12

1. DEFINITION OF VERTEX ALGEBRAS

on C

x

)

wrhose

value on^GC[z, ^

x]

equals

Res2=0 f(z)g(z)dz.

Similarly, formal power series in several variables define functionals on rational

functions. For this reason we sometimes refer to them as "formal distributions".

1.1.3. Delta-function. An important example of a formal power series in

two variables z,w is the formal delta-function S(z — w)1 which is by definition

(1.1.2) 6(z -w)=Y, z^w-™-1.

Its coefficients amn = 5m)_n_i are supported on the diagonal m + n = — 1, and

hence it can be multiplied by an arbitrary formal power series in one variable (i.e.,

depending only on z or only on w). Indeed, for such a series A(w), we obtain

A(w)5(z-w) = ^AkwkY^zm«~m~1= J2 Am+n+1zmwn,

keZ raGZ m,neZ

so each coefficient is well-defined. Furthermore, the formula above shows that as

formal power series in z,w,

(1.1.3) A(z)S{z -w) = A(w)6(z - w),

which motivates the terminology "delta-function".

We obtain from formula (1.1.3) that

(1.1.4) (z-w)S(z-w) = 0

and, by induction,

(1.1.5) (z - w)n+1dl5{z - w) = 0.

In fact, one has the following abstract characterization of 5(z — w) and its

derivatives.

1.1.4. Lemma([Kac3]). Letf{z1w) be a formal power series in

R^z±x ,t^±1]]

satisfying (z —

w)Nf(z,

w) — 0 for a positive integer N. Then f(z, w) can be written

uniquely as a sum

J V - 1

(1.1.6)

J2^wWJ(z-w),

9i(w) €

RUw*1]].

i=0

1.1.5. Proof. Formula (1.1.4) implies that (z — w)N5{z — w) — 0 for any pos-

itive integer N. Differentiating this formula with respect to w, we obtain by induc-

tion that any element of the form (1.1.6) satisfies the equation (z — w)Nf(z,w) — 0.

Conversely, suppose that f(z,w) satisfies this equation. Writing f(z,w) =

2-m ™ez

fn,mZnwrn7

we obtain the following relation on the coefficients /nm:

(1.1.7) (A i V /)

n

,

m

= 0, n , m e Z ,

where (A/)

n

,

m

= /

n

-i,m —/n,m-i- Note that each of the equations (1.1.7) involves

only the coefficients on the same "diagonal", i.e., those of the form fk,P-k with fixed

p G Z . Therefore a general solution is a sum f(z,w) — ^2pe% f(p\z,w), where the

series f^p\z, w) is "supported" on the pth diagonal, i.e., fn,L = 0 unless n + m = p.

Restricted to the pth diagonal, the system (1.1.7) becomes a difference equation

of order N. Hence the corresponding space of solutions is TV-dimensional. But we