Dynamics of 3D Susy Coulomb Branches

DYNAMICS OF 3D SUSY COULOMB BRANCHES

A Dissertation Presented to the Faculty of the Graduate School of Cornell University

in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

by Mario Claudio Martone January 2015 c 2015 Mario Claudio Martone

In this thesis we focus on the dynamics of Coulomb branches in 3 dimensional N = 2 SUSY theories. The first part of this thesis is dedicated for the most part to a review of previously known results, providing the necessary background to understand the generalization to more complicated theories. We review in details both the behavior of U(1) theories with generic number of flavors and SU(N) theories with fundamentals. We also provide insights on the mapping of Coulomb branch operators across dualities as vortex/monopole operators. In the second part of the thesis we investigate the IR dynamics of N = 2 SUSY gauge theories in 3D with antisymmetric matter. The presence of the antisymmetric fields leads to further splitting of the Coulomb branch. Counting zero modes in the instanton background suggests that more than a single direction along the Coulomb branch may remain unlifted. We examine the case of SU(4) with one or two antisymmetric fields and various flavors in detail. Using the results for the corresponding 4D theories, we find the IR dynamics of the 3D cases via compactification and a real mass deformation. We find that for the s-confining case with two antisymmetric fields, a second unlifted Coulomb branch direction indeed appears in the low- energy dynamics. We present several non-trivial consistency checks to establish the validity of these results. We also comment on the expected structure of general s-confining theories in 3D, which might involve several unlifted Coulomb branch directions. The original work presented in this thesis is mostly based on [1]. The paper will appear soon on JHEP. BIOGRAPHICAL SKETCH

Mario Martone is tremendously lucky of being born in the most beautiful and thrilling city in the world: Napoli, Italy. His loyalty and unconditional love for the city has been one of his most distinct traits in his adult life as an emigrant. Mario’s primary and secondary education took place in his beloved city where he attended the Scuola elementare Quarati to then move to the Scuola media Statale G. Lettieri. His love and interest for mathematics started at a very early age. At the first meeting with elementary school teachers, Mario’s math teacher told straight to his mother: signora, suo figlio éun genio! (madam, your son is a genius!). In his last year of elementary school Mario started play water-polo, another of his characteristic traits. He is now in his 21st consecutive year of playing the sport. After middle school Mario moved to the Liceo Classico A. Genovesi focusing his (non)-studies mostly on humanities, ancient Greek and Latin over anything else. During his high-school years he began his life-long commitment to revolutionary politics. During his junior year he was elected among the leaders of the school’s student movement and he and his amazing comrades led multiple protests and take-overs of the school.

Although clearly a science oriented person, Mario didn’t develop his love for physics until the last year of high-school when he read Time Travel in Einstein’s Universe: The Physical Possi- bilities of Travel Through Time. He was completely fascinated by the deep philosophical implications of modern physics and decided not to sell his soul by merely applying the beautiful laws of physics but try to contribute himself in the understanding of the fundaments of nature. He went on and pursued both a Laurea Triennale and a Laurea Specialistica (yes, Italy has a damn weird education system!) in physics. He was the ﬁrst of his class to graduate obtaining a 110/110 cum laude (again Italy, why do you grade people out of 110?) with a thesis on the Geometrical Formulation of Quantum Mechanics. Fascinated by differential geometry but slightly thrown off from the lack of physical relevance of his undergraduate research, Mario moved to heart of the Empire to pursue a PhD at Syracuse University in Non-commutaive Geometry under the wonderful guidance of Prof. A. P. Balachandran.

iii In his first years of graduate career Mario spent extended amount time in Brazil visiting the International Center for Condensed Matter Physics (ICCMP), Universidade de Bras´ılia, then Spain visiting the Universidad Carlos III de Madrid and then India being a guest of the Institute of Math- ematical Sciences in Chennai. During these years Mario’s research focused on understanding the microscopic structure of space-time and its implications on space-time symmetries and quantum field theories. After his marvelous time in India, Mario enthusiastically accepted the kind offer from Prof. Yuval Grossman of visiting Cornell University for a semester and learn more about flavor physics. This semester long collaboration turned in pursuing his second PhD at

Cornell, this time in High Energy Physics. Mario completed his first PhD with honors in January 2012 with a thesis on Quantum Fields on Noncommutative Spacetimes. Since then he has finally settled his research interests on Beyond the Standard Model Physics and Formal Aspects of Quantum Field Theory. On June 9th, 2014 Mario’s beloved twin brother Alfonso had his first daughter Clara. Clara is undoubtedly the most amazing, intriguing and beautiful niece on planet Earth. Shortly after starting her journey outside of Sara’s womb, Clara committed to soon thoroughly study her new favorite book, incidentally bought to her by her uncle Mario: non-Euclidean geometry for Babies. Unfortunately at the moment, Clara is still a little too busy learning about the world to focus on math. Mario completed his second PhD in physics with the particle theory group at Cornell Uni- versity in August 2014 under the guidance of Prof. Csaba Csaki who, during the summer of 2014, was also initiated into the Napoli love. Mario is tremendously excited to continue his research in particle physics at the University of Cincinnati.

iv To the people. From Gaza to Ferguson, MO.

v ACKNOWLEDGEMENTS

First and foremost I want to thank my great friend and colleague Flip Tanedo without whom I would most likely be traveling around the world by now instead on working on physics. He helped me and encouraged me enormously during my academic-wise darkest times at Cornell. Our physics interests might be slightly divergent but I will always be there for you. Thanks man!

Secondly I want to deeply thank my adviser Prof. Csaba Csaki.´ I wish I had started col- laborating with him earlier in my career. Luckily enough I don’t see our collaboration ending anytime soon. Despite having very different personalities, our interactions have been incredibly insightful. You taught me a lot Csaba and I would love to keep learning from you! I want to thank Prof. Yuval Grossman without whom I would have never joined the Cornell Particle Theory group. He has also forced me once more to choose on which side of history to be on and what my life priorities really are. I want to thank UC Davis Prof. John Terning and UC Irvine Prof. Yuri Shirman for this last year of exciting collaborations...and the best is yet to come!

Then I want to thank the amazing people who accompanied my journey at Cornell. I was incredibly lucky to meet each one of them: Cody Long, down to Earth like few (especially in the Ivy world!) and always available for a chat/support/laugh/drinks, in three words an amazing friend, David Marsh, my only true physicist-comrade who stood always on my side, Paul McGuirk, an amazing person who lives life by example and for that deserves my deepest respect, I wish him the best of luck in his upcoming out-of-physics career, Riccardo Pavesi, one of the most talented physicists I have met with an incredibly loving and gentle heart, Dean J. Robinson, it is pretty amazing how far different we operate and yet how beautifully we worked together, Javi Serra, our collaboration is yet to bloom but I am sure it will soon, and ﬁnally my two beautiful ofﬁce-mates Nic Rey-Le Lorier and Mathieu Cliche (Matt this acknowledgment dramatically underrepresents how much I am grateful to you for all the times you were there to help me). Thanks also to Brando Bellazzini, Josh Berger, Monika Blanke, Jack Collins, David

Curtin, Marco Farina, Naresh Kumar, Mike Saelim, Bibhushan Shakya, John Stout and Yuhsin

vi Tsai. And then there is my life outside of physics. First and foremost I want to thank my dearest

American comrades which luckily enough are too many to list. A special mention goes to the members of Student for Justice in Palestine who by being who they are have made the choice between my academic interests or ﬁght for what I believe is just, a piece of cake. No matter how many threats I will receive, today, like that November 17th 2012, I will choose to stand and march by you! Also my Ithaca family: Ayslin, Clare, Gino, Kayla, James, Paula, Shawnae. Unfortunately the Shawn Greenwood Working Group is not going through its best days, but for sure we have done some incredible work together honoring the memory of Shawn Greenwood, a yet another young black man murdered by a Police ofﬁcer. I love you all!

A massive thanks to all the members of the Cornell Water-polo team, ladies and boys alike. I will bring most of you in my heart for years to come. It has been an incredible journey to be involved with you for these last four years. Please continue with the champagne even without me! A special thanks to my closest US friends: Aisha, Anthony, Declan, Kevin, Julie, Max. Those countless drinks and amazing nights together contributed to make life in up-State NY so much better! Thanks to my girlfriend (WOW!) Ashley. The two of us have a hell lot to build together in Cincinnati.

And then there is my life outside of physics and outside of the US. Thank you to my family which represents still an incredibly solid pillar in my life despite thousands of miles among us. Thanks to Clara for being so incredibly cute. You contributed in giving me a purpose in life: make you the most spoiled niece ever. Finally I want to acknowledge the absolute centrality in my life of my napolitan brothers and comrades. Boccione, Carlo, Carolina, Claudia, Daniele, Gabriele, Germana, Julie, Luca,

Luca, Marcella, Pz8, Sciacallo, you are not only my closest friends and an inﬁnite source of support and love. But you provide the reference against which I deﬁne myself and the choices I make in my life.

vii TABLE OF CONTENTS

Biographical Sketch...... iii Dedication...... v Acknowledgements...... vi Table of Contents...... viii List of Figures...... xi

1 Introduction 1

I General aspects of Coulomb Branches5

2 General aspects of N = 2 SUSY in Three Dimensions6 2.1 3D Spinors and the N = 2 SUSY Algebra...... 6 2.2 Chiral superfield...... 7 2.2.1 Real and Complex Masses...... 8 2.3 Vector superfield...... 9 2.3.1 Central Charge...... 10 2.4 Topological U(1)J symmetry...... 12 2.5 Linear superfield...... 13 2.6 Chern-Simons terms...... 15

3 Coulomb branch and the U(1) case 17 3.1 3D N = 2 U(1) gauge theory and moduli space of vacua...... 17 3.2 Coulomb branch and preliminary considerations...... 19 3.3 One-loop correction...... 20 3.4 Loop-induced global charges...... 23 3.5 U(1) with ﬂavors...... 26 3.5.1 F = 1 ...... 27 3.5.2 F > 1 ...... 27 3.6 Mirror Symmetry...... 28

4 The non-abelian case 32 4.1 The non-abelian case...... 32 4.2 Coulomb branch pinch from matter...... 35 4.3 Instanton/Monopole background...... 37 4.3.1 BPS condition...... 38 4.3.2 Solution of the Bogomol’nyi equation...... 39 4.4 Non-perturbative corrections...... 40 4.5 Zero mode counting and general case...... 42 4.6 A taste of SU(N) dualities...... 46 4.6.1 F=N...... 47 4.6.2 F=N-1...... 47

viii 5 Monopole operators 48 5.1 BPS Vortices...... 49 5.1.1 Generic case...... 51 5.1.2 Fermionic zero-modes...... 52 5.2 A case of study: Mirror Symmetry...... 53 5.3 More general case...... 55 5.4 Non-abelian case...... 57

II The case with Antisymmetric matter 58

6 The Coulomb branch with antisymmetric matter 59 6.1 Coulomb branch coordinates...... 60 6.2 Dirac quantization condition of the monopole operators...... 62

7 Dimensional reduction of 4D dualities 66 7.1 General setting...... 66 7.2 S-conﬁning case...... 68 7.2.1 Check 1...... 69 7.2.2 Check 2...... 70

8 Duals of 3D theories with antisymmetrics 72 8.1 3D duality for SU(4) with 2 A and 2 (Q + Q¯)...... 72 8.2 Consistency checks of the duality...... 76 8.2.1 Matching charges of the Coulomb branch operators...... 76 8.2.2 Matching the quantum constraints on a circle...... 77 8.3 Duality for SU(4) with A and 3(Q + Q¯) ...... 77 8.4 Low energy description on the Coulomb branch...... 78

9 Classiﬁcation of 3D s-conﬁning theories 80

10 Conclusions 83

A Spinors and notation 85

B Elements of Group theory 87 B.1 Deﬁnition and Cartan subalgebra...... 87 B.2 Roots, weights and lots of SU(2)s...... 88 B.2.1 weights...... 88 B.2.2 Adjoint representation and roots...... 88 B.2.3 Simple roots...... 90 B.2.4 SU(2)s...... 91 B.3 A closer look at SU(N) ...... 92 B.3.1 Fundamental...... 93 B.3.2 Antisymmetric...... 95

ix C Zero modes and Callias index theorem 97 C.1 Callias Index Theorem...... 97 C.2 Fundamental representation...... 97 C.3 Adjoint representation...... 98 C.4 Antisymmetric representation...... 98

Bibliography 100

x LIST OF FIGURES

3.1 In presence of matter, the Coulomb branch of the U(1) theory splits into two independent branches and near the origin of the moduli space, where the Higgs branch intersects the Coulomb branches, the moduli space looks like the intersection of three cones...... 22 3.2 Vacuum polarization diagram which generates one-loop Chern-Simons terms.. 24 3.3 Vacuum polarization diagram which generates one-loop mixed Chern-Simons terms. ai labels a vector associated to a weakly gauge U(1) theory...... 25

4.1 In presence of matter, the Coulomb branches pinch off where they meet Higgs branches. At these points, σ · να = 0 and there is massless matter...... 36 4.2 In the theory with fundamentals, the Coulomb branch splits at each point where σi = 0. The total Coulomb branch is composed of (N − 1) regions...... 43

7.1 If we reduce s-conﬁning theories, there is no eηYe in the magnetic side...... 68

xi CHAPTER 1 INTRODUCTION

Quantum field theory is one of the richest subject in theoretical physics yet it still remains among the hardest to fully penetrate. Often in the last decades significant progresses in the understanding of gauge theory dynamics have come from looking at problems in simplified settings and only rarely by tackling straight the problem initially at hand. Supersymmetry and 3 dimensional physics, which represent the main focus of this thesis, have both being incredibly useful in providing simplified frames allowing considerable progress.

In the mid-1990s due to the seminal work of Seiberg and Intriligator on N = 1 theories [2–4], and Seiberg and Witten for N = 2 theories [5,6] the understanding of the dynamics of SUSY gauge theories was revolutionized. This brought to light a multitude of amazing properties which quantum field theory had: the interplay of holomorphy, global symmetries, instantons, anomalies, and monopoles can determine the IR behavior of a large class of theories. Depend- ing on the amount of matter, these theories manifest dynamical effects such as gaugino condensation, instanton generated superpotentials, confinement with or without chiral symmetry breaking, IR-free composite gauge groups, interacting non-Abelian quantum fixed points, monopole condensation as a dual of confinement, etc.

On the other side, the study of lower-dimensional field theory represents a simpler labora- tory where to study and attempt to understand more complicated dynamics in 4D, for example, as a tool for understanding possible mechanisms for confinement [7,8]. More recently a new mechanism for confinement from bion condensation was proposed [9, 10] and further studied in [11–14]. Since one may generate 3D theories from the compactification of 4D theories, one can expect that some of their behavior is reflective of 4D properties. Furthermore, the study of lower-dimensional field theory has provided incredible insights in the understanding of dualities. For instance the equivalence of the massive Thirring and the sine-Gordon models [15, 16] has a very transparent meaning in that it can be shown that massive Thirring model arises re- writing the initial sine-Gordon model in terms of its topological solutions (kinks). This duality

1 is not simply conjectural as it can be shown that the operator which creates the sine-Gordon kink satisfy the same equation of motions as the massive Thirring models. It is believed that many of the higher dimensional conjectured dualities, arise in a similar fashion but thus far all attempts of formally writing down dualities for D > 2 have failed.

Extending the study to 3D N = 2 SUSY gauge theories many of the phenomena appearing in 4D have also been observed [17–35]. In particular the combination of SUSY with lower- dimensional field theory provides one of the most promising setting where an equivalence like the massive Thirren and sine-Gordon models could be provided. Mirror symmetry, which will be extensively reviewed in this thesis work, has in fact many of the features of lower dimensional field theory [23, 56, 57]. The achievement of a formal duality map would provide a significant improvement in the understanding of gauge dualities.

Although we will touch on many of the described phenomena, our main focus in thesis will be slightly different. Over the past year, compelling derivations of many of the results obtained in [19–22,27,31,32] have emerged [36–38]. A careful sequence of compactification R4 → R3 × S 1 together with a real mass deformation yields purely three dimensional electric theories and allows one to determine the magnetic duals in a controlled way1 [36]. In this thesis we will present a first step to extend these results to theories with more general matter. In particular, we focus on the case with antisymmetric tensors. A careful counting of the fermionic zero modes in instanton backgrounds leads us to conclude that the dimension of the unlifted Coulomb branch can be larger once representations other than fundamentals are included. The reason for this is that the charges of the moduli parameterizing the Coulomb branch are modified in the presence of antisymmetric matter leading to fewer directions lifted by instanton effects. In the specific example of an SU(4) gauge group we identify two unlifted directions: the standard Y direction found in the presence of fundamental matter fields and a new Ye direction arising from the additional splitting of the Coulomb branch due to massless components of the antisymmetric matter fields—this direction is related to the unlifted Coulomb branch modulus of SO(6) theories with vectors. Using this insight we apply the program of [36] to the simplest examples of 4D models

1See Chapter7 for a self-contained review of dimensional reduction of 4D dualities. Also see [39].

2 with antisymmetric matter and known 4D dynamics: the s-confining [40–42] SU(4) gauge theories with either two antisymmetrics and three flavors or one antisymmetric and four flavors.

We find that for the case of two antisymmetrics, the corresponding s-confining 3D theory with two flavors indeed requires the Ye operator in the description of the IR dynamics. Several consistency checks are presented, including the matching of quantum numbers, reproducing the quantum modified constraints expected for theories on a circle, and connecting it to theories with fewer flavors, establishing a consistent, intricate web of IR dynamics of several SU(4) theories with antisymmetric matter and flavors. The experience acquired with these theories leads us to speculate on the classification of the Coulomb branch structure for general s-confining theories in 3D.

The thesis splits naturally into two parts. In the ﬁrst part we will mostly review known results. We will carefully and systematically present the material needed to tackle the analysis presented in the second part of the thesis. A review of N = 2 SUSY 3D theory is presented in Chapter2 where we outlined the differences between 3D N = 2 and 4D N = 1. Chapter 3 is dedicated to a thorough analysis of the dynamics of U(1) theories with a speciﬁc focus on the behavior of the Coulomb branch. The generalization to non-abelian theories (mostly

SU(N)) is performed in the subsequent chapter. There we will systematically explain the non- perturbative effects which drastically change the perturbative picture. Finally in Chapter5 we elaborate on how to interpret Coulomb branch operators across dualities. We will present a tentative a state-to-operator map between Coulomb branch operators and critical vortices.

In the second part of the thesis we will present the extension to the study of SU(N) theories with generic matter. Chapter6 contains a careful discussion of the structure of the Coulomb branch in the presence of antisymmetric matter. The counting of zero modes is performed and the general features outlined. In Chapter7 we will explain how to obtain 3D dualities from known 4D ones. This is mostly a review of [36]. We apply the program of [36] to the s-conﬁning SU(4) theories with antisymmetric matter in Chapter8 to ﬁnd the correct low-energy dynamics of those models, and perform various consistency checks. In Chapter9 we summarize our ex-

3 pectations for the general behavior of 3D s-conﬁning theories, and ﬁnally conclude in Chapter 10. In the appendices we will review 3D spinors in AppendixA, provide elements of group theory in AppendixB. Zero mode counting and the Callias index theorem is presented in Ap- pendixC.

4 Part I

General aspects of Coulomb Branches

5 CHAPTER 2 GENERAL ASPECTS OF N = 2 SUSY IN THREE DIMENSIONS

In this ﬁrst chapter we will introduce the formalism and the notation for the chapters to follow. In particular we will describe in details 3D N = 2 supersymmetry and its realization on super-space. For people already familiar with 4D SUSY, 3D N = 2 SUSY can be obtained by dimensional reduction of 4D N = 1 SUSY. Most of the material presented here follows the treatment in [19, 20, 43, 44]

2.1 3D Spinors and the N = 2 SUSY Algebra

A convenient representation of the Clifford algebra in 3D with metric ηi j = (−, +, +) is

i=0,1,2 γα,β = (iσ2, σ3, σ1). (2.1.1)

i j − i i jk Note that the generators of the Lorentz group, S = 2 γk, are pure imaginary and thus generate a real group. In fact in (2+1)-dimensions spinors come in representation of S pin(2, 1) ∼

SL(2, R), therefore the fundamental fermion representation is a 2-component Majorana fermion,

ψM. Under parity, this transforms as

P : ψM → ±γ1ψM. (2.1.2)

αβ As usual, spinor indices are contracted, raised and lowered with or αβ.

From the usual 4D N = 1 algebra, the 3D N = 2 SUSY algebra is

n o Qα, Qβ = [Pµ, Qα] = 0 n o ¯ i Qα, Qβ = 2γαβPi + 2iαβZ, (2.1.3) where the central charge Z is identiﬁed with momentum along xµ=2 in the 4D picture. The realization of (2.1.3) as differential operators acting on superspace follow from the 4D formalism:

∂ µ β ∂ β µ Qα = − iγ θ¯ ∂µ, Q¯α = − + iθ γ ∂µ (2.1.4) ∂θα αβ ∂θ¯α βα

6 Likewise the superspace derivatives, anti-commuting with (2.1.4), are ∂ ∂ D = + iγi θ¯β∂ , D¯ = − − iθβγi ∂ (2.1.5) α ∂θα αβ i α ∂θ¯α βα i From (2.1.5) one may read off the 3D N = 2 SUSY multiplets.

Finally we will deﬁne the following action of the differential operator (2.1.4) on a generic super ﬁeld O h ¯i δζO ≡ ζQ O, δζ¯O ≡ ζ¯Q O (2.1.6)

Once an explicit form for the action (2.1.6) is provided, it possible to explicitly check the closure of the SUSY algebra and, as we will do below, compute explicitly the central charge value. In fact h i n o α β ¯ δζ, δζ¯ O = ζ ζ¯ Qα, Qβ O (2.1.7)

For completeness spinor notations is reviewed in AppendixA.

2.2 Chiral superﬁeld

The ﬁrst multiplet which we introduce is the chiral multiplet contained in the chiral super-ﬁeld

Q which satisﬁes the constraint D¯ αQ = 0. Introducing the super-space coordinates (θ, θ,¯ y = x − iθγiθ¯) and using (2.1.5), we can derive an explicit expression for the chiral super-ﬁeld

2 Q(y, θ, θ¯) = φQ(y) + θψ(y) + θ F(y) (2.2.1) where φQ is a complex scalar, ψ is a Weyl fermion which decomposes into two independent real Majorana fermions in 3D, and F is an auxiliary ﬁeld.

From (2.1.4) and (2.1.6) we obtain the transformation properties of the chiral super-ﬁeld

δζφ = ζψ, δζ¯φ = 0,,

i δζψ = ζF, δζ¯ψ = −iγ ζ∂¯ iφ, (2.2.2)

i δζ F = 0, δζ¯ F = −iζγ¯ ∂iψ .

7 The action for chiral superﬁelds Q is Z Z 3 4 † 3 h 2 i S chiral = d x d θ K(Q, Q ) + d x d θ W(Q) + c.c. (2.2.3) for Kahler¨ potential K(Q, Q†) and superpotential W(Q). In particular for K(Q, Q†) = Q†Q the kinetic term is

2 † i 2 Lkin. = ∂iφQ + iψ γ ∂iψ + |F| , (2.2.4)

the subscript φQ indicates the lowest scalar component of the chiral superﬁeld Q.

2.2.1 Real and Complex Masses

In 3D there are two different types of mass terms one may write for a chiral superﬁeld Q. For a vector-like theory we can write down a holomorphic mass by adding a quadratic term to the super potential: W = m QQ¯ mC C (2.2.5) because mC is complex, this is known as a complex mass term and is the analog of the usual mass term in four dimensions. The mass term (2.2.5) is parity even.

Alternately, we can introduce a mass which is parity-odd. This is known as a real mass term and can be understood by modifying the Kahler¨ potential,

Z Z 2 ! 3 4 m θ2 † 3 m 2 αβ d xd θQe R Q ∼ d x R |φ | + im ψ¯ ψ . (2.2.6) 2 Q R α β

Again, observe that for the parity action (2.1.2), the complex mass preserves parity while the q m = m2 + m2 real mass breaks parity. The physical mass of the chiral superﬁeld is R C.

A real mass can be induced by weakly gauging an exact global symmetry and fixing the weakly gauged vector superfield Vb into a SUSY-preserving background configuration (see below) m σ = i R , A = λ = λ¯ = D = 0. (2.2.7) b g i,b b b b

8 Because the global symmetries of dual theories must coincide, one may perform a real mass deformation on one theory and straightforwardly map it to the dual theory. This is used extensively in this paper and is explained thoroughly in Section8.

2.3 Vector superﬁeld

The next multiplet is the vector multiplet which can be organized into a vector super-ﬁeld satisfying V = V†. The action we will write down is invariant under super-gauge transformation

V → V + Λ + Λ†, for any chiral super-field Λ. In a particular choice of gauge, the Wess-Zumino gauge, one finds the following field content:

1 V = −iθθσ¯ − θγiθ¯A + iθ2θ¯λ¯ − iθ¯2θλ + θ2θ¯2D, (2.3.1) i 2

i where we have explicitly separated the 3D vector Ai from the gauge scalar σ ∼ A2 and the γ are defined in (2.1.1). Unlike the chiral superfield, the 3D N = 2 vector superfield differs from its 4D N = 1 counterpart in that it carries components which may acquire vacuum expectation values (VEVs) that form the Coulomb branch of the moduli space. We will discuss extensively Coulomb vacua throughout this thesis.

The superymmetry transformations on the vector super-ﬁeld act as follow

i 1 δ A = − λ†γ ζ, δ σ = − λ†ζ ζ i 2 i ζ 2 i i jk δ λ = −γ F + D σ ζ, δ¯λ = 0. (2.3.2) ζ 2 i jk i ζ

A natural gauge invariant quantity which can be build out of the vector super-ﬁeld V is

1 W = − D¯ 2e−V D eV (2.3.3) α 4 α

Acting with (2.1.5) it is straightforward to check that Wα is a chiral super ﬁeld.

In terms of the the gauge-invariant combination (2.3.3), we can construct the supersymmet-

9 ric version of the Yang-Mills action

1 Z S = d3 x d2θ (Tr W Wα + c.c.) (2.3.4) YM g2 α Z ! 1 3 1 i j i 2 † i = d x Tr Fi jF + DiσD σ + D + λ γ Diλ . g2 4

We are now ready to write the supersymmetric action for a Yang-Mills theory with matter, that is coupling chiral super-ﬁelds to the vector super-ﬁelds. This is easily obtained modifying the Kahler potential K(Q, Q†) → Q†eV Q Z 3 4 † V S chYM = d xd θQ e Q (2.3.5) Z 3 2 † 2 † † i † † † † 2 = d x DiφQ + φQσ φQ + iφQDφQ + iψ γ Diψ − iψ σψ + iφQλ ψ‘ − iψ λφQ + |F| ,

where Di is the Dirac operator. From (2.3.5), we note that a non-vanishing VEV for the scalar component of the vector super-ﬁeld, hσi , 0, induces a supersymmetric mass term for Q, specifically, a real-mass term as we already pointed out in (2.2.7).

After imposing back the Wess-Zumino gauge, the SUSY transformations (2.2.2) get modiﬁed to

δζφ = ζψ, δζ¯φ = 0,,

i δζψ = ζF, δζ¯ψ = −iγ ζ¯ Diφ − iσφ , (2.3.6)

i i δζ F = 0, δζ¯ F = −iζγ¯ − iγ Diψ + iσψ − iλφ .

2.3.1 Central Charge

Plugging (2.3.6) into (2.1.7) we can check that the SUSY algebra is closed up to a gauge transformation. More interestingly if we consider a particular background conﬁguration for the vector super ﬁeld, namely ¯ σb = σeb, Ai,b = λb = λb = Db = 0 (2.3.7)

10 the action of (2.3.6) on the chiral super-ﬁeld gives

h i α ¯β i δζ, δζ¯ Q = −ζ ζ 2iγαβ∂i + 2iαβσeb Q (2.3.8)

Thus σeb in (2.3.7) provides the value for the central charge. In other words, one can think of the central charge as lying in a background vector multiplet. Furthermore (2.3.7) is a SUSY preserving background conﬁguration as can be explicitly checked by looking at (2.3.2) and observing that SUSY transformations only depend on ∂iσ.

One can think of the background configuration (2.3.7) as gauging a particular flavor symmetry and give it a very large kinetic term, so that it does not propagate and the field can be considered frozen in the given configuration. Because the kinetic term depends on the inverse of the gauge coupling 1/g2, this procedure is referred to as weakly gauging a flavor symmetry.

A weakly gauged flavor symmetry not only introduces a central charge but also a real mass for the chiral fields (2.2.7). The reverse can also be proven, that is any real mass in the theory can be interpreted as background configuration of a given weakly gauged flavor symmetry. This statement is of particular relevance for the work presented in this thesis. In fact we will only introduce real masses through weakly gauging global symmetries as this allows for a much simpler analysis when working with a dual pair. Knowing which background configuration provides a given real mass configuration in the electric theory, allows to straightforwardly guess the behavior of the magnetic side and the masses induced there .

q m ≥ Z m = m2 + m2 Massive representations satisfy , where in general C R. When the equality is satisfied the super-field is in a BPS configuration. It follows a very important result. In the absence of any mass term in the super-potential, that is if complex mass terms vanish, any chiral super-field charged under the weakly gauged flavor symmetry is in fact BPS.

Notice that because of the absence of anomalies, any global symmetry present in the theory can be weakly gauged.

11 2.4 Topological U(1)J symmetry

We have already mentioned that the existence of a scalar component σ in the vector super- multiplet, allows the possibility of a non-vanishing VEV for the vector super-ﬁeld. At a more careful analysis, σ is not, however, the only scalar that can acquire a vacuum expectation value. In fact a 3D vector carries a single propagating degree of freedom and may be dualized into a scalar, γ.

In 3D, the Hodge dual of the ﬁeld strength tensor, ?F, is a one-form which may locally be written with respect to a scalar γ,

[?F]i = dγ i . (2.4.1)

The ﬁeld γ is known as the dual photon and encodes the degrees of freedom of Ai. Due to charge quantization, the dual photon γ is periodic and thus takes values on S 1. From this it follows that the topology of the 3D N = 2 Coulomb branch for a U(1) gauge theory is R × S 1. It is useful to combine the scalars into a complex modulus,

φ = σ + iγ. (2.4.2)

(2.4.2) is suggestive that the vector super-ﬁeld can be dualized to a chiral super-ﬁeld, with φ being its lowest complex scalar component. We will outline this construction in the next section.

The existence of a dual photon, also implies the presence of an extra U(1) global symmetry. Eq (2.4.1), together with Maxwell’s equation d ? F = 0, in fact implies that the ?F is divergence

jk free. We can thus deﬁne the following conserved current Ji ≡ i jkF . This current is associated with an extra U(1) global symmetry that, because of its nature, gets the name of topological U(1) or U(1)J . The U(1)J plays a very important role in studying the dynamics of the low energy physics in 3D.

12 2.5 Linear superﬁeld

Lastly we can introduce the linear multiplet which can be arranged in the linear super-ﬁeld Σ.

A linear super-ﬁeld Σ satisﬁes D2Σ = D¯ 2Σ = 0 (2.5.1)

If written explicitly in terms of its components, the above condition can be understood as the

i supersymmetric generalization of current conservation ∂ Ji=0 [45]. Any divergence-free current

Ji is naturally included as one of the component of a linear super-ﬁeld and, similarly, each linear super-ﬁeld is associated to a symmetry of the theory whose conserved current is one of the component of the multiplet.

We have encountered a conserved current in the previous section where we showed that

i the dual field strength ?F = J is a divergence free one-form (2.4.1), ∂ Ji = 0, so that 3D gauge theories carry a global topological symmetry, U(1)J . We then expect that ?F should belong to a linear super-field Σ?F. This is in fact the case and the appropriate Σ?F can be directly constructed from the U(1) gauge theory vector super-field.

For each vector multiplet V, consider the new super-field Σ = DDV¯ . By construction Σ satisfies (2.5.1) and it is therefore a linear super-field. Explicitly, the linear super field associated to (2.3.1) is

i 1 i i 1 Σ ≡ − αβD¯ D V = σ + θλ¯ + θλ¯ + θγiθ¯J + iθθ¯D + θ¯2θγi∂ λ − θ2θγ¯ i∂ λ¯ + θ2θ¯2∂2σ. (2.5.2) 2 α β 2 i 2 i 2 i 4

The Σ introduced above contains ?F so it can be identified with Σ?F. Henceforth we will drop the extra label ?F and simply refer to the linear super-field containing dual field strength as Σ.

Using (2.5.2), we can write the YM action in terms of the linear super-ﬁeld Σ

1 Z S = d3 xd4θ Σ2. (2.5.3) YM e2

Once expanded in terms of its component, (2.5.3) is completely equivalent to (2.3.4)

13 Linear super-field are extremely useful in 3D SUSY. We will see in the next section that they provide the natural supersymmetric generalization of Chern-Simons terms, but even more im- portantly the use of Σ instead of V allows a formulation of U(1) theories in terms of a Chiral super-field Φ. The complex scalar (2.4.2) may then be naturally understood as the lowest component of Φ which arises as the dual to the linear superfield (2.5.2). Let us see this.

The Lagrangian density which provides the YM action (2.5.3) can analogously be written as a function of a general real superﬁeld Σ0 with the addition of a chiral superﬁeld Φ acting as a

Lagrange multiplier to enforce the linear superﬁeld conditions on Σ0, D2Σ0 = D¯ 2Σ0 = 0 ! Z Σ02 Σ0 L = d4θ − (Φ + Φ†) . (2.5.4) YM e2 2π

If the path integral over Σ0 is instead performed, one obtains a description of the physics in terms of the Chiral super-ﬁeld Φ. Integrating out Σ0 leads to the condition

Σ0 Φ + Φ† = 4π (2.5.5) e2 from which one may write LYM as a function of the chiral super-field Φ. This is now a dual description of the vector superfield—encoded into a linear superfield—in terms of a chiral superfield whose lowest components are

† † Φ + Φ |θ=0 = 2Re[φ] ≡ 2ϕ Φ − Φ |θ=0 = 2iIm[φ] ≡ 2iγ. (2.5.6)

Using (2.5.5), the duality straightforwardly reproduces the claimed identiﬁcation of (2.4.2) with the lowest component of Φ

2 jk ϕ ∼ 2πσ/e ∂iγ ∼ i jkF . (2.5.7)

In other words, the scalar component of the vector superfield is identified with the real part of the complex scalar in Φ, while its imaginary part can be identified with the dual photon.

14 2.6 Chern-Simons terms

In 3D, the Yang-Mills action is not the only gauge invariant combination of the gauge ﬁelds.

Chern-Simons terms can also be added to the action. While we do not discuss theories with non-vanishing Chern-Simons terms in any detail, they still play an important role in the theories discussed in this thesis as they are generated at one-loop. It is thus valuable to provide a brief discussion.

The supersymmetric generalization of Chern-Simons terms for generic non-abelian theories cannot be written down in super-space, yet it is possible to provide an explicit expression in terms of its component ﬁelds Z " # 3 i jk 2 † S = d x Tr A ∂ A + i A A A + 2Dσ − λ λ , (2.6.1) CS i j k 3 i j k

In the Abelian case, instead, supersymmetric CS terms can be written in a simple form involving both the vector and the linear multiplet: Z 3 4 S CS ≡ d x d θ ΣV. (2.6.2) such an expression can be generalized to mixed Chern-Simons terms Z Z mix 3 4 3 4 S CS ≡ d x d θ Σ1V2 = d x d θ Σ2V1 (2.6.3)

Mixed Chern-Simons terms play an important role in the discussion of the one-loop induced charges presented in the next chapter.

If any of the ﬁelds in (2.6.3) is taken to be in the background conﬁguration described above, one obtains a Fayet-Iliopoulos term Z 3 4 S FI = ξ d xd θ V (2.6.4)

where ξ is the scalar component of the linear super-ﬁeld Σb. This observation is very important. It shows that one possible way of obtaining a FI term is by weakly gauging the topological

U(1) instead of a ﬂavor symmetry. Since the conserved U(1)J current is contained into the

15 linear super-ﬁeld Σ, the U(1)J gauging introduces terms ∼ VJ Σ. VJ being the vector super-

field containing the photon for the now gauged U(1)J symmetry. If VJ is taken in a frozen background configuration, its scalar component provides a FI term. The relation between FI term and weakly gauging a U(1)J symmetry is thus analogous to the relation between central charges and weakly gauged flavor symmetries.

Before concluding this chapter is worth putting all the pieces together and explicitly write down the most general YM Lagrangian in 3D. For an abelian theory it has the form ! Z 1 k ξ L = d4θ − Σ2 − ΣV − V (2.6.5) e2 4π 2π

The ﬁrst term being the tree level supersymmetric YM action, the second being a supersymmetric CS-term while the last a FI contribution. For non abelian theories ξ must be set to zero and the Chern-Simons term must be written explicitly in terms of the component ﬁelds as in (2.6.1).

In the next chapter we will provide details on how each one of terms above affect the physics.

16 CHAPTER 3 COULOMB BRANCH AND THE U(1) CASE

In this chapter we will start discussing in details the moduli space of N = 2 3D theories with particular attention to Coulomb branches as they will play a crucial role in the discussion outlined in subsequent chapters. In the U(1) case things are simpler. We thus start discussing the abelian case. Part of the material presented in this chapter can be found in [19, 20, 44]

It is important to notice that in 3D the gauge coupling is not dimensionless. U(1) theories, though simpler, are not IR free and they will present interesting dynamics which affects the topology of the moduli space. We will see that in the presence of matter quantum corrections drastically change the classical picture. But let’s start with describing the latter ﬁrst.

