Localization of Dirac Operators on 4n+2 Dimensional Open Spin^c Manifolds

TL;DR

This paper introduces a topological index for Dirac operators on 4n+2 dimensional open Spin^c manifolds and relates it to the index of a Dirac operator on a characteristic submanifold through K-theory localization.

Contribution

It defines a new integer-valued topological index for Dirac operators on open Spin^c manifolds and establishes a localization formula linking it to submanifold indices.

Findings

Defined a topological index for Dirac operators on open Spin^c manifolds.

Established a relation between the index and the index on a characteristic submanifold.

Used K-class localization to connect the indices.

Abstract

An integer valued topological index of a Dirac operator is introduced for a pair of a 4n+2 dimensional open Spin^c manifold and a section of the determinant line bundle satisfying some property. We show a relation between the index and an index of a Dirac operator of its characteristic submanifold, by a localization of K-class.