214
10 Free BRST String Field Theory
Due to the definition of the BPZ product, the action is equivalent to a 2-point
correlation function on the disk.
The inner product satisfies the following identities:
A, B = (−1)
|A||B|
B, A,
Q B A, B = −(−1)
|A|
A, Q B B,
(10.15)
where |A| denotes the Grassmann parity of the operator A.
A first consistency check is to verify that the ghost number of the string can be
defined such that the action is not vanishing. Indeed, the ghost number anomaly on
the disk implies that the total ghost number must be N gh = 3. Since physical states
have N gh = 1, it is reasonable to take the string field to satisfy the same condition,
even off-shell:
N gh (() = 1.
(10.16)
This condition means that there is no ghost at the classical level beyond the one
of the energy vacuum | ↓↓, which has N gh = 1. Moreover, the BRST charge has
N gh (Q B ) = 1, such that the action has ghost number 3.
One needs to find the Grassmann parity of the string field. Using the properties
of the BPZ inner product, the string field should be Grassmann odd
|| = 1
(10.17)
for the action to be even. This is in agreement with the fact that the string field has
ghost number 1 and that the ghosts are Grassmann odd. One must impose a reality
condition on the string field (a complex field would behave like two real fields and
have too many states). The appropriate reality condition identifies the Euclidean and
BPZ conjugates:
|
‡
= |
t .
(10.18)
That this relation is correct will be checked a posteriori for the tachyon field in
Sect. 10.4.
Computation: Equation (10.17)
Q B = (−1)
||(|Q B |)
Q B , , = (−1)
||(1+||)
Q B , ,
= =Q B , , = −(−1)
||
Q B ,
where we used both properties in (10.15), together with the fact that ||(1 +
||) is necessarily even. In order for the bracket to be non-zero, one must have
||
= 1.
10 Free BRST String Field Theory
Due to the definition of the BPZ product, the action is equivalent to a 2-point
correlation function on the disk.
The inner product satisfies the following identities:
A, B = (−1)
|A||B|
B, A,
Q B A, B = −(−1)
|A|
A, Q B B,
(10.15)
where |A| denotes the Grassmann parity of the operator A.
A first consistency check is to verify that the ghost number of the string can be
defined such that the action is not vanishing. Indeed, the ghost number anomaly on
the disk implies that the total ghost number must be N gh = 3. Since physical states
have N gh = 1, it is reasonable to take the string field to satisfy the same condition,
even off-shell:
N gh (() = 1.
(10.16)
This condition means that there is no ghost at the classical level beyond the one
of the energy vacuum | ↓↓, which has N gh = 1. Moreover, the BRST charge has
N gh (Q B ) = 1, such that the action has ghost number 3.
One needs to find the Grassmann parity of the string field. Using the properties
of the BPZ inner product, the string field should be Grassmann odd
|| = 1
(10.17)
for the action to be even. This is in agreement with the fact that the string field has
ghost number 1 and that the ghosts are Grassmann odd. One must impose a reality
condition on the string field (a complex field would behave like two real fields and
have too many states). The appropriate reality condition identifies the Euclidean and
BPZ conjugates:
|
‡
= |
t .
(10.18)
That this relation is correct will be checked a posteriori for the tachyon field in
Sect. 10.4.
Computation: Equation (10.17)
Q B = (−1)
||(|Q B |)
Q B , , = (−1)
||(1+||)
Q B , ,
= =Q B , , = −(−1)
||
Q B ,
where we used both properties in (10.15), together with the fact that ||(1 +
||) is necessarily even. In order for the bracket to be non-zero, one must have
||
= 1.