Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{ \Y}$ and a flat U(1)-gerbe $c$ on $\hat{ \Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{ \Y}$ { \em twisted by} $c$. 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 $c$-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 $c$-twisted orbifold cohomology of $\hat{ \Y}$ as graded algebras.
Categories: math.AG math.DG math.KT math.SG