Discrete Torsion and Gerbes II

TL;DR

This work provides a rigorous geometric formulation of discrete torsion through 1-gerbes expressed as stacks, establishing a first-principles derivation of equivariant structures. It develops the stack and gerbe machinery (connective structures, curving, gauge transformations, and Čech cohomology) and proves that the set of equivariant structures on a gerbe with connection forms a torsor under a group that includes $H^2(\Gamma,U(1))$, with canonical identifications in special cases. The loop-space check and descent-theoretic constructions connect global orbifold data to local stack-theoretic data, clarifying how discrete torsion arises from lifting actions to gerbes. Overall, the paper provides a robust geometric origin for discrete torsion, clarifying the role of torsors, cohomology, and equivariance in B-field gerbes. The results have implications for the geometric understanding of orbifolds and their B-fields in string theory.

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.