4 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY
The precise meaning of formula (1.11), including the roles of operator-valued
rational functions and of their various formal series expansions, will be discussed
in the main text. In particular, the single identity (1.11) is equivalent to an
infinite family of identities for the component operators (see Corollary 8.8.17 in
[FLM]).
The definitions of a representation irz of the vertex operator algebra V and of
intertwining operators Iz between representations are again formed by combining
the corresponding definitions for Lie algebras with the Cauchy residue formula
(1.9). In the main text, we shall use the notations Y(y,z) for both &&z(v) and
TTZ(V), and the notation y(v,z) for Iz(?)- These operators are all called vertex
operators, in their appropriate settings. (The algebra and module operators
Y(-,z) are fixed, while the intertwining operators y(',z ) for a given triple of
modules form a vector space.) The Jacobi identity for algebras, modules or
intertwining operators is a precise statement of what is known in the physics
literature as the Ward identity on the three-punctured Riemann sphere, and
our choice of the notation Y for vertex operators reflects the shape of the tree
diagram associated with this geometric picture.
One of the basic features of the Jacobi identity for Lie algebras, modules and
intertwining operators is its symmetry with respect to the symmetric group S3.
To show an analogous symmetry in the case of vertex operator algebras, modules
and intertwining operators, it is useful to display first an §3-symmetry of the
Cauchy residue formula. For any algebraic expression z, we set 6(z) = Ylnezzn
provided that this sum makes sense. This expression is the formal analogue of
the ^-distribution at z 1; in particular, 6(z)f(z) = S(z)f(l) for any / for which
these expressions are defined. Consider the following formal Laurent series in
three commuting variables:
where we expand the negative powers {z\ Z2)71 as formal power series in the
second variable, 22 Then the residue formula (1.9) can be expressed in terms of
such formal series as follows:
ZQ1^ I Zl
z
) F(zo, zu z2) + z~x8 I Z^—pL J F(zo, zu z2)
= zl1* ( £ L ^ ) F(*o, *n *2), (1.13)
where F is a Laurent polynomial in z$,z\,Z2\ in the first term, F can be replaced
by F(zi Z2,zi,Z2), expanded as a Laurent series in Z2 involving only finitely
many negative powers of ^2, and analogously for the two other terms, where we
expand in large powers of z\ and z§, respectively. Thanks to the new variable 20,
the §3-symmetry properties become clear. (The symmetry becomes completely
apparent if we replace z\ by —Z\ and move the right-hand side to the left; the
^-functions then correspond to the symbolic relation ZQ -f- z\ 4- Z2 0.)
In the Jacobi identity for vertex operator algebras, modules and intertwin-
ing operators, special expressions are used in the role of F\ these expressions
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