An algebraic approach to a quartic analogue of the Kontsevich model

TL;DR

This paper introduces an algebraic approach to a quartic analogue of the Kontsevich model, revealing recursive structures and connections to blobbed topological recursion, advancing mathematical understanding of matrix models.

Contribution

It develops a new algebraic geometric solution method for the quartic Kontsevich model's initial recursive equations, linking it to blobbed topological recursion.

Findings

Recursive equations for cumulants are established.

A purely algebraic geometric solution strategy is developed.

The model is shown to obey blobbed topological recursion.

Abstract

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term $\mathrm{Tr}(\Phi^4)$. Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.