10
CRISTIANO HUSU
(cf. [6], Section 8.8, where "cross-bracket" refers to a certain component of
(1.19)). Then the Jacobi identity (1.15) can be written as
[Y(v2,z2) xZ21 Y(vuz1)] = z^6
{^-^2L)
Y{Y{v2,z2x)vx,zx). (1.20)
The cross-bracket can be defined more generally as
[/(*) x* 9{*i)\ = *?6
(^T1)
f(«)9{*i) + *?* i^f) 9(*i)f(«)
(1.21)
(for i ^ j), where
T7l€Z
n€Z
and where /
m
and 7
n
are formal Laurent series with coefficients in EndV in
variables other than zt-, Zj and ztj, such that the expressions
(zi~Zj)lf(zi)g(zj)
and (ZJ Zi)lg(zj)f(zi), for all / G Z, exist (cf. [6], Chapter 2). In particular,
we can form combinations of products of vertex operators and ^-functions
such as:
[Y(vn,zn) xZnl [Y(vn_uzn_x) x2n_M [Y(v2,z2) xZ21 Y(vx,zx)] •]]. (1.22)
Straightforward iteration of the Jacobi identity gives:
[Y(vn, zn) x2nl [F(un_!, Zn-t) x2n_1A [Y(v2, z2) x221 Y(vx, zx)\ •••]] =
= Y(Y(vn, znl)Y(vn.u zn.XtX) Y(v2, z2x)vu zx) f[ z^S ( ^ - ^ - ) .
j=2 \
ZX
J
.
( L 2 3 )
Unlike the Jacobi identity, however, (1.23) does not express a combination
of products of vertex operators in terms of only iterates of vertex operators;
products as well as iterates appear on the right. More specifically, (1.23)
suggests considering such expressions as
Y([Y(v
) *zn2 [• x*42 [Y{v3,z31) xZ32 Y(v2,z21)] -]]vx,zxy
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