On this page:
-differential modules
Pseudo-differential forms (PDF)
Special cocycles
A procedure for constructing

    -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 :

(16)

Let us consider Eq. (21) in the special case when satisfies:

(17)

(18)

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 ):

(19)

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:

(20)

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

(21)

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:

(22)

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:

(23)

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):