A virtual structure for symplectic Higgs bundles

We define a perfect obstruction theory for a moduli of symplectic Higgs sheaves $(E,\phi )$ on projective surfaces $S$. Key to this is a minimality assumption on $\text{ch}(E)$ that forces all $E$ to be locally free. This might have implications to define a virtual count and $Sp(r)$-Vafa-Witten invariants.