BPS cohomology for rank 2 degree 0 Higgs bundles (and more)

We give a formula comparing the E-series of the moduli stacks of rank 2 degree 0 semistable Higgs bundles in genus $g\ge 2$ to intersection E-polynomials of its coarse moduli space. A parellel formula holds in various 2-Calabi-Yau settings, for example for sheaves on K3 surfaces, or preprojective algebras of $g$-loop quivers. As a consequence we provide evidence for a conjecture of Davison on the BPS cohomology of Higgs bundles, which has implications for non-abelian Hodge theory for stacks. We apply the formula to cohomological $\chi$-independence tests for BPS cohomology of Higgs bundles and K3 surfaces.