Explicit large $N$ von Neumann algebras from matrix models

• URL: http://arxiv.org/abs/2402.10262v2
• Date: Thu, 07 Nov 2024 08:14:44 GMT
• Title: Explicit large $N$ von Neumann algebras from matrix models
• Authors: Elliott Gesteau, Leonardo Santilli,
• Abstract summary: We construct a family of quantum mechanical systems that give rise to an emergent type III$_$ von Neumann algebra in the large $N$ limit. We calculate the real-time, finite temperature correlation functions in these systems and show that they are described by an emergent type III$_$ von Neumann algebra at large $N$.
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• Abstract: We construct a large family of quantum mechanical systems that give rise to an emergent type III$_1$ von Neumann algebra in the large $N$ limit. Their partition functions are matrix integrals that appear in the study of various gauge theories. We calculate the real-time, finite temperature correlation functions in these systems and show that they are described by an emergent type III$_1$ von Neumann algebra at large $N$. The spectral density underlying this algebra is computed in closed form in terms of the eigenvalue density of a discrete matrix model. Furthermore, we explain how to systematically promote these theories to systems with a Hagedorn transition, and show that a type III$_1$ algebra only emerges above the Hagedorn temperature. Finally, we empirically observe in examples a correspondence between the space of states of the quantum mechanics and Calabi--Yau manifolds.

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