Note on higher spins and holographic symmetry algebra

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.

💡 Research Summary

The paper investigates how the presence of massless higher‑spin particles modifies the soft symmetry algebras that arise in celestial holography. In the standard setting, conformally soft gravitons generate a w₁₊∞ algebra, while conformally soft gluons generate an S‑algebra. The authors extend this picture to include higher‑spin fields (spin ≥ 2) and colored higher‑spin fields, revealing a richer structure consisting of two non‑commuting infinite‑dimensional algebras.

Starting from the known graviton OPE, the authors analytically continue the conformal weights to obtain a universal OPE (eq. 2.2) that applies to any positive‑helicity particle of arbitrary spin. The spin‑selection rule G_{σ₁}×G_{σ₂}∼G_{σ₁+σ₂−2} forces the inclusion of an infinite tower of spins once a spin‑3 field is introduced, ensuring closure of the operator product algebra. From this OPE they define conformally soft higher‑spin operators H_{k,σ} and, after a \bar z expansion, global modes H_{k,σ,n}. Their commutators (eq. 2.8) form a higher‑spin generalization of the w₁₊∞ algebra. By introducing light‑transformed generators T_{p,σ} they obtain a compact algebra (eq. 2.10) that reduces to the familiar w₁₊∞ when σ=2.

The paper then shows that the usual Poincaré algebra is enlarged to a higher‑spin Poincaré algebra. The soft operator H_{σ−1,σ} acts as a “higher‑spin super‑translation,” raising both the conformal dimension and the spin of a hard operator (eq. 3.3). Similarly, H_{σ−2,σ} provides higher‑spin Lorentz generators. Together they form a semi‑direct product analogous to the ordinary Poincaré algebra, with the translation sector remaining Abelian.

A central result is the identification of two distinct w‑algebras. The first, generated by the graviton soft modes T_{p,2}, reproduces the known w₁₊∞. The second, built from higher‑spin soft modes ˜w_{p,m}=T_{p,2p−2,m} (p=2,5/2,…), satisfies the same commutation relations and thus constitutes an independent w₍∞₎ algebra. These two copies do not commute, reminiscent of the “higher‑spin square” observed in tensionless string theory on AdS₃×S³×T⁴.

For colored higher‑spin particles the authors construct a higher‑spin extension of the S‑algebra. Starting from the standard gluon OPE, they analytically continue to arbitrary spins, defining soft operators R_{k,σ,a}. Their global modes obey a commutator (eq. 5.8) that, after a light‑transform, yields a simple algebra (eq. 5.10) identical in form to the original S‑algebra but now indexed by the spin σ. An additional subalgebra ˜S, generated by modes with σ=2p−1, is shown to be isomorphic to the S‑algebra, providing a second, non‑commuting copy.

To substantiate these algebraic claims, the authors turn to the Higher‑Spin Yang‑Mills (HSYM) theory introduced in arXiv:2210.07130. They compute the tree‑level four‑point MHV amplitude for two negative‑helicity colored higher‑spin particles and an arbitrary number of positive‑helicity ones. After translating the momentum‑space amplitude to celestial coordinates and performing the \bar z expansion, the leading term matches precisely the OPE (5.2) derived from the algebraic analysis. The subleading terms are fixed by SL(2,R)_R invariance, confirming that the HSYM amplitude respects the proposed higher‑spin S‑algebra.

Finally, the paper discusses how these structures extend to spacetimes with non‑zero cosmological constant, referencing the deformed holographic symmetry algebra of arXiv:2312.00876. The authors suggest that both w₁₊∞ and w₍∞₎ can be deformed in a compatible way, though a full asymptotic‑symmetry analysis remains future work. Appendices provide explicit checks of Jacobi identities and special‑case computations, reinforcing the internal consistency of the proposed algebras.

In summary, the work demonstrates that introducing higher‑spin (and colored higher‑spin) particles enriches the celestial soft symmetry landscape, yielding two intertwined infinite‑dimensional algebras for both graviton‑type and gluon‑type sectors. The explicit verification using HSYM amplitudes offers concrete evidence that these algebras are not merely formal but are realized in scattering processes, opening avenues for further exploration of non‑local higher‑spin theories within the celestial holography framework.