Chiral Higher Spin Gravity in 4D

• Chiral Higher Spin Gravity is a 4D theory that extends gravity with an infinite tower of massless fields, governed by a chiral helicity selection rule.
• The theory uses a covariant formulation with free differential algebras and homotopy methods to maintain locality despite exponential derivative interactions.
• It unifies self-dual Yang–Mills and gravity sectors, offering insights into holographic correspondences, three-dimensional bosonization, and UV-finite loop amplitudes.

Chiral Higher Spin Gravity is a unique class of local higher-spin gravities in four dimensions, built from an infinite tower of propagating massless fields and characterized by a chiral helicity selection rule. In the light-cone formulation it retains only one orientation of the cubic higher-spin couplings, while in the covariant formulation it is written directly as equations of motion in free differential algebra form. The theory admits both flat-space and $(A)dS_4$ realizations, with the $(A)dS_4$ system constructed as a smooth deformation of the flat theory by the cosmological constant $\lambda\sim 1/R$ (Sharapov et al., 2022). It contains self-dual Yang–Mills and self-dual gravity as closed subsectors, and its existence has been used to argue for closed chiral subsectors in Chern–Simons matter theories and for a structural explanation of three-dimensional bosonization (Sharapov et al., 2022).

1. Emergence, scope, and defining features

The modern four-dimensional theory was first isolated in light-cone gauge as a chiral higher-spin system whose interactions stop at cubic order and whose couplings are fixed by Lorentz invariance, locality, and the requirement of a genuine higher-spin self-interaction. In that formulation the spectrum is an infinite tower of massless integer-spin fields, commonly written as $s=0,1,2,\dots$, and the helicity selection rule is $\lambda_1+\lambda_2+\lambda_3>0$ for the chiral branch of cubic vertices (Skvortsov et al., 2022). This is the sense in which the theory is described as chiral: only one sign of helicity flow is retained in the fundamental interactions.

Several papers describe the theory as the smallest or minimal extension of gravity with propagating massless higher-spin fields. The minimality claim has a precise light-front meaning: once one includes one nontrivial massless higher-spin self-interaction, the algebraic consistency conditions force an infinite multiplet of spins together with a rigid pattern of couplings, while the interaction sector remains purely cubic in the light-cone action (Skvortsov et al., 2020). This same literature stresses that the theory is local for fixed external helicities and, unlike generic higher-spin constructions, admits a smooth deformation between flat space and $(A)dS_4$ (Sharapov et al., 2022).

The four-dimensional chiral theory is also presented as a higher-spin extension of both self-dual Yang–Mills and self-dual gravity. In this hierarchy, self-dual Yang–Mills and self-dual gravity are closed subsectors of Yang–Mills and general relativity, while Chiral Higher Spin Gravity contains their higher-spin analogues and unifies them within one chiral system (Dongen, 27 Dec 2025). This places the theory in a distinctive position: it is neither a parity-invariant higher-spin gravity nor a topological model, but a local, helicity-asymmetric interacting theory with propagating massless fields.

2. Covariant equations and the free differential algebra

The covariant formulation is written directly as equations of motion rather than as an action. Its fundamental variables are a one-form master field $\omega$, containing the dynamical higher-spin gauge fields, and a zero-form master field $C$, containing Weyl-tensor-like on-shell curvatures and the scalar. The nonlinear equations take the schematic form

$$d\omega = \mathcal{V}(\omega,\omega)+\mathcal{V}(\omega,\omega,C)+\mathcal{V}(\omega,\omega,C,C)+\cdots,$$

$$dC = \mathcal{U}(\omega,C)+\mathcal{U}(\omega,C,C)+\cdots.$$

Nilpotency of $(A)dS_4$0 implies the corresponding $(A)dS_4$1-relations among the multilinear maps, so algebraic consistency is built into the construction (Sharapov et al., 2022).

This FDA is described as the minimal model of the underlying dg-algebra or jet-space BV-BRST structure. In that sense, it is not merely a covariant rewriting of light-cone equations: it is intended to capture the full local content of the theory, including information relevant to counterterms, anomalies, and related local cohomological data (Skvortsov et al., 2022). The all-orders flat-space construction completed the earlier next-to-leading-order analysis and established a manifestly Lorentz-covariant FDA at every order in perturbation theory (Sharapov et al., 2022).

The $(A)dS_4$2 version is obtained by turning on the cosmological constant as a smooth deformation of the flat system. At the free level, the background is the $(A)dS_4$3 vacuum

$(A)dS_4$4

and the linearized equations become

$(A)dS_4$5

$(A)dS_4$6

In the limit $(A)dS_4$7, these reduce to the flat-space chiral system (Sharapov et al., 2022).

A notable structural departure from standard Vasiliev-type constructions is the choice of module for $(A)dS_4$8. The zero-form does not transform in the twisted-adjoint module; instead, the covariant construction uses a dual bimodule structure. This is presented as crucial for preserving locality and for making the flat limit smooth (Sharapov et al., 2022).

3. Homotopy algebra, duality, and convex geometry

The all-order construction is organized by homological perturbation theory. The relevant higher-spin gauge algebra is an even Weyl algebra $(A)dS_4$9, realized by a star product, while the $\lambda\sim 1/R$0-sector forms the dual bimodule. The multilinear products are generated from a dg-algebra built from polynomials in auxiliary variables, together with a contracting homotopy and the transfer of structure to a minimal model (Sharapov et al., 2022).

A central duality relation ties the two sectors together: $\lambda\sim 1/R$1 This means that the $\lambda\sim 1/R$2-vertices are obtained canonically from the $\lambda\sim 1/R$3-vertices rather than being built independently. The literature identifies this duality as one of the main reasons the covariant construction stays local (Sharapov et al., 2022).

Locality is highly nontrivial because the vertices are exponential in derivative contractions. The key mechanism is that the allowed contractions avoid the dangerous contractions among multiple $\lambda\sim 1/R$4-fields that would generate infinite derivative tails. As a result, each vertex has only finitely many derivatives for fixed external helicities, even though the theory contains an infinite tower of fields (Sharapov et al., 2022). In the geometric reformulation, the multilinear products are represented as integrals over compact configuration spaces of convex polygons or concave polygons called swallowtails; the cosmological term is identified with twice the signed area of the polygonal region (Sharapov et al., 2022).

The same algebraic structure is described as an $\lambda\sim 1/R$5-algebra of pre-Calabi–Yau type whose symmetrization yields the $\lambda\sim 1/R$6-structure governing the FDA. This pre-Calabi–Yau viewpoint is explicitly connected to noncommutative Poisson geometry, and the equations of motion are said to take Poisson sigma-model form (Sharapov et al., 2022). A representative cubic vertex is

$\lambda\sim 1/R$7

with the corresponding cubic $\lambda\sim 1/R$8-vertex generated by the duality map (Sharapov et al., 2022).

This formulation was later sharpened by a Stokes-theorem proof of the $\lambda\sim 1/R$9-relations. In that approach, all structure maps are built as configuration-space integrals over concave polygons, the first two maps are related to Shoikhet–Tsygan–Kontsevich formality, and the full $s=0,1,2,\dots$0-relations arise from boundary components of a closed form on an enlarged configuration space (Sharapov et al., 2023). This suggests a direct link between chiral higher-spin interactions and formality-type constructions.

4. Cubic data, amplitudes, and quantum structure

In the light-cone description, the cubic couplings are fixed by a universal gamma-function factor. One standard expression is

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

with the chiral selection rule $s=0,1,2,\dots$2 (Guarini, 6 Mar 2026). These couplings reproduce the covariant cubic amplitudes extracted directly from the FDA equations, and the covariant three-point amplitudes agree with the earlier light-cone results (Guarini, 6 Mar 2026).

The same literature emphasizes an unusual combination of nontrivial interactions and a trivial on-shell tree-level $s=0,1,2,\dots$3-matrix. In the flat-space theory all physical tree amplitudes vanish on shell; a typical $s=0,1,2,\dots$4-point tree amplitude contains an overall factor $s=0,1,2,\dots$5, so the amplitude disappears when the external leg is placed on shell (Skvortsov et al., 2020). In the higher-spin self-dual Yang–Mills truncation, Berends–Giele recursion in Lorenz gauge leads to currents with an overall factor $s=0,1,2,\dots$6, again implying that all tree-level amplitudes beyond cubic order vanish (Guarini, 6 Mar 2026).

At one loop, Chiral Higher Spin Gravity in flat four dimensions is proved to be UV-finite despite being naively non-renormalizable by power counting. The crucial structural input is that all physical tree amplitudes vanish on shell, which constrains the one-loop integrand through its residues. The resulting one-loop $s=0,1,2,\dots$7-matrix elements are UV-convergent and take the form

$s=0,1,2,\dots$8

where the first factor is the all-plus one-loop amplitude of pure QCD or self-dual Yang–Mills, $s=0,1,2,\dots$9 is a higher-spin dressing factor, and $\lambda_1+\lambda_2+\lambda_3>0$0 is the regularized sum over helicities (Skvortsov et al., 2020). With the standard higher-spin regularization,

$\lambda_1+\lambda_2+\lambda_3>0$1

so the total one-loop amplitude vanishes after regularization (Skvortsov et al., 2020).

This pattern extends to supersymmetric versions. $\lambda_1+\lambda_2+\lambda_3>0$2 and $\lambda_1+\lambda_2+\lambda_3>0$3 supersymmetric chiral higher-spin gravities have light-front superspace actions, explicit Feynman rules, vanishing on-shell tree amplitudes, and one-loop amplitudes that vanish after the same regularization logic. The $\lambda_1+\lambda_2+\lambda_3>0$4 model requires a doubled spectrum, whereas the $\lambda_1+\lambda_2+\lambda_3>0$5 spectrum is more economical (Tsulaia et al., 2022). Across these formulations, the quantum picture repeatedly presented in the literature is that higher-spin symmetry leaves little room for counterterms and strongly constrains loop corrections.

