-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):