Random Matrix, Singularities and Open/Close Intersection Numbers

TL;DR

This paper explores the connection between random matrix models and intersection numbers on moduli spaces, deriving explicit formulas and string equations for both closed and open surfaces, including non-orientable cases.

Contribution

It introduces new matrix models and duality identities that compute intersection numbers and Gromov-Witten invariants for various types of surfaces, including non-orientable and open surfaces.

Findings

Explicit all-genus intersection numbers via Bessel functions

Matrix models for non-orientable surfaces are developed

String equations for open and closed surfaces are derived

Abstract

The $s$-point correlation function of a Gaussian Hermitian random matrix theory, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the $p$-th spin curves through a duality identity. For one marked point, the intersection numbers are expressed to all order in the genus by Bessel functions. The matrix models for the Lie algebras of $O(N)$ and $Sp(N)$ provide the intersection numbers of non-orientable surfaces. The Kontsevich-Penner model, and higher $p$-th Airy matrix model with a logarithmic potential, are investigated for the open intersection numbers, which describe the topological invariants of non-orientable surfaces with boundaries. String equations for open/closed Riemann surface are derived from the structure of the $s$-point correlation functions. The Gromov-Witten invariants of $CP^1$ model are evaluated for one marked point as an application of the present method.