A fresh view on string orbifolds

TL;DR

The paper develops a spacetime orbifold construction for string theory that gauges a finite subgroup $\Gamma$ of the full spacetime gauge group $G$ while incorporating higher-form and 2-group structures to avoid global symmetries. It demonstrates the approach with two explicit examples: Type II on $S^1$ under a half-period shift and toroidal compactification under coordinate inversion, showing that higher-group constraints dictate gauge-group extensions and the twisted-sector content, with twisted-ground-state counts matching worldsheet expectations ($2$ for the half-period shift and $2^d$ for $T^d$). The framework aims to be duality-invariant and provides a principled way to reconcile worldsheet and spacetime perspectives, though some duality-related obstructions persist and require further refinement, such as nonabelian generalizations and non-invertible defects. The results illuminate how anomalies, Chern-Simons couplings, and higher-group data govern consistent orbifolds in quantum gravity and suggest avenues for a more complete, nonperturbative formulation of stringy orbifolds.

Abstract

In quantum field theory, an orbifold is a way to obtain a new theory from an old one by gauging a finite global symmetry. This definition of orbifold does not make sense for quantum gravity theories, that admit (conjecturally) no global symmetries. In string theory, the orbifold procedure involves the gauging of a global symmetry on the world-sheet theory describing the fundamental string. Alternatively, it is a way to obtain a new string background from an old one by quotienting some isometry. We propose a new formulation of string orbifolds in terms of the group of gauge symmetries of a given string model. In such a formulation, the `parent' and the `child' theories correspond to different ways of breaking or gauging all potential global symmetries of their common subsector. Through a couple of simple examples, we describe how the higher group structure of the gauge group in the parent theory plays a crucial role in determining the gauge group and the twisted sector of the orbifold theory. We also discuss the dependence of this orbifold procedure on the duality frame.

Figures (4)

• Figure 1: Two different ways of fusing three defects $\mathcal{L}_g$, $\mathcal{L}_h$, and $\mathcal{L}_k$ into a single defect $\mathcal{L}_{ghk}$ through a sequence of $3$-pronged junctions. The dashed lines represent the identity defect, that we added for later convenience. One can continuously deform the left configuration into the right one; along this transformation, there is a point where the defects $\mathcal{L}_g$, $\mathcal{L}_h$, $\mathcal{L}_k$, and $\mathcal{L}_{ghk}$ are connected to a single $4$-pronged junction. Passing across that point, a correlation function might get a non-trivial phase $\alpha(g,h,k)$. Equivalently, the left and the right-hand side are related by a local gauge transformation acting by a group element $h$ in the green region; this picture makes it clear that $\alpha(g,h,k)$ is a 't Hooft anomaly -- correlation functions pick up a non-trivial phase under gauge transformations of the background gauge field.
• Figure 2: The group $G_{res}$ is abelian. Thus, the fusion of any two defects $\mathcal{L}_g$ and $\mathcal{L}_h$, $g,h\in G_{res}$ is commutative $\mathcal{L}_{g}\mathcal{L}_{h}=\mathcal{L}_{h}\mathcal{L}_{g}$, which means that $\mathcal{L}_{g}$ and $\mathcal{L}_{h}$ can simply 'cross each other'. In this figure, the lines represent codimension $1$ defects.
• Figure 3: A triple intersection between three codimension $1$ defects, namely $\mathcal{L}_{g_i}$ (blue), $\mathcal{L}_{h_i}$ (red) and $\mathcal{L}_\mathsf{C}$ (grey). Due to the non-trivial $2$-group structure, the B-field on the two sides of the $\mathcal{L}_\mathsf{C}$ defect differs by a $2$-cochain that is Poincaré dual to the $(D-2)$-dimensional intersection of $\mathcal{L}_{g_i}$ and $\mathcal{L}_{h_i}$ (the yellow line).
• Figure 4: After gauging $2A_1\wedge B_1$, the $(D-2)$-dimensional intersection of a $\mathcal{L}_{g_i}$ (blue) and a $\mathcal{L}_{h_i}$ (red) defects must be the boundary of a $\mathcal{L}_Q$ defect (green). The presence of this $\mathcal{L}_Q$ defect can be deduced by the fact that a small Wilson line $e^{2\pi i\oint_{\gamma_1} \mathcal{A}_1}$ encircling the $\mathcal{L}_{g_i}$ and $\mathcal{L}_{h_i}$ intersection is non trivial. This means that, after gauging, the fusion between the $\mathcal{L}_{g_i}$, $\mathcal{L}_{h_i}$, $\mathcal{L}_Q$ defects is not commutative, i.e. the corresponding group is non-abelian. In this picture, the lines represent codimension $1$ defects.