Random matrix, singularities and open/close intersection numbers

Abstract

The s-point correlation function of a Gaussian Hermitian random matrix model, 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. For one marked point, the intersection numbers are expressed to all orders in the genus by Bessel functions. The matrix models for the Lie algebras of and provide the intersection numbers of non-orientable surfaces. The Kontsevich–Penner model, and the 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 the open/closed Riemann surface are derived from the structure of the s-point correlation functions. The Gromov–Witten invariants of the model are evaluated for one marked point as an application of the present method.