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

Abstract

We continue the study of positive geometries underlying the 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 φ 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.

Algebraic branch points at all loop orders from positive kinematics and wall crossing

“…However, it has been proven that the n-point amplitude in N = 4 pSYM has a finite number of branch points associated with solutions to the Landau equations [45], implying that the symbol alphabet could also be finite. Following this train of thought, several truncation procedures have been proposed, motivated by connections between stringy canonical forms and compactifications of configuration spaces [32,72,90,91].…”

“…One path is investigating the explicit calculations in ref. [90] that connect boundary points of Gr(k, n)/T to f i using generalized scattering equations.…”

Towards the Gravituhedron: new expressions for NMHV gravity amplitudes

“…Some progress has been made for more general φ p scalar theories [43][44][45][46][47][48][49][50]. Positive geometry was tied to certain one-loop integrands [51], string amplitudes [52][53][54][55][56] as well as cosmological [57,58] or CFT correlators [59].…”

One-loop integrand from generalised scattering equations

“…Moduli space of open string worldsheet is an associahedron, and the scattering equations of CHY act as diffeomorphism between the associahedron in the worldsheet and that described in the kinematic space. This led to a fascinating series of investigations into the connection between scattering amplitudes and positive geometries for various scalar theories [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. Stringy deformations of the scattering forms have been considered in [79].…”