  We study the local and topological features of Berry phases associated with
the adiabatic transport of vortices in a d-wave superconductor of lattice
fermions. At half filling, where the local Berry curvature must vanish due to
symmetries, the phase associated with the exchange of two vortices is found to
vanish as well, implying that vortices behave as bosons. Away from half
filling, and in the limit where the magnetic length is large compared to the
lattice constant, the local Berry curvature gives rise to an intricate flux
pattern within the large magnetic unit cell. This renders the Berry phase
associated with an exchange of two vortices highly path dependent. However, it
is shown that "statistical" fluxes attached to the vortex positions are still
absent. Despite the complicated profile of the Berry curvature away from half
filling, we show that the average flux density associated with this curvature
is tied to the average particle density. This is familiar from dual theories of
bosonic systems, even though in the present case, the underlying particles are
fermions.
