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

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

@article{PMIHES_2004__100__209_0,
     author = {Kapranov, Mikhail and Vasserot, Eric},
     title = {Vertex algebras and the formal loop space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {100},
     year = {2004},
     pages = {209-269},
     doi = {10.1007/s10240-004-0023-9},
     zbl = {1106.17038},
     mrnumber = {2102701},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2004__100__209_0}
}
Kapranov, Mikhail; Vasserot, Eric. Vertex algebras and the formal loop space. Publications Mathématiques de l'IHÉS, Volume 100 (2004) pp. 209-269. doi : 10.1007/s10240-004-0023-9. http://www.numdam.org/item/PMIHES_2004__100__209_0/

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