Relationship between a $\Phi^4$ matrix model and harmonic oscillator systems

TL;DR

This paper explores a Hermitian $\

Contribution

It provides explicit formulas for eigenstates of the Virasoro operators derived from a $\

Findings

Partition function satisfies the Schrödinger equation for the $N$-body harmonic oscillator.

Eigenstates can be explicitly expressed in terms of free energy.

Loop equations are derived using $U(1)^N$-symmetry.

Abstract

A Hermitian $\Phi^4$ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schr\"odinger equation of the $N$-body harmonic oscillator, and that eigenstates of the Virasoro operators can be derived from this partition function. We extend these results and obtain an explicit formula for such eigenstates in terms of the free energy. Furthermore, the Schr\"odinger equation for the $N$-body harmonic oscillator can also be reformulated in terms of connected correlation functions. The $U(1)^N$-symmetry allows us to derive loop equations.