Equivariant Extensions of Differential Forms for Non-compact Lie Groups

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 gauged 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.2010 Mathematics Subject Classification. 57R91, 57T10, 81T40, 81T70 .

On the equivariant de Rham cohomology for non-compact Lie groups

“…In [5] Witten showed that the absence of anomalies in gauged WZW actions on compact Lie groups was equivalent to the existence of closed equivariant extension of the WZW term on the Cartan complex, further showing that the existence or absence of anomalies is purely topological. The arguments of Witten could be extended without trouble to the non-compact case (see [2,Chapter 4]), and together with the main result of this paper, we conclude that the absence or existence of anomalies is purely topological fact, independent of the compacity of the Lie group.…”