Source and Response Soft Charges for Maxwell Theory on $AdS_d$

Erfan Esmaeili; M. M. Sheikh-Jabbari; Vahid Hosseinzadeh

REVIEW 3 major objections 4 minor 47 references

Source and Response Soft Charges for Maxwell Theory on $AdS_d$

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Maxwell theory on AdS_d has two boundary gauge sectors whose conserved soft charges close into an infinite-dimensional Heisenberg algebra.

desk verdict A coherent construction of source/response soft charges for Maxwell on AdS_d, with the response charges' physical status unsettled; worth refereeing, not yet a solid foundation for follow-up work. read the letter →

arxiv 1908.10385 v1 pith:4HKBHJWS submitted 2019-08-27 hep-th

Erfan Esmaeili , Vahid Hosseinzadeh , M. M. Sheikh-Jabbari This is my paper · ORCID

classification hep-th PACS 11.15.-q04.62.+v

keywords asymptoticsymmetriessoftchargesMaxwelltheoryanti-deSitterspaceHeisenbergalgebralargegaugetransformationsboundaryAdS/CFT

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

Maxwell theory on anti-de Sitter space in dimension $d>3$ is claimed to have two independent layers of boundary gauge symmetry, not just one. The first layer, called source, is the large gauge transformation left over after fixing radial gauge $A_\rho=0$; the second, called response, comes from the freedom to split the boundary gauge field into an exact part and a transverse part. After adding a boundary term so that the symplectic form is conserved, the two layers carry soft conserved charges that commute within each layer but not with each other, forming an infinite-dimensional Heisenberg algebra. If this is right, AdS electromagnetism has a canonically conjugate pair of boundary soft variables, and the flat space limit keeps only the source sector.

What carries the argument

The load-bearing object is the pair of boundary scalars $(\Phi,\Psi)$, defined through the radial-gauge expansion $A_\mu=\partial_\mu\Phi+\rho^{3-d}(\ell(\partial_\mu\Psi+\hat{A}_\mu))+\cdots$, with both $\Phi$ and $\Psi$ solving the scalar Laplace equation on the de Sitter boundary, $D^2\Phi=0=D^2\Psi$, and $\hat{A}_\mu$ transverse. The conserved symplectic form is $\Omega=\int_{\Sigma_\tau}\omega+(3-d)\oint_{\partial\Sigma_\tau}\sqrt{h}\,\tau^\mu(\delta\Psi D_\mu\delta\Phi+\delta\hat{A}_\mu\delta\Phi)$; this boundary term is what turns the non-conserved bulk symplectic structure into a well-defined one. The source and response charges are the boundary integrals $Q^S_\lambda=\oint\sqrt{h}\,\tau^\mu(\lambda_S D_\mu\Psi-\Psi D_\mu\lambda_S)$ and $Q^R_\lambda=\oint\sqrt{h}\,\tau^\mu(\lambda_R D_\mu\Phi-\Phi D_\mu\lambda_R)$. Their Poisson bracket is controlled by the Wronskian of the two independent solutions $\psi^\pm_{l,m_i}$ of the boundary Laplace equation, normalized so that $\psi^-\partial_\tau\psi^+-\psi^+\partial_\tau\psi^-=2i$; this normalization is the direct origin of the factor $2i$ in the Heisenberg algebra (5.4).

What would settle it

Compute the boundary symplectic flux $\omega_{\rm flux}=(3-d)\sqrt{h}\,(\delta\Psi D^2\delta\Phi+D^\mu\delta\hat{A}_\mu\,\delta\Phi)$ for a solution of Maxwell's equations in radial gauge whose boundary data satisfy all the falloffs (3.22)--(3.26) except $D^2\Phi=0$. If a solution with $D^2\Phi\neq 0$ has nonvanishing integrated flux between two $\tau$ slices, then the conserved symplectic form (4.7), and with it the Heisenberg algebra (5.4), holds only inside the chosen gauge class and fails for that physically allowed falloff.

