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Field-dependent diffeomorphisms and the transformation of surface charges between gauges

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In (A)dS3 gravity, a boundary charge can be switched on or off by changing gauge.

desk verdict A solid, worth-reading paper that identifies Weyl charges as 'kinematical' via an explicit BS/FG gauge comparison; the one real gap is an asserted all-orders check for the symplectic potential transformation. read the letter →

arxiv 2412.14992 v1 pith:6C5CCJHA submitted 2024-12-19 hep-th gr-qc

Luca Ciambelli , Marc Geiller This is my paper · ORCID
classification hep-thgr-qc
keywords kinematicalchargesdynamicalWeylchargeBondi-SachsgaugeFefferman-Grahamfield-dependentdiffeomorphismssurfaceAdS3gravity
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

Surface charges attached to asymptotic symmetries are usually classified as physical or pure gauge depending on whether they vanish. This paper argues that the physical class should be refined into dynamical and kinematical charges. The test case is (A)dS3 gravity with boundary Weyl rescalings: the Weyl charge vanishes in Bondi–Sachs gauge but not in Fefferman–Graham gauge for the Einstein–Hilbert Lagrangian, while for the metric Chern–Simons Lagrangian the pattern is reversed. The paper concludes that the Weyl charge is kinematical because it obeys no flux-balance law, arises from a corner term in the symplectic structure, and can be toggled on or off by a field-dependent diffeomorphism between gauges, unlike mass and angular momentum.

What carries the argument

The central object is the field-dependent diffeomorphism from Bondi–Sachs to Fefferman–Graham coordinates, constructed perturbatively in the radial coordinate $\rho$ through $u = t - \ell^2 e^{-\phi}\rho + O(\rho^2)$, $r = -\rho^{-1} + O(1)$, and $\theta = \varphi + O(\rho^2)$. Because the change of coordinates depends on the Weyl factor $\phi$, field-space variations of the old coordinates do not vanish and commutators such as $[\tilde\partial_\mu,\delta]$ fail to vanish. This forces corrected transformation laws: asymptotic Killing vectors transform as $\xi^\mu = \tilde J^\mu_\alpha \tilde\xi^\alpha + \tilde J^\mu_\alpha \mathcal L_\xi \tilde x^\alpha$, and the pre-symplectic potential transforms as $\Theta^\mu = |\tilde J|\tilde J^\mu_\alpha \tilde\Theta^\alpha - \tilde J^\mu_\alpha A^\alpha$, with the correction $A$ coming entirely from variations of the field-dependent coordinates. This non-tensorial mechanism is what carries the Weyl symmetry from one gauge to the other and turns the Weyl charge on or off.

What would settle it

Compute the order-$\rho^2$ contributions to both $|\tilde J|\tilde J^\rho_\alpha \tilde\Theta^\alpha_\mathrm{BS}$ and the correction $\tilde J^\rho_\alpha A^\alpha$ in the transformation formula (4.31) for the explicit diffeomorphism (4.15); if these terms do not cancel at that order and all higher orders, the claimed exact mapping between the two phase spaces, and hence the claim that the diffeomorphism toggles the Weyl charge, would fail.

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Extended reading notes

Core claim

The central claim is that a surface charge can be genuinely nonzero in one coordinate gauge and zero in another, and that this is a signal of a distinct kind of physical charge, called kinematical. Concretely, in Bondi–Sachs gauge the Einstein–Hilbert charge is $Q_\xi = fM + gN$, while in Fefferman–Graham gauge it is $Q_\xi = fM + gN + \ell^2(\dot{w}\phi - w\dot{\phi})$; for the metric Chern–Simons Lagrangian the Weyl contribution appears in Bondi–Sachs as $\frac{1}{\ell^2}fN + gM + w\phi'$ and disappears in Fefferman–Graham. The field-dependent diffeomorphism that maps between the two gauges is large in the sense that it activates or deactivates the Weyl charge, and it acts non-tensorially on variational forms because the coordinate change depends on the fields themselves.

