Duality theorems for étale gerbes on orbifolds

DOI10.1016/J.AIM.2013.10.002zbMATH Open1300.14058arXiv1004.1376OpenAlexW2963225619MaRDI QIDQ2445387

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Publication date: 14 April 2014

Published in: (Search for Journal in Brave)

Abstract: Let $G$ be a finite group and $Y$ a $G$-gerbe over an orbifold $B$. A disconnected orbifold $hatY$ and a flat U(1)-gerbe $c$ on $hatY$ 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 $hatY$ {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 $hatY$. 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 $hatY$ as graded algebras.

Full work available at URL: https://arxiv.org/abs/1004.1376

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