Determinants, analytic torsion, and functional analytic models for Dirac operators

University Of Arizona, Tucson AZ

Investigators

Abstract

DMS-0072551 Matthias Lesch Dirac type operators are symmetric first order elliptic differential operators arising naturally in Riemannian geometry. They are of considerable interest in theoretical physics. Given such an operator on a compact manifold with boundary it is possible to characterize those boundary conditions which lead to a self--adjoint elliptic problem. These so--called well--posed boundary value problems were introduced by R. T. Seeley. A recent account from a functional analytic perspective was given by the researcher in collaboration with J. Br\"uning. This is the starting point of the current project in which the researcher wants to investigate spectral invariants associated to Dirac operators on manifolds with boundary. More specifically, we want to present an analytic proof of the gluing formula for the analytic torsion in the presence of an arbitrary flat bundle. As a consequence one obtains an alternative approach to Cheeger--Muller type theorems on the relation between analytical and topological torsion in the most general setting, including manifolds with boundary. An invariant related to the analytic torsion is the zeta-regularized determinant. We want to study this determinant as a function of the boundary condition, generalizing earlier work of Burghelea-Friedlander--Kappeler and Scott-Wojciechowski. The third subproject, jointly with J. Bruening, deals with a functional analytic approach to boundary value problems for Dirac type operators. The ultimate goal is to find a simple functional analytic model for a Dirac type operator on a manifold with boundary which allows to derive basic results like heat trace expansions, index theorems and the spectral flow theorem. Such a model would hopefully lead to such theorems for manifolds with corners. Dirac type operators are symmetric first order elliptic differential operators arising naturally in Riemannian geometry. They are of considerable interest in theoretical physics. There are interesting relations between the spectrum of the operator (the 'measurable' quantities of the physical system) and the underlying geometry. In general the spectrum of the operator is hard to compute. However, it is possible to extract certain spectral invariants which are intimately related to the geometry. If the underlying geometry is a manifold with boundary then, at first, one has to impose boundary conditions in order to get a self--adjoint problem and hence a well--defined spectrum. In the current project the author wants to investigate certain spectral invariants (analytic torsion, determinant) for such boundary value problems. A special issue is the dependence of the invariants on the choice of the boundary condition.