Differential Geometry of Monopole Moduli Spaces
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Arxiv:1912.05718V1 [Math.DG] 12 Dec 2019
JNR MONOPOLES MICHAEL K. MURRAY AND PAUL NORBURY 1 Abstract. We review the theory of JNR, mass 2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that that rational map of a JNR monopole monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion we illustrate some examples of the energy-density at infinity of JNR monopoles. Dedicated to the memory of Sir Michael Atiyah Contents 1. Introduction 2 2. Hyperbolic monopoles 4 3. JNR monopoles 5 3.1. JNR spectral curve 5 3.2. Scattering 7 3.3. JNR rational map 7 4. A criterion to be a JNR spectral curve 8 4.1. (N; N) spectral curves 8 arXiv:1912.05718v1 [math.DG] 12 Dec 2019 4.2. Grids 9 4.3. JNR monopoles and grids 9 5. Holomorphic spheres 10 2010 Mathematics Subject Classification. 81T13, 53C07, 14D21. MKM acknowledges support under the Australian Research Council's Discovery Projects funding scheme (project number DP180100383). PN acknowledges support under the Aus- tralian Research Council's Discovery Projects funding scheme (project numbers DP170102028 and DP180103891). 1 2 MICHAEL K. MURRAY AND PAUL NORBURY 5.1. SO(3) action on JNR monopoles. 12 6. Projection to rational maps. 12 7. The energy density at infinity 14 References 16 1. -
Moduli of Monopole Walls and Amoebas Arxiv:1202.1294V1 [Hep-Th]
Moduli of Monopole Walls and Amoebas Sergey A. Cherkis Richard S. Ward Department of Mathematics Department of Mathematical Sciences University of Arizona, University of Durham, Tucson AZ, 85721-0089, USA Durham DH1 3LE, UK [email protected] [email protected] Abstract We study doubly-periodic monopoles, also called monopole walls, determining their spectral data and computing the dimensions of their moduli spaces. Using spectral data we identify the moduli, and compare our results with a perturbative analysis. We also identify an SL(2; Z) action on monopole walls, in which the S transformation corresponds to the Nahm transform. arXiv:1202.1294v1 [hep-th] 6 Feb 2012 Contents 1 Introduction and Motivation 2 1.1 Monopole Wall . 2 1.1.1 Asymptotic Conditions . 3 1.1.2 Singularities . 4 1.2 Moduli and Parameters . 5 2 String Theory Dualities 7 3 Spectral Approach 11 3.1 Spectral data . 11 3.1.1 x-spectral data . 11 3.1.2 y-spectral data . 12 3.1.3 z-spectral data . 13 3.1.4 Examples . 13 3.2 Newton Polygon . 14 3.2.1 Amoebas . 16 3.2.2 Examples . 16 4 Moduli and Asymptotics 22 4.1 Newton Polygon from the Boundary Data . 22 4.2 Number of Moduli . 22 5 Nahm Transform and SL(2; Z) Action 26 5.1 The Nahm transform . 26 5.2 SL(2; Z) Action . 27 6 Perturbative Approach 29 6.1 The case (Q−;Q+) = (0; 1)........................... 29 6.2 The case Q± =1 ................................ 32 6.3 Kuranishi Complex . 36 7 Conclusions 37 A A four-dimensional detour 38 B Theta-function Relations 39 1 1 Introduction and Motivation The classical dynamics of monopoles was found to be intimately related to quantum gauge theories in three [1, 2, 3] and four [4] dimensions. -
Spectral Curves, and the Equations of Bogomolny and Nahm
Monopoles on R3, Spectral Curves, and the Equations of Bogomolny and Nahm Alexander B. Atanasov Abstract This paper introduces the Bogomolny equations, describing SU(n) monopoles of Yang- Mills-Higgs theory in three dimensional Euclidean space. After restricting to the case of SU(2), we describe the geometry of the monopole solution space via a scattering approach of Hitchin [1]. We demonstrate how monopole solutions give rise to a spectral curve of eigenval- 1 ues on T CP and how this may more clearly be understood in terms of a rational map idea of Donaldson [2]. We then introduce the Nahm equations of [3] as an application of the ADHM 3 idea to finding solutions to the self-duality conditions in the reduced case of R , and illustrate the equivalence of the Bogomolny and Nahm equations. Finally, we generalize these ideas by introducing the Nahm transform: a nonabelian generalization of the Fourier transform that relates the self-dual vector bundles on one space to vector bundles on another. Introduction The goal of this paper is to give the reader a gentle introduction to the notable discoveries in the study of monopoles in R3. In section 1, we give a review of the mathematics of gauge theory. We make use of of these techniques in section 2 to give two derivations of the Bogomolny equations. The first approach derives the equations directly from the anti-self-duality (ASD) conditions for instanton solutions in 4 R by treating the fourth component of the connection 1-form, A4, as a scalar field φ and ignoring translations @4 along the x4 direction. -
Monopoles and Fueter Sections on Three-Manifolds
MONOPOLESANDFUETERSECTIONSONTHREE-MANIFOLDS aleksander doan A dissertation presented for the degree of Doctor of Philosophy Department of Mathematics Stony Brook University 2019 Well-versed in the expanses that stretch from earth to stars, we get lost in the space from earth up to our skull. Intergalactic reaches divide sorrow from tears. En route from false to true you wither and grow dull. — Wisława Szymborska, To My Friends1 Dedicated to my parents and grandparents, who taught me to navigate ’the space from earth up to my skull’ 1 Poems New and Collected, transl. Stanisław Bara´nczakand Clare Cavanagh, Harcourt Inc., 1998. ABSTRACT The subject of this dissertation is a relationship between two classes of solutions to elliptic differential equations on three-manifolds: monopoles and Fueter sections. The main result is a description of a wall-crossing phenomenon for a signed count of monopoles. As a corollary, we prove the existence of Fueter sections with singularities. We also study monopoles and Fueter sections using methods of complex geometry, on three-manifolds which are the product of a circle and a surface. Finally, we discuss the relevance of our results to gauge theory on higher-dimensional Riemannian manifolds with special holonomy. DECLARATION This dissertation is based on the articles [Doa17a; Doa17b;DW 17a;DW 17b; DW17c], the last three of which were written in collaboration with Thomas Wal- puski. In Section 1.6 and at the beginning of each chapter, we explain which parts were written in collaboration, and refer to the relevant sections of the articles. DISSERTATIONCOMMITTEE Prof. Simon Donaldson, Simons Center for Geometry and Physics (Ph.D. -
Monopoles and the Nahm Transform
Monopoles and the Nahm transform Jaklyn Crilly 27th October 2016 A thesis submitted for the degree of Bachelor of Science (Advanced) (Honours) of the Australian National University Declaration The work in this thesis is my own, except for where otherwise stated. -Jaklyn Crilly. iii Acknowledgments First and foremost I must thank my supervisor Peter Bouwknegt. I was very lucky to have met you at the end of my second year, and there are no words that can express how grateful I am for all of the support, opportunities and advice you have given me over the years. You have been both a great supervisor and mentor. Thankyou. Also a big thankyou to Bryan Wang for all the time he has put in to introducing me to the wonderful area of gauge theory and monopoles. I would also like to thank my fellow honours year students, my co-op fam, and of course the self-proclaimed MSI beerz crew for making my year so enjoyable. And lastly, I thank my family. In particular, to my parents for all of the love and support that they have given me throughout my life, and for always being there for me. I honestly cannot thank you both enough. v Contents Introduction 1 1 Preliminaries 4 1.1 Principal bundles . .4 1.1.1 Associated bundle . .6 1.1.2 Pullback and pushforward of a bundle . .7 1.2 Connection forms . .8 1.3 Gauge transformation . 11 2 Electric-magnetic duality & magnetic monopoles 13 2.1 Gauge theories and instantons . 13 2.2 Electric-Magnetic Duality . -
Michael Atiyah: Geometry and Physics
