⚛️ high-energy theory

Holographic zeros and poles from apparent singularities

This paper investigates holographic fluctuation equations with apparent singularities approaching the boundary, demonstrating that their merger can generate either zeros or poles in spectral functions while preserving the validity of the holographic product formula.

Original authors: Edwan Préau

Published 2026-09-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Edwan Préau

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 theoretical physics, there is a powerful idea known as the holographic principle. It suggests that a universe with gravity, like the one we inhabit, can be mathematically described by a simpler, gravity-free world living on its boundary. Imagine a three-dimensional room where every object and event is actually a projection of information stored on the two-dimensional walls. This concept has become a vital tool for physicists trying to understand complex systems, such as the behavior of hot, dense matter or the strange properties of superconductors. By translating difficult problems in our three-dimensional reality into calculations on a simpler boundary, scientists can uncover patterns that would otherwise remain hidden. A central task in this work is to calculate how disturbances in these systems ripple and interact, which is done by studying specific mathematical equations that describe how tiny fluctuations move through the system.

For decades, physicists have relied on a set of rules to predict the behavior of these fluctuations, particularly a formula that acts like a catalog of the system's possible vibrations. This catalog lists the frequencies at which the system naturally resonates, much like a musical instrument has specific notes it can play. The rules assumed that the mathematical equations describing these vibrations were smooth and free of any sudden, confusing breaks in the space between the boundary and the center of the system. However, recent investigations have revealed that many of these equations actually contain "apparent singularities." These are points where the math looks broken or infinite, but the physical reality remains perfectly smooth and continuous. It is as if a map shows a cliff where there is actually a gentle slope; the map is misleading, but the terrain is safe. The question remained: do these misleading points on the map change the actual music the system plays?

In a new study, a researcher at Utrecht University set out to answer this question by examining what happens when one of these apparent singularities moves closer and closer to the boundary of the system. The goal was to see if this movement creates new notes in the system's spectrum or silences existing ones. The study focused on a specific regime where the singularity approaches the edge, a scenario that had not been fully understood before. By breaking the problem into two parts—a region very close to the singularity and a region further away—the researcher could stitch the solutions together to see the full picture. This method allowed for a precise calculation of how the system's response changes as the singularity merges with the boundary.

The findings reveal that the presence of these apparent singularities does indeed alter the system's behavior, but in a very specific and predictable way. When the singularity approaches the boundary, it can generate either a new zero or a new pole in the system's response. In plain terms, a "zero" means the system stops responding to a particular frequency, effectively silencing a note, while a "pole" means the system responds with extreme intensity, amplifying that note. The study shows that whether a zero or a pole appears depends on the specific mathematical properties of the equation governing the system. Crucially, the research demonstrates that these new features are not chaotic or random; they are finite in number and occur only when the singularity reaches the boundary.

Perhaps the most significant outcome of this work is that it confirms a long-standing formula used to describe these systems remains valid, even with these apparent singularities present. The formula, which expresses the system's behavior as a product of its poles, still holds true. The only adjustment needed is to add a few extra factors to account for the new zeros or poles created by the singularities. This means that the fundamental structure of the holographic dictionary is robust. It can accommodate these mathematical quirks without breaking down, providing a more complete and accurate map of how these complex systems behave.

The study also uncovered a fascinating phenomenon known as "pole-skipping," where a line of zeros and a line of poles meet at a single point. This happens in specific cases where the system is charged, and it suggests a deep connection between the system's hydrodynamic behavior and its microscopic quantum nature. The researcher found that at these special points, the system's response becomes ambiguous, a feature that had been predicted by other methods but was now derived directly from the behavior of the singularities. This provides a new way to locate these special points, offering a fresh perspective on how energy and information move through holographic systems.

By applying this new understanding to concrete examples, such as the behavior of gauge fields and sound waves in charged black hole backgrounds, the study validated its general results. In every case examined, the predictions matched known behaviors, confirming that the method works. For instance, in the case of sound waves, the study showed that as the momentum of the wave approaches zero, the system's response remains finite, consistent with the laws of hydrodynamics. This consistency across different types of fluctuations reinforces the idea that the approach is sound and applicable to a wide range of physical scenarios.

Ultimately, this work bridges a gap between abstract mathematical properties and physical observables. It shows that even when the equations describing a system contain points that look singular, the physical predictions remain well-behaved and understandable. The research does not just solve a technical puzzle; it strengthens the foundation of the holographic principle, ensuring that the tools physicists use to explore the universe are reliable even in the most complex and charged environments. The ability to predict exactly when and how these singularities affect the system's spectrum opens the door to more precise calculations in the future, potentially helping to unravel the mysteries of quantum matter and the nature of spacetime itself.

