Vertex Algebras for Beginners ：Second Edition ( University Lecture Series )

Publication subTitle ：Second Edition

Publication series ：University Lecture Series

Author： Victor Kac  

Publisher： American Mathematical Society‎

Publication year： 1998

E-ISBN: 9781470421595

P-ISBN(Paperback): 9780821813966

Subject： O151.2 Linear Algebra

Keyword： Algebra and Algebraic Geometry

Language： ENG

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Vertex Algebras for Beginners

Description

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.

Chapter

Cover

Title

Copyright

Contents

Preface

Preface to the second edition

Chapter 1. Wightman axioms and vertex algebras

1.1. Wightman axioms of a QFT

1.2. d = 2 QFT and chiral algebras

1.3. Definition of a vertex algebra

1.4. Holomorphic vertex algebras

Chapter 2. Calculus of formal distributions

2.1. Formal delta-function

2.2. An expansion of a formal distribution a(z, w) and formal Fourier transform

2.3. Locality of two formal distributions

2.4. Taylor's formula

2.5. Current algebras

2.6. Conformal weight and the Virasoro algebra

2.7. Formal distribution Lie superalgebras and conformal superalgebras

2.8. Conformal modules and modules over conformal superalgebras

2.9. Representation theory of finite conformal algebras

2.10. Associative conformal algebras and the general conformal algebra

2.11. Cohomology of conformal algebras

Chapter 3. Local fields

3.1. Normally ordered product

3.2. Dong's lemma

3.3. Wick's theorem and a "non-commutative" generalization

3.4. Bounded and field representations of formal distribution Lie superalgebras

3.5. Free (super) bosons

3.6. Free (super) fermions

Chapter 4. Structure theory of vertex algebras

4.1. Consequences of translation covariance and vacuum axioms

4.2. Skewsymmetry

4.3. Subalgebras, ideals, and tensor products

4.4. Uniqueness theorem

4.5. Existence theorem

4.6. Borcherds OPE formula

4.7. Vertex algebras associated to formal distribution Lie superalgebras

4.8. Borcherds identity

4.9. Graded and Mobius conformal vertex algebras

4.10. Conformal vertex algebras

4.11. Field algebras

Chapter 5. Examples of vertex algebras and their applications

5.1. Charged free fermions and triple product identity

5.2. Boson-fermion correspondence and KP hierarchy

5.3.gl[sub(∞)] and W[sub(1+∞)]

5.4. Lattice vertex algebras

5.5. Simple lattice vertex algebras

5.6. Root lattice vertex algebras and affine vertex algebras

5.7. Conformal structure for affine vertex algebras

5.8. Super boson-fermion correspondence and sums of squares

5.9. Super conformal vertex algebras

5.10. On classification of conformal superalgebras

Bibliography

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

Back Cover

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