Introduction to Vertex Operator Algebras and Their Representations

The deep and relatively new field of vertex operator algebras is intimately related to a variety of areas in mathematics and physics: for example, the concepts of "monstrous moonshine," infinite-dimensional Lie theory, string theory, and conformal field theory. This book introduces the reader to the fundamental theory of vertex operator algebras and its basic techniques and examples. Beginning with a detailed presentation of the theoretical foundations and proceeding to a range of applications, the text includes a number of new, original results and also highlights and brings fresh perspective to important works of many researchers. After introducing the elementary "formal calculus'' underlying the subject, the book provides an axiomatic development of vertex operator algebras and their modules, expanding on the early contributions of R. Borcherds, I. Frenkel, J. Lepowsky, A. Meurman, Y.-Z. Huang, C. Dong, Y. Zhu and others. The concept of a "representation'' of a vertex (operator) algebra is treated in detail, following and extending the work of H. Li; this approach is used to construct important families of vertex (operator) algebras and their modules. Requiring only a familiarity with basic algebra, Introduction to Vertex Operator Algebras and Their Representations will be useful for graduate students and researchers in mathematics and physics. The book s presentation of the core topics will equip readers to embark on many active research directions related to vertex operator algebras, group theory, representation theory, and string theory.

Review :
" [The] authors give a systematic introduction to the theory of vertex operator algebras and their representations. Particular emphasis is put on the axiomatic development of the theory and the construction theorems for vertex operator algebras and their modules. The book provides a detailed study of most basic families of vertex operator algebras and their representation theory. A number of new, original results are presented . This excellent book is written in a self-contained manner with detailed proofs. It will be useful for graduate students and active researchers interested in the theory of vertex operator algebras and their applications." Mathematical Reviews "The book under review treats modules for vertex operator algebras and, more importantly, it gives an answer to the following important questions: "How do we construct modules for VOAs?" The answer to this question is the essense of this new exciting book. . . The book is written with care, clarity and pateience which is typical for both authros. It is self-contained with no details omitted. Misprints are most probably rare (if any). Even an advanced undergraduate can pick up a book and learn a whole new exciting subject. In my opinion this beautiful book has only one shortcoming - the list of references (around 600 items!). The authors were kind enough to give credit to almost everyone who ever contributed in some way to vertex operator algebra theory." ---Zentralblatt "The book gives a sound introduction in the theory of vertex algebras emphasizing in particular the construction of families of examples by means of a kind of representations developped by the second named author." ---Monatsheft fur Mathematik