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Strings and Monopoles in Strongly Interacting Gauge Theories ~ARCHWE by OF TECHNOLOGY Ethan Stanley Dyer JUL 0 1 2014 Submitted to the Department of Physics I LIBRARIES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 @ Massachusetts Institute of Technology 2014. All rights reserved. Signature redacted A uthorg ......................... Department of Physics May 8, 2014 Signature redacted Certified by.... Allan Wilfred Adams III Associate Professor Thesis Supervisor / -/ /I Signature redacted Accepted by.................... Krishna Rajagopal Associate Department Head for Education Room 14-0551 77 Massachusetts Avenue Cambridge, MA 02139 MTLibraries Ph: 617.253.2800 Email: [email protected] Document Services http://libraries.mit.edu/docs DISCLAIMER OF QUALITY Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. If you are dissatisfied with this product and find it unusable, please contact Document Services as soon as possible. Thank you. 2 Strings and Monopoles in Strongly Interacting Gauge Theories by Ethan Stanley Dyer Submitted to the Department of Physics on May 8, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis we discuss aspects of strongly coupled gauge theories in two and three dimensions. In three dimensions, we present results for the scaling dimension and transformation properties of monopole operators in gauge theories with large numbers of fermions. In two dimensions, we study (0,2) gauge theories as a tool for constructing string backgrounds with non trivial H-flux. We demonstrate how chiral matter content in the gauge theory allows the construction of infrared fixed points outside of the usual Calabi-Yau framework, and further derive consistency relations for a special class of torsional models. Thesis Supervisor: Allan Wilfred Adams III Title: Associate Professor 3 4 Acknowledgments There are many people without whom this thesis would not be possible. I would like to thank my family, especially my parents, Barbara and Sam Dyer; grandpar- ents, Betty and Ira Dyer; and girlfriend Gabrielle Lurie for their continual support of my interest in physics, and tolerance of long work hours. I would also like to thank my advisor, collaborator, and thesis committee member, Allan Wilfred Adams III, for his encouragement, critiques, and undying enthusiasm for all things physics, as well as for his help in navigating the world of academia. I am deeply indebted to my collaborators, Jaehoon Lee, Mark Mezei, Silviu Pufu, and Sho Yaida, who have been instrumental in shaping my graduate experience and research, and a pleasure to interact with. I am also grateful to my fellow center for theoretical physics(CTP) classmates who have made the past few years a joy both academically and socially, and the CTP faculty who have provided answers to countless questions, especially John McGreevy and Jesse Thaler who helped guide my research on numerous occa- sions. I would like to express my appreciation to the CTP administrative staff, Joyce Berggren, Scott Morley, and Charles Suggs who have helped me in many ways, and without whom I would most likely still be locked out of my office. I would like to give a special thanks to my committee members, Allan Adams, Hong Liu, and Michael Williams for their willingness to read this thesis. Lastly, I would like to acknowledge the United States taxpayers and private donors, without whose support physics could not go on. Thank you 5 6 Contents 1 Introduction 19 1.1 Monopoles and Confinement in Three Dimensions . . . . . . . . . . . 21 1.1.1 M onopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.1.2 Fate of the IR . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Strongly Coupled Gauge Theory for Chiral Strings . . . . . . . . . . . 27 1.2.1 Consistancy conditions from world-sheet and space-time . . . 31 1.2.2 Gauge Linear Sigma Models: The basic idea . . . . . . . . . . 33 1.2.3 Lorentz Symmetry, Supersymmetry, and Anomalies in Two Di- m ensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Monopole Operators in Strongly Coupled Gauge Theories 43 2.1 Introduction . . . . . . . . . . . . . . . . . . . 43 2.2 Monopole operators via the state-operator correspondence . . . . . . 47 2.2.1 Classical Monopole Backgrounds . . . . . . . . . . . . . . . . 48 2.2.2 Three Dimensional Gauge Theories with Fermions . . . . . . . 50 2.2.3 Quantum Monopole Operators . . . . . . . . . . . . . . . . . . 52 2.3 Free energy on S2 x R . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.2 Gauge Field Effective Action . . . . . . . . . . . . . . . . . . . 58 2.4 Functional determinants . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.4.1 The fermion determinant . . . . . . . . . . . . . . . . . . . . . 63 2.4.2 The Faddeev-Popov determinant . . . . . . . . . . . . . . . . 67 2.4.3 The gauge fluctuations determinant . . . . . . . . . . . . . . . 69 7 2.4.4 Combining the subleading terms in the free energy . . . . . . 81 2.4.5 Summary and an example . . . . . . . . . . . . . . . . . . . . 84 2.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.5.1 A systematic study of monopole stability in QCD . . . . . . . 88 3 2.6 Monopole operator dimensions . . . . . . . . . . . . . . . . . . . . . . 92 2.6.1 Monopole operator dimensions in QED . . . . . . . . . . . . . 93 2.6.2 Monopole operator dimensions in U(Nc) QCD . . . . . . . . . 93 2.7 Other quantum numbers of monopole operators . . . . . . . . . . . . 95 2.7.1 Quantum numbers of monopole operators in QED . . . . . . 96 2.7.2 Quantum numbers of monopole operators in U(Nc) QCD ... 106 2.8 Monopoles in general gauge theories . . . . . . . . . . . . . . . . . . . 109 2.8.1 Anomalous dimensions for general groups . . . . . . . . . . . . 110 2.8.2 Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.9 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.9.1 Summ ary . . . . . . . . . . . . . . . . . . . . . . . 131 2.9.2 Confinement and chiral symmetry breaking . . . . . 133 2.9.3 QED and and algebraic spin liquids . . . . . . . . . 136 3 Chiral Gauge Theory for Stringy Backgrounds 139 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.2 Generating dH in a (0,2) GLSM . . . . . . . . . . . . . . . . . . . . 142 3.2.1 Torsion in (0, 2) NLSMs . . . . . . . . . . . . . . . . . . . . . 142 3.2.2 Adding dH to a (0,2) GLSM by hand: the Green Schwarz m echanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.2.3 On the geometry of GS GLSMs . . . . . . . . . . . . . . . . . 147 3.2.4 Generating dH in a garden-variety (0, 2) GLSM . . . . . . . . 149 3.3 Verifying Quantum Consistency in a Special Class of Models . . . . . 154 3.3.1 The M odels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.3.3 Gauge Invariant Model . . . . . . . . . . . . . . . . . . . . . . 160 8 3.3.4 Anomalous Model with Green-Schwarz Mechanism . . . . . . 171 3.3.5 Multiple U(1)s . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 179 A Monopole Harmonics 183 A.1 Definition and Properties of Monopole Harmonics . . . . . . . . . . . 183 A.1.1 Scalar Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 183 A.1.2 Spin s Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 185 A.1.3 Spin 1/2 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . 186 A.1.4 Spin 1 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 187 B (0,2) Details 189 B.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.1.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.1.2 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 B .2 A ction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.3 OPEs . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . 194 B.3.1 Operator Product Expansion with single anomalous U(1) . . 194 B.3.2 Operator Product Expansion with multiple U(1)s . . . . . . . 196 B.4 Quantum Chirality . . . . . . . . . . . . . . . . . . . .. .. . . . . 197 9

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2-12 The SU(3) monopoles appearing as black dotted circles in Figure 2-11. Here, we consider .. construction. For non-abelian gauge theories we can consider more general monopoles known as may also carry a topological charge, Qtp. For abelian monopoles this is precisely the charge, QtP = q,.
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