Vertex algebras and the formal loop space
Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 209-269.

We construct a certain algebro-geometric version ℒ(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme 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 ℒ(X) supported in ℒ 0 (X). We also show that 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.

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Kapranov, Mikhail; Vasserot, Eric. Vertex algebras and the formal loop space. Publications Mathématiques de l'IHÉS, Tome 100 (2004), pp. 209-269. doi : 10.1007/s10240-004-0023-9. http://archive.numdam.org/articles/10.1007/s10240-004-0023-9/

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