Setup for cohomological perturbation theory
Definition of 
and 
Flat 
and expansion around it
Spectral sequence
First page
Definition of and
We
define odd coordinates 
so that:
Flat and expansion around it
Flat spacetime corresonds to
where:
Let us consider
as a small deformation of
:
to the first order in
. Such deformations form a linear space. They correspond to
odd vector fields
satisfying:
modulo the equivalence relation, corresponding to the action of diffeomorphisms:
where
is a ghost number zero vector field on
.
Therefore, the classification of nilpotent vector fields of the form (
13)
is equivalent to the computation of the
cohomology of the operator
on the space of vector fields.
In the rest of this paper we will compute the cohomology of on the space
of vector fields.
Spectral sequence
The grading operator:
defines a filtration on the algebra of functions on
,
and on the space of vector fields as a
-module. Let
be the space of vector fields with grade at least
. This filtration defines a spectral sequence converging to the cohomology of
.
First page
The first page of this spectral sequence is the cohomology of:
on the space of vector fields on
. For a set of coordinates
we denote
the space of functions of
and
the space of vector fields (
i.e. differentiations of
).
Let us introduce the following complexes:
Then,
with differential
decomposes as follows:
(We do not need to take care about the completions of the tensor products, since all our functions
are polynomials in
and
.)
The cohomology of
and
is well known,
see
e.g. the review part of
Mafra:2009wq:
and 
which are relevant to this work
will be computed in 〚Cohomology of 
in the space of vector fields〛.