Yukawa Couplings from D-branes on non-factorisable Tori Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Chatura Christoph Liyanage aus Bonn Bonn, 2018 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Priv. Doz. Dr. Stefan Förste 2. Gutachter: Prof. Dr. Albrecht Klemm Tag der Promotion: 05.07.2018 Erscheinungsjahr: 2018 CHAPTER 1 Abstract In this thesis Yukawa couplings from D-branes on non-factorisable tori are computed. In particular intersecting D6-branes on the torus, generated by the SO(12) root lattice, is considered, where the Yukawa couplings arise from worldsheet instantons. Thereby the classical part to the Yukawa couplings are determined and known expressions for Yukawa couplings on factorisable tori are extended. Further Yukawa couplings for the T-dual setup are computed. Therefor three directions of the SO(12) torus are T-dualized and the boundary conditions of the D6-branes are translated to magnetic fluxes on the torus. Wavefunctions for chiral matter are calculated, where the expressions, known from the factorisable case, get modified in a non-trivial way. Integration of three wavefunctions over the non-factorisable torus yields the Yukawa couplings. The result not only confirms the results from the computations on the SO(12) torus, but also determines the quantum contribution to the couplings. This thesis also contains a brief review to intersecting D6-branes on Z2 ×Z2 orientifolds, with applications to a non-factorisable orientifold, generated by the SO(12) root lattice. 3 Acknowledgements First of all I would like to thank my supervisor Priv.Doz. Stefan Förste for giving me the opportunity to work in this fascinating field of theoretical physics and for all the numerous discussions, which lead to fruitful collaborations. I am very grateful for his continuous support and unlimited patience. I could not have asked for more. I also thank Prof. Albrecht Klemm for having accepted to be the coreferee for my thesis. Further I would also like to thank Prof. Hans Peter Nilles for his great lectures, which arouse my interest for theoretical high energy physics. The past years in the BCTP have been a very nice time I gladly will remember. In this connection I want to thank Nana Cabo Bizet, Michael Blaszczyk, Stefano Colucci, Josua Faller, Cesar Fierro Cota, Andreas Gerhardus, Suvendu Giri, Abhinav Joshi, Joshua Kames-King, Dominik Köhler, Manuel Krauss, Sven Krippendorf, Victor Martin-Lozano, Damian Mayorga, Rahul Mehra, Christoph Nega, Saurab Nangia, Urmi Ninad, Paul Oehlman, Jonas Reuter, Fabian Rühle, Reza Safari, Thorsten Schimannek, Matthias Schmitz, Andreas Trautner, Clemens Wieck, Max Wiesner and all former and current members of the BCTP not only for their support and inspiring discussions but also for the providing a friendly atmosphere. I also thank Andreas Wisskirchen and the secretaries of the BCTP Christa Börsch, Dagmar Fassbender, Petra Weiss and Patricia Zündorf for the organizational and technical support. I will look back with pleasure to the "coffee breaks" and "profound" discussions with Joshua Kames-King, Christoph Nega, Thorsten Schimannek, Max Wiesner. I also want to thank Joshua Kames-King, Matthias Schmitz and Thorsten Schimannek for the great "gym" time. I also like to thank my friend and fellow student Tarek El Rabbat for all the nice and memorable time we shared since the beginning of our studies in Bonn. Further I also want to thank my former physics teacher Mr. Pick for having awoken my interest for fundamental physics and laying the foundation for my studies. I am also grateful for my friends and teammates from Clube do Leao and also Sven Freud and Hussein Al Abad for the strengthening time outside of the institute. Special thanks goes to my girlfriend Jana Lisa Wolff, who accompanied and supported me since the beginning of my time in the BCTP, and my brother Vidura Lawrence Liyanage, for his support and for proofreading the manuscript of this thesis. Most of all I thank my parents Padma and Lionel Liyanage for putting enormous effort in my education and giving me endless support in life, even though words can not express the deep gratitude I have for them. 5 Contents 1 Abstract 3 2 Introduction 1 3 Overview to string theory7 3.1 Superstrings . .7 3.1.1 Worldsheet and superstring action . .7 3.1.2 String quantization and D=10 string states . .9 3.2 Type II strings . 12 3.2.1 Modular invariance and GSO-projection . 12 3.2.2 Open strings and D-branes . 14 3.2.3 Circle compactification and T-duality . 16 4 Type IIA compactification on orientifolds 19 4.1 Geometry of orientifolds . 