Zhu's algebra and the $C_2$-algebra of a classically free vertex operator algebra

We prove that, for a classically free vertex algebra, the $C_2$-algebra is naturally isomorphic to the associated graded algebra of Zhu’s algebra. As an application, we show that non-trivial holomorphic VOAs of CFT type are not classically free. This includes the affine vertex operator algebra of $E_8$ at level one, as well as the Moonshine VOA.