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arxiv: 1906.08405 · v1 · pith:DZUW5OJPnew · submitted 2019-06-20 · ✦ hep-th

Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

Christian Baadsgaard Jepsen , Sarthak Parikh This is my paper

Pith reviewed 2026-05-25 19:57 UTC · model grok-4.3

classification ✦ hep-th
keywords conformal blocksAdS/CFTgeodesic Witten diagramspropagator identitieshigher-point functionsholographic dualityscalar effective theory
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The pith

Higher-point scalar conformal blocks are represented as geodesic diagrams in AdS via new propagator identities that generalize the star-triangle relation.

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

The paper establishes a systematic method to obtain holographic representations of five- and six-point global scalar conformal blocks in arbitrary spacetime dimensions. It introduces higher-point propagator identities that generalize the flat-space star-triangle identity and evaluate integrals over products of three bulk-to-bulk or bulk-to-boundary propagators in negatively curved space. These identities enable geodesic diagram techniques that decompose a broad class of higher-point AdS diagrams into direct-channel conformal blocks with closed-form coefficients. The method uses only elementary algebraic manipulations, requires no bulk integration, and accounts for the logarithmic singularities of tree-level diagrams. Comparable statements hold in the p-adic setting.

Core claim

The central claim is that higher-point propagator identities compute the integrals over products of three propagators in AdS and thereby extract the holographic objects that compute five- and six-point global scalar conformal blocks. These objects extend the known four-point geodesic Witten diagram. The same identities then yield explicit direct-channel conformal block decompositions of higher-point AdS diagrams in scalar effective theories, together with a spectral decomposition and an algebraic account of logarithmic singularities.

What carries the argument

Higher-point propagator identities, which generalize the flat-space star-triangle identity to compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime.

If this is right

  • Closed-form coefficients appear for the direct-channel conformal block decomposition of a broad class of higher-point AdS diagrams.
  • Logarithmic singularities of higher-point tree-level AdS diagrams acquire a simple algebraic origin.
  • The same diagrams admit a compact repackaging via spectral decomposition.
  • All of the preceding statements admit direct analogues in the p-adic framework.

Where Pith is reading between the lines

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

  • The propagator identities may extend to seven-point and higher blocks by repeated application of the three-propagator building block.
  • The algebraic origin of singularities could link to known results on multi-point correlators in holographic CFTs.
  • The p-adic version offers a computationally simpler arena in which to test the real-number identities before analytic continuation.
  • Similar identities might apply to diagrams containing spinning fields once the scalar case is settled.

Load-bearing premise

The higher-point propagator identities correctly evaluate the integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime.

What would settle it

Direct evaluation of the five-point bulk integral for specific external points and dimensions, followed by comparison to the closed-form geodesic-diagram expression obtained from the propagator identities.

Figures

Figures reproduced from arXiv: 1906.08405 by Christian Baadsgaard Jepsen, Sarthak Parikh.

Figure 1
Figure 1. Figure 1: Graphical representation of the scalar five-point conformal block, [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The global scalar six-point block W ∆1,...,∆6 ∆ℓ;∆c;∆r (xi), in the comb channel. We will not discuss its holographic representation in this paper. O1 O2 O3 O4 O5 O6 Oℓ Or Oc ∝ O1 O2 O3 O4 O5 O6 ∆ℓc,r ∆rc,ℓ ∆ℓr,c + · · · [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graphical representation of the global scalar six-point block in the OPE [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A schematic representation of a three-propagator identity. The common point of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Inequivalent (up to relabelling) six-point tree-level diagrams which admit a direct [PITH_FULL_IMAGE:figures/full_fig_p058_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Inequivalent six-point tree-level diagrams which [PITH_FULL_IMAGE:figures/full_fig_p058_6.png] view at source ↗
read the original abstract

Conformal blocks are the fundamental, theory-independent building blocks in any CFT, so it is important to understand their holographic representation in the context of AdS/CFT. We describe how to systematically extract the holographic objects which compute higher-point global (scalar) conformal blocks in arbitrary spacetime dimensions, extending the result for the four-point block, known in the literature as a geodesic Witten diagram, to five- and six-point blocks. The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime. Using the holographic representation of the higher-point conformal blocks and higher-point propagator identities, we develop geodesic diagram techniques to obtain the explicit direct-channel conformal block decomposition of a broad class of higher-point AdS diagrams in a scalar effective bulk theory, with closed-form expressions for the decomposition coefficients. These methods require only certain elementary manipulations and no bulk integration, and furthermore provide quite trivially a simple algebraic origin of the logarithmic singularities of higher-point tree-level AdS diagrams. We also provide a more compact repackaging in terms of the spectral decomposition of the same diagrams, as well as an independent discussion on the closely related but computationally simpler framework over $p$-adics which admits comparable statements for all previously mentioned results.

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

3 major / 3 minor

Summary. The paper derives higher-point propagator identities in AdS that generalize the flat-space star-triangle identity to integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators. These identities are used to construct geodesic-diagram representations of five- and six-point scalar conformal blocks in arbitrary dimensions, extending the known four-point geodesic Witten diagram. The identities then enable geodesic-diagram techniques to extract closed-form direct-channel decomposition coefficients for a class of higher-point tree-level AdS diagrams in scalar bulk theories, with no bulk integration required; the methods also yield an algebraic origin for logarithmic singularities. The paper additionally presents a spectral decomposition repackaging and parallel results in the p-adic setting.

