230
10 Free BRST String Field Theory
Computation: Equation (10.97)
T |c 0 L 0 |T =
1
α
d D k
(2π) D
d D k
(2π) D T (k)T (k
) −k
, 0|c −1 c 0 L 0 c 1 |k, 0
=
1
α
d D k
(2π) D
d D k
(2π) D T (k)T (k
)(α
k
2
− 1) −k
, 0|c −1 c 0 c 1 |k, 0
=
1
α
d D k
(2π) D d
D k
T (k)T (k
)(α
k
2
− 1) δ
(D) (k + k
),
where we used 0|c −1 c 0 c 1 |0 = 1 and k |k = (2π) D δ (D) (k + k ).
10.5 Closed String
The derivation of the BRST free action for the closed string is very similar. The
starting point is the equation of motion
Q B | = 0
(10.100)
for the closed string field |. The difference with (10.12) is that the BRST charge
Q B now includes both the left- and right-moving sectors. In the case of the open
string, the field was free of any constraint: we will see shortly that this is not the
case for the closed string.
The next step is to find an inner product ·, ·· to write the action:
S =
1
2
Q B .
(10.101)
Following the open string, it seems logical to give the string field the same ghost
number as the states in the cohomology:
N gh (() = 2.
(10.102)
In this case, the ghost number of the arguments of ·, ·· in (10.101) is N gh = 5. The
ghost number anomaly requires the total ghost number to be 6, that is,
N gh (·, ··) = 1.
(10.103)
There is no other choice because N gh (() must be integer. The simplest solution is
to insert one c zero-mode c 0 or ¯
c 0 , or a linear combination. The BRST operator
Q B contains both L
±
0 (see the decomposition (8.88)): the natural expectation (and
10 Free BRST String Field Theory
Computation: Equation (10.97)
T |c 0 L 0 |T =
1
α
d D k
(2π) D
d D k
(2π) D T (k)T (k
) −k
, 0|c −1 c 0 L 0 c 1 |k, 0
=
1
α
d D k
(2π) D
d D k
(2π) D T (k)T (k
)(α
k
2
− 1) −k
, 0|c −1 c 0 c 1 |k, 0
=
1
α
d D k
(2π) D d
D k
T (k)T (k
)(α
k
2
− 1) δ
(D) (k + k
),
where we used 0|c −1 c 0 c 1 |0 = 1 and k |k = (2π) D δ (D) (k + k ).
10.5 Closed String
The derivation of the BRST free action for the closed string is very similar. The
starting point is the equation of motion
Q B | = 0
(10.100)
for the closed string field |. The difference with (10.12) is that the BRST charge
Q B now includes both the left- and right-moving sectors. In the case of the open
string, the field was free of any constraint: we will see shortly that this is not the
case for the closed string.
The next step is to find an inner product ·, ·· to write the action:
S =
1
2
Q B .
(10.101)
Following the open string, it seems logical to give the string field the same ghost
number as the states in the cohomology:
N gh (() = 2.
(10.102)
In this case, the ghost number of the arguments of ·, ·· in (10.101) is N gh = 5. The
ghost number anomaly requires the total ghost number to be 6, that is,
N gh (·, ··) = 1.
(10.103)
There is no other choice because N gh (() must be integer. The simplest solution is
to insert one c zero-mode c 0 or ¯
c 0 , or a linear combination. The BRST operator
Q B contains both L
±
0 (see the decomposition (8.88)): the natural expectation (and