Supermanifolds

The following sections are included:

• Preface

• Contents

This book contains an account of the notions involved in constructing a theory of supermanifolds and the associated machinery and techniques of differential geometry, together with applications to various areas of physics, including supersymmetry and the quantization of systems with symmetry, and to classical geometry…

The following sections are included:

• The definition of a super algebra

• Homomorphisms and modules of super algebras

• Super matrices

• Super Lie algebras and super Lie modules

The following sections are included:

• Real Grassmann algebras

• The topology of superspace

• Complex Grassmann algebras

• Further super matrices

The following sections are included:

• Superdifferentiation and finite-dimensional Grassmann algebras

• Taylor expansion and Grassmann analytic continuation

• Supersmooth functions on

• Properties of supersmooth functions

• Other infinite-dimensional algebras

• Obtaining well defined odd derivatives with finite-dimensional Grassmann algebras

• The inverse function theorem

• Partitions of unity

• Superholomorphic functions of complex Grassmann variables

The following sections are included:

• G^∞ DeWitt supermanifolds

• The topology of supermanifolds

• More general supermanifolds

• The body of a supermanifold

• Complex supermanifolds

The following sections are included:

• G^∞ functions on supermanifolds

• Functions between supermanifolds

• Tangent vectors

• Vector fields

• Induced maps and integral curves

The following sections are included:

• Algebro-geometric supermanifolds

• Local coordinates on algebro-geometric supermanifolds

• Maps between algebro-geometric supermanifolds

The following sections are included:

• The construction of a split supermanifold from a vector bundle

• Batchelor's structure theorem for ( DeWitt, G^∞) supermanifolds

• A non-split complex supermanifold

• Comparison of the algebro-geometric and concrete approach

The following sections are included:

• The definition of a super Lie group

• Examples of super Lie groups

• The construction of a super Lie group with given super Lie ℝ-_s[L] -module

• The super Lie groups which correspond to a given super Lie algebra

• Super Lie groups and the algebro-geometric approach to supermanifolds

• Super Lie group actions and the exponential map

The following sections are included:

• Tensors

• Berezinian densities

• Exterior differential forms

• Super forms

The following sections are included:

• Integration with respect to anti commuting variables

• Integration on

• Integration on compact supermanifolds

• Rothstein's theory of integration on non-compact supermanifolds

• Voronov's theory of integration of super forms

• Integration on (1, 1)-dimensional supermanifolds

• Integration of exterior forms

The following sections are included:

• Fibre bundles

• The frame bundle and tensors

• Riemannian structures

• Even symplectic structures

• Odd symplectic structures

The following sections are included:

• Superfields and the superspace formalism

• Supergravity

• Super embeddings

The following sections are included:

• The superspace geometry of the spinning string

• The definition of a super Riemann surface

• The supermoduli space of super Riemann surfaces

• Contour integration on super Riemann surfaces

• Fields on super Riemann surfaces

The following sections are included:

• Path integrals and fermions

• Fermionic Brownian motion

• Brownian motion in superspace

• Stochastic calculus in superspace

• Brownian paths on supermanifolds

The following sections are included:

• Symplectic reduction

• BRST cohomology

• BRST quantization

• A topological example

The following sections are included:

• Supermanifolds and differential forms

• Supermanifolds and spinors

• Supersymmetric quantum mechanics and the Atiyah Singer Index theorem

• Further applications of supermanifolds

The following sections are included:

• Appendix A. Notation

• Bibliography

• Index