Pith. sign in

REVIEW 3 major objections 3 minor 64 references

Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Gravitational Wilson line networks in AdS2 are holographically reconstructed from boundary conformal blocks, and the 3-point scalar Witten diagram decomposes into vertex functions with running weights.

desk verdict Real new identities connecting Wilson line networks to Witten diagrams in AdS2, but the proof of the key three-bulk-point identity has a genuine analytic-continuation gap. read the letter →

arxiv 2412.03290 v3 pith:MU7QRHFW submitted 2024-12-04 hep-th

K.B. Alkalaev , V.S. Khiteev This is my paper · ORCID
classification hep-th MSC 81T4081T20
keywords WilsonlinenetworksAdS2holographyWittendiagramsglobalconformalblocksIshibashistatessl(2R)intertwinersHKLLreconstructionSvertexfunctions
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

The paper establishes a holographic reconstruction formula for gravitational Wilson line networks in AdS2: the n-point vertex function, defined by intertwiners and Ishibashi states, equals an integral of the global conformal block smeared with bulk-to-boundary propagators over Pochhammer contours. Working out this integral as a multidimensional series, the authors show that the 2-point vertex function is the scalar bulk-to-bulk propagator and that the 3-point vertex function with two boundary points is the geodesic Witten diagram. The central result is the decomposition of the full 3-point scalar Witten diagram into a leading AdS vertex function plus infinite towers of vertex functions whose weights run in steps of two. This matters because it places topological Wilson line networks and ordinary massive scalar dynamics in the same exact analytic family.

What carries the argument

The load-bearing object is the n-point AdS vertex function $V_{h_1\ldots h_n|\tilde h_1\ldots\tilde h_{n-3}}(x_1,\ldots,x_n)$: a matrix element of an $\mathfrak{sl}(2,\mathbb{R})$ Wilson line network built from 3-valent intertwiners and Ishibashi cap states, which satisfies the homogeneous Klein-Gordon equation in each argument. The key identity is the holographic reconstruction formula (Proposition 1, Eq. (3.10)): $$V = C_{\mathbf h} \prod_{k=1}^n \oint_{P[w_k,\bar w_k]} du_k\, K(x_k,u_k|1-h_k) F_{\mathbf h}(\mathbf u),$$ where the integral runs over Pochhammer contours around the branch points and $F_{\mathbf h}$ is the global conformal block in the comb channel. The evaluation uses Pochhammer contour integral representations of hypergeometric functions, Appell and Lauricella series, and the exact closed form of the 3-point Witten diagram, yielding the structured running-weight decomposition of Eq. (5.20).

What would settle it

Evaluate both sides of Eq. (5.20) numerically for a weight triple that violates the triangle inequalities with one weight h ≤ 1/2, such as h1=0.4, h2=0.4, h3=1.0, by direct integration of the 3-point Witten diagram and by summing the vertex functions; agreement outside the boundary asymptotic would support the analytic continuation, while any discrepancy would falsify Proposition 9 as stated.

Watch

Extended reading notes

Core claim

The paper's central claim is that scalar Witten diagrams in AdS2 are not separate from the topological Wilson line networks: they are the same objects. Specifically, Proposition 9 and Eq. (5.20) state that the 3-point Witten diagram with three bulk points equals a fixed coefficient times the 3-point AdS vertex function plus three discrete sums of vertex functions with intermediate weights h2+h3+2n, h1+h3+2n, and h1+h2+2n, with coefficients given explicitly in terms of gamma functions. The proof rests on Proposition 1, the HKLL-type reconstruction formula (3.10), which expresses any n-point vertex function as the product of bulk-to-boundary propagators integrated against the global conformal block, and on the exact 3-point Witten diagram expression of Jepsen and Parikh. The authors also show that with two boundary points the vertex function is exactly the geodesic Witten diagram, and that fewer boundary points revive infinite summation tails over running weights.

Load-bearing premise

The proof of the 3-point identity assumes that the Euclidean Klein-Gordon equation in AdS2 has a unique solution once the boundary value at one boundary point is fixed; this is established for h>1/2 and then extended to all weights by analytic continuation without a precise domain specification.

