Screening operators for $$\mathcal {W}$$ W -algebras

Abstract

Let $\mathfrak{g}$ be a simple finite-dimensional Lie superalgebra with a non-degenerate supersymmetric even invariant bilinear form, $f$ a nilpotent element in the even part of $\mathfrak{g}$, $\Gamma$ a good grading of $\mathfrak{g}$ for $f$ and $\mathcal{W}^{k}(\mathfrak{g},f;\Gamma)$ the $\mathcal{W}$-algebra associated with $\mathfrak{g},f,k,\Gamma$ defined by the generalized Drinfeld-Sokolov reduction. In this paper, we present each $\mathcal{W}$-algebra as the intersection of kernels of the screening operators, acting on the tensor vertex superalgebra of an affine vertex superalgebra and a neutral free superfermion vertex superalgebra. As applications, we prove that the $\mathcal{W}$-algebra associated with a regular nilpotent element in $\mathfrak{osp}(1,2n)$ is isomorphic to the $\mathcal{W}B_{n}$-algebra introduced by Fateev and Lukyanov, and that the $\mathcal{W}$-algebra associated with a subregular nilpotent element in $\mathfrak{sl}_{n}$ is isomorphic to the $\mathcal{W}^{(2)}_{n}$-algebra introduced by Feigin and Semikhatov.