⚛️ high-energy theory

Entanglement entropy in holographic CFTs with generic boundaries

This paper demonstrates that describing entanglement entropy in holographic conformal field theories with generic boundaries requires extending the dual bulk spacetime to complex coordinates, a finding supported by both analytical derivations and numerical checks using free Gaussian fermions.

Original authors: Fabio Ori, Peng-Xiang Hao

Published 2026-09-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabio Ori, Peng-Xiang Hao

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 modern physics, there is a powerful idea known as the holographic principle. It suggests that a three-dimensional universe, with all its gravity and complexity, can be completely described by a two-dimensional surface that surrounds it, much like how a hologram stores a 3D image on a flat piece of plastic. This concept, called the AdS/CFT correspondence, has become a cornerstone for understanding how gravity and quantum mechanics might fit together. In this framework, the "bulk" is the three-dimensional space where gravity lives, and the "boundary" is the two-dimensional surface where quantum particles dance. Usually, physicists assume that both the bulk and the boundary exist in a standard, real world where time flows forward and space stretches out in familiar directions. However, nature is often more subtle than our initial assumptions. In many quantum systems, especially those with edges or interfaces, the rules of time and space can become blurred, leading to scenarios where the boundary is not just a slice of time but a different kind of surface altogether. Understanding how to describe these unusual boundaries is crucial because they appear in theories about the early universe, black holes, and the very fabric of spacetime itself.

A team of researchers has now discovered that when the boundary of such a system has a specific, unusual shape—what physicists call a spacelike signature—the standard rules of the holographic universe break down. In their study, they found that to make the math work and match the predictions of the quantum theory on the surface, the interior space where gravity lives cannot remain purely real. Instead, the researchers showed that the interior space must be extended into the realm of complex numbers. This does not mean the space is imaginary in a fictional sense, but rather that the coordinates used to describe it must include complex values to remain consistent. When the boundary is shaped in this specific way, the objects that usually connect the surface to the interior, such as the surfaces used to measure quantum entanglement, become complex entities. They exist in a mathematical space that is a blend of real and complex dimensions, yet they remain firmly anchored to the real, physical boundary where the quantum theory lives.

The researchers demonstrated this by looking at two different scenarios: a static setup where the boundary is fixed in time, and a dynamic one where the boundary moves. In the static case, they examined a system where the boundary acts like a moment of preparation for a quantum state. When they tried to calculate the entanglement entropy—a measure of how much information is shared between different parts of the system—they found that the standard method of finding the shortest path through the interior space failed. The shortest path, which usually gives a real number, simply did not exist in the real world for this specific boundary shape. The only way to find a valid path was to allow the coordinates of the interior space to become complex. This led to a path that was mathematically complex, yet when they calculated the length of this path, the result perfectly matched the predictions made by the quantum theory on the boundary. The real part of this complex length corresponded to the expected entropy, while the imaginary part carried a specific, consistent value that the quantum theory also predicted.

To ensure this was not just a mathematical trick, the team performed a rigorous check using a computer simulation of a simple quantum system made of free fermions, which are basic particles that do not interact with each other. They simulated a system with a boundary that mimicked the unusual spacelike shape. The simulation produced results for the entanglement entropy that matched the holographic predictions almost exactly, including both the real and imaginary components. This agreement between the complex geometry of the interior and the behavior of the particles on the boundary provided strong evidence that the complex extension is a necessary feature of the theory, not an artifact of a specific calculation method. The researchers also looked at a more complex scenario involving a boundary that moves over time, similar to a mirror reflecting light. Even in this dynamic situation, they found that the connection between the boundary and the interior required the interior space to be complexified. The paths connecting the two remained complex, and their lengths continued to match the quantum predictions.

The study explicitly argues against the idea that these systems can be fully described within a standard, real-valued spacetime. Previous attempts to describe such boundaries often led to mathematical inconsistencies, such as paths that crossed over themselves in impossible ways or failed to provide a unique answer. By insisting that the interior space must be complex, the researchers resolved these inconsistencies. They showed that the complex nature of the interior is not a temporary fix but a fundamental requirement for describing these generic boundaries. The work suggests that whenever the causal structure of a boundary changes—when it stops behaving like a normal slice of time and takes on a different character—the holographic duality naturally extends into the complex plane. This finding opens a new window into how gravity and quantum mechanics interact in extreme or unusual conditions, suggesting that the geometry of our universe might be far more flexible and mathematically rich than previously thought. The researchers conclude that this complexification is a general feature of holography, one that ensures the physical meaning of the theory is preserved even when the boundaries become strange.

