Fat and Thin Emergent Geometries of Hermitian One-Matrix Models

TL;DR

This paper explores the geometric structures of Hermitian one-matrix models through spectral curves and their deformations, revealing dualities between different genus expansion versions.

Contribution

It introduces a novel approach to defining spectral curves from genus zero free energy functions and explores their dual deformations in Hermitian matrix models.

Findings

Spectral curves are characterized by formal power series with integral coefficients.

Two dual versions of spectral curves are constructed based on different genus expansions.

The work generalizes Catalan numbers within the context of matrix model geometries.

Abstract

We use genus zero free energy functions of Hermitian matrix models to define spectral curves and their special deformations. They are special plane curves defined by formal power series with integral coefficients generalizing the Catalan numbers. This is done in two different versions, depending on two different genus expansions, and these two versions are in some sense dual to each other.