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Description
Modular functors are collections of vector bundles with flat connections on (twisted) moduli spaces of curves, individually known as conformal blocks, that satisfy strong compatibility conditions with respect to natural maps between these moduli spaces. Such structures arise naturally in the representation theory of affine Lie algebras and quantum groups, where the conformal blocks are known to be semisimple.
Recently, Hodge structures on the genus-0 conformal blocks associated to affine Lie algebras have been studied by Belkale, Fakhruddin, and Mukhopadhyay through a motivic construction. In particular, they computed genus-0 Hodge numbers for $sl_n$.
I will discuss an axiomatic proof of the existence and uniqueness of such Hodge structures and of the semisimplicity of conformal blocks, for any modular functor (i.e. any modular fusion category). If the flat bundles of conformal blocks were rigid and semisimple, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. However, they are not rigid in general. I will explain how a different form of rigidity for modular fusion categories—Ocneanu rigidity—can be used, together with non-Abelian Hodge theory, to tackle these questions. Finally, I will discuss an application to the computation of Hodge numbers for $sl_2$ modular functors of odd level in higher genus and how these numbers are part of (new) cohomological field theories (CohFTs).
