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arxiv: 2605.05363 · v1 · submitted 2026-05-06 · ✦ hep-th

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Celestial dual of conformal gravity MHV amplitudes: an OPE analysis

Nirmal Ghorai , Partha Paul , Nemani V. Suryanarayana
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:20 UTC · model grok-4.3

classification ✦ hep-th
keywords celestial CFTconformal gravityMHV amplitudesOPEbms4 algebrafree field realizationvertex operatorscelestial dual
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0 comments X

The pith

A free-field 2d CFT realizes the celestial dual to conformal gravity MHV amplitudes through matching OPEs.

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

The paper aims to provide an explicit two-dimensional chiral conformal field theory description for the celestial dual of the MHV sector in conformal gravity. It builds a free-field model with three scalars and ghost pairs that realizes the chiral bms4 algebra. Vertex operators are proposed for the primary operators corresponding to positive-helicity gravitons and scalars. Their operator product expansions exactly reproduce the ones derived from the bulk theory's Mellin-transformed MHV amplitudes. This matters because it gives a concrete CFT picture for the symmetries and amplitudes in this gravitational theory on the celestial sphere.

Core claim

We construct a 2d chiral CFT free-field realisation of the relevant chiral bms4 algebra in terms of three free scalars and three ghost pairs, and propose vertex operators for the positive-helicity graviton primary G++_Δ(z, z-bar) as well as the scalar primary Φ_Δ(z, z-bar). We compute their OPEs. These OPEs reproduce exactly those obtained from the bulk conformal gravity MHV amplitudes, providing a concrete celestial dual description of its MHV sector.

What carries the argument

The 2d free-field realization of the chiral bms4 algebra using three scalars and three ghost pairs, with vertex operators for the graviton and scalar primaries that allow explicit OPE computation matching the bulk results.

If this is right

  • The MHV sector of conformal gravity admits a concrete 2d CFT dual that can be used to compute amplitudes via standard CFT methods.
  • The chiral bms4 symmetry, including its non-trivial central extension, is realized explicitly in the free-field theory.
  • Correlation functions of the celestial operators can be evaluated directly from the vertex operators and free fields.
  • The construction confirms the soft theorem analysis of the bulk symmetries for this extended gravity theory.

Where Pith is reading between the lines

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

  • This free-field model could be extended to describe non-MHV amplitudes or other sectors in conformal gravity.
  • Similar realizations might connect to celestial duals of Einstein gravity or other modified gravity theories.
  • The central extension could be used to derive constraints on higher-point functions or anomalies in the celestial CFT.

Load-bearing premise

That the proposed free-field realization using three scalars and ghost pairs, together with the chosen vertex operators, correctly identifies the primaries and captures the full celestial CFT dual to the bulk theory.

What would settle it

Explicit computation of the OPEs in the proposed 2d CFT yielding coefficients or structures that differ from those extracted from the bulk MHV amplitudes would disprove the duality proposal.

read the original abstract

In an earlier paper [arXiv:2511.03669] we extracted the OPE of celestial CFT operator duals of positive helicity graviton and scalar particles from the Mellin transformed relevant MHV amplitudes of conformal gravity, realised as the bosonic subsector of the Berkovits-Witten theory. A soft theorem analysis of bulk MHV amplitudes established that this conformal gravity exhibits a chiral $\mathfrak{bms}_4$ symmetry on the celestial sphere with the associated $\mathfrak{sl}(2,\mathbb{R})$ current algebra, which acquires a non-trivial central extension, unlike the Einstein gravity. Here we construct a $2d$ chiral CFT free-field realisation of the relevant chiral $\mathfrak{bms}_4$ algebra in terms of three free scalars ($\phi_i$) and three $(\beta_i,\gamma_i)$ ghost pairs, and propose vertex operators for the positive-helicity graviton primary $G^{++}_{\Delta}(z,\bar{z})$ as well as the scalar primary $\Phi_{\Delta}(z,\bar{z})$, and compute their OPEs. These OPEs reproduce exactly those obtained from the bulk conformal gravity MHV amplitudes, providing a concrete celestial dual description of its MHV sector.

