Relationship between a Φ⁴ matrix model and harmonic oscillator systems

2025/07/13 by Harald Grosse, Grosse, Harald, Naoyuki Kanomata +5
Computer Science · Engineering · Mathematics · #81R12 #81T32 #81T75 #Elasticity and Wave Propagation #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2507.09454

openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Hermitian Φ⁴ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schrödinger 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 Schrödinger 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.

Citations

• Real symmetric Φ⁴-matrix model as Calogero-Moser model
• Random Matrices

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