Algebra/Topology seminar

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.

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 such Hodge structures. Finally, we will discuss the computation of Hodge numbers for sl_2 modular functors (of odd level) and how these numbers are part of a cohomological field theory (CohFT).