thesis Lakshya Bhardwaj classifications of Quantum Field Theories.pdf - Classifications of Quantum Field Theories by Lakshya Bhardwaj A thesis presented

Thesis Lakshya Bhardwaj classifications of Quantum Field Theories.pdf

This preview shows page 1 out of 340 pages.

You've reached the end of your free preview.

Want to read all 340 pages?

Unformatted text preview: Classifications of Quantum Field Theories by Lakshya Bhardwaj A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2018 c Lakshya Bhardwaj 2018 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Amihay Hanany, Imperial College London Supervisor(s): Davide Gaiotto, Perimeter Institute Robert Myers, Perimeter Institute Internal Member: Robert Mann, University of Waterloo Internal-External Member: Ben Webster, University of Waterloo Other Member(s): Kevin Costello, Perimeter Institute ii This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions Chapter 2 of this thesis consists of material from the paper [1], co-authored with Michele Del Zotto, Jonathan J. Heckman, David R. Morrison, Tom Rudelius and Cumrun Vafa. Chapter 3 of this thesis consists of material from the paper [2], co-authored with David R. Morrison, Yuji Tachikawa and Alessandro Tomasiello. Chapter 4 of this thesis consists of material from the paper [3], co-authored with Davide Gaiotto and Anton Kapustin. Chapter 5 of this thesis consists of material from the paper [4]. Chapter 6 of this thesis consists of material from the paper [5], co-authored with Yuji Tachikawa. iv Abstract We discuss classifications of UV complete supersymmetric theories in six dimensions, and (spin)topological field theories admitting a finite global symmetry and possibly time-reversal symmetry in three dimensions. We also discuss a generalization of finite global symmetries and their gauging in two dimensions. First, we start with LSTs which are UV complete non-local 6D theories decoupled from gravity in which there is an intrinsic string scale. We present a systematic approach to the construction of supersymmetric LSTs via the geometric phases of F-theory. Our central result is that all LSTs with more than one tensor multiplet are obtained by a mild extension of 6D superconformal field theories (SCFTs) in which the theory is supplemented by an additional, non-dynamical tensor multiplet, analogous to adding an affine node to an ADE quiver, resulting in a negative semidefinite Dirac pairing. We also show that all 6D SCFTs naturally embed in an LST. Motivated by physical considerations, we show that in geometries where we can verify the presence of two elliptic fibrations, exchanging the roles of these fibrations amounts to T-duality in the 6D theory compactified on a circle. Second, we study the interpretation of O7+ -planes in F-theory, mainly in the context of the six dimensional models. In particular, we study how to assign gauge algebras and matter contents to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+ -planes. By including O7+ -planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of special unitary gauge algebra. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively. Third, we argue that it is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic ”shadow” theories, which are obtained from the original theory by ”gauging fermionic parity”. The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by the product of fermion parity and internal symmetry G. The quaternion group makes a surprise appearance. Fourth, we generalize two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We show how Turaev-Viro construction can be generalized to unoriented 3d TFTs. On the other hand, it is known that the ”fermionic” versions of oriented TFTs, known as Spin-TFTs, can be constructed in terms of ”shadow” TFTs which are ordinary oriented TFTs with an anomalous Z2 1-form symmetry. We generalize this correspondence to Pin+ -TFTs by showing that they can be constructed in terms of ordinary unoriented TFTs with anomalous Z2 1-form symmetry having a mixed anomaly with time-reversal symmetry. The corresponding Pin+ -TFT does not have any anomaly for time-reversal symmetry however and v hence it can be unambiguously defined on a non-orientable manifold. In case a Pin+ -TFT admits a topological boundary condition, one can combine the above two statements to obtain a TuraevViro-like construction of Pin+ -TFTs. As an application of these ideas, we construct a large class of Pin+ -SPT phases. Finally, we recall that it is well-known that if we gauge a Zn symmetry in two dimensions, a dual Zn symmetry appears, such that re-gauging this dual Zn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a nonanomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category. vi Acknowledgements I would like to thank my supervisor Davide Gaiotto for always pushing me to become an independent researcher and for teaching me the cutting-edge viewpoint on quantum field theory. A special thanks goes to Yuji Tachikawa who was always welcoming for discussions and guidance. Next, I would like to thank all of my collaborators who contributed their experience and expertise to our joint research projects. These are Michele Del Zotto, Davide Gaiotto, Jonathan Heckman, Anton Kapustin, Dave Morrison, Tom Rudelius, Yuji Tachikawa, Alessandro Tomasiello and Cumrun Vafa. I would also like to thank Rob Myers for serving as my co-supervisor, and Kevin Costello and Rob Mann for serving on my advisory committee. Finally, I am indebted to my parents for their love and support. vii Dedication This thesis is dedicated to Stephen Hawking and Joseph Polchinski - two brilliant physicists who unfortunately passed away recently. viii Table of Contents List of Tables xv List of Figures xvi 1 2 Introduction 1 1.1 6d supersymmetric QFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 3d topological field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Finite symmetries in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . 