Solving Topological 2D Quantum Gravity Using Ward Identities

TL;DR

This paper introduces a topological method for calculating correlation functions in (1,q) models of 2D quantum gravity, providing new recursion relations and algebraic structures that enhance understanding of non-critical string theory.

Contribution

It presents a novel topological procedure, new recursion relations, and an extended contact algebra for (1,q) models, advancing computational techniques in 2D quantum gravity.

Findings

Derived new recursion relations extending W constraints

Computed an extended contact algebra beyond Verlinde's algebra

Developed KdV-based techniques for correlation function calculations

Abstract

A topological procedure for computing correlation functions for any (1,q) model is presented. Our procedure can be used to compute any correlation function on the sphere as well as some correlation functions at higher genus. We derive new and simpler recursion relations that extend previously known results based on W constraints. In addition, we compute an effective contact algebra with multiple contacts that extends Verlindes' algebra. Computational techniques based on the KdV approach are developed and used to compute the same correlation functions. A simple and elegant proof of the puncture equation derived directly from the KdV equations is included. We hope that this approach can lead to a deeper understanding of D=1 quantum gravity and non-critical string theory. (Paper uses tex TeX macro package jytex and includes 8 Postscript figures in the text using dvips (and epsf). Instructions for processing are included.)