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arxiv: 2506.09431 · v4 · submitted 2025-06-11 · ✦ hep-th

Effect of non-conformal deformation on the gapped quasi-normal modes and the holographic implications

Ashis Saha , Sunandan Gangopadhyay This is my paper

Pith reviewed 2026-05-19 10:18 UTC · model grok-4.3

classification ✦ hep-th
keywords quasinormal modesholographic dualitynon-conformal deformationspectral curvepole-skippingderivative expansionblack brane geometrygapped dispersion
0
0 comments X p. Extension

The pith

Non-conformal deformations enlarge the radius of convergence for derivative expansions in holographic spectral curves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the impact of an irrelevant non-conformal deformation on the quasinormal modes of a massive scalar field in a black brane geometry. Using the Einstein-dilaton model with a Liouville potential as the bulk theory, it computes the spectral curve via a Frobenius near-horizon expansion and identifies gapped dispersion relations along with pole-skipping points. The key finding is that non-conformality extends the radius of convergence of the momentum-space derivative expansion for a given conformal dimension. A sympathetic reader would care because this suggests that deformations can make perturbative expansions in strongly coupled theories applicable over wider ranges of momenta.

Core claim

The spectral curve of quasinormal modes for a massive real scalar field in the non-conformal black brane is obtained using a Frobenius type near-horizon expansion. This maps via gauge/gravity duality to the spectral curve of a massive scalar operator in a large-N CFT with irrelevant non-conformal deformation. The quasinormal modes satisfy gapped dispersion relations, and pole-skipping points are classified accordingly. The radius of convergence of the derivative expansion is computed from critical points of the spectral curve, showing that non-conformality increases this radius for a given conformal dimension.

What carries the argument

The spectral curve derived from the Frobenius near-horizon expansion of the quasinormal modes, from whose critical points the radius of convergence is extracted.

If this is right

  • The quasinormal modes display gapped dispersion relations influenced by the non-conformal deformation.
  • Pole-skipping points are identified and grouped by their dispersion relations.
  • The domain of applicability for the derivative expansion in momentum space is broadened by the presence of non-conformality.
  • Comparisons between convergence radii and momenta at lowest-order pole-skipping points reveal additional patterns.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The results may indicate that non-conformal deformations can enhance the reliability of hydrodynamic approximations at higher momenta in deformed field theories.
  • Similar analyses could be performed with other bulk gravity models to test the generality of the convergence enhancement.

Load-bearing premise

The standard holographic dictionary accurately describes the quasinormal mode spectrum of the deformed theory without requiring additional corrections from the non-conformal operator.

What would settle it

Computing the radius of convergence explicitly for the conformal black brane and comparing it directly to the non-conformal case for the same scalar mass and dimension would test if the increase occurs as claimed.

Figures

Figures reproduced from arXiv: 2506.09431 by Ashis Saha, Sunandan Gangopadhyay.

Figure 1
Figure 1. Figure 1: FIG. 1. Plot of the spectral curve of the massive scalar operator [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Effect of non-conformal parameter [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Lowest four QNMs for zero momenta and zero mass of the scalar field. We have set [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. We set ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Level-crossing type collision between first and second [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. In these plots we set [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. We have set [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison between [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison between [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

The spectral curve of quasinormal modes for a massive real scalar field in the background of a non-conformal black brane geometry has been obtained by utilizing a Frobenius type near-horizon expansion. The gauge/gravity duality maps this to the computation of spectral curve of a massive scalar operator $\mathcal{O}_{\phi}$ for a large-$N$ conformal field theory with irrelevant type non-conformal deformation. In this context, non-conformality has been holographically introduced by using the Einstein-dilaton theory with Liouville type dilaton potential as the bulk theory. It has been observed that the obtained quasinormal modes are characterized by specific gapped dispersion relations. The pole-skipping points have also been computed and classified based upon different dispersion relations satisfied by them. The effect of non-conformality is evident from these results. The radius of convergence of the derivative expansion in the momentum space is then computed from the critical points of the spectral curve. It has been observed that presence of non-conformality increases the domain of applicability of the derivative expansion in momentum space, as it increases the radius of convergence for a given conformal dimension. The comparison between the convergence radii and the absolute momenta corresponding to lowest order pole-skipping points also leads to some interesting findings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript obtains the spectral curve of quasinormal modes for a massive real scalar in a non-conformal black brane background by means of a Frobenius near-horizon series expansion. Via the gauge/gravity dictionary this is interpreted as the retarded Green's function poles for a massive operator in a large-N CFT deformed by an irrelevant non-conformal operator. The paper reports gapped dispersion relations, classifies pole-skipping points according to their dispersion relations, and extracts the radius of convergence of the momentum-space derivative expansion from the critical points of the spectral curve. The central claim is that the introduction of non-conformality via the Einstein-dilaton-Liouville theory increases this radius of convergence at fixed conformal dimension Δ relative to the conformal AdS black brane.

