Dirac Operator and Spectral Geometry

Abstract

This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed. Elliptic boundary-value problems, index problems for closed manifolds and for manifolds with boundary, Bott periodicity and K-theory are then presented. This makes it clear why the Dirac operator is the most fundamental, in the theory of elliptic operators on manifolds. The topic of spectral geometry is developed by studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and heat-kernel asymptotics for operators of Laplace type on manifolds with boundary. The emphasis is put on the functorial method, which studies the behaviour of differential operators, boundary operators and heat-kernel coefficients under conformal rescalings of the background metric. In the second part, a number of relevant physical applications are studied: non-local boundary conditions for massless spin-1/2 fields, massless spin-3/2 potentials on manifolds with boundary, geometric theory of massive spin-3/2 potentials, local boundary conditions in quantum supergravity, quark boundary conditions, one-loop quantum cosmology, conformally covariant operators and Euclidean quantum gravity.