Free Variable Loop Equations for the 3-State Potts Model Coupled to 2D Gravity

TL;DR

This paper develops a new geometric free variable approach to derive generating functions for disk amplitudes in the 3-state Potts model coupled to 2D gravity, extending previous methods with a unified set of equations.

Contribution

It introduces a generalized method using graphical arguments to derive a single generating equation, expanding the class of constraints and determining the spectral curve.

Findings

Derived a closed set of equations for the spectral curve.

Extended the class of constraints beyond previous approaches.

Provided a unified generating equation for disk amplitudes.

Abstract

We revisit the long standing problem of the geometric free variable approach to computing the generating function for disk amplitudes in the matrix model formulation of the 3-state Potts model coupled to 2D discrete gravity. This method is a mild generalisation of Schwinger Dyson (loop-)equations approach, using graphical arguments to derive a single generating equation in non-commutative variables that contain the information of all loop equations. The method of constructing the generating equation is reviewed, we explain how a larger class of constraints than previously known can be constructed and we derive a closed set of equations that determine the spectral curve for the planar graph generating function.