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arxiv: 2604.05262 · v1 · submitted 2026-04-06 · ✦ hep-th · gr-qc· math-ph· math.MP

Recognition: 2 theorem links

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Residual Symmetries and Their Algebras in the Kerr-Schild Double Copy

B. P. Holton
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:36 UTC · model grok-4.3

classification ✦ hep-th gr-qcmath-phmath.MP
keywords Kerr-Schild double copyresidual symmetriesBRST cohomologyconformal Killing vectorsYang-Mills theorySchwarzschild solutionasymptotic flatnesshorizon regularity
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The pith

The Kerr-Schild double copy does not map residual symmetry structures between Yang-Mills theory and gravity.

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

The paper investigates whether the Kerr-Schild double copy relation extends beyond classical solutions to include a correspondence between the residual symmetries that preserve the ansatz form in gauge theory and gravity. On the Yang-Mills side, these residual symmetries constitute an infinite-dimensional algebra consisting of arbitrary functions along null directions. On the gravity side, the residual diffeomorphisms that preserve the Kerr-Schild form of the Schwarzschild solution generate a conformal algebra on the two-sphere, which splits into global Killing vectors plus an infinite set of proper conformal Killing vectors once asymptotic flatness and horizon regularity are required. A Weyl-compensated BRST complex is then used to show that the proper conformal Killing vectors are BRST-exact and therefore trivial in cohomology, so that only the expected global isometries survive as physical symmetries. This leaves a clear mismatch: the double copy enlarges the symmetry content at the level of the ansatz but, after the cohomological reduction, preserves only the physical symmetries of gravity.

Core claim

The KSDC correspondence does not provide a mapping between the residual symmetry structures of the Kerr-Schild ansatz in Yang-Mills theory and gravity. Residual symmetries on the gauge-theory side form an infinite-dimensional algebra of functions along null directions. On the gravitational side, residual diffeomorphisms preserving the Kerr-Schild Schwarzschild metric generate a conformal algebra on S^2 that decomposes into Killing vectors and proper conformal Killing vectors; after a Weyl-compensated BRST complex is constructed, the CKV sector is shown to be BRST-exact and trivial in cohomology under asymptotic flatness and horizon regularity, reducing the physical symmetry algebra to the is

What carries the argument

Weyl-compensated BRST complex that renders the sector of proper conformal Killing vectors BRST-exact and cohomologically trivial

If this is right

  • Residual symmetries in the Yang-Mills Kerr-Schild ansatz remain an infinite-dimensional algebra of functions along null directions.
  • Residual diffeomorphisms of the Kerr-Schild Schwarzschild metric generate a conformal algebra on S^2 containing both Killing vectors and proper conformal Killing vectors.
  • The proper conformal Killing vector sector becomes BRST-exact in the Weyl-compensated complex, reducing the physical symmetry algebra to global isometries.
  • The double copy therefore enlarges symmetry structure at the ansatz level while preserving only physical symmetries after cohomological reduction.

Where Pith is reading between the lines

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

  • The mismatch implies that any symmetry-level double copy would need additional structure, such as a different cohomology or a modified ansatz, beyond the classical KSDC.
  • The BRST reduction technique could be tested on other exact solutions, such as Kerr or multi-center metrics, to see whether the same pattern of enlargement followed by reduction holds.
  • The result isolates a concrete point where gauge-theory and gravitational symmetries diverge, which may constrain attempts to lift the double copy to a statement about underlying field theories.

Load-bearing premise

The Weyl-compensated BRST complex correctly identifies the CKV sector as BRST-exact and trivial in cohomology once asymptotic flatness and horizon regularity are imposed.

What would settle it

An explicit non-trivial proper conformal symmetry of the Schwarzschild metric that remains non-exact after the Weyl-compensated BRST procedure, or a direct one-to-one algebra isomorphism between the infinite null-direction functions of the Yang-Mills ansatz and the reduced gravitational symmetries.

