Supermanifolds: theory and applications

A. Rogers9810212283, 9789810212285, 9789812708854

This book aims to fill the gap in the available literature on supermanifolds, describing the different approaches to supermanifolds together with various applications to physics, including some which rely on the more mathematical aspects of supermanifold theory. The first part of the book contains a full introduction to the theory of supermanifolds, comparing and contrasting the different approaches that exist. Topics covered include tensors on supermanifolds, super fibre bundles, super Lie groups and integration theory. Later chapters emphasise applications, including the superspace approach to supersymmetric theories, super Riemann surfaces and the spinning string, path integration on supermanifolds and BRST quantization.

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Table of contents :
Contents……Page 8
Preface……Page 7
1. Introduction……Page 12
2.1 The definition of a super algebra……Page 18
2.2 Homomorphisms and modules of super algebras……Page 20
2.3 Super matrices……Page 23
2.4 Super Lie algebras and super Lie modules……Page 24
3.1 Real Grassmann algebras……Page 28
3.2 The topology of superspace……Page 32
3.3 Complex Grassmann algebras……Page 34
3.4 Further super matrices……Page 36
4. Functions of anticommuting variables……Page 42
4.1 Superdifferentiation and nite-dimensional Grassmann algebras……Page 44
4.2 Taylor expansion and Grassmann analytic continuation……Page 46
4.3 Supersmooth functions on Rsm,n……Page 49
4.4 Properties of supersmooth functions……Page 51
4.5 Other in nite-dimensional algebras……Page 55
4.6 Obtaining well defined odd derivatives with finitedimensional Grassmann algebras……Page 56
4.7 The inverse function theorem……Page 58
4.8 Partitions of unity……Page 60
4.9 Superholomorphic functions of complex Grassmann variables……Page 61
5. Supermanifolds: The concrete approach……Page 62
5.1 G DeWitt supermanifolds……Page 63
5.2 The topology of supermanifolds……Page 66
5.3 More general supermanifolds……Page 67
5.4 The body of a supermanifold……Page 70
5.5 Complex supermanifolds……Page 72
6. Functions and vector fields……Page 74
6.1 G functions on supermanifolds……Page 75
6.2 Functions between supermanifolds……Page 78
6.3 Tangent vectors……Page 80
6.4 Vector fields……Page 85
6.5 Induced maps and integral curves……Page 89
7.1 Algebro-geometric supermanifolds……Page 96
7.2 Local coordinates on algebro-geometric supermanifolds……Page 98
7.3 Maps between algebro-geometric supermanifolds……Page 100
8. The structure of supermanifolds……Page 102
8.1 The construction of a split supermanifold from a vector bundle……Page 103
8.2 Batchelor’s structure theorem for (Rm;n S , DeWitt, G ) supermanifolds……Page 105
8.3 A non-split complex supermanifold……Page 107
8.4 Comparison of the algebro-geometric and concrete approach……Page 109
9. Super Lie groups……Page 112
9.1 The definition of a super Lie group……Page 113
9.2 Examples of super Lie groups……Page 115
9.3 The construction of a super Lie group with given super Lie RS[L]-module……Page 118
9.4 The super Lie groups which correspond to a given super Lie algebra……Page 125
9.5 Super Lie groups and the algebro-geometric approach to supermanifolds……Page 129
9.6 Super Lie group actions and the exponential map……Page 132
10.1 Tensors……Page 136
10.2 Berezinian densities……Page 137
10.3 Exterior differential forms……Page 138
10.4 Super forms……Page 142
11. Integration on supermanifolds……Page 146
11.1 Integration with respect to anti commuting variables……Page 147
11.2 Integration on Rsm n……Page 151
11.3 Integration on compact supermanifolds……Page 155
11.4 Rothstein’s theory of integration on non-compact supermanifolds……Page 157
11.5 Voronov’s theory of integration of super forms……Page 163
11.6 Integration on (1, 1)-dimensional supermanifolds……Page 165
11.7 Integration of exterior forms……Page 166
12.1 Fibre bundles……Page 168
12.2 The frame bundle and tensors……Page 171
12.3 Riemannian structures……Page 172
12.4 Even symplectic structures……Page 173
12.5 Odd symplectic structures……Page 175
13. Supermanifolds and supersymmetric theories……Page 178
13.1 Super fields and the superspace formalism……Page 181
13.2 Supergravity……Page 186
13.3 Super embeddings……Page 189
14. Super Riemann surfaces……Page 192
14.1 The superspace geometry of the spinning string……Page 193
14.2 The definition of a super Riemann surface……Page 195
14.3 The supermoduli space of super Riemann surfaces……Page 197
14.4 Contour integration on super Riemann surfaces……Page 200
14.5 Fields on super Riemann surfaces……Page 202
15.1 Path integrals and fermions……Page 206
15.2 Fermionic Brownian motion……Page 208
15.3 Brownian motion in superspace……Page 210
15.4 Stochastic calculus in superspace……Page 212
15.5 Brownian paths on supermanifolds……Page 214
16. Supermanifolds and BRST quantization……Page 218
16.1 Symplectic reduction……Page 219
16.2 BRST cohomology……Page 222
16.3 BRST quantization……Page 224
16.4 A topological example……Page 226
17. Supermanifolds and geometry……Page 232
17.1 Supermanifolds and differential forms……Page 233
17.2 Supermanifolds and spinors……Page 235
17.3 Supersymmetric quantum mechanics and the Atiyah Singer Index theorem……Page 237
17.4 Further applications of supermanifolds……Page 244
Appendix A. Notation……Page 248
Bibliography……Page 250
Index……Page 260