Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants

摘要

The n-dimensional hypergeometric integrals associated with a hypersphere arrangement S are formulated by the pairing of n-dimensional twisted cohomology H-del(n)(X, Omega(.)(*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.