The 't Hooft equation as a quantum spectral curve

In an attempt to establish a link between different quantization methods, I examine the massless 't Hooft equation, which governs meson bound state wavefunctions in 2d SU(N) gauge theory in the large-N limit. The integral equation can also be obtained by lightcone quantizing a folded string in flat space. The folded string is a limiting case of a more general setup: a four-segmented string moving in AdS$_3$. I compute its classical spectral curve by using celestial variables and planar bipartite graphs (on-shell diagrams/brane tilings). The adjugate of the Kasteleyn matrix vanishes at two special points, which ensures that the string segments form a closed loop in target space. The Hamiltonian takes on a Ruijsenaars-Schneider form and the phase space is a coadjoint SL(2) orbit whose middle region has been removed. In AdS, the 't Hooft equation acquires an extra term, which has previously been proposed as an effective confining potential in QCD. After an integral transform, the equation can be inverted in terms of a finite difference equation. I show that this difference equation can be interpreted as the quantized (non-analytic) spectral curve of the string. I calculate the spectrum numerically, which interpolates between $M^2=n(n+1)$ in the tensionless limit and 't Hooft's nearly linear Regge trajectory at infinite AdS radius.