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Symmetries, Anomalies and Duality in Chern-Simons Matter Theories

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Symmetries, Anomalies and Duality in Chern-Simons Matter Theories Symmetries, Anomalies and Duality in Chern-Simons Matter Theories Po-Shen Hsin A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Nathan Seiberg September 2018 c Copyright by Po-Shen Hsin, 2018. All rights reserved. Abstract This thesis investigates the properties of quantum field theory with Chern-Simons interac- tion in three spacetime dimension. We focus on their symmetries and anomalies. We find many theories exhibit the phenomenon of duality - different field theories describe the same long-distance physics, and we will explore its consequence. We start by discussing Chern-Simons matter dualities with unitary gauge groups. The theories can couple to background gauge field for the global U(1) symmetry, and we produce new dualities by promoting the fields to be dynamical. We then continue to discuss theories with orthogonal and symplectic gauge groups and their dualities. For the orthogonal gauge algebra there can be discrete levels in addition to the ordinary Chern-Simons term, and the dualities require specific discrete levels as well as precise global forms of the gauge groups. We present several consistency tests for the dualities, such as consistency under deformation by the mass terms on both sides of the duality. When the matter fields are heavy the dualities reduce at long distance to the level-rank dualities between Chern-Simons theories, which we prove rigorously. We clarify the global form of the gauge groups, and we show the level-rank dualities hold generally only between spin topological quantum field theories. Next, we apply the dualities to describe the symmetry that only emerges in the infrared. For example, we argue that quantum electrodynamics (QED3) with two fermions has an emergent unitary O(4) symmetry. Similarly, we argue U(1)2 coupled to one scalar with the Wilson-Fisher interaction has an emergent SO(3) symmetry. We also investigate the microscopic symmetry and match its `t Hooft anomaly across the duality, thus providing another evidence for the duality. Finally, we comment on the time-reversal symmetry T in three spacetime dimension. We find examples where the square of T does not equal the fermion parity, but instead it iii is modified by the Z2 magnetic symmetry. This occurs in QED3 with two fermions. We also clarify the dynamics of QED3 with fermions of higher charges using the conjectured dualities. The results are generalized to theories with SO(N) gauge group. iv Acknowledgements I would like to thank the Physics department of Princeton University for support. I would like to thank my parents, without whom my life would not be possible. I would like to thank my advisor Nathan Seiberg for discussion and guidance. I would also like to thank Francesco Benini, Clay C´ordova for fruitful collaborations and many helpful discussions. I would like to thank my dissertation committee. I would like to thank Herman Verlinde for reading the manuscript. I would like to thank my friends and colleagues in the Physics Department and In- stitute of Advanced Study: Yuntao Bai, Ksenia Bulycheva, Laura Chang, Shai Chester, Will Coulton, Kolya Dedushenko, Kenan Diab, Yale Fan, Lin Fei, Tong Gao, Huan He, Luca Iliesiu, Jiming Ji, Jiaqi Jiang, Vladimir Kirilin, Dima Krotov, Hotat Lam, Aitor Lewkowycz, Aaron Levy, Jeongseog Lee, Jingjing Lin, Jingyu Luo, Zheng Ma, Lauren Mc- Gough, Alexey Milekhin, Kantaro Ohmori, Sarthak Parikh, Pavel Putrov, Shu-Heng Shao, Zach Sethna, Yu Shen, Siddharth Mishra Sharma, Joaquin Turiaci, Juven Wang, Jie Wang, Xin Xiang, Zhenbin Yang, Junyi Zhang and Yunqin Zheng. I would like to thank Cather- ine M. Brosowsky and Lisa Fleischer for dealing with the administration. I would like to thank Hsueh-Yung Lin for discussions about mathematics. I would also like to thank my two sisters, and my friends Yu-Hsuan Cheng, Chieh-Hsuan Kao and Ya-Ling Kao, for their friendship and support. v To my parents. vi Contents Abstract......................................... iii Acknowledgements...................................v 1 Introduction and Overview1 2 Level-rank duality and Chern-Simons Matter Theories 10 2.1 Preliminaries................................... 17 2.1.1 A brief summary of spinc for Chern-Simons theory.......... 17 2.1.2 Some facts about U(N)K;L ........................ 19 2.1.3 An almost trivial theory U(N)1 ..................... 22 2.1.4 A useful fact................................ 23 2.1.5 SU(N)K .................................. 25 2.2 Simple explicit examples of level/rank duality................. 27 2.2.1 A simple Abelian example U(1)2 ! U(1)−2 .............. 28 2.2.2 A simple non-Abelian example SU(N)1 ! U(1)−N ......... 30 2.2.3 Gauging the previous example U(N)1;±N+1 ! U(1)−N∓1 ...... 31 2.3 Level/rank duality................................ 32 2.3.1 The basic non-Abelian level/rank duality................ 32 vii 2.3.2 SU(N)K ! U(K)−N .......................... 