I will review a conjecture (joint work with Davide Gaiotto and Greg Moore) which gives a description of the hyperkähler metric on the moduli space of Higgs bundles, and recent joint work with David Dumas which has given evidence that the conjecture is true in the case of $SL(2)$-Higgs bundles.
32Q20 ; 53C07 ; 53C55 ; 53C26 ; 81T13 ; 81T60