Contents -- Preface -- 1. Introduction -- 2. Super algebras -- 2.1 The definition of a super algebra -- 2.2 Homomorphisms and modules of super algebras -- 2.3 Super matrices -- 2.4 Super Lie algebras and super Lie modules -- 3. Superspace -- 3.1 Real Grassmann algebras -- 3.2 The topology of superspace -- 3.3 Complex Grassmann algebras -- 3.4 Further super matrices -- 4. Functions of anticommuting variables -- 4.1 Superdifferentiation and nite-dimensional Grassmann algebras -- 4.2 Taylor expansion and Grassmann analytic continuation -- 4.3 Supersmooth functions on Rsm,n -- 4.4 Properties of supersmooth functions -- 4.5 Other in nite-dimensional algebras -- 4.6 Obtaining well defined odd derivatives with finitedimensional Grassmann algebras -- 4.7 The inverse function theorem -- 4.8 Partitions of unity -- 4.9 Superholomorphic functions of complex Grassmann variables -- 5. Supermanifolds: The concrete approach -- 5.1 G DeWitt supermanifolds -- 5.2 The topology of supermanifolds -- 5.3 More general supermanifolds -- 5.4 The body of a supermanifold -- 5.5 Complex supermanifolds -- 6. Functions and vector fields -- 6.1 G functions on supermanifolds -- 6.2 Functions between supermanifolds -- 6.3 Tangent vectors -- 6.4 Vector fields -- 6.5 Induced maps and integral curves -- 7. Supermanifolds: The algebro-geometric approach -- 7.1 Algebro-geometric supermanifolds -- 7.2 Local coordinates on algebro-geometric supermanifolds -- 7.3 Maps between algebro-geometric supermanifolds -- 8. The structure of supermanifolds -- 8.1 The construction of a split supermanifold from a vector bundle -- 8.2 Batchelor's structure theorem for (Rm -- n S , DeWitt, G ) supermanifolds -- 8.3 A non-split complex supermanifold -- 8.4 Comparison of the algebro-geometric and concrete approach -- 9. Super Lie groups -- 9.1 The definition of a super Lie group.

9.2 Examples of super Lie groups -- 9.3 The construction of a super Lie group with given super Lie RS[L]-module -- 9.4 The super Lie groups which correspond to a given super Lie algebra -- 9.5 Super Lie groups and the algebro-geometric approach to supermanifolds -- 9.6 Super Lie group actions and the exponential map -- 10. Tensors and forms -- 10.1 Tensors -- 10.2 Berezinian densities -- 10.3 Exterior differential forms -- 10.4 Super forms -- 11. Integration on supermanifolds -- 11.1 Integration with respect to anti commuting variables -- 11.2 Integration on Rsm n -- 11.3 Integration on compact supermanifolds -- 11.4 Rothstein's theory of integration on non-compact supermanifolds -- 11.5 Voronov's theory of integration of super forms -- 11.6 Integration on (1, 1)-dimensional supermanifolds -- 11.7 Integration of exterior forms -- 12. Geometric structures on supermanifolds -- 12.1 Fibre bundles -- 12.2 The frame bundle and tensors -- 12.3 Riemannian structures -- 12.4 Even symplectic structures -- 12.5 Odd symplectic structures -- 13. Supermanifolds and supersymmetric theories -- 13.1 Super fields and the superspace formalism -- 13.2 Supergravity -- 13.3 Super embeddings -- 14. Super Riemann surfaces -- 14.1 The superspace geometry of the spinning string -- 14.2 The definition of a super Riemann surface -- 14.3 The supermoduli space of super Riemann surfaces -- 14.4 Contour integration on super Riemann surfaces -- 14.5 Fields on super Riemann surfaces -- 15. Path integrals on supermanifolds -- 15.1 Path integrals and fermions -- 15.2 Fermionic Brownian motion -- 15.3 Brownian motion in superspace -- 15.4 Stochastic calculus in superspace -- 15.5 Brownian paths on supermanifolds -- 16. Supermanifolds and BRST quantization -- 16.1 Symplectic reduction -- 16.2 BRST cohomology -- 16.3 BRST quantization -- 16.4 A topological example.

17. Supermanifolds and geometry -- 17.1 Supermanifolds and differential forms -- 17.2 Supermanifolds and spinors -- 17.3 Supersymmetric quantum mechanics and the Atiyah Singer Index theorem -- 17.4 Further applications of supermanifolds -- Appendix A. Notation -- Bibliography -- Index.