Differential Equations Compatible with KZ Equations

TL;DR

This paper introduces a new system of differential equations compatible with the KZ equations, linking solutions and providing a novel determinant formula for hypergeometric solutions.

Contribution

It defines dynamical differential equations compatible with KZ equations and proves hypergeometric solutions satisfy both, offering a new determinant formula for solutions.

Findings

Hypergeometric solutions also satisfy the dynamical equations.

A new determinant formula for hypergeometric solutions is derived.

Compatibility between dynamical and KZ equations is established.

Abstract

We define a system of "dynamical" differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra $\mathbf{g}$. These are equations on a function of $n$ complex variables $z_i$ taking values in the tensor product of $n$ finite dimensional $\mathbf{g}$-modules. The KZ equations depend on the "dual" variable in the Cartan subalgebra of $\mathbf{g}$. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.