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