Candidate Phases for SU(2) Adjoint QCD$_4$ with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory

TL;DR

This work investigates the infrared behavior of SU(2) adjoint QCD with two Weyl flavors by embedding it into softly broken N=2 SYM and preserving all global symmetries and ’t Hooft anomalies. By tracking the deformation parameter M from the Seiberg–Witten Coulomb branch, the authors identify two robust IR candidates: an exotic Coulomb phase with unbroken chiral symmetry and a CP^1 phase featuring confinement and spontaneous SU(2)_R breaking, each consistent with anomaly constraints. They derive detailed low-energy descriptions: a dual U(1) gauge theory in the Coulomb regime, two CP^1 sigma-model copies in the confinement scenario, and a topological insulator structure on confining strings, with a potential second-order string-worldsheet transition under fermion mass deformations. The analysis links UV anomalies to IR realizations on non-spin manifolds, clarifying how background fields and spin^c structures shape the phase structure and suggesting that large-M dynamics may ultimately favor the CP^1 phase. Overall, the paper provides a coherent anomaly-driven map of candidate IR phases for adjoint QCD, anchored in exact results from Seiberg–Witten theory and topological considerations.

Abstract

We study four-dimensional adjoint QCD with gauge group SU(2) and two Weyl fermion flavors, which has an $SU(2)_R$ chiral symmetry. The infrared behavior of this theory is not firmly established. We explore candidate infrared phases by embedding adjoint QCD into $\mathcal{N}=2$ supersymmetric Yang-Mills theory deformed by a supersymmetry-breaking scalar mass M that preserves all global symmetries and 't Hooft anomalies. This includes 't Hooft anomalies that are only visible when the theory is placed on manifolds that do not admit a spin structure. The consistency of this procedure is guaranteed by a nonabelian spin-charge relation involving the $SU(2)_R$ symmetry that is familiar from topologically twisted $\mathcal{N}=2$ theories. Since every vacuum on the Coulomb branch of the $\mathcal{N}=2$ theory necessarily matches all 't Hooft anomalies, we can generate candidate phases for adjoint QCD by deforming the theories in these vacua while preserving all symmetries and 't Hooft anomalies. One such deformation is the supersymmetry-breaking scalar mass M itself, which can be reliably analyzed when M is small. In this regime it gives rise to an exotic Coulomb phase without chiral symmetry breaking. By contrast, the theory near the monopole and dyon points can be deformed to realize a candidate phase with monopole-induced confinement and chiral symmetry breaking. The low-energy theory consists of two copies of a $\mathbb{CP}^1$ sigma model, which we analyze in detail. Certain topological couplings that are likely to be present in this $\mathbb{CP}^1$ model turn the confining solitonic string of the model into a topological insulator. We also examine the behavior of various candidate phases under fermion mass deformations. We speculate on the possible large-M behavior of the deformed $\mathcal{N}=2$ theory and conjecture that the $\mathbb{CP}^1$ phase eventually becomes dominant.