Conformally Soft Higher Spin Particles

• Conformally soft higher spin particles are asymptotic modes that generalize soft graviton modes to arbitrary integer spins in flat spacetimes.
• They generate infinite-dimensional symmetry algebras such as w₁₊∞ and w∞ that act on radiative data and connect infrared phenomena with celestial CFT structures.
• Their construction via asymptotic expansions and celestial OPEs yields conserved charges and recursive evolution equations, validating key soft theorem insights.

Conformally soft higher spin particles are asymptotic modes associated with integer-spin primary fields that generalize soft graviton modes to arbitrary spin. These particles emerge naturally from large-radius expansions in (asymptotically) flat spacetime backgrounds or from celestial operator product expansions (OPEs) in the framework of celestial holography. Their associated charges generate infinite-dimensional symmetry algebras, including $w_{1+\infty}$, $w_\infty$, and, for colored higher-spin cases, extensions dubbed S-algebras. These symmetries act canonically on the radiative data of massless fields and underlie the corresponding towers of higher-spin soft theorems, establishing deep connections between infrared properties, asymptotic symmetries, and celestial CFT structures.

1. Definition and Construction of Conformally Soft Higher Spin Charges

In the context of gravity, starting from the Bondi gauge metric, one considers the asymptotic expansion of the Weyl scalar:

$$\Psi_0(u, r, z, \bar z) = \sum_{n=0}^\infty \frac{\Psi_0^{(n)}(u, z, \bar z)}{r^{5+n}}.$$

Each coefficient $\Psi_0^{(n)}$ decomposes into "global" Newman-Penrose charges and local parts constructed from differential operators acting on spin-$(n+2)$ fields. For every integer spin $s \geq -1$, one defines a helicity-$s$ charge $Q_s(u, z, \bar z)$—of celestial conformal weights $(\Delta, J) = (3, s)$—satisfying recursive evolution equations:

$\dot Q_s = D Q_{s-1} + \frac{s+1}{2} C Q_{s-2}.$

This recursion encompasses Bondi mass and angular momentum aspects for $s = -1, 0, 1$, the spin-2 superrotation charge, and conjecturally, for all higher $s$ arises from subsequent subleading terms in $\Psi_0$'s expansion.

To obtain renormalized "corner" charges, one inverts the time derivative and takes asymptotic limits at $u \to -\infty$:

$$q_s(z, \bar z) = \lim_{u \to -\infty} \sum_{n=0}^s (-u)^{s-n} (s-n)! D^{s-n} Q_n(u, z, \bar z).$$

These local charges generate infinite sets of symmetries upon smearing with test functions on the sphere, $Q_s(\tau)$.

In the celestial CFT picture, conformally soft higher-spin currents are constructed via dimension limits of spin-$\sigma$ primary fields:

$$H^{k, \sigma}(z, \bar z) = \lim_{\Delta \to k} (\Delta - k) G^\sigma_\Delta(z, \bar z), \quad k = \sigma, \sigma-1, \ldots,$$

with global mode expansions in the barred coordinate. Their nontrivial Laurent expansions and integral modes $H^{k,\sigma}_n$ provide concrete operator realizations of the underlying symmetries (Freidel et al., 2021, Banerjee et al., 3 Feb 2026).

2. Algebraic Structures: $w_{1+\infty}$ and $w_\infty$ Loop Algebras

The charges associated with conformally soft graviton modes (spin-2) canonically generate the $w_{1+\infty}$ algebra. More generally, the infinite tower of higher spin charges—including both graviton and higher-spin modes—realizes nested non-commuting loop algebras:

• The $w_{1+\infty}$ algebra, arising from graviton (spin-2) sector, is generated by suitably light-transformed currents $T^{p,2}_m$:

$$[T^{p,2}_m, T^{q,2}_n] = [m (q-1) - n (p-1)] T^{p+q-2, 2}_{m+n}.$$

• The conformally soft higher spin particles (with $\sigma=2p-2$) give rise to a second $w_\infty$ algebra generated by $w̃^p_m = T^{p,2p-2}_m$:

$$[w̃^p_m, w̃^q_n] = [m(q-1) - n(p-1)] w̃^{p+q-2}_{m+n}.$$

• These two algebras ($w_{1+\infty}$ for gravitons, and $w_\infty$ for higher spins) do not commute; their mixed commutators generally do not vanish, as higher-spin generators transform in the adjoint under $w_{1+\infty}$.

For colored (gauge) higher-spin particles, analogous constructions lead to S-algebras, extending the paradigms from gravity to gauge theory amplitudes and symmetry actions (Banerjee et al., 3 Feb 2026).

3. Ward Identities, Soft Theorems, and Action on Celestial Operators

The conservation of higher spin charges manifests as celestial Ward identities acting on conformal primary operators. In the conformal primary basis, quantum charge conservation splits at leading order into the soft insertion and the hard quadratic action:

$$\langle [q_s^{(1)}, \mathcal{S}] \rangle = -\langle [q_s^{(2)}, \mathcal{S}] \rangle,$$

leading to compact Ward identities governing negative-helicity soft insertions and encoding the subleading (and sub$^s$-leading) soft graviton theorems. The conformally soft charges act as differential operators (of increasing order for higher spin) on conformal graviton primaries and their descendants. The OPEs of two conformal primary operators generate, upon extraction of residues at soft dimensions, a finite sum closure:

$$q_s^1(z_1) G_{\Delta_2}^\pm(z_2) \sim \sum_{n=0}^s \ldots \delta^{(2)}(z_1, z_2) \, G_{\Delta_2+1-s}^\pm(z_2),$$

confirming compatibility with the Ward identities derived from the asymptotic symmetry charges (Freidel et al., 2021).

4. Colored Higher Spin Particles and S-Algebra Extensions

For colored higher-spin particles (e.g., higher-spin gluon multiplets), analogous conformally soft currents are defined from the celestial OPEs of gauge primaries. Mode expansions in this context yield global colored generators, and "light-transformed" versions $S^{p, \sigma, a}_m$ (with $a$ labeling color) generate the S-algebra:

$$[S^{p, \sigma_1, a}_m, S^{q, \sigma_2, b}_n] = -i f^{abc} S^{p+q-1, \sigma_1+\sigma_2-1, c}_{m+n}.$$

Both the standard S-algebra (from soft gluons) and a tilde-S algebra (from soft higher-spin colored particles) form infinite-dimensional extensions. Ward identities and OPEs involving colored higher-spin operators have been explicitly confirmed using MHV amplitudes of Higher-Spin Yang-Mills theory via Mellin transforms and celestial limits (Banerjee et al., 3 Feb 2026).

5. Deformations by Cosmological Constant and Generalizations

For nonzero cosmological constant $\Lambda$, the universal structure of these OPEs and algebras is deformed. The OPEs of positive-helicity graviton primaries acquire $\Lambda$-dependent double- and triple-pole contributions, leading to extended commutation relations:

$$[w^{p, \sigma_1}_{a, m}, w^{q, \sigma_2}_{b, n}] = [m(q-1) - n(p-1)] w^{p+q-2, \sigma_1+\sigma_2-2}_{a+b, m+n} - \Lambda [a(q - \sigma_2) - b(p - \sigma_1)] w^{p+q-1, \sigma_1+\sigma_2-2}_{a+b, m+n}.$$

This generalizes the $\Lambda$-deformed $w_{1+\infty}$ algebra (previously established for pure gravity) coherently to all higher spins, implying that the symmetry structure of the soft sector remains robust but undergoes precise parameter-deformations in asymptotically (A)dS spacetimes (Banerjee et al., 3 Feb 2026).

6. Significance, Consistency, and Open Directions

The emergence of an infinite tower of conformally soft higher spin particles and their closure into $w_{1+\infty}$, $w_\infty$, and S-algebras provides a unifying algebraic underpinning for the infinite soft theorems observed in both gravitational and gauge theory settings. These results substantiate the canonical realization of these algebras in the radiative phase spaces of massless fields, establish exact links to celestial CFTs, and suggest that a consistent chiral bulk composed of all integer spins necessarily admits these layered symmetry algebras.

A plausible implication is that any consistent theory of massless higher spin particles—gravitational or gauge, flat or with cosmological constant—carries at least two non-commuting infinite-dimensional symmetry algebras (graviton-generated and higher-spin-generated). Their charge actions, OPEs, and Ward identities are rigidly fixed by the underlying celestial framework.

Ongoing investigations address the interaction of these symmetries with matter, quantum corrections, central extensions, and the non-perturbative completion of the associated celestial CFTs (Freidel et al., 2021, Banerjee et al., 3 Feb 2026).