Abstract
We classify su(Nc) gauge theories on {mathrm{mathbb{R}}}^3times {mathbb{S}}^1 with massless fermions in higher representations obeying periodic boundary conditions along {mathbb{S}}^1 . In particular, we single out the class of theories that is asymptotically free and weakly coupled in the infrared, and therefore, is amenable to semi-classical treatment. Our study is conducted by carefully identifying the vacua inside the affine Weyl chamber using Verma bases and Frobenius formula techniques. Theories with fermions in pure representations are generally strongly coupled. The only exceptions are the four-index symmetric representation of su(2) and adjoint representation of su(Nc). However, we find a plethora of admissible theories with fermions in mixed representations. A sub-class of these theories have degenerate perturbative vacua separated by domain walls. In particular, su(Nc) theories with fermions in the mixed representations adjoint⊕fundamental and adjoint⊕two-index symmetric admit degenerate vacua that spontaneously break the parity mathcal{P} , charge conjugation mathcal{C} , and time reversal mathcal{T} symmetries. These are the first examples of strictly weakly coupled gauge theories on {mathrm{mathbb{R}}}^3times {mathbb{S}}^1 with spontaneously broken mathcal{C} , mathcal{P} , and mathcal{T} symmetries. We also compute the fermion zero modes in the background of monopole-instantons. The monopoles and their composites (topological molecules) proliferate in the vacuum leading to the confinement of electric charges. Interestingly enough, some theories have also accidental degenerate vacua, which are not related by any symmetry. These vacua admit different numbers of fermionic zero modes, and hence, different kinds of topological molecules. The lack of symmetry, however, indicates that such degeneracy might be lifted by higher order corrections. Finally, we study the general phase structure of adjoint⊕fundamental theories in the small circle and decompactification limits.
Highlights
Confining gauge theories that are analytically calculable in four dimensions are scarce
Seiberg-Witten theory on R4 [1, 2] and certain QCD-like theories on R3 × S1 [3] are the only two known examples. It has been known for a long time [4,5,6] that compactifying a gauge theory on a circle provides a mechanism for the gauge group to spontaneously break to its maximum abelian subgroup, and for the theory to admit monopole-instantons
In this work we have studied the general problem of classifying su(Nc) gauge theories on R3 × S1 endowed with nG ⊕ nR fermions
Summary
Confining gauge theories that are analytically calculable in four dimensions are scarce. These are stable correlated events made of two monopole-instantons and carry zero topological charge and two units of magnetic charges Once these molecules were identified, it was immediately realized that the confinement mechanism transcends the supersymmetric theory to QCD(adj) [3], which is a Yang-Mills theory endowed with Weyl fermions in the adjoint representation. There has been a tremendous amount of effort to study compactified gauge theories This includes confinement/deconfinement phase transitions [17,18,19], quantum/thermal continuity between compactified super Yang-Mills and hot pure YangMills [13, 20,21,22], the global structure [23], QCD under external magnetic field [24] and at finite density [25]. In appendices A to F we review Lie Algebra, explain the convention, and list a few useful results in group theory that we use throughout this work
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