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.

Describes a new equivalence between objects in differential and algebraic geometry Provides a foundation for the study of difference modules from both differential and algebraic geometry viewpoints Studies periodic monopoles via a systematic use of dimensional reduction to harmonic bundles

Auteur

Takuro Mochizuki has been awarded the 2022 Breakthrough Prize in Mathematics for advancing the understanding of holonomic D-modules through his research on harmonic bundles and twister D-modules, which he has studied at the "interface of algebraic geometry and differential geometry".

Contenu
. - Introduction. - Preliminaries. - Formal Difference Modules and Good Parabolic Structure. - Filtered Bundles. - Basic Examples of Monopoles Around Infinity. - Asymptotic Behaviour of Periodic Monopoles Around Infinity. - The Filtered Bundles Associated with Periodic Monopoles. - Global Periodic Monopoles of Rank One. - Global Periodic Monopoles and Filtered Difference Modules. - Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix).