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Local spacetime physics from the Grassmannian. (English) Zbl 1214.81181

Summary: A duality has recently been conjectured between all leading singularities of \(n\)-particle N\({}^{k - 2}\)MHV scattering amplitudes in \({\mathcal N} = 4\) SYM and the residues of a contour integral with a natural measure over the Grassmannian \(G(k, n)\). In this note we show that a simple contour deformation converts the sum of Grassmannian residues associated with the BCFW expansion of NMHV tree amplitudes to the CSW expansion of the same amplitude. We propose that for general \(k\) the same deformation yields the \((k - 2)\) parameter Risager expansion. We establish this equivalence for all MHV amplitudes and show that the Risager degrees of freedom are non-trivially determined by the \(GL(k - 2)\) “gauge” degrees of freedom in the Grassmannian. The Risager expansion is known to recursively construct the CSW expansion for all tree amplitudes, and given that the CSW expansion follows directly from the (super) Yang-Mills Lagrangian in light-cone gauge, this contour deformation allows us to directly see the emergence of local space-time physics from the Grassmannian.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14M15 Grassmannians, Schubert varieties, flag manifolds
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