On D-modules of categories I

TL;DR

This paper develops two methods to construct D-module structures on complexes computing periodic cyclic homology of stable infinity-categories over characteristic zero schemes, linking factorization homology, sheaves, Hochschild pairs, and dg Lie algebras.

Contribution

It introduces novel constructions of D-module structures on cyclic homology complexes using factorization homology and algebraic structures on Hochschild pairs.

Findings

Two distinct methods for D-module construction are provided.

Connections established between sheaves on free loop space and D-modules.

New insights into the algebraic and geometric structures underlying cyclic homology.

Abstract

In this paper, we will provide constructions of D-module structures on the complex computing the periodic cyclic homology of a stable infinity-category defined over a scheme of characteristic zero. We give two methods. The first one is based on a canonical extension of factorization homology to the mapping stack and relation between sheaves on free loop space and D-modules. The second one uses the algebraic structure on Hochschild pairs, its moduli-theoretic interpretation, Kodaira-Spencer morphisms, and the relation between dg Lie algebras and pointed formal stacks.