Hodge structures on conformal blocks

We prove existence and uniqueness of complex Hodge structures on modular functors. The proof is based on the non-Abelian Hodge correspondence and Ocneanu rigidity. Given a modular functor, we explain how its Hodge numbers fit into a Frobenius algebra and the Chern characters of its Hodge decompositions into a new cohomological field theory (CohFT). In the case of $\mathrm{S}\mathrm{U}(2)$ modular functors of level $2$ times an odd number, we give explicit formulas for all Hodge numbers, in any genus $g$.