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

TL;DR

This paper constructs the Shatashvili-Vafa G2 superconformal algebra as a quantum Hamiltonian reduction of the Lie superalgebra D(2,1;α) at α=1, revealing its structure, free field realization, and screening operators.

Contribution

It introduces a new realization of the G2 superconformal algebra via Hamiltonian reduction of D(2,1;α), including explicit free field and screening operator descriptions.

Findings

Complete family of W-algebras SW(3/2,3/2,2) constructed

Explicit free field realization of the G2 algebra provided

Screening operators explicitly described

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.