GUE via Frobenius Manifolds. II. Loop Equations

TL;DR

This paper derives loop equations for the GUE partition function, linking it to Gromov-Witten invariants and the NLS Frobenius manifold, revealing deep geometric connections.

Contribution

It establishes a new connection between GUE, Gromov-Witten invariants, and Frobenius manifolds through loop equations.

Findings

GUE partition function satisfies specific loop equations.

GUE partition function relates to the topological partition of the NLS Frobenius manifold.

The work extends the understanding of the geometric structure underlying random matrix models.

Abstract

A theorem of Dubrovin establishes the relationship between the GUE partition function and the partition function of Gromov-Witten invariants of the complex projective line. Based on this theorem we derive loop equations for the Gaussian Unitary Ensemble (GUE) partition function. We show that the GUE partition function is equal to part of the topological partition function of the non-linear Schr\"odinger (NLS) Frobenius manifold.