Chiral Higher-Spin Theories

• Chiral higher-spin theories are quantum field frameworks that restrict interactions to one helicity sector, ensuring integrability and a trivial quantum S-matrix.
• They employ diverse formulations—including light-cone, twistor, and L∞ models—to encode self-dual gauge algebra structures and holographic dualities.
• Their double-copy and deformation approaches yield controlled interactions, revealing novel connections to integrable systems and higher-spin gravities in varied spacetimes.

Chiral higher-spin theories are quantum field theories that describe an infinite tower of interacting massless (and, more recently, also massive) fields of all spins, where the interactions are constrained to only one helicity sector—typically “self-dual” or “anti self-dual.” This chirality gives rise to a set of unique features: integrability of the classical equations, profound connections to generalized gauge algebras and double-copy structures, vanishing or highly constrained S-matrices, and an array of mathematical formulations from light-cone to twistor and $L_\infty$-algebraic approaches. Chiral higher-spin gravities have been constructed in both flat and (A)dS backgrounds, and serve as closed subsectors, building blocks, and benchmarks for the full space of higher-spin theories in four and higher even dimensions.

1. Defining Features and Theoretical Framework

A chiral higher-spin theory is defined by restricting all nontrivial interaction vertices to a given helicity sector, for example, all cubic couplings involving fields $\{\lambda_1,\, \lambda_2,\, \lambda_3\}$ obey $\lambda_1+\lambda_2+\lambda_3>0$ (or the negative-conjugate for anti-chiral models) (Ponomarev et al., 2016, Jain et al., 1 May 2024). In the light-cone gauge, cubic couplings are uniquely fixed by Lorentz invariance and the closure of the Poincaré algebra to the Metsaev form: $$C^{(\lambda_1,\lambda_2,\lambda_3)} = \frac{l_p^{\lambda_1+\lambda_2+\lambda_3-1}}{\Gamma(\lambda_1+\lambda_2+\lambda_3)}$$ where $l_p$ is the Planck length (Ponomarev et al., 2016, Skvortsov et al., 2018). These couplings ensure that all tree-level amplitudes beyond three points vanish, and all allowed local three-point vertices are of strictly positive (for chiral) or strictly negative (for anti-chiral) helicity sum.

The chiral sector can be realized in the Chern-Simons/W algebra framework, twistor space as holomorphic or CR-holomorphic Chern-Simons models, or covariantly as minimal $L_\infty$-models of the jet-space BV-BRST theory (Skvortsov et al., 2022, Mason et al., 14 May 2025). In $d=4$, the light-cone and twistor approaches are precisely matched, reproducing all three-point vertices and giving a systematic route to spacetime action principles (Ponomarev et al., 2016, Tran, 2022, Mason et al., 14 May 2025). The higher-spin algebra underlying such theories emerges as an extension of the universal enveloping algebra of a chiral deformation of the Poincaré algebra, modulo quadratic constraints (Ponomarev, 2022).

Key structural themes are integrability, self-duality, universality of the cubic Lagrangian, and the emergence of vanishing higher-point amplitudes—properties tightly linked by constraint equations imposed by higher-spin symmetry.

2. Interactions, S-matrix, and Exact Quantum Structure

A defining property of chiral higher-spin theories in flat space is that all tree-level $n$-point amplitudes with $n>3$ vanish (Ponomarev et al., 2016, Ponomarev, 2017, Skvortsov et al., 2018, Skvortsov et al., 2020). This is a consequence of the algebraic structure of the cubic vertices, the factorization implied by the light-cone deformation procedure and a generalized color-kinematics duality. The vanishing of amplitudes extends beyond tree-level: all loop corrections also cancel due to a "coupling conspiracy" whereby every diagram inherits an overall factor,

$\nu_0 = \sum_\lambda 1 = 1 + 2\sum_{s\ge1} 1,$

which vanishes under zeta-function regularization, $\nu_0=1+2\zeta(0)=0$ (Skvortsov et al., 2018, Skvortsov et al., 2020, Skvortsov et al., 2020). As a result, the entire quantum S-matrix is trivial, $S=1$, and no UV divergences or counterterms arise despite the presence of higher-derivative vertices.

Even in cases with non-unitary spectrum or non-renormalizable power-counting, the infinite higher-spin symmetry enforces quantum consistency by forbidding the existence of covariant counterterms (Skvortsov et al., 2020). The one-loop amplitudes, prior to the final $\nu_0$ regularization, mirror the all-plus amplitudes in QCD and self-dual Yang-Mills theory, modulated by a theory-specific "higher-spin dressing factor" (Skvortsov et al., 2020).

In $AdS_4$, the chiral theory is likewise expected to be quantum finite and encode dualities to a higher-spin extension of logarithmic CFTs with $c=0$, consistent with the presence of logarithmic bulk modes and the collapse of the $W_3$ symmetry to Virasoro under chiral limits (Bagchi et al., 2011, Krishnan et al., 2017).

3. Algebraic and Geometric Formulations

Chiral higher-spin theories admit several compatible mathematical formulations:

• Light-cone/Light-front Hamiltonian: The action is constructed from massless fields in the light-cone gauge, where cubic vertices are determined by light-cone dynamical closure. The amplitude-like structure is $$C^{(s_1,s_2,s_3)} [12]^{s_1+s_2-s_3}[23]^{s_2+s_3-s_1}[13]^{s_1+s_3-s_2}$$ (Ponomarev et al., 2016).
• Twistor and CR-holomorphic Chern-Simons: The action is a holomorphic or CR-holomorphic Chern-Simons theory on twistor space, often noncommutative (Moyal-deformed), with higher-spin fields appearing as modes along the $U(1)$ fiber (Kaluza-Klein expansion) (Tran, 2022, Mason et al., 14 May 2025). Both flat and (A)dS backgrounds are realized by choosing appropriate infinity twistors, producing spacetimes such as $S^7$, $S^3\times\mathbb{R}^4$, or AdS$_{3+4}$.
• Free Differential Algebras and $L_\infty$-algebras: The minimal model of the extended BV-BRST theory is a free differential algebra whose multilinear products (vertices) are constructed via homological perturbation theory. These $A_\infty/L_\infty$ products can be written as integrals over the configuration space of convex polygons ("swallowtail" spaces), exhibiting cyclic pre-Calabi-Yau algebra structure and closely mirroring the formality theorem style of deformation quantization (Skvortsov et al., 2022, Sharapov et al., 2022, Sharapov et al., 2022). The equations of motion acquire a Poisson sigma-model form.

This multi-pronged mathematical underpinning provides not only technical control but also geometrically and algebraically clarifies the exceptional rigidity and integrability properties observed.

4. Double Copy, Hidden Symmetries, and Integrability

Chiral higher-spin theories possess and generalize features familiar from lower-spin double-copy structures:

• Gauge Algebra Factorization: The cubic interactions can be regarded as products of a kinematic factor—identical to the self-dual Yang-Mills—and a gauge-algebra-like structure constant dependent on kinematical invariants such as light-cone momentum (Ponomarev, 2017). These generalized structure constants obey Jacobi identities due to the closure of the light-cone Poincaré algebra.
• Generalized Double Copy: Theories can be squared (or combined with self-dual Yang-Mills or gravity factors) by taking the tensor product of their little-group spectra and factorizing the cubic interactions accordingly. The resulting "chiral higher-spin double copy" is Lorentz-invariant, and its vertices factorize as $$C^{(\lambda_1,\mu_1;\lambda_2,\mu_2;\lambda_3,\mu_3)} = C^{(\lambda_1,\lambda_2,\lambda_3)}C^{(\mu_1,\mu_2,\mu_3)}$$ (Ponomarev, 28 Sep 2024). Mixed products, such as $\mathrm{CHS} \times \mathrm{SDYM}$, produce extended spectra and interactions, governed by double-copy BCJ-type relations.
• Moyal/Poisson Deformation and Integrable Structure: The interactions may be viewed as arising from a Moyal deformation of the Poisson bracket in self-dual Yang-Mills or gravity, with the self-dual kinematic algebra appearing as an area-preserving diffeomorphism algebra. The associativity of the Moyal star-product yields Jacobi identities, enforcing vanishing of tree-level amplitudes beyond three points and encoding integrability (Monteiro, 2022).

