11.2 Off-Shell States
243
Following (7.176), the Hilbert space can be decomposed as
H = H ± ⊕ c
±
0 H ± ,
(11.37)
where
H ± := H ∩ ker b
±
0 = H 0 ⊕ c
∓
0 H 0 ,
H 0 := H ∩ ker b
−
0 ∩ ker b
+
0 .
(11.38)
In fact, we will find that a consistent description of the off-shell amplitudes for
the closed string requires imposing some conditions on the states even at the offshell level. The off-shell states will have to satisfy the level-matching condition and
to be annihilated by b
−
0 :
L
−
0 |φ = 0,
b
−
0 |φ = 0.
(11.39)
This implies that the off-shell states will be elements of H − ∩ ker L
−
0 . This will
appear as consistency conditions on the geometry of the moduli space and by
studying the propagator. In general, we shall work with H and indicate when
necessary the restriction to H − (keeping the condition ker L
−
0 implicit to avoid new
notations).
The Hilbert space H can be separated according to the ghost zero-modes
H ∼ H ↓↓ ⊕ H ↓↑ ⊕ H ↑↓ ⊕ H ↑↑ ,
(11.40)
with the following definitions:
H ↓↑ ∼ H 0 ,
H ↓↑ ∼ ¯
c 0 H ↓↓
H ↑↓ ∼ c 0 H ↓↓
H ↑↑ ∼ c 0 ¯
c 0 H ↓↓ .
(11.41)
Accordingly, every basis state can be split as
φ r = φ ↓↓,r + φ ↓↑,r + φ ↑↓,r + φ ↑↑,r
(11.42)
such that
b 0 |φ ↓↓,r = ¯
b 0 |φ ↓↓,r = 0,
b 0 |φ ↓↑,r = ¯
c 0 |φ ↓↑,r = 0,
c 0 |φ ↑↓,r = ¯
b 0 |φ ↑↓,r = 0,
c 0 |φ ↑↑,r = ¯
c 0 |φ ↑↑,r = 0.
(11.43)
Moreover, the basis can be indexed such that
|φ ↓↑,r = ¯
c 0 |φ ↓↓,r
|φ ↑↓,r = c 0 |φ ↓↓,r
|φ ↑↑,r = c 0 ¯
c 0 |φ ↓↓,r .
(11.44)