read the original abstract

The Kerr-Schild double copy (KSDC) is well-known for relating exact classical solutions between Yang-Mills theory and theories of gravity. However, whether this correspondence provides a more fundamental mapping between the underlying symmetries of gauge theory and gravity remains an underdeveloped area of research in the contemporary double copy program. In this paper, we demonstrate that the KSDC correspondence does not provide a mapping between the residual symmetry structures of the Kerr-Schild ansatz in Yang-Mills theory and gravity. On the gauge theory side, residual symmetries form an infinite-dimensional algebra of functions along null directions. On the gravitational side, residual diffeomorphisms preserving the Kerr-Schild form of the Schwarzschild metric generate a conformal algebra on $S^2$, which decomposes into Killing vectors and proper conformal Killing vectors (CKVs). While the Killing sector reproduces the expected global isometries, the CKV sector yields an infinite-dimensional algebra after imposing asymptotic flatness and horizon regularity. This appears to contradict the fact that the Schwarzschild solution admits no proper conformal symmetries. We resolve this apparent contradiction by constructing a Weyl-compensated BRST complex, showing that the CKV sector is BRST-exact and therefore trivial in cohomology, so that the physical symmetry algebra reduces to the global isometries of Schwarzschild. This demonstrates that the KSDC introduces an enlarged symmetry structure at the level of the ansatz, but preserves physical symmetries after a cohomological reduction, revealing a fundamental mismatch between Yang-Mills and gravity at the level of residual symmetries.

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

1 major / 0 minor

Summary. The manuscript argues that the Kerr-Schild double copy does not induce a direct mapping between residual symmetry structures in Yang-Mills theory and gravity. On the YM side, residual symmetries of the Kerr-Schild ansatz form an infinite-dimensional algebra of functions along null directions. On the gravity side, residual diffeomorphisms preserving the Kerr-Schild form of the Schwarzschild metric generate a conformal algebra on S^2 that decomposes into Killing vectors (reproducing the expected global isometries) and an infinite-dimensional proper conformal Killing vector (CKV) sector even after asymptotic flatness and horizon regularity are imposed. The apparent contradiction with the known absence of proper conformal symmetries in Schwarzschild is resolved by constructing a Weyl-compensated BRST complex in which the CKV sector is shown to be BRST-exact and thus trivial in cohomology, reducing the physical symmetry algebra to the global isometries. This establishes an enlarged symmetry structure at the level of the ansatz that is reduced cohomologically on the gravity side but not on the YM side.

Significance. If the Weyl-compensated BRST construction is correct, the result is significant for the double-copy program. It provides a concrete demonstration that the correspondence preserves physical symmetries after cohomological reduction while introducing a mismatch in the residual (ansatz-level) symmetries between gauge theory and gravity. The introduction of the compensated BRST complex as a technical tool for handling infinite-dimensional sectors under regularity conditions may prove useful in related studies of asymptotic symmetries and exact solutions.

major comments (1)
  1. The central claim that the CKV sector becomes cohomologically trivial rests on the Weyl-compensated BRST complex. The manuscript must supply an explicit verification that the modified BRST operator remains nilpotent after the Weyl compensation is added, that the CKV generators are BRST-exact (i.e., there exist operators whose BRST variation reproduces them), and that no extraneous equivalences are introduced that would alter the physical content. This calculation should be performed under the stated boundary conditions of asymptotic flatness and horizon regularity; without it the reduction to global isometries and the asserted mismatch cannot be confirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the Weyl-compensated BRST construction. We address the major comment below and will revise the manuscript to include the requested calculations.

read point-by-point responses

  1. Referee: The central claim that the CKV sector becomes cohomologically trivial rests on the Weyl-compensated BRST complex. The manuscript must supply an explicit verification that the modified BRST operator remains nilpotent after the Weyl compensation is added, that the CKV generators are BRST-exact (i.e., there exist operators whose BRST variation reproduces them), and that no extraneous equivalences are introduced that would alter the physical content. This calculation should be performed under the stated boundary conditions of asymptotic flatness and horizon regularity; without it the reduction to global isometries and the asserted mismatch cannot be confirmed.

    Authors: We agree that an explicit verification is required for rigor. The manuscript introduces the Weyl-compensated BRST operator and argues that the CKV sector is BRST-exact on the basis of the algebraic structure and the form of the compensation term, but does not expand the nilpotency check or the explicit coboundary operators in full detail under the boundary conditions. In the revised version we will add a dedicated subsection (or appendix) that performs these calculations explicitly: we verify that the modified operator Q_W satisfies Q_W^2 = 0 on the space of fields and gauge parameters respecting asymptotic flatness and horizon regularity; we construct the explicit operators whose Q_W-variation reproduces each CKV generator; and we confirm that the resulting equivalences do not enlarge the physical cohomology beyond the global isometries (i.e., no new relations are imposed on the Killing sector). These steps will be carried out directly on the Schwarzschild background with the stated regularity conditions, thereby confirming both the triviality of the CKV sector in cohomology and the mismatch with the Yang-Mills residual algebra. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central reduction uses newly constructed BRST complex independent of inputs

