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