On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916077187891200 |
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| author | Neretin, Yury A. |
| author_facet | Neretin, Yury A. |
| contents | 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) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09112 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions Neretin, Yury A. Classical Analysis and ODEs Mathematical Physics Representation Theory 33B15, 33C70 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) |
| title | On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions |
| topic | Classical Analysis and ODEs Mathematical Physics Representation Theory 33B15, 33C70 |
| url | https://arxiv.org/abs/2212.09112 |