1. DEFINITION OF VERTEX ALGEBRAS

This is analogous to the Sokhotsky-Plemelj formula

2ITIO(X) = — —,

v }

x + iO x-iO

well-known in complex analysis. Here in the right hand side we take the difference

between the boundary values of the function - on the real line from the upper

x

and lower half-planes (denoted ——- and —, respectively). The result is the

X ~\~ u\j . X ~~~

ZvJ

delta-distribution on the real line at 0 (up to the factor 2m).

1.1.8. Algebraic reformulation. For any C-algebra i?, we denote by R[[z]]

the space of iJ-valued formal Taylor series in z. Its elements are series ]Cn o

anZn•

where an G R for all n 0. This space is naturally an algebra.

The space R({z)) of i?-valued formal Laurent series in z is by definition the

space of series J^nez anZni where an G R for all n, and there exists N G Z such that

an = 0,Vn N (in other words, the series is finite in the "negative direction").

Note that R{(z)) is an algebra, and if R is a field, then R({z)) is also a field.

Denote by C((z))((w)) the space R((w)), where R = C((z)). In other words,

this is the space of Laurent series in w whose coefficients are Laurent series in z.

Then the series 6{z — w)- belongs to C((z))((w)) (actually, even to C[^""1][[ty]]).

This is an algebraist's way of saying that 6(z — w)- is the expansion of in

z — w

the domain \z\ \w\ (i.e., in positive powers of w/z). Similarly, 5(z — w)+ belongs

to C((tt/))((z)), and it is the expansion of in the domain \z\ \w\ (i.e., in

z — w

positive powers of z/w).

Denote by C((z,w)) the field of fractions of C[[z,it;]]; its elements may be

viewed as meromorphic functions in two formal variables. This field has two natural

topologies, in which the basis of open neighborhoods of 0 consists of all elements

of the form

wNf{zyw),N

G Z (resp.,

zNf(z,w),N

G Z), where f(z,w) does not

contain w (resp., z) in the denominator. The completions of C((z,w)) with respect

to these topologies are C((z))((w)) and C((w))((z)), respectively. Thus, C((z))((w))

contains expansions of meromorphic functions near the z axis, and C((w))((z))

contains expansions of meromorphic functions near the w axis (see the picture).