eBook ISBN:  9781470400712 
Product Code:  MEMO/104/494.E 
List Price:  $31.00 
MAA Member Price:  $27.90 
AMS Member Price:  $18.60 
eBook ISBN:  9781470400712 
Product Code:  MEMO/104/494.E 
List Price:  $31.00 
MAA Member Price:  $27.90 
AMS Member Price:  $18.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 104; 1993; 64 ppMSC: Primary 17; 81
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 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.
ReadershipProfessional mathematicians and graduate students working in algebra, representation theory, and finite groups.

Table of Contents

Chapters

1. Introduction

2. Vertex operator algebras

3. Duality for vertex operator algebras

4. Modules

5. Duality for modules


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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 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.
Professional mathematicians and graduate students working in algebra, representation theory, and finite groups.

Chapters

1. Introduction

2. Vertex operator algebras

3. Duality for vertex operator algebras

4. Modules

5. Duality for modules