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

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 Schrödinger-type equation for a non-interacting $N$-body Harmonic oscillator system.

16 pages, 3 figures