Adjoint QCD$_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension

TL;DR

The paper addresses the IR fate of 4d adjoint QCD$_4$ with SU(2) gauge group and two adjoint Weyl fermions by analyzing higher-form 't Hooft anomalies. It extends the symmetry-extension program to higher (2-group) symmetries, using higher group cohomology and cobordism to construct symmetric anomalous TQFTs and to test anomaly saturation; it proves a no-go result for fully saturating the Type I anomaly with odd class $k$ while obtaining constructive symmetric realizations for Type II, and it shows how a joint extension can saturate Type I (even class) together with Type II. The results constrain proposed IR phases and UV–IR dualities (e.g., Bi–Senthil's scenario) and illuminate connections to deconfined quantum criticality and 3+1D quantum spin/fermionic liquids. Overall, the work advances higher-symmetry/topological approaches to 4d gauge theories, clarifying which anomaly-saturating TQFTs are feasible and outlining implications for condensed matter analogs and nonperturbative IR dynamics.

Abstract

Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully trivialized, with the background 1-form field $A$, Pontryagin square $\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go scenario via symmetry-extension for specific higher anomalies. We comment on the implications and constraints on deconfined quantum critical points (dQCP), quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter, and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.

Figures (12)

• Figure 1: Candidate phases of fundamental QCD$_4$ and their possible dynamical fates. "ChSB" means the "chiral symmetry breaking phase." "Pure YM" means the pure Yang-Mills gauge theory with a SU($N_c$) gauge group. "CFT" means conformal field theory. "UV free" or "asym. free" means the asymptotic free. The question mark "$?$" means the detailed structure of the phase boundaries requires further studies.
• Figure 2: Candidate phases of adjoint QCD$_4$ with an SU(2) gauge group ($N_c=2$) and their possible dynamical fates. "ChSB" means the "chiral symmetry breaking phase." The Scenario \ref{['s1']}, \ref{['s2']}, \ref{['s3']}, and \ref{['s4']} are from the list summarized in Sec. \ref{['sec:fate-adjoint']}. The question marks "$?$, $??$, and $???$" means the detailed structure of the phase boundaries requires further studies.
• Figure 3: For the ${\cal N}=2$ supersymmetric Yang-Mills theory (SYM) or Seiberg-Witten theory Seiberg1994rs9407087, there is a moduli space of the supersymmetric vacua, labeled by the expectation value of the complex $u = {\rm Re} \; u + \space\mathrm{i}\space \operatorname{Im} u \equiv \langle {\mathrm{Tr}}(\phi^2) \rangle \in {\mathbb{C}}$ for the ${\cal N}=2$ chiral operator. The $\tau \equiv \tau_{{\rm U}(1)} \equiv \frac{\theta}{2\pi}+ \frac{2\pi \space\mathrm{i}\space}{e^2} =\space\mathrm{i}\space$ is the special point at the self-dual value of the coupling $\tau$. The coupling $\tau$ is for the Coulomb phase of U(1) gauge theory (the Coulomb branch for the moduli space of vacua). Even though the SYM is supersymmetric, Ref. [2018arXiv180609592C] enumerates the possible vacua by supersymmetry-breaking deformations. Let us relate the supersymmetric vacua to the adjoint QCD$_4$ vacua listed in the Scenarios 2018arXiv180609592C: $\diamond$ The vacua of magnetic monopole point ($u=\Lambda^2$) and dyon point ($u=-\Lambda^2$) is related by the broken symmetry $\mathbb{Z}_{8, {\rm A}}$ generator, which gives rise a Scenario \ref{['s1']}. $\diamond$ A generic vacua of $u \neq 0$, and $u \neq \pm \Lambda^2$ on the plane is related to a Scenario \ref{['s2']}. $\diamond$ The vacua of the $u=0$ with a self-dual coupling $\tau_{}$ is related to a Scenario \ref{['s4']}.
• Figure 4: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_2)$. Each dot indicates a $\mathbb{Z}_2$, the short straight line indicates a $\mathrm{Sq}^1$, the curved line indicates a $\mathrm{Sq}^2$.
• Figure 5: The $\mathcal{A}_2(1)$-module structure of $\operatorname{H}^{*+2}(M\mathbb{Z}_4,\mathbb{Z}_2)$. The dashed lines indicate a $(2,4)$-Bockstein. Each dot indicates a $\mathbb{Z}_2$, the short straight line indicates a $\mathrm{Sq}^1$, the curved line indicates a $\mathrm{Sq}^2$.