGv0^ v3
r
r0A = _ MG v0 л И
Г
R3A = _ MG v3 л И
Г
rAB = 2MG уА л VB
(I. 7.108a)
(I.7.108b)
(1.7.108c)
(I.7.108d)
We see that the intrinsic (coordinate choice independent) compo-
nents of the curvature are perfectly regular at r=2MG.
b)	We note that if we need the explicit expression of the
Christoffel connection	(which is necessary for the computation
295
of the geodesic of a test particle) we may either use the direct
definition (1.2.79) or to compute it from (I.2.90a) once the spin
connection is known.
The two procedures are obviously equivalent.
Finally we point out that the method given before for the construc-
tion of the vielbein fields in a symmetric time independent 4-D manifold
can be also used to construct vielbeins for other homogeneous spaces.
We leave to as an exercise for the reader to recover in this way the
general form of the Robertson-Walker metric, de Sitter metric and related
ones occurring in general relativity.
296
BIBLIOGRAPHY
A complete bibliography concerning the mathematical topics covered
or simply touched in this first part of the book would be extremely
massive. Therefore we just refer standard text-books and articles which
in our opinion are the most accessible to a physicist reader.
[1]	V. Arnold: Methodes Mathematiques de la Mecanique Classique,
Editions MIR - Moscow 1976.
[2]	A.O. Barut - R. Raczka: Theory of group representations and
applications, Polish Scientific Publishers, 1977.
[з]	W.M. Boothby: An Introduction to Differentiable Manifolds and
Riemannian Geometry, Academic Press N.Y. 1975.
[4]	E. Cartan: Le^on sur la geometric des espaces de Riemann,
Gauthier-Villars (1940).
[4a] L. Castellani, L.J. Romans and N.P. Warner: Symmetries of Coset
Spaces and Kaluza-Klein Supergravity, Ann. of Physics, 157 (1984)
394.
[5]	T. Eguchi - P.B. Gilkey - A.J. Hanson: Gravitation, Gauge
Theories and Different Geometry, Phys. Reports 66 (1980) 213-393.
[б]	H. Flanders: Differential Forms with Applications to the Physical
Sciences, Academy Press, New York - London 1963.
[7]	R. Gilmore: Lie Groups, Lie algebras and some of their Applica-
tions, Wiley 1974.
[в] W. Greub, S. Halperin and R, Vanstone: Connections, Curvature
and Cohomology, Academic Press 1973.
297
[9]	S. Helgason: Differential Geometry and Symmetric Spaces,
Academic Press.
[10]	S. lyanaga and Y. Kawada (eds.), Encyclopedic dictionary of
mathematics (MIT Press, 1977)'.
[11]	S. Kobayashi - K. Nomizu: Foundations of Differential Geometry,
(Interscience) 1963 Vol. I, 1969 Vol. II.
[12]	Y. Ne'eman and T. Regge: Riv. Nuovo Cimento 1_> 1 (1978).
[13]	R. Slansky: Group Theory for unified model building, Phys. Rep.
79, (1981) 1.
[14]	R. Stora: Lectures at the International School of Mathematical
Physics, Erice 1977.
[15]	A. Trautman: Differential Geometry for Physicists, Stony Brook
Lectures, Bibliopolis 1984.
[16]	P, van Nieuwenhuizen: An Introduction to simple supergravity
and the Kaluza-Klein program, Les Houches, eds. B.S. De Witt
and R. Stora, 1983.
[17]	S. Weinberg: Gravitation and Cosmology: Principles and applica-
tions of the General Theory of Relativity, J. Wiley and Sons 1972.
[18]	T.J. Willmore: An Introduction to Differential Geometry, Oxford
at the Clarendon Press, 1959.

PART TWO
THE ALGEBRAIC BASIS
of
SUPERSYMMETRY
what immortal hand or eye
could frame thy fearful symmetry?
William Blake

301
CHAPTER II.1
INTRODUCTION
In Part One we studied the Einstein theory of gravitation and we
elucidated its differential geometric structure, which is best
understood and most clear in the lahguage of exterior forms. These
latter, on the other hand, are naturally associated to the dual
formulation of Lie algebras via Maurer Cartan equations and to their
soft deformations. The Lie algebra one deals with in gravity is that
of the Poincare group (or (anti) de-Sitter group) which can be
viewed as the group of motions of flat Minkowski space (or (anti) de
Sitter space), namely the vacuum of gravity theorv.
Our treatment of supergravity will just be parallel to that of
gravity since we shall interpret it as describing the geometry of a
new kind of manifold, superspace, which extends the notion of space-
time.
We shall do this in terms of differential forms associated to a
suitable algebra. Such a programme, however, with all the needed new
concepts (in particular rheonomy) which are necessary, is postponed
302
to part three. Prior to that we still have to explain the meaning of
the prefix "super", clearly involving something quite essential. This
is the goal of the present part of the book.
Super stands for supersymmetry and this latter, as the word
suggests, is a new kind of symmetry, more general than the symmetries
used up to 1974, against which we shall demand the laws of nature to
be invariant, at least at a certain level. Indeed we shall realize
that supersymmetry is worthwhile in applications only if it is
spontaneously broken (this is a story told in part four).
Physically the novelty consists in the fact that the
supersymmetry transformations, against which supersymmetric field
theories are invariant, have the property of mapping Bose particles
into Fermi particles and viceversa: ordinary or bosonic symmetries
never mix particles of different statistics and they transform bosons
into bosons and fermions into fermions. Mathematically supersymmetry
corresponds to the replacement of Lie algebras by super Lie algebras.
These latter are algebraic structures, including ordinary Lie
algebras as subalgebras, where the Lie bracket is no longer always
antisymmetric but can be either symmetric or antisymmetric depending
on a grade (zero or one) which is assigned to all the elements of the
superalgebra.
First target for us will be the definition of super Lie algebras
and their classification. We shall see that in every dimension D the
Poincare Lie algebra admits a suitable super extension and that is
what makes the whole thing relevant to physics. Indeed the super
Poincare group, as we shall name the quoted superextension of
the Poincare group, can be thought of as the group of motions of a
supermanifold, the flat superspace, which contains flat space-time as
a subspace. The notion of super manifold corresponds to a structure
where the coordinates of a point rather than real numbers are
elements of a Grassmann algebra, even or odd. The even coordinates,
which are commuting, label space-time points, while the odd ones, anti-
commuting, have spinor rather than vector indices and their interpre-
tation is elucidated in working through this part of the book.
303
What we want to anticipate and we already did, is that
supergravity, in a sense to be precised later, describes the geometry
of superspace. Furthermore its basic fields will be the soft forms of
the Super Poincare group.
This statement suggests to the reader what is going to happen.
In the same way as Lie algebras, superalgebras also admit a dual
formulation in terms of forms and Maurer-Cartan equations. The
only difference is that the 1-forms have now a grading, namely they
are either bosonic or fermionic.
Bosonic 1-forms anticommute as usual, but fermionic 1-forms are
commutative objects.
Just as in the case of ordinary groups one can interpret the
basic forms of the dual formulation as left-invariant (alternatively
right-invariant) 1-forms on the supergroup manifold. This latter is a
supermanifold where the even coordinates are the parameters of the
ordinary Lie subgroup while the odd ones must be identified with the
parameters of the fermionic transformations, that is the proper
supersymmetries. Also the notion of coset manifold and of left-
invariant 1-forms defined over it extends to the super case. Actually
the above mentioned flat superspace can be viewed as the quotient G/H
where G is the super Poincare group and H is the Lorentz subgroup.
Analogously Minkowski space is the quotient of the Poincare group
divided by the Lorentz subgroup.
Functions defined over superspace are called superfields. Calling
the even and odd coordinates x^ and Sa, respectively, a superfield
Ф(х,6) can be expanded in powers of 6a with coefficients that are
functions only of x:
Ф(х,е) = ф(х) + Фа(х)еа + ФаВ(х)еабв + ...	(ii.i.i)
Since 6а is anti commutative the expansion will stop at a maximum
power equal to the number of components of 6a (4 in 4-dimensions,
304
since a four-dimensional spinor has 4 components, but 32, for exam-
ple, in 11 dimensions since an eleven-dimensional spinor has 32 compo-
nents). This shows that a superfield, namely a function over the
space whose geometry is described by supergravity, is actually a
collection of ordinary fields possessing different spins. Such a
collection is called a supermultiplet.
This ties up with another fact.
For superalgebras just as for Lie algebras one can define the concept
of a representation. The super Poincare algebra is non-compact and
as such its unitary representations are infinite dimensional. These
infinite dimensional representations, however, can be viewed as
generated by the Fock space associated to fields.
In the case of superalgebras each unitary representation,
instead of being generated by a single field, is generated by a
multiplet of fields of various spins encompassing both fermions and
bosons. Therefore a superfield or equivalently a supermultiplet is a
way of introducing a representation of the supersymmetry algebra.
Hence another task we shall be confronted with is the classifica-
tion of supermultiplets. To this effect we must point out that when we
said that the Poincare (or de Sitter) group admits a superextension we
were not quite right. Actually it admits an infinite sequence of exten-
sions, labeled by an integer N which tells you how many fermionic
generators are introduced into the game. Each of the extensions differs
in the structure of its representations and therefore we must speak not
just of supermultiplets but of N-extended supermultiplets. Actually the
game is even more complicated since the supermultiplet structure depends
not only on N but also on the dimension D of space-time. All these
complications are ultimately due to the structure of gamma matrix
algebra, which plays a fundamental role in supersymmetry and which
crucially depends on the number of space-time dimensions. The
identification code of the physically relevant superalgebras is
then provided by two numbers, D and N, which are always seen to
precede the word supersymmetry (or supergravity) in every context.
305
(Starting with D=6 it will actually happen that also the concept of
superalgebra is no longer sufficient and one has to resort to a wider
type of structures called free-differential algebras; this, however,
is a story which we leave for Part Three). This being the state of
affairs we shall study the properties of the N-extended supersymmetries
in D=4 for I S N S 8 and pave the way for the construction of higher
dimensional theories by considering Г-matrix algebra and spinor Fierz
identities in all dimensions 4 £ D £ 11.
The choice of the ranges of N and D is not random, rather it is
related to the deepest aspects of supersymmetry. Indeed, although
superextensions of the Poincare group exist for all N.s and all
D.s, it is not clear that the associated irreducible supermultiplets
should always correspond to permissible Lagrangian field-theories
invariant against the superalgebra transformations.
It comes out that increasing N or D we increase the highest spin
appearing in the lowest N-extended supermultiplet and for N>8 or D>11
no supermultiplet exists with maximum spin equal at most to s=2.
It is now a well-established result that no consistent
interacting field theory can be constructed for spins higher than
two, at least if they appear in a finite number. (Indeed string theory,
which contains all possible spins from s= minimum to infinity, to be
discussed in Part Six, seems to be the only way out of the problem of
coupling higher spins). Hence in the context of local field theory (the
string is an extended non-local object) we see that the limits N=8
in D=4 and N = 1 in D = 11 are immediately selected.
In these limitations is where we find the most appealing aspects
of supersymmetry: from them it follows that there is a finite number
of field-theoretic models which can be considered and which turn out
to be related among themselves either via truncation or via
spontaneous supersymmetry breaking or via spontaneous
compactification of extra dimensions. These relations and these
mechanisms will be discussed in Parts Three, Four and Five: here we
306
shall consider only the classification of the supermultiplets and
derive the field content of the possible field-theoretic models. They
fall in two main classes depending on whether the considered
supermultiplet does or does not contain the spin 2, that is the
graviton, and the spin 3/2, that is the gravitino. In a sense which
will be fully explained only in Part Three these fields can be
regarded as the gauge fields of the translations and of the
supersymmetries, respectively. Correspondingly the field theories
containing the graviton and the gravitinos (they are N in an
N-extended multiplet) will be supergravity models where the super
Poincare algebra is a local symmetry. On the other hand, in the
models based on so called matter multiplets, where the maximum spin
is 1 or lower, the super Poincare algebra is only a global symmetry,
namely the parameters of its transformations must be taken constant.
In this part of the book we shall discuss these types of models
postponing supergravity to Part Three.
The point of view we shall take is the following. Given the
left-invariant 1-forms which describe the geometry of flat
superspace, the multiplet of matter fields is described by a
multiplet of superfields related among themselves by supersymmetry
transformations. We interpret these latter as Lie derivatives of the
superfields in the direction of the supersymmetry generators and this
translates into certain conditions on the exterior differentials of
the superfields which we name rheonomic conditions.
The compatibility of these conditions with the closure of the
exterior derivative (d^=0) is equivalent to the condition that the
Lie derivatives do indeed close the super Poincare algebra and
generate, on the given superfields, a true representation. One finds
that the rheonomic conditions, inserted into the d^=0 equation
(Bianchi identity), imply further conditions on the x-space
derivatives of the superfields which are easily recognized to be
ordinary wave equations. This corresponds to the fact that the
particles appearing in a supermultiplet and carrying a representation
of the superalgebra are on-shell and not off-shell particles. We can
307
now try to obtain an action principle from which the rheonomic
conditions and the field equations come out on the same footing as
variational equations. This is indeed possible and it constitutes the
hard-core of the rheonomy method which we follow throughout this
book: before going to more complicated cases we shall illustrate it,
in this part of the book, on the simplest supersymmetric theories:
the Wess-Zumino model and the supersymmetric Yang-Mills theories.
The plan of Part Two is the following:
i)	In Chapter II.2 we introduce the concept of superalgebras
and we give the classification of the simple ones both classical and
exceptional. Introducing the concept of Grassman algebra and of
graded matrices we are able to interpret the classical algebras as
algebras of graded matrices satisfying suitable constraints.
Special attention is devoted to the orthosymplectic algebras Osp(4/N)
and to their real forms. Indeed they describe the N-superextensions
of the anti de Sitter algebra and by an Inonii Wigner contraction from
them we can obtain the N-superextensions of the Poincare algebra.
ii)	In Chapter II.3 we study the dual formulation of the
superalgebras in terms of exterior forms and Maurer Cartan equations.
After introducing the notion of superspace we construct the left-
invariant 1-forms and Maurer Cartan equations for the Osp(4/N)
supergroups and for their contractions, namely the N-extended
Poincare groups in D=4. We also discuss the generalization to the
supercase of the notion of Killing vectors and Lie derivatives which
is straightforward.
iii)	In Chapter II.4 we consider the Poincare supermultiplets,
namely the unitary field representations of the N-extended super
Poincare algebra, stressing the difference between the massless and
massive case. This study leads to a classification of the possible
N-extended models in D=4.
iv)	In Chapter II.5 we study the structure of supermultiplets
in anti de Sitter space, namely the unitary field representations of
the 0sp(4/N) algebras. The essential difference with respect to the
308
Poincare case is that the same anti de Sitter supermultiplet
encompasses particles not only of different spins but also of
different masses: the mass in fact is not an 0sp(4/N) invariant while
it is a super Poincare invariant. Furthermore the definition of what
one means by massless in anti de Sitter space needs special care:
indeed a sound definition can be based only on the enlargement of the
group of invariances of the associated field equation, typical of the
massless propagation. All these apparently strange features of anti
de Sitter supersymmetry are of the highest importance in the sequel
since both in the partial supersymmetry breaking mechanism and in the
spontaneous compactification of extra dimensions the general trend is
that of producing an anti de Sitter background. Henceforth the
classification of the small oscillations around these backgrounds can
be done only in terms of 0sp(4/N) supermultiplets. In particular the
concept of stability is different in anti de Sitter space since it
signifies a lower bound on the masses of the scalar particles which
is negative rather than being zero as in Minkowsky space. It follows
that saddle points of the scalar field potential can be locally
stable extrema and describe admissible vacua, contrary to naive
intuition. As a preparation to Part Four and Part 'Five we discuss
this important bound due to Breitenlohner and Freedman.
v)	In Chapter II.6 we study the field theory of the Wess-
Zumino model, namely the simplest supersymmetric theory corresponding
to the lowest representation of the N=1 Poincare superalgebra. As
already mentioned we introduce the concept of rheonomy and we see how
on one hand it relates to the theory of supersymmetry representations
discussed above while on the other hand it is used to construct the
action.
vi)	In Chapter II.7 we collect all the information we need on
Г-matrix algebra, spinors and their tensor products in dimensions
4< D Sil. Furthermore we consider the properties of super Poincare
algebra in the various dimensions 4< D <11.
vii)	Chapter II.8 will be devoted to the group-theoretical
study of Fierz identities for spinor 1-forms. This constitutes an
309
essential technical prerequisite for the analysis of supergravity
Bianchi identities and for the derivation of free-differential alge-
bras.
viii)	Finally in Chapter II.9 we construct the N=1 super-
symmetric Yang-Mills theory in D=4 and D=10.
310
CHAPTER II.2
SUPER LIE ALGEBRAS, SUPERMANIFOLDS AND SUPERGROUPS
II.2.1 - The definition of superalgebras and the example of
N-extended super Poincare algebra
We begin with the definition. A super Lie algebra is a vector
space A over the field of complex or real numbers which splits into
two subspaces G and LI , called respectively the even and odd
subspace
A = G Ф U (II.2.1)
Besides the operations of a vector space (sum and multiplication by a
scalar), in order to turn A into an algebra one must define a
further product operation which we shall call the Lie bracket and
denote by [ , }.
311
The following are the defining properties of the Lie bracket:
1) If X e G, Y e G are two elements of the even subspace then
their Lie bracket belongs to the same subspace and it is anti-
symmetric:
[X,Y} e G ;	[X.Y) = - [y,X)	(II.2.2)
Furthermore if X, Y, Z e G are three elements of the even subspace,
then the Jacobi identity is satisfied:
[x, [y.zH + [y,[z,x}} + [z, [x,y}} = о
(II.2.3a)
ii) If X e G and T e U, the Lie bracket of these
two elements
lies in the odd subspace and it is antisymmetric:
[X.T) e U ;	[X,T} = - [T,X}
(II.2.3b)
Furthermore if X, Y e G and T e tl we demand that the following
identity be satisfied:
[x,[yJ}} + [у,[у,х}} + [T,[X,Y}} = О	(II.2.4)
Eq. (II.2.4) can also be rewritten as follows:
[x, [y.T}} - [y,[x,T}} = [[x,Y},y}
(II.2.5)
iii) If ¥ e II, E e U are two elements of the odd subspace
then their Lie bracket is symmetric and lies in the even subspace:
312
[Ч>,Н} е G ;	[¥,“} = [=Л1
(II.2.6)
Moreover if ф, Е , Л 6 II are all odd the following is an identity:
[Ч',[-,Л}} + [Л,[Ч-,=В + [Е,[л,Т}} = о
while if гр» - e V are odd and X e G is even we have:
[X,[4-,E}} - [5, [x,T}} + [¥,[=,X}) = 0
(II.2.7)
(II.2.8)
iv) Finally the Lie bracket is distributive with respect to the
vector space operations, namely if a, [J e c (or R ) and А, В, C e A
then we have:
[aA + BB,c) = a[A,C} + В[В,С)	(II.2.9)
Let us discuss the meaning of these properties. Eqs. (II.2.2) and
(II.2.3) are equivalent to stating that G is closed under the Lie
bracket, namely it is a subalgebra. Not only. On this subspace the
properties of our Lie bracket are the same as the properties of the
Lie bracket of an ordinary Lie algebra. Hence the even subspace G is
an ordinary Lie algebra. Let us now consider Eqs. (II.2.3b) and
(II.2.4). They state that the odd subspace II is a carrier space for
a representation of the Lie algebra G, the Lie bracket [	,	}
defining the action of G on U. Indeed Eq. (II.2.4), once rewritten
in the form (II.2.5), is the statement that the action of elements of
G is consistent with the Lie bracket defined over G.
Eqs. (II.2.6) and (II.2.8) are really novel. They introduce
313
a symmetric Lie bracket, that is an anticommutator, over the odd-
subspace V and they state that the anticommutator of two odd
elements is an even one. In other words the odd elements are the
square-roots of the ordinary Lie algebra G.
We can now condense all the Eqs. (II.2.2 - II.2.8) in a much
more compact notation if we introduce the concept of grading. Let
be the set of integer numbers mod 2; representatives of the two
equivalence classes are 0 and 1. To each element A e A we associate
a grading a which is an element of :
X/А e A	a = grad A e	(II.2.10)
a is 1 if A lies in the odd subspace, while it is zero if A lies in
the even subspace:
A e U => a = 1 (mod2)
(II.2.Ila)
A e G => a = 0 (mod2) (II.2.lib)
Using this notation we can rewrite the defining properties of the Lie
bracket in the following way. First we note that, utilizing the
distributive property (II.2.9), the Lie bracket of two arbitrary
elements of the superalgebra, which in general do not have a definite
grading since they are the sum of an even and an odd part, can be
decomposed into a sum of terms which are Lie brackets of elements
possessing a definite grading. Then if А, В, C are elements of A
endowed with a definite grading, we can write:
[A,B) = (-)1+ab[B,A)
(II.2.12a)
314
[А, [в,С}} + (-)а(Ь+с)[в,[С,А)} + (-)Ь(а+с)[с, [А,В}} = О
(11.2.12b)
which summarizes Eqs. (II.2.2 - II.2.8).
As it happens for ordinary Lie algebras, superalgebras are most
conveniently described in terms of a basis of the vector space A.
Let {Тд} be such a basis (A=l,...,d) where d=dim A. Since A is the
direct sum of G and U, the basis {Тд} can be chosen in such a way
that it is the union of a basis for G and a basis for tl : in other
words the basis elements Тд have a definite grading. The super
algebra is completely specified if we give the Lie bracket of any two
basis elements, from now on referred to as generators:
[TA’V = С1вЧ	С11-2ЛЗ)
In Eq. (II.2.13) the summation convention on the index F is adopted
* *F
and Сдв are constants. They are the graded structure constants
of the superalgebra and satisfy the following properties, inherited
from eq.s (II.2.12):
C^F = <->1+abcMF	(II. 2.14a)
CALMcBCL + (-)a(b+C>CBLMcCAL + (-)b<a+C)cCLMcABL = ° <H-2.14b)
Furthermore if we adopt the convention that the capital latin index A
is replaced by a lower case latin index when the grading is even and
by a lower case Greek index when the grading is odd we have:
c\ = cbR = c\ = 0
ab ap aB
(II.2.15)
315
The only structure constants which can be different from zero are
c’’c ; c”y ; c"a = c”a	(II.2.16)
ab	aB	aB Ba
where:
i)	Gai,C	are the structure constants of the Lie algebra G
ii)	cagY are u xumatrices (u being the dimension of U )
which satisfy the Lie algebra and generate one of its
representations
iii)	cag3 are the symmetric structure constants to whose
existence the existence of the entire superalgebra is due.
Before proceeding to the classification of the superalgebras we want
to show that the concept is not empty. To this effect we introduce an
example which is of the highest relevance, namely the super Poincare
algebra. As the word suggests, this is a superalgebra A where the
ordinary subalgebra G is the familiar Lie algebra of the Poincare
group. This latter is ten dimensional and its natural basis is
provided by the 4 translation generators Pa plus the six Lorentz
generators the index a running from 0 to 3:
a = 0, 1, 2, 3
(II.2.17)
Utilizing the Minkowskian metric
ab
1,
0,
0,
0,
0 0 0 \
-1, 0, 0
0, -1, 0
0, 0, -1 /
(II.2.18)
316
to rise and lower the indices we can write the Poincare Lie algebra
in the following way:
Kb’Mcdl = *Mad + %dMbc - %dMac - ^аЛх?
(II.2.19a)
[Pa’PJ = 0
(II.2.19b)
[Mab’PJ “ " ^(rlcaPb ’ %bPa>
(II.2.19c)
This algebra is not semisimple and it is obtained as the semidirect
product of the simple Lorentz algebra S0(l,3) with its 4-dimensional
vector representation. To extend it to a superalgebra we need one of
its representations: we choose the 4-dimensional spinor representa-
tion.
Following a standard procedure we introduce the four gamma
matrices ya satisfying the Clifford algebra:
ha,Yb} = 2rlab
(II.2.20)
and we define the matrices (see ChapteriII.7 for further details on
the conventions)
Sab = 7 fra’41 = 7 Yab	(II.2.21)
o	4
which satisfy the Lorentz algebra in the form (II.2.19a):
fs . ,S ,1 = 4(n. s . + n ,S. - nKJS - П S. ,) (II.2.22)
L ab’ cdJ be ad ad be bd ac ac bd
(II.2.22)
317
An element of the carrier space for the spinor representation
(II.2.21) is a 4-component spinor. Hence we introduce new generators,
called Qa (a = 1,2,3,4) which transform as barred spinors under the
Lorentz algebra
[МаЬ’%1 “ Qa(Sab)aB 4 Qa ^ab^aB
and we declare that the action of the translation Pa on
Q is null
a
[p ,Q 1 = 0
L a aJ
(II.2.24)
In this way carries a representation of
and the structure constants have been
the Poincare adjoint: A = a, (ab).)
the full Poincare algebra
identified. (A runs on
c ’	= (ab)B	4	^ab^aB		(II. 2.25a)
c”Y = 0 aB				(II.2.25b)
It remains to	be	checked whether	we can construct	the structure
„••A constants Cag	. To	• this effect we	first recall that	the Poincare
Lie algebra (II.2.19) is constructed over the field of real numbers
and the generators Pa, Majj are antihermitean
M+, = - M .
ab ab
(II.2.26)
To extend it consistently, we must impose suitable reality conditions
also on the extra spinorial generators Q^. We do this by requiring
Qa (the spinor of which Qq is the Dirac conjugate) to be a Majorana
spinor. Hence we write:
318
—т
О = С Q
Q = У0б+
(II.2.27)
where С is the charge conjugation matrix (see Chapter II.7 again).
Equation (II.2.27) can be rewritten as follows
_	*	т
% = Wfc = QBCBa = Q c
(II.2.28)
The generators Sat, not only satisfy the Lorentz algebra but
fulfill the extra property of transforming Majorana spinors into
Majorana spinors. Indeed we have:
__  + + — +
= Q Wo = Q 7oWo =
T	T T	T
= - Q CYab = Q YabC = (yabQ) C
(II.2.29)
which follows from the two identities:
+
Wab\) = - 7ab
-1 T
С Y , C = - Y к
1 ab	ab
(II.2.30)
This is essential for equation (II.2.23) to be consistent.
We close the superalgebra by writing the anticommutator of two
spinorial generators:
{Qa,Q6J = i(CYa)a6Pa
(II.2.31)
In this way the structure constants Cag are identified:
31Э
С,7  ‘<сЛ«е - ««“'на
The fulfilment of the Jacobi identities (II.2.7) and
easily checked. We have
+ +
+ [0/%»^] = 0
since {Q,Q]=P and P^ commutes with (Eq. (II.2.24)).
get:
LPa’{%’M ~ Ка’^В’Ра}1 + {MPa’%l} = 0
since Pa commutes with Pb and Qa. Finally we should have
[“ab’Wl = 4’[Mab,Q6]} + <Q6.[Mab,Qa]}
To check the validity of (II.2.35) we just substitute Eqs. (II.2.23)
and (II.2.31). We get
" I (CKC)a6(ncaPb - %bPa> ?
+ (Ib2-36)
which is fulfilled if the following matrix identity holds
(II.2.32a)
(II.2.32b)
(II.2.8) is
(II.2.33)
Similarly we
(II.2.34)
(II.2.35)
i
320
- i (су 6. - Су. 6 ) = ST. (CyC)T+CyCS .	(II.2.37)
2 a bin ’b am ab ’	ab
Multiplying by C“1 and using Eqs. (II.2.30) plus the definition of
the matrix C:
Су С-1 = - ут	(II.2.38)
'a	'a
we find that equation (II.2.37) is the same as
- 4 (Y “ Yk<5 ) = [y ,s , ] =Y [y >Y r]	(II.2.39)
2 a bm b am L’m abJ 4 L m abJ
which is obviously true.
।
This check shows that when we decided that Q was a barred spinor
and we wrote the action of on the right rather than on the left
I
of Q we did not make an arbitrary choice but we just made the only
one consistent with equation (II.2.31). If we wanted we could use
Q instead of Q but then Eq. (II.2.31) would be replaced by:
(Q = i(Y C) DP	(II.2.40)
ча’чВ a aB a
The reason why we chose to work with Q rather than Q is related to
a	xa
the privileged role we want to give to the dual formulation of
the superalgebra. In Chapter II.3 we shall rewrite the super
Poincare algebra in terms of forms and the	generator will be
identified with the dual tangent vector to a 1-form which is an
unbarred Majorana spinor:
Aqb) = <5g	(II.2.41)

321
Since from our point of view the fundamental object is ф rather than
Q, we choose the first to be unbarred.
The superalgebra we have discussed is the N=1 Poincare super
algebra in 4-dimensions. We call it N=1 because it contains only one
spinorial generator and we say that it is 4-dimensional because the
vector indices a,b run from 0 to 3 while the spinor indices span the
4-dimensional spinor space. The generalization to other dimensions is
not automatic as it is for the pure Poincare algebra: indeed in
order to proceed one must make sure that in the chosen dimension D
the following properties hold true:
i) A charge conjugation matrix defined by Eq. (II.2.38) exists
ii) Majorana spinors, defined by Eq. (II.2.27) exist
iii) The matrix Cya is symmetric.
This does not happen in all dimensions. In those dimensions where it
happens we have N=1 extensions of the Poincare group; in the others
superextensions may still exist but they are more complicated: in any
case, as we shall see, in every dimension D>4 the algebraic
structure underlying supergravity theories is wider and more
complicated than the one presented here. Hence the only dimension
where the simple N=1 extension of the Poincare group leads to
interesting physical theories is precisely D=4 and this is the reason
why we chose it.
Remaining in D=4 we could ask what happens if, instead of one
spinorial generator we introduced several, labeled by an
additional index A which runs from 1 to N:
(A = 1, 2.....N)
(II.2.42)
In this case we would still set
322
!“*•$  i ft’*’*
-о
(11.2.43а)
(11.2.43b)
but we could replace Eq. (II.2.31) by:
{<£,ф =	+	+ KCY^Z^ (II.2.44)
where Z(+)AB= -Z(+)®^ and	-z^^BA are neu even generators
having the property of commuting with everything else:
= [C>z(-)1 - t^).^)] -
= tZS)«M = [^).Mab] = [Z&.Q3 =
= D&.U - [<>ab] - [z£,q3 - 0 (II.2.45)
For this reason they are called central charges. From the algebraic
point of view they are optional; we can either introduce Z+ or Z_ or
both or none of the two: the algebra closes in any case. When we
shall consider the multiplets, namely the representations of the
above algebras we shall see that the massless representation
beginning at spin 2 (the supergravity multiplet) is consistent only
if we include the spin 1 gauge field of Z (+)AB but not of
hence the algebra which leads to a supergravity theory is selected.
This will be further clarified when the algebra (H.2.44) will be
obtained as Inonu Wigner contraction of a simple algebra.
Eqs. (II.2.43) and (II.2.44), together with Eqs. (II.2.19)
323
define the N-extended Poincare superalgebras.
II.2.2 - Classification of the
simple superalgebras whose Lie
algebra is reductive
In this subsection we give the list of all the possible superalgebras
which have the following property:
a) A is simple in the sense that it
J.
contains no non trivial
ideals
The ordinary subalgebra G c A is reductive in the sense that
G = G^ ® G^	(II.2.46)
where Gj is a semisimple Lie algebra and G2 is an abelian one. In
other words G is the tensor product of some simple factors times a
certain number of U(l) factors.
For the reader's convenience we recall that an ideal J is a
subalgebra with the further property that the Lie bracket of any
element X e A with any element Ze J is still an element of J.
In full analogy with the treatment of ordinary Lie algebras the
algebras classified here are taken over the field of complex numbers:
it is then our privilege to choose a real form for them by introducing
suitable reality conditions. The classification theorem is due to
Scheunert, Nahm and Rittenberg: its proof being quite technical and
complicated we restrict ourselves to enunciating the thesis: the
interested reader is referred to the original article.
Theorem: The simple superalgebras whose Lie algebra is reductive are
the following ones:
324
A)	The infinite series of orthosymplectic algebras Osp(2p/N).
B)	The infinite series of superunitary algebras SU(m/N).
C)	The infinite series of P(n) and Q(n) algebras.
D)	The three exceptional algebras D(2,l,a), G(3) and F(4).
Let us describe these algebras one by one emphasizing once more that
they are classified as complex algebras although in case a) and b)
their name is taken from the name of their most important real form.
The basic idea for the construction is that of considering
complex matrices in dimension
d = m + N
(II.2.47)
where m and N are two integer numbers. Any d x d matrix can be writ-
ten in block form as follows
В \}m
Q =
£. /}N
N
(II.2.48)
where A is m x m, D is N x N, and В and C are m x N and N x m respec-
tively. The space of d x d matrices is a d^-dimensional vector space
which can be split, according to (II.2.1), into an even and odd sub-
space by defining:
/ A
0 \
Q e G <=> В = C = 0 => Q =
(II.2.49a)
325
I °	В \
QeU<=>A = D = O=>Q =
\ C	0 ,
(II.2.49b)
The Lie bracket can now be introduced, consistently with the grading
(II.2.49) and with all the axioms (II.2.2-II.2.9) of a superalgebra,
by means of the following recipe:
if QrQ2 e G : [QpQ2) = [Q1,Q2]	(II.2.50a)
if Qxe G, Q2 e Ю : [Ql,Q2) = [QpQ2]	(11.2.50b)
if Qp Q2e U : [QrQ2) = {QpQ2)	(II.2.50c)
where [	,	],	{ ,	} denote, respectively, the ordinary commutator
and anticommutator of matrices:
[Qx.Q2] = QxQ2 - ^2Q1
(II.2.51a)
{Qx,Q2} = QxQ2 + Q2Qi (II.2.51D1
Eqs. (II.2.50) can be summarized by stating that the Lie bracket of
any two matrices and Q2 of type (II.2.48) is a new matrix O3 of
the same type:
= Q3 =
(II.2.52)
326
where:
A3 - [ApAj + BXC2 + B2^1	(II.2.53a)
D3 = [DPD2] +	+ C2BX	(II.2.53b)
B3 = A1B2 " B2D1 " A2B1 + BID2
C3 D1C2 C2A1 “ D2C1 + C1A2
(11.2.53c)
(II.2.53d)
The superalgebra obtained in this way is called the general graded
Lie algebra GL(m/N): it is not simple. The simple algebras 0sp(2p/N),
SU(m/N), P(n), Q(n) are obtained as subalgebras of GL(m/N) by
imposing further conditions on the block matrices Q .
A)	The orthosymplectic algebras Osp(2p/N)
An element of 0sp(2p/N), which exists only when m=2p is even, is
a matrix Q e GL (m=2p/N) characterized by the following conditions:
A	.A	=	0
(2p)	(2p)
D	fi(N) +	fi(N)D =	°
C	=	Q(N)	П(2р)
(II.2.54a)
(II.2.54b)
(II.2.54c)
327
where the two matrices K(2p) an^ ^(n) have the following properties:
2	T
(2p)	(2p)	(2p)
(II.2.55a)
fi(N) fi(N)
(II.2.55b)
From (II.2.55a) we see that since fi(2p) antisymmetric the matrices
A span a symplectic subalgebra Sp(2p,C) of Osp(2p/N). On the other
hand £l(iq)> being symmetric, is an orthogonal metric and the submatri-
ces D span an orthogonal subalgebra O(N,C) of Osp(2p/N).
The ordinary Lie subalgebra of Osp(2p/N) is therefore
G = Sp(2p) ® 0(N)	(II.2.56)
which explains the name chosen for the superalgebra.
The off-diagonal matrices В and C which are related to each
other by Eq. (II.2.54) are acted on by the symplectic and orthogonal
algebra transforming respectively in the defining representations of
Sp(2p) and 0(N). Of particular interest to us will be the algebras
Osp(4/N) with 1 < N < 8. In this case one exploits the Lie algebra
isomorphism
Sp(4,C)	0(5,C)
(II.2.57)
and imposing suitable reality conditions one obtains a real
superalgebra 0sp(4/N) whose Lie algebra is SO(2,3) x SO(N).
SO(2,3) is the group of motions of anti de Sitter space (the
anti de Sitter group) containing the Lorentz generators Маь and the
non commuting anti de Sitter translations Pa: the off-diagonal
328
generators transform as vectors under SO(N) and as spinors under
S0(2,3) playing a role analogue to the role of the supersymmetry
generators in the N-extended Poincare superalgebra. Actually, as we
shall see, by an Inonii-Wigner contraction which sends the cosmolog-
ical constant to zero the Lie algebra Osp(4/N) reduces to the
N-extended Poincare superalgebra, the central charges being
the limit of the 0(N) generators.
B)	The superunitary algebras SU(p,q/N)
As we already mentioned the algebras we define here are complex
algebras: hence we are allowed to use, for their definition, only
those properties which do not distinguish between real and complex
numbers. Such a property in the case of the SU(m/N) family is the
following
Tr A = Tr D
(II.2.58)
which is conserved by the Lie bracket product (II.2.52) and
(II.2.53). The elements of GL(m/N) satisfying (II.2.58) span a simple
superalgebra whose Lie algebra is immediately seen to be:
G = SL(m,C) ® SL(N,C) ® GL(1,C)	(II.2.59)
In view of this the most appropriate name for this complex
superalgebra would be SL(m/N); the name superunitary SU(p,q/N)
originates from the possibility of imposing the following reality
conditions
-1 +
H, . AH, = - A
(m)	(m)
(II.2.60a)
329
H(N) D H(N) ~ d+
н/ ч ®H,L = - Cf
(m)	(N)
(II.2.60b)
(II.2.60c)
where H(m) and are hermitean
H, . = H, .
(m)	(m)
H(N) H(N)
(II.2.61)
and we admit p positive and q negative eigenvalues for while we
choose all positive eigenvalues for
Conditions (II.2.60) are preserved by the Lie bracket (II.2.52-
II.2.53) and imply that A and D span a U(p,q) (p+q=m) and U(N) Lie
algebra respectively. Condition (II.2.58) then tells us that the full
Lie algebra of our real superalgebra is
G = SU(p,q) ® SU(N) ® U(l)
(II.2.62)
This property of the real form (II.2.60) justifies the name SU(p,q/N)
given to the whole family of complex algebras.
Of special interest to us will be the case m=4.
Utilizing the isomorphism:
SU(2,2) 'V SO(2,4)
(II.2.63)
we can reinterpret the Lie algebra sector SU(2,2) x SU(N) x U(l) of
SU(2,2/N) as the anti de Sitter algebra in five space time dimensions
times an internal symmetry SU(N) x U(l): the off-diagonal generators
transforming as spinors under SO(2,4) will be the supersymmetry
330
generators. Hence SU(2,2/N) is the superalgebra of 5-dimensional
supergravity. We mention that SU(2,2/N) can also be interpreted as N-
extended superconformal algebra in D=4 and it is the basis of
conformal supergravity: however since this theory and its
applications are out of the scope and of the philosophy of this book
we will not discuss this point further.
Before leaving the SU(p,q/N) algebras we note that if p+q=m=N
then condition (II.2.58) does not define a simple algebra. Indeed in
this case the matrices of the type
Q =
(II.2.64)
span an abelian ideal of SL(N/N). To obtain a simple algebra we
take the quotient
SL(N/N)/Z(n) => Tr A = 0
(II.2.65)
and we obtain a superalgebra whose Lie algebra is
G = SL(N,C) ® SL(N,C)
(II.2.66)
with the GL(1,C) factor omitted. Correspondingly the real forms
SU(p,q/p+q) have, as Lie algebra
G = SU(p,q) ® SU(p+q)
(II.2.67)
with the U(l)-factor omitted.
331
C)	The superalgebras P(n) and Q(n)
The superalgebras P(n) are defined for m=n>3 as the set of
matrices (И.2.48) fulfilling the conditions:
AT + D = 0 Tr A = О	(II.2.68a)
BT = в ; Ст = - С	(II.2.68b)
while the Q(n) algebra are defined, under the same restrictions on
m and n by
A = D	;	Tr A = О	(II.2.69a)
r = г	•	Tr R = П	(II.2.69b)
In both cases the Lie algebra is
G = SL(n,C)
(II.2.70)
No physical application of these algebras has been found so far.
D)	The exceptional superalgebras D(2,l,g), G(3), F(4)
These exceptional superalgebras have so far found no interesting
applications and, therefore will be described very briefly.
332
1)	D(2,1,g): The Lie algebra is
G= SL(2,C) ® SL(2,C) ® SL(2,C)
(II.2.71)
and the odd-generators, which are 8, transform as the tensor product
of the fundamental 2-dimensional representations of the three
SL(2,C). Since in the anticommutator of two odd-generators there is
a number a which can be arbitrarily chosen, D(2,l, a) is actually a
one-parameter family of 17-dimensional simple superalgebras.
ii) G(3): This is a superalgebra whose Lie algebra is
G = SL(2,C) ® G2
(II.2.72)
The 14 odd generators transform in the 2 of SL(2,C) and in the 7 of
G2. Altogether the superalgebra has 3+14+14=31 generators.
iii) F(4): The superalgebra is
G = SL(2,C) ® SO(7,C)
(II.2.73)
The 16 odd generators transform in the 2^ of SL(2,C) and in the 8-
spinorial representation of SO(7,C).
The superalgebra is therefore 40-dimensional.
333
II.2.3 - Grassmann algebras
In order to exponentiate our superalgebras and obtain the
corresponding supergroups it is convenient to introduce the concept
of Grassmann algebra whose elements will be the parameters of the
supergroups.
A Grassmann algebra GAn is an extension of the field of complex
numbers defined through the following construction.
Let
7Ti 1=1. 2, 3,...,n	(II.2.74)
be n-objects, called generators of the Grassman
algebra, which
satisfy the following anticommutation relations:
ТГ.7Г. = —ТГ.7Т. => 7Г? = 0	(II.2.75)
1 j 311
and let us consider all the possible monomials ir. ... ir. .
Х1 к
The number N^ of different k-monomials is
к < к J	(II.2.76)
and the total number of monomials is
— У — 2m
monomials	s	(II. 2.77)
The Grassman algebra GAn generated by {tt^} is the 2n-dimensional
complex vector space spanned by all the linear combinations of the
2n-monomials тг. ...тг. . Note that we have included the k=0 mono-
X1 xk
mial which is by definition the complex number 1.
334
An element a e GAn of the Grassmann algebra is therefore written
as follows
i	i -i	Ilk
a = z + a.w +а..к 7rJ + a..,ir тг3тг
i 4	nk
(II.2.78)
where z, ait a^j, «ijk
are complex numbers. In particular if
“i = “ij = “ijk = • • • =
moreover that а; л
11 ... ik
GAn is an algebra
0 a is an ordinary complex number. Note
is by definition an antisymmetric tensor,
because the product of the generators
induces,
canonically, a product operation of the elements of GAr).
Explicitly we have
a(I).a<2) = a<3>
where
z(3) = Z(DZ(2)
(II.2.79)
(II.2.80a)
aP> =	+ z2 a<D
iii
(II.2.80b)
«<’> - «“’a!2» - «<2>a(1> ♦	+ z<2>a<>>
4	1 J	1 J	1J	1J
(II.2.80c)
a
(3)
(ijk)
(H.2.80d)
335
The product operation in GA(nj is associative and distributive but it
is not commutative. Every even monomial (k=2p) commutes with any
other monomial, odd or even (k=2p or
every element of GA(n) should be split
а =	+ о/ )
k=2p+l). This suggests that
into an even and an odd part:
= GA(+)
(n) UA(m)
® GA, >
(n)
(II.2.81)
where the even part is a linear combination of the even monomials and
the odd part a linear combination of the odd ones,
induces a Z,-grading of the Grassmann
f-) Z
GA; ( are the even and odd subspaces
(n)
are easily checked
This operation
algebra. Indeed if GA,*^ and
of
GA' '
(n)
GAn the following properties
GA^+?-GA.+?
(n)	(n)
GA<+>
(n)
(II.2.82а)
GA^+? • GA*: >
(.nJ (.n)
GAz ?
(n)
(II.2.82b)
GA, ?-GA< >
(n)	(n)
GA<+>
(n)
(II.2.82с)
Furthermore while
element a e GA(n)
anticommutative:
an even element
the product of two
odd
e GA+ commutes
elements а”, B"
with any
e GA^) is
а
а-В = - В a
(II.2.83)
Defining the grading a of an element
even and to be one if it is odd we can write
GA(n) to be zero
if
is
a e
336
а₽ = (-)ab6a
(II.2.83)
Equation (II.2.83) makes it clear that if we decide that the parameters
multiplying odd elements of a given superalgebra are odd elements of
a Grassmann algebra while those multiplying the even elements of the
superalgebra are even elements of the same Grassmann algebra then all
signs will be automatically taken care of and all factors (-)a^ will
disappear. However before showing how this happens we want to discuss
some more properties of the Grassmann algebras: in particular we want
to define complex conjugation. Let n=2p and let us label the
generators ttj in the following way
ira (a= l,2,...,p)	; 7r>a (•a=p+l,...,2p)	(II.2.84)
We define a mapping * which acts on the generators it in the following
way:
(tt )* = ir	(II.2.85a)
a
(tt )* = tt	(II.2.85b)
•a a
(ттлг^)* =	j ) * (^j) *	(II. 2.85c)
(атг±)* = a*(7rjL)*	(ll.2.85d)
where a* is the complex conjugate of the complex number a.
The mapping * extends canonically to all the elements of the
337
Grassmann algebra.
If a e GA
is given by (II.2.78) we have:
a* = z* + а*(Л* + a* (ir3)*^1)* + ...
1	Ij
(II.2.86)
The operation ★, which is called the complex conjugation in the even
Grassmann algebra
GA,. .,
(2p)
has the following formal properties:
я?’..-
Va e GA,. . : (a*)* = a	(II.2.87а)
(2p)
Vara2 e GA(2p) : (a^)* = a*aj	(II.2.87b)
Va eC, Va e GA,_ , : (aa)* = a*a*	(II.2.87c)
Eq.s (II.2.87) could also be regarded as the defining axioms.
Given the complex conjugation, the notions of reality and of norm
are defined in the same way as for complex numbers
-	__. Л
a = real => a = a
(II.2.88а)
|| a || 2 = a*a (II.2.88b)
It is important, however, to keep in mind that 11<x| |2 is an element
of the Grassmann algebra and it is not positive definite. For
instance the norm of an imaginary odd element is always zero
a = - a , a e GA<’3 y => || a || 2 = 0
(II.2.89)
338
Let now GA(n) be a Grassmann algebra with n generators: an analytic
function mapping GA(n) into itself:
f : GA z . GA f .
(n)	(n)
(II.2.90)
can be defined via a power series expansion:
CO
Va e GA(n) : f(a) = £ f/a)”1 e GA(n) (II.2.91)
m=0
where fm are the coefficients of a series with finite convergence
radius. If a is an even element the series (II.2.91) may extend to
infinity (we say may because a, although even, may be nilpotent
(am = 0 for some 0< m <")); however if a is odd the series neces-
sarily stops after the first element since a^=0.
A function of several variables mapping the tensor product
GA(n) x — x GA(n) into GA(n) can also be defined via a power se-
ries. In this case if the arguments of the function are odd the se-
ries does not stop at the first term but it degenerates into a poly-
nomial of finite degree equal to the number of arguments of the
function.
II.2.4 - Supermanifolds
Equipped with the notion of Grassmann algebras one can introduce
the concept of supermanifold. Without entering the endless
discussions and subtleties which this concept has stirred in the
mathematical and physical-mathematical literature one can take the
simple minded point of view that a supermanifold is a smooth space
339
whose points are labeled by two sets of coordinates: the bosonic and
the fermionic ones.
The bosonic coordinates are even elements of a Grassmann algebra
while the fermionic coordinates are odd elements of the same algebra
GA,», which, to avoid pitfalls, should be chosen to have an infinite
number of generators i>£. Since the concept of function on a
Grassmann algebra is well-defined, we can introduce charts, atlases
and transition functions: in a word the whole machinery of differen-
tial geometry.
Therefore by ^Р^Я we shall denote a supermanifold with p
bosonic dimensions and q-fermionic ones. The coordinates of a point
p e .^p/ч will be denoted by:
p=>{xa,ea)	(II.2.92)
where xa (a=l,2,...,p) are bosonic and Ga (a = l,...,q) are
fermionic.
The functions of several variables which map .^P^9 into GA^, :
ф:.^Р/ч=>СА	(II.2.93)
CO
are called the superfields. Utilizing the nilpotency of Ga the
superfield Ф (x, G) can be written as a polynomial in Ga, whose
coefficients are functions of the bosonic coordinates only:
a	“1 a2
Ф(х,6) = Hx) + Ф (x)Ga + <p (x)G G
u	^1^2
a. a
+ ... +	(x)G \..G 4	(IT.2.94)
1	4
As one sees, a superfield is just a bookkeeping device for а
collection of ordinary fields with different tensor structures. (We
340
emphasize that all the (x) are completely antisymmetric in
their indices because of the anticommutativity of the 0.s). In
supersymmetric theories where the fermionic coordinates 0a are
spinors, the fields in the collection have different spins, bosons
and fermions necessarily coexisting in the same .superfield.
The space of superfields, named С(.^Р^Ч) is acted on by
differential operators which are linear combinations of the
fundamental derivatives:
3
3xa
(II.2.95a)
а =-Э-
a aea
(II.2.95b)
The action of Эа and Эа
is defined through the following formulae
а Ф(х,б) = a ,p(x) + "Э ч> (x)0a + ...
a	a	a ot
(II.2.96a)
ЭаФ(х,0) = v>a(x) + 2^>ag(x)0e + 3v>apy(x)0₽0y + ... (II.2.96b)
and the following formal properties are easily verified:
a a. - a. a = о
a b ba
(II.2.97a)
a aQ - a„a = о
a В Ba
(II.2.97b)
341
Э Э„ + ЭОЭ =0	(11.2.97с)
а В В а
Эа(66ф(х,0)) = <5®Ф - ееЭаФ	(II.2.97d)
The differential operators
®	t = ta(x,0)3 + ta(x,0)9	(II.2.98)
	’ a	а
3	where ta and t?are respectively bosonic and fermionic superfields
span the tangent space to ^Р^Ч named т( ^P'4),
Ж	At each point p={x, 0}	Т(^Р^Ч) is a graded vector space with
J	p-bosonic and q-fermionic dimensions.
r	A graded vector space V(n/m) can be defined in the following
way. Let {ea, ea) be a collection of n elements ea (a=l,...,n) and m
s	elements ea (a = l,...,m) respectively called the bosonic and
fermionic fundamental vectors. An element v e V (n,m) is a linear
combination
V = Vae + v>“e	(II.2.99)
a	Ct
where
ua e GA(+)	(II.2.100a)
CO
va e GA(-)	(II.2.100b)
CO
342
In other words the Va components of a graded vector are even
elements of an infinitely generated Grassmann algebra GA„, while the
Va components of the same vector are odd elements of GA«,.
In complete analogy to ordinary vector space theory one can
introduce the notion of the dual space V*(n/m). It suffices to
introduce a dual basis of linear functionals {e*a, e*a):
Vw e V(n/m) : (w,e*) e GA^
(II.2.101a)
z-> b • г» b
<ea’e*> = 6a
(eQ.e*) = 0
(II.2.101b)
<Ve*> = 0
(e ,еЬ
a *

