von Neumann algebras in JT gravity

TL;DR

This work develops a nonperturbative operator-algebra framework for JT gravity with matter on a two-boundary AdS geometry. It defines left and right boundary von Neumann algebras generated by the boundary Hamiltonians and gravitationally dressed matter, proves they are commutants and both type II$_\infty$ factors, and shows their union yields the full operator algebra on the Hilbert space, thereby modeling the entanglement wedge beyond semiclassical limits. The analysis proceeds via a boundary-particle formalism, an energy-basis decomposition, and a gravity-dressed OPE/disk-conformal-block structure, culminating in a precise account of traces and representations. A key finding is that, unlike finite-$N$ holographic CFTs, the JT boundary algebras resist a naive Hilbert-space factorization due to their II-type nature, signaling nonlocality as a structural requirement for factorization and prompting discussion of potential nonlocal corrections or EOW-brane setups for any viable factorization picture.

Abstract

We quantize JT gravity with matter on the spatial interval with two asymptotically AdS boundaries. We consider the von Neumann algebra generated by the right Hamiltonian and the gravitationally dressed matter operators on the right boundary. We prove that the commutant of this algebra is the analogously defined left boundary algebra and that both algebras are type II$_\infty$ factors. These algebras provide a precise notion of the entanglement wedge away from the semiclassical limit. We comment on how the factorization problem differs between pure JT gravity and JT gravity with matter.

Figures (13)

• Figure 1: The left diagram depicts a four-point conformal block for a matter theory quantized on the hyperbolic disk. The matter operators are inserted on the boundary of the disk. By applying a simple set of rules to this diagram (and summing over the exchanged states/operators), we may obtain an expression for the disk four-point function in JT gravity coupled to matter. On the right, we separate the diagram into sub-diagrams to represent it as an expectation value of operators that act on a Hilbert space. The bottom sub-diagram represents a ket that corresponds to the Hartle-Hawking state with a matter operator insertion, and each of the two middle sub-diagrams represents an operator inserted on the right boundary. The top sub-diagram is the bra.
• Figure 2: The gray disk represents the hyperbolic disk, and the blue curve is the trajectory of the boundary particle, or the boundary of the physical manifold. The disk Euclidean path integral of JT gravity integrates over all geometries that can be realized as the interior of the blue curve. The condition $\phi(\tau + \beta) = \phi + 2 \pi$ ensures that the boundary curve wraps once around the disk.
• Figure 3: The $(\rho,\phi)$ coordinate system defined here is used throughout this paper. The $(r,u)$ coordinate system defined here is used in section \ref{['sec:trumpet']}. The thick black lines are geodesics with lengths $\rho$ and $r$.
• Figure 4: The Euclidean path integral prepares states in the Hilbert space $\mathcal{H}_0$, which is associated to the horizontal interval boundary. The red dots on the left represent the locations where $\varphi$ is inserted in the lower-half disk. Under the action of a symmetry transformation, the matter operators move along the red lines to their final positions, as shown. Here, $\theta$, $u$, and $x$ are positive.
• Figure 5: On the top-left, we depict the disk two-point function calculation, where operators (denoted by red dots) are inserted along the trajectory of the boundary particle at specified boundary times. To compute this two-point function, we pick two arbitrary boundary times (labeled by green crosses). The locations of the boundary particle at those times are given by $(\phi_1,\psi_1)$ and $(\phi_2,\psi_2)$. We explicitly integrate over these locations and use the boundary particle propagator to account for all of the possible trajectories (the integral is identical to \ref{['eq:unfixeddisk']} but with matter operator insertions). The gauge-fixing condition requires that the green crosses are equidistant from the center of the disk and have angular coordinates $\phi = 0$ and $\phi = \pi$ as shown. After gauge-fixing, the four-dimensional integral is reduced to a single dimensional integral over the renormalized length $\ell$. We also sum over a complete set of states of the matter theory. Thus, an orthonormal basis state of the Hilbert space is labeled by $\ell$ together with an orthonormal basis state of the matter Hilbert space.

Theorems & Definitions (3)

• Theorem
• Definition
• Definition