164
7 CFT Systems
The coefficients of the (z − w) −1 terms correspond to the ghost number of the b and
c fields (7.105). More generally, the ghost number N gh (O) of any operator O(z) is
defined by
j (z)O(w) ∼ N gh (O)
O(w)
z − w
.
(7.116)
The OPE for j with itself is
j (z)j (w) ∼
(z − w) 2 .
(7.117)
Finally, the OPE of the current with T reads
T (z)j (w) ∼
q λ
(z − w) 3 +
j (w)
(z − w) 2 +
∂j (w)
z − w
.
(7.118)
Due to the presence of the z −3 term, the current j (z) is not a primary field if q λ = 0,
that is, if λ = 1/2. In that case, its transformation under changes of coordinates gets
an anomalous contribution:
j (z) =
dw
dz
j
(w) +
q λ
2
d
dz
ln
dw
dz
=
dw
dz
j
(w) +
q λ
2
∂ 2
z w
∂ z w
.
(7.119)
This implies in particular that the currents on the plane and on the cylinder (w =
ln z) are related by
j (z) =
dw
dz
j
cyl (w) −
q λ
2
,
(7.120)
which leads to the following relation between the ghost numbers on the plane and
on the cylinder:
N gh = N
cyl
gh −q λ ,
N gh,L = N
cyl
gh,L −
q λ
2
,
N gh,R = N
cyl
gh,R −
q λ
2
.
(7.121)
For this reason, it is important to make clear the space with respect to which is given
the ghost number: if not explicitly stated, ghost numbers in this book are given on
the plane. 10
10 Other references, especially old ones, give it on the cylinder. This can be easily recognized if
some ghost numbers in the holomorphic sector are half-integers: for the reparametrization ghosts,
q λ is an integer such that the shift in (7.121) is a half-integer.