Rationality of Bershadsky-Polyakov vertex algebras

TL;DR

This paper proves the rationality of Bershadsky-Polyakov vertex algebras at specific levels, providing new examples of rational conformal field theories and confirming a conjecture by Kac-Wakimoto.

Contribution

It confirms the Kac-Wakimoto conjecture for the first non-trivial series of exceptional W-algebras, specifically for Bershadsky-Polyakov algebras at certain levels.

Findings

Proves rationality of $W_3^{(2)}$ at levels $k=p/2-3$ for $p=3,5,7,...$

Provides new examples of rational conformal field theories

Validates the Kac-Wakimoto conjecture for these algebras

Abstract

We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras $W_3^{(2)}$ at level $k=p/2-3$ with $p=3,5,7,...$. This gives new examples of rational conformal field theories.