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

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 SchwingerDyson 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. CTP # 2465 October 1995 THU-95/28 hep-th/9510199 This work was supported in part by the National Science Foundation under grants PHY/9200687 and PHY/9108311, by the U.S. Department of Energy (D.O.E.) under cooperative agreement DE-FC0294ER40818, by the divisions of applied mathematics of the D.O.E. under contracts DE-FG02-88ER25065 and DE-FG02-88ER25065, and by the European Community Human Capital Mobility programme. Address after January 1996: Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK. Address after January 1996: Department of Physics, Joseph Henry Laboratories, Princeton University, Princeton NJ 08544.