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Conformal and Einstein gravity from twistor actions

Tim Adamo, Lionel Mason

TL;DR

The paper tackles computing Einstein gravity amplitudes on backgrounds with cosmological constant by embedding Einstein data into conformal gravity and exploiting a twistorial action for conformal gravity. It develops a perturbative MHV framework on twistor space, where Feynman diagrams are summed with the matrix-tree theorem to yield a $\Lambda$-dependent MHV formula that reduces to Hodges' flat-space result as $\Lambda \to 0$. An off-shell Einstein twistor action is proposed, obtained by restricting to Einstein data, and the approach extends naturally to $\mathcal{N}=4$ supersymmetry. The results provide a concrete, gauge-independent derivation of Einstein MHV amplitudes in de Sitter/AdS backgrounds and reveal a determinant structure that generalizes Hodges' formula to nonzero $\Lambda$, with connections to twistor-string constructions. The work opens avenues for MHV-like formalisms in gravity and potential loop computations via an Einstein twistor framework.

Abstract

We use the embedding of Einstein gravity with cosmological constant into conformal gravity as a basis for using the twistor action for conformal gravity to obtain MHV scattering amplitudes not just for conformal gravity, but also for Einstein gravity on backgrounds with non-zero cosmological constant. The new formulae for the gravitational MHV amplitude with cosmological constant arise by summing Feynman diagrams using the matrix-tree theorem. We show that this formula is well-defined (i.e., is independent of certain gauge choices) and that it non-trivially reproduces Hodges' formula for the MHV amplitude in the flat-space limit. We give a preliminary discussion of a MHV formalism for more general amplitudes obtained from the conformal gravity twistor action in an axial gauge. We also see that the embedding of Einstein data into the conformal gravity action can be performed off-shell in twistor space to give a proposal for an Einstein twistor action that automatically gives the same MHV amplitude. These ideas extend naturally to N=4 supersymmetry.

Conformal and Einstein gravity from twistor actions

TL;DR

The paper tackles computing Einstein gravity amplitudes on backgrounds with cosmological constant by embedding Einstein data into conformal gravity and exploiting a twistorial action for conformal gravity. It develops a perturbative MHV framework on twistor space, where Feynman diagrams are summed with the matrix-tree theorem to yield a $\Lambda$-dependent MHV formula that reduces to Hodges' flat-space result as $\Lambda \to 0$. An off-shell Einstein twistor action is proposed, obtained by restricting to Einstein data, and the approach extends naturally to $\mathcal{N}=4$ supersymmetry. The results provide a concrete, gauge-independent derivation of Einstein MHV amplitudes in de Sitter/AdS backgrounds and reveal a determinant structure that generalizes Hodges' formula to nonzero $\Lambda$, with connections to twistor-string constructions. The work opens avenues for MHV-like formalisms in gravity and potential loop computations via an Einstein twistor framework.

Abstract

We use the embedding of Einstein gravity with cosmological constant into conformal gravity as a basis for using the twistor action for conformal gravity to obtain MHV scattering amplitudes not just for conformal gravity, but also for Einstein gravity on backgrounds with non-zero cosmological constant. The new formulae for the gravitational MHV amplitude with cosmological constant arise by summing Feynman diagrams using the matrix-tree theorem. We show that this formula is well-defined (i.e., is independent of certain gauge choices) and that it non-trivially reproduces Hodges' formula for the MHV amplitude in the flat-space limit. We give a preliminary discussion of a MHV formalism for more general amplitudes obtained from the conformal gravity twistor action in an axial gauge. We also see that the embedding of Einstein data into the conformal gravity action can be performed off-shell in twistor space to give a proposal for an Einstein twistor action that automatically gives the same MHV amplitude. These ideas extend naturally to N=4 supersymmetry.

Paper Structure

This paper contains 29 sections, 8 theorems, 180 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Einstein and Conformal Gravity on Asymptotically de Sitter Spaces
  3. Conformal gravity
  4. Einstein gravity amplitudes inside the conformal gravity S-matrix
  5. Relations between Einstein and conformal gravity polarization states
  6. Twistor Action for Conformal (Super-)Gravity
  7. Curved twistor theory
  8. Twistor action
  9. The $\mathcal{N}=4$ minimal twistor action
  10. The volume form
  11. Reduction to Einstein Gravity
  12. The Einstein degrees of freedom
  13. The action and MHV generating functional
  14. Perturbation theory and the MHV Amplitude
  15. A Feynman diagram calculus for the MHV amplitude
  16. ...and 14 more sections

Key Result

Theorem 1

There is a one-to-one correspondence between: (a.) Space-times $M$ with self-dual conformal structure $[g]$, and (b.) twistor spaces $\mathbb{P}\mathscr{T}$ (a complex projective 3-manifold) obtained as a complex deformation of $\mathbb{PT}$ and containing at least one rational curve $X_0$ with norm Here $\mathrm{D}^{3}Z$ is the tautologically defined section of $\Omega^3_{\mathbb{P}\mathscr{T}}(4

Figures (7)

  • Figure 1: Building blocks for Feynman diagrams
  • Figure 2: An isolated (a.) and un-isolated (b.) component.
  • Figure 3: Some diagrams for the 5-point amplitude which have a non-vanishing (a.), or excluded/vanishing (b.) contribution.
  • Figure 4: De Sitter space as the quadric$Q\subset\mathbb{RP}^{5}$and the identification of infinity.
  • Figure 5: De Sitter space on the affine Minkowski patch (a.), and the Poincaré patch (b.)
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1: Penrose Penrose:1976js, Ward Ward:1980am
  • Theorem 2: Mason Mason:2005zm
  • Lemma 5.1
  • Theorem 3: Weighted Matrix-Tree Theorem for Forests
  • Lemma A.1
  • Proposition A.1
  • Proposition A.2
  • Lemma A.2