Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants

TL;DR

This paper studies hypergeometric integrals linked to hypersphere arrangements, providing explicit cohomology representations, variational formulas involving Cayley-Menger determinants, and formulations of the Gauss-Manin connection in specific cases.

Contribution

It introduces explicit cohomology bases and variational formulas for hypergeometric integrals associated with hypersphere arrangements, connecting them with Cayley-Menger determinants.

Findings

Explicit representation of standard forms using NBC basis

Variational formulas involving Cayley-Menger minors

Gauss-Manin connection formulated in simple cases

Abstract

The n-dimensional hypergeometric integrals associated with a hypersphere arrangement are formulated by the pairing of n-dimensional twisted cohomology and its dual. Under the condition of general position there are stated some results which concern an explicit representation of the standard form by a special (NBC) basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms written by Cayley-Menger minor determinants. Gauss-Manin connection is also formulated and is explicitly presented in two simplest cases.