On axiomatic approaches to vertex operator algebras and modules
About this Title
Igor B. Frenkel, Yi-Zhi Huang and James Lepowsky
Publication: Memoirs of the American Mathematical Society
Publication Year 1993: Volume 104, Number 494
ISBNs: 978-0-8218-2555-6 (print); 978-1-4704-0071-2 (online)
MathSciNet review: 1142494
MSC: Primary 17B37
The notion of vertex operator algebra arises naturally in the vertex operator construction of the Monster—the 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 self-contained 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 far-reaching 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.
Professional mathematicians and graduate students working in algebra, representation theory, and finite groups.
Table of Contents
- 1. Introduction
- 2. Vertex operator algebras
- 3. Duality for vertex operator algebras
- 4. Modules
- 5. Duality for modules