2025/10/04 by Andrew R. Linshaw, Linshaw, Andrew, Arim Song +3
Mathematics · #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2510.03957
Let \mathfrakg be a basic Lie superalgebra and f be an odd nilpotent element in an \mathfrakosp(1|2) subalgebra of \mathfrakg. We provide a mathematical proof of the statement that the W-algebra Wk(\mathfrakg,F) for F=-(1)/(2)[f,f] is a vertex subalgebra of the SUSY W-algebra WN=1k(\mathfrakg,f), and that it commutes with all weight (1)/(2) fields in WN=1k(\mathfrakg,f). Note that it has been long believed by physicists \citeMadRag94. In particular, when f is a minimal nilpotent, we explicitly describe superfields which generate WkN=1(\mathfrakg,f) as a SUSY vertex algebra and their OPE relations in terms of the N=1 Λ-bracket introduced in \citeHK07. 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 WkN=1(\mathfraksl(2|1),fmin), WkN=1(\mathfrakosp(3|2),fmin), WkN=1(\mathfrakpsl(2|2),fmin), and WkN=1(D(2,1;α)⊕ ℂ,fmin), respectively, for the minimal odd nilpotent fmin, and examine some examples.