A fixed point formula for loop group actions

TL;DR

This paper derives a fixed point formula to compute the index of the Dirac operator on symplectic quotients arising from Hamiltonian loop group actions, linking index theory with fixed point data.

Contribution

It introduces a novel fixed point formula for the Dirac operator index in the context of Hamiltonian loop group actions, extending classical fixed point theorems.

Findings

Index expressed in terms of fixed point data

Applicable to symplectic quotients of loop group manifolds

Provides a new computational tool for Dirac operators

Abstract

We express the index of the Dirac operator on symplectic quotients of a Hamiltonian loop group manifold with proper moment map in terms of fixed point data.