If C is a smooth projective complex curve, then by classical nonabelian Hodge theory there is a diffeomorphism between the coarse moduli space of representations of the fundamental group of C, and the coarse moduli space of degree zero semistable Higgs bundles on C. In particular, the Borel-Moore homology of these two moduli spaces is isomorphic.

In this talk I will construct an isomorphism between the Borel-Moore homologies of the full stack of representations of the fundamental group and the full stack of degree zero semistable Higgs bundles.  In the absence of any kind of isomorphism between these two stacks, the isomorphism in BM homology has to take a roundabout route, via an isomorphism of the "BPS cohomology" of the two moduli problems.  This in turn is provided by the classical nonabelian Hodge theory, along with a freeness result regarding the BPS Lie algebra on both sides of nonabelian Hodge theory.  This is joint work in progress with Lucien Hennecart and Sebastain Schlegel-Mejia.