200
8 BRST Quantization
also Sect. 3.2.2). Imposing first the condition b
−
0 = 0 defines the semi-relative
cohomology. The relative cohomology is found by imposing b
±
0 = 0 and in fact
corresponds to the physical space (see [13, sec. 2.3] for more details). The rest of
the derivation follows straightforwardly because the two sectors commute: we find
that the cohomology is ghost-free and has no light-cone excitations
L
±
0 = N
0
± ¯
N
0
+ N
1
± ¯
N
1
+ N
b
± ¯
N
b
+ N
c
± ¯
N
c
= 0.
(8.92)
In general, it is simpler to work with a covariant expression and to impose the
necessary conditions. Taking a state |ψ ⊗ | ↓↓↓ with |ψ ∈ H m , we find that ψ is
a weight (1, 1) primary field of the matter CFT
(L
m
0 + ¯
L
m
0 − 2) |ψ = 0,
(L
m
0 − ¯
L
m
0 ) |ψ = 0,
∀n > 0 : L
m
n |ψ = ¯
L
m
n |ψ = 0.
(8.93)
An important point is that the usual mass-shell condition k 2 = −m 2 is provided
by the first condition only. This also shows that states in the cohomology naturally
appear with c ¯
c insertion since
| ↓↓↓ = c(0) ¯
c(0) |0 = c 1 ¯
c 1 |0 .
(8.94)
This hints at rewriting of scattering amplitudes in terms of unintegrated states (3.29)
only.
A state is said to be of level ((, ¯
) and denoted as ψ , ¯
if it satisfies
L 0 |ψ , ¯
= |ψ , ¯
,
¯
L 0 |ψ , ¯
= ¯
|ψ , ¯
.
(8.95)
Example 8.1: Closed String Tachyon
As an example, let us construct the state ψ 0,0 with level zero for a spacetime with
D non-compact dimensions. In this case, the transverse CFT contains D − 2 free
scalars that combine with X 0 and X 1 into D scalars X μ . The Fock space is built
on the vacuum |k, and we define the mass such that on-shell condition reduces
to the standard QFT expression
k
2
= −m
2 ,
m
2
:=
2
2 (N + ¯
N − 2),
(8.96)
where N and ¯
N are the matter level operators. The state in the remaining
transverse CFT (without the D − 2 scalars) is the SL(2, C) vacuum with L ⊥
0 =
¯
L ⊥
0 = 0 (this is the state with the lowest energy for a unitary CFT). In this case,
the on-shell condition reads
m
2
2
= −4 < 0.
(8.97)
8 BRST Quantization
also Sect. 3.2.2). Imposing first the condition b
−
0 = 0 defines the semi-relative
cohomology. The relative cohomology is found by imposing b
±
0 = 0 and in fact
corresponds to the physical space (see [13, sec. 2.3] for more details). The rest of
the derivation follows straightforwardly because the two sectors commute: we find
that the cohomology is ghost-free and has no light-cone excitations
L
±
0 = N
0
± ¯
N
0
+ N
1
± ¯
N
1
+ N
b
± ¯
N
b
+ N
c
± ¯
N
c
= 0.
(8.92)
In general, it is simpler to work with a covariant expression and to impose the
necessary conditions. Taking a state |ψ ⊗ | ↓↓↓ with |ψ ∈ H m , we find that ψ is
a weight (1, 1) primary field of the matter CFT
(L
m
0 + ¯
L
m
0 − 2) |ψ = 0,
(L
m
0 − ¯
L
m
0 ) |ψ = 0,
∀n > 0 : L
m
n |ψ = ¯
L
m
n |ψ = 0.
(8.93)
An important point is that the usual mass-shell condition k 2 = −m 2 is provided
by the first condition only. This also shows that states in the cohomology naturally
appear with c ¯
c insertion since
| ↓↓↓ = c(0) ¯
c(0) |0 = c 1 ¯
c 1 |0 .
(8.94)
This hints at rewriting of scattering amplitudes in terms of unintegrated states (3.29)
only.
A state is said to be of level ((, ¯
) and denoted as ψ , ¯
if it satisfies
L 0 |ψ , ¯
= |ψ , ¯
,
¯
L 0 |ψ , ¯
= ¯
|ψ , ¯
.
(8.95)
Example 8.1: Closed String Tachyon
As an example, let us construct the state ψ 0,0 with level zero for a spacetime with
D non-compact dimensions. In this case, the transverse CFT contains D − 2 free
scalars that combine with X 0 and X 1 into D scalars X μ . The Fock space is built
on the vacuum |k, and we define the mass such that on-shell condition reduces
to the standard QFT expression
k
2
= −m
2 ,
m
2
:=
2
2 (N + ¯
N − 2),
(8.96)
where N and ¯
N are the matter level operators. The state in the remaining
transverse CFT (without the D − 2 scalars) is the SL(2, C) vacuum with L ⊥
0 =
¯
L ⊥
0 = 0 (this is the state with the lowest energy for a unitary CFT). In this case,
the on-shell condition reads
m
2
2
= −4 < 0.
(8.97)