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.