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