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

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