Load-bearing premise

The central claim rests on the assertion, made without displaying the full calculation, that the perturbative Bondi–Sachs to Fefferman–Graham diffeomorphism maps the metric, residual Killing vectors, and pre-symplectic potential to all orders in $\rho$; the explicit check shown stops at order $\rho^0$, so the exactness of the charge toggle depends on an unshown all-orders cancellation.

Editorial extensions

If this is right

  • Mass and angular momentum are gauge-stable charges: the terms $fM + gN$ survive in both gauges and both Lagrangians, while the Weyl charge can be removed or restored by changing gauge.
  • The Weyl charge in Einstein–Hilbert gravity vanishes in Bondi–Sachs gauge and equals $\ell^2(\dot{w}\phi - w\dot{\phi})$ in Fefferman–Graham gauge, so a charge that is physical by the usual zero-vs-nonzero test can fail that test simply by choosing coordinates.
  • For the metric Chern–Simons Lagrangian the same toggle goes the other way, showing that two bulk formulations of the same on-shell metric can assign different kinematical charges to the same gauge.
  • The Bondi–Sachs to Fefferman–Graham map is a large diffeomorphism between gauges, not a residual symmetry within a gauge, because it changes the charge content of the solution space.
  • Field-dependent diffeomorphisms are not automorphisms of the variational bicomplex, so symplectic potentials and charge aspects transform with correction terms and behave like connections rather than tensors.

Reading between the lines

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

  • Beyond the paper, the three proposed markers of a kinematical charge—gauge-dependence, absence of a flux-balance law, and origin as a corner term—may be facets of one property: any charge removable by a corner ambiguity should be expected to toggle under a field-dependent gauge change.
  • Beyond the paper, one can test this criterion in four-dimensional partial Bondi gauge by computing the $\sqrt{q}$, $C$, and $D$ charges in two different boundary gauges; the paper names these as candidate kinematical charges, but the explicit two-gauge comparison is not performed there.
  • Beyond the paper, if field-space forms transform as connections, the Iyer–Wald charge aspect itself must obey a non-tensorial transformation law under the Bondi–Sachs to Fefferman–Graham diffeomorphism; extracting that law explicitly would give a sharper diagnostic of when a charge is kinematical.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper proposes a refinement of the usual classification of asymptotic charges as physical or pure gauge, distinguishing "dynamical" charges (those tied to flux-balance laws) from "kinematical" charges (those that can be turned off by a change of gauge and that arise from corner terms). The concrete test case is (A)dS3 gravity with boundary Weyl rescalings. In Bondi-Sachs gauge the authors compute the Iyer-Wald charges for the Einstein-Hilbert and metric Chern-Simons Lagrangians, finding respectively Q_xi = fM + gN and Q~_xi = (1/ell^2)fN + gM + w phi'; in Fefferman-Graham gauge the Weyl term moves from the Chern-Simons charge to the Einstein-Hilbert charge. They show that the Weyl field has no flux-balance law and that its charge contribution is a corner term, and they construct a field-dependent diffeomorphism between the two gauges. They derive non-tensorial transformation laws for asymptotic Killing vectors and for the pre-symplectic potential, and argue that this diffeomorphism is large and toggles the Weyl charge on or off.

Significance. If the all-orders statements in Section 4 are substantiated, this is a solid and interesting contribution to the asymptotic-charge literature. The direct computations in Sections 2 and 3 are the main strength: the charge formulas (2.15) and (3.19), the symplectic currents (2.8) and (3.12), and the flux laws (2.18) and (3.22) are explicit, and the paper is careful about integrability and about the relation between Iyer-Wald and Barnich-Brandt charges. The proposed dynamical/kinematical distinction, supported by the absence of a flux-balance law and the corner-term origin of the Weyl charge, is a plausible and useful organizing principle even if it is not yet a theorem. The technical observation that field-dependent diffeomorphisms make field-space forms transform as connections rather than tensors, with the explicit correction term (4.30), is a valuable point for future work.

major comments (4)
  1. [4.2.3, Eqs. (4.31)-(4.33)] The statement that (4.31) "yields the correct result to all orders in rho and for all the components of the potential" is not supported by the displayed calculation. The verification shown in (4.32)-(4.33) cancels only the O(rho^{-1}) pieces and does not display the O(1), O(rho), or higher coefficients, nor the angular components. Because the exactness of the phase-space mapping and the interpretation of the diffeomorphism as toggling the Weyl charge depend on this formula rather than only on the direct charge computations of Sections 2 and 3, this is a load-bearing omission. Please provide the all-orders verification, or an induction argument, or explicitly reduce the claim to the order needed for the boundary charges.
  2. [4.2.1, Eqs. (4.12)-(4.14)] The recursive construction of the perturbative diffeomorphism is asserted rather than demonstrated. After imposing g(3)_ab = 0 at (4.13), the text states that the subleading terms in rho can be made to vanish by tuning (R_n, T_n, F_n) for n >= 6, but no recursive algorithm or general argument is given. This matters because the later transformations of the vector field and of the symplectic potential use the full expansion (4.15). Please provide at least a schematic induction step, or state explicitly that the construction is a formal asymptotic expansion and specify which orders are needed for the charges.
  3. [4.2.2, Eqs. (4.18)-(4.22)] The vector-field transformation is checked only through O(rho). The conclusion that neglecting the field-dependent term in (4.18) would wrongly remove the Weyl symmetry is established at leading order, and that order is sufficient for the charge argument, but the sentence claiming that one can "explicitly verify" the expected result (3.13) is again an omitted check. Please display the higher-order terms or state clearly at which order the verification stops and why that order is sufficient.
  4. [Section 4 and abstract] The abstract says that the charge results "can also be derived" using the field-dependent diffeomorphism, but Section 4 transforms the metric, the asymptotic Killing vector, and the pre-symplectic potential only; the Iyer-Wald charge aspect (A.7) is not transported. The transformation of the charge aspect is mentioned in the conclusions as being "very intricate and lengthy," but it is not given. The toggle statement is therefore an inference from the direct computations of Sections 2 and 3 together with partial transformation data, not a complete derivation. Please either include the transformed charge aspect at the relevant order or reformulate the claim so that it matches what is actually shown.
minor comments (4)
  1. [Title page] The author affiliation contains a typo: "Fran ce" should read "France."
  2. [Eq. (4.3b)] The notation e^{phi_0 - phi} is written with unusual spacing as "e^{phi 0-phi}"; please clarify by proper superscript formatting.
  3. [Introduction, table] The summary table is useful, but the symbols f, g, w, M, N, and phi are not defined there; a cross-reference to Eqs. (2.9), (2.11), (3.13), and (3.15) would help the reader.
  4. [Section 4.2.2] In (4.19)-(4.22), the sign and ordering of the correction terms is clear in the final result, but the intermediate expressions would benefit from an explicit statement that h is always the original BS parameter before the redefinition (2.11), since the same letter is reused in (3.13) with a different meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: charges are direct Iyer-Wald evaluations; the kinematical/dynamical split is a proposed classification, not a self-referential derivation.

full rationale

The paper's central results are computed, not assumed. In §2 and §3 the charges (2.15) and (3.19) are obtained by substituting the explicit BS and FG solution spaces into the standard Iyer-Wald expressions (A.7)-(A.9); there is no fitted parameter, no quantity is defined as the target of a later 'prediction', and the gauge-dependence of the Weyl charge is an evaluated output. The 'kinematical' label is introduced as a proposed refinement of the concept of physical charge; the three properties (gauge-dependence, absence of flux-balance law, corner-term origin) are presented as characterizing features of the example and as criteria to be tested in future work (§5), not as an input that forces the computed charges. Section 4's field-dependent diffeomorphism is constructed order by order from the metric (§4.2.1), and the vector-field and symplectic-potential transformation laws (4.18) and (4.31) are derived from the Jacobian and its field-space variation; the displayed O(1) check (4.32)-(4.33) is a consistency test against the direct computation, not a fit. The paper's self-citations (e.g., [44], [106]) are used for technique attribution and context; the toggle conclusion does not rest on an unverified self-citation, since the perturbative map is exhibited explicitly. The unshown all-orders statement ('One can check that this yields the correct result to all orders in ρ') is a completeness/correctness gap, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim relies on the standard covariant phase space formalism, the identification of the EH solution space as a subspace of the metric CS solution space, and the imported field-dependent diffeomorphism transformation rules. The only genuinely chosen elements are the slicing redefinition and the corner symplectic potentials, both of which the paper exposes and which drive the kinematical classification.

