’t Hooft expansion of multi-boundary correlators in 2d topological gravity

Abstract

\n 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 KdV 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 boundaries 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.