Supersymmetries and dilatation
The vector field 
of Eq. (6) is manifestly supersymmetry-invariant.
In other words, it commutes with the super-Poincare algebra, which is generated by
and
. It is also invariant under dilatations, if we define the weight of 
to be twice the weight of 
, 
. It is perhaps less straightforward to see that there
are no other symmetries. For example, there are no conformal symmetries. (But the dilatation symmetry is present.)
We will now prove that there are no other symmetries.
This means that any infinitesimal symmetry can be brought to the form:
where
stand for terms of the higher order in the grading defined by Eq. (
14).
Commuting
with
,
we have to cancel the coefficients of all generators of
(see 〚
First page〛).
The vanishing of the coefficient of
implies that
(constant in
). Similarly,
,
,
.
The vanishing of the coefficient of
and
imply:
The vanishing of the coefficients of
and
imply
and
(do not depend on
).
.
The cohomology of
at the ghost number 
in the space of vector fields〛):
