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Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT

This paper establishes a formal connection between TQFT principles and AdS/BCFT by demonstrating that both the Ryu-Takayanagi prescription and the Complexity-Action proposal consistently extract universal boundary entropy from holographic setups involving end-of-the-world branes, including in the context of BTZ black holes.

Original authors: Fabiano F. Santos

Published 2026-08-12

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CC BY 4.0

Original authors: Fabiano F. Santos

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, three-dimensional hologram projected from a two-dimensional surface, like a movie playing on a flat screen that somehow contains all the depth and drama of a real world. This is the core idea of "holography" in modern physics, a concept that suggests the complex rules of gravity and space-time we experience might actually be a reflection of information stored on a distant boundary. Now, imagine that this boundary isn't a perfect, infinite circle, but rather a room with a wall. In physics, this is called a Boundary Conformal Field Theory (BCFT). It's like having a universe that stops abruptly at a wall, where the rules of the game change depending on how that wall is built. Scientists are obsessed with understanding these "walls" because they might hold the key to how information is stored, how black holes work, and even how the universe began. To make sense of these tricky boundaries, physicists often borrow tools from "Topological Quantum Field Theories" (TQFTs). Think of TQFTs as a set of Lego instructions that only care about the shape of the pieces and how they snap together, ignoring the specific colors or textures. This paper asks a fascinating question: Can we use these simple "shape-only" Lego rules to understand the messy, complex, and very real physics of these holographic walls?

The author of this paper, Fabiano F. Santos, has built a clever bridge between the abstract world of TQFTs and the concrete world of holographic gravity. Their main discovery is that the "End-of-the-World" (EOW) brane—the physical object representing that mysterious wall in the hologram—acts exactly like a special interface that carries a unique "signature" of information. They found that by carefully measuring the "complexity" of the holographic system (which is a way of counting how hard it is to build a specific quantum state), they can isolate a specific number called the "boundary entropy" (often written as $\log g$ ). This number is like a fingerprint of the wall itself; it tells you exactly what kind of boundary conditions are in place, independent of the messy details of the space around it.

To do this, the team used a method called "Complexity = Action," which suggests that the complexity of a quantum system is directly related to the amount of "work" (or action) required to build the geometry of space-time behind the scenes. They calculated this work for different scenarios, including a vacuum (empty space) and a thermal environment (like a black hole). They showed that if you subtract out the boring, standard parts of the calculation that depend on how you choose to measure things (renormalization), what remains is a clean, universal piece of data: the boundary entropy. This confirms that the "wall" isn't just a passive barrier; it actively stores information that changes the complexity of the entire system.

The paper goes a step further by looking at a specific type of black hole called a BTZ black hole. In this scenario, the "wall" (the EOW brane) interacts with the black hole's interior. The author found that as time goes on, the complexity of the system grows. This growth isn't just about the black hole getting bigger or hotter; it also encodes the specific details of the boundary condition. It's as if the black hole's "brain" is thinking about two things at once: the heat of the fire inside it, and the specific rules of the wall it's trapped against. The paper suggests that by measuring this complexity, we can actually "read" the properties of the boundary, separating the universal information (the $\log g$ ) from the local, messy details.

In short, this work doesn't just say "gravity is complicated." It provides a dictionary to translate the complex language of gravity into the simpler language of topological shapes. It proves that the "End-of-the-World" brane is a real, physical interface that carries a specific amount of information, and that we can extract this information by looking at how the complexity of the universe grows over time. While the math is heavy, the idea is playful: the universe has a "wall," and that wall has a secret code written on it that we can finally start to read.

Technical Summary: Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT

Problem Statement
The paper addresses the need to systematically understand the intersection between the organizational principles of Topological Quantum Field Theories (TQFTs) and the dynamical framework of the AdS/BCFT (Anti-de Sitter/Boundary Conformal Field Theory) correspondence. Specifically, it seeks to clarify how "end-of-the-world" (EOW) branes, which introduce non-trivial boundary conditions and localized degrees of freedom, influence holographic observables. While TQFTs provide a rigorous language for state preparation, cobordism composition, and gluing via functors, AdS/BCFT is a metric-dependent gravitational theory. The central problem is to establish a dictionary that maps TQFT axioms (such as cobordism composition and interface contraction) to the geometric and variational data of AdS/BCFT, particularly focusing on how these structures manifest in holographic complexity and boundary entropy.

Methodology
The author employs a comparative analysis between TQFT axioms (Atiyah–Segal) and the semiclassical gravity description of AdS/BCFT. The methodology proceeds through the following steps:

Conceptual Mapping: The paper constructs a "dictionary" mapping TQFT operations (disjoint union, orientation reversal, gluing of cobordisms) to holographic processes. In this framework, the EOW brane $Q$ is interpreted as a physical interface carrying boundary-state data, analogous to how a boundary in a TQFT prepares a state or contracts indices.
Geometric Decomposition: The author analyzes the decomposition of gravitational manifolds into patches (e.g., excising worldlines or tubular neighborhoods). They distinguish between "artificial" internal interfaces, where Gibbons–Hawking–York (GHY) terms cancel, and "physical" interfaces (EOW branes, joints, null boundaries), where localized action terms survive.
Complexity = Action (CA) Calculation: Applying the Complexity=Action proposal, the author computes the on-shell action of the Wheeler–DeWitt (WDW) patch. This includes bulk volume terms, boundary terms, joint terms at intersections, null counterterms, and the specific action localized on the EOW brane.
Renormalization and Relative Complexity: To isolate universal contributions, the author utilizes a relative renormalization prescription. They compare configurations with different brane tensions (and thus different boundary conditions) while keeping the global topology and cutoff data fixed. This allows for the subtraction of scheme-dependent local terms to isolate the universal interface contribution.
Thermal Extension: The analysis is extended from vacuum AdS $_3$ to thermal geometries, specifically the planar BTZ black hole with an EOW brane. The author evaluates the late-time growth of the WDW action, accounting for the horizon, the thermal interior, and the interaction between the brane and the null boundaries of the WDW patch.

Key Contributions and Results

TQFT-Gravity Dictionary: The work establishes a formal correspondence where the EOW brane and its associated joint terms realize the TQFT concept of "sewing" or gluing. The brane is not merely a geometric truncation but encodes the variational data and degrees of freedom necessary for the composition of gravitational amplitudes.
Extraction of Boundary Entropy ( $\log g$ ): In the vacuum case, the author demonstrates that the Euclidean action and the Ryu–Takayanagi prescription consistently isolate the universal boundary entropy, identified as the Affleck–Ludwig entropy ( $\log g_B$ ). They show that $\log g_B$ corresponds to the finite overlap between the vacuum and the boundary state prepared by the brane.
Complexity as a Probe of Boundary Data: Applying the CA proposal, the author shows that a relative renormalization of the WDW action extracts the same universal contribution ( $\log g_B$ ) from the holographic complexity. They derive that the renormalized difference in complexity, $\Delta C$ , is directly proportional to the boundary entropy:
$\Delta C(T) = \frac{\log g}{\pi \hbar}$
This result holds after consistently canceling metric-dependent residues and scheme-dependent counterterms.
Thermal Complexity and Black Hole Interiors: In the thermal BTZ setting, the late-time growth rate of complexity is shown to encode two distinct pieces of information:
1. The universal thermal contribution associated with the black hole interior, proportional to $2M$ (or $T_H S_{BH}$ ).
2. A contribution sensitive to the holographic boundary condition, determined by the brane tension and the interface structure.
  The total growth rate is expressed as:
  $\frac{d C_{A}}{dt} = \frac{1}{\pi \hbar} (2M + S_{bdry})$
  This indicates that holographic complexity measures not only the extent of the thermal interior but also the specific manner in which the system is terminated at the boundary.

Significance and Claims
The paper claims to provide a unified organizational principle for understanding holographic observables in systems with boundaries. By treating the EOW brane as a physical interface analogous to TQFT sewing data, the author argues that the "computational cost" of preparing a boundary state (complexity) is fundamentally linked to the universal information stored at that interface (boundary entropy).

The significance lies in the ability to separate universal, topological-like data (boundary entropy) from local, metric-dependent data through a consistent renormalization scheme. The author emphasizes that while AdS/BCFT is not a strict TQFT due to its metric dependence, the TQFT axioms serve as a powerful framework for organizing the composition of states and interfaces. The work suggests that holographic complexity acts as a probe for boundary degrees of freedom, extending the understanding of the CA conjecture to include the specific influence of boundary conditions and brane tensions in both vacuum and thermal regimes.

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