free parameters (3)
  • Weyl symmetry parameter redefinition h -> w (eq. 2.11) = none (choice of parameterization)
    The field-dependent redefinition (2.11) is chosen so that the Weyl transformations act simply (2.12c) and the charges become integrable. It is not forced by the equations and affects which charges appear in the EH vs CS expressions.
  • Corner symplectic potentials ϑ (BS and FG) = none (choice of convention)
    The corner term ϑ = -1/2 φ' δφ (BS) or ϑ = ℓ² φ̇ δφ (FG) is chosen by hand so that Iξ(δϑ) reproduces the Weyl contribution to the charge. Adding or not adding this corner term to the symplectic structure moves the Weyl charge between gauges, which is the basis for the kinematical classification.
  • Weyl freedom φ0 in the BS to FG diffeomorphism (section 4.2.1) = 0 (set to zero in the main computation; can be turned on)
    The free function R1 = ℓ² e^{-φ0} in the perturbative diffeomorphism encodes the Weyl rescaling between the gauges. Turning on φ0 = φ maps the BS gauge with φ = 0 to an FG gauge with non-vanishing Weyl factor, demonstrating that the diffeomorphism is large and toggles the Weyl charge.
assumptions (4)
  • domain assumption Iyer-Wald charge formula (A.7) defines the surface charges.
    The paper computes all charges using the Iyer-Wald expressions (A.7); any other covariant phase space prescription could in principle assign different values to the corner terms.
  • domain assumption The solution space of EH AdS3 gravity is also a solution space of the metric CS Lagrangian.
    Used in sections 2 and 3 to compare EH and CS charges; relies on the Cotton tensor vanishing for constant-curvature metrics (footnote 4).
  • domain assumption The modified bracket (2.13) for field-dependent vector fields and the Compère-Long transformation formula (4.16) are valid.
    These tools from the asymptotic symmetry literature are imported without proof; they are essential for computing the algebra of residual symmetries and for mapping the Killing vector between gauges.
  • domain assumption New-gauge coordinates are field-independent, i.e., [∂α, δ] = 0 (section 4.2.3).
    Used to compute δJ in (4.28); if the FG coordinates themselves depended on the fields, the non-tensorial transformation law (4.31) would need further corrections.

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Pith. "Pith review of Field-dependent diffeomorphisms and the transformation of surface charges between gauges." pith.science (2026). https://pith.science/paper/6C5CCJHA

@misc{pith2026241214992,
  author       = {Pith},
  title        = {Pith review of: Field-dependent diffeomorphisms and the transformation of surface charges between gauges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6C5CCJHA}},
  note         = {Machine review of arXiv:2412.14992}
}
abstract

When studying gauge theories in the presence of boundaries, local symmetry transformations are typically classified as gauge or physical depending on whether the associated charges vanish or not. Here, we propose that physical charges should further be refined into "dynamical" or "kinematical" depending on whether they are associated with flux-balance laws or not. To support this proposal, we analyze (A)dS$_3$ gravity with boundary Weyl rescalings and compare the solution spaces in Bondi-Sachs and Fefferman-Graham coordinates. Our results show that the Weyl charge vanishes in the Bondi-Sachs gauge but not in the Fefferman-Graham gauge. Conversely, the charges arising from the metric Chern-Simons Lagrangian behave in the opposite way. This indicates that the gauge-dependent Weyl charge differs fundamentally from charges like mass and angular momentum. This interpretation is reinforced by two key observations: the Weyl conformal factor does not satisfy any flux-balance law, and the associated charge arises from a corner term in the symplectic structure. These properties justify assigning the Weyl charge a kinematical status. These results can also be derived using the field-dependent diffeomorphism that maps between the two gauges. Importantly, this diffeomorphism does not act tensorially on the variational bi-complex due to its field dependency, and is able to "toggle" charges on or off. This provides an example of a large diffeomorphism $\textit{between}$ gauges, as opposed to a residual diffeomorphism $\textit{within}$ a gauge.

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