Unification of Residues and Grassmannian Dualities

TL;DR

This work unifies the residues contributing to tree-level amplitudes in N=4 SYM into a single algebraic variety on the Grassmannian, revealing a particle interpretation that allows higher-point amplitudes to be built by adding one particle at a time. It establishes a deep link between the Grassmannian contour approach and the connected prescription of twistor string theory via Veronese mappings and a deformation parameter t, proving t-independence for NMHV amplitudes and formulating a general duality between two seemingly different formulations. The NMHV and N^2MHV analyses provide explicit constructions, recursion relations, and geometric residue classifications (including non-consecutive minors and Veronese constraints), along with detailed examples at n=7,8. The results suggest a universal, deformation-controlled relationship between grassmannian-based leading singularities and twistor-string descriptions, with implications for locality, soft limits, and potential all-loop extensions.

Abstract

The conjectured duality relating all-loop leading singularities of n-particle N^(k-2)MHV scattering amplitudes in N=4 SYM to a simple contour integral over the Grassmannian G(k,n) makes all the symmetries of the theory manifest. Every residue is individually Yangian invariant, but does not have a local space-time interpretation--only a special sum over residues gives physical amplitudes. In this paper we show that the sum over residues giving tree amplitudes can be unified into a single algebraic variety, which we explicitly construct for all NMHV and N^2MHV amplitudes. Remarkably, this allows the contour integral to have a "particle interpretation" in the Grassmannian, where higher-point amplitudes can be constructed from lower-point ones by adding one particle at a time, with soft limits manifest. We move on to show that the connected prescription for tree amplitudes in Witten's twistor string theory also admits a Grassmannian particle interpretation, where the integral over the Grassmannian localizes over the Veronese map from G(2,n) to G(k,n). These apparently very different theories are related by a natural deformation with a parameter t that smoothly interpolates between them. For NMHV amplitudes, we use a simple residue theorem to prove t-independence of the result, thus establishing a novel kind of duality between these theories.

Figures (6)

• Figure 1: The two classes of solutions to setting minors $(2\,3\,4\,5), (1\,3\,4\,5),$ and $(2\,3\,5\,6)$, to zero. In solution A, line $[3\,5]$ lies on the plane $[1\,2\,4]$ and passes through the point $2$; for B, the line $[3\,5]$ lies on the plane $[6\,2\,4]$ and passes through the point $4$.
• Figure 2: The two classes of solutions to setting minors $(4\,5\,6\,7)=(5\,6\,7\,8)=(6\,7\,8\,1)=0$, where the possible configurations for the line $[6\,7]$ are indicated.
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