Rationality, regularity, and 𝐶₂-cofiniteness

We demonstrate that, for vertex operator algebras of CFT type, C 2 C_2 -cofiniteness and rationality is equivalent to regularity. For C 2 C_2 -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and C 2 C_2 -cofinite, V L + V_L^+ is C 2 C_2 -cofinite, and the fusion rules are finite.