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 constructs a free-field realization of the chiral bms_4 algebra in a two-dimensional CFT using three free scalar fields φ_i and three (β_i, γ_i) ghost pairs. It proposes explicit vertex operators for the positive-helicity graviton primary G^{++}_Δ(z,¯z) and the scalar primary Φ_Δ(z,¯z), computes their OPEs, and demonstrates that these OPEs exactly reproduce the coefficients extracted from the Mellin-transformed MHV amplitudes of conformal gravity in the authors' prior work.

Significance. If the free-field realization is shown to be independent and the vertex operators are verified to be the correct primaries, the work supplies a concrete 2d CFT model for the celestial dual of the MHV sector of conformal gravity, including the chiral bms_4 symmetry with its non-trivial central extension. The explicit construction and exact OPE matching constitute a clear strength for celestial holography.

major comments (2)
  1. [§2] §2: The free-field realization must be shown to reproduce the full chiral bms_4 algebra, including the central extension of the sl(2,R) current algebra. Explicit computation of the OPEs or commutation relations among the realized generators (supertranslations, superrotations, and currents) is required to confirm the central charge matches the bulk value; without this, the claim that the construction realizes the symmetry algebra remains unverified.
  2. [§3] §3: The vertex operators for G^{++}_Δ and Φ_Δ are proposed in a form that yields the desired OPEs, but it is not demonstrated that these operators are primaries transforming correctly under the full set of realized bms_4 generators (including the central extension). The operators appear selected to match the known bulk OPE data from arXiv:2511.03669 rather than being fixed by requiring the correct transformation laws; an explicit check of their conformal dimensions, charges, and OPEs with the generators is needed to establish them as the correct celestial duals.
minor comments (2)
  1. The abstract refers to the 'bosonic subsector of the Berkovits-Witten theory' without a brief reminder of its relation to conformal gravity; adding one sentence of context would improve accessibility.
  2. [§4] Notation for the ghost pairs (β_i, γ_i) and the precise form of the vertex operators should be cross-checked for consistency between the algebra realization in §2 and the OPE computations in §4.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications on the existing construction and committing to explicit verifications in the revised version.

read point-by-point responses

  1. Referee: [§2] §2: The free-field realization must be shown to reproduce the full chiral bms_4 algebra, including the central extension of the sl(2,R) current algebra. Explicit computation of the OPEs or commutation relations among the realized generators (supertranslations, superrotations, and currents) is required to confirm the central charge matches the bulk value; without this, the claim that the construction realizes the symmetry algebra remains unverified.

    Authors: In §2 we define the free scalar and ghost fields and construct the generators of the chiral bms_4 algebra (supertranslations, superrotations, and the sl(2,R) currents) explicitly in terms of these fields. The algebra relations, including the non-trivial central extension of the sl(2,R) currents, follow directly from the canonical OPEs of the free fields and are arranged to reproduce the central charge extracted from the bulk conformal gravity analysis in our prior work. To make the verification fully explicit, we will expand §2 with a complete tabulation of the relevant OPEs among all generators. revision: yes

  2. Referee: [§3] §3: The vertex operators for G^{++}_Δ and Φ_Δ are proposed in a form that yields the desired OPEs, but it is not demonstrated that these operators are primaries transforming correctly under the full set of realized bms_4 generators (including the central extension). The operators appear selected to match the known bulk OPE data from arXiv:2511.03669 rather than being fixed by requiring the correct transformation laws; an explicit check of their conformal dimensions, charges, and OPEs with the generators is needed to establish them as the correct celestial duals.

    Authors: The vertex operators are fixed by the celestial dictionary to carry the conformal dimensions and charges appropriate to the positive-helicity graviton and scalar primaries. Their OPEs with the full set of bms_4 generators (including the action of the central extension) are computed in the manuscript and confirm the expected primary transformation laws. These checks are consistent with the dimensions and charges obtained from the bulk MHV amplitudes. We will add the explicit OPE computations with the generators to the revised manuscript to demonstrate the primary property directly. revision: yes

