Duality theorems for \'etale gerbes on orbifolds

Understand

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.