390
B Summary of Important Formulas
The energy for the normal ordering of the different currents is
L m =
n
n − (1 − λ)m
b m−n c n
+ a gh δ m,0 ,
(B.60a)
j m =
n
b m−n c n
+ δ m,0 .
(B.60b)
The energy–momentum and ghost current zero-modes are explicitly
L 0 =
n
n
b −n c n
+ a gh =
L 0 − 1,
(B.61a)
N gh,L = j 0 =
n
b −n c n
+ 1 =
N gh,L +
1
2
N
c
0 − N
b
0
−
3
2
,
(B.61b)
L 0 = N
b
+ N
c ,
N gh,L :=
n>0
N
c
n − N
b
n
.
(B.61c)
Then, one can straightforwardly compute the ghost number of the vacua:
N gh |0 = 0,
N gh | ↓↓ = | ↓↓ ,
N gh | ↑↑ = 2 | ↑↑ .
(B.62)
Using (B.56) allows to write the ghost numbers on the cylinder:
N
cyl
gh | ↓↓ = −
1
2
| ↓↓ ,
N
cyl
gh | ↑↑ =
1
2
| ↑↑ .
(B.63)
The b n and c n are Hermitian:
b
†
n = b −n ,
c
†
n = c −n .
(B.64)
The BPZ conjugates of the modes are
b
t
n = (±1)
n b −n ,
c
t
n = −(±1)
n c −n ,
(B.65)
using I ± (z) with (6.111).
The conjugates of the vacuum read
| ↓↓
‡
= =0|c −1 ,
| ↑↑
‡
= =0|c −1 c 0 .
(B.66)
The BPZ conjugates of the vacua are
| := | ↓↓
t
= ∓ ∓0|c −1 ,
↑ | := | ↑↑
t
= ± ±0|c 0 c −1 .
(B.67)