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Unification of residues and Grassmannian dualities

2011; Springer Nature; Volume: 2011; Issue: 1 Linguagem: Inglês

10.1007/jhep01(2011)049

1127-2236

Nima Arkani–Hamed, Jacob L. Bourjaily, Freddy Cachazo, Jaroslav Trnka,

Nonlinear Waves and Solitons

The conjectured duality relating all-loop leading singularities of n-particle N k−2MHV scattering amplitudes in $$ \mathcal{N} = 4 $$ 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 N2MHV 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) → 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.