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 $\prod |x_k|^{σ-1}\,|1-x_k|^{τ-1} \prod|x_k-x_l|^{2θ}$ 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)

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6