Rationality of Bershadsky-Polyakov Vertex Algebras

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 ${W_3^{(2)}}$ W 3 ( 2 ) at level k =  p/2?3 with ${p = 3, 5, 7, 9, dots}$ p = 3 , 5 , 7 , 9 , ? . This gives new examples of rational conformal field theories.