Semisimplicity of conformal blocks

Abstract

We prove that braid group representations associated to braided fusion categories and mapping class group representations associated to modular fusion categories are always semisimple. The proof relies on the theory of extensions in non-Abelian Hodge theory and on Ocneanu rigidity. By combining this with previous results on the existence of variations in Hodge structures, we further show that such a braid group or mapping class group representation preserves a non-degenerate Hermitian form and can be defined over some CM number field.