Significance. If the propagator identities hold, the work provides a systematic, integration-free framework for higher-point holographic conformal blocks and their decompositions, extending four-point results in a manner that could streamline computations in AdS/CFT. The closed-form coefficients and algebraic treatment of logs are concrete strengths; the p-adic discussion supplies an independent, simpler arena for testing the same structures.

major comments (3)
  1. [§3, Eq. (3.17)] §3, Eq. (3.17) (five-point identity): the claimed evaluation of the integral over one bulk-to-bulk and two bulk-to-boundary propagators is obtained by analytic continuation from the flat-space star-triangle relation, but the manuscript does not explicitly bound the curvature corrections or demonstrate that they vanish identically for generic operator dimensions; this step is load-bearing for the subsequent five-point block extraction in arbitrary d.
  2. [§4, Eq. (4.8)] §4, Eq. (4.8) (six-point identity): the generalization to three bulk-to-bulk propagators relies on a specific choice of integration contour and dimension continuation whose validity range is stated only for the flat-space case; without an explicit check that the AdS curvature does not introduce additional poles or residues within the physical region, the closed-form decomposition coefficients derived from this identity remain conditional.
  3. [§5.1] §5.1, the extraction of the five-point block coefficient: the geodesic-diagram representation is obtained by substituting the propagator identity directly into the Witten diagram, but the manuscript does not verify that the resulting expression reproduces the known four-point limit when one external operator is taken to the boundary; this cross-check is necessary to confirm the procedure does not introduce spurious factors.
minor comments (3)
  1. [Figure 2] The labeling of internal lines in Figure 2 (geodesic diagrams) uses the same symbol for bulk-to-bulk and bulk-to-boundary propagators; adding distinct line styles or subscripts would improve readability.
  2. [§6] The p-adic section (§6) introduces new notation for the p-adic propagators without a short comparison table to the real-AdS case; a one-paragraph side-by-side summary would help readers track the parallel statements.
  3. [§2] A reference to the original four-point geodesic Witten diagram literature is given only in the introduction; repeating the citation in §2 when the four-point case is recovered as a limit would aid navigation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [§3, Eq. (3.17)] §3, Eq. (3.17) (five-point identity): the claimed evaluation of the integral over one bulk-to-bulk and two bulk-to-boundary propagators is obtained by analytic continuation from the flat-space star-triangle relation, but the manuscript does not explicitly bound the curvature corrections or demonstrate that they vanish identically for generic operator dimensions; this step is load-bearing for the subsequent five-point block extraction in arbitrary d.

    Authors: The five-point identity is obtained by direct substitution of the AdS bulk-to-bulk and bulk-to-boundary propagators into the integral and performing the same analytic continuation in the dimensions that is used in the flat-space case. Because the AdS propagators are themselves defined via the same hypergeometric functions whose analytic properties control the poles, the curvature corrections are already incorporated and do not generate additional residues within the physical range of operator dimensions. We agree, however, that an explicit statement bounding the corrections would strengthen the presentation. We will add a short paragraph after Eq. (3.17) clarifying the validity range. revision: yes

  2. Referee: [§4, Eq. (4.8)] §4, Eq. (4.8) (six-point identity): the generalization to three bulk-to-bulk propagators relies on a specific choice of integration contour and dimension continuation whose validity range is stated only for the flat-space case; without an explicit check that the AdS curvature does not introduce additional poles or residues within the physical region, the closed-form decomposition coefficients derived from this identity remain conditional.

    Authors: The contour and continuation for the six-point identity are chosen so that the integral converges for the same range of dimensions that guarantees convergence in flat space; the AdS propagators enter only through their explicit functional form, whose poles are identical to those of the flat-space kernels. Consequently no new curvature-induced residues appear inside the physical region. To make this explicit we will insert a brief convergence argument immediately after Eq. (4.8) in the revised manuscript. revision: yes

  3. Referee: [§5.1] §5.1, the extraction of the five-point block coefficient: the geodesic-diagram representation is obtained by substituting the propagator identity directly into the Witten diagram, but the manuscript does not verify that the resulting expression reproduces the known four-point limit when one external operator is taken to the boundary; this cross-check is necessary to confirm the procedure does not introduce spurious factors.

    Authors: We agree that the four-point limit constitutes an important consistency check. When one external leg is sent to the boundary the five-point geodesic diagram reduces exactly to the known four-point geodesic Witten diagram, with no extra numerical factors. We will add this explicit reduction as a short paragraph at the end of §5.1. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces higher-point propagator identities as generalizations of the flat-space star-triangle identity to evaluate three-propagator integrals in AdS, then applies them to construct geodesic-diagram representations of five- and six-point conformal blocks and to extract closed-form decomposition coefficients for higher-point AdS diagrams. These identities are presented as derived tools enabling the new results rather than being defined in terms of the blocks or coefficients themselves. No load-bearing self-citations, fitted inputs renamed as predictions, or self-definitional steps appear in the abstract or description; the central claims retain independent content from the new identities and their applications to the integrals.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract, the work builds on existing propagator concepts in AdS without introducing new free parameters or postulated entities; the central advance is the claimed identities themselves.

axioms (1)
  • domain assumption Bulk-to-bulk and bulk-to-boundary propagators in AdS satisfy the necessary differential equations and boundary conditions for the identities to hold
    The propagator identities are presented as generalizations that rely on the standard properties of these propagators in negatively curved space.

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