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
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) £
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
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
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