Twisted K-theory of differentiable stacks

TL;DR

This paper develops a comprehensive framework for twisted K-theory on differentiable stacks, establishing fundamental properties and connecting it to various existing theories using groupoid presentations and KK-theory.

Contribution

It introduces a unified approach to twisted K-theory for stacks, including properties, product structures, and a Fredholm model, extending existing theories to a broader context.

Findings

Established Mayer-Vietoris property for twisted K-theory

Proved Bott periodicity in the stack context

Unified framework encompassing various twisted K-theories

Abstract

In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_\alpha \otimes K^j_\beta \to K^{i+j}_{\alpha +\beta}$ are derived. Our approach provides a uniform framework for studying various twisted $K$-theories including the usual twisted $K$-theory of topological spaces, twisted equivariant $K$-theory, and the twisted $K$-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted $K$-groups can be expressed by so-called "twisted vector bundles". Our approach is to work on presentations of stacks, namely \emph{groupoids}, and relies heavily on the machinery of $K$-theory ($KK$-theory) of $C^*$-algebras.