Generating Functions in Two Dimensional Quantum Gravity

TL;DR

This paper introduces a method to compute generating functions in two-dimensional quantum gravity using 1-matrix models without the double scaling limit, enabling analysis across different topologies and extracting universal terms.

Contribution

It presents a novel approach to calculate generating functions directly from 1-matrix models, applicable to higher genus surfaces, and identifies universal terms in the solutions.

Findings

Exact generating functions for simple and double torus computed.

Universal terms identifiable in the double scaling limit.

Regular part of spherical generating function is bilinear in source couplings.

Abstract

We solve general 1-matrix models without taking the double scaling limit. A method of computing generating functions is presented. We calculate the generating functions for a simple and double torus. Our method is also applicable to more higher genus. Each generating function can be expressed by a ``specific heat'' function for sphere. Universal terms, which are survived in the double scaling limit can be easily picked out from our exact solutions. We also find that the regular part of the spherical generating function is at most bilinear in coupling constants of source terms.