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Self-dual pp-wave solutions in chiral higher-spin gravity

Tung Tran

TL;DR

This work constructs a class of exact self-dual pp-wave solutions in chiral higher-spin gravity at zero cosmological constant ($\Lambda=0$) by employing a light-cone/Kerr– Schild-inspired ansatz. Central to the method is a harmonic-profile framework for the spin-2 sector, extended to all spins via a linearized and then non-linear free differential algebra (FDA) structure, with higher-order vertices shown to vanish on these backgrounds. The analysis demonstrates that the resulting configurations satisfy both linear and non-linear equations of motion, effectively yielding free-field dynamics on a self-dual background sourced by a positive-helicity spin-2 field, and it derives the corresponding effective action that reduces to kinetic terms on the SD pp-wave geometry. The results suggest a robust integrability structure for chiral HSGRA in flat space and point to potential extensions to nearby theories and to quantum-consistent constructions, including twistor-space approaches and Green–Schwarz-type mechanisms for anomaly cancellation.

Abstract

We show that chiral higher-spin gravity with a vanishing cosmological constant admits a class of exact self-dual pp-wave solutions derived from harmonic scalar functions and two principal spinors. These solutions satisfy both the linear and non-linear equations of motion, as they annihilate all higher-order vertices, leading to the equations of motion for free fields on a self-dual background sourced by a positive-helicity spin-2 field. Our method employs a simple light-cone ansatz for positive-helicity chiral higher-spin fields, along with a modified Kerr-Schild ansatz adapted for the self-dual gravity framework.

Self-dual pp-wave solutions in chiral higher-spin gravity

TL;DR

This work constructs a class of exact self-dual pp-wave solutions in chiral higher-spin gravity at zero cosmological constant () by employing a light-cone/Kerr– Schild-inspired ansatz. Central to the method is a harmonic-profile framework for the spin-2 sector, extended to all spins via a linearized and then non-linear free differential algebra (FDA) structure, with higher-order vertices shown to vanish on these backgrounds. The analysis demonstrates that the resulting configurations satisfy both linear and non-linear equations of motion, effectively yielding free-field dynamics on a self-dual background sourced by a positive-helicity spin-2 field, and it derives the corresponding effective action that reduces to kinetic terms on the SD pp-wave geometry. The results suggest a robust integrability structure for chiral HSGRA in flat space and point to potential extensions to nearby theories and to quantum-consistent constructions, including twistor-space approaches and Green–Schwarz-type mechanisms for anomaly cancellation.

Abstract

We show that chiral higher-spin gravity with a vanishing cosmological constant admits a class of exact self-dual pp-wave solutions derived from harmonic scalar functions and two principal spinors. These solutions satisfy both the linear and non-linear equations of motion, as they annihilate all higher-order vertices, leading to the equations of motion for free fields on a self-dual background sourced by a positive-helicity spin-2 field. Our method employs a simple light-cone ansatz for positive-helicity chiral higher-spin fields, along with a modified Kerr-Schild ansatz adapted for the self-dual gravity framework.
Paper Structure (34 sections, 3 theorems, 102 equations, 1 figure, 1 table)

This paper contains 34 sections, 3 theorems, 102 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation.
  3. Review
  4. Free differential algebra
  5. Chiral HSGRA
  6. Chiral FDA data
  7. Vertices
  8. $\star$-product
  9. Higher-order vertices.
  10. Kerr-Schild ansatz and exact solutions
  11. Cartan's structure equations for empty self-dual background
  12. Deformed self-dual pp-wave background
  13. Deformed Cartan's structure equation.
  14. Tetrad and dyad bases.
  15. Kerr-Schild ansatz.
  16. ...and 19 more sections

Key Result

Proposition 3.1

Let be the field content of the spin-2 sector. Then, all higher-order vertices $\mathcal{V}_{n\geq 4}(\boldsymbol\Omega,\boldsymbol\Omega,\underline{\mathsf{C}},\ldots,\underline{\mathsf{C}})$ and $\mathcal{U}_{n\geq 3}(\boldsymbol\Omega,\underline{\mathsf{C}},\ldots,\underline{\mathsf{C}})$ vanish. Thu

Figures (1)

  • Figure 1: The horizontal/vertices axes represent the number of dotted and un-dotted spinorial indices that a tensorial field has. Here, [red] are fields with positive helicity, and [blue] are fields with negative helicity. The arrows indicate the directions in which auxiliary fields will be generated by acting $\nabla$ on the previous ones. All fields generated this way are referred to as chiral FDA data.

Theorems & Definitions (8)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition B.1
  • Definition B.2