There's a literature about dessins d'enfants (including my previous question here), and one amazing thing about them is that absolute Galois group Gal Q acts on cartographic group which, I believe, is isomorphic to letters_2 = <<A, B>> (group, freely generated by two noncommuting letters).

The funny thing about the latter group is that there is a flat connection coming from string theory defined on its group algebra, C[letters_2], which I think has the name of Knizhnik-Zamolodchikov. So, it that latter connection somehow related to Galois group?