Delocalized Chern character for stringy orbifold K-theory

Abstract:

In this paper, we define a stringy product on $K^*_{orb}(\mathfrak {X}) \otimes \mathbb {C}$, the orbifold K-theory of any almost complex presentable orbifold $\mathfrak {X}$. We establish that under this stringy product, the delocalized Chern character \[ ch_{deloc} : K^*_{orb}(\mathfrak {X}) \otimes \mathbb {C} \longrightarrow H^*_{CR}(\mathfrak {X}), \] after a canonical modification, is a ring isomorphism. Here $H^*_{CR}(\mathfrak {X})$ is the Chen-Ruan cohomology of $\mathfrak {X}$. The proof relies on an intrinsic description of the obstruction bundles in the construction of the 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 Pontryagin product (the latter is also called the fusion product in string theory).