35 2.3.3 U(N)K;N+K ! U(K)−N;−(N+K) .................... 37 2.3.4 U(N)K;K−N ! U(K)−N;K−N ...................... 38 2.4 Boson/fermion duality in Chern-Simons-Matter theories............ 39 2.5 New boson/boson and fermion/fermion dualities................ 45 2.5.1 Boson/boson dualities.......................... 45 2.5.2 Fermion/fermion dualities........................ 46 2.5.3 Self-duality of QED with two fermions................. 47 2.6 Appendix A: Often used equations....................... 51 3 Chern-Simons-matter dualities with SO and Sp gauge groups 53 3.1 Dualities between USp(2N) Chern-Simons-matter theories.......... 57 3.1.1 RG flows.................................. 60 3.1.2 Coupling to background gauge fields.................. 61 3.1.3 Small values of the parameters...................... 63 3.1.4 New fermion/fermion and boson/boson dualities............ 63 3.2 Dualities between SO(N) Chern-Simons-matter theories........... 65 3.2.1 Flows................................... 66 3.2.2 Global symmetries............................ 67 3.2.3 Small values of N and k ......................... 72 3.2.4 The k = 1 case.............................. 74 3.2.5 The N = 1 case.............................. 75 3.2.6 The k = 2 case.............................. 75 3.2.7 The N = 2 case.............................. 76 viii 3.3 Relation to theories of high-spin gravity.................... 77 3.4 Level-rank dualities with orthogonal and symplectic groups.......... 78 3.4.1 Level-rank dualities of 3d TQFTs.................... 78 3.4.2 Matching the symmetries......................... 80 3.4.3 Level-rank dualities of spin-TQFTs................... 85 3.4.4 More non-spin level-rank dualities.................... 88 3.5 T -invariant TQFTs from level-rank duality................... 89 3.6 Appendix A: Notations and useful facts about Chern-Simons theories.... 90 4 Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups 93 4.1 Chern-Simons Theories with Lie Algebra so(N)................ 104 4.1.1 Groups, Bundles, and Lagrangians................... 104 4.1.2 Ordinary Global Symmetries and Counterterms............ 110 4.1.3 One-form Global Symmetries...................... 115 4.1.4 't Hooft Anomalies of the Global Symmetries............. 118 4.1.5 Chiral Algebras.............................. 120 4.2 Level-Rank Duality................................ 122 4.2.1 Conformal Embeddings and Non-Spin Dualities............ 123 4.2.2 Level-Rank Duality for Spin Chern-Simons Theory.......... 130 4.2.3 Consistency Checks............................ 133 4.3 Chern-Simons Matter Duality.......................... 136 4.3.1 Fermion Path Integrals and Counterterms............... 136 4.3.2 Dualities with Fundamental Matter................... 143 ix 4.3.3 Phase Diagram of Adjoint QCD..................... 148 4.4 Appendix A: Representation Theory of so(N)................. 155 4.5 Appendix B: Z2 Topological Gauge Theory in Three Dimensions....... 156 4.6 Appendix C: P in−(N) and O(N)1 from SO(N)................ 159 4.7 Appendix D: The Chern-Simons Action of O(N) for Odd N ......... 162 4.8 Appendix E: Derivation of Level-Rank Duality for Odd N or K ....... 165 4.8.1 Even N and odd K ............................ 166 4.8.2 Odd N; K ................................. 168 4.9 Appendix F: Low Rank Chiral Algebras and Level-Rank Duality....... 170 4.9.1 Chiral Algebras Related to so(2).................... 170 4.9.2 Chiral Algebras Related to so(4).................... 172 4.10 Appendix G: Projective Representations in SO(4)4 with CM 6= MC ..... 175 4.11 Appendix H: O(2)2;L as Family of Pfaffian States............... 177 4.12 Appendix I: Duality Via One-form Symmetry................. 179 5 Global symmetries, anomalies, and duality in (2 + 1)d 183 5.0.1 Global symmetries............................ 186 5.0.2 Anomalies................................. 189 5.0.3 Outline.................................. 195 5.1 't Hooft Anomalies and Matching........................ 196 5.1.1 Global symmetry............................. 197 5.1.2 Background fields............................. 198 5.1.3 Symplectic gauge group......................... 205 5.2 Quantum Global Symmetries from Special Dualities.............. 206 x 5.2.1 U(1)k with one Φ............................. 207 5.2.2 U(1) Nf with Nf Ψ.......................... 208 −N+ 2 5.2.3 SU(2)k with one Φ............................ 209 5.2.4 SU(2)k with 2 Φ............................. 209 5.2.5 SU(2) Nf with Nf Ψ......................... 210 −N+ 2 5.2.6 Examples with Quantum SO(3) Symmetry and 't Hooft anomaly matching.................................. 211 5.3 Example with Quantum O(4) Symmetry: QED with Two Fermions..... 212 5.3.1 QED3 with two fermions......................... 213 5.3.2 Mass deformations............................ 216 5.3.3 Coupling to a (3 + 1)d bulk....................... 218 5.4 Example with Global SO(5) Symmetry..................... 220 5.4.1 A family of CFTs with SO(5) global symmetry............ 220 5.4.2 Two families of CFTs with SO(3) × O(2) global symmetry...... 223 5.4.3 A family of RG flows with O(4) global symmetry..........
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