---
title: Cubic Chiral Higher Spin Gravity
url: https://www.emergentmind.com/topics/cubic-chiral-higher-spin-gravity
type: topic
---

# Cubic Chiral Higher Spin Gravity

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 [1710.00270].

## 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 [1710.00270].

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 \(s=3,4,\dots\) [2002.08487]. 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 [1710.00270].

“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 [2002.08487]. 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 \(\bar{\mathbb P}_{ij}\equiv \bar q_i\beta_j-\bar q_j\beta_i\), \(\mathbb P_{ij}\equiv q_i\beta_j-q_j\beta_i\) [1710.00270]. 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 [2603.06044].

## 2. Universal cubic action from light-cone consistency

The structural starting point is the free light-cone action
$$
S_2=-\frac12\sum_\lambda \int d^4x\;
\partial_a\Phi^{-\lambda}\,\partial^a\Phi^\lambda \, ,
$$
together with interaction deformations of the dynamical Poincaré generators \(H\), \(J\), and \(\bar J\) [1710.00270]. 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 \([H_3,J_3]=0\), with all quartic and higher deformations vanishing [1710.00270].

This truncation fixes the spin dependence of the cubic couplings universally. In one normalization the coefficients are written as
$$
C^{\lambda_1,\lambda_2,\lambda_3}
=2\,\ell^{\Lambda-1}(\Lambda-1)! \, ,
$$
while in another common normalization they are
$$
C_{\lambda_1,\lambda_2,\lambda_3}
=\frac{\kappa\, l_p^{\,\Lambda-1}}{\Gamma(\Lambda)} \, ,
$$
with \(\Lambda=\lambda_1+\lambda_2+\lambda_3\) [1710.00270][2002.08487]. 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 \(\bar{\mathbb P}^\Lambda/(\beta_1^{\lambda_1}\beta_2^{\lambda_2}\beta_3^{\lambda_3})\), multiplied by \(\mathrm{Tr}[\Phi^{\lambda_1}\Phi^{\lambda_2}\Phi^{\lambda_3}]\) in the color-valued version [1805.00048]. Standard examples already appear inside the same universal family: the chiral Einstein–Hilbert vertex \(C^{+2,+2,-2}=\kappa\,l_p\) and the chiral \(R^3\) vertex \(C^{+2,+2,+2}=\kappa\,l_p^5/5!\) [2002.08487].

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 [1710.00270].

## 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 \(A\equiv\{\lambda,q\}\) labels a Fourier mode \(\Phi^A\), the structure constants are defined by factoring out the self-dual Yang–Mills kinematic vertex:
$$
f^{A_1}{}_{A_2A_3}
\equiv \frac12\,h^{A_1}{}_{A_2A_3}\,
\frac{\beta_2\beta_3}{\beta_1\bar{\mathbb P}_{23}} \, .
$$
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 [1710.00270].

In coordinate space the algebra becomes especially transparent after a helicity-dependent rescaling and the introduction of a generating function \(\tilde E(x;z)\). The commutator closes as
$$
[\tilde E_2,\tilde E_3]
=
\sinh\!\Big[\frac{\ell}{z}
\big(\bar\partial_2\partial_3^+
-\partial_2^+\bar\partial_3\big)\Big]
\,\tilde E_2\,\tilde E_3 \, ,
$$
which is an “exponential of the Poisson bracket” on the \((x^-,x)\)-plane at fixed \((x^+,\bar x)\) [1710.00270]. A consistent Poisson contraction exists,
$$
[\tilde E_1,\tilde E_2]_P
=
\frac{\ell}{z}
\big(\bar\partial_1\partial_2^+
-\partial_1^+\bar\partial_2\big)
(\tilde E_1\tilde E_2) \, ,
$$
and the associated cubic theory has only two-derivative vertices [1710.00270].

The field equations themselves can be written as a higher-spin analogue of self-dual Yang–Mills:
$$
\Box\Phi
-
2g\,\partial^+\!\left(
\Big[\frac{\bar\partial}{\partial^+}\Phi,\Phi\Big]
\right)=0 \, .
$$
Defining \(A^x\equiv \Phi\) and \(A^-\) through \(\partial^+A^-+\partial^{\bar x}A^x=0\), the equation becomes
$$
\partial^-A^x-\partial^xA^-+g[A^-,A^x]=0 \, ,
$$
namely the vanishing of the \((+\bar x)\)-component of a non-Abelian field strength. Undoing light-cone gauge in the fiber gives the self-dual Yang–Mills system
$$
F_{ij}=\partial_iA_j-\partial_jA_i+g[A_i,A_j],\qquad
F=i*F \, ,
$$
but now with a gauge algebra involving space-time derivatives [1710.00270].

This self-dual reformulation is accompanied by standard integrable structures. The equations are equivalent to a flatness condition \([\hat K^-,\hat K^x]=0\) and to the Wess–Zumino sigma-model equation
$$
\partial^- (g^{-1}\partial g)-\partial (g^{-1}\partial^- g)=0 \, .
$$
The hidden symmetry algebra is infinite-dimensional; for the chiral higher-spin theory it takes the loop form
$$
\mathfrak F_{HS}=hs(x^-,x)\otimes \mathbb{C}[\zeta,\zeta^{-1}] \, .
$$
A Plebanski-type reformulation also exists:
$$
(\partial^+\partial^-+\partial^{\bar x}\partial^x)\omega
+
g[\partial^x\omega,\partial^-\omega]=0 \, ,
$$
equivalently a higher-spin generalization of Plebanski’s first heavenly equation [1710.00270].

## 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 \(w(y,\bar y)\), containing the positive-helicity potentials, and a zero-form \(C(y,\bar y)\), containing the negative-helicity sector, the scalar, and all on-shell descendants [2204.10285]. In this formulation the dynamics is organized by \(L_\infty\) structure maps. The free equations are the \(l_1\) data; the associative higher-spin product and the coadjoint action furnish \(l_2\); and the essential cubic vertices appear as local \(l_3\) maps \(\mathcal V(w,w,C)\) and \(\mathcal U(w,C,C)\) [2204.10285].

These cubic maps admit explicit integral representations over the two-simplex \(\Delta_2\). Their kernels are local poly-differential operators acting on generating functions, and the dotted-index sector is encoded by a star-product factor
$$
S=\exp[p_{01}+p_{02}+p_{03}+p_{12}+p_{13}+p_{23}] \, .
$$
This makes the cubic sector a concrete example of a pre-Calabi–Yau \(A_\infty\)-algebra whose cyclic pairing converts the \(\mathcal V\)-vertices into the companion \(\mathcal U\)-vertices. In the resulting “Poisson sigma-model form,” the equations become
$$
dC^k=T^{kj}(C)W_j,\qquad
dW_k=\partial_kT^{ij}(C)W_iW_j \, ,
$$
with \(T\) determined by the cyclic \(A_\infty\) data [2209.15441].

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 \(CCC\), \(\omega CC\), and \(\omega\omega C\), whereas \(\omega\omega\omega\) vertices vanish identically because the corresponding three-form is killed by self-duality identities [2603.06044]. The covariant extraction reproduces the light-cone amplitudes, including the characteristic \(1/\Gamma(\Lambda)\) 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 [2209.00925]. In flat space the resulting three-point amplitude is
$$
\mathcal M_3^{\text{flat}}(h_1,h_2,h_3)
\propto
C_{h_1,h_2,h_3}
[12]^{h_1+h_2-h_3}
[23]^{h_2+h_3-h_1}
[31]^{h_3+h_1-h_2} \, ,
$$
and the same framework admits a smooth \((A)dS\) deformation through the infinity-twistor star product [2209.00925]. A related CR-holomorphic Chern–Simons formulation on non-projective twistor space interprets higher-spin fields as Kaluza–Klein modes along an \(S^1\) fiber and organizes a broader Moyal-deformed family of chiral higher-spin theories [2505.09419].

## 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
$$
\mathcal O_s\!\left[f_{A_1A_2}{}^E f_{A_3A_4E}\right]
+
\mathcal O_t\!\left[f_{A_2A_3}{}^E f_{A_1A_4E}\right]
+
\mathcal O_u\!\left[f_{A_3A_1}{}^E f_{A_2A_4E}\right]
=0 \, ,
$$
where the operators \(\mathcal O_{s,t,u}\) act on the self-dual Yang–Mills kinematic numerators [1710.00270]. The channel dependence is controlled by the same Schouten identity in spinor-helicity variables,
$$
\bar{\mathbb P}_{43}\bar{\mathbb P}_{12}
+
\bar{\mathbb P}_{42}\bar{\mathbb P}_{31}
+
\bar{\mathbb P}_{41}\bar{\mathbb P}_{23}
=0 \, ,
$$
so the Jacobi identity of the higher-spin structure constants is equivalent to light-cone consistency at four points [1710.00270].

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 [1710.00270].

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 \(n\)-point amplitude with one off-shell leg takes a universal form proportional to the off-shellness \(p_1^2\), and therefore vanishes when the last leg is put on shell [2002.08487]. A Berends–Giele analysis of the higher-spin self-dual Yang–Mills truncation yields currents
$$
\mathcal J^{(n-1)}_{12\ldots n}(y)^{B'}
=
(y\!\cdot\! q)\,
\Big(\prod_{i=1}^n g_i(y)\Big)\,
q_B\,(p_{12\ldots n})^{BB'}\,
J(1,2,\ldots,n)\,
p_{12\ldots n}^2 \, ,
$$
so that the same \(p_{12\ldots n}^2\) factor enforces vanishing of all on-shell tree amplitudes for \(n\ge 4\) [2603.06044]. Nonzero three-point amplitudes require complexified momenta or split signature \((2,2)\), exactly as in other self-dual theories [2603.06044].

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 [1805.00048].

## 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 [1710.00270]. 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
$$
\nu_0 \equiv \sum_\omega 1
=
1+2\sum_{\lambda>0}\lambda
=
1+2\zeta(0)=0 \, ,
$$
so the regulated loop corrections vanish [2002.08487]. In the same scheme the one-loop vacuum partition function reduces to
$$
Z_{\text{1-loop}}=1 \, ,
$$
and more generally the full perturbative flat-space S-matrix is \(S=1\) [1805.00048]. 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 \(U(N)\), \(SO(N)\), and \(USp(N)\), 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 [2002.08487]. Minimal choices such as \(U(1)\) or \(SO(1)\) produce, respectively, one copy of each integer spin or only the even-spin subsector [2002.08487].

Beyond flat space, chiral higher-spin gravity exists in \((A)dS_4\) and has a local light-cone cubic action there as well [2002.08487]. 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 [2405.00773]. 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 [2605.30276].

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 \(L_\infty/A_\infty\) 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 [2204.10285].

Source: https://www.emergentmind.com/topics/cubic-chiral-higher-spin-gravity