188
8 BRST Quantization
where the longitudinal mass and total level operator are
m
2
,L = −p
2
,L ,
L
0 = N
0
+ N
1
+ N
b
+ N
c
∈ N.
(8.34)
A state |ψ is said to be on-shell if it is annihilated by L 0
on-shell:
L 0 |ψ = 0.
(8.35)
The absolute BRST cohomology H abs (Q B ) defines the physical states (Sect. 3.2)
and is given by the states ψ ∈ H that are Q B -closed but not exact
H abs (Q B ) :=
|ψ ∈ H
Q B |ψ = 0, |χ ∈ H
|ψ = Q B |χ
.
(8.36)
Since Q B commutes with L 0 , (8.25), the cohomology subspace is preserved under
time evolution.
Before continuing, it is useful to outline the general strategy for studying the
cohomology of a BRST operator Q in the CFT language. The idea is to find an
operator —called contracting homotopy operator—which, if it exists, trivializes
the cohomology. Conversely, this implies that the cohomology is to be found within
states that are annihilated by or for which is not defined. Then, it is possible
to restrict Q on these subspaces: this is advantageous when the restriction of the
BRST charge on these subspaces is a simpler. In fact, we will find that the reduced
operator is itself a BRST operator, for which one can search for another contracting
homotopy operator. 1
Given a BRST operator Q, a contracting homotopy operator for Q is an
operator such that
{Q, ,} = 1.
(8.37)
Interpreting Q as a derivative operator, corresponds to the Green function or
propagator. The existence of a well-defined with empty kernel implies that the
cohomology is empty because all closed states are exact. Indeed, consider a state
|ψ ∈ H, which is an eigenstate of and closed Q B |ψ = 0. Inserting (8.37) in
front of the state gives
|ψ = {Q B , ,} |ψ = Q B
|ψ
.
(8.38)
If is well-defined on |ψ and |ψ /
∈ ker , then |ψ is another state in H, which
implies that |ψ is exact. Hence, the BRST cohomology has to be found inside the
subspaces ker or on which is not defined.
1 A similar strategy shows that there is no open string excitation for the open SFT in the tachyon
vacuum.
8 BRST Quantization
where the longitudinal mass and total level operator are
m
2
,L = −p
2
,L ,
L
0 = N
0
+ N
1
+ N
b
+ N
c
∈ N.
(8.34)
A state |ψ is said to be on-shell if it is annihilated by L 0
on-shell:
L 0 |ψ = 0.
(8.35)
The absolute BRST cohomology H abs (Q B ) defines the physical states (Sect. 3.2)
and is given by the states ψ ∈ H that are Q B -closed but not exact
H abs (Q B ) :=
|ψ ∈ H
Q B |ψ = 0, |χ ∈ H
|ψ = Q B |χ
.
(8.36)
Since Q B commutes with L 0 , (8.25), the cohomology subspace is preserved under
time evolution.
Before continuing, it is useful to outline the general strategy for studying the
cohomology of a BRST operator Q in the CFT language. The idea is to find an
operator —called contracting homotopy operator—which, if it exists, trivializes
the cohomology. Conversely, this implies that the cohomology is to be found within
states that are annihilated by or for which is not defined. Then, it is possible
to restrict Q on these subspaces: this is advantageous when the restriction of the
BRST charge on these subspaces is a simpler. In fact, we will find that the reduced
operator is itself a BRST operator, for which one can search for another contracting
homotopy operator. 1
Given a BRST operator Q, a contracting homotopy operator for Q is an
operator such that
{Q, ,} = 1.
(8.37)
Interpreting Q as a derivative operator, corresponds to the Green function or
propagator. The existence of a well-defined with empty kernel implies that the
cohomology is empty because all closed states are exact. Indeed, consider a state
|ψ ∈ H, which is an eigenstate of and closed Q B |ψ = 0. Inserting (8.37) in
front of the state gives
|ψ = {Q B , ,} |ψ = Q B
|ψ
.
(8.38)
If is well-defined on |ψ and |ψ /
∈ ker , then |ψ is another state in H, which
implies that |ψ is exact. Hence, the BRST cohomology has to be found inside the
subspaces ker or on which is not defined.
1 A similar strategy shows that there is no open string excitation for the open SFT in the tachyon
vacuum.