8.3 BRST Cohomology: Two Flat Directions
199
This form can be made covariant: taking a state of the form |ψ ⊗ | ↓↓ with |ψ ∈
H m , acting with Q B , implies the equivalence with the old covariant quantization
L
m
0 − 1
|ψ = 0,
∀n > 0 : L
m
n |ψ = 0.
(8.86)
This means that ψ must be a weight 1 primary field of the matter CFT.
Remark 8.2 (Open String) The results of this section provide, in fact, the cohomology for the open string after taking p L = p (instead of p L = p/2 for the closed
string).
8.3.4 Cohomology for Holomorphic and Anti-holomorphic Sectors
It remains to generalize the computation of the cohomology when considering both
the holomorphic and anti-holomorphic sectors.
In this case, the BRST operator is
Q B = c 0 L 0 − b 0 M +
Q B + ¯
c 0 ¯
L 0 − ¯
b 0 ¯
M +
Q B .
(8.87)
It is useful to rewrite this expression in terms of L
±
0 , b
±
0 and c
±
0
Q B = c
+
0 L
+
0 − b
+
0 M
+
+ c
−
0 L
−
0 − b
−
0 M
−
+
Q
+
B ,
(8.88)
where
L
+
0 =
L
⊥+
0 −
m 2
2
2
− 2
+
L
+
0 ,
L
−
0 = L
⊥−
0 +
L
−
0
(8.89)
and
M
±
:=
1
2
(M ± ¯
M).
(8.90)
Because of the relations L
±
0 = {Q B , b
±
0 }, we find that states in the cohomology
must be on-shell L
+
0 = 0 and must satisfy the level-matching condition L
−
0 = 0 5
L
+
0 |ψ = L
−
0 |ψ = 0.
(8.91)
Again, it is possible to reduce the cohomology by imposing conditions on
the zero-modes such that the above conditions are automatically satisfied (see
5 In the current case, the propagator is less easily identified. We will come back on its definition
later.
199
This form can be made covariant: taking a state of the form |ψ ⊗ | ↓↓ with |ψ ∈
H m , acting with Q B , implies the equivalence with the old covariant quantization
L
m
0 − 1
|ψ = 0,
∀n > 0 : L
m
n |ψ = 0.
(8.86)
This means that ψ must be a weight 1 primary field of the matter CFT.
Remark 8.2 (Open String) The results of this section provide, in fact, the cohomology for the open string after taking p L = p (instead of p L = p/2 for the closed
string).
8.3.4 Cohomology for Holomorphic and Anti-holomorphic Sectors
It remains to generalize the computation of the cohomology when considering both
the holomorphic and anti-holomorphic sectors.
In this case, the BRST operator is
Q B = c 0 L 0 − b 0 M +
Q B + ¯
c 0 ¯
L 0 − ¯
b 0 ¯
M +
Q B .
(8.87)
It is useful to rewrite this expression in terms of L
±
0 , b
±
0 and c
±
0
Q B = c
+
0 L
+
0 − b
+
0 M
+
+ c
−
0 L
−
0 − b
−
0 M
−
+
Q
+
B ,
(8.88)
where
L
+
0 =
L
⊥+
0 −
m 2
2
2
− 2
+
L
+
0 ,
L
−
0 = L
⊥−
0 +
L
−
0
(8.89)
and
M
±
:=
1
2
(M ± ¯
M).
(8.90)
Because of the relations L
±
0 = {Q B , b
±
0 }, we find that states in the cohomology
must be on-shell L
+
0 = 0 and must satisfy the level-matching condition L
−
0 = 0 5
L
+
0 |ψ = L
−
0 |ψ = 0.
(8.91)
Again, it is possible to reduce the cohomology by imposing conditions on
the zero-modes such that the above conditions are automatically satisfied (see
5 In the current case, the propagator is less easily identified. We will come back on its definition
later.