Binary Geometries, Generalized Particles and Strings, and Cluster Algebras

Understand

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.