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

TL;DR

This paper develops a complex analytic equivariant cohomology theory based on double free loop spaces, connecting it to Grojnowski's elliptic cohomology and extending previous constructions to a holomorphic setting.

Contribution

It constructs a new complex analytic equivariant elliptic cohomology theory from double free loop spaces, linking it to Grojnowski's theory and employing an inverse limit approach.

Findings

The theory is defined on finite, torus-equivariant CW complexes.

The construction's fiber over a complex elliptic curve matches Grojnowski's elliptic cohomology.

The approach generalizes Kitchloo's method for single free loop spaces.

Abstract

We construct a complex analytic version of an equivariant cohomology theory which appeared in a recent paper of Rezk, and which is roughly modeled 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 $\mathcal{C}$.