150
7 CFT Systems
Note that classically h k = 0 since ∼ ¯
h [10, p. 81]. The weight is invariant under
k → −k. Finally, the OPE between two vertex operators is
V k (z, ¯
z)V k (w, , ¯
w) ∼
V k+k (w, ¯
w)
(z − w) −kk 2 /2
,
(7.47)
where only the leading term (non-necessarily singular) is displayed. In particular,
correlation functions should be computed for kk < 0 in order to avoid exponential
growth.
Computation: Equation (7.38)
T (z)∂X(w) = −
2 : ∂X(z)∂X(z) : ∂X(w)
∼ −
2
2 :∂X(z)∂X(z): ∂X(w) ∼
∂X(z)
(z − w) 2 .
The result (7.38) follows by Taylor expanding the numerator.
Computation: Equation (7.39)
T (z)∂X(w) =
1
4 :∂X(z)∂X(z): :∂X(w)∂X(w):
∼
1
4
:∂X(z)∂X(z): :∂X(w)∂X(w): + :∂X(z)∂X(z): :∂X(w)∂X(w):
+:∂X(z)∂X(z): :∂X(w)∂X(w): + perms
∼ 2 ×
1
4
1
(z − w) 4 − 4 ×
1
2 2
1
(z − w) 2 :∂X(z)∂X(w):
∼
1
2
1
(z − w) 4 −
2
2
1
(z − w) 2
:∂X(w)∂X(w): + (z − w) :∂
2 X(w)∂X(w):
.
Computation: Equation (7.42)
T (z)∂
n X(w) ∼ ∂
n−1
w
∂X(z)
(z − w) 2
∼ n!
∂X(z)
(z − w) n+1
∼
n!
(z−w) n+1
· · · +
1
(n−1)! (z − w) n−1 ∂ n−1 (∂X(w))
+
1
n! (z − w) n ∂ n (∂X(w))
.