2D Gravity and Random Matrices

TL;DR

This paper reviews recent advances in 2D gravity coupled with conformal matter using random matrix models, discussing solutions, non-perturbative issues, and connections to topology and moduli space.

Contribution

It provides a comprehensive overview of matrix model techniques, their solutions, and their relation to continuum theories and topological aspects of 2D gravity.

Findings

Matrix models can be solved using orthogonal polynomials for d<1 matter.

Non-linear differential equations like Painlevé describe sum over topologies.

Perturbation series are not Borel summable, affecting non-perturbative definitions.

Abstract

We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising--like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlev\'e equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.