  An interesting theme in complex differential geometry is to find a
correspondence between algebraic objects and differential geometric objects.
One of the most attractive is the non-abelian Hodge theory of Simpson. In this
paper, pursuing an analogue of the non-abelian Hodge theory in the context of
$q$-difference modules, we study Kobayashi-Hitchin correspondences between
doubly periodic monopoles and parabolic $q$-difference modules, depending on
twistor parameters.