(11.2.101c)
a
and to define the elements of V*(n/m) as the linear combinations
V* e V*(n/m) v* = e% + e“v	(II.2.102)
x a 75 ex
It follows that:
Vv e V(n/m) , Vw* e V*(n/m) :
(v,w*) = (Ла + Vawa) e GAOT	(II.2.103)
Notice that to avoid ordering problems we have written the
coefficients of the vectors on the left and the coefficients of the
dual vectors on the right.
As usual we can define differential 1-forms on	as
343
elements of the dual vector space to Т(.л/Р/4). The dual basis to the
derivatives Эа and Эа is provided by the differentials dxa and d0a.
Writing
(Э ,dxb) = 6b
a	a
(Э ,dxb) = 0
a’
(II.2.104а)
(Э ,d0B) = 0
(Э ,d06) =
a	a
(II.2.104b)
one can define a 1-form co e T*(^#P^<1) via the following equation
co = dxaco (x,0) + d0aco (x,0)
a	a
(II.2.105)
where coa(x,0), <oa(x,0) are respectively bosonic and fermionic
superfields.
p-forms can now be introduced as elements of the exterior
product of p copies of the cotangent vector space Т*(.4?Р^Ч).
Consistency is obtained if we set the following rules for the
exterior product.
dxa л dxb = - dxb л dxa	(II.2.106a)
dxa л d0e = - d0e . dxa	(II.2.106b)
d©a л d0B = d0B л d0a
(II.2.106c)
344
and if we define
u/p) = W	(x,0)dx 1 л dx л ... л dx P +
a • * •»a
1 p
“1 a9	a.
+ W	(x,0)d0 л dx л л dx p +
aia2-"ap
a.	a
+ ... + w d0 1................d0 p	(II.2.107)
a... .a
where again w_ ... „ _ ... (x,0) are fermionic or bosonic super-
1 m m+1 "p
fields depending on whether the number of Greek indices is odd or
even: in this way the usual grading of the exterior product of forms
is respected:
(p) л ы(ч) = (-)РЧШ<Ч) л U<P)
(II.2.108)
Equation (II.2.106c), however, suggests that (II.2.108) can be
generalized. Indeed the above choice of the fermionic or bosonic
character of “aj.-.c^a^i_____apiS one ^or a b°sonic P~f°rm
It is perfectly legitimate, however, to consider fermionic
p-forms: such are, for instance, the coordinate differentials d©a
and, in general, all the p-forms carrying free fermionic indices in
an odd number.
When the forms are fermionic they must commute with opposite
sign with respect to the bosonic forms of the same degree. Hence we
declare that each form co has two gradings: a grading a=0,l which
tells you whether it is bosonic or fermionic and a grading p which is
its degree. Equation (II.2.108) is then replaced by
Ы(Р) л Щ(Ч) = (xab+pq(q)
(a)	(b)	’ U(b)
w
(P)
(a)
(II.2.109)
345
This is as much as we need, for the moment, of supermanifolds: let us
turn to the exponentiation of superalgebras.
II.2.5 - Supergroups and graded matrices
We come back to the definition of the simple superalgebras
discussed in Section II.2.2. The two classical infinite families
Osp(m/N) and SU(m/N), which are the most relevant to our purposes,
are described in terms of ordinary matrices Q (see Eq. (11.2.48))
whose Lie bracket, however, is the unusual one defined by Eqs. (II.2.52)
and (II.2.53). This Lie bracket can be understood if we perform the
following construction. Consider GL(m/N), namely the algebra of
(m+N) x (m+N) complex matrices, which is closed under (II.2.52), and
let {ta, ta} be a basis of GL(m/N). {ta} (a=l,2,..., m^+№) is a
basis of the even subspace:
(II.2.110)
while {taJ (a = l,2,...,2mN) is a basis of the odd subspace:
(II.2.Ill)
Any matrix Q e GL(m/N) can be written as
Q =
Qat + Qat
a a
(II.2.112)
346
where Qa, Qa e C are complex numbers and the Lie bracket of two
matrices Qj and Q2 is, according to (II.2.52-53), the following one:
[QPQ2} =	+	~
+ QlQ2{ta’t6}
(II.2.113)
At this point notice that the right-hand side of Eq. (II.2.113)
would be the ordinary commutator of
Qi
and
q2
[QpQ2] = QiQ2 - Q2Qx
(II.2.114)
if Qa( ^ ) and ft?'),
instead of being complex numbers, were,
respectively, even and odd elements of a Grassmann algebra GA^ .
In
view of this, to every superalgebra and, in particular, to GL(m/N) we
associate a graded vector space spanned by the linear combinations of
the even generators with even elements of a GA„, and of the odd
generators with odd elements of the same GA^,. The ordinary commutator
of elements of the associated graded vector space provides, in view
of our previous observation, an isomorphic realization of the super-
algebra.
The advantage of this point of view is that we are now able to
define the supergroup corresponding to a given superalgebra as the
exponentiation of the associated graded vector space. Formally we can
write:
A => A = graded vector space where complex
numbers are replaced by elements
of the Grassmann algebra	(II.2.115a)
347
- exp ( A ) = supergroup associated to A
(II.2.115b)
In the case of GL(m/N) our construction introduces the notion of a
graded matrix: an element of the associated graded vector space is a
matrix whose entries are elements of the Grassmann algebra: even in
the diagonal blocks (A,D) odd in the off-diagonal ones (B,C). Such
objects are worth considering for their own sake: indeed they can be
viewed as GA« -linear operators on graded vector spaces and the
supergroups Osp(m/N) and SU(m/N) can be viewed as groups of graded
matrices. The product operation is the ordinary product of graded-
matrices.
Let
/ A
Q =
(11.2.116)
be a graded matrix. A,D are m x m and N x N matrices, with commuting
entries while I, П are m x N and Nxm matrices, respectively, with
anticommuting entries.
The product QjQ2 is defined as for ordinary matrices. We have:
QxQ2 = Q3
(II.2.117a)
A3 = A1A2 + *"1П2
D3 = П1Г2 + D1D2
(II.2.117b)
23 - A1L2 + 1^2
П3 = ПХА2 +
(II.2.117c)
348
The operations of transposition, hermitean conjugation plus the
definition of supertrace and superdeterminant are given below:
(II.2.118a)
Str Q = Tr A - Tr D
S det Q = (det A)(det D')
(II.2.118b)
(II.2.118c)
where D'is defined by the following equations:
I	1	0	1
\	0	1	,
A'	E*	\
,	П'	D'	,
These definitions are designed in such a way that the following
properties valid for ordinary matrices hold true also for graded
matrices:
(QXQ2)T = Q2qI
(II.2.120a)
(QXQ2)+ = Q2Qj
(II.2.120b)
349
strG^op = StrCQ^p
(II.2.120c)
Sdet(Q1O2) = (SdetQ1)(SdetQ2)
(II.2.120d)
Sdet(expQ) = exp(Str Q)
(H.2.120e)
The proof is a straightforward exercise and it is left to the reader:
we just point out that in checking Eq. (II.2.120b) one must utilize
Eq. (II.2.85) to evaluate the complex conjugate of the product of two
entries. In terms of graded matrices the supergroups Osp(m/N) and
SU(m/N) have a simple interpretation.
Let
(II.2.121a)
/ "w
\ 0
(II.2.121b)
be two graded matrices of even-type whose diagonal blocks are defined
in Eqs. (II.2.55) and (II.2.61) respectively.
Й is called an orthosymplectic metric while H is named a
superhermitean one. The reason is that they can be utilized to define
two quadratic forms on a graded vector space V(m/N) of which the
first is the generalization of a symplectic plus an orthogonal
350
quadratic form while the second is the generalization of two
hermitean forms.
Given any two elements V, w e V(n/m) we set
ft(v,w) = vTfiw = \>°\Лг	+ vawbn
(m)aB	(N)ab
(II.2.122a)
- »+i>, . h“)*A(<(rt + (V«)<A(i)ib
(II.2.122b)
and we can define the complex orthosymplectic group Osp(m/N;C) as the
group of graded matrices 0 which preserve the orthosymplectic
quadratic form Й:
fl(0v,0w) = fi(v,w)
(II.2.123)
Equation (II.2.123) implies
0T fi 0 = Й
(II.2.124)
which can be assumed as the defining property of the orthosymplectic
graded matrices. Setting
0 = ехр(Л)
(II.2.125)
and considering A infinitesimal we see that (II.2.124) is equiva-
lent to
ПЛП 1 = - AT
(II.2.126)
351
This can be taken as the defining property of the complex
orthosymplectic algebra Osp(m/N;C). Condition (II.2.126), once
written in explicit block form, coincides with Eqs. (II.2.54).
Similarly we can define the superunitary group SU(m/N) as the
group of graded matrices which, besides having superdeterminant
equal to 1
s det = 1
(II.2.127)
have the property of preserving the superhermitean quadratic form H:
H(^W,'^w) = H(v,w)
(II.2.128)
Equation (II.2.128) implies
= H
(II.2.129)
and setting
°1I = exp(I)
(II.2.130)
at the infinitesimal level we get:
HZH 1---
(II.2.131a)
Str Z = 0
(II.2.131b)
the last equation following from Eq. (II.2.127).
352
These conditions are the same as Eqs. (II.2.58) and (II.2.60).
The real form of the complex orthosymplectic group Osp(4/N;C) which
is relevant to the construction of supergravity theories, henceforth
called Osp(4/N), is the intersection of Osp(4/N;C) with SU(2,2/N):
Osp(4/N) = Osp(4/N;C) П SU(2,2/N)
(II.2.132)
In other words an element Л of the Osp(4/N) algebra satisfies the
following two conditions
S'	T
ПАЙ = - Л1
(II.2.133a)
/4	~-l	+
НЛН = - лт
(II.2.133b)
where has two positive and two
It is to a more detailed study
section of this chapter is devoted.
negative eigenvalues.
of this supergroup that the last
II.2.6 - Osp(4/N) as the N-extended supersymmetry algebra
in anti de Sitter space
As we anticipated the simple superalgebra Osp(4/N) plays a
special role in supergravity because it is the generalization to the
case of an anti de Sitter space of the N-extended super Poincare
algebra (II.2.44). To obtain its explicit form we consider equations
(II.2.133) and we make the following choice for the matrices and
H:
353
л I Y0 0	\
H =______________________
\ °	~^(N) /	(11-2.134)
where C is the charge-conjugation matrix defined by Eq. (II.2.38), Yq
is the gamma matrix in the time-direction and ^(n) is the unit-
matrix in N-dimensions.
The most general graded matrix Л which satisfies Eqs.
(II.2.133) has the following form:
r
(II.2.135)
where ea^= -e^a are the parameters of the Lorentz subalgebra and
ea may be interpreted as the parameters of the anti de Sitter
boosts. Indeed the 4x4 matrices
,	1 ab i a
L=4E Yab-2GYa
(II.2.136)
generate the anti de Sitter group SO(2,3). Furthermore the
antisymmetric parameters едВ= -eba correspond to the generators of
SO(N) while the £д are Majorana spinors
CA = с Сд
(II.2.137)
which play the role of supersymmetry parameters.
354
Writing Л as a GA*,-linear combination of matrices:
Л = - (eabM + eaP + е^т + Q.?A)	(II.2.138)
ab a AB A
we single out the definition of the generators of the superalgebra.
Calculating the commutator
[ЛГЛ2] = Л3	(II.2.139)
and reexpanding the result along the generators Mab, Pa, TAB and QA
we obtain the following commutation relations:
fMab’Mcdl = 1 (rlbcMad + %dMbc-%dMac-riacMbd)	(H-2.140a)
[P ,P.] = - 2M .	(II.2.140b)
L a bJ	ab
[M , ,P 1 = - - (Г) P. - П. P )	(II.2.140c)
L ab’ cJ 2 ac b °c a
[M ,,QaQ] = 7- Q. (y . ) 0	(II.2.140d)
L ab ABJ 4 'Aa 'abaB
Гр ,Q,_1 = - A q (y )	(H.2.140e)
L a’4A6J 2 чАР’а'аВ
355
[’«•’J		1	“Л	-	<I1-2-14M>
tQAa,CW = 1(Cya)a6SABPa " <Cyab)aefiABMab
-4СЛ	(11.2.140g)
Гт T 1	=	-	(6 T	+	6	T -6T	-6T)	(II.2.140h)
L AB’ CDJ	4	1 BC AD	AD	BC	BD	AC	AC	BDJ
In Eqs. (II.2.140 a-b-c) we recognize the anti de Sitter algebra
SO(2,3) of Eqs. (1.3.173): Eqs. (II.2.140 d-e-f) tell us that the
supersymmetry generators Q^a transform under SO(2,3) as a 4-di-
mensional spinor and under SO(N) as a vector. Finally Eq.
(II.2.140g) shows that the Q.s are square roots of all the bosonic
generators: translations, Lorentz rotations and SO(N) rotations.
The relation between the Osp(4/N) algebra (II.2.140) and the
super Poincare algebra (II.2.19), (II.2.43), (II.2.44), (II.2.45)
can be obtained through the following rescaling procedure which
an Inonii-Wigner contraction. Let us redefine our
the following way:
= (M )new	(II.2.141a)
ab7
generators in
(Mab>°ld
.old
(TAB}
= J_ (T )new	(II.2.141b)
2e AB
I
i
!	fp = _L (p )new	(11.2.141c)
J.	а 2ё a
rf‘
I
I
356
(Q )O1<1 = _J— (Q )nEW
k4A<?	l4Aa'
where ё is a dimensionful parameter which we shall
inverse radius of the anti de Sitter space: if ё*0
(II.2.141) are simply a change of basis.
In the new basis the commutation relations (II.
tMab’McdJ 4 Чс^Ш’ШЛЛ?
[Pa,Pb] = - 8 ё2МаЬ
fMab’PJ = - I (Vb - %cPa>
^ab’^W = 4 ^Aa^ab^B
[Pa’<U = "
tTAB‘%J = ~i e ^CA^Ba " 5СВ^Ас?
(II.2.141d)
interpret as the
, then equations
2.140) become:
(II.2.142a)
(II.2.142b)
(II.2.142c)
(I1.2.142d)
(II.2.142e)
(I1.2.142f)
357
= ^Ара - WZ/ - 4 Wab
(II.2.142g)
l-TAB’TCD-l = (6BCTAD + 6ADTBC " 6BDTAC " 6ACTBD^ (II.2.142h)
and are equivalent to the old ones. The limit e->0, however is
singular and it gives rise to a new non semisimple algebra. We easily
see what happens. The translations Pa become abelian and commute with
the spinorial charges 0да; similarly the SO(N) generators become
abelian central charges (they commute with everything else) and the
Lorentz generator in the anticommutator of two Q.s drops out. The
result is exactly the N-extended super Poincare algebra with the
ident i f icat i on:
Z*“ = - 4 lim T
+	e->0 ‘
(II.2.143a)
(II.2.143b)
In this way we obtain an algebraic justification for discarding
the pseudoscalar central charges Z^®. Since the N-extended super
Poincare group with scalar central charges (Z^®) is the Inonii-Wigner
contraction of Osp(4/N) we refer to it as the Osp(4/N) group (or
algebra). For every 1 < N < 8 we shall have two theories of super-
gravity depending on whether a certain parameter is zero or non zero.
In the first case the vacuum state is Minkowski space and the theory
can be regarded as the gauging of the Osp(4/N) algebra (Poincare
358
supergravity) in the second case the vacuum state is anti de Sitter
space and the theory can be viewed as the gauging of the non
contracted	Osp(4/N) algebra (de Sitter supergravity). In the
Poincare case the N(N-l)/2 vectors associated to the charges
are abelian while in the anti de Sitter case they gauge the SO(N)
subgroup. The most intriguing feature of the extended theories (N>1)
is the rigid relation between the cosmological constant and the SO(N)
gauge coupling constant:
о = ё • л = -as2	(II.2.144)
8SO(N) е , л 4е
which is a direct consequence of the Osp(4/N) algebra (II.2.142).
Eq. (II.2.144) implies that a realistic coupling constant
gSO(N) ~ 1 yields a non-realistic curvature radius of the Universe
(R=10~33cm) and vice-versa: hence it is very embarassing. Unfortu-
nately it seems also very central to the whole business since it
reappears both in Kaluza-Klein theory where one tries to obtain the
gauge group from the extra dimensions and in the partial breaking of
supersymmetry in 4-dimensions where one tries to obtain an effective
N=1 Lagrangian from a higher N Lagrangian. The only theory which is
free from this disease is N=1 supergravity based on either Osp(4/1)
or 0sp(4/l).
Moreover this theory is the only one which admits chiral
fermions. For these two facts it is the natural candidate for a
description of particle phenomenology at low-energies and as such it
has many advantages (see Part Four): it is unsatisfactory, however,
because it provides a too low degree of unification (it has too many
freedoms) and because it is not finite nor renormalizable. The
central problem in the programme of superunification is therefore
"how to obtain N=1 supergravity as a low-energy approximation of a
more constrained theory". In this effort, higher supersymmetries,
higher dimensions and recently strings have all come into play. The
patient reader of this book will discover how and will see the
359
advantages and disadvantages of each structure. He will realize that
at every stage a universal threat is waiting in ambush: anti de
Sitter space.
Such universality means only two things: either there is a
unique way out which selects Minkowski space and that is the truth
precisely because it is unique (this may be the compactification of
the heterotic string) or we are biased by a pseudo problem, anti de
Sitter space being acceptable if suitably reinterpreted (see Part
Five). In both cases, as a trap to avoid or as a path to follow the
properties of supersymmetry in anti de Sitter space are an essential
ingredient for the student of supergravity. More of it is going to
come in the next chapters.
360
CHAPTER II.3
SUPER MAURER-CARTAN EQUATIONS AND THE GEOMETRY OF SUPERSPACE
II.3.1 - Maurer-Cartan equations of supergroups on supergroup
manifolds
In Chapter 1.3 we considered the dual formulation of Lie
algebras in terms of Maurer Cartan equations.
The starting point was the construction, on the Lie group-
manifold G, of the left-invariant (alternatively right-invariant
1-forms):
°(L) = g’ldg ;	°(r) = gd®
where
(II.3.1)
g = g(tl.....V
(II.3.2)
361
is any matrix representation of the group element identified by
parameters t^,...,tn (supposing G of dimension n).
In want of better g can always be taken in the adjoint (regular)
representation.
Since c(l) (°r °(R)) are Lie algebra valued we can expand them
along a basis of matrix generators {TAJ
°(L) °(L)TA 5	°(R) - 0(R)TA	(II.3.3)
and by exterior differentiation of (II.3.1) we obtain the Maurer Cartan
equations:
d0(L) + 2 ^ВС°(Ь) л °(L)	°
(II.3.4a)
d4) Ч <Л) ^°(R) = °	(Ib3’4b)
where CABC are the structure constants of the Lie algebra spanned
by the matrix generators Тд
Гт T 1 = C’*CT	(II.3.5)
L1A‘ B-1 AB C
The tangent vectors TA^R^ and TA^L) are dual respectively to the
left and right-invariant 1-forms
A (T^) = 6A
(L)UB ' В
(11.3.6a)
_A pt(L). _ <.A
WTb } " бв
(II.3.6b)
362
They are differential operators on the group manifold G and are,
respectively, the generators of the right and left translations
and	defined by equations (1.3.4-5). Both Тд^ and Тд^
L, ,
(a)
satisfy the Lie algebra (II. 3.5):
i(R)
c
(II.3.7a)
r*(L) +(L)
L A ’ В
,”C *(L)
'AB C
(II.3.7b)


Furthermore (see Eq. (1.3.23)) they commute among themselves
[t{R),T^L)] = О (II.3.8)
As we saw in Chapter II.1 the elements of the classical and
exceptional supergroups are represented by graded matrices whose
entries are Grassmann algebra elements rather than real or complex
numbers.
In addition we saw that one can introduce the notion of a
supermanifold, whose coordinates are Grassmann algebra elements, and
one can straightforwardly extend the calculus of exterior forms from
manifolds to supermanifolds.
Hence the elements of a supergroup can be regarded as the points
of a supergroup manifold and the notion of left-invariant or right-
invariant 1-forms (II.3.1) can be canonically extended to
supergroups. It suffices to replace the matrix g in (II.3.1) by a
graded matrix g:
3(L) = 8-1dg
S(R) = 2 d8-1
(II.3.9)
363
Expanding the graded matrix valued 1-form о т along a basis of
(n)
matrix generators of the supergroups (see Eqs. (II.2.110-111-112)):
o = oA I = oa t + o“ t
(r)	(r)	(r) a (r)
(II.3.10)
we obtain a natural separation into bosonic and fermionic 1-forms.
According to Eq. (II.2.109) we have
О ,L. л O ,L.	— О .L. л О .L.
IrI <r)	(r) iRl
(II.3.11a)
o«T . 66r = aeT .s“	(ы.з.иь)
(r) Ф Ф Ф
oa л o6 = - ов л Oa	(11.3.11c)
ф	ф	ф	ф
к	к	к	к
Therefore from the Maurer Cartan equation
do +oT лОт =0
ф ф Ф	(II.3.12)
which follows from (II.3.9) upon exterior differentiation, by
inserting Eq. (II.3.10) we get
И^ИМ****!*’-
364
daa t
zL) a
'R-'
+ do
- о6,	= °
zL\ Cl p
kR'	(II.3.13)
which can be rewritten as:
= 0
(II.3.14)
In (II.3.14) CAgC are the graded structure constants defined by
Eq. (II.2.13).
Once more as in the case of ordinary Lie algebras eq. (II.3.14)
can be taken as the definition of the superalgebra. Indeed if we are
able to write a super Maurer-Cartan equation (II.3.14) which is
integrable:
ddoA = - cf daB л ac = c\ CB aF л aG л $c e о
•IjC	’ВС FG
(II.3.15)
then the constants CABC satisfy the graded Jacobi identities
(II.2.14) and define a superalgebra.
II.3.2 - Maurer-Cartan equations of Osp(4/N) and Osp(4/N)
As an exercise let us write the Maurer Cartan equations
associated to the 0sp(4/N) superalgebra (II.2.140) whose Inonii-Wigner
365
contraction is the N-extended super Poincare algebra
(11.2.19,43,44).
The procedure is very easy. Consider the graded matrix
(II.2.135) which represents an element of the 0sp(4/N) algebra and
let the parameters (eaB, ea, сЛ®, 6B) be finite rather than
infinitesimal.
, ab a AB _B, , , ab a AB „В.
(G ,£ ,£ ,C ) b- (n ,x ,n ,G )
(II.3.16)
., ab a AB _B.	., ab a AB -B,	,	„
Л(£ ,£ ,£ ,? ) i- A(n ,x ,n ,£r)	(II.3.17)
A generic element of the 0sp(4/N) supergroup can be written as
the exponential of Л:
0(П,х,е) = ехр[Л(п,х,6)]
(II.3.18)
and the parameters n, x, 0 can be regarded as the coordinates of the
supergroup manifold. The left-invariant 1-form
8 = c»_1(n,x,e)dO(n,x,e)
(II.3.19)
is an element of the 0sp(4/N) superalgebra and as such it can be
written in the form (II.2.135)
(II.3.20)
366
where ыаЬ=-ыЬа, Va, A^-A^ are bosonic one-forms while фА is a
Majorana spinor fermionic 1-form
,A _ztA4T
Ф = С(ф )
(II.3.21)
As a result of its being left-invariant the graded-matrix valued
1-form O(L) satisfies Eq. (II.3.15).
Performing the multiplication o(bj A 0(l) we obtain:
dS(L) + 6(L) л °(L) -
(II.3.22)
where the 2-forms Rab, Ra, RAB, pA are defined below
Rab = dwab - wac	л w’b + Va л Vb +	4 фд - YaV	(II.3.23a)
Ra = dva - tcab A	Vb - pA A Ya4>A	(II.3.23b)
pA = d^A - 1 шаЬ	. Yab*A - j YA Va 4 Am a	(11.3.23c)
„АВ ,.AB . 1 .AC .CB , „ yA ,B
R =dA +yA	лА + 2 ф ~ ф
(II.3.23d)
367
To obtain this result the only ingredient which we have used, besides
the multiplication of gamma matrices, is the following Fierz identity
(ФА л	ЛА Ч (Ъ)01Ч ~ Va
Ad 4 d A 4 э d Э A
, 1 , .аВт	।	1 z .aB? ,
+ 4 <Y5Ya> *B - y5Va + 4 <Ya> 4 - Va
’ | (Yab)Ct4 л Y^A	(II.3.24)
plus the observation that фв фд, фв Л фА, фв Л уа фА are anti-
symmetric in (А-’-’-В) while фд Л уа фв and фд Л уаВ фв are symmetric
in the same indices.
Eq. (II.3.24) will be derived in Chapter (II.8): for the moment
the reader should take it as given.
Comparing Eqs. (II.3.20) and (II.3.22) with Eqs. (II.2.135) and
(II.2.138) we see that ojab, Va, А^в, ф^ are, respectively, the
1-form coefficients of the Lorentz generators MaB, the translations
Pa, the SO(N) generators ТдВ and of the supersymmetry charges QA.
In the previous chapter we prepared for the Inonii Wigner
contraction by defining a new basis of generators (MaB)new, (TAB)new,
(Pa)new, (QAa^neW related to the old ones by the rescalings
(II.2.141).
Through the identification
(Л.)old(TA)old = (a* )new(TA)neW
(II.3.25)
we see that (II.2.141) are equivalent to the following rescalings of
the left-invariant 1-forms:
z ab.old . ab.new
(w )	= (w )
ab.old .ab.new
< ) - )
(II.3.26a)
368
(AAB)Old = 2_(AAB)new (RAB)Old = 2e(RAB)neW	(II.3.26b)
(Va)old = 2e(Va)neW => (Ra)old = 2e(Ra)neW	(II.3.26c)
(<pA)old = /2ё (/)new => (pA)old = /2ё (pA)new	(II.3.26d)
In terms of the new quantities, depending on the rescaling
parameter e, the Maurer Cartan equations of 0sp(4/N) read;
Ra = DVa -|фд . -Ад = О	(II.3.27a)
Rab = kab + 452Va . Vb + ёфА . ya\ = О	(II.3.27b)
pA =	" iaVA - V3 = °	(11.3.27c)
RM E +	= 0	(II.3.27d)
where we have used the definitions
„ab _ . ab ac ’b
R = dw - w л ш	(II.3.27e)
369
DVa a dva - шаЬ л V.
b
(II.3.27f)
(11.3.27g)
1 a.b
D*A E *A - Г - \Л
(II.3.27h)
^A S + aAAB - *B
(II.3.271)
The interpretation of Eqs. (II.3.28) is quite evident. Ra® is the
Riemann curvature 2-form associated to the spin connection oja®; DVa
is the Lorentz covariant derivative of the vierbein Va, while Бфд is
the Lorentz covariant derivative of the spinor form фд. is the
field strength 2-form of the SO(N) gauge field A^g while Цфд is the
derivative of фд covariantized not only with respect to S0(l,3) but
also with respect to SO(N).
In the limit ё -» 0 in which the 0sp(4/N) algebra contracts to
the N-extended super Poincare algebra, the Maurer Cartan equations
(II.3.27) become
Ra E DVa - i л y\ = 0
(II.3.28a)
Rab = Rab = 0
(II.3.28b)
PA E ^a = °
(II.3.28c)
RAB E dA^ + 2ФА . фВ = 0
(II.3.28d)
370
II.3.3 - Osp(4/N) Maurer-Cartan equations as the structure equations
of rigid superspace
The Maurer-Cartan equations (II.3.27) or (II.3.28) acquire a
geometrical meaning, which is the starting point for the construction
of supersymmetric field-theories (including supergravity), if we
restrict the space on which they hold from the supergroup manifold
Osp(4/N) to the supercoset manifold
(AdS)4/4N = OsP<4/N>
S0(l,3) ® SO(N)
(II.3.29)
Equation (II.3.29) defines a supermanifold which we call the N-
extended anti de Sitter superspace. Its coordinates are the 4 bosonic
parameters {xa} associated to the translations generators Pa and the
4N fermionic coordinates (tT) associated to the supersymmetry
charges (Од). The ring of functions on (AdS)^/^ is named the ring
of anti de Sitter N-extended superfields
ф = Ф(ха,еА)
Similarly
(M)4/4N = Osp(4/N)
SO(1,3) ® SO(N)
defines the supermanifold which we name N-extended Minkowski
superspace. Its coordinates are {xa, 6^} as in the previous case: the
difference however is the following. The bosonic submanifold of
371
(М)4/4Ы
is flat Minkowski space while the bosonic submanifold of
(AdS)4/4N is anti de Sitter space which has constant negative
curvature. The ring of functions on (M)4^4^ is by definition the ring
of Minkowski N-extended superfields.
We construct the explicit form of the left-invariant 1-forms
(Va, фд, фд, o>ab, A^B) extending to the supercosets the techniques
described in Chapter 1.6 for the ordinary cosets. First we note
that a convenient parametrization of the supercoset (II..3.29) is the
following one:
0(xa,eA) = о (e)0R(x)
г Jd
(II.3.30)
where 0(0) is a parametrization of the coset
Г
Osp(4/N)/Sp(4) ® SO(N)
(II.3.31)
whose coordinates are purely fermionic and 0R(x) is a
parametrization of the coset
Sp(4) ® SO(N)
SO(3,1) ® SO(N)
(II.3.32)
whose coordinates are purely bosonic and which coincides with anti
de Sitter space. That (II.3.30) is a parametrization of the coset
(II.3.29) follows from a simple argument. Let g be an arbitrary
Osp(4/N) element. Acting with g on O(x,0) we get
gO(x,Q) = g0F(e)0B(x) = oF(0’)g(e)0B(x)
(II.3.33)
where g(6) e1 Sp(4) ® SO(N) is the compensator of the transformation
g on the coset element OR(0).
372
On the other hand since (>B(x) is a parametrization of the coset
(II.3.32) we have:
g(6)0B(x) = 0B(x')h(6,x)
(II.3.34)
where h(0,x) e S0(l,3) ® SO(N) is the compensator of the transforma-
tion g(0) on the coset element 0_(x).
D
Therefore we get
gO(x,0) = 0(x' ,0')h(x, 0)
(II.3.35)
which is the correct behaviour for a parametrization of the coset
(II.3.29).
We construct (?B(x) and Op(0) separately. Recalling Eqs.
(II.2.139), (II.3.20) and (II.3.26) we set
I сЧ2ё
0B(x) = exp
(II.3.36)
where the relation between the parameters ta and the coordinates xa
is still to be established. Using the simple relation
(t Ya) - t ta 1)(4х4)
(II.3.37)
we obtain
exp(ietaya)
= cos
(II.3.38)
By means of the position
373
2 ё ха ta . ГТ
	 - —- sin(e V t )
/ 1 + 4е2|х|2------/t2
(II.3.39)
which defines
the coordinate xa we get
(II.3.40)
The inverse matrix
0B(-x) :
is easily seen to be identical with
D
Of/w =
D
CB(-x) =
(II.3.41)
Next we construct 0^(6) :

