Exact holographic thermal spectral functions: OPE, non-perturbative corrections, and black hole singularity

This paper demonstrates that the exact thermal spectral functions of even-dimensional holographic CFTs factorize into perturbative and non-perturbative components, utilizing exact WKB techniques to derive a full transseries expansion that explicitly links the non-perturbative sector to the imprints of the black hole singularity in complex time correlators.

The Gist

Imagine you are trying to understand the inside of a black hole, but you are stuck outside the event horizon. You can't see inside, and you can't send a probe in (it would get crushed). However, you have a magical mirror on the outside wall of the universe. This mirror reflects the "shadows" of everything happening inside the black hole.

This paper is about figuring out exactly what those shadows look like, specifically when the black hole is hot (thermal) and how the light (or information) bounces around inside it.

Here is the breakdown of the paper's journey, using simple analogies:

1. The Setup: The Hot Black Hole and the Mirror

In the world of physics, there's a famous idea called AdS/CFT correspondence. Think of it like a hologram.

• The Black Hole (The Bulk): This is the 3D (or higher-dimensional) object in space. It has a surface (the horizon) and a terrifying center (the singularity) where physics breaks down.
• The Hologram (The Boundary): This is a lower-dimensional surface surrounding the black hole. It contains a "Quantum Field Theory" (a set of rules for particles).
• The Connection: Everything that happens inside the black hole is encoded in the patterns on the hologram. If you study the "music" (correlations) played on the hologram, you can deduce what the black hole looks like inside.

The authors are studying the "music" played when the black hole is hot. They want to know: Can we hear the black hole's singularity (the crushing center) just by listening to the music on the outside?

2. The Problem: The Song is Too Complicated

When you listen to the music (the "spectral function"), it's a mix of two things:

1. The Predictable Part (Perturbative): This is like the standard notes of a song. You can calculate these easily using known rules (like how a guitar string vibrates). In physics, this is the "OPE" (Operator Product Expansion) part. It tells you about the surface of the black hole and the horizon.
2. The Mysterious Part (Non-Perturbative): This is the "ghost notes" or the echo that comes from deep inside the black hole, near the singularity. Standard math tools can't calculate this. It's like trying to predict the sound of a drum by only looking at the drumhead, ignoring the hollow space inside.

For a long time, physicists couldn't separate these two parts cleanly. They were stuck in a soup of complex equations.

3. The Breakthrough: Splitting the Song

The authors discovered a magical trick for a specific type of black hole (even dimensions) and specific types of particles. They found that the total "song" (the spectral function) can be factored (split) perfectly into two independent pieces:

$$\text{Total Song} = (\text{Predictable Surface Part}) \times (\text{Mysterious Interior Part})$$

• The Surface Part: This is fixed by looking at the edge of the universe. It's boring but necessary.
• The Interior Part: This is the gold mine. It contains all the secrets about the black hole's interior, including the singularity.

The Analogy: Imagine a complex cake. Usually, the flour, sugar, and eggs are mixed together so you can't tell them apart. The authors found a way to separate the cake into the "flour layer" (which they already understood) and the "secret spice layer" (which tells you about the black hole's core).

4. The Tool: The "Exact WKB" Flashlight

To understand the "secret spice layer," they used a technique called Exact WKB.

• Standard WKB: Imagine trying to walk through a dark forest with a dim flashlight. You can see the path a few steps ahead, but it's an approximation.
• Exact WKB: The authors turned the flashlight into a super-laser that sees every step, every branch, and every shadow perfectly, even in the dark.

They used this "super-laser" to trace the path of waves as they travel from the black hole's surface, dive into the horizon, bounce off the singularity, and come back out.

5. The Discovery: The "Bouncing" Singularity

By using this perfect math, they calculated exactly how the "ghost notes" behave. They found that the information about the singularity shows up as specific singularities (sharp spikes or breaks) in the complex time plane.

The Metaphor:
Imagine you are in a cave with a very strange echo.

