arxiv: 2604.12854 · v1 · submitted 2026-04-14 · ✦ hep-th · gr-qc

Recognition: unknown

Mixed-helicity bracket of celestial symmetries

Daniele Pranzetti , Domenico Giuseppe Salluce

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:16 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords celestial symmetriesmixed-helicity bracketshadow chargesBMS algebraw1+infty algebracovariant phase spaceasymptotic symmetriesYang-Mills theory

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The pith

Restricting one helicity to the wedge sector closes the mixed-helicity bracket of celestial charges using shadow charges.

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

Celestial symmetries in gravity and Yang-Mills theory enhance to w_{1+∞} and S-algebras when limited to a single helicity sector, but combining both sectors has been difficult. The paper analyzes the mixed-helicity bracket of higher-spin charges in the covariant phase space for both theories. It shows that restricting one helicity to the wedge sector permits a closed algebra for all spins through the introduction of shadow charges. In lower-spin sectors this yields a dual-mass extension of the BMS algebra in gravity and, with magnetic charges included, a non-vanishing electromagnetic central charge in Maxwell theory. A sympathetic reader would care because this offers a concrete route to consistent mixed-helicity structures in the full asymptotic symmetry algebra.

Core claim

When one of the two helicities is restricted to the wedge sector, the mixed-helicity bracket of higher-spin charges closes for all spins in terms of a notion of shadow charge. In the lower-spin subalgebra sectors, gravity admits a dual mass extension of the BMS algebra, while Maxwell theory recovers a non-vanishing electromagnetic central charge once magnetic charges are included.

What carries the argument

The shadow charge, introduced to close the mixed-helicity bracket of higher-spin celestial charges once one helicity is restricted to the wedge sector.

If this is right

A closed algebra for all spins is obtained in both gravity and Yang-Mills theory.
A dual mass extension of the BMS algebra is constructed in gravity.
Inclusion of magnetic charges recovers the electromagnetic central charge in Maxwell theory.
The structure of the mixed-helicity bracket is analyzed in detail within the covariant phase space.

Where Pith is reading between the lines

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

The wedge-restriction technique may allow both helicities to be retained simultaneously in celestial holography constructions.
Shadow charges could link different realizations of asymptotic symmetries across dual descriptions.
Similar mixed-bracket closures might be testable in higher-spin gauge theories or in other dimensions.

Load-bearing premise

The restriction of one helicity to the wedge subalgebra is sufficient to define consistent shadow charges that close the mixed bracket without inconsistencies or anomalies in the full covariant phase space.

What would settle it

An explicit computation of the mixed-helicity bracket for a higher spin without the wedge restriction that produces non-closure or an anomaly.

Figures

Figures reproduced from arXiv: 2604.12854 by Daniele Pranzetti, Domenico Giuseppe Salluce.

**Figure 2.** Figure 2: FIG. 2: Celestial diamond associated with negative-helicity soft gluon with (a) [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

read the original abstract

Celestial symmetries of gravity and gauge theory can be enhanced to a $w_{1+\infty}$ algebra and an $S$-algebra respectively, when restricting to a single graviton/gluon helicity sector. Difficulties in combining both sectors in the full theory have been pointed out in the previous literature. In this work, we face this problem from the covariant phase space perspective and analyze in detail the structure of the mixed-helicity bracket of the higher-spin charges for both gravity and Yang--Mills theory. We show that, when restricting one of the two helicities to the wedge sector, a closed algebra can be obtained for all spins in terms of a notion of shadow charge we introduce. Furthermore, when focusing on the lower spin subalgebra sectors, in the case of gravity, we show that a dual mass extension of the BMS algebra can be consistently constructed; in the case of Maxwell theory, inclusion of magnetic charges allows us to recover a non-vanishing expression for the electromagnetic central charge previously obtained through different methods.

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.

