The Shatashvili-Vafa $G_{2}$ superconformal algebra as a Quantum Hamiltonian Reduction of $D(2,1;\alpha)$

Abstract

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 $D(2,1;\alpha)$ for $\alpha=1$. 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 $D(2,1;\alpha)$. As a corollary we find a free field realization of the Shatashvili-Vafa $G_{2}$ algebra, and an explicit description of the screening operators.