Introduction to vertex operator algebras and their representations
After \textit{R. E. Borcherds}' seminal contribution [``Vertex algebras, Kac-Moody algebras, and the Monster'', Proc. Natl. Acad. Sci. USA 84, 3068--3071 (1986; Zbl 0613.17012)] several books and reviews have been written about vertex algebras, vertex operator algebras and related structures. In the retrospective the most important and by far the most exciting aspects of this theory were obtained by \textit{I. Frenkel, J. Lepowsky} and \textit{A. Meurman}, in [Vertex operator algebras and the Monster (Pure and Applied Mathematics 134) (Boston: Academic Press) (1988; Zbl 0674.17001)], where vertex operator algebra structure on \(V^\natural\) (the Moonshine Module) was used in an essential way for a construction of the largest sporadic finite simple group, the Monster! Even though considered a major breakthrough the book needed a proper algebraic and axiomatic framework. That was partially achieved by \textit{I. Frenkel, Y.-Z. Huang} and \textit{J. Lepowsky} in [On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc. 494 (1993; Zbl 0789.17022)], where among other things, the authors introduced the notion of module for vertex operator algebras. Another circle of ideas, also based on Frenkel, Lepowsky and Meurman's construction, led to a generalization of the notion of vertex operator algebra [cf. \textit{C. Dong} and \textit{J. Lepowsky}, Generalized vertex algebras and relative vertex operators (Prog. Math. 112) (Boston: Birkhäuser) (1993; Zbl 0803.17009)]. A few years later Kac [cf. \textit{V. Kac}, Vertex algebras for beginners (University Lecture Series 10) (Providence: AMS) (1996; Zbl 0861.17017), 2nd ed. (1998; Zbl 0924.17023)] offered yet another approach to vertex algebras which largely overlaps with Frenkel, Huang and Lepowsky's book and eventually led to the theory of \textit{conformal algebras}. These structures are in many ways closer to Lie algebras rather than vertex operator algebras. Most recently [cf. Vertex algebras and algebraic curves (Math. Surv. Monogr. 88) (2001; Zbl 0981.17022)], \textit{E. Frenkel} and \textit{D. Ben-Zvi} developed a general theory of VOA on smooth algebraic curves, which was meant to be an introduction to a version of the geometric Langlands correspondence introduced by Beilinson and Drinfeld. Finally, \textit{Yi-Zhi Huang}'s book [Two-dimensional conformal geometry and vertex operator algebras (Prog. Math. 148) (Boston: Birkhäuser) (1997; Zbl 0884.17021)], captures the most important conformal geometric features of the vertex operator theory in genus zero. It seems that all mentioned approaches, with the exception of Frenkel, Huang and Lepowsky's book, largely ignored the representation theory of vertex operator algebras (i.e., vertex algebra and vertex operator algebra modules). The book under review treats modules for vertex operator algebras and, more importantly, it gives an answer to the following important question: ``How do we construct modules for VOAs?'' The answer to this question is the essence of this new exciting book. It is safe to say that the book is largely based on an important earlier work by H. Li on local systems of vertex operators. Lepowsky and Li expanded further the idea of local systems, and they made it accessible to a broad mathematical audience. The book under review also gives detailed introduction to formal calculus and classification of irreducible representations for the most important class of vertex operator algebras, those associated to certain highest weight modules for affine Kac-Moody Lie algebras. Let us elaborate briefly what a reader can find in this book. In Chapter~1, which is the introduction, the authors motivate and introduce the notion of vertex operator algebras. Chapter~2 deals with formal calculus, of crucial importance in the vertex operator algebra theory (it has some overlaps with Chapter~2 in Frenkel, Lepowsky and Meurman's book). In Chapter~3 Lepowsky and Li laid out the foundations of the theory: main axioms, commutativity and associativity properties. Chapter~4 deals with modules for vertex algebras and related results such as commutativity and associativity. Chapter~5 is the most vital part of the book. Here the ``local systems'' are introduced and construction of modules and vertex (operator) algebras is elaborated. Finally, Chapter~6 deals with representation theory of familiar vertex operator algebras: Virasoro vertex operator algebra, Heisenberg vertex operator algebras and vertex operator algebras stemming from certain representations of affine Kac-Moody Lie algebras. The book is written with care, clarity and patience which is typical for both authors. It is self-contained (\textit{ab ovo}), with no details omitted. Misprints are most probably rare (if any). Even an advanced undergraduate can pick up a book and learn a whole new exciting subject. In my opinion this beautiful book has only one shortcoming -- the list of references (around 600 items!). The authors were kind enough to give credit to almost everyone who ever contributed in some way to vertex operator algebra theory. Unfortunately, in doing that lot of references are now only vaguely related to the main topic of the book. The truth is that the authors themselves, with just a few other mathematicians (Y. Z. Huang, C. Dong, etc.), shaped the vertex operator algebra landscape during the last decade or so. In doing this they transformed the theory of vertex (operator) algebras from ``something needed for a construction of the Monster'', to a new, fresh, exciting, fast developing area of mathematics.
scientific article; zbMATH DE number 2061773