388
B Summary of Important Formulas
In particular, L 0 commutes with the zero-modes:
[L 0 , b 0 ] = 0,
[L 0 , c 0 ] = 0.
(B.46)
The anomalous global U(1) symmetry for the ghost number N gh is generated by
the ghost current:
j = −:bc:,
N gh,L =
dz
2π i
j (z),
(B.47)
such that
N gh (c) = 1,
N gh (b) = −1.
(B.48)
Remember that N gh = N gh,L in the left sector, such that we omit the index L. The
modes of the ghost current are
j m = −
n
: b m−n c n : = −
n
: b n c m−n :,
N gh,L = j 0 = −
n
: b −n c n :.
(B.49)
The commutator of the current modes with itself and with the Virasoro modes are
[j m , j n ] = m δ m+n,0 ,
[L m , j n ] = −nj m+n −
3
2
m(m + 1)δ m+n,0 .
(B.50)
Finally, the commutators of the ghost number operator are
[N gh , b(w)] = −b(w),
[N gh , c(w)] = c(w).
(B.51)
The level operators N b and N c and the number operators N b
n and N c
n are defined
as
N
b
=
n>0
n N
b
n ,
N
c
=
n>0
n N
c
n ,
(B.52a)
N
b
n = :b −n c n :,
N
c
n = :c −n b n :.
(B.52b)
The commutator of the number operators with the modes is
[N
b
m , b −n ] = b −n δ m,n ,
[N
c
m , c −n ] = c −n δ m,n .
(B.53)