Circularity Check

0 steps flagged

Independent 2d free-field realization computes OPEs that match prior bulk extraction

full rationale

The paper constructs a chiral bms4 algebra realization using three free scalars and ghost pairs, proposes vertex operators for G^{++}_Δ and Φ_Δ based on the algebra, and computes their OPEs. These are shown to reproduce the OPEs extracted from bulk conformal gravity MHV amplitudes in the authors' prior work. The 2d construction relies on standard CFT free-field methods and the algebra generators rather than being defined in terms of or fitted to the target OPE coefficients. The self-citation supplies an external benchmark for the match but does not reduce the derivation to its inputs by construction. No self-definitional steps, fitted predictions, or load-bearing uniqueness theorems from self-citation are present.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters are introduced. The construction relies on standard 2d CFT axioms and the bms4 algebra taken from prior work; no new entities are postulated.

axioms (2)
  • standard math Standard axioms and OPE rules of 2d conformal field theory for free scalars and ghost systems
    Invoked to define the free-field realization and compute the OPEs of the vertex operators.
  • domain assumption The chiral bms4 algebra with non-trivial central extension as derived from soft theorem analysis in prior work
    This is the target symmetry that the 2d CFT must realize.

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discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Pasterski, S

    S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D96(2017) 065026 [arXiv:1701.00049]

    work page arXiv 2017

  2. [2]

    Lectures on the Infrared Structure of Gravity and Gauge Theory

    A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” Princeton University Press, 2018, ISBN 978-0-691-17973-5 [arXiv:1703.05448 [hep-th]]

    work page Pith review arXiv 2018

  3. [3]

    Conformal basis for flat space amplitudes,

    S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96(2017) 065022 [arXiv:1705.01027]

    work page arXiv 2017

  4. [4]

    Null Infinity and Unitary Representation of The Poincare Group,

    S. Banerjee, “Null Infinity and Unitary Representation of The Poincare Group,” JHEP 01(2019), 205 doi:10.1007/JHEP01(2019)205 [arXiv:1801.10171 [hep-th]]

    work page doi:10.1007/jhep01(2019)205 2019

  5. [5]

    Raclariu,Lectures on celestial holography,2107.02075

    A. M. Raclariu, “Lectures on Celestial Holography,” [arXiv:2107.02075 [hep-th]]

    work page arXiv

  6. [6]

    Pasterski,Lectures on celestial amplitudes,Eur

    S. Pasterski, “Lectures on Celestial Amplitudes,” Eur. Phys. J. C81(2021) 1062 [arXiv:2108.04801]

    work page arXiv 2021

  7. [7]

    Gluon Amplitudes as 2d Conformal Correlators,

    S. Pasterski, S. H. Shao and A. Strominger, “Gluon Amplitudes as 2d Conformal Correlators,” Phys. Rev. D96(2017) 085006 [arXiv:1706.03917]

    work page arXiv 2017

  8. [8]

    On BMS Invariance of Gravitational Scattering

    A. Strominger, “On BMS Invariance of Gravitational Scattering”, JHEP07(2014) 152 doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]]

    work page doi:10.1007/jhep07(2014)152 2014

  9. [9]

    BMS supertranslations and Weinberg’s soft graviton theorem,

    T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP05, 151 (2015) doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]]

    work page doi:10.1007/jhep05(2015)151 2015

  10. [10]

    Semiclassical Virasoro symmetry of the quantum gravityS-matrix,

    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravityS-matrix,” JHEP08, 058 (2014) doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]]

    work page doi:10.1007/jhep08(2014)058 2014

  11. [11]

    Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,

    T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP08, 050 (2017) doi:10.1007/JHEP08(2017)050 [arXiv:1701.00496 [hep-th]]. – 24 –

    work page doi:10.1007/jhep08(2017)050 2017

  12. [12]

    Asymptotic symmetries and subleading soft graviton theorem,

    M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D90, no.12, 124028 (2014) doi:10.1103/PhysRevD.90.124028 [arXiv:1408.2228 [hep-th]]

    work page doi:10.1103/physrevd.90.124028 2014

  13. [13]

    Burg-Metzner-Sachs symmetry, string theory, and soft theorems,

    S. G. Avery and B. U. W. Schwab, “Burg-Metzner-Sachs symmetry, string theory, and soft theorems,” Phys. Rev. D93, 026003 (2016) doi:10.1103/PhysRevD.93.026003 [arXiv:1506.05789 [hep-th]]

    work page doi:10.1103/physrevd.93.026003 2016

  14. [14]

    Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime,

    M. Campiglia, “Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime,” JHEP11, 160 (2015) doi:10.1007/JHEP11(2015)160 [arXiv:1509.01408 [hep-th]]

    work page doi:10.1007/jhep11(2015)160 2015

  15. [15]

    Noether’s second theorem and Ward identities for gauge symmetries,

    S. G. Avery and B. U. W. Schwab, “Noether’s second theorem and Ward identities for gauge symmetries,” JHEP02, 031 (2016) doi:10.1007/JHEP02(2016)031 [arXiv:1510.07038 [hep-th]]

    work page doi:10.1007/jhep02(2016)031 2016

  16. [16]

    MHV graviton scattering amplitudes and current algebra on the celestial sphere,

    S. Banerjee, S. Ghosh and P. Paul, “MHV graviton scattering amplitudes and current algebra on the celestial sphere,” JHEP02, 176 (2021) doi:10.1007/JHEP02(2021)176 [arXiv:2008.04330 [hep-th]]

    work page doi:10.1007/jhep02(2021)176 2021

  17. [17]

    (Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes,

    S. Banerjee, S. Ghosh and P. Paul, “(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes,” JHEP09(2022), 236 doi:10.1007/JHEP09(2022)236 [arXiv:2108.04262 [hep-th]]

    work page doi:10.1007/jhep09(2022)236 2022

  18. [18]

    Sub-subleading soft gravitons and large diffeomorphisms,

    M. Campiglia and A. Laddha, “Sub-subleading soft gravitons and large diffeomorphisms,” JHEP01, 036 (2017) doi:10.1007/JHEP01(2017)036 [arXiv:1608.00685 [gr-qc]]

    work page doi:10.1007/jhep01(2017)036 2017

  19. [19]

    Sub-subleading soft gravitons: New symmetries of quantum gravity?,

    M. Campiglia and A. Laddha, “Sub-subleading soft gravitons: New symmetries of quantum gravity?,” Phys. Lett. B764, 218-221 (2017) doi:10.1016/j.physletb.2016.11.046 [arXiv:1605.09094 [gr-qc]]

    work page doi:10.1016/j.physletb.2016.11.046 2017

  20. [20]

    Weinberg, Phys

    S. Weinberg, “Infrared photons and gravitons,” Phys. Rev.140(1965), B516-B524 doi:10.1103/PhysRev.140.B516

    work page doi:10.1103/physrev.140.b516 1965

  21. [21]

    Evidence for a New Soft Graviton Theorem,

    F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]]

    work page arXiv

  22. [22]

    csl2 symmetry ofR 1,3 gravity,

    N. Gupta, P. Paul and N. V. Suryanarayana, “ csl2 symmetry ofR 1,3 gravity,” Phys. Rev. D108, no.8, 086029 (2023) doi:10.1103/PhysRevD.108.086029 [arXiv:2109.06857 [hep-th]]

    work page doi:10.1103/physrevd.108.086029 2023

  23. [23]

    Celestial operator products of gluons and gravitons,

    M. Pate, A. M. Raclariu, A. Strominger and E. Y. Yuan, “Celestial operator products of gluons and gravitons,” Rev. Math. Phys.33, no.09, 2140003 (2021) doi:10.1142/S0129055X21400031 [arXiv:1910.07424 [hep-th]]

    work page doi:10.1142/s0129055x21400031 2021

  24. [24]

    Guevara, E

    A. Guevara, E. Himwich, M. Pate and A. Strominger, “Holographic symmetry algebras – 25 – for gauge theory and gravity,” JHEP11(2021), 152 doi:10.1007/JHEP11(2021)152 [arXiv:2103.03961 [hep-th]]

    work page doi:10.1007/jhep11(2021)152 2021

  25. [25]

    w(1+infinity) and the Celestial Sphere,

    A. Strominger, “w(1+infinity) and the Celestial Sphere,” [arXiv:2105.14346 [hep-th]]. A. Strominger, “w 1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries,” Phys. Rev. Lett.127, no.22, 221601 (2021) doi:10.1103/PhysRevLett.127.221601

    work page doi:10.1103/physrevlett.127.221601 2021

  26. [26]

    Celestial operator product expansions and w 1+∞ symmetry for all spins,

    E. Himwich, M. Pate and K. Singh, “Celestial operator product expansions and w 1+∞ symmetry for all spins,” JHEP01, 080 (2022) doi:10.1007/JHEP01(2022)080 [arXiv:2108.07763 [hep-th]]

    work page doi:10.1007/jhep01(2022)080 2022

  27. [27]

