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Braiding Knots with Topological Strings

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Braiding Knots with Topological Strings Braiding Knots with Topological Strings Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn von Jie Gu aus Ningbo, China Bonn, March 2015 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Albrecht Klemm 2. Gutachter: Priv. Doz. Stefan F¨orste Tag der Promotion: 12.08.2015 Erscheinungsjahr: 2015 Abstract For an arbitrary knot in a three{sphere, the Ooguri{Vafa conjecture associates to it a unique stack of branes in type A topological string on the resolved conifold, and relates the colored HOMFLY invariants of the knot to the free energies on the branes. For torus knots, we use a modified version of the topological recursion developed by Eynard and Orantin to compute the free energies on the branes from the Aganagic{Vafa spectral curves of the branes, and find they are consistent with the known colored HOMFLY knot invariants `ala the Ooguri{Vafa con- jecture. In addition our modified topological recursion can reproduce the correct closed string free energies, which encode the information of the background geometry. We conjecture the modified topological recursion is applicable for branes associated to hyperbolic knots as well, encouraged by the observation that the modified topological recursion yields the correct planar closed string free energy from the Aganagic{Vafa spectral curves of hyperbolic knots. This has implications for the knot theory concerning distinguishing mutant knots with colored HOMFLY invariants. Furthermore, for hyperbolic knots, we present methods to compute colored HOM- FLY invariants in nonsymmetric representations of U(N). The key step in this computation is computing quantum 6j{symbols in the quantum group (sl ). Uq N iii Acknowledgements I will always be in debt to my direct supervisor Dr. Hans Jockers, without whose continuous support and encouragement I could not have made the transition from an astroparticle physicist to a string theoretician, gone through the PhD studies, and in the end finished this dissertation. I am grateful for his seemingly inexhaustible patience to bear with my numerous and sometimes stupid questions. I also thank him for proofreading my thesis. I would like to thank my supervisor Prof. Dr. Albrecht Klemm, who not only provided financial support for my PhD studies and various academic trips, but also inspired me through his impressive passions for researches and his insights. I would like to thank Prof. Manuel Dress. He kindly agreed to write the third reference letter for my post{doctoral application, and also gave me many good advices concerning the BCGS (Bonn{Cologne Graduate School) program. I would like to thank Priv. Doz. Stefan F¨orsteto be my second thesis referee, as well as Prof. Dr. Ian Brock, Prof. Dr. Daniel Huybrechts for taking time to be on my thesis committee. I would like to thank Dr. Masoud Soroush for the collaboration on my first string paper. In addition I am grateful for him inviting me to give a talk at Brandeis and his hospitality during my visit to Boston. I thank Jonas Reuter as well as Prof. Dr. Marcos Mari~noin Geneva for an interesting collaboration. I would also like to thank many other fellows in the BCTP (Bethe Center for Theoretical Physics) for many warm memories we shared together: for instance, the Wednesday seminars, the company of Thorsten Schimannek and Pwl Oilmann in the dead of many nights in the institute, and many a nonacademic discussion including some very cunning plans with Raghuveer Garani. During my PhD studies, I was supported by the Honors Branches of the BCGS program, and the successive secretaries of the BCGS program were all very helpful, especially Frau Doris Thrun, who has now retired. I was also very lucky to have the service of the amazing group of the BCTP secretaries, and all of their names deserve to be spelled out: Christa B¨orsch, Dagmar Fassbender, Petra Weiß, and Patricia Z¨undorf. I also should not forget the IT magician in the institute, Dr. Andreas Wisskirchen. I leave my last thanks to my parents, to whom I owe everything. v Contents 1 Introduction 1 1.1 String Theory as a Logical Next Step . .1 1.2 A Useful Exercise of Topological String . .4 2 Topological String and Field Theories 7 2.1 2d Topological Sigma Models . .7 2.1.1 2d = (2; 2) Nonlinear Sigma Model . .7 N 2.1.2 Topological Twists and Topological QFT of the Witten Type . 10 2.1.3 Topological A{Twist . 13 2.1.4 Topological B{Twist . 16 2.1.5 A Diversion: Complex Structure Moduli Space of Calabi{Yau Threefold . 