174
7 CFT Systems
One can straightforwardly compute the ghost number of the vacua:
j 0 | ↓↓ = (λ − 1) | ↓↓ =
−
q λ
2
−
1
2
| ↓↓ ,
(7.165a)
j 0 | ↑↑ = λ | ↑↑ =
−
q λ
2
+
1
2
| ↑↑ .
(7.165b)
This confirms that the SL(2, C) vacuum has vanishing ghost number since | ↓↓
contains exactly λ − 1 ghosts:
j 0 |0 = 0.
(7.166)
Using (7.121) allows to write the ghost numbers on the cylinder:
j
cyl
0 | ↓↓ = −
1
2
| ↓↓ ,
j
cyl
0 | ↑↑ =
1
2
| ↑↑ .
(7.167)
That both ghost numbers have same magnitude but opposite signs could be
expected: since the ghost number changes as N gh → −N gh when b ↔ c, the mean
value of the ghost number should be zero.
Remark 7.6 (Ghost Number Conventions) Since the ghost number is an additive
quantum number, it is always possible to shift its definition by a constant. This can
be used to set the ghost numbers of the vacua to some other values. For example, [1,
p. 116] adds q λ /2 to the ghost number in order to get N gh = ±1/2 on the plane
(instead of the cylinder). We do not follow this convention in order to keep the
symmetry between the vacuum ghost numbers on the cylinder.
Computation: Equation (7.159)
Start with (7.129) and use (6.157):
L 0 = −
n
n :b n c −n : = −
n≤−λ
n b n c −n +
n>−λ
n c −n b n
=
n≥λ
n b −n c n +
n>−λ
n c −n b n
=
n≥λ
n b −n c n +
n>0
n c −n b n +
0
n=−λ+1
n c −n b n
=
n≥λ
n b −n c n +
n>0
n c −n b n +
λ−1
n=0
n b −n c n + a λ
7 CFT Systems
One can straightforwardly compute the ghost number of the vacua:
j 0 | ↓↓ = (λ − 1) | ↓↓ =
−
q λ
2
−
1
2
| ↓↓ ,
(7.165a)
j 0 | ↑↑ = λ | ↑↑ =
−
q λ
2
+
1
2
| ↑↑ .
(7.165b)
This confirms that the SL(2, C) vacuum has vanishing ghost number since | ↓↓
contains exactly λ − 1 ghosts:
j 0 |0 = 0.
(7.166)
Using (7.121) allows to write the ghost numbers on the cylinder:
j
cyl
0 | ↓↓ = −
1
2
| ↓↓ ,
j
cyl
0 | ↑↑ =
1
2
| ↑↑ .
(7.167)
That both ghost numbers have same magnitude but opposite signs could be
expected: since the ghost number changes as N gh → −N gh when b ↔ c, the mean
value of the ghost number should be zero.
Remark 7.6 (Ghost Number Conventions) Since the ghost number is an additive
quantum number, it is always possible to shift its definition by a constant. This can
be used to set the ghost numbers of the vacua to some other values. For example, [1,
p. 116] adds q λ /2 to the ghost number in order to get N gh = ±1/2 on the plane
(instead of the cylinder). We do not follow this convention in order to keep the
symmetry between the vacuum ghost numbers on the cylinder.
Computation: Equation (7.159)
Start with (7.129) and use (6.157):
L 0 = −
n
n :b n c −n : = −
n≤−λ
n b n c −n +
n>−λ
n c −n b n
=
n≥λ
n b −n c n +
n>−λ
n c −n b n
=
n≥λ
n b −n c n +
n>0
n c −n b n +
0
n=−λ+1
n c −n b n
=
n≥λ
n b −n c n +
n>0
n c −n b n +
λ−1
n=0
n b −n c n + a λ