Topological string theory and applications Zhihao Duan To cite this version: Zhihao Duan. Topological string theory and applications. Mathematical Physics [math-ph]. PSL Research University, 2019. English. NNT : 2019PSLEE011. tel-02341882 HAL Id: tel-02341882 https://tel.archives-ouvertes.fr/tel-02341882 Submitted on 31 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prepar´ ee´ a` l’Ecole´ Normale Superieure´ Topological string theory and applications Soutenue par Composition du jury : Zhihao Duan Stefan Hohenegger Le 8 juillet 2019 Universite´ Claude Bernard Lyon 1 Rapporteur Amir-Kian Kashani-Poor Ecole´ Normale Superieure´ Directeur de these` Ecole´ doctorale no564 Kimyeong Lee Physique en ˆIle-de-France Korea Institute for Advanced Study Examinateur Andre Lukas University of Oxford Rapporteur Boris Pioline Specialit´ e´ Universite´ Paris-Sorbonne President´ Physique Theorique´ Abstract This thesis focuses on various applications of topological string theory based on different types of Calabi-Yau (CY) manifolds. The first type considered is the toric CY manifold, which is intimately related to spectral problems of difference operators. The particular example considered in the thesis closely resembles the Harper-Hofstadter model in con- densed matter physics. We first study the non-perturbative sectors in this model, and then propose a new way to compute them using topological string theory. In the second part of the thesis, we consider partition functions on elliptically fibered CY manifolds. These exhibit interesting modular behavior. We show that for geometries which do not lead to non-abelian gauge symmetries, the topological string partition functions can be reconstructed based solely on genus zero Gromov-Witten invariants. Finally, we discuss ongoing work regarding the relation of the topological string partition functions on the so-called Higgsing trees in F-theory. 1 Acknowledgements First of all, I’m always in debt to my thesis advisor, Amir-Kian Kashani-Poor, who has taught me not only both mathematics and physics, but also influenced deeply my way of doing research. During numerous discussions, I’m especially grateful for his patience when answering my stupid questions and emphasis on the importance of looking at examples and doing computations. I also thank him for giving feedbacks on my thesis. I would like to thank my rapporteurs Stefan Hohenegger and Andre Lukas for spending their time reading my manuscript, and to Kimyeong Lee and Boris Pioline as well for accepting to be my jury members. I would like to thank Jie Gu, Yasuyuki Hatsuda and Tin Sulejmanpasic for the collabo- ration on my first paper, during which I have learnt quite a lot. I’m grateful to all of them for their amazing work and writing the reference letter for me. Besides, I appreciate Jie and Tin’s continuous encouragement during the most difficult period of my thesis. I‘m grateful to the Laboratoire de Physique Th´eoriquede l’ENS´ (now part of LPENS) for providing a stimulating and comfortable atmosphere. I benefit a lot from daily con- versations with permanent members such as Constantin Bachas, Camille Aron, Benjamin Basso, Fernandes Paulos Miguel, Giuseppe Policastro and Jan Troost. As for other fellows at the LPT, to start with I feel obliged to mention my office mates Tony Jin, Alexandre Krajenbrink, Ioannis Lavdas and Deliang Zhong. We started the PhD at the same time, share the same office and now we are all leaving for our Postdoc. Together with Dongsheng Ge and Songyuan Li, I would definitely remember all the fun and joy we had by heart. I wish all my best to Dongsheng and Songyuan, who will surely find fantastic Postdoc positions next year. Besides, I thank Cl´ement Le Priol, Gwenael Ferrando, Etienne Granet for all the meals we had together and teaching me a lot of French. I would also like to thank Louise Budzynski, Arnaud Fanthomme, Bruno Le Floch, Tristan Gautie, Apratim Kaviraj, Augustin Lafay and Takahiro Nemoto, with whom my life at the LPT becomes even more colorful. I would like to thank my dear friends in Paris, including Guilhem Brunet, Yichang Cai, Peng Cheng, Yawen Cheng, Yifan Chen, Ji Dai, Senwen Deng, Hao Fu, Louis Gar- rigue, Jewel Kumar Ghosh, Mengzi Huang, Zhining Liu, Ciambelli Luca, Charles Marteau, David Martin, Jian Qian, Zicheng Qian, Jiaxin Qiao, Liang Qin, Yichen Qin, Qiuyang Qu, Mengxi Ren, Martin Rieu, Assaf Schapira, Steven Schulz, Jing Tian, Alexandra Vino- 2 gradova, Benjamin Walter, Keyu Wang, Songbo Wang, Shiyi Wei, Baojun Wu, Jimin Wu, Shengquan Xiang, Fan Yang, Kang Yang, Yizhen Zhao, Chao Zhang, Hao Zhang, Haowen Zhang, Yi Zhang, Tunan Zhu... I still remember all the happy moment we had together, and with your company I never feel lonely. Finally, I would like to thank my parents, for their love and unconditional support. i Contents 1 Introduction1 1.1 String theory as a candidate for a “theory of everything”..........1 1.2 Topological string theory as a bridge between mathematics and physics..4 2 Topological Field Theory8 2.1 Two dimensional N = (2, 2) supersymmetry.................8 2.2 Topological field theories of Witten type................... 12 2.3 Topological twisting............................... 14 2.4 Interlude: moduli space of complex structures................ 22 3 Topological String Theory: Coupling to Gravity 25 3.1 Topological string theory............................ 25 3.2 Mirror symmetry................................ 33 4 Resurgence and Quantum Mirror Curve: A Case Study 39 4.1 The Harper-Hofstadter problem........................ 40 4.2 Trans-series expansion and one-instanton sector............... 46 4.3 Two-instanton sector.............................. 54 4.4 Instanton fluctuation from topological string................. 59 4.5 Interlude: holomorphic anomaly at work................... 64 5 Elliptic Genera and Topological Strings: Overview 67 5.1 Elliptic fibrations and four-cycles....................... 68 5.2 The base degree k partition function as Jacobi form............. 72 6 Geometries without codimension-one singular fibers: Reconstruction 77 6.1 The structure of the topological string free energy.............. 78 6.2 Zk of negative index.............................. 87 6.3 Partition function from genus zero GW Invariants.............. 89 7 Geometries with codimension-one singular fibers: Higgsing Trees 100 7.1 Higgsing trees.................................. 100 ii CONTENTS 7.2 a2 model..................................... 102 7.3 g2 model..................................... 105 8 Conclusion and Outlook 108 A Modular Forms 110 A.1 Elliptic modular forms............................. 110 A.2 Jacobi modular forms.............................. 113 A.3 Weyl-invariant Jacobi forms.......................... 116 B Toric Geometries 121 B.1 Fans....................................... 121 B.2 Polytopes.................................... 124 B.3 Blow-ups..................................... 126 B.4 Toric Diagrams................................. 128 C Computations in One-Instanton Sector 132 C.1 Instanton solution................................ 132 C.2 The moduli-space metric............................ 133 C.3 The one-instanton determinant........................ 134 D Principal Parts of Z3 and Z4 for the Massless E-string 139 D.1 Base degree 3.................................. 139 D.2 Base degree 4.................................. 141 E Some Enumerative Invariants 143 E.1 Genus zero GW invariants for massless E strings............... 143 E.2 GV invariants for a2 and g2 models...................... 143 Bibliography 148 iii Chapter 1 Introduction 1.1 String theory as a candidate for a “theory of everything” The Standard Model of fundamental particles is one of the most successful theories in physics. Based on the quantum theory of gauge fields, it provides an extremely precise account of the behavior of nature at length scales down to at least l ≈ 1 TeV−1. With the discovery of the long-anticipated Higgs particles at the LHC in 2012, it seems that finally the dust has settled. However, the Standard Model in fact is still far from a microscopically complete theory to describe our world. As is well known, it can only incorporate three out of four funda- mental forces in nature. In other words, gravity, the fundamental interaction that plays a crucial role in our daily life, is still left untouched. Gravity is known to be best described by Einstein’s theory of General Relativity. Therefore, one might be tempted to simply unify these two successful theories. However, naive attempts to construct a field theory model for gravity fail for the following reason. According to our current understanding, the Standard Model is an effective theory. This means that we first start from a very high energy, say the Planck scale, then lower the energy level down to the energy scale accessible to large colliders. Through this procedure, high energy modes in the system are integrated out, and if the outcome can be absorbed into redefinition of certain parameters, it is still under control. We call this type of theories