In this talk, I will talk about a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics.
(1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians.
(2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is a product indexed by prime modules of a quantum affine
algebra. We give a general formula of u-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type A) and Auslander-Reiten
quivers of Grassmannian cluster categories.
This is joint work with Nick Early.