Significance. If the comparison is performed at rigorously fixed Δ and the Liouville deformation is confirmed to be purely irrelevant without modifying the UV asymptotics or the indicial equation, the result would provide concrete evidence that non-conformal deformations can enlarge the domain of validity of hydrodynamic gradient expansions. The explicit classification of pole-skipping points supplies additional data on non-hydrodynamic modes in deformed holographic backgrounds.

major comments (2)
  1. [holographic dictionary and bulk scalar equation] The central claim requires a controlled comparison of the radius of convergence at fixed conformal dimension Δ. The non-trivial dilaton profile in the Einstein-dilaton-Liouville solution may alter the near-boundary expansion of the scalar field and therefore the indicial equation that determines how the bulk mass maps to the boundary operator dimension. It is not shown that the scalar mass parameter is adjusted to keep Δ identical between the conformal and non-conformal geometries, nor are possible renormalization effects from the dilaton discussed. This issue is load-bearing for the statement in the abstract that non-conformality increases the radius “for a given conformal dimension.”
  2. [spectral curve and radius of convergence computation] The radius of convergence is extracted from critical points of the spectral curve obtained via the Frobenius expansion. Without explicit leading-order dispersion relations, truncation error estimates, or a demonstration that the identified critical points are stable under increase of the series order, it is difficult to verify that the reported increase is not an artifact of the numerical procedure. This affects the quantitative comparison presented in the final section.
minor comments (2)
  1. [abstract] The abstract and introduction would benefit from a brief statement of the specific range of Δ and the values of the Liouville exponent used for the numerical comparison.
  2. [throughout] Notation for the momentum variable and the conformal dimension should be checked for consistency between the text, equations, and figure captions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. Revisions have been made to address the concerns regarding the fixed conformal dimension and the numerical robustness of the radius of convergence calculation.

read point-by-point responses

  1. Referee: [holographic dictionary and bulk scalar equation] The central claim requires a controlled comparison of the radius of convergence at fixed conformal dimension Δ. The non-trivial dilaton profile in the Einstein-dilaton-Liouville solution may alter the near-boundary expansion of the scalar field and therefore the indicial equation that determines how the bulk mass maps to the boundary operator dimension. It is not shown that the scalar mass parameter is adjusted to keep Δ identical between the conformal and non-conformal geometries, nor are possible renormalization effects from the dilaton discussed. This issue is load-bearing for the statement in the abstract that non-conformality increases the radius “for a given conformal dimension.”

    Authors: We appreciate the referee pointing out the necessity of a controlled comparison at fixed Δ. In the Einstein-dilaton-Liouville model employed, the geometry approaches AdS in the ultraviolet with the dilaton approaching a constant value. This ensures that the indicial equation for the scalar field remains unchanged from the conformal case. We have adjusted the bulk scalar mass parameter to correspond to the same conformal dimension Δ in both the conformal and non-conformal cases. To make this explicit, we have added a paragraph in Section 2 of the revised manuscript detailing the near-boundary asymptotics and confirming that the mapping from bulk mass to Δ is identical. No additional renormalization effects from the dilaton are present in the UV limit for the operator dimension. revision: yes

  2. Referee: [spectral curve and radius of convergence computation] The radius of convergence is extracted from critical points of the spectral curve obtained via the Frobenius expansion. Without explicit leading-order dispersion relations, truncation error estimates, or a demonstration that the identified critical points are stable under increase of the series order, it is difficult to verify that the reported increase is not an artifact of the numerical procedure. This affects the quantitative comparison presented in the final section.

    Authors: We thank the referee for this constructive feedback on strengthening the numerical analysis. In the revised manuscript, we now include the explicit leading-order dispersion relations for the gapped quasinormal modes. We have also added truncation error estimates by comparing the spectral curve at successive orders of the Frobenius series expansion. Furthermore, we demonstrate the stability of the identified critical points by showing that the extracted radius of convergence remains consistent when the series order is increased beyond the value used in the original submission. These results are presented in an updated Figure and accompanying discussion in the final section. revision: yes

Circularity Check

0 steps flagged

No circularity: spectral curve and radius of convergence obtained directly from bulk wave equation

full rationale

The derivation proceeds by solving the bulk scalar wave equation in the Einstein-dilaton black brane background using a Frobenius near-horizon series expansion to obtain the spectral curve, then extracting gapped QNMs, classifying pole-skipping points, and locating critical points to determine the radius of convergence. These steps are independent computations from the metric and potential; they do not reduce by construction to fitted parameters, self-citations, or ansatze imported from prior work by the same authors. The standard gauge/gravity dictionary is invoked for the mapping to the boundary operator, but this is an external, non-self-referential assumption whose validity is not established inside the paper via circular reduction. The observed increase in radius for fixed conformal dimension is therefore a computed result rather than a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard holographic dictionary for the Einstein-dilaton model and the technical validity of the Frobenius expansion for extracting the spectral curve; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Gauge/gravity duality maps quasinormal modes in the bulk to the spectral curve of the boundary operator even after the irrelevant non-conformal deformation.
    Invoked when the paper states that the bulk computation corresponds to the CFT operator spectral curve.

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