4 F-theory and the Classification of Little Strings 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 LSTs from the Bottom Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 LSTs from F-theory 2.3.1 2.4 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Geometry of the Gravity-Decoupling Limit . . . . . . . . . . . . . . . . 13 Examples of LSTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 Theories with Sixteen Supercharges . . . . . . . . . . . . . . . . . . . . 15 2.4.2 Theories with Eight Supercharges . . . . . . . . . . . . . . . . . . . . . 18 Constraints from Tensor-Decoupling . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.1 Graph Topologies for LSTs . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Inductive Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.3 Low Rank Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Atomic Classification of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Classifying Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.1 Low Rank LSTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.2 Higher Rank LSTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.3 Loop-like Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ix 2.8 2.9 Embeddings and Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8.1 Embedding the Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8.2 Embedding the Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9.2 Examples Involving Curves with Torsion Normal Bundle . . . . . . . . . 40 2.9.3 Towards T-Duality in the More General Case . . . . . . . . . . . . . . . 42 2.10 Outliers and Non-Geometric Phases . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10.1 Candidate LSTs and SCFTs . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10.2 Towards an Embedding in F-theory . . . . . . . . . . . . . . . . . . . . 46 2.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 The Frozen Phase of F-theory 50 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Frozen seven-branes and their properties . . . . . . . . . . . . . . . . . . . . . . 52 3.3 3.4 3.5 3.2.1 Basics of orientifold planes . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Frozen divisors in F-theory . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.4 NS5- and D6-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.5 Shared gauge algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.6 NS5-branes and O-planes . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.7 Smooth transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.8 Tangential intersections and O8-planes . . . . . . . . . . . . . . . . . . 66 Anomaly analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.1 Anomaly cancellation with frozen singularities . . . . . . . . . . . . . . 68 3.3.2 Matter content with frozen singularities . . . . . . . . . . . . . . . . . . 72 Noncompact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4.1 so-sp chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4.2 su-su chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Compact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.1 The F−4 model and its flip . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.2 The CP1 × CP1 model and its flips . . . . . . . . . . . . . . . . . . . . 80 x 4 State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter 86 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 4.2.1 One-form symmetries and their anomalies . . . . . . . . . . . . . . . . . 88 4.2.2 Shadow of a product theory . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.3 Gu-Wen and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.4 A Hamiltonian perspective . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.5 Open questions and future directions . . . . . . . . . . . . . . . . . . . . 100 Spherical fusion categories and fermions . . . . . . . . . . . . . . . . . . . . . . 100 4.3.1 Categories of boundary line defects . . . . . . . . . . . . . . . . . . . . 102 4.3.2 Example: toric code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.3 Example: bosonic SPT phases and group cohomology . . . . . . . . . . 111 4.3.4 Example: G-equivariant Z2 gauge theory from a central extension . . . . 114 4.3.5 Example: Z2 -equivariant toric code vs Ising . . . . . . . . . . . . . . . . 118 4.3.6 Example: Ising pull-backs . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3.7 Gauging one-form symmetries in the presence of gapped boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3.8 Example: 1-form symmetries in the toric code . . . . . . . . . . . . . . 123 4.3.9 The Π-product of shadows . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.10 Triple products and quaternions . . . . . . . . . . . . . . . . . . . . . . 129 4.3.11 The group structure of fermionic SPT phases . . . . . . . . . . . . . . . 130 4.3.12 Π-categories and Π-supercategories . . . . . . . . . . . . . . . . . . . . 132 4.4 4.5 Spherical fusion categories and state sum constructions . . . . . . . . . . . . . . 132 4.4.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.4.2 Review of the Turaev-Viro construction . . . . . . . . . . . . . . . . . . 