Finite braid group orbits on character varieties

The moduli space of rank n local systems on a topological surface admits an action of a huge group, namely the mapping class group. The finite orbits of this action then correspond to very special points in the moduli space. This is a generalization of the classical question of finding algebraic solutions to the Painlevé VI equation, which corresponds to n = 2 and the surface being the 4-punctured sphere.

I’ll discuss joint work with Aaron Landesman and Daniel Litt, in which we classify all finite orbits in the case when n=2 and the Riemann surface is a punctured P^1, under a mild assumption. This extends previous work of Dubrovin—Mazzocco and Lisovvyy—Tykhyy, though our techniques are rather different and essentially Hodge theoretic.