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
Previous Page Next Page