# On Axiomatic Approaches to Vertex Operator Algebras and Modules

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*Igor B. Frenkel; Yi-Zhi Huang; James Lepowsky*

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.

#### Table of Contents

# Table of Contents

## On Axiomatic Approaches to Vertex Operator Algebras and Modules

- Contents v6 free
- Historical note vii8 free
- 1. Introduction 110 free
- 2. Vertex operator algebras 918 free
- 2.1. Formal calculus 918
- 2.2. Definition of vertex operator algebras 1120
- 2.3. Consequences of the definition 1322
- 2.4. Elementary categorical notions 1524
- 2.5. Tensor products 1726
- 2.6. The Virasoro algebra and primary fields 1726
- 2.7. S[sub(3)]-symmetry of the Jacobi identity 1827
- 2.8. Quasi-vertex operator algebras 2029

- 3. Duality for vertex operator algebras 2130
- 3.1. Expansions of rational functions 2130
- 3.2. Rationality of products and commutativity 2433
- 3.3. Rationality of iterates and associativity 2534
- 3.4. The Jacobi identity from commutativity and associativity 2736
- 3.5. Several variables 2837
- 3.6. The Jacobi identity from commutativity 3039
- 3.7. Proof of the tensor product construction 3241

- 4. Modules 3342
- 4.1. Definition 3342
- 4.2. Consequences of the definition 3443
- 4.3. Elementary categorical notions 3544
- 4.4. Primary fields 3645
- 4.5. Rationality, commutativity, associativity and the Jacobi identity 3645
- 4.6. Tensor product modules for tensor product algebras 3645
- 4.7. Irreducibility and tensor products 3746

- 5. Duality for modules 4251
- 5.1. Duality for one module element and two algebra elements 4251
- 5.2. Adjoint vertex operators and the contragredient module 4453
- 5.3. Properties of contragredient modules 5261
- 5.4. Intertwining operators 5463
- 5.5. Adjoint intertwining operators 5968
- 5.6. Duality for two module elements and one algebra element 6271

- References 6473