Tropical critical points of the superpotential of a flag variety

In this paper we investigate the idea of a tropical critical point of the superpotential for the full flag variety of type A. Recall that associated to an irreducible representation of G=SLn(C) are various polytopes whose integral points parameterize a basis for the representation, e.g. the Gelfand-Zetlin polytope. Such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function, and in fact, the function involved coincides with the superpotential from the Landau-Ginzburg model for G/B coming from mirror symmetry. In mirror symmetry a special role is played by the critical points of the superpotential, and motivated by this, we give a definition of the tropical critical points and use it to find a canonical point in each polytope. We then characterise the highest weights for which this tropical critical point is integral and therefore corresponds to a basis vector of the corresponding representation. Finally we give an interpretation of the tropical critical point by constructing a special vector in the representation using Borel-Weil theory and conjecturing a correspondence between this vector and the tropical critical point.