10 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY
The fundamental property of the $-function is:
f(z)6(z) = f(l)6(z) for f(z) 6 F ^ z " 1 ] , (2.1.8)
proved trivially by observing its truth for f(z) = zn and using linearity. This
property has many variants; in general, whenever an expression is multiplied by
the 6-function, we may formally set the argument appearing in the ^-function
equal to 1, provided the relevant algebraic expressions make sense. For instance,
given a formal Laurent series in two variables
X(Zl,z2) (End
W)[[z1,z-\z2,z-1}}
(2.1.9)
with coefficients which are operators on a vector space W, such that
lim X(zi,z2) exists, i.e., X(z\, z2)\Zl=z2 exists (2.1.10)
Zi—*Z2
(that is, when X(zi,z2) is applied to any element of W, setting the variables
equal leads to only finite sums in W), we have
X(zuz2)S (j±\ = X(z2,z2)S (?±\ . (2.1.11)
As in this case, all limits, products of formal Laurent series and other operations
will be understood in a purely algebraic sense.
In the theory of vertex operator algebras we shall often use three-variable
generating functions of the following sort:
- - U fzi~z2\ _ V ( * i - * 2 ) n _ v ^ ,
v n
( n \
x m
X U 7 n£7L 0 m(=N, n£% V '
(2.1.12)
There are two basic properties of the S-function involving such expressions:
--,'(S£2)"»-,'(a5a) 2113)''
These are easily proved by direct expansion. Note that the three terms in (2.1.14)
are formal power series in (i.e., involve only nonnegative integral powers of) z2,
z\ and zo, respectively. In particular, the two terms on the left-hand side are
unequal formal Laurent series in three variables, even though at first glance they
appear equal.
The following residue notation will be useful:
Res2 \\^vnzn = ! ; _ ! . (2.1.15)
VnGZ /
For instance, for the expression in (2.1.13),
R e s ^ s f *6 fZ2 + z°\
=
i. (2.1.16)
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