't Hooft expansion of multi-boundary correlators in 2D topological gravity

TL;DR

This paper develops a method to compute multi-boundary correlators in 2D topological gravity using a 't Hooft expansion, solving the Korteweg-De Vries equation, and reproduces known results in related models.

Contribution

It introduces an algebraic approach to compute multi-boundary correlators' 't Hooft expansion directly from lower boundary cases in 2D topological gravity.

Findings

Explicit computation of 1, 2, 3 boundary correlators' 't Hooft expansion.

Method reproduces known results in Jackiw-Teitelboim gravity.

Independent calculation of three-boundary correlator in the Airy case.

Abstract

We study multi-boundary correlators of Witten-Kontsevich topological gravity in two dimensions. We present a method of computing an open string like expansion, which we call the 't Hooft expansion, of the $n$-boundary correlator for any $n$ up to any order by directly solving the Korteweg-De Vries equation. We first explain how to compute the 't Hooft expansion of the one-boundary correlator. The algorithm is very similar to that for the genus expansion of the open free energy. We next show that the 't Hooft expansion of correlators with more than one boundary can be computed algebraically from the correlators with a lower number of boundaries. We explicitly compute the 't Hooft expansion of the $n$-boundary correlators for $n=1,2,3$. Our results reproduce previously obtained results for Jackiw-Teitelboim gravity and also the 't Hooft expansion of the exact result of the three-boundary correlator which we calculate independently in the Airy case.