Notes on flat-space limit of holographic defect correlators in position space

Qi Chen; Xinan Zhou; Yi-Xiao Tao

Notes on flat-space limit of holographic defect correlators in position space

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A position-space formula extracts flat-space scattering form factors from holographic defect correlators.

desk verdict Solid position-space derivation of the flat-space limit for holographic defect correlators, with real checks and a clearly scoped probe-brane assumption. read the letter →

arxiv 2507.14421 v1 pith:45KZLKFT submitted 2025-07-19 hep-th

Qi Chen , Yi-Xiao Tao , Xinan Zhou This is my paper

classification hep-th MSC 81T4081T30

keywords holographiccorrelatorsdefectCFTflat-spacelimitAdS/CFTMellinspaceformfactorspositiongiantgravitons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives a position-space rule for what happens to two-point functions in holographic defect CFTs when the AdS radius is taken to infinity. The claim is that the flat-space limit of the correlator of two operators placed away from the defect is governed by a form factor: a particle scattering off the extended object dual to the defect. The paper shows this scaling limit is related to the flat-space form factor by an integral transform involving a modified Bessel function, and that the Mellin-space version follows from it. Three top-down examples, Wilson loops in N=4 super Yang-Mills, surface defects in six-dimensional (2,0) theory, and giant gravitons, reproduce the known flat-space graviton form factors. The formula matters because it gives a route from computed defect correlators to stringy and M-theory corrections, including cases where no Mellin-space description exists.

What carries the argument

The engine is a saddle-point evaluation of the defect two-point function written as an AdS LSZ integral. After Schwinger-parameterizing the bulk-to-boundary propagators, the integral over the AdS$_{p+1}$ worldvolume and the nearby transverse directions becomes a Fourier transform to a flat-space form factor, with a two-by-two matrix $L$ controlling the phases. The flat-space limit is the configuration in which $L$ develops a zero eigenvalue, realized when a point on the defect is null-separated from both operators and their mirror images. The remaining radial integral is evaluated in terms of the modified Bessel function $K_{(p-1)/2}$, producing the transform (3.35), and the same structure is recovered from the Mellin prescription by a saddle point on the Mellin variables followed by a Wick rotation.

What would settle it

Take a known one-loop or string-corrected defect correlator, such as the next correction to the Wilson-loop two-point function in N=4 SYM, apply the scaling limit (3.34) and the Bessel transform (3.35), and compare with the independently known flat-space string form factor at the same order; any mismatch would show that the probe-brane localization assumption breaks down.

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Extended reading notes

Core claim

The central result is equation (3.35): for a probe-brane defect of dimension $p$ in a $d$-dimensional defect CFT, the scaling limit $\mathcal{G}_{\mathrm{f.s.}}(\eta,\chi)$ of the two-point function of operators inserted away from the defect is $$\mathcal{G}_{\mathrm{f.s.}}(\eta,\chi) = (-i)^{\Delta_1+\Delta_2-1}\,\frac{\$pi^{{(p+1)/2}}$}{\Gamma(\Delta_1)\Gamma(\Delta_2)\,$2^{{(\Delta_1+\Delta_2-p-2)/2}}$} \int_0^\infty du\, $u^{{\Delta_1+\Delta_2-2}}$(u\eta)^{(1-p)/2} K_{(p-1)/2}(u\eta)\,\mathcal{F}_{\mathrm{flat}}(S,Q),$$ with $S=-(2+\chi)u^2/\ell_s^2$ and $Q=2u^2/\ell_s^2$, valid for $0\le p\le d-1$. The relevant limit is a Lorentzian scaling limit in which the cross ratio $\Theta=1-(\xi+\chi)^2/4$ tends to zero while $\eta^2=-2R^2\Theta/\ell_s^2$ stays fixed; geometrically, a point on the defect is null-separated from both operator insertions and from their mirror images across the defect. The paper proves that this position-space prescription is equivalent to the Mellin-space flat-space-limit formula conjectured earlier, and records the inverse transform that recovers the flat-space form factor from the correlator. Explicit checks on Wilson-loop, surface-defect and giant-graviton correlators reproduce the known flat-space graviton form factors in the supergravity limit.

Load-bearing premise

The derivation assumes the defect is light enough that it does not bend the spacetime geometry, so the flat-space limit is controlled by a thin layer within one string length of the defect's worldvolume; if the defect back-reacts, the formula would need modification.

