---
title: Chiral Higher Spin Gravity in 4D
url: https://www.emergentmind.com/topics/chiral-higher-spin-gravity-6d8bc1d8-6d3c-41a8-96e1-3ab672c2711f
type: topic
---

# Chiral Higher Spin Gravity in 4D

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 \(\lambda\sim 1/R\) [2205.15293]. 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 [2205.07794].

## 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 [2204.10285]. 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 [2002.08487]. 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 \((A)dS_4\) [2209.01796].

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 [2512.22711]. 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 \(d\) implies the corresponding \(L_\infty\)-relations among the multilinear maps, so algebraic consistency is built into the construction [2205.07794].

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 [2204.10285]. 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 [2205.07794].

The \((A)dS_4\) 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 \((A)dS_4\) vacuum
\[
\omega_0 = \tfrac14 \omega^{AB} y_A y_B + \tfrac12 e^{AA'} y_A y_{A'} + \tfrac14 \omega^{A'B'} y_{A'} y_{B'} ,
\]
and the linearized equations become
\[
\nabla\omega = e^{BB'}\big(y_B \partial_{B'} + \partial_B y_{B'}\big)\omega + H^{B'B'}\, C(y=0),
\]
\[
\nabla C = e^{BB'}\big(y_B \partial_{B'} - \partial_B y_{B'}\big)C.
\]
In the limit \(\lambda\to 0\), these reduce to the flat-space chiral system [2205.15293].

A notable structural departure from standard Vasiliev-type constructions is the choice of module for \(C\). 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 [2205.15293].

## 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 \(A_2^e\), realized by a star product, while the \(C\)-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 [2209.15441].

A central duality relation ties the two sectors together:
\[
\langle \mathcal{V}(\omega,\omega,C,\ldots,C)\,|\,C\rangle
=
\langle \omega\,|\,\mathcal{U}(\omega,C,\ldots,C)\rangle.
\]
This means that the \(\mathcal{U}\)-vertices are obtained canonically from the \(\mathcal{V}\)-vertices rather than being built independently. The literature identifies this duality as one of the main reasons the covariant construction stays local [2205.15293].

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 \(C\)-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 [2205.15293]. 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 [2209.15441].

The same algebraic structure is described as an \(A_\infty\)-algebra of pre-Calabi–Yau type whose symmetrization yields the \(L_\infty\)-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 [2209.01796]. A representative cubic vertex is
\[
\mathcal{V}_1(\omega,\omega,C) = p_{12}\int_{\Delta_2} \exp\!\Big[(1-t_1)p_{01}+(1-t_2)p_{02}+t_1 p_{13}+t_2 p_{23}+(1+t_1-t_2)p_{12}\Big],
\]
with the corresponding cubic \(\mathcal{U}\)-vertex generated by the duality map [2205.15293].

This formulation was later sharpened by a Stokes-theorem proof of the \(A_\infty\)-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 \(A_\infty\)-relations arise from boundary components of a closed form on an enlarged configuration space [2312.16573]. 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
\[
C_{\lambda_1,\lambda_2,\lambda_3}
=
\frac{\kappa\,(l_p)^{\lambda_1+\lambda_2+\lambda_3-1}}{\Gamma(\lambda_1+\lambda_2+\lambda_3)},
\]
with the chiral selection rule \(\lambda_1+\lambda_2+\lambda_3>0\) [2603.06044]. 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 [2603.06044].

The same literature emphasizes an unusual combination of nontrivial interactions and a trivial on-shell tree-level \(S\)-matrix. In the flat-space theory all physical tree amplitudes vanish on shell; a typical \(n\)-point tree amplitude contains an overall factor \(p_1^2\), so the amplitude disappears when the external leg is placed on shell [2004.10797]. In the higher-spin self-dual Yang–Mills truncation, Berends–Giele recursion in Lorenz gauge leads to currents with an overall factor \(p_{12\ldots n}^2\), again implying that all tree-level amplitudes beyond cubic order vanish [2603.06044].

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 \(S\)-matrix elements are UV-convergent and take the form
\[
A_{\text{1-loop}}^{\text{HSG}}
=
A_{\text{QCD, 1-loop}}^{++\cdots +}
\times
D^{\text{HSG}}
\times
\nu_0,
\]
where the first factor is the all-plus one-loop amplitude of pure QCD or self-dual Yang–Mills, \(D^{\text{HSG}}\) is a higher-spin dressing factor, and \(\nu_0=\sum_\lambda 1\) is the regularized sum over helicities [2004.10797]. With the standard higher-spin regularization,
\[
\nu_0 = 1 + 2\sum_{\lambda>0}\lambda = 1 + 2\zeta(0)=0,
\]
so the total one-loop amplitude vanishes after regularization [2004.10797].

This pattern extends to supersymmetric versions. \(\mathcal N=1\) and \(\mathcal N=4\) 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 \(\mathcal N=1\) model requires a doubled spectrum, whereas the \(\mathcal N=4\) spectrum is more economical [2209.13907]. 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 [2204.10285]. 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 [2603.06044].

Exact solutions provide another sharp probe of the covariant FDA. In the flat limit \(\Lambda=0\), 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
\[
\mathcal V_{n\ge 3}=0,
\qquad
\mathcal U_{n\ge 2}=0,
\]
so the full nonlinear FDA collapses to the linear one [2501.06445]. The resulting metric is of pp-wave type,
\[
ds^2_{SD}=-du\,dv+\langle\lambda\,H_2\rangle\,dv^2+dzd\tilde z,
\]
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 [2501.06445].

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 \(+1\) gauge field, sources the opposite-helicity spin-one field and a singlet scalar, and leaves all higher-spin fields with \(s>1\) unsourced [2403.17148]. The associated reduced effective field theory starts with the Chalmers–Siegel term
\[
\mathcal{L}_{CS} = \mathrm{Tr}\!\left( Y^{AB} F_{AB}(A) \right),
\]
supplemented by higher-derivative couplings fixed by the chiral higher-spin embedding [2403.17148]. 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 \((A)dS_4\) 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 [2205.15293]. 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 \(U(1)\) 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 [2205.15293]. 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 [2405.00773].

This holographic picture has also been extended in more speculative directions. A non-relativistic chiral massive higher-spin gravity in a deformed \(AdS_3\) 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 \(4d\) theory, and a heuristic mass–spin relation
\[
s=\alpha_0+\alpha' m^{\frac{2}{z}}
\]
is used to argue that higher-spin interactions are suppressed at large spin [2511.20818]. This suggests one route by which chiral higher-spin structures may survive outside the original relativistic \((A)dS_4\) framework.

The phrase “chiral higher spin gravity” also appears in three-dimensional literature with a different meaning. In \(\mathrm{AdS}_3\), 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 \(W_3\times \widehat{\mathfrak{sl}(3)}_k\) and a phase space with 19 functions [1703.01769]. It also appears in work on holographic chiral induced \(W\)-gravities with one classical \(W_3\) algebra plus one affine current algebra [1412.2510], and in topologically massive higher-spin gravity at a special chiral point \(\mu\ell=1\) for spin three [1107.0915]. 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.

Source: https://www.emergentmind.com/topics/chiral-higher-spin-gravity-6d8bc1d8-6d3c-41a8-96e1-3ab672c2711f