What is the Simplest Gauge-String Duality?

TL;DR

This work proposes the Gaussian Hermitian one-matrix model in the conventional 't Hooft limit as the simplest gauge–string duality and derives a concrete string description of its single-trace correlators. Correlators are shown to be sums over Belyi maps from worldsheet Riemann surfaces to ${\mathbb P}^1$, with a target space identified with the eigenvalue spectral curve, yielding an AdS/CFT–like dictionary built from holomorphic maps that act as stringy Witten diagrams. The construction employs Strebel differential gluing of Feynman ribbon graphs and Razamat's integer-length modification, which localises contributions to arithmetic Riemann surfaces and explicit Belyi maps. A strong link to the A-model topological string on ${\mathbb P}^1$ is demonstrated in the planar limit, including exact identifications of planar Gaussian correlators with genus-zero topological-string amplitudes and a consistent degree-selection rule. Together, these results provide a tractable, intrinsic realisation of gauge–string duality in a non-gravitational setting and hint at deep connections between matrix models, moduli-space geometry, and topological strings.

Abstract

We make a proposal for the string dual to the simplest large $N$ theory, the Gaussian matrix integral in the 'tHooft limit, and how this dual description emerges from double line graphs. This is a specific realisation of the general approach to gauge-string duality which associates worldsheet riemann surfaces to the Feynman-'tHooft diagrams of a large N gauge theory. We show that a particular version (proposed by Razamat) of this connection, involving integer Strebel differentials, naturally explains the combinatorics of Gaussian matrix correlators. We find that the correlators can be explicitly realised as a sum over a special class of holomorphic maps (Belyi maps) from the worldsheet to a {\it target space} ${\mathbb P}^1$. We are led to identify this target space with the riemann surface associated with the (eigenvalues of the) matrix model. In the process, an AdS/CFT like dictionary, for arbitrary correlators of single trace operators, also emerges in which the holomorphic maps play the role of stringy Witten diagrams. Finally, we provide some evidence that the above string dual is the conventional A-model topological string theory on ${\mathbb P}^1$.