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Towards a Holographic dual of Carrollian BCFT

This paper establishes the foundation for a holographic dual of Carrollian BCFT by demonstrating that the Boundary Carrollian Conformal Algebra emerges as the symmetry of flat spacetime at null infinity when constrained by a specific end-of-the-world brane, thereby constructing a flat-space analogue of the AdS $_3$ /BCFT $_2$ correspondence.

Original authors: Pronoy Chakraborty, Ritankar Chatterjee, Priyadarshini Pandit

Published 2026-08-11

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CC BY 4.0

Original authors: Pronoy Chakraborty, Ritankar Chatterjee, Priyadarshini Pandit

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 movie screen. For decades, physicists have been trying to figure out how the "movie" playing inside the screen (gravity and space-time) is actually just a projection of a simpler, two-dimensional story happening on the edge of the screen. This idea is called the Holographic Principle. The most famous version of this is the AdS/CFT correspondence, which says that a universe with a specific kind of curved geometry (like a saddle shape) is mathematically identical to a quantum theory living on its boundary. It's like realizing that a 3D video game is actually just a 2D code running on a computer chip.

But what about our actual universe? It doesn't look like a saddle; it looks flat. Scientists have been trying to build a holographic map for flat space, but it's been tricky. To do this, they use a strange, futuristic concept called Carrollian physics. Imagine a world where the speed of light is zero. In this world, time stands still for everything, and space becomes the only thing that really "moves." It sounds impossible, but mathematically, it's a powerful tool to describe the edges of our flat universe. Now, add a twist: what if that universe has a wall? In physics, a "boundary" is like a wall where the universe stops. The paper you are about to read explores what happens when you put a wall in this zero-speed-of-light universe and try to find its holographic twin.

The Paper: Building a Holographic Wall in a Zero-Speed Universe

In this work, the authors, Pronoy Chakraborty, Ritankar Chatterjee, and Priyadarshini Pandit, take a first step toward building a holographic map for a Carrollian Boundary Conformal Field Theory (BCCFT). Think of this as trying to write the "source code" for a universe that has a wall and moves at zero speed. Their main goal is to find the "bulk" (the 3D interior) that corresponds to this 2D boundary theory.

The Big Discovery: The Tensionless Wall
The authors start by asking: "What kind of wall (or End-of-the-World brane) do we need in a flat 3D universe to create the right kind of boundary?" They find that the answer is a very specific, invisible wall. It's not a heavy, solid wall; it's a tensionless one.

To understand this, imagine a drumhead. If you pull it tight, it has tension. If you let it go completely slack, it has zero tension. The authors discovered that the only wall that works for this flat, zero-speed universe is one that is completely slack—like a piece of paper floating in a vacuum with no weight or pull on it. They proved this in two ways:

Looking inside: They analyzed the wall directly in flat space and found it must have zero tension to exist without breaking the rules of the universe.
Looking from the past: They started with the famous "curved" universe (AdS) where walls have tension, and then slowly "flattened" the universe. As they flattened it, the tension of the wall had to shrink to zero to keep the math working. The result was the same: a tensionless wall.

The Symmetry Puzzle: Breaking the Rules
In physics, "symmetry" means the rules stay the same even if you rotate or move things. A flat universe usually has a huge set of symmetries (like being able to spin in any direction). But when you put a wall in the way, you break some of those rules.

The authors found that placing this tensionless wall at specific spots (where the angle is 0 or 180 degrees) breaks the universe's symmetry down to a smaller, specific set. They call this new set of rules the Boundary Carrollian Conformal Algebra (BCCA).

The Twist: In normal, relativistic physics (where light has a speed), putting a wall usually just cuts the symmetry in half in a predictable way. But in this zero-speed Carrollian world, the authors found that the type of wall matters. If you put the wall in a different spot (like a time boundary instead of a space boundary), you get a completely different set of rules. This suggests there isn't just one "flat holography," but many different versions depending on how you build your wall.

