Stringy Canonical Forms

TL;DR

The paper introduces stringy canonical forms as α′-deformations of canonical forms for polytopes, unifying regulated integrals with positive geometries and string theory. It demonstrates that the α′→0 limit recovers the canonical function (or dual-volume) of the Minkowski sum of Newton polytopes, while the α′→∞ regime through saddle points yields scattering-equation pushforwards to these polytopes, reproducing CHY-type structures. By applying the framework to ABHY associahedra, cluster algebras, and Grassmannians, it connects open-string Koba–Nielsen integrals and their factorization to geometric realizations, and extends these ideas to complex/closed-string analogues and tropical compactifications. The work also develops dual u-variables and big-polyhedron formalisms, enabling a universal, geometry-driven perspective on convergence, residues, and factorization across a wide class of stringy integrals, including cluster and Grassmannian variants, with numerous examples and conjectures for future study. Overall, it provides a principled, polyhedral, and tropical-geometric foundation for understanding string amplitudes as intrinsic pushforwards of canonical forms, with broad implications for positive geometries and beyond.

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 $α'$. 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 $α' \to 0$, 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 $α'$, 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 $α' \to 0$ limit of tree-level string amplitudes, and scattering equations that appear when studying the $α' \to \infty$ 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.

Figures (8)

• Figure 1: A pentagon ${\mathcal{P}}$ containing the origin, its normal fan ${\mathcal{N}}({\mathcal{P}})$ with maximal cones labeled by vertices of ${\mathcal{P}}$, and the dual polytope ${\mathcal{P}}^\circ$ with facets labeled by vertices of ${\mathcal{P}}$.
• Figure 2: The regions $\exp(C_v)$ for the normal fan of Figure \ref{['fig:pentex']}. The curve separating $\exp(C_1)$ from $\exp(C_5)$ is $xy=1$.
• Figure 3: The domains of linearity of $X+Y - \min(0,X) - \min(0,Y) - \min(0,X,X+Y)$.
• Figure 4: The ABHY associahedron ${\cal A}_2$ (pentagon) as a Minkowski sum.
• Figure 5: The generalized associahedron (or cyclohedron) for ${\cal A}(B_2)$ as a Minkowski sum.

Theorems & Definitions (18)

• Claim 1
• Remark 2.1
• Example 2.1: Interval
• Example 2.2: Triangle, quadrilateral and pentagon
• Claim 2
• Claim 3
• Example 4.1
• Claim 4
• Example 7.1: Interval
• Example 7.2: Pentagon