Parafermion vertex operator algebras and W-algebras

We prove the conjectual isomorphism between the level $k$ ${\widehat{sl}}_2$-parafermion vertex operator algebra and the $(k+1,k+2)$ minimal series $W_k$-algebra for all integers $k\ge 2$. 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 $SU(k)_1 \otimes SU(k)_1/SU(k)_2$.