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 be...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Cham
Springer Nature
2022
Springer Springer International Publishing AG Springer International Publishing |
| Edition | 1 |
| Series | Lecture Notes in Mathematics |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3030945006 9783030945008 9783030944995 3030944999 |
| ISSN | 0075-8434 1617-9692 |
| DOI | 10.1007/978-3-030-94500-8 |
Cover
Table of Contents:
- 3.3.2 Good Filtered Bundles over Formal Difference Modules -- 3.3.3 The Induced Endomorphisms on the Graded Pieces -- 3.4 Geometrization of Formal Difference Modules -- 3.4.1 Ringed Spaces -- 3.4.2 Some Formal Spaces -- 3.4.3 Difference Modules and OH∞,q(H∞,q)-Modules -- 3.4.4 Lattices and the Induced Local Systems -- 3.5 Filtered Bundles in the Formal Case -- 3.5.1 Pull Back and Descent of OH∞,p(H∞,p)-Modules -- 3.5.2 Filtered Bundles -- 3.5.2.1 Subbundles and Quotient Bundles -- 3.5.2.2 Basic Functoriality -- 3.5.2.3 Pull Back -- 3.5.2.4 Push-Forward -- 3.5.2.5 Descent -- 3.5.3 Basic Filtered Objects with Pure Slope -- 3.5.4 Good Filtered Bundles over OH∞,q(H∞,q)-Modules with Level ≤1 -- 3.5.5 Good Filtered Bundles over OH∞,q(H∞,q)-Modules -- 3.5.5.1 An Equivalence -- 3.5.5.2 Some Properties -- 3.5.6 Global Lattices on the Covering Space -- 3.5.7 Local Lattices -- 3.5.8 Complement for Good Filtered Bundles with Level ≤1 -- 3.6 Formal Difference Modules of Level ≤1 and Formal λ-Connections -- 3.6.1 Formal λ-Connections -- 3.6.2 Some Sheaves of Algebras on H∞,q -- 3.6.3 From Formal λ-Connections to Formal Difference Modules -- 3.6.4 Equivalence -- 3.6.4.1 Simpler Cases of Proposition 3.6.8 -- 3.6.5 Example 1 -- 3.6.5.1 -- 3.6.5.2 -- 3.6.6 Example 2 -- 3.6.6.1 -- 3.6.6.2 -- 3.6.7 Comparison of Good Filtered Bundles -- 3.6.8 Comparison of the Associated Graded Pieces -- 3.6.9 Some Functoriality -- 3.7 Appendix: Pull Back and Descent in the R-Direction -- 3.7.1 Examples -- 4 Filtered Bundles -- 4.1 Outline of This Chapter -- 4.2 Filtered Bundles in the Global Case -- 4.2.1 Subbundles and Quotient Bundles -- 4.2.2 Degree and Slope -- 4.2.3 Stability Condition -- 4.2.4 Good Filtered Bundles of Dirac Type and Parabolic Difference Modules -- 4.2.4.1 Polystable Parabolic Difference Modules -- 4.2.4.2 Equivalence -- 4.3 Filtered Bundles on Ramified Coverings
- 1.7.1.5 Kobayashi-Hitchin Correspondence in the Case λ=0 -- 1.7.1.6 OM0Z(H0∞)-Modules and C(w)-Modules with an Automorphism -- 1.7.2 The Correspondences in the General Case -- 1.7.2.1 Preliminary Consideration -- 1.7.2.2 Mini-complex Structure Corresponding to the Twistor Parameter λ -- 1.7.2.3 Another Coordinate System and the Compactification of Mλ -- 1.7.2.4 Mini-holomorphic Bundles Associated with Monopoles -- 1.7.2.5 Meromorphic Extension and Filtered Extension at Infinity -- 1.7.2.6 Kobayashi-Hitchin Correspondence of Periodic Monopoles of GCK Type -- 1.7.2.7 Difference Modules and OMλZ (Hλ∞)-Modules -- 1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type -- 1.8.1 Setting -- 1.8.2 Decomposition of Mini-holomorphic Bundles -- 1.8.3 The Induced Higgs Bundles -- 1.8.3.1 Preliminary (1) -- 1.8.3.2 Preliminary (2) -- 1.8.3.3 The Induced Higgs Bundles -- 1.8.4 Asymptotic Orthogonality -- 1.8.5 