From loop spaces and connections to Gauss-Manin

We explain how the formalism of "loop spaces and connections" in derived algebraic geometry produces a Gauss-Manin connection on periodic cyclic homology that is manifestly Morita invariant, flat at the chain level, and satisfies Griffiths transversality. More precisely, we show that given a dualisable dg category C in characteristic zero, periodic cyclic homology of C admits a natural structure of filtered D-module on the classifying space of infinitesimal automorphisms of the category C. This is part of joint work with Nick Rozenblyum.