Loop Spaces and Connections

TL;DR

This paper explores the geometry of loop spaces in derived algebraic geometry, establishing a categorification of cyclic homology and relating equivariant sheaves on loop spaces to sheaves with flat or arbitrary connections.

Contribution

It provides a new categorification linking S^1-equivariant quasicoherent sheaves on loop spaces to sheaves with flat connections, extending the geometric understanding of cyclic homology.

Findings

Relates S^1-equivariant sheaves to sheaves with flat connections.

Recovers D_X-modules precisely via the Hodge filtration.

Connects Omega S^2-equivariant sheaves to sheaves with arbitrary curvature.

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 S^1-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 D_X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover D_X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S^3-equivariance.