Some curious aspects of the spectral flow for the Vafa-Witten equations (with and without mass term)

The Vafa Witten equations on 4-manifolds are the variational equations of a functional that generalizes one of the Chern-Simons functionals for SL(2;C) connections on 3-manifolds (and it reduces to that on products of a 3-manifold with the circle).  Being that the moduli space of solutions is often non-compact, one might wonder whether or not non-convergent sequences have bounded spectral flow.  As it turns out (for reducible solutions), sometimes they do and sometimes they don’t.  I will try to explain how and why this phenomena comes about.