-differential modules
Pseudo-differential forms (PDF)
Special cocycles
A procedure for constructing 
-differential modules
Suppose that 
is a -differential module. This means that
is a representation of the Lie superalgebra 
, with the
differential 
which agrees with the differential 
of 
.
(The differential of , which we denote 
, does not participate
in these definitions, but will play its role.)
For every
we denote
and
the corresponding elements of
, and
and
their action in
. With the notations of Eqs. (
7) and (
14),
Eq. (
12) implies, for all
:
Let us consider Eq. (
21) in the special case when
satisfies:
for all
. Then:
Eq. (
18) is a special requirement on
and
.
It is by no means automatic. Intuitively, it may be understood as an interplay between
and
(and
):
We do not requite that
act in
.
Instead, we want Eqs. (
17) and (
18)
(or, equivalently, Eqs. (
17) and (
19)).
We will now consider two examples of -differential modules.
Pseudo-differential forms (PDF)
Suppose that a supermanifold
comes with an infinitesimal action of
,
i.e. a homomorphism:
This is only a homomorphism of Lie superalgebras;
we forget, for now, about the differential
.
We denote
the deRham differential on
, and
.
Then Eq. (
12) implies
Let us consider Eq. (
21) in the special case when
is closed:
Consider a linear subspace
consisting of all vectors
such that exists
some other vector
satisfying:
Suppose that
is “non-degenerate” in the sense that the map from
to PDFs on
given by
is injective.
Then Eq. (
22) defines an odd nilpotent operator:
Moreover,
is closed under the operation of commutator of vector fields.
Indeed:
Therefore
is a differential Lie superalgebra.
Suppose that:
Then Eq. (
21) implies that:
In other words,
defined by the equation:
is a cocycle in the Cartan’s model of equivariant cohomology.
Notice that apriori there is no action of on , and we have never used it.
Special cocycles
Similarly, suppose that
is mapped into some Lie superalgebra
:
and
is a representation of
.
Consider the Chevalley-Eilenberg cochain complex
of
with coefficients in
.
The cone
acts on
;
for each
we denote
and
the action of the corresponding elements of
.
Eq. (
21) still holds, now
is a cochain:
We are allowing arbitrary dependence of cochains on the ghosts of
, not only polynomials.
We will say that a cocycle
is
special if exists
satisfying Eq. (
22)
for all
.
Moreover, we require:
A procedure for constructing
We will now describe a procedure for constructing an embedding:
This is not really an “algorithm” because, as we will see, it may fail at any step.
Suppose that we can choose, for each
, some
so that exist
such that:
where
and
the structure constants of
. Then verify the existence
of
such that:
The mutual consistency of these two equations follows from Eq. (
10).
Then continue this procedure order by order in the number of indices:
If we are able to satisfy these equalities, order by order in the number of indices, then we can put,
in notations of Eqs. (
13) and (
14):
