  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.
