Integrability of $\Phi^4$ Matrix Model as $N$-body Harmonic Oscillator System

TL;DR

This paper demonstrates that a Hermitian matrix model with a quartic potential and a specific kinetic term can be mapped to an integrable system of non-interacting harmonic oscillators, revealing its underlying integrability.

Contribution

It introduces a novel matrix model with a quartic potential and shows its partition function satisfies an integrable Schrödinger equation for harmonic oscillators.

Findings

Partition function solves an integrable Schrödinger-type equation.

Model generalizes Kontsevich matrix model with a quartic potential.

Establishes connection between matrix models and harmonic oscillator systems.

Abstract

We study a Hermitian matrix model with a kinetic term given by $ Tr (H \Phi^2 )$, where $H$ is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential $\Phi^3$ replaced by $\Phi^4$. We show that its partition function solves an integrable Schr\"odinger-type equation for a non-interacting $N$-body Harmonic oscillator system.