Spatial compactification on ℝ3×SL1\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1 $$\end{document} at small S1\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb{S}}^1 $$\end{document}-size L often leads to a calculable vacuum structure, where various “topological molecules” are responsible for confinement and the realization of the center and discrete chiral symmetries. Within this semiclassically calculable framework, we study how distinct theories with the same SUNc/ℤk\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k $$\end{document} gauge group (labeled by “discrete θ-angles”) arise upon gauging of appropriate ℤk\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathbb{Z}}}_k $$\end{document} subgroups of the one-form global center symmetry of an SU(Nc) gauge theory. We determine the possible ℤk\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathbb{Z}}}_k $$\end{document} actions on the local electric and magnetic effective degrees of freedom, find the ground states, and use domain walls and confining strings to give a physical picture of the vacuum structure of the different SUNc/ℤk\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k $$\end{document} theories. Some of our results reproduce ones from earlier supersymmetric studies, but most are new and do not invoke supersymmetry. We also study a further finite-temperature compactification to ℝ2×Sβ1×SL1\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathbb{R}}}^2\times {\mathbb{S}}_{\beta}^1\times {\mathbb{S}}_L^1 $$\end{document}. We argue that, in deformed Yang-Mills theory, the effective theory near the deconfinement temperature βc ≫ L exhibits an emergent Kramers-Wannier duality and that it exchanges high- and low-temperature theories with different global structure, sharing features with both the Ising model and S-duality in N=4\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal{N}=4 $$\end{document} supersymmetric Yang-Mills theory.
