10.5 Closed String
231
by analogy with the open string) is that the gauge fixed equation of motion (to be
discussed below) should be equivalent to the on-shell equation L
+
0 = 0 (see also
Sect. 8.3.4). This is possible only if the insertion is c
−
0 . With this insertion, ·, ·· can
be formed from the BPZ product:
A, B = =A|c
−
0 |B .
(10.104)
Then, the action reads
S =
1
2
|c
−
0 Q B | .
(10.105)
However, the presence of c
−
0 has a drastic effect because it annihilates part of the
string field. Decomposing the Hilbert space as in (7.175)
H = H
−
⊕ c
−
0 H
− ,
H
−
:= H ∩ ker b
−
0 ,
(10.106)
the string field reads
| = | − + c
−
0 |
− ,
, − ,
− ∈ H
− ,
(10.107)
such that
c
−
0 | = c
−
0 | − .
(10.108)
The problem in such cases is that the kinetic term may become non-invertible. This
motivates to project out the component
− by imposing the following constraint on
the string field:
b
−
0 | = 0.
(10.109)
The constraint (10.109) is stronger than the constraint L
−
0 = 0 for states in the
cohomology (Sect. 8.3.1), so there is no information lost on-shell by imposing it.
For this reason, we will also impose the level-matching condition:
L
−
0 | = 0,
(10.110)
such that
∈ H
−
∩ ker L
−
0 .
(10.111)
This will later be motivated by studying the propagator and the off-shell scattering
amplitudes. To avoid introducing more notations, we will not use a new symbol for
this space and keep implicit that ∈ ker L
−
0 .