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We consider $\text{SU}(2)$ quantum chromodynamics in $1+1$ dimensions with asingle quark in the spin $J$ representation of the gauge group and study thetheory in the large $J$ limit where the gauge coupling $g^2 \to 0$ and $J \to\infty$ with $\lambda = g^2 J^2$ fixed. We work with a Dirac spinor field forarbitrary $J$, and with a Majorana spinor for integer $J$ since the integerspin representations of $\text{SU}(2)$ are real, and analyse the two casesseparately.  The theory is reformulated in terms of global color non-singlet fermionbilocal operators which satisfy a $W_\infty \times \text{U}(2J+1)$ algebra. Inthe large $J$ limit, the dynamics of the bilocal fields is captured byfluctuations along a particular coadjoint orbit of the $W_\infty$ algebra. Weshow that the global colour-singlet sector of the bilocal field fluctuationssatisfy the same integral equation for meson wavefunctions that appears in the't Hooft model. For Majorana spinors in the integer spin $J$ representation,the Majorana condition projects out half of the meson spectrum, as a result ofwhich the linear spacing of the asymptotic meson spectrum for Majorana fermionsis double that of Dirac fermions. The Majorana condition also projects out thezero mass bound state that is present for the Dirac quark at zero quark mass.  We also consider the formulation of the model in terms of local chargedensities and compute the quark spectral function in the large $J$ limit: wesee evidence for the absence of a pole in the quark propagator.

Meson spectrum of $\text{SU}(2)$ QCD$_{1+1}$ with Quarks in Large Representations