Chiral higher-spin equations of motion can be reformulated as zero-curvature/self-dual equations, displaying infinite-dimensional hidden symmetry algebras analogous to those of self-dual Yang-Mills: $$\mathfrak{hs}(x^-,x)\otimes\mathbb{C}[\zeta,\zeta^{-1}]$$ or their Poisson contractions. These algebraic structures guarantee integrability and an infinite number of non-local conserved charges (Ponomarev, 2017).

5. Holography, Chiral Subsector Projections, and Celestial Algebras

Chiral higher-spin gravity is conjectured to be holographically dual to hidden closed subsectors of Chern-Simons-matter theories. The duality is realized by projecting three-dimensional Chern-Simons vector models to their chiral/anti-chiral sector in an extremal limit of the ’t Hooft coupling (e.g., $\tilde{\lambda}\to\mp i$), leaving only net positive or negative helicity components in current correlators. The bulk theory is then dual to this chiral subsector, with a precise correspondence between boundary correlation functions and bulk cubic vertices constrained by the helicity selection rule $\lambda_1+\lambda_2+\lambda_3 > 0$ (or $<0$) (Jain et al., 1 May 2024). The presence of logarithmic bulk modes and the contraction of the $W_3$ algebra to Virasoro at the chiral point further motivate a duality to extensions of LCFT$_2$ at vanishing central charge (Bagchi et al., 2011, Krishnan et al., 2017).

The celestial holography program links chiral higher-spin theories to non-unitary $W_{1+\infty}[\mathfrak{g}]$ algebras in the sense that their generator dimensions can be negative (Tran, 1 Jul 2025). At the sphere at null infinity (celestial twistor sphere), the corresponding chiral algebra $\mathfrak{ca}$ is associative at first quantum order only in anomaly-free spectra, but can be restored by enlarging the algebra with additional axionic currents. Correlation functions of chiral CFTs provide a direct route to higher-spin form factors; surprisingly, some higher-spin celestial correlators yield non-vanishing, rational loop-level amplitudes for certain helicity sectors (e.g. $(-,+,+,…,+)$), suggesting possible extensions of the "strictly trivial" S-matrix picture in certain quantum backgrounds (Tran, 1 Jul 2025).

6. Massive and Higher-Dimensional Generalizations

Recent work has extended the chiral approach to massive higher-spin fields in four dimensions, where chiral fields in $(2s,0)$ representations admit simple Lagrangians, leading to Feynman rules and scattering amplitudes analogous to those for massive scalars, with spin captured by symmetrized factors of massive spinor-helicity wavefunctions (2207.14597). The resulting amplitudes, in gauge or gravitational backgrounds, factor into a universal "spin pre-factor" and the corresponding scalar amplitude, matching on-shell recursion results for parity-invariant theories in sectorial configurations.

For chiral higher-spin fields in six and higher even dimensions, a formulation in terms of paired 0- and 2-forms valued in symmetric $SL(2,\mathbb{H})$ tensors is applicable, with the structure extending naturally to arbitrary even dimensions and to the partially-massless case. Interacting examples are constructed by coupling the singleton sector to a background of higher-spin gauge connections valued in an arbitrary Lie algebra, leading to a generalization of the four-dimensional chiral higher-spin symmetry and spectrum (Basile, 19 Sep 2024).

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In conclusion, chiral higher-spin theories provide a self-consistent, integrable, and quantum finite sector of higher-spin gravity and gauge theory, encoding a wealth of exact structural and algebraic information. They offer a platform for double-copy constructions, controlled quantization, and testable exact holography, with deep connections to integrable systems, algebraic deformation theory, and celestial operator algebras. The understanding of higher-spin dynamics, classical and quantum, in both flat and curved (A)dS backgrounds is significantly enriched by the chiral paradigm, which continues to drive advances in both formal and applied mathematical physics.