Contents

Preface to the Second Edition ix

Preface to the First Edition xi

Introduction 1

Chapter 1. Definition of Vertex Algebras 11

1.1. Formal distributions 11

1.2. Locality 16

1.3. Definition of a vertex algebra 20

1.4. First example: commutative vertex algebras 22

1.5. Bibliographical notes 23

Chapter 2. Vertex Algebras Associated to Lie Algebras 25

2.1. Heisenberg Lie algebra 25

2.2. The vertex algebra structure on ir 28

2.3. Checking vertex algebra axioms 32

2.4. Affine Kac-Moody algebras and their vertex algebras 38

2.5. The Virasoro vertex algebra 42

2.6. Bibliographical notes 46

Chapter 3. Associativity and Operator Product Expansion 47

3.1. Goddard's uniqueness theorem 47

3.2. Associativity 48

3.3. Operator product expansion 51

3.4. Examples of OPE 56

3.5. Bibliographical notes 61

Chapter 4. Applications of the Operator Product Expansion 63

4.1. A Lie algebra attached to a vertex algebra 63

4.2. U(V) and a completion of the universal enveloping algebra 65

4.3. An associative algebra attached to a vertex algebra 69

4.4. Strong reconstruction theorem 70

4.5. Correlation functions 72

4.6. Bibliographical notes 74

Chapter 5. Modules over Vertex Algebras and More Examples 75

5.1. Modules over vertex algebras 76

5.2. Vertex algebras associated to one-dimensional integral lattices 80

5.3. Boson-fermion correspondence 85

5.4. Lattice vertex algebras 88