0F(6) = exp
0
(II.3.42)
374
The exponentiation of this off-diagonal graded matrix can be formally
performed in the same way as in Chapter 1.6 we exponentiated the off-
diagonal bosonic matrix:
(II.3.43)
(II.3.44)
where the square root of the matrices
+	= <5°^ + e“ 0^ = (i + м)а6
M M.
(II.3.45a)
6^ +	= 6^ + 6A0B = (1 + N)*33
(II.3.45b)
is defined by the power series expansion
2.13	54
+ — x---x
16	128
(II.3.46)
What changes with respect to the case of ordinary cosets is that
д
being the 0 .s anticommuting the series stops at a certain point,
all the subsequent terms being identically zero. In particular since
2
2
8
375
there are 4N-different £>.s and since both M and N are quadratic in
6.s the power series expansion (II.3.46) stops at the 2N-power.
In the N=8 case which is the highest of physical relevance we
arrive at the 16-th power. In general we can write
м м M M
(i +	- fl +1 eM0M - | e :0 x0 2e 2 + ...
MM MM
!• 1«3,5*7..  (2№H) Q Ip 1 e 2Ne 2N
2-4-6«8-10 ... (4N)
(i + /W = 6A? + | ©A10B - |	+ ...
_ 1»1*3*5*7 ... (2N+1) qAqA2-A2 _ __ -A2N0B
2*4’6-8’10 ... (4N)
(II.3.47a)
(II.3.47b)
Recalling the rescaling prescription (II.3.26), it is convenient to
rewrite Eq. (II.3.44) as follows:
J,
(VWb>
and we get
(II.3.48)
376
0^(0) = OF(-0) =	! (i + 2 5eMeM)’5	/2е6В	\
	I	- /Ге (ЭА	(<5AB + 2 ё ^S®)*5 у
(II.3.49)
We are now in a position to calculate the left-invariant 1-forms:
a=o 1(x,e)dO(x,e) = о 1(x)0“l(e)d0„(e)0_(x) +
IJ	T	T	1J
+ 0 X(x)dO (x)
В	D
(II.3.50)
Comparing Eq. (II.3.20) with the definition of Va, ыа^, A^B
(II.3.51)
we obtain
, 1 - 2 ё i xay
> = ( . . 'a
A /1 + 4 e2|x| 2
(1 + 2 5 0,0)410.
MM A
(II.3.52a)
377
_ Ъ	с
.	r rl-2ieyjc	/ 1 + 2i е ух
Va = - ^ Tr уа -	X- d ( .......=
4е	I L /1+4ё2|х|2	'/1 + 4ё2|х|2
- — Тг
4е
1 - 2i е уЬх^
1 + 4ё2|х|2
(1 + 2ёем®м)!г<1(1 + 2ёеМём)!г
- 256„de„ (l+2ieyCx )
М и I	с
(П.3.52Ъ)
— Ъ	— с
, I , г 1 - 2i е у х, / 1 + 2i е у х \ i
1 Tr ab	d /	с \
' L /1 +4 е2|х|2	\ /1 + 4 е21х|2 ' '
. г , 1 - 2i е уЬх. 1	ъ	с
+ iTr Г TWIT (1 + 25W d(1 + 2*W
z ’ l + 4ez|x| J
- 2EeNd ®N I	(11.3.52c)
AM = 4 Г(6АВ + 2ёёА0С)Ч1(6СВ+2ё6С6В) - 2e©Ad©B I (II.3.52d)
e	J
л	A
As we see the spinor 1-form фА contains only the differential d© of
the fermionic coordinate ©\ while Va contains both dxa and d©-terms.
The dxa-part of Va is the vierbein of an anti de Sitter space
whose Riemann tensor is
Rab
• •mn
4e2
rab
о
mn
(II.3.53)
Indeed on the submanifold = 0{^d©A =0} we have:
378

.А	ЛАВ _
ф =	А =0	;
Va -	v?(x)dxb	;	wab = wab(x)dxc (II.3.54)
о	с
and the Maurer Cartan equations (II.3.27) reduce to:
i	O„
1	Dv = 0	(II.3.55a)
I
Rab = -4g2 Va л Vb	(II.3.55b)
which are the structure equations of an anti de Sitter space with the
Riemann tensor (II.3.53). This justifies our claim that the bosonic
,	submanifold of the supercoset manifold (П.3.29) is anti de Sitter
space. With a similar argument we can show that the bosonic
,)	submanifold of (II. 3.31) is Minkowski space, characterized by a
vanishing Riemann tensor:
i
।	Rab = 0	(II.3.56)
1	mn
J1}
'fl	It suffices to perform the limit ё -» 0 on the Eqs. (II.3.52) and
on the Maurer-Cartan Eqs. (II.3.27).
'I	Actually in the contraction limit ё * 0 the left-invariant
j	1-forms Va, шаЬ, AAB take the very simple form
фА = d 0A	(II.3.57a)
Va = 2dxa + - 0Aya d 0A
2
(II.3.57b)
379
Ж	wab = 0	(11.3.57с)
А^ = d бА0В - 0Ad0B	(II.3.57d)
which allows an immediate verification of the Maurer Cartan equations
(II.3.28).
The (4 ® 4N) 1-forms {Va, ф^} are the components of an
anholonomic cotangent frame on the N-extended superspace (either
Minkowski or anti de Sitter).
Given any function on	(AdS^/^N), namely any superfield ф(х,0)
’	its exterior differential can be written as:
d(j>(x,0) = Ф^ + ЛдфА	(II.3.58)
where the superfields Фа(х,0) are named the inner components of d<J
ff	while the superfields Лд(х,0) are christened the outer components
of the same.
Calling Da and 5Aa the tangent vectors dual to (Va, Фда):
V-'
Va(D, ) = 6a ; Va(D ) = О	(II.3.59a)
D D
ФА(ЛЬ) = 0
we can write
. Act / ~	<	г A p.a
* (DBB) =
(II.3.59b)
фа(х,0) = »аФ(х,0)
(11.3.60a)
380
MX’S) = 5АаФ(х’6)
(11.3.60b)
In the case (II.3.57) of Minkowski superspace the explicit form of Da
and DAa is easily obtained. Recalling equations (II.2.104) we can
write
1	Й	1
Da “ 7 9a = a =* Фа(х’0) = 7 Фа (x’6)	(II.3.61a)
a 2 a a a a	2 a
°Aa = Aa + 2 (0А^> A => KAa = i Ф + i (II.3.61b)
3G	Эх	Э0
where we have used
de <^BB) - бвбв
d0Aa<J^ = <6S
(II.3.62)
The differential operators Da and DAa are named the invariant
derivatives of superspace. They should not be confused with the
generators {Pa, QAa) of the supersymmetry algebra (II.2.142). In the
next section we discuss why.
II.3.4 - Killing vectors on superspace, that is the generators of the
supersymmetry algebra of superisometries
The reason why Da and DAa should not be taken for the trans-
lation and supersymmetry generators of the supersymmetry algebra
381
(II.2.142) is best understood by recalling that superspace is a coset
rather than a group manifold. On a group-manifold we have two sets of
tangent vectors, both satisfying the Lie algebra: the generators of
right-translations (dual to the left-invariant 1-forms) and the gen-
erators of the left-translations (dual to the right-invariant
1-forms).
On a coset manifold, instead, the symmetry between left and
right is broken by the very fact that G/H can be chosen to be either
a right or a left coset space. In this book we have adopted the
convention that we consider every coset manifold to be a right-coset
space and this is the reason why we restrict our attention to left-
invariant 1-forms. Under these conditions what happens is that the
generators of left-translations (which are not dual to the left-in-
variant 1-forms!) become the Killing vectors of the coset manifold
and satisfy the complete Lie algebra. On the other hand the genera-
tor of the right-translations are now restricted only to the direc-
tions of the vielbein (in our case the Va and фа directions) and in
general are not required to satisfy any algebra. The Da and
D^a vectors are the remnants on the coset of the full set of right-
translations existing only on the supergroup manifold. The Killing
vectors (Pa, QAa, Mab» ^AB^> instead, have still to be constructed.
To obtain their form it suffices to apply the techniques of
Chapter 1.6 and calculate the action of a Lie algebra element on the
matrix 0(x,6) parametrizing the coset.
Recalling Eq. (1.6.72) we can set
Тд(?(х,6) =KA0(x,6) - 0(х,6)ТЛ/д(х,6) (II.3.63)
where Тд is any of the generators (Pa, QAa , Mat,, TAB), Кд is the
associated Killing vector, is either Маь or TAB and WA(x, 6) is
the S0(l,3) • SO(N) compensator. Utilizing the fundamental represen-
tation (II.2.135) by explicit evaluation of (II.3.63) one can work
out the form of the differential operators (Го, QAa, MaB, ^AB^‘
382
the (ё4 0) case this is quite a bit of work which we do not feel the
need to do explicitly. In the contracted case (ё = 0) we can attempt
a direct evaluation of the Killing vectors (P , Q., M . , T. ) = {K.}
SL А Эи Aij	A
relying on their alternative definition as isometries of the vielbein
defined by Eqs. (II.3.57a-b). Hence we write:
va = wabvb
K. A
(II.3.64a)
-> фв
KA
1 ,,ab ,B
---- W, Y кФ
4 A ab
(II.3.64b)
where SL denotes the Lie derivative WAa^ is a suitable S0(l,3) com-
pensator and furthermore we impose that the {Кд} satisfy the appro-
priate super algebra (II.2.142) (with ё=0). (We note that in the
limit ё -* 0 the group SO(N) degenerates into a bunch of N U(l).s
whose action is zero on everything).
To solve this problem we make the ansatz
P
a
= аЭ
a
(II.3.65a)
QAa $	+ Y0ABYgaSa
Mab " 6<Xa9b - Xb3a) + n°AYab
(II.3.65b)
(II.3.65c)
383
ТАВ ’ °
(II.3.65d)
where а, В, Y, 6, П are numerical coefficients to be determined.
Considering first the commutation relations (II.2.142) we obtain
(11.3.66a)
2 BY = ia
(II.3.66b)
Then we impose the invariance conditions of the vielbeins, that is
. -	.	,.ab ..ab _
Eqs. (II.3.64). We begin with Pa and and we assume Wg =	= 0
The definition of the Lie derivative (£t = _Jd + d^j ) extends
trivially to supermanifolds and hence we find:
+ d(a6*) = 0
(II.3.67)
Similarly
Fb
(II.3.68a)
„ ,Aa
QBB
_ . Act , f
dip + d(-
de0^)
= d(6j6j) = 0
(II.3.68b)
384
On the other hand:
Я_ Va = _ I dva + d(_ I Vй) =
^Bg	QBg|
= iBdoV .
V
aeB₽
d0A + (2yd0BYa - - 6d0BYa)o
2	P
= 4 ₽ + 2Y) (d0V)R
2	P
(II.3.69)
Hence in order for Qpp to be a Killing vector we must have
у =

(II.3.70)
In the above equations note that we used the commutation rule:
э/ае|(de л э/аеIл ш + de л э/зор
(II.3.71)
which is correct since both the 1-form de and the vector Э/Э0 are
of the fermionic type. In general for graded differential forms and
vectors we have:
t
, (a)	(b) _	, (a)
(c) W(p) л W(q) " t(c) Ы(р)
(a)
,„(b) + ( >ac+p (a)
%) + (-)	“(P)
,,(b)
(c) W(q)
(II.3.72)
where (c),	(a),	(b) are the gradings and (p) and (q) the degrees of
the differential forms.
Calculating the Lie derivative of Va and along Maj, we can
cd
determine the compensating function Wa^ . We find
385
(|аеАусаеА) + d(
£-	М
ab
i aAz ab c. Jr.A	-c i xA ab cJL
= 4 0 (Y Y )d© + d(2x[a6b] - g 0Y Y 6 )
In the term under exterior derivative the only surviving current is
Indeed the matrix Ya^c = const eat>cd 454д is antisymmetric and
can sit in between two anticommuting 0.s while ya is symmetric and
gives a vanishing contribution in between 0.s (see Chapter II.8).
Taking this into account we get:
,c i xA ab c . Ji . _ .	r-c i xA abc . _A
' =	0 у Yd© + 2dxr 0	- — 0 у d 0r =
= I OA[Yab,YC]d0A + 2dx[a6‘] =	=>
(II.3.74)
Summarizing: Minkowski or anti de Sitter N-extended superspaces are
supermanifolds whose supervielbein (Va, i|)A) admits a group of iso-
metries isomorphic to the 0sp(4/N) or 0sp(4/N) supergroup. The
isometries are generated by the Killing vectors discussed above. The
Maurer Cartan equations (II.3.28) or (II.3.27) are to be reinter-
preted as the structure equations of the supermanifold, (pA, Ra)
being the supertorsion and (Ra^, RA®) the supercurvatures.
The vectors Da> dual to the supervielbein are the left-
invariant generators of the right translations on the supermanifold.
(Their covariant Lie derivative along all the Killing vectors is
zero). Any constraint imposed on a superfield Ф(х,0) by means of the
differential operator Da and D^a has an invariant character with re-
spect to the superisometries: typically a supersymmetric field equa-
386
tion is obtained by applying to ф(х,0) some operator ° constructed
with Da and Бда.
In Table II.3.1 we summarize the explicit form of all the relevant
forms and operators in Minkowski superspace. The reader will in par-
ticular note that we have
= i(CY WABDa = " 1(Суа)а6бАВРа
“ 1(CY ^ag^ABFa
4to’5B? = 0	(II.3.75)
The fact that the anticommutator of the DAa.s is almost identical to
that of the QAa.s is often a source of confusion with respect to
which operator should be named the supersymmetry generator. We hope
to have clarified the matter. The DAa.s which used to generate the
right-supersymmetries on the group-manifold are not Killing vectors
of superspace! Rather they generate fermionic translations which are
invariant under supersymmetry!
Finally let us introduce some names. The 1-form Va is the
vierbein while the 1-form is christened the gravitino 1-form. The
reason is simple: with a procedure similar to the one discussed in
Chapter 1.3, on superspace we can introduce new sets of vielbeins
(Va , фА ) and connections (ша^ , AA® ) which do not satisfy the
Maurer-Cartan equations (II.3.27) (the soft forms). They describe
dynamical fields for which we are going to write action principles
(supergravity theories). While Va is associated to a spin 2-parti-
cle, (the graviton) фА turns out to be associated to N spin 3/2 par-
ticles (the gravitinos).
When the curvatures Ra^, Ra,pA, RA® are non zero, the closure
of the original Maurer equations is reflected into the Bianchi iden-
tities:
387
DRa + Rab л Vb - 1фА л уафд = О
(11.3.76а)
DRab - 8e2R^a л + 2ефд л Т^Рд = О
(11.3.76b)
DpA  5RAB - фв - iaVA - Ra - 1 Rab - Wa = 0	(11.3.76c)
dRab + 4 *а ~ pb = 0	(II.3.76d)
which in the Poincare limit e •+ 0 become
DRa + Rab ~ Vb - ilpA - YapA = 0	(II.3.77a)
DRab = 0	(II.3.77b)
’A ’ °
(II.3.77c)
dRAB + 4 *[А л PB] = °
(II.3.77d)
As we shall see the Bianchi identities play a crucial role in the
construction of the supergravity models.
388
TABLE II.3.1
Minkowski N-extended superspace
A)	Maurer Cartan equations of 0sp(4/N)
Ra = DV3 - |	- Y^A = 0
Rab = Rab = 0
PA = ВфА = 0
rAB = dAAB + 2 фА ~ фв = 0
В)	Explicit form of the vielbein and connections
Va = 2dxa + | 0A л ya d 0й ; фА = d 0A
ab n	.AB _	л rP
w = 0	; A = d © 0 - 0 d 0
C)	Invariant derivatives in superspace
D)	Explicit form of the Killing vectors
P = - - —	T = °
a 2 3xa	AB
Q = —--------i (0Aya)
э0Аа 4	Эха
Mab = 7 (xa3b “ xb3a) + 4 (®AYab
oU
389
E)	Explicit form of the W-compensators
Wcd = W^d = W‘d = 0
a Aa AB
cd
W V
ab
cd
ab
= - 6
F)	Invariance equations
£/Pa / = У = 0 => Ы = [%’DJ =
(%а)	(Qoa)
TaB	TAB	= tTAB‘DJ = [Pa’5AJ = ^WaJ =
= W= °
/ - Vb] ; ЧЛ= < Wa
390
CHAPTER II.4
POINCARE SUPERMULTIPLETS
II.4.1. - How to construct the unitary irreducible representations of
the N-extended Poincare superalgebra
It is well known that the physical concept of a particle of mass
m and spin s corresponds to the mathematical concept of a unitary
irreducible representation of the Poincare group, the mass and the
spin being related to the eigenvalues of the two Casimir operators.
Since the group is non compact, the unitary representations are in-
finite-dimensional and are implemented on a Hilbert space .
The super Poincare group is also non compact and its unitary
irreducible representations are infinite-dimensional: however they
decompose into a finite number of unitary irreducible representations
of the Poincare subgroup. In other words every irreducible unitary
representation of the super Poincare group contains a finite number
of particles. This finite collection of particles is called a
supermultiplet: each supermultiplet corresponds to a conceivable
391
supersymmetric field-theory. This relation will be established in
chapter II.6 by studying a particular example: the Wess-Zumino multi-
plet. Here we confine ourselves to a pure algebraic study and we
begin by writing the N-extended super Poincare algebra which corre-
sponds to setting ё + 0 in Eqs. (II.2.142):
[M , ,M .1 = - (П, M . + T) ,M.	- Tk ,M - Г) M. .)
L ab cdJ 2 be ad ad oc bd ас ас ТЬкг
[Ра’РЬ-1 ”	° tTAB‘^Cy] fTAB,TCD-l ” °
[М ,,P 1 = - - (n P. - П, P )
L ab cJ 2 ac k be a
[Mab’QB₽^ 4 <W7al?ci6
Vb? ~ 1(Cy )ag6ABPa ~ 4 CagTAB
Since the operator
M2 = P pa
a
(II.4.1e)
(II.4.2)
commutes with all the generators (Maj,, Pa, 0да» Тдр)» then the mass
squared is a supersymmetric invariant. Hence all the particles within
the same irreducible supermultiplet have the same mass. Two types of
392
irreducible representations arise: the massive and massless represen-
tations, corresponding respectively to m^O and m^=0. As we are going
to see they have a rather different structure and need a separate
study. The method utilized in both cases, however, is that of in-
duced representations: we consider states |p, { }> which are eigen-
states of the momentum operator
palp> { } > = - ipjp, < > >
(II.4.3)
and we fix our attention on a special reference frame where the mo-
mentum pa has the maximum number of vanishing components. In the
massive case we can go to the rest frame where
Pa = (m, 0, 0, 0)
(II.4.4)
while in the massless case the best we can do is to choose the light-
cone frame, where
P = (- -, 0, 0, -)
a 2’	> 2'
(II.4.5)
The crucial difference resides in the different little groups of the
two vectors (II.4.4) and (II.4.5): the first is invariant under an
SU(2) group, while the second admits only an S0(2). Let us see how
these differences extend to the supersymmetry algebra. To this ef-
fect we choose a у-matrix basis. The best suited to our purposes is
the following:
Remember that Ра=1/2Эа, hence it is an antiherroitean operator and
its eigenvalues are purely imaginary.
393
Yo
(II.4.6a)
Y5 = 1	1 41	0	C =	1 ~ia2	о	(II.4.6b)
	0	-D		°	i02 1	
where are the Pauli matrices and C is the charge conjugation ma-
trix.
In this basis a Majorana spinor,
charge QAa, has the following structure,
the upper and lower components of
such as the supersymmetry
Calling Q(uP> and (/down)
A	A
(II.4.7)
the Majorana condition Q=Q^C reads
!<down) = ioXUP))*
(II.4.8)
So Qa is described by a 2-component complex Weyl spinor. It is con-
venient to introduce a Greek index p=l,2 taking the first two values
of a and utilize Van der Waerden notations for Weyl spinors:

c
0
-1
(1C2\v	(II.4.9a)
394
(II.4.9b)
ХУ =	, ё = еу; %
ft	Й
<Ч> = « >{>
(II.4.9с)
Then the supercharge QA can be written as follows
I % \
(II.4.10)
In these notations the basic anticommutation relation (II.4.1e),
rewritten in terms of the unbarred Q's, becomes
= -		(II.4.11a)
{W =4	£ T pv AB	(II.4.11b)
		
{Q:y,Q*s = - where	4 е^т XAB	(11.4.11c)
°a = (11’		(II.4.12)
This is the common starting point for the analysis of the various
types of representations (or multiplets). We shall consider the follow-
ing cases:
395
a)	massive multiplets without central charges. These are repre-
sentations characterized by m^*0 and by the fact the central charge
operator Тдд vanishes on all the states of the representation.
b)	massive multiplets with central charges where т^*0 and Тдд
acts non trivially on the states.
c)	massless multiplets characterized by m^=0.
II.4.2 - Massive multiplets without central charges
In this case we can set Тдв=0 and we can go to the reference
frame (II.4.4). Eqs. (II.4.11) reduce to:
m^xC2^p*<5AB
=0
(II.4.13a)
(II.4.13b)
Recalling Eq. (II.4.8) it is convenient to introduce 2N operators
BAp =
and their hermitean conjugates
4  л<й 
so that
(II.4.14)
(II.4.15)
*V VU +
=EBBP
(II.4.16)
396
In terms of the B.s the anticommutation relations (II.4.13) become:
	m.	<i1-4-17*1
	(CEB„J  »	(II.4.17b)
and have an obvious interpretation.
The (ВАц, BajP are pairs of creation-annihilation operators
associated to 2N fermionic harmonic oscillators.
The standard way of constructing irreducible representations of
Clifford algebras such as (II.4.17) is via the notion of Fock-space.
One introduces a Clifford vacuum state |Q> which, besides ful-
filling Eq. (II.4.3)
pjft > -im|Q >	; p|fi > = О	(II.4.18)
has the further property of being annihilated by all the destruction
operators BA^:
Вд > = 0	<	> = 1	(II.4.19)
Then the 2^ orthogonal states:
|fl > = |0 >	(II.4.20a)
|Ap > = Вд |0 >	(II.4.20b)
397
IW2 > = В11У1 B12U2I° > (II.4.20c)
IVvVN > = В11Р1---В1Л1° > (II.4.20d)
span a 2^-dimensional vector space J^(Q) which supports a unitary
representation of the Clifford algebra (II.4.17).
The important question is the spin content of these states.
Indeed the little group of the timelike vector (II.4.4) is the spin
group SU(2)s generated by
Ji ° 1 £ijk Mjk
(1, j, к = 1, 2, 3)
(II.4.21a)
[Jpjj] =ieljkJk	(II.4.21b)
and to each multiplet of states |m, s^, sj> which form an irreducible
representation of SU(2)s:
J2|m,s2,s3 > = s(s+l)|m,s2,s3 >
।	2	I 2
J3|m,s ,s3 > = s3|m,s ,s3 >
(II.4.22a)
(II.4.22b)
we can associate a unitary irreducible representation of the Poincare
group, namely a particle of mass m and spin s. It suffices to boost
|m, s^, sj> with the Lorentz boosts Мор
Hence if we classify the irreducible representations of SU(2)s
contained in the vector space J^(B) what we have found is the spec-
trum of particles of mass m and spin s contained in the unitary irre-
398
ducible representation of supersymmetry associated to the Clifford
vacuum |B>.
To accomplish this we begin by noting that the commutation rela-
tions of the spin generators with the В-operators are:
[j.,BA ] = - (О.) В
L i’ Apr 2	1 PV AV
(II.4.23)
This shows that the index p behaves as a 2-component spinor index of
SU(2)s. Secondly we notice that the Clifford Algebra (II.4.17) admits
an SU(2) ® U(N) group of automorphisms. Indeed if we define new B.s
according to
B' = A A VB
Ap A p Bv
(11.4.24)
where A e U(N), Л e SU(2), then the anticommutation relations of the
B.s are identical with those of the old ones:
a	=
4-	+
{%’BAv} = {BAU’BBV} = °	(II.4.25b)
The automorphism group SU(2) coincides with the spin group SU(2)s
while the group U(N) is additional, namely is not part of the super-
symmetry algebra: it is its group of outer automorphisms.
Indeed it can be seen that the infinitesimal U(N) transforma-
tions
6QA = aABQB + 1 SABY5QB
(II.4.26a)
399
t5
aAB “ aAB ” aBA
SAB SAB SBA
(II.4.26b)
(II.4.26c)
are automorphisms of (II.4.1).
It follows that the states of the 22N-(jimenSionai Hilbert space
.#'(й) can be arranged into irreducible SU(2)s ® SU(N) * U(l) repre-
sentations. One begins by assigning the Clifford vacuum |fi> to some
irreducible representation of SU(2)s ® SU(N) ® U(l)
|fi > = |jQ, YQ, X, t >	(II.4.27)
where Jq is the spin
J2|JO, Yo, X, t > = JO(JO+1)|JO, Yo, X, t >	(11-4.28)
X is the third component of the angular momentum
- j0 < X £ j0
J3lJ0’ Y0’ C >	^lJ0’ Y0’ c *
(II.4.29a)
(II.4.29b)
Yq is a set of labels which characterizes an irreducible representa-
tion of SU(N) ® U(l) and t is an index labeling the states within
such a representation.
Then one observes that the operators
400
f "t "t
Vl-.-Vn ° ЧЪ 4^2‘"BVn	(H.4.30)
which applied to the vacuum create the orthogonal basis (II.4.20) are
completely antisymmetric in the exchange of the pairs of indices
Alh ^^2*	~ •"
This means that if the symmetry of the indices p^... pn is given by
the following Young tableau:
 I * “ * I — n - 2p
(II.4.31)
then the symmetry of the indices Аг... An must be that of the comple-
mentary tableau
- 2p = 2J
(II.4.32)
Now the tableau (II.4.31) corresponds to an irreducible SU(2) repre-
sentation whose spin is (for SU(2)M=1):
n - 2p n
J --------- = - ~ P	(II.4.33)
2	2
while the tableau (II.4.32) corresponds to an irreducible representa-
tion of SU(N) whose dimension is
401
D(2J,p)
2J+ 1
2J+p + 1
N+ 1
P
N
2J + p
(II.4.34)
Hence the operator (II.4.30) can be decomposed into a sum of opera-
tors belonging to irreducible representations of SU(2) ® SU(N):
[n/2]	(J=^-p)
K. .	= I K.
A.p,...Ay Ln A.p....A p
11 nn p=0	11 n n
As we see the value of the spin varies in the range
[ —]< J <—
2 L 2J ~	“ 2
(II.4.35)
(II.4.36)
and to each pair (n, J) there is associated a unique SU(N) represen-
tation. If n is even J varies on integer values while if n is odd J
varies on half-integer values. This means that the even products of
supersymmetry charges are bosonic operators while the odd products of
Q.s are fermionic operators: a very reasonable result.
Finally we observe that the number n labels the U(l) irreducible
representations. This can be seen by observing that the generators of
the U(N)-automorphism group are given by quadratic expressions in the
B-oscillators.
Indeed the generators of the infinitesimal U(N) transforma-
tions:
6 BA = BR = \B BR + “ hRB BA
Ap A Bp A Bp N B Ap
(II.4.37a)
i B _ z. А. л
hA - - (hB >
(II.4.37b)
402
h В = h B - * 6 Bh„C
A A n A C
are the following
(II.4.37c)
V  i - i \E"
(II.4.38a)
- Bl B.
2 Ap Ap
(II.4.38b)
as can be checked by computing the commutators:
[1LNTnM, b+ ] = const h B B*
L M N ’ ApJ	A Bp
(II.4.39a)
Blj = const Blp	(II.4.39b)
Now if we apply W to a state |A^p|... Anpn> we obtain
.. I,	.	. n - N I.
W A.p. . . .A p > = --- A.p... .A p
1 11 nn 2	1" n n
(II.4.40)
and we see that the U(l)-charge W is the number operator which counts
how many B.s we have applied to the Clifford vacuum.
It follows that the decomposition (II.4.35) corresponds to a
decomposition into irreducible SU(2) ® SU(N) ® U(l) representations.
The representations of SU(2) ® SU(N) ® U(l) which one finds in
an irreducible supermultiplet are therefore those one obtains by
taking the tensor products of the (jQ,YQ)-representation of the
Clifford vacuum with the representations corresponding to the various
j.(J,n) operators.
N
2
403
From Eqs. (II.4.35) and (II.4.36) it is immediately evident that
the maximum spin which can be carried by an operator	is
JMAX 2
(II.4.41)
The operator	is unique and belongs to the singlet representa-
tion of SU(N)
N
Its W-charge is W=0.
More generally the SU(N) representations which corresponds to a
given spin J are:
(J,n=2J)
K.
„(J,n=2J+2p)
IX
(J,n=N) (II.4.42)
lx
and by taking the tensor product of each of these representations
with the representation (Yq) of the vacuum we obtain the SU(N) repre-
sentations corresponding to the spin J' appearing in the decomposi-
tion of J+Jq:
|J-Jo| < J' < J + Jo
(II.4.43)
404
The best thing to do, at this point, is to consider explicit exam-
ples.
N=1 Supersymmetry
The maximal spin carried by the operators K^»n^ is
JMAX = f	(II.4.44)
and the maximal n is П|^ду=2. Indeed in this case we have just two
operators.
„(J=4, n=l)	t
1	~ a)
V	и
K(J=0, n=2) = ePW
P v
(II.4.45)
Since SU(1) is no group, the vacuum carries only the spin quantum
number Jo. Applying the K.s to the vacuum we obtain the following
states:
|JO> ) - |fi > = Clifford vacuum
(II.4.46a)
K(J=°, n=2)(fi > = K(j=0,
n=2)।.
J0
(II.4.46b)
-
— Г\.
У
n >
(II.4.46c)
As we see the typical N=1 massive multiplet has the following struc-
ture
[(J), 2(J-y, (J-l)]
(II.4.47)
405
and one can check the equality of the Fermi and Bose degrees of free-
dom. If J is odd integer the fermions are J and J-l, while the two
(J-l/2) are bosons. If J is integer the opposite is true. In any case
one has
[2J+1] + [2(J-1) + 1] = 2[2(J-y + l]
(II.4.48)
which is the desired check.
There is one exception to the general pattern (II.4.47). It cor-
responds to the choice Jq=0 for the Clifford vacuum. In that case the
state Jq-1/2 does not exist and we obtain the so called Wess-Zumino
or scalar multiplet
[(У, 0+, O’]	(II.4.49)
Notice that, since к(3=0,п=2) carr£es negative parity the states
|Jq>(+) and |Jq>(") have always opposite parity. In the Wess Zumino
case they are a scalar and a pseudoscalar particle.
The first low lying N=1 massive multiplets are displayed in
Table II.4.I
406
TABLE II.4.I
MASSIVE N=1 MULTIPLETS
J=	5/2	2	3/2	1	1/2	0
Wess Zumino mult.					1	2
Vector multiplet				1	2	1
gravitino multiplet			1	2	1	
graviton multiplet		1	2	1		
	1	2	1			
N=2 Supersymmetry
In this case	an<^ nMAX=Z|- The operators are
К
(J=l)
AB
Ap
BBV
(J=l)
(1=0)
(II.4.50a)
e
К
Ka “ %
K d=^)
(II.4.50b)
v(J=!s) _
K. .	— b
Bf B+ B+ - (K<J=5*>
Ap BBV BCX K (1=4)
(II.4.50c)
407
5
K(J=0) _ 1Л) t Rt _ KO0)
КШ e % BB\> - K(1=1)
(II.4.50d)
(J=0) _ JJV pa t +
m	ap bbv
'a
"и
(J=0)
(1=0)
(H.4.50e)
К
Since the automorphism group SU(N=2) is isomorphic to the spin group
SU(2)g, the operators К can be labeled by two quantum numbers: the
spin J, relative to SU(2)g and the isospin I relative to the internal
SU(N=2). The assignments are those displayed in Eqs. (II.4.50).
Now let the Clifford vacuum be labeled by spin Jq and isospin Iq
Ё
I
TJ?
> = |J0,I0 >	(II.4.51)
Applying the operators (II.4.50) to (II.4.51) and performing the ad-
dition of the spins and isospins we obtain the states of the various
irreducible multiplets.
Let us check the first lowest cases
О iO—
The result is displayed in Table II.4.II
408
TABLE II.4.II
LOWEST N=2 MASSIVE MULTIPLET WITHOUT CENTRAL CHARGES
(VECTOR MULTIPLET)
J	SU(2)isospin	Dimension of SU(2) irreps	Total multiplicity
1	1=0	1	1
1/2	1=1/2, 1=1/2	2 + 2	4
0	1=1, 1=0, 1=0	3 + 1 + 1	5
One can easily check the equality of the Bose and Fermi states:
(2 + 1) + 5 = 4(2% + 1)
Bose	Fermi
(II.4.52)
ii) Jo=l/2,, 1^0
The result is displayed in Table II.4.Ill
409
TABLE II.4.Ill
NEXT LOWEST LYING N=2 MASSIVE MULTIPLET WITHOUT CENTRAL CHARGES
(GRAVITINO MULTIPLET)
J	SU(2) isospin	Dimension of SU(2) irreps	Total multiplicity
3/2	1=0		1
1	1=1/2, 1=1/2	2 + 2	4
1/2	1=1, 1=0, 1=0, 1=0	3 + 1 + 1 + 1	6
0	1=1/2, 1=1/2	2 + 2	4
Again one checks the equality of the Bosons and the Fermions
4x3 + 4 = 1 x 4 + 6 x 2
Bosons	Fermions
(II.4.53)
Finally let us observe that if Iq / 0 all we have to do is to take at
each J-level the tensor product of the representations displayed at
Tables II.4.II and II.4.Ill with Iq. In other words we take the ten-
sor product of the full multiplet Iq=0 with the given isospin repre-
sentation of the Clifford vacuum.
In an analogous way but with increasing complications as the dimen-
sionality of SU(N) grows we can derive the structure of the massive
multiplets for all the N-extended supersymmetries.
410
For N=4, for instance, the lowest lying massive multiplet without
central charges extends up to	^ts structure is displayed in
the following table.
TABLE II.4.IV
LOWEST LYING N=4 MASSIVE MULTIPLET
J	SU(4) Young tableaux								Dimension of SU(4) irrepses	Total multiplicity
2		—							1	1
3/2		__	e						4 + 4	8
1		—	—	Ф	z				6+15+6	27
0		—	—	Ф	—		LL	® 1	1+10+20+ 10 + 1	42
411
II.4.3 - Massive Multiplets with Central Charges
In this section we analyse the modifications due to non-vanish-
ing central charges. When the antisymmetric tensor TAB is non zero,
by means of the definition (II.4.14) the algebra (II.4.11) reduces to
the form
{ВАц‘4}	" %V6AB	(II.4.54a)
	m AB pv	(II.4.54b)
4*bbV	4 * = — Z E m AB pv	(II.4.54c)
where Z^g is the eigenvalue of ТдВ on the states of the representa-
tion. Eqs. (II.4.54) no longer describe a Clifford algebra of har-
monic oscillators and this very fact invalidates the whole construct-
tion of the previous section.
However there is a suitable transformation which converts Eqs.
(II.4.54) into a standard Clifford algebra and reduces the present
case to the previous one. First one notes that by means of an SU(N)
transformation we can always reach a basis where the antisymmetric
tensor Zj^g is in its canonical symplectic form
Note that T is real in the contraction 0sp(4/N)	0sp(4/N) but
after we have set ё = 0 nothing prevents us to take T^-comp lex.
In the language of Chapter II.2 this corresponds to allowing a
Z^g-generator.
412
Zp Z2»--- ^[n/2] bein8 real numbers and e being the matrix ic2:
0	1
(II.4.56)
As we see the odd case reduces to the even one plus the addition of an
extra supercharge which behaves as if the central charge did not
exist. For this reason we concentrate on the N=even case and we intro-
duce an index a which runs on N/2 values rather than N and an index
a which runs on the same range. In this way A=(a,a) and the anti-
commutation relations (II.4.54) can be rewritten as follows
{ВаР’ВЬ\>} “ ^ab^pv
(II.4.57a)
413
{в. ,в£ } = 6.{6 , ар b\)	ab рР		(II.4.57b) (II.4.57c)		
{Вар’4} =	(B- ,B^,J = 0 ap by			
{W =	{в ,B ,} = 0 ap bv			(I1.4.57d)
{ВаЛ’ =	Ze 6 , = {B+ ,Вл } a pv ab	ap bv			(II.4.57e)
{Bap’V =	-Ze <5*. = {bI ,B^,} a pv ab	av bv			(II.4.57f)
We define a new set of operators		(where a are	constants)	
(±) (±) + Sap = “ (Bap + %VBaV>				(II.4.58a)
(±) (±)* SaJ= a	(Bap 1 EpPBiv)			(II.4.58b)
and we find				
(+) (+) { s . s.„} ay bV	(+) (-) = { Sap’ SbV}	(-) (-) = { Sap’ V	= 0	(II.4.59a)
(+) (-). { Sap’ SbJ}	(-) (+)+ = { S , s +} ap bv	= 0		(II.4.59b)
414
(+) (+)□. (+) 9
{ s , s. '} = 2| a I (1 + Z )<5 ,6
ap bv 1	1 ' a ab pv
(-) (-)+ (-) ,
{ S , , S.,} = 2| a I (1-Z )6 .6
ap bv ii' a ab pv
Eqs. (II.4.59) describe a standard Clifford algebra of 2N harmonic
oscillators identical with the Clifford algebra (II.4.17), whose rep-
resentations we constructed in last section. Indeed it suffices to
set
V = (1 + Za) ; V = (1 - Za) (II.4.60)
and Eqs. (II.4.59) coincide with Eqs. (II.4.17). Therefore nothing
has to be done and the multiplets are the same. There is, however, a
caveat of great moment which concerns the possible zeros in Eqs.
(II.4.59d). If 2д=1, namely if some of the symplectic eigenvalues of
ZAB are e4ual to 4m, then the oscillators decouple. Indeed
since they commute we can set them to zero on all the states of the
representation. In this way the effective number of oscillators is
not 2N rather it is
2(N - q)
where q is the number of Za which takes the value 1. The highest pos-
sible value for q is [N/2] so that the effective number of oscilla-
tors can be reduced to a minimum value
N if N » even
2“.ff 	- [fb
N+l if N = odd
(II.4.61)
415
Therefore it seems that we may conclude the following: "A representa-
tion of N-supersymmetry with central charges coincides with a repre-
sentation of Neff-supersyinmetry without central charges". This howev-
er is not yet quite correct because of PCT symmetry. Indeed under a
PCT transformation we have
*
0. H 10,
’Ap ’Ap
which implies that
Therefore under PCT the central charges reverse their sign. This
means that the Clifford vacuum, which is supposed to be an eigen-
state of the central charge cannot be self-conjugate under PCT. Hence
we must double the Clifford vacuum, adjoining its PCT conjugate. This
results into a doubling of the Neff-supersymmetry multiplet.
The correct conclusion is therefore the following.
"A representation of N-supersymmetry with central charges coincides
with a doubled representation of Nefj-supersymmetry without central
charges".
In particular the lowest lying massive multiplet with central charges
has maximum spin = N/4.
For instance introducing central charges the lowest lying repre-
sentation of N=2 supersymmetry displayed in Table II.4.II can be
shortened as follows.
416
TABLE II.4.V
LOWEST LYING MASSIVE MULTIPLET OF N=2 SUPERSYMMETRY
J	SU(2) isospin	Dimension of SU(2) irreps	Total multiplicity
1/2	1=1/2	2	2
0	1=1/2, 1=1/2	2 + 2	4
Table II.4.V describes what is called a hypermultiplet. It is due to
central charges if for N=2 supersymmetry there still exist, as in the
N=1 case, massive representations having maximum spin J =1/2. In
MAX
phenomenological applications these representations are used to accomo-
date the leptons, the quarks and the Higgs particles. Beyond N=2 the
leptons, the quarks and the Higgses must sit in the same multiplet which
contains also the gauge bosons.
This places extremely severe restrictions on the permitted gauge
groups.
II.4.4 - Massless Multiplets
When the mass is zero we choose the light-cone reference frame
where the momentum vector Pa takes the form (II.4.5). Putting the
central charges to zero the algebra (II.4.II) becomes
417
= - ^+02>Лв
= °
(11.4.62а)
(11.4.62b)
where о+ is the following singular matrix
In
a+ = | (0o + °3) =
t
terms of the Вдц
' 1	0
0	0
(Вдц) operators (see
(II.4.63)
Eqs. (II.4.14) and
(И.4.15), Eqs. (II.4.62) reduce to
{BA1’BB1} = 6AB
{B.9,B* } = {B,,,B* } = 0
A2 B2 A2 Bl
(II.4.64a)
(II.4.64b)
<Vb»> - <«’ - 0 (II-‘-“c>
The outcome is that the 2N harmonic oscillators of the massive case,
reduce to N in the massless one. Indeed the Вд2» ®A2 °Perators can be
set to zero on the Clifford vacuum and forgotten. To simplify the no-
tation we drop the index 1 of Вд^ and we are left with the Clifford
algebra
(II.4.65a)
<BA’BB} = {BM} = °
(II.4.65b)
{BA’BI} = 6AB
418
whose fundamental representation is 2^-dimensional and which admits a
SU(N) ® U(l) group of automorphisms. The generators of SU(N) ® U(l)
are represented by the following bilinear forms
В 1	+	+	1u +	+
TA = <BABB - W - 6A (Vm - W	(II.4.66a)
«4^ =4В^М+?	(II.4.66b)
The action of the U(l) generator on Вд:

