Topological terms and anomaly matching in effective field theories on $\mathbb{R}^3\times S^1$: I. Abelian symmetries and intermediate scales

TL;DR

This paper calculates topological terms in IR effective theories for certain $SU(N)$ gauge theories on $R^3 imes S^1$, demonstrating how they match UV anomalies, revealing new forms of anomaly matching involving background TQFTs.

Contribution

It explicitly derives topological terms in IR effective theories for $SU(N)$ gauge theories on $R^3 imes S^1$ and shows their role in anomaly matching, introducing novel background TQFT descriptions.

Findings

Topological terms match all global anomalies in IR theories.

Cubic and mixed $U(1)$ anomalies are matched by background TQFTs.

Coupling of 0-form and 1-form symmetries is described consistently throughout RG flow.

Abstract

We explicitly calculate the topological terms that arise in IR effective field theories for $SU(N)$ gauge theories on $\mathbb{R}^3 \times S^1$ by integrating out all but the lightest modes. We then show how these terms match all global-symmetry 't Hooft anomalies of the UV description. We limit our discussion to theories with abelian 0-form symmetries, namely those with one flavour of adjoint Weyl fermion and one or zero flavours of Dirac fermions. While anomaly matching holds as required, it takes a different form than previously thought. For example, cubic- and mixed-$U(1)$ anomalies are matched by local background-field-dependent topological terms (background TQFTs) instead of chiral-lagrangian Wess-Zumino terms. We also describe the coupling of 0-form and 1-form symmetry backgrounds in the magnetic dual of super-Yang-Mills theory in a novel way, valid throughout the RG flow and consistent with the monopole-instanton 't Hooft vertices. We use it to discuss the matching of the mixed chiral-center anomaly in the magnetic dual.