5. Self-dual sectors, propagators, and exact solutions

Self-dual Yang–Mills and self-dual gravity are not merely analogies but closed subsectors of the chiral theory. The covariant variables separate positive-helicity data into a one-form sector and negative-helicity data into a zero-form sector, and this split is adapted to twistor-inspired first-order field equations (Skvortsov et al., 2022). In a recent amplitude analysis, propagators for arbitrary spin were derived in Feynman or Lorenz gauges, and the corresponding cubic amplitudes from the covariant equations were shown to match the known light-cone amplitudes (Guarini, 6 Mar 2026).

Exact solutions provide another sharp probe of the covariant FDA. In the flat limit $\lambda_1+\lambda_2+\lambda_3>0$6, Chiral Higher Spin Gravity admits exact self-dual pp-wave solutions constructed from harmonic scalar functions and two principal spinors. The key ansatz aligns all positive-helicity fields with the same principal spinor and all negative-helicity fields with the complementary one; on this configuration the interacting vertices satisfy

$\lambda_1+\lambda_2+\lambda_3>0$7

so the full nonlinear FDA collapses to the linear one (Tran, 11 Jan 2025). The resulting metric is of pp-wave type,

$\lambda_1+\lambda_2+\lambda_3>0$8

and the interpretation given in the paper is that the full interacting theory reduces to free higher-spin fields propagating on a self-dual pp-wave background sourced by a positive-helicity spin-2 field (Tran, 11 Jan 2025).

A distinct exact solution is the BPST instanton. In the higher-spin embedding, the instanton is an exact low-spin solution of Chiral Higher Spin Gravity: it turns on the helicity $\lambda_1+\lambda_2+\lambda_3>0$9 gauge field, sources the opposite-helicity spin-one field and a singlet scalar, and leaves all higher-spin fields with $(A)dS_4$0 unsourced (Skvortsov et al., 2024). The associated reduced effective field theory starts with the Chalmers–Siegel term

$(A)dS_4$1

supplemented by higher-derivative couplings fixed by the chiral higher-spin embedding (Skvortsov et al., 2024). This is one of the clearest examples in the literature of a low-spin configuration surviving exactly inside a fully interacting higher-spin theory.

6. Holography, hidden sectors, and variant usages of the term

A recurring claim in the four-dimensional literature is that the very existence of a complete, local, Lorentz-covariant chiral theory in $(A)dS_4$2 implies a closed subsector in Chern–Simons matter theories. The proposed bulk–boundary relation is not to the full Chern–Simons vector model, but to a chiral or anti-chiral subsector isolated by helicity decomposition and related limits on the CFT side (Sharapov et al., 2022). This is why the theory is repeatedly described as a controlled local subsector of higher-spin holography.

The same body of work connects the chiral theory to three-dimensional bosonization. The chiral and anti-chiral pieces together generate all cubic vertices, while the gluing between them is controlled by an additional parameter interpreted as a bulk $(A)dS_4$3 electromagnetic duality phase. In this description, correlators interpolate between bosonic and fermionic Chern–Simons vector models, and the parity-violating parameter of the CFT is mirrored by the bulk duality phase (Sharapov et al., 2022). The phrase “hidden sectors” is used on the CFT side for the corresponding chiral and anti-chiral subsectors that survive after suitable operator rescalings and chiral limits (Jain et al., 2024).

This holographic picture has also been extended in more speculative directions. A non-relativistic chiral massive higher-spin gravity in a deformed $(A)dS_4$4 spacetime has been proposed via a Lifshitz deformation and null reduction of the four-dimensional chiral massless theory; in that setting the vertices are less constrained than in the original $(A)dS_4$5 theory, and a heuristic mass–spin relation

$(A)dS_4$6

is used to argue that higher-spin interactions are suppressed at large spin (Mitra et al., 25 Nov 2025). This suggests one route by which chiral higher-spin structures may survive outside the original relativistic $(A)dS_4$7 framework.

The phrase “chiral higher spin gravity” also appears in three-dimensional literature with a different meaning. In $(A)dS_4$8, it has been used for Chern–Simons higher-spin gravity with a Drinfeld–Sokolov reduction on one side and a fully general sector on the other, leading in the spin-3 case to an asymptotic symmetry algebra $(A)dS_4$9 and a phase space with 19 functions (Krishnan et al., 2017). It also appears in work on holographic chiral induced $\omega$0-gravities with one classical $\omega$1 algebra plus one affine current algebra (Poojary et al., 2014), and in topologically massive higher-spin gravity at a special chiral point $\omega$2 for spin three (Bagchi et al., 2011). These three-dimensional usages are structurally distinct from the four-dimensional local chiral higher-spin gravity whose covariant FDA, convex-geometry organization, and Chern–Simons-matter holography dominate the current four-dimensional research literature.