Supersymmetries in the theory of W-algebras

TL;DR

The paper proves a precise relationship between W-algebras and SUSY W-algebras for odd nilpotent elements $f$ in an $\mathfrak{osp}(1|2)$ subalgebra: the ordinary W-algebra $W^k(\frak g,F)$ with $F=-\tfrac12[f,f]$ embeds as a coset inside the SUSY W-algebra $W^k_{N=1}(\frak g,f)$ and, for generic $k$, splits as $W^k_{N=1}(\frak g,f) \cong W^k(\frak g,F)\otimes\mathcal F(\frak g^f_0)$. The authors construct an explicit isomorphism between Miura-map targets, compare SUSY and non-SUSY screening operators, and extend the principal-case isomorphism to general cases. They classify minimal odd nilpotents, describe the minimal SUSY W-algebras via superfield formalism, and relate them to $N=2,3,4$ superconformal algebras, providing rich examples and coset/orbifold results. The work unifies SUSY and non-SUSY W-algebra structures, enables coset constructions, and yields a framework for generating $N=2,3,4$ SUSY VOAs as conformal extensions of minimal W-algebras with broad implications for representation theory and superconformal models.

Abstract

Let $\mathfrak{g}$ be a basic Lie superalgebra and $f$ be an odd nilpotent element in an $\mathfrak{osp}(1|2)$ subalgebra of $\mathfrak{g}$. We provide a mathematical proof of the statement that the W-algebra $W^k(\mathfrak{g},F)$ for $F=-\frac{1}{2}[f,f]$ is a vertex subalgebra of the SUSY W-algebra $W_{N=1}^k(\mathfrak{g},f)$, and that it commutes with all weight $\frac{1}{2}$ fields in $W_{N=1}^k(\mathfrak{g},f)$. Note that it has been long believed by physicists \cite{MadRag94}. In particular, when $f$ is a minimal nilpotent, we explicitly describe superfields which generate $W^k_{N=1}(\mathfrak{g},f)$ as a SUSY vertex algebra and their OPE relations in terms of the $N=1$ $Λ$-bracket introduced in \cite{HK07}. In the last part of this paper, we define $N=2,3$, and small or big $N=4$ SUSY vertex operator algebras as conformal extensions of $W^k_{N=1}(\mathfrak{sl}(2|1),f_{\text{min}})$, $W^k_{N=1}(\mathfrak{osp}(3|2),f_{\text{min}})$, $W^k_{N=1}(\mathfrak{psl}(2|2),f_{\text{min}})$, and $W^k_{N=1}(D(2,1;α)\oplus \mathbb{C},f_{\text{min}})$, respectively, for the minimal odd nilpotent $f_{\text{min}}$, and examine some examples.

Theorems & Definitions (65)

• Theorem 1.1: Theorem \ref{['thm:nonSUSY vs SUSY']}
• Definition 1.2
• Example 2.1
• Proposition 3.1
• Remark 3.2
• Remark 3.3
• Proposition 3.4
• Lemma 4.1
• proof
• Proposition 4.2