A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras

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.

💡 Research Summary

The thesis presents a comprehensive geometric construction of cyclic cocycles for convolution algebras twisted by gerbes over discrete translation groupoids. The problem originates from the fact that Connes’ map from equivariant cohomology to periodic cyclic cohomology, as well as the Tu‑Xu map for gerbe‑twisted algebras, rely on the existence of an invariant connection on the gerbe. When the groupoid action is not proper, such a connection cannot be built, leaving the cyclic theory of the twisted algebra inaccessible by classical means.

To overcome this obstacle, the author adopts a simplicial approach inspired by Dupont’s simplicial de Rham complex. The nerve of the translation groupoid, denoted NG, is a simplicial manifold whose n‑simplices are given by Gⁿ × M. On each simplex a differential form of degree three, η, is constructed from local data (2‑forms B_i and 1‑forms A_{ij}) that satisfy the usual Čech‑de Rham compatibility conditions. The closed form η represents the Dixmier‑Douady class of the gerbe and serves as a “twisting” element in the simplicial de Rham complex (Ω^*(NG), d + η∧).

With η in hand, the author generalizes the Jaffe‑Lévy‑Oliveira (JLO) formula, which originally produces cyclic cocycles from a Dirac‑type operator D on a smooth algebra. Here D is defined on matrix algebras of smooth functions on Δⁿ × M, and the JLO-type functional is modified to incorporate η: \