136 4.4.3 Adding a flat connection . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.4.4 Gauging standard global symmetries . . . . . . . . . . . . . . . . . . . . 140 4.4.5 Adding a 2-form flat connection . . . . . . . . . . . . . . . . . . . . . . 141 4.4.6 Example: toric code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4.7 Gu-Wen Π-category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.4.8 State sums and spin-TFTs . . . . . . . . . . . . . . . . . . . . . . . . . 146 String net models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 xi 5 Example: toric code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.5.2 A fermionic dressing operator . . . . . . . . . . . . . . . . . . . . . . . 154 4.5.3 Fermionic dressing for general Π-categories . . . . . . . . . . . . . . . . 156 4.5.4 Including global symmetries . . . . . . . . . . . . . . . . . . . . . . . . 157 4.5.5 Example: the shadow of Gu-Wen phases . . . . . . . . . . . . . . . . . 157 Unoriented 3d TFTs 159 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.2 Turaev-Viro construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.3 5.4 5.5 6 4.5.1 5.2.1 Boundary line defects and spherical fusion category . . . . . . . . . . . 161 5.2.2 Oriented Turaev-Viro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.2.3 Twisted spherical fusion category and orientation reversing defects . . . . 168 5.2.4 Unoriented Turaev-Viro . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2.5 Example: Bosonic SPT phases . . . . . . . . . . . . . . . . . . . . . . . 175 Pin+ -TFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3.1 Review of Spin case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.3.2 Fermion in Pin+ -theories . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.3.3 Shadows of Pin+ -TFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.3.4 Product of Pin+ -TFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Fermionic SPT phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.4.1 Gu-Wen phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.4.2 Anomaly for Pin+ -shadows . . . . . . . . . . . . . . . . . . . . . . . . 188 5.4.3 Group structure of Gu-Wen phases . . . . . . . . . . . . . . . . . . . . . 190 5.4.4 ZR 2 version of Ising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.4.5 Pin+ -SPT phases with no global symmetry . . . . . . . . . . . . . . . . 192 Conclusion and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 194 On finite symmetries and their gauging in two dimensions 196 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.2 Re-gauging of finite group gauge theories . . . . . . . . . . . . . . . . . . . . . 199 6.3 6.2.1 Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.2.2 Non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Symmetries as categories in two dimensions . . . . . . . . . . . . . . . . . . . . 201 xii 6.4 6.5 6.6 6.7 6.3.1 Basic notions of symmetry categories . . . . . . . . . . . . . . . . . . . 201 6.3.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3.3 More notions of symmetry categories . . . . . . . . . . . . . . . . . . . 210 6.3.4 Groups and representations of groups as symmetry categories . . . . . . 212 Gaugings and symmetry categories . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.4.1 Module and bimodule categories . . . . . . . . . . . . . . . . . . . . . . 216 6.4.2 Duality of C(G) and Rep(G) . . . . . . . . . . . . . . . . . . . . . . . . 217 6.4.3 Gauging by an algebra object . . . . . . . . . . . . . . . . . . . . . . . 219 6.4.4 Symmetries of the gauged theory from bimodules for the algebra object . 221 6.4.5 Gauging of C(G) to get Rep(G) and vice versa . . . . . . . . . . . . . . 224 6.4.6 Gaugings and module categories . . . . . . . . . . . . . . . . . . . . . . 226 6.4.7 (Re-)gauging and its effect on the symmetry category . . . . . . . . . . . 228 6.4.8 The effect of the gauging on Hilbert space on S 1 . . . . . . . . . . . . . 229 More examples of symmetry categories and their gauging . . . . . . . . . . . . . 233 6.5.1 Symmetry category with two simple lines . . . . . . . . . . . . . . . . . 233 6.5.2 Symmetry category of SU(2) WZW models and other RCFTs . . . . . . 233 6.5.3 Gauging a subgroup of a possibly-anomalous group . . . . . . . . . . . . 235 6.5.4 Integral symmetry categories of total dimension 6 . . . . . . . . . . . . . 238 6.5.5 Integral symmetry categories of total dimension 8 . . . . . . . . . . . . . 239 6.5.6 Tambara-Yamagami categories . . . . . . . . . . . . . . . . . . . . . . . 240 2d TFT with C symmetry and their gauging . . . . . . . . . . . . . . . . . . . . 243 6.6.1 2d TFTs without symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.6.2 TFT with C symmetry on a cylinder . . . . . . . . . . . . . . . . . . . . 247 6.6.3 TFT with C symmetry on a general geometry . . . . . . . . . . . . . . . 251 6.6.4 Gauged TFT with the dual symmetry . . . . . . . . . . . . . . . . . . . 257 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 References 262 APPENDICES 276 xiii A Appendix to Chapter 2 277 A.1 Brief Review of Anomaly Cancellation in F-theory . . . . . . . . . . . . . . . . 277 A.2 Matter for Singular Bases and Tangential Intersections . . . . . . . . . . . . . . 278 A.3 Novel DE Type Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 A.4 Novel Non-DE Type Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 A.5 Novel Gluings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 A.6 T-Duality ...
View Full Document

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture