Twistor-strings, Grassmannians and leading singularities

We derive a systematic procedure for obtaining explicit, l-loop leading singularities of planar N = 4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum space channel diagram. 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 Nk-2MHV amplitudes may be mapped into the Grassmannian G(k, n). For a leading singularity, the image of this map is a 2(n - 2)-dimensional subcycle of G(k, n) and, when 'primitive', it is of exactly the type found from the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung & Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce restrictions on the possible leading singularities of multi-loop (NMHV)-M-p amplitudes. In particular, we argue that no new leading singularities can arise beyond 3 p loops.