3.1 3D N = 2 U(1) gauge theory and moduli space of vacua

As we saw in the previous chapter, the most generic Lagrangian for a U(1) gauge theory can be written as ! Z 1 k ξ L = d4θ − Σ2 − ΣV − V (3.1.1) e2 4π 2π where (3.1.1) is written in terms of both the vector (V) and linear super ﬁeld (Σ) discussed extensively in the previous chapter.

The ﬁrst contribution is the tree level gauge U(1) Lagrangian. The second term is the su- persymmetrised version of the Chern-Simons term. The last term is instead the more familiar Fayet-Iliopoulos term. In most of the examples that we will discuss, we will set both k = 0 and

ξ = 0 yet it is instructive to have a grasp on the effect of CS terms. In particular we will see below that despite being absent at tree level, CS terms could be generated radiatively at one loop.

Now consider adding matter ﬁelds Qi with U(1) charges ni and generic real masses mR,i.

17 Following [44], the semi-classical potential is  2 2 X X e  2  2 2 V =  2πni Qi − ζ − kσ + (m ,i + niσ) Qi (3.1.2) 32π2   R i i where σ is the lowest component of the linear super ﬁeld (2.5.2). Here we set Wtree = 0. If a non- trivial super-potential is present, (3.1.2) is easily modiﬁed by adding the F-term contribution.

Consistency of the theory only allows certain value for the CS sector k 1 X k + n2 ∈ (3.1.3) 2 i Z i we will only work with theories where k is allowed to be set to zero. For more details, we refer to [20, 44].

Few observations are due. A non-vanishing value of the scalar component σ gives an effective real mass to the matter ﬁelds Qi,

mi(σ) = mR,i + niσ. (3.1.4)

Thus for generic values of the real masses mR,i, Qi is massless only at σQi = −ni/mR,i. Away from

σQi , Qi is massive and can be integrated out generating one loop CS and FI terms (see below for more details). It follows that in (3.1.2) both k and ξ must be substituted by their effective values

1 X 2 k = k + n sign m + n σ (3.1.5) e f f 2 i R,i i i 1 X ξ = ξ + n m sign m + n σ (3.1.6) e f f 2 i i R,i i i

Semi-classical Supersymmetric vacua need to minimize (3.1.2), this implies X 2 2πni Qi = ξe f f + ke f f σ, &(mR,i + niσ)Qi = 0, for all i. (3.1.7) i

Solutions of (3.1.7) can be labeled as follows [44]:

1. Higgs vacua or Higgs branches, with hQii , 0 for some i. This implies that σ = σQi for

generic real masses mR,i. Below we will set mR,i = 0, then Higgs vacua are only allowed at the origin of the Coulomb branch (see below).

18 2. Coulomb vacua or Coulomb branches, solutions with σ , 0. In particular we will call Coulomb vacua only those solutions in which a continuous set of σ values is allowed. In particular

this implies hQii = 0 and ξe f f = ke f f = 0.

3. Topological vacua, cases in which hQii = 0 but only discrete values of σ are allowed. We will not discuss this case any further.

3.2 Coulomb branch and preliminary considerations

Let us recall the explicit expression for the 3D vector super-ﬁeld

1 V = −iθθσ¯ − θγiθ¯A + iθ2θλ¯ − iθ¯2θλ + θ2θ¯2D, (3.2.1) i 2 where we have explicitly separated the 3D vector Ai from the gauge scalar σ ∼ A3. While in N = 1 4D case no vacuum expectation value is allowed for the vector super-ﬁeld, N = 2 3D

SUSY admits Coulomb vacua. In fact if hQii = 0 and ke f f = ξe f f = 0, the σ component of the vector super-ﬁeld (3.2.1) may acquire a vacuum expectation value (VEV).

Because of the existence of the dual photon γ (2.4.1) the topology of the Coulomb branch is classically R × S 1. The transformation properties under the previously introduced topological U(1) symmetry, suggest that quantum corrections modify this picture considerably. Let’s see this.

The U(1)J charge is defined as Z Z 2 0 2 F12 qJ = d x J = d x . (3.2.2) J 2π it then follows that in a U(1) gauge theory with F flavors, matter fields are invariant under the action of the topological U(1). The dual photon is instead charged under the U(1)J , and its action corresponds to a shift of γ and thus to going around the circle. A VEV of (2.4.2) breaks the U(1)J spontaneously with the dual photon arising as the Goldstone boson associated to the breaking.

19 We have already seen that classically the moduli space has a Coulomb branch with topology

R × S 1 and a Higgs branch that is parametrized by gauge neutral composite operators. The two branches intersect in one point, at σ = 0. At the intersection point the value of γ is unconstrained and we can move along the circle through the action of the U(1)J . On the other side the operators describing the Higgs branch are invariant under this action. It then follows that the intersection point of the Coulomb and the Higgs branch must be shrank to a point by quantum corrections, changing the topology of the Coulomb branch from the initial cylinder to two independent cones.

From its definition (3.2.2), qJ can be interpreted as magnetic charge, in 3D B ∼ F12. As we will describe in next chapters in more details, fields charged under the topological U(1) are in fact monopole operators and their action on the vacuum, with some non-trivial subtleties, create BPS vortices. For the moment we will not delve more into this connection and move to other properties of U(1) theories. We will come back often in subsequent sections to the connection between fields charged under the topological U(1) and monopole operators. We move now to compute explicitly the one-loop correction of the metric on the moduli space, providing a more formal derivation of the argument outlined above.

3.3 One-loop correction

In order to compute quantum correction on the moduli space metric, it is more useful to use a mixed variables basis: (σ, γ). The former being the lowest component of the linear super-ﬁeld Σ and the latter the imaginary part of the lowest component of the chiral super-ﬁeld Φ [19]. To account for quantum corrections we need to generalize slightly the duality described in Section 2.5.

The most general effective Lagrangian for a linear superﬁeld is a real function of Σ Z 4 Leff = d θ f (Σ). (3.3.1)

20 and for a generic f (Σ),(2.5.5) generalizes to

∂ f (Σ0) Φ + Φ† = 2π (3.3.2) ∂Σ0 from which one may write now the Lagrangian in terms of the chiral super-ﬁeld Φ Z 4 † Leff = d θ K(Φ + Φ ), (3.3.3) where K is the Legendre transform of f .

Including now one-loop contributions, the low-energy effective action reads ! Σ2 Σ f (Σ) = + Σ log (3.3.4) e2 e2

Plugging (3.3.4) back into (3.3.2), that is performing the Legendre transform, we thus obtain

Σ 1 1 Φ = + log(Σ/e2) + + iγ (3.3.5) 2e2 2 2

The metric on the moduli space is deﬁned as

ds2 = K00(Φ + Φ¯ )dΦdΦ¯ . (3.3.6)

Thus we need to compute the second derivative of the Legendre transform in (3.3.3). This can be done noticing that the inverse of a Legendre transform is again a Legendre transform, or K0(Φ + Φ¯ ) = Σ01. Differetiating once more with respect to the chiral ﬁeld Φ, we obtain

!−1 !−1 ∂(Φ + Φ¯ ) 1 1 K00(Φ + Φ¯ ) = = + (3.3.7) ∂Σ 2e2 Σ which can be plugged back into (3.3.6)

!−1 ! ! ! ! 1 1 1 1 1 1 1 1 ds2 = + + dσ + idγ + dσ − idγ . (3.3.8) 2e2 σ 2 2e2 σ 2 2e2 σ

(3.3.8) can be written in a more concise form as

! !−1 1 1 1 1 1 ds2 = + dσ2 + + dγ2 (3.3.9) 4 2e2 σ 2e2 σ

1In this paragraph we are using interchangeably the same letter for the ﬁelds and their VEVs, that is Φ ≡ hΦi.

21 Figure 3.1: In presence of matter, the Coulomb branch of the U(1) theory splits into two independent branches and near the origin of the moduli space, where the Higgs branch intersects the Coulomb branches, the moduli space looks like the intersection of three cones.

From (3.3.9), we can explicitly see the phenomenon outlined in the previous section. For large |σ| the metric (3.3.9) reduces to the canonical form. Yet σ, the scalar component of Σ, is no longer a good variable throughout the Coulomb branch. In fact once the one-loop correction is taken into account, the origin of the Coulomb branch, σ = 0, becomes a singularity. On the other side from (3.3.5) we can see that the chiral super ﬁeld Φ becomes −∞ as we approach the singularity. It thus follows that the Coulomb branch of a U(1) theory splits into two holomorphically independent regions which are therefore parametrized by two chiral superﬁeld V± ∼ exp ±Φ.

The expression of V± in terms of the chiral super-ﬁeld Φ is only true in the semi-classical limit, that is for large VEVs and far away from the origin, this explains the sign ‘∼’. In other words

V± satisfy the constraint V+V− = 1 only semi-classically. Once quantum correction are taken into account, this constraint is modiﬁed and the two chiral ﬁelds become unconstrained V−V+ ∼ 0. Near the origin, where the Higgs branch and the Coulomb branch intersect, the moduli space looks like the intersection of three cones, see Fig. 3.1.

Both V+ and V− are charged under the topological U(1), while they are singlet under all the other global and gauge symmetries. From their semi-classical deﬁnition, it follows that they have qJ respectively +1 and −1. As we will describe in the next section, such operators can acquire charges under the other global U(1)’s through one-loop induced Chern-Simons terms.

22 3.4 Loop-induced global charges

One-loop induced U(1) charges are extensively used to understand the low-energy behavior of the 3D N = 2 theory, including in the non-abelian case. We will careful explain here how it works and the crucial role played by Chern-Simons terms in this construction.

The presence of a Chern-Simons term in the Lagrangian

µνρ µνρ LCS ∼ kTr Aµ∂νAρ = kTr AµFµν (3.4.1) modiﬁes Gauss’ law (i.e. the A0 equation of motion) as follows 1 k − ∂ F = ρ − F (3.4.2) e2 i 0i matter 2π 12

δLmatter where ρmatter = is the matter contribution to the electric charge density, while a non- δA0 vanishing k value introduces a “magnetic” contribution.

The most practical way of getting rid of the gauge redundancy working within the canonical

2 scheme, is to impose the Coulomb gauge, i.e. A0 = 0. Thus A0 becomes a non-dynamical ﬁeld and Gauss’ law has to be imposed as a constraint implying that any ﬁeld with charge qJ acquires an electric charge

CS qelec = −kqJ. (3.4.3)

This effects it is also present for theories with no tree level CS term, that is if k = 0. This is because, as we have already mentioned a few times, Chern-Simons terms are generated radia- tivelly [20, 46, 47]. Let’s see now how this works in details.

If all real masses mR,i are set to zero, matter ﬁelds are only massless at σ = 0, while moving out in either direction of the Coulomb branch generates a real mass mi(σ) = niσ. It follows that when we compute the effective action for the light degrees of freedom for σ , 0, we need to integrate out these heavy ﬁelds. Then the vacuum diagram in Fig. 3.2 induces a term 1 Z d3q d3 p S [A] = n2 A (−p)A (−q)[−iΠlm(p)] (3.4.4) e f f 2 i (2π)3 (2π)3 l m

2 δL Recall that because of the antisymmetry of Fi j, A0 has no momentum conjugate, that is Π0 ≡ = 0. Thus δ∂0 A0 the Coulomb gauge allows to canonically quantize the other Aµ components, circumventing the problem.

23 A` Am

Figure 3.2: Vacuum polarization diagram which generates one-loop Chern-Simons terms.

lm where ni is the U(1) charge of the heavy fermion running in the loop and Π is deﬁned as " # Z d3k −i −i Πlm(p) = −ie2 Tr γl γm (3.4.5) (2π)3 (p + k) − m(σ) k − m(σ)

The mass m(σ) really represents the real mass of the fermion running in the loop which is induced by moving out in the Coulomb branch, that is mi(σ) = niσ. In the formula above we suppressed the index to make the formula less cumbersome and avoid confusion.

For large m,

lm lmk Π → −i pk sign[m(σ)] + ... (3.4.6)

The dots indicate terms which need to be regularized. Plugging (3.4.6) back into (3.4.4), we see that each electrically charged fermion of the theory, generates an effective CS term at one loop

2 with coefﬁcient keff,i = ni sign(niσ). Carefully integrating out all the massive fermions along the Coulomb branch we have 1 X k (σ) = k + n2sign(n σ) (3.4.7) eff 2 i i i where ni are the electric charges of matter ﬁelds. This result exactly reproduces the mR,i = 0 limit of (3.1.5).

From (3.4.7) and (3.4.3) we can read off the one-loop electric charge induced to the U(1)J charged ﬁeld   X CS  1 2  q = −k qJ = k + n sign(n σ) qJ . (3.4.8) elec e f f  2 i i  i

In presence of non-vanishing real masses mR,i,(3.4.8) is easily generalized     X X CS  1 2   1 2  q = k + n sign(m (σ)) qJ ≡ k + n sign(m + n σ) qJ . (3.4.9) elec  2 i i   2 i R,i i  i i

24 a` Am

Figure 3.3: Vacuum polarization diagram which generates one-loop mixed Chern-Simons terms. ai labels a vector associated to a weakly gauge U(1) theory.

This formula gives the one-loop induced electric charge. In a similar way ﬁelds which have a non-vanishing qJ charge also acquire global U(1) charges at one-loop. One-loop global U(1) charges are generated by mixed Chern-Simons terms (2.6.3) which arise if we weakly gauge the particular U(1). Again let us see this explicitly.

Weakly gauging a particular U(1) global symmetry requires the introduction of an interaction among the weakly gauged vector boson, that we will call ai, and the heavy fermions charged under such U(1). Thus when we integrate out heavy ﬁelds, loop diagrams involving both Ai and ai (see Fig. 3.3) need to be also taken into account. Going through a similar calculation as before, they induce a one loop mixed Chern-Simons terms (2.6.3)

Lmixed ∼ mix i jk CS keff (σ)Tr aiF jk (3.4.10) where 1 X kmix(σ) = n n sign(n σ) (3.4.11) eff 2 iei i i eni representing the charge of the i-th matter ﬁeld under the weakly gauged global U(1).

kmix U For eff , 0, the mixed CS terms will modify Gauss’ law for the weakly gauged (1) symmetry (i.e. the a0 equation of motion), which in turns will induce a U(1) charge for ﬁelds which have a non-vaninishing qJ or F12 charge. Explicitly   1 X  q =  n n sign(n σ) qJ (3.4.12) e 2  iei i  i

In summary, so far we have seen that U(1) theories with matter:

25 Have Higgs vacua parametrized by gauge invariant operators. • The Coulomb branch splits into two one-dimensional branches parametrized by two • quantum mechanically un-constrained operators which are given semi-classically by

V± = exp(±Φ).

In addition to the global symmetries of the initial theory there is an extra U(1), the U(1)J . • Matter ﬁelds are not charged under the topological U(1) while V± are.

Through one-loop generated CS terms, ﬁelds charged under the U(1)J , in particular V±, • acquire both electric (3.4.9) and global (3.4.12) U(1) charges.

In the next section we will see that the loop-induced charges play a crucial role to deﬁne the correct dynamics of the low-energy physics of U(1) theories with ﬂavors.

3.5 U(1) with ﬂavors

Consider a U(1) theory with F ﬂavors. The quantum dynamics of such theories is constrained by the global quantum numbers of the ﬁelds

SU(F) SU(F) U(1)A U(1)J U(1)R

Q 1 1 0 0 (3.5.1) Q¯ 1 1 0 0

M ≡ QQ¯ 2 0 0 Furthermore the theory has two Coulomb branches described by the previously introduced operators V±. Both V+ and V− are charged under the topological U(1) thus they acquire charges under the global U(1) at one loop. From the table above and (3.4.12) we can read off the U(1) charges of the two Coulomb branch operators

U(1)A U(1)J U(1)R

V+ −F +1 F (3.5.2)

V− −F −1 F

26 The opposite charge for the U(1)R is due to the fact that the fermionic component in the chiral super-ﬁeld has R-charge −1.

As already mentioned in previous sections, in 3D even the U(1) theory is not IR free, thus we expect the theory to ﬂow to a non-trivial IR ﬁxed point with a non-vanishing super-potential.

The super-potential has to respect all the global symmetries and have R-charge 2. This strongly constrains the possible dynamics.

3.5.1 F = 1

For F = 1 the Higgs branch and the two Coulomb branches are all complex manifolds of complex dimension 1. In this case the only Superpotential allowed by the symmetries in (3.5.1) and

(3.5.2) is

W = −MV+V− (3.5.3) thus SQED in 3D with one ﬂavor ﬂows to the XYZ model and the three cones which intersects near the origin of the two Coulomb branches are actually related by triality exchange symmetry.

3.5.2 F > 1

Now consider a U(1) gauge theory with a generic F. Again we can perform a similar analysis, using the symmetries in (3.5.1) and (3.5.2) to infer the low energy behavior. This gives the super-potential

1/F W = −F(V+V− det M) (3.5.4) this super-potential provides the correct description of the low-energy theory away from the origin but it provides a singular scalar potential at the origin. This suggests that we need additional degrees of freedom to be able to describe the theory in a generic point of the moduli space.

27 This description will be provided in the next section using results from mirror symmetry. A careful analysis of U(1) theories provides a good background to understand the behavior of the Coulomb branch in non-abelian theory with ﬂavors which will be described in the next chapter. In the second part of this thesis we will further generalize the analysis to theories with antisymmetric tensors. Furthermore, mirror symmetry also sheds light in the connection between monopole operators and Coulomb branches. This will be addressed in Chapter5.

3.6 Mirror Symmetry

3D U(1) theories with F fundamentals in N = 4 have been studied and understood by a series of beautiful papers [18, 23, 48–50]. They are a subset of a bigger list of theories which show a very peculiar kind of duality named “Mirror symmetry”. N = 4 theories in 3D have a global

SO(4) SU(2)L×SU(2)R, with SU(2)R the R symmetry of N = 1 in 6D or N = 2 in 4D, and SU(2)L associated with rotations in the three directions reduced in going from 6 to 3 dimensions. Mirror symmetry exchanges [18]:

SU(2)L and SU(2)R. • The Coulomb and the Higgs branch. • Mass and Fayet-Iliopoulos terms. •

By promoting some of the 3D N = 4 coupling constants to background super-ﬁelds and analyzing their transformation properties under the global SO(4), it is possible to prove a few non-renormalization theorems. Namely that [18]

Higgs branches are not renormalized by quantum effects while Coulomb branches are. • Mass terms only affect the metric on the Coulomb branch. • Fayet-Ilipoulos terms only affect the metric on the Higgs branch instead. •

28 Mirror symmetries are very interesting dualities. In fact from the properties above and the inter-change of the Higgs/Coulomb branch, it follows that the quantum modiﬁcations of the moduli space in one side of the duality should be already present at the classical level in the other side. We will see examples of this phenomenon shortly, for instance the extensively discussed V± splitting of the Coulomb branch of U(1) theories it is evident classically looking at the Higgs branch of the Mirror dual. In the following chapters we will also see that Mirror symmetry can be understood as particle-vortex duality and we will explicitly build the map. This will provide a case study to see in action how Coulomb branch operators could show up as particle-like operators in the dual side. Therefore it is important to discuss these dualities in details in order to lay the ground for the subsequent discussion on monopole operators and to understand their interpretation across the duality. Also by breaking N = 4 to N = 2, we can use mirror symmetry to provide the low energy description of U(1) theories with more than a single ﬂavor.

Flowing from N = 4 to N = 2 in 3D is in many ways analogous to the ﬂow from N = 2 to N = 1 in 4D. This can be achieved noticing that the N = 4 vector multiplet contains a N = 2 vector and chiral multiplet while N = 4 hyper-multiplets contain a chiral and an anti-chiral

N = 2 multiplets. Giving a mass to Ψ, the chiral component of the N = 4 vector multiplet

i) Breaks N = 4 to N = 2.

ii) The chiral component of the N = 4 vector multiplet decouples leaving only the N = 2 vector multiplet in the low-energy theory.

iii) Each hypermultiplets now should be read as a ﬂavor of N = 2.

Thus a mass for Ψ is enough to ﬂow exactly to the kind of N = 2 theories which we are interested in. U(1) theories with F ﬂavors in N = 4 are known to have a “mirror dual” in terms

F of an U(1) /U(1) gauge theory with F hypermultiplets Qi with charge 1 under the i-th U(1) and (-1) under the (i + 1)-th U(1), while the sum of the F U(1) is ungauged. We will not discuss the N = 4 case further, nor provide any check for the duality above, for that we refer to the

29 original literature [18, 23, 48–50]. We will now discuss how to ﬂow to the N = 2 case and gain understanding on the theories discussed above.

Because the chiral component of the N = 4 vector super-ﬁeld Ψ is neutral, we are allowed to add an extra singlet S which couples to Ψ through

Wmass = mSΨ (3.6.1) thus providing a mass for Ψ. In N = 4,(3.6.1) can be seen as a dynamical Fayet-Iliopoulos term3. Mirror symmetry maps FI terms into mass term. Following [18], (3.6.1) induces a superpotential term to the mirror dual F mirror X Wmass = Siqieqi (3.6.2) i where the qi and eqi represents respectively the chiral and the anti-chiral N = 2 multiplets contained in each Qi hyper-multiplet.

Let us first compare the Coulomb vauca of the electric theory with the Higgs ones of the mirror dual. Solving both the D-term and the F-term equations, in the magnetic side, we are left with only two Higgs branches parametrized by the gauge invariant operators B+ = q1q2...q3 and B− = eq1eq2...eq3. B± parametrize two 1-dimensional branches which only intersect at the origin B+ = B− = 0. Mirror symmetry identifies V+ and V− with B+ and B−. As anticipated, in the mirror dual the quantum splitting of V± is already visible at the classical level. In Chapter5 we will further discuss the identification V+ ∼ B+ and V− ∼ B− in terms of particle vortex duality but let us briefly look at the Coulomb branch of the magnetic theory.

The Coulomb branch of the magnetic theory must be identiﬁed with the Higgs branch of F2 Mi the electric theory. The latter is a parametrized by the meson operators ¯i = subjected to M j Ml = M j Ml 2F − 1 the classical constraints ¯i k¯ k¯ ¯i. The Higgs branch of the electric theory is thus dimensional. On the magnetic side the F singlets Si and the F − 1 dual photons make up for the dimensions. Furthermore we can explicitly identify operators on both side of the duality. We

3 In N = 4 the most generic FI term ξ is a SU(2)R triplet which in N = 2 language can be taken as a real and a complex parameter ξ ≡ (ξr, ξc). Only the real part ξr translates directly to FI term in N = 2, while ξc can appear in the super-potential or more generally in quantities which are constrained by holomorphicity.

30 Mi will only outline the identiﬁcation for the diagonal element of the meson operators ¯i which is straightforward. Mirror symmetry exchanges mass and FI terms. Mass terms in the electric ∼ m Mi F S theory are of the form i ¯i while the singlets i are dynamical FI terms in the magnetic Mi ∼ S theory. Thus we are led to make the identiﬁcation ¯i i.

In summary a N = 2 U(1) theory with F ﬂavors is dual to U(1)F /U(1) gauge theory with F

ﬂavors (qi,eqi) with charge 1 under the i-th U(1) and (-1) under the (i+1)-th U(1), F chiral singlets

mirror PF Si and super-potential Wmass = i Siqieqi. The sum of the F U(1) is ungauged. Away from the origin of the moduli space the gauge group is fully broken and we derive the description previously obtained in (3.5.4).

We can show that this mirror dual is also compatible with the derived exact result for the

F = 1 case (3.5.3). To do this we will start from the F = 2 and ﬂow to F = 1 adding a complex mass deformation. The F = 2 theory is dual to a N = 2 U(1) theory with super-potential

WF=2 = S1q1eq1 + S1q1eq1. (3.6.3)

2 A complex mass in the electric theory consists in the addition of an operator of the form mM2 which maps to a complex FI term in the dual ∼ mS2 which needs to be added to (3.6.3). Solving the equation of motion ∂W/∂S2 = 0 induces a VEV to q2eq2 which breaks the gauge group and leaves three one-dimensional branches parametrized by (S1, q1,eq1) with super-potential

WF=1 = S1q1eq1 (3.6.4)

identifying the three ﬁelds with (V+, V−, M), reproduces exactly the result we obtained in (3.5.3) [18].

In this chapter we have acquired some familiarity with the low-energy behavior of 3D N = 2 U(1) theories. We have also outlined in some details, some of the techniques involved in the analysis of Coulomb and Higgs branches. We are now ready to generalize what presented here to the non-abelian case. In the next chapter we will mainly deal with SU(N) theories with fundamentals which also present new interesting features worth a close and careful analysis.

31 CHAPTER 4 THE NON-ABELIAN CASE

We now move to the study of the general non-abelian case, focusing for the most part on the

SU(N) case. As we will see many of the features encountered in the previous chapter while discussing the behavior of the Coulomb branch for U(1) theories, carry over to SU(N). In fact on a generic point of the Coulomb branch SU(N) → U(1)N−1 and we can describe those branches using operators which are the natural generalization of V±.

Despite many similarities, the non-abelian case allows for non-perturbative corrections to the super-potential which dramatically change the behavior of the Coulomb Branch. The last part of this chapter is devoted to a careful discussion of non-perturbative corrections. For the moment we will only consider theories with fundamental and anti-fundamental matter ﬁelds.

The material presented here, though, represents the fundaments for what we are going to discuss in the second part of this thesis.

4.1 The non-abelian case

The most general Lagrangian for a non-abelian simple gauge group differs from (3.1.1). ξ is set to zero by gauge invariance, while the parity anomaly constraint on CS terms (3.1.3) takes the form 1 X k + d (R ) ∈ (4.1.1) 2 3 i Z i In what follows we will set k = 0. This will be always possible as we will only consider theories that are also well deﬁned in 4D and respect the 4D gauge anomaly condition.

In presence of matter, both Higgs and Coulomb vacua are allowed and will be discussed extensively. Let us ﬁrst look at Coulomb vacua. The 3D vector multiplet is the natural general-

32 ization of (3.2.1) " # 1 V = −iθθσ¯ a − θγiθ¯Aa + iθ2θλ¯ a − iθ¯2θλa + θ2θ¯2Da T a, (4.1.2) i 2 where each of the component generalizes to a Lie-algebra value function. T a are the generators of the gauge group. Henceforth we will suppress the gauge indices as it will be clear from the context whether a given function is matrix valued or a simple function.

In absence of Chern-Simons terms and additional matter ﬁelds, σ has no potential. Thus σ is a ﬂat direction and can be diagonalized by a gauge transformation. The general scalar VEV can

i be written as hσi = ς Hi, where Hi are elements of the Cartan subalgebra. This VEV breaks the gauge group to its maximal Abelian subgroup, e.g. SU(N) → U(1)N−1. In order to remove the leftover gauge redundancy, one must impose additional constraints. The most common choice is to assume

αk · σ ≥ 0 (k = 1, ..., rG) , (4.1.3) where the αk are the simple roots and rG is the rank of the group. These conditions restrict σ to a Weyl chamber. Unless otherwise stated, we focus on SU(N), which is broken to U(1)N−1 in the bulk of the Coulomb branch. In this case, the Weyl chamber is described by the subspace

σ = diag(σ1, σ2, ..., σN), σ1 ≥ σ2 ≥ · · · ≥ σN . (4.1.4)

The eigenvalues σi are not completely independent since σ is traceless, σ1 + ··· + σN = 0. Please

i notice that, although related, σi , ς , in fact the indices in the former case run up to N while only up to N − 1 in the latter. The explicit relation among the two will be described in more details in AppendixB.

For generic values of hσi, the off-diagonal vector bosons become massive so only N − 1 massless vector bosons are left in the theory which are identiﬁed with the photons of the N − 1 unbroken U(1)s. Each photon can be dualized to a scalar γi as we have discussed extensively in the last two chapters. Thus the light degrees of freedom also include N − 1 dual photons which can be naturally parametrized by a diagonal traceless matrix X γ ≡ diag(a1, a2, ..., aN), with ai = 0 (4.1.5) i

33 As it happened in the U(1) case, σ and γ can be combined into a complex scalar which can be seen as the lowest component of a chiral super-ﬁeld Φ. Φ is dual to a linear super-ﬁeld, again as in the U(1) case.

The set of simple roots, αi, provide the most natural basis for the N − 1 U(1)s that generalize

V±

2 Yi ∼ exp Φ · αi/g3 , (4.1.6) where g3 is the 3D gauge coupling and αi are the simple roots. Φ is the chiral superﬁeld whose lowest component contains both the dual photon and the scalar σ. The sign convetion is chosen in such a way that Yi is large for large VEVs of αi · Φ where we expect semi-classical considerations to be valid and (4.1.6) to apply. Closer to the origin of the Coulomb branch, in the region of strong coupling regime, we expect quantum correction to modify the topology of the

Coulomb moduli, this is the explanation of the ∼ sign in (4.1.6). Notice that for each unbroken

U(1), we only have introduced one operator, while in the U(1) case we had both V±. This is due to the fact that because of (4.1.3) Φ · αi has a deﬁned sign and cannot vary from −∞ to +∞.

By now we know well that for each U(1) factor present in the gauge group, there exists an

i i extra, topological, global U(1)J . Call the i-th topological U(1), U(1)J . U(1)J shifts the i-th dual

i photon and therefore induces a charge qJ = 1 to the Yi. Yi is also charged under the other global U(1)s as, depending on the matter content of the theory, one-loop mixed-CS should induce charges as in (3.4.12).

We will extensively discuss that Yi is also associated to a particular 3D instanton conﬁg- uration. The global U(1) charges induced by one-loop effects (3.4.12) are consistent with the counting of the zero-modes in the instanton background.

34 4.2 Coulomb branch pinch from matter

Consider now adding matter to the theory. For simplicity we will mostly specialize to the

SU(N) case but what we describe below can be easily generalized to any non-abelian gauge group. We will also only consider theories with vector like ﬂavor, that is a set of (Q, Q¯) pairs. We will need to wait until the second part of this thesis to discuss what happens when general representations are added to the theory.

Matter fields, Qi and Q¯ i, couple to the lowest component of the vector superfield, σ, accord- ing to 2 a a α β ∼ σ T Q , (4.2.1) Q β i where the T a are group generators in the representation of the Q fields, α and β are gauge indices, and i is a flavor index. On the Coulomb branch, the scalar VEV is an effective real mass for the matter fields,

2 a a α β α 2 h i ∼ h i · α σ T βQ i σ να Q i (no sum over ) (4.2.2)

th where να is the weight of the α vector in the Q representation (see AppendixB for details).

Due to this real mass, the Higgs branch only intersects the Coulomb branch when σ · να = 0.

At these points, the low energy theory is governed by a particular U(1) ⊂ SU(N) with F massless ﬂavors. We expect the Coulomb Branch to look locally like the σ = 0 region of the theories described in the previous chapter. After we include quantum effects, we therefore expect the Coulomb branch to ‘pinch’ and split into two distinct regions, see Fig. 4.1. As in the U(1) case,we need different variables describing either side of the split at the intersection of the Coulomb and Higgs branch. The operators V± are here identiﬁed with a particular combination of the chiral operators Yi in (4.1.6) on the two sides of each singularity making the moduli Yis not viable global coordinates for the Coulomb branch.

The fact that we need different Yi operators is also a consequence of the facts that the Yi one- loop global U(1) charges jump at the singularities. This is because pinches take place where

35 Higgs Branches

˜ Yi Yi0 Yi0

σ ν = 0 σ ν = 0 h 0i · α1 h 0i · α3 Coulomb Branches

Figure 4.1: In presence of matter, the Coulomb branches pinch off where they meet Higgs branches. At these points, σ · να = 0 and there is massless matter.

matter ﬁelds become massless, that is where one or many of the m(σ)i’s in (3.1.4) become zero making the function sign mi(σ) in (3.4.9) discontinuous. A similar conclusion can be obtained counting the fermion zero modes in 3D instanton backgrounds which also jump where matter

ﬁelds become massless. We will provide details on how this happens below.

Even though we just claimed that single Yis cannot be deﬁned globally, there exists, however, a set of globally deﬁned operators that parameterize the Coulomb branch. One such operator is1

YN−1 Y = Yi. (4.2.3) i=1

The continuity of Y introduces an extra constraint among the Yk at each split. The number of globally deﬁned moduli needed to parameterize the Coulomb branch is determined both by the number of singularities and the number of moduli whose charges jump at each of these singularities (more details below).

1 This statement requires some care. Even in absence of real masses, Y might not be globally deﬁned in SU(N) theories with N > 4 and with generic matter representations. In this thesis we will not discuss such cases, we can thus take Y to be continuous and the statement above is true for vanishing real masses. The case of bigger SU(N) groups for N > 4 will be discussed in a upcoming publication [51].

36 The phenomena thus far described are mostly a straightforward generalization of our discussion of the U(1) case, now involving matrix valued function. We will now discuss the major novelty of non-abelian groups. Non-trivial vacuum solutions are possible are in fact allowed for gauge groups with a more complicated topology. Such solutions could affect the low-energy theory through instanton generated contribution to the super-potential. For the SU(N) theories with matter this is in fact the case and non-perturbative corrections need to be taken into account.

4.3 Instanton/Monopole background

Instanton solutions are finite action solutions, this implies that their behavior at infinity is fixed to be a pure gauge configuration. Topologically this means that the point at infinity is identified and instanton solutions in d-dimensions can be seen as maps from S d−1 → G and are thus classified by πd−1(G). We are interested in the three dimensional case. 3D intantons are classified by

π2(G). We have already discussed that in the bulk of the Coulomb branch the scalar component of the vector superﬁeld triggers the breaking SU(N) → U(1)N−1. In the SU(N) case we then ex-

N−1 N−1 N−1 pect to ﬁnd (N − 1) independent instanton solutions as π2(SU(N)/U(1) ) π1(U(1) ) Z .

To construct the allowed topological conﬁgurations explicitly, note that 3D instanton solutions can be obtained by starting with 4D monopoles and compactifying the Euclidean time coordinate. There are many ways to see this. One way is through homotopy theory. In fact

4D monopoles are classified by π2(G), as are 3D instantons. Below we will construct 3D instanton solutions explicitly showing that they are in fact simply 4D ’t Hooft-Polyakov monopoles with the scalar component σ of the vector superfield (4.1.2) playing the role of the adjoint Higgs scalar in 4D monopole configurations. Let us start with the simplest non-abelian case, G = SU(2).

37 4.3.1 BPS condition

Consider a 3D N = 2 SU(2) gauge theory without matter, analogous to a 4D Georgi-Glashow model. From topological considerations we know that each equivalent class of π2(SU(2)/U(1)) ∼

Z contains a single conﬁguration which is topologically stable, labeled by its winding number. We want here to construct such solutions explicitly with a particular care for the fundamental, n = 1, conﬁguration. The derivation reported here follows closely [52].

Consider the energy associated to a given instanton/monopole solution: Z " # 3 1 a a 1 a a E = d x B B + (Diσ )(Diσ ) (4.3.1) 2g2 i i 2

a 1 a where Bi = 2 i jkF j k. The above equation can be re-written as follow Z " ! ! # 3 1 1 a a 1 a a 1 a a E = d x B − Diσ B − Diσ + B Diσ . (4.3.2) 2 g i g i g i

With a bit of algebra, the last term can be cast in terms of the topological charge of the solution under study Z ! Z 3 1 a a 1 2 a a d x Bi Diσ = d S i(Bi σ ) ≡ σ0QM, (4.3.3) g g S ∞ where σ0 is the asymptotic value of the scalar component of the vector super-ﬁeld, hσi = σ0. The topological charge of a given solution (which reduces to the magnetic charge of the 4D monopole solution) is proportional to the integer number which labels the solution. For a wind-

n ing number n solution, its charge is QM = n(4π/g).(4.3.1) then becomes Z " ! !# 4π 3 1 1 a a 1 a a 4π E = σ0n + d x B − Diσ B − Diσ ≥ σ0n (4.3.4) g 2 g i g i g

For a given equivalence class which contains all the solutions with the same winding number n, the stable solution is the one which minimize (4.3.4). This is attained at

1 a a B − Diσ = 0 (4.3.5) g i

(4.3.5) is called Bogomol’nyi equation and instanton/monopole solutions which satisfy it are named BPS. We can now solve (4.3.5) and construct the BPS solution explicitly.

38 4.3.2 Solution of the Bogomol’nyi equation

We have already mentioned that requiring the instanton solution to be a ﬁnite action solution,

a a ﬁxes the asymptotic value both for σ and Ai  a  a x  σ = σ0 r → ∞  r  x j (4.3.6)  Aa =  i ai j r In order to solve (4.3.5), we can write the most generic solution which satisfy the boundary conditions (4.3.6) in a convenient way

n j σa = σ naH( f ), Aa = ai j F(r). (4.3.7) 0 i r

In fact in terms of H(r) and F(r) boundary conditions simplify, as H(∞) = F(∞) = 1 while to avoid the singularity at the origin H(0) = F(0) = 0.

Plugging back into (4.3.5) and deﬁning ρ = gσ0r, we can ﬁnd the expression

1 ρ H(r) = coth ρ − , F(r) = 1 − , (4.3.8) ρ sinh ρ substituting back into (4.3.7), provides the sought solution.

The SU(2) instanton solution generalizes straightforwardly to SU(N) [53,54] by simply embedding the above solutions in the N − 1 SU(2)s contained in SU(N). This means that there are

(N − 1) independent instantons which are labeled by integers (n1, n2, ..., nN−1), indicating the instanton charges. Each index indicating a different SU(2) embedding. The asymptotic behavior is XN−1 ! j XN−1 ! 1 1 rˆ 1 σ = σ0 − nI αI · HI + O , Ai = i j3 nI αI · HI + O , (4.3.9) ∞ r r2 ∞ r r2 I=1 I=1 P I where σ0 ≡ diag(σ1, σ2, ..., σN) = σ0HI. Here HI are the generators of the Cartan sub-algebra,

αI are simple roots, and αI · HI = diag(0, 0..., 1, −1, ..., 0) (see AppendixB for details). Further P deﬁning g0 ≡ nI αi · HI = diag n1, n2 − n1, ..., −nN−1 , one may write (4.3.9) in a more concise form ! j ! g0 1 rˆ 1 σ = σ0 − + O , Ai = i j3 g0 + O (4.3.10) ∞ r r2 ∞ r r2

39 4.4 Non-perturbative corrections

The above described instanton solutions generate non-trivial contributions to the superpotential of 3D N = 2 theories. This was shown by Afﬂeck, Harvey and Witten in the ’80s in a beautiful paper [43]. We will brieﬂy discuss how this happens. For the sake of simplicity we will consider an SU(2) → U(1) theory. Furthermore it is more convenient to describe the

U(1) low-energy physics in terms of σ and the dual photon. The low-energy degrees of freedom will thus be a complex scalar and a Weyl fermion which can be all arranged as previously described in a dualized chiral super-ﬁeld Φ.

In the discussion of instanton generated contributions to the super-potential, a special role is played by fermionic zero modes in the given instanton background. It is known that bosonic zero modes are associated with generators of the symmetries in the theory which don’t preserve the instanton solution. In fact by construction such generators give rise to a different solution with the same energy. In a similar fashion, fermionic zero modes can be obtained acting with super-charges on the instanton vacuum. In 3D N = 2, there are 4 independent supercharges and we thus expect 4 fermionic zero modes in each instanton background. Looking carefully at the gaugino supersymmetry transformations (2.3.2)

i ∗ i δζλ = −γ (Diσ + Bi) ζ, δζ¯λ = −γ (Diσ − Bi) ζ¯ (4.4.1) follows that a BPS instanton/monopole solution satisfying (4.3.5), is preserved by two of the super-charges. This, on the one side clariﬁes further why such solutions are named BPS, and on the other shows that a 3D instanton background has in fact only two zero-modes. It then follows that the gaugino two-point function has a non VEV h0| σ · λ(x) σ · λ(y) |0i , 0. The natural interpretation of this fact is that instanton generate a mass for the gauginos which should be accounted for when we write down the effective theory.

Instantons also generate a non-vanishing potential for the dual photon γ. This happens in a much subtler way, as this potential comes from taking into account the classical instanton interaction which, in 3D, is Coulombic 1/|x| and cannot be neglected. In [7] Polyakov showed

40 that instanton generate an interaction

V(γ) ∼ e−S 0 exp (iγ) (4.4.2)

where S 0 ∼ 4π σ0/g is the one-instanon action.

Putting all the pieces together, the effects of instanton contribution can be written in terms of a single term in the super-potential

Winst = exp (−Φ) (4.4.3) where Φ is the dualized chiral super-ﬁeld. (4.4.3) can be also written using the notation introduced earlier as

−1 Winst = Y (4.4.4) where Y is deﬁned in (4.1.6).

(4.4.4) also generalizes straightforwardly to the SU(N) case: the ith fundamental instanton conﬁguration for which nk = δik, generates a contribution to the super-potential

−1 Winst ∼ Yi . (4.4.5)

th thus the operator Yi in (4.1.6) is associated with the i fundamental instanton. In SU(N) there are N −1 fundamental instanton solutions, it follows that the full super-potential is given by the sum over (N − 1) instanton contributions

N−1 X −1 Winst = Yi . (4.4.6) i

−1 In absence of matter this is in fact exact. Furthermore each Yi contribution lifts the corresponding Coulomb branch ﬂat directions. In the absence of matter ﬁelds, the Coulomb branch is thus completely lifted.

A crucial ingredient for the generation of (4.4.5) is that each fundamental monopole has only two fermionic monopoles. Matter ﬁelds generically have extra fermionic zero modes in addition of the two gauginos. If this happens the gaugino two point function vanishes and no contribution to the super-potential is generated.

41 4.5 Zero mode counting and general case

As we have thus far described, the Coulomb branch dynamics for generic non-abelian theories is fairly complex. At the classical level, for G = SU(N), we have N − 1 one-dimensional branches described by the Yis. Once quantum corrections are included we need to account both for the pinching of the Coulomb branch if any of the matter fields becomes massless and the instanton-generated super-potential. The former “generates” more independent Coulomb branch operators as some of the Yi might be discontinuous at the pinch and thus splitting into two operators. The latter instead reduces the number of flat directions thus decreasing the di- mensionality of the Coulomb branch. We also mentioned in passing that matter fields could prevent the generation of some of the terms in the super-potential.

In this final section we will carefully put all the pieces together to describe the dynamics of the Coulomb branch of a SU(N) theory with F flavors with the following field content

SU(N) U(1)A U(1)B U(1)R

Q 1 1 0 , (4.5.1) Q¯ 1 −1 0 λ adj. 0 0 1 where we write the global U(1) symmetry charges of the matter superfields Q, Q¯ as well as the gaugino λ. The flavor quantum numbers are not relevant for the immediate discussion. Let us first look at the splitting of the various branches due to massless matter.

The Coulomb branch has singularities when a real mass for the matter ﬁelds vanishes. Since

th the real mass for the i component of a fundamental is proportional to σi, the Coulomb branch splits into (N − 1) regions, as shown in Fig. 4.2.

σ1 > ... > σi > 0 > σi+1 > ... > σN.

We will refer at the subspace above as the ith region of the Coulomb branch. The holomorphic structure is different in each region and there is no reason to assume that a given Yk is continuous across a singularity.

42 ¯ 2 ¯i 3 ¯i 4 ¯i (M,B, B) Qj Q2 Qj Q3 Qj Q4

I II III

σi = 0 σ2 = 0 σ3 = 0 σ4 = 0

Figure 4.2: In the theory with fundamentals, the Coulomb branch splits at each point where σi = 0. The total Coulomb branch is composed of (N − 1) regions.

In order to further evaluate which Coulomb branch operator is discontinuous at each splitting, we need to account for the one-loop generated charges (see Section 4.2). This time we will use zero modes counting instead of explicitly computing the one-loop generated mixed

CS terms. In fact the induced global U(1) charges of Yi can be also computed as follows: if a fermion charged under a global U(1) has zero modes in the i-th instanton/monopole background, Yi also acquires charges under that U(1). Because the zero mode counting changes in different Coulomb branch regions (see below), the charges of the Yk under the global U(1)s change as well. We therefore conclude that if, for a given index i, the number of zero modes in an nk = δik instanton/monopole background jumps while crossing from one region to another, the operator Yi is discontinuous at the pinch and we need two independent operators, Yi and

0 Yi , to describe the Coulomb branch on each side of the pinch. It can be shown that this picture is perfectly consistent with the induced global charges computed through one-loop generated mixed CS terms. It is now time to address the counting of zero modes.

We review the counting of fermion zero modes using the Callias index theorem in Ap- pendixC. Here we present a more intuitive description. Each instanton/monopole solution is associated with a particular SU(2) subgroup of SU(N), say the ith such subgroup. A quark decomposes into a doublet and (N − 2) singlets with respect to this subgroup. On the Coulomb branch, this SU(2) is broken to a U(1) subgroup by the corresponding adjoint VEV which also

43 sets the size of the instanton, 1 1 − i = σi σi+1 . (4.5.2) ρmon 2 The σ VEVs induce a real mass for the SU(2) doublet (Qi, Qi+1) in Q. Since Q is in the fundamental representation, i 1 m = σi + σi+1 . (4.5.3) eﬀ 2 If the effective doublet mass is larger than the inverse size of the instanton (4.5.2), then the quarks do not have zero modes in the ith instanton background. On the other hand, if the inverse size of the instanton is larger than the effective real mass, each quark has ni zero modes, where ni is the instanton charge. For details see, for example, [52]. (4.5.2) and (4.5.3) imply

th that each fundamental contributes ni zero modes in the i instanton background if and only if

σi > 0 > σi+1.

This argument applies to any representation with an appropriate generalization of (4.5.3), as can be veriﬁed by the Callias index theorem. It is important to note that the number of zero modes depends on the fermion representation. In particular, matter ﬁelds in larger representations may simultaneously have zero modes under different instantons. Carrying out a similar

th calculation for gauginos, one ﬁnds 2ni zero modes under the i instanton/monopole indepen- P dent of the size of the σi. Thus there are 2 i ni gaugino zero modes everywhere on the Coulomb branch.

So far we have shown that the Coulomb branch of an SU(N) theory with F ﬂavors splits into

th (N − 1) regions and in the i region, where σi > 0 > σi+1, there are 2Fni zero modes coming from P the fundamental and anti-fundamental fermions, as well 2 i ni gaugino zero modes. Thus the

th U(1) charges for the Yk in the i region are given by

U(1)A U(1)B U(1)R → σi > 0 > σi+1 > σi+2 Yk,i 0 0 −2 (4.5.4)

Yi 2F 0 F − 2

−1 We see that the presence of matter zero modes prevents the appearance of the Yi term and the

44 effective superpotential in the ith region is

X 1 W = . (4.5.5) Yk k,i Since this result applies to all values of i, the quantum numbers of some of the Y operators change at each singularity. In particular, at a singularity deﬁned by σi+1 = 0, two operators,

Yi and Yi+1 are discontinuous and one must introduce two new operators. In order to evaluate the total number of un-lifted ﬂat directions, we need to compute both the total number of Coulomb branch coordinates accounting for the splittings and subtract the ones which are lifted by instanton generated super-potential.

It is helpful to illustrate these results in an explicit yet simple example: SU(4) gauge group with F ﬂavors. In this case, we can ﬁll up a table accounting for the behavior of Coulomb branch coordinates in the three separate regions:

Region Zero Modes Coulomb operators W

−1 −1 I σ1 > 0 > σ2 > σ3 > σ4 2Fn1 Y1, Y2, Y3 Y2 + Y3 (4.5.6) −1 −1 II σ1 > σ2 > 0 > σ3 > σ4 2Fn2 Ye1, Ye2, Y3 Ye1 + Y3 0 −1 0−1 III σ1 > σ2 > σ3 > 0 > σ4 2Fn3 Ye1, Y2, Ye3 Ye1 + Y2

There are seven distinct Yk operators in total. Four of these operators appear in the dynamical superpotential and lift parts of the Coulomb branch. The continuity of a globally defined Y operator imposes two constraints relating the operators. It follows that the unlifted Coulomb branch is parameterized by a single operator. It is naturally to parametrized the Coulomb branch using the only coordinate which is continuous everywhere. This single un-lifted flat direction is described by the operator Y, defined in (4.2.3)[20].

Carefully accounting for all the splittings it is possible to show that the result above extends to any SU(N) theory with F > N − 1 ﬂavors which too has a one dimensional Coulomb branch parametrized by the operator Y.

45 4.6 A taste of SU(N) dualities

We concluded the previous chapter discussing U(1) dualities and low-energy behavior of such theories with arbitrary number of ﬂavors F. It is natural to ask whether similar results are known in the most general SU(N) case. Surprisingly enough a thorough understanding of N = 2 SU(N) with F ﬂavors was missing until recently [36].

For arbitrary ﬂavors F, a N = 2 SU(N) without tree level super-potential is dual to an

2 i U(F − N) with F ﬂavors (q,eq), F meson operators M j, a singlet (b,eb) and at highly non-trivial super-potential

Wdual = Mqeq + Ybeb + eηXe− + Xe+ (4.6.1) where X− and X+ are two operators parametrizing the U(F − N) Coulomb branch. They are the natural generalization of the previously introduced V+ and V−. In the magnetic theory Y is not associated to any Coulomb branch but it simply shows up as a chiral operator whose quantum charges match the ones of Y. The fact that the Y should not be interpreted as a Coulomb branch operator it is evident in the way the dual (4.6.1) is constructed [36]. The details of this construction are beyond the scope of this thesis and we refer to the original paper [36] for details. We just want to highlight that in the SU(N) case as well the physical interpretation of Coulomb branch operators appearing in the magnetic three level super-potential is puzzling.

We will shed some light on the topic in the next chapter, although there is no analog of Mirror symmetry for non-abelian theory so the correct physical interpretation of the Y is far less clear than V±.

Before moving on to the next chapter, let us look at two particular example, SU(N) with N and N − 1 ﬂavors. The duals of these two theories was presented in [20].

46 4.6.1 F=N

Consider a SU(N) theory with N ﬂavors, (Q, Q¯). From our analysis of the Coulomb branch we expect a single direction to remain unlifted. Such direction should be parametrized by the i i ¯ operator Y. On the other side the Higgs branch allow both for meson M j ≡ Q Q j and baryon

[1 2 N] B = Q Q ...Q and anti-baryon operator B¯ = Q¯ [1Q¯ 2...Q¯ N]. Furthermore a Witten index analysis (see [44] or Chapter9 for details) suggests that this theory should s-conﬁne, we thus expect the dual to have no gauge group.

In [20] it was shown that a SU(N) with N ﬂavors is dual to a theory of chiral operators with the following super potential

WF=N = Y(BB¯ − det M). (4.6.2)

The super-potential (4.6.2) is ﬁxed by the global symmetries of the theory. We refer to the original literature for consistency checks. It is important to point out that while the presence of

M, B and B¯ in the low-energy super-potential can be interpreted as a sign of conﬁnement. The physical interpretation of the chiral super-ﬁeld Y is again puzzling. In the next chapter we will present a suggestive interpretation of Y as a monopole operator.

4.6.2 F=N-1

After a similar analysis which led to (4.6.2), in [20] it was argued that for F = N −1 ﬂavors in the correct low-energy description the Higgs and the Coulomb branch merge quantum mechanically

WF=N−1 = λ(Y det M − 1) (4.6.3) where λ is just a Lagrange multiplier to enforce the constraint. Once again the operator Y shows up in the super-potential for theories without gauge groups.

47 CHAPTER 5 MONOPOLE OPERATORS

In many of the dualities described in the previous chapters, there is a non-trivial identification between Coulomb branch operators in the electric theory and simple chiral, meson-like, operators in the magnetic side. In some of the cases the magnetic theory does not even present a gauge group (e.g. U(1) with F = 1 or SU(N) with F = N). The physical interpretation of this matching could be somewhat puzzling. Despite a few consistency checks and matching of quantum numbers of the operators on both sides of the duality, we have not provided any details shedding light into this identification. In this chapter we will attempt to provide some clarifications.

A crucial role for the interpretation of Coulomb branch operators across the duality is played by the topological global symmetry, U(1)J , whose associated conserved current is Ji ≡ ∗Fi =

jk i jkF . See [20, 44] or Chapter2 for details. The associated charge is Z Z 2 0 2 F12 qJ = d x j = d x . (5.0.1) J 2π

Non-vanishing qJ charge could have physically different implications:

From the definition above, it follows that operators with qJ , 0 introduce a magnetic flux. • Operators that introduce a magnetic flux through any sphere S 2 surrounding a spacetime

point x, are called monopole operators [36, 44, 55–59]. In particular the operator O1(x) ∼

exp [iγ(x)] creates a magnetic ﬂux of charge qJ = 1, where γ is the periodic scalar dual to photon of any unbroken U(1)s.

In particular regions of the Higgs branch of theories with non-vanishing FI term ξ, there • are non-trivial vortex solutions. Their BPS mass is m = |ξqJ |. Notice that FI terms can

be naturally accommodated in the vector-superﬁeld for a weakly gauged U(1)J . This

suggests that operators with qJ , 0 create BPS states which in fact acquire the appropriate

real mass once U(1)J is weakly gauged (more details below).

48 We have already seen that the operators describing the un-lifted Coulomb directions both for the U(1) case (V±) and SU(N) (Y) have a non zero qJ . In fact they can be interpreted as monopole operators. More precisely Coulomb branch operators should be interpreted as low- energy description of appropriate monopole operators. Here we will provide arguments to clarify this interpretation. It is worth pointing out that the in SU(N) case the gauge group is simple and does not contain any U(1) factors nor topological U(1)J . In the non-abelian case, Monopole operators can only appear on the Coulomb branch where SU(N) is broken to U(1) subgroups.

In the U(1) case we could go beyond heuristic arguments. As anticipated, we will show that the correct interpretation of the Coulomb branch operators in the magnetic side is in terms of vortex operators. By that we mean that the correct interpretation of V± acting on the “magnetic vacuum” is that it creates electric vortices. It is then natural to interpret mirror symmetry as a particle-vortex duality where the magnetic theory is obtained by “re-writing” the electric theory in terms of vortex operators. Furthermore, the connection between vortices and Coulomb branches can be made explicit. Building on what we have seen in Chapter3, we will thus start with an analysis of U(1) theories.

Things to add

How to generalize to non-abelian case? Just sketch what I understand mentioning the • dirac quantization condition in some details. ”An explicit connection is yet missing”.

5.1 BPS Vortices

Consider 3D N = 2 SQED with a non vanishing FI term ξ. For simplicity we consider at ﬁrst the case with a single charged matter ﬁeld Q, we will then generalize to generic matter content.

We seek topologically stable solutions. From (3.1.7), setting ξe f f = ξ, generic Higgs vacua need

49 to satisfy p |Q|2 ∼ ξ → Q ∼ ξeiα (5.1.1) where the phase α is arbitrary. Any topological solution with finite energy needs to approach such vacuum configuration at r → ∞. This condition effectively identifies the point at the in-

finity thus the boundary of R2 becomes isomorphic to the circle S 1. Also the vacuum manifold, eiα, is a circle. Therefore topological stable solution can be classified as maps from a circle into a circle. A more formal way of expressing that is to say that solutions of this kind are in one-to- one correspondence with elements of π1(U(1)) Z and identified by their winding number n. We will call such solutions vortices.

The strategy implemented to find vortex solutions is similar to the one used to find instanton/monopole solutions in the last chapter. Homotopy theory describes the overall behavior of a given vortex, yet within a given equivalence class only the solution with minimal energy is stable. The energy of a vortex solution is " # Z 2 2 2 1 i j ne 2 2 2 E = d x Fi jF + e |Q| − ξ + |DiQ| (5.1.2) vor 4e2 2 where ne is the charge of the field Q and we set σ = −A0 so such fields don’t show up in the energy. Integrating by parts and neglecting boundary terms (5.1.2) can be re-written as [52]   Z  " #2  Z 2 1 1 2 2 2 Evor = d x B + nee |Q| − ξ + |(D1 + iD2)Q| + neξ d xB. (5.1.3) 2 e 

Some observations are in order. First of all the last term is nothing but 2πqJ showing that vortex solutions are in fact charged under the topological U(1)J . This also provides the correct interpretation of the winding number n ∼ neqJ . Furthermore in (5.1.3) the ﬁrst two terms are both positive. It follows that in a given equivalence class the minimal value of the energy is reached at

2 2 B + nee |Q| − ξ = 0, (D1 + iD2)Q = 0. (5.1.4) which can be readily solved leading to:

vor −ineqJ α p vor Q = e ξ + ..., Ai = qJ + ... (5.1.5)

50 where the dots indicate terms which vanish at r → ∞.

When the minimum of the energy is attained we can read immediately the mass of the vortex mvor = neξqJ ≡ Z. It follows that the solution that we have just constructed not only is topologically stable but also it is also persevered by half of the super-charges. Eq (5.1.4) is the equivalent of the Bogomol’nyi equation for vortices.

