Gauge theory geography: charting a path between semiclassical islands

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.