The 't Hooft model on (A)dS$_2$

Abstract

We solve large-$N$ two-dimensional QCD with fundamental quarks---the 't Hooft model---on rigid AdS$_2$ and dS$_2$. Their isometries preserve the null gauge $A_-=0$ and in affine coordinates the curvature enters the light-front Hamiltonian only through the special-conformal generator of the null line. The $SL(2,\mathbb{R)}$ Casimir then results in an exact meson equation: 't Hooft's equation plus a Legendre operator whose sign is determined by the curvature. On AdS$_2$ it is related to equations proposed in the context of folded/segmented strings and light-front holography. The pion is massless in the chiral limit and obeys a Gell-Mann--Oakes--Renner relation that fixes the chiral condensate, the dimensions cross over from Regge to logarithmic growth once the flux tube exceeds the curvature radius, and the flat-space integrable structure survives. We also solve the problem of a quark interacting with an antiquark placed at the boundary: beyond a critical coupling the string pulls it back to the boundary, resulting in a scattering phase. On dS$_2$ and due to the expansion of space, the mesons are unstable resonances even at $N=\infty$ and eventually dissociate. Their widths are given by Boltzmann factors at the de Sitter temperature.