Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes

The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas: (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 no longer a finite, but rather an infinite 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. (3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras.