Differential Borel equivariant cohomology via connections

Understand

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.