On rationality of C-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then rational, i.e. has semi-simple representation theory, with one dimensional level zero algebra. Here $\Omega$ denotes vectors in are annihilated by lowering real part grading. We apply our result to family rank Weyl conformal element $\omega_\mu$ parameterized $\mu \in \mathbb{C}$, non-integer values $\mu$, these algebras, which graded, In addition, generalize this appropriate arbitrary ranks.