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 $p\in {\mathbb{Z}}_{\ge 2}$ or the even symplectic fermion VOAs $S{F}_d^{+}$ 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 $N\ge 4$ 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 $N\ge 4$. 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 $\mathbb{E}={\int }_{\mathbb{X}\in \mathrm{Mod}(\mathbb{X})}\mathbb{X}\otimes {\mathbb{X}}^{\prime }$ as an object of $\mathrm{Mod}({\mathbb{V}}^{\otimes 2})$.

21 pages, comments are welcome