A perfect obstruction theory for SU(2)-Higgs pairs

We present a new method for constructing virtual cycles for rank-2 Higgs sheaves $(E,\phi )$ on a smooth projective surface $S$. Using this, we redefine the $\mathbf{S}\mathbf{U}(2)$-perfect obstruction theory previously constructed by Tanaka-Thomas. The key step in our construction involves modifying the ${\mathbf{C}}^{\times }$-localisation formula of Graber-Pandharipande by replacing the torus action with an involution $(E,\phi )\mapsto (E^{\ast },-{\phi }^{\ast })$.