Fermionic fields
The first terms in both
and
are of grade 1, and the rest of the terms are of grade 3.
Grade 3 terms are determined by the grade 1 terms
Let us first assume that the grade 1 terms are zero,
i.e 
and
.
Considering the coefficient of
, we deduce that
satisfies:
The coefficients
and
come with gauge transformations:
Considering the coefficient of
,
we conclude that
(and similarly
)
are constants, and we ignore them.
Grade 1 terms
Requiring 
Requiring
, the “Maxwell bishop move”:
we conclude that
(and similarly
) should satisfy the Maxwell equations:
Requiring 
We will now consider the anticommutator of
with
. It is convenient to
start by completing
to
.
We have:
The term
is removed by further modifying (
cp Eq. (
36)):
to:
Then, when we commute with
, this modification produces:
which should cancel
. Indeed, let us denote:
The total contribution is:
We observe:
In fact:
To cancel Eq. (
60) we must impose the Dirac equation on
, in the following sense.
Require that exists
such that:
Then we cancel Eq. (
60) by choosing
To summarize:
and a similar formula for
.
Comparison to SUGRA
The only fermionic superfields of Berkovits:2001ue are 
and 
. The top component of
corresponds to 
,
and the top component of 
to
.
parameterized by even functions 
. We will now add the terms
parameterized by odd functions.
According to 〚Computation of 
〛 these terms are:
. This implies
(see 〚Higher spin conformal Killing tensors〛)
that modulo finite dimensional subspaces (which we ignore):
