Notes on Confinement on ${\mathbf{R}}^3\times {\mathbf{S}}^1$: From Yang-Mills, super-Yang-Mills, and QCD(adj) to QCD(F)

Abstract

This is a pedagogical introduction to the physics of confinement on $R^3\times S^1$, using $SU(2)$ Yang-Mills with massive or massless adjoint fermions as the prime example; at the end, we also add fundamental flavours. The small-$S^1$ limit is remarkable, allowing for controlled semiclassical determination of the nonperturbative physics in these, mostly non-supersymmetric, theories. We begin by reviewing the Polyakov confinement mechanism on $R^3$. Moving on to $R^3\times S^1$, we show how introducing adjoint fermions stabilizes center symmetry, leading to abelianization and semiclassical calculability. We explain how monopole-instantons and twisted monopole-instantons arise. We describe the role of various novel topological excitations in extending Polyakov's confinement to the locally four-dimensional case, discuss the nature of the confining string, and the $\theta$-angle dependence.~We study the global symmetry realization and, when available, present evidence for the absence of phase transitions as a function of the $S^1$ size. As our aim is not to cover all work on the subject, but to prepare the interested reader for its study, we also include brief descriptions of topics not covered in detail: the necessity for analytic continuation of path integrals, the study of more general theories, and the 't Hooft anomalies involving higher-form symmetries.

arXiv comment125 pages, 30 figures, references to important work from the 1990s added

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• The gaugino condensate from asymmetric four-torus with twists
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