7.2 First-Order bc Ghost System
169
while those between the L n and the ghost modes are
[L m , b n ] =
m(λ − 1) − n
b m+n ,
[L m , c n ] = −(mλ + n)c m+n ,
(7.137)
in agreement with (6.115). If n ∈ Z, each ghost field has zero-modes b 0 and c 0
which commutes with L 0
[L 0 , b 0 ] = 0,
[L 0 , c 0 ] = 0.
(7.138)
The commutator of the current modes reads
[j m , j n ] = m δ m+n,0 .
(7.139)
Then, the commutator with the Virasoro operators is
[L m , j n ] = −nj m+n +
q λ
2
m(m + 1)δ m+n,0 .
(7.140)
Finally, the commutators of the ghost number operator with the ghosts are
[N gh , b(w)] = −b(w),
[N gh , c(w)] = c(w).
(7.141)
Computation: Equation (7.134)
[b m , c n ] =
C 0
dw
2π i
w
−1
C w
dz
2π i
z
−1 w
n+λ z
m−λ+1 b(z)c(w)
∼
C 0
dw
2π i
w
−1
C w
dz
2π i
z
−1 w
n+λ z
m−λ+1
z − w
=
C 0
dw
2π i
w
m+n−1
= δ m+n,0 .
Computation: Equation (7.141)
[N gh , b(w)] =
dz
2π i
j (z)b(w) ∼ −
dz
2π i
b(w)
z − w
= −b(w).
The computation for c is similar.