Symmetries and Mass Splittings in QCD$_2$ Coupled to Adjoint Fermions

Abstract

Two dimensional QCD coupled to fermions in the adjoint representation of the gauge group $SU(N)$, a useful toy model of QCD strings, is supersymmetric for a certain ratio of quark mass and gauge coupling constant. Here we study the theory in the vicinity of the supersymmetric point; in particular we exhibit the algebraic structure of the model and show that the mass splittings as one moves away from the supersymmetric point obey a universal relation of the form ${M_i}^2(B)-{M_i}^2(F)=M_i\delta m+O(\delta m^3)$. We discuss the connection of this relation to string and quark model expectations and verify it numerically for large $N$. At least for low lying states the $O(\delta m^3)$ corrections are extremely small. We also discuss a natural generalization of QCD$_2$ with an infinite number of couplings, which preserves SUSY. This leads to a Landau -- Ginzburg description of the theory, and may be useful for defining a scaling limit in which smooth worldsheets appear.