Principal SUSY and nonSUSY W-algebras and their Zhu algebras
Naoki Genra, Arim Song, Uhi Rinn Suh
TL;DR
The paper proves that for a simple basic Lie superalgebra $\mathfrak{g}$ with an odd principal nilpotent $f$ and $F=-\frac{1}{2}[f,f]$, the nonSUSY W-algebra $W^k(\mathfrak{g},F)$ is isomorphic to the SUSY W-algebra $W^k(\bar{\mathfrak{g}},f)$ via screening operators, thereby imparting a natural $N=1$ SUSY structure to $W^k(\mathfrak{g},F)$. It then constructs finite SUSY W-algebras $U(\widetilde{\mathfrak{g}},f)$ from the SUSY Takiff algebra and proves an isomorphism with the Zhu algebra $Zhu_H W^k(\bar{\mathfrak{g}},f)$, linking affine and finite SUSY W-structures. In the principal case, the paper establishes $U(\mathfrak{g},F) \cong U(\widetilde{\mathfrak{g}},f)$ and provides explicit realizations via low-rank examples, highlighting the role of BRST cohomology, Miura maps, and screening operators in unifying SUSY and nonSUSY W-algebras. The results integrate free-field realizations with Zhu-algebra techniques to illuminate the representation theory of SUSY W-algebras and their finite counterparts, with corollaries for principal SUSY finite W-algebras.
Abstract
This paper consists of two parts. In the first part, we prove that when $\mathfrak{g}$ is a simple basic Lie superalgebra with a principal odd nilpotent element $f$, the W-algebra $W^k(\mathfrak{g}, F)$ for $F=-\frac{1}{2}[f,f]$ is isomorphic to the SUSY W-algebra $W^k(\bar{\mathfrak{g}},f)$ via screening operators, which implies the supersymmetry of $W^k(\mathfrak{g}, F)$. In the second part, we show that a finite SUSY W-algebra, which is a Hamiltonian reduction of $U(\widetilde{\mathfrak{g}})$ for the SUSY Takiff algebra $\widetilde{\mathfrak{g}}=\mathfrak{g}\otimes \wedge(θ)$ is isomorphic to the Zhu algebra of a SUSY W-algebra. As a corollary, we show that a finite SUSY principal W-algebra is isomorphic to a finite principal W-algebra.