Relation Between Hyperelliptic Integrals

TL;DR

This paper investigates a property of hyperelliptic integrals related to their derivatives and demonstrates how this property underpins solutions to the level zero Knizhnik-Zamolodchikov equation.

Contribution

It identifies a simple derivative property of hyperelliptic integrals that explains their role in solving a specific differential equation.

Findings

Derivatives of hyperelliptic integrals are linear combinations of the same integrals.

The property explains the solution structure of the level zero Knizhnik-Zamolodchikov equation.

The observed property holds for hyperelliptic surfaces of arbitrary genus.

Abstract

A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We check that this property is responsible for the solution to the level zero Knizhnik-Zamolodchikov equation given in terms of hyperelliptic integrals.