Abstract: In this talk, we will discuss moduli spaces of Higgs pairs on surfaces from which Vafa-Witten invariants are derived. We will focus on the associated Hitchin maps and introduce a necessary and  sufficient condition on the nonemptyness of generic fibers. We then show how to calculate monopole branches for surfaces of Picard number one assuming the nonemptyness of generic fibers. This is joint work with Xiaoyu Su.