C O N T E N T S

Historical note vii

1. Introduction 1

2. Vertex operator algebras 9

2. 1. Formal calculus 9

2. 2. Definition of vertex operator algebras 11

2. 3. Consequences of the definition 13

2. 4. Elementary categorical notions 15

2. 5. Tensor products 17

2. 6. The Virasoro algebra and primary fields 17

2. 7. 83-symmetry of the Jacobi identity 18

2. 8. Quasi-vertex operator algebras 20

3. Duality for vertex operator algebras 21

3. 1. Expansions of rational functions 21

3. 2. Rationality of products and commutativity 24

3. 3. Rationality of iterates and associativity 25

3. 4. The Jacobi identity from commutativity and associativity 27

3. 5. Several variables 28

3. 6. The Jacobi identity from commutativity 30

3. 7. Proof of the tensor product construction 32

4. Modules 33

4. 1. Definition 33

4. 2. Consequences of the definition 34

4. 3. Elementary categorical notions 35

4. 4. Primary fields 36

4. 5. Rationality, commutativity, associativity and the Jacobi

identity 36

4. 6. Tensor product modules for tensor product algebras 36

4. 7. Irreducibility and tensor products 37

5. Duality for modules 42

5. 1. Duality for one module element and two algebra elements 42

5. 2. Adjoint vertex operators and the contragredient module 44

5. 3. Properties of contragredient modules 52

5. 4. Intertwining operators 54

5. 5. Adjoint intertwining operators 59

5. 6. Duality for two module elements and one algebra element 62

References 64