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3D-3D Correspondence for Seifert Manifolds

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3D-3D Correspondence for Seifert Manifolds 3d-3d Correspondence for Seifert Manifolds Thesis by Du Pei In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2016 Defended May 26th ii © 2016 Du Pei ORCID: https://orcid.org/0000-0001-5587-6905 All rights reserved except where otherwise noted. iii To my family. iv ACKNOWLEDGEMENTS I have spent five amazingly enriching years at Caltech and I thank all of my friends who have added to my experience along this wonderful journey. First, I consider myself extremely fortunate to have Professor Sergei Gukov as my advisor. Going far beyond his duty in academic mentorship, Sergei has provided me with immeasurable help, guidance and encouragement and made every effort to carve me into a good researcher. In every way, he epitomizes the greatest virtues of an advisor and a friend. I am grateful to Professor Anton Kapustin, Yi Ni, Hirosi Ooguri, John Schwarz and Cumrun Vafa, not just for serving on my thesis/candidacy committee but also for the enlightening academic exchanges we had during the whole five years. I wish to thank my collaborators including Jørgen Ellegaard Andersen, Martin Fluder, Daniel Jafferis, Monica Kang, Ingmar Saberi, Wenbin Yan and Ke Ye. It is so fun working with you guys and I have learned so much from you! And I also want to thank Ning Bao, Francesco Benini, Sungbong Chun, Hee-Joong Chung, Tudor Dimofte, Lei Fu, Abhijit Gadde, Temple He, Enrico Herrmann, Sam van Leuven, Qiongling Li, Wei Li, Vyacheslav Lysov, Brendan McLellan, Daodi Lu, Tristan Mckinney, Satoshi Nawata, Hyungrok Kim, Murat Koloğlu, Petr Kravchuk, Tadashi Okazaki, Chris Ormerod, Tony Pantev, Chan Youn Park, Daniel S. Park, Abhishek Pathak, Ana Peón-Nieto, Jason Pollack, Pavel Putrov, Brent Pym, Grant Remmen, Tom Rudelius, Laura Schaposnik, Shu-Heng Shao, Caili Shen, Chia-Hsien Shen, Jaewon Song, Bogdan Stoica, Kung-Yi Su, Piotr Sulkowski, Yifan Wang, Richard Wentworth, Dan Xie, Masahito Yamazaki, Huang Yang, Xinyu Zhang and Peng Zhao for interesting discussions and conversations, and Carol Silberstein for helping with all kinds of logistic issues. This dissertation contains the fruits of not just my five years as a graduate student at Caltech but also the twenty-seven years since I was born, during the entirety of which I was consistently provided an extraordinary education about math, physics, nature, languages and life. There are no words that could ever describe my gratitude towards those whose effort and dedication made all of this possible. Among them, my middle school teacher Jun Shao and my high school teacher Jiangtao Gao had the greatest impact on my decision to become a physicist, and I wish to give them special thanks for setting me on this beautiful journey. v Lastly, I would like to thank my family. My dad Bailin Pei has the biggest intellectual influence on me, and it is he who taught me how to think critically, showed me the excitement of discovery and, step by step, led me onto the road of pursuing knowledge. My mom Fei Chen always gives full respect and unconditional support for my decisions. Her care, love and trust have sustained me through every difficulty and hardship. Her optimism, kindness and profound enthusiasm for life have helped shape me into the person I am. Beside my parents, I owe an equally immense debt to my wife Honggu Fan. She was on the course of becoming an exceptional scholar but, without any hesitation, chose to sacrifice her own career to ensure that I could stay fully committed to the pursuit of mine. Ever since our baby was born, she has spend every second awake doing what is the best for me and our child. I can hardly imagine anyone who could better exemplify the best attributes of a wife or these of a mother. Also, I would like to thank my parents-in-law, Hua Fan and Shifang Chen, and my cousins, aunts, uncles and grandparents for the enjoyable time we have shared. And I want to thank Rae Pei, my baby daughter, for being amazingly easygoing and agreeable when I was working on research and writing the thesis. vi ABSTRACT In this dissertation, we investigate the 3d-3d correspondence for Seifert manifolds. This correspondence, originating from string theory and M-theory, relates the dy- namics of three-dimensional quantum field theories with the geometry of three- manifolds. We first start in Chapter 2 with the simplest cases and demonstrate the extremely rich interplay between geometry and physics even when the manifold is just a 1 direct product M3 = Σ × S . In this