Michael Atiyah: Geometry and Physics Nigel Hitchin Mathematical Institute, Woodstock Road, Oxford, OX2 6GG [email protected] December 31, 2020 1 Introduction Over the years, since I was a student, I experienced the many ways in which Michael Atiyah brought mathematics to life by importing new ideas, reformulating them in his own way, communicating them in his inimitable style and using these fresh insights to advance his own research. While in Oxford he would take every opportunity to travel to the USA and listen to what was currently interesting to him, then explain the results with his added contributions at his regular Monday seminar, hoping that some of these themes would be taken up by his audience. He was at heart a geometer, but in the mid 1970s he became convinced that theo- retical physics was by far the most promising source of new ideas. From that point on he became a facilitator of interactions between mathematicians and physicists, attacking mathematical challenges posed by physicists, using physical ideas to prove pure mathematical results, and feeding the physicist community with the parts of modern mathematics he regarded as important but were unfamiliar to them. Here is what he wrote in a retrospective view [24], producing two lists of the historical influences in both directions. It gives an impression of the weighting he applied to areas of the two disciplines. Mathematics ahead of physics: curved space, Lie groups, higher dimensions, fibre bundles, spinors, exterior algebra, non-commutative algebra, Hilbert space, special holonomy. Physics ahead of mathematics: infinite-dimensional representations, Maxwell theory, Dirac theory, supersymmetry, quantum cohomology, conformal field theory, quantum 1 field theory in 3 and 4 dimensions. -
JNR MONOPOLES Dedicated to the Memory of Sir Michael Atiyah
JNR MONOPOLES MICHAEL K. MURRAY AND PAUL NORBURY 1 Abstract. We review the theory of JNR, mass 2 hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that the rational map of a JNR monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion we illustrate some examples of the energy density at infinity of JNR monopoles. Dedicated to the memory of Sir Michael Atiyah Contents 1. Introduction 2 2. Hyperbolic monopoles 4 3. JNR monopoles 5 3.1. JNR spectral curve 5 3.2. Scattering 7 3.3. JNR rational map 7 4. A criterion to be a JNR spectral curve 8 4.1. (N; N) spectral curves 8 4.2. Grids 9 4.3. JNR monopoles and grids 9 5. Holomorphic spheres 10 2010 Mathematics Subject Classification. 81T13, 53C07, 14D21. MKM acknowledges support under the Australian Research Council's Discovery Projects funding scheme (project number DP180100383). PN acknowledges support under the Aus- tralian Research Council's Discovery Projects funding scheme (project numbers DP170102028 and DP180103891). 1 2 MICHAEL K. MURRAY AND PAUL NORBURY 5.1. SO(3) action on JNR monopoles. 12 6. Discrete Nahm equations 13 7. Projection to rational maps. 15 8. The energy density at infinity 16 References 18 1. Introduction The first author was introduced to monopoles by Sir Michael Atiyah in 1980 when Atiyah proposed that generalising Nigel Hitchin's then recent article [14] from SU(2) to an arbitrary compact Lie group would make a good DPhil project. -
Two Integrable Systems Related to Hyperbolic Monopoles*
ASIAN J. MATH. © 1999 International Press Vol. 3, No. 1, pp. 325-332, March 1999 012 TWO INTEGRABLE SYSTEMS RELATED TO HYPERBOLIC MONOPOLES* R. S. WARD* Abstract. Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This article describes two integrable systems which are closely related to hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or discrete Nahm equation), and a soliton system in (2-1-1)- dimensional anti-deSitter space-time. 1. Introduction. Sir Michael Atiyah has made important contributions in sev- eral areas of mathematical physics, and these include the study of Bogomolny-Prasad- Sommerfield (BPS) monopoles; see, for example, Atiyah and Hitchin (1979). Static BPS monopoles are solutions of a nonlinear elliptic partial differential equation on some three-dimensional Riemannian manifold. Most work on monopoles has dealt with the case when this manifold is Euclidean space R3: the equations are then completely-integrable, and can be handled by