Supermanifolds

This book presents a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the super-analogs of all the basic structures of ordinary manifold theory. The exposition opens with the theory of analysis over supernumbers (Grassmann variables), Berezin integration, supervector spaces and the superdeterminant. This basic material is then applied to the theory of supermanifolds, with an account of the super-analog of Lie derivatives, connections, metrics, curvature, geodesics, Killing flows, conformal groups, etc. The book goes on to discuss the theory of super Lie groups, super Lie algebras, and invariant geometrical structures on coset spaces. Complete descriptions are given of all the simple super Lie groups. The final chapter contains an account of the Peierls bracket for superclassical dynamical systems, super Hilbert spaces, path integration for fermionic quantum systems, and simple models of Bose-Fermi supersymmetry. Many exercises are included to supplement the material in the text.