Global KdV Flows and Stable 2D Quantum Gravity

TL;DR

This paper explores the global KdV flows in 2D quantum gravity models, demonstrating their existence and uniqueness across various critical models through analytical and numerical methods.

Contribution

It introduces a framework for understanding KdV flows as deformations of linear problems in stable 2D quantum gravity models, establishing their existence and uniqueness.

Findings

Flow existence conditions are satisfied for relevant models.

Numerical evidence of flows between key solutions.

Conjecture of universal applicability to all m-critical models.

Abstract

The string equation for the $[{\tilde P},Q]=Q$ formulation of non--perturbatively stable 2D quantum gravity coupled to the $(2m-1,2)$ models is studied. Global KdV flows between the appropriate solutions are considered as deformations of two compatible linear problems. It is demonstrated that the necessary conditions for such flows to exist are satisfied. A numerical study reveals such flows between the pole--free solutions of pure gravity ($m=2$), the Lee--Yang edge model ($m=3$) and topological gravity ($m=1$). We conjecture that this is the case for all of the $m$--critical models. As the $m=1$ solution is unique these global flows define a {\sl unique} solution for each $m$--critical model.