Twistor-Strings, Grassmannians and Leading Singularities

TL;DR

This work develops a twistor-space framework to compute leading singularities of planar N=4 SYM across all loops by mapping momentum-space channel diagrams to moduli spaces of nodal curves in twistor space. It establishes a genus-g twistor-string moduli space mapping into the Grassmannian G(k,n), where leading singularities correspond to specific 2(n-2)-dimensional cycles, matching the Arkani-Hamed–Cachazo–Cheung–Kaplan Grassmannian residue formula for primitive KS configurations. It then derives a loop bound: no new leading singularities appear beyond 3p loops for N^pMHV amplitudes, by analyzing twistor-space supports and Grassmannian residues. The work also outlines a conjectured twistor-string for pure N=4 SYM and highlights the Grassmannian duality as a central organizing principle for all-loop leading singularities.

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.

Figures (20)

• Figure 1: The $\rm MHV$ tree amplitude is supported on a line in dual twistor space.
• Figure 2: The box coefficient $A^{(0)}_{\rm MHV}R_{n;ab}$ is supported on three, pairwise intersecting lines in $\mathbb{PT}^*$. The curve is connected, but is not irreducible, corresponding to the fact that it gives the twistor support of the leading singularity of a 1-loop amplitude, rather than the loop amplitude itself. The marked points may be located anywhere along the three lines, except that $W_n$ lies at the intersection of two lines, as shown. Note that state $n$ is attached to the 3-particle $\overline{\rm MHV}$ amplitude (denoted by a filled blob in the diagram on the left).
• Figure 3: The $\mathbb{PT}^*$ support of the two classes of contribution to the ${\rm N}^2{\rm MHV}$ tree amplitude. Each term is supported on two planes in $\mathbb{PT}^*$, with marked points lying on three pairwise intersecting lines in each plane. The intersection of the two planes is a common edge of the triangles. We have taken this figure from Korchemsky:2009jv, except that we have redrawn it to make the $\mathbb{PT}^*$ structure more transparent.
• Figure 4: Type A diagrams correspond to a momentum space leading singularity in the pentabox channel shown on the left of this figure. The rest of the figure illustrates the explicit calculation of the leading singularity in this channel.
• Figure 5: The pentabox corresponding to the type B contributions to the N$^2$MHV tree amplitude.