Symmetries and Strings of Adjoint QCD${}_2$

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 $T_k\sim |m| \sin(\pi k /N)$ 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.