Duality theorems for étale gerbes on orbifolds

Let G be a finite group and Y a G-gerbe over an orbifold B. A disconnected orbifold Yˆ and a flat U(1)-gerbe c on Yˆ 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 Yˆtwisted by c. We prove several results verifying this duality in the contexts of non-commutative 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 Yˆ. When Y is symplectic, we show, by a combination of techniques from non-commutative geometry and symplectic topology, that the Chen-Ruan orbifold cohomology of Y is isomorphic to the c-twisted orbifold cohomology of Yˆ as graded algebras.