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

TL;DR

The paper addresses how discrete 't Hooft anomalies behave under spontaneous breaking, especially via domain walls, by establishing a canonical CPT-based wall symmetry and a 4-periodic hierarchy that constrains possible wall theories.It develops a general anomaly-matching program using anomaly inflow and cobordism, and introduces Smith maps that relate higher-dimensional bulk anomalies to lower-dimensional wall/junction anomalies, enabling practical reductions for Abelian groups and Z2 systems.Through explicit examples (Majorana fermions, CP^1 and S^4 sigma models, spin CP^1) and a rigorous mathematical treatment of bordism groups, the work provides new consistency checks for IR phases in 2+1D QCD and offers a framework to simplify discrete anomaly computations via wall physics.Overall, the work connects domain-wall physics, CPT structure, and cobordism-based classifications to deliver a coherent mechanism for discrete anomaly matching and dimensional reduction in interacting quantum field theories.This approach has potential implications for understanding symmetry-protected topological phases, decorated domain walls, and crystalline symmetries in strongly correlated systems.

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.

Figures (3)

• Figure 1: The $\mathbb{Z}_2$ domain wall is created by imposing frustrated boundary conditions for the order parameter $\phi$ along a coordinate $x_\perp$ or breaking the symmetry with a spatially-varying potential. Indeed, the global symmetry $U$ does not act on the wall but if we combine it with $CP_\perp T$, a canonical symmetry which involves a reflection in the normal coordinate $x_\perp$, then we obtain a symmetry of the domain wall degrees of freedom, which is anti-unitary if $U$ is unitary and vice versa.
• Figure 2: With two real order parameters $\phi_{1,2}$ fully breaking a $\mathbb{Z}_4$ symmetry we have four ground states, labelled $0,1,2,3$. These can be identified with the four signs of the VEVs of the two order parameters $\phi_{1,2} = \pm$. Choosing a boundary condition for $\phi_{1,2}$ such that they wind the unit circle at infinity along a pair of spatial coordinates we obtain a codimension-2 junction where four domain walls coalesce. The global $\mathbb{Z}_4$ symmetry does not act on this junction but we can combine it with a $\pi/2$ rotation to obtain a $\mathbb{Z}_4$ symmetry of the junction. In this case, we did not use $CPT$, so both $\mathbb{Z}_4$ symmetries are unitary.
• Figure 3: The injectivity proof for the Smith isomorphism theorem in a nutshell: one uses a nullbordism $Z$ of the image of the Smith map $(X,A) \mapsto (Y,A|_Y)$ to construct a bordism $(W,\tilde{A})$, depicted above, of $(X,A)$ to $(X',0)$ (blue), with the latter carrying no $\mathbb{Z}_2$ bundle. Indeed the green curve Poincaré dual to $\tilde{A}$ only meets the boundary of the bordism along $X$.

Theorems & Definitions (4)

• Theorem 4.1
• proof
• Theorem 4.2
• proof