Appears in collection : Resurgence in Mathematics and Physics

Recently, we obtained equivalences between monopoles with periodicity and difference modules of various types, i.e., periodic monopoles and difference modules, doubly periodic monopoles and q-difference modules, and triply periodic monopoles and difference modules on elliptic curves.In this talk, we shall give a review on the equivalences. If possible, we would like to discuss some deeper aspects of the correspondences from non-abelian Hodge theoretic viewpoints.