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    Vertex Algebras and Algebraic Curves (Mathematical Surveys & Monographs) (Hardback) By (author) Edward Frenkel, By (author) David Ben-Zvi


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    DescriptionVertex algebras were first introduced as a tool used in the description of the algebraic structure of certain quantum field theories. It became increasingly important that vertex algebras are useful not only in the representation theory of infinite-dimensional Lie algebras, where they are by now ubiquitous, but also in other fields, such as algebraic geometry, theory of finite groups, modular functions, and topology. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of moduli spaces of algebraic curves. The authors make the first steps toward reformulating the theory of vertex algebras in a way that is suitable for algebraic-geometric applications. The notion of a vertex algebra is introduced in the book in a co-ordinate independent way, allowing the authors to give global geometric meaning to vertex operators on arbitrary smooth algebraic curves, possibly equipped with some additional data. To each vertex algebra and a smooth curve, they attach an invariant called the space of conformal blocks. When the complex structure of the curve and other geometric data are varied, these spaces combine into a sheaf on the relevant moduli space. From this perspective, vertex algebras appear as the algebraic objects that encode the geometric structure of various moduli spaces associated with algebraic curves. Numerous examples and applications of vertex algebras are included, such as the Wakimoto realization of affine Kac-Moody algebras, integral solutions of the Knizhnik-Zamolodchikov equations, classical and quantum Drinfeld-Sokolov reductions, and the $W$-algebras. The authors also establish a connection between vertex algebras and chiral algebras, introduced by A. Beilinson and V. Drinfeld.

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  • Full bibliographic data for Vertex Algebras and Algebraic Curves

    Vertex Algebras and Algebraic Curves
    Authors and contributors
    By (author) Edward Frenkel, By (author) David Ben-Zvi
    Physical properties
    Format: Hardback
    Number of pages: 348
    Width: 184 mm
    Height: 260 mm
    Thickness: 19 mm
    Weight: 839 g
    ISBN 13: 9780821828946
    ISBN 10: 0821828940

    BIC E4L: MAT
    Nielsen BookScan Product Class 3: S7.8
    B&T Book Type: NF
    B&T Merchandise Category: TXT
    B&T Modifier: Region of Publication: 01
    B&T Modifier: Academic Level: 01
    BIC subject category V2: PBW
    B&T General Subject: 710
    BIC subject category V2: PBF
    DC21: 512.5
    BISAC V2.8: MAT002010, MAT002050
    BIC subject category V2: PBMW
    LC subject heading:
    DC22: 512/.55
    LC subject heading:
    LC classification: QA326 .F76 2001
    Illustrations note
    index, bibliography
    American Mathematical Society
    Imprint name
    American Mathematical Society
    Publication date
    01 November 2001
    Publication City/Country
    Table of contents
    Definition of vertex algebras; vertex algebras associated to Lie algebras; associativity and operator product expansion; rational vertex algebras; vertex algebra bundles; action of internal symmetries; vertex algebra bundles - examples; conformal blocks I; conformal blocks II; free field realization I; free field realization II; the Knizhnik-Zamolodchikov equations; solving the KZ equations; quantum Drinfeld-Sokolov reduction and $mathcal{W}$-algebras; vertex Lie algebras and classical limits; vertex algebras and moduli spaces I; vertex algebras and moduli spaces II; chiral algebras.