Twistor-strings and gravity tree amplitudes

TL;DR

This paper explores the relationship between twistor-string theory and gravity tree amplitudes, revealing how Hodges matrices encode correlators and connect Einstein and conformal gravity amplitudes through graph-theoretic methods.

Contribution

It provides a systematic understanding of how twistor-string theory relates to Einstein and conformal gravity amplitudes using Hodges matrices and the Matrix-Tree theorem.

Findings

Hodges matrices encode world sheet correlators in twistor-string theory.

Reduced determinants of Hodges matrices yield Einstein gravity amplitudes.

Weighted Laplacian matrices and the Matrix-Tree theorem explain connected tree contributions.

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.