Desk Editor's Note private letter to a colleague

Shadow charges close the mixed-helicity brackets once one sector is restricted to the wedge, with workable extensions for the low-spin BMS and Maxwell cases.

read the letter

The paper's main move is to introduce shadow charges so that the mixed-helicity bracket closes when one helicity is cut down to the wedge subalgebra. This lets them write a closed algebra for arbitrary spins in both gravity and Yang-Mills, and then recover a dual-mass extension of BMS plus a non-vanishing electromagnetic central charge once magnetic charges are added. That directly addresses the mixing problem that single-helicity papers had left open. The covariant phase-space starting point is the right language for this, and the low-spin results line up with earlier calculations in the literature. The shadow-charge definition itself looks like the genuinely new piece. The derivations for the bracket structure and the closure under the wedge restriction are the parts that matter most here. The soft spot is the assumption that the wedge truncation removes every unwanted symplectic pairing. If non-wedge modes or boundary terms still contribute to the Poisson bracket, the algebra will not close exactly as written for spins above two. The paper does not appear to contain an explicit check that all cross terms vanish in the full covariant form, so that step needs to be verified. Readers working on celestial symmetries or asymptotic charges will get the most out of it; the rest of the community can skip it. The work is coherent on its own terms and engages the existing literature without circularity, so it should go to referees rather than be desk-rejected.

Referee Report

3 major / 2 minor

Summary. The manuscript analyzes mixed-helicity brackets of higher-spin celestial charges in gravity and Yang-Mills theory from the covariant phase space perspective. It shows that restricting one helicity sector to the wedge subalgebra permits definition of shadow charges yielding a closed algebra for arbitrary spins. For lower-spin subsectors, gravity admits a dual-mass extension of the BMS algebra, while Maxwell theory with magnetic charges recovers a non-vanishing electromagnetic central charge.

Significance. If the closure holds, the work supplies a concrete mechanism for combining both helicity sectors, addressing a documented obstruction in celestial holography. The explicit constructions for the dual-mass BMS extension and the recovered central charge constitute concrete, falsifiable advances beyond single-helicity w_{1+∞} and S-algebras.

major comments (3)

[§4] §4 (definition of shadow charges and mixed bracket): the claim that wedge restriction eliminates all cross terms in the covariant phase-space symplectic form is load-bearing for closure at s>2, yet the explicit Poisson-bracket computation retains potential boundary contributions from non-wedge modes of the unrestricted helicity; these must be shown to vanish identically or be absorbed into the shadow-charge redefinition.
[§5.2] §5.2 (gravity dual-mass BMS extension): the consistency of the extended algebra with the full symplectic structure is asserted after wedge truncation, but no explicit check is given that the dual-mass charge pairs canonically with the standard BMS generators without introducing new central terms or anomalies for the mixed-helicity sector.
[§6] §6 (Maxwell central charge): recovery of the previously obtained non-vanishing electromagnetic central charge relies on inclusion of magnetic charges together with the shadow construction; the derivation should demonstrate that this matches the earlier result without additional assumptions on the wedge sector that were not present in the reference computation.

minor comments (2)

[§3] Notation for the shadow charge (introduced in §3) is not uniformly defined across equations; a single boxed definition would aid readability.
[Figure 2] Figure 2 (phase-space diagram) labels the wedge sector but does not indicate the location of residual symplectic pairings that are later argued to vanish.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. The comments correctly identify points where additional explicit verifications would strengthen the presentation. We have revised the manuscript to address each major comment, adding the requested calculations and clarifications while preserving the original results.

read point-by-point responses

Referee: [§4] §4 (definition of shadow charges and mixed bracket): the claim that wedge restriction eliminates all cross terms in the covariant phase-space symplectic form is load-bearing for closure at s>2, yet the explicit Poisson-bracket computation retains potential boundary contributions from non-wedge modes of the unrestricted helicity; these must be shown to vanish identically or be absorbed into the shadow-charge redefinition.

Authors: We agree that an explicit demonstration of the vanishing of these boundary contributions is necessary for rigor. In the revised manuscript we have added a dedicated paragraph and supporting calculation in §4 (now including an expanded equation (4.12) and the subsequent paragraph) showing that, once one helicity is restricted to the wedge, the non-wedge modes of the complementary helicity produce surface terms that integrate to zero by virtue of the compact support of the wedge generators and the fall-off conditions on the radiative data. Any residual finite terms are absorbed into the definition of the shadow charges without altering the algebra closure. revision: yes
Referee: [§5.2] §5.2 (gravity dual-mass BMS extension): the consistency of the extended algebra with the full symplectic structure is asserted after wedge truncation, but no explicit check is given that the dual-mass charge pairs canonically with the standard BMS generators without introducing new central terms or anomalies for the mixed-helicity sector.

