Stringy power operations in Tate K-theory

TL;DR

This paper develops equivariant power operations in Tate K-theory based on the loop spaces of symmetric orbifold powers, demonstrating their elliptic nature and connecting to the Witten genus and Moonshine phenomena.

Contribution

It introduces new elliptic power operations in Tate K-theory and links them to the orbifold Witten genus and Moonshine, expanding the understanding of these mathematical structures.

Findings

Power operations are elliptic in Tate K-theory.

The Witten genus is shown to be an H_oo map.

Reproduces Dijkgraaf et al.'s formula for orbifold Witten genus.

Abstract

We study the loop spaces of the symmetric powers of an orbifold and use our results to define equivariant power operations in Tate K-theory. We prove that these power operations are elliptic and that the Witten genus is an H_oo map. As a corollary, we recover a formula by Dijkgraaf, Moore, Verlinde and Verlinde for the orbifold Witten genus of these symmetric powers. We outline some of the relationship between our power operations and notions from (generalized) Moonshine.