Memoirs of the American Mathematical Society 1993; 64 pp; softcover Volume: 104 ISBN10: 0821825550 ISBN13: 9780821825556 List Price: US$29 Individual Members: US$17.40 Institutional Members: US$23.20 Order Code: MEMO/104/494
 The notion of vertex operator algebra arises naturally in the vertex operator construction of the Monsterthe largest sporadic finite simple group. From another perspective, the theory of vertex operator algebras and their modules forms the algebraic foundation of conformal field theory. Vertex operator algebras and conformal field theory are now known to be deeply related to many important areas of mathematics. This essentially selfcontained monograph develops the basic axiomatic theory of vertex operator algebras and their modules and intertwining operators, following a fundamental analogy with Lie algebra theory. The main axiom, the "Jacobi(Cauchy) identity", is a farreaching analog of the Jacobi identity for Lie algebras. The authors show that the Jacobi identity is equivalent to suitably formulated rationality, commutativity, and associativity properties of products of quantum fields. A number of other foundational and useful results are also developed. This work was originally distributed as a preprint in 1989, and in view of the current widespread interest in the subject among mathematicians and theoretical physicists, its publication and availability should prove no less useful than when it was written. Readership Professional mathematicians and graduate students working in algebra, representation theory, and finite groups. Table of Contents  Introduction
 Vertex operator algebras
 Duality for vertex operator algebras
 Modules
 Duality for modules
 References
