Self-dual classical higher-spin multicopy

Abstract: We show that the self-dual classical double copy can be straightforwardly extended to the higher-spin case when formulated in terms of light-cone gauge prepotentials. This allows us to construct a higher-spin extension for any self-dual spacetime that admits Kerr-Schild form. We also discuss the counterpart of this procedure at the level of Weyl tensors. We find that depending on the class of the original gravitational background higher-spin Weyl tensors may follow various multicopy patterns.

• The paper introduces a universal mapping from gravitational Kerr-Schild backgrounds to higher-spin self-dual prepotentials, simplifying solution construction.
• Methodology employs harmonic prepotentials in light-cone gauge, demonstrating the vanishing of non-linear interactions in the self-dual sector.
• The study reveals that higher-spin Weyl tensors generally lose double-copy structures, with exceptions in algebraically special spacetimes.

Self-Dual Classical Higher-Spin Multicopy: Extensions and Weyl Structures

Introduction

The paper "Self-dual classical higher-spin multicopy" (2604.13646) investigates the extension of the self-dual classical double copy to higher-spin theories, specifically within the context of light-cone gauge prepotentials. The double copy formalism, originally derived from string-theoretic relations (KLT relations) and further established in both perturbative S-matrix theory and classical field equations, provides a mapping between gravitational and Yang-Mills solutions. The present work generalizes these ideas from lower-spin cases (helicity $\pm1$ and $\pm2$) to an infinite spectrum of massless higher-spin fields described by chiral, self-dual actions. The approach emphasizes constructing explicit classical solutions and elucidates structural relations among higher-spin Weyl tensors.

Self-Dual Classical Double Copy and Light-Cone Prepotentials

The formalism starts by recalling the formulation of self-dual Yang-Mills and gravity equations in light-cone coordinates via prepotentials, $\Phi^{+s}$, carrying helicity $+s$. The key property exploited is that, for backgrounds admitting a Kerr-Schild form, the higher-spin, self-dual field equations admit harmonic prepotential solutions: both the non-linear terms (e.g., cubic interactions in self-dual gravity) and their higher-spin analogues identically vanish. Thus, any self-dual Kerr-Schild background can be directly multicopied by equating all higher-spin prepotentials (up to constants) to the gravitational one:

$$\Phi^{h} = c^{h} \Phi^{+2}, \quad \forall h\in\mathbb{Z}$$

This prescription bypasses the complexity of direct higher-spin solution construction, as the multicopied solutions automatically satisfy the non-linear self-dual equations for all spins in the considered light-cone gauge formulation.

Extension to Higher-Spin Theories: The Multicopy Procedure

The paper thoroughly details the consistent extension of the classical double copy to higher spins, resulting in what the authors call the "self-dual multicopy" for prepotentials. The main results in this sector include:

• Universality of the Light-Cone Prepotential Mapping: Every self-dual Kerr-Schild background leads to a one-parameter family of classical higher-spin solutions, indexed by helicity, via the universal prepotential ansatz.
• Harmonicity and Vanishing Non-linearities: For multicopied configurations, all higher-derivative, non-linear interaction terms vanish due to the Kerr-Schild constraint, rendering the equations linear and leading to harmonic prepotentials for all spins.
• Generality of the Construction: The procedure is independent of explicit Lagrangian formulations for higher-spin analogs but is robust wherever the light-cone gauge description is valid and interactions are mediated by Poisson-chiral contractions (as in [Ponomarev:2017nrr]).

Structure of Higher-Spin Weyl Tensors Under Multicopy

A crucial part of the paper is the investigation of higher-spin curvatures, specifically the Weyl tensors, under the self-dual multicopy. Weyl tensors are computed as higher-derivative operators acting on the prepotentials, generalizing the well-known spinor decomposition of electromagnetic and gravitational curvatures. The analysis in this work reveals nontrivial patterns:

• General Non-Factorization: For generic self-dual Kerr-Schild backgrounds, higher derivatives of the gravitational prepotential are not algebraically reducible to (finite) lower-order derivatives. Consequently, the higher-spin Weyl tensors do not universally inherit simple double-copy structures from their gravitational counterparts.
• Type-N Spacetimes: In special cases—when the background is of Petrov type N—the Weyl tensors for all spins factorize maximally. Explicitly, $$\bar{C}_{\dot\alpha(2s)}\sim (\mu_{\dot\alpha})^{2s}$$ for a principal spinor $\mu_{\dot\alpha}$, thus preserving a perfect multicopy analogy. However, the coefficients are not solely determined by the gravitational field, and new algebraically independent data arise at each order.
• Type-D Spacetimes and Generalizations: For backgrounds with two pairs of principal spinors, such as type D (e.g., the Eguchi-Hanson instanton), the higher-spin Weyl tensors universally feature the structure $$[\mu_{\dot\alpha}\mu_{\dot\alpha}\tau_{\dot\alpha}\tau_{\dot\alpha}] \cdot F_{\dot\alpha(2s-4)}$$, where $F$ is a symmetric function of the spinors. In special sub-cases, $F$ reduces to explicit powers, enabling closed-form factorization and full algebraic control.
• Implication for Differential Structures: The formal replacement of principal spinors by differential operators (as in the differential Kerr-Schild double copy) shows that, in the self-dual sector, all multicopied higher-spin Weyl tensors are of differential Petrov type N.

Implications and Future Directions

The results in this work have several important theoretical consequences:

• Systematic Solution Generating Technique: The multicopy prescription is an efficient generating tool for higher-spin solutions, sidestepping the ill-developed notion of higher-spin geometry by leveraging underlying gauge-theoretic structures.
• Limitations of Weyl Double Copy for Higher Spins: The analysis sharply distinguishes the behavior of lower-spin classical double copy from the higher-spin case, showing that for generic backgrounds, multicopy structures are lost at the curvature level, except in algebraically special classes.
• Perspective on Interactions: The chiral, self-dual truncation is essential. The role of derivative contractions and the canonical structure of interactions in the multicopied theory indicates deep constraints and perhaps a path toward classifying consistent higher-spin self-dual interactions via inherited lower-spin data.
• Bridge to Geometry and Multipole Expansions: The multicopy may facilitate the study of geometric aspects (e.g., symmetries, horizons, asymptotics) in higher-spin contexts, potentially advancing the program of "higher-spin geometry" via imported lower-spin constructions.

Conclusion

The paper demonstrates the straightforward extension of the self-dual classical double copy to an infinite tower of higher-spin massless fields, at the level of both field equations and prepotentials, through a universal harmonic mapping for Kerr-Schild backgrounds. However, this simplicity does not, in general, persist at the level of higher-spin Weyl tensors, except in algebraically special cases (e.g., type N or specific type D backgrounds) where multicopy structures survive and closed-form factorization is possible. These results mark a significant step in understanding classical higher-spin dynamics, providing a systematic method for constructing explicit solutions and clarifying the intricate relations between spin, field equations, and curvature structures in self-dual sectors. Future work may solidify the connection of these multicopy methods to S-matrix approaches, geometric frameworks, and the broader program of higher-spin theory in both flat and curved backgrounds.