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

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.