Covariant variation and its applications

Jiang Long; Wen-Bin Liu

REVIEW 2 major objections 5 minor 73 references

Covariant variation and its applications

T0 review · 2 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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

Pith's one-line read A metric-compatible rewrite of the Lie derivative fails to close, and that failure is the electromagnetic helicity flux on null boundaries.

desk verdict Solid differential-geometry packaging of metric-compatible transport whose non-closure is the helicity/little-group rotation; algebra is clean, physical reach is real but scoped to cases with a symplectic form. read the letter →

arxiv 2607.28420 v1 pith:KOUQFDYK submitted 2026-07-30 hep-th gr-qc

Wen-Bin Liu , Jiang Long This is my paper · ORCID

classification hep-thgr-qc

keywords covariantvariationmetricLiederivativeKosmannhelicityfluxsuperrotationnullhypersurfacep-formanomalyofcommutator

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

Ordinary Lie transport does not keep a fixed metric, so it does not commute with raising indices or with the Hodge star. This paper defines a covariant variation that adds a compensating term built from the metric change, so the operator annihilates the metric, preserves contractions and Hodge duality, and still acts as a derivation on tensors. The price is that two such variations do not close: their commutator differs from the variation along the Lie bracket by an antisymmetric endomorphism called the anomaly. Along superrotations (and similar generators) at null hypersurfaces, that anomaly rotates radiative polarization and is realized by a helicity flux on the boundary phase space. The same construction extends to mixed spacetime–Lorentz fields via the Kosmann derivative, to multi-parameter families that include the affine connection and Lie derivative, and to tensor and p-form data in arbitrary dimension.

What carries the argument

Covariant variation ∆_X (equivalently the metric Lie derivative from the orthonormal-frame lift): the Lie derivative plus the slotwise action of the metric compensator S_X = (1/2)g^{-1} L_X g. Its anomaly A_{X,Y} = −[S_X, S_Y] is the object that becomes the helicity flux.

What would settle it

Compute the double commutator of two superrotation fluxes on the radiative Maxwell (or p-form) data at null infinity or a finite horizon and check whether the extra term equals the predicted helicity operator o(Y,Z) acting by the volume form (or so(d−2) rotation) on the angular indices; disagreement would falsify the identification.

Watch

Extended reading notes

Core claim

The operator ∆_X = L_X + ρ(S_X) = ∇_X + ρ(A_X), with S_X the symmetric part of ∇X and A_X its antisymmetric part, is metric-compatible on ordinary tensors, but [∆_X, ∆_Y] − ∆_[X,Y] equals ρ(−[S_X, S_Y]). On four-dimensional null boundaries that endomorphism is a helicity rotation of the radiative field, matching the action of the electromagnetic helicity flux obtained from the stress-tensor superrotation charge.

Load-bearing premise

A boundary symplectic structure, fall-offs, gauge conditions, and endpoint behaviour must exist so the kinematic anomaly can be turned into an actual Hamiltonian helicity flux; for general tensors that phase space is assumed rather than derived from a bulk action.

Editorial extensions

If this is right

Superrotation flux algebras on null boundaries must include a helicity (or little-group) extension generated by the anomaly of ∆.
The same anomaly supplies spin charges for massive fields at timelike infinity and candidate operators in AdS boundary reductions.
Total covariant variation preserves the vielbein for every vector field, so mixed spacetime–Lorentz fields (e.g. Rarita–Schwinger) can be varied without soldering mismatch.
In d = 2p + 2 the anomaly of a radiative p-form is an so(2p)-valued polarization rotation whose Hamiltonian is a weighted occupation-number difference of conjugate modes.

Reading between the lines

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

If the stress-tensor flux always generates covariant variation on free boundary data, many asymptotic symmetry algebras may be forced to carry classical helicity/spin central terms once metric variation is kept.
The topological origin of the helicity flux (Chern–Simons-type pullbacks) is likely boundary- and degree-dependent: only special (p, d) pairs recover a single duality generator, while generic h_AB needs extra background structure.
The two- and three-parameter derivation families offer a uniform language for comparing affine, Lie, Kosmann, and density-weighted boundary operators in other gauge theories.

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.

