Completed K-theory and Equivariant Elliptic Cohomology

TL;DR

This paper develops an integral model of completed equivariant K-theory for S^1-stacks, connecting elliptic cohomology at the Tate curve with the work of Freed-Hopkins-Teleman, and resolving previous contradictions.

Contribution

It constructs a carefully completed, integral model of S^1-equivariant K-theory for stacks, enabling a consistent link to elliptic cohomology and Freed-Hopkins-Teleman theory.

Findings

Provides an integral, convergent model of equivariant K-theory.

Resolves contradictions in twist, degree, and cup product.

Links elliptic cohomology at the Tate curve with representation theory.

Abstract

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. Several authors (cf [ABG],[KM],[L]) have suggested that an equivariant version ought to be related to the work of Freed-Hopkins-Teleman ([FHT1],[FHT2],[FHT3]). However, a first attempt at this runs into apparent contradictions concerning twist, degree, and cup product. Several authors (cf. [BET],[G],[K]) have solved the problem over the complex numbers by interpreting the $S^1$-equivariant parameter as a complex variable and using holomorphicity as the technique for completion. This paper gives a solution that works integrally, by constructing a carefully completed model of $K$-theory for $S^1$-equivariant stacks which allows for certain ``convergent" infinite-dimensional cocycles.