Loop spaces and connections (Q2892846)

The authors study the geometry of loop spaces in derived algebraic geometry. They establish relation between equivariant sheaves on the free loop space of a stack and sheaves with connection on the stack. This may be viewed as categorification of the well-known relation in topology and algebra between equivariant homology of free loop spaces (cyclic homology) and topological cohomology (singular or de Rham cohomology). Section 1 contains an overview of the paper. In section 2 the authors review the basic notions of derived algebraic geometry and \(\infty\)-categories. Section 3 is devoted to derived analog of affinization and group actions on affine derived schemes. In section 4 the authors study equivariant sheaves on loop spaces of schemes. In section 5 the Koszul dualization of the previous results is provided. In section 6 the authors generalize their results to loop spaces of geometric stacks.