⚛️ high-energy theory

Effective dynamics and quantum information in de Sitter wedge holography

This paper proposes a non-unitary conformal field theory dual to a de Sitter wedge holography setup in an Anti-de Sitter bulk, supporting the duality through partition function and entanglement entropy calculations while verifying the first law of entanglement entropy and deriving the Page curve via the island prescription.

Original authors: Sabyasachi Maulik, Soumen Pari

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sabyasachi Maulik, Soumen Pari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, multi-layered cake. Usually, physicists study the "Anti-de Sitter" (AdS) cake, which has a very specific, negative-curvature shape that makes math behave nicely. But our real universe looks more like a "de Sitter" (dS) cake, which is expanding and has positive curvature. The problem? The math for the expanding cake is notoriously messy and hard to slice.

In this paper, the authors Sabyasachi Maulik and Soumen Pari try a clever trick called Wedge Holography. Instead of trying to slice the whole expanding cake at once, they imagine taking a slice of the nice AdS cake and sandwiching it between two special "branes" (think of them as magical, floating slices of bread). These slices of bread are shaped like de Sitter space.

Here is the big idea: They propose that the physics happening inside this sandwich (the "wedge") is secretly the same as a different kind of physics happening on the very edge where the two slices of bread touch. It's like saying the flavor of the whole sandwich is actually just a code written on the crust where the bread meets.

The "Ghostly" Code

When the authors tried to decode this crust, they found something weird. The "code" they found (which they call a Conformal Field Theory, or CFT) behaves like a ghost.

In normal physics, probabilities always add up to 100% (you either win or lose, but the math stays real). But in their calculation, the central numbers describing this code came out as imaginary numbers (involving the square root of -1).

What this means: The paper suggests that the theory living on this edge is non-unitary. In plain English, it's a "ghostly" theory where the usual rules of probability don't work the way they do in our everyday world. It's not a "broken" theory, but a very strange one that fits the expanding universe better than the standard AdS models.

The Gravity Trap: The Volcano

One of the biggest questions in this field is: "Can we get normal gravity (like Einstein's) to stick to these branes, or does it leak away into the higher dimensions?"

The authors tested this by looking at how "ripples" (gravitons) move through the sandwich. They found that the answer depends entirely on how you hold the edges of the sandwich:

The "Hard" Edge (Dirichlet): If you clamp the edges down tight, the ripples cannot stick. There is no "massless" (normal) gravity mode. Gravity leaks away.
The "Soft" Edge (Neumann): If you let the edges wiggle freely, a massless graviton mode appears. This is the particle that carries gravity.

They visualized this using a Volcano Potential. Imagine a valley with steep sides. The "soft" edge setup creates a volcano shape where the gravity particle (the zero-mode) gets trapped at the bottom of the crater, stuck to the brane. This suggests that if we want to explain why gravity feels strong on our universe (the brane), we need the "soft" boundary conditions.

The Island of Information

The paper also tackles a famous puzzle: The Black Hole Information Paradox. If a black hole evaporates, does the information inside get destroyed? (Physics says no, but math often says yes).

To test this, they set up a simplified model:

The "Bath": One side of the sandwich (the UV brane) acts like a bucket collecting radiation.
The "Black Hole": The other side (the IR brane) acts like the evaporating black hole.

They watched how the "entanglement entropy" (a measure of how much information is shared) changed over time.

Phase 1 (No Island): At first, the entropy just keeps growing forever, like a balloon inflating. This would mean information is lost.
Phase 2 (The Island): Suddenly, a new path opens up. A surface connects the radiation bucket to the black hole side, creating an "Island." This island acts like a secret vault that recovers the lost information.

When they compared the two paths, the "Island" path eventually became the winner. The entropy stopped growing and flattened out, forming a Page Curve. This suggests that even in this strange, ghostly de Sitter setup, information is not lost; it's just hidden in an island that we can't see until the right time.

