Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants

Abstract

The $n$-dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$-dimensional twisted cohomology $H_\nabla^n (X, \Omega^\cdot (*S))$ and its dual. Under the condition of general position we present an explicit representation of the standard form by a special (NBC) basis of the twisted cohomology (contiguity relation in positive direction), the variational formula of the corresponding integral in terms of special invariant $1$-forms $\theta_J$ written by Calyley-Menger minor determinants, and a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. Gauss-Manin connections are formulated and are explicitly presented in two simplest cases. In the appendix contiguity relation in negative direction is presented in terms of Cayley-Menger determinants.