Twisted Equivariant Tate K-Theory

TL;DR

This paper introduces a new loop space-based definition of twisted equivariant Tate K-theory for finite groups, connecting gerbe transgression techniques with existing theories and applications in Moonshine and quasi-elliptic cohomology.

Contribution

It develops a loop space approach to twisted equivariant Tate K-theory using gerbe transgression, extending prior work to finite groups and relating to Moonshine and quasi-elliptic cohomology.

Findings

Constructs a gerbe on orbifold loop space from a cocycle on the quotient orbifold.

Defines twisted equivariant Tate K-theory via loop space gerbes.

Establishes connections to Ganter's Moonshine and Huan's quasi-elliptic cohomology.

Abstract

Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$. This gives a loop based definition of twisted equivariant Tate K-theory for finite groups that conforms to the definition that Luecke provides for compact, connected Lie groups. We relate our construction to Ganter's work on Moonshine and Huan's quasi-elliptic cohomology.