⚛️ high-energy theory

Testing holographic computation of entanglement pseudo-entropy in dS\textsubscript{3}/ICFT\textsubscript{2}

This paper establishes a bottom-up holographic model for dS/ICFT correspondence using a Coleman-De Luccia instanton to analyze pseudo-entropy prescriptions, identify potential paradoxes with universal interface CFT properties, and extract the g-function by comparing results with the analytically continued AdS/ICFT correspondence.

Original authors: Liang Li

Published 2026-08-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Liang Li

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

In the vast landscape of theoretical physics, there is a persistent puzzle regarding how to describe the universe we live in. For decades, physicists have relied on a powerful idea called the holographic principle, which suggests that a three-dimensional world of gravity can be mathematically translated into a two-dimensional surface where quantum particles interact. This translation works beautifully for universes that curve inward, like a saddle, but our actual universe curves outward, expanding forever. This outward curvature, known as de Sitter space, has stubbornly resisted the same kind of translation. The mathematical tools that work for the inward-curving universes often break down or produce nonsensical results when applied to our own. The central question remains: can we build a reliable bridge between the gravity of our expanding cosmos and the quantum rules that govern its smallest parts?

A researcher has taken a significant step toward answering this by constructing a simplified model to test these translation rules. They focused on a specific scenario where two different expanding universes, each with its own rate of expansion, are glued together by a thin, energetic membrane. In the language of quantum theory, this setup is expected to correspond to a "boundary" where two different types of quantum theories meet. The researcher used this model to test two competing methods that physicists have proposed for calculating a specific quantity called pseudo-entropy. This quantity is a measure of how much information is shared between different parts of the quantum system, but because the universe is expanding, the numbers involved are complex and behave differently than in static systems.

The study began by building the geometry of this glued universe. Imagine two spheres of different sizes, representing the two expanding regions. The researcher used a mathematical technique involving a "bubble" of true vacuum forming inside a "fake" vacuum to create a shape where these two spheres are joined at a specific radius by a membrane. This membrane acts as a wall separating the two regions. Once this shape was established, they turned to the problem of calculating the pseudo-entropy. They needed to find the shortest path, or geodesic, that connects two points on the boundary of this universe. In a normal, static universe, this path is a simple curve. In an expanding universe, however, the path becomes tricky; it is a "timelike" path that moves forward in time, and it does not naturally close on itself.

To solve this, the researcher tested two different prescriptions, or sets of rules, for how to close these paths. The first rule suggested that one should smoothly connect the time-moving path in our universe to a space-moving path in a theoretical, static version of the universe. The second rule suggested a different approach: instead of trying to match the shapes, one should extend the calculation into a complex mathematical space, effectively moving the path into an imaginary direction to close the loop.

When the researcher applied the first rule to their model, they found a serious problem. As they changed the size of the region they were measuring, the rules forced the path to bend in a way that contradicted what the quantum theory on the other side of the membrane should predict. Specifically, the calculation suggested that the information shared between the two sides of the membrane would change in a way that defied the basic laws of locality, which state that distant objects should not instantly influence each other. The researcher concluded that this first rule, while elegant in simpler cases, leads to inconsistencies when applied to a universe with a membrane interface.

They then turned to the second rule. This approach, which involves extending the calculation into the complex plane, proved to be much more robust. When they used this method, the results matched what one would expect from the quantum theory side. The calculated information shared between the regions behaved correctly, respecting the boundaries and the nature of the interface. Furthermore, this method allowed them to extract a specific number, known as the g-function, which encodes the unique properties of the membrane itself. This number acts like a fingerprint for the interface, telling us exactly how the two different quantum theories are joined together.

The study also revealed a subtle difference between this expanding universe model and the well-understood models of inward-curving universes. In the inward-curving models, it is possible to choose a region so small that the connecting path never touches the membrane. However, in this expanding model, the geometry is such that the path always intersects the membrane, no matter how small the region is chosen. This suggests that the interface in an expanding universe is fundamentally more accessible to observation than in its inward-curving counterparts.

By rigorously testing these methods, the researcher has provided a strong argument for which mathematical tools are reliable for studying the holographic nature of our expanding universe. They have shown that one of the popular methods for calculating these quantities leads to contradictions, while another method holds up and reveals new details about the structure of the universe. This work does not solve the entire mystery of how gravity and quantum mechanics fit together in our universe, but it clears away a significant obstacle, offering a clearer path forward for understanding the deep connection between the shape of space and the flow of information.

