Rationality of W-algebras: principal nilpotent cases

TL;DR

This paper proves the rationality of all minimal series principal W-algebras discovered in 1992, establishing a new family of rational, C_2-cofinite vertex operator algebras through analysis of Zhu's algebra and Drinfeld-Sokolov reduction.

Contribution

It demonstrates the rationality of minimal series principal W-algebras and links Zhu's algebra with the quantized Drinfeld-Sokolov reduction, providing new insights into their structure.

Findings

Proved rationality of all minimal series principal W-algebras.

Determined the maximal spectra of associated graded Zhu's algebras.

Established that Zhu's algebra functor commutes with reduction functor.

Abstract

We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.