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

Abstract

We continue the study of positive geometries underlying the {\it Grassmannian string integrals}, which are a class of "stringy canonical forms", or stringy integrals, over the positive Grassmannian mod torus action, $G_{+}(k,n)/T$. The leading order of any such stringy integral is given by the canonical function of a polytope, which can be obtained using the Minkowski sum of the Newton polytopes for the regulators of the integral, or equivalently given by the so-called scattering-equation map. The canonical function of the polytopes for Grassmannian string integrals, or the volume of their dual polytopes, is also known as the generalized bi-adjoint ${\varphi }^3$ amplitudes. We compute all the linear functions for the facets which cut out the polytope for all cases up to $n=9$, with up to k=4 and their parity conjugate cases. The main novelty of our computation is that we present these facets in a manifestly gauge-invariant and cyclic way, and identify the boundary configurations of $G_{+}(k,n)/T$ corresponding to these facets, which have nice geometric interpretations in terms of $n$ points in $(k-1)$-dimensional space. All the facets and configurations we discovered up to $n=9$ directly generalize to all $n$, although new types are still needed for higher $n$.

34 pages+ appendices, lots of figures, with typo fixed. Ancilliary files for all the facets up to $k=3$, $n=10$ and $k=4$, $n=9$, as well as ampltitudes up to $n=8$ are included with the submission

References in corpus (3)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Labelled tree graphs, Feynman diagrams and disk integrals

Cited by in corpus (13)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Smoothly Splitting Amplitudes and Semi-Locality
• Tropical fans, scattering equations and amplitudes
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbol Alphabets from Tensor Diagrams
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Planar Matrices and Arrays of Feynman Diagrams: Poles for Higher $k$
• One-loop integrand from generalised scattering equations
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• CHY Theory for Several Fields
• BCFW like recursion for Deformed Associahedron