A GEOMETRIC APPROACH TO FREE VARIABLE LOOP EQUATIONS IN DISCRETIZED THEORIES OF 2D GRAVITY

We present a self-contained analysis of theories of discrete 2D gravit y coupled to matter, using geometric methods to derive equations for g enerating 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 eq uations and is closely related to recent work describing the master fi eld in terms of noncommuting variables; the important differences are that we derive a single equation for the generating function using pur ely graphical arguments, and that the approach is applicable to a broa der class of theories than those described by matrix models. Several e xample applications are given here, including theories of gravity coup led to a single Ising spin (c = 1/2), multiple Ising spins (c = 1/2k), a general class of two-matrix models which includes the Ising theory and its dual, the three-state Ports model, and a dually weighted graph model which does not admit a simple description in terms of matrix mo dels.