A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras

TL;DR

This paper develops a geometric method to construct cyclic cocycles on twisted convolution algebras associated with gerbes over discrete translation groupoids, extending prior work to non-proper actions.

Contribution

It introduces a simplicial approach and a JLO formula-based morphism to compute cyclic cocycles for twisted convolution algebras without invariant connections.

Findings

Constructed cyclic cocycles using simplicial forms and JLO formula

Extended cyclic cohomology methods to non-proper groupoid actions

Linked gerbe data to cyclic cohomology via new geometric constructions

Abstract

In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper \'etale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups. Our focus will be the convolution algebra with a product defined by a gerbe over a discrete translation groupoid. When the action is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial notions related to ideas of J. Dupont to construct a simplicial form representing the Dixmier-Douady class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix algebras. Finally, we define a morphism from this complex to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.