Desk Editor's Note

read the letter

The punchline is simple: they define ∆_X = L_X + ρ(S_X) = ∇_X + ρ(A_X), prove it preserves g, contractions and Hodge star, and show the commutator fails by the endomorphism A_{X,Y} = −[S_X,S_Y]. Along superrotations (and analogous null generators) that endomorphism is exactly a helicity or so(d−2) little-group rotation on the radiative data. That matching is explicit in 4d Maxwell and controlled for free p-forms.

What is actually new is the systematic reading of the anomaly as the source of the helicity flux, the total mixed-index operator £_X that kills the vielbein, and the two- and three-parameter families that put ∇, L and ∆ on one footing, with a clean curvature formula. The identification with the metric Lie derivative (Bourguignon–Gauduchon) and Kosmann is correctly cited; they are not reinventing those lifts, they are using them. The algebra in §§3–4 and Apps. B–C is careful: derivation properties, Jacobi/Bianchi identities, and the action on forms all check out by direct computation. Horizon and higher-d sections state their restrictions (div-free Y; fall-off plus transversality) rather than hiding them.

The soft spot is exactly the one already flagged: for a general Young-type tensor the boundary symplectic form is postulated, so the kinematic anomaly becomes a Hamiltonian flux only under that extra assumption. For Maxwell and free p-forms the symplectic structure is derived, so the claim is solid there. The stress-tensor → ∆ conjecture in the outlook is left open; that is honest, not a hole in the present results. Self-citation to their earlier flux papers is heavy but expected—this is the geometric reorganization of that series.

This is for people already working asymptotic symmetries, soft theorems or celestial holography who want a uniform language for metric-compatible transport and spin-type extensions. It deserves a serious referee. I would engage with the operator theory and the controlled cases; I would not treat the general-tensor flux as automatic.

Referee Report

2 major / 5 minor

Summary. The paper defines a metric-compatible covariant variation ∆_X = L_X + ρ(S_X) = ∇_X + ρ(A_X) on ordinary spacetime tensors, shows that it preserves the metric, contractions, volume form and Hodge duality, and derives the classical non-closure anomaly A_{X,Y} = −[S_X, S_Y]. It identifies ∆_X with the metric Lie derivative of Bourguignon–Gauduchon, compares it with the Kosmann derivative, introduces a total covariant variation £_X for mixed spacetime/Lorentz fields, and embeds all three standard operators (∇, L, ∆) in two- and three-parameter families of tensor derivations. The main physical claim is that the anomaly along superrotations (and analogous null generators) acts as a helicity or little-group rotation on radiative data and is realized by a helicity flux on the boundary phase space, recovering and extending the authors’ earlier 4d Maxwell, gravity, higher-spin and fermionic results, with further applications to finite horizons (div-free sector), general tensors in arbitrary dimensions, and p-forms in d = 2p + 2.

Significance. If the geometric identification and the anomaly formula hold—as the direct computations in §§3–4 and Apps. B–C indicate—the paper supplies a clean, representation-theoretic home for several previously ad-hoc compensating transformations used in asymptotic symmetry and flux-algebra literature. The explicit matching of [∆_Y, ∆_Z] to the electromagnetic helicity flux in 4d (Motivation and §5.1), the so(m)-valued little-group rotation for general tensors (§5.2), and the controlled p-form Hamiltonian from the free action (§5.3) are concrete strengths. The two- and three-parameter families and the metric-Lie unification are useful organizing results even without the flux applications. The work is incremental relative to the authors’ prior series, but the differential-geometric packaging is new and potentially reusable.

major comments (2)

[§5.2] §5.2, around Eqs. (5.48)–(5.52): for a general rank-k tensor of unspecified Young type the boundary symplectic form Ω_κ is postulated rather than derived from a bulk action or reduced phase space. The kinematic anomaly (5.46) is then promoted to a Hamiltonian helicity flux only under that extra assumption (plus fall-off ∆ = m/2, transversality, and endpoint conditions). The paper flags this as kinematical, but the central physical claim that “the anomaly gives a helicity flux” is then secure only for the cases where a symplectic structure is supplied (Maxwell, free p-form). Either derive or cite a reduced symplectic structure for at least one nontrivial mixed-symmetry example, or explicitly restrict the flux claim to theories with known radiative phase spaces.
[§5.1] §5.1, Eqs. (5.27)–(5.30) and the accompanying footnote: the finite-horizon analysis is restricted to divergence-free Y^A because the Gaussian-null asymptotic Killing vector cannot cancel ∇·Y while preserving the horizon gauge. The anomaly and candidate flux O_h are therefore obtained only on that subalgebra. This is stated clearly, but the abstract and introduction still advertise “an electromagnetic helicity flux o at null hypersurfaces” without the div-free caveat. The horizon claim should be qualified in the abstract/intro at the same level of precision used in §5.1, or an alternative bulk vector that restores the full algebra (and matches ∆) should be exhibited.

