B Summary of Important Formulas
391
We have the following relations:
| = ∓ | ↓↓
‡ ,
↑ | = ∓ | ↑↑
‡ .
(B.68)
The ghost are normalized with
↑ | ↓↓ = =↓ |c 0 | ↓↓ = =0|c −1 c 0 c 1 |0 = 1,
(B.69)
which selects the minus sign in the BPZ conjugation. The conjugate of the ghost
vacuum is
0
c
| = =0|c −1 c 0 c 1 .
(B.70)
Considering both the holomorphic and anti-holomorphic sectors, we introduce
the combinations:
b
±
n = b n ± ¯
b n ,
c
±
n =
1
2
(c n ± ¯
c n ).
(B.71)
The normalization of b ±
m is chosen to match the one of L ±
m (B.75) and the one of c ±
m
such that
{b
+
m , c
+
n } = δ m+n ,
{b
−
m , c
−
n } = δ m+n .
(B.72)
We have the following useful identities:
b
−
n b
+
n = 2b n ¯
b n ,
c
−
n c
+
n =
1
2
c n ¯
c n .
(B.73)
B.4
Bosonic String
The BPZ conjugates of the scalar and ghost modes are
(α n )
t
= −(±1)
n α −n ,
(b n )
t
= (±1)
n b −n ,
(c n )
t
= −(±1)
n c −n .
(B.74)
Combinations of holomorphic and anti-holomorphic modes are
L
±
n = L n ± ¯
L n ,
b
±
n = b n ± ¯
b n ,
c
±
n =
1
2
(c n ± ¯
c n ).
(B.75)