A construction of complex analytic elliptic cohomology from double free loop spaces. (26th August 2021)

• Record Type: Journal Article
• Authors: Spong, Matthew
• Is Part Of: Compositio mathematica. Volume 157:Number 8(2021)
• Page Start: 1853
• Page End: 1897
• Publication Date: 2021-08-26
• Subjects: elliptic cohomology -- double loop spaces -- double loop groups; 55N34 -- 22E67; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007363
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\mathcal {C}$ .