  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.
