8.3 BRST Cohomology: Two Flat Directions
197
This implies that the combination in parenthesis is Q 0 -closed and, for the same
reason as above, it is exact
Q 1 |ψ 1 + Q 2 |ψ 0 = Q 0 |ψ 2 ,
|ψ 2 = −
B
L
0
Q 1 |ψ 1 + Q 2 |ψ 0
.
(8.78)
Computation: Equation (8.77)
{Q 0 , Q 1 } |ψ 1 = Q 0 Q 1 |ψ 1 − Q
2
1 |ψ 0 = Q 0 Q 1 |ψ 1 + {Q 0 , Q 2 } |ψ 0 .
The first equality follows from (8.76), and the second by using (8.67). The final
result is obtained after using that |ψ 0 is Q 0 -closed.
Iterating this procedure leads to a series of states
|ψ k+1 = −
B
L
0
Q 1 |ψ k + Q 2 |ψ k−1
.
(8.79)
We claim that a state in the relative cohomology |ψ ∈ H 0 (
Q B ) is built by summing
all these states
|ψ =
k∈N
|ψ k .
(8.80)
Indeed, it is easy to check that |ψ is
Q B -closed
Q B |ψ = 0.
(8.81)
We leave aside the proof that ψ is not exact (see [5]). Note that ψ and ψ 0 have the
same ghost numbers
N gh (ψ) = N gh (ψ 0 ) = 1
(8.82)
since N gh (BQ j ) = 0.
In fact, since ψ 0 does not contain longitudinal modes, it is annihilated by Q 1
and Q 2 (these operators contain either a ghost creation operator together with a
light-cone annihilation operator, or the reverse)
Q 1 |ψ 0 = Q 2 |ψ 0 = 0.
(8.83)
As a consequence, one has ψ k = 0 for k ≥ 1 and ψ = ψ 0 .