Editorial extensions

If this is right

The same scaling limit can be applied to any probe-brane holographic defect correlator, so flat-space form-factor data can constrain stringy corrections beyond supergravity even where the Mellin representation is ill-defined.
For defect dimensions $0\le p\le d-1$ the formula gives one universal prescription, including the point-like $p=0$ case of giant gravitons where no defect Mellin space is available.
The inverse transform expresses the flat-space form factor directly as a contour integral of the scaling-limit correlator, giving a practical extraction recipe for string-theory form factors from correlator computations.
In the supergravity regime the formula reproduces the known flat-space graviton form factors for Wilson loops, surface defects in the 6d (2,0) theory, and giant gravitons, confirming that defect two-point functions encode scattering off extended objects.
The derivation also identifies a Regge-type limit for codimension-one defects, where one operator approaches the mirror image of the other and only a single Mandelstam variable remains independent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the formula extends to backreacting, non-probe defects with suitable modifications, it would give a flat-space handle on dynamical brane geometries, a regime the paper explicitly leaves open.
The Bessel-kernel structure suggests a position-space bootstrap condition: imposing this scaling limit as a boundary condition on the defect OPE might determine loop-level correlators without computing them fully, though the paper does not develop this.
The same mechanism may transfer to higher-point defect correlators, where a universal Mellin-space prescription is already known; a position-space counterpart would remove the restriction to two insertions.
For the special $p=-1$ case of CFTs on real projective space, a separate analysis is likely needed, and a position-space flat-space formula there would open a route not available in Mellin space.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

These notes deliver a real result. The new object is eq. (3.35): a position-space integral transform that extracts the flat-space form factor from a Lorentzian scaling limit of holographic defect two-point functions, valid for defect dimensions 0 ≤ p ≤ d−1. That includes p=0 giant gravitons, a case where the Mellin-space prescription isn't available, and the authors make good use of that advantage. They also show the position-space formula reproduces the Mellin-space conjecture of [31], and they test the formula against three independent tree-level examples: the Wilson loop, the M2 surface defect, and the giant graviton correlator. The checks are real, not fitted: the extracted Fflat matches flat-space form factors computed from string-theory D-brane scattering.

The derivation is a careful adaptation of the Okuda-Penedones defect-free analysis, with the new kinematic ingredient that the flat-space limit localizes on a point of the defect null-separated from both insertions and their mirror images. The saddle-point analysis and the analytic continuation discussion in Section 3.2 are clear, and the cross-ratio parameterization is useful on its own.

The soft spot is the one the authors flag themselves. The derivation assumes a probe brane: the flat-space limit is dominated by a region within ℓs of the AdS_{p+1} subspace, and the bulk two-point function is replaced by the flat-space form factor. That is exactly the right assumption for the examples considered, and the paper is explicit about it. But there is no estimate of how backreaction would modify the formula, and all three checks live in the probe-brane regime. So the formula should be treated as established for probe defects, not for backreacted ones. That is a scoping limitation, not an internal inconsistency. The equivalence proof in Section 3.3 is explicit in one direction, and the reverse direction is plausible but not fully detailed; a thorough referee might ask for that to be filled in.

Citation practice looks fair; the debt to [29] and [31] is acknowledged. No code or data, but the derivations are reproducible by hand. This paper is for people working on stringy corrections and loop-level defect correlators. I'd bring it to the reading group and would cite it. It deserves a serious referee, and I'd expect it to pass with minor comments.

Referee Report

0 major / 6 minor

Summary. This paper derives a position-space prescription for taking the flat-space (large AdS radius) limit of holographic two-point functions in defect CFTs. Working in a probe-brane setup where the defect is dual to an AdS_{p+1} subspace, the authors show by saddle-point analysis that a Lorentzian scaling limit of the defect correlator, with the cross ratio Θ scaled to zero at fixed η (Eqs. (1.1)-(1.2), (3.34)), is related to the flat-space 1-to-1 form factor through the Bessel-kernel integral transform in Eq. (3.35). They discuss the Euclidean-to-Lorentzian analytic continuation in Section 3.2, prove equivalence with the Mellin-space prescription of Alday and Zhou in Section 3.3, and test the formula against tree-level supergravity results for Wilson loops, surface defects in 6d (2,0) theories, and giant graviton correlators, reproducing the independent flat-space form factors. The paper also notes that the derivation assumes a probe brane without backreaction and flags this as an open direction.

Significance. The result is a nontrivial and useful extension of the Okuda-Penedones/Penedones flat-space limit to defect CFTs. Its main value is that the position-space formula covers configurations where a defect Mellin representation is not available, in particular the p=0 giant graviton system in N=4 SYM, and it provides a parameter-free derivation rather than a fit. The equivalence proof with the Mellin-space conjecture gives a coherent picture of the two prescriptions. The checks against three independent flat-space form factors are substantive: they involve nontrivial kinematical and R-symmetry dependence and use the inverse transform (3.40), not just the forward formula. The probe-brane scoping is explicit and is the only substantive limitation; it is acknowledged in Sections 3.1 and 5 and does not affect the central claim as stated. The paper is clearly written and the main derivation is transparent.

minor comments (6)

[Section 3.1, Eq. (3.20)] The saddle-point evaluation of the second Schwinger parameter is asserted rather than justified; because the integration domain is finite and the saddle-point width depends on y, a sentence explaining why endpoints and small-u regions are subleading would make the derivation fully rigorous.
[Section 3.1, p=0 paragraph] The extension to p=0, which the paper advertises as a case where the position-space formula is particularly useful, is supported only by the statement that the relevant integrand is the same as in Eq. (3.7) with p=0; a short explicit derivation for the geodesic defect would fully support the claimed range 0≤p≤d-1.
[Section 4 general comments] The dictionary between the CFT defect dimension p and the flat-space brane dimension q (q=2,3,4 in the three examples) is not stated; a sentence defining q and explaining why q differs from p+1 for the giant graviton case would help the reader connect the CFT and flat-space data.
[Section 4] The three examples verify the flat-space form factor only up to an overall normalization. Given that the prefactors in Eq. (3.35) are derived explicitly, it would strengthen the checks to fix one normalization convention and verify the overall constant in at least one example.
[Section 3.1 and Section 5] The probe-brane assumption is stated in Section 3.1 and its limitations are acknowledged in Section 5, but the paper does not estimate the size of corrections when the defect backreacts; a brief remark identifying the parametric smallness of the backreaction would be helpful.
[Throughout] There are several typos, e.g., 'limt' for 'limit' in Section 3.1, 'behevior' for 'behavior' near the discussion of Regge behavior in Section 4.1, and 'Mandalstam' for 'Mandelstam' in Section 3.1; these should be corrected in the final version.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the position-space flat-space limit formula is independently derived from Witten-diagram saddle-point analysis and checked against externally known flat-space amplitudes.

full rationale

The central result (3.35) is obtained by starting from the probe-brane defect two-point function (3.1), approximating the bulk correlator near the AdS_{p+1} brane by the flat-space form factor (3.2), and then carrying out Schwinger-parameter, Gaussian, and saddle-point integrals (3.3)-(3.25). The final integral transform follows from the kinematics of the flat-space-limit configuration (3.16)-(3.24) and from the y-integral leading to the Bessel K function; no parameter is fitted to the correlators whose flat-space limit is being predicted. The inverse transform (3.40) is derived from a standard Bessel orthogonality relation rather than imposed. The examples in Section 4 extract Fflat from independent position-space Witten-diagram results and compare with flat-space p-brane form factors (4.17), (4.28), (4.42), with the giant-graviton case (Section 4.3) providing a check where no Mellin representation exists. The same-author Mellin conjecture [31] appears in Section 3.3 as a consistency comparison; it is not an input to the derivation of (3.35), so the paper's derivation chain does not reduce to its own assumptions. The probe-brane localization assumption is a physical scoping condition, explicitly flagged in Section 5 for backreacted defects, not a circular redefinition of the result.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The formula rests on standard holographic and flat-space limit assumptions, plus a probe-brane locality assumption. No free parameters are fitted and no new entities are introduced.

assumptions (7)

domain assumption AdS/CFT LSZ reduction: holographic correlators are integrals of bulk-to-boundary propagators with a bulk correlator.
Invoked in Section 2, eq. (2.3), and Section 3.1, eq. (3.1). Standard holographic dictionary.
domain assumption In the limit |Xi - Xj| << R, the AdS bulk correlator reduces to the flat-space correlator.
Used in Section 2, around eq. (2.5). Standard flat-space limit of AdS amplitudes.
domain assumption The flat-space limit receives contributions only from a region localized near the AdS_{p+1} subspace within a distance of order ell_s.
Stated in Section 3.1, after eq. (3.1). This probe-brane assumption is the main scope restriction; the paper notes in Section 5 it may fail for backreacting defects.
domain assumption The bulk theory has an intrinsic length scale ell_s independent of R.
Section 2, first paragraph. Needed to define the large R/ell_s expansion.
domain assumption Euclidean correlators can be continued to Lorentzian signature via the complex path (3.42) with alpha from 0 to pi/2.
Section 3.2, eq. (3.42), Figures 2 and 3. Standard Wick rotation following [34]; branch choices are assumed.
domain assumption Scattered particles are massless in the flat-space limit.
Stated in Section 5. The derivation assumes operator dimensions stay fixed as R/ell_s goes to infinity.
domain assumption The Mellin-space flat-space limit formula (3.68) conjectured in [31] is used in the equivalence proof.
Section 3.3, eq. (3.68), is substituted into the Mellin representation to reproduce the position formula; the position formula itself is derived independently in Section 3.1.

