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arxiv: 2602.03365 · v2 · pith:UJBMNV4Unew · submitted 2026-02-03 · ✦ hep-th

Note on higher spins and holographic symmetry algebra

Shamik Banerjee , Suman Guchait , Raju Mandal , Sudhakar Panda This is my paper

Pith reviewed 2026-05-25 07:29 UTC · model grok-4.3

classification ✦ hep-th
keywords higher spinsoft symmetry algebraw infinityholographic symmetryconformally softMHV amplitudeS algebra
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The pith

Higher spins add non-commuting w_∞ to soft symmetry algebra

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

The paper shows that adding higher spin particles extends the soft symmetry algebra by a w_∞ subalgebra from conformally soft higher spin modes. This subalgebra does not commute with the w_{1+∞} from soft gravitons. Similar non-commuting S-algebra appears for colored higher spins. The claim is verified with tree-level 4-point MHV amplitudes in higher spin Yang-Mills. The note also addresses the deformed case for non-zero cosmological constant.

Core claim

In the presence of higher spin particles the soft symmetry algebra has a subalgebra isomorphic to w_∞ which is generated by the conformally soft higher spin particles. This w_∞ subalgebra does not commute with the w_{1+∞} subalgebra generated by the conformally soft gravitons. The same holds for colored higher spin particles with a subalgebra isomorphic to the S-algebra.

What carries the argument

w_∞ subalgebra generated by conformally soft higher spin particles that does not commute with the graviton w_{1+∞} subalgebra

If this is right

  • Soft symmetry algebra gains a non-commuting higher spin subalgebra.
  • Colored higher spins produce an analogous non-commuting S-algebra.
  • The structure persists in the deformed algebra for non-zero cosmological constant.
  • The algebra is confirmed by explicit 4-point amplitude calculations.

Where Pith is reading between the lines

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

  • This non-commutativity may impose new constraints on multi-particle scattering in higher spin theories.
  • The result points to a richer structure for holographic symmetries in theories with higher spins.
  • One could test this by computing higher-point or loop amplitudes involving these soft modes.

Load-bearing premise

Higher spin particles act as conformally soft modes closing into w_∞ without extra relations that would change the non-commutativity with gravitons.

What would settle it

An explicit commutator calculation between higher spin and graviton soft generators showing they commute or have different closure would falsify the non-commutativity claim.

read the original abstract

In this paper we discuss a higher spin extension of the holographic symmetry algebra for graviton and gluon. Our primary observation is that in the presence of higher spin particles the soft symmetry algebra has a subalgebra isomorphic to $w_{\infty}$ which is generated by the \textit{conformally soft higher spin particles}. This $w_{\infty}$ subalgebra does not commute with the $w_{1+\infty}$ subalegbra generated by the conformally soft gravitons. The same thing holds for the colored higher spin particles. One gets a subalgebra isomorphic to the $S$-algebra which is generated by the conformally soft colored higher spin particles. We further verify the soft algebra for colored higher spin particles using the (tree-level) $4$-point MHV amplitude of the higher spin Yang-Mills theory constructed in arXiv:2210.07130. At the end we also discuss the higher spin extension of the deformed holographic symmetry algebra for non-zero cosmological constant as constructed in arXiv:2312.00876.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript is a short note claiming that the presence of higher-spin particles extends the holographic soft symmetry algebra by introducing a w_∞ subalgebra generated by conformally soft higher-spin modes; this subalgebra does not commute with the w_{1+∞} subalgebra generated by conformally soft gravitons. An analogous non-commuting S-algebra is identified for colored higher-spin particles. The colored case is verified by reference to the tree-level 4-point MHV amplitude of higher-spin Yang-Mills theory from arXiv:2210.07130, and the note closes with a brief discussion of the higher-spin extension of the deformed algebra at non-zero cosmological constant from arXiv:2312.00876.

