Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism

TL;DR

This paper explores how domain walls in theories with spontaneously broken discrete symmetries encode anomaly matching conditions, providing new insights into the Smith isomorphism and simplifying anomaly computations.

Contribution

It introduces a novel analysis of anomaly matching via domain wall physics in discrete symmetry breaking, extending the Smith isomorphism to abelian groups.

Findings

Demonstrates the role of domain walls in anomaly matching.

Reveals a mod 4 periodic structure in Z/2 and T cases.

Provides new consistency checks for 2+1D QCD phases.

Abstract

Symmetries in Quantum Field Theory may have 't Hooft anomalies. If the symmetry is unbroken in the vacuum, the anomaly implies a nontrivial low-energy limit, such as gapless modes or a topological field theory. If the symmetry is spontaneously broken, for the continuous case, the anomaly implies low-energy theorems about certain couplings of the Goldstone modes. Here we study the case of spontaneously broken discrete symmetries, such as Z/2 and T. Symmetry breaking leads to domain walls, and the physics of the domain walls is constrained by the anomaly. We investigate how the physics of the domain walls leads to a matching of the original discrete anomaly. We analyze the symmetry structure on the domain wall, which requires a careful analysis of some properties of the unbreakable CPT symmetry. We demonstrate the general results on some examples and we explain in detail the mod 4 periodic structure that arises in the Z/2 and T case. This gives a physical interpretation for the Smith isomorphism, which we also extend to more general abelian groups. We show that via symmetry breaking and the analysis of the physics on the wall, the computations of certain discrete anomalies are greatly simplified. Using these results we perform new consistency checks on the infrared phases of 2+1 dimensional QCD.