24

CRISTIANO HUSU

T h e o r e m 1.11 Let Vi,v2,- • -vn be a family of vectors in a vertex operator

algebra and let 1 m j n; then

Y(vn,vn_1,'--v1;zn,zn_1,---z1;{zij 3 i j};m;\jj]) =

= ¥(¥(••• Y(Y(vn,znm)Y(vn^

(1.72)

• • • V 2 , 2 2 l H , Z l ) -

where both sides of this identity are existing expressions. In particular,

%

l

» » - i r • • «i;*n«n-i,• • • *i; tei 3 * j}) =

= Y(Y(- • • Y(Y(vn, «„,»_i)wB_i, a^-Ln-a) • • • t2, *2iM, *i)- (1.73)

• n ^(^r^V

ltin \ *M-1 /

We call the identity (1.73) the multi-operator Jacobi identity.

Proof The existence of the right-hand side of(1.72) is a direct application

of Lemma 1.9. The existence of the left-hand side of (1.72) is a consequence

of Lemma 1.9 and the following identities, which also show the equality of

the two expressions.

Using (1.23) and the linearity of vertex operators, we obtain

Y(vn, v

n

_i,..., vi; znj z

n

_i,..., zX] {z{j\i j}; 2; [j, j]) =

= Y([Y(vn,znl) xZn2 [Y(vn^lyzn.1A) x2n_12 • • • [Y(v3,z31) xZ32 Y(v2,z2i)] • • •

• • • • ] ] ^ i ) n ^ (

£ L

^ ) =

j=2 \ Z\ J

= r(y(r(n,,jnJ)y(vi,vy) • • • *3, z32K *2iK *i)- (i-74)

(n^(^))(n^(^e)).