The Holographic Match
The most exciting part is the match. The authors showed that the symmetries of this flat universe with a tensionless wall are exactly the same as the symmetries of the BCCFT on the boundary.

They checked this by looking at the "global" symmetries (the big, obvious rules) and the "asymptotic" symmetries (the rules that apply far away at the edge of the universe).
In both cases, the math lined up perfectly. The "bulk" (the 3D flat space with the wall) and the "boundary" (the 2D Carrollian theory) are two sides of the same coin.

What They Didn't Solve (Yet)
The paper is careful to say this is just the beginning. They have built the foundation and found the right wall, but they haven't written the full "dictionary" yet.

They haven't fully figured out how to calculate specific things, like how particles interact on this boundary (correlation functions).
They haven't calculated the "boundary entropy" (a measure of information stored on the wall), which is a key test for holographic theories in other contexts.
They suggest that future work needs to explore what happens if the wall is placed along a "null" direction (a light-like path), which might lead to a totally different kind of symmetry (just one copy of a famous algebra called Virasoro instead of the complex BCCA).

Why It Matters
This paper suggests that if we want to understand quantum gravity in our flat universe using holography, we might need to think about "tensionless" walls and zero-speed physics. It opens a door to a new kind of map where the universe isn't just a projection of a curved space, but a projection of a flat space with a very specific, slack boundary. It's a bit like realizing that to understand the 3D shape of a shadow, you don't need a curved light source; you just need to understand the flat object casting it and the specific angle of the light. The authors have found the right angle and the right object, and now the real work of reading the shadow begins.

Technical Summary: Towards a Holographic dual of Carrollian BCFT

Problem Statement
While the holographic principle has been successfully realized in asymptotically Anti-de Sitter (AdS) spacetimes via the AdS/CFT correspondence, its extension to asymptotically flat spacetimes (AFS) remains less understood. Recent developments in Carrollian holography propose that quantum gravity in AFS is dual to a Carrollian Conformal Field Theory (CCFT) living on the null boundary. However, while Carrollian CFTs have been studied extensively, Carrollian CFTs with boundaries (BCCFTs) remain largely unexplored. The specific problem addressed in this work is the identification of a holographic dual for 2d Carrollian BCFTs. Specifically, the authors aim to construct a flat-space analogue of the AdS $_3$ /BCFT $_2$ correspondence by identifying the appropriate bulk geometry (a restricted segment of 3d flat spacetime) and verifying that its symmetries match the recently discovered Boundary Carrollian Conformal Algebra (BCCA).

Methodology
The paper employs a dual-pronged approach, analyzing the system both intrinsically within flat spacetime and by taking the flat-space limit of the known AdS $_3$ /BCFT $_2$ setup.

Symmetry Analysis of BCCA: The authors begin by reviewing the BCCA, the symmetry algebra of a Carrollian manifold with boundaries. They analyze how the introduction of boundaries on a relativistic cylinder (reducing two copies of the Virasoro algebra to one) translates to the Carrollian limit. They identify that different boundary conditions on the Carrollian manifold lead to distinct realizations of the algebra.
Intrinsic Construction in Flat Space: The authors introduce an End-of-the-World (EOW) brane in 3d flat spacetime defined by the equation $x^2 = r \sin \phi = 0$ . This brane intersects the future null infinity ( $\mathcal{I}^+$ ) at the angular boundaries $\phi = 0, \pi$ . They analyze the properties of this brane using the Einstein-Hilbert action with a Gibbons-Hawking-York boundary term, determining its tension and intrinsic geometry.
Flat Limit of AdS $_3$ : To validate the intrinsic findings, the authors take the flat-space limit ( $\ell \to \infty$ ) of the AdS $_3$ /BCFT $_2$ correspondence. They consider an AdS $_3$ bulk restricted by an EOW brane with profile $r \sin \phi = k$ , where the tension $T$ is dialed to zero ( $T \sim 1/\ell^2$ ) to ensure the brane passes through the central Minkowski diamond in the limit.
Symmetry Derivation: The authors derive the global and asymptotic symmetries of the flat spacetime restricted by the EOW brane. They perform an Inönü-Wigner contraction on the global symmetries of AdS $_3$ ($so(2,2)$) and the asymptotic symmetries (Virasoro) to demonstrate that they reduce to the global and asymptotic symmetries of the restricted flat spacetime, respectively.
Representation Theory: The paper constructs the representation theory of the resulting global symmetry algebra ($iso(1,1)$) by inducing it from the 3d Poincaré algebra ($iso(2,1)$) and by taking the Carrollian limit of the highest-weight representations of $so(2,1)$.

