Duality theorems for étale gerbes on orbifolds
Xiang Tang, Hsian-Hua Tseng
TL;DR
This work establishes a rigorous duality between a finite G-gerbe Y over an orbifold B and a dual pair (hat{Y}, c) formed from Out(G)-bands and a flat U(1)-gerbe c. It leverages Morita equivalence in noncommutative geometry to relate the crossed product algebras and categories of (twisted) sheaves, yielding a ring isomorphism between Chen–Ruan orbifold cohomology of Y and the c-twisted orbifold cohomology of hat{Y}. The authors further connect these structures to Hochschild cohomology and deformation quantization, showing that HH^• of the crossed products matches twisted de Rham cohomology of inertia stacks, and extend the duality to Gromov–Witten theory with concrete verification for gerbes arising from group extensions BH→BQ. The results provide a conceptual bridge linking representation theory, noncommutative geometry, and symplectic/Gromov–Witten topology, with a clear path for generalizing the duality to broader G-gerbe contexts and higher-dimensional Lie groups.
Abstract
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.