Overview

This is an updated and expanded second edition of a successful and well-reviewed text presenting 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 (Grassman variables), Berezin integration, supervector spaces and the superdeterminant. This basic material is then applied to the theory of supermanifolds, with an account of super-analogs of Lie derivatives, connections, metric, 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 book then turns to applications. Chapter 5 contains an account of the Peierals bracket for superclassical dynamical systems, super Hilbert spaces, path integration for fermionic quantum systems, and simple models of Bose–Fermi supersymmetry. The sixth and final chapter, which is new in this revised edition, examines dynamical systems for which the topology of the configuration supermanifold is important. A concise but complete account is given of the pathintegral derivation of the Chern–Gauss–Bonnet formula for the Euler–Poincaré characteristic of an ordinary manifold, which is based on a simple extension of a point particle moving freely in this manifold to a supersymmetric dynamical system moving in an associated supermanifold. Many exercises are included to complement the text.

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9780511564000

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'Supermanifolds is destined to become the standard work for all serious study of super-symmetric theories of physics.' Nature ...destined to become the standard work for all serious study of super-symmetric theories of physics. Nature The material is not presented in the manifestly rigorous style of modern mathematics, but in a style which in my opinion is entirely appropriate for the presentation of serious new mathematics to theoretical physicists; explanations are always admirably lucid, and the motivation for each step is clearly described. Alice Rogers, Mathematical Reviews I recommend the book to those hardy souls who wish to venture out into the stormy waters of current particle theory. Arthur S. Wightman, Physics Today

'Supermanifolds is destined to become the standard work for all serious study of super-symmetric theories of physics.' Nature

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