Symmetries and Strings of Adjoint QCD${}_2$

TL;DR

The paper shows that massless adjoint QCD in 1+1D with SU(N) gauge group hosts a large zoo of non-invertible topological lines (~2^{2N}) that RG-invariantly constrain infrared physics, forcing deconfinement of fundamental Wilson lines for all N. By leveraging non-Abelian bosonization, RCFT fusion categories, and alpha-induction, the authors map the UV theory to a coset TQFT Spin(N^2-1)_1/SU(N)_N and classify possible IR phases using modular invariants and module categories, yielding precise results for k-string tensions at small mass and explicit vacua counting (universes) for N≤5. They show that small quartic fermion deformations can break non-invertible lines and induce confinement in a controlled way, while certain deformations preserve a subset of lines, predicting partial confinement. The work provides a detailed, symmetry-based framework linking fusion categories, modular invariants, and IR TQFTs to strongly coupled 2d gauge dynamics, with precise, testable predictions such as T_k ~ |m| sin(πk/N) for small m and the existence of massless worldsheet modes at intermediate masses. Overall, the study demonstrates how category-theoretic symmetries deliver strong IR constraints and analytic results in a tractable non-supersymmetric 2d gauge theory, with implications for confinement, deconfinement, and emergent supersymmetry on flux tubes.

Abstract

We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group $SU(N)$. The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits $\sim 2^{2N}$ non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all $N$ these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, $m\ll g_\mathrm{YM}$, the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the $k$-string tension for $N\leq 5$. Our results suggest that the $k$-string tension, $T_k$, is $T_k\sim |m| \sin(πk /N)$ for all $N$. We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for $SU(N\leq5)$ gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of $N$. The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

Figures (12)

• Figure 1: The symmetry operator $U(g)$ for an element $g$ of the symmetry group $G$ causes the symmetry action on a local operator when the symmetry operator passes through the local operator.
• Figure 2: The non-invertible duality line $\mathcal{N}$ in the critical Ising CFT maps the spin operator $\sigma$ to the disorder operator $\mu$ which lives at the edge of the $\mathbb{Z}_2$ symmetry line $\eta$.
• Figure 3: The $G_k$ WZW model can be obtained by considering the 2+1-d CS theory on $I\times \Sigma_2$, where $I$ is the interval and $\Sigma_2$ is the 2d spacetime of the WZW model. The boundary conditions $\mathcal{B}$ and $\overline{\mathcal{B}}$ are conjugate of each other. (a): a local operator $\mathcal{O}_\mu$ in the Verma module $\mathrm{V}_\mu\otimes \overline{\mathrm{V}}_\mu$ can be realized as the Wilson line $L^\text{CS}_\mu$ in the representation $\mu$ bridging the two boundaries. (b): a Verlinde line $L_\mu$ is identified with the bulk Wilson line $L_\mu^\text{CS}$ along the 2d spacetime $\Sigma_2$
• Figure 4: Local operator $\mathcal{O}_{\mu,{\bar{\mu}}}^m$
• Figure 5: Topological line $\alpha^+_\mu$

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Example : $\hat{\mathfrak{su}}(2)_k$ Modular Invariants