Zhu algebras of superconformal vertex algebras

Abstract

The purpose of this note is to demonstrate the advantages of Y.-Z.~Huang's definition of the Zhu algebra (Comm.\ Contemp.\ Math., 7 (2005), no.~5, 649--706) for an arbitrary vertex algebra, not necessarily equipped with a Hamiltonian operator or a Virasoro element, by achieving the following two goals: (1) determining the Zhu algebras of $N=1, 2, 3, 4$ and big $N=4$ superconformal vertex algebras, and (2) introducing the Zhu algebras of $N_K=N$ supersymmetric vertex algebras.