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

Abstract

We study confining strings in $\mathcal{N}=1$ 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.

53 pages, 23 figures. Version to appear in JHEP. Added references, updated the title, and added a section explaining the differences from confinement in the Polyakov model

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• Monopole-vortex continuity of $\mathcal{N}=1$ super Yang-Mills theory on ${\mathbb{R}}^2\times S^1\times S^1$ with 't Hooft twist
• Domain walls in super Yang-Mills: worldvolume TQFTs and deconfinement from semiclassics on ${\mathbb{R}}^3\times {\mathbb{S}}^1$
• On the perturbative aspects of deformed Yang-Mills theory