Editorial extensions

If this is right

  • The extrapolate dictionary becomes two-way: every n-point global conformal block in the comb channel is the boundary value of a bulk vertex function, and the vertex function is obtained from the block by the reconstruction formula (3.10).
  • The 2-point and 3-point scalar Witten diagrams in AdS2 are reproduced by Wilson line networks: the bulk-to-bulk propagator is the 2-point vertex function, the geodesic Witten diagram is the 3-point vertex function with two boundary points, and the full 3-point Witten diagram is the linear combination (5.20).
  • Via Eq. (5.23), the 3-point Witten diagram identity becomes an integral relation between AdS vertex functions only, which the authors suggest admits a purely group-theoretic derivation within the Wilson line network.
  • The triangle inequalities on the conformal weights select which term of the decomposition (5.20) controls the boundary asymptotic of the 3-point Witten diagram; violating them shifts the leading term to the running-weight towers.

Reading between the lines

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

  • One natural extension, not pursued in the paper, is to n=4: the 4-point global conformal block should admit a bulk realization as a 4-point AdS vertex function, and the corresponding 4-point Witten diagrams might decompose into these vertex functions with running intermediate weights.
  • The running-weight sums in (5.20) resemble a shadow or degeneracy decomposition; if they can be resummed, the 3-point Witten diagram would be expressible as a single integral kernel that directly links the scalar three-point function to the Wilson line network, possibly illuminating the role of the Pochhammer contour as a shadow integral.
  • A numerical check of Eq. (5.20) for weights outside the h>1/2 domain, where the uniqueness argument in Appendix D.6 is only justified by analytic continuation, would clarify the exact domain of validity of the decomposition; the paper does not specify the convergence region of the multidimensional series in Proposition 2.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper constructs an HKLL-type integral representation for n-point gravitational Wilson line network matrix elements in AdS2, evaluated between Ishibashi states of sl(2,R), and identifies the integrand as the product of bulk-to-boundary propagators times the boundary global conformal block. The integral is evaluated as a multidimensional series, and for n=2 and n=3 the resulting expressions are used to relate Wilson line networks to scalar Witten diagrams. The main claims are Proposition 1 (holographic reconstruction from global conformal blocks), Proposition 2 (explicit multidimensional series), Propositions 4-6 (relations between AdS vertex functions and geodesic or partially boundary Witten diagrams), and Proposition 9 (the full three-point bulk Witten diagram as a structured linear combination of three-point AdS vertex functions with running weights).

Significance. If the central identities hold for general weights, this is a significant result: it connects topological Wilson line networks, which are solutions of sl(2,R) BF/Chern-Simons gravity, to local massive-scalar dynamics in AdS2 through exact analytic relations. The derivations are detailed, and the comparisons with the independent exact Witten-diagram expression of Jepsen and Parikh [39] provide a strong cross-check that does not rely on fitted parameters. The lower-point relations with geodesic Witten diagrams are also supported by independent calculations in the literature. The main weakness is that the proof of Lemma 8, which is the load-bearing step for Proposition 9, is complete only in a restricted weight range; as written, the claimed general-weight identity is not established.

