Circle compactification and 't Hooft anomaly

TL;DR

This paper develops a systematic method to derive 't Hooft anomalies in circle-compactified theories from their uncompactified counterparts, with applications to sigma models and QCD.

Contribution

It introduces a new procedure for deriving anomalies in compactified theories without relying on one-form symmetries, using twisted boundary conditions.

Findings

Explicit anomaly calculations for $ ext{CP}^{N-1}$ sigma model and $ ext{QCD}$.

Demonstrates anomalies can persist under circle compactification.

Provides a framework for analyzing anomalies in compactified strongly-coupled theories.

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 $\theta=\pi$. 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.