Non-linear Lie conformal algebras with three generators (Q1030515)

In the paper under review the authors study certain nonlinear Lie conformal algebras (and associated vertex algebras) with three generators, which generalize the current Lie conformal algebra of \(\mathfrak{sl}_2\) at level \(k\). Such generalizations include rank one lattice vertex algebras and a new family of vertex algebras, denoted \(V_{-1}^d\) (\(d \in \mathbb{N}\)), constructed in the paper under review. For \(d=1\), this family includes the affine vertex algebra of \(\mathfrak{sl}_2\) at the critical level \(k=-2\). Generalizing the Wakimoto realization of critical level representations, the authors give a free-field realization of vertex algebras \(V_{-1}^d\). They also determine the Zhu algebra of \(V_{-1}^d\), and it turns out that it is one of the associative algebras introduced in [\textit{S. P. Smith}, Trans. Am. Math. Soc. 322, No. 1, 285--314 (1990; Zbl 0732.16019)].