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**Vertex algebras and algebraic curves.
2nd revised and expanded ed.**
*(English)*
Zbl 1106.17035

Mathematical Surveys and Monographs 88. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3674-9/pbk). xiv, 400 p. (2004).

This is a substantially rewritten and expended second edition of [E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves. Mathematical Surveys and Monographs. 88. Providence, RI: American Mathematical Society (AMS). (2001; Zbl 0981.17022)]. The principal changes with respect to the first edition are the following: Throughout the book the authors have dropped the requirement for a vertex algebra to be \(\mathbb{Z}\)-graded with finite-dimensional graded components. The exposition of associativity and operator product expansions is changed. A more detailed discussion of the Lie algebra \(U(V)\), attached to a vertex operator algebra \(V\), is included as a separate section. Further, the authors define a topological associative algebra \(\tilde{U}(V)\) and prove an equivalence between the category of \(V\)-modules and the category of smooth \(\tilde{U}(V)\)-modules. In a separate section the authors introduce twisted modules associated to a vertex algebras equipped with an automorphism of finite order. A direct algebraic proof of coordinate-independence of the connection on the vertex algebra bundle is given. The authors also add new example of chiral algebras, which do not arise from vertex algebras, and explain how to attached to modules and twisted modules over vertex algebras certain modules over the corresponding chiral algebras.

Finally, a new chapter on factorization algebras, which provide a purely geometric reformulation of the definition of vertex algebras, is added. In this chapter the authors also discuss chiral Hecke algebras and the geometric Langlands conjecture.

Finally, a new chapter on factorization algebras, which provide a purely geometric reformulation of the definition of vertex algebras, is added. In this chapter the authors also discuss chiral Hecke algebras and the geometric Langlands conjecture.

Reviewer: Volodymyr Mazorchuk (Uppsala)

### MSC:

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

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

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

14D20 | Algebraic moduli problems, moduli of vector bundles |

14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |

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

14H81 | Relationships between algebraic curves and physics |

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