⚛️ high-energy theory

Monitoring a de Sitter universe through an anti-de Sitter window

This paper proposes a holographic duality where AdS $_3$ gravity coupled to dS $_2$ end-of-the-world branes corresponds to a unitary CFT $_2$ with non-unitary boundary conditions, thereby realizing de Sitter holography as the unitary time evolution of a state prepared by a Euclidean path integral with complex $g$ -functions.

Original authors: Sarah Hoback, Daniel L. Jafferis, Zixia Wei

Published 2026-07-01

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CC BY 4.0

Original authors: Sarah Hoback, Daniel L. Jafferis, Zixia Wei

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

The Big Question: How do you put an elephant in a refrigerator?

The authors start with a joke: "How do you put an elephant in a refrigerator?" The answer is simple: Open the door, put the elephant in, and close the door. But the last step—closing the door—is actually very hard to do technically.

They use this joke to describe their scientific problem: How do you put a "de Sitter" universe (a type of expanding universe like ours) inside a "holographic refrigerator" (a mathematical framework called AdS/CFT) and then close the door?

In physics, we have a very successful way to describe universes that curve inward (Anti-de Sitter or AdS) using a 2D quantum theory (like a computer program). This is called the AdS/CFT correspondence. However, our actual universe curves outward (de Sitter or dS), and we don't have a good way to describe it using this same "refrigerator" method.

The Solution: A Complex "Ghost" Door

The authors propose a clever trick. Instead of trying to force the elephant (the dS universe) into the fridge using normal, real-world physics, they use complex numbers.

In math, "complex" numbers involve the square root of negative numbers. They aren't "real" in the everyday sense, but they are incredibly useful for solving problems.

The Setup: Imagine a black hole in a 3D universe (AdS). Behind the black hole's event horizon (the point of no return), they place a 2D "brane" (a membrane) that acts like a tiny, expanding universe (dS).
The Problem: When they try to calculate the math for this setup using standard "real" numbers, the equations break. There is no solution. It's like trying to fit a square peg in a round hole.
The Fix: They change the rules slightly. They allow the math to happen in a "complex" space. In this space, the equations do have solutions. These solutions are "complex saddles"—mathematical paths that don't exist in our real world but exist in the complex mathematical landscape.

The "Non-Unitary" Boundary Condition

Here is the most important part:

The "refrigerator" (the 2D quantum theory) is unitary. This means it follows the strict rules of quantum mechanics: information is never lost, and everything is predictable and stable.
The "elephant" (the dS universe) is described by non-unitary boundary conditions. This is like a special instruction manual for the refrigerator that looks "weird" or "broken" because it involves complex numbers.

The Analogy:
Imagine you have a perfect, high-tech video game console (the Unitary CFT). You want to play a game that involves a glitchy, broken level (the dS universe).

Usually, you can't play a broken level on a perfect console.
The authors say: "What if we create a special, imaginary 'patch' (the complex boundary condition) that tells the console how to run this broken level?"
The console itself remains perfect and follows all the rules. The "brokenness" is only in the specific instructions (the boundary condition) used to start the game.

What They Found

By using this "complex patch," they discovered:

The Elephant Fits: They successfully mapped the expanding dS universe behind the black hole horizon to a specific state in the perfect 2D quantum theory.
Time Works: Even though the math used complex numbers to get there, the result describes a real, physical process. The dS universe evolves in time just like a normal universe would.
A New Particle: The math predicts a special "boundary-changing operator" (a specific type of particle or signal) with a negative energy value ( $h = -c/8$ ). This is a signature that the boundary condition is "non-unitary," but it doesn't break the main quantum theory. It lives in the "boundary sector," not the main bulk of the universe, so the overall system stays safe and stable.

The Conclusion

The paper claims to have solved the "elephant in the refrigerator" problem.

The Elephant: A de Sitter universe (like ours).
The Refrigerator: A standard, unitary quantum theory (AdS/CFT).
The Trick: You don't force the elephant in with real physics. You use a "complex" mathematical door (a non-unitary boundary condition) to prepare the state.

Once the state is prepared, it lives in a perfectly normal, unitary quantum world. The dS universe is essentially a "guest" that was invited in through a complex backdoor, but once inside, it behaves normally.

In short: They found a way to describe our expanding universe using a standard quantum theory by allowing the "entry instructions" to be complex and slightly "unreal," while keeping the universe itself real and consistent. They successfully "closed the door" on the problem.

Technical Summary: Monitoring a de Sitter universe through an anti-de Sitter window

Problem Statement
The paper addresses the challenge of constructing a holographic description of de Sitter (dS) spacetime. While the AdS/CFT correspondence provides a well-understood framework for anti-de Sitter (AdS) gravity via a dual unitary Conformal Field Theory (CFT), extending this to dS space remains difficult. Existing proposals, such as dS/CFT (where the dual CFT is non-unitary) or holographic screens within dS, have limitations. Specifically, non-perturbative evidence suggests that closed dS universes may correspond to one-dimensional Hilbert spaces (pure states), yet constructing these states within a unitary framework is nontrivial. The authors pose the question of how to embed a dS region into an AdS spacetime and identify its counterpart in the CFT Hilbert space, noting that while the embedding is straightforward, the identification of the dual state is technically complex.