    Deforming soft algebras for gauge theory,

    W. Melton, S. A. Narayanan and A. Strominger, “Deforming soft algebras for gauge theory,” JHEP03, 233 (2023) doi:10.1007/JHEP03(2023)233 [arXiv:2212.08643 [hep-th]]

    work page doi:10.1007/jhep03(2023)233 2023

  28. [28]

    Perturbatively exact w 1+∞ asymptotic symmetry of quantum self-dual gravity,

    A. Ball, S. A. Narayanan, J. Salzer and A. Strominger, “Perturbatively exact w 1+∞ asymptotic symmetry of quantum self-dual gravity,” JHEP01, 114 (2022) doi:10.1007/JHEP01(2022)114 [arXiv:2111.10392 [hep-th]]

    work page doi:10.1007/jhep01(2022)114 2022

  29. [29]

    Celestialw 1+∞ Symmetries from Twistor Space,

    T. Adamo, L. Mason and A. Sharma, “Celestialw 1+∞ Symmetries from Twistor Space,” SIGMA18, 016 (2022) doi:10.3842/SIGMA.2022.016 [arXiv:2110.06066 [hep-th]]

    work page doi:10.3842/sigma.2022.016 2022

  30. [30]

    Celestial holography meets twisted holography: 4d amplitudes from chiral correlators

    K. Costello and N. M. Paquette, “Celestial holography meets twisted holography: 4d amplitudes from chiral correlators,” JHEP10, 193 (2022) doi:10.1007/JHEP10(2022)193 [arXiv:2201.02595 [hep-th]]

    work page doi:10.1007/jhep10(2022)193 2022

  31. [31]

    Associativity of One-Loop Corrections to the Celestial Operator Product Expansion,

    K. Costello and N. M. Paquette, “Associativity of One-Loop Corrections to the Celestial Operator Product Expansion,” Phys. Rev. Lett.129, no.23, 231604 (2022) doi:10.1103/PhysRevLett.129.231604 [arXiv:2204.05301 [hep-th]]

    work page doi:10.1103/physrevlett.129.231604 2022

  32. [32]

    Bootstrapping two-loop QCD amplitudes,

    K. J. Costello, “Bootstrapping two-loop QCD amplitudes,” JHEP08, 011 (2025) doi:10.1007/JHEP08(2025)011 [arXiv:2302.00770 [hep-th]]

    work page doi:10.1007/jhep08(2025)011 2025

  33. [33]

    Supersymmetry and the celestial Jacobi identity,

    A. Ball, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, “Supersymmetry and the celestial Jacobi identity,” JHEP04, 099 (2024) doi:10.1007/JHEP04(2024)099 [arXiv:2311.01364 [hep-th]]

    work page doi:10.1007/jhep04(2024)099 2024

  34. [34]

    An infinite family of w 1+∞ invariant theories on the celestial sphere,

    S. Banerjee, H. Kulkarni and P. Paul, “An infinite family of w 1+∞ invariant theories on the celestial sphere,” JHEP05(2023), 063 doi:10.1007/JHEP05(2023)063 [arXiv:2301.13225 [hep-th]]

    work page doi:10.1007/jhep05(2023)063 2023

  35. [35]

    MHV gluon scattering amplitudes from celestial current algebras,

    S. Banerjee and S. Ghosh, “MHV gluon scattering amplitudes from celestial current algebras,” JHEP10(2021), 111 doi:10.1007/JHEP10(2021)111 [arXiv:2011.00017 [hep-th]]

    work page doi:10.1007/jhep10(2021)111 2021

  36. [36]

    Descendants in celestial CFT and emergent – 26 – multi-collinear factorization,

    S. Ebert, A. Sharma and D. Wang, “Descendants in celestial CFT and emergent – 26 – multi-collinear factorization,” JHEP03, 030 (2021) doi:10.1007/JHEP03(2021)030 [arXiv:2009.07881 [hep-th]]

    work page doi:10.1007/jhep03(2021)030 2021

  37. [37]

    MHV gluon scattering in the massive scalar background and celestial OPE,

    S. Banerjee, R. Mandal, A. Manu and P. Paul, “MHV gluon scattering in the massive scalar background and celestial OPE,” JHEP10(2023), 007 doi:10.1007/JHEP10(2023)007 [arXiv:2302.10245 [hep-th]]

    work page doi:10.1007/jhep10(2023)007 2023

  38. [38]

    All-order celestial OPE in the MHV sector,

    T. Adamo, W. Bu, E. Casali and A. Sharma, “All-order celestial OPE in the MHV sector,” JHEP03, 252 (2023) doi:10.1007/JHEP03(2023)252 [arXiv:2211.17124 [hep-th]]

    work page doi:10.1007/jhep03(2023)252 2023

  39. [39]

    All-order celestial OPE from on-shell recursion,

    L. Ren, A. Schreiber, A. Sharma and D. Wang, “All-order celestial OPE from on-shell recursion,” JHEP10, 080 (2023) doi:10.1007/JHEP10(2023)080 [arXiv:2305.11851 [hep-th]]

    work page doi:10.1007/jhep10(2023)080 2023

  40. [40]

    Loop-level gluon OPEs in celestial holography,

    R. Bhardwaj, L. Lippstreu, L. Ren, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, “Loop-level gluon OPEs in celestial holography,” JHEP11, 171 (2022) doi:10.1007/JHEP11(2022)171 [arXiv:2208.14416 [hep-th]]

    work page doi:10.1007/jhep11(2022)171 2022

  41. [41]

    Celestial gluon and graviton OPE at loop level,

    H. Krishna, “Celestial gluon and graviton OPE at loop level,” JHEP03, 176 (2024) doi:10.1007/JHEP03(2024)176 [arXiv:2310.16687 [hep-th]]

    work page doi:10.1007/jhep03(2024)176 2024

  42. [42]

    Celestial recursion,

    Y. Hu and S. Pasterski, “Celestial recursion,” JHEP01(2023), 151 doi:10.1007/JHEP01(2023)151 [arXiv:2208.11635 [hep-th]]

    work page doi:10.1007/jhep01(2023)151 2023

  43. [43]

    Celestial OPE in Self-Dual Gravity

    S. Banerjee, H. Kulkarni and P. Paul, “Celestial OPE in Self-Dual Gravity”, Phys. Rev. D109(2024) 086017 doi:10.1103/PhysRevD.109.086017 [arXiv:2311.06485 [hep-th]]

    work page doi:10.1103/physrevd.109.086017 2024

  44. [44]

    Towards celestial CFT dual of 4d conformal gravity: I,

    N. Ghorai, P. Paul and N. V. Suryanarayana, “Towards celestial CFT dual of 4d conformal gravity: I,” [arXiv:2511.03669 [hep-th]]

    work page arXiv

  45. [45]

    Celestial Dual for Maximal Helicity Violating Amplitudes,

    W. Melton, A. Sharma, A. Strominger and T. Wang, “Celestial Dual for Maximal Helicity Violating Amplitudes,” Phys. Rev. Lett.133, no.9, 091603 (2024) doi:10.1103/PhysRevLett.133.091603 [arXiv:2403.18896 [hep-th]]

    work page doi:10.1103/physrevlett.133.091603 2024

  46. [46]

    Conformal supergravity in twistor-string theory,

    N. Berkovits and E. Witten, “Conformal supergravity in twistor-string theory,” JHEP 08, 009 (2004) doi:10.1088/1126-6708/2004/08/009 [arXiv:hep-th/0406051 [hep-th]]

    work page doi:10.1088/1126-6708/2004/08/009 2004

  47. [47]

    Conformal Gravity from Gauge Theory,

    H. Johansson and J. Nohle, “Conformal Gravity from Gauge Theory,” [arXiv:1707.02965 [hep-th]]

    work page arXiv

  48. [48]

    Unraveling conformal gravity amplitudes,

    H. Johansson, G. Mogull and F. Teng, “Unraveling conformal gravity amplitudes,” JHEP09(2018), 080 doi:10.1007/JHEP09(2018)080 [arXiv:1806.05124 [hep-th]]

    work page doi:10.1007/jhep09(2018)080 2018

  49. [49]

    Quantization and representation theory of finite W algebras,

    J. de Boer and T. Tjin, “Quantization and representation theory of finite W algebras,” Commun. Math. Phys.158, 485-516 (1993) doi:10.1007/BF02096800 [arXiv:hep-th/9211109 [hep-th]]

    work page doi:10.1007/bf02096800 1993

  50. [50]

    The Relation between quantum W algebras and Lie – 27 – algebras,

    J. de Boer and T. Tjin, “The Relation between quantum W algebras and Lie – 27 – algebras,” Commun. Math. Phys.160, 317-332 (1994) doi:10.1007/BF02103279 [arXiv:hep-th/9302006 [hep-th]]

    work page doi:10.1007/bf02103279 1994

  51. [51]

    Di Francesco, P

    P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” Springer-Verlag, 1997, ISBN 978-0-387-94785-3, 978-1-4612-7475-9 doi:10.1007/978-1-4612-2256-9

    work page doi:10.1007/978-1-4612-2256-9 1997

  52. [52]

    Polchinski,String theory

    J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge University Press, 2007, ISBN 978-0-511-25227-3, 978-0-521-67227-6, 978-0-521-63303-1 doi:10.1017/CBO9780511816079

    work page doi:10.1017/cbo9780511816079 2007

  53. [53]

    Conformal blocks of Wess-Zumino-Witten model from its free-field representation,

    A. Morozov and H. Sifat, “Conformal blocks of Wess-Zumino-Witten model from its free-field representation,” [arXiv:2511.23132 [hep-th]]

    work page arXiv

  54. [54]

    Correspondences among CFTs with different W-algebra symmetry,

    T. Creutzig, N. Genra, Y. Hikida and T. Liu, “Correspondences among CFTs with different W-algebra symmetry,” Nucl. Phys. B957, 115104 (2020) doi:10.1016/j.nuclphysb.2020.115104 [arXiv:2002.12587 [hep-th]]

    work page doi:10.1016/j.nuclphysb.2020.115104 2020

  55. [55]

    Conformal field theories via Hamiltonian reduction,

    M. Bershadsky, “Conformal field theories via Hamiltonian reduction,” Commun. Math. Phys.139(1991), 71-82 doi:10.1007/BF02102729

    work page doi:10.1007/bf02102729 1991

  56. [56]

    Hidden SL(n) Symmetry in Conformal Field Theories,

    M. Bershadsky and H. Ooguri, “Hidden SL(n) Symmetry in Conformal Field Theories,” Commun. Math. Phys.126, 49 (1989) doi:10.1007/BF02124331

    work page doi:10.1007/bf02124331 1989

  57. [57]

    All chiralW-algebra extensions ofso(2,3),

    N. Gupta and N. V. Suryanarayana, “All chiralW-algebra extensions ofso(2,3),” JHEP08, 137 (2024) doi:10.1007/JHEP08(2024)137 [arXiv:2304.14938 [hep-th]]

    work page doi:10.1007/jhep08(2024)137 2024

  58. [58]

    A chiral Λ-bms4 symmetry of AdS4 gravity,

    N. Gupta and N. V. Suryanarayana, “A chiral Λ-bms4 symmetry of AdS4 gravity,” Nucl. Phys. B1010, 116759 (2025) doi:10.1016/j.nuclphysb.2024.116759 [arXiv:2211.13176 [hep-th]]

    work page doi:10.1016/j.nuclphysb.2024.116759 2025

  59. [59]

    A Mathematica package for computing operator product expansions,

    K. Thielemans, “A Mathematica package for computing operator product expansions,” Int. J. Mod. Phys. C2, 787-798 (1991) doi:10.1142/S0129183191001001

    work page doi:10.1142/s0129183191001001 1991

  60. [60]

    Introduction to conformal field theory: with applications to String theory,

    R. Blumenhagen and E. Plauschinn, “Introduction to conformal field theory: with applications to String theory,” Lect. Notes Phys.779, 1-256 (2009) doi:10.1007/978-3-642-00450-6

    work page doi:10.1007/978-3-642-00450-6 2009

  61. [61]

    Quasi-primary Fields and Associativity of Chiral Algebras,

    P. Bowcock, “Quasi-primary Fields and Associativity of Chiral Algebras,” Nucl. Phys. B356, 367-386 (1991) doi:10.1016/0550-3213(91)90314-N – 28 –

    work page doi:10.1016/0550-3213(91)90314-n 1991