(II.4.67)
coincides with the action of the
es the S0(2) little group of the
helicity operator J3, which generat-
momentum vector (II.4.5). Indeed:
[jQ,B. ] = ~ (O4) bm.	[Ji>Bt] = " - Bt
L 3 ApJ 2 3zpv Av	L 3’ AJ 2 A
Hence we can introduce an invariant operator
X = j3 - W
(II.4.68)
(II.4.69)
commuting with Jj, Вд and Вд"*" whose eigenvalues characterize the ir-
reducible massless representations of the supersymmetry algebra.
The Clifford vacuum |й> is assigned to an irreducible represen-
tation of S0(2) ® SU(N); namely it is an eigenstate of Jj, with
helicity Л^д^ and it transforms in some representation R of SU(N):
419
lfi>=lXMAX’R>	(II.4.70a)
Лз1ХМАХ’ R > = XMAxlXMAX’ R >	(II.4.70b)
Since |fi> is annihilated by all the B^.s the eigenvalue of W is also
fixed to be N/4. This shows that fixing the S0(2) ® SU(N) representa-
tion of the Clifford vacuum we automatically fix also its U(l)~
quantum number. Hence we have
X = XMAX " 4
The 2^-dimensional basis of the representation is provided by the
states
>	(II.4.71a)
B?|S2 >	(II.4.71b)
A1
B^ B? |fi >	(11.4.71c)
A1 A2
в|...в!|П>	(II.4.71d)
Ai V
which, in view of Eq. (II.4.68), have helicities
XMAX’ XMAX 2’ S1AX 1,””XMAX“ 2
(II.4.72)
420
and which belong to the following SU(N) representations
R, R® □ , R® Q ..,
r ® : n
(II.4.73)
As it happened for the massive representations with central charges
the 2^-dimensional representation in general is not PCT self-conjuga-
te. Indeed, under PCT we have X + -X so that PCT invariance implies
X=0 or the doubling of a supermultiplet, namely the addition to a
multiplet with superhelicity X of a new multiplet with superhelicity
-X.
Therefore the only case when a supermultiplet is automatically
self-conjugate is when Х^ду = N/4. In this case it has 2^-states.
If X*0, namely if Х^ду * N/4 we must add a PCT conjugate multi-
plet with X1мдх = N/2 - Х^ду and in this case we will have a total
number of 2^+' states.
Furthermore when we are in the conditions of self-conjugacy
(Х]^дх = N/4) the SU(N) representation R must be a self-conjugate
one:
R = R
In the case instead that X^y * N/4 the R1 representation of the
additional Х'мдх = N/2 - Х^ду multiplet is
R' = R
It is clear that if N is odd a representation is never PCT self-con-
jugate. PCT self-conjugation happens only for N = 4n.
Equipped with this lore we can easily write down the structure
of massless multiplets for all N-extended supersymmetries 1< N<8.
421
The limit N=8 finds its justification precisely here.
Indeed as we have pointed out in the introduction to part two, up to
this day no convincing way of constructing interacting local field
theories with spin higher than S=2 has been found. On the contrary
there are several partial theorems claiming the internal inconsisten-
cy of such theories. We stress the word local which excludes from
these arguments the non local string field theory containing fields
of any spin up to infinity.
Hence in the context of local field theory we cannot consider
representations of supersymmetry where the maximal helicity is
^MAX'*^- This imposes the bound N=8 since for N > 9, every multiplet
involves at least a 5/2 helicity.
In the following Tables we display all the massless multiplets
1< N <8. In all cases the R-representation of the Clifford vacuum is
R=l. As before R*X means a trivial tensoring of the displayed multi-
plets with R.
TABLE II.4.VI
MASSLESS MULTIPLETS WITH *^=1 (VECTOR MULTIPLETS)
	1	1/2	0
			
1	1	1	
2	1	2	1 + 1
3	1	3 + 1	3 + 3
4	1	4	4
self conjugate
multiplet
422
TABLE II.4.VII
MASSLESS MULTIPLETS WITH /^=3/2 (GRAVITINO MULTIPLETS)
	3/2	1	1/2	0
i	1	1		
2	1	2	1	
3	1	3	3	1 + 1
4	1	4	6 + 1	4 + 4
5	1	5 + 1	10 + 5	20 + 20
6	1	6	15	20
self
adjoint
multiplet
423
TABLE II.4.VIII
MASSLESS MULTIPLETS WITH XMAX=2 (GRAVITON MULTIPLETS)
n\j	2	3/2	1	1/2	0
1	1	1			
2	1	2	1		
3	1	3	3	1	
4	1	4	6	4	1 + 1
5	1	5	10	10 + 1	5 + 5
6	1	6	15 + 1	20 + 6	15 + 15
7	1	7 + 1	21 + 7	35 + 35	35 + 35
8	1	8	28	56	70
self adjoint
multiplet
The underlined numbers in Tables (II.4.VI-VIII) are the dimensionali-
ties of the SU(N) representations the spin J-particles are assigned to.
The derivation of the Tables is almost immediate: we just show
how it works in one example. Take, for instance, the n=6,
multiplet. If the Clifford vacuum |X=2,1_> is an SU(6) singlet then
by successive application of the Вд operators we obtain the following
states:
424
|Х=3/2, 6 >, |Л=1,_15>, |Х = ^,20>, |Х=0,_15>
|Х = - Ч, Ь >, |Х = - 1, _1 >
Consider now the multiplet generated by
X' = S - X =
MAX 2 MAX
Applying the Вд to |X=1,1_> we obtain the states
|X = ^, 6 >, |X=0, 25 >, |X=- 4, 20 >, |X = - 1, 15 >
|X=- 3/2, 6 >,
|X=- 2, 2 >
Putting everything together we see that for each of the SU(6)
representations displayed in the N=6 row of Table II.4.VIII we have
both the X=J and X=-J helicity state which are necessary to
build up a spin J massless particle.
Massless spin-one particles are the gauge bosons of Yang-Mills
theory, while the massless spin-two particle is the graviton. Corre-
spondingly Table II.4.VI lists the spectra of the N-extended super-
symmetric versions of Yang-Mills theory, while Table II.4.VIII dis-
plays the field content of the N-extended supergravities. By this
name we mean the supersymmetric versions of Einstein gravitational
theory which can be identified as N=0 supergravity.
How to construct supersymmetric field theories whose line-
arization yields the spectra displayed in Tables II.4.VI and
II.4.VIII is the question addressed in the sequel of the book. Prior
to that, however, we still need to see which modifications are intro-
duced in the supersymmetry algebra (II.2.142) representations by a
non vanishing constant e.
This is the topic of the next chapter.
425
CHAPTER II.5
SUPERMULTIPLETS IN ANTI DE SITTER SPACE
II.5.1 - Free field equations and the concept of mass in anti
de Sitter space
Anti de Sitter space is the bosonic submanifold of the N-extend-
ed anti de Sitter superspace (II.3.29). It is the following coset
manifold
Sp(4,H)/SO(l,3) % S0(2,3)/S0(l,3)
(II.5.1)
and in the normalization which we use the intrinsic components of its
By Sp(4,H) we understand that particular real form of the
Sp (4, C) Lie algebra which is defined by the condition
Л H(4)= -A+ (see Eqs. (II.2.121b) and (II.2.133b)).
426
Riemann tensor are
pab - /г2яаЬ - Q-2.xaxb ra,b.
R--cd ~ " 4e 6cd = " 2e (6e6d ’ ^d6^	(II.5.2)
This result is easily deduced from Eqs. (II.3.55).
In order to study the unitary irreducible representations of the
Osp(4/N) superalgebra (II.2.142) we need first some information on
the representation of Sp(4,H) ~ SO(2,3). Indeed, as a unitary irre-
ducible representations of Poincare supersymmetry is composed of a
finite number of unitary irreducible representations of the Poincare
subalgebra (each of them infinite dimensional) in the same way, a
unitary irreducible representation of 0sp(4/N) is made out of a
finite number of unitary representations of SO(2,3), also infinite
dimensional. In perfect analogy with the Poincare case, a unitary
irreducible representation of SO(2,3) is what one calls a particle in
anti de Sitter space. The new features are related with the concept
of mass. Indeed the operator PaPa is an invariant neither for the
full Osp(4/N) algebra nor for the SO(2,3) subalgebra. Hence in AdS
a particle is not characterized by the eigenvalue of PaPa, rather by
the eigenvalue of the true second order Casimir of S0(2,3) which in
our normalizations has the following expression:
C2
2 м ьМаЬ
ab
1
4e2
P Pa
a
(II.5.3)
This result is easily retrieved by noticing that if we introduce
indices Л=0,1,2,3,4 and we set
M.v = - Мгл ; M . = M .	; M. = — P
ЛЕ Lh	ab ab	4a a
4e
(II.5.4)
then the 10 generators satisfy the SO(2,3) Lie algebra in its stan-
dard form:
tMAI'MlJ = I (ПЛ + >М2Г- П£ДМЛГ“ «Д>	(П.5.5а)
ПЛЕ = (+’	+)
(11.5.5b)
and С2 defined by Eq. (II.5.3) coincides with the standard quadratic
invariant
C2 = - 2	(II.5.6)
The problem is how to relate the eigenvalues of C? t<T something which
we can call the mass and the spin of a particle. The answer to such a
question is our present goal. It is mainly a matter of comparison. On
one side, as we shall see in the next section, we can construct an
irreducible unitary representation of S0(2,3) via the Wigner induced
representation method, starting from an irrep of the maximal compact
subgroup S0(2) ® S0(3) ® S0(2,3).
The S0(2) quantum number Eq is the eigenvalue of the hamiltonian
operator and, as such, is worth the name of energy (the minimal
energy of the representation); the number J labelling the S0(3) irrep
is instead what we call the spin. On the other hand an irreducible
unitary representation must also be identified with the Hilbert space
spanned by the finite norm solutions of a free field equation suita-
ble to the spin S particle we consider.
We know how to write field equations for arbitrary spin fields
on an arbitrary curved space-time . In particular we can choose
Л = AdS = anti de Sitter space
and we have the result we look for, namely an equation
428
,	2 ,
(sAs) V(S)
(II.5.7)
where l/\J is a second order invariant differential operator (it
commutes with the Lie derivatives along the SO(2,3) Killing vectors)
which acts on the space of spin-s wave-functions and whose eigenvalue
we can call the mass-squared.
Since there is just one quadratic Casimir operator we must have
R7I	2
Led. . = a C„ + В = m
(s)	2 s s
(II.5.8)
where a and Ps are constants. Moreover, since C2 is a function of Eq
and J, that is the labels of the vacuum state in the induced repre-
sentation procedure, then equation (II.4.8) provides a relation be-
tween these labels and ms^. Needless to say we must choose J=S and
(II.4.8) becomes a relation between Eq and ms^, relation which is
different for different spins.
The delicate point in this game is the choice of the origin of
2
the	scale or, in other words, the definition of massless
particles. Indeed we know that m^=0 is a singular value for Poincare
representations, corresponding to a reduction of the number of states
(multiplet shortening) and the same must be true of anti de Sitter
representations. The best way to understand this shortening is from a
symmetry point of view. At ms2=0 the wave equation (II.4.7) must ac-
quire a larger symmetry than the Poincare or anti de Sitter symme-
try. This larger symmetry is conformal symmetry for s=0 and s=l/2
while it is a gauge symmetry for s>l; in anycase it is responsible
for a reduction of the dynamical degrees of freedom and the associated
particle is worth the name of massless.
Since we are interested in particles of spin s=0,^,1,3/2 and 2,
we shall explicitly consider the wave equations of those five kinds of
particles.
429
The s=0 particle
Let @a be the Lorentz covariant derivative, defined as follows. Giv-
en the covariant differential of a field f(x), belonging to some rep-
resentation of S0(l,3)
©f(x) = df(x) + wabtabf(x)	(II. 5.9)
(where tab are the appropriate Lorentz generators) @af(x) is given
by:
Q f(x) = I @f(x)
a Da I	(II.5.10)
Da being the tangent vector dual to the vierbein Va:
Va(Db) = 6a
(II.5.11)
Having defined the covariant Laplacian by:
= ©a@
cov	a
(II.5.12)
it is well known that the following equation
( □	+^)v(x) = 0
cov 3
(II.5.13)
in addition to invariance under whatever isometries the metric
„	_ vavb л	may possess, has further invariance under scale
=р\) у V ab
430
transformations which are instead broken by the equation
(i—Lv+	*(x) =  m0 *(x)
(II.5.14)
This means that (II.5.14) is the correct wave-equation for a scalar
particle of mass mg.
If we choose anti de Sitter space as a background we find
<b	I <^b - 6b6a> = "	(II.5.15)
and (II.5.14) becomes:
(Qcov + m0 - 8g2) *(x> = 0
(II.5.16)
The s=l/2 particle
In the spin 1/2 case we have
S»X = dX - * wabY .X
4 ab
,Й>]Х = - 7Л J
L a bJ 4 ab mn
and the wave equation
(II.5.17a)
(II.5.17b)
For a short derivation of this result see Part One.
431
i ya 9 X = О	(II.5.18)
a
is scale invariant in addition to being invariant under any possible
isometry. Hence the correct spin 1/2 equation is
a h
(II.5.19)
,	Д
ml/2 being the mass. Squaring the = Y operator, using
lyabY
mn ab
(II.5.20)
and inserting the explicit form of the anti de Sitter Riemann-tensor
(see Eq. (II.5.2)) we get
(□	+ ПЧ - 12 e2)X = 0
COV Sj
(II.5.21)
The spin 1 particle
The wave equation of a spin 1 particle, which is described by a vec
tor field Wa, is
@a( Й W - ^.W ) = - m?W.
a b ba	lb
(II.5.22)
Indeed when m*2=0 Eq. (II.5.22) becomes gauge invariant under the
transformation
W i—» W + 4^(x)
a a 2
(II.5.23)
432
where <p(x) is any scalar function.
In the massive case from Eq.
(II.5.22) we derive the transversality constraint
QSV = 0
a
and on the transverse field Eq.
(II.5.24)
(II.5.22) reduces to
W
cov a
+ 25?“" W
ab m
2
ml
(II.5.25)
w
a
Substituting (II.5.2) into (II.5.25) the spin 1 field equation in
anti de Sitter space becomes:
(□	+ m? - 12e2)W = 0
cov 1	a
(II.5.26)
The spin 3/2 particle
A spin 3/2 particle is described by a spinor-vector field xa- Its
wave-equation is the Rarita-Schwinger equation given by:
abed
£
Y5YbVcXd m3/2 Xa
(II.5.27)
where the anti de Sitter derivative Vc is defined as follows:
VcXd = S?cXd + ieYcXd
(II.5.28)
When the mass mj^ Is non-zero, from (II.5.27) one deduces both the
irreducibility-transversality constraints
433
	YaXa - 0	(II.5.29a) = 0	(II.5.29b)
and	the Dirac equation: i^X_ = ~ (m„/? " 2ё)Хя	. a	3'2	a	(II.5.30)
On	the other hand since in anti de Sitter space the derivatives Vc
are	commutative: [V ,V.] = -	_ 2e2Y . = (2e2- 2e2)Y . = 0 L c dJ	2 cd’ab	cd	cd	(II.5.31)
at	тз/2=0 the Rarita Schwinger equation (II.5.27) acquires the fol-
lowing gauge invariance:
X '—* X + v Л
ла a a
Indeed we have:
Eabcdy Y.V V.X = 0
5 b c d
(II.5.32)
(II.5.33)
Applying i^ to both sides of Eq. (II.5.30) we find
,Pvl 1 pq i.rs . pq „am ,	,-.2
(LN ------Y 3 Y )X ~Y 3 X_= _(m„/9 - 2e) x_
VK^cov If pq rs ла ' pq tn 3/2	a
(II.5.34)
and substituting (II.5.2) in (II.5.34) we arrive at
434
	flcov " 16e + (m3/2	2e) ^Xa ~ °	(II.5.35) The spin 2 particle
A	spin 2 field is a symmetric tensor	and its wave-equation
is	the linearized Einstein equation : 2 ^covhab V{a ^"^blm + ®a ^bhmm - 2	h =-im2 h.	(II.5.36) as ms	2 (2) ab
When	(II.5.37) yields the irreducibility-transversality con-
straints:
	h = 0	;	S” h =0 mm	ma	(II.5.37)
and the wave-equation
	(I—11	- 8e2)h , = - n/L.h . cov	ab	(2) ab	(II.5.38)
On	the other hand at m(2)2=0 Eq. (II.5.36) acquires the following
gauge-invariance
	h	।—> h + 9 r t i did	mn	tm rij	(II 5 39)
and	Eqs. (II.5.37) can be imposed as gauge fixings. The same is true
of	the spin 3/2 and spin 1 equation. The irreducibility-transver-
sality constraints, become in the massless case gauge fixing choices.
435
Our results can now be summarized by saying that the second
order wave-equation of a massless spin s particle in anti de Sitter
space is:
(- I I + 4ё2а = 0
cov	s s
where the numbers as are
(II.5.40)
aQ = 2	;	a1/2 = 3	;	= 3	(II.5.41а)
a3/2 = 3	;	“2 = 2	(II.5.41b)
This result combined with the results of next section allows to
express the C2 Casimir operator (II.5.6) in terms of the covariant
Laplacian Ccov.
II.5.2 - Unitary irreducible representations of SO(2,3)
We address now the problem of constructing the unitary irreduci-
ble representations of the anti de Sitter group from a purely alge-
braic point of view. Our starting point is a convenient decomposition
of the SO(2,3) Lie algebra (II.2.142a) + (II.2.142b) with respect to
its maximal compact subgroup
Go = S0(2) 8 S0(3) c SO(2,3)
(II.5.42)
Since Мдь’ = -Ma|, and Pa = -Pa^ are antihermitean we define
436
Н = - — Р.	(II.5.43а)
2ё 0
J+ = - /2 (iM23 + М13)	(II.5.43b)
J_ = - /2’(iM23 - M13)	(II.5.43c)
J3 = -2iM12	(II.5.43d)
H is hermitean, compact and generates the S0(2) subgroup. It can be
identified with the hamiltonian of the system and its eigenvalues are
worth the name of energy E:
H|ip > = е|ф >
}*<
(E = E)
(II.5.44)
З4.» J_, J3 commute with H and generate the spin subgroup
S0(3) -ь SU(2).
[H.jJ =	= [H.jJ = О	(II.5.45a)
(II.5.45b)
[J3’J±1 = 4 J±
(II.5.45c)
= J3
(II.5.45d)
437
The remaining 6 generators spanning the coset SO(2,3)/SO(2) ® S0(3)
can be arranged into the combinations
'«i	-/ "	.’,£ _. . -ж »ы.
Kt =	- 2M-. + — P.			
1 к? = 1	~ 2M0i	2e x - — P, 2e 1	i = 1, 2, 3	(II.5.46a)
Ki = - <Kt)+	(II.5.46b)
which have the following commutation relations with H (as can be
checked from Eqs. (II.2.142):
[h,kJ] = ± kJ
(II.5.47)
Hence the K^ , act as raising and lowering operators for the energy
eigenvalues E.
Furthermore we can rearrange the Kj-, which under SU(2) trans-
form as vectors, in the following way
K(l+i2)	(K1 +
kJ = — (kJ - ikJ)
1-12	^2	1	2
(II.5.48a)
(II.5.48b)
кз = кз
(II.5.48c)
The commutation relations with J3 are
[J3>K1±i2] “ ± Ki±i2
(II.5.49a)
438
[j3,K3] о
(II.5.49b)
This shows that Kj+^2 raises both the energy and the third component
of the spin, while kJ_£2 raises E and lowers Jj. Finally Kj+ raises
E but leaves J3 unchanged.
In this basis the Casimir invariant (II.5.3) takes the form
C = H2 + J2 + -
2	2	1 1
(II.5.50)
Let Jff be the Hilbert space carrying the typical unitary irreducible
representation we are looking for. It is convenient to label the
states |ф> e by the eigenvalues of H, and Jj:
H|(...)Ejm > = E|(...)Ejm >
(II.5.51a)
J2|(...)Ejm > = j(j+l)|(...)Ejm >
(II.5.51b)
J3[ (... )Ejm > = m| (... )Ejm >
(11.5.51c)
where (...) denotes an as yet unspecified representation label.
The representations we are interested in must have an energy
spectrum bounded from below. Hence we introduce a multiplet of vacuum
states |(E0,s) Eq s m> which form an SU(2) irreducible representa-
tion of spin s
integer
J = s(s + 1)
s
half integer
(II.5.52a)
439
- s < m < s	(II.5.52b)
and are eigenstates of H with eigenvalue Eq>0
H| (Eo,s)Eosm> = Eol(Eo,s)Eosm>	(II.5.53)
Furthermore, by definition, the vacuum is annihilated by all the en-
ergy lowering operators:
K~|(Eo,s)Eosm > = О	(II.5.54)
Eq and s label the irreducible representation generated by applying
to |(Eq,s) Eq s m> the raising operators , Kj as many times as we
like, and regarding the Hilbert space spanned by such ket vectors as
the carrier space. For this reason Eq and s have been inserted in the
slot we had prepared for the representation labels.
Evaluating C2 on the vacuum |(Eq,s) Eq s m> we get
C2 = Eq(E0 - 3) + s(s + 1)	(II.5.55)
This result follows from Eqs. (II.5.53), (II.5.54) and from the com-
mutation relation:
Lkt,k+] = 26 ..H + 2i e..A
(II.5.56)
whose validity can be checked by use of Eqs. (II.2.142).
The explicit structure of the Hilbert space Jf is then given by
440
Х = ф jf	(II.5.57)
n=0 n
where is the span of all vectors of the form
	Д	Cnin7n(Kl> <K2> <K3) K,s)Eosn>> nl+n2+n3=n 12 3 (cnxn2n3eC)	(II. 5.58)
•^n	is the Hilbert subspace of states whose energy E is Eq+п. In-
deed	for any |фп> e	we get Н|фп > = (Е0+п)|фп >	(II.5.59)
It is also clear that .#"n is a finite dimensional vector space
The	dimjTn < “	(II.5.60) crucial point is that, in order for	to be a true Hilbert
space its states must have a positive norm
	I’P > = ®0 l*n >	(II.5.61a) 1Ы12 = Ф ll^l|2>0 И II2 = < ’^nl’J'n >	(II.5.61b) n=0
This is guaranteed if the scalar products in all the	subspaces
are positive definite
441
V ф e
n n
< ф Illi > > О
vn' rn
(II.5.62)
In this case the Hilbert space Ж is composed of those series
(II.5.61a) whose norm is convergent
© II Фп H2 < “
n=0
(II.5.63)
We may also tolerate the presence of zero-norm states. If these exist
we define an Hilbert space -^phys comP°sed of the equivalence clas-
ses of all states modulo the zero norm states
X =	= {|i^i> eJf", || ф ||2 = 0} (II.5.64)
pnys
Such a situation is typical of all massless theories and in particu-
lar of gauge theories. The zero-norm states which are removed by the
standard procedure (II.5.64) are gauge degrees of freedom and their
subtraction leads to a shortening of the representation.
What we can never accept is the presence of negative norm states
(ghosts).
Hence before declaring that we have found the unitary irreduci-
ble representations of SO(2,3) we must ascertain under which condi-
tions the space does not contain negative norm states and is
therefore a Hilbert space. These conditions are simply expressed as
lower bounds on the energy label Eq, relative to the spin s.
Let us first state these bounds and then give a sketch of their
derivation
a)	For s^> 1_ there are no ghosts if and only if
Eo > s + i
(II.5.65)
442
When Eo> s+1 there are no zero-nortn states and no representation
shortening occurs. The representation is massive. For Eq= s+1 we
have zero-norm states which can be decoupled. The corresponding
representation is massless and it is described by the appropriate
massless wave-equation.
b)	For s=l/2 there are no ghosts if and only if
Eo > x	(II.5.66)
Decoupling of zero-norm states takes place for Eq=3/2 and Eq=1.
The first value corresponds to a massless representation described by
the massless wave equation (II.5.18), while the limiting value Eq=1
is the so called Dirac singleton for which no field-theoretic inter-
pretation has been found and which has no counterpart in Poincare
theory.
c)	For s=0 there are no ghosts if and only if
Eo >	(II.5.67)
The zero-norm states are found for the special values Eq=2, Eq=1 and
Eq=1/2. Both values Eq=1 and Eq=2 yield the standard massless
representation described by the conformal invariant wave equation
(II.5.13), while the lowest value Eq=1/2 is again a Dirac singleton
representation with no counterpart in the Poincare case and no field
theory interpretation.
Indeed the best way to convince yourself that (Eq =3/2; s= 1/2)
and (Eq = 2, 1,; s = 0) are massless representations is to check that on
the corresponding Hilbert spaces one can not only implement the SO(2,3)
group but also the full conformal group SO(2,4).
Such a check was done by Fronsdal [14] and we refer the reader
to its work.
443
Finally before sketching the derivation of these results given
in [14] and reviewed by Nicolai in [26] we would like to answer a
question we are sure the reader is presently concerned with.
If Eq=2,1 correspond to the massless s=0 case and yield the
same value -2 for C2, what do the other permissible values 2>Eq>1/2
correspond to? The answer is: to negative but permissible squared-
2
mass values. Indeed in anti de Sitter space m is allowed to be nega-
tive provided it is not too negative. This bound which we are going to
discuss again in short is of extreme practical importance in super-
gravity. It says that in anti de Sitter space a saddle point of a
potential can still be stable provided the slope of the descent is
not too extreme.
The reader will appreciate the value of this fact when he realizes
that in extended supergravities no potential is bounded from below and
no extrema are found except saddle points. Coming back to the boring
task of proving the bounds a), b), c) we just illustrate the procedure
with an example which is also the easiest: case a). Let s £ 1 and
and consider the action of Kj+£2 on vacuum I (Eq,s) Eq s m> .
We can write
K;+i2|(Eo,s)Eosm
= R+ <sml,11s+1, m+1> I(Eq,s)Eq+1, s+1, m+1>
+ R0<sml,l|s, m+1 > I (E0,s)Eq+1, s, m+1 >
+ R <sml,l|s-l, m+1 > I (Eq,s)E0+1 , s-l, m+1 >
(II.5.68)
where <s m 1,1| s', m+l> are the Clebsch-Gordan coefficients relating
the product of a spin (s,m) state with a spin (1,1) state to a spin
(s', m+1) state. In our normalizations, which are Nicolai's normaliza-
tions [26] we have
444
(II.5.69a)
n*l> = -
2s(s+l)
(II.5.69b)
<sml,l|s-l, m+l> = ((s~m)
2s(2s+l)	(II.5.69c)
Rj., Rq and R_ are the reduced matrix elements and equation (II.5.68)
is a straightforward application of the Wigner-Eckart theorem of
quantum mechanics.
|R+I2. |R0l2 and |R-P are easily calculated choosing in se-
quence m=s, m=s-l, m=s-2 and utilizing the commutator
^Kl-i2’Kl+i2^ = “ 2^H + J3^	(II.5.70a)
Kl-i2 ° " (Kl+i2)+	(II.5.70b)	5
J
in the evaluation of | |Кц. 2|(rq,s) Eq s m>||2.
The result is	\
5
.	,2	1
lR+l = 2(E0 + s)	(II.5.71a)
Л
j
lRo!2 = 2(E0 “	(II.5.71b)	i
lR_|2 = 2(Eq - s - 1)	(11.5.71c) .	\
445
So we see that (II.5.65) is a necessary condition for the absence of
ghosts. To prove that it is also sufficient is a much harder story
and we refer the reader to the literature [26]. From (II.5.71c)
however the appearance of null-norm states, characteristic of
massless representations is immediately evident in the case
The other bounds are proved with similar techniques.
Let us now call D(Eq,s) the unitary irreducible representations of
SO(2,3) with the proper bounds on Eq implemented and let us finally
come to the relation between this energy label and the mass-squared
m^. Taking into account Eq. (II.5.55), the second order field equa-
tion of a spin s field must be of the following form:
{C2(Dcov) " E0(E0 " 3) “ s(s +	= °	(II.5.72)
where C.(Q ) is the expression of the C„ Casimir (II.5.3) in
2 cov	2
terms of second order differential operators on the coset manifold
AdS = SO(2,3)/SO(1,3)
(II.5.73)
Since П is a second order differential operator which is invar-
cov
iant and since there is no more than one quadratic Casimir, we must
have
C,(D ) = aC + b
2 cov	cov s
(II.5.74)
where a and bs are constants. Now since Pa can be identified with the
tangent vector Da dual to the vierbein the normalization coefficient
a is fixed by inspection of (II.5.3):
446
C2(Deov) = ~~2 Dcov +bs	(II-5.75)
4e
The constant bs can now be fixed by comparison of (II.5.72) with the
field equation (II.5.40).
We take the appropriate massless value Eq= s+1 and we obtain
’ ^2 °cov " 2<® + l)(s-l) + bs = - ^2 Qcov + 4e2as
(II.5.76)
Hence we deduce
j
b = 2(sz - 1) + a
°	о
(II.5.77)
so that
C2(Dcov> = -~Z2 °cov + 2(s2"1) + % = E0(E0’3) + s<s+1>
(II.5.78)
We can now derive the Eq/ш relation for each spin. Comparing succes-
sively (II.5.78) with (II.5.26), (II.5.35) and (II.5.38) by use of
(II.5.41) we find:
2
m0
-2 = (EO-2)(EO-1)
4e
2
21Z2 = E (E -31 + ’ - (F	312
4g2	E0(E0 3) + 4	(E0 2>
2
ml	9	3 2
^2 = E0(E0-3) + |=
4e	4
(II.5.79a)
(II.5.79b)
(II.5.79c)
447
_ 2
^m3/2" 2е)	9	3 2
—= eo<eo-3>+F<eo-?>
4e
2	2
m^/Ae = E0(E0-3)
(II.5.79d)
(II.5.79e)
These relations are summarized for the reader's benefit in Table
II.5.I.
As we are going to see each 0sp(4/N) representation decomposes
into a certain number of SO(2,3) representations with energy labels
related by integral or half-integral shifts: these relations insert-
ed into Table II.5.I reflect into mass relations among the fields
belonging to the same multiplet.
448
TABLE II.5.1
ENERGY - LABEL - MASS RELATIONS
m02 = 4e2 (Eq-2) (Eq-1)
m/ = 4e2 (E0-2) (Eq-1)
m22 = 4e2 Eq(Eq-3)
Bosons
|m1/2|	= 2e(E0-3/2)
Fermions
|m3/2-2e| = 2e(E0-3/2)
II.5.3 - Unitary irreducible representations of 0sp(4/N)
We turn to the full 0sp(4/N) algebra (II.2.142) and we study its
unitary irreducible representations. Since the bosonic subgroup is
SO(2,3) ® SO(N), a typical representation will decompose into a cer-
tain number of D(Eq,s) representations of SO(2,3) which are further-
more assigned to specific representations of SO(N). Such a collec-
tion of particles is what we shall call an anti de Sitter supermulti-
plet. To derive the structure of the supermultiplets we begin by
choosing a gamma-matrix basis. For reasons of convenience we do not
take the same utilized in Chapter II.4 (see Eqs. (II.4.6)), rather we
adopt the following
449
	' 1	0 \		1 °	
Y0 =		=		(II.5.80a)
	° -1		\ °i	°	
	/ 0	1			
y5 =				(II.5.80b)
	11	0			
which corresponds to an		exchange of Y(	) with 75.	1	n this basis the
charge conjugation matrix is given by				
(II.5.81)
and, as a consequence, the supersymmetry charge Од has the same
structure as it had in Eq. (II.4.7), namely
^UP)
A
(down)
4A
(II.5.82a)
Q(down) = ^(UP)/ (II. 5.82b)
It is convenient to introduce a set of 2N operators адц (and their
hermitean conjugates аДр) similar to the Вдц introduced in Chapter
II.4. We set
= /2ё а
a.
Ap
(II.5.83)

450
and we compute the commutation relations of the ад^ with the genera-
tor of the S0(2) ® S0(3) subalgebra. Using (II.2.142) and (II.5.43)
we get		
[h,	] L	AuJ	= 14 2 AU	(II.5.84a)
[H, ал ] L	AyJ	1 		a. 2 Ay	(II.5.84b)
[Ji’	= — (o.) a, 2 i yv Av	(11.5.84c)
[j., ct 1 L 1’ ApJ	= — a. (o.) 2 Av ivy	(II.5.84d)
(where J1=1/V2(J++J_) and J2=-iA/2(J+-J_)).
It appears that адм like К^+ is a raising operator for the ener-
gy eigenvalues. Differently from Kj*, however, адц raises E by half a
unit rather than a unit and it transforms as an SU(2) spinor rather
than as a vector.
Moreover адц is an SO(N) vector.
Introducing hermitean SO(N)-generators:
tAB	2iT g AB	(II.5.85)
we get		
^AB’	acJ = 1(<5ACaBp " 6BCaAu)	(II.5.86)
In terms of these operators, the basic anticommutator (II.2.142g)
becomes:
451
{а, , at } = i6._ К+(о.а_)
Ap’ Bv AB i 2 pv
{%’ aBv} = i6AB K?°i02\v
{%’ “Bv1 = 6AB(V + °ivJi> - 1б^АВ
(II.5.87a)
(II.5.87b)
(II.5.87c)
Eqs. (II.5.87) reveal that the адц can be thought as the square roots
of the SO(2,3) energy and spin "boosts" К +.
To build up a unitary irreducible representation of 0sp(4/N) we
introduce a multiplet of vacuum states
|(E0, s, Y)EQSYmy >	(II.5.88)
which span an irreducible representation of S0(2) ® SU(2) ® SO(N). Eq
is, as usual, the eigenvalue of H, s is the spin and Y denotes the
collection of labels which identify an irreducible SO(N) representa-
tion. Furthermore as m labels the states within the spin s represen-
tation, in the same way у labels the states in the SO(N) irrep.
By hypothesis the states (II.5.88) are annihilated not only by
the Kj" operators but also by all the
aAiJ(EO’ s’ Y>EosYlny > = 0
(II.5.89)
From (II.5.87) it follows that all even symmetric combinations of the
operators адц can be expressed through the even elements of the
0sp(4/N) algebra, namely the generators of SO(2,3) ® SO(N). There-
fore applying such combinations to the vacuum state we simply build
up the unitary irreducible representation
452
PQ = D(E0,s) ® Y
(II.5.90)
of SO(2,3) ® S0(N). Such a representation is one of the finite number
of particles into which the 0sp(4/N) irrep breaks up. We want to find
out the other particles Pj which sit with Pq in the same supermul-
tiplet. In view of the previous observation, to find them out, we
just need to consider the action on the vacuum state of the antisym-
metric combinations of ад^ operators: we call В the set of such com-
binations. В is the following union
В = Bo u B1u...uB2N
(II.5.91)
where Bn denotes the set of antisymmetric combinations of n адм oper-
ators. For instance
«0 = { 1 }
(II.5.92a)
(II.5.92b)
B2 { “Ap’ “BxJ }
(11.5.92c)
2N
Clearly Bn contains \ ) operators and hence the set В contains a
total of 2^N operators. Each of them applied to the vacuum (II.5.89)
creates a superposition of states |(Eq,s,Y) EQ's'Y'm'y'? which can be
taken as the vacua of new SO(2,3) ® SO(N) representations D(Eq,,s') ®
Y'.
453
These are the other particles completing the supermultiplet.
The fact that the set В is finite means that the number of par-
ticles in a supermultiplet is also finite.
The 0sp(4/N) supermultiplets are of the form:
DCE^1), s(1)) Ф ...®D(E<r), s(r)) with r < «>
(II.5.93)
where we suppress SO(N) labels for simplicity. An important differ-
ence between Poincare and anti de Sitter representations is the fol-
lowing: in general we shall have Eq^^ * Eq(^) * •••• * Eq^17) so that
the particles within the same supermultiplet having different energy
labels have also different masses. This is no scandal since PaPa is
not an invariant operator.
In perfect analogy with the discussion of the previous section
the relevant questions are two
i) Which bounds have to be imposed on the energy label Eq
of the vacuum state (II.5.88) in order for no ghost to be present
in the spectrum and therefore for the representation to be truly uni-
tary?
ii) For which critical values of Eq are there zero-norm states
which can be decoupled producing a shortening of the multiplet?
The answer to the second question is that on top of the massless
representations there are, for N >2 short massive representations
where Eq is related in a convenient way to the SO(N) quantum num-
bers. These short representations are somehow the counterpart of the
Poincare massive multiplets with central charges.
To see how all this works we consider in some detail the super-
multiplets of 0sp(4/l) and 0sp(4/2).
454
II.5.4. - 0sp(4/l) supermultiplets
In the N=1 case there is no SO(N) group; the vacuum state is
labeled only by Eq and by its spin s.
The sector is composed of the two operators
B1 = {au P = 1>2}
(II.5.94)
while the B2 sector contains just one operator
pv t t
e a a
у v
62
(II.5.95)
The B^-sector operators increase the energy label by half a unit and
carry spin 1/2. B2 is a scalar operator (it does not change the spin)
and it increases the energy label by a unit. Considering the action
of Оц on the vacuum we can write the following linear combination
af (Eq,s)Eq s m > =
D x 1 1 I . 1	. ls In,	a.1	a.1	. 1 s .
R. < sm----s + — , m + —> (En,s)En + — , s+— , m + — > +
+	22	2	2	0	0 2	2	2
R_
1 1 I
s m-----s
2 2
J.
2
,n + 7> I(EO’S)EO + |
"2 ’ 1П+2 > (И-5.96)
corresponding to the angular momentum addition rule
(-1) ® (s)
2
<.*!>
(s-A>
(II.5.97)
Inserting the value of the Clebsh Gordan coefficients
455
£ £
2 2
|s-i.
fs tmt 1.^
1 2s + 1 J
£
2
(II.5.98a)
££
2 2
1
m*2
, s - m
l2s + 1J
(II.5.98b)
£
2
and utilizing the anticommutator (II.5.87c) to calculate the norm
l|a| |(Eo,s)Eosm > ||2 = < Eo s m(E0,s) |a1a|(Eo,s)Eosm >
(II.5.99)
we can deduce the values of the reduced matrix elements R+ and R_. We
get
lR+|2 = Eo + s	(II.5.100a)
lR_|2 = Eo " s " 1	(II.5.100b)
Hence to avoid ghosts in the 0sp(4/l) representation
Eo > s + 1
(II.5.101)
which is the same bound which guarantees the absence of ghosts in the
SO(2,3) representation build on the same vacuum state. Furthermore
for Eq> s+1 Eqs. (II.5.100) show that to the representation D(Eq,s)
we must add the representations D(Eq+1/2, s+1/2) and D(Eq+1/2,s-1/2).
In the case Eq= s+1 the vacuum D(Eq, s) is a massless particle:
Eqs. (II.5.100) show that in this case we have multiplet shortening
since the representation D(Eq+1/2, s-1/2) drops out (|R_p=0). Fur-
456
thermore the surviving representation D(Eq+1/2, s+1/2) is also a
massless particle. Thus masslessness implies multiplet shortening as
in the Poincare case.
In the special case s=0 these arguments break down and at this
level we derive no bound on E^, since D(Eq+1/2, 0-1/2) is absent
anyhow.
At the next level we must consider the action of the B2 operator
which raises the energy by one unit and leaves the spin unchanged.
Applying B2 to the vacuum we may obtain a superposition of the ground
state of a new representation D(Eq+1,s) with first level excited
states of the representation D(Eq,s) itself.
So we can write:
[<4> ct+] I (Eo,s)Eosm > =
Rol(Eo+1’s)Eo+1’ S’ s > + aKi+i2l(E0’s)E0’ S’ S-1 > +
gkijl (E0,s)E0, s, s >
(II.5.102)
To calculate the coefficients Rq, a and p one observes that the lower-
ing operator annihilates the state |(Eq+1,s) Eq+1,s,s>. Hence we
apply to both sides of (II.5.102) and calculate the norm of the
resulting state Ki-[aj+a2+]| (Eq s m> by use of the super Lie algebra
commutation rules (II.5.142). After a lengthy but straightforward
calculation one arrives at
a = 6 = 0 for s = 0
у/ g	S
ct = - ----- ,	6 ---------otherwise
Eo-1	V1
(II.5.103)
457
Moreover one obtains
bo 1
This result proves that the representation D(Eq=1,s) is absent if
either s=0 and Eq=1/2 or otherwise if Eq= s+1 for s >1/2.
We arrive at the following complete classification of 0sp(4/N)
multiplets due to Heidenreich
Type A; Wess Zumino multiplets (Eq>1/2)
D(E„,O) Ф D(E„+- , -) Ф D(E_+ 1,0)
U	и 2	2	u
Type B: Massive higher spin multiplets (Eq>s+1, s 1/2)
D(E0,s) Ф D(Eq+| , s + |) Ф D(E0+j , s-|)
Ф D(EQ+ l.s)
Type C; Massless higher spin representations (s >1/2)
3	1
D(s+1, s) Ф D(s+— , s+—)
Type D: The Dirac singleton
D(| , 0) Ф D(1 , |)
Apart from the Dirac singleton which has no counterpart in Poincare
theory there is a perfect analogy between the field content of the
0sp(4/l) and the N=1 Poincare multiplets. One should compare these
results with Tables II.4.I, II.4.VI and II.4.VIII. As in the
458
Poincare case 0sp(4/l) massive multiplets have the following struc-
ture
[(s+|), 2(s), (s-|)]
except for the lowest lying Wess-Zumino multiplet which exists in
both theories:
[(|), 2(0)]
Furthermore the massless multiplets are in both cases of the same
type:
[(s), (s-|)]
The difference resides in the following: the particles sitting in an
AdS massive multiplet have all different masses while in a Poincare
multiplet they are all degenerate in mass. With the help of Table
II.5.I one can easily retrieve the masses within a given multiplet.
Examples are given in Tables II.5.II,III,IV,V. The massless multi-
plets are the same as for the Poincare superalgebra.
459
TABLE II.5.II
OSP(4/1) WESS-ZUMINO MULTIPLET WITH Е0=ш>1/2 (w/1,2)
No	Spin	Energy	Mass
1	1/2	to + 1/2	ml/2 = 2e(w-l)
2	0	tO + 1 co	_2 m()+ = 4e (co— 1) _2 - 4e (co-2) (co-1)
TABLE II.5.Ill
OSP(4/1) WESS-ZUMINO MULTIPLET WITH E0=l
No	Spin	Energy	Mass	
1	1/2	3/2	|mi/2| = 0	
		2	m0+	= 0
2	0	1	m0-	= 0
460
TABLE II.5.IV
OSP(4/1) WESS-ZUMINO MULTIPLET WITH Eo=2
No	Spin	Energy	Mass
1	1/2	5/2	lml/2l =
		3	-2 m0+	= 4e
2	0	2	m0-	= 0
TABLE II.5.V
OSP(4/1) MASSIVE VECTOR MULTIPLET (E0=w>3/2)
No	Spin	Energy	Mass
1	1	W + 1/2	m]2	= 4e2(w-3/2) (<i»-l/2)
2	1/2	0) + 1 0)	lml/2l = 2e(w-l/2) 1ml/21 = 2e(w-3/2)
0		to + 1/2	m02	= 4ё2(ш-3/2) (ш-1/2)
461
Let us finally make some comments on the Wess-Zumino multiplet. As we
see from the Tables we have three possibilities depending on whether
Eq=1,2 or is none of the two.
In the first case all particles in the multiplet are massless,
in the second, one particle is massless (the scalar) and the other
two are massive; finally in the third they are all massive. Multi-
plets of the second type do indeed occur in Kaluza-Klein super-
gravity.
Furthermore for 1/2<Eq<1 and 1<Eq<2 the multiplet contains a
negative squared mass particle.
Nonetheless the energy is positive and the values of m^ in the
range
,	2
-7<J4<0	(II.5.105)
4
do not signal any instability of the field theory in which they
appear. In particular suppose that we have n-scalar fields coupled to
gravity (see Eq. (1.5.23)) and that the scalar potential W(<))1) has
an extremum
Ф1 = Фо л| i i = °
9ф =Ф0	(II.5.106)
corresponding to anti de Sitter space (see Eq. (1.5.36))
- 2e2 S И(ф ) < 0
def 0
(II.5.107)
In this case the linerized field equation of the scalar fields be-
comes (see Eq. (1.5.37))

