Zhu's rationality conjecture on $C_2$-cofiniteness

Let $V$ be a vertex operator algebra graded by the nonnegative integers, and let $R_V$ denote its $C_2$-algebra.

Zhu's conjecture. If $V$ is rational, then $R_V$ is finite-dimensional.

Finite-dimensionality of $R_V$ is the $C_2$-cofiniteness condition and is important in vertex operator algebra theory. The conjecture is attributed to Zhu and remains open in general; the paper proves it for classically free vertex operator algebras.