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Pith. "Pith review of Notes on flat-space limit of holographic defect correlators in position space." pith.science (2026). https://pith.science/paper/45KZLKFT

@misc{pith2026250714421,
  author       = {Pith},
  title        = {Pith review of: Notes on flat-space limit of holographic defect correlators in position space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45KZLKFT}},
  note         = {Machine review of arXiv:2507.14421}
}

read the original abstract

We study the large AdS radius limit of correlation functions in holographic defect CFTs. For two-point functions of operators inserted away from the defect, we derive a position space formula relating a certain scaling limit of the correlators to the flat-space scattering form factors. We show that our position space prescription is equivalent to the flat-space limit formula recently conjectured in Mellin space and also test our result in a few nontrivial theories.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Gravity Corrections to Giant Graviton Correlators
hep-th 2026-07 conditional novelty 6.0 of 10

One-loop AdS supergravity correction to maximal giant graviton–supergraviton correlators is reconstructed in closed Mellin and position form via defect unitarity, including leading-twist anomalous dimensions.

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S. Giombi, H. Khanchandani, and X. Zhou, “Aspects of CFTs on Real Projective Space,” J. Phys. A 54 no. 2, (2021) 024003, arXiv:2009.03290 [hep-th]

2021 arXiv
[50]

Aspects of higher-point functions in BCFT d,

J. Chen and X. Zhou, “Aspects of higher-point functions in BCFT d,” JHEP 09 (2023) 204, arXiv:2304.11799 [hep-th]. – 30 –

2023 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [31]

Flat-space limit of defect correlators and stringy AdS form factors,

L. F. Alday and X. Zhou, “Flat-space limit of defect correlators and stringy AdS form factors,” JHEP 03 (2025) 182, arXiv:2411.04378 [hep-th]

arXiv 2025

[2] [29]

String scattering in flat space and a scaling limit of Yang-Mills correlators,

T. Okuda and J. Penedones, “String scattering in flat space and a scaling limit of Yang-Mills correlators,” Phys. Rev. D 83 (2011) 086001, arXiv:1002.2641 [hep-th]

arXiv 2011

[3] [1]

Mellin amplitudes for AdS5 × S5,

L. Rastelli and X. Zhou, “Mellin amplitudes for AdS5 × S5,” Phys. Rev. Lett. 118 no. 9, (2017) 091602, arXiv:1608.06624 [hep-th]

arXiv 2017

[4] [2]

How to Succeed at Holographic Correlators Without Really Trying,

L. Rastelli and X. Zhou, “How to Succeed at Holographic Correlators Without Really Trying,” JHEP 04 (2018) 014, arXiv:1710.05923 [hep-th]

arXiv 2018

[5] [3]

The Witten Diagram Bootstrap for Holographic Defects,

A. Gimenez-Grau, “The Witten Diagram Bootstrap for Holographic Defects,” arXiv:2306.11896 [hep-th]

arXiv

[6] [4]

Defect two-point functions in 6D (2,0) theories,

J. Chen, A. Gimenez-Grau, and X. Zhou, “Defect two-point functions in 6D (2,0) theories,” Phys. Rev. D 109 no. 6, (2024) L061903, arXiv:2310.19230 [hep-th]

arXiv 2024

[7] [5]

Unitarity Method for Holographic Defects,

J. Chen, A. Gimenez-Grau, H. Paul, and X. Zhou, “Unitarity Method for Holographic Defects,” arXiv:2406.13287 [hep-th]

arXiv

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Correlators of N=4 Supersymmetric Yang-Mills Theory on Real Projective Space at Strong Coupling,

X. Zhou, “Correlators of N=4 Supersymmetric Yang-Mills Theory on Real Projective Space at Strong Coupling,” Phys. Rev. Lett. 133 no. 20, (2024) 201602, arXiv:2408.04926 [hep-th]

arXiv 2024