Hypergeometric-type integrals and the sl(2,C) Knizhnik-Zamolodchikov equation. (English) Zbl 0722.33007

K. Aomoto [J. Math. Soc. Japan 39, 191-208 (1987; Zbl 0619.32010)] has studied the integral \[ I_{\alpha}=\int...\int \prod^{m}_{i=1}dt_ i\prod_{i<j}(t_ i-t_ j)^ c\prod^{m}_{i=1}\prod^{n-1}_{a=1}(t_ i-z_ a)^{r_ a}(t_ 1-z_{a_ 1})^{-1}\times...\times (t_ m-z_{a_ m})^{-1}, \] where \(\alpha =(\alpha_ 1,...,\alpha_{n-1})\) and \(\alpha_ a=\#\{i|\) \(a_ i=a\}\). This integral satisfies the equations \[ (1)\quad \sum_{j}(r_ j+\alpha_ jc/2)I_{\alpha_ 1,...,\alpha_ j+1,...,\alpha_{n-1}}=0, \]
\[ (2)\quad dI_{\alpha}=\sum_{a\neq b}\alpha_ a(r_ a+\alpha_ bc/2)(I_{\alpha}-I_{\alpha_{ab}})d \log (z_ a-z_ b), \] where \(\alpha_{ab}\) is obtained from \(\alpha\) by changing \(\alpha_ a\) by \(\alpha_ a-1\) and \(\alpha_ b\) by \(\alpha_ b+1\). The main result of this paper is a proof of equivalence of the equations (1) and (2) with well-known Knizhnik-Zamolodchikov equation for the Lie algebra sl(2,C).
Reviewer: A.Klimyk (Kiev)


33C70 Other hypergeometric functions and integrals in several variables
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
33C80 Connections of hypergeometric functions with groups and algebras, and related topics


Zbl 0619.32010
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