Depending on the sign of the central charge Z = neξqJ different supercharges preserve the vortex solution. Although in many respects similar, we will distinguish the two cases. Assum- ing ξ > 0, we will call a vortex BPS if Z > 0 while anti-BPS if Z < 0. This implies that for a BPS

(anti-BPS) solution, sign(qJ ) = sign(ne) (sign(qJ ) = −sign(ne)).

5.1.1 Generic case

Consider now a theory with generic matter, Qi with electric charges ni. The vacuum condition (5.1.4) generalizes to [59]   X  2  | |2 −  B + e  ni Qi ξ = 0 (5.1.6) i

(ni) (ni) BPS → Dz Qi = 0, or ANTI-BPS → Dz¯ Qi = 0.

where we deﬁned Dz,z¯ = D1 ± iD2.

Consider now the following identity

Z −→ Z 2 (n j) 2 2 (n j) 2 2 d x|D Q j| = d x |2Dz,z¯ Q j| ± n j|Q j| F12 (5.1.7)

(n j) (n j) 1 where we used [Dz , Dz¯ ] = 2 n jF12. Since the LHS is non-negative, (5.1.6) has Q j , 0 solutions only if the RHS has the right sign, that is sign(qJ ) = sign(ni) for BPS and sign(qJ ) = −sign(ni) for anti-BPS. Furthermore, we already saw that for BPS solutions sign(qJ ) = sign(ξ) while for anti-BPS solutions sign(qJ ) = −sign(ξ). Putting these conditions together we obtain that for either BPS or anti-BPS solutions Qi = 0 if sign(ni) = −sign(ξ).

51 An immediate consequence is that in a theory with matter ﬁelds with both positive (Qi) and

¯j negative (Q¯ ) electric charges, BPS solutions are only possible if either Qi=0 or Q¯ i = 0. Which implies that critical (BPS or anti-BPS) vortices are allowed only in Higgs vacua where all the ¯j ¯ ¯j meson operators Mi = QiQ vanish.

5.1.2 Fermionic zero-modes

In the identifying the map between Coulomb branch operators and critical vortices, a crucial role is played by fermionic zero modes (more details below). To be specific consider a theory with N+ matter fields with ni > 0 and N− fields with ni < 0. Furthermore for ξ , 0, consider BPS and anti-BPS vortices in the vacuum with hσi = 0 and r vac ξ Qi = δi,1. (5.1.8) 2πn1

To avoid subtleties let us assume that n1 = 1 [59]. We have already seen that only the N+ matter

ﬁelds Qi for which ni > 0 have bosonic zero modes in (5.1.8) (precisely ni zero modes for each

ﬁeld Qi) while the ﬁelds with ni < 0 have none. What about fermi zero modes?

For the sake of simplicity we can assume that the vortex solution under study has only the

“core” Q1 conﬁguration turned on

vor vor Qi (z, z¯) = Q1 (z, z¯)δi,1 (5.1.9)

vor vac where ﬁniteness of the energy of the solution implies that Qi (r → ∞) → Qi .

The solution (5.1.9) is only annihilated by two of the supercharges. The action of the remaining two supercharges creates two fermionic zero modes. These two are associated with the Q1

† and therefore can be labeled as Ψ1 and Ψ1. The treatment of the zero modes associated with the remaining ﬁelds Qi is somewhat involved, thus we will only summarize the important results. For a far more detailed analysis we refer to the original literature [44, 59].

52 So long as fermionic zero modes go, both ni > 0 and ni < 0 ﬁelds contribute. In particular each ﬁeld with charge ni has |ni| fermionic zero modes in the vortex background (5.1.9).

Fermionic zero modes associated with ﬁelds with positive charge ni can be labeled as u¯i,p while for ni < 0 di,p. In both cases p = 1, ..., |ni| yet the spin of the mode is different, namely u¯i,p has spin − 1 − − 1 p 2 while di,p has spin (p 2 ). After quantization, the fermionic zero modes thus far analyzed P give a tower of 2 |ni| states.

To conclude this section let us point out that not all the u¯i,p and di,p are normalizable. In fact the |ni|-th mode has a logarithmically divergent norm. There is a tentative physical interpretation which is very interesting. Namely non-normalizable zero modes map states (in particular BPS doublets) into different Hilbert spaces. Once the state-to-operator map is implemented this reﬂects in the corresponding operators to satisfy O1O2 ∼ 0 in the chiral ring. See below or [44,59] for more details.

5.2 A case of study: Mirror Symmetry

We concluded Chapter3 with a review of a very interesting set of 3D SQED dualities which go under the name of mirror symmetry [18,23,48–50]. Mirror symmetry relates the Coulomb branch of the electric theory with the Higgs branch of the magnetic theory and vice-versa. Here we want to argue that the correct interpretation of these dualities is a particle/vortex duality [20].

In other words the quantum number of particles in the one side of the duality match with the vortices in the other side leading naturally to the interpretation that semi-classical topological stable solutions are mapped by mirror symmetry to fundamental, particle-like, degrees of freedom. We will focus on the simplest case, F = 1, but our discussion naturally generalizes to F > 1.

In 3D a N = 2 U(1) theory with one ﬂavor is dual to a W = XYZ model with the following

53 mapping

X → M ≡ QQ¯, (5.2.1)

Y → V+, (5.2.2)

Z → V−. (5.2.3)

We know from the analysis in previous section that critical vortices in the electric theory only exist if hMi = 0. Furthermore the BPS (anti-BPS) vortices are charged under the topological

U(1)J with qJ = 1 (qJ = −1) and V± are the only operators in the theory with non-vanishing qJ . It is thus plausible to interpret low-energy particle-like states V+|0i (V−|0i) as BPS (anti-BPS) vortices in the UV theory. This also provides a physical interpretation for the Coulomb branch in the electric theory in terms of vortex condensation.

Besides just matching quantum numbers, we can provide further checks of the validity of the proposed identiﬁcation:

Let us consider weakly gauging U(1)J and freeze the corresponding vector super-ﬁeld in • the conﬁguration (see also (2.3.7))

σJ = ξ, Ai,J = λJ = λ¯ J = DJ = 0. (5.2.4)

In the magnetic side of the duality, this gives a real mass ξqJ to the “particles” created by

V±. In the electric side, instead, (5.2.4) introduces a FI term through mixed Cher-Simons

terms (see Section 2.6 for details). This turns on a central charge Z = ξqJ which induces a

real mass to the critical vortices. The behavior on both sides thus perfectly matches.

Consider a non-vanishing ξ. In the magnetic side V± are massive and only hV±i = 0 vac- • uum are allowed. This exactly matches what we expect in the electric side as a non-

vanishing FI term lifts the Coulomb branch imposing again hV±i = 0.

We have explained that if hMi , 0, vortices are no longer critical as their mass is no • longer equal to the central charge. Consider a non-vanishing VEV for M in the magnetic

theory. The super-potential W = MV+V− provides a complex mass to both V+ and V−. The

54 p 2 2 particles’ mass is now (ξqJ ) + |hMi| , |Z|. The behavior of particles in the magnetic side thus reﬂects the behavior expected from the analysis of electric vortices.

5.3 More general case

Recently a general map between Coulomb branch operators for U(1) theories and BPS vortices was proposed [59]. Consider a BPS vortices solution with charge qJ and the fermion zero modes in such background. After quantization we have

† {ΨA, ΨB} = δAB, A, B = 1, ..., N (5.3.1)

N The operators ΨAs create a tower of 2 states. From the previous analysis of fermionic zero P modes in a vortex background, N = i |ni| where ni are the charges of the matter ﬁelds in the theory and the sum runs through all the ﬁelds with both positive and negative charges.

Consider the states at the top and the bottom of this tower

† |Ω±iqJ , ΨA|Ω+iqJ = ΨB|Ω−iqJ = 0 (5.3.2) from (5.3.2) we can derive the following relation

Y † Y |Ω+iqJ ∼ ΨA|Ω−iqJ , & |Ω−iqJ ∼ ΨA|Ω+iqJ (5.3.3) A A

Also writing as |0iqJ the bosonic BPS vacuum, then

 ∓ 1 Y  2 |Ω i ∼  Ψ  |0i ± qJ  A qJ (5.3.4) A

From the charges of the fermionic zero-modes, we can compute the quantum numbers of

|Ω±iqJ =±1 which match the quantum number of the Coulomb branch operator V±. Thus we could identify

† |Ω+iqJ =±1 ≡ V±|0i, |Ω−iqJ =±1 ≡ V∓|0i (5.3.5)

55 In [44, 59] it was shown that the above map correctly reproduces the behavior of U(1) theories with (almost) generic matter content. Through a series of checks and counting of zero modes the picture is perfectly consistent and it provides a physical insights in the behavior of the Coulomb branches in the electric theory.

The state-to-operator map (5.3.5) tells us that an unlifted Coulomb branch only exists if the P top and the bottom states in the tower of 2 |ni| states generated by the quantized fermi zero- modes are both spin 0 and gauge invariant. The former condition is not at all trivial as we have seen that fermionic the zero modes u¯i,p and di,p have a complicated spin structures. Yet we can immediately provide an example where the picture is consistent.

Consider a U(1) theory with N pair (Qi, Q¯ i) where Qi and Q¯ i have opposite charges. In this − 1 theory it is possible to set k = 0 and we will do so. For each u¯i,p with spin (p 2 ) there is a di,p − − 1 | i | i with spin (p 2 ) thus both Ω+ and Ω− have s = 0.(5.3.5) suggests that both V− and V+ are un-lifted matching our analysis in Chapter3.

Let us now analyze the simplest case, a U(1) theory with a single ﬂavor, in a bit more details.

We will specifically look at the BPS, qJ = 1, configuration. Only the field Q with ne = 1 can acquire a VEV and thus Q ≡ Q1. There are two normalizable fermionic zero modes Ψ1 and

† Ψ1 which can be obtained by simply acting with the two super-symmetric charges, Ψ1 ∼ Q+ † ¯ ¯ and Ψ1 ∼ Q−. The other matter ﬁeld Q with opposite charge n¯e = −1 gives rise to two non- † normalizable zero modes, Ψ2 and Ψ2. We can thus construct explicitly the 4 states in the tower, starting from |Ω+i      Q−|Ω+i  |Ω i →   → Q−Ψ |Ω i ≡ |Ω−i (5.3.6) +   2 +  Ψ2|Ω+i  Following (5.3.5) we can identify

† |Ω+i ≡ V+|0i & |Ω−i ≡ V−|0i (5.3.7)

This map shows that the states created by the two operators V± belong to different Hilbert spaces suggesting that V± satisfy the relation V+V− ∼ 0 which in the electric theory only arises after quantum corrections have been taken into account.

56 5.4 Non-abelian case

In the non-abelian case no concrete mapping is allowed. Yet a valuable interpretation of the unlifted operator Y is given by monopoles.

57 Part II

The case with Antisymmetric matter

58 CHAPTER 6 THE COULOMB BRANCH WITH ANTISYMMETRIC MATTER

We now take the ﬁrst steps towards extending the results of the previous chapters to a theory that incorporates more general matter representations. Depending on the choice of representations, the Coulomb branch may be described by more than a single chiral operator. We concentrate on a simple example: SU(4) N = 2 SUSY gauge theory with matter in (anti-)fundamental

(Q, Q¯) and antisymmetric tensor representations (A). Many of our conclusions, however, are general. In this section we do not ﬁx the exact number of antisymmetrics and fundamental ﬂavors and instead focus on the general properties of the Coulomb branch in the presence of these two representations.

As explained in the previous section, the number of independent chiral operators which describe the Coulomb branch depends on the number of pinches that occur when a matter ﬁeld becomes massless. As before, we restrict the scalar VEV to a single Weyl chamber:

σ1 ≥ σ2 ≥ σ3 ≥ σ4, and σ1 + σ2 + σ3 + σ4 = 0 . (6.0.1)

For simplicity we suppress the ﬂavor indices since gauge interactions are ﬂavor diagonal. The classical vacua satisfy

α α αβ σαQ = 0 −σαQ = 0 σα + σβ A = 0, (6.0.2) where α, β ∈ {1, ··· 4} and there is no sum over repeated indices. The Higgs branches pinch the

Coulomb branches at σα = 0 and at σα + σβ = 0. In light of (6.0.1), the non-trivial Coulomb branch singularities are σ2 = 0, σ3 = 0 and σ2 + σ3 = 0. The last singularity is due to the presence of antisymmetric matter. The Coulomb branch splits into four distinct regions. The

Callias index theorem tells us the number of fermion zero modes in each region and thus determines the nonperturbative contributions to the superpotential and the number of independent

Yi operators. The results are summarized in the Section 6.1, with a detailed derivation given in AppendixC. The main result is that the each antisymmetric tensor provides two additional

59 zero modes: if σ2 + σ3 > 0 the zero modes are in the ﬁrst fundamental instanton, while for

σ2 + σ3 < 0 they are in the third.

6.1 Coulomb branch coordinates

The intuitive picture for the zero mode counting works out the same way as before. For example, consider the zero modes under the ﬁrst fundamental instanton/monopole, corresponding to the SU(2) embedded into the upper left corner of SU(4). The background VEV for the adjoint can be written as a sum of two contributions: σ − σ σ − σ σ + σ σ + σ diag 1 2 , − 1 2 , 0, 0 diag 1 2 , 1 2 , σ , σ . (6.1.1) 2 2 2 2 3 4 the ﬁrst determines the instanton size while the second is invariant under the SU(2) rotations and contributes to the effective real mass [53, 54, 60]. The antisymmetric of SU(4) decomposes

1 into two doublets and two singlets. The doublet masses are 2 (σ1 + σ2) + σ3,4. Comparing the doublet masses with the inverse instanton/monopole size (4.5.2) we obtain the following results: SU(4) with FA + F + Coulomb Region Zero Modes W operators

−1 −1 I σ1 > 0 > σ2 > σ3 > σ4 (2F + 2FA) n1 Y1, Y2, Y3 Y + Y 2 3 (6.1.2) −1 II σ1 > |σ3| > σ2 > 0 > σ3 > σ4 2FAn1 + 2Fn2 Ye1, Ye2, Y3 Y3 0 0−1 III σ1 > σ2 > |σ3| > 0 > σ3 > σ4 2Fn2 + 2FAn3 Y1, Ye2, Ye3 Y1 0 0 0 0−1 0−1 IV σ1 > σ2 > σ3 > 0 > σ4 (2F + 2FA) n3 Y1, Y2, Y3 Y1 + Y2

In regions II and III, the zero modes of the fundamentals and antisymmetrics are misaligned so that fewer directions are lifted and more than a single operator Y is required to globally describe the Coulomb branch. Indeed, only four of the nine Yi operators are lifted. Continuity of the globally deﬁned Y in (4.2.3) imposes three constraints. We thus require two operators to describe the Coulomb branch. The ﬁrst is the usual Y that we have already introduced.

60 The second is required to describe the novel properties in regions II and III. One operator that is continuous between those two regions is Ye2. However, since Y itself is already continuous across these regions, a combination of Ye2 and Y can also be used. We demonstrate below that the most useful deﬁnition for the second coordinate of the Coulomb branch is the combination q q q 2 0 2 Ye = YYe2 = Ye1Ye2 Y3 = Y1Ye2 Ye3 . (6.1.3)

This has the correct quantum numbers to be identiﬁed with a meson of the dual description1.

The physics of these two variables can be understood as follows. In the presence of only fundamental matter, the non-perturbative contributions to the superpotential push the eigenvalues of the adjoint to be as far apart as possible. This leads to as many massless matter ﬁelds as possible with a non-vanishing adjoint and the conﬁguration    σ       0  ↔   Y   . (6.1.4)  0       −σ  In the presence of antisymmetric tensors, however, the dynamics is somewhat different. The antisymmetric provides two doublets under the kth SU(2) subgroup corresponding to the simple root αk. These behave like the doublets from a fundamental representation except the corresponding adjoint contributions to real masses are replaced by (σi + σk, σi + σk+1), where i , k nor k + 1, rather than (σk, σk+1) for fundamentals. The effective masses of the doublets provided by the antisymmetric can be ordered within a Weyl chamber similarly to the masses of the doublets from a fundamental representation. The argument leading to (6.1.4) can then be repeated, except the different structure of the effective real mass leads to a different unlifted direction in the Coulomb branch,    σ       σ  ↔   Ye   . (6.1.5)  −σ       −σ 

1The same operator has been shown to be un-lifted on the Coulomb branch of SO(6) with F vectors [58].

61 When both anti-symmetric tensors and fundamentals are present, the un-lifted Coulomb branch is two dimensional and can be parametrized generically as      σ1         Y ↔ diag σ1 − σ2, 0, 0, σ2 − σ1     σ2     with  .    (6.1.6)  −    σ2      Ye ↔ diagσ , σ , −σ , −σ   2 2 2 2  −σ1 

Since regions I and IV have a zero mode structure similar to the case with only fundamental matter, those regions are always described by the operator Y. In regions II and III, however, the antisymmetric wants to align the adjoint into the Ye direction.

The arguments presented here are semi-classical. Depending on the speciﬁc matter content of the theory, there may be important additional dynamical effects that modify the semi- classical picture [61]. In the next chapters we investigate the dynamics of the SU(4) theory with one and two antisymmetric tensors. We show that in theories with one anti-symmetric, only the direction Y is present, suggesting that Ye=0. Both Y and Ye are needed to describe the dynamics of theories with two antisymmetric tensors. We show this both by considering the dimensional reduction of 4D theories with one and two antisymmetric tensors, and by studying the decoupling of an antisymmetric from a 3D theory with two antisymmetrics, which sets Ye = 0.

As already mentioned above, both operators parametrizing the Coulomb branch, namely

Y and Ye are the low energy limits of monopole operators. Such an identiﬁcation has multiple subtleties. We address them next.

6.2 Dirac quantization condition of the monopole operators

Monopoles are subject to the Dirac quantization condition which restricts the minimal magnetic charge that can appear in a theory. This, in turn, restricts the form of the monopole operators

62 that describe the un-lifted Coulomb branch directions. If there are unbroken non-Abelian subgroups, then magnetic charges smaller than the na¨ıve minimal value might be allowed, as long ∼ −1 as the monopole carries non-Abelian magnetic charge that is screened at r Λconﬁne. This is analogous to the well known case of GUT monopoles, see e.g. [62]. We therefore need to check that both Y and Ye are the appropriate “minimal” monopole operators allowed by Dirac quantization. In particular, the deﬁnition of Ye might raise concerns since the square root would naively imply the presence of a monopole with a fractional charge.

As it is indicated in (6.1.6), Y and Ye are associated to the breaking SU(4) → SU(2) × U(1)1 ×

U(1)2 and SU(4) → SU(2) × SU(2) × U(1)3 respectively. We can explicitly identify the generators of each one of the unbroken U(1)s     1  1           0   −1      Y : Q1 =   Q2 =   , (6.2.1)  0   −1           −1  1   1       1    Ye : Q3 =   , (6.2.2)  −1       −1 where Qi generates the corresponding U(1)i. While the Y operator is associated to a monopole

2 operator with a minimal Dirac charge under U(1)1, the square root in the deﬁnition of Ye in (6.1.3) suggests that this operator corresponds to half of the minimal charge and therefore could be not allowed [58].

