The Shatashvili-Vafa $G_{2}$ 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 $G_{2}$ holonomy, i.e., the Shatashvili-Vafa $G_{2}$ 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 $SW(\frac{3}{2},\frac{3}{2}, 2)$ (extensions of the $N=1$ superconformal algebra by two primary supercurrents of conformal weight $\frac{3}{2}$ 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 $G_{2}$ algebra, and an explicit description of the screening operators.

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