Differential Borel equivariant cohomology via connections

TL;DR

This paper develops a new framework for differential equivariant cohomology using connections on principal bundles, linking it to classical equivariant cohomology theories and the Chern-Weil homomorphism.

Contribution

It introduces a differential cohomology theory for quotient stacks with group actions, integrating connections and providing natural factorization of the Chern-Weil homomorphism.

Findings

Defines differential equivariant cohomology for quotient stacks

Establishes natural maps to equivariant forms and cohomology

Shows the Chern-Weil homomorphism factors through this new theory

Abstract

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern-Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.