On commutator-based linearization of vector-spinor nonlinear supersymmetry and Rarita-Schwinger fields

We discuss the linearization of vector-spinor (spin-3/2) nonlinear supersymmetry (vsNLSUSY) transformations for both $N = 1$ and $N$-extended SUSY in flat spacetime based on the commutator algebra by using functionals (composites) of spin-3/2 Nambu-Goldstone (NG) fermions, which are expressed as simple products of powers of the spin-3/2 NG fermions and a fundamental determinant in the vsNLSUSY theory. We define basic component fields by means of those functionals in a linearized vsSUSY theory including spin-3/2 fields, general auxiliary ones and a $D$-term. The general forms of linear (rigid) vsSUSY transformations for the component fields are determined uniquely from the commutator-based linearization procedure. By considering appropriate recombinations of the functionals of the spin-3/2 NG fermions for $N = 1$ SUSY, we find that variations of the recombinations under the vsNLSUSY transformations include linear spin-1/2 SUSY ones of spin-(3/2, 1) fields with $U(1)$ gauge invariance. The spinorial gauge invariance of the Rarita-Schwinger action in the linearization process is also discussed together with the $U(1)$ gauge invariance.