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The monodromy representation and twisted period relations for Appell’s hypergeometric function \(F_4\). (English) Zbl 1327.32001

It is known that the Appell’s hypergeometric series in two complex variables admits certain integral representations that can be understood as 2-dimensional extensions of the conventional contour integrals. Using the notion of the twisted \(k\)-chain as the \(k\)-chain loaded with a branch of the multivalued function in two complex variables used in one of the above mentioned integral representations, such integrals can be viewed as the pairings between 2-forms and elements of the 2nd homology group. The authors construct a basis of the homology group and, using a suitably defined “intersection form”, compute the relevant “intersection numbers” of its elements. The latter are important for an explicit characterization of the eigenspaces of the monodromy matrices of Appell’s local differential system. The twisted cohomology group is described using the “twisted” exterior derivative, \(\nabla=d_t+\omega\wedge\), with a 1-form \(\omega\) reproducing the singularity structure of Appell’s system. The authors also introduce an integral pairing between 2-forms, also called “intersection form”, and compute it for four particular forms associated with the Appell’s system. Finally, considering the “twisted periods”, i.e. the pairings between the elements of the twisted 2-chains and the twisted 2-forms, the authors find a relation between the period matrices and thus the quadratic relations between the Appell’s series called the “twisted period relations”.

MSC:

32A05 Power series, series of functions of several complex variables
33C65 Appell, Horn and Lauricella functions
32G20 Period matrices, variation of Hodge structure; degenerations
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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References:

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