Semisimplicity of conformal blocks

TL;DR

This paper proves that certain algebraic representations related to braided and modular fusion categories are always semisimple, and they preserve Hermitian forms, with implications for their algebraic structure and field of definition.

Contribution

It establishes the semisimplicity of braid and mapping class group representations from fusion categories and shows they preserve Hermitian forms and are defined over CM fields.

Findings

Representations are always semisimple.

They preserve a non-degenerate Hermitian form.

They can be defined over CM number fields.

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.