Cubic Chiral Higher Spin Gravity

• Cubic Chiral Higher Spin Gravity is a 4D flat-space theory defined by an infinite tower of massless integer-spin fields interacting via a single chiral family of cubic vertices.
• The model uniquely fixes its interaction terms using Lorentz invariance, reformulating cubic equations as a self-dual Yang–Mills system with integrable properties.
• It features rich algebraic structures, including higher-spin gauge algebras and L∞ formulations, that guide both amplitude constructions and holographic applications.

Cubic Chiral Higher Spin Gravity is a four-dimensional flat-space theory of interacting massless higher-spin fields in which the interaction sector is reduced to a single chiral family of cubic vertices and all quartic and higher vertices are absent by construction. In light-cone gauge it contains an infinite tower of integer-spin fields, including the graviton, and is fixed uniquely, up to an overall normalization and one length scale, by Lorentz invariance. Its cubic equations admit a reformulation as self-dual Yang–Mills with a higher-spin gauge algebra in place of the ordinary color algebra, which places the theory in the same integrable class as self-dual Yang–Mills and self-dual gravity (Ponomarev, 2017).

1. Definition, spectrum, and chiral selection

The theory is formulated on four-dimensional Minkowski space with mostly plus signature, conveniently written in light-cone coordinates

$ds^2 = 2\,dx^+dx^- + 2\,dx\,d\bar x \, .$

Its elementary variables are massless light-cone fields $\Phi^\lambda$ labelled by helicity $\lambda\in\mathbb{Z}$, with positive and negative helicities treated as independent fields in the chiral description (Ponomarev, 2017).

The spectrum is an infinite tower of massless integer-spin states. Later descriptions emphasize that it starts with the scalar $s=0$, includes the spin-1 Yang–Mills field and the spin-2 graviton, and continues through all higher spins $s=3,4,\dots$ (Skvortsov et al., 2020). Spin-2 is not appended as an external sector: it is part of the same higher-spin tower, and the cubic theory includes universal minimal couplings to gravity at that order (Ponomarev, 2017).

“Chiral” means that only one parity-conjugate family of cubic interactions is retained. In light-cone language this is implemented by keeping only one of the holomorphic or antiholomorphic cubic structures. The fixed selection rule is that only vertices with total helicity

$\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$

are present in the chiral theory; in one common presentation the only $\Lambda=0$ exception is the scalar self-interaction (Skvortsov et al., 2020). The corresponding three-point kinematic factor is

$$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$

with

$$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$

and $$\bar{\mathbb P}_{ij}\equiv \bar q_i\beta_j-\bar q_j\beta_i$$, $\Phi^\lambda$0 (Ponomarev, 2017). Some covariant and twistor treatments write the retained branch with the opposite bracket convention. This suggests a convention dependence in the holomorphic versus antiholomorphic labeling; the invariant statement is that only one parity-conjugate family of three-point structures is kept (Guarini, 6 Mar 2026).

2. Universal cubic action from light-cone consistency

The structural starting point is the free light-cone action

$\Phi^\lambda$1

together with interaction deformations of the dynamical Poincaré generators $\Phi^\lambda$2, $\Phi^\lambda$3, and $\Phi^\lambda$4 (Ponomarev, 2017). Lorentz invariance constrains the cubic Hamiltonian density to split into holomorphic and antiholomorphic parts. Light-cone locality, understood as polynomial dependence on transverse momenta, implies that only one chiral half can be kept consistently at all orders. Choosing the chiral truncation sets the opposite-chirality couplings to zero, and then the consistency condition reduces to $\Phi^\lambda$5, with all quartic and higher deformations vanishing (Ponomarev, 2017).

This truncation fixes the spin dependence of the cubic couplings universally. In one normalization the coefficients are written as

$\Phi^\lambda$6

while in another common normalization they are

$\Phi^\lambda$7

with $\Phi^\lambda$8 (Ponomarev, 2017, Skvortsov et al., 2020). Both presentations encode the same universal dependence on the total helicity and the same statement: once a nontrivial higher-spin self-interaction is present, Lorentz invariance determines the full cubic tower.

The light-cone action is local and stops at cubic order. In momentum space its interaction term is built from the basic light-cone factor $\Phi^\lambda$9, multiplied by $\lambda\in\mathbb{Z}$0 in the color-valued version (Skvortsov et al., 2018). Standard examples already appear inside the same universal family: the chiral Einstein–Hilbert vertex $\lambda\in\mathbb{Z}$1 and the chiral $\lambda\in\mathbb{Z}$2 vertex $\lambda\in\mathbb{Z}$3 (Skvortsov et al., 2020).

The significance of this result is twofold. First, the theory is “minimal” in the precise light-cone sense that no higher-order contact terms are required to maintain Poincaré symmetry. Second, the truncation isolates a self-dual higher-spin sector rather than a parity-invariant completion, so consistency is achieved by sacrificing parity and unitarity rather than locality at cubic order (Ponomarev, 2017).

3. Higher-spin gauge algebra and self-dual reformulation

A central feature of cubic chiral higher-spin gravity is that its cubic vertex determines a nontrivial gauge algebra on the space of off-shell modes. If $\lambda\in\mathbb{Z}$4 labels a Fourier mode $\lambda\in\mathbb{Z}$5, the structure constants are defined by factoring out the self-dual Yang–Mills kinematic vertex:

$\lambda\in\mathbb{Z}$6

For the chiral higher-spin theory this yields explicit momentum-dependent structure constants, and the Jacobi identity follows from the same light-cone Lorentz-invariance condition that controls the four-point off-shell amplitude (Ponomarev, 2017).

In coordinate space the algebra becomes especially transparent after a helicity-dependent rescaling and the introduction of a generating function $\lambda\in\mathbb{Z}$7. The commutator closes as

$\lambda\in\mathbb{Z}$8

which is an “exponential of the Poisson bracket” on the $\lambda\in\mathbb{Z}$9-plane at fixed $s=0$0 (Ponomarev, 2017). A consistent Poisson contraction exists,

$s=0$1

and the associated cubic theory has only two-derivative vertices (Ponomarev, 2017).

The field equations themselves can be written as a higher-spin analogue of self-dual Yang–Mills:

$s=0$2

Defining $s=0$3 and $s=0$4 through $s=0$5, the equation becomes

$s=0$6

namely the vanishing of the $s=0$7-component of a non-Abelian field strength. Undoing light-cone gauge in the fiber gives the self-dual Yang–Mills system

$s=0$8

but now with a gauge algebra involving space-time derivatives (Ponomarev, 2017).

This self-dual reformulation is accompanied by standard integrable structures. The equations are equivalent to a flatness condition $s=0$9 and to the Wess–Zumino sigma-model equation

$s=3,4,\dots$0

The hidden symmetry algebra is infinite-dimensional; for the chiral higher-spin theory it takes the loop form

$s=3,4,\dots$1

A Plebanski-type reformulation also exists:

$s=3,4,\dots$2

equivalently a higher-spin generalization of Plebanski’s first heavenly equation (Ponomarev, 2017).

4. Covariant, homotopy-algebraic, and twistor formulations

The original light-cone construction has been recast covariantly as a free differential algebra whose fields are a one-form $s=3,4,\dots$3, containing the positive-helicity potentials, and a zero-form $s=3,4,\dots$4, containing the negative-helicity sector, the scalar, and all on-shell descendants (Skvortsov et al., 2022). In this formulation the dynamics is organized by $s=3,4,\dots$5 structure maps. The free equations are the $s=3,4,\dots$6 data; the associative higher-spin product and the coadjoint action furnish $s=3,4,\dots$7; and the essential cubic vertices appear as local $s=3,4,\dots$8 maps $s=3,4,\dots$9 and $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$0 (Skvortsov et al., 2022).

These cubic maps admit explicit integral representations over the two-simplex $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$1. Their kernels are local poly-differential operators acting on generating functions, and the dotted-index sector is encoded by a star-product factor

$\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$2

This makes the cubic sector a concrete example of a pre-Calabi–Yau $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$3-algebra whose cyclic pairing converts the $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$4-vertices into the companion $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$5-vertices. In the resulting “Poisson sigma-model form,” the equations become

$\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$6

with $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$7 determined by the cyclic $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$8 data (Sharapov et al., 2022).

A later covariant amplitude analysis recast the same cubic content in two-component spinor form and classified all admissible cubic amplitudes by master-field composition. The nonvanishing classes are $\Lambda \equiv \lambda_1+\lambda_2+\lambda_3 > 0$9, $\Lambda=0$0, and $\Lambda=0$1, whereas $\Lambda=0$2 vertices vanish identically because the corresponding three-form is killed by self-duality identities (Guarini, 6 Mar 2026). The covariant extraction reproduces the light-cone amplitudes, including the characteristic $\Lambda=0$3 normalization.

Twistor theory provides another covariant encoding. A holomorphic Chern–Simons theory on twistor space with a higher-spin star product reproduces all known flat-space cubic vertices, with couplings fixed by the same helicity-dependent rule (Tran, 2022). In flat space the resulting three-point amplitude is

$\Lambda=0$4

and the same framework admits a smooth $\Lambda=0$5 deformation through the infinity-twistor star product (Tran, 2022). A related CR-holomorphic Chern–Simons formulation on non-projective twistor space interprets higher-spin fields as Kaluza–Klein modes along an $\Lambda=0$6 fiber and organizes a broader Moyal-deformed family of chiral higher-spin theories (Mason et al., 14 May 2025).

5. Amplitudes, generalized BCJ relations, and tree-level triviality

Because the theory contains only cubic vertices, off-shell amplitudes are naturally represented as sums over cubic diagrams. In this setting a generalized color-kinematics structure emerges: the four-point off-shell amplitude satisfies

$\Lambda=0$7

where the operators $\Lambda=0$8 act on the self-dual Yang–Mills kinematic numerators (Ponomarev, 2017). The channel dependence is controlled by the same Schouten identity in spinor-helicity variables,

$\Lambda=0$9

so the Jacobi identity of the higher-spin structure constants is equivalent to light-cone consistency at four points (Ponomarev, 2017).

This algebraic control motivates generalized double-copy constructions. Colored and Poisson chiral higher-spin amplitudes can be generated from self-dual Yang–Mills numerators by helicity-dependent squaring rules, whereas the full Metsaev-type theory may require supplementing self-dual Yang–Mills with additional cubic vertices to reproduce all helicity weights (Ponomarev, 2017).

At the level of actual scattering, the theory is highly constrained. In flat space all physical tree amplitudes beyond three points vanish on shell. In the light-cone formulation the $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$0-point amplitude with one off-shell leg takes a universal form proportional to the off-shellness $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$1, and therefore vanishes when the last leg is put on shell (Skvortsov et al., 2020). A Berends–Giele analysis of the higher-spin self-dual Yang–Mills truncation yields currents

$$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$2

so that the same $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$3 factor enforces vanishing of all on-shell tree amplitudes for $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$4 (Guarini, 6 Mar 2026). Nonzero three-point amplitudes require complexified momenta or split signature $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$5, exactly as in other self-dual theories (Guarini, 6 Mar 2026).

The resulting S-matrix behavior is therefore characteristic of self-dual sectors: the cubic couplings are nontrivial, yet the on-shell tree S-matrix is trivial beyond three points (Skvortsov et al., 2018).

6. Quantum properties, gaugings, and holographic role

The chiral theory is parity violating and, taken by itself, non-unitary. Keeping only one chirality means that the cubic Hamiltonian is not real, so the time-evolution operator is non-unitary; restoring parity by adding the opposite-chirality sector encounters locality obstructions beyond cubic order (Ponomarev, 2017). This obstruction is precisely how the theory evades standard flat-space no-go constraints: it keeps locality in the self-dual sector while abandoning parity invariance and a conventional unitary completion.

Quantum mechanically, the flat-space cubic theory displays striking cancellations. One-loop two-, three-, and four-point amplitudes are UV-finite in the light-cone quantization, and every loop amplitude carries the regularized factor

$$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$6

so the regulated loop corrections vanish (Skvortsov et al., 2020). In the same scheme the one-loop vacuum partition function reduces to

$$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$7

and more generally the full perturbative flat-space S-matrix is $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$8 (Skvortsov et al., 2018). The mechanism combines the chiral helicity selection rule with the infinite higher-spin symmetry.

The cubic theory also admits Chan–Paton gaugings. Consistent gauge groups include $$A_3(\lambda_1,\lambda_2,\lambda_3)\propto [12]^{\lambda_1+\lambda_2-\lambda_3} [23]^{\lambda_2+\lambda_3-\lambda_1} [31]^{\lambda_3+\lambda_1-\lambda_2} \, ,$$9, $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$0, and $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$1, with spin-parity assignments fixed by the light-cone algebra closure conditions; these internal charges do not modify the universal kinematic structure of the cubic vertices (Skvortsov et al., 2020). Minimal choices such as $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$2 or $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$3 produce, respectively, one copy of each integer spin or only the even-spin subsector (Skvortsov et al., 2020).

Beyond flat space, chiral higher-spin gravity exists in $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$4 and has a local light-cone cubic action there as well (Skvortsov et al., 2020). Recent holographic work argues that its existence implies closed chiral and anti-chiral subsectors inside Chern–Simons matter theories, with cubic correlators matching the higher-spin selection rules after an appropriate complex ’t Hooft-coupling limit (Jain et al., 2024). A higher-spin extension of self-dual Yang–Mills, realized as a contraction of chiral higher-spin gravity, has already been used to derive explicit AdS three- and four-point correlators and a self-dual holographic dictionary (Skvortsov et al., 28 May 2026).

In this broader setting, cubic chiral higher-spin gravity functions as a universal self-dual building block. It is simultaneously a minimal light-cone solution of higher-spin Poincaré consistency, a covariant $$[ij]=\sqrt{2\beta_i\beta_j}\,\bar{\mathbb P}_{ij},\qquad \langle ij\rangle=-\sqrt{2\beta_i\beta_j}\,\mathbb P_{ij},$$5 system with explicit local cubic cocycles, a twistor-holomorphic theory, and an integrable sector whose algebraic structures continue to organize amplitude and holographic constructions well beyond their original flat-space derivation (Skvortsov et al., 2022).