On the Random Matrix Model of the Virasoro Minimal String

TL;DR

This paper presents a non-perturbative approach to the Virasoro Minimal String using double-scaled orthogonal polynomials, revealing key features of the model's spectral properties and confirming the special nature of the (c=25, ĉ=1) case.

Contribution

It introduces an alternative non-perturbative definition of the Virasoro Minimal String via orthogonal polynomials, enabling detailed spectral analysis and solutions of the string equation.

Findings

The (c=25, ĉ=1) case shares features with supersymmetric JT gravity.

Solutions to the full string equation are constructed explicitly.

Spectral density and spectral form factor are computed beyond perturbation.

Abstract

The model of two dimensional quantum gravity defining the "Virasoro Minimal String", presented recently by Collier, Eberhardt, M\"{u}hlmann, and Rodriguez, was also shown to be perturbatively (in topology) equivalent to a random matrix model. An alternative definition is presented here, in terms of double-scaled orthogonal polynomials, thereby allowing direct access to non-perturbative physics. Already at leading order, the defining string equation's properties yield valuable information about the non-perturbative fate of the model, confirming that the case $(c{=}25,{\hat c}{=}1)$ (central charges of spacelike and timelike Liouville sectors) is special, by virtue of sharing certain key features of the ${\cal N}{=}1$ supersymmetric JT gravity string equation. Solutions of the full string equation are constructed using a special limit, and the (Cardy) spectral density is completed to all genus and beyond. The distributions of the underlying discrete spectra are readily accessible too, as is the spectral form factor. Some examples of these are exhibited.