(II.5.108)
462
Comparing (II.5.108) with the standard Eq. (II.5.16) we deduce the
structure of the mass matrix (see Eq. (1.5.42))
K2.. = 6 3.3.W+6..8e2 = (6 3 . 3 .W - 46. .W) I
4 1J 4	1J |ф=Ф0
(II.5.109)
Stability of the solution corresponding to the extremum (II.5.106) is
guaranteed if the eigenvalues m^ of the mass-matrix (II.5.109)
satisfy the lower bound (II.5.105) which reads
2 „
m. >
i “
_2
e
и(Ф0)
2
(II.5.110)
Eq. (II.5.110) is called the Breitenlohner-Freedman bound. It has the
very pleasant implication that a saddle point of the scalar poten-
tial can be stable. Indeed calling the eigenvalues of the Hessian
matrix the bound (II.5.110) means
3
Xi > Z И(Ф0) < 0
(II.5.Ill)
requiring only the steepest descent to be not too steep. This proper-
ty is a blessing for extended supergravities whose scalar potentials
are found to be unbounded from below and to admit at most saddle
point extrema.
In Fig. II.5.I we have depicted the behaviour of squared masses
in the Wess-Zumino multiplet.
463
FIG. II.5.I
The behaviour of the mass-squared in the Wess-Zumino multiplet
464
Using Table II.5.II one can check that whatever the value of Eq=o)
the following mass sum rule is satisfied
2	2	2
“0++“0--2 ml/2 = °	(II.5.112)
This is an example of a more general relation satisfied by the masses
of any supersymmetric theory in an unbroken phase
2
Str m = 0
(II.5.113)
where Str denotes the supertrace of the squared mass matrix. This
relation will be discussed in Chapter IV.5.
U.S.5
Remarks about the N-extended
case and the example of the
Osp(4/2) multiplets
The construction of unitary
0sp(4/N) with N>2 can be done in
complicated. One’s work, however,
following observations
irreducible representations for
a similar way but it is much more
is considerably facilitated by the
a)	Any irreducible 0sp(4/N) supermultiplet is a direct sum of
0sp(4/l) multiplets since 0sp(4/l) c: 0sp(4/N).
b)	In the limit ё -» 0 0sp(4/N) tends to the N-extended Poincare
group 0sp(4/N) and the 0sp(4/N) multiplets tend to Poincare multi-
plets. Since all masses are proportional to 5, they all tend to zero,
so that in the limit ё -> all 0sp(4/N) representations tend to mass-
less Poincare representations. This implies that all 0sp(4/N) multi-
465
plets can be obtained from products of (possibly) reducible representa-
tions of S0(N) with massless multiplets of Poincare supersymmetry.
Notwithstanding these facts, when N>2 in the analysis of
0sp(4/N) multiplets there arise new features. The unitary bound
(II.5.101) is no longer sufficient and it must be replaced by a rela-
tion
EQ > s + 1 + f(Y)
(II.5.114)
which involves also the quantum numbers Y of the SO(N) representa-
tion. Furthermore it may happen and indeed it does happen that for
certain critical values of Eq, not corresponding to massless parti-
cles, the conditions of multiplet shortening (zero-norm states) are
nevertheless met. This phenomenon has been studied by Freedman and
Nicolai [12] in the particular case of N=3. It occurs also in N=2
and seemingly for all N.s. Here as an illustration we give a short
description of the 0sp(4/2) multiplets which will be useful in the
discussion of Kaluza-Klein supergravity.
Recalling Eqs. (II.5.84), (II.5.86) and (II.5.87) we note that
in the N=2 case the index A takes only 2-values and that the S0(2)
generator can be identified with a hypercharge Y
Introducing the linear combinations:
f +	.	+ X
fal ± ia2y)
(II.5.116a)
a+ = ч—— (a. ± iaQ )
P	2p
(II.5.116b)
466
(II.5.116c)
we find
[H, a*]	=	-a*	;	[j,,	a*] = -	(a.)	a*	(II.5.117a)
L	2	U	Li’pJ	2	i	pV	v
[y, 51]	=	±	51	(II.5.117b)
L UJ	u
from which one reads off the energy, spin and hypercharge raising and
lowering properties of the operators a“, a^
To construct unitary irreducible representations we consider a
vacuum state (II.5.88) where Y=y denotes the vacuum eigenvalue of the
hypercharge
Y|(EosY), Eosmy> = y| (Eo s Y)E0 s my >
(II.5.118)
As usual the particles in the multiplet are obtained by successive
applications to the vacuum |E(q s Y) Eq s m y> of the antisymmetric
combinations of the raising operators a~ (the B^-sectors). These
combinations may be classified according to their spin and hyper-
charge content
a with As = —	, AY = ± 1
U	2
e^Va+a+ with As = 0	, AY = 2
U V
(II.5.119a)
(II.5.119b)
467
ePVav	with	As = 0	,	AY = О	(II.5.119c)
ePVa"a“	with	As = 0	,	AY = -	2	(II.5.119d)
with As = 1	, AY = 0	(ll.5.119e)
epVap5*a" with As = 1/2	, AY = ± 1	(II.5.119f)
epVepaa~a"3+a+	with As = 0	,	AY = О (II.5.119g)
Ц V p О	6
from which one can obtain the N=2 multiplets and their energy, spin
and hypercharge assignements.
The unitary bound is found with arguments similar to those em-
ployed in the previous section and reads
|E0| > |y| + s + 1 (11(5.120)
There are only three types of massive unshortened representations if
one restricts oneself to particles of maximum spin-2 which are ob-
tained by applying the operators (II.5.119) to vacuum states of spin
0, 1/2, 1 respectively.
They are displayed in Tables II.5.VI, VII, VIII.
In addition to these unshortened representations, there are also
short multiplets. The first kind of shortened representations are the
massless multiplets which have the same structure as in Poincare
supersymmetry and where all energy labels obey Eq= s+1 for s > 1/2
and Eo= 1,2 for s=0. The second kind in general will contain massive
468
particles. A careful analysis which we omit reveals that for
(ye Z):
E0 ’ |y|
1
2
(11.5.121)
s = 0
several of the operators (II.5.119) create zero-norm states and that
there are no s=l representations.
The resulting multiplets are the anti de Sitter analogues of the
N=2 hypermultiplets of Table II.4.V and their structure is displayed
in Table II.5.IX.
Finally we note that there are further short multiplets which
have no analogue in N=2 Poincare supersymmetry. These are obtained
by saturating the unitarity bound (II.5.120) with s>l/2.
469
TABLE II.5.VI
N=2 VECTOR MULTIPLETS IN ADS SPACE
No	Spin	Energy	Hyp.	Mass
1	1	E0+l	У	16EO(EO-1)
	1/2	Eq+3/2	У-1	-4E0
4	1/2	E0+3/2	y+1	-4E0
	1/2	Eq+1/2	y-1	4E0-4
	1/2	Eq+1/2	У+1	4E0-4
	0	E0+2	У	16Eq(Eq+1)
5	0	E0+l	У-2	16E0(E0-l)
	0	E0+l	У+2	16EO(EO-1)
	0	E0+l	У	16EO(EO-1)
	0	EO	У	16(E0-2)(E0-l)
470
TABLE II.5.VII
N=2 GRAVITON MULTIPLETS IN ADS SPACE
No	Spin	Energy	Hyp-	Mass
1	2	E0+l	У	16(Eq+1)(Eq-2)
	3/2	E0+3/2	y-1	-4E0-4
4	3/2	Eq+3/2	y+i	-4E0-4
	3/2	Eq+1/2	y-i	4Eq-8
	3/2	Eq+1/2	y+1	4 Eq-8
	1	Eq+2	У	16(Eq+1)Eq
6	1	E0+l	y-2	16(E0-1)Eq
	1	E0+l	y+2	16(Eq-1)E0
	1	E0+l	У	16(Eq-1)Eq
	1	E0+l	У	16(Eq-1)Eq
	1	E0	У	16(E0-1)(Eq-2)
	1/2	Eq+3/2	y-1	4E0
4	1/2	Eq+3/2	y+1	4E0
	1/2	Eq+1/2	y-1	-4Eq+4
	1/2	Eq+1/2	y+1	-4Eq+4
1	0	Eq+1	У	16(Eq-1)Eq
471
TABLE II.5.VIII
N=2 GRAVITINO MULTIPLETS IN ADS SPACE
No	Spin	Energy	Hyp-	Mass
1	3/2	Eq+1	У	4Eq-6
	1	Eq+3/2	y-1	16(Eo-1/2)(Eo+1/2)
	1	Eq+3/2	y+1	16(Eq-1/2)(Eq+1/2)
4	1	Eq+1/2	y-1	16(Eq-3/2)(Eq-1/2)
	1	Eq+1/2	y+1	16(E0-3/2)(Eq-1/2)
	1/2	E0+2	У	4 Eq+2
	1/2	Eq+!	У-2	-4E0+2
6	1/2	E0+!	У	-4E0+2
	1/2	Eq+1	У+2	-4E0+2
	1/2	Eq+!	У	-4Eq+2
	1/2	E0	У	4E0-6
	0	Eq+3/2	y-1	16(E0-1/2)(Eq+1/2)
	0	Eq+3/2	y+1	16(E0-1/2)(Eq+1/2)
4	0	Eq+1/2	y-1	16(E0-1/2)(Eq-3/2)
	0	Eq+1/2	y+1	16(Eq-1/2)(Eq-3/2)
1	0	Eq+1	У	16(Eq-1)Eq
TABLE II.5.IX
N=2 HYPERMULTIPLETS IN ADS SPACE
	No	Spin	Energy	Hyp.	Mass
	1	1/2	E0+l/2	y-1	4E0-4
y>0	2	0	E0+l	y-2	16Eq(Eq-1)
		0	E0	У	16(E0-l)(E0'2)
	1	1/2	Eq+1/2	y+1	4E0-4
y<0	2	0	Eq+1	y+2	16Eo(Eo-l)
		0	e0	У	16(Eo-l)(Eo-2)
473
CHAPTER II.б
SUPERSYMMETRIC FIELD THEORIES: THE EXAMPLE OF
THE WESS ZUMINO MULTIPLET
II.6.1 - Supersymmetric field-theories corresponding to an irreduci-
ble representation of the supersymmetry algebra
In ChaptersII.5 and II.6 we have considered the representations
of the supersymmetry algebra (II.2.142) both in the Poincare (e=0)
and in the anti de Sitter case (e^O).
Let us concentrate for the moment on the Poincare case, which
is simpler. Since the mass-squared operator PaPa is a Casimir, name-
ly commutes with all the operators of the supersymmetry algebra, an
irreducible representation can be characterized by its eigenvalue m4,
which can be either zero (massless representations) or non zero
(mass ive representations).
The carrier space of our irreducible representation is an infi-
nite dimensional vector space Ж, spanned by ket vectors |p,J,Y,A,t>
which are eigenstates of the momentum operator (with mass m^)
474
₽alp>J»Y^« t > = Palp.J»Y,X,t >
(II.6.la)
2	2
p = m
(II.6.lb)
and which belong to a finite set of irreducible representations of
the little group Gq of the world-vector Pa. Gq is the direct product
of S0(3) with SU(N) in the massive case and of S0(2) with SU(N) in
the massless case
GQ = S0(3) ® SU(N)	m2 + О	(II.6.2a)
2
GQ = S0(2) ® SU(N)	m = О	(II.6.2b)
J = "the spin", Y = "the hypercharge" are the eigenvalues of the S0(3)
(SO(2)) and SU(N) groups, respectively, and characterize the Gq irre-
ducible representation. A and t label the states within the J=J and
Y=Y representations, respectively.
The action of the supersymmetry generators QAa on the
|p,J,Y,X,t> states is that of mapping different Gq representations
into each other at a fixed value of the momentum Pa.
This structure of the supersymmetry representations admits the
following field-theoretic interpretation. To each of the different Gq
representations appearing in an irreducible representation of SUSY (by
this name we shall hereafter refer to the supersymnetry algebra) we
(J Y)	(J Y1
associate a space-time field Ф^ ’	(x).	Ф.4 ’	(x) is supposed to
belong to an irreducible representation of the Lorentz group S0(l,3)
and of the automorphism group SU(N): i is an index which runs in
(j Y)
such a representation. Furthermore Ф^ ’ '(x) is supposed to be "on
shell", namely to satisfy a wave-equation
I	.	47!
g|(J,Y) $U,Y)(x) = 0	(II.6.3)
4-
* where the matrix-valued differential operator И	is
<	S0(l,3) ® SU(N) invariant. (For instance И	can be the Klein-
Й	Gordon operator □ when acting on scalar fields or the Dirac opera-
tor when acting on spinor fields). If we expand ®£^’^\x) in a
Fourier series
Ф?’¥)(х) = [ (e-iP^ata.Y.pl+h.c.)	(II.6.4)
J (2tt)4	1
the wave equation reduces, in momentum space, to a condition of the
following type:
₽-j’Y)(p)aT(p,J,Y) = О	(II.6.5)
where Pjj^’^(p) is a p-dependent matrix which projects out cer-
tain components of the creation-annihilation operators a^, a'j- The
kernel of the matrix	is an irreducible representation of the
transverse group Gq which we demand to be identical with the (J,Y)
representation to which the field ®j^»Y\x) has been associated.
In this way the states |p,J,Y,A,t> can be thought as the result
of applying to the vacuum |o> the creation operators a'j(J,Y,p) sub-
ject to the on-shell condition (II.6.5).
It is then natural to proceed to the construction of a
Lagrangian for the finite multiplet of fields {ф£^’^^(х)} from which
the wave-equations (II.6.3) should follow as variational equations.
At the same time we can try to implement the SUSY generators as in-
finitesimal transformations on the fields ®£^’^'(x) in such a way
that the action remains invariant against such transformations. As we
are going to see, such a programme can be carried through. However, we
476
should keep in mind that the carrier space of the SUSY representa-
tion is composed only of the ”on-shell states" subject to the condi-
tion (II.6.5). This implies that if we compute the commutator of two
transformations on a field Ф^^’^Сх) this, in general, is not a
transformation belonging to the same SUSY algebra unless Ф|^»^)(х)
satisfies its wave-equation, namely is "on-shell".
This simple argument justifies what will be seen to be general
features of supersymmetric field theories:
a)	The action A=f Sf (x)d—x is invariant against supersymmetry
transformations which close an algebra only "on-shell", namely upon
implementation of the field equations, (i.e. the verification of
Jacobi identities requires the use of the field equations).
b)	The number of "on-shell" Bose and Fermi degrees of freedom is
equal since they must be mapped into each other by the supercharge
Q&„. To verify this identity one must count the degrees of freedom
properly discarding those which are suppressed by the field equa-
tions.
This situation has led people to consider the following ques-
tion.
Question:	Can we add to the list of fields {®£^’^^(x)J, which are
associated to the (J.Y)-representations by means of the
above described procedure and which are called physical
fields, a new set of fields (Ap^ \x)), named auxil-
iary fields, which fulfill the following properties?
Properties: a) On the enlarged set {Ф£^’^^(х), Ap \x)} we can
define SUSY transformations which close the algebra
without use of the field equations.
b)	There is an action A1 = J 4^хУ(х) which is invariant
against the above transformations.
c)	The field equations of the auxiliary fields are
477
А<ГУ,)(Х) =H<Y'>(^J-Y)(x))
(II.6.б)
namely the auxiliary field can be expressed as func-
tions of the physical ones.
d)	Upon insertion of Eq. (II.6.6) the new transformation
rules reduce to the old ones on the physical fields.
If the answer to this question is affirmative we say that we have an
off-shell representation of the SUSY algebra and the enlarged multi-
plet {4>(J»Y)(x), A(J ,Y )(x)} is christened an "off-shell multiplet.
Remarkably enough in certain cases off-shell representations of
the supersymmetry algebra do indeed exist, although this is far from
being the general pattern. Actually, as we emphasize in Chapter IV.7,
the existence of off-shell formulations seems to be an accident lim-
ited to the N=2 and N=1 SUSY algebras. Far from being essential under
any respect, the auxiliary fields when they exist provide a conven-
ient way to deal with several questions.
We consider now a simple example where all these concepts are
illustrated.
11,6.2 - The Wess-Zumino model: the simplest example of a supersymme-
tric field theory
We consider the N=1 SUSY algebra. From Chapter II.5 we learn
that the lowest lying irreducible representation of this algebra is
composed of the following states
478
2	2
р = in
(|p,j = o+> ;	|p,j = o” >)
2	2
p = m C|p»J = 4,'t> ; |p,J = ’i , 4- > )
(11.6,7)
corresponding to the on shell states of a multiplet [l/2,0+,0'J
formed by a scalar A(x), a pseudoscalar field B(x) and a Majorana
spinor X(x), Indeed if we assume that the wave equations of A(x),
B(x) and X(x) are the usual ones:
(□ + m2)A(x) = 0
(II.6.8a)
(□ + m2)B(x) = 0
(II.6.8b)
(i\ + т)Л = 0 => ( □ + m2)X = 0
(II.6.8c)
where
□ = ЭаЭ ; X = уаЭ
a	a
(II.6.9)
and if we expand the fields in a Fourier series:
A(x) = [ d P- [e ip Xa^(p) + eip Xa(p)]
J (2tt)4
(II.6.10a)
B(x) = [ [e'iP- b+(p) + е£Р-х b(p)]
J (2it)4
(II.6.10b)
I
479
A(x) = f —d P [e ip xu(p) + eip xv(p)]
J (2tt)4
- T
C(u(p)) = v(p) (Majorana condition)
then the explicit form of equation (II.6.5) is
(11.6.10c)
(II.6.10d)
2	2.+	x +	9	9
(-p + tn )a (p) = 0 =ф> a (p) = 0 if pZ / m (Ц.6. Ila)
2	2+	к +	2	2
(-p + m )bT(p) = 0 =фь (p) = 0 if p j m (II.6.11b)
(/ + m)u(p) = 0 =ф> u(p) = 0 if p2 / m2	(II.6.11c)
In the frame where (pg=m, p=0) equation (II.6.11c) reduces to
m(l + Yg)u(p) =	1 2m	°	/ Mp) \
	I 0	°	\ X_(p) /
(II.6.12)
where X-f-(p) denotes the upper two component of the spinor u(p) while
X-(p) denotes its lower two components.
Hence we have
X^/p) = 0
(II.6.13)
and we could show that X-(p) transforms as a J=l/2 spinor represen-
tation of the S0(3) little group of pa=(m,o).
Hence the states |p, J=0+,m>, |p, J=0 ,m>, |p, J=l/2,m> can be
480
thought as the result of applying a+(p),	b+(p), u(p) to the vacuum
|o> and subsequently implementing equations (II.6.11).
We are then confronted with the rather easy task of writing a
Lagrangian from which equations (II.6.8) follow as variational equa-
tions. This latter is of the form
y(wz) = ( э АЭаА+ а„Э ВЭаВ+1аоХуаЭ Л)
La	z a	3 a
- m2(aiA2 + a2B2) + ma3 XX	(II.6.14)
where а|,а2,аз are arbitrary numerical coefficients.
Note that the classical spinor field Xa(x) is an odd element of a
Grassmann algebra, namely it is taken anticommuting:
x“(x)X6(x) = - X6(x)Xa(x)	(II.6.15)
The next problem to solve is that of defining the action of the gen-
erators Pa,Qa,Jab as operators on the fields A,B,X. We set:
/ A(x) \	/ A(x) \ (e3? ) B(x) = - ea3	B(x) a 1	2	a I у Л(х)	у X(x) у A(x) \ (eabM ,)	1 = — eab(x a -X Э ) ab	2	a b ba в(х) у (eabM )X(x) = | eab(x 3. - x. Э )X(x) + ab	2	a d d a	(II.6.16a) A(x) \ I B(x) /	(II.6.16b) |YabA(x)	(II.6.16c)
481
(eQ)A(x) - ЁХ(х)	(ll.6.16d)
(eQ)B(x) = - | ёу5Л(х)	(I1.6.16e)
(Qe)X(x) = ~ ЭаАуае + ~ ЭаВуау5е
?	- — m(A-iB)Yj£	(II.6.16f)
Eqs. (II.6.16a-c) are very familiar equations. They correspond to the
usual definition of the Poincare generators in a relativistic field
theory. Pa generates the infinitesimal translations:
Pa : ф(х) -> ф(х + e)	(II.6.17)
and Ma{, generates the infinitesimal Lorentz rotations.
Furthermore Ma^ has an orbital part 1/2(ха^ь-х^Эа) which is the
same for all fields and a spin part which depends on the spin of the
,	field it acts on.
For the scalar field A,В the spin part is zero, while for the
spinor A it is 1/4 Yab- Since the Lagrangian (II.6.14) is manifestly
a Lorentz scalar nothing has to be done in order to prove the invar-
iance of the action
A(WZ) _ | ^<wz>(x)d4x (II.6.18)
under the transformations (II.6.16a-c): it is manifest. Our non triv-
ial claim is that for suitable values of the constants a^fa^ra^, the
action (II.6.18) is invariant also against the transformations
(II.6.lbd-f).
482
Calculating the variation of the Lagrangian we obtain, by means
of a partial integration:
6-Z(WZ) = 2а1ЭаАЭабА + 2а2ЭаВЭа<5В * ia ЛуаЭ 6Л +
+ ia,<5Xya9 X - 2m2(a1A6A + аэВ<5В) + 2m а, Х6Л =
= 2а.ЭаАЗ <5A + 2а_ЭаВЭ 6B + ia_<5Xya3 Л -
la 2 а 3 1 а
- ia_3 Луа6Л + 2m2(a,A6A + a_B6B) +
э a	j.	2
+ 2m а3 Ш	(II.6.19)
and substituting Eqs. (I1.6.16d-f) we get:
6^(WZ) = a№aX - 1а2ЭаВ^5ЭаХ + | ^a^V +
+	2 ^a^VV + | a33bXYbAaaA -
	7 a33b^bYaY5e3 В - i- а_тАёуаЭ X + i- a mA3 Xyae -
-	1 а,тВёу5уаЭ X + 1 a, т ВЭ Луау5е -
-	m2a1AeX + im2a2Bey5X + ima^S AXyae +
+ ma33aBXy Yj.e - m а3АЛе + im а3ВХу5€	(II.6.20)
483
Using the properties of the gamma matrices which are extensively
discussed in Chapter II. 7, equation (II.6.20) can be rewritten as:
<sy- = (a, + а_)Э АёЭ X - i(a + а_)Э Вёу-Э X +
x j а а	д j а о a
+ a33aAEyab3bX - ia33aBEYaby53bX -
2	_	2	—
-	m (- Э] + a3)AeX + im (- a2 + а3)Вёу5Х -
-	1та_(АёуаЭ X + 3 АёуЯХ) - ma (Вёу,у 3 X +
j	a a	о j d d
(II.6.21)
+ эавёу5УаЛ)
We see that <5 У is zero up to a total divergence if
al a2 a3
Indeed the terms
9аА£уаЬэЬЛ 5 9aBEY Y59bA	(II.6.22)
partially integrate to zero because of the antisymmetry of Yab;
ah
Aey 3a3bX = 0	(II.6.23)
and the last two addends are, themselves a total derivative
- ma33a(iAeYaX + Веу5У./) = ^д^)
(II.6.24)
484
Непсе we have proved that the Lagrangian
y(WZ) = af9aA9 A + 3aB3 B _	_
a	а	л
2	2	2	-
- m (A + В ) - mX X}
(II.6.25)
is invariant under the supersymmetry transformations (II.6.16).
y-(WZ) describes a free system composed of a scalar a pseudoscalar
and a Majorana spinor field.
It remains to be seen that the transformations (II.6.16) close
the SUSY algebra in the sense discussed above, namely upon use of the
field equations.
Let us consider the commutator of two supersymmetries on the
field A(x). We find
[E1O,tQ2]A(x) = - (e2Yae1 - ^АрЭ.А + ™ (E^ - E^pA -
- I (E2W1 - W5E2)8aB - f (W1 - W?®
= | е^ае2ЭаА = ±е^аЕ2РаА	(II.6.26)
On the other hand, since
[e1Q,e2Q] = E“e2(Qa,Qe}	(II.6.27)
we have verified that the anticommutation relation
(Qa,Qe) = i(CYa)ap>a
(II.6.28)
485
holds on the field A(x).
Similarly we can verify that it holds on the field B(x). However
if we compute the same commutator on the field X(x) we obtain:
[E]Q,E q]X(x) = - у уа(е_ё1Э Л-y Е,ё,у,Э X)
A	[t	л— J- cl	X ,z cl
+ (е2Ё1Л у5е2е1у5Х) (E1«-+e2)
(II.6.29)
Since we antisymmetrize in the indices (1 >-> 2) we can use the fol-
lowing Fierz rearrangement
Im- . 1 mn -
Er,E.i =--Y e.y e„ + — Y e,Y E,
[2 1J ц 1 m 2	8	1 ™ 2
(II.6.30)
and relying on
VmY5 = - Ym ' Vmn*5 = \nn	(II.6.31)
we get
г— л — „1 i a nk , —	mm,-
[eiQ,e2qJ = - Y Y aaAE1YmE2 - - Y XE1YmE2 =
“ 2 E1Y е2ЭтЛ - 4 E1Y E2Ym(1 ^X + m X) =
= ±E1YmE2P X - | E YmE2Ym(i^ X + m X)	(II.6.32)
-L	III	Q X Д III
We see that the anticommutation relation (II.6.28) is realized on the
field X(x) only upon use of the Dirac equation (II.6.8c) in line with
the general discussion of the previous section.
Now the Wess-Zumino model is one of those few systems for which
the auxiliary field programme can be carried through. We start by
486
observing that the essential reason why the supersymmetry algebra
cannot close off-shell is the mismatch between the Bose and Fermi
degrees of freedom when the Dirac equation (II.6.8c) is relaxed.
Indeed the 4-components of the spinor X(x) are not matched by the 2
scalar fields A(x) and B(x). This suggests that we should be able to
close the algebra if we add one more scalar field F(x) and one more
pseudoscalar G(x).
The idea of how this can be done comes from the observations that
in the commutator of two Q.s on A(x) (and similarly on B(x)) the non
derivative term of eQ, namely:
-	m(A - iBy5)e
yields a vanishing contribution.
This happens for purely algebraic reasons, irrespectively of
whether the coefficients of the linear combination are A(x), B(x) or
any other two scalar fields. Therefore we make the identifications
-	m A(x) = F(x)
-	m B(x) = - G(x)
(11.6.33a)
(11.6.33b)
assuming that this will be the form of the auxiliary field equations
(II.6.6) in our specific case. This leads to the following rewriting
of the supersymmetry transformation rules (H.6.16d-e)
(eQ)A(x) = ± eX(x)
(£Q)B(x) = - J ey5X(x)
(II.6.34a)
(II.6.34b)
487
(QE)X(x) = Э AyaE + | Э By у e +
/ d	/ d a. J
+ | (F + iGy5)E
(II.6.34c)
which automatically close the anticommutation relation (II.6.27) on
the fields A(x) and B(x). Our goal is to achieve this closure also
on the spinor field X(x). To this effect we must postulate the trans-
formation rules of the auxiliary fields F(x) and G(x). Without loss
of generality we can write
Ё0 F(x) = E?
(II.6.35a)
eQ G(x) = i Ey^w
(II.6.35b)
where E, and ш are some Majorana spinors to be specified. From Eqs.
(II.6.33), it follows that "on-shell", namely when Eq. (ll.6.8c) is
implemented, £ and w are given by
>>	in >
£ = ш = _ - X
(II.6.36)
Off-shell, E. and Ш can be anything which reduces to (II.6.36) upon
use of (II.6.8c). We try the following
€ =	| $ X
(II.6.37)
namely we add to Eqs. (II.6.34) the following rules for F and G
(sQ) F(x) - | | X
(II.6.38a)
488
(eQ) G(x) = - ey5 $ X
(II.6.38b)
If we recalculate the commutator [e^Q,e2Q] on X(x) we arrive at the
same equation as (II.6.32) with mA, however, replaced by -i#X. Hence
the algebra closes on X(x) without use of the Dirac equation.
It remains to be seen that the algebra closes also on the auxil-
iary fields F(x) and G(x). We find
[5QJ2QjF(x) = |	’ J	+
-	| Е2^е1ЭаГ + 1 Е2УЧЕ19аС
-	(e^ep	(II.6.39)
Since the double derivatives Э Э A and Э Э В are symmetric only the
am	am
б31" part of the product ya ym survives.
Using
E2E1 E1E2 E2Y5E1 E1Y5E2 E2YaY5El ElYaY5E2 °	(II.6.40)
we finally get:
[e1Q,e2q]f(x) = у e1Yae23aF(x) = ЗЕ^ер^х)
(II.6.41)
which is the desired closure of the algebra. An analogous calculation
proves the result also on the field G(x).
If we utilize the auxiliary field formulation the Lagrangian
(II.6.25) must be replaced by another one which incorporates F(x) and
G(x), yields (II.6.33) as field equations and reduces to (II.6.25)
489
upon substitution of (II.6.33).
These requirements are fulfilled by
yy(WZ) = a|gaAg g + эавэ B-iXfcX-mXX
1 a	a
2	2
+ 2m(AF - BG) + F + G ]
(II.6.42)
which is invariant against the off-shell closed supersymmetry trans-
formation rules (II.6.34) + (II.6.38).
II.6.3 - Superfield interpretation of the Wess-Zumino model and
rheonomy
In Chapter II.3, studying the dual formulation of the super-
algebras we realized that the super Poincare group can be viewed as
the group of superisometries of a specific supermanifold, namely
Minkowski superspace	(see Table II. 1).
However, in our treatment of the supersymmetric fie Id-theories the
notion of superspace has not yet popped out. It is time to see what
the geometry of this manifold, be it soft or rigid, has to do with
supersymmetric x-space Lagrangians such as (II.6.42) or (II.6.25).
Ultimately Physics takes place in x-space and no one has so far
devised any "gedanken experiment" to measure one's coordinates in super-
space, the B.s remaining a rather defying notion to human imagination.
The final yield of supersymmetry is indeed nothing more than some rela-
tivistic field theory like (II.6.25) whose field content is
490
very specific and which has very strictly fixed ratios of all masses
and coupling constants.
However since we are stubborn geometrists and since we ought to
develop systematic algorithms in place of trial and error procedures,
we like to think that the supersymmetry rules (ll.6.16a-f) are the
reflection on x-space of something going on in superspace which is
geometrically well defined.
This something clearly ought to be a "coordinate transforma-
tion" :
. a „a.	, a' .a'.
(x , e ) > (x , e )
(II.6.43)
In chapter 1.1 we saw that the coordinate transformations are
generated by the Lie derivative operator. Given an infinitesimal
tangent vector
I =
(II.6.44)
and a function Ф(х) (i.e. a 0-form), we have
й^Ф(х) = Ф(х+О - Ф(х) = £^ф = ^jd<J> + d( ^ф)	(II.6.45)
Hence we see that the transformations generated by Pa and Ma^ are
interpretable as coordinate transformations
а а >-a
x x + £
(II.6.46)
with £a given by
491
a la
, = — e
(II.6.47)
in the case of the translation and with ga given by:
(II.6.48)
in the case of the Lorentz rotations. Actually, in the case of the
Lorentz rotation on X(x), to the Lie derivative term X(x) we have
to add also the rotation of the spinor indices: this is familiar and
does not spoil the interpretation. Indeed what we have to do is to
introduce the Lorentz-Lie algebra valued compensators W^a^ satisfying
the consistency conditions:
(II.6.49)
(see Eq. (1.6.123)) and use as generators of the coordinate transfor-
tions covariant Lie derivatives rather than ordinary ones:
бф(х) = (Я? + АаЬЖх)
(II.6.50)
The unanswered question is which kind of coordinate transformations
are the supertransformations (II.6.16d-f): obviously they should be
associated with 6.s translations
(II.6.51)
but the trouble is that our fields are x-space fields and do not
depend on 6.s.
492
The way out is to change the rules of the game and say that
actually A(x), B(x), X(x) do depend on 6.s. In other words we regard
A(x), B(x) and X(x) as the boundary values at 0a=O of three N=1
superfields A(x,6), B(x,6), Л(х,6)
A(x) = A(x, 6 = 0)
(II.6.52a)
B(x) = B(x, 6 = 0)
(II.6.52b)
X(x) = (x, 6 = 0)
(II.6.52c)
(Since we deal with superspace, the 6.s have no capital Latin
index).
What we would like to have is a well-defined rule which uniquely
associates to a set of boundary values {A(x), B(x), A(x)} a set of
superfields {A(x,6), B(x,6), X(x,6).
Having such a rule which, for reasons to be explained later, we
call a rheonomic extension mapping,
rh:{A(x), B(x), X(x)} -> {a(x,6), B(x,6), X(x,6)}	(II.6.53)
we can devise the following sequence of operations
493
{л^Сх.б), В^х.б), Х-^х.б)}------------> {А2(х,6), В2(х,0), Х2(х,6)}
е - е + е
(boundary
 (rh)	" value
6 = 0)
{Aj(x), B1(x), AjCx)} -----------*--- {A2(x), B2(x), Л2(х)}
(eQ - transf.)
(II.6.54)
Namely, after extending the x-space
their corresponding superfields, we
in superspace, represented by a Lie
E. In this way we obtain a new set
fields (A^x), B[(x), Xj(x)J to
perform a coordinate transformation
derivative along a fermionic vector
of superfields
A2(x,6)	= A^x.6) +	^(х.б)	(II.6.55a)
B2(x,6)	= B1(x,6) +	Я.В^х.6)	(II.6.55b)
X2(x,6)	= ^(x.6) +	^Х^х.б)	(II.6.55c)
of which we	can take the	boundary values A2(x), B2(x),	Л2(х) at 0=0.
Our programme is successful if we find:
6eA(x) = A2(x) - AT(x) = eQA(x) = i eX(x)	(II.6.56a)
494
6 В(х) =	В?(х) - В (х) = eQB(x)	= - ^ еуЛСх)	(II.6.56b)
SgACx) =	Х2(х) - Х1(х) = eQX(x)	=
=	± (Э А+±у Э В)у е - |	m(A-IByJe	(II.6.56с)
Z “ j а <л 2	3
Under these circumstances the supersymmetry transformations can be
really interpreted as the image in x-space of the fermionic transla-
tions in superspace.
In this way the problem of interpreting the supersymmetry trans-
formations is converted into the problem of defining the rheonomic
extension mapping. Thinking a little bit one should be convinced that
such a problem is a classical one: it is a Cauchy problem. Indeed a
rule which associates a function of two variables, say f(x,y), to
any function of a single variable f(x) can be nothing else but a
partial differential equation which admits f(x) as a complete set of
initial data.
An outstanding example is provided by the analytic functions
f(z). These are complex functions of two real variables
f(z) = u(x,y) + iv(x,y)
(II.6.57)
whose real and imaginary part satisfy the Cauchy Riemann equations:
Эу _ Эи	.	Эи _ _ Эу
Эу Эх	’	Эу Эх
(II.6.58)
495
Calling "inner" the direction of the x-axis and "outer" the direction
of the y-axis
у "outer"
we see that the Cauchy-Riemann equations relate the "outer deriva-
tives" to the the "inner derivatives". As a result if we know
u(x)=u(x,y=O) and v(x)=v(x,y=O), namely if we know the boundary
value of f(x,y) at y=0, we can solve the Cauchy Riemann equations and
we construct the complete analytic function f(x).
It follows that the "rhenomic extension mapping" must be de-
fined as the solution of certain differential equations, named
"rheonomic conditions", which are imposed on the superfields A(x,6)
B(x,e), A(x,6) and which must have the structure of Cauchy-Riemann
equations. Namely the "rheonomic conditions" should relate the "inner
derivatives" (derivatives Э/Эха) to the "outer derivatives" (deriva-
tives Э/Э6“).
Indeed in the present analogy the role of the real axis is
played by the x-space submanifold while the role of the imaginary
axis is played by the B.s
б“ "outer"
"inner"
(II.6.60)
We now show that the appropriate "rheonomic conditions" are obtained
by requiring that the result of the sequence of operations (II.6.54)
496
is the desired one, namely Eqs. (II.6.56).
Let A(x,6), B(x,6), X(x,6) be the superfields we are looking for
and let us consider their exterior differentials dA, dB, dX. Since a
complete cotangent basis to superspace is provided by the vielbein Va
and the gravitino ф (see Table II.3.1) we can write
dA =	Фа	Va	+	фу	(II.6.61a)
dB =	Па	Va	+	фп	(II.6.61b)
dX =	Ла	Va	+	Мф	(11.6.61c)
where the intrinsic components {Фа, Па, Ла, х»	М) (х, q аге
Majorana spinors and М is а 4x4 matrix in spinor space) can be easily
computed by projection along the tangent vectors Da and Da dual to Va
and ф“:
Фа Daj	dA = D A = — a	2	Э A(x,6) a	(II.6.62a)
Па = Da	[ dB = D В = — a	2	3aB(x,6)	(II.6.62b)
Ла = Da	I dX = D X = - a	2	Э X(x,6) a	(II.6.62c)
Ct X =	I dA = DaA = (-	—+— у еэ ) A(x,e> , ~	a	(I1.6.62d)
497
Г)“ = - I dB = DaB = (4r+- Ya63 )“b(x,6)	(II.6.62e)
Da |	36 4 a
m“P = R I dX“ = D6x“ = (—4-i ёуаЭ )eXa(x,6)	(II.6.63)
DP I	36 4 a
Let us now assume that every transformation of the algebra (II.6.16)
is a superspace coordinate transformation generated by some tangent
vector:
Instead of the holonomic basis (3/3xa, 3/96a) we can use the
anholonomic one (Da,Da) and we can name ga and the component of g
along Da and Da respectively:
I = ?aDa +	(II.6.65)
The eaPa-transformations are identified with coordinate transforma-
tions having constant ga=ea components and vanishing ga=0 compo-
nents. Viceversa the eQ-transformations are identified with coordi-
nate transformation characterized by a vanishing ga=0 and a
constant ga=ea component.
Since and D* commute with the Killing vectors (P^, Qa> M^)
(see Table II.3.1) these particular translations, which we want to
identify with the supersymmetry algebra transformation of the fields,
are invariant under the superisometry group. Such a property guarantees
that the action of eQ on A(x,6), B(x,6), X(x,6) defined at 6 = 0
will retain the same form after it is lifted in superspace by means of
an isometry.
As it happens already in Minkowski space, the Lorentz rotations
498
are coordinate transformations whose components £a, 5“ are not con-
stant. Rather they are linear functions of the coordinates xa and
linear or quadratic functions of the coordinates 6a.
ra - ab i a abc^
? - c1E ^4- c26y eebc
e
abM ,
ab
(=« - ПЬ,Х
c - сзе
(II.6.66)
Indeed the coefficients c^ c2> c^ can be fixed so that
? = (.I) +£D =--e (x Э. - x. Э ) + — 6y ,	(и g 67)
a a 2 a b Ъ а ц ab gg	<11.0.0/7
Now we can compute the Lie derivative of A, В and X along the
vector (£a=0, £a=ea) which, supposedly, represent an eQ-supertrans-
formation. We find
V J “ ?	(,.v* * »> -
-	* ‘Чч* - г“<4,е,>р> - «	<11.6.6»
Similarly we obtain
Я^В = ЕП
Я^Х = Me
Hence if we want to reproduce the transformation rules
(II.6.69)
(I1.6.16d-f)
we must impose
Г)(х,6) = у X(x,6)
(II.6.70a)
499
х(х,е) = - | у5л(х,б)
(II.6.70b)
М = i Э А(х,6)уа + 4 Э В(х,6)уау5 - - m(A- ±В)у.-	(11.6.70с)
/а	2 а	2	->
Recalling Eqs. (II.6.62) we see that these conditions are differential
equations on the superfields and have the structure of the Cauchy-
Riemann equations.
Indeed we find:
4- A(x,6) = - i (Yae)“3 A + j X	(II.6.71a)
30“	4	a 2
Ifi = - i (Ya6)a3 В - i y4A	(II.6.71b)
ЭА	4	a z
Л = T Э X“(6Ya)P + i (3 Aya+ j 3aBYaY5)“P
4 a	2 a 2 a
- | m(AH - iBY5)“3
(11.6.71c)
Eqs. (II.6.71) express the 3/30 derivatives of all the superfields in
terms of their 3a-derivatives or of the fields themselves.
This allows an easy order by order solution in 0. At first order
we find
A(x,6) = A(x) +0X(x) + ...
(II.6.72a)
B(x,0) = B(x) -0y5X(x) + ...
(II.6.72b)
500
Л(х,0) = Л(х) +| (1ЭаА(х)уа+ЭаВ(х)уау5)е -
- i ш(А(х) - 1В(х)у5)0 + ...
(11.6.72с)
Then we can utilize the result to determine the second order terms.
For instance we get
A(x,6) = A(x) + — 0Л(х) + — 0уау^0Э B(x) -
2	8	a
- — m(00A(x) - i6y 6B(x)) + ...
8	**
(II.6.73a)
i -	1 _ A 5
B(x,6) = B(x) - | 6ysX(x) - £ 5y Y 69 A(x) +
z	о	a
+ -i m(i0y 0A(x) + 60B(x)) + ...
8 J
(II.6.73b)
Eqs. (II.6.72-73) give the explicit form of the rheonomic extension
mapping. By construction it is guaranteed, upon insertion into the
diagram (II.6.54), to reproduce the supersymmetry transformation rules
of the Wess-Zumino model.
II.6.4 - The integrability of the rheonomic conditions and the Bianchi
identities
The rheonomic conditions are partial differential equations and as
such, in order to be integrable, must satisfy suitable integrability
conditions. The fact that we have integrated them explicitly shows
501
that they were integrable; however, it is still important to understand
why. Actually if one considers carefully the integration of Eq.
(II.6.71c) one discovers that the procedure we have utilized does not
work consistently for any set of initial data A(x), B(x) , X(x) : it
works only if X(x) satisfies the constraint
iya8aX(x) + mX(x) = 0
(II.6.74)
which is nothing else but the field equation (II.6.8c).
This is the counterpart, in the rheonomic picture, of the fact
that the algebra (II.6.16) closes only on-shell. The best way to see
this phenomenon is to consider the rheonomic conditions (II.6.70) for
what they really are, namely exterior differential equations.
Eqs. (II.6.70) can be rewritten as:
dA =	0>aVa	+	у фХ	(II.6.75a)
dB =	П /	-	i фу	X	(II.6.75b)
a 2	3
dX =	+	Мф	(II.6.75c)
M =	1ФаУа	+	naYay5 - 2 m^A- iBY5)	(II.6.75d)
Let us first consider Eq. (II.6.75a). This equation does not impose
any constraint on the superfield A(x,e). Indeed (II.6.75) is the most
general expression we can write for dA. The only information it con-
tains is of a linguistic type: we have decided to call Л(х,6) one
half of the spinor derivative of A(x,0).
The first non-trivial information comes with Eq. (II.6.75b).
502
Indeed by means of it we have required a relation between the spinor
derivatives of A(x,6) and B(x,0). The best way of appreciating the
meaning of this equation is to consider A(x,6) and B(x,0) as the real
and imaginary parts of a complex superfield z(x,6):
z(x,6) = A(x,6) + iB(x,6)
(II.6.76)
Then equations (H.6.75a-b) can be rewritten as
dz = ZVa+ - ф X	(II.6.77)
a 2 • •
where
Z = Ф + i П = D z(x,e)
a a a a
and X., ф. are the left-handed chiral projections of A and ф
X. =
1+YS
(-----)X
2
_ 1 + YS
Ф. = Ф(------)
2
(II.6.78)
Equation (H.6.77) is not the most general expression we can write
for the exterior differential of a complex superfield z(x,0): in
general we should have also a term ф"х‘ where x is some right-handed
spinor and ф' is the right-handed chiral projection of the gravitino
1-form
Ф = Ф(-
'5.
2
(II.6.79)
The absence of the term ф‘Х‘ means a constraint on the superfield
z(x,6). This contraint is
503
(II.6.80)
1-У5
D)a z(x,6) = 0
and the superfields satisfying Eq. (II.6.80) are named chiral super-
fields. One can write their most general form, obtained by solving
the differential equation (II.6.80), via power series expansion:
z(x,e) = z(x) +-e.x.(x) --е\ае.э z(x) + - e.e ,^(x) -
2 *	4	’ a 8**
U(x)0.e. - e.e.e e □ z(x) . (n.6.81)
о	32
In (II.6.81) the complex scalar fields z(x), J^(x) and the left-
handed spinor field y5X0(x) = Л0(х) are completely arbitrary. (The
dots denote the chiral projections on the 0-coordinates:
(II.6.82a)
(II.6.82b)
Comparing Eq. (II.6.81) with the linear combination A(x,0) + iB(x,0)
of Eqs. (II.6.73a) + (II.6.73b) we see that the two agree if the
complex field Jf’(x) is identified with
J^(x) = -mA(x) + imB(x)
(II.6.83)
Recalling moreover Eqs. (II.6.33) we see that, "off-shell" we can
write
504
#'(х) = F(x) + i G(x)
(II.6.84)
,X'(x) is the auxiliary field. This means that by imposing the rheo-
nomic conditions (II.6.75a-b) we have not yet enforced the field
equation (II.6.8c): this happens only when we impose Eqs. (II.6.75c-
d) as well. Such a conclusion can be drawn by a simple inspection of
the integrability conditions of the system (II.6.75).
Integrability means d^=0 (closure of the exterior differential
operator). Utilizing the Maurer Cartan equations of Table II.3.1 and
computing the second differential of (II.6.75) we find:
0 = d2A = — Ф ф л уаф + d$ д Va
2 a '	a
1т»	..a	It - — ФА A. V	ф 2 a	2	zs Мф	(II.6.85а)
0 = d2B = d П д va + i П i » a	2 a	 №ф + 7 Ф - 2	j а	. Va +
+ | Ф - У5МФ		(II.6.85b)
Since Va and ф are independent, we must impose the independent can-
cellation of the Va Л Vb, Va Л ф“, and фа Л фЬ terms: this leads to
the following equations
D[a*b] = D[a ПЬ] = °
Va 4 \	’ Va = 4 Va
(II.6.86a)
(II.6.86b)
505
я	л S 1
М = 1Ф у + П Y Y + Т (F+iYeG)	(11.6.86с)
а а 2 J
where F and G are arbitrary superfields. Eqs. (II.6.86a-b) are quite
simple to interpret and are automatically satisfied. Indeed we have
Ф a	= D A=>Di a	1	а ФЬ] =	[d ,d, ]a = 1 a bJ	0	(II.6.87a)
П a	= D B=>D, a	[а ПЬ] =	[d ,d,]b = 1 a bJ	0	(II.6.87b)
Л a	= D X ; a	±X = 2	D A=>D Ф a	a а	= D D A = - Л = a a	2 a	
and similarly we see that DaIIa = -1/2 У5Ла is true.
The non trivial information sits in Eq. (II.6.86c). It implies
that once we have imposed Eqs. (II.6.75a-b), namely the supersymmetry
transformation rule of the Bose fields, that of the Fermi fields is
also determined, up to the arbitrary fields F(x) and G(x). In prac-
tice the form of M is determined by writing it as a linear combina-
tion of the 16 independent gamma matrices (1, yg, 75 ya, ya, yat>) and
then enforcing the two linear equations:
ф Мф = i^a0 ~ уаф 5 Ф ~ У5МФ = - ПаФ ~ уаф (II.6.88)
F and G exist because the homogeneous system has a vanishing
determinant.
The supersymmetry transformation of F and G, namely the form of
their exterior derivatives, can be determined by looking at the
integrability of equation (II.6.75d). We find
506
о = d2x = ал va
а
л уаф + i(d$a - dn^JYgip
+ - Л ф
2 а
+ (dF + iYg d G) л ф
(II.6.89)
Expanding along the Va, ф basis we obtain the result
dF = ^Va + ±ф л уаЛа	(II.6.90a)
dG = '$ Va - ф л у5уаЛ	(II.6.90b)
a	a
namely we retrieve the supersymmetry transformation rules (II.6.38a-
b). Indeed for a coordinate transformation (5a=0, 5a=ea) the Lie
derivative of F reads:
V -JdF -IEA  5'Л.>
