Equivariant Elliptic Cohomology and Mapping Stacks I

TL;DR

This paper introduces a novel cohomology theory called elliptic Hochschild homology for stacks, explores its properties, computes examples, and connects it to Grojnowski's equivariant elliptic cohomology over complex quotient stacks.

Contribution

It develops elliptic Hochschild homology, establishes its fundamental properties, and links it to existing equivariant elliptic cohomology theories.

Findings

Defined elliptic Hochschild homology for stacks.

Computed examples demonstrating the theory.

Connected the periodic cyclic version to Grojnowski's equivariant elliptic cohomology.

Abstract

We introduce a new cohomology theory for stacks called elliptic Hochschild homology, prove some fundamental properties and compute it in some classes of examples. We then introduce its periodic cyclic version and show that, over the complex numbers and for a quotient stack, this recovers Grojnowski's equivariant elliptic cohomology of the analytification.