  Consistency conditions for the local existence of massless spin 3/2 fields
has been explored that the field equations for massless helicity 3/2 are
consistent iff the space-time is Ricci-flat and that in Minkowski space-time
the space of conserved charges for the fields is its twistor space itself.
After considering the twistorial methods to study such massless helicity 3/2
fields, we derive in flat space-time that the charges of spin-3/2 fields
defined topologically by the first Chern number of their spin-lowered self-dual
Maxwell fields, are given by their twistor space, and in curved space-time that
the (anti-)self-duality of the space-time is the necessary condition. Since in
N=1 supergravity torsions are the essential ingredients, we generalize our
space-time to that with torsion (Einstein-Cartan theory) and have investigated
the consistency of existence of spin 3/2 fields in it. A simple solution is
found that the space-time has to be conformally (anti-)self-dual, left-(or
right-)torsion-free. The integrability condition on $\alpha$-surface shows that
the (anti-)self-dual Weyl spinor can be described only by the covariant
derivative of the right-(left-)handed-torsion.
