Pierre Godfard: Hodge structures on conformal blocks

Abstract:

Modular functors are families of vector bundles with flat connection on (twisted) moduli spaces of curves, with strong compatibility conditions with respect to natural maps between the moduli spaces. Such structures arise naturally in the representation theories of affine Lie algebras and of quantum groups.

In this talk, we will discuss Hodge structures on such flat bundles. If these flat bundles were rigid, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. Unfortunately, these bundles are in general not rigid. However, 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 sl₂ modular functors (of odd level) and how these numbers are part of a cohomological field theory (CohFT). Motivicity of certain families of modular functors in genus 0 will also be reviewed.