Stabilised Matrix Models for Non-Perturbative Two Dimensional Quantum Gravity

TL;DR

This paper analyzes stabilized hermitian one matrix models for two-dimensional quantum gravity across all multicritical points, revealing non-perturbative effects and their relation to integrable hierarchies and orthogonal polynomials.

Contribution

It provides a comprehensive analysis of stabilised matrix models at all multicritical points and clarifies the role of supersymmetry and non-perturbative effects in 2D quantum gravity.

Findings

Non-perturbative effects lead to loss of direct relation to KdV hierarchy at even multicritical points.

Orthogonal polynomials satisfy a Hartree-Fock equation.

Supersymmetry breaking is not the mechanism for non-perturbative effects in these models.

Abstract

A thorough analysis of stochastically stabilised hermitian one matrix models for two dimensional quantum gravity at all its $(2,2k-1)$ multicritical points is made. It is stressed that only the zero fermion sector of the supersymmetric hamiltonian, i.e., the forward Fokker-Planck hamiltonian, is relevant for the analysis of bosonic matter coupled to two dimensional gravity. Therefore, supersymmetry breaking is not the physical mechanism that creates non perturbative effects in the case of points of even multicriticality $k$. Non perturbative effects in the string coupling constant $g_{str}$ result in a loss of any explicit relation to the KdV hierarchy equations in the latter case, while maintaining the perturbative genus expansion. As a by-product of our analysis it is explicitly proved that polynomials orthogonal relative to an arbitrary weight $\exp (-\beta V(x))$ along the whole real line obey an Hartree-Fock equation.