Explicit large $N$ von Neumann algebras from matrix models

TL;DR

This work shows how large-$N$ quantum systems formulated via matrix-model partition functions give rise to emergent type $III_1$ von Neumann algebras for real-time, finite-temperature correlators. A general, rigorous construction based on CCR/Araki–Woods theory ties the algebra type directly to the spectral density, enabling explicit determination of when type $III_1$ arises from continuous spectra. The authors develop a broad class of toy models—unitary matrix models and their representations—together with a probe framework to compute Källén–Lehmann densities and saddle-point eigenvalue densities, then connect these to large-$N$ algebras. They further promote the third-order large-$N$ phase transitions to first-order Hagedorn transitions by extending the Hilbert space to include a flavor rank sum, showing that the type-$III_1$ algebra emerges only above the Hagedorn temperature, with several concrete examples illustrating the phenomenon and revealing Calabi–Yau connections. Altogether, the paper provides a systematic, computable route from matrix-model gauge theories to emergent interior-time structure in holography, and links spectral data to operator-algebraic classification with potential holographic implications.

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.

Figures (24)

• Figure 1: In this work we explore the implications of writing the finite temperature partition function of a theory in terms of representations of the global symmetries. We construct a quantum mechanics from these representations and determine the large $N$ von Neumann algebra of operators.
• Figure 2: Below the Hagedorn temperature ($T<T_H$) the partition function is finite and the algebra of single-trace operators is a type I von Neumann algebra. Above the Hagedorn temperature ($T>T_H$), the partition function diverges and the von Neumann algebra becomes type III$_1$.
• Figure 3: The steps through which this work associates von Neumann algebras to quantum mechanical systems with large $N$ factorization.
• Figure 4: Chart of the main concepts explored in this section.
• Figure 5: Illustration of a constant-$a$ slice of the parameter space. The intersection of the curves $\gamma_c$ and $\gamma_{\ast}$ produces a phase transition. The solution $\gamma_{\ast}$ to \ref{['eq:genericSPEgamma']} is valid only for values of $\beta$ such that $\gamma_{\ast} > \gamma_c$, on the left of the dashed vertical line.

Theorems & Definitions (136)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Haag_1992
• Definition 3.1
• Definition 3.2
• Proposition 3.3: Derezinski
• Definition 3.4
• Theorem 3.5: Derezinski
• Definition 4.1