Dual Superconformal Invariance, Momentum Twistors and Grassmannians

TL;DR

The paper develops a momentum-twistor formulation of dual superconformal invariance in planar $N=4$ SYM, showing that tree amplitudes and box coefficients can be generated by contour integrals over cycles in a Grassmannian ${\rm G}(k,n)$ with momentum-twistor delta constraints. It unifies NMHV and N$^2$MHV structures through $R$-invariants expressed as linear dependencies of momentum supertwistors and clarifies their geometric meaning, including connections to Hodges' polytope picture. The authors extend Arkani-Hamed et al.'s Grassmannian program to momentum twistors, derive explicit contour prescriptions for NMHV and four-musbox coefficients, and discuss all-loop leading singularities and the potential to encode loop information within this framework. The work also relates the Grassmannian construction to polytopes, offering a geometric interpretation that links dual conformal invariants, residue calculus, and polytope volumes. Overall, the approach provides a manifestly dual-conformal, dihedrally symmetric, and signature-free framework for organizing planar amplitudes in $\mathcal{N}=4$ SYM and suggests avenues toward a holistic all-loop description.

Abstract

Dual superconformal invariance has recently emerged as a hidden symmetry of planar scattering amplitudes in N=4 super Yang-Mills theory. This symmetry can be made manifest by expressing amplitudes in terms of `momentum twistors', as opposed to the usual twistors that make the ordinary superconformal properties manifest. The relation between momentum twistors and on-shell momenta is algebraic, so the translation procedure does not rely on any choice of space-time signature. We show that tree amplitudes and box coefficients are succinctly generated by integration of holomorphic delta-functions in momentum twistors over cycles in a Grassmannian. This is analogous to, although distinct from, recent results obtained by Arkani-Hamed et al. in ordinary twistor space. We also make contact with Hodges' polyhedral representation of NMHV amplitudes in momentum twistor space.

Figures (7)

• Figure 1: The Penrose diagram of Minkowski space. Spacelike infinity $i_0$ is a single point, and $i_0$ and $i^\pm$ are all identified in the conformal compactification. They thus correspond to the same point on the Klein quadric, and the same line $I$ in twistor space. The rest of the conformal boundary -- null infinity -- corresponds to points $X$ on the Klein quadric that obey $X^{\alpha\beta}I_{\alpha\beta}=0$, or twistor lines that intersect the distinguished line $I$.
• Figure 2: A scattering amplitude in momentum space, together with the corresponding array of (generically skew) intersecting lines in momentum twistor space. The diagram illustrates the labelling of region momenta $x_i$. Our conventions are such that $x_{ij} = \sum_{k=i}^{j-1} p_k$ and therefore $X_i\cap X_{i+1}=W^i$. Note that the array of twistor lines corresponds precisely to the polygonal contour of the Wilson loop in $x$-space, with edges and vertices interchanged.
• Figure 3: The Riemann sphere given by $\{ {\bf T\cdot W}_\alpha=0\}\subset\mathbb{CP}^5$. The six marked points are the intersections of this $\mathbb{CP}^1$ with the hyperplanes $T_i=0$. The homology class of the displayed contour is invariant (up to a reversal in orientation) under cyclic permutations of the external states.
• Figure 4: The four-mass box coefficient.
• Figure 5: Adding a new particle truncates the previous polytope along a new plane, shown here for $5\to6$ particles, projected into the plane $Z\cdot W^5=0$. The vertex where planes $Z\cdot W^5=0$, $Z\cdot W^1=0$, $Z\cdot W^2=0$ and $Z\cdot W^3=0$ meet becomes spurious when particle 6 is added, and the volume of the resulting polytope stays finite even when this vertex moves to infinity.