8.3 BRST Cohomology: Two Flat Directions
187
they will be identified with the light-cone and perpendicular directions in the target
spacetime (and, correspondingly, with unphysical and physical states).
The Hilbert space of the theory is decomposed as
H := H ⊗ H ⊥ ,
H :=
dk
0
F 0
k
0
⊗
dk
1
F 1
k
1
⊗ H gh ,
(8.27)
where F 0 (k 0 ) and F 1 (k 1 ) are the Fock spaces (7.83a) of the scalar fields X 0 and
X 1 , and H gh is the ghost Hilbert space (7.169). As a consequence, a generic state of
H reads
|ψ = |ψ ⊗ |ψ ⊥ ,
(8.28)
where ψ ⊥ is a generic state of the transverse matter CFT H ⊥ and ψ is built by
acting with oscillators on the Fock vacuum of H
|ψ = c
N c
0
0
m>0
(α
0
−m )
N 0
m (α
1
−m )
N 1
m (b −m )
N b
m (c −m )
N c
m |k
0 , k
1 , ↓↓
|k
0 , k
1 , ↓↓ := |k
0
⊗ |k
1
⊗ | ↓↓ ,
N
0
m , N
1
m ∈ N,
N
b
m , N
c
m = 0, 1.
(8.29)
Since the Virasoro modes commute with the ghost number, eigenstates of the
Virasoro operators without zero-modes
L 0 , given by the sum of (7.68) and (7.160),
can also be taken to be eigenstates of N gh . It is also useful to define the Hilbert space
of states lying in the kernel of b 0
H 0 = H ∩ ker b 0
(8.30)
such that
H = H 0 ⊕ c 0 H 0 .
(8.31)
The full L 0 operator reads
L 0 = L
m
0 + L
gh
0 = (L
m
0 − 1) + N
b
+ N
c ,
(8.32)
using (7.129) for L
gh
0 . A more useful expression is obtained by separating the two
sectors and by extracting the zero-modes using (7.67)
L 0 =
L
⊥
0 − m
2
,L
2
− 1
+
L
0 ,
(8.33)