Significance. If the non-commutativity and subalgebra identifications hold, the result would enlarge the known structure of celestial soft algebras to include higher-spin generators, with possible implications for the operator product expansions and Ward identities in celestial CFT. The explicit amplitude-based verification cited for the colored sector supplies a concrete check that strengthens the colored claim relative to the uncolored one.

major comments (2)
  1. [paragraph stating the primary observation (following the abstract)] The central claim that the conformally soft higher-spin generators close into a w_∞ subalgebra whose brackets with the graviton w_{1+∞} generators are non-vanishing is presented as following directly from the soft-factor construction, yet the note contains no explicit computation or tabulation of the relevant commutators (or structure constants) among these generators. Without such a derivation, the assertion that no additional relations alter the non-commutativity rests entirely on the external references.
  2. [paragraph on verification using the 4-point MHV amplitude] The verification of the S-algebra for colored higher spins is performed solely by invoking the 4-point MHV amplitude of arXiv:2210.07130. The note does not reproduce the soft factors, compute the relevant brackets, or demonstrate closure onto the S-algebra independently; this leaves the colored claim dependent on the cited amplitude without an internal consistency check.
minor comments (1)
  1. [Abstract] Typo in the abstract: 'subalegbra' should read 'subalgebra'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our short note and for the constructive comments. We address each major point below, clarifying the scope of the note while agreeing to add targeted explanations where helpful.

read point-by-point responses

  1. Referee: The central claim that the conformally soft higher-spin generators close into a w_∞ subalgebra whose brackets with the graviton w_{1+∞} generators are non-vanishing is presented as following directly from the soft-factor construction, yet the note contains no explicit computation or tabulation of the relevant commutators (or structure constants) among these generators. Without such a derivation, the assertion that no additional relations alter the non-commutativity rests entirely on the external references.

    Authors: The non-commutativity follows from the standard soft-factor construction already developed for celestial algebras in the cited literature (e.g., the methods of arXiv:2312.00876). The higher-spin soft factors introduce additional terms whose action on graviton modes yields non-vanishing brackets by direct substitution into the general commutator formula. As this is a brief note, we did not repeat that standard derivation. We will add a short paragraph in the revised version that explicitly recalls the relevant soft-factor form and shows why the bracket is non-zero, without expanding into a full tabulation. revision: yes

  2. Referee: The verification of the S-algebra for colored higher spins is performed solely by invoking the 4-point MHV amplitude of arXiv:2210.07130. The note does not reproduce the soft factors, compute the relevant brackets, or demonstrate closure onto the S-algebra independently; this leaves the colored claim dependent on the cited amplitude without an internal consistency check.

    Authors: The cited 4-point MHV amplitude is constructed to obey the Ward identities generated by the colored higher-spin soft operators; matching these identities to the S-algebra provides the verification. Reproducing the soft factors or brackets would duplicate material from arXiv:2210.07130. We will insert one clarifying sentence stating that the amplitude satisfies the S-algebra Ward identities, thereby making the consistency check explicit within the note. revision: partial

Circularity Check

1 steps flagged

Verification of colored higher-spin S-algebra relies on self-cited 4-point amplitude; primary w_∞ observation stated without explicit closure derivation in note

specific steps
  1. self citation load bearing [Abstract]
    "We further verify the soft algebra for colored higher spin particles using the (tree-level) 4-point MHV amplitude of the higher spin Yang-Mills theory constructed in arXiv:2210.07130."

    The verification of the claimed S-algebra subalgebra and its non-commutativity for colored higher-spin particles is performed exclusively via the amplitude construction in arXiv:2210.07130 (overlapping authors), without an independent computation of the full commutation relations or closure inside this note.

full rationale

The paper presents its central claims as a 'primary observation' from the soft symmetry algebra and soft-factor construction, with an additional verification step for the colored case. The only explicit load-bearing citation with author overlap is used for verification rather than as the sole justification for the uncolored w_∞ non-commutativity claim. No equations or reductions within the provided text show the result being equivalent to inputs by construction or fitted parameters renamed as predictions. The self-citation is therefore minor and not fully load-bearing for the core uncolored result, yielding a low-moderate score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The note rests on the existence of conformally soft higher-spin modes whose OPEs or soft factors close into w_∞ without further relations, on the validity of the 4-point MHV amplitude in the cited higher-spin Yang-Mills theory, and on the prior construction of the deformed algebra for nonzero cosmological constant. No new free parameters are introduced in the abstract itself.

axioms (2)
  • domain assumption Conformally soft higher-spin particles exist and generate a closed w_∞ subalgebra under the soft symmetry action.
    Invoked in the primary observation paragraph of the abstract.
  • domain assumption The 4-point MHV amplitude of higher-spin Yang-Mills (arXiv:2210.07130) correctly encodes the soft algebra for colored higher spins.
    Used for verification step.

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