Random Matrix, Singularities and Open/Close Intersection Numbers

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 $C{P}^1$ model are evaluated for one marked point as an application of the present method.

arXiv comment48 pages

References in corpus (17)

• Topological Strings and Integrable Hierarchies
• Universal singularity at the closure of a gap in a random matrix theory
• Level Spacing of Random Matrices in an External Source
• Characteristic polynomials of real symmetric random matrices
• Duality in random matrix ensembles for all Beta
• Intersection theory on moduli of disks, open KdV and Virasoro
• Intersection theory from duality and replica
• Open intersection numbers, Kontsevich-Penner model and cut-and-join operators
• The Witten equation, mirror symmetry and quantum singularity theory
• Vertices from replica in a random matrix theory
• On an Airy matrix model with a logarithmic potential
• The intersection numbers of the p-spin curves from random matrix theory
• Solution of W-Constraints for R-Spin Intersection Numbers
• Open intersection numbers and the wave function of the KdV hierarchy
• Descendent integrals and tautological rings of moduli spaces of curves
• Formal pseudodifferential operators and Witten's r-spin numbers
• A geometric parametrization for the virtual Euler characteristic for the moduli spaces of real and complex algebriac curves

Cited by in corpus (1)

• Open intersection numbers and the wave function of the KdV hierarchy