Technical Summary: Holographic Zeros and Poles from Apparent Singularities

Problem Statement
In the holographic dictionary, thermal two-point functions (and their associated spectral functions, $\rho(\omega)$ ) for boundary operators are computed from the solutions to bulk fluctuation equations. These equations are typically Fuchsian differential equations defined on black hole backgrounds. A significant result in the field, the holographic product formula [12], expresses the spectral function as a product over its poles, assuming the absence of bulk singularities between the boundary and the horizon. However, many holographic fluctuation equations possess "apparent singularities"—points where the potential diverges, yet the solutions remain analytic. The presence of these singularities violates the assumptions of the standard product formula, raising the question of whether additional zeros or poles arise in the spectral function and how the formula generalizes.

Methodology
The paper analyzes general holographic fluctuation equations in the regime where an apparent singularity, located at a radial coordinate $r_*$ , approaches the boundary at $r=0$ . The analysis employs a matched asymptotic expansion technique, dividing the bulk into two regions:

Outer Region ( $r \gg r_*$ ): The equation is expanded in powers of $r_*$ , treating the singularity as removed. The leading-order equation is independent of $r_*$ but possesses "fused" characteristic exponents at the boundary, denoted by $\tilde{\nu}$ .
Inner Region ( $r \sim r_*$ ): The radial coordinate is rescaled as $\zeta = r/r_*$ . In this limit, the equation reduces to a hypergeometric form with regular singularities at $\zeta=0$ (boundary) and $\zeta=1$ (the apparent singularity). Due to the "apparent" nature of the singularity at $\zeta=1$ , the hypergeometric solutions reduce to Jacobi polynomials rather than infinite series.

The solutions in the inner and outer regions are matched in an overlap region to determine the scaling of the source ( $\psi_-$ ) and vacuum expectation value ( $\psi_+$ ) coefficients as $r_* \to 0$ . The behavior of the spectral function is then deduced from the scaling of the source term, as $\rho \propto 1/|\psi_-|^2$ .

Key Contributions and Classification
The paper classifies the behavior of the spectral function based on the parameters of the hypergeometric inner equation. Two distinct classes of equations are identified:

The $(-)$ Class: Defined by the condition that one of the hypergeometric parameters $a_\pm$ is a negative integer. For this class, the spectral function exhibits a universal scaling behavior as $r_* \to 0$ :
$\rho(\omega) \sim r_*^{2(\tilde{\nu} - \nu)}$
where $\nu$ is the characteristic exponent at the boundary for finite $r_*$ , and $\tilde{\nu}$ is the fused exponent at $r_*=0$ .
- If $\tilde{\nu} > \nu$ , the spectral function develops a zero.
- If $\tilde{\nu} < \nu$ , the spectral function develops a pole.
- If $\tilde{\nu} = \nu$ , the spectral function remains finite.
The $(+)$ Class: Defined by the alternative condition on the hypergeometric parameters. For this class, the behavior is not universal at leading order; the source scaling requires higher-order calculations in the $r_*$ expansion. The paper demonstrates through examples that this class can yield either poles or finite limits, depending on the specific system.

Results and Applications
The author applies this formalism to three specific holographic systems on Reissner-Nordström black brane backgrounds:

Probe Gauge Field: The longitudinal sector exhibits an apparent singularity. The analysis confirms the potential belongs to the $(-)$ class with $\tilde{\nu} > \nu$ , reproducing the known result that the spectral function has simple zeros at $\omega = \pm k$ .
Charged Gauge Field: The longitudinal sector for charged fluctuations also falls into the $(-)$ class, generating zeros at $(\omega \pm \mu)^2 = k^2$ . A special case where $\omega \pm \mu = 0$ is identified as belonging to the $(+)$ class, where the spectral function exhibits a pole, indicating a pole-skipping point where lines of zeros and poles intersect.
Sound-Channel Metric Fluctuations: Depending on the boundary dimension $d$ , these fluctuations fall into the $(+)$ class (for $d \ge 5$ ) or the $(-)$ class (for $d=3$ ). For $d \ge 5$ , the spectral function remains finite as momentum $k \to 0$ , while the real part of the correlator diverges. For $d=3$ , the spectral function vanishes as $k^4$ .

Significance and Claims
The primary significance of this work is the generalization of the holographic product formula [12] to systems with apparent bulk singularities. The author argues that while apparent singularities can generate additional zeros (and potentially poles) in the spectral function, these occur only when the singularity merges with the boundary. Since there is a finite number of such singularities, the product formula remains valid if modified to include these additional zeros in the numerator.

Furthermore, the paper establishes a systematic method to determine the existence and location of zeros and poles in holographic spectral functions without solving the full bulk equations exactly. It also highlights that the merging of apparent singularities with the boundary can reveal pole-skipping points, offering an alternative method to identify such phenomena compared to standard hydrodynamic approaches. The work is modest in its scope, noting that the generalization to coupled fluctuation equations remains a challenge for future research.

More like this