Discrete gauge anomalies revisited

TL;DR

This paper revisits discrete gauge anomalies in 3+1D chiral fermion theories with specific symmetry groups, reformulates anomaly cancellation conditions, and explores gapped states with anomalous global symmetries.

Contribution

It provides a new formulation of anomaly cancellation conditions for discrete gauge anomalies and clarifies the role of symmetry extensions in these anomalies.

Findings

Reformulation of anomaly cancellation conditions

Comparison with previous results by Ibanez and Ross

Construction of gapped states with anomalous $oldsymbol{ ext{Z}_n}$ symmetry

Abstract

We revisit discrete gauge anomalies in chiral fermion theories in $3+1$ dimensions. We focus on the case that the full symmetry group of fermions is $\mathrm{Spin}(4)\times\mathbb{Z}_n$ or $(\mathrm{Spin}(4)\times\mathbb{Z}_{2m})/\mathbb{Z}_{2}$ with $\mathbb{Z}_2$ being the diagonal $\mathbb{Z}_2$ subgroup. The anomalies are determined by the consistency condition --- based on the Dai-Freed theorem --- of formulating a chiral fermion theory on a generic spacetime manifold with a structure associated with either one of the above symmetry groups and are represented by elements of some finite abelian groups. Accordingly, we give a reformulation of the anomaly cancellation conditions, and compare them with the previous result by Ib\'{a}\~{n}ez and Ross. The role of symmetry extensions in discrete symmetry anomalies is clarified in a formal fashion. We also study gapped states of fermion with an anomalous global $\mathbb{Z}_n$ symmetry, and present a model for constructing these states in the framework of weak coupling.