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

TL;DR

This paper introduces and evaluates a family of complex beta integrals extending the Dotsenko-Fateev twin of the Selberg integral to complex spaces with additional parameters, providing new explicit formulas.

Contribution

It defines and computes a new class of complex beta integrals that generalize the Dotsenko-Fateev twin of the Selberg integral with extra parameters.

Findings

Derived explicit formulas for complex beta integrals.

Extended the Dotsenko-Fateev integral to complex spaces with parameters.

Connected the integrals to products of Gamma functions.

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 $\mathbb{C}^m\times \mathbb{C}^{m+1}\times \dots \times \mathbb{C}^n$, which for $m=n$ gives the complex twin of the Selberg integral mentioned above (with three additional integer parameters)