Chern-Weil homomorphism in twisted equivariant cohomology

TL;DR

This paper extends the Chern-Weil homomorphism to twisted equivariant cohomology, clarifies the role of coefficients, and explores applications to Courant algebroids and generalized complex structures with Hamiltonian group actions.

Contribution

It introduces an extension of the Chern-Weil homomorphism to twisted equivariant cohomology and clarifies the necessary coefficient conditions for cohomology theory.

Findings

Extended Chern-Weil homomorphism to twisted equivariant cohomology.

Identified coefficients as completed polynomial algebra over dual Lie algebra.

Connected equivariant cohomology of Courant algebroids with generalized complex structures.

Abstract

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of $G$. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite dimensional approximations of the Borel construction $M\times_G EG_k$, whenever the generalized complex manifold $M$ possesses a Hamiltonian $G$ action.