Modularity of VOA conformal blocks beyond the strongly rational, self-dual case

Ascertain whether the vector bundles of conformal blocks V constructed from vertex operator algebras, as in the works of Ben-Zvi–Frenkel and Damiolini–Gibney–Tarasca, define a modular functor outside the strongly rational, self-dual setting; equivalently, determine whether V is a modular functor for general vertex operator algebras.

Background

The authors compare their Heisenberg-picture description of conformal blocks with the geometric construction of conformal blocks for vertex operator algebras (VOAs), which yields projectively flat bundles over moduli spaces of curves. For strongly rational, self-dual VOAs, these bundles are known to form a modular functor; beyond this regime, the status is unclear.

Establishing modularity of VOA conformal blocks in the non-rational setting would bridge the algebraic factorization-homology framework and the geometric Knizhnik–Zamolodchikov picture, enabling a broader application of the paper’s results.