Anomaly of non-Abelian discrete symmetries

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 $G=S_n\simeq A_n⋊Z_2$ and $G=\Delta (6n^2)\simeq \Delta (3n^2)⋊Z_2$, 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.

Comments: 25 pages, 1 figure