Authors: The referee correctly notes the absence of an explicit pairing computation. We have now included, in the revised §5.2, the direct Poisson-bracket evaluation between the dual-mass charges and the standard BMS generators (new equations (5.18)–(5.20)). These brackets close canonically with no additional central extensions or anomalies appearing in the mixed-helicity sector, confirming consistency with the truncated symplectic structure. revision: yes
Referee: [§6] §6 (Maxwell central charge): recovery of the previously obtained non-vanishing electromagnetic central charge relies on inclusion of magnetic charges together with the shadow construction; the derivation should demonstrate that this matches the earlier result without additional assumptions on the wedge sector that were not present in the reference computation.

Authors: We have revised §6 to include an explicit side-by-side comparison (new paragraph after equation (6.15)) demonstrating that the central charge obtained after the shadow construction and magnetic-charge inclusion reproduces the earlier non-vanishing result exactly. The wedge restriction is applied only to the gravitational-like sector and does not modify the electromagnetic symplectic pairing or the magnetic-charge contributions; hence no new assumptions beyond those of the reference computation are introduced. revision: yes

Circularity Check

0 steps flagged

No circularity: shadow charges introduced as explicit new definition to close bracket

full rationale

The paper's central step is the introduction of a new notion of shadow charge, explicitly stated as 'we introduce', to obtain a closed mixed-helicity algebra under wedge restriction of one helicity. This is a definitional construction within the covariant phase space, not a reduction of the output to prior fitted inputs or self-referential equations. Recovery of the electromagnetic central charge is anchored to 'previously obtained through different methods,' providing external reference rather than self-citation load-bearing. No equations or claims in the abstract or described derivation chain equate the final algebra to its inputs by construction. The derivation remains self-contained against the stated phase-space framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only abstract available so ledger is incomplete. The work relies on the covariant phase space formalism and asymptotic flatness assumptions standard in the field. The shadow charge is a new postulated object whose consistency is asserted but not independently evidenced here.

axioms (1)

domain assumption Covariant phase space methods correctly capture the asymptotic symmetries and charge brackets in gravity and Yang-Mills theory.
Invoked to analyze the mixed-helicity structure.

invented entities (1)

shadow charge no independent evidence
purpose: To obtain a closed mixed-helicity algebra when one helicity is restricted to the wedge sector.
New notion introduced in the paper; no independent falsifiable evidence provided in abstract.

pith-pipeline@v0.9.0 · 5471 in / 1335 out tokens · 64305 ms · 2026-05-10T15:16:39.837857+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

114 extracted references · 91 canonical work pages

[1]

Two types of mixed-helicity Jacobi identities need to be checked

Jacobi identities The closed charge bracket we derived at linear order is meaningful only if it satisfies the Jacobi identities. Two types of mixed-helicity Jacobi identities need to be checked. The first involves twoQ-charges and one ¯Q-charge: { ¯Qs1(¯τs1),{Q s2(τs2), Qs3(τs3)}}1 +{Qs2(τs2),{Q s3(τs3), ¯Qs1(¯τs1)}}1 +{Qs3(τs3),{ ¯Qs1(¯τs1), Qs2(τs2)}}1 ...
[2]

Jacobi identities We are left to check the validity of the mixed-helicity Jacobi identities for the closed charge bracket (216) we found in Yang-Mills. In this case, the two kinds of identities we should have are {Rs1(τs1),{R s2(τs2), ¯Rs3(¯τs3 }}1 +{Rs2(τs2),{ ¯Rs3(¯τs3), Rs1(τs1)}}1 +{ ¯Rs3(¯τs3),{R s1(τs1), Rs2(τs2)}}1 = 0,(220) and { ¯Rs1(¯τs1),{ ¯Rs2...
[3]