minor comments (5)

[Abstract, §1] Abstract and §1: “anomaly” is used as shorthand for classical non-closure. A one-sentence clarification that this is not a quantum chiral anomaly (despite the Adler–Bell–Jackiw citations) would prevent misreading.
[§4.4] §4.4, Eq. (4.28): the curvature formula for the two-parameter family is useful; a short remark that the flat points (1,±1) are the only ones for the faithful full-tensor representation (already proved later in the paragraph) could be moved earlier for readability.
[App. B.2] Appendix B.2: several inequivalent conventions for ∆_X Γ are listed without a preferred choice. Since the main text never needs a definite ∆Γ, a sentence stating that the main results are independent of this ambiguity would help.
Typos/notation: “T otal” and “F amilies” in §4 headings (stray spaces); occasional “ind=” for “in d=”; reference [29] is listed as 2607.07360 (same-month arXiv) and should be checked for final citation data.
[§6] §6, bullet on AdS_3 Proca: the operator (6.1) is intriguing but unsupported in the present text; either move to a companion note or give a one-line derivation sketch so the claim is falsifiable.

Circularity Check

2 steps flagged · score 3.0 of 10

Math core is self-contained; physical helicity story largely reorganizes the authors’ prior flux algebras by defining ∆ to match those actions.

renaming known result [Sec. 2.1, eqs. (2.15)–(2.17) and (2.19)–(2.25)]
"This inspires us to propose a covariant variation ∆_Y A_A = δ_Y A_A − 1/2 δ_Y γ_AB A^B ... which matches the action of superrotation flux on A_A [iF_Y, A_A] = ∆_Y A_A. ... The appearance of helicity flux in (2.19) is actually the consequence of non-closure of the covariant variation [∆_Y, ∆_Z] A_A = ∆_{[Y,Z]} A_A − ... which exactly agrees with the action of the emerging helicity flux on the Maxwell field −o(Y,Z) ε_AB A^B = [iO_{o(Y,Z)}, A_A]."

∆_Y is defined to equal the already-computed infinitesimal action of the superrotation flux F_Y on A_A. Once that identification is made, commutator associativity forces the operator anomaly on A to equal the action of whatever central/extension term sits in [F_Y, F_Z]. The helicity flux and the factor o(Y,Z) were already obtained in the authors’ prior flux algebra; calling them “the consequence of non-closure of ∆” renames that known extension in the new operator language rather than deriving it independently from the anomaly alone.
self citation load bearing [Sec. 2.1 eq. (2.19); applications relying on [8,9,11–14,40]]
"Moreover, a surprising helicity flux emerges from the quantum commutator of two superrotation fluxes [8] [F_Y, F_Z] = i F_{[Y,Z]} + i O_{o(Y,Z)}, where o(Y,Z) is defined as o(Y,Z) = 1/4 ε_AB Θ_AC(Y) Θ^C_B(Z)."

The load-bearing physical claim that a helicity flux appears in the superrotation algebra is justified by citation to the authors’ own prior paper [8] (and analogous results [11–14,40]). The present work supplies a clean geometric packaging (∆ and its anomaly) for that extension, but the existence and form of O_o as a physical flux are not re-derived from a new bulk computation here; they enter as self-cited input that the anomaly is then shown to match.

full rationale

The operator ∆_X := L_X + ρ(S_X) = ∇_X + ρ(A_X), its metric compatibility, the anomaly A_{X,Y} = −[S_X, S_Y], the two- and three-parameter families, and the metric-Lie/Kosmann identification are ordinary differential-geometric constructions derived from the definitions (Secs. 3–4, Apps. B–C). No parameter is fitted to data, and no external uniqueness theorem is imported to force the choice. Circularity is limited to the physical narrative in Sec. 2 (and similarly Sec. 5.1): ∆ is introduced so that [iF_Y, A_A] = ∆_Y A_A matches the superrotation flux action already computed in the authors’ prior work, after which the statement that the helicity extension in [F_Y, F_Z] is the anomaly of ∆ is essentially the representation property of that same action—not an independent first-principles derivation of the flux. Sec. 5.2–5.3 then use the kinematic anomaly plus a (sometimes postulated) symplectic form to build helicity Hamiltonians; that is assumption-heavy but not circular. Overall this is mild reorganization via self-citation, not a by-construction fake prediction. Score 3.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central operator claim rests on standard pseudo-Riemannian geometry plus the known metric/orthonormal-frame lift. Physical flux claims add asymptotic fall-offs, a boundary symplectic form, and (for general tensors) an assumed radiative phase space not derived from a universal action.

