Large-\N master-field convergence conjecture for Yang\Mills theory on surfaces

Let $G_N$ be a classical compact matrix Lie group of size $N$, and let $\Sigma$ be a compact surface, the Euclidean plane $\mathbb{R}^2$, or the Poincar\e9 disc $\mathbb{D}_{\mathfrak{h}}$. For a loop $\ell$ of $\Sigma$, let $W_\ell$ be its normalized Wilson loop under the Yang\Mills measure $\mathrm{YM}_\Sigma$. Master-field convergence conjecture. There is a constant $\Phi_\Sigma(\ell)$ such that

$$W_\ell\to\Phi_\Sigma(\ell) \quad\text{in probability as }N\to\infty.$$

The functional $\Phi_\Sigma$ is called the master field on $\Sigma$. This conjecture is proved for the sphere in some cases and, in the paper, for the torus; convergence for general compact surfaces and all loops remains open.