Tropical critical points of the superpotential of a flag variety

TL;DR

This paper explores tropical critical points of the superpotential in flag varieties, linking mirror symmetry, geometric crystals, and representation bases, and characterizes when these points are integral for basis correspondence.

Contribution

It introduces a definition of tropical critical points for flag varieties, constructs a canonical point in each polytope, and relates these points to basis vectors via Borel-Weil theory.

Findings

Defined tropical critical points in the context of flag varieties

Identified conditions for tropical critical points to be integral

Connected tropical critical points to basis vectors in representations

Abstract

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 characterize 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.