What They Didn't Find (And What They Ruled Out)

It's important to know what this paper didn't do:

No Real Black Holes Yet: The "black hole" in their model wasn't a real, dynamic black hole eating stars. It was a simplified stand-in (the IR brane). The authors admit this is a "toy model" and that a full, realistic black hole evaporation story needs more work.
No Unitary Fix: They did not prove that our universe is a normal, unitary place. In fact, their results explicitly suggest the opposite: the dual theory is non-unitary (ghostly). They don't claim to have "solved" the information paradox for our real universe, only shown that a Page curve can appear in this specific, simplified wedge setup.
No Massless Gravity with Hard Edges: They explicitly ruled out the idea that you can get localized gravity on the brane if you use "Hard" (Dirichlet) boundary conditions. If you clamp the edges, gravity leaks.

The Bottom Line

The authors have built a mathematical playground where they can study the expanding universe using a sandwich of higher dimensions. They found that:

The "code" on the edge is ghostly (non-unitary), which fits the weird nature of de Sitter space.
Gravity can be trapped on the brane, but only if the edges are allowed to wiggle (Neumann conditions), creating a "volcano" that holds the gravity particle.
Even in this weird, ghostly setup, information seems to be saved via an Island, producing a Page curve that looks just like what we hope to see in real black holes.

It's a promising step, but the authors are careful to say this is a simplified model. They haven't solved the mystery of our real universe yet, but they've built a very cool, very strange laboratory to keep looking for the answer.

Technical Summary: Effective dynamics and quantum information in de Sitter wedge holography

Problem and Setup
This paper investigates the holographic dictionary and quantum information properties of a "de Sitter (dS) wedge" setup. The authors consider a $(d+1)$ -dimensional Anti-de Sitter (AdS) bulk spacetime bounded by two End-of-the-World (ETW) branes. Unlike standard wedge holography where the branes induce flat or AdS geometries, the induced geometry on each brane in this construction is $d$ -dimensional de Sitter spacetime ( $dS_d$ ). The wedge is foliated by $dS_d$ slices, with the radial coordinate $\rho$ restricted between the two branes at positions $\rho_1$ and $\rho_2$ .

The central proposal is that this bulk configuration is holographically dual to a Conformal Field Theory (CFT) living on the codimension-two intersection of the branes (a $(d-1)$ -dimensional sphere). This setup serves as a laboratory to explore the dS/CFT correspondence, which is generally less understood than AdS/CFT, and to study quantum information quantities like entanglement entropy and the Page curve in a de Sitter context.

Methodology
The authors employ a combination of classical gravitational analysis, holographic renormalization, and extremal surface calculations:

Effective Action and Partition Function: By integrating over the radial direction of the wedge, the authors derive the effective gravitational action on the branes. They evaluate the Euclidean on-shell action for various spacetime dimensions ( $d=3, 4, 5$ ) to extract the partition function of the dual CFT.
Mass Spectrum Analysis: The stability and localization of gravity are probed by analyzing the spectrum of graviton fluctuations on the branes. The authors impose both Dirichlet and Neumann boundary conditions (BCs) on the branes and solve the resulting Sturm-Liouville equations for the radial wavefunctions.
Holographic Entanglement Entropy (HEE): Using the Ryu-Takayanagi (RT) prescription, the authors compute the entanglement entropy for spherical subregions on the boundary. They verify the "first law of entanglement entropy" by comparing the variation of entropy with the variation of the modular Hamiltonian for both massless and massive graviton fluctuations.
Timelike Entanglement: The paper calculates the "timelike entanglement entropy" (or pseudo-entropy) between the two codimension-two defects at the tips of the wedge, utilizing extremal surfaces that connect timelike-separated regions.
Information Recovery (Page Curve): To study information recovery, the authors model the IR brane as a gravitating sector (analogous to an evaporating black hole) and the UV brane as a non-gravitating radiation bath. They compute the generalized HRT entropy for two competing configurations: a "No-island" phase (disconnected surface) and an "Island" phase (connected surface extending to the IR brane).

Key Contributions and Results

Dual CFT and Non-Unitarity: The computation of the gravitational partition function and holographic entanglement entropy yields central charges that are purely imaginary. For instance, in $d=3$ , the central charge is $c = \frac{3i}{2G_N}(\cosh \rho_2 - \cosh \rho_1)$ . This result supports the proposal that the dual theory is a non-unitary CFT, consistent with expectations for dS/CFT correspondences.
Graviton Mass Spectrum and Localization:
- Dirichlet BCs: The spectrum contains an infinite tower of massive modes, but no massless mode exists. Consequently, gravity cannot be localized on the brane under Dirichlet conditions.
- Neumann BCs: The spectrum admits a massless graviton mode ( $m=0$ ) in addition to a tower of massive Kaluza-Klein modes. The analysis of the Schrödinger-like potential (a "volcano potential") shows that the massless zero-mode wavefunction is localized near the outer brane, indicating that localized Einstein gravity emerges on the brane only with Neumann boundary conditions.
First Law of Entanglement Entropy: The authors verify the first law of entanglement entropy ( $\Delta S = \Delta \langle H \rangle$ $Δ S = Δ ⟨ H ⟩$ ) for the dS wedge.
- For massive fluctuations, both the change in entanglement entropy and the change in the modular Hamiltonian vanish at first order, satisfying the law trivially.
- For massless fluctuations, explicit calculation shows that the variations match, confirming the consistency of the holographic dictionary. The negative variation of the modular Hamiltonian is noted as consistent with the non-unitary nature of the boundary theory.
Timelike Entanglement Entropy: The calculation of the timelike entanglement entropy between the two defects yields a purely imaginary value. This arises because the extremal surface connecting the defects remains entirely timelike, resulting in a Lorentzian induced metric with an overall factor of $i$ .
Page Curve and Island Phase:
- In the No-island phase, the radiation entropy grows monotonically with time, reflecting the continuous increase of entanglement without a mechanism for information recovery.
- In the Island phase, a connected extremal surface extends from the UV brane to the IR brane. The area of this surface remains finite and constant at late times.
- The competition between these two phases results in a Page curve where the entropy initially grows (following the no-island branch) and then saturates (following the island branch) after the Page time. This demonstrates the emergence of a unitary Page curve within the dS wedge framework, despite the non-unitary nature of the dual CFT.

Significance and Claims
The paper claims to provide a detailed examination of the basic holographic dictionary for de Sitter wedge holography. Its primary significance lies in:

Establishing a concrete link between the bulk wedge geometry and a non-unitary CFT on the codimension-two defect, evidenced by the imaginary central charges derived from independent calculations (partition function and entanglement entropy).
Clarifying the conditions for localized gravity in this setup, specifically showing that Neumann boundary conditions are required to recover a massless graviton mode.
Demonstrating that the "island prescription" can be successfully applied to a de Sitter wedge to recover a Page curve, thereby offering a mechanism for information recovery in a de Sitter context without requiring a dynamical evaporating black hole in the bulk (the IR brane serves as an effective gravitating sector).
Highlighting the distinction between standard entanglement entropy and timelike/pseudo-entropy in Lorentzian signatures, noting the purely imaginary nature of the latter in this specific geometric setup.

The authors conclude that while the setup successfully reproduces the Page curve, the non-unitary character of the dual CFT and the absence of a dynamical black hole in the bulk mean this should be viewed as an extremal-surface transition in the de Sitter wedge rather than a complete model of black hole evaporation. They suggest future work could involve introducing explicit black holes to the bulk geometry.

The "Ghostly" Code

The Gravity Trap: The Volcano

The Island of Information

What They Didn't Find (And What They Ruled Out)

The Bottom Line

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