The $SW(3/2,2)$ superconformal algebra via a Quantum Hamiltonian Reduction of $osp(3|2)$

TL;DR

This paper demonstrates that the non-linear $SW(3/2,2)$ superconformal algebra can be constructed via quantum Hamiltonian reduction of $osp(3|2)$, providing explicit free field realizations and linking it to special holonomy sigma models.

Contribution

It introduces a new realization of the $SW(3/2,2)$ algebra through quantum Hamiltonian reduction of $osp(3|2)$ and offers explicit free field representations.

Findings

Realization of $SW(3/2,2)$ via quantum Hamiltonian reduction.

Explicit free field realization using screening operators.

Connection to $Spin(7)$ superconformal algebra at $c=12$.

Abstract

We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are extensions of the $N=1$ superconformal algebra by a primary supercurrent of conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field realization of the algebra in terms of the screening operators. At central charge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to the superconformal algebra associated to sigma models based on eight-dimensional manifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa $Spin(7)$ superconformal algebra.