Scattering equations: from projective spaces to tropical grassmannians

Abstract

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on CP 1 , to higher-dimensional projective spaces CP k−1 . The standard, k = 2 Mandelstam invariants, s ab , are generalized to completely symmetric tensors s a 1 a 2 ...a k subject to a 'massless' condition s a 1 a 2 ···a k−2 b b = 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.

Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

“…where we have denoted the boundary configuration as γ, and the corresponding facet S[γ]. This is a direct consequence of our SE map, and has also been shown in the context of higher-k CHY formula in [14]. Therefore, once we have obtained a boundary configuration we can directly read off the facet by computing the residue of d log R k,n .…”

“…On the other hand, there are other methods for studying these facets, configurations and the related fan N (P(k, n)), which can be derived from the tropical positive Grassmannian. The generalized amplitudes A k,n can be directly obtained from tropical positive Grassmannian, as shown in [14][15][16][17], where cluster variables are used to find a natural triangulation of the fan N (P(k, n)). Each ray of the fan, which corresponds to a vertex of the dual polytope P • (k, n), is associated with a pole of A k,n .…”

“…The latter are natural integrals over positive Grassmannian mod torus action, G + (k, n)/T (string integrals correspond to k = 2), which were first studied very recently in [12]. At leading order, the Grassmannian string integral is computed by the canonical function of a class of polytopes, which are closely related to the positive tropical Grassmannian [13], as shown in [14][15][16][17]). It has also been shown that the canonical function can also be obtained using a CHY formula which are higher-k generalization of the bi-adjoint φ 3 amplitudes (see also [18][19][20]).…”

“…In this paper, we continue to investigate the leading order of Grassmannian string integral for G + (k, n)/T as α ′ → 0, which is the canonical function of polytope denoted as P(k, n), and we call it the generalized amplitudes A k,n = Ω(P(k, n)) since these functions generalize bi-adjoint φ 3 amplitudes for k = 2 case [14]:…”

On polarized scattering equations for superamplitudes of 11D supergravity and ambitwistor superstring

“…More recently an apparently different approach to 11D supergravity and 10D SYM amplitudes was proposed in [28]. It is based on the so-called polarized scattering equation, which can be considered as a kind of square root of the CHY scattering equations [35,36] (actually present already in [37,38,39]; see [40] for recent development and more references). The polarized scattering equation for 6D amplitudes was proposed in [41] while the 11D and 10D polarized scattering equations are among the beautiful findings of [28].…”