2.VERTEX OPERATOR ALGEBRAS

2.1. Formal calculus

We shall work in an algebraic setting over an arbitrary field F of characteristic

0. The set of nonnegative integers will be denoted by N. The symbols

Z,ZQ,Z\,

...

will designate commuting formal variables. For a vector space V", we set

V[z) = J2 vnZn\vn £ V, all but finitely many vn = 0 (2.1.1)

IneN J

V\z,z~x\ = ^2,vnzn\vn e V, all but finitely many vn - 0 (2.1.2)

Inez J

^[[*]] = | £ t ^ n e ^ } (2.1.3)

InGN J

V((z)) =

\Y1

v^zU\vn

eV, vn = 0 for sufficiently small n \ , (2.1.4)

Inez J

n M "

1

] ] - j ] » * » G v l , (2.1.5)

and we shall also use analogous notation for several variables.

The following formal version of Taylor's theorem is easily verified by direct

expansion: For f(z) —

^vnzn

£ V[[z,

z'1)},

ez°-tf{z) = f{z + z0), (2.1.6)

where an expression of the form ex denotes the formal exponential series, and

where on the right-hand side, each binomial (z + ZQ)11 is understood to be ex-

panded in nonnegative integral powers of the second summand, namely, z$. We

shall repeatedly use this binomial expansion convention.

We introduce a basic generating function, the "formal (^-function at z — 1" :

*0O = X

n

. (2.1.7)

nG2

9