Quasi-quantum groups from Kalb-Ramond fields and magnetic amplitudes for strings on orbifolds

TL;DR

This paper explores how Kalb-Ramond fields influence the algebraic structure of orbifold string theories, revealing a connection to quasi-quantum groups and clarifying the role of discrete torsion.

Contribution

It introduces the operators that generate the quasi-quantum group D_ω[G] in orbifold string theory with Kalb-Ramond fields, linking cohomological data to algebraic structures.

Findings

Operators generate the quasi-quantum group D_ω[G]

The 3-cocycle ω relates to the Kalb-Ramond field H

Discrete torsion appears as an ambiguity in group action lift

Abstract

We present the general form of the operators that lift the group action on the twisted sectors of a bosonic string on an orbifold ${\cal M}/G$, in the presence of a Kalb-Ramond field strength $H$. These operators turn out to generate the quasi-quantum group $D_{\omega}[G]$, introduced in the context of orbifold conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche. The 3-cocycle $\omega$ entering in the definition of $D_{\omega}[G]$ is related to $H$ by a series of cohomological equations in a tricomplex combining de Rham, Cech and group coboundaries. We construct magnetic amplitudes for the twisted sectors and show that $\omega=1$ arises as a consistency condition for the orbifold theory. Finally, we recover discrete torsion as an ambiguity in the lift of the group action to twisted sectors, in accordance with previous results presented by E. Sharpe.