Scattering forms and the positive geometry of kinematics, color and the worldsheet
Abstract
The search for a theory of the SMatrix 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 biadjoint 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 biadjoint CHY formula. We also find "scattering forms" on kinematic space for YangMills theory and the Nonlinear Sigma Model, which are dual to the fully colordressed 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 welldefined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for colorkinematics duality.
 Publication:

Journal of High Energy Physics
 Pub Date:
 May 2018
 DOI:
 10.1007/JHEP05(2018)096
 arXiv:
 arXiv:1711.09102
 Bibcode:
 2018JHEP...05..096A
 Keywords:

 Scattering Amplitudes;
 Differential and Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Combinatorics
 EPrint:
 77 pages, 25 figures