Chern character, Hopf algebras, and BRS cohomology

Abstract

We present the construction of a Chern character in cyclic cohomology, involving an arbitrary number of associative algebras in contravariant or covariant position. This is a generalization of the bivariant Chern character for bornological algebras introduced in a previous paper, based on Quillen superconnections and heat-kernel regularization. Then we adapt the formalism to the cyclic cohomology of Hopf algebras in the sense of Connes-Moscovici. This yields a Chern character for ``equivariant K-cycles'' over a bornological algebra A, generalizing the Connes-Moscovici characteristic maps. In the case of equivariant spectral triples verifying some additional conditions, we also exhibit secondary characteristic classes. The latter are not related to topology but rather define characteristic maps for the higher algebraic K-theory of A. In the classical case of spin manifolds, we compute and interpret these secondary classes in terms of BRS cohomology in Quantum Field Theory.