⚛️ high-energy theory

Near-horizon modifications in finite $N$ holography

By performing explicit bulk reconstructions in near-horizon modified AdS $_2$ and BTZ black hole backgrounds, this paper demonstrates that extending the AdS/CFT dictionary to finite $N$ induces non-perturbative bulk micro-causality violations characterized by a throat-dependent non-locality scale and a dip-ramp-plateau structure in the spectral form factor.

Original authors: Rishkrith Bairy, Mrityunjay Nath, Debajyoti Sarkar

Published 2026-06-23✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Rishkrith Bairy, Mrityunjay Nath, Debajyoti Sarkar

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant hologram. In this view, a complex 3D world (like a black hole) is actually a projection of information living on a 2D surface, much like a 3D movie is projected from a flat screen. This idea is called the AdS/CFT correspondence.

For a long time, scientists have studied this hologram assuming the "screen" is infinitely large and perfect (a concept called "large N"). But in reality, the screen is finite. The authors of this paper, Rishkrith Bairy, Mrityunjay Nath, and Debajyoti Sarkar, wanted to understand what happens when we look at the hologram with a "finite" screen size.

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: The "Infinite Time" Glitch

In the perfect, infinite version of this hologram, if you try to reconstruct what is happening deep inside a black hole (the "bulk"), you need to look at the edge of the screen (the "boundary") at times that go on forever.

Think of it like trying to hear a whisper from the center of a canyon. In a perfect canyon, the echo takes forever to die out. To understand the whisper, you have to listen for an infinite amount of time.

However, in our real, finite universe, there is a limit to how much information can be stored. The "echo" doesn't last forever; it eventually fades or fluctuates. If you try to use the infinite-time recipe to describe a finite system, the math breaks down. It's like trying to measure a finite cup of water with an infinite ruler; the numbers get messy and nonsensical. This suggests that the rules of "locality" (things only affecting their immediate neighbors) break down near the edge of a black hole when we account for the finite size of the universe.

2. The Proposed Fix: The "Smooth Throat"

The authors ask: What if the black hole doesn't actually have a sharp, infinite edge (a horizon)?

Usually, a black hole is drawn with a point of no return where space-time stretches infinitely. The authors propose a different shape: a Damour-Solodukhin (DS) wormhole.

Imagine a black hole not as a bottomless pit, but as a smooth, narrow throat (like the neck of a bottle).

In the old view: The throat goes down forever.
In this new view: The throat gets very narrow, but then it curves back up or stops. It's a smooth tunnel, not a bottomless hole.

This "throat" is controlled by a tiny parameter called $\lambda$ . This parameter is incredibly small, but it acts like a safety valve. Because the throat is finite, the "echo" (or the time it takes for information to travel) is also finite. You don't need to listen forever; the signal stops naturally because the geometry of space itself forces it to.

3. The Connection: Cutting the Tape vs. Building a Wall

The paper shows that two different ways of fixing the "infinite time" problem are actually the same thing:

The Boundary Fix: You manually tell the math to "stop listening" after a certain time (a "cut-off").
The Bulk Fix: You change the shape of the black hole to have a smooth throat (the DS wormhole).

The authors prove that if you build a black hole with this smooth throat, the math automatically stops at the right time, just as if you had manually cut the tape. The "throat" acts as a natural, geometric version of the "cut-off." It explains why the universe stops the signal: because the space itself has a limit.

4. The Evidence: The "Dip-Ramp-Plateau" Dance

To prove this idea works, the authors looked at how particles (probes) move inside these wormhole black holes. They calculated a specific pattern called the Spectral Form Factor (SFF).

Think of the SFF as a heartbeat monitor for the quantum system.

Dip: The signal drops quickly at the start.
Ramp: The signal slowly climbs back up in a straight line.
Plateau: The signal levels off at the top.

This "Dip-Ramp-Plateau" pattern is a famous signature of quantum chaos and is usually seen in complex systems like random matrices (think of it as the chaotic shuffling of a deck of cards).

The authors found that when they simulated particles moving in their "smooth throat" wormholes, the SFF showed this exact pattern.

When they looked at a single, specific throat size, the pattern was a bit wiggly.
But when they averaged over many slightly different throat sizes (like averaging out the noise), the "Ramp" became perfectly straight and clean.

5. The Conclusion

The paper concludes that these "wormhole" geometries are a valid way to describe what happens to black holes when we account for the finite size of the universe (finite $N$ ).

Instead of a black hole being a place where physics breaks down due to infinite stretching, it might be a smooth, finite tunnel. This smoothness naturally limits how far information can travel, fixing the "infinite time" glitch and producing the chaotic patterns (the Ramp) that we expect to see in a real, finite quantum universe.

In short: The authors replaced the "bottomless pit" of a black hole with a "smooth tunnel." This simple change fixes the math, stops the signals from going on forever, and produces the exact chaotic fingerprints we expect from a finite universe.

Technical Summary: Near-horizon modifications in finite N holography

Problem Statement
The paper addresses the tension between the semiclassical emergence of local bulk operators in the AdS/CFT correspondence and the expectations of finite $N$ holography. In the standard HKLL (Hamilton, Kabat, Lifschytz, Lowe) reconstruction, bulk fields are expressed as smeared boundary operators. However, in black hole backgrounds, the smearing integral requires access to arbitrarily late boundary times as the bulk point approaches the horizon. In a finite $N$ theory with finite entropy $S$ , boundary correlators exhibit fluctuations of order $O(e^{-S/2})$ at timescales $t \sim S$ . These fluctuations cause the HKLL smearing integrals to diverge, signaling a breakdown of the naive semiclassical reconstruction. Previous work (e.g., [13]) proposed resolving this by imposing an ad-hoc late-time cut-off on the boundary smearing integral, effectively "excising" the near-horizon region. The central question is whether this boundary excision has a geometric realization in the bulk that naturally enforces such a cut-off without external intervention.