However, as described above, in the presence of unbroken non-Abelian subgroups, the Dirac quantization condition can be more subtle. We argue that Dirac quantization is, in fact, obeyed by the state corresponding to the operator Ye. The key aspect is that while the symme-

2 The magnitude of the magnetic charge depends on the normalization of the U(1) generators. However, this normalization also affects the minimal allowed electric charge so that a more meaningful quantity is the magnetic charge in units of the minimal electric charge. This quantity is independent of the normalization of the generators.

63 try breaking pattern corresponding to Ye is locally SU(4) → SU(2) × SU(2) × U(1), there is a non-trivial identiﬁcation between a discrete U(1) rotation and one of the elements of the Z2 × Z2 center of SU(2) × SU(2). From the explicit deﬁnition above, it is clear that exp (iπQ3) coincides with the diag(−1, −1 − 1, −1) element of the center of SU(2) × SU(2). In such a case, one need not go around the entire U(1) factor to obtain a closed loop; the loop can be closed by going halfway around and then closing through the Z2 center. In this case the monopole picks up a discrete magnetic Z2 charge under the SU(2) × SU(2) non-Abelian group. For cases like this, the proper formulation of the Dirac quantization condition has been given by Preskill [62]: a magnetic charge is allowed if one can combine the charge matrix with some of the diagonal non-Abelian generators to obtain the full magnetic charge matrix M which has only integer elements. In our case the proper choice is    1      1  0  M = Q + T (1) + T (2) =   . 3 3 3   (6.2.3) 2  0       −1 

This is in accordance with our deﬁnition that the operator Ye carries charge 1/2 under Q3.

A more physical way of describing the above argument is to only consider the theory at large distances. When considering the breaking pattern described by Ye, an SU(4) fundamental decomposes into a (2, 1) ⊕ (1, 2) under the unbroken SU(2) × SU(2). At low energies, both of the SU(2)s are still strongly interacting and the only physical states are composite objects. This requires at least two doublets originating from two fundamentals under SU(4). The minimal charge under the U(1)3 is thus double the one na¨ıvely given by a fundamental, and therefore the minimal Dirac charge at low energy is in fact a half of the na¨ıve result. This provides an additional explanation for the presence of the square root in the deﬁnition of Ye. This is similar to the argument that is usually given for why the fractional electric charges of quarks do not forbid the appearance of a minimally charged Dirac monopole in GUT extensions of the Standard

Model. It is also clear that such an argument cannot be applied for the breaking associated with Y: when considering a fundamental, during the breaking of SU(4) to SU(2) × U(1) × U(1), one

64 component of the fundamental is only charged under the U(1) with the minimal charge. The possible presence of this state leads to the usual Dirac quantization without any subtleties.

65 CHAPTER 7 DIMENSIONAL REDUCTION OF 4D DUALITIES

In the previous chapter we presented strong evidences that the Coulomb branch of theories with antisymmetric matter behave in a considerably different fashion compared with the case with ﬂavors. In order to provide even stronger evidence of the validity of the analysis previously presented, we need to connect with theories whose behavior is established and check that our picture reduces to the correct one. Given the lack of known 3D dualities involving antisymmetric matter we will use a different strategy and we will dimensionally reduce 4D dualities with antisymmetric tensors. Thus checking explicitly, in the next chapter, that the analysis presented in the previous chapter matches the low-energy behavior of the dimensionally reduced 4D theories.

Deriving new 3D dualities from known 4D dualities is more subtle that it could be expected at ﬁrst. An accurate analysis of the subtleties and how to get around them, was carried out recently [36, 58]. In this chapter we will take a short digression from our study of 3D theories with antisymmetric matter, summarizing the main features of the dimensional reduction of 4D dualities focusing in particular on dimensional reduction of s-conﬁning theories.

7.1 General setting

Consider a 4D duality between an ‘electric’ theory A4 and a ‘magnetic’ theory B4. One may then dimensionally reduce each theory in two steps:

1. Compactify a space-like dimension: R4 → R3 × S 1.

2. Reduce the size of the compact dimension to zero R3 × S 1 −−−→ R3 r→0

This procedure indeed reduces the 4D theories into 3D theories. However the 3D theories so obtained, (A3, B3), are not duals of one another. This can be understood intuitively when A4 and

66 B4 are Seiberg duals. These are related by an infrared duality that is valid at energies much lower than the conﬁnement scales of the dual theories1,

2 − 8π g2(µ)b E ΛA4 , ΛB4 Λ = µ e . (7.1.1)

After compactiﬁcation, the effective 3D gauge coupling depends on the size of the compact 2 2 → → dimension, g4 = 2πrg3. The limit r 0 thus corresponds to ΛA4 , ΛB4 0. This means that the

IR limit E ΛA4,B4 is not valid in the na¨ıve compactiﬁcation and the duality does not hold.

In order to find a 3D version of 4D Seiberg duality, one must examine the compactified theory at finite radius, 1 E Λ , Λ . (7.1.2) A4 B4 r

The theories on R3 × S 1 are intrinsically different from the na¨ıve dimensional reduction because the compact S 1 direction allows an extra topological configuration that is distinct from the purely 3D topology described in Section3 and AppendixC. This additional configuration is a a Kaluza-Klein (KK) monopole that wraps around the circle with a twist [63–65]. Like the other instanton/monopole configurations, the KK monopole also generates a non-perturbative contribution to the superpotential [43], ! b σ1 − σN 1 − WS = ηY, η = Λ , Y = exp 2 + i(a1 aN)) , (7.1.3) g3 where Y is the chiral operator introduced in (4.2.3). The superpotential WS 1 must be included in the compactified theory to preserve the duality. In the r → 0 limit, η → 0 and this term in the superpotential disappears [66]. In SQCD with F > N, the resulting IR duality between intermediate compactified theories is

SU(N) with F( + ) SU(F − N) with F( + ) ←→ , (7.1.4) W = ηY W = qMq¯ + eηYe

− with ηeη = (−1)F N. The superpotential term from the KK monopole configuration can also be decoupled. This can be done by decoupling one flavor in the compactified theory with a real mass deformation [36]. 1We implicitly assume that both theories are asymptotically free. This assumption plays no role in what follows and the conclusions apply also to the case where the magnetic theory is IR free.

67 s-conﬁned IR theory s-conﬁning UV theory with superpotential W

4D 3D s-conﬁned IR theory s-conﬁning UV theory with superpotential + ηY W , no η0Y 0

Figure 7.1: If we reduce s-conﬁning theories, there is no eηYe in the magnetic side.

7.2 S-conﬁning case

S-conﬁning theories are a particular set of dual 4D supersymmetric theories where the electric theory, A4, conﬁnes and the magnetic degrees of freedom describe the resulting composite degrees of freedom. The dual theory, B4 has no gauge group. These have the following properties [41, 42]:

1. The IR physics is described exactly by gauge invariant composites.

2. A conﬁning superpotential is dynamically generated.

3. The origin of the classical moduli space, where all the global symmetries of the electric

theory are unbroken, is also an IR vacuum in the quantum theory.

For these theories, the dimensional reduction algorithm follows as it did above with the R3 × S 1 magnetic theory having no eηYe term as the magnetic theory has no gauge group, as shown in Fig. 7.1. One may perform consistency checks for the resulting 3D duality. In what follows, we set Λ = 1 for notational simplicity.

68 7.2.1 Check 1

From the procedure above, SQCD with F = N + 2 ﬂavors on R3 × S 1 is dual to SU(2) with F = N + 2 ﬂavors and a superpotential

W = qMq¯ + eηYe. (7.2.1)

Deforming the electric theory by a complex mass (2.2.5) for one ﬂavor then leads to SQCD with

0 0 F = N + 1 and W = (η/m)Y|F=N+1 ≡ η Y . From Fig. 7.1 we expect that this theory is described in the IR by the 4D s-conﬁning superpotential with no extra eηYe term:

SU(N) with W = η0Y0 & F = N + 1 ←→ W = BMB¯ − det M. (7.2.2)

To see how this correspondence arises in a 3D description, we note that a holomorphic mass for the (N + 2)th electric quark flavor maps onto a tadpole for the (N + 2)th diagonal element of the magnetic meson. As a result the last flavor of dual quark acquire a VEV qN+2q¯N+2 = −m completely breaking magnetic SU(2) group. After identifying the remaining dual quarks with the baryons of N + 1 flavor theory, the low energy superpotential becomes2

W = BMB¯ + eηYe. (7.2.3)

An interpretation of eηYe term generated by the KK instanton is somewhat non-trivial because SU(2) is completely broken. Comparing to 4D physics, we know that this term is proportional to det M since it must reproduce the effect of 4D instanton. It is, however, instructive to obtain this result from a purely 3D perspective. To this end, consider a limit of small m and large M, such that3 rankM = N + 1. In this regime, the low energy physics is described by a single ﬂavor

SU(2). It is known that such a theory has a quantum modiﬁed constraint Yelow = 1/(qN+2q¯N+2) [20]. Matching quantum KK instanton-monopole operators of the high and low energy theories, we

ﬁnd

Ye = det M Yelow . (7.2.4)

2Unless explicitly noted, from now on M refers only to meson composites of light electric quarks. 3Since classical constraints are not modiﬁed quantum mechanically in this theory, one must have rank(M) ≤ N in the vacuum. Thus, a superpotential for M must be generated dynamically.

69 Finally, holomorphy guarantees that this result is also valid in the large m regime where qN+2q¯N+2 = −m. Thus, after appropriate rescalings, the superpotential (7.2.3) becomes

W = BMB¯ − detM , (7.2.5) in full agreement with (7.2.2).

7.2.2 Check 2

A second non-trivial check is to start instead with SU(N) with (N + 1) ﬂavors and W = ηY.

Because of the lack of the KK monopole, such theory should be dual to an s-conﬁned theory with super-potential

i j ¯ Wmag = B Mi B j − det M. (7.2.6)

¯ Adding a real mass to Q and Q and taking mR → ∞ decouples the instanton term in the electric theory (for more details see [36] ). In order not to induce extra Chern-Simons term, we need to mQ = −mQ¯ assign R R. Real masses belong to vector super-ﬁeld and can be obtained through weakly mQ = −mQ¯ gauging a particular choice of ﬂavor symmetries (see the discussion in Section 2.3). R R can be easily obtained weakly gauging SU(N + 1)L × SU(N + 1)R D × U(1)B. The masses induced in the magnetic theory are:       −  mR  mR              0   0   0 , 0 m =   m ¯ =   m =   (7.2.7) B  .  B  .  M    .   .  , 0 0           0   0 

Identifying the (N + 1) × (N + 1)-th component of the meson ﬁeld with the Coulomb branch operator of the electric theory, and decoupling the massive ﬁelds, we obtain

Wmag = Y(BB¯ − det M) (7.2.8) which matches the known IR description of the SU(N) with N ﬂavors. Therefore again the initial assumption of no eηYe in the magnetic theory is consistent from a fully 3D perspective.

70 We can now apply the dimensional reduction procedure to theories with antisymmetric mat- ters. Focusing on s-conﬁning theories simpliﬁes life considerably as the magnetic is described in terms of singlets and it is thus much easier to match operators across the duality. The next chapter is devoted to a careful analysis of that.

71 CHAPTER 8 DUALS OF 3D THEORIES WITH ANTISYMMETRICS

We now discuss the dynamics of the 3D SU(4) theories with antisymmetric tensors. First we consider a 3D N = 2 SUSY gauge theory with two antisymmetric tensors and two ﬂavors. The corresponding theory in 4D has three ﬂavors and the anomaly-free symmetries:

0 SU(4) SU(2) SU(3)L SU(3)R U(1) U(1) U(1)R

A 1 1 0 -3 0 (8.0.1) 1 Q 1 1 1 2 3 1 Q 1 1 -1 2 3

It is well-known that this theory is s-conﬁning in 4D [40–42]: it is dual to a theory of gauge singlets with the following non-trivial superpotential, which is smooth everywhere including the origin:

1 2 3 2 W = T M − 12THHM¯ − 24M M − 24HHM¯ (8.0.2) dyn Λ7 0 0 0 2 2 where the composite ﬁelds have the following charge assignment under the global symmetries:

0 SU(4) SU(2) SU(3)L SU(3)R U(1) U(1) U(1)R ¯ 2 M0 = QQ 1 1 0 4 3 2 ¯ 2 M2 = QA Q 1 1 0 -2 3 (8.0.3) 2 2 H = AQ 1 1 2 1 3 ¯ ¯ 2 2 H = AQ 1 1 -2 1 3 T = A2 1 1 1 0 -6 0

8.1 3D duality for SU(4) with 2 and 2 ( + )

Based on the rules established in [36], and reviewed in the previous chapter, one can obtain a dual pair on R3 × S 1 by adding to the superpotential an ηY term generated by KK monopoles on both sides of the duality. In the presence of antisymmetric tensors, this statement requires

72 more care. In a KK monopole background, matter in a generic representation can provide extra zero modes and prevent a contribution to the superpotential. It is known that fundamentals have no zero modes in the KK monopole background but an antisymmetric generically does. Yet this only happens for SU(N) with N > 4 but not for SU(4). This statement can be checked in two equivalent ways.

1. One can explicitly compute the number of the antisymmetric zero modes in the KK

monopole conﬁguration using the appropriate index theorem on R3 × S 1 [67] which has been very nicely ﬂeshed out recently by Poppitz and Unsal [60]. The result of the index theorem counting is that the antisymmetric has no KK monopole zero modes in any of the four regions.

2. One can exploit the important property of the ordinary SU(N) instanton: compactiﬁed on a circle it can be thought of as a composite of the (N−1) fundamental instanton/monopoles

and the KK monopole [63–65, 68–70]. Thus the total number of zero modes in R3 × S 1 for all N independent monopole solutions (instanton/monopoles plus the KK monopole) should match the number of zero modes of the one 4D instanton given by the Atiyah- Singer index theorem. Therefore the number of zero modes in the KK monopole for a

given representation is given by the difference of the 4D instanton zero modes and the sum of zero modes in the (N − 1) independent 3D instanton/monopoles. The latter can be obtained directly from the 3D Callias index theorem. We can see from (6.1.2) that

in each of the four regions of the Coulomb branch, the total number of zero modes of the antisymmetric matches the number of zero modes in the four dimensional instanton solution. Thus, for SU(4) the antisymmetric does not have any zero modes in the KK

monopole which, in turn, generates a non-trivial superpotential in each of the 4 regions of the Coulomb branch.

To obtain the pure 3D dual pair one can remove the ηY term by introducing a real mass term for one ﬂavor. This can be most easily carried out on the dual pair by weakly gauging the SU(3)L × SU(3)R D × U(1) subgroup of the global symmetries. One can then introduce

73 constant scalar backgrounds in the U(1) and along the λ8 direction of the diagonal SU(3) in such a way that the first two generations remain massless while the third generation quarks Q3 ¯ and Q3 acquire real masses mR and −mR respectively. Such a background configuration breaks the global SU(3)L × SU(3)R × U(1) to SU(2)L × SU(2)R × U(1)1 × U(1)2. Furthermore in the limit mR → ∞, the ηY term decouples [36]. Thus on the electric side we flow to an SU(4) gauge theory with no superpotential and the following matter content:

0 SU(4) SU(2) SU(2)L SU(2)R U(1)1 U(1)2 U(1) U(1)R

A 1 1 0 0 -3 0 (8.1.1) 1 Q 1 1 1 0 2 3 1 Q 1 1 0 -1 2 3

It is just a 3D SU(4) theory with two antisymmetric tensors and two ﬂavors.

In order to find the effect of these background scalars on the dual we need to carefully identify their effects on the composites. We find that the following dual fields remain massless:

ia ia 33 M0 → M0 i, a = 1, 2, M0 (8.1.2)

ia ia 33 M2 → M2 i, a = 1, 2, M2 (8.1.3)

HIi → HI3 (8.1.4)

H¯ Ia → H¯ I3 (8.1.5)

As expected, almost all composites containing a third generation quark or antiquark decouple,

33 33 except for M0 and M2 : since the real masses of the quark and the antiquark are of opposite sign, these fields remain in the spectrum. Using the field identification HI3 ≡ hI, H¯ I3 ≡ h¯ I ,

74 33 33 M0 ≡ Me0, M2 ≡ Me2 we ﬁnd the low-energy matter content of the dual theory to be

0 SU(2) SU(2)L SU(2)R U(1)1 U(1)2 U(1) U(1)R

2 M0 1 1 -1 4 3 2 M2 1 1 -1 -2 3 2 h 1 1 2 0 1 3 (8.1.6) 2 h 1 1 0 -2 1 3 T 1 1 0 0 -6 0

2 Me0 1 1 1 -2 2 4 3 2 Me2 1 1 1 -2 2 -2 3 with the superpotential (we don’t explicitly write an overall scale needed on dimensional grounds)

2 Wdyn = Me0 3T det M0 − 12Thh − 24 det M2 + Me2 2M0 M2 + hh (8.1.7)

Most of the chiral operators in (8.1.6) and (8.1.7) are easily identified as meson operators of the electric theory (8.1.1), yet for both Me0 and Me2 such identification fails. This is not a new feature. In the SU(N) SQCD case, one of the meson operators of the magnetic theory is identified with the chiral operator describing the unlifted region of the Coulomb branch of the electric theory. Such identification is explained in more detail in [36]. We claim that similar dynamics takes place in the present case and that the mesons Me0 and Me2 are identified with the Coulomb branch moduli Y and Ye of the electric theory:

Me0 → Y, Me2 → Ye . (8.1.8)

75 8.2 Consistency checks of the duality

8.2.1 Matching charges of the Coulomb branch operators

The ﬁrst check on the proposed duality is simply a matching of the quantum numbers of the

Coulomb branch operators. Using the zero mode counting summarized in table (6.1.2) together with the quantum numbers of the elementary ﬁelds in (8.1.1) we can explicitly compute the charge assignment for the Coulomb branch operators, which correspond to the global charges carried by the fermionic zero modes in a given one-instanton background. The resulting charges are

0 U(1)1 U(1)2 U(1) U(1)R

14 Y1 -2 2 4 3

Y2 0 0 0 -2

Y3 0 0 0 -2

Ye1 0 0 12 2

2 Ye2 -2 2 -8 3 (8.2.1) Ye3 0 0 12 2

0 Y1 0 0 0 -2 0 Y2 0 0 0 -2 0 14 Y3 -2 2 4 3

Q 2 Y = i Yi -2 2 4 q 3 2 Ye = YYe2 -2 2 -2 3

We can see that the charges of Y, Ye indeed match with those of Me0, Me2 from table (8.1.6).

