A geometric approach to free variable loop equations in discretized theories of 2D gravity

TL;DR

This paper introduces a geometric method to derive equations for generating functions in discrete 2D gravity theories with matter, extending previous matrix model techniques to broader classes of theories.

Contribution

It generalizes Schwinger-Dyson based techniques using graphical arguments, applicable to a wider range of discrete gravity theories beyond matrix models.

Findings

Derived a single graphical equation for generating functions.

Applied method to theories with Ising spins and Potts models.

Included models not describable by matrix models.

Abstract

We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity theories which correspond to matrix models, our method is a generalization of the technique of Schwinger-Dyson equations and is closely related to recent work describing the master field in terms of noncommuting variables; the important differences are that we derive a single equation for the generating function using purely graphical arguments, and that the approach is applicable to a broader class of theories than those described by matrix models. Several example applications are given here, including theories of gravity coupled to a single Ising spin ($c = 1/2$), multiple Ising spins ($c = k/2$), a general class of two-matrix models which includes the Ising theory and its dual, the three-state Potts model, and a dually weighted graph model which does not admit a simple description in terms of matrix models.