• If you clap your hands, the echo comes back normally.
• But if you clap at a very specific, weird time, the echo doesn't just come back; it seems to "bounce" off the back wall of the cave in a way that creates a sharp, distinct sound.

The authors found that the "echo" from the black hole singularity creates these sharp, distinct sounds at specific times.

• The Result: They mapped out exactly when these echoes happen. They found a lattice (a grid) of these "echo times" in the complex time plane.
• The Significance: This proves that the singularity (the point where physics breaks down) leaves a fingerprint on the outside world. It's not hidden forever; the universe "remembers" the singularity through these complex time echoes.

6. The "Zero Momentum" Twist

There was one tricky case: what if the particles aren't moving sideways at all (zero momentum)?

• The Problem: In this case, the "forest" of the black hole interior changes shape. The path the wave takes merges with the singularity in a weird way.
• The Fix: The authors had to use a different kind of math (matched asymptotics) to handle this. They found that the "echoes" here behave slightly differently (using fractional powers instead of whole numbers), but the main message remains: The singularity is still there, and it's still leaving a trace.

Summary

This paper is a detective story.

1. The Crime: We can't see inside a black hole.
2. The Clue: The "music" played on the holographic boundary.
3. The Method: The authors separated the music into "surface noise" and "interior secrets."
4. The Tool: They used a super-precise mathematical flashlight (Exact WKB) to trace the path of the secrets.
5. The Verdict: They proved that the black hole's crushing center (singularity) leaves a specific, calculable fingerprint on the outside world. It's like hearing a specific echo that tells you, "Yes, there is a crushing wall at the back of this cave, and here is exactly where it is."

This is a major step forward in understanding how the universe hides its most dangerous secrets and how we might one day decode them.

Technical Summary

1. Problem Statement

The paper investigates the analytic properties of thermal spectral functions in holographic Conformal Field Theories (CFTs) with a classical gravity dual (specifically, large central charge $c$ and a sparse low-lying spectrum). The primary objectives are:

• To understand the structure of thermal two-point functions beyond the standard thermal strip (Im$(t) \in (-\beta/2, \beta/2)$).
• To establish a precise factorization of the exact spectral function into perturbative (OPE-controlled) and non-perturbative (bulk interior-controlled) components.
• To compute the full transseries expansion of the non-perturbative sector at large momentum.
• To identify the "imprints" of the black hole curvature singularity within the dual CFT observables, specifically in the complex time plane of thermofield double correlators.

2. Methodology

The authors employ a multi-faceted approach combining conformal field theory techniques, holographic duality, and advanced mathematical physics methods:

• Thermal OPE Analysis: They analyze the Operator Product Expansion (OPE) of thermal correlators. They distinguish between contributions from the stress-tensor sector (which are perturbative and determined by near-boundary data) and the double-twist sector (which typically vanishes in the momentum space OPE for specific dimensions).
• Exact Virasoro Block Method: Utilizing recent developments, the authors express the exact retarded Green's function in terms of semiclassical Virasoro blocks. This allows for an exact solution to the connection problem between the AdS boundary and the black hole horizon without solving the wave equation in closed form.
• Exact WKB Analysis: To extract the large-momentum asymptotics of the non-perturbative sector, they employ Exact WKB analysis (resurgent analysis). This involves:
  • Mapping the bulk scalar wave equation on a planar Schwarzschild-AdS background to a Schrödinger-type equation.
  • Analyzing the monodromy of the wave equation around singular points (boundary, horizon, and singularity).
  • Computing Voros periods (Borel-summed WKB periods) and applying Stokes automorphisms to handle critical phases where Stokes lines connect turning points.
  • Using median Borel summation to ensure real-valued results at critical phases (specifically for real momentum).
• Complex Time Singularities: They perform Fourier transforms of the asymptotic spectral function to locate singularities in the complex time plane of the spatially averaged thermofield double correlator.

