Unitary Matrix Models and 2D Quantum Gravity (Virasoro constraints modified)

TL;DR

This paper explores the connection between unitary matrix models and 2D quantum gravity, showing how integrable hierarchies relate to string theory and providing non-perturbative solutions that unify different descriptions of the gravitational system.

Contribution

It demonstrates that the KdV and modified KdV hierarchies describe the same 2D gravitational system and links non-perturbative solutions of matrix models to string susceptibility equations.

Findings

Non-perturbative solutions map to non-singular solutions of the renormalisation group equation.

Large N solutions in external fields match pure closed-string solutions.

KdV and modified KdV hierarchies are equivalent descriptions of the same system.

Abstract

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system -- open-closed string theory. Non-perturbative solutions of the multi-critical unitary matrix models map to non-singular solutions of the `renormalisation group' equation for the string susceptibility, $[\tilde{P},Q]=Q$. We also demonstrate that the large N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of $[\tilde{P},Q]=Q$.