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Extended reading notes

Core claim

The paper's central claim is that the boundary phase space of Maxwell theory on $AdS_d$ ($d>3$) is governed by two scalars, $\Phi$ and $\Psi$, together with a transverse vector $\hat{A}_\mu$. $\Phi$ is the leading pure-gauge part of $A_\mu$ and is shifted by source gauge transformations, while $\Psi$ comes from the decomposition $A_\mu=\ell(\partial_\mu\Psi+\hat{A}_\mu)$ and is shifted by response gauge transformations. On the chosen falloff class the bulk symplectic form is not conserved by itself, but the addition of the boundary term (4.7) makes it conserved and yields the boundary action (4.9). The conserved charges $Q^S_\lambda[\Psi]$ and $Q^R_\lambda[\Phi]$ then obey an infinite-dimensional Heisenberg algebra, with the bracket $\{Q^S_{\sigma,l,m_i},Q^R_{\sigma',l',m_i'}\}=2i\,\delta(\sigma\sigma'+1)\delta_{l,l'}\delta_{m_i,m_i'}$ and vanishing brackets within each sector. The paper also establishes that these soft charges have zero bulk energy, commute with AdS translation charges, transform correctly under the Lorentz subgroup, and that in the $\ell\to\infty$ flat limit only the source charges survive while the response charges become subleading.

Load-bearing premise

The argument stands or falls with the chosen boundary falloff class (3.22)--(3.26) and the gauge constraints $D^2\Phi=0$ and $D_\mu\hat{A}^\mu=0$: these are what make the symplectic flux vanish, and any physically allowed solution that falls off differently will not carry the same source and response charges or the same algebra.

Editorial extensions

If this is right

Every physical configuration in this falloff class carries both a source and a response charge for each boundary harmonic, and the two are canonically conjugate: in a quantum theory the source and response soft sectors obey an uncertainty-type relation.
All electric multipoles in AdS have the same near-boundary falloff $\rho^{3-d}$, so a boundary observer can read off every multipole charge from ordinary $O(1)$ large gauge transformations, unlike in flat space where higher multipoles fall off faster.
The soft charges commute with all AdS-translation charges, including the Hamiltonian, so they are genuinely soft: they label zero-energy sectors of the boundary phase space.
In the $\ell\to\infty$ flat space limit the response charges disappear, so the infinite Heisenberg algebra degenerates to the abelian algebra of flat-space source charges at spatial infinity.

Reading between the lines

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

If the paper's boundary picture survives in AdS/CFT, the dual CFT should contain a pair of boundary operators with opposite scaling dimensions whose charge brackets have a constant central term; a contact term of this kind in boundary current correlators would be a direct signature of the Heisenberg sector.
The same de Sitter slicing and source/response decomposition can be attempted for p-form gauge fields and for gravity; for gravity, a nonzero Heisenberg-type extension would refine the standard statement that the AdS$_d$ asymptotic symmetry algebra is just $SO(d-1,2)$, so searching for the analogue of (5.4) is a sharp test.
The antipodal matching that is forced in flat space is optional in AdS; a concrete extension is to decide whether physical boundary states must be projected onto CPT-even combinations of soft modes, and to check which charge combinations remain well defined after that projection.

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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 paper constructs a conserved boundary symplectic form for Maxwell on AdS_d, identifies source and response boundary gauge transformations, and derives an infinite-dimensional Heisenberg algebra between their charges. The flat-space limit recovers the known source charges at spatial infinity, which is a real consistency check. The construction is internally coherent, and the response sector is genuinely new relative to the flat-space literature.

The novelty is real: the Heisenberg algebra in (5.4) and the boundary symplectic form (4.7) are not in the cited papers. The flat-space limit, where only the source charges survive, is a nontrivial check and the paper does it carefully. The authors also show that the soft charges commute with the AdS-translation charges (so they have zero bulk energy) and that Lorentz charges act as expected.

The soft spots are real but not all equal. First, the boundary falloffs in (3.22)-(3.26) are chosen so that the symplectic flux vanishes; the paper does not defend them against other relaxed falloff classes, and the reader's conditional verdict is fair on that count. Second, and more serious, is the status of the response transformations. The transformation lambda_R shifts Psi and Ahat_mu in a way that leaves the boundary gauge field A_mu invariant at every order in rho. The paper treats this as a large gauge symmetry, but it could equally be a redundancy of the Hodge-type decomposition A_mu = l(d_mu Psi + Ahat_mu). If it is a redundancy, the charge Q_R should vanish on physical states, and the Heisenberg bracket {Q_S, Q_R} would be an artifact. The paper does not provide a principle that fixes the split or shows that lambda_R is a global rather than a gauge symmetry. The fact that Q_S[Psi] is not invariant under lambda_R makes the charges representation-dependent unless such a principle exists. This is the main question a referee should push on.

A smaller issue: the integrability of the AdS-translation charges is delegated to a 'lengthy but straightforward' calculation in appendix B that is not shown. This is fixable, but it is load-bearing for the isometry sector.

