Algebraic curve for a cusped Wilson line

We consider the classical limit of the recently obtained exact near-BPS result for the anomalous dimension of a cusped Wilson line with the insertion of an operator with L units of R-charge at the cusp in planar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 4 SYM. The classical limit requires taking both the ’t Hooft coupling and L to infinity. Since the formula for the cusp anomalous dimension involves determinants of size proportional to L, the classical limit requires a matrix model reformulation of the result. Building on results of Gromov and Sever, we construct such a matrix model-like representation and find the corresponding classical algebraic curve. Using this we find the classical value of the cusp anomalous dimension and the 1-loop correction to it. We check our results against the energy of the classical solution and numerically by extrapolating from the quantum regime of finite L.