Notes on twisted period relations for the Wirtinger integral

TL;DR

This paper investigates the structure of twisted homology and cohomology groups related to the Wirtinger integral, leading to simplified twisted period relations for the Gauss hypergeometric function.

Contribution

It provides a detailed analysis of the twisted homology and cohomology groups, resulting in more straightforward forms of the twisted period relations.

Findings

Simplified forms of twisted period relations

Deeper understanding of the structure of twisted homology groups

Enhanced mathematical framework for hypergeometric functions

Abstract

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. The twisted homology and cohomology groups associated with this integral yield twisted period relations. By studying the structure of the twisted homology and cohomology groups in detail, we obtain simple forms of the relations.