assumptions (5)

standard math Levi-Civita connection and standard Lie derivative on tensor algebras over a pseudo-Riemannian manifold (M,g).
Used throughout to define S_X = (1/2)g^{-1} L_X g and A_X = ∇_{[μ}X_{ν]}.
domain assumption Metric Lie derivative / orthonormal-frame lift of Bourguignon–Gauduchon (and reductive lifts) exists and acts as ∆_X on ordinary tensors and K_X on Lorentz/spinor fields.
Invoked in §4.3 to geometrize ∆ and K; cited as prior geometry rather than re-proved in full.
domain assumption Asymptotic superrotation (or Gaussian-null horizon) vector fields and radiative fall-offs/transversality for boundary fields.
§2 and §5; needed to reduce bulk ∆_ξ to boundary operators and close angular transformation laws.
domain assumption A boundary symplectic form exists (Maxwell/p-form from free actions; general tensors postulated as Ω_κ ∝ ∫ δT ∧ δṪ) with suitable u-endpoint conditions so anomalies integrate to Hamiltonian fluxes.
§5.1–5.3; without it the anomaly remains kinematic.
ad hoc to paper For finite horizons, restrict to divergence-free Y^A so bulk generators match the intrinsic covariant variation without an extra conformal weight term.
§5.1 explicitly notes no choice of f makes ξ divergence-free when ∇·Y ≠ 0 in Gaussian null gauge; subalgebra restriction is imposed by hand.

invented entities (3)

Covariant variation ∆_X independent evidence
purpose: Metric-compatible derivation on ordinary tensors combining L_X with ρ(S_X).
Named and axiomatized here as the working operator; coincides with the known metric Lie derivative on ordinary tensors.
Total covariant variation £_X
purpose: Joint spacetime+Lorentz metric-compatible operator on mixed bundles, preserving the vielbein.
Defined in §4.2 as K_X + ρ(S_X) = ∆_X + ρ_L(λ_X); packaging appears new even if ingredients are standard.
Anomaly endomorphism A_{X,Y} = −[S_X,S_Y] independent evidence
purpose: Measures non-closure of ∆ (and K, £) and supplies helicity/little-group rotations.
Classical curvature of the metric lift; physical identification with fluxes is the paper’s application focus.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Covariant variation and its applications." pith.science (2026). https://pith.science/paper/KOUQFDYK

@misc{pith2026260728420,
  author       = {Pith},
  title        = {Pith review of: Covariant variation and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOUQFDYK}},
  note         = {Machine review of arXiv:2607.28420}
}

abstract

We define a covariant variation of tensor fields by combining its Lie derivative with the metric variation. This operator preserves the metric, contractions, and Hodge duality, but its commutator is not closed due to an anomaly. We derive its algebraic and geometric properties, and compare it with the Kosmann derivative. Combining the covariant variation with Kosmann derivative gives total covariant variation for the fields with both spacetime and Lorentz structure, all of which belong to the metric Lie derivative. Moreover, we introduce families of extended operators which contain the affine connection, Lie derivative, and covariant variation. From the anomaly of the covariant variation along the superrotation, an electromagnetic helicity flux appears at null hypersurfaces in four dimensions. We also apply the covariant variation and its anomaly to tensor fields in arbitrary spacetime dimensions, and especially focus on the $p$-forms in $d=2p+2$ dimensions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 44 linked inside Pith

[1]

Calkin,An Invariance Property of the Free Electromagnetic Field,American Journal of Physics33(1965) 958

M.G. Calkin,An Invariance Property of the Free Electromagnetic Field,American Journal of Physics33(1965) 958

1965
[2]

Deser and C

S. Deser and C. Teitelboim,Duality Transformations of Abelian and Nonabelian Gauge Fields,Phys. Rev. D13(1976) 1592

1976
[3]

Afanasev and Y.P

G.N. Afanasev and Y.P. Stepanovsky,The Helicity of the free electromagnetic field and its physical meaning,Nuovo Cim. A109(1996) 271

1996
[4]

Fernandez-Corbaton, X

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M.L. Juan and G. Molina-Terriza,Electromagnetic Duality Symmetry and Helicity Conservation for the Macroscopic Maxwell’s Equations,Phys. Rev. Lett.111(2013) 060401 [1206.0868]

arXiv 2013
[5]

Bliokh, A.Y

K.Y. Bliokh, A.Y. Bekshaev and F. Nori,Dual electromagnetism: Helicity, spin, momentum, and angular momentum,New J. Phys.15(2013) 033026 [1208.4523]. – 42 –

arXiv 2013
[6]

Elbistan, C

M. Elbistan, C. Duval, P.A. Horvathy and P.M. Zhang,Duality and helicity: a symplectic viewpoint,Phys. Lett. B761(2016) 265 [1608.01131]

arXiv 2016
[7]

Agullo, A

I. Agullo, A. del Rio and J. Navarro-Salas,Classical and quantum aspects of electric-magnetic duality rotations in curved spacetimes,Phys. Rev. D98(2018) 125001 [1810.08085]

arXiv 2018
[8]

Liu and J

W.-B. Liu and J. Long,Symmetry group at future null infinity II: Vector theory,JHEP07 (2023) 152 [2304.08347]

arXiv 2023

Show all 73 references

[9]

W.-B. Liu, J. Long and X.-H. Zhou,Electromagnetic helicity flux operators in higher dimensions,JHEP04(2025) 026 [2407.20077]

2025 arXiv
[10]

Henneaux and C

M. Henneaux and C. Teitelboim,Duality in linearized gravity,Phys. Rev. D71(2005) 024018 [gr-qc/0408101]

2005 arXiv
[11]

Liu and J

W.-B. Liu and J. Long,Symmetry group at future null infinity III: Gravitational theory, JHEP10(2023) 117 [2307.01068]

2023 arXiv
[12]

Liu and J

W.-B. Liu and J. Long,Holographic dictionary from bulk reduction,Phys. Rev. D109(2024) L061901 [2401.11223]

2024 arXiv
[13]

W.-B. Liu, J. Long and X.-H. Zhou,Quantum flux operators in higher spin theories,Phys. Rev. D109(2024) 086012 [2311.11361]

2024 arXiv
[14]

Guo, W.-B

S.-M. Guo, W.-B. Liu and J. Long,Quantum flux operators in the fermionic theory and their supersymmetric extension,JHEP03(2025) 205 [2412.20829]

2025 arXiv
[15]

Long and R.-Z

J. Long and R.-Z. Yu,Gravitational helicity flux density from two-body systems,Class. Quant. Grav.42(2025) 045005 [2403.18627]

2025 arXiv
[16]

Z.-Y. Heng, J. Long, R.-Z. Yu and X.-H. Zhou,Electromagnetic helicity flux density for radiative systems,Class. Quant. Grav.43(2026) 015016 [2507.14966]

2026
[17]

Seraj and B

A. Seraj and B. Oblak,Precession Caused by Gravitational Waves,Phys. Rev. Lett.129 (2022) 061101 [2203.16216]

2022 arXiv
[18]

Oblak and A

B. Oblak and A. Seraj,Orientation memory of magnetic dipoles,Phys. Rev. D109(2024) 044037 [2304.12348]

2024 arXiv
[19]

Maleknejad,Photon chiral memory effect stored on celestial sphere,JHEP06(2023) 193 [2304.05381]

A. Maleknejad,Photon chiral memory effect stored on celestial sphere,JHEP06(2023) 193 [2304.05381]

2023 arXiv
[20]

Nieh and M.L

H.T. Nieh and M.L. Yan,An Identity in Riemann-cartan Geometry,J. Math. Phys.23 (1982) 373

1982
[21]

Nieh and M.L

H.T. Nieh and M.L. Yan,Quantized Dirac Field in Curved Riemann-cartan Background. 1. Symmetry Properties, Green ’s Function,Annals Phys.138(1982) 237

1982
[22]

Nieh and C.N

H.T. Nieh and C.N. Yang,A torsional topological invariant,Int. J. Mod. Phys. A22(2007) 5237

