Screening operators for $\mathcal {W}$-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 (affine) $\mathcal {W}$-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.