Cyclic cocycles on twisted convolution algebras

TL;DR

This paper develops methods to construct cyclic cocycles on twisted convolution algebras associated with gerbes over groupoids, extending to non-proper cases using simplicial techniques and a JLO formula.

Contribution

It introduces a novel approach using simplicial curvature forms and JLO formulas to study cyclic cohomology of twisted convolution algebras beyond proper groupoids.

Findings

Constructed cyclic cocycles on twisted convolution algebras

Extended techniques to non-proper groupoids using simplicial methods

Defined a morphism to compute periodic cyclic cohomology with a simplicial curvature form

Abstract

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras. The results in this article were originally published in the author's Ph.D. thesis.