Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 051, 28 pages      arXiv:1207.4712
Contribution to the Special Issue on Progress in Twistor Theory

Gravity in Twistor Space and its Grassmannian Formulation

Freddy Cachazo a, Lionel Mason b and David Skinner c
a) Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, Ontario N2L 2Y5, Canada
b) The Mathematical Institute, 24-29 St. Giles', Oxford OX1 3LB, UK
c) DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

Received November 21, 2013, in final form April 23, 2014; Published online May 01, 2014

We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy, with the same initial conditions. As part of the proof, the behavior of the new formula under large BCFW deformations is studied. An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity. In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian. The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV and an $\overline{{\rm MHV}}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.

Key words: twistor theory; scattering amplitudes; gravity.

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