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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 Ω  S  3 →Ω  S  2 → S  1 , and relate Ω  S  2 ‐equivariant sheaves on the loop space with sheaves on  X  with arbitrary connection, with curvature given by their Ω  S  3 ‐equivariance.

Loop spaces and connections