3. Key Contributions and Results

A. Exact Factorization of the Spectral Function

For even-dimensional holographic CFTs ($d \in 2\mathbb{Z}_{>1}$) with scalar primaries of integer dimension $\Delta_\phi = d/2 + \mathbb{Z}_{\geq 0}$, the authors prove a precise factorization of the exact thermal spectral function $\rho(\omega, k)$:
$$\rho_{\text{exact}}(\omega, k) = \rho_{\text{pert}}^{(\text{OPE})}(\omega, k) \times \rho_{\text{np}}(\omega, k)$$

• Perturbative Part ($\rho_{\text{pert}}$): Controlled by the stress-tensor exchange. It is fixed entirely by a near-boundary expansion and corresponds to the vacuum spectral function plus finite perturbative corrections. For specific integer dimensions, this series truncates.
• Non-Perturbative Part ($\rho_{\text{np}}$): Encodes information about the bulk interior (horizon and singularity). It is given by:
  $$\rho_{\text{np}}(\omega, k) = \frac{\sinh(\beta\omega/2)}{\cosh(\beta\omega/2) + (-1)^{\Delta_\phi - d/2} \cos(2\pi\sigma)}$$
  Here, $\sigma$ is the intermediate momentum in the Virasoro block, related to the monodromy of the bulk wave equation around the boundary and horizon.

B. Exact WKB Analysis and Transseries Expansion

The authors perform an exact WKB analysis to determine the asymptotic behavior of the monodromy invariant $\cos(2\pi\sigma)$ at large timelike momentum ($\omega \to \infty$) with non-vanishing spatial momentum ($k = \zeta \omega$).

• They derive the full transseries expansion for the non-perturbative spectral function:
  $$\rho_{\text{np}}(w, \zeta w) \sim 1 + \sum_{r,s,q} e^{(-r\pi - sv)w} \dots$$
  where $v$ is a classical WKB period dependent on the momentum ratio $\zeta$.
• Critical Phases: They address the subtleties of Borel summability at critical phases (where Stokes lines connect turning points) by utilizing median Borel sums, ensuring the result is manifestly real.
• Vanishing Momentum Limit: They show that the $\zeta \to 0$ limit is singular in the exact WKB framework (turning points merge with the singularity). They perform a separate matched asymptotic analysis for this case, recovering results consistent with [ACCM25] but highlighting that the perturbative corrections involve fractional powers of frequency ($w^{-4/3}$) rather than integer powers.

C. Imprints of the Black Hole Singularity

The non-perturbative corrections dictate the location of singularities in the complex time plane of the spatially averaged correlator $C_\zeta(t)$.

• The singularities occur at:
  $$t_{rsq} = i\frac{\beta}{2} + \left(r + s\frac{v}{\pi}\right)i\frac{\beta}{2} + q\frac{\beta}{2}\left(1 - \frac{v}{\pi}\right)$$
• Significance: These singularities lie outside the fundamental thermal strip. Their real parts are related to the critical time $t_c$ associated with "bouncing" geodesics that hit the black hole singularity. The parameter $v$ is directly linked to the classical WKB period near the singularity. This establishes a direct, quantitative link between the non-perturbative spectral data and the geometry of the black hole interior, specifically the curvature singularity.

4. Significance

• Resolution of the Connection Problem: The paper provides a rigorous framework for connecting boundary CFT data to bulk interior geometry using exact WKB methods, moving beyond standard quasinormal mode approximations.
• Non-Perturbative Control: It offers the first complete transseries description of holographic thermal spectral functions at finite momentum, separating perturbative OPE data from non-perturbative bulk physics.
• Singularity Encoding: The work provides concrete evidence that the black hole curvature singularity leaves a distinct "fingerprint" in the analytic structure of CFT correlators (specifically in the complex time plane), supporting the idea that the singularity is encoded in the dual field theory.
• Universality: The factorization claim and the method of using Virasoro blocks to access non-perturbative data are argued to be potentially universal for holographic theories, extending beyond planar black holes.

In summary, this paper bridges the gap between the perturbative OPE regime and the non-perturbative bulk geometry, demonstrating how the black hole singularity manifests in the analytic structure of thermal correlators through exact WKB techniques and resurgent transseries.