Categorified Chern character and cyclic cohomology

TL;DR

This paper explores the categorification of the Chern character via Hopf cyclic cohomology, connecting derived algebraic geometry, loop space geometry, and the development of new coefficient modules.

Contribution

It introduces mixed anti-Yetter-Drinfeld contramodules and links Hopf cyclic cohomology with categorified Chern character theory.

Findings

Defines mixed anti-Yetter-Drinfeld contramodules

Establishes a relationship between $S^1$-equivariant sheaves and $D_X$-modules

Provides a new framework for Hopf cyclic cohomology in categorified settings

Abstract

We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories. The cohomology is then obtained as a $Hom$ in this dg-category between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.