An alternate description of equivariant connections

TL;DR

This paper establishes an equivalence between the category of bundles with connection on the differential quotient stack and G-equivariant bundles with G-invariant connection on a manifold, providing a new perspective on equivariant connections.

Contribution

It introduces an alternative description of equivariant connections by relating differential quotient stacks to G-equivariant bundles with invariant connections.

Findings

Proves the equivalence of categories between bundles on quotient stacks and G-equivariant bundles.

Provides a new framework for understanding equivariant connections in differential geometry.

Enhances the theoretical foundation for studying connections in the presence of symmetries.

Abstract

In this note, we consider a Lie group G acting on a manifold M. We prove that the category of bundles with connection on the differential quotient stack is equivalent to the category of G-equivariant bundles on M with G-invariant connection.