Anomaly of non-Abelian discrete symmetries

TL;DR

This paper investigates the anomaly structure of non-Abelian discrete symmetries, identifying which parts are anomaly-free and how the anomaly manifests within the group structure.

Contribution

It provides a detailed analysis of the subgroup structure related to anomalies in non-Abelian discrete groups, including specific examples like symmetric and delta groups.

Findings

Anomaly-free elements form a normal subgroup of the original group.

The anomalous part of the group is isomorphic to a cyclic group.

The derived subgroup is crucial for understanding the anomaly structure.

Abstract

We study anomalies of non-Abelian discrete symmetries; which part of non-Abelian group is anomaly free and which part can be anomalous. It is found that the anomaly-free elements of the group $G$ generate a normal subgroup $G_0$ of $G$ and the residue class group $G/G_0$, which becomes the anomalous part of $G$, is isomorphic to a single cyclic group. The derived subgroup $D(G)$ of $G$ is useful to study the anomaly structure. This structure also constrains the structure of the anomaly-free subgroup; the derived subgroup $D(G)$ should be included in the anomaly-free subgroup. We study the detail structure of the anomaly-free subgroup from the structure of the derived subgroup in various discrete groups. For example, when $G=S_n \simeq A_n \rtimes Z_2$ and $G=\Delta(6n^2) \simeq \Delta(3n^2) \rtimes Z_2$, in particular, $A_n$ and $\Delta(3n^2)$ are at least included in the anomaly-free subgroup, respectively. This result holds in any arbitrary representations.