Gravitational Algebras with Two Areas

TL;DR

The paper develops an algebraic framework for spacetimes with two homologous extremal surfaces by applying the split property and operator-valued weights to construct gravitational algebras as crossed products. It shows that when both area modes are accessible, the middle region supports a type II∞ algebra whose entropy reproduces the generalized entropy, and extends the construction to cases with only the area sum or area difference. For area sum, the middle region algebra remains II∞, while area difference requires OVWs to define a consistent modular flow and yields a Petz-like expression for entropy differences. The results provide an algebraic lens on entanglement-wedge phase transitions, offering a potential bridge to boundary decoding complexity and nonisometric bulk codes.

Abstract

We study gravitational algebras on spacetimes with two extremal surfaces. In the example of a long wormhole with two asymptotic AdS boundaries and two compact extremal surfaces, we discuss the assignment of gravitational algebras to various regions bounded by the extremal surfaces and/or asymptotic boundaries. Using the split property, we construct type II algebras from the crossed product in the left exterior, right exterior, the middle ``python's lunch'' region, and their complement regions. We also study the case where only the area sum operator or area difference operator is included as part of the gravitational algebra. This can be achieved by picking the appropriate microcanonical ensemble, and these gravitational algebras can either be type II or type III depending on the region. In the case where we include only the area difference mode, the crossed product gives rise to a weight that restricts to a trace on the middle region. Differences of relative entropies with respect to this weight give differences in generalized entropies. This provides an algebraic understanding of the order parameter that controls the phase transitions between entanglement wedges. We emphasize the role of operator-valued weights used in our construction.

Figures (12)

• Figure 1: The Penrose diagram of a long wormhole supported by a thin shell (the red line). Geometries are described by the Schwarzschild metric in the left and right exteriors (denoted as L and R). There are two minimal surfaces (blue dots, denoted as $A_1$ and $A_2$), and the region they bound (denoted as M), is referred to as python's lunch. The time-reflection symmetric Cauchy slice in this geometry is denoted as the black dashed line.
• Figure 2: The time reflection symmetric Cauchy slice of the long wormhole spacetime in Figure \ref{['fig:PL-Penrose']}. The two blue circles are the two minimal extremal surfaces.
• Figure 3: Timeslices in two-sided AdS-Schwarzschild black hole. The green line denotes the zero-extrinsic timeslice which is smooth at the bifurcation horizon. The red line is a timeslice intersecting boundaries at $t_L=t_R=0$, which has a kink at the horizon.
• Figure 4: Spacetime viewed as the development of Cauchy slices with different kink angles at the extremal surface (the black dot). Causality ensures that the domains of dependence of $A$ and $\bar{A}$ are independent of $t$, while the geometry outside depends on $t$ in general.
• Figure 5: An example of split inclusion (left) and contact inclusion (right). For the split inclusion, there is a type I factor $\mathcal{A}$ such that $\mathcal{N}\subset \mathcal{M}$. $\mathcal{A}$ is not associated to any geometric region despite how it is shown.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Theorem 6
• Lemma 7