REVIEW
Holographic zeros and poles from apparent singularities
High Energy Physics - Theory · Submitted 2 Sep 2026 · 36 references
Pith's verdict: A clean, self-contained analytic framework for spectral function zeros from apparent singularities in holographic equations, with explicit assumptions and worked examples; the main residual risk is scope (non-confluent, single-set assumptions) which the paper flags.
T0 · AI review
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Referee report
The paper
The paper claims
This paper shows that when an apparent singularity of a holographic fluctuation equation approaches the boundary, the spectral function can develop a zero or a pole, with universal scaling for one class of potentials, and the holographic…
The reading
What carries the argument
The central technique is an inner/outer matching approach. In the inner region (|r| ≪ r₁, where r₁ is the next singularity), the leading-order potential takes the form $V₀^{{in}}$(ζ) = (p₀ + $p₁ζ^{{m}}$ + $p₂ζ^{{2m}}$)/[ζ²($ζ^{{m}}$−1)²] with ζ = r/rₚ. This corresponds to a hypergeometric equation after the change of variable u = $ζ^{{m}}$. The apparent nature of the singularity at u=1 forces the hypergeometric parameters to satisfy specific integrality conditions, reducing the inner solution to Jacobi polynomials (given in eqs. (3.23)–(3.24)). Matching the inner solution to the outer solution in the overlap region then relates the near-boundary source and vev, leading to the scaling of the spectral function.
Extended reading notes
Core claim
The author discovers that when an apparent singularity of a holographic fluctuation equation merges with the boundary at r=0, the spectral function ρ(ω) can acquire a zero or a pole. The behavior is governed by the class of the potential: for the (−) class, ρ scales as $r^{{2(ν̃−ν)}}$ (a universal power law that yields a zero when ν̃ > ν, a pole when ν̃ < ν, and O(1) when ν̃ = ν); for the (+) class, ρ tends to a finite constant or a pole depending on subleading corrections. The author also establishes that the holographic product formula of [12] continues to apply in the presence of apparent singularities, with the zeros appearing as additional factors in the numerator.
Load-bearing premise
The analysis assumes that the limit rₚ → 0 is non-confluent (all parameters remain finite) and that the potential can be written as g(r)/H(r)² with a single symmetric set of apparent singularities; if either condition fails—for example, if parameters diverge or multiple distinct singularities approach the boundary simultaneously—the universal scaling and the formula for additional zeros may not hold.
Editorial extensions
If this is right
- The holographic product formula of [12], which writes the spectral function as a product over its poles, remains valid when the bulk fluctuation equation contains apparent singularities, with additional factors for the finite number of zeros that may appear.
- For the (−) class of potentials, the spectral function always follows the universal scaling ρ ∼ κ^{2n₊−(n₁−1)}, where κ is the parameter that vanishes as the singularity approaches the boundary, making the zero/pole structure completely determined.
- The method provides a concrete way to compute the location and nature of new zeros (and occasionally poles) in holographic spectral functions without solving the full fluctuation equation.
- The examples studied—probe gauge field, charged gauge field, and sound-channel metric fluctuations—confirm that the formalism reproduces known results (e.g., simple zeros at ω = ±k, poles at ω = ∓μ₃) and predicts new ones.
- The classification into (−) and (+) classes, together with the half-integer case, gives a complete taxonomy of possible near-boundary behaviors when an apparent singularity merges with the boundary.
Reading between the lines
- The matching analysis can identify pole-skipping points (where a line of zeros intersects a line of poles) in a new way: the example in Section 5.2.1 shows that ω = ∓μ₃ is exactly such a point, which standard methods usually locate from poles rather than zeros.
- Because the product formula still holds, one can write down effective low-energy expansions for the spectral function even in theories with apparent singularities, similar to the hydrodynamic approximations in [13] but now accounting for the extra zeros.
- The assumption of a single symmetric set of apparent singularities may be the most restrictive; if multiple sets or a non-monomial structure appear (as possible in more complicated backgrounds), a full numerical or analytic treatment would be needed, though the present method could still be extended.
- The technique might be adapted to fermionic spectral functions (the paper mentions straightforward generalization), potentially offering new insights into holographic Fermi surfaces and non-Fermi liquids.
Original abstract
We study general holographic fluctuation equations with apparent singularities, and the associated spectral functions. We consider the regime where an apparent singularity approaches the boundary, for which the near-boundary behavior of solutions can be determined via an inner/outer matching type of calculation. Depending on properties of the equation, we find that the merging of an apparent singularity with the boundary may generate either a zero or a pole for the corresponding spectral function. Our results also indicate that the recently introduced holographic product formula still applies in the presence of apparent singularities.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study general holographic fluctuation equations with apparent singularities, and the associated spectral functions. We consider the regime where an apparent singularity approaches the boundary, for which the near-boundary behavior of solutions can be determined via an inner/outer matching type of calculation. Depending on properties of the equation, we find that the merging of an apparent singularity with the boundary may generate either a zero or a pole for the corresponding spectral function. Our results also indicate that the recently introduced holographic product formula still applies in the presence of apparent singularities.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study general holographic fluctuation equations with apparent singularities, and the associated spectral functions. | contribution 0.97 | 0.89 | 0.05 | asserted 0.76 |
| s1 | We consider the regime where an apparent singularity approaches the boundary, for which the near-boundary behavior of solutions can be determined via an inner/outer matching type of calculation. | method 0.91 | 0.48 | 0.05 | asserted 0.90 |
| s2 | Depending on properties of the equation, we find that the merging of an apparent singularity with the boundary may generate either a zero or a pole for the corresponding spectral function. | result 1.00 | 0.95 | 0.75 | hedged possibility 0.54 |
| s3 | Our results also indicate that the recently introduced holographic product formula still applies in the presence of apparent singularities. | result 0.87 | 0.76 | 0.15 | hedged possibility 0.99 |
What was examined, and what was not
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References
Discussion
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