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NAHM's EQUATIONS and JACOBI FORMS by Jason Quinones

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MATHEMATICAL ASPECTS OF FIELD THEORY: NAHM’S EQUATIONS AND JACOBI FORMS by Jason Quinones __________________________ Copyright © Jason Quinones 2020 A Dissertation Submitted to the Faculty of the DEPARTMENT OF MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 2020 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by: Jason Quinones titled: and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. _________________________________________________________________ Date: ____________Jul 30, 2020 Christoph Keller _________________________________________________________________ Date: ____________Jul 30, 2020 David A Glickenstein _________________________________________________________________ Date: ____________Jul 30, 2020 Sergey Cherkis Nicholas Ercolani _________________________________________________________________ Date: ____________Jul 30, 2020 Nicholas Ercolani Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. _________________________________________________________________ Date: ____________Jul 30, 2020 Christoph Keller Mathematics 3 Acknowledgements I would like to thank my supervisors Christoph Keller and Sergey Cherkis for their guidance and mentorship. I am grateful for their support and their willingness to invest time on advising me. I would also like to express my thanks to Prof. Harry Braden for helpful conservations and notes on the topic of monopoles. I am very appreciative of my fellow students for their help and encour- agement, I would like to especially thank Rebekah Cross, Matthew Wheeler, and Thomas Harris. I would like to acknowledge support from the Math Department and the Disability Resource Center here at the University of Arizona. I am especially thankful to David Glickenstein, who was of great help through administrative issues, and to Catherine Mazzola for her coordination of interpreting services. I would like to thank the interpreters Michael Friedmann, Dawn Wise, Heather Donnel, and James Carpenter for their hard work. Finally, I would like to express my gratitude and appreciation towards my family for their support. Table of Contents Abstract 7 1 Introduction 8 1.1 Overview of Nahm's Equations . .9 1.2 Atiyah's Conjecture . 16 1.3 Objectives for Nahm's Equations . 20 1.4 Motivation: Siegel Modular Forms and Black Holes . 22 1.5 Objectives for Jacobi Forms . 26 I 31 2 Nahm's Equations 32 2.1 Introduction . 32 2.2 Boundary Conditions . 34 2.2.1 The Complex Equation . 35 2.2.2 The Real Equation . 40 3 Spectral Curve 43 3.1 MiniTwistor Space . 44 3.2 Spectral Curve of Nahm data . 46 3.3 Eigenline Bundle over Spectral Curve . 50 3.4 Spectral Method for Nahm's Equations . 54 3.4.1 Solving Nahm's Equations . 54 3.4.2 Basis Constructions . 61 3.4.3 Perturbation Expansion . 70 3.5 Nahm Solutions . 74 3.5.1 n=2 Exact Solution . 75 4 TABLE OF CONTENTS 5 3.5.2 Perturbative Expansion of the n=3 Solution . 76 4 Dirac Monopole and ADHMN 80 4.1 Introduction . 80 4.2 Multimonopole Configuration . 81 4.3 Dirac Operators . 82 4.3.1 Dirac Operators Coupled to a Monopole . 82 4.3.2 Dirac Operator Coupled to a Nahm Solution . 83 4.3.3 Zero Modes of Dirac Operators . 84 4.3.4 Down Transform . 87 4.3.5 Up Transform . 89 4.4 Spectral Curve of Monopole Data . 90 4.5 Dirac Zero Modes from Rows of Polynomials . 93 5 Conclusions for Part I 98 II 100 6 Polar Terms of Weak Jacobi Forms 101 6.1 Polar Part of Weak Jacobi Forms . 101 6.1.1 A Lower Bound for P (m)................ 110 6.1.2 An Upper Bound for P (m)................ 110 7 Slow Growth around yb 117 7.1 Slow Growth Forms . 121 7.2 Theta Quotients . 133 7.3 Slow growth of single theta quotient . 136 7.4 Slow growth of multiple theta quotients . 138 8 Slow Growth around qayb 143 8.1 Overview and Results . 143 8.2 Analytical Results . 147 8.2.1 Atkin-Lehner Involution . 147 8.2.2 Index 6 . 147 8.2.3 Index 8 . 150 8.3 Numerical Results . 152 Appendix A CODE FOR IMPLEMENTATION 155 TABLE OF CONTENTS 6 References 159 Abstract This thesis consists of two projects wherein we explore some mathematical aspects of field theory. In the first project, we address Nahm's equations, which is an integrable system with a Lax pair. We consider boundary conditions on Nahm's equa- tions that correspond to the Dirac multimonopole in Yang-Mills theory. The algebro-geometric integration method is to construct solutions via a linear flow in the Jacobian of the spectral curve associated to the Lax pair. We construct a frame of sections of this linear flow, which allows us to obtain exact solutions to Nahm's equations for arbitrary rank n. Nahm's equations with our boundary conditions correspond to the Dirac multimonopole via the ADHMN construction. The ADHMN construction requires us to find normalizable zero modes of Dirac operators. We again use the frame of sections of the linear flow on the Jacobian of the spectral curve to construct these normalizable zero modes. In the second project, we consider weak Jacobi forms of weight 0. The polar coefficients of such weak Jacobi form are known to uniquely determine the weak Jacobi form, and we improve on the number of polar coefficients that determine the weight 0 form. Weak Jacobi forms of weight 0 may be exponentially lifted to Siegel mod- ular forms, which appear in the string-theory of black holes. In connection to this, the growth of a certain sum about a term qayb in the Fourier-Jacobi expansion of the underlying weak Jacobi form is of interest to us. We discover that the weak Jacobi forms which are quotients of theta functions give us a large class of forms that are slow growing about their most polar term. Additionally, the characteristics of growth behavior for a weak Jacobi form about a term yb are known, here we investigate growth behavior about an arbitrary qayb term and find several analogues. 7 Chapter 1 Introduction Gauge theory is a powerful mathematical framework for studying a wide vari- ety of physical phenomena. Its roots developed as early as the second half of the 19th century with Maxwell's equations describing electromagnetism, and entered its golden period in the 1970s and 1980s, culminating with the estab- lishment of the standard model of particle physics. However, the standard model does not adequately explain some fundamental physical phenomena, such as gravity. Attempts to go beyond the standard model have proved quite challenging, given our available methods and computing power. In- stead, we can look at simpler models in an attempt to better understand the mathematical structures at hand. It is in this spirit that this thesis is undertaken. In this thesis, we describe two different projects, unified by the common goal of understanding some mathematical aspect of simple field theories. In the first, we consider Nahm's equations, which arise from Yang-Mills gauge theory. We construct exact solutions to those equations, subject to a simple boundary condition. Chapters 2, 3, and 4 are dedicated to this project. In the second, we consider weak Jacobi forms and the growth behavior of a certain sum of their Fourier-Jacobi coefficients. This growth behavior is of interest to us as it has connections to the holographic behavior of a 2D conformal field theory. Chapters 6, 7, and 8 are dedicated to this project. In the following sections of this chapter, we elaborate on the motivations behind each project and summarize our findings. 8 CHAPTER 1. INTRODUCTION 9 1.1 Overview of Nahm's Equations Nahm's equations for antihermitian matrix-valued functions T1(s), T2(s), T3(s) with s in some interval in R are dT1 = T2T3 T3T2; ds − dT2 = T3T1 T1T3; (1.1) ds − dT3 = T1T2 T2T1: ds − Nahm's equations are an integrable system, a system of nonlinear ordi- nary differential equations with sufficiently many conserved quantities to be solved by means of algebraic geometry. The starting point for this algebro- geometric integration method is to discover a Lax representation with spec- tral parameter ζ for matrix-valued functions L and M such that the system is equivalent to d L(s; ζ) = [L(s; ζ);M(s; ζ)]: (1.2) ds This Lax equation implies the spectrum of L(s; ζ) is independent of the vari- able s, indeed trLk are the conserved quantities of the system. For Nahm's equations, ζ P1 and we obtain an algebraic curve S in T P1 2 det(η1 L(s; ζ)) = 0; (1.3) − @ for η T P1. One then defines for generic L(s; ζ) a line bundle F s over S @ζ 2 with fiber at (ζ; η) the η-eigenspace of L(s; ζ) and one obtains from the flow of L(s; ζ) a linear flow F s in the Jacobian Jac(S) [1]. The method concludes with writing a solution for the system in terms of an appropriate basis of the linear flow. For some overviews on integrable systems from this point of view, see [2] and [3]. One goal of this paper is to carry out the algebro-geometric method of integration explicitly and obtain exact solutions to Nahm's equations in boundary conditions we specify. In general, exact solutions have proved dif- ficult to construct but one classical solution is provided by the substitutions Ti(s) = fi(s)ρi for ρi a constant matrix and fi(s) a scalar function, in which CHAPTER 1.
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