Loop spaces and connections

David Nadler We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well-known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently DX-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover DX-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration ΩS3 → ΩS2 → S1, and relate ΩS2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their ΩS3-equivariance. 1