ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

Understand

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes.

• We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type.
• Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras.
• In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.