Scattering forms and the positive geometry of kinematics, color and the worldsheet

Abstract

The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar N = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint φ 3 scalar theory, we establish a direct connection between its "scattering form" and a classic polytope-the associahedron-known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the "canonical form" associated with this "positive geometry". Fundamental physical properties such as locality and unitarity, as well as novel "soft" limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact-"Color is Kinematics"-whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality. arXiv:1711.09102v2 [hep-th]

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

“…with X 1 = s 345 , X 2 = s 123 . The Newton polytopes from the regulator R 3,5 are two intervals The facets of P (3,5), which correspond to the poles of A 3,5 , are…”

“…The list of these 16 facets, which can be expanded in X's and c's, is a rather compact and invariant way to describe the polytope. Note that there are three kinds of facets for P (3,6), which are all three-dimensional polytopes and we draw them in figure 3.…”

“…where equivalently the canonical function can be written as a generalization [3] of the CHY formula for bi-adjoint φ 3 amplitudes [4,5] (see also [6,7]).…”

“…As discussed in [1], open-string amplitudes can be thought of as the stringy canonical form for the so-called ABHY associahedron [3]; in the α ′ → 0 limit we obtain the fieldtheory, bi-adjoint φ 3 amplitudes as its canonical function, which can also be obtained using the pushforward/CHY formula from the scattering-equation map for the associahedron [3]. As shown in [1], the two most natural extensions of string amplitudes are the so-called cluster string integrals and the Grassmannian string integrals.…”

“…It is convenient to organize them into cyclic classes: when k and n are coprime, the length for each cyclic class is always n, but if there is a common factor of k and n, the length of some classes may be smaller. For (k, n) = (3, 7), (3,8), (3,10) and (4,9), the number of facets/configurations is 7 × 6, 8 × 15, 10 × 314 and 9 × 2155, respectively. For (3,6), (3,9) and (4,8), we have 6 × 2 + 2 × 2, 9 × 50 + 3 × 7 and 8 × 42 + 4 × 3 + 2 × 6, respectively.…”