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We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which areextensions of the $N=1$ superconformal algebra by a primary supercurrent ofconformal weight $2$ can be realized as a quantum Hamiltonian reduction of theLie superalgebra $osp(3|2)$. In consequence we obtain an explicit free fieldrealization of the algebra in terms of the screening operators. At centralcharge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to thesuperconformal algebra associated to sigma models based on eight-dimensionalmanifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa $Spin(7)$superconformal algebra.

The $SW(3/2,2)$ superconformal algebra via a Quantum Hamiltonian Reduction of $osp(3|2)$