Lifting of to superalgebra

Let be the part of involving only the ghosts of even generators of (essentially, all the ghosts of odd generators all put to zero). The of Eq. (16) is annihilated by . What happens if we act on with the full , including the terms containing odd indices? Can we extend to a cocycle of ?

To answer this question, let us look at the spectral sequence
corresponding to .
At the first page we have (in these formulas we put ):

Our belongs to . The first obstacle lives in

We actually know that the SUGRA solution exist. Therefore this obstacle automatically vanishes.
But there is another obstacle, which arizes when we go to the second page. It lives in:

We used the fact that relative
cochains are -invariant, therefore the cocycles automatically fall into the finite-dimensional
. In fact, this obstacle does not have to be zero, because
there is something that can cancel it. Remember that is generally speaking not annihilated
by , but rather satisfies Eq. (7). And, in fact, there is a nontrivial cocycle:

We conjecture
that the supersymmetric extension of indeed exists, but instead of
satisfying satisfies:

This conjecture should be verified by explicit computations, which we leave for future work.
It may happen that the obstacle which would take values in the cohomology group of Eq. (18)
actually vanishes for some reason.

We will now describe of Eq. (18).