A generalization of decomposition in orbifolds

TL;DR

The paper extends decomposition in orbifolds to cases with discrete torsion, showing that a decomposition-like structure persists even when trivially acting subgroups host torsion and one-form symmetries are partially broken. It formulates a general conjecture, distinguishing three regimes via the maps iota^*ω and β(ω), and provides a constructive method to assemble the component theories using projective representations and cocycles. The authors validate the framework across a broad spectrum of explicit examples—including abelian and nonabelian trivially-acting subgroups—demonstrating consistent genus-one partition functions and clarifying the role of quantum-symmetry-like behavior. The work unifies the plus-times with abelian orbifold phenomena and connects to open-string/gerbe interpretations, offering a robust toolkit for analyzing orbifolds with discrete torsion in two dimensions.

Abstract

This paper describes a generalization of decomposition in orbifolds. In general terms, decomposition states that two-dimensional orbifolds and gauge theories whose gauge groups have trivially-acting subgroups decompose into disjoint unions of theories. However, decomposition can be, at least naively, broken in orbifolds if the orbifold has discrete torsion in the trivially-acting subgroup. (Formally, this breaks finite global one-form symmetries.) Nevertheless, even in such cases, one still sees rudiments of decomposition. In this paper, we generalize decomposition in orbifolds to include such examples of discrete torsion, which we check in numerous examples. Our analysis includes as special cases (and in one sense generalizes) quantum symmetries of abelian orbifolds.