Twistor-strings, Grassmannians and leading singularities

Abstract

We derive a systematic procedure for obtaining explicit, -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 N k−2 MHV 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 N p MHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops. arXiv:0912.0539v3 [hep-th] 30 Dec 20091 With Penrose conventions for twistor space, MHV amplitudes -those whose 'pure glue' sector involves two negative and arbitrarily many positive helicity gluons -are supported on holomorphic lines in dual twistor space PT * . We abuse notation by taking PT * to be variously a copy of CP 3 , CP 3|4 or the neighbourhood of a line in either of these spaces, according to context. We will often describe PT * in terms of its homogeneous coordinates W α = (λ A , µ A ) and χ a in the supersymmetric case. This space was called twistor space by Witten in [1].

On soft theorems and form factors in N = 4 $$ \mathcal{N}=4 $$ SYM theory

“…and is also interesting in connection with different twistor string theories formulations proposed recently [43][44][45][46][47]. Besides, it allows study of scattering amplitudes with the use of powerful tools from combinatorics and algebraic geometry [40].…”

Six-point next-to-maximally helicity-violating amplitude in maximally supersymmetric Yang-Mills theory

“…One possible extension is the notion of pseudo-conformality: were we to regulate the integral functions W by off-shell continuation, they would become dual conformal invariant, as they are sums of box integrals with the appropriate prefactors. Additional evidence towards a kind of dual conformal invariance comes from the observation [33,34] that leading singularities are dual conformal invariant.…”

“…The structure of leading singularities suggests [33] that new structures beyond the R invariants of one and two loops will be generated at three loops for amplitudes with ten or more external legs, but that no new structures will appear beyond that order. It also suggests that no new invariants should appear beyond two loops for amplitudes with seven or more, but fewer than ten, external legs; and that no new invariants will appear beyond one loop for the six-point amplitude.…”

“…It is possible to argue that, to all orders in the loop expansion, four-dimensional cuts of any planar scattering amplitude in N = 4 SYM, in particular non-MHV amplitudes, have this symmetry. Hints in this direction also come from the Grassmannian interpretation of leading singularities; in that framework it was shown [33,34] that leading singularities are dual conformally invariant.…”

The Yangian origin of the Grassmannian integral

“…Even though each choice of contour appears to produce a dual superconformally invariant leading singularity, the dual superconformal symmetry is not immediately apparent in the initial integral formula. Many developments based on this formula have been pursued [40][41][42][43]. In particular a T-dual version was proposed [44] which is of exactly the same form but in terms of momentum twistors [45], the twistors associated to the dual coordinate space.…”