Key Contributions and Results

Identification of the Bulk Dual: The authors identify the bulk dual of a 2d BCCFT as 3d flat spacetime restricted by a tensionless EOW brane located at $x^2 = 0$ (or more generally $x^2 = -a$ ). This brane intersects the null infinity at $\phi = 0, \pi$ .
Symmetry Matching:
- Global Symmetries: The presence of the EOW brane breaks the 3d Poincaré algebra ($iso(2,1)$) down to a 2d Poincaré algebra ($iso(1,1)$). The authors demonstrate that this $iso(1,1)$ algebra is isomorphic to the global subalgebra of the BCCA. This result is confirmed by taking the flat limit of the $so(2,1)$ global symmetries of the AdS $_3$ /BCFT $_2$ setup.
- Asymptotic Symmetries: By imposing the boundary condition that symmetry generators must not distort the brane at $\phi = 0, \pi$ (even at subleading orders in $1/r$ ), the authors show that the asymptotic symmetry algebra of the restricted flat spacetime is exactly the BCCA. This algebra is generated by specific linear combinations of the standard BMS $_3$ generators ( $O_n$ and $P_n$ ) that survive the boundary constraints.
Representation Theory: The paper constructs the irreducible representations of the global $iso(1,1)$ algebra. These are shown to correspond to massive particles restricted to the $x^2=0$ hypersurface. The authors verify that these representations emerge consistently from both the induced representation of the 3d Poincaré algebra and the flat limit of the highest-weight representations of $so(2,1)$.
Alternative Boundary Configurations: The authors explore a different boundary condition: a temporal boundary at $u=0$ (corresponding to a light-cone $t=r$ EOW brane). They find that this configuration preserves a single copy of the Virasoro algebra rather than the BCCA, highlighting that the choice of boundary in Carrollian spacetime fundamentally alters the resulting symmetry algebra, unlike the universal reduction seen in relativistic BCFTs.

Significance and Claims
The paper claims to lay the foundation for a "flat version of AdS $_3$ /BCFT $_2$ correspondence" through the Carrollian framework. Its primary significance lies in:

Establishing a concrete bulk realization (tensionless EOW brane in AFS) for Carrollian BCFTs.
Demonstrating that the asymptotic symmetries of this restricted flat spacetime coincide exactly with the BCCA, thereby providing the necessary symmetry match for a holographic duality.
Showing that the results are robust, as they can be derived both intrinsically and as a consistent flat-space limit of the well-understood AdS $_3$ /BCFT $_2$ correspondence.
Highlighting the unique feature of Carrollian holography where different boundary choices lead to distinct symmetry algebras (BCCA vs. Virasoro), suggesting a richer landscape of AFS/BCCFT correspondences compared to the relativistic case.

The authors position this work as an initial step, noting that a complete formulation requires further development of the BCCFT itself (including correlation functions and boundary states) and the construction of a holographic dictionary (e.g., flat-space analogues of geodesic Witten diagrams and the Ryu-Takayanagi conjecture for entanglement entropy).

The Paper: Building a Holographic Wall in a Zero-Speed Universe

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