Vector Bundles of Coinvariants for Admissible Affine Vertex Operator Algebras

Source abstract

Simple affine vertex operator algebras $L_k(\mathfrak{g})$ at non-integral admissible levels $k$ are generally neither $C_2$-cofinite nor rational. Despite the absence of these standard finiteness conditions, we prove that the sheaf of coinvariants (and, dually, of conformal blocks) of ordinary modules in the category $\mathcal{O}_k$ forms a vector bundle over the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed genus-zero curves. The proof relies on establishing the finite-dimensionality of genus-zero coinvariants, a vanishing theorem that isolates ordinary admissible modules at the boundary, and a smoothing theorem enabled by partial strong identity elements in the mode transition algebra. Consequently, admissible affine vertex operator algebras supply a rich class of non-rational, non-$C_2$-cofinite examples in which vector-bundle behavior is preserved for a well-behaved subcategory of modules.