Supersymmetries in the theory of W-algebras

Let g be a basic Lie superalgebra and f be an odd nilpotent element in an osp(1|2) subalgebra of g. We provide a mathematical proof of the statement that the W-algebra Wᵏ(g,F) for F=-1/2[f,f] is a vertex subalgebra of the SUSY W-algebra WN=1ᵏ(g,f), and that it commutes with all weight 1/2 fields in WN=1ᵏ(g,f). Note that it has been long believed by physicists {MadRag94}. In particular, when f is a minimal nilpotent, we explicitly describe superfields which generate WᵏN=1(g,f) as a SUSY vertex algebra and their OPE relations in terms of the $N=1$ $Λ$-bracket introduced in {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 WᵏN=1(sl(2|1),fₘᵢₙ), WᵏN=1(osp(3|2),fₘᵢₙ), WᵏN=1(psl(2|2),fₘᵢₙ), and WᵏN=1(D(2,1;α)⊕ C,fₘᵢₙ), respectively, for the minimal odd nilpotent fₘᵢₙ, and examine some examples.