Equivariant extensions of differential forms for non-compact Lie groups

TL;DR

This paper explores the relationship between Cartan and equivariant cohomology for manifolds with Lie group actions, extending Witten's anomaly results to non-compact groups using spectral sequences and equivariant De Rham complexes.

Contribution

It generalizes Witten's anomaly equivalence to non-compact Lie groups by connecting Cartan and equivariant cohomology via spectral sequences and equivariant De Rham complexes.

Findings

Cohomology of the Cartan complex is on the 0-th row of a spectral sequence converging to equivariant cohomology.

Generalization of Witten's result to special linear groups with real coefficients.

Establishment of a link between anomaly absence and equivariant extensions in non-compact settings.

Abstract

Consider a manifold endowed with the action of a Lie group. We study the relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and we show that the cohomology of the Cartan complex lies on the 0-th row of the second page of a spectral sequence converging to the equivariant cohomology. We use this result to generalize a result of Witten on the equivalence of absence of anomalies in gauge WZW actions on compact Lie groups to the existence of equivariant extension of the WZW term, to the case on which the gauge group is the special linear group with real coefficients.