Technical Summary: Testing Holographic Computation of Entanglement Pseudo-Entropy in dS3/ICFT2

Problem Statement
The correspondence between de Sitter (dS) spacetime and Conformal Field Theory (dS/CFT) remains less understood than its Anti-de Sitter (AdS) counterpart. A central issue is the computation of entanglement entropy in the dual CFT, which possesses complex conformal dimensions and an imaginary central charge, rendering the theory non-unitary. Consequently, the entropy is interpreted as "pseudo-entropy," taking complex values. While several prescriptions exist for calculating the holographic extremal surfaces (geodesics in dS3) required for this computation, they yield identical results for pure dS spacetime but diverge when the geometry is deformed. The paper addresses the lack of a consensus on the correct prescription for extremal surfaces in deformed dS spacetimes, specifically those involving interfaces, and seeks to determine which prescription aligns with the universal properties of Interface CFTs (ICFTs).

Methodology
The author constructs a bottom-up model for the dS/ICFT correspondence using the Coleman-De Luccia (CDL) instanton, which describes vacuum decay in the presence of gravity.

Geometry Construction: They consider a scenario where an inflaton potential with two local minima (fake and true vacua) leads to a bubble nucleation. In the thin-wall approximation, the region where the inflaton interpolates is absorbed into a brane with tension $\sigma$ . The resulting Euclidean geometry consists of two spheres with different radii ( $l_+$ and $l_-$ ) glued at a specific radius $R_0$ .
Lorentzian Evolution: Unlike standard analytic continuation which yields an open universe, the author employs the Hamiltonian formalism (specifically the FMP formalism) to derive the Lorentzian time evolution of the CDL state. This results in a spacetime where two de Sitter regions with different radii are joined by a brane that traces out a dS slice. The brane expands until future infinity, where the true vacuum occupies exactly half of the time slice.
Holographic Dictionary: Based on the dS/CFT conjecture, the author proposes that this bulk geometry is dual to an Interface CFT (ICFT) joining two non-unitary CFTs with different central charges ( $\tilde{c}_+$ and $\tilde{c}_-$ ).
Testing Prescriptions: The author applies two specific prescriptions for computing the holographic pseudo-entropy (geodesic length) to subregions centered at the fake vacuum, the true vacuum, and the interface:
- Prescription 1: Smoothly connect the timelike geodesic in Lorentzian dS to a spacelike geodesic in Euclidean dS at the $T=0$ slice, imposing a "smoothness" condition on the tangent vectors.
- Prescription 2: Extend the integral contour for the geodesic length into the complex plane (specifically the imaginary direction of the coordinate $\rho$ ) to close the path, keeping the timelike part on the real axis.

Key Contributions and Results

Inconsistency of Prescription 1: When applied to a subregion centered at the fake vacuum, Prescription 1 leads to a contradiction with the expected universal properties of ICFTs. As the subregion size shrinks, the "smoothness" condition forces the spacelike geodesic to intersect the brane, causing the real part of the entropy to probe interface data. This contradicts the locality of the interface in the CFT, where a small subregion far from the interface should not be sensitive to it. Furthermore, removing the smoothness condition fails to reproduce the universal $\pi \tilde{c}/6$ term in the real part of the entropy.
Validity of Prescription 2: Prescription 2 yields results consistent with the locality of the interface. For subregions centered at the fake vacuum, the geodesic does not intersect the brane, and the entropy depends only on the central charge of that side, matching the analytic continuation of AdS/ICFT results.
True Vacuum Subregions: For subregions centered at the true vacuum, the geodesic inevitably intersects the brane. Prescription 2 successfully calculates the entropy, showing that the UV-divergent term is determined by the side of the subregion, while the finite imaginary terms encode the brane's presence. A notable difference from AdS/ICFT is that in dS/ICFT, the geodesic always intersects the brane for true-vacuum-centered subregions, regardless of size, due to the global geometry.
g-function Extraction: By analyzing a subregion perpendicular to the interface (centered at the brane), the author successfully extracts the $g$ -function (boundary entropy) using Prescription 2. This requires considering complex solutions for the intersection points. The resulting $g$ -function depends on the brane tension and the radii of the two dS spaces, encoding the conformal data of the interface.

Significance
The paper establishes a concrete dS/ICFT dictionary and uses it as a laboratory to test holographic entropy prescriptions. The primary significance lies in ruling out Prescription 1 (smooth connection) for deformed dS spacetimes due to its inconsistency with ICFT universality and locality. Conversely, it validates Prescription 2 (complex contour extension) as a viable method that reproduces expected universal terms and correctly captures interface data (the $g$ -function). The work highlights that while dS/ICFT shares similarities with AdS/ICFT, the global structure of de Sitter space imposes unique constraints (such as the inevitability of brane intersection for certain subregions) that distinguish the two holographic dualities. The author concludes that further study of extremal surfaces in complexified or holomorphic geometries is necessary to fully understand entropy in de Sitter spacetime.

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