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Multidimensional hypergeometric functions in conformal field theory, algebraic \(K\)-theory, algebraic geometry. (English) Zbl 0747.33002

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 281-300 (1991).
[For the entire collection see Zbl 0741.00019.]
This is a survey article, and at the same time an interesting mathematical essay around the theme of hypergeometric functions, or the hyperplane configurations. It reveals how they are related to so many different topics of mathematics, old and new, such as Beta function, polylogarithms, Bloch-Wigner function, special value of Dedekind zeta- function, local systems, Gauss-Manin connection, cohomology of logarithmic differentials, determinant of matrix of period integrals, monodromy representation of Knizhnik-Zamolodchikov equation, braid groups, conformal field theory, quantum groups.
An elaborated reference list, to which the author himself made important contributions, is an important guide to further study on these subjects.
Reviewer: A.Kaneko (Komaba)

MSC:

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Citations:

Zbl 0741.00019
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