Loop Equations and the Topological Phase of Multi-Cut Matrix Models

TL;DR

This paper analyzes the double scaling limit of two-cut Hermitian matrix models, revealing a pure topological phase with recursion-determined correlation functions and connections to 2D gravity and dense polymers.

Contribution

It provides an exact solution including all odd scaling operators and derives loop equations as Virasoro constraints for the two-cut matrix model.

Findings

Discovery of a pure topological phase with recursion relations

Exact solution including all odd scaling operators

Relation of macroscopic loop amplitudes to 2D gravity and dense polymers

Abstract

We study the double scaling limit of mKdV type, realized in the two-cut Hermitian matrix model. Building on the work of Periwal and Shevitz and of Nappi, we find an exact solution including all odd scaling operators, in terms of a hierarchy of flows of $2\times 2$ matrices. We derive from it loop equations which can be expressed as Virasoro constraints on the partition function. We discover a ``pure topological" phase of the theory in which all correlation functions are determined by recursion relations. We also examine macroscopic loop amplitudes, which suggest a relation to 2D gravity coupled to dense polymers.