Instantons, Symmetries and Anomalies in Five Dimensions

TL;DR

The paper identifies a finite-order mixed 't Hooft anomaly in five-dimensional Yang–Mills between the instanton $U(1)^{(0)}_I$ and the center one-form symmetry $\\mathbb{Z}^{(1)}_N$, detected by coupling to a background $\\mathcal{B}$ and described by anomaly inflow from a 6d theory. This obstruction persists under circle reduction and finite temperature, constraining possible IR phases and guiding symmetry-breaking scenarios; it extends to theories with fundamental matter through twisted bundles and to supersymmetric theories with symmetry enhancements at conjectured 5d RG fixed points such as $E_{N_f+1}$, with nuanced possibilities like $SO(3)$ versus $SU(2)$ instanton symmetry enhancements. The analysis yields a framework to relate UV completions, phase structure (deconfined, gapped, or gapless), and potential Higgs branches, illustrating how higher-form and ordinary global symmetries interplay in 5d gauge dynamics. Overall, the work provides a robust anomaly-based method to organize and predict symmetry realizations and phase structure in 5d gauge theories and their SUSY extensions.

Abstract

All five-dimensional non-abelian gauge theories have a $U(1)_I$ global symmetry associated with instantonic particles. We describe an obstruction to coupling $U(1)_I$ to a classical background gauge field that occurs whenever the theory has a one-form center symmetry. This is a finite-order mixed 't Hooft anomaly between the two symmetries. We also show that a similar obstruction takes place in gauge theories with fundamental matter by studying twisted bundles for the ordinary flavor symmetry. We explore some general dynamical properties of the candidate phases implied by the anomaly. Finally, we apply our results to supersymmetric gauge theories in five dimensions and analyze the symmetry enhancement patterns occurring at their conjectured RG fixed points.