Binary Geometries, Generalized Particles and Strings, and Cluster Algebras

TL;DR

The paper introduces binary geometries as a rigid, $U(\Phi)$-based realization of the compatibility structure of generalized associahedra for finite-type cluster algebras, using $u$-variables that satisfy $u_a+\prod_b u_b^{b||a}=1$. It then constructs cluster configuration spaces $U(\Phi)$ and their positive parts $U^+(\Phi)$, develops open and closed cluster string integrals with $\,\alpha'$-regulated canonical forms, and demonstrates elegant factorization properties by removing Dynkin diagram nodes, recovering ABHY polytopes in the $\alpha'\to 0$ limit. For type ${\cal A}$, these structures reproduce gauge-invariant descriptions of open/closed string moduli and their amplitudes; for other Dynkin types they yield generalized string amplitudes with rich finite-$\alpha'$ factorization patterns and connections to one-loop bi-adjoint $\phi^3$ theory. The work also provides finite-field point counts to extract topological data of $U(\Phi)$, presents explicit formulas for several types, and outlines substantial open questions about the physical interpretation and further mathematical development of cluster string amplitudes. Overall, the framework offers a unifying geometric and algebraic approach to scattering across Dynkin types, with deep factorization and combinatorial structure and broad avenues for future exploration.

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 $α'$, obtained simply by removing nodes of the Dynkin diagram. As $α'\to 0$ 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 $φ^3$ theory through one-loop order.