Higher-Spin Charge Brackets

• Higher-spin charge brackets are algebraic operations defined on conserved higher-rank currents, forming Lie algebras in free theories and requiring L∞ deformations in interacting regimes.
• They are realized in diverse frameworks, including vector models, canonical AdS formulations, and celestial CFTs, each demonstrating unique closure properties and central extensions.
• Their study uncovers invariant correlator structures and reconciles soft operator OPEs, guiding the understanding of asymptotic symmetries and higher-spin dynamics.

Higher-spin charge brackets arise whenever one associates charges to higher-rank conserved currents, asymptotic gauge symmetries, or celestial symmetries, and studies their commutator, Poisson bracket, Dirac bracket, or homotopy-Lie deformation. In the free or $N=\infty$ limit they can form an honest Lie algebra, while in interacting, asymptotic, or mixed-helicity settings the naive binary bracket may fail to close and must be completed either by higher $L_\infty$ products, central extensions, or shadow-charge constructions (Gerasimenko et al., 2021, Campoleoni et al., 2016, Pranzetti et al., 14 Apr 2026, Pranzetti et al., 30 Oct 2025, Henneaux et al., 2012).

1. Free higher-spin charges and the undeformed bracket

In vector models one denotes by $J_s(x)$ the rank-$s$ conserved current of the free or large-$N$ theory, with $s=0,1,2,\dots$, and by $Q_s$ the corresponding higher-spin charge, where the smearing parameter is a traceless conformal Killing tensor. In the free theory, or at $N=\infty$, these charges form an honest Lie algebra $\mathfrak{hs}$, with bracket

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$

induced by the commutator in the underlying associative higher-spin algebra $L_\infty$0 (Gerasimenko et al., 2021).

This undeformed bracket provides the reference point for later constructions. A small parameter

$L_\infty$1

controls the breaking of current conservation and simultaneously deforms the charge algebra together with its action on currents (Gerasimenko et al., 2021). In this regime, the higher-spin charge bracket ceases to be exhausted by an ordinary Lie commutator.

A closely related but distinct realization appears in Hamiltonian higher-spin gauge theory on Anti de Sitter backgrounds. There one works with the smeared gauge generator

$L_\infty$2

where $L_\infty$3 and $L_\infty$4 are first-class constraints and $L_\infty$5 is the boundary term that makes $L_\infty$6 differentiable (Campoleoni et al., 2016). In that setting the bracket is a Poisson bracket of asymptotic surface charges rather than a commutator inside an associative algebra.

This suggests that the expression “higher-spin charge bracket” is not tied to a single formalism. Rather, it refers to the algebraic law obeyed by higher-spin charges in the relevant phase space or operator algebra, with the precise realization depending on whether one is in a CFT, an AdS canonical description, or a celestial framework.

2. $L_\infty$7 deformation and slightly broken higher-spin symmetry

A systematic description of slightly broken higher-spin symmetry is obtained by assembling the parameters $L_\infty$8 and the currents $L_\infty$9 into a graded vector space

$J_s(x)$0

equipped with graded-antisymmetric multilinear maps

$J_s(x)$1

satisfying the generalized Jacobi identities (Gerasimenko et al., 2021). In this formulation, the higher-spin charge bracket is the binary operation $J_s(x)$2, but closure is controlled by the full tower $J_s(x)$3.

For on-shell charges and currents one can work with $J_s(x)$4, while the non-conservation of currents is encoded in higher products. The binary bracket on two charges is

$J_s(x)$5

where

$J_s(x)$6

is the first Hochschild cocycle of the one-parameter family of associative products $J_s(x)$7 (Gerasimenko et al., 2021). On a charge and a current,

$J_s(x)$8

The first obstruction to ordinary Jacobi closure is measured by $J_s(x)$9. For three charges one finds

$s$0

and the Jacobiator of $s$1 is compensated according to

$s$2

(Gerasimenko et al., 2021). Higher products $s$3 are obtained by antisymmetrizing the higher $s$4 products $s$5.

The same deformation is visible in current non-conservation. When interactions are turned on,

$s$6

and this non-conservation forces the appearance of higher $s$7 when one computes commutators of two deformed transformations on $s$8 and demands closure (Gerasimenko et al., 2021).

A recurrent misconception is that slightly broken higher-spin symmetry should still be governed by an ordinary Lie algebra with modified structure constants. The $s$9 description shows more precisely that the deformed binary bracket is only one component of the algebraic structure; the full symmetry data include higher multilinear operations required by generalized Jacobi identities.

3. Canonical surface charges and Poisson brackets in AdS

In the canonical Hamiltonian formalism for symmetric massless higher-spin fields on AdS backgrounds, the surface charge is the boundary term in the generator $N$0. Campoleoni, Henneaux, Hörtner, and Léonard give closed expressions

$N$1

where $N$2 and $N$3 are explicit sphere integrals built from the asymptotic deformation parameters, the canonical momentum $N$4, and the extra canonical variable $N$5 (Campoleoni et al., 2016).

Finiteness requires boundary conditions on canonical fields and momenta at large AdS radius $N$6. The leading angular components behave as

$N$7

with faster decay for components carrying radial indices. The leading tensors obey tracelessness and conservation conditions, and the deformation parameters approach traceless conformal Killing tensors of the boundary metric (Campoleoni et al., 2016).

Because $N$8 generates the gauge transformation through

$N$9

the charge bracket takes the form

$s=0,1,2,\dots$0

where $s=0,1,2,\dots$1 is the composite deformation parameter and $s=0,1,2,\dots$2 is a possible central term (Campoleoni et al., 2016). In the strictly linearised Fronsdal theory on AdS one finds that for $s=0,1,2,\dots$3 the surface terms cancel identically, so $s=0,1,2,\dots$4, whereas for $s=0,1,2,\dots$5 an $s=0,1,2,\dots$6-independent remainder survives as the classical central term. Nontrivial brackets among higher-spin charges, and mixing of different spins, only arise once one includes the cubic and higher vertices of the full interacting theory (Campoleoni et al., 2016).

The resulting closed form is

$s=0,1,2,\dots$7

with $s=0,1,2,\dots$8 the higher-spin algebra bracket of two conformal-Killing-tensor parameters (Campoleoni et al., 2016).

This sharply separates three regimes. In $s=0,1,2,\dots$9 at the free level the asymptotic algebra is Abelian. In $Q_s$0 the free theory already carries a classical central term. In the interacting theory the same boundary conditions that ensure finiteness also allow the nonlinear gauge-variation terms to produce a closed, non-Abelian charge algebra identified with the bulk higher-spin algebra (Campoleoni et al., 2016).

4. Three-dimensional asymptotic $Q_s$1 and super-$Q_s$2 brackets

In three dimensions the higher-spin charge bracket is particularly explicit in the Chern-Simons formulation. For $Q_s$3-extended higher-spin AdS$Q_s$4 supergravity one works with gauge connections valued in

$Q_s$5

imposes highest-weight (Drinfeld-Sokolov) gauge, and identifies the residual gauge parameters preserving the asymptotic form of the connection (Henneaux et al., 2012).

The canonical boundary charge is

$Q_s$6

and Fourier expansion leads to modes

$Q_s$7

(Henneaux et al., 2012). Their Poisson brackets have the form

$Q_s$8

For the $Q_s$9 case, the low-spin sector includes

$N=\infty$0

$N=\infty$1

with central charge $N=\infty$2 (Henneaux et al., 2012). For higher-spin integer generators $N=\infty$3,

$N=\infty$4

while the bracket of two higher-spin generators closes nonlinearly on lower-spin composites and central terms (Henneaux et al., 2012).

The bosonic truncation sets all supercharges to zero and recovers the non-linear $N=\infty$5 algebra. A finite-spin Drinfeld-Sokolov truncation to spins $N=\infty$6 retains $N=\infty$7, $N=\infty$8, and $N=\infty$9, reproducing the non-linear $\mathfrak{hs}$0-extended super-Virasoro algebra (Henneaux et al., 2012).

This three-dimensional picture matches the canonical analysis of higher-spin surface charges in AdS$\mathfrak{hs}$1, where the charge bracket is centrally extended already at the classical level and becomes the classical $\mathfrak{hs}$2-algebra once interactions are reinstated (Campoleoni et al., 2016). It provides a concrete instance in which higher-spin charge brackets are neither Abelian nor merely linear in the generators.

5. Celestial higher-spin brackets, mixed helicity, and shadow closure

At future null infinity $\mathfrak{hs}$3 in Bondi coordinates $\mathfrak{hs}$4, gravity admits higher-spin asymptotic charge aspects $\mathfrak{hs}$5 and their conjugates $\mathfrak{hs}$6. Assuming $\mathfrak{hs}$7, one defines the asymptotic charge aspect

$\mathfrak{hs}$8

of helicity $\mathfrak{hs}$9, and the smeared charges

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$0

(Pranzetti et al., 14 Apr 2026).

Up to linear order in the radiative fields, the mixed-helicity Dirac bracket is

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$1

and for general spin and arbitrary smearing functions one obtains an open expression that does not close onto any smeared charge $$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$2 (Pranzetti et al., 14 Apr 2026). This is the celestial mixed-helicity obstruction.

Closure can be restored by combining two ingredients. First, one restricts one helicity sector to the global wedge modes,

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$3

Second, one introduces Green functions $$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$4 solving

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$5

and defines anti-derivative maps $$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$6, $$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$7 together with the shadow map

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$8

which leads to the shadow charge

$$[Q_{s_1},Q_{s_2}]_0 = Q_{s_1}\star Q_{s_2}-Q_{s_2}\star Q_{s_1} \equiv \mathrm{ad}_{Q_{s_1}}Q_{s_2},$$9

(Pranzetti et al., 14 Apr 2026).

With positive-helicity charges wedge-restricted and shadow maps included, the mixed-helicity bracket closes at linear order: $L_\infty$00 In the same-helicity sector one recovers the usual $L_\infty$01 bracket

$L_\infty$02

(Pranzetti et al., 14 Apr 2026).

The lower-spin subalgebras make the physical content explicit. In gravity, restricting to $L_\infty$03 with additional holomorphicity conditions yields a dual-mass-extended BMS algebra,

$L_\infty$04

while in Maxwell theory inclusion of magnetic charges produces the electromagnetic central extension

$L_\infty$05

(Pranzetti et al., 14 Apr 2026). On a compact sphere without punctures the only harmonic $L_\infty$06 are constants and dual mass decouples, whereas punctures allow nontrivial dual mass (Pranzetti et al., 14 Apr 2026).

6. Charge brackets, celestial OPEs, and correlator invariants

The higher-spin charge bracket also has a direct operator-product interpretation. In the celestial framework, dual formulations of tree-level soft theorems imply a correspondence between the charge bracket and the celestial OPE. One defines the higher-spin charge bracket through its action on radiative fields: in gravity, for example,

$L_\infty$07

and similarly in Yang-Mills (Pranzetti et al., 30 Oct 2025). Equating the energy-basis and celestial formulations gives the dictionary

$L_\infty$08

which identifies the charge bracket with the OPE action of soft operators (Pranzetti et al., 30 Oct 2025).

In the mixed-helicity sector of celestial OPEs, a double-soft ambiguity appears when two operators are taken conformally soft. The prescription fixed by the charge-bracket correspondence is: “Always take the soft limit of the operator in the first OPE entry before taking the second limit” (Pranzetti et al., 30 Oct 2025). The same framework yields an algorithm for shadow-transformed celestial OPEs: start from the basic bracket action $L_\infty$09, replace operators by shadow kernel integrals where needed, interchange shadow integrals and bracket, use the conformal integral identity when the triple integral appears, and take the conformally soft residue at the end if a second operator is also soft (Pranzetti et al., 30 Oct 2025).

In slightly broken higher-spin symmetry, the bracket data determine invariant correlator structures. One seeks functionals

$L_\infty$10

annihilated by all $L_\infty$11. A cohomological classification shows that for each $L_\infty$12 there is exactly one nontrivial invariant, with no obstructions, so the space of invariant $L_\infty$13-point structures is one-dimensional (Gerasimenko et al., 2021). The generating functional can be written as

$L_\infty$14

where “principal part” means expansion in $L_\infty$15 with only the poles in $L_\infty$16 retained (Gerasimenko et al., 2021).

A central consequence is that each $L_\infty$17 receives only finitely many corrections in $L_\infty$18 in perturbation theory, and there can be no nontrivial functions of cross-ratios multiplying the free-theory conformal structures; the correlator is a fixed linear combination of the free and parity-odd structures with $L_\infty$19-dependent coefficients (Gerasimenko et al., 2021). In three-dimensional Chern-Simons vector models, the deformed bracket of two spin-$L_\infty$20 charges takes the form

$L_\infty$21

and the unique three-point invariant reproduces

$L_\infty$22

with exactly the combination required by the large-$L_\infty$23 3d bosonization duality (Gerasimenko et al., 2021).

Taken together, these developments show that higher-spin charge brackets serve both as symmetry algebras and as dynamical organizing principles. In AdS they control asymptotic surface symmetries; in celestial CFT they determine soft-operator OPEs and their shadow transforms; and in slightly broken higher-spin systems they govern the finite, unambiguous set of deformed correlator structures.