2007
[23]

Long and X.-H

J. Long and X.-H. Zhou,Reduction of topological invariants on null hypersurfaces,JHEP01 (2026) 116 [2509.06073]

2026
[24]

Adler,Axial vector vertex in spinor electrodynamics,Phys

S.L. Adler,Axial vector vertex in spinor electrodynamics,Phys. Rev.177(1969) 2426

1969
[25]

Bell and R

J.S. Bell and R. Jackiw,A PCAC puzzle:π 0 →γγin theσmodel,Nuovo Cim. A60(1969) 47

1969
[26]

Bourguignon and P

J.P. Bourguignon and P. Gauduchon,Spinors, Dirac operators and metric variations. (In French),Commun. Math. Phys.144(1992) 581. – 43 –

1992
[27]

Godina and P

M. Godina and P. Matteucci,Reductive G structures and Lie derivatives,J. Geom. Phys.47 (2003) 66 [math/0201235]

2003 arXiv
[28]

Giotopoulos,Covariant Lie derivatives and (Super-)gravity,Lett

G. Giotopoulos,Covariant Lie derivatives and (Super-)gravity,Lett. Math. Phys.116(2026) 60 [2507.00140]

2026
[29]

ˇDalak and F

F. ˇDalak and F. Valach,On the generalised Lie derivative of (s)pinor fields,2607.07360

arXiv
[30]

Lichnerowicz,Spineurs harmoniques,C

A. Lichnerowicz,Spineurs harmoniques,C. R. Acad. Sci. Paris257(1963) 7

1963
[31]

Kosmann,D´ eriv´ ees de Lie des spineurs,Annali di Matematica Pura ed Applicata91 (1971) 317

Y. Kosmann,D´ eriv´ ees de Lie des spineurs,Annali di Matematica Pura ed Applicata91 (1971) 317

1971
[32]

Jackiw and N.S

R. Jackiw and N.S. Manton,Symmetries and Conservation Laws in Gauge Theories,Annals Phys.127(1980) 257

1980
[33]

Ortin,A Note on Lie-Lorentz derivatives,Class

T. Ortin,A Note on Lie-Lorentz derivatives,Class. Quant. Grav.19(2002) L143 [hep-th/0206159]

2002 arXiv
[34]

Obukhov and G.F

Y.N. Obukhov and G.F. Rubilar,Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism symmetry,Phys. Rev. D74(2006) 064002 [gr-qc/0608064]

2006 arXiv
[35]

Fatibene and M

L. Fatibene and M. Francaviglia,General theory of lie derivatives for lorentz tensors, 0904.0258

arXiv
[36]

Jacobson and A

T. Jacobson and A. Mohd,Black hole entropy and Lorentz-diffeomorphism Noether charge, Phys. Rev. D92(2015) 124010 [1507.01054]

2015 arXiv
[37]

Prabhu,The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class

K. Prabhu,The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class. Quant. Grav.34(2017) 035011 [1511.00388]

2017 arXiv
[38]

Aneesh, S

P.B. Aneesh, S. Chakraborty, S.J. Hoque and A. Virmani,First law of black hole mechanics with fermions,Class. Quant. Grav.37(2020) 205014 [2004.10215]

2020 arXiv
[39]

Elgood, P

Z. Elgood, P. Meessen and T. Ortin,The first law of black hole mechanics in the Einstein-Maxwell theory revisited,JHEP09(2020) 026 [2006.02792]

2020 arXiv
[40]

Liu and J

W.-B. Liu and J. Long,Extrapolating the massive fields to future timelike infinity,JHEP12 (2025) 042 [2508.15619]

2025
[41]

Campiglia and A

M. Campiglia and A. Laddha,Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D90(2014) 124028 [1408.2228]

2014 arXiv
[42]

Campiglia and A

M. Campiglia and A. Laddha,New symmetries for the Gravitational S-matrix,JHEP04 (2015) 076 [1502.02318]

2015 arXiv
[43]

Comp` ere, A

G. Comp` ere, A. Fiorucci and R. Ruzziconi,Superboost transitions, refraction memory and super-Lorentz charge algebra,JHEP11(2018) 200 [1810.00377]

2018 arXiv
[44]

Liu and J

W.-B. Liu and J. Long,Symmetry group at future null infinity: Scalar theory,Phys. Rev. D 107(2023) 126002 [2210.00516]

2023 arXiv
[45]

O. Hohm, C. Hull and B. Zwiebach,Generalized metric formulation of double field theory, JHEP08(2010) 008 [1006.4823]

2010 arXiv
[46]

Aldazabal, D

G. Aldazabal, D. Marques and C. Nunez,Double Field Theory: A Pedagogical Review,Class. Quant. Grav.30(2013) 163001 [1305.1907]

2013 arXiv
[47]

Penrose and W

R. Penrose and W. Rindler,SPINORS AND SPACE-TIME. VOL. 2: SPINOR AND TWISTOR METHODS IN SPACE-TIME GEOMETRY, Cambridge Monographs on Mathematical Physics, Cambridge University Press (4, 1988), 10.1017/CBO9780511524486. – 44 –

1988 doi
[48]

Habermann,The graded algebra and the derivative of spinor fields related to the twistor equation,Journal of Geometry and Physics18(1996) 131

K. Habermann,The graded algebra and the derivative of spinor fields related to the twistor equation,Journal of Geometry and Physics18(1996) 131

1996
[49]

Godina and P

M. Godina and P. Matteucci,The Lie derivative of spinor fields: Theory and applications, Int. J. Geom. Meth. Mod. Phys.2(2005) 159

2005
[50]

Le˜ ao, W.A

R.F. Le˜ ao, W.A. Rodrigues and S.A. Wainer,Concept of Lie Derivative of Spinor Fields A Geometric Motivated Approach,Adv. Appl. Clifford Algebras27(2017) 209

2017
[51]

Helfer,Spinor Lie derivatives and Fermion stress–energies,Proc

A.D. Helfer,Spinor Lie derivatives and Fermion stress–energies,Proc. Roy. Soc. Lond. A 472(2016) 20150757 [1602.00632]

2016 arXiv
[52]

Grana, R

M. Grana, R. Minasian, M. Petrini and D. Waldram,T-duality, Generalized Geometry and Non-Geometric Backgrounds,JHEP04(2009) 075 [0807.4527]

2009 arXiv
[53]

Angus, K

S. Angus, K. Cho and J.-H. Park,Einstein Double Field Equations,Eur. Phys. J. C78 (2018) 500 [1804.00964]

2018 arXiv
[54]

Ballesteros, E

R. Ballesteros, E. Lescano and J.A. Rodriguez,Kosmann derivative and momentum maps from a duality covariant framework,JHEP05(2026) 310 [2602.11267]

2026
[55]

Tamburino and J.H

L.A. Tamburino and J.H. Winicour,Gravitational Fields in Finite and Conformal Bondi Frames,Phys. Rev.150(1966) 1039

1966
[56]

Booth,Spacetime near isolated and dynamical trapping horizons,Phys

I. Booth,Spacetime near isolated and dynamical trapping horizons,Phys. Rev. D87(2013) 024008 [1207.6955]

2013 arXiv
[57]

Donnay, G

L. Donnay, G. Giribet, H.A. Gonzalez and M. Pino,Supertranslations and Superrotations at the Black Hole Horizon,Phys. Rev. Lett.116(2016) 091101 [1511.08687]

2016 arXiv
[58]

Donnay, G

L. Donnay, G. Giribet, H.A. Gonz´ alez and M. Pino,Extended Symmetries at the Black Hole Horizon,JHEP09(2016) 100 [1607.05703]

2016 arXiv
[59]

Labastida,Massless Particles in Arbitrary Representations of the Lorentz Group, Nucl

J.M.F. Labastida,Massless Particles in Arbitrary Representations of the Lorentz Group, Nucl. Phys. B322(1989) 185

1989
[60]

de Medeiros and C

P. de Medeiros and C. Hull,Geometric second order field equations for general tensor gauge fields,JHEP05(2003) 019 [hep-th/0303036]

2003 arXiv
[61]

Bekaert and N

X. Bekaert and N. Boulanger,Tensor gauge fields in arbitrary representations of GL(D,R). II. Quadratic actions,Commun. Math. Phys.271(2007) 723 [hep-th/0606198]

2007 arXiv
[62]

Campoleoni, D

A. Campoleoni, D. Francia, J. Mourad and A. Sagnotti,Unconstrained Higher Spins of Mixed Symmetry. I. Bose Fields,Nucl. Phys. B815(2009) 289 [0810.4350]

2009 arXiv
[63]

Buchbinder and A

I.L. Buchbinder and A. Reshetnyak,General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. I. Bosonic fields,Nucl. Phys. B862(2012) 270 [1110.5044]

2012 arXiv
[64]

Campoleoni, D

A. Campoleoni, D. Francia and C. Heissenberg,Asymptotic Charges at Null Infinity in Any Dimension,Universe4(2018) 47 [1712.09591]

2018 arXiv
[65]

Campoleoni, A

A. Campoleoni, A. Delfante, D. Francia and C. Heissenberg,Finite actions and asymptotic charges at null infinity for any spin,Phys. Lett. B870(2025) 139908 [2507.19310]

2025
[66]

Deser, A

S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim,Duality, selfduality, sources and charge quantization in Abelian N form theories,Phys. Lett. B400(1997) 80 [hep-th/9702184]

1997 arXiv
[67]

Bandos, K

I. Bandos, K. Lechner, D. Sorokin and P.K. Townsend,On p-form gauge theories and their conformal limits,JHEP03(2021) 022 [2012.09286]. – 45 –

2021 arXiv
[68]

Oliveri and S

R. Oliveri and S. Speziale,A note on dual gravitational charges,JHEP12(2020) 079 [2010.01111]

2020 arXiv
[69]

Yano,The Theory of Lie Derivatives and Its Applications, vol

K. Yano,The Theory of Lie Derivatives and Its Applications, vol. 3 ofBibliotheca Mathematica, North-Holland Publishing Co., Amsterdam (1957)

1957
[70]

Nomizu and K

K. Nomizu and K. Yano,On infinitesimal transformations preserving the curvature tensor field and its covariant differentials,Annales de l’Institut Fourier14(1964) 227

1964
[71]

Kobayashi and K

S. Kobayashi and K. Nomizu,Foundations of Differential Geometry. Vol. I, no. 15 in Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, a division of John Wiley & Sons, New York–London (1963)

1963
[72]

Kol´ aˇ r, P.W

I. Kol´ aˇ r, P.W. Michor and J. Slov´ ak,Natural Operations in Differential Geometry, Springer-Verlag, Berlin–Heidelberg (1993), 10.1007/978-3-662-02950-3

1993 doi
[73]

Vignolo and L

S. Vignolo and L. Fabbri,On the lie derivative of an affine connection and its geometrical meaning,International Journal of Geometric Methods in Modern Physics22(2025) 2550085. – 46 –

2025

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [1]

Calkin,An Invariance Property of the Free Electromagnetic Field,American Journal of Physics33(1965) 958

M.G. Calkin,An Invariance Property of the Free Electromagnetic Field,American Journal of Physics33(1965) 958

1965

[2] [2]

Deser and C

S. Deser and C. Teitelboim,Duality Transformations of Abelian and Nonabelian Gauge Fields,Phys. Rev. D13(1976) 1592

1976

[3] [3]

Afanasev and Y.P

G.N. Afanasev and Y.P. Stepanovsky,The Helicity of the free electromagnetic field and its physical meaning,Nuovo Cim. A109(1996) 271

1996

[4] [4]

Fernandez-Corbaton, X

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M.L. Juan and G. Molina-Terriza,Electromagnetic Duality Symmetry and Helicity Conservation for the Macroscopic Maxwell’s Equations,Phys. Rev. Lett.111(2013) 060401 [1206.0868]

arXiv 2013

[5] [5]

Bliokh, A.Y

K.Y. Bliokh, A.Y. Bekshaev and F. Nori,Dual electromagnetism: Helicity, spin, momentum, and angular momentum,New J. Phys.15(2013) 033026 [1208.4523]. – 42 –

arXiv 2013

[6] [6]

Elbistan, C

M. Elbistan, C. Duval, P.A. Horvathy and P.M. Zhang,Duality and helicity: a symplectic viewpoint,Phys. Lett. B761(2016) 265 [1608.01131]

arXiv 2016

[7] [7]

Agullo, A

I. Agullo, A. del Rio and J. Navarro-Salas,Classical and quantum aspects of electric-magnetic duality rotations in curved spacetimes,Phys. Rev. D98(2018) 125001 [1810.08085]

arXiv 2018

[8] [8]

Liu and J

W.-B. Liu and J. Long,Symmetry group at future null infinity II: Vector theory,JHEP07 (2023) 152 [2304.08347]

arXiv 2023