Delocalized Chern character for stringy orbifold K-theory

TL;DR

This paper introduces a new stringy product on orbifold K-theory, showing it forms a ring isomorphism with Chen-Ruan cohomology after a canonical modification, and explores its properties in equivariant K-theory.

Contribution

It defines a novel stringy product on orbifold K-theory and proves its isomorphism with Chen-Ruan cohomology, providing new insights into orbifold invariants.

Findings

The stringy product is a ring isomorphism with Chen-Ruan cohomology after modification.

The stringy product differs from the Pontryagin (fusion) product in equivariant K-theory.

Intrinsic description of obstruction bundles facilitates the proof.

Abstract

In this paper, we define a stringy product on $K^*_{orb}(\XX) \otimes \C $, the orbifold K-theory of any almost complex presentable orbifold $\XX$. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here $ H^*_{CR}(\XX)$ is the Chen-Ruan cohomology of $\XX$. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryajin product (the latter is also called the fusion product in string theory).