Modular functors are families of finite-dimensional representations of Mapping Class Groups of surfaces, with strong compatibility conditions. As Mapping Class Groups of surfaces are isomorphic to fundamental groups of moduli spaces of curves, modular functors can alternatively be seen as families of vector bundles with flat connection on (twisted) moduli spaces of curves, with strong compatibility conditions with respect to some natural maps between the moduli spaces.
The data necessary to define such families of flat vector bundles is encapsulated in a type of braided tensor categories called modular categories. Some modular functors arise naturally in the representation theory of affine Lie algebras and examples of modular categories can be constructed as suitable representation categories of quantum groups.
In this talk, we will first motivate and define the notion of a Hodge structure on a modular functor. We will then explain how a rigidity result for modular categories and non-Abelian Hodge theory can be used to prove an existence and uniqueness result for such Hodge structures. Finally, we will discuss the cohomological field theories (CohFTs) associated to Hodge structures on modular functors, and the computation of Hodge numbers for $sl_2$ modular categories (of odd level).