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).
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