Methodology
The authors investigate this problem by comparing two approaches:

Boundary Cut-off/Excision: They review the procedure of truncating the HKLL smearing integral at a time $t_{cut}$ , which corresponds to subtracting a specific semiclassical contribution localized near the Poincaré boundary at late times. This introduces a non-locality of order $O(e^{-S/2})$ .
Geometric Realization via Damour-Solodukhin (DS) Wormholes: They propose that the near-horizon modification required to implement this cut-off is geometrically realized by DS-type wormholes. In these geometries, the would-be black hole horizon is replaced by a smooth throat controlled by a parameter $\lambda$ $λ$ .
- 2D Analysis (AdS $_2$ ): They perform explicit bulk reconstruction in the DS $_2$ wormhole background. By solving the wave equation, they show that the tortoise coordinate $r_*$ becomes finite ( $r_*^{max}$ ) due to the throat, naturally bounding the support of the smearing integral. They match the geometric scale $r_*^{max}$ to the boundary cut-off scale $t_{max}$ to estimate $\lambda \sim e^{-S/4}$ (implying $\lambda^2 \sim e^{-S/2}$ ).
- 3D Analysis (AdS $_3$ ): They generalize the analysis to BTZ black hole mimickers (DS $_3$ wormholes). The radial wave equation reduces to a general Heun equation. They derive the exact smearing kernel in terms of Heun functions and perform a small- $\lambda$ expansion to demonstrate that the DS field decomposes into the standard BTZ local field plus a correction of order $O(\lambda^2)$ .
- Spectral Analysis: To probe the non-perturbative spectral structure, they reformulate the radial dynamics in 3D as a Schrödinger problem with an effective potential. Using the Bohr-Sommerfeld quantization condition, they numerically compute the discrete WKB spectrum of probe modes. They then calculate the Spectral Form Factor (SFF) $g(\beta, t)$ for this spectrum, both at fixed $\lambda$ and averaged over an ensemble of $\lambda$ values.

Key Contributions and Results

Geometric Excision: The paper demonstrates that DS wormholes provide a natural, geometric realization of the finite $N$ excision process. The smooth throat imposes a finite optical depth, which automatically bounds the HKLL smearing integral, recovering the same non-locality estimates ( $O(e^{-S/2})$ ) as the ad-hoc boundary cut-off method.
Non-locality Estimate: By matching the maximal tortoise depth of the wormhole to the boundary time scale $t_{max} \sim S$ , the authors derive the scaling $\lambda^2 \sim e^{-S/2}$ , consistent with previous estimates in the literature [17, 18].
3D Bulk Reconstruction: The authors explicitly construct the bulk-to-boundary dictionary in 3D using Heun functions. They show that the deviation from the local BTZ reconstruction is perturbatively controlled by the throat parameter $\lambda$ , appearing as a correction term $\lambda^2 \delta \Phi$ .
Spectral Form Factor (SFF) Structure: The numerical analysis of the probe spectrum in the DS $_3$ $_{3}$ background reveals a "dip-ramp-plateau" (DRP) structure in the SFF.
- The discrete spectrum arises naturally from the confining potential of the throat, without imposing an artificial stretched horizon boundary condition.
- At fixed $\lambda$ , the SFF exhibits non-universal oscillations.
- Upon averaging over a distribution of the throat parameter $\lambda$ (analogous to parameter averaging in brick wall models), the oscillations are suppressed, revealing a clean, near-linear ramp followed by a plateau.
Comparison to Brick Wall Models: The results parallel recent findings in BTZ black holes with stretched horizons or brick walls [27, 28, 31]. The DS throat acts as a purely geometric regulator that reproduces the same qualitative SFF phenomenology (DRP structure) expected for finite $N$ black hole microphysics.

Significance and Claims
The paper claims that DS wormhole geometries serve as a "bulk completion" of the cut-off/excision picture, offering a geometric motivation for the non-perturbative modifications required at finite $N$ . The work establishes that:

The breakdown of semiclassical locality near horizons can be modeled by smooth throat geometries where the optical depth is finite.
These geometries naturally generate the discrete spectra and late-time spectral correlations (manifested as the linear ramp in the SFF) associated with quantum chaotic systems and finite entropy.
The DRP structure in the SFF can emerge from deterministic spectra with specific level organization (induced by the throat) and does not necessarily require full Random Matrix Theory (RMT) universality, though $\lambda$ -averaging helps isolate the coarse DRP envelope.

The authors conclude that while these wormhole geometries successfully reproduce the expected phenomenology of finite $N$ black hole microphysics (specifically the SFF structure), probing the true chaotic structure of quantum gravity would require further deformations to the effective potential and a study of unfolded spectral properties, which they identify as a potential future direction.

1. The Problem: The "Infinite Time" Glitch

2. The Proposed Fix: The "Smooth Throat"

3. The Connection: Cutting the Tape vs. Building a Wall

4. The Evidence: The "Dip-Ramp-Plateau" Dance

5. The Conclusion

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