Torus fibrations and localization of index III

TL;DR

This paper advances the theory of index localization for Dirac-type operators on open manifolds with torus fiber structures, introducing equivariant localization methods and proving a key conjecture in symplectic geometry.

Contribution

It introduces two equivariant localization techniques for Dirac-type operators and applies them to prove Guillemin-Sternberg's quantization conjecture for torus actions.

Findings

Established two new equivariant localization methods.

Proved Guillemin-Sternberg's quantization conjecture in the torus action case.

Extended index theory to open manifolds with torus fiber structures.

Abstract

This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Dirac-type operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds. We introduce two equivariant versions of the localization. As an application we give a proof of Guillemin-Sternberg's quantization conjecture in the case of torus action.