My overall take: this is a serious contribution to the asymptotic-symmetry program, and the authors are honest about the arbitrary split at the start and about the omitted calculation. But the central claim is conditional. I would not yet build a follow-up on the response charges. I would send the paper to a serious referee and ask specifically whether lambda_R changes any gauge-invariant observable; that determines whether the Heisenberg algebra is physical or a gauge artifact.

Referee Report

3 major / 4 minor

Summary. The paper studies asymptotic symmetries and conserved charges for Maxwell theory on AdS_d (d>3) in de Sitter slicing. After fixing radial gauge and imposing a specific falloff class, the authors decompose the boundary gauge field into a scalar Φ, a scalar Ψ, and a transverse vector Â_μ. They identify two sets of boundary gauge transformations, 'source' (λ_S) and 'response' (λ_R), construct a conserved symplectic form by adding a boundary term, and compute the associated charges Q_S and Q_R. Their central claim is that these charges form an infinite-dimensional Heisenberg algebra, Eq. (5.4). They also analyze charges for the AdS isometry group, show that improved AdS-translation charges are integrable only after field-dependent gauge transformations, and take a large-AdS-radius limit in which only the source charges survive, matching known flat-space results. The paper closes with a discussion of AdS/CFT implications.

Significance. If the construction is sound, the paper would provide the first systematic account of asymptotic symmetry charges for Maxwell theory on AdS_d in d>3, with a new 'response' sector absent in flat space. The explicit boundary action (4.9), the conserved symplectic form (4.7), the analysis of AdS isometry charges, and the flat-space limit are concrete and potentially useful for holographic studies of soft modes. The authors also make a falsifiable prediction: only source charges survive in the flat limit. These are nontrivial strengths. The central difficulty is the physical status of the response transformations, which the paper itself says 'seems arbitrary' in Sec. 3.2; until that ambiguity is resolved, the Heisenberg algebra remains an assertion about a field redefinition redundancy rather than a demonstrated symmetry of the physical theory.

major comments (3)

[Sec. 3.2, Eqs. (3.24)-(3.26)] The λ_R transformations leave the boundary gauge field A_μ invariant everywhere: δ_λ Ψ = λ_R and δ_λ Â_μ = -∂_μ λ_R combine to δ_λ A_μ = 0. Thus λ_R is a pure redundancy of the decomposition (3.24), not a transformation of any bulk or boundary gauge-invariant field. Nevertheless, the paper assigns these transformations nonzero charges Q_R (Eq. 5.2) and finds {Q_S, Q_R} ≠ 0 (Eq. 5.3). Moreover, Q_S_λ[Ψ] is not invariant under λ_R: a shift Ψ → Ψ + λ_R changes Q_S_λ[Ψ] by ∮ √h τ^μ(λ D_μ λ_R - λ_R D_μ λ), which is generically nonzero. The paper acknowledges in Sec. 3.2 that 'at this stage the separation of A_μ into Ψ and Â_μ parts seems arbitrary', but does not supply a criterion fixing the (Ψ, Â_μ) split. Unless the authors show that λ_R is a global symmetry rather than a gauge redundancy, or demonstrate that Q_R vanishes on the physical phase space after imposing the corresponding first-class constraint, the Heisenberg algebra (5.4) appears to be an artifact of an unfixed field-redefinition redundancy. This is a load-bearing point that must be addressed explicitly.
[Appendix B, Eqs. (B.9)-(B.11)] The claim that the improved AdS-translation charges are integrable relies on the assertion that the boundary one-form B in Eq. (B.10) vanishes after 'a lengthy but straightforward calculation'. This calculation is not presented. Since integrability of these charges is used in Sec. 6.3 to conclude that Q_S and Q_R commute with the Hamiltonian and hence have zero bulk energy (Eq. (6.36b)), the omitted proof should be supplied or at least sketched in sufficient detail for the reader to verify that all boundary terms cancel under the stated falloffs and constraints.
[Sec. 3.2, Eqs. (3.22)-(3.26)] The boundary falloffs are imposed rather than derived. In particular, the choices A_μ ~ O(1) + O(ρ^{3-d}), D^2 Φ = 0, D^μ Â_μ = 0, and D^2 λ_S = D^2 λ_R = 0 are selected so that the symplectic flux vanishes, but alternative relaxed boundary conditions (for example, standard Dirichlet conditions that remove the large gauge transformations, or other subleading falloffs) would change or eliminate the source and response charges. The paper should state more prominently that the central result is conditional on this specific falloff class and discuss how robust the Heisenberg algebra is under small perturbations of the boundary conditions.

minor comments (4)

[Sec. 2.1, Eq. (2.8)] The metric expression is garbled: it should read ds^2 = ℓ^2 dρ^2/(ρ^2+ℓ^2) + ρ^2 h_μν dx^μ dx^ν; as printed, the formula is missing the denominator and the plus signs are misplaced.
[Sec. 4.1, Eq. (4.8)] The passage from (4.7) to (4.8) is not fully explained: the bulk term in (4.7) is written as δF^{ab} δA_b, while after substituting A_a = ∂_a Φ + Ā_a one obtains the second line of (4.8) only after using the equations of motion. A brief indication of this step would improve readability.
[Sec. 7, Eq. (7.20)] The dictionary Ψ_★ = (3-d)Ψ and E^{>0}_ν = (3-d)Â_ν in the flat limit is given without derivation; a short explanation of the origin of the factor (3-d) would help readers connect the flat-space and AdS normalizations.
[Appendix C] The notation ∂Σ_τ is used for the boundary of a constant-time slice in the bulk, but in Eq. (C.8) it also denotes the sphere at the AdS boundary. These two objects are different co-dimension surfaces; a distinct notation (e.g., ∂Σ_τ^B) would avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the soft-charge algebra is derived from the stated symplectic structure; the only self-citation is a non-load-bearing flat-space cross-check.

full rationale

No significant circularity: the central claim (existence of source and response soft charges and the Heisenberg algebra (5.4)) is derived from a stated symplectic structure and boundary falloff class, not assumed as an input. The boundary fields Φ and Ψ are introduced via the decomposition (3.24) of A_μ, and the response transformation λ_R is defined as the ambiguity in that decomposition (3.26); the charges (5.2) are then obtained from δQ = Ω(·, δλ) using the conserved symplectic form (4.7), which was fixed by the bulk variation (4.2)-(4.3). The algebra (5.3)-(5.4) is the symplectic pairing of the corresponding Hamiltonian vector fields, so (5.4) is an output of the calculation, not an input. The falloff conditions (3.22) are indeed chosen so that flux vanishes, but that is a boundary-condition choice, not a circular reduction, and it is stated transparently rather than smuggled in. The only self-citation of note is ref. [26] by the first author, used in Section 7 to cross-check the flat-space symplectic form; since the same flat-space result is independently proposed in ref. [22] and the central AdS construction does not rely on [26], this citation is not load-bearing. The skeptical concern that λ_R is a pure redundancy of the Ψ/Â split is a physical correctness question about whether Q_R survives reduction, not a circularity of the derivation.

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

The central construction is driven by boundary-condition choices rather than by fitted constants. There are no fitted free parameters; the only inputs are the falloff class (3.22), the decomposition (3.24), and the gauge constraints (3.25, 4.4). No new physical particles, forces, or dimensions are postulated; the response sector is a boundary gauge ambiguity in that decomposition. The main burden is that the boundary falloffs are assumed, so the phase space is constructed rather than derived from the bulk action alone.

assumptions (5)

domain assumption The Maxwell field propagates on a fixed AdS_d background with metric (2.8); gravitational back-reaction is neglected.
The action (3.1) and all charge computations treat gab as a fixed background defined in Section 2.
domain assumption The radial gauge A_ρ=0 is globally accessible for d>3.
Fixed in Eq. (3.15); the paper notes this gauge is not available in d=3, so the analysis excludes d=3.
ad hoc to paper Boundary fields satisfy D^2Φ=0, D^2Ψ=0, and D_μ Ahat_μ=0, and gauge parameters satisfy D^2λS=0=D^2λR.
Imposed in Eqs. (3.25), (3.26), and (4.4) to make the action variation and symplectic flux vanish; they select the phase space the charges live on.
standard math The boundary vector A_μ admits the exact-plus-transverse decomposition A_μ=ℓ(∂_μΨ+Ahat_μ) with transverse part Ahat_μ.
Used in Eq. (3.24); the ambiguity in this decomposition is parameterized by the response gauge parameter λR.
domain assumption A conserved symplectic form can be defined on non-globally-hyperbolic AdS by adding a boundary term to the bulk symplectic current.
Section 4.1 constructs Ω in Eq. (4.7) on constant-τ slices; this relies on the Wald-type symplectic formalism rather than on Cauchy-surface global hyperbolicity.

