Associated Varieties of Simple Affine VOAs \(L_k(sl_3)\) and W-algebras \(W_k(sl_3,f)\)

Abstract

In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra \(V^k(sl_n)\) for \(k=-n+\frac{n-1}{q}\) is generated by two singular vectors of conformal weight 3q if \(n=3\), and by one singular vector of conformal weight 2q if \(n\geqslant 4\). We next determine the associated varieties of the simple vertex operator algebras \(L_k(sl_3)\) for all the non-admissible levels \(k=-3+\frac{2}{2m+1}\), \(m\geqslant 0\). The varieties of the associated simple affine W-algebras \(W_k(sl_3,f)\), for nilpotent elements f of \(sl_3\), are also determined.