full rationale

The paper's derivation chain proceeds from the Kerr-Schild ansatz residual symmetries (infinite-dimensional on YM side, conformal algebra on gravity side) to an apparent contradiction with Schwarzschild isometries, resolved by introducing a Weyl-compensated BRST complex whose cohomology trivializes the CKV sector. This construction is presented as original work in the abstract and does not reduce any claimed prediction or physical symmetry algebra to a fitted parameter, self-definition, or prior self-citation by construction. No load-bearing step quotes or relies on the authors' earlier results as an unverified uniqueness theorem or ansatz; the BRST nilpotency and exactness claims are derived within the present manuscript rather than imported. The argument remains self-contained against external benchmarks such as known Schwarzschild isometries and standard BRST cohomology, yielding a normal non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard BRST cohomology, the definition of residual symmetries for the Kerr-Schild ansatz, and the construction of a Weyl-compensated complex; no free parameters are introduced.

axioms (2)
  • standard math BRST cohomology correctly identifies trivial symmetries in the presence of Weyl compensators.
    Invoked to declare the CKV sector BRST-exact.
  • domain assumption Residual diffeomorphisms preserving the Kerr-Schild form generate a conformal algebra on S^2 when asymptotic flatness and horizon regularity are imposed.
    Used to obtain the infinite-dimensional algebra that is subsequently reduced.
invented entities (1)
  • Weyl-compensated BRST complex no independent evidence
    purpose: To render the proper conformal Killing vector sector BRST-exact and therefore trivial in cohomology.
    Constructed in the paper to resolve the apparent contradiction between the generated algebra and the known absence of proper conformal symmetries of Schwarzschild.

pith-pipeline@v0.9.0 · 5577 in / 1606 out tokens · 43653 ms · 2026-05-10T18:36:41.416035+00:00 · methodology

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Works this paper leans on

45 extracted references · 38 canonical work pages

  1. [1]

    Adamo and A

    T. Adamo and A. Ilderton,Classical and quantum double copy of back-reaction,JHEP09 (2020) 200 [arXiv:2005.05807]

    work page arXiv 2020

  2. [2]

    Alkac, M

    G. Alkac, M. K. Gumus and M. Te,The Kerr-Schild double copy in Lifshitz spacetime,JHEP 05(2021) 214 [arXiv:2103.06986]

    work page arXiv 2021

  3. [3]

    Anastasiou, L

    A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy,Yang-Mills origin of gravitational symmetries,Phys. Rev. Lett.113(2014) no. 23, 231606 [arXiv:1408.4434

    work page arXiv 2014

  4. [4]

    Anastasiou, L

    A. Anastasiou, L. Borsten, M. J. Duff, M. J. Hughes, A. Marrani, S. Nagy and M. Zoccali Twin supergravities from Yang-Mills theory squared,Phys. Rev. D96(2017) no. 2, 026013 [arXiv:1610.07192]

    work page arXiv 2017

  5. [5]

    Anastasiou, L

    A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy and M. Zoccali,Are all supergravity theories Yang-Mills squared?,Nucl. Phys. B934(2019), 606-633 [arXiv:1707.03234]

    work page arXiv 2019

  6. [6]

    Anastasiou, L

    A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy and M. Zoccali,Gravity as gauge theory squared: a ghost story,Phys. Rev. Lett.121(2018), no.21, 211601 [arXiv:1807.02486]

    work page arXiv 2018

  7. [7]

    Ay´ on-Beato, M

    E. Ay´ on-Beato, M. Hassa¨ ıne and D. Higuita-Borja,Role of symmetries in the Kerr-Schild derivation of the Kerr black hole,Phys. Rev. D94(2016), no.6, 064073 [arXiv:1512.06870]

    work page arXiv 2016

  8. [8]

    A. Ball, A. Bencke, Y. Chen and A. Volovich,Hidden symmetry in the double copy,JHEP 10(2023), 085 [arXiv:2307.01338]

    work page arXiv 2023

  9. [9]

    Barnich, F

    G. Barnich, F. Brandt and M. Henneaux,Local BRST cohomology in Einstein Yang-Mills theory,Nucl. Phys. B455(1995), 357-408 [arXiv:hep-th/9505173]

    work page arXiv 1995

  10. [10]

    Barnich, F

    G. Barnich, F. Brandt and M. Henneaux,Local BRST cohomology in gauge theories,Phys. Rept.338(2000), 439-569 [arXiv:hep-th/0002245]

    work page arXiv 2000

  11. [11]

    Z. Bern, J. J. M. Carrasco and H. Johansson,Perturbative quantum gravity as a double copy of gauge theory,Phys. Rev. Lett.105(2010), 061602 [arXiv:1004.0476]

    work page arXiv 2010

  12. [12]

    Z. Bern, T. Dennen, Y.T. Huang and M. Kiermaier,Gravity as the square of gauge theory, Phys. Rev. D82(2010), 065003 [arXiv:1004.0693]

    work page arXiv 2010

  13. [13]

    Z. Bern, C. Cheung, R. Roiban, C.H. Shen, M.P. Solon and M. Zen,Scattering amplitudes and the conservative Hamiltonian for binary systems at third post-Minkowskian order,Phys. Rev. Lett.122(2019), no.20, 201603 [arXiv:1901.04424] – 25 –

    work page arXiv 2019

  14. [14]

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban,The duality between color and kinematics and its applications,J. Phys. A57(2024), no.33, 333002 [arXiv:1909.01358]

    work page arXiv 2024

  15. [15]

    Campiglia and A

    M. Campiglia and A. Laddha,Asymptotic symmetries of gravity and soft theorems for massive particles,JHEP12(2015), 094

    2015

  16. [16]

    Campiglia and S

    M. Campiglia and S. Nagy,A double copy for asymptotic symmetries in the self-dual sector, JHEP03(2021), 262 [arXiv:2102.01680]

    work page arXiv 2021

  17. [17]

    G. L. Cardoso, S. Nagy and S. Nampuri,A double copy forN= 2supergravity: a linearised tale told on-shell,JHEP10(2016), 127 [arXiv:1609.05022]

    work page arXiv 2016

  18. [18]

    G. L. Cardoso, S. Nagy and S. Nampuri,Multi-centeredN= 2BPS black holes: a double copy description,JHEP04(2017), 037 [arXiv:1611.04409]

    work page arXiv 2017

  19. [19]

    S. M. Carroll,Spacetime and Geometry: An Introduction to General Relativity,Camb. Uni. Press(2019)

    2019

  20. [20]

    Catren,Geometric foundations of classical Yang-Mills theory,Stud

    G. Catren,Geometric foundations of classical Yang-Mills theory,Stud. Hist. Phil. Sci. B39 (2008), 511-531 [doi:10.1016/j.shpsb.2008.02.002]

    work page doi:10.1016/j.shpsb.2008.02.002 2008

  21. [21]

    Cheung and J

    C. Cheung and J. Mangan,Covariant color-kinematics duality,JHEP11(2021), 069 [arXiv:2108.02276]

    work page arXiv 2021

  22. [22]

    Cheung, A

    C. Cheung, A. Helset and J. Parra-Martinez,Geometry-kinematics duality,Phys. Rev. D 104(2022), no.4, 045016 [arXiv:2202.06972]

    work page arXiv 2022

  23. [23]

    Chevalley, S

    C. Chevalley, S. EilenbergCohomology theory of Lie groups and Lie algebras,Trans. Am. Math. Soc.63(1948), 85-124

    1948

  24. [24]

    B. Coll, S. R. Hildebrandt and J. M. M. Senovilla,Kerr-Schild symmetries,Gen. Rel. Grav. 33(2000), 649-670 [arXiv:gr-qc/0006044]

    work page arXiv 2000

  25. [25]

    J. A. de Azc´ arraga and J. M. Izquierdo,Lie groups, Lie algebras, cohomology and some applications in physics,Camb. Univ. Press(1995)

    1995

  26. [26]

    D. A. Easson, G. Herczeg, T. Manton and M. Pezzelle,Isometries and the double copy, JHEP09(2023), 162 [arXiv:2306.13687]

    work page arXiv 2023

  27. [27]

    Gieres, J

    F. Gieres, J. M. Grimstrup, H. Nieder, T. Pisar and M. Schweda,Symmetries of topological field theories in the BV framework,Phys. Rev. D66(2002) [arXiv:hep-th/0111258]

    work page arXiv 2002

  28. [28]

    Godazgar, C

    M. Godazgar, C. N. Pope, A. Saha and H. Zhang,BRST symmetry and the convolutional double copy,JHEP11(2022), 038 [arXiv:2208.06903]

    work page arXiv 2022

  29. [29]

