Binary Geometries, Generalized Particles and Strings, and Cluster Algebras

TL;DR

This paper introduces binary geometries and cluster string integrals linked to Dynkin diagrams, providing a geometric framework for scattering amplitudes that generalizes particle and string interactions without relying on a worldsheet.

Contribution

It defines binary positive and complex geometries for all Dynkin types and constructs cluster string integrals, extending scattering amplitude frameworks beyond traditional methods.

Findings

Binary geometries provide a rigid realization of associahedra.

Cluster string integrals exhibit factorization properties at finite '.

In the ' e0 0 limit, integrals reduce to associahedron canonical forms.

Abstract

We introduce the notion of "binary" positive and complex geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. We also define open and closed "cluster string integrals" associated with these "cluster configuration spaces". The binary geometry of type ${\cal A}$ gives a gauge-invariant description of the usual open and closed string moduli spaces for tree scattering, making no explicit reference to a worldsheet. The binary geometries and cluster string integrals for other Dynkin types provide a generalization of particle and string scattering amplitudes. Both the binary geometries and cluster string integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram. As $\alpha'\to 0$ these cluster string integrals reduce to the canonical forms of the ABHY generalized associahedron polytopes. For classical Dynkin types these are associated with $n$-particle scattering in the bi-adjoint $\phi^3$ theory through one-loop order.