(II.6.91)
and an analogous calculation yields the rule of G.
Summarizing: we have shown that the system of exterior differen-
tial equations
dA = Ф Va + - фХ
a 2
dB = П Va
a
(II.6.92a)
(II.6.92b)
- i ф . у5л
dX = AaVa + (1Фауа-ПаУ5уа)ф + A (F+iGy5)	(11.6.92c)
507
+ 1Ф - YaAa
(II.6.92d)
dF = & Va
a
dG = ^va - ф л у\аЛа	(II.6.92e)
is completely integrable, that is consistent with the equations
ddA = ddB = ddX = ddF = ddG = 0, which, by extension of language, we
call Bianchi identities.
The integrability implies that given an arbitrary set of func-
tions A(x), B(x), X(x), F(x) and G(x) we can always reconstruct a set
of superfields A(x,6), B(x,6), X(x,6), F(x,6), G(x,6) which admits
these functions as boundary values at 6=0 and which is a solution of
the system (II.6.92).
Furthermore the coordinate transformations (g—, g— = e—) on the
superfields A(x,6), B(x,e), X(x,6), F(x,6), G(x,6) induce the super-
symmetry transformation rules (II.6.34) + (II.6.38) on their boundary
values.
As we see, finding a Bianchi identity consistent parametrization
of dA, dB, dA is fully equivalent to the determination of a set of
transformation rules which closes the algebra of supersymmetry. The
verification of the Bianchi identities is an alternative and, usual-
ly, much more efficient way of checking the Jacobi identities.
Finally let us remark some features which have emerged from this
analysis
i) The existence of auxiliary fields is crucially linked to an
undeterminacy in the solution of the fermion exterior derivative
parametrization when the Bose derivatives have been fixed. If this
undeterminacy is absent the auxiliary field programme fails.
ii) When we implement the auxiliary field equations
508
F mA 5 G = mB	(II.6.93)
by comparing (II.6.92a-b) with (II.6.92d-e) we obtain the field
equation
1уЗЛа + 2 X " °	(II.6.94)
Hence as we have claimed at the beginning of this section the differ-
ential system (II.6.75), corresponding to the rheonomic conditions
(II.6.71) is integrable if and only if Eq. (II.6.94) holds true.
An equivalent way of saying all this is the following. The space
of solutions of the Lagrangian (II.6.25) is mapped into itself by a
group of transformations isomorphic to the super Poincare group. The
space of configurations is not. Viceversa the space of configurations
associated to the action (II.6.42) is already mapped into itself by
the super Poincare group.
II.6.5 - The rheonomic action principle
In this section we finally come to what is the heart of the
rheonomy approach, namely the formulation of a new action principle
which yields, as equations of motion both the ordinary field equa-
tions (II.6.8) and the rheonomic conditions (II.6.75).
Our action will be of the form
A(WZ) f у(A> x, Ф , П )
a a
4
(II.6.95)
where У (the Lagrangian) is a superspace 4-form constructed out of
509
the superfields А, В, A and of the Va components Фа, Па of the Bose
fields, which are treated as independent (first order formalism).
The integration manifold Мд is a Bose 4-dimensional hypersur-
face. The novelty with respect to all the other approaches is the
following: in the rheonomy framework the surface Мд is itself a
dynamical variable, namely, when we derive the equations of motion we
must vary A^Z) both in the fields (А, В, А, Фа, Па) and in the
surface Мд. However if .У' is an exterior form, the variation in Мд
can be substituted by a diffeomorphism on the superfields A, B, A,
Фа, Па. Hence the way of taking into account both the field varia-
tions and the surface variation is to write the ordinary field equa-
tions obtained by varying A^Z) in the superfields and then imple-
ment them on the whole superspace.
Since У is a 4-form and the superfields are 0-forms the varia-
tions 6У/6А, 6.У76В, ... are also 4-forms which can be projected
along the various sectors Va V*5 Vе V^, Va Vе ф, ....
In the V V V V sector called the inner sector we retrieve the
field equations (II.6.8) while in all the others, called the outer
sectors we obtain an information equivalent to the rheonomic condi-
tions. Furthermore the restriction to x-space (e=d6=0) of the
rheonomic action (II.6.95) should be the supersymmetric action
(II.6.25).
In this way to the requirement of invariance of the x-space ac-
tion against supersymmetry transformations we substitute the rheonomy
principle for a superfield action. From the point of view of the
x-space restriction of this superfield action rheonomy means that the
manifold of its classical solutions is mapped into itself by the
super Poincare group. This is certainly true of the manifold of
solutions associated to a supersymmetric action: the converse, howev-
er, might not be true and indeed, in Chapter III.5 we shall discuss a
counterexample. This being the programme let us show that an action
with all the advocated properties does indeed exist.
510
(II.6.96а)
A(WZ) = f y(WZ)
4
^(WZ) = {[$a(dA- | фХ) +Па (<1В+|фу5Х) XyadX -
-	утХ(А+1Ву )уаф] , v\vC л Vde . .
4	5 J	abed
— (AdB-BdA) „флуафл V (Ф Фа + П Па) +
2	v а 18 а а
+ -7- m(mA2 + mB2+XX)]va л v\ Vе Vde . .
32	J	abed
+	[у dA . Ху5УаЬф +| dB . ЛуаЬф + ^ Хт5ТаХф . уьф
-	т(А2- В2)^цу ф тАВф „ у ,ф]уал Vb (II. 6.96b)
о	j ао ц	ао
To prove our statements, we write down the variational equations
6^(WZ)/6$a = 6^№)/6Па = 6У№/6Х = 6^WZ/6A = 6УШ/6В = 0 and we
study their projections. This analysis is done in detail for paedagogi-
cal reasons: from it the reader can learn the techniques used over and
over throughout the book in the construction of all the models. Later
we shall briefly discuss the arguments which lead to postulate the
Lagrangian (II.6.96b).
We have:
ЭУ 1 T ..
—~ = (dA-- фл X)
ЭФа	2
vb,vc
vde
abed
--Ф e. .. .v1 л V-i ^Vk^V1 = o
4 a ijkl
(II.6.97a)
511
Л 7'	1 — bed
oil
- 7 П e. .V1^ V-
4 a ijkl
(II.6.97b)
0
77 = - I YadX ~ Vb . Vе л Vdeabcd +| yaXip . УЬФ . Vе , Vdeabcd
ОЛ
_ A mYa (A _ iBY5)ф . vb . Vе . v\bcd - i XeabcdVa . Л Л Vd
- T V5 V - dA - Va ~ Vb -| УаЬФ ~ dB . Vй . Vb
+ у Y5YaXip ~ Ybip ~ Va л Vb -1 (Фа - inaY5)<P ~ Vb л vc л Vdeabcd = 0
(II.6.97c)
= - dV vb - vc~ vdeabcd Фаф. Yb*vc. vdeabcd
- i тХУаф . Vb . Vе . Vdeabcd - i m2 A	/ , Vd
_ 3i T dA л va vb _ 3xY Y ф л ф л Уаф - Vb
2	51ab	э ab
- | imAijiY5Yab^-V3-Vb-|mB^Yab>>-Va^ Vb
+ 3idB<sijj~Ya<P-. Va-| Вф,^аф-.’ИаФ = 0
(II.6.97d)
512
<5У
<Sb
= - dll л V л V л V eabcd - П ф л уф а V л V .eabcd
abed	2 а Ъ с а
ш г а. h с d
-Ху5у ФаЛу aV eabcd
2
ID
16
Be vaлvb a vc zs vd
abed
-	4 Ф л л va A vb + 3il у , Ф A Va A vb
z aD	ab
+	J “В ФУ5УаЬФ A va A vb -1 тАф А УаЬФ A Va A Vb
-	31 dA а ф а уЭф A Va + -^ Аф А уаф А ф А у^ф = 0
(II.6.97e)
It is worth recalling that in deriving these equations whenever we
met a derivative either of Va or of ф we made use of the Maurer
Cartan equations of Table II.3.1. Before going on we anticipate a
result shown in Chapter II.8. In N=1 supersymmetry we have the Fierz
identity
УэФаФаУЭФ = 0 => фАуафАфАУаФ = 0	(II.6.98)
This implies that the last addend of both Eqs. (I1.6.97d) and
(II.6.97e) vanishes identically and that our equations have non triv-
ial projections only on the following sectors:
i)	V	V	V	V
ii)	ф	V	V	V
iii)	ф	ф	V	V
iv)	ф	ф	ф	У
513
To analyse the information contained in each sector we write the
expansion of every differential along the cotangent basis (this
expansion involves unknown coefficients)
a —	h “
dA = H V + ФХ	;	ЙФ = Ф , V + ip E
а л	a ab	a
dB = К Va + ipn	;	dh = П . Vb + ip £2
a	a ab
dX = A Va + Mip	(II.6.98)
a
and we insert it into the field equations. Then we collect the terms
proportional to the four monomials i), ii), iii), iv) and we set them
independently to zero. This produces a set of algebraic equations
which relates the components (Ha, Ka, ^a’ ^ab’ ^ab’ X, П, М» ^a’ ^a)
among themselves and with the fields (А, В, А, Фа, Па). The best way
of displaying these equations is by means of the replacement
a.	a t....t, a. .. .a
V 1________v p + e 1	4’P 1 p
We obtain
i)	V V V V - sector:
6У/6Фа : Ha = Фа => Фа = DaA = | ЭаА	(II.6.99a)
6>/6JIa : К = П => П - D В = | Э В	(II. 6.99b)
cl	cl	ci	ci	) Ct
<5 У /6 X : - 7 YaA - 7 x = 0 => (iya3 X + m X) = О (II.6.99c)
2 a 4	a
514
2	2
6У/6А : Ф + S- А = D ф + В- А = у (□+ш2)А = 0	(ll.6.99d)
2	2
6У76Б : Паа + В_ В = Dana + В_ в = 1 (П+и^в = о (11.6.99е)
11) ф V V V sector:
6^/6Фа : X = | X	(11.6.100а)
6У'/6Па : п = - | У5Х	(11.6.100b)
6У76А : Za =|Лв	(11.6.100с)
6У76В : «а = - | у5ла	(II.6.100d)
6У76Х : М = 1Ф Ya + п YaY5 - у (A-iBy,)	(Н.б.ЮОе)
a	a	z	Э
iii) ф ф V V - sector:
6^/6фа	no terms	(11.6.101а)
67/6Па	no terms	(II.6.101b)
515
6 У /61
all terms cancel identically using
previous results
(11.6.101c)
6У/6А
all terms cancel identically using
previous results
(H.6.101d)
6У/6В	all terms cancel identically using
previous results
(H.6.101e)
iv) ф ф ф V - sector:
6У76Ф
a		
6У76П a	no terms	(II.6.102a)
6 У76Х		
6У/6А	all terms cancel identically using	
6 У76В	previous results	(II.6.102b)
This proves what we claimed. Indeed in the V V V V sector, after
identifying Фа, Па with the derivatives of A and В we retrieve the
field equations (II.6.99c-e). In the ф VVV sector, instead, we
retrieve the rheonomic conditions (II.6.100a, II.6.100b, И.б.ЮОе).
The other "outer" projections do not contain any further information
and are simply a consistency check.
For paedagogical sake let us consider explicitly a couple of
these projections. All the others work in an analogous way.
For instance let us take the ф V V V projection of the бУ/бХ-
equation (II.6.97c). We have
516
" 7 Yaи^(-3!)<51 - ~-mya(A- iBy-Ж-ЗОб11
z.	а 4	j	а
-f wv^-lwnr11
“ 2 (Фа“ 1Па'У5)’J'6a("3!) = О	(II.6.103)
Using
etmabYab = 21у5уСт	(II.6.104a)
уСут = 6tm + Ytm	(II.6.104b)
equation (II.6.103) can be rewritten as
31уСМф-31уС{(1ФУ + Пуу5)	(A-iBY5)H = O (XI.6.105)
from which the result (Il.b.lOOe) follows.
Similarly if we take the 2<J)-projection of Eq. (II.6.97d) we
find:
Vi -	^1^2 Vi -	^1^9
ТУЛ^*баЬ - T^VabM^ai
3	-	tlt2ab 3i - t.t-ab
- -1тАфлу5УаЬ^ - V «п Вфyab^
t t ma
+ 3i П ф ~у ф£ =0	(II.6.106)
ma
and substituting for the matrix M its expression (Il.b.lOOe) we get
an identical cancellation of all the terms.
Finally, as promised, let us discuss how the action (II.6.96)
517
has been guessed. First notice that the part of the Lagrangian which
does not involve the gravitino 1-form ip, namely
Уо = ($adA + nadB - i XyadX)Vbл VC л Vde , , - - [ф Фа + П Па +
u	4	abed g 1 a a
у (mA2 + mB2 + XX) ] Va л Vb л Vе л Vde v
4	abed
(II.6.107)
is simply the first order transcription of the 2nd-order action
(II.6.25). How to reformulate the Lagrangians of matter fields in
first order formalism was discussed in Chapter 1.5.
It remains to be understood how one determines the terms involv-
ing at least one ф-form.
The procedure is conceptually simple: one writes an ansatz
containing all the terms which have the following properties
i)	are at most quadratic in the fields А, В, X, Па, Фа.
ii)	are Lorentz-invariant and have negative parity (are pseudo-
scalars like the action (II.6.107)).
iii)	have the correct scale weight яЛ.
By scale weight we mean the following. Since the Maurer Cartan equa-
tions of superspace Ф the rheonomic conditions (II.6.75) are invar-
iant under the following scale transformations:
Va - £Va
ф >-• /Я ф
cm:
X 1 x	m
A »-* ----- Д щ -------
/Г /Т
(II.6.108)
where Я is any real constant, the action must have the same invar-
iance, namely its terms must all scale in the same way.
518
Writing arbitrary coefficients for all the terms which are per-
missible we work out the field equations and we impose that the
rheonomic conditions should come out as solution of the outer projec-
tions. This fixes all the coefficients and the action (II.6.96) is
uniquely singled out.
All the mechanisms, concepts and techniques discussed in this
chapter will be essential for the development of supergravity theory
in Part Three.
519
CHAPTER II.7
Г-MATRIX ALGEBRA AND SPINORS IN 4< D <11
II.7  1 - The construction of Г-matrices
In order to describe spinor fields and hence supersymmetric
theories one needs the Dirac gamma matrices. These form the Clifford
algebra
{Га’ ГЬ} “ 2 ЛаЬ	(II. 7.1)
where паь is the invariant metric of the D-dimensional Lorentz group
SO(1,D-1):
ПаЬ = diag(+,
(II.7.2)
To study the general properties of the Clifford algebra (II.7.1) one
520
can use group-theoretical techniques: we prefer a very pedestrian
approach based on the direct construction of the gamma matrices.
We begin by fixing our conventions. The matrix Гд=Г° correspond-
ing to the plus sign in the signature (II.7.2) is hermitean:
= Г
0	0
(II.7.3)
the matrices Г^=-Г1 (i=l,2,... ,D-1) corresponding to the minus signs
in the signature (II.7.3) are antihermitean:
rt = - Г
1
(II.7.4)
We subdivide the range of dimensions in the even and odd sector
D = 2v= even
In this case the representation of the Clifford algebra has
dimension
D
.. r ,2 „V
dim Г = 2	=2
a
(II.7.5)
In other words the gamma {Га} are 2V x 2V matrices.
The proof is easily obtained by iteration. Suppose that we have
the gamma matrices ya corresponding to the case v' = v - 1
{y ,. YvJ = 2	(a' = 0, 1...........D-3)
1 a d	a b
(II.7.6)
and that they are 2V dimensional. We write down the following 2V + I
dimensional matrices
521
(II.7.7)
and we verify that they satisfy the Clifford algebra (II.7.1); fur-
thermore they have the correct hermiticity properties:
Г = - Г
1D-2	D-2
r+ = - Г
JD-1 D-l
(II.7.8)
The matrices (II.7.7) can be interpreted as the following tensor
product of the ya -matrices with the Pauli sigma-matrices:
Г ,=y ,®O,	; rn„ = 1]®io	; Г = fl®io	(II.7.9)
a a 1	D— Z	j	L>—i	z
To complete the proof of our statement we just have to show that for
V=2, corresponding to D=4, we have a 4-dimensional representation of
the gamma matrices. This is a well-known result; for example one can
use the representation:
(II.7.10)
In D=2v one can construct the matrix
fD+l " % r0rir2””’rD-l
(II.7.11)
522
where ap is a normalization factor to be fixed in such a way that
Г2
IH-1
By direct
= 1
(II.7.12)
evaluation one can verify that
{Г
a
Г1Н-1
(II.7.13)
namely Гц+i is the generalization of the yg-matrix of
The normalization ap is easily derived. We have
4-dimensions.
r0ri”"’rD-l
4 D(D-1)
I г г . г г
'd-I D-2’	’ 120
(II.7.14)
and therefore, imposing Eq. (II.7.12) we find:
j -y D(D-l)
%(-)
D"1 = 1
(II.7.15)
This implies
% = 1
V = 2ц + 1 = odd
if
v = 2Ц " even
(II.7.16)
With the same
token
we can show that rD+l is hermitean.
Indeed
i
+	* I D(D-^ D-l
ГГН-1 " %	Г0Г1......ГП-1 “ Г1)+1
(II.7.17)
523
D = 2v+l = odd
In this case the Clifford algebra (II.7.1) is represented by
2V x 2V matrices. It suffices to take the gamma matrices Га< corre-
sponding to the even case D' = D-l and add to them the matrix
iPu'+l = ^D-l which is antihermitean and anticommutes with all the
other ones.
II.7.2 - The charge conjugation matrix
Since Га and their transposed
algebra it follows that there must
connecting these two representations
ment relies on Schur's lemma and it
Introducing the notation
Г	=ГГ Г ,...,Г т =— У
а1.....an Lal а2 anJ n! Р
ГаТ satisfy the same Clifford
be a similarity transformation
of the same algebra. Such state-
is proved in the following way.
«Р
(-) Г	Г
aP(D......aP(n)
(II.7.18)
denotes the sum over permutations, we can easily convince
constitutes a
where Ep
ourselves that the union 1, Г , Г ,...,
а1 ala2
finite group of 2 -dimensional matrices.
Furthermore the groups generated by Га,
-Га or ГаТ
Hence by Schur's lemma two irreducible representations
are the same.
of the same
Г
а
group, with the same dimension and defined over the same vector
space, must be equivalent, that is there must be a similarity trans-
formation which connects the two. The matrix realizing such a simi-
larity is called the charge conjugation matrix. Instructed by this
discussion we define the charge conjugation matrix by the following
equations
524
(11.7.19а)
-1 т
CZJ. Г С,* = Г
(+) а (+) а
(11.7.19b)
С(_) connects the representation generated by Га to that generated
by “Га^, while C(+) relates the Га and Га^ representations. In even
dimensions both and exist, while in odd dimensions only
one is possible. Indeed in odd dimensions Гр-i is proportional to
ГфГ1 — Гр-2 so that the C(-) and C(+) of D-l dimensions yield the
same result for Гр-j, namely either +Г])_1 or -Гд.р This decides
which C exists in a given odd dimension.
Another important property of the charge conjugation matrix fol-
lows by iterating (II.7.19). Using Schur's lemma one finds that
= aC(_)T (idem for C(-)) so that iterating again a^=l. In other
words and C(_) are either symmetric or antisymmetric.
It is very important to decide which is the case in every dimen-
sion.
We do not dwell on the derivation which can be based either on
general arguments or on an explicit construction in a gamma matrix
basis. We simply collect the results in Table II.7.1.
525
TABLE II.7.I
CHARGE CONJUGATION MATRICES IN 4< D <11
D	c*(+) = C(+) (real)	C’\-) = C(-) (real)
4	C(+)T = 'C(+)5 C(+)2 = -1	C(-)T = -C(-)5 C(-)2 = -1
5	C(+)T = "C(+); C(+)2 = 4	
6	C(+)T = ”C(+)’ C(+)2 = -1	C(_)T = C(.}; С(.)2 = 1
7		C(-)T = C(-); C(-)2 = 1
8	C(+)T = C(+); C(+)2 = 1	C(.)T = c(.); C(_)2 = 1
9	C(+)T = C(+); C(+)2 = 1	
10	C(+)T = C(+); C(+)2 = 1	C(-)T = C(-); C(-)2 =
11		C(-)T = C(-); C(-)2 =
526
II.7.3 - Majorana, Weyl and Majorana-Weyl spinors
The Dirac conjugate of a spinor i|> is defined by
* “ *4	(II.7.20)
and the charge conjugate of i|> is given by:
ФС = С фТ	(II.7.21)
where C is the charge conjugation matrix. When we have the option in
(II.7.21) we can use either or C(_j. By definition a Majorana
spinor A satisfies the following condition
which means that A is its own conjugate. Eq. (II.7.22) is not always
self consistent. Indeed by iterating it a second time
X = С Xcl = С Го C rj X = С rj С Го X	(II.7.23)
we get the consistency condition
T
C r0 C = r0	(11.7.24)
in which we have used the reality of C(C* = C).
It can be shown that there are two possible solutions to equation
(II.7.24): either	is antisymmetric, or	is symmetric.
Hence looking at Table II. 7.1 we conclude that Majorana spinors exist
only in
527
0=4,8,9,10,11	(II.7.25)
In D = 4,10,11 they are defined using C(_), which is antisymmetric
while in D = 8,9 they are defined using which is symmetric.
Majorana spinors do not exist in:
D = 5, 6, 7
(II.7.26)
Weyl spinors, on the contrary exist in every even dimension; by defi-
nition they are eigenstates of the rp+^-matrix, corresponding to the
+1 or -1 eigenvalue:
(II.7.27)
As a matter of convention the spinors belonging to the positive
eigenvalue are named "left-handed", while those belonging to the
negative one are called "right-handed".
In some special dimensions we can define Majorana-Weyl spinors
which are both eigenstates of Гр+| and satisfy Eq. (II.7.22). In
order for this to be possible we must have
Л
•P = rw ф
(II.7.28)
Using (II.7.24) equation (II.7.28) becomes
(II.7.29)
Since Гр-ц is hermitean this relation can also be written as
528
-1	т т т т
С 4+1 С = rS 41 Го = - rD+l
(II.7.30)
Recalling that ijj+j is defined by equation (II.7.11), we can verify
in which dimensions this relation holds.
If C = C(+) we have
-1	T T T	j
C Г1)+1 C = aD Г0 ri’’"’rD-l =	Г1)+1 (II.7.31)
while if C = C(.j we find
_1	1 D(D-1)+D C rD+l C = <->	41	(II.7.32)
Hence we get	
d	= 4	c 1	T Г5 C = Г5			
d	= 8	c"1	T Г9 C = Г9			
d	= 10	с’1	T rll c = - rll		(II	.7.33)
and we	see that	in	the range 4< D <11 the	only dimension	for	which
Majorana-Wey1 spinors can be defined is D=10.
The results are summarized in Table II.7.II.
529
TABLE II.7.II
SPINORS IN 4< D <11
D	Dirac	Majorana	Weyl	Maj orana-Weyl
4	Yes	Yes	Yes	No
5	Yes	No	No	No
6	Yes	No	Yes	No
7	Yes	No	No	No
8	Yes	Yes	Yes	No
9	Yes	Yes	No	No
10	Yes	Yes	Yes	Yes
11	Yes	Yes	No	No
530
II. 7.4 - Useful formulae in Г-matrix algebra
In every dimension it is important to know which Г
“ 1''' n
matrices are symmetric and which are antisymmetric. By this we mean
the following
T
(С Г ) = С Г	=> symmetric
< 3	И a a a a £1
In	In
(С Г	) = - С Г	=> antisymmetric
аГ" n	al-‘-an	(II.7.34)
C being either or C(_). In odd-dimensions there is no ambiguity;
in those even dimensions where Majorana spinors exist we choose C to
coincide with the charge conjugation matrix entering the Majorana
condition. Finally in D=6 where no criterion is available we select
C = C(-)'
With these conventions and using the general inversion formula
Г	= (_)П(п-1)/2 г
in a «. .m	mm....m.
In	n n-1	1
(II.7.35)
we obtain the results of Table II.7.Ill
531
TABLE II.7.Ill
SYMMETRIC AND ANTISYMMETRIC Г-MATRICES
D	Symmetric	Antisymmetric
4	Ya’ Yab	1 . Y5. Y5 Ya
5	Га a 1 2	u , ra
6	U > Гу, Г7Га a » Га a a 12	12 3	Га» ГуГа» Га a 1 2
7	1 , Га a a 12 3	Г Г 1 а’ 1 а а 1 2
8	D , Га, ГоГя a > Га a a	> d . »ci	ci • . a 13	14	ГдГа, Га а , ГдГд д , Гд д д У а' а а ' У а а ' а а а 12	12	12 3
9	и Г Г	Г U » 1 а» 1 a ..a ’ 1 a ..a 14	15	Г	Г 1 а а » 1 а . .а 12	13
10	^a’ Га a » ПЛ’ ГпГа >a 12	14 Г 1 a . .a 1 s	»Гд	а>Га	э’ГцГдд а«»а	d . .d	X L d d 13	14	12 ГцГа ..а 1	3
11	Г Г	Г 1 a* 1 a a * 1 a ..a 12	IS	и , га _а , га а 1 ’ ’ 3	1	4
532
Note that in D = 2v +1 = odd we consider Г-matrices with, at most,
[D/2] indices. Those with more indices are redundant since they are
proportional to the ones considered. Indeed we can utilize the
duality relation:
const
(II.7.36)
Finally we write some
formulae of invaluable help in practical calcu-
lations :
a,...a c,..c
pl n 1 q
Г v v
с, . .c b,..b
1 q 1 n
inf(n,m)	[a...a. Z	C (q,n,m) 6 г b k=l	Lbr-bk	rak+l"aJ b, ,..b ] k+1 mJ
	(II.7.37a)
Ck(q,n,m) =	
q 4(q-l)	|(к-(-1)П-1) = (-) 2 (-) (	_	/ D-n-m+k \ kHk)q!k!( q )
	(II.7.37b)
a,...a b	a,..a	,	[a,..a , a 1 Г 1 n = Г 1 n rb - n Г 1 n"L 6" D	(II.7.37c)
ba,..a	,	а...а	Га, a„..a 1 r 1 n = p,b r 1 П _ n I 1 p 2 nJ b	(II.7.37d)
533
Furthermore if 0
ap-
is an irreducible (3/2)m(l/2) m Spinor>
m
namely an antisymmetric spinor tensor satisfying the Г-trace condition
1
r e = о
a, . . .a
1 m
(II.7.38)
then we have:
a.	..a c. . ,c c...c b,. .b
pl nl	q g 1 ql	tn
q y(q+l)	[а ..а	а х1...а]Ь,..Ь
_	2	n!	pL 1 n-q g n-q+1 nJ 1 tn
<n-q>!	(II.7.39a)
b.	...b c...c сл..с a1t..a
g 1 ml 4pl ql n _
4 I(4+1) n! gbr-bm^al"aq aq+r"aJ
(п-q)!	(II.7.39b)
сл . .c
' 1	4 rr	e, t
Lcl--cqal"an bl"bmJ
(-)
(q-1)
(q+n)!(n+m) !(D-n-2m)!
n!(n+m+q)!(D-n-q-2m)!
Г г	6,	, 1
a. .. a b. . .b
L 1 n 1 tTi
(II.7.39c)
c.	.. .c
6r.	. г	]Г 1	4
b...b a...a с...с I
L 1 in 1 n 1 qJ
Ч о (q+n)!(n+m)1(D-n-2m)! _
—-----------------------------• 6 [г , Г	-i
n! (n+m+q) ! (D-n-2m-q) !	*- 1 * * m al ’ ’ arr
(II.7.39d)
534
Moreover in D=4 we write the explicit form of the duality relation on
Yab	EabcdYcd = 2 1 Л Yab	(II.7.40)
and	we conclude the chapter with another useful formula valid in
every dimension:
	Г 1	4 Г	Г	= I(n) Г	(II.7.41) ar.an cr.cq q	a^.^
In (II.7.41) the coefficient In^^ is determined by the recurrence
relation:
	z(n) = ^(n)^ _ (q_1)(D_q+2)r<n)2)	(11.7.42a)
	IqR) = 1	(II.7.42b) Ij = D - 2n	(11.7.42c)
535
CHAPTER II.8
FIERZ IDENTITIES AND GROUP THEORY
II8.1 - Introduction
This chapter is very technical but nonetheless very important for
all what follows. It deals with a very specific problem which arises
in the development of both globally and locally supersymmetric field
theories.
As we saw in Chapter II.6, in order to construct the action of a
supersymmetric field-theory model we have, in general, to solve exterior
form equations on superspace which arise either as Bianchi identities
Or as field equations associated to a Lagrangian which is itself an
exterior form.
A complete cotangent frame on superspace is provided by the
Q	A
vielbein V and the gravitino 1-form ф which is a spin 1/2 repre-
sentation of the Lorentz group SO(1,D-1) and has, moreover, an index A
enumerating the supersymmetries (A=l,2,...,N).
Henceforth an arbitrary p-form	on superspace can be ex-
panded as follows
536
= I (ш.
q=0	'
®	A-iCC-
>(pJ ,	. V ...V Р-ч.ф 1 1
а1-‘ар-Ч 1а1“-%аЧ
,А а
Ф q q
(II.8.1)
and our exterior-form equations
are implemented by requiring that the
coefficient of each independent monomial
a.. .a A,a,. .A a a	a A.a,
I 1	n11 m m = v .. лу . Ф 1 1
(II.8.2)
vanishes independently.
a.... a A.a....A a
The relevant point is that fl n	m m is a tensor
product of irreducible representations of SO(l,d-l) and SO(N) which is
antisymmetric in:
{a1+> a2aj	(II.8.3)
and symmetric in
{a1A1 +> A2a2 *>...*> Anan}	(II.8.4)
Эр . . a^A^ap .
This implies that we can decompose Q	into irreduci-
ble representations via a Clebsch-Gordan series and that only certain
representations occur, the others being ruled out by Eqs. (II. 8.3-4).
This decomposition, with explicitly calculated coefficients, provides
a systematic way to perform the calculations we shall be confronted
with. On one side, the absence of certain irreducible representations
in the decomposition of the spinor tensor-products is the origin of all
the "miraculous" Fierz identities one needs to derive supersymmetric
theories; on the other side, using the procedure of projecting every
ap. .a AjOp . .А а
form equation on the irreducible components of fi
one is sure to deal with a set of independent equations and is free
from the danger of overcounting.
537
A
In principle one should consider tensor products of ф .s with
arbitrary number of them but in practice the number of fermions is
limited to a maximum of 4. Indeed every supersymmetric Lagrangian for
scaling reasons is at most quartic in the gravitino 1-forms. Hence we
shall be mainly interested in the decomposition of the product of 2ip.s
(which is rather simple) of 3ip.s (which is the highest needed in the
analysis of Bianchi identities, these latter being 3-forms) and occa-
sionally of 4ф.5.
11.8.2 - The structure of forms on N-extended D=4 superspace
As we saw in Chapter II.3, rigid anti de Sitter and Minkowski
superspaces are the homogeneous supermanifolds (II.3.29-30) possessing
four bosonic coordinates x11, associated to the translations, and 4N
fermionic coordinates 6^a, associated to the N-supersymmetries. The
soft version of these manifolds have the same number of coordinates.
The cotangent space to superspace has 4 + 4N dimensions and it is
д	A
spanned by V and ф , as we already remarked. To illustrate the
method let us begin with the three-forms and let us call D(3) the
linear space spanned by them.
The dimension of D(3) can be easily computed. Let us denote by
z^ = (x\ 0^) the superspace coordinates. The most general 3-form
can be written as:
r>(3)	/->(3) , A , X j П	ГТТ R СЛ
Jr =	~ dz ~ dz	(II. 8.5)
where the superspace wedge product obeys the standard commutation rule:
, A , £ e -.1 + AX, L , Л	гтт я
dz dz = (-) dz л dz	(.lI-8-Ь)
538
Instead of the coordinate differentials dzA = (dx11, d0Aa) we can use
the intrinsic basis (Va, фаА) , and this is what we shall do systema-
tically, Hence we find:
= fi^vaAvb„vc + va vb фаА +
abc	ab(aA)	v
+	ya ^oiA ,BB q(3)	^aA ,BB ,yC
a(aA)(BB)V	+ (aA) (BB) (уС)*	"Ф ~Ф
(II.8.7)
Since the Va anti commute among themselves and with the фаА while
the latter commute with each other we easily compute the dimension of
the monomials appearing in the expansion (II.8.7). We get:
dim(Va. Vb „ Vе) = 4
dim(va ~ Vb л ф^В) =	• 4N = 24N
dim(Va „ ф“\ ф№) = 4 • —-t4^ + 1) = 8N(4N + 1)
dim(ip“A л ф№ л фуС) = | N(4N + I) (4N + 2)
(II. 8.8a)
(II.8.8b)
(11.8.8c)
(II.8.8d)
Hence the space of 3-forms
in N-extended d=4 superspace has the
following dimension:
2	2	N
dim D(3) = (32N +8N+4) Ф (32N + 24N+ 76)y
(II.8.9)
Bosonic
Fermionic
For the first interesting cases we have Table II. 8.1. The basic idea
of our technique is to write a basis of 3-forms which is composed of
irreducible representations of the H=SO(1,3) ® 0(N) group. Explicitly
we want a decomposition of the following type:
539
Vе + ,Va^Vb^ipaA +
abc	ab|aA	v
!Cfl.,Va. X(1J
a I (1)
(3) = (i)
(i)“
(II.8.10)
TABLE II.8.1
DIMENSIONS OF 3-FORMS IN D=4 N-EXTENDED SUPERSPACE
N	dim D(3)	taA dim ф	Zl.otA ,BB4 dim(ip хч ф )	,. r .otA , I3B ,YC, dim^ лф лф )
1	44+ 44	4	10	20
2	148+ 168	8	36	120
3	316 + 436	12	78	364
4	548+ 912	16	136	816
5	844+ 1660	20	210	1540
6	1204+ 2744	24	300	2600
7	1600+ 4228	28	406	4060
8	2116 + 6176	32	528	6160
where X1 are the 2-form irreducible representations appearing in the
,	. .	, ,aA ,BB
decomposition of ф л ф
,.«A .BB _ „аА.ВВ (i)
V л ф - I- A
(II.8.11)
and	are the 3-form irreducible representations appearing in the
decomposition of фаА^ Ф^В л ф^,
ф^.ф^.ф^ = f“A’eB’YC =(i)
(II.8.12)
540
Since H is a direct product and the fermionic wedge product is symme-
tric, we use the following procedure. First we classify all the S0(l,3)
representations appearing in the tensor product of two or three spin 1/2
representations and all the 0(N) representations occurring in the tensor
product of two or three 0(N) vectors.
TABLE II.8.II
REPRESENTATION OF SO(1,3)
Representation type	Dimension	Corresponding tensor, spinor or spinor tensor
[1,1]	6	Xab = -Xba (antisymm- tensor)
[1,0] (+}	4	x/+’ (vector)
[1,0]	4	x/ ’ (axial vector)
[o,o]	1	X^ (scalar)
[0,0]	1	X^ ) (pseudoscalar)
[3/2, 3/2]	8	5ab = ~Тэа’ уЬ Eab = ° firred‘ ®Pinor’ -tensor)
[3/2; 1/2]	12	ya 5a^+^ = 0 (irred. spinor-vector)
[3/2,1/2]	12	= ( ya E ) = 0 (irred. spinor-axial vector)
[l/2,l/2](+)	4	(Majorana spinor)
[1/2,1/2]	4	) (Majorana pseudospinor)
(фаА is at the same time an S0(l,3) spinor and an 0(N) vector). Then
we consider all products of these S0(l,3) and 0(N) representations which
are completely symmetric in the exchange (aA*+ |3B +>yC). (This procedure
is also used for the coupling of spin and isospin in Nuclear Physics.)
541
We begin by tabulating the relevant representations of the proper
Lorentz group S0(l,3) with their dimensions.
In Table II.8.II the numbers on the extreme left are the eigen-
values of the Casimir operators of S0(l,3) whose rank is 2 and the plus
and minus superscripts refer to the parity eigenvalues. The dimensions
can be calculated using standard formulae in group-theory.
They can be also obtained by more elementary means. For instance,
~ has dimension 8 because it is a spinor-tensor (4x6 = 24) satisfying
(4x4=16) conditions (y^	= 0).
Table II.8.II exhausts the list of relevant representations because
of the following decomposition rules:
[1/2,1/2] ® [1/2,1/2] = [1,1] ® [1,0] Ф [1,0]	® [0,0]	® [0,0]’
symmetric	antisymmetric
(II.8.13a)
[1/2,1/2] ® [1,1]	=	[3/2,3/2]	Ф	[3/2,1/2]	Ф [1/2,1/2]	(II.8.13b)
[1/2,1/2] ® [1,0]	=	[3/2,1/2]	ф	[1/2,1/2]	(II.8.13c)
[1/2,1/2] ® [0,0]	=	[1/2,1/2]	(II.8.13d)
Equations (II. 8.12) say that any 4x4 matrix can be expanded in a com-
plete Dirac basis. Indeed, if we have the wedge product ф л ф
AB
where фд and фв are Majorana spinor 1-forms:
♦a  *A • C<*A>T
(II.8.14)
фд ~ Фр is a matrix in spinor space and can be expanded as follows:
1	(+)	('I a i ab
*АЛ*В = 7 XBA+Y5 XBA+Y5Ya XBA+Y XBA5 + 4 YabXBA
(II.8.15)
542
where Eqs. (II.8.13c) and (II.8.13d) correspond to the familiar decom-
position of a spinor-vector (or a spinor-tensor) into a traceless part
plus a trace. Let us, for instance, consider a spinor-vector £ . We
can write:
e = ecl2) +±Y e(4)
a a 4 'a
(II.8.16)
where
r(12) r 1 bf
4	= ?a - 4 V 4
(II.8.17)
is the irreducible [3/2,l/2]
part satisfying:
ас(12)
Y C = 0
a
(II.8.18)
TABLE II.8.Ill
BOSONIC 2-FORMS
Represen- tation	Current	Symmetry			Reality
[1,1]	„ab 7 i	, XAB “ *A 2 Yab*B	xab"	+XBA sym-		, ab.t vab . XAB = ’XAB lm-
[i,o](+)	XBA=VY4	W xa aab	(+) = + xa BA	sym.	(+) f (*) XAB^ “ " XAB lm"
[1,0]	(-) - 5 a XAB = VY Y *B	(-) x!b	(-)	antisym.	(-) + (-) (XAB5 = XAB real
[o,o](+)	(+) _ XAB = '•'a ' *B	w XAB	(+) = " XBA	antisym.	(+) + (+) (XAb) = XAB real
[0,0]	(-)	- s xab = Vy4	(-) XAB	(-) XBA	antisym.	(-)+ (-) XAB =‘ XAB lm‘
543
and
e(4) = y • e	(II.8.19)
is the [1/2,1/2] part.
Similarly given a spinor-tensor we can write:
c	г(8) ч mc	1 ,	(4) ^ab ~ ^ab ” Y[aY ^b]m ” 12 ^ab^	(II.8.20)
where
Д8) r	1	m n- ^ab " ^ab + Y[aY ^b]m + 6 YabY Y ^mn	(II.8.21a)
_(12)	1 a m n- S = Y ?am ~ 7 Y Y Y ?mn	(II.8.21b)
-(4)	m nr ?	= Y Y ?mn	(11.8.21c)
are the irreducible representations [ 3/2,3/2], [3/2,l/2] and
[1/2,1/2], respectively. From this point on we shall define the
irreducible representations assuming Eqs. (II.8.16) and (II.8.20) as
the standard expansion of any spinor-vector or spinor-tensor.
Coming now to the relevant representations of the group 0(N),
ctA
since ф is an 0(N) vector, they are those contained in the tensor
product of two or three vectors.
In general we have:
□ *□= I I I®
(II.8.22a)
Ф
□ «□«□ = Illi ® [
(II.8.22b)
544
TABLE II.8.IV
0(N) REPRESENTATIONS
				□					
Type	•	□	—	□				Illi	1 j 1
Dimension	1	N	2 N(N -3N+2)/6	N(N-l)/2	2 N(N		-4)/3	2 N(N +3N)/6 -2N/3	N(N+l)/2-l
N= 1 N= 2 N= 3 N= 4 N= 5 N= 6 N= 7 N= 8	1 1 1 1 1 1 1 1	1 2 3 4 5 6 7 8	0 0 1 4 10 10® 10 35 56	0 1 3 6 10 15 21 28		0 2 5 16 35 64 105 160		0 2 7 16 30 40 77 112	0 2 5 9 14 20 27 35
where we have used the standard Young tableaux notation and each of them
is meant to represent the corresponding 0(N) irreducible (traceless)
tensor. The dimensionality of these representations is given by standard
formulas.* Our findings are shown in Table II.8.IV. In general given a
rank 2 0(N) tensor, like the bosonic 2-forms of Table II.8.Ill, we shall
write its decomposition into irreducible components according to the
following conventions:
t,D = t,	6<Dt. . + t,	(II.8.23)
AB [aTb] N ab a
DE
where
= I ftAB+ tBA) " N бАВгММ	(II.8.24a)
See for instance M. Hamermesh, Group Theory (Addison-Wesley, Reading
1962).
545
t-' гмм
(II.8.24b)
VaTI^AB^Ba)	(11.8.24с)
В
A rank 3 0(N) tensor will, on the other hand, be decomposed in the
following way:
(II.8.25)
II. 8.3 - Fierz decompositions in the N=1 D=4 superspace
Once every product of фд has been decomposed both with respect
to S0(l,3) and 0(N), the standard Fierz identities correspond to the
exclusion of all representations which are not fully symmetric and to
the determination of the decomposition coefficients. We begin by con-
sidering the N=1 example explicitly.
Ci В
Recalling Table II.8.1, we see that, in this case, tp has
10 components. Looking at Table II.8.II, on the other hand, we realize
that
10 = dim[l,l] + dim[l,0]<+) = 6+4	(II.8.26)
which are precisely the representations occuring in the symmetric pro-
duct decomposition. In the N=1 case no contribution from the antisym-
metric [l.o]^, [0,0](+) and [o,o](~) representations is allowed.
Equation (II.8.15) therefore reduces to
546
, -г 1 a
Ф	- Ф	= - Y
4
+ - Y /
a 4 ab
(II.8.27)
Going now to the ipa ~ ipY sector we read from Table II.8.1 that it
has 20 components; moreover, from Table II.8.II it is evident that the
only way to obtain 20 is by setting
20 = 8 + 12
(II.8.28)
This means that the only representations being completely symmetric in
a,B,Y are [3/2,3/2] and [з/2,1/2]. This is the origin of all Fierz
identities.
The explicit construction is the following. If we have the wedge
product of three ip.s (tpф л tp) we can start by decomposing two of them
according to (II.8.27).
In this way we end up with the following spinor-vector and spinor-
tensor:
e = ф „ ф „ у ф
d	а
(II.8.29a)
e , = v a y .
ab 2	'ab r
(II.8.29b)
Because of the previous discussion we can set
6 =
a a
= =(8) _ Vr =(12)
ab “ab ^[a b]
(II.8.30a)
(II.8.30b)
where
ya E^^O)
(satisfying
is an irreducible [3/2,l/2] representation (satisfying
and hQP is an irreducible [3/2,3/2] representation
(8)
Y, ’ = 0). We compute the coefficient by applying
d ao
Eq. (II.8.27) once again in the definition of 6a- We get
547
Ф гч ф гч У3!/? = Q
-(12)	1,7	1 b -
“а	'ab1^
(II.8.31)
and hence
ф л ф л у ф = а =92) = -21уЬф ф л 1у ф = -2i=(12) (II.8.32)
а.	а.	2. ao	a
TABLE II.8.V
IRREDUCIBLE BASIS OF N=l, d=4 SUPERSPACE
Ф - ф = i Y * 7 V i, Xab
r 4 a 4 ab
флфуаф = фхч X^= -2i=a(12)
H Yab Ф = Ф ~ xab = %bC8) - У[Л](12)
With Eq. (II.8.32) we have completed the construction of an irreducible
basis for N=1 D=4 superspace.
The results are summarized in Table II.8.V.
II.8,4 - The N=2, 0=4 case
We consider now the N=2 superspace. In this case ф0^ ф^ has
dimension 36 and „ ф^ л ф^ has dimension 120 (see Table II. 8.1).
548
Looking at Table II.8.Ill and at Eq. (II.8.15) we easily obtain the
otA 6B
representation content of ф ~ ф.
-	1	(+)	5	5 a (-]a
= К x:* * x. ♦ Y\a х:а)ЕвА
4wxf<l^ w (ii-8-33)
where we have used the standard decomposition (II.8.24), the symmetry
properties given in Table II.8.Ill and the fact that, in the 0(2) case
we can write
RS
t. = e t.—.	(II.8.34)
t = — e t.
Щ 2 EABr:
Indeed we have:
(♦)	ui	П
X.	=	[0,0] 1	’	® —
(-)	(	1	П
X. = [o,o]	® —
4	= [l,o]	® —
<:ии- [1-01'*)e
=	1x1	=	1
=	1X1	=	1
=	4x1	=	4
Д =	4X2	=	8
14
549
X?. = [1,о]® •	=	4x1	=	4
A = [1,1] ®
НИ
6x2	=	12
6x1
6 (II.8.35)
22
х*ь = [1,1] ® •
The possible representations appearing in the decomposition of
л л (j/C are now obtained by looking at the product of ф0^ ,
which is [1/2,1/2] ® | | times each of the X we have so far introduced.
Let us define:
6ABC =	V A	(II.8. 36a)
НВС = *A A A	(II. 8. 36b)
(-1	-	5 a
6ABC =	*A	~*B -	A	(II. 8. 36C)
6 ABC =	*A	~ Me	(II.8.36d)
'“Lc 	*Л	- *B -	(II.8.36.)
Starting from the top we find:
6ABC = Ф/2’1/2! ® □ ) ® (E1.!] ® I I I ® [1,1] ® •) =
= {[1/2,1/2] ® [1,1]} ®{JJ®( I I I ® •)} =
= {[3/2,3/2] ® [3/2,1/2] (+:) ® [3/2,1/2] ( } + [1/2,1/2] M +
550
°{ЩЗ •
(11.8.37)
At this point we recall that the representations [3/2,3/2] and
[3/2,1/2]W are fully symmetric in (a *—* 6	y) so that they
can be patched together only with | | | | and | | (the latter
being the trace of the former).
In this way we have singled out four elements of the irreducible
basis, namely
	ib	= [3/2,3/2]®					= 8 x 2 = 16	(II. 8. 38a)
	A| В | C [3/						
		2,3/2] ® Ц					= 8x 2 = 16	(II. 8.38b)
“ABC = [3/2,l/2]l+->® (Еда = [3/2,1/2] (+) ®				1		= 12 x2 = 24	(11.8.38c)	
						= 12 x 2 = 24	(II.8.38d) 80	
On the other hand the representations
[1/2,1/2] are not fully symmetric
[3/2,1/2]	[l/2,l/2](+) and
in spinor space and therefore
can be patched together only with the
representation of 0(2)
This latter, however, for N=2 is equivalent to
symmetric pair AB can be replaced by an едв tensor,
remaining elements of the irreducible basis are:
since the anti
Hence the re-
= a = [3/2,1/2] (
12x 2 = 24
(II. 8. 39a)
551
(J
[1/2,1/2]	®
=4x2= 8
(11.8.39b)
[1/2,1/2] ( ]
=4X2= 8
40
(11.8.39с)
Indeed 40+ 80= 120, and therefore (II.8.38) and (II.8.39) exhaust the
list of representations contained in л л ipY<" . All the 6's intro-
duced in Eqs. (II. 8.36) can be expanded in the basis provided by the
E's. The coefficients are obtained via a lengthy but straightforward
algebra. This will be omitted; the result is shown in Table II.8.IV.
TABLE II.8.VI
IRREDUCIBLE BASIS FOR N=2, d=4 SUPERSPACE
ФАЛ
(/xi + y^xJ + yV^)
8 EBA
1 Wa г •>
2 X”6AB)
i-Yab(Xab
4 из
|x-
1 w
ФД~ФВ~ФС 2 EBC “A
i О
ФА ~ФВ лУ5ФС " 2 EBC
\ 4 - ^4 = I EBC< - 7 ^A + 1>
- a,	. Q?a 3 . л (*^a 1	5 Уа
Фд - Фв л Y ФС " -1 | A | B [ c~| ’ 4 1 6{AB “ C) + 3 £A{BY " C) ~
1 ar W .5 U ,
4 Y LEA{B "C) + 1 Y EA{B “C}J
552
TABLE II.8.VI (cont'd)
i ,	ab,
1 ~ | A | В | C |
. r _ab	1
1 6{Ab“c)	2 Y
[a((t)b]
|a|b|c|
la	ie у\[а(?ь] _
4 °{AB “C? ) ’ 3 A{BY У ~C)
i ab/*=)	. 5^ .
12 eA{BY ( “Cj" 1 Y “Cp
Note: All spinors and spinor-tensors are Majorana.
II.8.5 - The N=3, D=4 case
With the same techniques we can obtain the decomposition into
irreducible representations of the Зф-sector of N=3 superspace. This
decomposition is utilized in deriving the results of Chapter IV.7.
From Table II. 8.1 we see that the dimension of the N=3 3ip-sector is
364. The irreducible basis is found to be the following
(+) “A = (-) “A =		[1/2,1/2]t+) ® □ [1/2,1/2]	® □	=	4x3	= =	4x3	=		12 12	(II.8.40a) (II.8.40b)
(+)	A C	_ = [1/2,1/2]C+) ®	—	— = 4x5 =	20	(11.8.40c)
(-)	A C	= [1/2,1/2]	® в 1			1 = 4x5 =	20	(II.8.40d)
553
“ A
В
С.
“И
Уа
“И
[1/2,1/2] ®
[3/2,1/2] W »□
[3/2,1/2] (® □
~~| a| b| [cj	= [3/2,1/2]( 5 ®		□
m a	= [3/2,1/2] ® |	|	1 !
IA | В | C ]			
=ab	= [3/2,1/2] ® |		1 1
1Aj B|C|			
[3/2,3/2] ® □
„аЬ
~А
4X1=	4
12 х 3 = 36
12x3= 36
12 х 5 = 60
12 х 7 = 84
8х 7 = 56
8x3= 24
364
(И.
(П.
(II-
(II.
(II.
(И-
(II.
. 40e)
. 40f)
• 40g)
. 40h)
. 40i)
,40j)
.40k)
and the coefficients of the decomposition are displayed in Table
II.8.VII.
TABLE II.8.VII
THE IRREDUCIBLE BASIS FOR N=3, D=4 SUPERSPACE
554
TABLE II.8.VII (cont'd)

