Landau-Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety

TL;DR

This paper constructs a Landau-Ginzburg mirror for partial flag varieties by developing solutions to quantum differential equations and qKZ difference equations using hypergeometric functions, linking geometry, quantum cohomology, and integrable systems.

Contribution

It introduces a new basis of solutions for quantum differential equations of partial flag varieties via hypergeometric functions, connecting them with qKZ equations and K-theory.

Findings

Constructed hypergeometric solutions labeled by K-theory elements.

Established a Landau-Ginzburg mirror for partial flag varieties.

Derived a formula for the fundamental Levelt solution.

Abstract

We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the $K$-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and qKZ difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit. Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety.