Twistor-Strings, Grassmannians and Leading Singularities

TL;DR

This paper develops a systematic method to compute leading singularities of planar N=4 super Yang-Mills amplitudes using twistor space and connects these to Grassmannian geometry, providing insights into the structure of multi-loop amplitudes.

Contribution

It introduces a procedure to derive explicit L-loop leading singularities in twistor space from momentum space diagrams and links them to Grassmannian residues, advancing the understanding of amplitude structures.

Findings

Explicit formulas for L-loop leading singularities in twistor space.

Mapping of twistor-string moduli space into Grassmannian G(k,n).

Restrictions on leading singularities beyond 3p loops.

Abstract

We derive a systematic procedure for obtaining an explicit, L-loop leading singularities of planar N=4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum space channel diagrams. The expressions are given as integrals over the moduli of connected, nodal curves in twistor space whose degree and genus matches expectations from twistor-string theory. We propose that a twistor-string theory for pure N=4 super Yang-Mills, if it exists, is determined by the condition that these leading singularity formulae arise as residues when an unphysical contour for the path integral is used, by analogy with the momentum space leading singularity conjecture. We go on to show that the genus g twistor-string moduli space for g-loop N^{k-2}MHV amplitudes may be mapped into the Grassmannian G(k,n). Restricting to a leading singularity, the image of this map is a 2(n-2)-dimensional subcycle of G(k,n) of exactly the type found from the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung and Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce restrictions on the possible leading singularities of multi-loop N^pMHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops.