Candidate phases for SU(2) adjoint QCD$_4$ with two flavors from $\mathcal{N}=2$ supersymmetric Yang-Mills theory

Abstract

We study four-dimensional adjoint QCD with gauge group SU(2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and two Weyl fermion flavors, which has an SU(2)_R <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> supersymmetric Yang-Mills theory deformed by a supersymmetry-breaking scalar mass M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:msub> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> symmetry that is familiar from topologically twisted {\mathcal N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> theories. Since every vacuum on the Coulomb branch of the {\mathcal N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> itself, which can be reliably analyzed when M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mrow> <mml:mi>ℂ</mml:mi> <mml:mi>ℙ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> sigma model, which we analyze in detail. Certain topological couplings that are likely to be present in this \mathbb{CP}^1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mrow> <mml:mi>ℂ</mml:mi> <mml:mi>ℙ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> behavior of the deformed {\mathcal N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternati

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