392
B Summary of Important Formulas
The closed string inner product is defined from the BPZ product by an additional
insertion of c
−
0
A, B = =A|c
−
0 |B ,
(B.76)
while the open string inner product is equal to the BPZ product
A, B = =A|B .
(B.77)
The vacuum for the matter and ghosts is
|k, 0 := |k ⊗ |0 ,
|k, ↓↓ := |k ⊗ | ↓↓ .
(B.78)
The vacuum is normalized as
open: k, ↓ |c 0 |k, ↓↓ = =k
, 0|c −1 c 0 c 1 |k, 0 = (2π)
D δ
(D) (k + k
),
(B.79a)
closed: k, ↓↓ |c 0 ¯
c 0 |k, ↓↓↓ = =k
, 0|c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 |k, 0 = (2π)
D δ
(D) (k + k
).
(B.79b)
B Summary of Important Formulas
The closed string inner product is defined from the BPZ product by an additional
insertion of c
−
0
A, B = =A|c
−
0 |B ,
(B.76)
while the open string inner product is equal to the BPZ product
A, B = =A|B .
(B.77)
The vacuum for the matter and ghosts is
|k, 0 := |k ⊗ |0 ,
|k, ↓↓ := |k ⊗ | ↓↓ .
(B.78)
The vacuum is normalized as
open: k, ↓ |c 0 |k, ↓↓ = =k
, 0|c −1 c 0 c 1 |k, 0 = (2π)
D δ
(D) (k + k
),
(B.79a)
closed: k, ↓↓ |c 0 ¯
c 0 |k, ↓↓↓ = =k
, 0|c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 |k, 0 = (2π)
D δ
(D) (k + k
).
(B.79b)