major comments (3)
  1. [Appendix D.6, Eqs. (D.37)-(D.42)] The uniqueness proof for Lemma 8 treats only the case h>1/2. In the Bessel-mode solution (D.40), the term C2(k) u^{1/2} J_{1/2-h}(|k|u) contributes C2(k) u^{1-2h} to u^{-h}F(k,u), which tends to zero as u->0 when h<1/2; hence the boundary condition u^{-h}F -> Fz[g] does not determine C2(k). The sentence "any restrictions on the weights can be removed by analytic continuation" does not close this gap, because both W F(1) in (5.17) and V_{h1h2h3} in (4.6) are defined as hypergeometric/Appell-type series, and the coefficients in (5.10) contain denominators h1(h1-1)-(h2+h3+2n)(h2+h3+2n-1) that vanish at resonance points. Proposition 9, Eq. (5.20), is therefore proven only for weights in the open region h_i>1/2 away from those resonances, rather than for general weights as claimed.
  2. [Section 3.2, Eq. (3.14)] Proposition 2 presents the n-point AdS vertex function as an explicit multidimensional series, but no convergence domain is stated. The derivation in Appendix B repeatedly evaluates Pochhammer-contour integrals, expands Lauricella functions as series, and changes summation variables; without a stated region of absolute convergence or an explicit analytic-continuation prescription, the identities (3.10) and (3.14) are formal for arbitrary real weights. This is load-bearing for Proposition 9, which uses vertex functions with running weights h2+h3+2n whose arguments lie outside any small neighborhood of the original series convergence domain.
  3. [Appendix D.6, Eqs. (D.43)-(D.46)] The proof of Proposition 9 rearranges multiple infinite sums and applies the identity (A.24) after changing summation variables. The paper does not justify the absolute convergence or uniform convergence needed to interchange these sums, nor does it justify passing the boundary limits through the infinite sums in Eq. (5.21). This is a separate technical gap from the uniqueness issue in Lemma 8, since (5.20) is an identity between infinite series and the coefficients in (5.10) are not absolutely summable for all parameter ranges without additional assumptions.
minor comments (3)
  1. [Section 2, Eqs. (2.11)-(2.12)] The notation T_h^m in (2.12) uses factorials such as (-h)! for non-integer weights h, since the paper allows h in R. The authors should state the Gamma-function convention or restrict the notation to integer or half-integer cases where factorials are unambiguous.
  2. [Section 4.2, Eq. (4.11)] The second representation of the 3-point AdS vertex function is written as a double sum with the condition k<s. The origin of this strict inequality and its role in avoiding singular or duplicate terms should be explained, because the same sum does not appear in the first representation (4.6).
  3. [Section 5.1, Eqs. (5.3) and (5.6)] The comparison between the 3-point AdS vertex function and the geodesic Witten diagram is asserted by saying the expressions coincide, but the hypergeometric parameters and arguments are written in slightly different orders in (5.3) and (5.6). A short sentence indicating which identity (e.g., Pfaff transformation) maps one form to the other would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Witten-diagram identities are established by explicit calculation against independent results, and the self-cited prior work is used as verifiable input, not as the target result.

full rationale

The paper's derivation chain is not circular. Proposition 1 (Eq. (3.10)) is a reconstruction formula obtained by substituting the integral representation of Wilson matrix elements (3.6) into the definition (2.15) and using the extrapolate dictionary (2.17) to identify the summed boundary asymptotics as the global conformal block; the dictionary itself is an independently stated result from [30] and is re-verified asymptotically in Section 3.3. Propositions 3-9 are proven by direct series manipulation of explicit expressions, with the 3-point Witten diagram input taken from the independent papers [39] and [43]; the coefficients (5.8), (5.10) are computed, not fitted, and no quantity is defined in terms of the identity being proved. The reliance on [30] for the Klein-Gordon property of Wilson matrix elements (Eq. (5.16)) is not circular because that property is explicit and checkable, and the current paper reproduces the relevant matrix elements in Section 2. The only substantive caveat is in Appendix D.6, Eqs. (D.37)-(D.42): the uniqueness argument for Lemma 8 is completed only for h>1/2, and the sentence 'any restrictions on the weights can be removed by analytic continuation' does not specify the domain or exclude resonance denominators in (5.10) and (5.18). That is a completeness/correctness gap in the h<1/2 regime, not a circular reduction of the claimed result to its input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on no fitted or ad hoc numerical parameters; the only hand-chosen convention is alpha = 1, beta = 0 for integer negative weights. The main assumptions are the flat-connection BF/JT setting, the restriction to sl(2,R) discrete-series modules, the quoted Klein-Gordon property of Wilson matrix elements, the exact 3-point Witten diagram expression from [39], and a uniqueness and analytic-continuation step in the proof of Lemma 8. No new physical entities are introduced.

