Periodic monopoles and difference modules This book studies a class of monopoles defined by certain mild conditions called periodic monopoles of generalized Cherkis–Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure revealing a kind of Kobayashi–Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity. The theory of periodic monopoles of GCK type has applications to Yang–Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given including major generalizations of results due to Charbonneau Cherkis Hurtubise Kapustin and others and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette Donaldson Hitchin and Simpson. This work will be of interest to graduate students and researchers in differential and algebraic geometry as well as in mathematical physics.