Methodology
The authors propose a construction where AdS $_3$ gravity coupled to 2D end-of-the-world (EOW) branes is dual to a unitary holographic CFT $_2$ equipped with non-unitary conformal boundary conditions.

Gravitational Setup: They consider a two-sided BTZ black hole in AdS $_3$ with a dS $_2$ EOW brane situated behind the event horizon. The theory is defined by an Einstein-Hilbert action with a negative cosmological constant, coupled to a brane with tension $T > 1$ . The brane satisfies Neumann boundary conditions (NBC).
Path Integral Obstruction: In the Euclidean path integral, no real saddle points exist where a dS brane intersects the asymptotic AdS boundary. Standard Euclidean solutions (like the hyperbolic ball) do not accommodate a dS brane intersecting the boundary.
Complex Saddle Points: To resolve this, the authors perform the gravitational path integral over complex geometries. They utilize a "wedge trick," constructing solutions by gluing a half-AdS ball to a "complex disk wedge" or "complex cylinder wedge." These wedges involve complexifying the radial coordinate $\rho$ , allowing the dS brane to intersect the Euclidean boundary at complex angles.
Boundary Conditions: The complex saddles are dual to boundary conditions in the CFT with complex $g$ -functions. These boundary conditions appear in conjugate pairs, denoted $B$ and $\bar{B}$ , where the boundary state $|B\rangle$ and its conjugate $\langle \bar{B}|$ are related by Hermitian conjugation but are not identical (unlike unitary boundary states).

Key Results

Dual to Unitary Evolution: The authors demonstrate that an AdS $_3$ black hole microstate with a dS $_2$ brane behind the horizon corresponds to the unitary time evolution of a pure CFT state. This state is prepared by a Euclidean path integral on a cylinder with boundary conditions $B$ and $\bar{B}$ at the two ends:
$|\Psi(t)\rangle = e^{-itH_{\text{CFT}}} e^{-L H_{\text{CFT}}/2} |B\rangle$
Here, $L$ relates to the inverse temperature, and the state evolves unitarily within the standard CFT Hilbert space.
Complex Saddles and $g$ -functions: For a disk boundary $\Sigma$ , the complex saddles yield complex $g$ -functions ( $g_+$ and $g_-$ ) satisfying $g_- = g_+^*$ . These correspond to non-unitary boundary conditions in the BCFT.
Boundary Condition Changing (BCC) Operator: In the case of a cylinder with $B\bar{B}$ boundaries, the connected saddle point dominates. The analysis of the holographic energy-momentum tensor reveals the existence of a boundary condition changing primary operator connecting $B$ and $\bar{B}$ with conformal dimension:
$h_{\text{bcc}} = -c/8$
The negative dimension indicates the non-unitarity of the open-string sector (the boundary conditions), but the operator resides in the boundary sector rather than the bulk spectrum, preserving the unitarity of the underlying parent CFT.
Global Quench Analogy: The setup generalizes the Calabrese–Cardy global quench. While standard quenches use unitary boundary states ( $|B\rangle^\dagger = \langle B|$ ), this construction uses $|B\rangle^\dagger = \langle \bar{B}| \neq \langle B|$ . This results in an effective temperature $\beta^{-1} = L^{-1}$ , which is twice that of the standard case, driven by the $h = -c/8$ operator.

Significance and Claims
The paper claims to realize dS holography as state preparation within a unitary AdS/CFT Hilbert space.

Agnostic to dS Stability: The construction is agnostic to the ultimate fate (stability or decay) of the real Lorentzian dS region. The dS physics is encoded in the procedure used to prepare the CFT state. Once prepared, the state evolves unitarily in the AdS/CFT dual, possessing a real Lorentzian gravity dual governed by quantum gravity laws.
Parasitic dS: Conceptually, the work positions dS/CFT not as an autonomous non-unitary quantum theory, but as a "parasite" or a specific boundary condition that prepares a state in the Hilbert space of a unitary theory (the AdS/CFT dual).
Resolution of the "Elephant in the Refrigerator": The authors metaphorically claim to have solved the problem of embedding dS in AdS by allowing the dS brane to enter through "complex saddles" rather than real Euclidean geometries, effectively "closing the door" on the non-unitary dual theory by embedding it within a unitary framework.

The paper emphasizes that while the boundary conditions are non-unitary (complex $g$ -functions, negative scaling dimensions), the parent CFT remains unitary, and the Hilbert space is standard. The non-unitarity is confined to the specific boundary conditions used to prepare the state.