arXiv subjects
Publications and source records attributed to Matthew Spong.
Following an outline of Rezk, we give a construction of complex-analytic $G$-equivariant elliptic cohomology for an arbitrary compact Lie group $G$ and we prove some of its fundamental properties. The construction is parametrised over the orbit category of the groupoid of principal $G$-bundles over 2-dimensional tori and generalises Grojnowski's construction of equivariant elliptic cohomology.
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 $\mathcal{C}$.
In 1994, Grojnowski gave a construction of an equivariant elliptic cohomology theory associated to an elliptic curve over the complex numbers. Grojnowski's construction has seen numerous applications in algebraic topology and geometric representation theory, however the construction is somewhat ad hoc and there has been significant interest in the question of its geometric interpretation. We show that there are two global models for Grojnowski's theory, which shed light on its geometric meaning. The first model is constructed as the Borel-equivariant cohomology of a double free loop space, and is a holomorphic version of a construction of Rezk from 2016. The second model is constructed as the loop group-equivariant K-theory of a free loop space, and is a slight modification of a construction given in 2014 by Kitchloo, motivated by ideas in conformal field theory. We investigate the properties of each model and show that each is isomorphic to Grojnowski's theory, for any given elliptic curve.