354
17 Superstring
Berkovits’ original SFT cannot describe the Ramond sector in the large Hilbert
space, but it is possible to couple Berkovits’ action for the NS field in the large
Hilbert space to a Ramond field in the small Hilbert space. Another limitation of
Berkovits’ approach is that it works only for the open and heterotic superstrings
(but not for type II).
We assume that the problems with PCO are absent (in Berkovits’ and supermoduli constructions) or that they have been defined using vertical integration.
In the rest of this chapter, we will discuss the kinetic term for each of the first
three approaches. At the level of the free action, the open and heterotic super-SFT
differs only in the bosonic factors as in Chap. 10.
17.3.1 String Field and Propagator
As in the bosonic case, it is natural to consider a string field gathering all possible
states
= −1 + −1/2 ,
(17.59)
where −1 and −1/2 are, respectively, the NS and R string fields. If the field is in
the small Hilbert space, it satisfies
η 0 | = 0.
(17.60)
The propagator was found in (17.46) to be
= b
+
0 b
−
0
G
L 0 + ¯
L 0
δ(L
−
0 ),
G =
1
NS,
X 0 R.
(17.61)
As for the bosonic case, the constraints
b
−
0 | = L
−
0 | = 0,
b
+
0 | = 0
(17.62)
must be imposed on the field to ensure that the propagator is invertible.
For similar reasons, the PCO insertion implies that the propagator is not invertible
since X 0 has zero-modes: this means equivalently that it has a non-empty kernel offshell or that it contains derivatives. Two different solutions can be chosen to address
this issue: imposing constraints as for the level-matching condition, or introducing
auxiliary fields.
17 Superstring
Berkovits’ original SFT cannot describe the Ramond sector in the large Hilbert
space, but it is possible to couple Berkovits’ action for the NS field in the large
Hilbert space to a Ramond field in the small Hilbert space. Another limitation of
Berkovits’ approach is that it works only for the open and heterotic superstrings
(but not for type II).
We assume that the problems with PCO are absent (in Berkovits’ and supermoduli constructions) or that they have been defined using vertical integration.
In the rest of this chapter, we will discuss the kinetic term for each of the first
three approaches. At the level of the free action, the open and heterotic super-SFT
differs only in the bosonic factors as in Chap. 10.
17.3.1 String Field and Propagator
As in the bosonic case, it is natural to consider a string field gathering all possible
states
= −1 + −1/2 ,
(17.59)
where −1 and −1/2 are, respectively, the NS and R string fields. If the field is in
the small Hilbert space, it satisfies
η 0 | = 0.
(17.60)
The propagator was found in (17.46) to be
= b
+
0 b
−
0
G
L 0 + ¯
L 0
δ(L
−
0 ),
G =
1
NS,
X 0 R.
(17.61)
As for the bosonic case, the constraints
b
−
0 | = L
−
0 | = 0,
b
+
0 | = 0
(17.62)
must be imposed on the field to ensure that the propagator is invertible.
For similar reasons, the PCO insertion implies that the propagator is not invertible
since X 0 has zero-modes: this means equivalently that it has a non-empty kernel offshell or that it contains derivatives. Two different solutions can be chosen to address
this issue: imposing constraints as for the level-matching condition, or introducing
auxiliary fields.