Quantum higher Teichmüller theory, as described by Fock and Goncharov, endows a quantum character variety on a surface $S$ with a cluster structure. The latter allows one to construct a canonical representation of the character variety, which happens to be equivariant with respect to an action of the mapping class group of $S$. It was conjectured by the authors that these representations behave well with respect to cutting and gluing of surfaces, which in turn yields an analogue of a modular functor. In this talk, I will show how the quantum group and its positive representations arise in this context. I will also explain how the modular functor conjecture is related to the closedness of positive representations under tensor products as well as to the non-compact analogue of the Peter-Weyl theorem. If time permits, I will say a few words about the proof of the conjecture.

This talk is based on joint works with Gus Schrader.