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Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants. (English) Zbl 1436.33012

Summary: 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.

MSC:

33C70 Other hypergeometric functions and integrals in several variables
14F40 de Rham cohomology and algebraic geometry
14H70 Relationships between algebraic curves and integrable systems
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