Outer Automorphism Anomalies

TL;DR

This paper analyzes anomalies related to outer automorphisms like charge conjugation and parity in gauge theories, revealing their anomalous nature and implications for symmetry breaking and anomaly matching.

Contribution

It identifies and characterizes anomalies of outer automorphisms in classical gauge groups, clarifies their inequivalence, and explores their effects on symmetry and anomaly matching.

Findings

Certain charge conjugations are anomalous and not true symmetries.

Outer automorphisms can be spontaneously broken.

Subgroups commuting with automorphisms are explicitly identified.

Abstract

We discuss anomalies associated with outer automorphisms in gauge theories based on classical groups, namely charge conjugations for $SU(N)$ and parities for $SO(2r)$. We emphasize the inequivalence (yet related by a flavor transformation) between two versions of charge conjugation for $SU(2k)$, $SO(2r)$, and $E_6$ symmetries. The subgroups that commute with the outer automorphisms are identified. Some charge conjugations can lead to a paradox, which is resolved by the observation that they are anomalous and hence not symmetries. We then discuss anomaly matching conditions that involve the charge conjugations or parities. Interesting examples are given where the charge conjugation is spontaneously broken.