Non-Equivalence of Smooth and Nodal Conformal Block Functors in Logarithmic CFT

Let V be an N-graded, C₂-cofinite vertex operator algebra (VOA) admitting a non-lowest generated module in Mod( V) (e.g., the triplet algebras Wₚ for p∈ Z≥ 2 or the even symplectic fermion VOAs SFd⁺ for d∈ Z₊). We prove that, unlike in the rational case, the spaces of conformal blocks associated to certain V-modules do not form a vector bundle on M̄0,N for N≥ 4 by showing that their dimensions differ between nodal and smooth curves. Consequently, the sheaf of coinvariants associated to these V-modules on M̄0,N is not locally free for N≥ 4. It also follows that, unlike in the rational case, the mode transition algebra A introduced by Damiolini-Gibney-Krashen is not isomorphic to the end E=∫X∈ Mod( X) X⊗ X' as an object of Mod(V⊗ 2).