Von Neumann Algebras in Double-Scaled SYK

TL;DR

The paper establishes a rigorous algebraic framework for double-scaled SYK with matter chords, proving that the resulting double-scaled chord algebra is a Type II$_1$ factor with the empty state $\Omega$ furnishing a faithful, finite trace. It demonstrates that $\Omega$ is cyclic and separating, enabling a Tomita–Takesaki modular structure and revealing $\,\mathcal{A}_L$ and $\mathcal{A}_R$ as mutual commutants. Through triple scaling and various limits, it connects the abstract algebra to JT gravity, the Hilbert space of baby universes, and Brownian DSSYK, and provides analytic solutions for the 0- and 1-particle spectra. The work also discusses how finite-temperature behavior emerges from the operator algebra, the potential for phase transitions and different algebraic types (e.g., Type III$_1$ in certain regimes), and outlines future directions for understanding entropy, dilaton profiles, and scrambling in this holographic context.

Abstract

It has been argued that a finite effective temperature emerges and characterizes the thermal property of double-scaled SYK model in the infinite temperature limit. Meanwhile, in the static patch of de Sitter, the maximally entangled state satisfies a KMS condition at infinite temperature, suggesting the Type II$_1$ nature of the observable algebra gravitationally dressed to the observer. In this work, we analyze the double-scaled algebra generated by chord operators in the double-scaled SYK model and demonstrate that it exhibits features reflecting both perspectives. Specifically, we prove that the algebra is a Type II$_1$ factor, and that the empty state with no chord satisfies the tracial property, in agreement with expectations from earlier work. We further show that this state is cyclic and separating for the double-scaled algebra, based on which we explore its modular structure. We then explore various physical limits of the theory, drawing connections to JT gravity, the Hilbert space of baby universes, and Brownian double-scaled SYK. We also present analytic solutions to the energy spectrum in both the zero- and one-particle sectors of the left/right chord Hamiltonian.