The nonabelian Hodge correspondence gives a diffeomorphism between the
moduli of flat connections and the moduli of higgs bundles on a smooth
Riemann surface. The two moduli, however, are completely different as
algebraic varieties. Thus natural structures on one side of the
correspondence, such as Hitchin’s integrable system, become rather
mysterious on the other side.
I will discuss joint work with Zsuzsanna Dancso and Vivek Shende, which
defines a ‘microlocal model’ of the correspondence in a neighborhood of a
singular fiber of the Hitchin system. In particular, we prove a conjecture
of Hausel and Proudfoot relating the weight and perverse filtrations on
either side of this microlocal correspondence.