The monodromy representation and twisted period relations for Appell's hypergeometric function F_4

TL;DR

This paper investigates the monodromy representation and twisted period relations of Appell's hypergeometric function F_4 by analyzing its differential system, integral solutions, and associated twisted (co)homology groups.

Contribution

It introduces explicit integral representations of solutions and derives the monodromy and period relations using twisted (co)homology, advancing understanding of F_4's structure.

Findings

Explicit integral representations for solutions of F_4

Monodromy representation of the differential system

Twisted period relations for fundamental solutions

Abstract

We consider the system $\mathcal{F}_4(a,b,c)$ of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$. We find the integral representations for four linearly independent solutions expressed by the hypergeometric series $F_4$. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of $\mathcal{F}_4(a,b,c)$ and the twisted period relations for the fundamental systems of solutions of $\mathcal{F}_4$.