Positive Geometry of the S-Matrix

Abstract

The search for a theory of the S-Matrix has revealed unprecedented structures underlying amplitudes. In this text, we present a new framework for understanding a class of amplitudes that includes Yang-Mills, Non-linear Sigma Model, the bi-adjoint cubic scalar, planar N=4 super Yang-Mills, and more. We introduce positive geometries, which are generalizations of convex polytopes to geometries with higher order (i.e. non-linear) boundaries. Our construction provides a unique differential form called the canonical form of the positive geometry, whose pole structure is completely controlled by the geometric boundaries. The central claim of this text is that positive geometries play a fundamental role in our class of scattering amplitudes, whereby the corresponding canonical form determines a physical quantity. Our primary examples are (1) the bi-adjoint cubic scalar for which the positive geometry is the famous associahedron polytope whose canonical form gives the color-ordered tree amplitude, and (2) planar N=4 super Yang-Mills for which the positive geometry is the amplituhedron whose canonical form gives the scattering integrand. One recurrent theme in our text is that physical properties of amplitudes like local poles and factorization are direct consequences of the boundary structure. We are therefore led to the point of view that locality and unitarity are emergent properties of the positive geometry. Furthermore, we discuss unexpected connections between positive geometry and on-shell diagrams, BCFW recursion, scattering equations, color-kinematics duality and the open string.