(+)
C
_ 1 vaf6UC) .(AB) V
4 ’ 1 (BM) °(CM)J ~M '
В
± 5 a,.(AC)	. (AB)/^
47 7 (6(BM)+6(CM? =M
—	a.b
^A " ^B " yab^C = ~ I—I—I—I
Ы В | C |
- „ab
+ °{AB“C) ’
5, l„JM.x(AC)
217 7 [a %] (<S(BM)
6
(AB)
(CM)
555
TABLE II.8.VII (cont'd)
+ =	)	1 (6CAC)
“ ТЭ * W6(BM)
_cj •>
1	_ (AC) x(AB).
6 YSYab =M(6(BM) + 6 (CMp
(ab)/;)
(CM)' ~M
II.8,6 - The N=2,D=5 case
As we discussed in Chapter II.7 Majorana spinors do not exist in
five dimensions. Rather we can introduce a doublet of pseudo-Maj оrana
spinor 1-forms фд which transform as 4-component S0(l,4)	Sp(4)
spinors and obey the conjugation rule:
(T)C = C(T)T = eAB *B	(II-8-41)
The rigid D=5 N=2 superspace can be identified with the homogeneous
space:
G/H = SU(2,2/1)/SO(1,4) ® U(l)	(II.8.42)
д	CtA
which has five bosonic coordinates x and eight fermionic ones 6
(Pseudo-Majorana). Its cotangent space is spanned by Va and ф . As
in the D=4, N=2 case ф ф has 36 components and ip ^ф ~ф'
120 components. We must arrange these latter into irreducible repre-
556
sentations of H=SO(1,4)®0(2) rather than into irreducible represen-
tations of H = S0(l,3) ® 0(2) .
To do this we list the relevant S0(l,4) representations. Our task
is simplified by the local isomorphism S0(l,4)tUsp(2,2) which is the
spinor representation of the Lorentz group.
Usp(2,2) has rank 2 and we can characterize each of its represen-
tations by two numbers directly related to the symmetry of the spinor
indices.
	TABLE II.8.VIII
IRREDUCIBLE REPRESENTATIONS OF USP(2,2)® SO(1,4)
Rep. type Dimension	Corresponding spinor, tensor, spinor-tensor
[0,0]	1 [1,0]	4 [2,0]	10	scalar Xevc „ (charge conjug. mat.) •	Ct spinor ф ...	.aS .Ba symmetric bispinor A = + A	antisymmetric „ab tensor X antisymmetric (traceless) bispinor
[1,1]	5 [2,1]	16 [3,0]	20	,aB_	.Ba,.aB	_ .	Ya r a.aB A = - A (A Cag= 0) n, vector X =	A Irreducible spinor-vector Ea; ГаЕ = 0 1 .1 ,	-	—ab	-ba -b~	_ Irreducible spinor tensor -	= - n ; Г - , = 0
We now decompose the product
V*B = eBcV£C = еВС^ФС	(П.8.43)
where C is the charge conjugation matrix. We want the fully symmetric
part of the tensor product
557
([1,0] в □ ) в ([1,0] ® □ ) =
= ([1,0] ® [1,0]) »(□»□) =
= ([2,0] ® [о,о] ® [1,1]) ® ( I I I ® — ® •)
(II.8.44)
[0,0] and [1,1] are antisymmetric in spinor indices and can be patched
together only with the antisymmetric 0(2) representation. Hence two
elements of the irreducible basis for the ip.a л space are given by:
X0=^A^A = WlC*A= M®- = 1
(II.8.45a)
Ха=^А-ГЧ= WBCr4=
=5x1=5
(II.8.45b)
The representation [2,0] is symmetric and can be coupled only with
or with its trace	Hence the third element of the
a R
ф,	irreducible basis is given by:
A В
XAB = *A | ^4 = ЕаУсC| ГаЧ = t2’°J ® ( Ш ® •) =
= 10x (2 + 1) = 30	(II.8.46)
1 e 5 ® 30 = 36 is then the dimension this superspace sector. The decom-
position coefficients follow from standard Г-matrix algebra. We obtain:
lb „	= T Г к xTn + -	+ 1 X®)
v A VB 4 ab AB 8 AB1 a
(II.8.47)
558
The representations appearing in the 1рда ~	sector are now
easily identified by considering the product of the highest representa-
tion X „ab with ф.а. We have
BC	A
! U i Т^Ь I	I vflb
W?r ^C = VXBC
= ([1,0] ®  ) ® ([2,0] ®
] ® •))
= ([1,0] ® [2,0]) ® [ Ц ® ( j I I ® •)]
= ([3,0] ® [2,1] ® [1,0]) ® (
)
Ф
Ф
(II.8.48)
Since [3,o] is symmetric it can be coupled only with | j | and
its trace |	[ .
Therefore the irreducible basis contains the follow-
ing two elements:
„ab
"lAl Blc
(ГbEab	= 0); [3,0] ®
I A I B I c
EAb ’ (rbEAb= °); [3’°18 □
20 x 2 = 40
20 x 2 = 40
(II.8.49a)
(II.8.49b)
The representations [2,1] and [1,0] are not fully symmetric and there-
fore can couple only with
559
The remaining elements of the irreducible basis are then:
Ha ; (Г Ea = 0); A	a A	[2,1] ®	—J		1	= 16 X 2 = 32	(II.8.50a)
=A ’	[1,0] ®				= 4x2=8	(II.8.50b)
and we have
40 + 40 + 32 + 8 = 120
(II.8.51)
as expected.
The actual decomposition coefficients can be calculated with a
somewhat lengthy but straightforward algebra. They are given in Table
II.8.IV.
TABLE II.8.IX
IRREDUCIBLE BASIS OF N=2, D=5 SUPERSPACE
Ф. = - г . xab + 1 6 (г xa+ их®)
A B 4 ab BA g AB a
lpA-tPB-^c 2 6BC^A"X 2 6bc"a
VVr4 = 1 бвА' ка = -1 Wha4 r4)
ivvra4 = vxbc
„ab
36
„ab
AB“C
„ab
BC“A
„ab
CA“B
- 6
_ 6 (1 p[a=bj _ J_ rab„ .
ABl3 1 “C io “CJ
+ 1 6 rl r[a=b] 1 rab“ 1
2 °BCl3 ’ 10 “AJ
560
For paedagogical purposes we just give a sketch of the derivation of
this Table.
Since
VVPbaMm
(II.8.52)
the second equation of Table II. 8. IX is just a definition
=A = 1	(II. 8.53)
Considering now the third equation of the same Table, since
Vr4 = pABVr4	(II. 8.54)
we can set
VVMWa	<П-8-55)
where
e: = Wr4	(II-8-56)
is a spinor-vector. In general it can be decomposed as:
a C16)a	1 a (4)
ea = 6,	+ 4 г e.	(ii.8.57)
A	A 5 A
By definition we set
(16)
= i 6Aa	(II.8.58)
A A
then, in order to obtain the third equation of Table II.8.IX we just
have to prove 6д^ = -iEA, which is indeed true since
561
rWra*M =
(II.8.59)
The fourth equation of Table II.8.IX is more complicated.
Setting
OABC i A 7 В rab.C
6ab = 7 лГ ф
flABC rb„ABC
a	ab
„АВС гаоАВС
и = 1 и
(II.8.60)
(II.8.61)
(II.8.62)
and using the Fierz rearrangement formula given by the first equation
of Table II. 8. IX we can easily prove
.ABC 1 cCBA 31 XAB C , ra
= — e + — о л ~ ф., /ч г ф., -
а	о a c VMM
, _r,C 7	_ , i XAB_ ,C 7	„a,
Г ф л ф., zv Г ф,.--6 Г ф * ф..л Г ф„
а М rvM 4 а М ТМ
(II.8.63)
Now from the representation analysis done in the text we know that
ABC
6^ must contain only the representations
~ab
~ |AlB l£j
_ab _a =
“A ’ “A =A
(II.8.64)
- i 6ABT
8 i
Hence we can write a decomposition of the type:
.ABC
ab
.|a|b|c|
“ab
+ a<5
AB„C
“ab
DXBC_A xCA-B
B<5 = , + yo = ,
ab ' ab
, xABr _C , xBCp .A „p -В xCAa
(аб Г[а%] ♦ Ь6 Г[а=ь] + сГ[а=ь]6 )
562
+ (a'6AB i ГаЬЕС + b'6BC I ГаЬЕА + c'6CA j ГаЬ=В) (II.8.65)
IAI b|c~|
The coefficient in front of E&b	is just a definition because
this representation appears only once. The coefficients in front of
Д
^ab are now fixed by the fact that
<sBCeABC = о	di. 8.66)
and the requirement that
SB, XAB„C „-А xBC ,CA_B .	,TT o
e (aC =ab * B“ab6 *	c^)	(II.8.67)
be fully symmetric in (S,A,C). Indeed the representation [3,0] is
symmetric in (a-*—* |3y) and hence must be also symmetric in (S,A,C).
At this point two of the coefficients (а,В,у) are determined. The
ab
remaining one is absorbed in the definition of Ед which appears
nowhere else. The remaining coefficients are determined by Eq. (II.8.66)
and by comparison with (II.8.63). In fact from (II.8.63) we get
ABC
e1——— = о
a
(II.8.68)
,	XAB ,CA -BC	_ -ABC	„ ,
and the remaining о ,6 , о terms of 6 appear to be given
—	3.
in terms of Фсл^мл^а^М ^C л ~ ^M’ which is precisely what
we need.
The result we have been discussing will be utilized in Chapter
III.5 where D=5 supergravity is explicitly constructed.
563
II8.7 - Systematics of Fierz identities in eleven dimensions
The theory of supergravity in eleven dimensions occupies a
special and privileged position among all the other supergravities.
It is the maximally extended theory, it has a simple and beautiful
structure and it spontaneously compactifies to 4-dimensions, giving
rise to N=8 supergravity and to several other interesting models.
Because of this it will be discussed over and over in this book.
Chapter III.7 is devoted to its explicit construction and the whole
of Part Five deals with the spontaneous compactifications of this
theory.
As a necessary technical preparation for its development, in
this section we undertake the decomposition into irreducible components
of the 2, 3 and 4 sectors of the D=ll superspace.
In D=ll we have at most N=1 supersymmetry which is associated to
a Majorana 32-component spinor 1-form фа. All decompositions are
therefore decompositions of tensor products of S0(l,10) irreducible
representations, S0(l,10) being the Lorentz group in eleven dimensions.
We start by giving the dimensionality of the S0(l,10) representa-
tions appearing in the symmetric product of two, three and four gravi-
tino 1-forms ip (i|) is a spin 1/2 Majorana 1-form).
The eleven-dimensional Lorentz group S0(l,10) has, like S0(l,9),
rank 5 and therefore its irreducible representations are labeled by 5
integer of half-integer numbers.
In the integer case we are dealing with a bosonic representation
and the 5-numbers
with the number of
the representation
^2» Х^, Хд, X,- labeling it can be identified
boxes in each row of a Young tableau. In this way
2	3
(1) (0) corresponds, for instance, to the tableau
2	3
tensor Ta^a^. Analogously (2) (0)
that is to the tensor T . ? while
—	a a
__	a3a4
namely to an antisymmetric
corresponds to the tableau
a^aj
a2 a4
(1^) is a skew-symmetric 5-index
tensor
al ’'' a5
564
TABLE II.8.X
DIMENSION OF SO(1,10) IRREPS APPEARING IN THE SYMMETRIC PRODUCTS
OF 2,3,4 IRREPS (1/2)5
Type	Bose irreps Dimension	Type	Fermi irreps Dimension
(0)5	1	(1/2)5	32
(1) (0)4	11	(3/2)(1/2)4	320
(l)* 2(0)3	55	(3/2)2(l/2)3	1408
(1)3(O)2	165		
(l)4(0)	330		
(I)5	462	(3/2)5	4224
(2)(0)4	65		
(2) (1) (0)3	429		
(2)2(0)3	1144		
(2) (I)4	4290		
(2)2(1)3	17160		
(2)5	32604		
In the half-integer case the representation is of the Fermi type.
The corresponding object is a spinor tensor having in its vectorial
indices the symmetry of the Young tableau X^ - 1/2, x^ - 1/2, X^ - 1/2,
X4 - 1/2, X,-- 1/2. Moreover, it is irreducible in the sense that what-
ever trace can be obtained by contracting it with r-matrjCes is zero.
For instance the irrep (3/2)(1/2)4 is a spinor tensor with the
symmetry (1)(0)4 in its Bose indices, namely 5 . The irreducibility
means ГаЕ =0. Analogously (3/2)2(l/3)3 is a spinor tensor with
2	3	"
Bose indices of the type (1) (0) , namely - (skew symmetric).
ala2
a2„
The irreducibility condition is Г =	= 0.
dld2
The use of numerology provides an easy tool to WOrk out the repre-
sentations appearing in each symmetric product. We find
565
{(1/2)5 ® (i/2)5}syjn = Cl) СО)4 ® (l)2C0)3 ® (I)5 ,
(32x 33/2= 528= 11 + 55+ 462) ;	(II. 8.69a)
{(1/2)5 ® (1/2)5 ® (l/2)5}sym = (1/2)5 ® (3/2)(1/2)4 ®
2	3	5
® (3/2)Z(l/2) ® (3/2j	,
((32 x 33 x 34)/(3x 2) = 32+ 320+ 1408+ 4224) ; (II. 8.69b)
{(1/2)5 ® (1/2)5 ® (1/2)5 ® (l/2)5}sym = (0)5 ® (1)3(0)2
® (l)4(0) ® (I)5 ® (2)(0)4 ® (2) (1) (0)3 ® (2) (I)4
® (2)2(0)3 ® (2)2(1)3® (2)5 ,
((32 x 33 x 34 x 35)/(4 x 3 x 2) = 1 + 165 + 330 + 462 + 65 +
+ 429+ 4290+ 1144+ 17160+ 32604) ,	(II. 8.69c)
These decompositions are made explicit in the following way. Let ф
-	+ T
be the Majorana gravitino 1-form and ф=ф Гр = ф C be its bar conju-
gate. Then we can write the Fierz decompositions given in Table
TT о vt w -(32) -(320) „(1408) „(4224)	.
II.8.XI, where s ,	, s , n4	are, respectively,
a aia2 ai---as
the irreducible representations (1/2)3, (3/2)(1/2)4, (3/2)2(1/2)3,
(3/2)5 listed in Table II.8.X. Similarly, X(1), X(65)
х(ззо) x(462)	x(429) x(1144;) )C(4290)	x(17160)
aj-.-a^j’ aj.-.aj’ axa2 ’ ага2 ’ b1...b5’ aj...a5
a3 a3a4 aa	blb2
tively, the bosonic irreducible representations (0)3, (2)(0)4,
(l3){0)2, (l4)(0), (I)5, (2)(l)(0)3, (2)2(0)3, (2)(1)4, (2)2(1)3, also
listed in Table I1.8.X. Moreover, we have
a, X(165) ,
b ala2a3
are, respec-
X(462J	= e	y(462)
ar--a6 al---a6b1-- -bsOj-.-bj
(II.8.70)
566
TABLE II.8.XI
EXPLICIT FIERZ DECOMPOSITION OF D=ll SUPERSPACE
Ф-Ф = — (Г ф,Гаф - - г ф. rai\ + — г a
32 a	2 ala2	5! al’’’a5
’ a5
.(320) + 1 r .(32)
а	и a“
Ф ~ ф ~ гаФ
флфлГ ф= -f1408^ _ 2 r ala2	ala2	9	-(320) + 1 r =(32) [a^] И aia2
ф.ф.Г	ф=Е«224) + al-"a5	aVa5	2Г г ~(14Cf |_aia2a3 a4asJ
+	5	-(320) _l_r	.(32) 9 Lar  -a4“a5J	77 ai---as
флГ ФлфлГ Ф
а1 a2
x(65) +	x(l)
al 11 ala2
a2
Ф-Г ’р.ф^Г ф
ala2 a3
„(429)
»A
a3
x(165)
ala2a3
Ф - ra a Ф - Ф
ala2
x(1144)
ala2
x(330)
ala2a3a4
a3a4
£6f x(ff> - A ^xd)
9 Lai a2J 11 a3a4
[a3 a4]
E	i
al-”a6bV“5
x(462)
;b1...b5xb1...b5
„(4230)
•r->s
a6
+ 15 6	x(330)
*7 a6[al a2 -a5]
Ф - ra a a
1“	5	6 7
i F	„(330)
56 аГ • -a7br • -b4X- -b4
567
TABLE II.8.XI (cont'd)
i e	„(4290) + x(17160) _ 180 fia6a7 (167)
300 bl.. .bjaj.. •a5[a6%1.. ,b5 aj.-.as 21 [ala2 a3a4asJ
а7]	a6a7
- i 1200 6
ta6 ~(462)
lal a2---as]
a7]
The decomposition of Table II.8.XI is a substitute for all Fierz iden-
tities which correspond to the appearance of the same irreps in several
different products of fermionic currents.
II.8.8 - Irreducible representations of S0(l,9) and the irreducible
basis of the D=10 superspace
Supergravity and super Yang-Mills theory in ten space dimensions
are as important as D=ll supergravity. Indeed ten are the critical
dimensions of superstring theory and the matter coupled supergravity in
D=10 is the field theoretic limit of this non local theory. Superstring
generated supergravities are treated in Part Six; D=10 super Yang-Mills
will be presented in Chapter II.9. Here we study the irreducible
representations of the 10-dimensional Lorentz group S0(l,9) and the
decomposition into an irreducible basis of the D=10 superspace.
In ten dimensions we have Majorana Weyl spinors. We shall deal
with fermionic Majorana 0-forms which are respectively Weyl and anti-
Weyl.
C(X)T = X ; C(x)T = x	(II.8.71a)
| (H ♦ fn)X = X
(II. 8.71b)
568
I П - rn)x = x
(11.8.71c)
The spinor X (called the gauging) will turn out to be the supersymme-
tric partner of the Yang-Mills field A= A dx^ and because of that
carries, in general, an index running in the adjoint representation of
some internal symmetry group G. x which has the opposite chirality
and which we name the gravitello sits in the graviton multiplet: it is
part of the pure supergravity theory and accordingly it carries no
internal symmetry index.
The SO(1,9) gamma matrices:
{га*гЬ} = 2riab (a =0,1..........9)	(II.8.72)
are 32 x 32 as the 11-dimensional ones. The charge conjugation matrix
is antisymmetric
2	*	T
C = -1	; С = С ; С = -С	(II.8.73a)
-1 T
СГаС = -Г	(II.8.73b)
and Гц is "symmetric" in the C-sense:
-1 т	T
СГцС = -Гц (СГц)1 = С Гц	(II.8.74)
This is what allows the definition of Majorana-Weyl spinors. In parti-
cular the D=10 superspace has 10 bosonic coordinates {xa} and 16
fermionic coordinates {ба} corresponding to the independent components
of a Majorana-Weyl spinor 6:
| (fl + rn)6 = e
(II.8.75)
569
It follows that the cotangent space to superspace is spanned by the
zenhbein Va and gravitino 1-form фа which is also Majorana-Weyl:
c(WT = Ф
| (-a + гп)ф = у
(II.8.76)
Setting
.. .a
г 1	1
i [a, a. al
n _ rL Ij, 2 p n'
(II.8.77)
we find
аг-
СГ
(II.8.78)
where
n = 3,4,7,8
(II.8.79a)
n = 1,2,5,6,9,10
(II.8.79b)
Chapter II. 7)
symmetric, while if
In general we
As usual (see
S = 1
n
have
if
we
SR = 0, we say
say that it is
ar.
that Г
symmetric.
. a
n .
is anti-
Г Г
111*ал...а
1 n
cost(n)
e
a.
b ...-b.
p n+1	10
..a b ....b.„
n n+1	10
(II.8.80)
where cost(n) is a number
that, if we consider the bilinear forms
depending on n. Formula (II.8.80) implies
al-.-%	_ ax...a^
I	= ХГ X
(II.8.81a)
aj.-.ajj	ap-.a-
X	= i|i « Г ф
(II.8.81b)
№
S a....a —.
n 1 n^T
S = 0
n
S = 1
n
570
where X is a Majorana Weyl 0-form and ф a Majorana Weyl 1-form,
a^.. . a^	a-p .. an
then I	is non-vanishing only when both Г	and
aj...a~	ap..a^
ГцГ	are antisymmetric, while X	is non-vanishing only
ai...an	аГ--ап
when both Г	and Г^Г	are symmetric. A look at formu-
lae (II.8.79) is sufficient to conclude that the only non-vanishing I-
current is
j.ala2a3
a.a?a,
ХГ X
(II.8.82)
while the only non-vanishing X-currents are
Xa = ф л Гаф
(II. 8. 83a)
аГ-а5 - аГ-а5
X	= ф л Г ф =
1
__ с
5! al-’-a5bl---b5
b ...b
X
(II.8.83b)
ala2a3
This result is understood by recalling that I	has
(10»9*8)/(3*2) = 120 components, which is precisely the number of compo-
nents of the antisymmetric object XaX^: 16 • 15/2= 120. On the other
hand X has 10 components and the antiselfdual X	has 1/2
a	al • • • a5
(10*9ф8"7*6)/(5"4"3*2) = 126 components which together make the 136 of
the symmetric object
dim ф“ л ф6 = | 16 x17 = 136
(II.8.84)
On the other hand the dimension of the Зф-sector is easily calcu-
lated:
j- ,a ,6 iY 16*17’18	01,
dim ф л ф /v ф - ----------- = 816
(II.8.85)
3-2
571
Since the gravitino does not carry SO(N) indices the representations
relevant to our analysis are only those of the Lorentz group as in the
D=ll case of the previous section.
S0(l,9) has rank 5: hence its representations are labelled by 5
numbers X^,	^3» X^, X^, where X^-X^+^. For bosonic representa-
tions X^ are all integer and stand for the number of boxes in the rows
of a Young tableau. The representation is therefore a tensor and [x^,
X^, Xj, X^, Xj_] give the symmetry of its indices. In the fermionic
case X^ are all half-integer and the representation is a spinor-tensor
whose bosonic indices have the symmetry [X^ - 1/2, X^ - 1/2, X - 1/2,
Хд - 1/2, X,-- 1/2]. The spinor-tensor also fulfills a convenient trace
condition, obtained by contraction with a Г-matrix, which guarantees its
irreducibility. In both the bosonic and the fermionic case, X^ are
also the eigenvalues of a complete set of Casimir operators.
We have computed the dimensionality of the relevant representa-
tions using standard formulae in group theory and our results are summa-
rized in Table II.8.XII where, for writing convenience we have arranged
the indices as in a Young tableau rotated of 90°.
For instance, when we write a tensor of the following type:
T
aVan
its symmetry is that of the following tableau:
•’’an-l
l'‘‘bm-l
(II.8.86)
572
The spinor-tensor E	have, in their bosonic indices the same
al‘‘‘an
br..b
m
properties as I	;
al' ’' n
b... .b
1 m
trace condition
satisfy a
moreover, in order to be irreducible they
with Г-matrices:
a
Г n
	= 0
a, ... a
1	n
Ь-...b
1	m
(II.8.87)
Now we have to explain why the Г-matrix trace conditions do indeed con-
vert a spinor tensor into an irreducible representation. This is simply
a counting argument. On one hand we have the dimension of the irreduci-
ble representation which was computed from group theory. On the other
hand we have a spinor tensor. If we do not impose any Г-trace condition
it has 16 x (dimension of the boson rep.) components. We just have to
show that the Г-trace condition subtracts the correct number of compo-
nents. To do that for the cases listed above we introduce the following
recurrence relations.
Let 6	be an antisymmetric tensor spinor. We write
1 n
(II.8.88)
Let П
aVan
b
be a tensor
spinor
with
al
traceless symmetry
an
in bosonic indices.
We set
г
П
ar • • *41-1*41
b
(nt 1)
(n- 1)
, (II.8.89)
ai-..an-ib
n
b
e
573
Eq. (II. 8.89) is justified by the fact that after elimination of the
index an the remaining tensor is, as far as the bosonic indices are
concerned, the sum of the following two tableaux:
The normalization factors in Eq. (II.8.89) are obviously arbitrary and
have been chosen in a particular way only for later convenience. Now
using Eq. (II.8.89) with a little algebra one can show the following
identity:
rm П
a
... .a
1	n-1
(n-i)rm e
al"
a
n-1
(n-l)6
ar •
n-1
(II.8.91)
m
Eqs. (II.8.88) and (II.8.91) are the fundamental tools of our counting
argument. Let us for instance consider the spinor tensor 6
a,... ar
aj- 1	5
It has (126 + 126) x 16 = 4032 components. The condition Г b0	=0
аГ ’ a5
corresponds to 16 x210 = 3360 constraints and indeed we find 4032 - 3360 =
672. The same argument goes through for all the remaining [3/2,3/2,...
1/2] representations.
Coming now to the [s/2,3/2,...,1/2,...] representations we start
by considering, for instance, the spinor tensor IT	. It has
al---a4
m
a4
16 x 1728= 27648 components. If we impose the condition Г П	=0
aVa4
D
it would seem from Eq. (II.8.89) that we subtract a spinor tensor
574
П	and a spinor tensor 6	, , namely 16 x 945 + 16 x 210
a^...a$	r	a^... a^b’
b
components. Because of the identity (II.8.91) however, this overcounts
the constraints and we have to take 16 x120 of them (the components of
6	) back. We get
ala2a3
27648 - 16 x 945 - 16 x 210 + 16 x 120 = 11088
(II.8.92)
which is the correct dimension of the representation [s/2,3/2,3/2,3/2,
1/2]. In the same way we can check all the remaining numbers of the
Table. Now that we have classified the irreducible representations,
every spinor tensor will be decomposed into irreducible components.
What we just need are the Clebsch-Gordan coefficients which were ob-
tained via an iterative procedure starting from the recursion relations
(II.8.88) and (II.8.89). We omit the extremely long but straightforward
computations. The result is summarized in Table II.8.XIII.
Equipped with this lore we can now derive the irreducible basis of
the 2ф and Зф-sectors.
ct В
We have already pointed out that ф ф has 136 components cor-
responding to the two currents Xa, X ''a5 defined in Eqs. (II. 8.83).
The exact decomposition coefficients are very easily computed and are
given by:
Ф
ф = — Г Xa + —i— Г
16 a 32’5!	al’’’“5
.a.
x I" 5
(II.8.93)
The ф°\ф®~фУ form, instead, has 816 components which are distributed
among the two spinor-tensors:
Га*
(II. 8.94a)
н 4' A
(II.8.94b)
575
TABLE II.8.XII
REPRESENTATIONS OF SO(1,9)
Rep. type	Dimension	Corresponding tensor/spinor-tensor	
[2,2,2,1,1]	6930 ® 6930	CT os’4 CT 03 V4 VI	self- or antiself- dual in al•••a5
[2,2,2,l,0]	10560	o' 03 O' 0? VJ £*	
[2,2,2,0,0]	4125	O' Os’4 O' 03 VJ VJ	
[2,2,1,1,1]	3696 ® 3696	T aVa5 blb2	self- or antiself- dual in al-.-a5
[2,2,l,l,0]	5940	T al"-a4 blb2	
[2,2,l,0,0]	2970	Tala2a3 blb2	
[2,2,0,0,0]	770	T ala2 blb2	
[2,1,1,1,1]	1050 ® 1050	T al’’’a5 b	self- or antiself- dual in aVa5
[2,l,l,l,0]	1728	T aVa4 b	
[2,l,l,0,0]	945	Tala2a3 b	
[2,l,0,0,0]	320	T ala2	
		b	
576
TABLE II. 8. XII (cont'd)
Rep. type		Dimension	Corresponding tensor/spinor-tensor	
[2,0,0,0,0]		54	T a	
			b	
[1,1,1,1,1]		126 ® 126	T al'"a5	self- or antiself- dual in aVa5
[1,1,1,1,0]		210	T ar..a4	
[1,1,1,0,0]		120	4 W ft)	
[1,1,0,0,0]		45	T ala2	
[1,0,0,0,0]		10	T a	
Г1 L 1 1 I2 » 2 * 2 » 2 »	f]	5280	a5 ~	• p “a	-а’	"ar b1 5	b1	= 0 •a5
rs_ a A 3 l2 » 2 ’ 2 9 2 9	11	11088	O* {D Ju CT* {D	= 0 ••a4
г a A A A L2 » 2 * 2 9 2 ’	11	8800	ra3„ =	: Г £ al"'a3	ar b	b	= 0 •a3
[A A A A l2 » 2 ’ 2 * 2 *	fl	3696	=	a2= %a2’ =ala2~ b z	b Z	0
г a A A A L2 » 2 9 2 ’ 2 *	11	720	= ; Га= = 0 a	a b	b	
[A A A A l2 » 2 9 2 9 2 9	7]	672	a5 =	; Г E аГ--а5	аГ	= 0 •a5
rA A A A l2 » 2 9 2 9 2 9	11	1440	=	; Г 4E ar..a4	ar	= 0 ••a4
lA A A A l2 » 2 9 2 9 2 9	1!	1200	a3 -	; Г = а^...а3	а^а	= 0 l2a3
577
TABLE II.8.XII (cont'd)
Rep. type	Dimension	Corresponding tensor/spinor-tensor
Г-1 3 1-2 * 2 *	1 2 *	1,1] 2 ’ 2J	560	a2 =	; Г	=0 ala2	ala2
L2 2	1 2 ’	1,1]	144	5 ; Га= = 0 a	a
Г-,"» L2 ’ 2 ’	1 2 ’	1,1] 2 ’ 2J	16	E (Majorana Weyl spinor)
TABLE II.8.XIII
DECOMPOSITION COEFFICIENTS FOR FERMIONIC REP.
e = e(16) e = e(i44) + r e(i6) a	a	10 a 6	= 6(560) _ 1 r e(144) _ ab	ab	4 (a b] 6	=e(1200)+l	e(560) ala2a3 ala2a3 2	al а2аз' = e(1440) _ f e(1200)	А- г 90 ab _lrr e^44)-A_r 56 la1a2 a3J	72o apa2a3 _ 1	e(560) +
1 a2'”a4J	5 lala2 аза4-
* — rr 84 L + - rr 2 La  Ur[ 1 + 	 30240	e(144)+__l_r	e(16) ala2a3 a4^	5040 аГ”а4 e(1440) ! - 5 rr e(1200) T - J a2...a5J	6 [a1a2 a3-..a5J e(560) + J_	e(144) + al”-a3a4a5J 336 lar--a4a5J • г	o(16) aVa5
578
TABLE II.8.XIII (cont'd)
e
a
b
e(72°) _ I rr 61g44)
a 6 La bj
b
e
ala2
b
e(3696)
ala2
b
* 4 rLa el7l0) - z (Г, e(560) - г
4 [a2 a2J 6 b ага2
3 fr fi(144)
80	a^0 b
e(560) _
1 a2]b
e
a.a.a,
b1
e(8800)
ala2a3
-	rr e(369^
Lal a2a3J
-	1 (r e(12°0)
4 bar..a3
e
г fi(144>
b [a16a2]
- -L г e(720)
28 [aja2 a3]
In r fi(144\
3 Va^aJ
+ r[a16a2120°l3]b) -
. a.
-1 (Гг	+ r r e(560]
9 La1a2 a3Jb Ь[ах a2a3J
iv e(56°]
2 b[aj a2a3]
3 fr fi(144)	(144)	.(144).
280 (a ... a 6b ГЬ[а a 6a ]	* ^La/a/aJ }
J. О	1.	О
e(11088) + 5 r e(8800)	_ 2	Д3696)
al‘‘‘a4	3 [aj a2-..a4]	з tala2 a3a4-J
b	b	b
.a
_ _L Гг e(7?0) 3 rr e(1440) rr e(144°) i
42 [ar..a3a4] 8СГЬ6аг..а4 ^аД a ]b}
b
.a
.a
- (rbFa e(1200) 1 - Гг e(120ft
24 b|a1 a2...a4J	[a^ a3a4Jb
+ - г г e(1200) т) + — (г. r e(56°]
7 bjaj а2...а4Р	13S bLa^ a^]
г	fl(56°) * 10 П
Г[а1...а3 a4]b 7 nl
r e(560b
b[al a2 a3a4J
_ A_ (r ed44)
360	b[aj...a3 a4]
Г 6<14^
ar..a4 b
15, _ r г „(144)
7 b[aira2a3ea4]
579
TABLE II.8.XIII (cont'd)
e	= e(5280) a .. .ac a..... a b1 5 b1	_ 15 r e(11088) _ 25 r	e(8800) 5	4 tal а2‘”а5^	12 tala2 a3’ ‘ ’as] + + ^ГГ	e1369^ + — rr	e(7?0) - 12 Lai• • • a3 a4a5J 500 Lar--a4 a5J b	b - _L (Г e(672) + rr e^672^ lh) - 12lbai...a5	[aia2...a5]b . 205	e(1440)	+	0(144O)	+ 96	Ь[а1а2...а5]	[aia2 a3...a5Jb (1440) ,	25 ,	„(1200) ^bfa^a,.. .aj	96 b[a1a9 a,.. .a,] .L £.	О	x £> о	о „(1200)	„(1200) tai’••a3 a4a5^b b tal a2 a3‘‘•asl + — (r r	e(560] + гг	+ 216 b[ar..a3 a4a5J	lar..a4 a5Jb fi(560^ .	1 rr г	+ 3nb[aira2a3ea4a5]') 1344 ( b[ar..a a] + г	<144) ♦ 4n г г	e^1^) a1...a5b	Ь(ат a2...a4 a5J
In principle, H contains the representations (672), (1440),
al‘ ’ >a5
(560), (144) and (16). Many of them, however, have to be zero because
we have, at most, 816 components. Actually the only way to obtain 816
with the above numbers is by summing (672) and (144). Therefore, we
can conclude
Ф - Ф - г
al"
ф = =(672)	+ _1_ rr
a5 al''*a5	336 Lal---a4a5J
ф л ф л Га1р = а =£144)
(II. 8.95a)
(II.8.95b)
580
where a is a coefficient to be determined.
A further justification of (3.10) (Fierz identities) is the fol-
lowing. As we already know	ф is antiselfdual and there-
al"•,a5
fore has 126 components. Hence	is an antiselfdual spinor
al’’'a5
tensor which, therefore, has at most 16 x 126= 2016 components. Consi-
dering the decomposition of the Table we see that:
4032 = 672 + 1440 + 1200 + 560 + 144 + 16
= (672 + 1200 + 144) + (1440 + 560 + 16)
= 2016 + 2016	(II.8.96)
Hence we conclude that the most general antiselfdual spinor tensor
g(antidual) a superposition either of (672), (1200) and (144) or of
al ‘ ‘a5
(1440), (560) and (16). By explicit inversion of Table II. 8.XIII we
can show that, for an antiselfdual spinor tensor the components (1440),
(560) and (16) are zero so that the first is the correct expansion. It
is then sufficient to note that the representations (672) = [3/2,3/2,
3/2,3/2,3/2] and (144)= [3/2,1/2,1/2,1/2,1/2] are certainly fully sym-
metric in (a< — >6<-->y) because they correspond to the highest
spins in the products
[1/2,1/2,1/2,1/2,1/2] ® [1,1,1,1,1]
[1/2,1/2,1/2,1/2,1/2] ® [l,0,0,0,0]
Hence (672) and (144) are indeed
фа,ф6.фУ.
the two irreducible components of
The coefficient а
of Eq. (II.8.95b) is easily computed using
Eq. (II.8.93) once again in
a5
Г Ч л Ф л Г
аГ-
1
ф = — г ьг
5	336
-а4“а5]
(II.8.97)
581
TABLE II.8.XIV
IRREDUCIBLE BASIS FOR N=l, D=10 SUPERSPACE
a,..
ф л ф = 1/16 Г Xa+ 1/32-5! Г	X
a	aVas
Ф л ф . Гаф = ф . Xa = 1/336 S^144)
(672)
The result is
+ 1/336 Г[аг
s(144)
..a4-a5]
(II.8.98)
Our discussion is summarized in Table II.8.XIV.
582
CHAPTER 11.9
SUPER YANG-MILLS THEORIES
II.9.1 - Introduction
Supersymmetric field theories were defined in Chapter II.6, where
we treated the example of the Wess-Zumino model: they correspond to the
available multiplets of a given N-extended supersymmetry and can be
constructed with procedures analogous to those outlined in that chapter.
Super Yang-Mills theories are the supersymmetric field theories corres-
ponding to the massless vector multiplets whose field content, for the
various values of N is displayed in Table II.4.VI.
As evident from the Table, such theories exist only for N^4
since beyond that limit no representation of supersymmetry is found
which does not involve spins higher than one. For N^4 the spin
1-field can be identified with a gauge field B^a(x), the index a
running in the adjoint representation of some gauge group 'S whose Lie
algebra is
583
(II.9.1)
fh , hD] = KYO h
L a BJ aB a
All the other fields in the multiplet will carry the same index, namely
the whole multiplet is in the adjoint representation of 'S. In parti-
Ct
cular besides the gauge bosons В (x) we always have spin 1/2 particles
ct
X (x) belonging to adj 'S which are usually named gauginos. For N> 2
the vector multiplet contains also scalar particles deprived of a
specific name.
The Lagrangian of a super Yang-Mills theory is a supersymmetric
extension of the Lagrangian
у = _ pa pa
(II.9.2)
4 pv pv
it maintains local gauge invariance under ^-transformations and adds
to it invariance against global supersymmetries. Super Yang-Mills
theories may be used to construct supersymmetric grand unified models
of particle interactions. If N< 2 we have at our disposal both the
vector multiplets and the scalar multiplets (Wess-Zumino or hypermulti-
plets): therefore we may consider supersymmetric models where we couple
the gauge multiplet of some group rS to a certain number of scalar
multiplets assigned to given representations of 'S. In the scalar sec-
tor we can fit the quarks, the leptons and the Higgses while, in the
gauge sector we have the gauge bosons together with their superpartners.
In the process superpartners are obviously introduced for the quarks,
the leptons and the Higgses as well.
They are usually called squarks, sleptons and Higgsinos. Intro-
ducing suitable scalar field potential, both the supersymmetry and the
‘S gauge symmetry can be broken spontaneously with a mass generation for
the otherwise massless gauginos. The topic of spontaneous supersymmetry
breakdown will be addressed in Chapter IV.S directly in the framework of
local supersymmetry, namely after coupling of the scalar and vector
multiplets to supergravity. Indeed, as further discussed in that chap-
ter it is only by use of local supersymmetry that insolvable problems of
584
globally supersymmetric models were solved. Let us now observe that
for N>3 scalar multiplets do not exist so that quarks, leptons and
Higgses must be fitted in the vector multiplet itself: this requires a
sufficiently large group 'S able to accommodate the full spectrum of
elementary particles in its adjoint representation. Furthermore S?
must break conveniently and suitable mass splittings and supersymmetry
breakings must occur. Such breaking mechanism are out of the scope of
the present chapter. Here we want to study the rheonomic formulation
of super Yang-Mills theories as a further example of supersymmetric
field theories and as a preparation to their coupling to supergravity
studied in one specific example in Chapter IV.7.
In particular we want to discuss the question of auxiliary fields
for these models showing that they exist in the N=1 and N=2 cases, while
they are absent in the N=4 (and N=3 case). Such a discussion will be
resumed in Chapter IV.7. Then we shall present the rheonomic actions
of the N=1 and N=2 models. The N=4 action could be retrieved by dimen-
sional reduction from the N=1 super Yang-Mills action in ten dimensions.
This latter model is important for its connection with String Theory:
we present it here as a preparation for PART SIX.
II.9.2 - Super Yang-Mills theories in D=4
Our aim is that of supersymmetrizing the Yang-Mills Lagrangian
(II.9.2). The Yang-Mills field B^(x) is a ^-Lie algebra valued
1-form. Indeed we can write
В = B“(x)hadxp	(II.9.3)
where h are the generators of the 1^-group (see Eq. (II.9.1)).
The field strength of В is the following 2-form
dB + В л В
(II.9.4)
585
and it can be decomposed along the h according to

