Conformal blocks are quasi-geometric

We prove that the bundles with flat connections on configuration spaces associated to braided fusion categories, as well as the bundles with flat connections on moduli spaces of curves (conformal blocks) associated to modular fusion categories, are defined over number fields. The proof relies on Ocneanu rigidity. This result answers a conjecture of Etingof and Varchenko. Furthermore, we show that for a fixed braided or modular category, all the associated bundles with flat connections and their compatibilities (i.e., the braided or modular functor) can be defined over the same number field.