# Anomalies in the Space of Coupling Constants and Their Dynamical Applications II

## TL;DR

This paper explores anomalies in four-dimensional gauge theories, revealing their role in phase transitions, interfaces, and symmetries, and extends previous work to include matter fields and broader systems.

## Contribution

It identifies new mixed anomalies involving global symmetries and the theta parameter in gauge theories, broadening understanding of their phase structure and dynamics.

## Key findings

- Mixed anomaly between one-form symmetry and theta periodicity in pure Yang-Mills.
- Anomalies involving flavor symmetry and theta periodicity in matter-coupled theories.
- Unified explanation for phase transitions and interface phenomena in these theories.

## Abstract

We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $\theta$-parameter. This anomaly is at the root of many recently discovered properties of these theories, including their phase transitions and interfaces. These new anomalies can be used to extend this understanding to systems without discrete symmetries (such as time-reversal). We also study $SU(N)$ and $Sp(N)$ gauge theories with matter in the fundamental representation. Here we find a mixed anomaly between the flavor symmetry group and the $\theta$-periodicity. Again, this anomaly unifies distinct recently-discovered phenomena in these theories and controls phase transitions and the dynamics on interfaces.