Vertex algebras and algebraic curves.

*(English)*Zbl 0997.17015
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 299-339, Exp. No. 875 (2002).

In this Bourbaki exposé the theory of vertex algebras is reviewed with a particular emphasis on their algebro-geometric interpretations and applications. The author gives an overview of results that are developed further in the book [D. Ben-Zvi and E. Frenkel, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs. 88. Providence, American Mathematical Society (2001; Zbl 0981.17022)]. It can be highly recommended as an introduction to this book and to the results presented there.

After recalling the axiomatic definition of a vertex algebra, the most important properties and examples are given. Beside the Heisenberg, affine Kac-Moody and the Virasoro algebra, the \({\mathcal W}\)-algebras are also considered. It is shown how to make a conformal vertex algebra coordinate independent by attaching to it a vector bundle with a flat connection on a formal disc, equipped with an intrinsic operation.

A definition of the space of conformal blocks associated to a conformal vertex algebra and an algebraic curve is given. Varying the curve and the other data on it, a sheaf on the corresponding moduli space is obtained. It is shown that this sheaf carries the structure of a twisted \({\mathcal D}\)-module. From the study of these \({\mathcal D}\)-modules, geometric information about the moduli spaces can be obtained. The vertex algebras appear as the local objects controlling deformations of curves with various extra structures. The relation between vertex algebras and the Beilinson-Drinfeld chiral algebras is discussed.

As examples of the relevance of vertex algebras to algebraic geometry, the constructions due to Beilinson and Drinfeld in the context of the conjectural geometric Langlands correspondence and the construction of the chiral de Rham complex on an arbitrary smooth projective variety due to F. Malikov, V. Schechtman and A. Vaintrob [Commun. Math. Phys. 204, 439–473 (1999; Zbl 0952.14013)] are explained.

For the entire collection see [Zbl 0981.00011].

After recalling the axiomatic definition of a vertex algebra, the most important properties and examples are given. Beside the Heisenberg, affine Kac-Moody and the Virasoro algebra, the \({\mathcal W}\)-algebras are also considered. It is shown how to make a conformal vertex algebra coordinate independent by attaching to it a vector bundle with a flat connection on a formal disc, equipped with an intrinsic operation.

A definition of the space of conformal blocks associated to a conformal vertex algebra and an algebraic curve is given. Varying the curve and the other data on it, a sheaf on the corresponding moduli space is obtained. It is shown that this sheaf carries the structure of a twisted \({\mathcal D}\)-module. From the study of these \({\mathcal D}\)-modules, geometric information about the moduli spaces can be obtained. The vertex algebras appear as the local objects controlling deformations of curves with various extra structures. The relation between vertex algebras and the Beilinson-Drinfeld chiral algebras is discussed.

As examples of the relevance of vertex algebras to algebraic geometry, the constructions due to Beilinson and Drinfeld in the context of the conjectural geometric Langlands correspondence and the construction of the chiral de Rham complex on an arbitrary smooth projective variety due to F. Malikov, V. Schechtman and A. Vaintrob [Commun. Math. Phys. 204, 439–473 (1999; Zbl 0952.14013)] are explained.

For the entire collection see [Zbl 0981.00011].

Reviewer: Martin Schlichenmaier (Mannheim)

##### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

14H60 | Vector bundles on curves and their moduli |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

14H10 | Families, moduli of curves (algebraic) |

17B68 | Virasoro and related algebras |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |