14 CRISTIANO HUSU
expressions, it too exists, and so does (1.28), by the Jacobi identity. Since it
is illuminating to have a direct proof that (1.28) exists, we also present such
an argument based on Lemma 1.1. We use Lemma 1.1 to rewrite (1.28) as
Y(Y(Y(v3,Z32)v2,zn)vllZl)z;H (*±£»)
Z
-H (*+ ** + **)
.Wffclifcy
(L40)
The coefficient of
z2x~x z^1"3 z3±~k,
i , j , k G Z in (1.40) equals
Y(Y(Y(v3,z32)v2,z2i)vlyzl)(z1 + z21)\z1 + z2l +
z32)j(z21
+
z32)k.
(1.41)
Expanding Y(v3, z32) according to (1.4) and using (1.5), the coefficient of
zl32
(for each / G Z) in (1.41) can be written as a finite linear combination of
expressions
Y(Y{wJz21)vuz1)(z1+z21)mz^
(1.42)
where w G V and m, n are fixed integers. The same argument shows that
these exist. (Cf. [6], Chapter 8).
Lemma 1.3 gives the existence of (1.25) (1.29).
Remark 1.4 Using (1*19) the expression (1.29) can be formally written as
a sum of eight terms, each one of which is a product of three 8-functions and
the three vertex operators Y{yi,Zi), i = 1,2,3. The existence of (1.29) does
not imply the existence of each of these eight summands, and, in particular,
the distributive law does not apply since the summands do not all exist. For
instance, the two summands obtained by applying (1*19) to
[Y(v2, z2) x221 Y(v1,z1)}Y(v3, zz)z32H f *
3 1
" *
2 1
) z;i6 (?L1*) (1.43)
\ z32 / \ z13 J
do not exist separately.
Previous Page Next Page