Parafermion vertex operator algebras and $W$-algebras

Abstract

We prove the conjectural isomorphism between the level $k$ $\widehat {\mathfrak {sl}}_2$-parafermion vertex operator algebra and the $(k+1,k+2)$-minimal series $W_k$-algebra for all $k \ge 2$. As a consequence, we obtain the conjectural isomorphism between the $(k+1,k+2)$-minimal series $W_k$-algebra and the coset vertex operator algebra $SU(k)_1\otimes SU(k)_1/SU(k)_2$.