On the Schwinger-Dyson Equations for a Vertex Model Coupled to 2D Gravity

TL;DR

This paper analyzes a two-matrix model coupled to 2D gravity, deriving closed-form Schwinger-Dyson equations for correlators in the planar limit, thus providing a solution for a class of vertex models on fluctuating geometries.

Contribution

It introduces a solvable matrix model with a quartic interaction describing vertex models coupled to 2D gravity, and derives decoupled Schwinger-Dyson equations in the planar limit.

Findings

Decoupling of loop equations in the planar limit

Closed equations for correlators and eigenvalue density

Perturbative corrections to the free case outlined

Abstract

We consider a two matrix model with gaussian interaction involving the term $tr ABAB$, which is quartic in angular variables. It describes a vertex model (in particular case - of F-model type) on the lattice of fluctuating geometry and is the simplest representative of the class of matrix models describing coupling to two-dimensional gravity of general vertex models. This class includes most of physically interesting matrix models, such as lattice gauge theories and matrix models describing extrinsic curvature strings. We show that the system of loop (Schwinger-Dyson) equations of the model decouples in the planar limit and allows one to find closed equations for arbitrary correlators, including the ones involving angular variables. This provides a solution of the model in the planar limit. We write down the equations for the two-point function and the eigenvalue density and sketch the calculation of perturbative corrections to the free case.