particular case, by examining the problem from various vantage points, we generalize the celebrated relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) Chern-Simons theory on Σ × S1 and 4) the index theory of the moduli space of flat connections to a completely new set of relations between 1) the “equivariant Verlinde algebra” for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) complex Chern-Simons theory on Σ × S1 and 4) the equivariant index theory of the moduli space of Higgs bundles. In Chapter 3 we move one step up in complexity by looking at the next simplest example of M3 = L(p; 1). We test the 3d-3d correspondence for theories that are labeled by lens spaces, reaching a full agreement between the index of the 3d N = 2 “lens space theory” T[L(p; 1)] and the partition function of complex Chern-Simons theory on L(p; 1). The two different types of manifolds studied in the previous two chapters also have interesting interactions. We show in Chapter 4 the equivalence between two seemingly distinct 2d TQFTs: one comes from the “Coulomb branch index” of the class S theory on L(k; 1) × S1, the other is the “equivariant Verlinde formula” on Σ × S1. We check this relation explicitly for SU(2) and demonstrate that the SU(N) equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg dualities. In the last chapter, we directly jump to the most general situation, giving a proposal for the 3d-3d correspondence of an arbitrary Seifert manifold. We remark on the huge class of novel dualities relating different descriptions of T[M3] with the same M3 and suggest ways that our proposal could be tested. vii PUBLISHED CONTENT AND CONTRIBUTIONS This thesis is based on preprints [1], [2] and [3], which are adapted for Chapters 2, 3 and 4 respectively, as well as ongoing work such as [4] and [5]. Each of them is the outcome of active and fruitful interactions between me and my collaborators. [1] Sergei Gukov and Du Pei. “Equivariant Verlinde formula from fivebranes and vortices” (2015). arXiv: 1501.01310 [hep-th]. url: http://arxiv. org/abs/1501.01310. [2] Du Pei and Ke Ye. “A 3d-3d appetizer” (2015). arXiv: 1503 . 04809 [hep-th]. url: http://arxiv.org/abs/1503.04809. [3] Sergei Gukov, Du Pei, Wenbin Yan, and Ke Ye. “Equivariant Verlinde alge- bra from superconformal index and Argyres-Seiberg duality” (2016). arXiv: 1605.06528 [hep-th]. url: http://arxiv.org/abs/1605.06528/. [4] Daniel Jafferis, Sergei Gukov, Monica Kang, and Du Pei. “Chern-Simons theory at fractional level” (work in progress). [5] Jørgen Ellegaard Andersen and Du Pei. “Verlinde formula for Higgs bundle moduli spaces” (work in progress). viii TABLE OF CONTENTS Acknowledgements . iv Abstract . vi Published Content and Contributions . vii Table of Contents . viii List of Illustrations . ix List of Tables . x Chapter I: Introduction . 1 1 Chapter II: M3 = Σ × S and the equivariant Verlinde formula . 4 2.1 From the Verlinde formula to its equivariant version . 4 2.2 Fivebranes on Riemann surfaces and 3-manifolds . 9 2.3 Branes and vortices . 16 2.4 Equivariant integration over Hitchin moduli space . 24 2.5 β-deformed complex Chern-Simons . 29 2.6 A new family of 2d TQFTs . 37 2.7 t-deformation and categorification of the Verlinde algebra . 49 Chapter III: M3 = L(p; 1) and a “3d-3d appetizer” . 63 3.1 Testing the 3d-3d correspondence . 63 3.2 Chern-Simons theory on S3 and free chiral multiplets . 65 3.3 3d-3d correspondence for lens spaces . 70 Chapter IV: When a lens space talks to Σ × S1 . 84 4.1 Equivalence between two TQFTs . 84 4.2 Equivariant Verlinde algebra and Coulomb branch index . 87 4.3 A check of the proposal . 102 4.4 SU(3) equivariant Verlinde algebra from the Argyres-Seiberg duality 107 Chapter V: Generalization and discussion . 126 5.1 Going one step further . 126 5.2 The theory L(p; q) and its dualities . 127 5.3 Coupling lens space theories to the star-shaped quiver . 130 Appendix A: . 133 A.1 Complex Chern-Simons theory on lens spaces . 133 ix LIST OF ILLUSTRATIONS Number Page 2.1 A genus-2 Riemann surface decomposed into two pairs of pants. 8 2.2 CP1 as a circle fibration. 17 2.3 The lens space L(k; 1) as a torus fibration. 17 2.4 The NS5-D3-(1,k) brane system in type IIB string theory. 18 2.5 The (1,k)-brane in figure 2.4 as a bound state of an NS5-brane and k D5-branes. 20 2.6 The NS5-D2-NS5-D4 brane system in Type IIA string theory ob- tained by dimensionally reducing the system in figure 2.4. 21 2.7 The “fusion tetrehedron”. 57 4.1 Illustration of Argyres-Seiberg duality. 109 4.2 Illustration of geometric realization of Argyres-Seiberg duality for T3 theory.
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