18 2.2 Topological String Theory . 21 2.2.1 General Ideas . 21 2.2.2 Type A Closed String . 23 2.2.3 Type A Open String . 26 2.2.4 Type B Closed String . 30 2.2.5 Type B Open String . 33 2.2.6 Mirror Symmetry . 33 2.3 Chern{Simons Theory . 37 2.3.1 Basic Definitions . 37 2.3.2 HOMFLY Knot Invariants . 39 2.3.3 Surgery and Nonperturbative Solutions . 40 2.3.4 Symmetry Properties . 44 2.3.5 Free Energies 1=N Expansion . 45 2.3.6 Relation to Topological String Theory . 46 2.4 Matrix Models . 47 2.4.1 Basic Definitions . 47 2.4.2 Saddle Point Approximation . 50 2.4.3 Relation to Chern{Simons Theory . 53 2.4.4 Relation to Topological String Theory . 55 2.4.5 Large N Duality . 56 2.5 Topological Recursion . 58 2.5.1 Loop Equations and Their Solutions . 58 2.5.2 Topological Recursion and Symplectic Invariants . 60 2.5.3 Properties of Symplectic Invariants . 63 2.5.4 Resolvents, Unknot, and Special Lagrangian Branes . 66 2.5.5 The BKMP Theorem . 68 2.5.6 Ooguri{Vafa Conjecture and Open Problems . 70 vii 3 Topological String and Knot Invariants 71 3.1 Spectral Curves . 71 3.1.1 Brini{Eynard{Mari~noSpectral Curve . 71 3.1.2 Aganagic{Vafa Spectral Curve . 73 3.2 Annulus Kernel . 80 3.2.1 Problems with Bergman Kernel . 80 3.2.2 Constructing the Annulus Kernel . 84 3.2.3 Calibration and Poles . 86 3.2.4 Stretched Annulus Instantons . 89 3.3 Modified Topological Recursion . 91 3.3.1 Normalization and Free Energies . 91 3.3.2 Computing Correlation Differentials . 93 3.3.3 Computing Free Energies . 95 3.4 Implications for and from Knot Theory . 96 4 Knot Invariants and RCFT 99 4.1 Braiding Method . 100 4.1.1 Basic Ideas . 100 4.1.2 Eigenstates of Braiding Operators . 101 4.1.3 Summary . 107 4.2 Quantum Group and Quantum 6j{Symbols . 109 4.3 Computing 6j{Symbols . 115 4.3.1 Phase Convention . 117 4.3.2 Eigenvector Method . 118 4.3.3 Bootstrap: Reduction to Cores . 119 4.3.4 Bootstrap: Crushing the Cores . 121 4.4 Results . 124 4.4.1 Quantum 6j{Symbols of the Two Kinds . 124 4.4.2 HOMFLY Invariants Colored in (2,1) Representation . 128 5 Conclusions and Prospect 135 A Composite Representations of U(N) 137 B Calibrated Annulus Kernel 141 Bibliography 145 List of Figures 159 List of Tables 161 viii CHAPTER 1 Introduction 1.1 String Theory as a Logical Next Step The Standard Model of fundamental particles is one of the most successful theory of physics. On the one hand, it is a very comprehensive theory, covering length scales from down to the size of quarks up to the macroscopic world. On the other hand, it is also extremely accurate. For almost every experiment particle physicists could think of, the Standard Model can produce results consistent with the measurements, often with stunningly high precisions. With the discovery of the Higgs{like particle at the LHC in 2012, the last piece of the Standard Model fell into place. However, the Standard Model is short of being a complete fundamental theory of physics. As a relativistic quantum field theory, the Standard Model incorporates only three out of the four fundamental interactions, leaving out the gravitational interaction. When the energy scale of the theory is pushed toward the Planck scale, the quantum fluctuation of matter fields would become so violent that the spacetime structure, due to the laws of general relativity, would be induced into a state of wild quantum fluctuation as well. It is therefore logically necessary1 to develop a quantum theory for the gravitational field at high energy scale as part of a complete fundamental theory, of which the Standard Model is only the low energy approximation. However the marriage between quantum field theory and general relativity faces great problems, as the Hilbert{Einstein action of the gravitational field in general relativity is non{renormalizable, and as a consequence a naive quantum field theory of gravity would lose predicative power. Therefore to quantize general relativity, one has to generalize the Standard Model, or rather quantum field theories in general, by relaxing some of the fundamental assumptions. One way to generalize the Standard Model is to enhance its symmetry. The Lie algebra of symmetry the S{matrices respect in a relativistic quantum field theory has to be the direct product of the Poincar´ealgebra and some Lie algebra describing the internal symmetry, whose generators are Lorentzian scalars [1]. One possible way of enhancing the symmetry is to allow graded Lie algebra, and the only consistent graded Lie algebra extension is the supersymmetry algebra [2]. A The supersymmetry generators Qα convert a bosonic field to a fermionic field and vice versa.
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