Confinement on $\mathbb{R}^3 \times \mathbb{S}^1$ and Double-String Collapse

We study confining strings in ${\cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $\mathbb{R}^{1,2} \times \mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which are lines in $\mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks' chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c \ge 5$, the two domain walls confining unit $N$-ality quarks attract and form non-BPS bound states, collapsing to a single flux line. We determine the $N$-ality dependence of the string tensions for $2 \le N_c \le 10$. Compared to known scaling laws, we find a weaker, almost flat $N$-ality dependence, which is qualitatively explained by the properties of BPS domain walls. We also quantitatively study the behavior of confining strings upon increasing the $\mathbb{S}^1$ size by including the effect of virtual "$W$-bosons" and show that the qualitative features of double-string confinement persist.