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

TL;DR

This work analyzes Appell's hypergeometric system $ ext{F}_4(a,b,c)$ by constructing integral representations for four independent solutions and formalizing twisted (co)homology frameworks. It develops a twisted 2-cycle picture, computes an intersection form, and derives the monodromy representation via explicit matrices that are valid even for integer parameters. The paper then builds twisted cohomology and obtains a diagonal intersection matrix, enabling explicit twisted period relations that yield quadratic relations among ${F}_4$-solutions with shifted parameters. These results advance understanding of the analytic structure of $ ext{F}_4$ and provide a robust, parameter-independent framework for monodromy and period relations, with potential extensions to Lauricella systems and Pfaffian formulations.

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 $\mathcal{F}_4$.

Figures (4)

• Figure 1: Domains of the integrals
• Figure 2: Domains of integrals
• Figure 3: Cycles $\mathit{\Delta}_5,\dots,\mathit{\Delta}_8$
• Figure 4: Pole divisor of $\omega_x$

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 3.1
• Proposition 3.1
• Proposition 4.1
• Lemma 4.1
• Remark 4.1
• Lemma 4.2
• Lemma 4.3
• Remark 4.2
• Proposition 4.2