Unification of residues and Grassmannian dualities
By N. Arkani-Hamed, J. Bourjaily, F. Cachazo, J. Trnka. 2009
170 citations recorded by the source corpus.
The conjectured duality relating all-loop leading singularities of n-particle Nk−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.
Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/260aeeccce8d119e3d7d56f67b0d70363c96530b
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Citations in the corpus
- Positive geometries and canonical forms (2017, methodology): “This method has been applied ubiquitously in the literature on scattering amplitudes, from the earliest understanding of the relations between leading singularities and the connection between the twistor-string and Grassmannian formalisms [1, 32, 33], to the recent application to scattering equation”
- Evidence for a New Soft Graviton Theorem (2014, background): “(68) In order to compare to the six-point amplitude we take the explicit form given in [22] as M6 = D1 +D1 flip +D2 +D3 +D3 flip +D6 (69) with(6) D1 = 〈23〉〈1|2 + 3|4]7(〈1|2 + 3|4]〈5|3 + 4|2][51]− [12][45]〈51〉s234) 〈15〉〈16〉[23][34]2〈56〉s234〈1|3 + 4|2]〈5|3 + 4|2]〈5|2 + 3|4]〈6|3 + 4|2]〈6|2 + 3|4] +(1 ↔”
- The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM (2010, background): “It is natural to try and connect different n’s by choosing a contour of integration that allows a “particle interpretation”, by which we mean simply that the variety defining the contour for the scattering amplitudes of (n+1) particles differs from the one for n particles only by specifying the extr”
- Motivic amplitudes and cluster coordinates (2013, background): “This Grassmannian, describing the external kinematic data of an amplitude, may look unrelated to those which star in [27–32] and involve also ‘internal’ data related to loop integration variables.”
- Scattering amplitudes and Wilson loops in twistor space (2011, background): “The Yangian invariant quantities appearing in amplitudes of planar N = 4 SYM and the numerous non-trivial relations among them are captured by a Grassmannian contour integral [1, 49, 55, 58, 92], as we discuss briefly in section 7.”