Higher Chiral Algebras in a Polysimplicial Model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916790669410304 |
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| author | Felder, Laura O. Gui, Zhengping Young, Charles A. S. |
| author_facet | Felder, Laura O. Gui, Zhengping Young, Charles A. S. |
| contents | Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this paper we give an algebraic construction of a chiral algebra on $\mathbb{A}^n$; this can be seen as an algebraic construction of a higher-dimensional vertex algebra.
We introduce a model, in dg commutative algebras, of the derived algebra of functions on the configuration space of $k$ distinct labelled marked points in $\mathbb{A}^n$. Working in this model -- which we call the polysimplicial model -- we obtain a dg operad of chiral operations on a degree-shifted copy of the canonical sheaf. We prove that there is a quasi-isomorphism, to this dg operad, from the Lie-infinity operad. This result makes the shifted canonical sheaf into a first example of a homotopy polysimplicial chiral algebra on $\mathbb{A}^n$, in a sense which generalizes to higher dimensions Malikov and Schechtman's notion of a homotopy chiral algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Chiral Algebras in a Polysimplicial Model Felder, Laura O. Gui, Zhengping Young, Charles A. S. Quantum Algebra High Energy Physics - Theory Mathematical Physics Algebraic Geometry Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this paper we give an algebraic construction of a chiral algebra on $\mathbb{A}^n$; this can be seen as an algebraic construction of a higher-dimensional vertex algebra. We introduce a model, in dg commutative algebras, of the derived algebra of functions on the configuration space of $k$ distinct labelled marked points in $\mathbb{A}^n$. Working in this model -- which we call the polysimplicial model -- we obtain a dg operad of chiral operations on a degree-shifted copy of the canonical sheaf. We prove that there is a quasi-isomorphism, to this dg operad, from the Lie-infinity operad. This result makes the shifted canonical sheaf into a first example of a homotopy polysimplicial chiral algebra on $\mathbb{A}^n$, in a sense which generalizes to higher dimensions Malikov and Schechtman's notion of a homotopy chiral algebra. |
| title | Higher Chiral Algebras in a Polysimplicial Model |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Algebraic Geometry |
| url | https://arxiv.org/abs/2506.09728 |