Technical Summary: Entanglement Entropy in Holographic CFTs with Generic Boundaries

Problem Statement
Standard holographic duality (AdS/CFT) and its extension to Boundary Conformal Field Theories (AdS/BCFT) are typically formulated for real Lorentzian bulk geometries. In the AdS/BCFT correspondence, the entanglement entropy of a boundary region is computed via the Ryu-Takayanagi (RT) formula, involving extremal surfaces anchored on the boundary and potentially on an "end-of-the-world" (EOW) brane. However, this framework encounters inconsistencies when applied to BCFTs with boundaries of generic spacetime signatures, particularly spacelike boundaries. In such cases, the standard prescription for extremal surfaces can lead to self-intersecting RT surfaces or fail to reproduce the correct field-theoretic predictions derived from twist operator correlators. The central problem addressed is how to consistently describe holographic entanglement entropy for BCFTs with spacelike or time-dependent boundaries without violating the physical interpretation of the boundary theory.

Methodology
The authors propose that a consistent description of CFTs with generic boundaries requires extending the dual bulk spacetime to complex coordinates. The methodology involves:

Complexification of the Bulk: The spacetime manifold $M_{d+1}$ and the EOW brane $B_d$ are analytically continued from real coordinates ( $x^\mu \in \mathbb{R}$ ) to complex coordinates ( $x^\mu \in \mathbb{C}$ ). Crucially, the complexified brane remains anchored on the real asymptotic boundary, preserving the physical meaning of the BCFT.
Extremization in Complex Geometry: The holographic entanglement (pseudo) entropy is calculated by finding complex extremal surfaces (geodesics in AdS $_3$ ) that connect the boundary region to a complex island on the analytically continued brane.
Selection of Dominant Saddles: When multiple complex saddle points exist, the dominant contribution is selected by minimizing the real part of the area functional. If real parts are degenerate, time-ordering prescriptions (similar to those used for temporal entanglement) are employed to select the correct saddle.
Verification: The proposal is tested in three distinct settings:
- Static Spacelike Branes: In Poincaré AdS $_3$ with a constant-time brane ( $t=0$ ).
- Finite-Cutoff Branes: In AdS $_3$ with a de Sitter (dS $_2$ ) brane carrying dynamical gravity.
- Time-Dependent Boundaries: Using holographic moving mirrors with spacelike trajectories.
Numerical Cross-Check: The holographic results are compared against numerical simulations of a (1+1)-dimensional system of free Gaussian fermions with a simulated spacelike boundary.

Key Contributions and Results

Complex RT Surfaces and Branes: The authors demonstrate that for spacelike boundaries, the extremization of the geodesic length functional naturally forces the anchor points on the EOW brane to lie in the complex plane. Consequently, the RT surfaces themselves become complex.
- In the constant-time brane example, the disconnected geodesic length acquires an imaginary part ( $i\pi$ ), matching the phase structure of the twist-operator correlators in the dual BCFT $_2$ .
- In the de Sitter brane example, the complex saddles reproduce the island formula predictions. The authors show that the "real" saddles previously identified in similar contexts are unphysical (corresponding to local maxima or self-intersecting surfaces), leaving the complex family as the only valid solution.
Agreement with BCFT Predictions: The complex holographic calculation yields an entanglement (pseudo) entropy that exactly matches the field-theoretic result derived from twist-anti-twist correlators in the BCFT. This includes both the real logarithmic scaling and the specific imaginary constant terms arising from time-ordering prescriptions.
Time-Dependent Generalization: The framework extends to moving mirrors. When the mirror trajectory corresponds to a spacelike boundary, the resulting geodesic lengths are complex, and the imaginary parts are fixed by the causal structure preserved under conformal transformations.
Numerical Validation: The paper provides a third numerical check using free Gaussian fermions. The simulation reproduces the asymptotic regimes of the holographic entropy (both real and imaginary parts) for small and large entangling regions, confirming the validity of the complex saddle point approach.

Significance and Claims
The paper claims that the emergence of complex bulk geometries is not merely an artifact of a specific analytic continuation but a general feature of holographic duality required to describe CFTs with generic boundaries.

Resolution of Inconsistencies: The complexification resolves the inconsistencies (such as self-intersecting surfaces) that arise when applying standard real Lorentzian prescriptions to spacelike boundaries.
Unified Framework: The work suggests that complex bulk configurations represent a broader feature of holography whenever real ones fail to capture relevant boundary observables. It bridges the gap between standard entanglement entropy and "pseudo-entropy" (associated with non-unitary transition matrices), viewing the latter as a transition in the boundary region size or signature.
Physical Interpretation: Despite the complex nature of the bulk objects, the framework preserves the physical interpretation of the BCFT because the complexified bulk remains anchored to a real asymptotic boundary.
Future Directions: The authors suggest that this approach can be applied to higher-dimensional islands and other contexts involving complex metrics, such as timelike entanglement, dS holography, and black hole singularities, provided consistency criteria (like the KSW condition) are met.

In summary, the paper argues that to consistently describe holographic entanglement for boundaries with generic signatures, one must accept that the dual bulk geometry and its relevant extremal surfaces are inherently complex, yet this complexity is necessary to faithfully reproduce the physical predictions of the boundary field theory.

Technical Summary: Entanglement Entropy in Holographic CFTs with Generic Boundaries

More like this