Random Surfaces and Quantum Gravity

The Cargese Workshop Random Surfaces and Quantum Gravity was held from May 27 to June 2, 1990. Little was known about string theory in the non-perturbative regime before Oetober 1989 when non-perturbative equations for the string partition functions were found by using methods based on the random triangulations of surfaees. This set of methods pro vides a deseription of non-eritical string theory or equivalently of the coupling of matter fields to quantum gravity in two dimensions. The Cargese meeting was very successful in that it provided the first opportunity to gather most of the active workers in the field for a fuH week of lectures and extensive informal discussions about these exeiting new developments. The main results were reviewed, recent advances were explained, new results and conjectures (which appear for the first time in these proceedings) were presented and discussed. Among the most important topics discussed at the workshop were: The relation of KdV theory to loop equations and the Virasoro algebra, new results in Liouville field theory, effective (1 + 1) dimensional theory for 2 - D quantum gravity coupled to c = 1 matter and its fermionization, proposal for a new geometrical interpretation of the string equation and possible definition of quantum Riemann surfaces, discussion of the string equation for the multi-matrix models, links with topological field theories of gravity, issues in using target space supersymmetry to define good theories, definition of the partition function via analytic continuation, new models of random surfaces

Proceedings of a NATO ARW held in Cargese, France, May 27--June 2, 1990

Diffusion Equation, Continuum Limit and Universality in 2D Quantum Gravity.- On Triangles and Squares.- Non-Perturbative Effects in 2D Gravity and Matrix Models.- Integrable Models of 2D Quantum Gravity.- Topological Strings and Loop Equations.- The Two-Matrix Model.- Action Principle and Large Order Behavior of Non-Perturbative Gravity.- W -Geometry.- Strings Embedded in Dynkin Diagrams.- Phase Transitions in One-Matrix Models.- Matrix Models of 2D Gravity and Isomonodromic Deformation.- The Strength of Nonperturbative Effects in String Theory.- Approximating the Critical String Measure Using Dynamically Triangulated Surfaces.- Matrix Models, String Field Theory and Topology.- Multipoint Correlation Functions in One-Dimensional String Theory.- The Penner Model and D = 1 String Theory.- Non-Perturbative String Theory.- Bosonic Strings and String Field Theories in One-Dimensional Target Space.- A Few Remarks on Low Dimensional String Theory.- Excitations and Interactions in a One-Dimensional String Theory.- Random Surfaces in Dimensions Larger Than One.- On Gauge Invariances in Stochastic Quantization.- The Quantum Group Structure of Quantum Gravity in Two Dimensions.- Notes on Quantum Liouville Theory and Quantum Gravity.