Abstract
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors {mathrm{s}}_{a_1{a}_2dots {a}_k} subject to a ‘massless’ condition {mathrm{s}}_{a_1{a}_2dots {a}_{k-2}bb}=0 and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.
Highlights
Sab are known as Mandelstam invariants and denotes the SL(2, C)-invariant combination of the homogeneous variables of points a and b
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on CP1, to higher-dimensional projective spaces CPk−1
We find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones
Summary
We have introduced the shorthand notation |abc| for the determinant in S3 At first sight, these are 2n equations for 2n variables. As it is familiar in the k = 2 case, these equations are covariant under SL(3, C) transformations which is the automorphism group of CP2 This means that 8 equations are redundant and that we can use the group to fix the positions of 4 points (each having two coordinates) to generic, i.e., non-collinear, positions. This makes it clear that n ≥ 4 in order to have a stable CP2, i.e., one in which all the automorphism group is fixed.
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