Adjoint Majorana QCD$_2$ at Finite $N$

The mass spectrum of $1+1$-dimensional $\mathrm{SU}(N)$ gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large $N$ limit using Light-Cone Quantization. Here we extend this approach to theories with small values of $N$, exhibiting explicit results for $N=2, 3$, and $4$. In the context of Discretized Light-Cone Quantization, we develop a procedure based on the Cayley-Hamilton theorem for determining which states of the large $N$ theory become null at finite $N$. For the low-lying bound states, we find that the squared masses divided by $g^2 N$, where $g$ is the gauge coupling, have very weak dependence on $N$. The coefficients of the $1/N^2$ corrections to their large $N$ values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a Kac-Moody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to $g^2 N / \pi$, we find evidence that the spectrum exhibits boson-fermion degeneracies, in agreement with the supersymmetry of the model at any value of $N$.