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=-12f, f is a vertex subalgebra of the SUSY W-algebra W₍=₁ᵏ (g, f), and that it commutes with all weight 12 fields in W₍=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 Wᵏ₍=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 Wᵏ₍=₁ (sl (2|1), f₌₈₍), Wᵏ₍=₁ (osp (3|2), f₌₈₍), Wᵏ₍=₁ (psl (2|2), f₌₈₍), and Wᵏ₍=₁ (D (2, 1;α) C, f₌₈₍), respectively, for the minimal odd nilpotent f₌₈₍, and examine some examples.