EXTENSIONS OF THE JACOBI IDENTITY 15
As we have mentioned, the identity obtained by equating (1.28) with
(1.29) is called the three-operator (or three-vertex operator) Jacobi identity:
zm & ( ) [Y(v3iz3) xz3i \y(v2,Z2) x221
Y(v1,z1))}+
\ z32 J
+z£6 P
2 1
~
Z 3 1
) [Y(v2,z2) x221 [Y(v3,z3) x231 Y(v1,z1)}} =
\ Z23 /
^
221
/ j
= 2
\ ZX J
(1-44)
This achieves the goal of expressing a combination of products of vertex
operators Y(vi,Zi) and suitable ^-functions in terms of the product of an
"iterated" vertex operator and the resulting ^-functions. Moreover, we recall
that both sides of (1.44) equal the expression (1.27).
Remark 1.5 Applying the three-operator Jacobi identity we see that the
identity of Proposition A.3.3 in [6], namely
Y, o-Y(Y(Y(v3,z32)v2,Z2i)vuzl)z^S (— J
rG{l,(123),(132)} V Z21 '
•Ii^(Zj-^)=0,
(1.45)
can be written in the form
£ *[Y(v3,z3) x231 [Y(v2,z2) xZ21 Y{vuZl)))z3^ ( *
3 1
~ *
2 1
) = 0, (1.46)
where S3 is the symmetric group on the 3 indices.
The case n = 4 in (1.23) and (1.24) shows all the ideas and subtleties of
the general case, which will be discussed in the next subsection. The identity
(1.23) gives
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