W-algebras, Gaussian Free Fields and $\mathfrak{g}$-Dotsenko-Fateev integrals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912594963464192 |
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| author | Cerclé, Baptiste |
| author_facet | Cerclé, Baptiste |
| contents | Based on the intrinsic connection between Gaussian Free Fields and the Heisenberg vertex algebra, we study some aspects of the correspondence between probability theory and $W$-algebras. This is first achieved by providing a construction of the $W$-algebra associated to a complex simple Lie algebra $\mathfrak g$ by means of Gaussian Free Fields.
This correspondence in turn allows to translate algebraic statements into actual constraints for free-field correlation functions. This leads to new integrability results for Dotsenko-Fateev integrals associated to $\mathfrak g$, such as Ward identities and the derivation of a new Fuchsian differential equation for deformations of $B_2$-Dotsenko-Fateev integrals arising from the Mukhin-Varchenko conjecture.
Along the proof of this statement we also provide new results on representation theory of $W$-algebras such as the description of some singular vectors for the $W$-algebra associated to $\mathfrak g=B_2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | W-algebras, Gaussian Free Fields and $\mathfrak{g}$-Dotsenko-Fateev integrals Cerclé, Baptiste Mathematical Physics Probability Quantum Algebra Representation Theory 17B69, 60G60, 81R10, 81T40 (Primary) 17B20, 17B68 (Secondary) Based on the intrinsic connection between Gaussian Free Fields and the Heisenberg vertex algebra, we study some aspects of the correspondence between probability theory and $W$-algebras. This is first achieved by providing a construction of the $W$-algebra associated to a complex simple Lie algebra $\mathfrak g$ by means of Gaussian Free Fields. This correspondence in turn allows to translate algebraic statements into actual constraints for free-field correlation functions. This leads to new integrability results for Dotsenko-Fateev integrals associated to $\mathfrak g$, such as Ward identities and the derivation of a new Fuchsian differential equation for deformations of $B_2$-Dotsenko-Fateev integrals arising from the Mukhin-Varchenko conjecture. Along the proof of this statement we also provide new results on representation theory of $W$-algebras such as the description of some singular vectors for the $W$-algebra associated to $\mathfrak g=B_2$. |
| title | W-algebras, Gaussian Free Fields and $\mathfrak{g}$-Dotsenko-Fateev integrals |
| topic | Mathematical Physics Probability Quantum Algebra Representation Theory 17B69, 60G60, 81R10, 81T40 (Primary) 17B20, 17B68 (Secondary) |
| url | https://arxiv.org/abs/2412.12657 |