The Shatashvili-Vafa G₂ superconformal algebra as a quantum Hamiltonian reduction of D(2, 1; α)

We obtain the superconformal algebra associated to a sigma model with target a manifold with G₂ holonomy, i.e., the Shatashvili-Vafa G₂ algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra D(2, 1; α) for α = 1. We produce the complete family of W-algebras SW (3/2, 3/2, 2) (extensions of the N = 1 superconformal algebra by two primary supercurrents of conformal weight 3/2 and 2 respectively) as a quantum Hamiltonian reduction of D(2, 1; α). As a corollary we find a free field realization of the Shatashvili-Vafa G₂ algebra, and an explicit description of the screening operators.