The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM

TL;DR

The paper provides an explicit all-tree formula for N=4 SYM amplitudes across all N^{(k-2)}MHV sectors by recasting Witten’s twistor-string connected prescription as a contour integral with Veronese-map–defined constraints. A deformation parameter t_ll^j smoothly connects this twistor-string form to the Grassmannian integrand L_{n,k}, while preserving parity and soft-limit properties; the authors build the general contour by adding particles sequentially and verify the NMHV, N^2MHV, and N^3MHV cases, including detailed NMHV/N^2MHV examples and higher-point structures. They also establish equivalence with the Dolan–Goddard link-variable formulation by transforming δ(F_ll^j)’s and comparing prefactors, thereby unifying twistor-string and Grassmannian approaches. The framework yields a geometrically transparent, particle-adding procedure with manifest parity and soft limits, offering a systematic route to all tree amplitudes in planar N=4 SYM and deepening the link between twistor-string theory and modern Grassmannian methods.

Abstract

We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those given by Dolan and Goddard, and generalizes the results of earlier work for NMHV and N^2MHV to all N^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.