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

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.