Table Of ContentStrings 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.
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Department of Physics
May 8, 2014
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Allan Wilfred Adams III
Associate Professor
Thesis Supervisor
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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
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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
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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
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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
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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
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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
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Description: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,.
