7.2 First-Order bc Ghost System
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7.2.3 OPE
The OPEs between the b and c fields are found from the propagator (7.108):
c(z)b(w) ∼
1
z − w
,
b(z)c(w) ∼
z − w
,
(7.109a)
b(z)b(w) ∼ 0,
c(z)c(w) ∼ 0.
(7.109b)
The OPE of each ghost with T confirms the conformal weights in (7.98):
T (z)b(w) ∼ λ
b(w)
(z − w) 2 +
∂b(w)
z − w
,
(7.110a)
T (z)c(w) ∼ (1 − λ)
c(w)
(z − w) 2 +
∂c(w)
z − w
.
(7.110b)
The OPE of T with itself is
T (z)T (w) ∼
c λ /2
(z − w) 4 +
2T (w)
(z − w) 2 +
∂T (w)
z − w
,
(7.111)
where the central charge is
c λ = 2(−1 + 6λ − 6λ
2 ) = −2
1 + 6λ(λ − 1)
.
(7.112)
Introducing the ghost charge:
q λ = (1 − 2λ),
(7.113)
the central charge can also be written as
c λ = (1 − 3q
2
λ ).
(7.114)
This parameter will appear many times in this section and its meaning will become
clearer as we proceed.
The OPE between the ghost current (7.103) and the b and c ghosts read
j (z)b(w) ∼ −
b(w)
z − w
,
(7.115a)
j (z)c(w) ∼
c(w)
z − w
.
(7.115b)