Unification of residues and Grassmannian dualities

  • N. Arkani-Hamed
  • J. Bourjaily
  • F. Cachazo
  • J. Trnka


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.


Supersymmetry and Duality Supersymmetric gauge theory Extended Supersymmetry String Duality 


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Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • N. Arkani-Hamed
    • 1
  • J. Bourjaily
    • 1
    • 3
  • F. Cachazo
    • 1
    • 2
  • J. Trnka
    • 1
    • 3
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Perimeter Institute for Theoretical PhysicsN WaterlooCanada
  3. 3.Department of PhysicsPrinceton UniversityPrincetonU.S.A.

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