Duality theorems for étale gerbes on orbifolds
arXiv:1004.1376
Abstract
Let be a finite group and $\Y$ a -gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the -gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} . We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of -twisted sheaves on $\hat{\Y}$. When $\Y$ is symplectic, we show, by a combination of techniques from noncommutative geometry and symplectic topology, that the Chen-Ruan orbifold cohomology of $\Y$ is isomorphic to the -twisted orbifold cohomology of $\hat{\Y}$ as graded algebras.
59 pages