  We study $1+1$-dimensional $\text{SU}(N)$ gauge theory coupled to one adjoint
multiplet of Majorana fermions on a small spatial circle of circumference $L$.
Using periodic boundary conditions, we derive the effective action for the
quantum mechanics of the holonomy and the fermion zero modes in perturbation
theory up to order $(gL)^3$. When the adjoint fermion mass-squared is tuned to
$g^2 N/(2\pi)$, the effective action is found to be an example of
supersymmetric quantum mechanics with a nontrivial superpotential. We separate
the states into the $\mathbb{Z}_N$ center symmetry sectors (universes) labeled
by $p=0, \ldots, N-1$ and show that in one of the sectors the supersymmetry is
unbroken, while in the others it is broken spontaneously. These results give us
new insights into the $(1,1)$ supersymmetry of adjoint QCD$_2$, which has
previously been established using light-cone quantization. When the adjoint
mass is set to zero, our effective Hamiltonian does not depend on the fermions
at all, so that there are $2^{N-1}$ degenerate sectors of the Hilbert space.
This construction appears to provide an explicit realization of the extended
symmetry of the massless model, where there are $2^{2N-2}$ operators that
commute with the Hamiltonian. We also generalize our results to other gauge
groups $G$, for which supersymmetry is found at the adjoint mass-squared $g^2
h^\vee/(2\pi)$, where $h^\vee$ is the dual Coxeter number of $G$.
