17.3 Superstring Field Theory
357
where satisfies the same constraints as (in particular, it contains more
components than the of the previous section). Then, the equations of motion are
Q B | = 0,
Q B |
= 0.
(17.78)
This shows that both fields are free and decoupled and that the spectrum is doubled.
To push the interpretation further, one needs to consider the interactions.
Amplitudes involve only the states contained in , and thus, the interactions are
built solely in terms of . Then, the equations of motion have the form:
Q B
| − G |
= 0,
Q B |
= |J () ,
(17.79)
where J (() is a source term due to the interactions. An equation for only is
obtained by multiplying the second with G
Q B | = G |J (() .
(17.80)
Once is determined by solving this equation, the auxiliary field
is completely
fixed by the second equation up to free field solutions. This shows that
describes
only free fields even when is interacting. Note that this implies that the degrees of
freedom contained in
do not even couple to the gravitational field! This can also
be shown at the level of Feynman diagrams.
Remark 17.8 The field
is not an auxiliary field strictly speaking since it is
propagating (its equation of motion is not algebraic).
17.3.4 Large Hilbert Space
The last formulation of the kinetic term considers the NS string field to be in the
large Hilbert space, i.e. η 0 = 0. The Ramond field must be described with one of
the two previous approaches.
Writing the action requires to use a NS field 0 with picture number 0. The
kinetic term becomes
S 0,2 = −
1
2
0 , η 0 Q B , , 0 ,
(17.81)
where ·, ··· is the inner product in the large Hilbert space (contains a ξ 0 insertion).
This action has an enlarged gauge invariance:
δ | 0 = Q B | 0 + η 0 | 1 ,
(17.82)
and the equation of motion reads
Q B η 0 | 0 = 0.
(17.83)
357
where satisfies the same constraints as (in particular, it contains more
components than the of the previous section). Then, the equations of motion are
Q B | = 0,
Q B |
= 0.
(17.78)
This shows that both fields are free and decoupled and that the spectrum is doubled.
To push the interpretation further, one needs to consider the interactions.
Amplitudes involve only the states contained in , and thus, the interactions are
built solely in terms of . Then, the equations of motion have the form:
Q B
| − G |
= 0,
Q B |
= |J () ,
(17.79)
where J (() is a source term due to the interactions. An equation for only is
obtained by multiplying the second with G
Q B | = G |J (() .
(17.80)
Once is determined by solving this equation, the auxiliary field
is completely
fixed by the second equation up to free field solutions. This shows that
describes
only free fields even when is interacting. Note that this implies that the degrees of
freedom contained in
do not even couple to the gravitational field! This can also
be shown at the level of Feynman diagrams.
Remark 17.8 The field
is not an auxiliary field strictly speaking since it is
propagating (its equation of motion is not algebraic).
17.3.4 Large Hilbert Space
The last formulation of the kinetic term considers the NS string field to be in the
large Hilbert space, i.e. η 0 = 0. The Ramond field must be described with one of
the two previous approaches.
Writing the action requires to use a NS field 0 with picture number 0. The
kinetic term becomes
S 0,2 = −
1
2
0 , η 0 Q B , , 0 ,
(17.81)
where ·, ··· is the inner product in the large Hilbert space (contains a ξ 0 insertion).
This action has an enlarged gauge invariance:
δ | 0 = Q B | 0 + η 0 | 1 ,
(17.82)
and the equation of motion reads
Q B η 0 | 0 = 0.
(17.83)