Vertex Operator Algebras and the Monster,134 (Pure and Applied Mathematics) [Hardcover]

Several books elucidating the properties and theory of vertex operator algebras (VOA) are now available but this is one of the early ones. While formal in its approach, and using notation that can be very difficult to read, this book nevertheless gives the reader keen insights into the theory, this coming from the summaries and motivations that occur at the beginning of every chapter. The authors of the book are responsible for most of the early development of the theory of vertex operator algebras, and have chosen in this book to discuss it using the (formal) calculus of power series. The physicist reader, particular the one involved in string theory or quantum field theory, will perhaps for this reason appreciate the book more than the one who is not. However the authors respect enough mathematical rigor to satisfy those readers who are approaching the book from a purely mathematical viewpoint. The book can be time-consuming to study at various places, again because of the choice of notation, but on the whole is worthwhile to study for those readers who have a genuine interest in what the subject is all about, even if they do not intend to pursue research into this area.

Chapter 1 contains a very quick overview of Lie algebras and modules over Lie algebras. Readers should pay close attention to the notion of an induced module since it used throughout the book and may be new to them, as it was to this reviewer. The most important part of this chapter is the discussion on affine Lie algebras, which are infinite-dimensional generalizations of ordinary Lie algebras. Their construction begins with a Lie algebra G and an invariant symmetric bilinear form on G. The authors then introduce, but unfortunately do not motivate why, the commutative associative algebra of Laurent polynomials in an indeterminate t. The affine Lie algebra is formed by tensoring the Laurent polynomials with G and then taking the direct sum with a one-dimensional space. The role of this space is made somewhat more clear when the authors show that the affine Lie algebra is a central extension of the tensor product. A bit more insight into the term "affine" comes from remembering what an affine map is: a variable term shifted by a constant. String theorists will understand the origin of the Laurent polynomials, for in the "stringy" context the variable t corresponds to a rotation in the unit circle, and the Lie group corresponding to the Lie algebra is the loop group. Readers not familiar with string theory will have to view t as a formal variable.

Also very important for the rest of the book is the authors' discussion on an analog of affinization called "twisting by an automorphism" (actually by an involution which is a special type of automorphism whose square is the identity). This involution must also be an isometry with respect to the bilinear form of the Lie algebra. The Lie algebra can then be written as the direct sum of (orthogonal) subalgebras symmetric and antisymmetric under the action of the involution. Instead of the Laurent polynomials in t one takes the square root of t. After defining an automorphism that switches the sign of the indeterminate and an automorphism that fixes the central element, the authors define the twisted affine algebra as the subalgebra that is fixed under this automorphism. The twisted affine algebra reduces to the untwisted one when the automorphism is the identity.

The physicist reader will find the discussion of the Heisenberg and Virasoro algebras familiar even though it is couched in formal mathematical language. The Heisenberg Lie algebra is one whose commutant is equal to its (one-dimensional) center. The authors consider integer-graded Heisenberg Lie algebras and derive the Heisenberg commutation relations and the canonical realization of the Heisenberg commutation relations, the latter of which is done via a "position-space" representation. The familiar vacuum vector, so ubiquitous in quantum field theory, makes it appearance here. The Virasoro algebra is defined as the central extension of the Lie algebra of derivatives of Laurent polynomials in t and its inverse.

Chapters 2 and 8 are a detailed overview of formal power series, which in the physics literature are known as operator product expansions, and are used to make sense of how to "multiply" quantum fields at the same point. The reader must pay close attention to the notation that is deployed, for the mathematical objects can be different when going from "parentheses" to "brackets". The main points to be remembered from these chapters are how to express the algebra of affine Lie algebras in terms of formal variables, for both the twisted and untwisted cases. The authors offer enough details so as to be able to give an axiomatic treatment of (untwisted) operator algebras in the middle of chapter 8.

Fortunately the axiomatic treatment in these chapters is well motivated by the concrete examples in chapters 3-7. The authors first consider the affine Kac-Moody algebra sl(2)^ in chapter 3 and its representation by twisted vertex operators. To obtain a twist, they define an involution of sl(2) that preserves the bilinear form given by the trace of a product of two matrices of sl(2). Both the twisted and untwisted affine Lie algebras are considered by giving their bases, as well as the Heisenberg (sub) algebra of sl(2)^ and sl(2)^(twist). The structure of both sl(2)^ and sl(2)^(twist) are expressed in terms of formal variables and another twisted version of sl(2) is considered using a different involution. A representation of this new twisted affine algebra sl(2)^(twist2) is the main topic of chapter 3.

In chapter 4, the authors use untwisted vertex operators for representing sl~(2) and sl~(2)(twist1). Here they begin with an integer-graded untwisted affine Lie algebra H~ corresponding to an abelian Lie algebra H. One then focuses attention on the Heisenberg subalgebra H^(Z) (instead of H^(Z + ½) as in chapter 3) of H~. Technical considerations involving how the action of the Heisenberg algebra can be extended to the action of sl~(2) and sl~(2) (twisted) on the H~module (denoted by M(1) from chapter 1) leads the authors to consider the tensor product of M(1) with the group algebra of H (suitably graded), before they can get a definition of a vertex operator that has the correct commutation properties. Taking this tensor product requires some changes in the normal ordering procedures, as the authors discuss in detail. Taking H to be a one-dimensional space (just as was done for sl~(2)(twist2) in chapter 3), the authors give explicit vertex operator representations for sl~(2) and sl~(2)(twist1). The reader should take note of the fact that the representation spaces are constructed using the root and weight lattices of sl(2). The authors then show that the twisted and untwisted vertex operator modules are graded-isomorphic. This implies their graded dimensions are the same, and this leads to the famous Dedekind eta-function, which the authors write down at the end of the chapter.

The authors give a quick overview of the theory of central extensions of Abelian groups in chapter 5. Even though readers may already know the material in this chapter, the introduction to this chapter should be mandatory reading because of the insight it offers not only to the topic at hand but also why it is used in later chapters.

In chapter 6 readers will see how to use lattices to obtain Lie algebras, with the eventual goal in understanding how to obtain vertex operator representations of the affine algebras corresponding to these Lie algebras. The discussion on lattices should be familiar to most readers, and given a positive definite even lattice L with a symmetric form the authors first consider the Lie algebra H obtained by tensoring this lattice with the field of definition. Using a (unique) central extension L^ of L by a cyclic group with two elements, and choosing a section of this central extension the authors concentrate attention on a subset of L^ whose central elements have length 2. A set of symbols is defined whose elements are labeled by this subset and whose sign changes when the index is operated on by a member of the cyclic group. The authors do not motivate very well at all the choice of this symbol set, and so readers will at this place in the book have to accept that its introduction will be fruitful. Formal linear combinations of these symbols are taken using the elements from the field of definition F, and the resulting collection is direct summed with H to give a finite-dimensional vector space G. A nonassociative algebra product is then put on G, along with a nonsingular bilinear form, and the authors then prove that G is a Lie algebra and the bilinear form is symmetric and G-invariant. At this point readers can better understand the use of the symbol set, for a special case of this construction gives sl(2, F). The authors then construct the untwisted affine Lie algebra G~ associated with G and its bilinear form.

This relatively abstract construction is made more concrete by considering the situations where G is simple. These cases are labeled by the dimension n of H, and are those that have equal root lengths and indecomposable root systems: An, Dn, and En. The consideration of an automorphism group of G allows the author to construct a twisted affine Lie algebra corresponding to G. Vertex operator realizations for An, Dn, and En are then outlined as promised in chapter 7.

This review is for now based on the reading of the introduction and Chapters 1-8.