Discrete Torsion and Gerbes II

TL;DR

This paper provides a geometric and categorical framework for understanding discrete torsion in orbifolds through the theory of gerbes and stacks, offering a first-principles derivation and classification of equivariant structures.

Contribution

It develops the theory of gerbes in terms of stacks, and classifies equivariant structures, linking them to cohomological groups like H^2(G,U(1)).

Findings

Gerbes are described using stacks, enhancing their geometric understanding.

Equivariant structures on gerbes form a torsor under a cohomological group.

Special cases allow canonical identification of equivariant structures with the group.

Abstract

In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms of objects known as stacks (essentially, sheaves of categories), and develop much of the basic theory of gerbes in such language. Then, once the relevant technology has been described, we give a first-principles geometric derivation of discrete torsion. In other words, we define equivariant gerbes, and classify equivariant structures on gerbes and on gerbes with connection. We prove that in general, the set of equivariant structures on a gerbe with connection is a torsor under a group which includes H^2(G,U(1)), where G is the orbifold group. In special cases, such as trivial gerbes, the set of equivariant structures can furthermore be canonically identified with the group.