# A de Rham model for complex analytic equivariant elliptic cohomology

## TL;DR

This paper develops a new cocycle model for complex analytic equivariant elliptic cohomology, connecting it with loop group representations and providing a refined orientation theory for elliptic cohomology.

## Contribution

It introduces a de Rham-based cocycle model that refines existing theories and relates elliptic classes to positive energy representations and the MString orientation.

## Key findings

- Constructed a cocycle model refining Grojnowski and Devoto theories.
- Developed Mathai--Quillen cocycles for elliptic Euler and Thom classes.
- Established a unique equivariant MString orientation via elliptic classes.

## Abstract

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins' "theorem of the cube" construction of the ${\rm MString}$-orientation of elliptic cohomology.