On the relative dual of an $S^1$-gerbe over an orbifold

TL;DR

This paper constructs a dual orbifold with an $S^1$-gerbe and demonstrates an equivalence in sheaf categories, Morita equivalence of algebras, and isomorphism of K-theory and cohomology groups between the original and dual pairs.

Contribution

It introduces a new effective orbifold with an $S^1$-gerbe as a relative dual and proves categorical and algebraic equivalences with the original gerbe over an orbifold.

Findings

Categories of sheaves are isomorphic.

Twisted groupoid algebras are Morita equivalent.

K-theory and cohomology groups are isomorphic.

Abstract

We construct a new effective orbifold $\widehat{\Y}$ with an $S^1$-gerbe $c$ to study an $S^1$-gerbe $\mathfrak{t}$ on a $G$-gerbe $\Y$ over an orbifold $\B$. We view the former as the relative dual, relative to $\B$, of the latter. We show that the two pairs $(\Y, \mathfrak{t})$ and $(\widehat{\Y}, c)$ have isomorphic categories of sheaves, and also the associated twisted groupoid algebras are Morita equivalent. As a corollary, the K-theory and cohomology groups of $(\Y, \mathfrak{t})$ and $(\widehat{\Y}, c)$ are isomorphic.