Stringy Canonical Forms

TL;DR

This paper introduces stringy canonical forms, a deformation of canonical forms for polytopes inspired by string amplitudes, revealing deep connections between polytope geometry, string theory, and scattering amplitudes.

Contribution

It defines stringy canonical forms as a new extension of canonical forms for polytopes, linking them to string amplitudes and scattering equations, and explores their properties and applications.

Findings

Stringy canonical forms reduce to usual canonical forms as α' approaches 0.

They have simple poles with residues related to facets of the polytope.

Application to the associahedron reproduces the Koba-Nielsen string integral.

Abstract

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce "stringy canonical forms", which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter $\alpha'$. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As $\alpha' \to 0$, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite $\alpha'$, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the $\alpha' \to 0$ limit of tree-level string amplitudes, and scattering equations that appear when studying the $\alpha' \to \infty$ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A for string amplitudes), and other natural integrals over the positive Grassmannian.