free parameters (1)
  • Cap-state coefficients (alpha, beta) for integer negative weights = alpha = 1, beta = 0
    In Eq. (2.7) the authors fix alpha = 1 and beta = 0 to factor out singular submodules and define AdS vertex functions for h in Z_-; the generic-weight central claims do not depend on this choice.
assumptions (5)
  • domain assumption The gravitational connection obeys the zero-curvature condition dA + A and A = 0, the equation of motion of the BF formulation of JT gravity.
    Section 2 uses this to make Wilson lines path-independent, so networks depend only on endpoints; this is the physical setting for the whole paper.
  • domain assumption Only sl(2,R) negative and positive discrete-series modules and comb-channel intertwiners are used; continuous series are excluded.
    Section 2, items 2 and 5: the explicit intertwiner (A.25) and Wilson matrix elements are restricted to discrete series, excluding representations that would require more complicated intertwiners.
  • domain assumption Each Wilson line matrix element and therefore each 3-point AdS vertex function satisfies the homogeneous Klein-Gordon equation with mass squared h_i(h_i-1).
    Eq. (5.16) and the proof of Lemma 7 quote this from [18,25,30]; it is essential to the uniqueness argument in Lemma 8.
  • domain assumption The exact expression for the 3-point Witten diagram from Jepsen and Parikh [39] is correct in the present conventions.
    Eq. (5.17) and Appendix D.6 use that expression as the starting point for the decomposition into AdS vertex functions.
  • standard math The Euclidean Klein-Gordon Cauchy problem in AdS2 with a single boundary value at one boundary point has a unique solution for h > 1/2 when the boundary function is absolutely integrable and continuous; other weights are covered by analytic continuation.
    Appendix D.6, around Eqs. (D.37)-(D.42), solves via Fourier and Bessel functions and then extends by analytic continuation without specifying the domain; this premise is load-bearing for Proposition 9.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams." pith.science (2026). https://pith.science/paper/MU7QRHFW

@misc{pith2026241203290,
  author       = {Pith},
  title        = {Pith review of: Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MU7QRHFW}},
  note         = {Machine review of arXiv:2412.03290}
}
abstract

We find a holographic reconstruction formula for gravitational Wilson line network operators in AdS$_2$ evaluated between Ishibashi states of the algebra $sl(2,\mathbb{R})$. It is given in integral form where the integrand is the global conformal block multiplied by a smearing function which is the product of the scalar bulk-to-boundary propagators. The integral can be explicitly calculated as multidimensional series of which arguments are rational functions of endpoint coordinates. In the case of two and three endpoints the resulting expressions allow one to establish a number of relations between the gravitational Wilson line networks and Witten diagrams for massive scalar fields in AdS$_2$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 28 canonical work pages

  1. [39]

    C. B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268, [ arXiv:1906.08405]

    arXiv 2019

  2. [1]

    Achucarro and P

    A. Achucarro and P. K. Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys. Lett. B180 (1986) 89

    work page 1986

  3. [2]

    Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System , Nucl

    E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System , Nucl. Phys. B 311 (1988) 46

    1988

  4. [3]

    Fukuyama and K

    T. Fukuyama and K. Kamimura, Gauge Theory of Two-dimensional Gravity , Phys. Lett. B 160 (1985) 259–262

    work page 1985

  5. [4]

    Penrose, Angular momentum; an approach to combinatorial space time

    R. Penrose, Angular momentum; an approach to combinatorial space time . Quantum Theory and Beyond. Cambridge University Press, Cambridge, 1971

    work page 1971

  6. [5]

    G. W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory , Commun. Math. Phys. 123 (1989) 177

    1989

  7. [6]

    Ammon, A

    M. Ammon, A. Castro, and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110, [ arXiv:1306.4338]

    arXiv 2013

  8. [7]

    de Boer and J

    J. de Boer and J. I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS 3, JHEP 04 (2014) 089, [ arXiv:1306.4347]

    arXiv 2014

Show all 64 references
  1. [8]

    de Boer, A

    J. de Boer, A. Castro, E. Hijano, J. I. Jottar, and P. Kraus, Higher spin entanglement and WN conformal blocks, JHEP 07 (2015) 168, [ arXiv:1412.7520]

    2015 arXiv

  2. [9]

