CHAPTER 1

Definition of Vertex Algebras

In this chapter we give the definition of vertex algebras and establish their

first properties. We first develop a formalism of formal power series needed to

introduce the notion of locality of vertex operators. Then we give several definitions

of locality, algebraic as well as analytic. After that we give the axioms of vertex

algebra and discuss the simplest example, that of a commutative vertex algebra.

More interesting examples are given in the next chapter and in Chapters 5 and 15.

1.1. Formal distributions

Let R be a C-algebra.

1.1.1. Definition. An i?-valued formal power series (or formal distribution)

in variables zi, Z2,..., zn is an arbitrary (finite or infinite) series

(111) A(ZU. • • , Zn) = Y, • • • - E

Ail,~;in4l

• • • 4 " .

ii€Z ineZ

where each A^ i G R. These series form a vector space, which is denoted by

R[[*t\ ,4%

If P{zu...,zn) £

R[{zf\...,zt1])

and Q(wu...,wm) G R[[wt\...,w±%

then their product is a well-defined element of R[[zf *, ...,

z^1, wf1,..., w^1]].

In general, a product of two elements of

iZf^f1,..., z^1]]

does not make sense,

since individual coefficients of the product are infinite sums of coefficients of the

factors. However, the product of a formal power series by a Laurent polynomial (i.e.,

a series (1.1.1) such that A;lv..^n = 0 for all but finitely many n-tuples i i , . . . ,i

n

)

is always well-defined.

1.1.2. Power series as distributions. Given a formal power series in one

variable, f(z) = V^a^ 2 , we define its residue (at 0) as

Res f(z)dz — Res^=o f(z)dz — a_i.

Note that if R = C and f(z) is the Laurent series of a meromorphic function defined

on a disc around 0, having poles only at 0, then

Res2=0 f(z)dz = — / f(z)dz

where the integral is taken over a closed curve winding once around 0. To simplify

notation, we will henceforth suppress the factor 2ni from all contour integrals.

Any formal power series f(z) = Ylnez fn2-™ m ^[I2^1]] defines a linear func-

tional on the space of Laurent polynomials C[z, z~x] (in other words, a distribution

l i

http://dx.doi.org/10.1090/surv/088/01