Read a review which appeared in the Bulletin of the American Mathematical Society .
The notion of a vertex algebra is introduced in the book in a coordinate-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. Among other topics discussed in the book are vertex Poisson algebras, Virasoro uniformization of the moduli spaces of pointed curves, the geometric Langlands correspondence, and the chiral de Rham complex. The authors also establish a connection between vertex algebras and chiral algebras, recently introduced by A. Beilinson and V. Drinfeld.
This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.
This book may be used by the beginners as an entry point to the modern theory of vertex algebras, and by more experienced readers as a guide to advanced studies in this beautiful and exciting field.
For a brief summary of this book, see the text of the talk given by E. Frenkel at Seminaire Bourbaki in June of 2000.
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