Binary geometries, generalized particles and strings, and cluster algebras

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 $\mathcal{A}$ gives a gauge-invariant description of the usual open and closed string moduli spaces for tree scattering, making no explicit reference to a world sheet. 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 ${\alpha }^{\prime }$, obtained simply by removing nodes of the Dynkin diagram. As ${\alpha }^{\prime }\to 0$ these cluster string integrals reduce to the canonical forms of the Arkani-Hamed-Bai-He-Yan (ABHY) generalized associahedron polytopes. For classical Dynkin types these are associated with $n$-particle scattering in the biadjoint ${\varphi }^3$ theory through one-loop order.