Intersection numbers of twisted homology and cohomology groups associated to the Riemann-Wirtinger integral

TL;DR

This paper investigates the intersection forms on twisted homology and cohomology groups related to the Riemann-Wirtinger integral, providing explicit formulas and applications to monodromy, connection, and contiguity problems.

Contribution

It introduces explicit formulas for intersection numbers of twisted homology and cohomology groups associated with the Riemann-Wirtinger integral, advancing understanding of their monodromy and connection properties.

Findings

Derived explicit formulas for intersection numbers.

Applied formulas to monodromy representation analysis.

Explored connection and contiguity relations.

Abstract

The Riemann-Wirtinger integral is an analogue of the hypergeometric integral, which is defined as an integral on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to study the monodromy representation, connection problems, and contiguity relations.