The Drinfelâ€™d-Kohno theorem is about a geometric relation between classical and quantum $R$-matrices for quantum groups: the monodromies of the solutions of Knizhnik-Zamolodchikov equations (which are equations of Fuchsian type) for a given classical $r$-matrix give rise to the corresponding quantum $R$-matrix; more precisely it gives the same representation of the braid group.

References

Toshitake Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Annales de lâ€™institut Fourier 37, no. 4 (1987), p. 139-160, numdam

Toshitake Kohno, Conformal field theory and topology, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs 210. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002. x+172 pp.

P. Etingof, O. Shiffman, Lectures on quantum groups