Anomalies, Extensions and Orbifolds

TL;DR

We analyze gauge anomalies in orbifold CFTs, showing anomalies appear as violations of modular invariance in the twisted-partition traces $Z_{g_1,g_2}$ and are classified by $H^3(G,U(1))$. The authors develop a framework of group extensions $1\to K\to\Gamma\to G\to1$, including Tachikawa's construction, to trivialize the anomaly and study how $K$-actions on $G$-twisted sectors induce discrete-torsion–like choices (quantum symmetries). They provide a general method to compute the extended $\Gamma$-orbifold partition function, then decompose it into orbifolds by non-anomalous subgroups of $G$, often yielding multiple copies of non-anomalous theories. Through extensive explicit examples for $G=\mathbb{Z}_2$, $\mathbb{Z}_2\times\mathbb{Z}_2$, and $\mathbb{Z}_2\times\mathbb{Z}_4$, they illustrate when extensions produce a single cured theory or a direct sum of non-anomalous orbifolds, and how discrete torsion and quantum symmetry govern the outcome. The results support a general conjecture that consistent extensions yield (possibly multiple copies of) orbifolds by non-anomalous subgroups, revealing non-uniqueness of minimal extensions and highlighting the role of decomposition in resolving orbifold anomalies.

Abstract

We investigate gauge anomalies in the context of orbifold conformal field theories. Such anomalies manifest as failures of modular invariance in the constituents of the orbifold partition function. We review how this irregularity is classified by cohomology and how extending the orbifold group can remove it. Working with such extensions requires an understanding of the consistent ways in which extending groups can act on the twisted states of the original symmetry, which leads us to a discrete-torsion like choice that exists in orbifolds with trivially-acting subgroups. We review a general method for constructing such extensions and investigate its application to orbifolds. Through numerous explicit examples we test the conjecture that consistent extensions should be equivalent to (in general multiple copies of) orbifolds by non-anomalous subgroups.

Figures (3)

• Figure 1: A three-way junction.
• Figure 2: 4-way junctions can be resolved in two distinct but related ways.
• Figure 3: We evaluate the modular $T$ transformation of an anomalous partial trace.