162
7 CFT Systems
Sometimes, if = +1, one denotes b and c, respectively, by β and γ . If b and c
are ghosts arising from Faddeev–Popov gauge fixing, then = 1 if λ is integer; and
= −1 if λ is half-integer (“wrong” spin–statistics assignment).
The U(1) global symmetry (7.94) reads infinitesimally
δb = −ib,
δc = ic,
δ ¯
b = −i ¯
b,
δ ¯
c = i ¯
c.
(7.102)
It is generated by the conserved ghost current with components:
j (z) = −:b(z)c(z) :,
¯
j(¯ z) = −: ¯
b(¯ z) ¯
c(¯ z) :
(7.103)
and the associated charge is called the ghost number
N gh = N gh,L + N gh,R ,
N gh,L =
dz
2π i
j (z),
N gh,R = −
d¯ z
2π i
¯
j(¯ z).
(7.104)
This charge counts the number of c ghosts minus the number of b ghosts, such that
N gh (c) = 1,
N gh (b) = −1,
N gh ( ¯
c) = 1,
N gh ( ¯
b) = −1.
(7.105)
The propagator can be derived from the path integral
d
b d
c
δ
δb(z)
b(w)e
−S[b,c]
= 0
(7.106)
which gives the differential equation
δ
(2) (z − w) +
1
2π
b(w) ¯
∂c(z) = 0.
(7.107)
Using (B.2), the solution is easily found to be
c(z)b(w) =
1
z − w
.
(7.108)
Remark 7.5 The propagator is constructed with the path integral. For convenience,
the zero-modes are removed from the measure: reintroducing them, one finds
that the propagator is computed not in the conformal vacuum (which has no
operator insertion), but in a state with ghost insertions. This explains why the
propagator (7.108) is not of the form (6.62b). However, this form is sufficient to
extract the OPE as changing the vacuum does not introduce singular terms.