Equivariant Chern Weil Forms and The Families Index Theorem

TL;DR

This paper extends the families index theorem to an equivariant setting by applying the equivariance --> families principle to Clifford module bundles with a compact Lie group action, providing new insights into equivariant index theory.

Contribution

It introduces an equivariant version of Bismut's families index theorem for Clifford module bundles under compact Lie group actions.

Findings

Established an equivariant families index theorem

Extended Bismut's theorem to differential Borel quotients

Provided new tools for equivariant geometric analysis

Abstract

We apply the equivariance --> families principle to a geometric family of Clifford module bundles with an action of a compact Lie group G to prove an equivariant version of Bismut's families index theorem on the differential Borel quotient of the geometric family.