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

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