Periodic Cyclic Homology and Equivariant Gerbes

Abstract

This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $\alpha \in H^3_G (M, {\mathbb Z})$ we introduce a notion of localized equivariant twisted cohomology $H^\bullet ({\bar{\Omega}}^\bullet (M, G, L)_g, d^\alpha_{G^g})$, indexed by $g\in G$. We prove that there exists a natural family of chain maps, indexed by $g\in G$, inducing a family of morphisms from the equivariant periodic cyclic homology $HP^G_\bullet ( C^\infty (M, \alpha ) )$, where $C^\infty (M, \alpha )$ is a certain smooth algebra constructed from an equivariant bundle gerbe defined by $\alpha \in H^3_G (M,{\mathbb Z} )$, to $H^\bullet ( {\bar{\Omega}}^\bullet (M, G, L)_g, d^\alpha_{G^g})$. We formulate a conjecture of Atiyah-Hirzebruch type theorem for equivariant twisted $K$-theory.