Computing topological invariants with one and two-matrix models

TL;DR

This paper extends matrix models to compute intersection numbers of p-spin curves and orbifold Euler characteristics, using duality and double scaling limits, with a generalization to time-dependent two-matrix models.

Contribution

It introduces a generalized matrix model framework for calculating topological invariants of moduli spaces, including new models for orbifold Euler characteristics.

Findings

Derived intersection numbers as polynomials in p

Established a matrix model for orbifold Euler characteristics

Extended to time-dependent two-matrix models with logarithmic potentials

Abstract

A generalization of the Kontsevich Airy-model allows one to compute the intersection numbers of the moduli space of p-spin curves. These models are deduced from averages of characteristic polynomials over Gaussian ensembles of random matrices in an external matrix source. After use of a duality, and of an appropriate tuning of the source, we obtain in a double scaling limit these intersection numbers as polynomials in p. One can then take the limit p to -1 which yields a matrix model for orbifold Euler characteristics. The generalization to a time-dependent matrix model, which is equivalent to a two-matrix model, may be treated along the same lines ; it also yields a logarithmic potential with additional vertices for general p.