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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeOct 27th 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItex* [[Mikhail Kapranov]], [[Eric Vasserot]], _Vertex algebras and formal loop space_, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209--269 (2004) [doi](https://doi.org/10.1007/s10240-004-0023-9) [arXiv:math/0107143](https://arxiv.org/pdf/math/0107143.pdf) > We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety $X$. This is an ind-scheme containing the scheme $\mathcal{L}_0(X)$ of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on $\mathcal{L}(X)$ supported in $\mathcal{L}_0(X)$. We also show that $\mathcal{L}(X)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras. <a href="https://ncatlab.org/nlab/revision/diff/chiral+de+Rham+complex/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/chiral+de+Rham+complex/9">v9</a>, <a href="https://ncatlab.org/nlab/show/chiral+de+Rham+complex">current</a>

   • Mikhail Kapranov, Eric Vasserot, Vertex algebras and formal loop space, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209–269 (2004) doi arXiv:math/0107143

   We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety $X$. This is an ind-scheme containing the scheme $\mathcal{L}_0(X)$ of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on $\mathcal{L}(X)$ supported in $\mathcal{L}_0(X)$. We also show that $\mathcal{L}(X)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.

   diff, v9, current