The asymptotic metric structure of the Hitchin moduli space

I will report on some recent work with Swoboda, Weiss and Witt which aims to determine the detailed asymptotic structure of the hyperkähler metric on the moduli space of solutions of the Hitchin equation over a Riemann surface. This partially vindicates a conjectural (stringy) picture by Gaiotto-Moore-Neitzke.