Vertex algebras for beginners. (English) Zbl 0861.17017

Kac, Victor

University Lecture Series. 10. Providence, RI: American Mathematical Society (AMS). viii, 141 p. (1996).

This is a very good introductional book on vertex algebras. Vertex algebras were introduced by R. E. Borcherds in connection with the moonshine representation of the Monster group [Proc. Natl. Acad. Sci. USA 84, 3068-3071 (1986; Zbl 0613.17012)]. In this book, the author presents a simplification of Borcherds’ axioms of a vertex algebra and shows that the new axioms can be deduced from Wightman’s axioms of a quantum field theory. Moreover, the author lays a rigorous ground for the notion of operator product expansion. The book discusses many properties of a vertex algebra such as: consequences of translation covariance, quasi-symmetry, the uniqueness of a local field determined by its action on the vacuum vector, Borcherds’ identity, gradings of the algebra, etc. Furthermore, the author presents constructional examples of vertex algebras from: free bosonic and fermionic fields, \(\widehat {gl}_\infty\) and \(W_{1+ \infty}\), integral lattices, affine Lie algebras and conformal superalgebras. The classification of simple conformal superalgebras is discussed.

Reviewer: Xu Xiaoping (Kowloon)

MSC:

17B69	Vertex operators; vertex operator algebras and related structures
17-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

formal distributions; delta functions; local fields; vertex algebras; operator product expansion; conformal superalgebras

Citations:

Zbl 0613.17012