Asymptotic Higher Spin Symmetries II: Noether Realization in Gravity

In this paper we construct a non-perturbative action of the higher spin symmetry algebra on the gravitational phase space. We introduce a symmetry algebroid $\mathcal{T}$ which allows us to include radiation in an algebraic framework. We show that $\mathcal{T}$ admits a non-linear realization on the asymptotic phase space generated by a Noether charge defined non-perturbatively for all spins. Besides, this Noether charge is conserved in the absence of radiation. Moreover, at non radiative cuts, the algebroid can be restricted to the wedge symmetry algebra studied in ArXiv:2409.12178. The key ingredient for our construction is to consider field and time dependent symmetry parameters constrained to evolve according to equations of motion dual to (a truncation of) the asymptotic Einstein’s equations. This result then guarantees that the underlying symmetry algebra is also represented canonically.

💡 Research Summary

The paper presents a non‑perturbative construction of the higher‑spin symmetry algebra acting on the gravitational phase space at future null infinity. The authors introduce a new mathematical object, the “symmetry algebroid” 𝒯, which extends the usual asymptotic symmetry algebras (BMS, BMSW, GBMS, etc.) to include an infinite tower of higher‑spin generators (spin s ≥ 0). The key insight is that the symmetry parameters themselves must obey a set of dual equations of motion that are directly tied to the radiative shear C(u,z,ż) on 𝓘⁺.

Dual equations of motion. For each spin‑s parameter τₛ(u,z,ż) (with spin –s), the authors impose
∂ᵤτₛ = D τₛ₊₁ − (s+3) C τₛ₊₂, s ≥ 0,
where D is a covariant derivative on the celestial sphere deformed by the shear. This system is the exact dual of the evolution equations satisfied by the higher‑spin charge densities eQₛ(u,z,ż):
∂ᵤeQₛ = D eQₛ₋₁ + (s+1) C eQₛ₋₂.
The duality guarantees that the generating functional Q