Spin-3/2 Fermions in Twistor Formalism

TL;DR

This paper explores the conditions for the existence of massless spin-3/2 fields using twistor formalism, establishing links between space-time properties, conserved charges, and torsion in Einstein-Cartan theory.

Contribution

It introduces a twistor-based framework for analyzing massless spin-3/2 fields, including in torsionful space-times, and identifies conditions for their consistent existence.

Findings

Conservation charges are given by twistor space in flat space-time.

Self-duality of space-time is necessary for charge definition in curved space.

Conformal (anti-)self-duality and torsion-free conditions are key for field existence.

Abstract

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.