**University Lecture Series**

Volume: 10;
1998;
201 pp;
Softcover

MSC: Primary 17;
Secondary 81

Print ISBN: 978-0-8218-1396-6

Product Code: ULECT/10.R

List Price: $38.00

Individual Member Price: $30.40

**Electronic ISBN: 978-1-4704-2159-5
Product Code: ULECT/10.R.E**

List Price: $38.00

Individual Member Price: $30.40

# Vertex Algebras for Beginners: Second Edition

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*Victor Kac*

This is a revised and expanded edition of Kac's original introduction to algebraic aspects of conformal field theory, which was published by the AMS in 1996. The volume serves as an introduction to algebraic aspects of conformal field theory, which in the past 15 years revealed a variety of unusual mathematical notions. Vertex algebra theory provides an effective tool to study them in a unified way.

In the book, a mathematician encounters new algebraic structures that originated from Einstein's special relativity postulate and Heisenberg's uncertainty principle. A physicist will find familiar notions presented in a more rigorous and systematic way, possibly leading to a better understanding of foundations of quantum physics.

This revised edition is based on courses given by the author at MIT and at Rome University in spring 1997. New material is added, including the foundations of a rapidly growing area of algebraic conformal theory. Also, in some places the exposition has been significantly simplified.

#### Table of Contents

# Table of Contents

## Vertex Algebras for Beginners: Second Edition

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface 18 free
- Preface to the second edition 310 free
- Chapter 1. Wightman axioms and vertex algebras 512 free
- Chapter 2. Calculus of formal distributions 1724
- 2.1. Formal delta-function 1724
- 2.2. An expansion of a formal distribution a(z, w) and formal Fourier transform 1926
- 2.3. Locality of two formal distributions 2431
- 2.4. Taylor's formula 2936
- 2.5. Current algebras 3138
- 2.6. Conformal weight and the Virasoro algebra 3441
- 2.7. Formal distribution Lie superalgebras and conformal superalgebras 3946
- 2.8. Conformal modules and modules over conformal superalgebras 5057
- 2.9. Representation theory of finite conformal algebras 5663
- 2.10. Associative conformal algebras and the general conformal algebra 6168
- 2.11. Cohomology of conformal algebras 6774

- Chapter 3. Local fields 8188
- Chapter 4. Structure theory of vertex algebras 103110
- 4.1. Consequences of translation covariance and vacuum axioms 103110
- 4.2. Skewsymmetry 105112
- 4.3. Subalgebras, ideals, and tensor products 106113
- 4.4. Uniqueness theorem 108115
- 4.5. Existence theorem 110117
- 4.6. Borcherds OPE formula 111118
- 4.7. Vertex algebras associated to formal distribution Lie superalgebras 113120
- 4.8. Borcherds identity 116123
- 4.9. Graded and Mobius conformal vertex algebras 119126
- 4.10. Conformal vertex algebras 125132
- 4.11. Field algebras 129136

- Chapter 5. Examples of vertex algebras and their applications 133140
- 5.1. Charged free fermions and triple product identity 133140
- 5.2. Boson-fermion correspondence and KP hierarchy 137144
- 5.3.gl[sub(∞)] and W[sub(1+∞)] 143150
- 5.4. Lattice vertex algebras 148155
- 5.5. Simple lattice vertex algebras 152159
- 5.6. Root lattice vertex algebras and affine vertex algebras 158165
- 5.7. Conformal structure for affine vertex algebras 161168
- 5.8. Super boson-fermion correspondence and sums of squares 168175
- 5.9. Super conformal vertex algebras 178185
- 5.10. On classification of conformal superalgebras 185192

- Bibliography 193200
- Index 199206
- Back Cover Back Cover1209

#### Readership

Graduate students, research mathematicians and physicists working in mathematical aspects of quantum field theory.

#### Reviews

Very good introductional book on vertex algebras.

-- Zentralblatt MATH

Essential reading for anyone trying to learn about vertex algebras … well worth buying for experts.

-- Bulletin of the London Mathematical Society