Loop spaces and connections

ISSN： 1753-8416

Source： Journal of Topology, Vol.5, Iss.2, 2012-06, pp. : 377-430

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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 _X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover _X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration S³ S²S¹, and relate S²-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their S³-equivariance.