Circle compactification and 't Hooft anomaly

TL;DR

This work presents a systematic method to derive 't Hooft anomalies for circle-compactified theories starting from (D+1)-dimensional theories with mixed anomalies, even when no one-form symmetry is present, by using twisted boundary conditions and appropriate background gauge fields.The procedure is explicitly implemented for the two-dimensional ℤ_N-twisted CP^{N-1} model and for massless ℤ_N-QCD, deriving concrete mixed anomalies in the circle-compactified theories and showing how the twisting and shift symmetries preserve anomaly information.The results support the view that anomalies can survive circle compactification through a faithful symmetry intertwined with the center symmetry, providing nonperturbative constraints on vacuum structure, phase transitions, and adiabatic continuity between regimes, with implications for confinement and finite-temperature dynamics.

Abstract

Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an 't Hooft anomaly involving one-form symmetries as in pure $SU(N)$ Yang-Mills theory at $θ=π$. Recent development about large-$N$ volume independence, however, gives us a circumstantial evidence that 't Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an 't Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider $\mathbb{Z}_N$-twisted $\mathbb{C}P^{N-1}$ sigma model and massless $\mathbb{Z}_N$-QCD, and compute their anomalies explicitly.