2 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY

In the physics literature the main ingredients of the definition of the physical

counterpart of vertex operator algebras were discovered in relation first to the

dual resonance model and then to conformal field theory (see for instance the

Introduction in [FLM] for a discussion of the history). One of the focal points of

the axiomatic formulation of conformal field theory was the paper [BPZ], in which

the role of the Virasoro algebra was especially emphasized. The modern notion

of chiral algebra accepted now in the physics literature essentially coincides with

our notion of vertex operator algebra; see e.g. [MS]. In particular, the mutual

locality, or "commutativity," of operators and the "associativity" of the operator

product expansion are necessary properties of chiral algebras. Our argument that

the latter follows from the former under certain natural conditions can be used

to simplify the verification of the axioms in concrete examples [FLM]; see also

[G]. Many important discoveries involving representations of chiral algebras and

the associated intertwining operators, which are known in the physics literature

as chiral vertex operators, have recently been made in such works as [TK], [V]

and [MS]. The latter paper extends and develops the axiomatic approach to

conformal field theory, and it also contains an extensive review of the relevant

physics literature in the five-year period since [BPZ].

The present paper starts from a rigorous definition of vertex operator algebra

(the same as that introduced in [FLM]), a definition implicit, but not completely

explicit, in the physics literature, and it serves the purpose of building a founda-

tion for the rich structures associated to conformal field theory and mentioned

above. The scope of this work is to present the "monodromy-free" fundamen-

tals and basic results of a rapidly-developing theory; we treat the situations in

which the matrix coefficients of compositions of vertex operators are essentially

single-valued rational functions. This paper overlaps, elaborates and extends the

axiomatic material presented in [FLM], especially in Chapter 8 and the Appen-

dix.

Here we explain some of the basic axioms of vertex operator algebras and their

relation to classical mathematical notions, in particular, to Lie algebras. Let V

be a vector space over a field F, assumed for later purposes to have characteristic

0, and let

*&(•)• :V®V-+ V (1.1)

be a linear map satisfying the identity

ad(u)a,d(v) — a,d(v)a,d(u) =• ad(ad(tt)v)

for any ti, v G V. Then if we require that

ad(-u) = 0 implies v = 0, (1-3)

the pair (V, ad) is nothing but a Lie algebra having zero center, with ad denoting

the adjoint representation, i.e., the Lie bracket is given by:

[u,v] = ad(u)v. (1-4)

Then (1.2) is one of the equivalent forms of the Jacobi identity, which together

with (1.3) also implies the skew-symmetry of the bracket.

(1.2)