Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Mathematical physics --- Supermanifolds (Mathematics)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Supermanifolds (Mathematics) --- Mathematical physics.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Field theory (Physics) --- Global differential geometry --- String models --- Supermanifolds (Mathematics) --- Congresses. --- Congresses. --- Congresses. --- Congresses.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Supermanifolds (Mathematics) --- Cosmology --- Gravitational fields --- Singularities (Mathematics) --- Space and time --- Topology --- Congresses. --- Cosmology. --- Gravitation --- Space-time model

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1/1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them. This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory.

Global differential geometry. --- Differential Geometry. --- Mathematical Physics. --- Quantum Field Theories, String Theory. --- Geometry, Differential --- Supermanifolds (Mathematics) --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Differentiable manifolds --- Differential geometry. --- Mathematical physics. --- Quantum field theory. --- String theory. --- Models, String --- String theory --- Nuclear reactions --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Physical mathematics --- Physics --- Differential geometry --- Mathematics --- Geometry, Differential. --- Elementary particles (Physics). --- Elementary Particles, Quantum Field Theory. --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics