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Celestial $w_{1+\infty}$ Symmetries from Twistor Space

Tim Adamo, Lionel Mason, Atul Sharma

TL;DR

This work shows that the self-dual sector of four-dimensional gravity can be encoded as loop Poisson diffeomorphisms on twistor fibers, realized by the $Lw_{1+ obreak+ obreakinfty}$ algebra. A twistor sigma model provides a local realization of these symmetries through its OPEs, and its vertex operators reproduce the celestial soft-graviton symmetries observed in celestial holography, including the leading and subleading towers. The authors then lift the construction to the four-dimensional ambitwistor string to accommodate both self-dual and anti-self-dual sectors within a quantum framework, enabling a unified celestial OPE description of gravity. The results forge a concrete bridge between Penrose's non-linear graviton, twistor/ambitwistor string theory, and celestial CFT, with clear paths toward quantization and extensions to gauge theories such as Yang–Mills.

Abstract

We explain how twistor theory represents the self-dual sector of four dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra $Lw_{1+\infty}$ of the algebra $w_{1+\infty}$ of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in tree-level perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector which describes maps from the Riemann sphere to the asymptotic twistor space defined from characteristic data at null infinity ${\mathcal I}$. We show that the OPE of the sigma model naturally encodes the Poisson structure on twistor space and gives rise to the celestial realization of $Lw_{1+\infty}$. The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of $w_{1+\infty}$ and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of $Lw_{1+\infty}$, one for each of the self-dual and anti-self-dual sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor string.

Celestial $w_{1+\infty}$ Symmetries from Twistor Space

TL;DR

This work shows that the self-dual sector of four-dimensional gravity can be encoded as loop Poisson diffeomorphisms on twistor fibers, realized by the algebra. A twistor sigma model provides a local realization of these symmetries through its OPEs, and its vertex operators reproduce the celestial soft-graviton symmetries observed in celestial holography, including the leading and subleading towers. The authors then lift the construction to the four-dimensional ambitwistor string to accommodate both self-dual and anti-self-dual sectors within a quantum framework, enabling a unified celestial OPE description of gravity. The results forge a concrete bridge between Penrose's non-linear graviton, twistor/ambitwistor string theory, and celestial CFT, with clear paths toward quantization and extensions to gauge theories such as Yang–Mills.

Abstract

We explain how twistor theory represents the self-dual sector of four dimensional gravity in terms of the loop group of Poisson diffeomorphisms of the plane via Penrose's non-linear graviton construction. The symmetries of the self-dual sector are generated by the corresponding loop algebra of the algebra of these Poisson diffeomorphisms. We show that these coincide with the infinite tower of soft graviton symmetries in tree-level perturbative gravity recently discovered in the context of celestial amplitudes. We use a twistor sigma model for the self-dual sector which describes maps from the Riemann sphere to the asymptotic twistor space defined from characteristic data at null infinity . We show that the OPE of the sigma model naturally encodes the Poisson structure on twistor space and gives rise to the celestial realization of . The vertex operators representing soft gravitons in our model act as currents generating the wedge algebra of and produce the expected celestial OPE with hard gravitons of both helicities. We also discuss how the two copies of , one for each of the self-dual and anti-self-dual sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor string.

Paper Structure

This paper contains 16 sections, 1 theorem, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\boldsymbol{Lw_{1+\infty}}$ and self-dual gravity
  3. Poisson diffeomorphisms and $\boldsymbol{Lw_{1+\infty}}$
  4. Realization on twistor space
  5. Twistor sigma model and MHV amplitudes
  6. Holomorphic curves and twistor sigma model
  7. $\boldsymbol{\mathscr{I}}$ and asymptotic twistor space
  8. From the sigma model to the MHV amplitude
  9. From twistorial to celestial $\boldsymbol{Lw_{1+\infty}}$
  10. $\boldsymbol{Lw_{1+\infty}}$ charges and algebra
  11. Vertex operators and currents and soft limits
  12. Soft graviton symmetries
  13. The lift to 4d ambitwistor string
  14. Lifting to ambitwistor space
  15. The 4d ambitwistor string
  16. ...and 1 more sections

Key Result

Theorem 2.1

There is a $1:1$ correspondence between self-dual Ricci-flat holomorphic metrics on regions in $\mathbb{C}^4$, and complex deformations $\mathbb{P}\mathscr{T}$ of twistor space $\mathbb{PT}$ that preserve the fibration $p\colon \mathbb{P}\mathscr{T}\rightarrow\mathbb{CP}^1$ and the Poisson structure

Figures (1)

  • Figure 1: The deformed twistor space $\mathbb{P}\mathscr{T}$ in terms of a patching fibred over $\mathbb{CP}^1$.

Theorems & Definitions (1)

  • Theorem 2.1: Penrose Penrose:1976js