We take another approach to Hitchin's strategy of computing the cohomology of
moduli spaces of Higgs bundles by localization with respect to the
circle-action. Our computation is done in the dimensional completion of the
Grothendieck ring of varieties and starts by describing the classes of moduli
stacks of chains rather than their coarse moduli spaces.
As an application we show that the n-torsion of the Jacobian acts trivially
on the middle dimensional cohomology of the moduli space of twisted
SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for
the motive of the moduli space of Higgs bundles of rank 4 and odd degree. This
provides new evidence for a conjecture of Hausel and Rodr\'iguez-Villegas.
Along the way we find explicit recursion formulas for the motives of several
types of moduli spaces of stable chains.Comment: 44 page