Spectral Invarinats of Deformed Dirac Operators on Open G-Manifolds

Northeastern University, Boston MA

Investigators

Abstract

ABSTRACT: DMS 0204421. In this project we will continue to study the deformed equivariant Dirac operators on open manifolds. We will use the nice spectral properties of these operators to construct new invariants of manifolds. In particular, we introduce the regularized cohomology of equivariant holomorphic vector bundles over open Kaehler manifolds. One of the goals of the project is to extend the vanishing theorems and the semi-continuity theorem to non-compact setting. The applications will include the non-compact versions of Witten's holomorphic Morse inequalities and of the formula for the cohomology of the Mumford quotient, as well as new proofs of these results in the compact case. We also introduce an analogue of the Quillen metric on the determinant of the regularized cohomology and are planning to study this metric. This will shed a new light on the properties of the Quillen metric on compact manifolds. In particular, in a joint project with A. Abanov we suggest a mathematically rigorous description of the non-linear sigma-model describing the superconductivity. Jointly with P.-E. Paradan we a planning to use the deformed Dirac operator in the study of discrete series representations of Lie groups. The elliptic operators on compact manifolds have very nice properties. In this project we introduce a class of operators on non-compact manifolds with similar properties. The study of these operators not only leads to a generalization of many theorems from compact manifolds to non-compact ones, but also provides new results and methods in the theory of compact manifolds. The applications of these results and methods range from the mathematical theory of superconductivity to representation theory and complex geometry.