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The heat kernel of a Sturm-Liouville operator with logarithmic potential can be described by using the Wiener integral associated with a real hyperplane arrangement. The heat kernel satisfies an infinite-dimensional analog of the Gauss-Manin connection (integrable system), generalizing a variational formula of Schläfli for the volume of a simplex in the space of constant curvature.
The cohomological structure of hypersphere arragnements is given. The Gauss-Manin
connections for related hypergeometrtic integrals are given in terms of invariant forms.
They are used to get the explicit differential formula for the volume of a simplex whose
faces are hyperspheres.
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