[Paper Review] Pfaffian of Appell's hypergeometric system $F_4$ in terms of the intersection form of twisted cohomology groups

Yoshiaki Gotō, Jyoichi Kaneko+1 more|arXiv (Cornell University)|Feb 3, 2015

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper derives a simplified Pfaffian system for Appell's hypergeometric function $F_4$ using twisted cohomology and a double cover of the complement of its singular locus. By pulling back the connection to this cover and choosing a suitable frame, the connection matrix becomes logarithmic and integrable, and its structure is fully expressed via the intersection form of twisted cohomology groups, eliminating matrix representations in favor of intrinsic geometric data.

ABSTRACT

We study a Pfaffian of the system of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$ by twisted cohomology groups associated with integrals representing solutions to this system. We simplify its connection matrix by the pull-back under a double cover of the complement of the singular locus. We express the simplified connection matrix in terms of the intersection form of the twisted cohomology groups.

Motivation & Objective

• To simplify the connection matrix of Appell’s $F_4$ hypergeometric system, which is initially complex and non-logarithmic.
• To construct a double cover of the base space that trivializes the monodromy and simplifies the connection structure.
• To express the pull-back connection in terms of the intersection form of twisted cohomology groups, enabling a geometric, matrix-free representation.
• To establish a frame in the pull-back bundle for which the intersection pairing satisfies $d_x \mathcal{I}_c(\varphi_i, \varphi_j^\vee) = 0$, ensuring compatibility with the connection.
• To generalize insights from $F_4$ to higher-rank Lauricella systems by identifying conditions under which a similar simplification is possible.

Proposed method

• Utilizes twisted cohomology groups $H^2(\Omega^\bullet(\mathbb{C}_x^2), \nabla)$ to represent solutions of the $F_4$ system.
• Applies a double cover $\mathbb{C}^2 \to \mathbb{C}^2$ defined by $(y_1,y_2) \mapsto (y_1(1-y_2), y_2(1-y_2))$ to simplify the connection.
• Pulls back the original connection $\nabla_X$ to a new connection $\nabla_Y$ on the cover, yielding a logarithmic form.
• Constructs a frame of the pull-back bundle such that the intersection pairing $\mathcal{I}_c$ satisfies $d_x \mathcal{I}_c(\varphi_i, \varphi_j^\vee) = 0$ for all $i,j$.
• Represents the pull-back connection $\nabla_Y$ using the intersection form $\mathcal{I}_c$ and inverse of submatrices $\widehat{C}_1, \widehat{C}_2, \widehat{C}_{44}$, avoiding explicit matrix entries.
• Employs eigenvectors and eigenvalues of the coefficient matrices $\widehat{\Xi}^i$ to express the connection in terms of $\mathcal{I}_c$ and the frame $\widehat{\varphi}_i$.

Experimental results

Research questions

• RQ1 Can the connection matrix of the $F_4$ system be simplified via a covering map to eliminate apparent singularities?
• RQ2 Does there exist a frame in the twisted cohomology bundle over a covering space for which the intersection pairing is closed under the exterior derivative?
• RQ3 Can the Pfaffian connection of $F_4$ be expressed purely in terms of the intersection form without matrix representations?
• RQ4 How does the monodromy of $F_4$ relate to the intersection form and vanishing cycles on the double cover?
• RQ5 What conditions allow the generalization of this method to higher-rank Lauricella systems?

Key findings

• The pull-back connection $\nabla_Y$ on the double cover is integrable: $d\widehat{\Xi} = \widehat{\Xi} \wedge \widehat{\Xi} = 0$.
• The connection $\nabla_Y$ is expressed in terms of logarithmic 1-forms $dy_1/y_1$, $dy_2/y_2$, etc., with coefficients derived from the intersection form $\mathcal{I}_c$.
• The simplified connection matrix $\widehat{\Xi}$ is expressed via $\mathcal{I}_c$ and inverse submatrices $\widehat{C}_1$, $\widehat{C}_2$, $\widehat{C}_{44}$, avoiding explicit matrix entries.
• The frame $\widehat{\varphi}_1, \dots, \widehat{\varphi}_4$ satisfies $d_x \mathcal{I}_c(\widehat{\varphi}_i, \widehat{\varphi}_j^\vee) = 0$ for all $i,j$, ensuring compatibility with the connection.
• The coefficient matrices $\widehat{\Xi}^1, \widehat{\Xi}^2, \widehat{\Xi}^3$ are expressed in terms of $\widehat{C}$, eigenvectors, and eigenvalues, with explicit formulas involving $\lambda_i$ and $c_i$ parameters.
• The connection is fully reconstructed from the intersection form $\mathcal{I}_c$, demonstrating that the geometric structure of twisted cohomology fully encodes the differential system.