Quantum mechanics of the vacuum state in two-dimensional QCD with adjoint fermions

A study of two-dimensional QCD on a spatial circle with Majorana fermions in the adjoint representation of the gauge groups SU(2) and SU(3) is performed. The main emphasis is put on the symmetry properties related to the homotopically nontrivial gauge transformations and the discrete axial symmetry of this model. Within a gauge-fixed canonical framework, the delicate interplay of topology on the one hand and Jacobians and boundary conditions arising in the course of resolving Gausss law on the other hand is exhibited. As a result, a consistent description of the residual ZN gauge symmetry [for SU(N)] and the axial anomaly emerges. For illustrative purposes, the vacuum of the model is determined analytically in the limit of a small circle. There, the Born-Oppenheimer approximation is justified and reduces the vacuum problem to simple quantum mechanics. The issue of fermion condensates is addressed and residual discrepancies with other approaches are pointed out.