Curvature Decay -- 1.8.6 The Filtered Extension in the Case λ=0 -- 1.8.7 The Filtered Extension for General λ -- 1.8.7.1 Ramified Covering Space -- 1.8.7.2 Approximation -- 1.8.7.3 Formal Completion of Asymptotic Harmonic Bundles at Infinity -- 1.8.7.4 The Formal Structure of PhEλ at Infinity -- 2 Preliminaries -- 2.1 Outline of This Chapter -- 2.2 Mini-Complex Structures on 3-Manifolds -- 2.2.1 Mini-Holomorphic Functions on RC -- 2.2.2 Mini-Complex Structure on Three-Dimensional Manifolds -- 2.2.3 Tangent Bundles -- 2.2.4 Cotangent Bundles -- 2.2.5 Meromorphic Functions -- 2.3 Mini-Holomorphic Bundles -- 2.3.1 Mini-Holomorphic Bundles -- 2.3.2 Metrics and the Induced Operators -- 2.3.3 Splittings -- 2.3.4 Scattering Maps -- 2.3.5 Dirac Type Singularity of Mini-Holomorphic Bundles -- 2.3.6 Kronheimer Resolution of Dirac Type Singularity -- 2.3.7 Precise Description of Dirac Type Singularities -- 2.3.8 Subbundles and Quotient Bundles
- 2.3.9 Basic Functoriality -- 2.4 Monopoles -- 2.4.1 Monopoles and Mini-Holomorphic Bundles -- 2.4.2 Euclidean Monopoles -- 2.4.3 Dirac Type Singularity -- 2.4.3.1 Dirac Monopoles (Examples) -- 2.4.4 Basic Functoriality -- 2.5 Dimensional Reduction from 4D to 3D -- 2.5.1 Instantons Induced by Monopoles -- 2.5.2 Holomorphic Bundles and Mini-Holomorphic Bundles -- 2.6 Dimensional Reduction from 3D to 2D -- 2.6.1 Monopoles Induced by Harmonic Bundles -- 2.6.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with a Higgs Field -- 2.6.3 Mini-Holomorphic Sections and Monodromy -- 2.6.4 Appendix: Monopoles as Harmonic Bundles of Infinite Rank -- 2.7 Twistor Families of Mini-Complex Structures on RC and (R/TZ)C -- 2.7.1 Preliminary -- 2.7.2 Spaces -- 2.7.3 Twistor Family of Complex Structures -- 2.7.4 Family of Mini-Complex Structures -- 2.7.5 The Mini-Complex Coordinate System (t0,β0) -- 2.7.6 The Mini-Complex Coordinate System (t1,β1) -- 2.7.7 Coordinate Change -- 2.7.8 Compactification -- 2.7.9 Mini-Holomorphic Bundles Associated with Monopoles -- 2.7.9.1 Compatibility with the Dimensional Reduction from 4D to 3D -- 2.8 OMλ-Modules and λ-Connections -- 2.8.1 Dimensional Reduction from OMλ-Modules to λ-Flat Bundles -- 2.8.1.1 Setting -- 2.8.1.2 Some Vector Fields and Forms -- 2.8.1.3 A General Equivalence -- 2.8.1.4 Mini-Holomorphic Bundles and Flat λ-Connections -- 2.8.1.5 λ-Flat Bundles of Infinite Rank -- 2.8.1.6 Remark -- 2.8.2 Comparison of Some Induced Operators -- 2.8.2.1 Comparison of Mini-Holomorphic Bundles Induced by Harmonic Bundles -- 2.8.3 OMλ-Modules and λ-Connections -- 2.8.3.1 Setting -- 2.8.3.2 A General Equivalence -- 2.8.3.3 Mini-Holomorphic Bundles and Meromorphic Flat λ-Connections -- 2.8.3.4 Another Description of the Construction -- 2.9 Curvatures of Mini-Holomorphic Bundles with Metric on Mλ
