Loop Spaces and Connections
Abstract
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 Omega S3 --> Omega S2 --> S1, and relate Omega S2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S3-equivariance.