Discrete Torsion and Gerbes I

TL;DR

This work provides a geometric interpretation of discrete torsion as an orbifold Wilson surface for the NS-NS $B$-field by modeling the $B$-field as a connection on a $1$-gerbe and analyzing how the orbifold group $\Gamma$ acts on this gerbe. It develops an equivariant-structure framework, showing that discrete torsion arises from the nontrivial lifting of $\Gamma$ to the $1$-gerbe, even in backgrounds with nonzero $H$-flux, and extends the viewpoint to potentially analogous structures for other tensor fields via higher cohomology groups. The paper situates these ideas within Čech cohomology and gerbe theory, discusses gauge transformations of gerbes, and argues for a broad program to derive discrete-torsion analogues for higher-form potentials, conjecturing that such data are governed by $H^{p+1}(\Gamma, U(1))$. These insights clarify the geometric origin of discrete torsion and offer a principled path toward understanding orbifold actions on general tensor fields in string theory.

Abstract

In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as connections on gerbes. Discrete torsion then naturally appears in describing the action of the orbifold group on (1-)gerbes, just as orbifold Wilson lines appear in describing the action of the orbifold group on the gauge bundle. Our results are not restricted to trivial gerbes -- in other words, our description of discrete torsion applies equally well to compactifications with nontrivial H-field strengths. We also describe a speciric program for rigorously deriving analogues of discrete torsion for many of the other type II tensor field potentials, and we are able to make specific conjectures for the results.