Zhu algebras of superconformal vertex algebras
Ryo Sato, Shintarou Yanagida
TL;DR
This work advocates Huang's Zhu algebra \\widetilde{A}(V) for arbitrary vertex algebras and applies it to superconformal theories, computing the Zhu algebras for N=1,2,3,4 and big N=4, and introducing SUSY Zhu algebras for NK=N SUSY vertex algebras. The main results identify these Zhu algebras with universal enveloping algebras of centralizers of minimal nilpotent elements, up to explicit structural factors (e.g., Clifford and polynomial algebras) in certain cases, and relate them to finite W-algebras and Ramond-sector twists. The paper also develops a SUSY Zhu framework, showing how odd SUSY translations induce differentials on \\widetilde{A}_\\gamma(\\mathbb{V}) and how these reduce to the C2-Poisson limit, thereby providing a natural SUSY extension of the Zhu construction. Together, these results illuminate the representation theory of superconformal and SUSY vertex algebras and point to directions for non-linear generalizations and further links with chiral homology and W-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 Hamiltonian operators or Virasoro elements, 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.