Equivariant elliptic cohomology and the 2-loop groupoid

TL;DR

This paper constructs a complex-analytic G-equivariant elliptic cohomology theory for compact Lie groups, generalizing previous work and establishing its fundamental properties within a 2-dimensional torus bundle framework.

Contribution

It introduces a new construction of equivariant elliptic cohomology parametrized over the 2-torus groupoid, extending Grojnowski's approach and following Rezk's outline.

Findings

Provides a new complex-analytic G-equivariant elliptic cohomology construction

Proves fundamental properties of the constructed cohomology theory

Generalizes existing models to arbitrary compact Lie groups

Abstract

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.