Cyclic cocycles on twisted convolution algebras

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.

💡 Research Summary

The paper addresses the problem of constructing cyclic cocycles for convolution algebras twisted by gerbes over discrete translation groupoids, a setting where the usual approach based on invariant connections fails because the groupoid is not proper. In the proper étale case, Tu and Xu (and earlier Mathai‑Stevenson) constructed a map from the periodic cyclic cohomology of a gerbe‑twisted convolution algebra to twisted cohomology groups using an invariant connection on the gerbe. When the groupoid is improper, such a connection does not exist, and the standard Jaffe‑Lesniewski‑Osterwalder (JLO) formula cannot be applied directly.

To overcome this obstacle, the author develops a simplicial framework. The nerve of the translation groupoid (M\rtimes\Gamma) yields a simplicial manifold ((M\rtimes\Gamma)\bullet). On this simplicial space the author works with Dupont’s complex of compatible differential forms (\Omega^\ast((M\rtimes\Gamma)\bullet)), which is a bicomplex mixing de Rham differentials on the base manifolds with simplicial differentials on the standard simplices. By introducing a formal variable (u) (of degree (-2)) the author defines a “twisted” simplicial complex ((\Omega^\ast((M\rtimes\Gamma)_\bullet)