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Twistor-strings and gravity tree amplitudes

Tim Adamo, Lionel Mason

TL;DR

This work analyzes how Einstein gravity tree amplitudes can be realized within the Berkovits–Witten twistor-string framework, by contrasting conformal gravity amplitudes with Einstein amplitudes and identifying Hodges’ matrix and its generalizations as the central organizing tool for worldsheet contractions. The authors show that, although BW-CSG does not generically yield Einstein amplitudes, a connected-tree restriction to the worldsheet contractions allows a partial twistor-string interpretation that reproduces Hodges’ MHV formula and illuminates the Cachazo–Skinner $N^{k}MHV$ structures via extended deteminants and a weighted Matrix-Tree theorem. The key mechanism is that diagonal entries of generalized Hodges matrices correspond to worldsheet propagators, while reduced determinants sum the connected tree contributions, thereby isolating Einstein content in the MHV sector. However, beyond MHV the connected-tree approach does not fully recover the determinant structures and Vandermonde factors of the Cachazo–Skinner formula, signaling the need for a more complete twistor-string theory (potentially an $\,\mathcal{N}=8$ formulation) to capture the full gravity amplitude landscape.

Abstract

Recently we discussed how Einstein supergravity tree amplitudes might be obtained from the original Witten and Berkovits twistor-string theory when external conformal gravitons are restricted to be Einstein gravitons. Here we obtain a more systematic understanding of the relationship between conformal and Einstein gravity amplitudes in that twistor-string theory. We show that although it does not in general yield Einstein amplitudes, we can nevertheless obtain some partial twistor-string interpretation of the remarkable formulae recently been found by Hodges and generalized to all tree amplitudes by Cachazo and Skinner. The Hodges matrix and its higher degree generalizations encode the world sheet correlators of the twistor string. These matrices control both Einstein amplitudes and those of the conformal gravity arising from the Witten and Berkovits twistor-string. Amplitudes in the latter case arise from products of the diagonal elements of the generalized Hodges matrices and reduced determinants give the former. The reduced determinants arise if the contractions in the worldsheet correlator are restricted to form connected trees at MHV. The (generalized) Hodges matrices arise as weighted Laplacian matrices for the graph of possible contractions in the correlators and the reduced determinants of these weighted Laplacian matrices give the sum of of the connected tree contributions by an extension of the Matrix-Tree theorem.

Twistor-strings and gravity tree amplitudes

TL;DR

This work analyzes how Einstein gravity tree amplitudes can be realized within the Berkovits–Witten twistor-string framework, by contrasting conformal gravity amplitudes with Einstein amplitudes and identifying Hodges’ matrix and its generalizations as the central organizing tool for worldsheet contractions. The authors show that, although BW-CSG does not generically yield Einstein amplitudes, a connected-tree restriction to the worldsheet contractions allows a partial twistor-string interpretation that reproduces Hodges’ MHV formula and illuminates the Cachazo–Skinner structures via extended deteminants and a weighted Matrix-Tree theorem. The key mechanism is that diagonal entries of generalized Hodges matrices correspond to worldsheet propagators, while reduced determinants sum the connected tree contributions, thereby isolating Einstein content in the MHV sector. However, beyond MHV the connected-tree approach does not fully recover the determinant structures and Vandermonde factors of the Cachazo–Skinner formula, signaling the need for a more complete twistor-string theory (potentially an formulation) to capture the full gravity amplitude landscape.

Abstract

Recently we discussed how Einstein supergravity tree amplitudes might be obtained from the original Witten and Berkovits twistor-string theory when external conformal gravitons are restricted to be Einstein gravitons. Here we obtain a more systematic understanding of the relationship between conformal and Einstein gravity amplitudes in that twistor-string theory. We show that although it does not in general yield Einstein amplitudes, we can nevertheless obtain some partial twistor-string interpretation of the remarkable formulae recently been found by Hodges and generalized to all tree amplitudes by Cachazo and Skinner. The Hodges matrix and its higher degree generalizations encode the world sheet correlators of the twistor string. These matrices control both Einstein amplitudes and those of the conformal gravity arising from the Witten and Berkovits twistor-string. Amplitudes in the latter case arise from products of the diagonal elements of the generalized Hodges matrices and reduced determinants give the former. The reduced determinants arise if the contractions in the worldsheet correlator are restricted to form connected trees at MHV. The (generalized) Hodges matrices arise as weighted Laplacian matrices for the graph of possible contractions in the correlators and the reduced determinants of these weighted Laplacian matrices give the sum of of the connected tree contributions by an extension of the Matrix-Tree theorem.

Paper Structure

This paper contains 15 sections, 10 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Twistor-string Formula for Conformal and Einstein Gravity
  3. Twistor-strings for conformal supergravity
  4. Reduction to Einstein gravity
  5. Evaluating the Twistor-string Tree Formula
  6. The $O(\Lambda^0)$ contribution in BW-CSG
  7. Connected trees and Feynman diagrams
  8. The MHV Amplitude
  9. The $O(\Lambda^0)$ case
  10. Hodges' formula from the $O(\Lambda)$ contribution
  11. The N$^{k}$MHV Amplitude
  12. Discussion
  13. Graph Theory
  14. Proof of Lemma \ref{['xilemma']}
  15. Cachazo-Skinner Formula for $\mathcal{N}=4$ Supergravity

Key Result

Lemma 3.1

The contractions $\langle Y_{i}\tau_{j}\rangle_{1}$ vanish and so do not contribute to the correlator $\mathcal{C}_{n,1}$.

Figures (3)

  • Figure 1: Building blocks for Feynman diagrams
  • Figure 2: Feynman diagrams for the 5-point MHV correlator which are admitted (a.), and excluded (b.) from the semi-classical connected tree formalism.
  • Figure 3: The graph G features all possible contractions

Theorems & Definitions (12)

  • Lemma 3.1
  • Lemma 4.1
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 5.1
  • Definition A.1
  • Theorem 1: Matrix-Tree Theorem
  • Lemma A.1: c.f., Stanley:2012 Lemma 9.7
  • ...and 2 more