    Hegde, P

    A. Hegde, P. Kraus, and E. Perlmutter, General Results for Higher Spin Wilson Lines and Entanglement in Vasiliev Theory , JHEP 01 (2016) 176, [ arXiv:1511.05555]

    2016 arXiv

  3. [10]

    Bhatta, P

    A. Bhatta, P. Raman, and N. V. Suryanarayana, Holographic Conformal Partial Waves as Gravitational Open Wilson Networks , JHEP 06 (2016) 119, [ arXiv:1602.02962]

    2016 arXiv

  4. [11]

    Besken, A

    M. Besken, A. Hegde, E. Hijano, and P. Kraus, Holographic conformal blocks from interacting Wilson lines , JHEP 08 (2016) 099, [ arXiv:1603.07317]

    2016 arXiv

  5. [12]

    Besken, A

    M. Besken, A. Hegde, and P. Kraus, Anomalous dimensions from quantum Wilson lines , arXiv:1702.06640

    arXiv

  6. [13]

    Hikida and T

    Y. Hikida and T. Uetoko, Correlators in higher-spin AdS 3 holography from Wilson lines with loop corrections, PTEP 2017 (2017), no. 11 113B03, [ arXiv:1708.08657]

    2017 arXiv

  7. [14]

    Anand, H

    N. Anand, H. Chen, A. L. Fitzpatrick, J. Kaplan, and D. Li, An Exact Operator That Knows Its Location, JHEP 02 (2018) 012, [ arXiv:1708.04246]

    2018 arXiv

  8. [15]

    Hikida and T

    Y. Hikida and T. Uetoko, Superconformal blocks from Wilson lines with loop corrections , JHEP 08 (2018) 101, [ arXiv:1806.05836]

    2018 arXiv

  9. [16]

    Hikida and T

    Y. Hikida and T. Uetoko, Conformal blocks from Wilson lines with loop corrections , Phys. Rev. D 97 (2018), no. 8 086014, [ arXiv:1801.08549]

    2018 arXiv

  10. [17]

    Besken, E

    M. Besken, E. D’Hoker, A. Hegde, and P. Kraus, Renormalization of gravitational Wilson lines , JHEP 06 (2019) 020, [ arXiv:1810.00766]

    2019 arXiv

  11. [18]

    Bhatta, P

    A. Bhatta, P. Raman, and N. V. Suryanarayana, Scalar Blocks as Gravitational Wilson Networks, JHEP 12 (2018) 125, [ arXiv:1806.05475]

    2018 arXiv

  12. [19]

    D’Hoker and P

    E. D’Hoker and P. Kraus, Gravitational Wilson lines in AdS 3, arXiv:1912.02750. – 47 –

    1912 arXiv

  13. [20]

    Castro, N

    A. Castro, N. Iqbal, and E. Llabr´ es, Wilson lines and Ishibashi states in AdS 3/CFT2, JHEP 09 (2018) 066, [ arXiv:1805.05398]

    2018 arXiv

  14. [21]

    Kraus, A

    P. Kraus, A. Sivaramakrishnan, and R. Snively, Late time Wilson lines , JHEP 04 (2019) 026, [arXiv:1810.01439]

    2019 arXiv

  15. [22]

    Blommaert, T

    A. Blommaert, T. G. Mertens, and H. Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 12 (2018) 022, [ arXiv:1806.07765]

    2018 arXiv

  16. [23]

    Hulik, J

    O. Hulik, J. Raeymaekers, and O. Vasilakis, Multi-centered higher spin solutions from WN conformal blocks, JHEP 11 (2018) 101, [ arXiv:1809.01387]

    2018 arXiv

  17. [24]

    L.-Y. Hung, W. Li, and C. M. Melby-Thompson, Wilson line networks in p-adic AdS/CFT, JHEP 05 (2019) 118, [ arXiv:1812.06059]

    2019 arXiv

  18. [25]

    Castro, P

    A. Castro, P. Sabella-Garnier, and C. Zukowski, Gravitational Wilson Lines in 3D de Sitter , JHEP 07 (2020) 202, [ arXiv:2001.09998]

    2020 arXiv

  19. [26]

    Alkalaev and V

    K. Alkalaev and V. Belavin, More on Wilson toroidal networks and torus blocks , JHEP 11 (2020) 121, [ arXiv:2007.10494]

    2020 arXiv

  20. [27]

    Belavin and J

    V. Belavin and J. R. Cabezas, Wilson lines construction of osp(1 |2) conformal blocks , Nucl. Phys. B 985 (2022) 115981, [ arXiv:2204.12149]

    2022 arXiv

  21. [28]

    Belavin, P

    V. Belavin, P. Oreglia, and J. R. Cabezas, Wilson lines construction of sl3 toroidal conformal blocks, Nucl. Phys. B 990 (2023) 116186, [ arXiv:2301.04575]

    2023 arXiv

  22. [29]

    Castro, I

    A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory, Phys. Rev. Lett. 131 (2023), no. 17 171602, [ arXiv:2304.02668]

    2023 arXiv

  23. [30]

    Alkalaev, A

    K. Alkalaev, A. Kanoda, and V. Khiteev, Wilson networks in AdS and global conformal blocks , Nucl. Phys. B 998 (2024) 116413, [ arXiv:2307.08395]

    2024 arXiv

  24. [31]

    Hamilton, D

    A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality , Phys. Rev. D 73 (2006) 086003, [ hep-th/0506118]

    2006 arXiv

  25. [32]

    K. B. Alkalaev and V. A. Belavin, From global to heavy-light: 5-point conformal blocks , JHEP 03 (2016) 184, [ arXiv:1512.07627]

    2016 arXiv

  26. [33]

    Rosenhaus, Multipoint Conformal Blocks in the Comb Channel , JHEP 02 (2019) 142, [arXiv:1810.03244]

    V. Rosenhaus, Multipoint Conformal Blocks in the Comb Channel , JHEP 02 (2019) 142, [arXiv:1810.03244]

    2019 arXiv

  27. [34]

    Fortin, W.-J

    J.-F. Fortin, W.-J. Ma, and W. Skiba, All Global One- and Two-Dimensional Higher-Point Conformal Blocks, arXiv:2009.07674

    2009 arXiv

  28. [35]

    Fortin, W.-J

    J.-F. Fortin, W.-J. Ma, S. Parikh, L. Quintavalle, and W. Skiba, One- and two-dimensional higher-point conformal blocks as free-particle wavefunctions in AdS ⊗m 3 , JHEP 01 (2024) 031, [arXiv:2310.08632]

    2024 arXiv

  29. [36]

    A. L. Fitzpatrick, J. Kaplan, D. Li, and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, JHEP 07 (2017) 092, [ arXiv:1612.06385]

    2017 arXiv

  30. [37]

    Nakayama and H

    Y. Nakayama and H. Ooguri, Bulk Locality and Boundary Creating Operators , JHEP 10 (2015) 114, [arXiv:1507.04130]

    2015 arXiv

  31. [38]

    Nakayama and H

    Y. Nakayama and H. Ooguri, Bulk Local States and Crosscaps in Holographic CFT , JHEP 10 (2016) 085, [ arXiv:1605.00334]. – 48 –

    2016 arXiv

  32. [40]

    Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves , JHEP 05 (2019) 006, [ arXiv:1812.01006]

    X. Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves , JHEP 05 (2019) 006, [ arXiv:1812.01006]

    2019 arXiv

  33. [41]

    Giombi, H

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

    2021 arXiv

  34. [42]

    D’Hoker, D

    E. D’Hoker, D. Z. Freedman, and L. Rastelli, AdS / CFT four point functions: How to succeed at z integrals without really trying , Nucl. Phys. B 562 (1999) 395–411, [ hep-th/9905049]

    1999 arXiv

  35. [43]

    Hijano, P

    E. Hijano, P. Kraus, E. Perlmutter, and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks , JHEP 01 (2016) 146, [ arXiv:1508.00501]

    2016 arXiv

  36. [44]

    A. H. Chamseddine and D. Wyler, Topological Gravity in (1+1)-dimensions, Nucl. Phys. B 340 (1990) 595–616

    1990

  37. [45]