- 2.9.1 Contraction of Curvature and Analytic Degree -- 2.9.2 Chern-Weil Formula -- 2.9.3 Another Description of G(h) -- 2.9.4 Change of Metrics -- 2.9.5 Relation with λ-Connections -- 2.9.5.1 λ-Flat Bundles of Infinite Rank with a Harmonic Metric -- 2.9.5.2 Remark -- 2.9.6 Dimensional Reduction of Kronheimer -- 2.9.7 Appendix: Ambiguity of the Choice of a Splitting -- 2.10 Difference Modules and OMλZ(Hλ∞)-Modules -- 2.10.1 Difference Modules with Parabolic Structure at Finite Place -- 2.10.2 Construction of Difference Modules from OMλZ(Hλ∞)-Modules -- 2.10.3 Construction of OMλZ(Hλ)-Modules from Difference Modules -- 2.10.4 Appendix: Mellin Transform and Parabolic Structures at Finite Place -- 2.10.4.1 Mellin Transform -- 2.10.4.2 Algebraic Nahm Transform for Filtered λ-Flat Bundles (Special Case) -- 2.11 Filtered Prolongation of Acceptable Bundles -- 2.11.1 Filtered Bundles on a Neighbourhood of 0 in C -- 2.11.1.1 G-Equivariance -- 2.11.1.2 Subbundles, Quotient and Splitting -- 2.11.1.3 Basic Functoriality -- 2.11.1.4 Pull Back -- 2.11.1.5 Push-Forward -- 2.11.1.6 Descent -- 2.11.1.7 Some Examples -- 2.11.2 Acceptable Bundles on a Punctured Disc -- 2.11.2.1 Basic Functoriality -- 2.11.2.2 Pull Back and Descent -- 2.11.3 Global Case -- 2.11.3.1 Filtered Bundles -- 2.11.3.2 Acceptable Bundles -- 3 Formal Difference Modules and Good Parabolic Structure -- 3.1 Outline of This Chapter -- 3.2 Formal Difference Modules -- 3.2.1 Formal Difference Modules of Level ≤1 -- 3.2.2 Formal Difference Modules of Pure Slope -- 3.2.3 Slope Decomposition of Formal Difference Modules -- 3.3 Good Filtered Bundles of Formal Difference Modules -- 3.3.1 Filtered Bundles over C((yq-1))-Modules -- 3.3.1.1 G-Equivariance -- 3.3.1.2 Submodules, Quotient Modules and Splittings -- 3.3.1.3 Basic Functoriality -- 3.3.1.4 Pull Back -- 3.3.1.5 Push-Forward -- 3.3.1.6 Descent
- 4.3.1 The Case λ=0
- Intro -- Preface -- Acknowledgements -- Contents -- 1 Introduction -- 1.1 Background and Motivation -- 1.2 Monopoles of GCK-Type -- 1.3 Previous Works on Monopoles and Algebraic Objects -- 1.3.1 SU(2)-Monopoles with Finite Energy on R3 -- 1.3.2 The Correspondence due to Charbonneau and Hurtubise -- 1.3.3 Remark -- 1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles -- 1.4.1 Harmonic Bundles and Their Underlying λ-Flat Bundles -- 1.4.2 Kobayashi-Hitchin Correspondences in the Smooth Case -- 1.4.3 Tame Harmonic Bundles and Regular Filtered λ-Flat Bundles -- 1.4.4 Wild Harmonic Bundles and Good Filtered λ-Flat Bundles -- 1.5 Equivariant Instantons and the Underlying Holomorphic Objects -- 1.5.1 Instantons and the Underlying Holomorphic Bundles -- 1.5.2 Instantons and Harmonic Bundles -- 1.5.3 Instantons and Monopoles -- 1.5.4 Instantons and Monopoles as Harmonic Bundles of Infinite Rank -- 1.5.4.1 Instantons as Harmonic Bundles of Infinite Rank -- 1.5.4.2 The Underlying λ-Flat Bundles of Infinite Rank -- 1.5.4.3 Monopoles as Harmonic Bundles of Infinite Rank -- 1.6 Difference Modules with Parabolic Structure -- 1.6.1 Difference Modules -- 1.6.2 Parabolic Structure of Difference Modules at Finite Place -- 1.6.3 Good Parabolic Structure at ∞ -- 1.6.4 Parabolic Difference Modules -- 1.6.5 Degree and Stability Condition -- 1.6.6 Easy Examples of Stable Parabolic Difference Modules (1) -- 1.6.6.1 The Case Where (∞) Is Even -- 1.6.6.2 The Case Where (∞) Is Odd -- 1.6.7 Easy Examples of Stable Parabolic Difference Modules (2) -- 1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles -- 1.7.1 The Correspondence in the Case λ=0 -- 1.7.1.1 Mini-complex Structure -- 1.7.1.2 Mini-holomorphic Bundles Associated with Monopoles -- 1.7.1.3 Dirac Type Singularity -- 1.7.1.4 Meromorphic Extension and Filtered Extension at Infinity