UNIVERSITY OF CALIFORNIA SAN DIEGO Aspects of Supersymmetric Conformal Field Theories in Various Dimensions A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Emily M. Nardoni Committee in charge: Professor Kenneth Intriligator, Chair Professor Patrick Diamond Professor John McGreevy Professor James McKernan Professor Justin Roberts 2018 Copyright Emily M. Nardoni, 2018 All rights reserved. The dissertation of Emily M. Nardoni is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California San Diego 2018 iii DEDICATION To my family. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . .x List of Tables . xii Acknowledgements . xiii Vita ............................................. xv Abstract of the Dissertation . xvi Chapter 1 Introduction . .1 1.1 Outlook . .1 1.2 Outline . .7 Chapter 2 Technical Introduction . .9 2.1 Supersymmetry: The Basics . .9 2.1.1 The supersymmetry algebra . .9 2.1.2 Irreducible representations of supersymmetry in 4d . 11 2.2 Some Properties of Supersymmetric Gauge Theories . 14 2.2.1 N = 1 supersymmetric actions and the power of holomorphy 14 2.2.2 NSVZ b-function . 16 2.2.3 Moduli space of vacua . 17 2.2.4 A short introduction to anomalies in QFT . 18 2.3 Superconformal Field Theories . 19 2.3.1 A word on conformal symmetry . 19 2.3.2 A tour of SCFTs in various dimensions . 20 2.3.3 Facts about fixed points . 21 v 2.3.4 The a-theorem, and a-maximization . 24 2.3.5 Chiral ring . 26 2.3.6 4d N = 1 superconformal index . 26 Chapter 3 Deformations of WA;D;E SCFTs . 30 3.1 Introduction . 30 3.1.1 The chiral ring of WA;D;E(X;Y) for matrix fields X and Y .. 32 3.1.2 WA;D;E in 4d SQCD with fundamental plus adjoint matter . 34 3.1.3 WA;D;E + DW RG flows . 36 3.1.4 WA;D;E flat direction flows . 38 3.1.5 Outline . 39 3.2 Technical Review . 40 3.2.1 The a-theorem, and a-maximization . 40 3.2.2 Duality for the 4d SCFTs . 41 3.2.3 Moduli spaces of vacua of the theories . 44 3.3 Example and Review: A and Ak One-Adjoint Cases . 46 3.3.1 A Ak flow and Ak duality . 46 ! b 3.3.2 WA + DW deformations and Ak Ak <k RG flows . 48 b k ! 0 3.3.3 Comments on SU(Nc) vs U(Nc) RG flows . 50 3.3.4 SU(Nc) flat direction deformations . 52 3.4 The WDk+2 Fixed Points and Flows . 52 3.4.1 Previously proposed dualities for WDk+2 ........... 53 3.4.2 Matrix-related flat directions at the origin . 56 3.4.3 SU(Nc)-specific (as opposed to U(Nc)) flat directions . 63 3.4.4 Dk+2 RG flows from relevant DW deformations . 67 3.4.5 The SU(Nc) version of the RG flows . 71 3.4.6 The Dodd Deven RG flow and the hypothetical D theory 72 ! even0 3.5 The WE7 Fixed Point and Flows . 74 3.5.1 Previously proposed dualities for WE7 ............ 75 3.5.2 Matrix-related flat directions at the origin . 77 3.5.3 SU(Nc)-specific (as opposed to U(Nc)) flat directions . 79 3.5.4 Case studies in E7 RG flows from DW deformations . 82 vi 3.6 Conclusions, Future Directions, and Open Questions . 89 3.6.1 Recap: puzzles and open questions for the Deven and E7 theories 89 3.6.2 Future directions: aspects of the WE6 and WE8 theories . 90 Chapter 4 Landscape of Simple Superconformal Field Theories in 4d . 93 4.1 Introduction . 93 4.2 A Landscape of Simple SCFTs . 95 4.3 T0 —Minimal c, Minimal a with U(1) ................ 99 4.4 H0∗—Minimal a ............................ 100 4.5 H1∗—Minimal a with SU(2) ...................... 101 4.6 Discussion . 102 Chapter 5 Interlude: An Introduction to the Theories of Class S ........... 103 5.1 Introduction . 103 5.2 4d SCFTs from 6d . 104 Chapter 6 4d SCFTs from Negative Degree Line Bundles . 109 6.1 Overview and Summary of Results . 109 (m) 6.2 Constructing the BBBW Duals from TN Building Blocks . 111 (m) 6.2.1 TN review . 111 6.2.2 Gluing procedure . 113 6.2.3 Construction of Cg>1;n=0 and computation of a and c .... 114 6.2.4 Operator dimensions and large-N .............. 118 (m) 6.3 Cg=0;n=0 from the Higgsed TN ................... 119 (m) 6.3.1 Constructing Higgsed TN theories . 119 6.3.2 Computation of atrial and ctrial ................. 121 6.3.3 Comments on ruling out g = 0 SCFTs . 122 (mi) 6.4 General Cg;n from TN Building Blocks . 123 6.4.1 Computing atrial and ctrial for g = 1............. 123 6 6.4.2 Comments on operators . 126 6.4.3 Computing atrial and ctrial for the torus . 127 6.5 Future Directions . 128 vii Chapter 7 Structure of Anomalies of 4d SCFTs from M5-branes, and Anomaly Inflow 130 7.1 Introduction . 131 7.2 Structure of Class S Anomalies . 135 7.2.1 Anomalies of the (2;0) theories . 136 7.2.2 Structure of class S anomalies . 138 7.3 Anomalies of (p;q) Punctures in Class S .............. 143 7.3.1 Anomalies of (p;q) punctures . 143 7.3.2 Effective nv;nh for N = 1 theories . 145 7.4 Inflow for Flat M5-branes: A Review . 148 7.4.1 Anomaly inflow . 149 7.4.2 M5-brane inflow . 150 7.5 Class S Anomalies from Inflow . 154 7.5.1 Inflow for curved M5-branes . 155 7.5.2 Sources for connection forms . 160 7.5.3 Variation of the action . 165 7.5.4 Total 4d anomaly from inflow . 172 7.6 Conclusions . 174 7.6.1 Summary . 174 7.6.2 Outlook . 176 Appendix A WA;D;E SCFTs . 178 A.1 ADE Facts . 178 A.2 RG Flows Whose Deformations Seem Irrelevant . 179 A.2.1 E6 A5 ............................ 179 ! A.2.2 E7 D6 ............................ 181 ! A.2.3 E8 D7 ............................ 181 ! Appendix B Relevant Formulae for Gluing Negative-Degree Line Bundles . 184 B.1 Conventions and Main TN Formulae . 184 B.2 Relevant BBBW Results . 185 (m) B.3 ’t Hooft Anomalies for Gluing TN Building Blocks . 187 viii Appendix C Class S Anomaly Conventions . 189 C.1 Anomaly Polynomials and Characteristic Classes . 189 C.2 Anomalies for Regular N = 2 Punctures . 191 C.3 Integral Identities . 193 Bibliography . 194 ix LIST OF FIGURES Figure 1.1: The coupling l as a function of energy scale m plotted for an infrared free theory that breaks down at m LLandau....................5 ∼ Figure 1.2: The coupling l as a function of energy scale m for an asymptotically free theory that develops a dynamical scale at m LQCD.............6 ∼ Figure 1.3: The coupling l as a function of energy scale m for an asymptotically free theory that develops an interacting fixed point at low energies. .6 Figure 3.1: Deforming the WA;D;E 2d N = 2 SCFTs corresponds to adjoint Higgsing of the ADE group, hence cutting the Dynkin diagrams, as illustrated here for the flows in (3.3). This gives the vacua associated with 1d representations of the F-terms. 32 Figure 3.2: Flows among the fixed points of SQCD with two adjoints. 37 2 Figure 3.3: (aUV aIR)=N plotted for x in the conformal window. The WD theory is − f 4 5 IR-free for x 1. The WD theory requires x > 3:14 for the TrX term in WD ≤ 6 6 to be relevant, while the corresponding term in the Brodie dual is relevant if x < 8:93. The A theory is asymptotically free in both plotted domains. 60 Figure 3.4: Using the Q-flat directions to bypass the strong coupling regime. 63 b Figure 3.5: Flat directions for Dodd, Nc = 2m+kn, integrating out massive fields (denoted by dotted lines) and fields eaten by the Higgs mechanism (not shown). 65 Figure 3.6: Flat directions for Deven, with Nc = 2m + kn. Again we integrate out massive fields (denoted by dotted lines), and those eaten by the Higgs mechanism (not shown). 67 Figure 3.7: We enhance the k odd Dk singularity to a Dk+1 singularity in one of two ways. 73 2 Figure 3.8: (aUV aIR)=N for WE in the UV and A in the IR. The E7 deformation term − f 7 in the UV theory is relevant for x & 4:12, while the corresponding term in b Kutasov-Lin dual is relevant if x . 26:11. The Aˆ theory is UV-free in this whole range. 78 Figure 3.9: Flat directions for E7, Nc = 2m + 3n, integrating out massive fields (denoted by dotted lines) and fields eaten by the Higgs mechanism (not shown). The subscripts label the gauge groups and their matter. 81 x Figure 4.1: The central charges of the 35 “good” theories. The ratios a=c all lie within the range (0:8246;0:9895). The mean value of a=c is 0:8733 with standard deviation 0:03975. 97 Figure 4.2: A subset of the fixed points that can be obtained from SU(2) Nf = 1 ad- joint SQCD with singlets. Note that the graph is not arranged vertically by decreasing a central charge, because the deformations we consider involve coupling in the singlet fields.