Periodic Monopoles and Difference Modules

This book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized CherkisKapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of KobayashiHitchin 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 YangMills 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 graduatestudents and researchers in differential and algebraic geometry, as well as in mathematical physics.