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Pith. "Pith review of Source and Response Soft Charges for Maxwell Theory on $AdS_d$." pith.science (2026). https://pith.science/paper/4HKBHJWS

@misc{pith2026190810385,
  author       = {Pith},
  title        = {Pith review of: Source and Response Soft Charges for Maxwell Theory on $AdS_d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4HKBHJWS}},
  note         = {Machine review of arXiv:1908.10385}
}

abstract

We study asymptotic symmetries and their associated charges for Maxwell theory on anti de Sitter (AdS) background in any dimension. This is obtained by constructing a conserved symplectic structure for the bulk and a theory on the boundary, which we specify. We show that the boundary phase space is described by two scalars and two sets of "source" and "response" boundary gauge transformations. The bulk dynamics is invariant under these two sets of boundary transformations. We study the (soft) charges associated with these two sets and show that they form an infinite dimensional Heisenberg type algebra. Studying the large AdS radius flat space limit, we show only the source soft charges survive. We also analyze algebra of charges associated with SO(d-1,2) isometries of the background $AdS_d$ space and study how they act on our source and response charges. We briefly discuss implication of our results for the AdS/CFT.

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Scalar Asymptotic Charges and Dual Large Gauge Transformations,

M. Campiglia, L. Freidel, F. Hopfmueller, and R. M. Soni , “Scalar Asymptotic Charges and Dual Large Gauge Transformations,” JHEP 04 (2019) 003, 1810.04213

2019 arXiv
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Asymptotic structure o f a massless scalar ﬁeld and its dual two-form ﬁeld at spatial inﬁnity,

M. Henneaux and C. Troessaert, “Asymptotic structure o f a massless scalar ﬁeld and its dual two-form ﬁeld at spatial inﬁnity,” JHEP 05 (2019) 147, 1812.07445

2019 arXiv
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BMS 4 algebra, its stability and deformations,

H. R. Safari and M. M. Sheikh-Jabbari, “BMS 4 algebra, its stability and deformations,” JHEP 04 (2019) 068, 1902.03260

2019 arXiv
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Multipole charge conservation and implicat ions on electromagnetic radiation,

A. Seraj, “Multipole charge conservation and implicat ions on electromagnetic radiation,” JHEP 06 (2017) 080, 1610.02870

2017 arXiv
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Gravitational mu ltipole moments from Noether charges,

G. Compere, R. Oliveri, and A. Seraj, “Gravitational mu ltipole moments from Noether charges,” JHEP 05 (2018) 054, 1711.08806. – 29 –

2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Lectures on the Infrared Structure of Gr avity and Gauge Theory,

A. Strominger, “Lectures on the Infrared Structure of Gr avity and Gauge Theory,” 1703.05448

arXiv

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Ne w Symmetries of Massless QED,

T. He, P. Mitra, A. P. Porfyriadis, and A. Strominger, “Ne w Symmetries of Massless QED,” JHEP 10 (2014) 112, 1407.3789

arXiv 2014

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Gravitational Memory, BMS Supertranslations and Soft Theorems,

A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01 (2016) 086, 1411.5745

arXiv 2016

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Soft Hair o n Black Holes,

S. W. Hawking, M. J. Perry, and A. Strominger, “Soft Hair o n Black Holes,” Phys. Rev. Lett. 116 (2016), no. 23, 231301, 1601.00921

arXiv 2016

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Near horizon soft hair as microstates of three dimensional black holes,

H. Afshar, D. Grumiller, and M. M. Sheikh-Jabbari, “Near horizon soft hair as microstates of three dimensional black holes,” Phys. Rev. D96 (2017), no. 8, 084032, 1607.00009

arXiv 2017

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A. Ashtekar and S. Das, “Asymptotically Anti-de Sitter s pace-times: Conserved quantities,” Class. Quant. Grav. 17 (2000) L17–L30, hep-th/9911230

arXiv 2000

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Asymptotic Symmetries, Holography and Topological Hair

R. K. Mishra and R. Sundrum, “Asymptotic Symmetries, Hol ography and Topological Hair,” JHEP 01 (2018) 014, 1706.09080

work page Pith review arXiv 2018

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AdS Asymptotic Symmetries from CFT Mirrors

R. K. Mishra, A. Mohd, and R. Sundrum, “AdS Asymptotic Sym metries from CFT Mirrors,” JHEP 03 (2019) 017, 1809.07331

work page Pith review arXiv 2019