Binary geometries, generalized particles and strings, and cluster algebras

The authors study the fundamental properties of scattering amplitudes particles in any spacetime dimension. They introduce binary geometries, giving a completely rigid geometric realization combinatorics generalized associahedra attached to Dynkin diagram. Furthermore, they define open and closed ``cluster string integrals'', which provide generalization particle amplitudes, enjoy remarkable factorization at finite ${\ensuremath{\alpha}}^{\ensuremath{'}}$.