Vertex algebras and the formal loop space

Mikhail Kapranov; Eric Vasserot

Publications Mathématiques de l'IHÉS (2004)

  • Volume: 100, page 209-269
  • ISSN: 0073-8301

Abstract

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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.

How to cite

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Kapranov, Mikhail, and Vasserot, Eric. "Vertex algebras and the formal loop space." Publications Mathématiques de l'IHÉS 100 (2004): 209-269. <http://eudml.org/doc/104201>.

@article{Kapranov2004,
abstract = {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.},
author = {Kapranov, Mikhail, Vasserot, Eric},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Ind-pro-schemes; chiral de Rham complex},
language = {eng},
pages = {209-269},
publisher = {Springer},
title = {Vertex algebras and the formal loop space},
url = {http://eudml.org/doc/104201},
volume = {100},
year = {2004},
}

TY - JOUR
AU - Kapranov, Mikhail
AU - Vasserot, Eric
TI - Vertex algebras and the formal loop space
JO - Publications Mathématiques de l'IHÉS
PY - 2004
PB - Springer
VL - 100
SP - 209
EP - 269
AB - 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.
LA - eng
KW - Ind-pro-schemes; chiral de Rham complex
UR - http://eudml.org/doc/104201
ER -

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