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Periodic Monopoles and Difference Modules

  • Book
  • © 2022

Overview

Authors:
  1. Takuro Mochizuki

    1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

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  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2300)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Introduction

    • Takuro Mochizuki
    Pages 1-42

  3. Preliminaries

    • Takuro Mochizuki
    Pages 43-105

  4. Formal Difference Modules and Good Parabolic Structure

    • Takuro Mochizuki
    Pages 107-150

  5. Filtered Bundles

    • Takuro Mochizuki
    Pages 151-176

  6. Basic Examples of Monopoles Around Infinity

    • Takuro Mochizuki
    Pages 177-193

  7. Asymptotic Behaviour of Periodic Monopoles Around Infinity

    • Takuro Mochizuki
    Pages 195-226

  8. The Filtered Bundles Associated with Periodic Monopoles

    • Takuro Mochizuki
    Pages 227-257

  9. Global Periodic Monopoles of Rank One

    • Takuro Mochizuki
    Pages 259-269

  10. Global Periodic Monopoles and Filtered Difference Modules

    • Takuro Mochizuki
    Pages 271-287

  11. Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix)

    • Takuro Mochizuki
    Pages 289-310

  12. Back Matter

    Pages 311-324
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Keywords

About this book

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 graduatestudents and researchers in differential and algebraic geometry, as well as in mathematical physics.

Authors and Affiliations

  • Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Takuro Mochizuki

About the author

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".

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