Relations in the cohomology ring of the moduli space of rank 2 Higgs   bundles
  The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus.   This paper accomplishes the same thing for the non-compact moduli spaces of Higgs bundles, in the sense of Hitchin and Simpson. There are many more independent relations than for stable bundles, but in a sense the answer is simpler, since the formulas are completely explicit, not recursive. The results of Kirwan on equivariant cohomology for holomorphic circle actions are of key importance.   Together, Parts I and II describe the cohomology rings of spaces of rank 2 Higgs bundles at essentially the same level of detail as is known for stable bundles. 
Tamas Hausel and Michael Thaddeus