What is the Simplest Gauge-String Duality?

TL;DR

This paper proposes a simple gauge-string duality for the Gaussian matrix model, showing how its correlators can be represented as sums over holomorphic maps, and suggests the dual is an A-model topological string on ${ m P}^1$.

Contribution

It introduces a concrete realization of gauge-string duality for the Gaussian matrix model using Strebel differentials and Belyi maps, linking correlators to topological string theory.

Findings

Correlators expressed as sums over Belyi maps.

Identification of the dual as A-model topological string on ${ m P}^1$.

Emergence of an AdS/CFT-like dictionary for the matrix model.

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 ${\mathbb P}^1$.