  On a polarised surface, solutions of the Vafa-Witten equations correspond to
certain polystable Higgs pairs. When stability and semistability coincide, the
moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action
with compact fixed locus. Applying virtual localisation we define invariants
constant under deformations.
  When the vanishing theorem of Vafa-Witten holds, the result is the (signed)
Euler characteristic of the moduli space of instantons. In general there are
other, rational, contributions. Calculations of these on surfaces with positive
canonical bundle recover the first terms of modular forms predicted by Vafa and
Witten.
