A construction of complex analytic elliptic cohomology from double free loop spaces

Abstract

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\\mathcal {C}$, the fiber of our construction over $\\mathcal {C}$ is isomorphic to Grojnowski s equivariant elliptic cohomology theory associated to $\\mathcal {C}$.