Modular functors are 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. Such structures arise naturally in the representation theory of affine Lie algebras or equivalently in suitable representation categories of quantum groups. The flat bundles forming a modular functor are referred to as conformal blocks.
In this talk, we will discuss Hodge structures on such flat bundles. If these flat bundles where rigid, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. However, that is not the case in general. We will explain how a different kind of rigidity for modular functors can be used to prove an existence and uniqueness result for Hodge structures on conformal blocks. Finally, we will discuss the computation of Hodge numbers for SL2 modular functors (of odd level) and how these numbers are part of a cohomological field theory (CohFT).