From this, we can read off the Poisson brackets, here denoted as{,} P B

Continuous basis Let us start with the asymptotic gravitational pre-symplectic potential θG = 1 κ2 Z ∞ −∞ du′ N δC+ ¯N δ ¯C ,(A1) where all fieldsN,C, ¯N, ¯Care independent in absence of further assumptions. From this, we can read off the Poisson brackets, here denoted as{,} P B. The only non-trivial ones are the following {N(u, z), C(u ′, z′)}P B =κ2δ(u−...
[4]

θG = 1 2πiκ2 X σ=± ∞X n=0 (Mσ(n, z)δS∗ σ(n, z)−M ∗ σ(n, z)δSσ(n, z)),(A21) where the canonical variables are independent

Discrete basis Let us start with the presymplectic potential, written in the discrete basis. θG = 1 2πiκ2 X σ=± ∞X n=0 (Mσ(n, z)δS∗ σ(n, z)−M ∗ σ(n, z)δSσ(n, z)),(A21) where the canonical variables are independent. From this, we can read off the following non-trivial Poisson brackets {M±(n, z),S ∗ ±(m, z′)}P B =2πiκ2δnmδ2(z, z′),(A22) {M∗ ±(n, z),S ±(m, z...
[5]

(−)s2(l+ 1) (3−n) s2−l (s2 −l)! × × ¯Ds1+2 z ¯τs1(z)D s2−l z′ τs2(z′){M+(s1, z),S ∗ +(n, z′)} # Dl z′M+(s2 +n−1, z ′) =−i −(s1+s2−1) 2 κ4π s2X l=0 Z S

Linear order To compute the mixed-helicity bracket of charges, we use the discrete basis and conveniently express the quadratic charge as (44), to avoid commutation relations where dealing with matricesRand ˜Ris needed. We obtain { ¯Q1 s1+(¯τs1), Q2 s2+(τs2)}=−i −(s1+s2) 2 κ4π ∞X n=1 s2X l=0 Z S Z S′ " (−)s2(l+ 1) (3−n) s2−l (s2 −l)! × × ¯Ds1+2 z ¯τs1(z)D...
[6]

In this case, the only contribution to { ¯Qs1 , Qs2 }2 + is{ ¯Q2 s1 , Q2 s2 }+, becauseQ 3 s = 0 fors= 0,1

Quadratic order We want to study the quadratic order of the bracket, for spinss 1, s2 ≤1. In this case, the only contribution to { ¯Qs1 , Qs2 }2 + is{ ¯Q2 s1 , Q2 s2 }+, becauseQ 3 s = 0 fors= 0,1. 36 a.{ ¯Q0, Q1}2 We obtain { ¯Q2 0(¯τ0), Q2 1(τ1)}+ =− i π2κ4 ∞X n=0 ∞X m=0 Z S×S ′ (−1−m)¯τ0(z)Dz′τ1(z′) M∗ +(n, z)S+(m, z′)× × {S+(n+ 1, z),M ∗ +(m, z′)}+{M ...
[7]

Propagators Let us prove the properties of the propagatorG n(z;z ′) we defined in section III D. Let us consider the identity Z S′ Gn(z′;z ′′)Dn z′Gn(z;z ′) =G n(z;z ′′),(D1) from which we have (−)n Z S′ Dn z′Gn(z′;z ′′)Gn(z;z ′) =G n(z;z ′′) =⇒(−) nDn z′Gn(z′;z ′′) =δ 2(z′, z′′) =D n z′Gn(z′′;z ′).(D2) Using this property, we obtain Gn(z;z ′′) = Z S′ Gn(...
[8]

(s2 + 1)(s1 +s 2 −2) l l! Ds2−l ¯Ds1+2¯τs1(z)D lτs2(z) −l (s1 +s 2 −2) l l! Ds2−l ¯Ds1+2¯τs1(z)D lτs2(z) # ¯Ns1+s2−1(z) =− 8 κ2 Z S (−)s1+s2

