Toda systems and hypergeometric equations

TL;DR

This paper investigates solutions to $SU(n)$ Toda systems with three singular sources, establishing conditions for their relation to hypergeometric equations, constructing solutions, and classifying $SU(3)$ cases under certain assumptions.

Contribution

It provides new existence, construction, and classification results linking $SU(n)$ Toda systems to hypergeometric equations, especially for the $SU(3)$ case.

Findings

Necessary conditions relate Toda systems to hypergeometric equations.

Constructed solutions satisfying interlacing conditions.

Classified $SU(3)$ solutions under a reality assumption.

Abstract

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions for $SU(n)$ Toda systems that satisfy the necessary conditions and also the interlacing conditions from Beukers and Heckman. Finally, for $SU(3)$ Toda systems satisfying the necessary conditions, we classify, under a natural reality assumption, that all the solutions are related to hypergeometric equations. This proof uses the Pohozaev identity.