    Gonzo and C

    R. Gonzo and C. Shi,Geodesics from classical double copy,Phys. Rev. D104(2021), no.10, 105012 [arXiv:2109.01072]

    work page arXiv 2021

  30. [30]

    Henneaux and C

    M. Henneaux and C. Teitelboim,Quantization of Gauge Systems,Princeton University Press (1992), [https://doi.org/10.2307/j.ctv10crg0r]

    work page doi:10.2307/j.ctv10crg0r 1992

  31. [31]

    Holton,Limits of symmetry in Schwarzschild: CKVs and BRST triviality in the Kerr-Schild double copy(2025) [arXiv:2509.25801]

    B. Holton,Limits of symmetry in Schwarzschild: CKVs and BRST triviality in the Kerr-Schild double copy(2025) [arXiv:2509.25801]

    work page arXiv 2025

  32. [32]

    Holton,Residual symmetries and BRST cohomology of Schwarzschild in the Kerr-Schild double copy(2025) [arXiv:2509.24112]

    B. Holton,Residual symmetries and BRST cohomology of Schwarzschild in the Kerr-Schild double copy(2025) [arXiv:2509.24112]

    work page arXiv 2025

  33. [33]

    Liang and S

    Q. Liang and S. Nagy,Convolutional double copy in (anti) de Sitter space,JHEP04(1992), 139 [arXiv:2311.14319] – 26 –

    work page arXiv 1992

  34. [34]

    Lionetti,Asymptotic symmetries and soft theorems in higher-dimensional gravity,EPJ Web Conf.270(2022), 00034 [arXiv:2209.10889]

    S. Lionetti,Asymptotic symmetries and soft theorems in higher-dimensional gravity,EPJ Web Conf.270(2022), 00034 [arXiv:2209.10889]

    work page arXiv 2022

  35. [35]

    A. Luna, S. Nagy and C. White,The convolutional double copy: a case study with a point, JHEP09(2020), 062 [arXiv:2004.11254]

    work page arXiv 2020

  36. [36]

    McLoughlin, A

    T. McLoughlin, A. Puhm and A. M. Raclariu,The SAGEX review on scattering amplitudes chapter 11: soft theorems and celestial amplitudes,J. Phys. A,55(2022), no.44, 443012 [arXiv:2203.13022]

    work page arXiv 2022

  37. [37]

    Monteiro, D

    R. Monteiro, D. O’Connell and C. D. White,Black holes and the double copy,JHEP12 (2014), 056 [arXiv:1410.0239]

    work page arXiv 2014

  38. [38]

    Monteiro, D

    R. Monteiro, D. O’Connell and C. D. White,Gravity as a double copy of gauge theory: from amplitudes to black holes,Int. J. Mod. Phys. D24(2015), no.09, 1542008 [doi:10.1142/S0218271815420080]

    work page doi:10.1142/s0218271815420080 2015

  39. [39]

    Obata,Conformal transformations of Riemannian manifolds,J

    M. Obata,Conformal transformations of Riemannian manifolds,J. Differential Geom.04 (1970), 311-333

    1970

  40. [40]

    A. K. Ridgway and M. B. Wise,Static spherically symmetric Kerr-Schild metrics and implications for the classical double copy,Phys. Rev. D2016(1970), no.4, 044023 [arXiv:1512.02243]

    work page arXiv 1970

  41. [41]

    Schottenloher,A Mathematical Introduction to Conformal Field Theory,Lecture Notes in Physics, Springer-Verlag.759(2008)

    M. Schottenloher,A Mathematical Introduction to Conformal Field Theory,Lecture Notes in Physics, Springer-Verlag.759(2008)

    2008

  42. [42]

    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) [arXiv:1703.05448]

    work page Pith review arXiv 2018

  43. [43]

    Weyl,Raum, Zeit, Materie

    H. Weyl,Raum, Zeit, Materie. Lectures on General Relativity,Berlin: Springer.(1993)

    1993

  44. [44]

    R. M. Wald,General Relativity,Chicago Univ. Pr.(1984) [doi:10.7208/chicago/9780226870373.001.0001]

    work page doi:10.7208/chicago/9780226870373.001.0001 1984

  45. [45]

    Zucchini,The Gauging of BV algebras,J

    R. Zucchini,The Gauging of BV algebras,J. Geom. Phys.60(2010), 1860-1880 [arXiv:1001.0219] – 27 –

    work page arXiv 2010