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

Abstract

Let $\mathbb V$ be an $\mathbb N$-graded, $C_2$-cofinite vertex operator algebra (VOA) admitting a non-lowest generated module in $\mathrm{Mod}(\mathbb V)$ (e.g., the triplet algebras $\mathcal{W}_p$ for $p\in \mathbb{Z}_{\geq 2}$ or the even symplectic fermion VOAs $SF_d^+$ for $d\in \mathbb{Z}_+$). We prove that, unlike in the rational case, the spaces of conformal blocks associated to certain $\mathbb V$-modules do not form a vector bundle on $\overline{\mathcal{M}}_{0,N}$ for $N\geq 4$ by showing that their dimensions differ between nodal and smooth curves. Consequently, the sheaf of coinvariants associated to these $\mathbb V$-modules on $\overline{\mathcal{M}}_{0,N}$ is not locally free for $N\geq 4$. It also follows that, unlike in the rational case, the mode transition algebra $\mathfrak A$ introduced by Damiolini-Gibney-Krashen is not isomorphic to the end $\mathbb E=\int_{\mathbb X\in \mathrm{Mod}(\mathbb X)}\mathbb X\otimes \mathbb{X}'$ as an object of $\mathrm{Mod}(\mathbb{V}^{\otimes 2})$.