;	= dBa + - Ka„ BB л В7
2 -By
(II.9.5)
Because of its definition ST satisfies the Bianchi identity
= d.^ +	= 0
(II.9.6)
or in components
= d.^a + Ka„ BB = О	(II.9.7)
BY
In a non-supersymmetric framework where the base manifold of the fiber
bundle Р=Р(М^,^) is Minkowski or anti de Sitter space, a complete
cotangent frame is provided by the vierbein Va and we can write:
^=FabVa.Vb ; (Fab^bha)	(II.9.8)
Fab)a being the intrinsic components of the field-strength Fa.
Treating these intrinsic components as independent fields we were able
to write the Yang-Mills action in the following form (see Eqs. (1.5.70)):
AY.M. = |^(^Fab . VC л Vd)eabcd -
- — Tr(F . Fab)e. . V л Vj л Vk л V1]
12 ah ijkl
(II.9.9)
which yields (II.9.8) as an equation of motion and reproduces the Yang-
Mills propagation equation:
V F ,
a ab
= 0
(II.9.10)
586
Introducing supersymmetry we replace the fiber bundle P=	by
a fiber bundle whose base manifold is either N-extended Minkowski super-
4/4N	4/4N
space M or N-extended anti de Sitter superspace AdS . For
simplicity we put ё = 0 and we consider the fiber bundle:
P = P(M4/4N, 3?)	(II.9.11)
The cotangent space, now, is spanned by Vй and the gravitino 1-forms
фд: correspondingly Eq. (II.9.8) is replaced by
= F , Va л Vb + CA ф. л Vm + ф, л NA%d	(11,9.12)
ab	* m A	VA	B	1	’
A	AB
where £ is a О-form vector-spinor and N is a О-form matrix in
spinor space. The idea of rheonomy is that the outer components £A
AB	m
and N must be related to the fundamental space-time fields or to
their spatial derivatives. This is what brings in the other components
of the supermultiplet displayed in Table II.4.VI.
In fact the Bianchi identity (II.9.6) implies that £Am, NAB can-
not be zero if F , / 0.
ab
The process of constructing the super Yang-Mills theory consists
A	AB
therefore of parametrizing Z and N by means of the fields at our
disposal in a rheonomic and Bianchi consistent way. Once this is done
one looks for a geometrical variational principle capable of yielding
the rheonomic conditions as equations of motion. This variational prin-
ciple usually turns out to be unique. Throughout the process one uses
the Maurer Cartan equations of superspace given in Table III.3.1.
In general the rheonomic conditions are consistent with the
Bianchis only on-shell, namely, inserted in the Bianchis, they yield
the equations of motion of the physical fields. The auxiliary fields
can then be defined as the additional components of the curvature F,
A AB
or the covariant derivatives D£ , dN which appear when one removes
those rheonomic conditions (and only those) responsible for placing the
physical fields on-shell. It is in the number and type of these latter
that the N=4 theory differs from her sisters N=1 and 2.
587
Let us consider these theories individually and for simplicity
let us assume an abelian gauge group rS so that the Bianchi identity
(II.9.6) reduces to dF=0.
N=1 Theory: The vector multiplet contains, besides the spin 1
field В (x), just a Majorana spinor X(x). The rheonomic conditions
are
F = F , Va „ Vb - 2i X у ф л Vго
ab	m
DX = Л Vго - - Faby . Ф
m 4 ab
where is the space-time derivative of X:
(II.9.13a)
(II.9.13b)
(II.9.14)
(II.9.15)
(II.9.16)
The Bianchi identities
dF = DDX = 0
imply the Dirac equation
YmA = 0
' m
However, if one imposes only Eq. (II.9.10a), which is the equation of
motion of the F . field, one obtains from the Bianchis
ab
DX =	+ i Y5^D - | FabYab^	(II.9.17a)
ao.	<k.-d|W	di.s.iTb)
where D is an auxiliary pseudoscalar.
588
N-2 Theory: The reason why in the N=1 theory we could set the
AB
matrix N of Eq. (II.9.12) equal to zero is that, while computing
dF we can take advantage of the identity у^Фл флутф= 0. In fact we
get
d(X УтФ~Ут) = DXym л ф л Vm - 1 XyJ л ф л утф
= DX л утФ - Vm	(II.9.18)
and therefore we do not have to worry about three ф terms appearing
in dF. For N>2, however, this is no longer true. In fact the basic
Fierz identity is
Va - ^в л Y4 = -2*b VA ~ Фв * 2у5фв л фд л у5фв
(II.9.19)
and, accordingly, the matrix NAB
has to be different from zero in
order for
- фд dN фв
to cancel the флфлф term arising from d(XAy^A л Vm) . NAB is the
place where the scalar fields of the extended super Yang-Mills theories
sit.
In the N=2 case, besides the 0(2) doublet of spinor Хд, we have
a scalar and a pseudo scalar 0(2) singlets denoted respectively ф and
ТГ.
The Bianchi consistent rheonomic conditions are the following:
F = FabVa . Vb - 2i ХЛфд . Vго - |гВ(ффлфв+ШфАЛу5Фв)
(II.9.20a)
DXA = AV - 1 F YaV + | eAB(iy ф Фп + У5УПФВПП) (II.9.20b)
111	Д ctU	/Т	Il D	V 11 I?
589
d<j> =	+ 2еАВХдфв	(11.9.20c)
dir = П Vm + 2ieABX,y_ipD	(II.9.20d)
m	Add
As in the previous case if we insert Eqs. (II.9.20) into the Bianchi
identities
dF = DDXa = dd$ = ddrr = 0	(II.9.21)
we find that they are satisfied only when Хд is on-shell:
утЛА = 0	(II.9.22)
However, if we impose only conditions (II.9.20a,c,d), which will be
shown to he the equations of motion respectively of the fields F b,
$a and П^, then Eq. (II.9.20b) is replaced by
DXa = AV - 1 FabyaV + | eAB(iYnVn * VnVn) * ^4
(II.9.23)
where MAB is a matrix in spinor space which transform Majorana spinors
into Majorana spinors and which, in order to preserve the Bianchi iden-
tities, satisfies the following constraints:
*A ' ^%в =	' VC -	* Y5Mb/ = о	(II.9.24)
The most general solution of Eqs. (II.9.24) is the following
MAB = iPABY5 * eABS 1	(II.9.25)
AB
where S is a scalar field and P a pseudoscalar one, symmetric and
traceless in 0(2) space:
590
рАВ _ рВА	рАА _ Q
(II.9.26)
In this way the actual set of auxiliary fields which is brought in con-
sists of two pseudoscalars and a scalar, making up the three Bose degrees
of freedom necessary to close the off-shell algebra. This set of auxi-
liary fields is also consistent with the fact that when the N=2 super-
symmetry is truncated to N=l, the N=2 irreducible vector multiplets
breaks down into the N=1 YM multiplet (1,1/2) times the WZ multiplet
+ -	AB
(1/2,0 ,0 ). Of the two pseudoscalars contained in P one is then
the auxiliary field of the (1,1/2) system while the other one, together
with S, completes the auxiliary set of the (1/2,0 ,0 ) system.
N=4 Theory: In the N=4 case the on-shell counting of states re-
veals that we must have 3 scalar and 3 pseudoscalar fields, besides the
four Majorana spinors Хд and the Yang-Mills connection B. Therefore
we introduce two zero-form, anti-symmetric, self-dual tensors in 0(4)
AB AB
space, denoted ф and it , which transform under the Lorentz group as
a scalar and a pseudoscalar, respectively:
ЛАВ Ф		.BA Ф	1 ABCD = — e	.CD Ф
AB TT		BA TT	2	CD IT
(II.9.27)
The rheonomic conditions
consistent with the Bianchi identities
dF = DDX = ddфAB = ddirAB = 0
(II.9.28)
are then given by the following equations:
F = FabVa . Vb - 2i XaY\ . Vm - | (фАВфА„фв + 1/ВфдЛу5Фв)
DXA = AV - 1 FabyaV ♦ 1 dY4<B +Y5Y%bO (II.9.29b)
591
d4>AB =	.AB а	л A [A, Bl	1 Ф V + 4(XL ф J + — a	2	ABCD- , , e W	(11.9.29c)
d/B =	JBa л-а[а , В1 Па V + 41 (XL у$ф J +	1 ABCD" , . 2 £ Wd’	(II.9.29d)
The difference between the N=4 theory and the N=1 and 2 case is the
following: on one hand Eqs. (II.9.29) inserted into the Bianchis
(II.9.28) place the spinor on-shell as before
= О	(II. 9.30)
on the other hand, if one assumes only Eqs. (II.9.29,a,c,d) and sets
DXA = AAv” - A Fabyab/ ♦ A (iYVB ♦ у\МВ)фв ♦ /4
(II.9.31)
the constraints on the matrix MAB enforced by the Bianchis (II.9.28)
are
*A л V^B = 0	(II.9.32a)
ф[А л МВ1Сфс + | eABCDipc л = 0	(II.9.32b)
^А л	* у еАВСПф л ум фр = 0	(11.9.32с)
which have the unique solution MAB= 0. This means that as soon as we
implement the constraints (II.9.28a,c,d) we are already on-shell and
we have already lost all the auxiliary fields. In the dynamical theory
the constraints (II.9.28a,c,d) correspond to the equations of motion of
AB AB
the non-propagating fields Fab, $a • Па which are algebraically
solved in terms of the other fields through these very same equations
of motion. For this reason Eqs. (II.9.28a,c,d) and their analogous
(II.9.20a,c,d), (II.9.13a) are named "second order constraints”. We
see that there exist auxiliary fields when the second order constraints
592
inserted into the Bianchi identities do not imply the propagation equa-
tions (do not imply the shell). This happens in the N=1 and N=2 theories
but not in the N=3 and N=4 cases.
In Chapter IV.7 we shall add further remarks on this point and
trace back the absence of auxiliary fields for N>3 to the different
structure of the superalgebra automorphism groups. We can now write the
rheonomic actions for the N=1 and N=2 theories. The N=3 theory will be
considered in chapter IV.7, while coupling it to the corresponding super-
gravity.
N=1 super Yang-Mills theory in D=10 is discussed in next section.
From its dimensional reduction one could obtain the N=4 model.
N=1 Super Yang-Mills action (any gauge group )
A^^ =	- VC л Vd - i Хкф л у8Ьф л Vе л Vd -
-	i FabVC л VdXY^ л Vго - | Ху^Х л Vb л Vе л Vd}Eabcd -
-	— Tr(F . Fab)V£ л VJ л Vk л A. . „ -
12 ab '	ijk£
-	— Tr X Yr X ф л уаЬф - V л V, +
4	»5 т । v a ь
+ 2 Tr X YjY Ф - V™ л. &]	(II.9.33)
N=2 Super Yang-Mills action for an abelian gauge group
= U(l) a U(l) <8 ... U(l)
A(N=2)
f { ? - {- V4bcd + 4*Ч Va л V& + eAB (i^A л Vb -
M4
- ТГФл-Фп)}-- (F краЬ-| Ф Фа--П Па)е. .. A'1 . Vj W +
*A *BJJ 2 ab 2 a 2 a ' ijkl
593
+ раЬуС~ V4bc<r [2iX\nV ^4	Чг W1 -
-|$av\vC.Vdeabcd(d<f>-2XAeAB)-
- | Па . Vb л Vе . Vdeabcd «hr - 21еАВХАуЛ) ♦
4 .7 a„,A ,,b ,,c ,,d o. AB,,, ~A ab.
+ — 1Хду DX ,V ,V .Veabcd-2ie (d^.X^y Фд +
+ idir л ЧаЬФА) a v\ Vb - iirdcf) л ^дА уафд - Va +
1 ,.,2	2. AB CDy ,	7	,	1 . AB CD7 ,	7	,
+	-(1Ф-7Т)Е e ф^ф^ф^у^--фте с VWV
+	2еАВ(ФФА -	ФВ +	Ч л	Y5 V л ХСу5уЙфС л Va -
-	2ЛАу_ХВ —	ф, л	у , фк	Va a. Vb - iXAXB - ф. л у . ф а. V л V eabcd	+
’5	2	A	air В	2 A 'abvB с d
+ 21ЕАВЕС1)Хду5УаХсФв .	уьфп A Va A vb t 21Л%^В л у5уьфп A	va A vb t
+ 1 \ЧЬМв^ФА.vb- iXAyabx/лу/лVaлVb}
(II.9.34)
These actions were obtained with the same procedure described in sec-
tion II.6.5 of Chapter II.6 for the Wess-Zumino model. Trying to
retrieve them may be a useful exercise for the reader.
In the Lagrangian II.9.34 we note the absence of any scalar
potential term. No Higgs phenomenon is possible. This is simply due
to the abelian character of the chosen gauge group. If '& were non-
abelian we would have to change both the Lagrangian and the solution
(II.9.20) of the Bianchi identities by terms proportional the gauge
coupling constant g.
The process of converting a theory of abelian vector multiplets
into a theory of non abelian ones, process popularly called "gauging",
will be studied in Chapter IV.7 in the context of local supersymmetry.
594
II.9.3 - The Action Principle for N=l, D=10 super Yang-Mills theory
In D=10, as we saw in previous chapters (II. 7 and II.8), the
gravitino 1-form can be chosen Majorana Weyl
С фТ = ф ; Гп ф = ф	(II.9.35)
and the Maurer-Cartan equations of flat N=1 superspace have the usual
form
Ra = DVa - 1 ф л Гаф = 0 (a = 0,1,...,9)	(II.9.36a)
R = 0	(II.9.36b)
p = Оф = 0	(11.9.36c)
in terms of such a gravitino.
We have not studied the representations of N=1 Poincard super-
symmetry in D=10 and therefore we ignored, up to the moment, which
multiplets do exist in such a dimension.
The existence of a certain representation can, however, be inferr-
ed if we are able to write Bianchi consistent rheonomic parametrizations
for a certain multiplet of fields. Before attempting any such deriva-
tion we must ascertain that we start from a multiplet having equal number
of on-shell Bose and Fermi degrees of freedom. Such a situation is rea-
lized by the couple (B,X) where B, being a Bose 1-form describes a
spin 1 particle and X, being a Majorana-Weyl spinor 0-form, describes
a spin 1/2 particle. The transverse group in D=10 is SO(D-2)= S0(8) and
an S0(8) vector has exactly 8 components like an S0(8) spinor. This
makes sure that В has as many on-shell degrees of freedom as X.
Taking both В and X in the adjoint representation of a group
we can write the field strength
.F = dB + В л В
(II.9.37)
595
together with the Bianchi identity
* ВлУ = 0
and we can check that
F . Va . Vb - 21ХГ ф л Vm
ab	m
DX = dX + [A,X] = Л Vm - — Г . ipFab
i ’ J m 4 abv
(11.9.38)
(II.9.39a)
(II.9.39b)
is a Bianchi consistent rheonomic parametrization which implies the
field equations:
DaF , = 0	; ГаЛ = 0
ab	a
(II.9.40)
This shows that В and X form
symmetry: the vector multiplet.
(II.9.39) as equations of motion
cedures and reads
a representation of N=1 D=10 super-
The rheonomic action which yields
can be constructed with the usual pro
д(%10) = |Tr{^ai\va\....v4
Y.M. J 1	ar..a1(
a- a«	a7	a1 M
+ 2iF 1 2 X Г ф л Vm л V 3 a .. . л V WE
IB	;
596
a....a,	a	a1n
+ Л Г 1	-----V 10e
m	avaio
1	-	1' ’	5
- 84i(^^B+4 ВлВл.В)флГ	ip^V л...л.У
3	a,	a.
(II.9.41)
In the derivation of (II.9.41) one makes heavy use of the Fierz identi-
ties discussed in Chapter II.8.
Crucial is the identity
A 4 . Гт & аФ=0	(II.9.42)
Г ' ’ 4
which the reader can deduce from Eq. (II.8.97) through multiplication
by ф. This identity implies that the 7-form
aj...a_
П=флГ ф л V л ... л V	(II.9.43)
al	a5
is closed
dfi = О	(II.9.44)
and is what guarantees that the last term in the action (11.9.41) is
gauge invariant. The coefficient of is the Chem-Simons form of
the gauge field playing an important role in D=10 Supergravity and in
the cancellation of anomalies in String Theory. As we see it emerges
at a very early stage in the rheonomic formulation of D=10 Super Yang-
Mills and it is intimately related to the appearance of the closed form
in superspace. In Chapter! III.4 we relate the existence of to
the structure of D=10 free differential algebras and in PART SIX a lot
more about the Chem-Simons forms will be said.
597
HISTORICAL REMARKS AND REFERENCES FOR PART TWO
The birth of "supersymmetry" may be traced back to 1970-1971,
when the spinning string was invented by Neveu, Schwarz and Ramond. In
the Neveu-Schwarz model one finds an algebra of constraints which con-
tains both commutators and anticommutators [25] and is, therefore, a
super Lie algebra. It is an infinite dimensional one, being the super
extension of the Virasoro algebra to be discussed in Part six. Nowadays
it is interpreted as the algebra of "superconformal" transformations in
2-dimensions.
The discovery of the 4-dimensional super Poincare algebra came
shortly after. First, in chronological order, comes the 1971 paper by
Gol'fand and Likhtman [17] followed by the 1974 paper of Wess and
Zumino [31] who, unaware of the russian results, discovered the field
theoretic model of Chapter II.6 and who can be correctly regarded as
the initiators of the whole field of supersymmetry. One should also
quote the papers by Volkov and Akulov who, prior to Wess and Zumino,
found a model possessing a non linear supersymmetry [30].
The notion of superspace and of superfields was also introduced
in the years 1974-1975 which witnessed the development of globally
supersymmetric field theories.
Salam and Strathdee considered the construction of unitary irre-
ducible representations of supersymmetry [27] and arrived at the notion
of Poincare supermultiplets (discussed in Chapter II.4) and at the con-
cept of superspace (see Chapters II.2 and II.3).
598
The supersymmetric extension of the Yang-Mills theories, discussed
in Chapter II.9, is due to Ferrara and Zumino [в] and to Salam and
Strathdee [28]. The first attempts at the development of a supersymme-
tric theory of weak and electromagnetic interactions, were made by
Fayet who introduced a supersymmetric SU(2) x U(l) model [б]. In 1976,
the same year supergravity was discovered, leading to an impressive
number of new developments, Fayet and Ferrara wrote a Physics Report [7]
summarizing the results obtained in global supersymmetry, which were
already quite respectable.
i)	The techniques of superfields had been applied with success to
perturbative quantum calculations. This resulted in the algorithm of
supergraphs which subsequently underwent an impressive development and
was used to prove many of the miracolous quantum cancellations implied
by supersymmetry. On this subject, which we do not touch, we quote the
lectures by Grisaru at the Trieste schools [18,19,20] and the book on
superspace by Gates, Grisaru, Rocek and Siegel [15].
ii)	The problem of spontaneous breakdown of supersymmetry, to be
satisfactorily solved only in the context of supergravity (see Chapter
IV.5), had already been posed and various ill-fated attempts had been
made at identifying the neutrino with the Goldstone fermion.
iii)	The basis of a supersymmetry phenomenology, which was fully
developed only after 1981, when the general N=1 Supergravity matter
couplings were available (see Chapters IV.1-IV.5) had anyhow been laid
down.
The mathematical classification of simple super algebras discussed
in chapter II.2 is due to Freund and Kaplanski [13] and to Scheunert,
Nahm and Rittenberg [29]. This is also a 1976 result. Furthermore in
1977 Nahm wrote a very important paper [24] where he classified the
supersymmetry multiplets in all dimensions and discovered that the first
levels of the spinning string formed supersymmetric multiplets. Such a
599
result further extended by Gliozzi, Olive and Scherk [16] was to have a
big influence on the development of superstring theory.
Finally we should mention the fundamental 1976 paper by Haag,
Lopuszanski and Sohnius [22] where the following result was established.
In 4-dimensions the largest possible symmetry of the S-matrix is not the
direct product of the Poincare group with some internal group, as it was
claimed by a famous theorem due to Coleman and Mandula [4] rather it is
the superconformal algebra SU(2,2/N). The key point in overcoming the
no go theorem is precisely the "super" character of the symmetry algebra.
Allowing for anticommutators besides commutators one can effectively
unify space-time symmetries with internal ones. This possibility has
been one of the main motivation for supersymmetry. As one can see, the
algebraic development of this new symmetry principle, which was the main
topic of PART TWO, is essentially a yield of the years 1974-1976.
Let us now discuss the literature related to other topics also
discussed in PART TWO.
The systematic derivation of Poincare supermultiplets given in
Chapter 11.4 follows closely the exposition by Ferrara and Savoy at the
'81 Trieste School [9] . Similarly Chapter II.5, discussing anti de
Sitter supermultiplets, is mainly based on Nicolai's lectures at the
'84 Trieste School [26]. The study of the unitary irreducible represen-
tations of the anti de Sitter group SO(2,3) has been a programme
carried through by Fronsdal and collaborators over an almost twenty
years period beginning in 1965 [14].
The supersymmetric extension of his work is extremely recent. In
1982 Heidenreich derived the Osp(4/1) multiplets [23] and in 1984 Nicolai
and Freedmann studied the multiplet shortening phenomenon for 0sp(4/N)
[12]. General techniques to construct representations were developed by
Gunaydin and Saclioghu [21] while a study of the Osp(4/2) multiplets was
given by Ceresole, Fre and Nicolai [з], following the analogous treat-
ment of the Osp(4/8) multiplets performed by Biran, Casher, Englert,
Nicolai, Freedman and Spindel [12,1].
The criteria for stability in anti de Sitter space (see Chapter
II.5) have been derived by Breitenlohner and Freedman in their very
important 1982 paper [2].
600
As far as the whole of the rheonomy approach is concerned this is
a contribution of the Torino group (Regge, D'Auria, Fre], In particular
the development of the rheonomy approach for super Yang-Mills theories
is due to Fre [10,11].
The N=1 0=10 super Yang-Mills model was obtained as field theore-
tic limit of the open spinning string by Gliozzi, Olive and Scherk [1б].
Its rheonomic formulation is due to da Silva, D’Auria and Fre [5].
601
PART TWO REFERENCES
[1]	B. Biran - A. Casher - F. Englert - M. Rooman - P. Spindel:
Phys. Lett. 134B (1984) 179.
[2]	P. Breitenlohner - D.Z. Freedman: Stability in Gauged Extended
Supergravity - Annals of Physics 144 (1982) 249.
[з]	A. Ceresole - P. Fre - H. Nicolai: Multiplet structure and spec-
tra of N=2 supersymmetric compactifications - Classical and
Quantum Grav. 2^ (1985) 133.
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