Dynamics of QCD$_{3}$ with Rank-Two Quarks And Duality

TL;DR

This work analyzes 2+1D SU(N) Yang–Mills–Chern–Simons theories with a Dirac fermion in rank-two symmetric or antisymmetric representations, identifying a critical level $k_{ ext{crit}}$ below which an intermediate quantum phase with emergent TQFTs appears, and at $k=0$ a baryon condensate breaks $U(1)_B$ with a Nambu–Goldstone boson coupled to a TQFT. For large $k$ the theory flows to two semiclassical phases described by $SU(N)_{k ext{±}T(R)}$, while for $0<k<k_{ ext{crit}}$ an intrinsically quantum phase connects these extremes; the authors propose new fermion–fermion dualities linking these theories to dual $U(T(R) ext{±}k)$ descriptions and give detailed phase diagrams. They validate the picture with nontrivial consistency checks, including special-case dualities (e.g., $SU(4)_k$ antisymmetric $ ightsquigarrow Spin(6)_k$ with $N_f=2$) and matching of gravitational and baryon counterterms across dual paths, and they explore domain walls in corresponding 4D theories, where wall theories reflect the 3D phase structure. The results illuminate nonperturbative IR dynamics of two-index matter in 3D gauge theories and connect to 4D domain-wall physics and time-reversal anomalies, providing a framework for further dualities and consistency tests in non-supersymmetric gauge theories.

Abstract

Three-dimensional gauge theories coupled to fermions can develop interesting nonperturbative dynamics. Here we study in detail the dynamics of $SU(N)$ gauge theories coupled to a Dirac fermion in the rank-two symmetric and antisymmetric representation. We argue that when the Chern-Simons level is sufficiently small the theory develops a quantum phase with an emergent topological field theory. When the Chern-Simons level vanishes, we further argue that a baryon condenses and hence baryon symmetry is spontaneously broken. The infrared theory then consists of a Nambu-Goldstone boson coupled to a topological field theory. Our proposals also lead to new fermion-fermion dualities involving fermions in two-index representations. We make contact between our proposals and some recently discussed aspects of four-dimensional gauge theories. This leads us to a proposal for the domain wall theories of non-supersymmetric gauge theories with fermions in two-index representations. Finally, we discuss some aspects of the time-reversal anomaly in theories with a one-form symmetry.

Figures (8)

• Figure 1: Phase diagram of $SU(N)$ gauge theory with a symmetric fermion for $k\geq\frac{N+2}{2}$. The solid circle represents a phase transition between the asymptotic phases. For sufficiently large $k$ we know for certain that the phase transition is associated with a CFT.
• Figure 2: Phase diagram of $SU(N)$ with an antisymmetric fermion for $k\geq\frac{N-2}{2}$. The solid circle represents a phase transition between the asymptotic phases. For sufficiently large $k$ we know for certain that the phase transition is associated with a CFT.
• Figure 3: Phase diagram of $SU(N)$ with a symmetric fermion for $0<k<\frac{N+2}{2}$. The solid circles represent a phase transition between the asymptotic phases and the intermediate quantum phase. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by $m_\psi=-m_{\hat{\psi}}$ and $m_\psi=-m_{\tilde{\psi}}$.
• Figure 4: Phase diagram of $SU(N)$ with an antisymmetric fermion for $0<k<\frac{N-2}{2}$. The solid circles represent a phase transition between the asymptotic phases and the intermediate quantum phase. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by $m_\psi=-m_{\hat{\psi}}$ and $m_\psi=-m_{\tilde{\psi}}$.
• Figure 5: Phase diagram of $SU(N)$ gauge theory with a symmetric fermion for $k=0$. The circle $S^1$ represents the corresponding sigma model. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by $m_\psi=-m_{\hat{\psi}}$ and $m_\psi=-m_{\tilde{\psi}}$.