On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions

Abstract

The Selberg integral has a twin (`the Dotsenko--Fateev integral') of the following form. We replace real variables $x_k$ in the integrand $\prod |x_k{|}^{\sigma -1}|1-x_k{|}^{\tau -1}\prod |x_k-x_l{|}^{2\theta }$ of the Selberg integral by complex variables $z_k$, integration over a cube we replace by an integration over the whole complex space ${\mathbb{C}}^n$. According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces ${\mathbb{C}}^m\times {\mathbb{C}}^{m+1}\times \cdots \times {\mathbb{C}}^n$, which for $m=n$ gives the complex twin of the Selberg integral mentioned above (with three additional integer parameters)

arXiv comment15pages; typos are corrected; some proofs are extended

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