# Theta, Time Reversal, and Temperature

## TL;DR

This paper investigates the implications of a discrete 't Hooft anomaly at = in $SU(N)$ gauge theories, revealing constraints on vacua, phase diagrams, and possible nontrivial domain walls and gapless phases.

## Contribution

It demonstrates the presence of a 't Hooft anomaly at = in $SU(N)$ gauge theories and explores its consequences for vacua, phase structure, and domain walls, including scenarios with gapless phases.

## Key findings

- Anomaly at = constrains vacua and phase diagrams.
- Vacuum at = cannot be trivial and gapped.
- Possible existence of nontrivial domain walls and gapless phases.

## Abstract

$SU(N)$ gauge theory is time reversal invariant at $\theta=0$ and $\theta=\pi$. We show that at $\theta=\pi$ there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at $\theta=\pi$ the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at $\theta=0$ is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at $\theta=\pi$, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for $SU(2)$ gauge theory. The underlying symmetry at $\theta=\pi$ is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two $O(2)$-symmetric fixed points. It may also be that the four-dimensional theory around $\theta=\pi$ is gapless, e.g. a Coulomb phase could match the underlying anomalies.