19 4.1.1 Torus . 19 4.1.2 Orientifolds for type IIA compactification . 23 4.1.3 Fixed point resolution . 27 4.1.4 3-cycles on tori and orbifolds . 29 6 × × 4.1.5 Example: TSO(12)= (Z2 Z2 ΩR) ........................ 33 4.2 Intersecting D-branes on Z2 × Z2 × ΩR-Orientifolds . 40 4.2.1 Massless states from type IIA closed strings . 41 4.2.2 Massless spectrum from intersecting D6-branes . 43 4.2.3 Consistency conditions and anomaly cancellation for D6-branes . 47 6 × × 4.3 Model building on TSO(12)=(Z2 Z2 ΩR)........................ 49 4.3.1 Towards realistic four dimensional particle physics . 49 6 × × 4.3.2 Supersymmetric toy model on TSO(12)=(Z2 Z2 ΩR)............. 52 5 Yukawa couplings from D6-branes on T6 55 SO(12) 5.1 Yukawa couplings from D6-branes . 55 5.1.1 Yukawa couplings from worldsheet instantons . 55 5.1.2 Yukawa couplings on the torus . 56 5.2 Yukawa couplings on T 2 ................................. 57 5.2.1 Computing Yukawa couplings on T 2 ....................... 57 5.2.2 Example: Branes with non coprime intersection numbers in T 2 ......... 62 6 5.3 Yukawa couplings on TSO(12) ............................... 64 5.3.1 Labeling inequivalent intersections . 64 6 5.3.2 Computing Yukawa couplings on TSO(12) .................... 66 7 5.3.3 Example . 70 6 Yukawa couplings from D9-branes on the dual T6 77 SO(12) 6 6 6.1 From TSO(12) to the dual TSO(12) ............................. 77 6.1.1 Buscher rules . 77 6 6.1.2 T-dualizing TSO(12) with D6-branes . 79 6 6.2 Magnetic fluxes on the dual TSO(12) ............................ 82 6.2.1 Symmetry breaking via magnetic fluxes . 82 6.2.2 Wilson loops and quantization condition . 83 6.2.3 Boundary conditions for bifundamentals . 86 6.3 Wavefunctions for chiral matter on the T 6 ........................ 90 6.3.1 Massless chiral fermions . 90 6.3.2 Light chiral scalars . 93 6.3.3 Massless chiral fields and Wilson lines . 95 6.3.4 Counting numbers of independent zeromodes . 96 6.3.5 Normalization factor for chiral wavefunctions . 99 6.4 Yukawa couplings from overlapping wavefunctions . 102 6.4.1 Yukawa couplings from magnetic fluxes . 102 6.4.2 Computing Yukawa couplings . 103 6.4.3 Quantum contribution to Yukawa couplings . 109 7 Conlusion 111 Bibliography 113 A Massless type IIA closed string states on Z2 × Z2 × ΩR-orientifolds 121 A.1 Four dimensional fields from massless type IIA strings . 121 A.1.1 Type IIA compactified on a T 6 .......................... 122 A.2 Z2 × Z2-orbfiold projection . 124 A.2.1 Untwisted states . 124 A.2.2 Twisted states . 126 A.3 Orientifold projection . 128 A.3.1 Untwisted states . 128 A.3.2 Twisted states . 129 B Labels for intersection points on T6 131 SO(12) C Quotient lattices and integral matrices 137 D Lattices for gauge indices and irreducible subsets 139 List of Figures 143 List of Tables 145 8 CHAPTER 2 Introduction Motivation Physics at the microscopic level is described with great accuracy by the Standard Model (SM) of particle physics. The embedding space for particles of the SM is given by a four dimensional Minkowski space with an SU(3)C × SU(2)L × U(1)Y (2.1) gauge symmetry. Elementary particles are treated as irreducible representations of their little group and gauge groups and are described as fields of a quantum field theory [1–3]. To preserve the gauge symmetry all fields must be apriori massless and the algebra for the little group of massless particles in four dimensional Minkowski space is su(2) × su(2). It is distinguished between matter fields and gauge fields, where the first ones have spin 1/2 and the second ones have spin 1. Matter fields occur in three families of the following bifundamental representations of SU(3) × SU(2) (1; 2) ⊕ (3; 2) ⊕ (1; 1) ⊕ (1; 1) ⊕ 3; 1 ⊕ 3; 1 ; (2.2) −1=2 1=6 1 0 −2=3 1=3 where the subscript denotes the U(1)Y charge. The fields belonging to (1; 2)−1=2 ⊕ (3; 2)1=6 transform in 1 1 the chiral representation ( 2 ; 0) of su(2) × su(2) and the other fields in the antichiral representation (0; 2 ). Further matter fields charged under the SU(3) are identified with quarks, where fields not charged under 1 1 the SU(3) are called leptons. Gauge fields transform in the vector representation ( 2 ; 2 ) of their little group and in the adjoint representation of the gauge groups. Hence the SM contains 12 gauge fields belonging to the representation (1; 1)0 ⊕ (1; 3)0 ⊕ (8; 1)0 (2.3) of SU(3) × SU(2) × U(1). In order to allow the particles to gain mass, the SU(2)L × U(1)Y factor hast to be spontaneously broken to a U(1)el gauge symmetry.