Gauge theory geography: charting a path between semiclassical islands

TL;DR

This paper explores two semiclassical limits of $SU(2)$ Yang-Mills theory on a torus with twists, revealing that different classical ground states share symmetry properties and analyzing their quantum stability through one-loop potentials.

Contribution

It introduces a detailed analysis of semiclassical limits of twisted Yang-Mills theories, highlighting the symmetry properties and stability of ground states across different regimes.

Findings

Classical ground states have identical transformation properties under key symmetries.

The one-loop potential informs about the quantum stability of these states.

Results suggest new features in gauge theories with twisted compactifications.

Abstract

We study two semiclassical limits of $SU(2)$ Yang-Mills theory on a spatial torus with a 't Hooft twist: the ``femtouniverse,'' where all $\mathbb{T}^3$ directions are small, and deformed Yang-Mills theory on $\mathbb{T}^2 \times \mathbb{S}^1$, with small $\mathbb{S}^1$ and large or infinite $\mathbb{T}^2$. Carefully defining the symmetries, we show that the classical ground states, while different, have the same transformation properties under the 1-form center symmetry and parity. We argue that this is behind the identical multi-branch $\theta$-dependent vacuum structure of these theories. We then calculate the one-loop potential for the $\mathbb{S}^1$-holonomy in the presence of twists on $\mathbb{T}^2$. We use it to study the quantum stability of the semiclassical ground states in gauge theories with massive or massless adjoint fermions on spatial $\mathbb{T}^2 \times \mathbb{S}^1$, with a twist in the $\mathbb{T}^2$. The results point towards some interesting features worthy of further study.