Riemann-Wirtinger integrals on the product of two one-dimensional complex tori

TL;DR

This paper extends the concept of Riemann-Wirtinger integrals to the product of two complex tori, analyzing their cohomological structure and deriving related differential equations.

Contribution

It introduces a new generalization of the Riemann-Wirtinger integral on product tori and studies its associated twisted cohomology and differential equations.

Findings

Structure of the twisted cohomology group characterized

Differential equations satisfied by the integral derived

Framework for analyzing integrals on product tori established

Abstract

The Riemann-Wirtinger integral is an analogue of the hypergeometric integral defined on a one-dimensional complex torus. As a generalization, we define the Riemann-Wirtinger integral on the product of two one-dimensional complex tori. We study the structure of the twisted cohomology group associated with the Riemann-Wirtinger integral and derive a system of differential equations satisfied by this integral.