A remark on the three approaches to 2D Quantum gravity

TL;DR

This paper explores the connections between the one-matrix model, Liouville gravity, and topological gravity, showing their generating functions are analytically related and verifying recursion relations in the matrix model.

Contribution

It demonstrates that the generating function of the one-matrix model aligns with Liouville gravity and is an analytic continuation of topological gravity's generating function, using Kontsevich's theorem.

Findings

The generating function of the one-matrix model coincides with that of Liouville gravity.

The generating function is an analytic continuation of topological gravity.

Recursion relations are verified in the p-critical matrix model.

Abstract

The one-matrix model is considered. The generating function of the correlation numbers is defined in such a way that this function coincide with the generating function of the Liouville gravity. Using the Kontsevich theorem we explain that this generating function is an analytic continuation of the generating function of the Topological gravity. We check the topological recursion relations for the correlation functions in the $p$-critical Matrix model.