geometrical techniques. But the mono- pole equations on hyperbolic space H3 are also integrable, as was pointed out by Atiyah (1984a,b); and in some ways, hyperbolic monopoles are simpler than Euclidean ones. Hyperbolic monopoles tend to Euclidean monopoles as the curvature of the hyperbolic space tends to zero, although this is a delicate fact which was only recently established (Jarvis and Norbury 1997). This note describes two integrable systems which are intimately related to, and were motivated by, hyperbolic monopoles. The first is a discrete system (or integrable mapping), which in a certain sense is dual to the hyperbolic monopole system; this is the discrete Nahm or Braam-Austin equation. -
On Hyperbolic Monopoles
On Hyperbolic Monopoles Joseph Y.C. Chan a thesis submitted to The University of Melbourne towards the degree of Doctor of Philosophy supervised by Dr. Paul Norbury Associate Professor Department of Mathematics and Statistics The University of Melbourne April 25, 2017 Contents Chapter 1. Introduction1 Part 1. Euclidean Monopoles 9 Chapter 2. SU(2) monopoles 11 Chapter 3. The Nahm equations 21 Chapter 4. Monopoles of the classical groups 27 Part 2. Instantons 33 Chapter 5. Instantons 35 Chapter 6. The ADHM construction 41 Chapter 7. Holomorphic bundles over P2 47 Part 3. Hyperbolic Monopoles 53 Chapter 8. Circle-invariant instantons 55 Chapter 9. Discrete Nahm equations and boundary values 67 Chapter 10. Symmetric examples and ansätze 71 Chapter 11. A calculation by localisation 75 Chapter 12. The generalised discrete Nahm equations 83 Chapter 13. The rational map and boundary values revisited 89 Chapter 14. Spectral curves 97 Chapter 15. The classical Lie groups 103 Chapter 16. Conclusion 107 Bibliography 109 i CHAPTER 0. CONTENTS Declaration The contents of Chapters1 to 10 are expository and the remaining chapters of the thesis from Chapter 11 onwards are my original contributions. Chapter 14 is a collaboration with Michael Murray. Letter But, at night, under the full moon, they (mathematicians) dream, they float among the stars and wonder at the miracle of the heav- ens. They are inspired. Without dreams there is no art, no math- ematics, no life. - Sir Michael Atiyah. One begins the undertaking that is doctoral study with the illusion that one will produce a Great Work. One finishes stripped of those illusions. -
Monopoles and Fueter Sections on Three-Manifolds
Monopoles and Fueter Sections on Three-Manifolds ADissertationpresented by Aleksander Doan to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Stony Brook University May 2019 Stony Brook University The Graduate School Aleksander Doan We, the dissertation committe for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Simon Donaldson - Dissertation Advisor Professor, Department of Mathematics John Morgan - Chairperson of Defense Professor, Department of Mathematics Kenji Fukaya Professor, Department of Mathematics Mohammed Abouzaid Professor, Department of Mathematics, Columbia University This dissertation is accepted by the Graduate School Dean of the Graduate School ii MONOPOLES AND FUETER SECTIONS ON THREE-MANIFOLDS aleksander doan A dissertation presented for the degree of Doctor of Philosophy Department of Mathematics Stony Brook University 2019 Well-versed in the expanses that stretch from earth to stars, we get lost in the space from earth up to our skull. Intergalactic reaches divide sorrow from tears. En route from false to true you wither and grow dull. — Wisława Szymborska, To My Friends1 Dedicated to my parents and grandparents, who taught me to navigate ’the space from earth up to my skull’ 1 Poems New and Collected, transl. Stanisław Bara´nczakand Clare Cavanagh, Harcourt Inc., 1998. ABSTRACT The subject of this dissertation is a relationship between two classes of solutions to elliptic differential equations on three-manifolds: monopoles and Fueter sections. The main result is a description of a wall-crossing phenomenon for a signed count of monopoles. As a corollary, we prove the existence of Fueter sections with singularities.