Introduction to Vertex Operator Algebras and Their Representations

Vertex operator algebra theory has deep roots and, deep applications in mathematics and, physics. Both string theory in physics and, “monstrous moonshine” in mathematics played crucial roles in the development of the theory, as we shall sketch briefly.

In this chapter we shall present the elementary formal calculus that is basic in vertex operator algebra theory. Formal calculus has been treated in Chapters 2 and, 8 of [FLM6] and, in [FHL]. For our present purposes and, to make this work self-contained, we present here a full but slightly different treatment.

In this chapter we begin presenting the axiomatic theory of vertex operator algebras. The notion of vertex algebra was introduced in [B1] and, a variant of the notion, that of vertex operator algebra, was developed in [FLM6] and, [FHL]. The notion of vertex operator algebra is the mathematical counterpart of the notion of “operator algebra,” or “chiral algebra,” in conformai field theory, as formalized in [BPZ], and, many of the algebraic considerations in this chapter were introduced and, exploited in more physical language and, settings in the vast physics literature on conformai field theory and, string theory. Our treatment is based on [B1], [FLM6], [FHL], [DL3] and, [Li3]. In Section 3.3 we include a discussion of the relations between the mathematical treatment of “associativity” presented here and, operator product expansions in the sense of conformai field theory and, string theory.

In this chapter we discuss the notion of module for a vertex (operator) algebra V. A V-module is defined, as expected, to be a vector space W equipped with a linear map such that all the defining properties of a vertex algebra that make sense hold. (Actually, this is the typical principle for defining the notion of module for categories of algebras in general—Lie algebras, associative algebras, etc. A module is a vector space equipped with a linear action of the algebra such that all the algebra axioms that make sense hold.) Specifically, these defining properties are the truncation condition, the vacuum property and, the Jacobi identity. (The creation property, for instance, would not make sense, so it will not be an axiom.) Accordingly, almost all of the assertions in Chapter 3 that make sense also hold and, a large amount of material in Chapter 3 carries over in the obvious ways, often without change. In this chapter, we carry out many of these analogues, and we also discuss some additional concepts.

In this chapter, the focal chapter of the work, we shall define and, study the notion of representation of a vertex algebra. Just as in classical algebraic theories, in particular, the theories of Lie algebras and, of associative algebras, this notion of representation is distinct from the notion of module, but the two notions will turn out to be essentially equivalent.

We have developed the fundamental theory of vertex operator algebras and, modules in Chapters 1 through 5. The reader has certainly noticed that so far we have not constructed or exhibited any examples of vertex (operator) algebras other than the vertex (operator) algebras based on commutative associative algebras. Unlike in classical algebraic theories such as the theory of Lie or associative algebras, nontrivial examples of vertex (operator) algebras and, modules for them cannot be easily presented right after the definitions. But now, with the general representation theory having been developed in Chapter 5, we are fully prepared to present an array of interesting examples of vertex operator algebras and, modules, by systematically invoking this general representation theory.