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 Ŷ and a flat U(1) -gerbe c on Ŷ 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 Ŷ twisted 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 Ŷ. 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 Ŷ as graded algebras.