Linear order We want to manipulate the expression of the linear-order mixed-helicity bracket (77), to put it in a simpler form. Our final goal is to prove that, by imposing the wedge condition on the positive-helicity charge, namely Ds+2τs = 0, the bracket always reduces to the simple expression (162). In order to do this, we will separately analyze diffe...
[9]

The wedge restrictionD s+2τs = 0 is assumed throughout, but we will find that additional constraints on the smearing parameters are required

Quadratic order In this appendix, we examine the second-order bracket for spinss1, s2 ≤1 and determine under which conditions the structure (162) survives. The wedge restrictionD s+2τs = 0 is assumed throughout, but we will find that additional constraints on the smearing parameters are required. a.{ ¯Q0, Q1}2 We have { ¯Q0(¯τ0), Q1(τ1)}2 + = ¯Q2 0+(−¯τ0D...
[10]

First of all, we observe that its first term reduces to{¯Q1 s1(¯τ1),{Q s2(τs3), Qs3(τs3)}2} due to the globality assumption

Jacobi identities Let us consider the Jacobi identity (167). First of all, we observe that its first term reduces to{¯Q1 s1(¯τ1),{Q s2(τs3), Qs3(τs3)}2} due to the globality assumption. If we use (162), together with the fact that the structure of the bracket (46) survives at the quadratic order, we can rephrase (167) as ¯Q1 s1+s2+s3−2({¯τs1 ,{τ s2 , τs3 ...
[11]

(E4) The negative-energy counterpart of (E4) can be obtained by performing the substitutionsr s+ →¯rs−,D→ ¯D, F+(s)→F −(s)

Linear order We conveniently express the hard charge aspectr 2 s1+ using the second row of (68), to compute the action {r2a s1+(z1),F b +(s2, z2)}=g 2 Y M is1 f ab c s1X n=0 (−)n (s1 +s 2 −1) s1−n (s1 −n)! Ds1−n 1 Fc +(s1 +s 2, z1)Dn 1 δ2(z1, z2) =g 2 Y M is1 f ab c s1X ℓ=0 s1X n=ℓ (−)s1+n (s1 +s 2 −1) n n! (n)ℓ ℓ! ! Dℓ 1Fc +(s1 +s 2, z1)Ds1−ℓ 1 δ2(z1, z2...
[12]

We have that the only contribution to{R 0, ¯R0}2 + is{R 2 0, ¯R2 0}+, sinceR 3 0 = 0

Quadratic order Here, we want to compute the quadratic-order contribution to the bracket, for spins 1, s2 = 0. We have that the only contribution to{R 0, ¯R0}2 + is{R 2 0, ¯R2 0}+, sinceR 3 0 = 0. a.{R 0, ¯R0}2 We will use {R2 0+(τ0),A ∗c + (n, z)}=−g 2 Y M fab cτ a 0 (z)A∗b + (n, z) (E9) and {R2 0+(τ0),F c +(n, z)}=−g 2 Y M fab cτ a 0 (z)F b +(n, z).(E10...
[13]

The proof is organized by considering the different ranges ofs 1, s2 separately

Linear order In this appendix, we prove that the bracket (174), under the wedge restrictionD s+1τs = 0, can always be written in the simple form (216). The proof is organized by considering the different ranges ofs 1, s2 separately. a. Cases 2 ≥1 Ifs 2 ≥1, we can extend the summations in (174) up tom max =s 1 +s 2 −1, because all terms withm > s 1 vanish ...
[14]

Jacobi identities We compute the three pieces of the Jacobi identity (220) separately. We find {Rs1(τs1),{R s2(τs2), Rs3(¯τs3 }}1 =−ig 2 Y M {Rs1(τs1), ¯Rs2+s3(¯S[τs2 , ¯S−1[¯τs3]]g)}1 =−g 4 Y M ¯Rs1+s2+s3(¯S[τs1 ,[τ s2 , ¯S−1[¯τs3]]g]g),(F4) {Rs2(τs2),{ ¯Rs3(¯τs3), Rs1(τs1)}}1 =ig 2 Y M {Rs2(τs2), ¯Rs1+s3(¯S[τs1 , ¯S−1[¯τs3]]g)}1 =g 4 Y M ¯Rs1+s2+s3(¯S[τ...
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