7.2 First-Order bc Ghost System
167
The definitions follow from the commutators (7.134). Then, the level operators N b
and N c are obtained by summing over n:
N
b
=
n>0
n N
b
n ,
N
c
=
n>0
n N
c
n .
(7.127)
The Virasoro operators are
L m =
n
n − (1 − λ)m
: b m−n c n : =
n
(λm − n) : b n c m−n :.
(7.128)
Of particular importance is the zero-mode
L 0 = −
n
n : b n c −n : =
n
n : b −n c n :.
(7.129)
We will give the expression of L 0 in terms of the level operators below, see (7.159).
To do this, we will first need to change the normal ordering, which first requires to
study the Hilbert space.
The modes of the ghost current are
j m = −
n
: b m−n c n : = −
n
: b n c m−n :.
(7.130)
Note that the zero-mode of the current also equals the ghost number
N gh,L = j 0 = −
n
: b −n c n :.
(7.131)
When both the holomorphic and anti-holomorphic sectors enter, it is convenient
to introduce the combinations
b
±
n = b n ± ¯
b n ,
c
±
n =
1
2
(c n ± ¯
c n ).
(7.132)
The normalization of b ±
m is chosen to match one of L ±
m (6.118). The normalization
of c ±
n is chosen such that (7.135) holds. Note the following useful identities:
b
−
n b
+
n = 2b n ¯
b n ,
c
−
n c
+
n =
1
2
c n ¯
c n .
(7.133)