The equivariant index of twisted dirac operators and semi-classical limits

TL;DR

This paper investigates the asymptotic behavior of distributions associated with twisted Dirac operators on spin manifolds under group actions, revealing functorial properties for reduced spaces in the semi-classical limit.

Contribution

It introduces a new analysis of the semi-classical limits of distributions linked to twisted Dirac operators on possibly non-compact manifolds with group actions, and explores their functorial implications.

Findings

Asymptotic formulas for the distributions Theta(k) as k becomes large.

Extension of semi-classical analysis to non-compact spin manifolds.

Functorial relationships for reduced spaces derived from the asymptotic behavior.

Abstract

Consider a spin manifold M, equipped with a line bundle L and an action of a compact Lie group G. We can attach to this data a family Theta(k) of distributions on the dual of the Lie algebra of G. The aim of this paper is to study the asymptotic behaviour of Theta(k) when k is large, and M possibly non compact, and to explore a functorial consequence of this formula for reduced spaces.