**Contemporary Mathematics**

Volume: 121;
1991;
146 pp;
Softcover

MSC: Primary 17; 81;

**Print ISBN: 978-0-8218-5128-9
Product Code: CONM/121**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

**Electronic ISBN: 978-0-8218-7709-8
Product Code: CONM/121.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

# Spinor Construction of Vertex Operator Algebras, Triality, and \(E^{(1)}_{8}\)

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*Alex J Feingold; Igor B Frenkel; John F X Ries*

The theory of vertex operator algebras is a remarkably rich new
mathematical field which captures the algebraic content of conformal field
theory in physics. Ideas leading up to this theory appeared in physics as
part of statistical mechanics and string theory. In mathematics, the axiomatic
definitions crystallized in the work of Borcherds and in Vertex Operator
Algebras and the Monster, by Frenkel, Lepowsky, and Meurman. The
structure of monodromies of intertwining operators for modules of vertex
operator algebras yields braid group representations and leads to natural
generalizations of vertex operator algebras, such as superalgebras and
para-algebras. Many examples of vertex operator algebras and their
generalizations are related to constructions in classical representation
theory and shed new light on the classical theory.

This book accomplishes several goals. The authors provide an explicit spinor
construction, using only Clifford algebras, of a vertex operator superalgebra
structure on the direct sum of the basic and vector modules for the affine
Kac-Moody algebra \(D^{(1)}_n\). They also review and extend
Chevalley's spinor construction of the 24-dimensional commutative
nonassociative algebraic structure and triality on the direct sum of the
three 8-dimensional \(D_4\)-modules. Vertex operator para-algebras,
introduced and developed independently in this book and by Dong and Lepowsky,
are related to one-dimensional representations of the braid group. The authors
also provide a unified approach to the Chevalley, Griess, and \(E_8\)
algebras and explain some of their similarities. A third goal is to provide a
purely spinor construction of the exceptional affine Lie algebra
\(E^{(1)}_8\), a natural continuation of previous work on spinor and
oscillator constructions of the classical affine Lie algebras. These
constructions should easily extend to include the rest of the exceptional
affine Lie algebras. The final objective is to develop an inductive technique
of construction which could be applied to the Monster vertex operator algebra.

Directed at mathematicians and physicists, this book should be accessible to
graduate students with some background in finite-dimensional Lie algebras and
their representations. Although some experience with affine Kac-Moody algebras
would be useful, a summary of the relevant parts of that theory is included.
This book shows how the concepts and techniques of Lie theory can be
generalized to yield the algebraic structures associated with conformal field
theory. The careful reader will also gain a detailed knowledge of how the
spinor construction of classical triality lifts to the affine algebras and
plays an important role in a spinor construction of vertex operator algebras,
modules, and intertwining operators with nontrivial monodromies.

#### Reviews & Endorsements

A successful attempt to describe a common background of recent investigations.

-- Zentralblatt MATH

# Table of Contents

## Spinor Construction of Vertex Operator Algebras, Triality, and $E^{(1)}_{8}$

- Contents ix10 free
- Chapter 0: Introduction 112 free
- Chapter 1: Summary 1324
- Chapter 2: Affine Algebras and Representations 3445
- Chapter 3: Spinor Construction of Vertex Operator Superalgebras 4455
- Chapter 4: Spinor Construction of the Chevalley Algebra and Triality for D4 7485
- Chapter 5: Spinor Construction of Triality for D(1)4 91102
- Chapter 6: Spinor Construction of a Vertex Operator Para-algebra for D(1)4 106117
- Chapter 7: Spinor Construction of E8 128139
- Chapter 8: Spinor Construction of Vertex Operator Algebras for E(1)4 132143
- References 144155