Equivariant twisted Real $K$-theory of compact Lie groups

TL;DR

This paper generalizes the Freed-Hopkins-Teleman Theorem to include Real structures on compact Lie groups, developing new tools like Real Dixmier-Douady bundles and Real cohomology to establish a broader framework in equivariant twisted K-theory.

Contribution

It introduces a Real-structured extension of the FHT, including definitions of Real Dixmier-Douady bundles and Real cohomology, expanding the theoretical landscape of equivariant twisted K-theory.

Findings

Generalization of FHT to Real Lie groups

Development of Real Dixmier-Douady bundles

Introduction of Real third cohomology group

Abstract

Let $G$ be a compact, connected, and simply-connected Lie group viewed as a $G$-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted $K$-homology of $G$ and its Verlinde algebra. In this paper we give a generalization of FHT in the presence of a Real structure of $G$. Along the way we develop preliminary materials necessary for this generalization, which are of independent interest in their own right. These include the definitions of Real Dixmier-Douady bundles, the Real third cohomology group which is shown to classify the former, and Real $\text{Spin}^c$ structures.