    Isler and C

    K. Isler and C. A. Trugenberger, A Gauge Theory of Two-dimensional Quantum Gravity , Phys. Rev. Lett. 63 (1989) 834

    1989

  38. [46]

    Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions , Phys

    C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions , Phys. Lett. B 126 (1983) 41–45

    1983

  39. [47]

    Jackiw, Lower Dimensional Gravity , Nucl

    R. Jackiw, Lower Dimensional Gravity , Nucl. Phys. B 252 (1985) 343–356

    1985

  40. [48]

    W. J. Holman and L. C. Biedenharn, Complex angular momenta and the groups su(1, 1) and su(2), Annals of Physics 39 (1966), no. 1 1 – 42

    1966

  41. [49]

    Banados, Global charges in Chern-Simons field theory and the (2+1) black hole , Phys

    M. Banados, Global charges in Chern-Simons field theory and the (2+1) black hole , Phys. Rev. D 52 (1996) 5816–5825, [ hep-th/9405171]

    1996 arXiv

  42. [50]

    Ishibashi, The Boundary and Crosscap States in Conformal Field Theories , Mod

    N. Ishibashi, The Boundary and Crosscap States in Conformal Field Theories , Mod. Phys. Lett. A 4 (1989) 251

    1989

  43. [51]

    Saran, Transformations of certain hypergeometric functions of three variables , Acta Mathematica 93 (1955), no

    S. Saran, Transformations of certain hypergeometric functions of three variables , Acta Mathematica 93 (1955), no. none 293 – 312

    1955

  44. [52]

    Witten, Anti-de Sitter space and holography , Adv

    E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150]

    1998 arXiv

  45. [53]

    Kabat and G

    D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge fields in AdS , Phys. Rev. D 87 (2013), no. 8 086004, [ arXiv:1212.3788]

    2013 arXiv

  46. [54]

    Castro and P

    A. Castro and P. J. Martinez, Revisiting Extremal Couplings in AdS/CFT , arXiv:2409.15410

    arXiv

  47. [55]

    Nishida and K

    M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field , PTEP 2017 (2017), no. 5 053B06, [ arXiv:1609.04563]

    2017 arXiv

  48. [56]

    E. Dyer, D. Z. Freedman, and J. Sully, Spinning Geodesic Witten Diagrams , JHEP 11 (2017) 060, [arXiv:1702.06139]

    2017 arXiv

  49. [57]

    Sleight and M

    C. Sleight and M. Taronna, Spinning Witten Diagrams , JHEP 06 (2017) 100, [arXiv:1702.08619]

    2017 arXiv

  50. [58]

    M. P. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory in D = (2+1) , Class. Quant. Grav. 6 (1989) 443. – 49 –

    1989

  51. [59]

    Bateman and A

    H. Bateman and A. Erd´ elyi,Higher transcendental functions . California Institute of technology. Bateman Manuscript project. McGraw-Hill, New York, NY, 1953

    1953

  52. [60]

    W. N. Bailey, The generating function of jacobi polynomials , Journal of the London Mathematical Society s1-13 (1938), no. 1 8–12, [https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-13.1.8]

    1938 doi

  53. [61]

    Carlson, Lauricella’s hypergeometric function fd , Journal of Mathematical Analysis and Applications 7 (1963), no

    B. Carlson, Lauricella’s hypergeometric function fd , Journal of Mathematical Analysis and Applications 7 (1963), no. 3 452–470

    1963

  54. [62]

    Fronsdal, Elementary particles in a curved space

    C. Fronsdal, Elementary particles in a curved space. ii , Phys. Rev. D 10 (1974) 589–598

    1974

  55. [63]

    A. V. Ramallo, Introduction to the AdS/CFT correspondence, Springer Proc. Phys. 161 (2015) 411–474, [arXiv:1310.4319]

    2015 arXiv

  56. [64]

    Vretblad, Fourier Analysis and Its Applications

    A. Vretblad, Fourier Analysis and Its Applications . Graduate Texts in Mathematics. Springer New York, 2006. – 50 –

    2006

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.