Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fiber bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics.
Author(s): Josi A. de Azcárraga, Josi M. Izquierdo
Series: Cambridge monographs on mathematical physics
Publisher: Cambridge University Press
Year: 1998
Language: English
Pages: 474
City: Cambridge [England]; New York, NY, USA
Cover......Page 1
Title......Page 6
Copyright......Page 7
Contents ......Page 10
Preface ......Page 14
1 Lie groups, fibre bundles and Cartan calculus ......Page 20
1.1 Introduction: Lie groups and actions of a Lie group on a manifold ......Page 21
1.2 Left- (X') and right- (XR) invariant vector fields on a Lie group G ......Page 27
1.3 A summary of fibre bundles ......Page 31
1.4 Differential forms and Cartan calculus: a review ......Page 50
1.5 De Rham cohomology and Hodge-de Rham theory ......Page 62
1.6 The dual aspect of Lie groups: invariant differential forms. Invariant integration measure on G ......Page 73
1.7 The Maurer-Cartan equations and the canonical form on a Lie group G. Bi-invariant measure ......Page 77
1.8 Left-invariance and bi-invariance. Bi-invariant metric tensor field on the group manifold ......Page 81
1.9 Applications and examples for Lie groups ......Page 84
1.10 The case of super Lie groups: the supertranslation group as an example ......Page 89
1.11 Appendix A: some homotopy groups ......Page 92
1.12 Appendix B: the Poincare polynomials of the compact simple groups ......Page 97
Bibliographical notes for chapter 1 ......Page 100
2.1 Connections on a principal bundle: an outline ......Page 103
2.2 Examples of connections ......Page 114
2.3 Equivariant forms on a Lie group ......Page 120
2.4 Characteristic classes ......Page 123
2.5 Chern classes and Chern characters ......Page 130
2.6 Chern-Simons forms of the Chern characters ......Page 134
2.7 The magnetic monopole ......Page 139
2.8 Yang-Mills instantons ......Page 145
2.9 Pontryagin classes and the Euler class ......Page 150
2.10 Index theorems for manifolds without boundary ......Page 156
2.11 Index theorem for the spin complex. Twisted complexes ......Page 163
Bibliographical notes for chapter 2 ......Page 168
3.1 Some known facts of 'non-relativistic' mechanics: two-cocycles ......Page 171
3.2 Projective representations of a Lie group: a review of Bargmann's theory ......Page 178
3.3 The Weyl-Heisenberg group and quantization ......Page 182
3.4 The extended Galilei group ......Page 187
3.5 The two-cocycle ambiguity and the Bargmann cocycle for the Galilei group ......Page 189
3.6 Dynamical groups and symplectic cohomology: an introduction ......Page 192
3.7 The adjoint and the coadjoint representations of the extended Galilei group and its algebra ......Page 200
3.8 On the possible failure of the associative property: three-cocycles ......Page 205
3.9 Some remarks on central extensions ......Page 208
3.10 Contractions and group cohomology ......Page 211
Bibliographical notes for chapter 3 ......Page 216
4.1 Exact sequences of group homomorphisms ......Page 218
4.2 Group extensions: statement of the problem in the general case ......Page 220
4.3 Principal bundle description of an extension G(K, G) ......Page 226
4.4 Characterization of an extension through the factor system co ......Page 228
4.5 Group law for G in terms of the factor system ......Page 230
Bibliographical notes for chapter 4 ......Page 232
5.1 Cohomology of groups ......Page 234
5.2 Extensions G of G by an abelian group A ......Page 240
5.3 An example from supergroup theory: superspace as a group extension ......Page 242
5.4 F(M)-valued cochains: cohomology induced by the action of G on a manifold Mm ......Page 244
Bibliographical notes for chapter 5 ......Page 248
6.1 Cohomology of Lie algebras: general definitions ......Page 249
6.2 Extensions of a Lie algebra W by an abelian algebra , : HP (W, W) ......Page 253
6.3 Three cases of special interest ......Page 255
6.4 Local exponents and the isomorphism between Ho (G, R) and H0(S6,R) ......Page 257
6.5 Cohomology groups for semisimple Lie algebras. The Whitehead lemma ......Page 261
6.6 Higher cohomology groups ......Page 263
6.7 The Chevalley-Eilenberg formulation of the Lie algebra cohomology and invariant differential forms on a Lie group G ......Page 265
6.8 The BRST approach to the Lie algebra cohomology ......Page 268
6.9 Lie algebra cohomology vs. Lie group cohomology ......Page 272
Bibliographical notes for chapter 6 ......Page 281
7.1 The information contained in the G-kernel (K, o) ......Page 283
7.2 The necessary and sufficient condition for a G-kernel (K, a) to be extendible ......Page 287
7.3 Construction of extensions ......Page 289
7.4 Example: the covering groups of the complete Lorentz group L ......Page 294
7.5 A brief comment on the meaning of the higher cohomology groups H"(G,A) ......Page 298
Bibliographical notes for chapter 7 ......Page 299
8.1 A short review of the variational principle and of the Noether theorem in Newtonian mechanics ......Page 300
8.2 Invariant forms on a manifold and cohomology; WZ terms ......Page 305
8.3 Newtonian mechanics and Wess-Zumino terms: two simple examples ......Page 309
8.4 Cohomology and classical mechanics: preliminaries ......Page 316
8.5 The cohomological descent approach to classical anomalies ......Page 322
8.6 A simple application: the free Newtonian particle ......Page 327
8.7 The massive superparticle and Wess-Zumino terms for supersymmetric extended objects ......Page 330
8.8 Supersymmetric extended objects and the supersymmetry algebra ......Page 334
8.9 A Lagrangian description of the magnetic monopole ......Page 343
Bibliographical notes for chapter 8 ......Page 348
9.1 Introduction ......Page 350
9.2 The group of mappings G(M) associated with a compact Lie group G, and its Lie algebra W(M) ......Page 354
9.3 Current algebras as infinite-dimensional Lie algebras ......Page 357
9.4 The Kac-Moody or untwisted affine algebra ......Page 359
9.5 The Virasoro algebra ......Page 364
9.6 Chevalley-Eilenberg cohomology on Diff S' and two-dimensional gravity ......Page 370
9.7 The conformal algebra ......Page 373
Bibliographical notes for chapter 9 ......Page 377
10.1 The group of gauge transformations and the orbit space of Yang-Mills potentials ......Page 379
10.2 Theory with Dirac fermions: the abelian anomaly and the index theorem ......Page 385
10.3 The action of gauge transformations on the space of functionals ......Page 391
10.4 Cohomology on G(M), 9(M) and its BRST formulation ......Page 393
10.5 Theory with Weyl fermions: non-abelian gauge anomalies and their path integral calculation ......Page 398
10.6 The non-abelian anomaly as a probe for non-trivial topology ......Page 406
10.7 The geometry of consistent non-abelian gauge anomalies ......Page 413
10.8 The cohomological descent in the trivial P (G, M) case: cochains and coboundary operators ......Page 417
10.9 The descent equations. Cocycles and coboundaries ......Page 422
10.10 A compact form for the gauge algebra descent equations ......Page 426
10.11 Specific results in D = 2, 4 spacetime dimensions ......Page 427
10.12 On the existence of consistent anomalous theories ......Page 435
10.13 Appendix: calculating Chern-Simons forms ......Page 437
Bibliographical notes for chapter 10 ......Page 439
List of symbols ......Page 443
References ......Page 448
Index ......Page 463