76 8.2.2 Matching the quantum constraints on a circle

The second consistency check is to reproduce the dynamics of the theory on a circle (R3 × S 1). To obtain the description on the circle from 3D theories, we need to add the contribution of the KK monopole1 (see the previous chapter or the original literature [63–65]). In our model, a calculation a` la Polyakov gives the KK monopole contribution of the form

8 WKK = ηY ∼ Λ Y . (8.2.2)

This modiﬁes the superpotential to

2 Wdyn = Y 3T det M0 − 12Thh − 24 det M2 + ηY + Ye 2M0 M2 + hh (8.2.3)

On the other hand, the theory on the circle can be obtained directly from the 4D description by compactifying one of the spatial directions both for the electric and magnetic sides of the 4D duality. It is well-known that the physics of the 4D SU(4) gauge theory with 2 flavors and 2 antisymmetrics is described by a set of gauge invariant fields satisfying two constraints, one of which is quantum modified while the other is not [40–42]. These constraints can be captured by the superpotential

2 8 W = λ 3T det M0 − 12Thh¯− 24 det M2 − Λ + µ 2M0 M2 + hh¯ (8.2.4) where λ and µ are Lagrange multipliers enforcing the constraints. A comparison of (8.2.3) and

(8.2.4) suggests an identiﬁcation of the Lagrange multipliers with the Coulomb branch moduli, λ → Y and µ → Ye. Indeed, integrating out Y and Ye from (8.2.3) reproduces the constraints imposed by (8.2.4).

8.3 Duality for SU(4) with and 3( + )

We can also connect the theory investigated above to the sequence obtained from SU(4) with a single antisymmetric tensor and ﬂavors. The 4D s-conﬁning version of this sequence with one

1The KK monopole contribution can be lifted by matter zero modes. But, as described in Section 8.1, in the presence of antisymmetric tensors, this only happens for SU(N), with N > 4.

77 antisymmetric is the model with 4 flavors. Carrying through the steps of compactification on a circle and adding a real mass we arrive the 3D s-confining version of the theory with a single antisymmetric and three flavors, given by the superpotential

W = Y T det M + HMH¯ (8.3.1) where T = A2, M = QQ¯, H = AQ2 and H¯ = AQ¯ 2. Note, that this theory has a single Coulomb branch operator Y appearing in its low energy dynamics. Giving a complex mass µ to one more

ﬂavor from this theory we obtain a low energy description with a quantum modiﬁed constraint given by the superpotential h i W = λ Y(T det M + HH¯ ) − µ (8.3.2)

As a consistency check we can verify whether this quantum modiﬁed constraint is obtained

2 when a holomorphic mass term µA2 is added to one of the antisymmetric ﬁelds of the model discussed in section 8.1:

2 Wdyn = Y 3T det M0 − 12Thh − 24 det M2 + Ye 2M0 M2 + hh + µT22 . (8.3.3)

Indeed the T22 equation of motion will provide precisely the constraint in (8.3.2) while together with the equations of motion for the other ﬁelds containing A2 the entire superpotential will be set to zero. In particular, any reference to the variable Ye disappears (without having to take the equation of motion for Ye). This suggests that the Coulomb branch dynamics in regions II and III changes when one goes from two antisymmetrics to a single one: in the latter case the fundamental will dominate the dynamics and a single variable will be sufﬁcient to describe the entire Coulomb branch. We propose a tentative explanation for it in the next section.

8.4 Low energy description on the Coulomb branch

Another important check of the duality involves the analysis of the low energy physics on the

Coulomb and Higgs branches of the theory. While a detailed analysis will be presented in a future work [51], let us point out some qualitative features of this regime. On the electric side,

78 at large Ye the SU(4) gauge group is broken to an SU(2)×SU(2)×U(1). Let us concentrate on the dynamics of the SU(2) × SU(2) SO(4) group. Components of the fundamental ﬁelds charged under this group obtain large real masses and are expected to decouple from the low energy physics while the light degrees of freedom in the anti-symmetric tensors transform as vectors of the SO(4).

On the magnetic side there is no gauge group and thus no Coulomb branch. Ye only appears as a chiral superfield, therefore we expect that the effect of decoupling the fundamentals will be taken care of by the non-trivial superpotential in (8.1.7). This expectation is in fact accurate. At large Ye, all composite fields containing electric fundamental degrees of freedom acquire a mass through (8.1.7) and become heavy. Integrating them out we find that the IR degrees of freedom are Y, Ye, and T, while the superpotential vanishes. This also agrees with the results of [32] where it was found that in the SO(4) theory with two vectors, the IR superpotential vanishes, the classical moduli space is not modified quantum mechanically, but the classical singularity at the origin is smoothed out. The massless degrees of freedom in these two theories match once the dependence of the SO(4) coupling on Ye is taken into account.

It is also interesting to consider the Coulomb branch dynamics of a theory with a single antisymmetric. In the electric theory, along the Ye direction, an effective superpotential is generated for the modulus YSO of the unbroken SO(4) subgroup of the form [32] 1 W = 2 . (8.4.1) YSOT

Matching YSO to the operators of the full SU(4) theory one can see that the ﬁelds are pushed towards the Y branch while the Ye direction is lifted. This resolves a puzzle we encountered in the previous subsection – while the counting of the instanton-monopole zero modes suggest a possibility of two unlifted moduli, the correct magnetic description (8.3.1) found through the reduction of the 4D theory involves only Y. Such considerations are independent of the number of ﬂavors present in the theory. It is therefore reasonable to assume that the presence of a single unlifted Coulomb branch direction parametrized by Y is a generic feature of SU(4) theories with a single antisymmetric tensor.

79 CHAPTER 9 CLASSIFICATION OF 3D S-CONFINING THEORIES

We can now briefly outline how the previous discussion generalizes to all three dimensional s-confining theories. We only make preliminary comments and leave a general classification of

3D s-conﬁnement to subsequent work [51].

Anomaly matching techniques cannot be used in 3D1, yet it has been argued in [44,72] that a necessary condition for confinement in 3D is that the Witten index Tr(−1)F equals 1. The Witten index [73] for an SU(N) gauge theory with Chern-Simons number k and generic matter content in the r f representation has been recently computed to be [44, 74]: (k0 + N − 1)! X Tr(−1)F = , k0 = |k| − T (adj.) + T (r ) (9.0.1) (N − 1)!k0! 2 2 f f where the sum is over the matter fields in representation r f , and T2(r f ) is the Dynkin index normalized to 1/2 for fundamentals. (9.0.1) implies that 3D s-confining theories satisfy [44, 72]:

0 X k = 0 ⇒ |k| = T2(adj.) − T2(r f ). (9.0.2) f Let us now focus on the case with vanishing Chern-Simons index, k = 0. In such a case, (9.0.2) reduces to the condition for a 4D theory to exhibit a quantum modified constraint. Such theories can often be obtained by decoupling one flavor from 4D s-confining theories. In order to derive an explicit expression for the superpotential 3D s-confining theories, one must first dimensionally reduce the 4D s-confining theory which has an extra flavor compared to its 3D s-confining partner. Next one must make a real mass deformation to remove one flavor and set the Witten index to 1. This also removes the ηY term from the electric superpotential. During this process, the IR description of the 3D theory is always under control and one is be able to explicitly write down the superpotential.

Based on the examples presented in this paper as well as the original case of 3D SQCD [19,

20, 36], we expect the following structure to emerge. The matter content of the s-conﬁning

1In odd dimensions, gauge invariance can require the addition of a classical Chern-Simons term which breaks parity. This is referred to as the parity anomaly [46, 47, 71].

80 3D theories correspond to 4D models with quantum moduli spaces described by several constraints among the gauge invariant fields, one of which is quantum modified, while the others are not. We expect that the number of unlifted Coulomb branch directions should match the number of constraints present in the 4D theory. In other words, the surviving Coulomb branch operators are identified with the Lagrange multipliers of the 4D theory once these are promoted to dynamical fields.

The SU(4) model with 2 + 2( + ) in this paper is part of a sequence of theories SU(N) + + 2( + ) which satisﬁy (9.0.2) and are expected to s-conﬁne. The 4D description is well known, and the number of constraints are expected to indicate the number of unlifted Coulomb branch operators:

SU(2N) with + + 2 + → N constraints (9.0.3) SU(2N + 1) with + + 2 + → N constraints (9.0.4)

For SU(4), SU(5) and SU(6) the Coulomb branch operators corresponding to the directions of the various constraints can be identiﬁed in the following way: √ SU(4) Y4 → diag(σ, 0, 0, −σ) Ye → YY2 → diag(σ, σ, −σ, −σ) √ SU(5) Y5 → diag(σ, 0, 0, 0, −σ) Ye5 → YY2Y3 → diag(σ, σ, 0, −σ, −σ) √ (9.0.5) SU(6) Y6 → diag(σ, 0, 0, 0, 0, −σ) Ye6 → YY2Y3Y4 → diag(σ, σ, 0, 0, −σ, −σ)

1 ˆ 2 3 Y6 → (Ye1 Y3) → diag(σ, σ, σ, −σ, −σ, −σ) with an obvious generalized pattern for even and odd N. The quantum numbers of these operators exactly match the quantum numbers of the “extra” meson operators which remain massless after the real mass deformation and suggest that these Coulomb branch operators are identiﬁed with the extra mesons.

We now argue for the validity of these operators. All of these operators correspond to monopole conﬁgurations and satisfy the Dirac quantization condition. The SU(5) case is very similar to SU(4) with respect to these monopole operators. In fact, it can be checked that the

2 2 direction described by Ye5 breaks SU(5) → SU(2) × U(1) /Z2. Half-integer charged monopoles

81 are allowed and therefore justify the square root in the deﬁnition of the monopole operator. A similar discussion applies for Ye6.

New features arise when considering Yˆ6 operator in an SU(6) theory. Along the this direction, the symmetry breaking is locally SU(6) → SU(3)2 ×U(1), with the U(1) direction generated by ! 1 1 1 1 1 1 Q = diag , , , − , − , − . (9.0.6) 6 3 3 3 3 3 3

There are multiple identiﬁcations among the U(1) orbit and the Z3 × Z3 center of SU(3) × SU(3),

2 thus modifying the global structure of the breaking into SU(6) → SU(3) × U(1) /Z3. We therefore expect fractionally charged monopole, in particular with 1/3 of the minimal Dirac charge.

82 CHAPTER 10 CONCLUSIONS

In this thesis we described the initiation of the study of the dynamics of 3D N = 2 supersymmetric theories with matter ﬁelds in generic representations. Already in the simplest extension of 3D SQCD obtained by adding antisymmetric matter, there are interesting new features.

While previously studied theories have at the most one unlifted Coulomb branch direction, the physics is very different when antisymmetric tensors are added to the theory because multiple directions along the Coulomb branch could remain unlifted. We have described the case of

SU(4) with ﬂavors and one or two anti-symmetric tensors. By performing a careful analysis of fermionic zero modes in monopole backgrounds and matching to the dynamics of 4D theories, it is possible to show that theories with two antisymmetric tensors have two unlifted Coulomb branch directions. It is also possible to identify the additional Coulomb branch modulus Ye with a fractionally charged monopole operator.

It is possible to analyze the flow to a 3D s-confining theory with two flavors and two antisymmetric tensors by performing a dimensional reduction of 4D s-confining dualities and decoupling a flavor through a real mass deformation. The quantum numbers of the two Coulomb branch operators, Y and Ye, exactly match the quantum numbers of certain meson operators which naturally appear in the superpotential obtained by the dimensional reduction. This is strong evidence that the Coulomb branch analysis is correct. Furthermore it is possible to identify both Y and Ye as monopole operators associated to particular monopole configurations in the UV. In particular Ye is associated with a fractionally charged monopole and is allowed because of the non-trivial global topology of the unbroken group along the particular Coulomb branch direction described by Ye.

We provided multiple checks for these dualities. The 3D magnetic dual correctly reproduces the behavior expected on R3 × S 1, which can be obtained by compactifying the 4D SU(4) theory with two antisymmetric tensors and two ﬂavors. It is known that in 4D, this theory is described

83 in the IR by a quantum modified and a classical constraint. From a 3D perspective, the quantum modification of one of the classical constraints arises from the ηY operator generated by the dynamics of the KK monopole. A similar analysis in SU(4) s-confining theories with a single antisymmetric suggests that only a single Coulomb branch direction remains in such a theory, while the Ye direction is lifted. While a detailed explanation of this phenomenon is beyond the scope of this thesis, we provided a tentative explanation of such dynamics looking at the ADS-like superpotential generated along the Ye direction.

Finally we also presented initial comments on the description of more general 3D s- conﬁning theories. The details of the general analysis will be presented in future works.

84 APPENDIX A SPINORS AND NOTATION

In this chapter we will review the basics of fermions in 3 dimensional Minkowski and set up the notation which is used throughout the text.

In three dimensional Minkowski space with metric ηi j = (−, +, +), a convenient choice for the generator of the Clifford algebra is        0 1   1 0   0 1  γ0 β =   , γ1 β =   , γ2 β =   . α   α   α   (A.0.1)  −1 0   0 −1   1 0 

In fact the above matrices satisfy the following relations

i j i j i j i jk {γ , γ } = 2η , γ , γ = −2 γk (A.0.2)

0 1 2 where γk = (−γ , γ , γ ). We can then build the generators of the sponsorial representation of the Lorentz group in 3D i i S i j = [γi, γ j] = − i jkγ (A.0.3) 4 2 k (A.0.3) shows that the sponsorial representation of the Lorentz group is real, more precisely

S pin(2, 1) SL(2, R). The fundamental sponsorial representation is a two-dimensional Majo- rana fermion and it will be indicated with a fermion with lower component ψα. A spinor with a upper index, ψα transforms instead in the anti-fundamental representation. These two representation are actually equivalent and indices can be raise and lowered by the invariant symbols    0 1  αβ   εαβ = ε =   (A.0.4)  −1 0 

∗ †α furthermore complex conjugation switches how a spinor transforms (ψα) = ψ .

As usual it is important to establish a convention for raising and lowering indices

α αβ β ψ = ε ψβ, ψα = ψ εβα (A.0.5)

85 from which follows the convention for contracting sponsorial indices, the upper index will always come before the lower index. With this convention, the difference between a spinor in the fundamental or in the anti-fundamental is merely that the former transform by left matrix multiplication while the latter by multiplication on the right.

Similarly we deﬁne the product of spinor as

α βα ψ1ψ2 = ψ1 ψ2α = ε ψ1αψ2β. (A.0.6)

In our treatment of N = 2 3D SUSY we often use the 4D N = 1 intuition. It is thus helpful to briefly review how 4D spinors decompose in terms of 3D representations. This can be easily achieved noticing that S pin(3, 1) SL(2, C) which is the complexification of the spinor group in 3D. It thus follows that a fundamental (Weyl) fermion in 4D can be written as complexification of 3D fundamental fermion. Therefore a 4D Weyl spinor decompose into two, two dimensional Majorana fermions.

86 APPENDIX B ELEMENTS OF GROUP THEORY

In this appendix we will present a quick review of the essential tools of simple Lie groups representation theory. Given that this thesis focuses mostly on the SU(N) case, we will present a more detailed analysis of the SU(N) group. Great references to learn more about group theory in a “particle physics” oriented way are [75].

B.1 Deﬁnition and Cartan subalgebra

Consider a simple Lie group G. Here we will only consider ﬁnite-dimensional groups, so elements of a group G can be simply thought as matrices. Furthermore any group element can be written as an exponential of other matrices

a g = exp (iω Ta) (B.1.1)

a dimG where ω = {ω } ∈ R . Tα are called the generators of the group G. For the SU(N) case, it is straightforward to obtain that Ta are the set of N × N traceless hermitian matrices. The group action can be also deﬁned by the action of its generators through exponentiation. For the most part we will in fact discuss the action of the Tas rather than the elements of the group G.

The set of the Tas forms a Lie algebra, and the Lie algebra structure is given by their commutator

c [Ta, Tb] = i fab Tc. (B.1.2)

c The real coefﬁcients fab are called the structure constants of the group.

f c = 0 In any given basis, the subset of generators which commute, that is a¯b¯ , form a sub- algebra of the Lie-algebra which plays a very special role in representation theory. Such sub- algebra is called Cartan sub-algebra and we will indicate the generators which belong to it with a different letter, Hi

h i † Hi, H j = 0, Hi = Hi. (B.1.3)

87 The rank of a group G, indicated as rG, is deﬁned as the dimension of its Cartan sub-algebra. So in (B.1.3), i = 1, ..., rG.

By construction, it follows that in any given representation R, there exists a basis in which all the elements of the Cartan sub-algebra can be diagonalized. Thus the Cartan sub-algebra can be analogously deﬁned as the maximum set of diagonal generators Ta. This is a more handy deﬁnition if we are working with explicit representation of the generators Ta, which is often the case.

B.2 Roots, weights and lots of SU(2)s

B.2.1 weights

Consider a given irreducible representation R. We can write its elements as vectors |vA, Ri where A = 1, ..., dimR. Let us assume that the vectors |vA, Ri, are taken in the basis in which the elements of the Cartan {Hi} are diagonal. By construction we have

A A A Hi|v , Ri = µi |v , Ri (B.2.1)

Because the weight µA (almost) uniquely identiﬁes the corresponding state |vA, Ri, we can label the states of the representation by their eigenvalues |vA, Ri ≡ |µAi, Notice that µA is a rG dimensional vector. The eigenvalues µA are called the weights of the representation R. Furthermore

† the fact that the His are hermitian, Hi = Hi, implies that the weights are real numbers.

B.2.2 Adjoint representation and roots

The set of all generators {Ta} themselves provide a representation of the group G, as the commutator provide a natural action of the generators on themselves. We know this representation

88 very well as gauge boson transform under it. The representation provided by the Tas is called adjoint representation and the action of the generators Tas is deﬁned

→ 0 Tk¯ Tk¯ = [Ta, Tk¯] (B.2.2)

Purely for notational purposes, when we want to think at generators as element of the adjoint representation, we will indicate them with as |Tai, then (B.2.2) can be rewritten as

Ta|Tk¯i ≡ |[Ta, Tk¯]i (B.2.3)

Now consider the elements of the Cartan Hi. Hi can always be chosen to be hermitian, thus

† in what follows Hi = Hi. In a basis in which Hi are diagonal we have, by construction,

Hi|Tai = βi|Tai ≡ |[Hi, Ta]i (B.2.4)

βi, for all Hi and Ta, are the roots of the Lie group. The roots are nothing but the weights of the adjoint representation, so the βis are also real. A given element Ta can be also identiﬁed through the eigenvalue βi. It is therefore natural to label the generators by the root associated to them.

Thus |Tai → |Tβi. Recall that β is in fact a rG-dimensional vector, the i-th component given by the action of Hi on |Tβi.

Let us notice that although {|Tai} and {|Tβi} both provide a possible choice for a basis for the Lie-algebra generators, there is a conceptual difference among the two choices that is worth highlighting. The {|Tβi} label the speciﬁc combinations of generators which diagonalize the

† Hi while {|Tai} is usually taken as the basis where the generators are hermitian |Ta i = |Tai.

† Generally the {|Tβi} are not hermitian, |Tβ i , |Tβi. We are already familiar with this. In the SU(2) SU(2) case, there exists a basis of hermitian operators given by the Pauli matrices {|Ta i} ≡ {|σii}. Such basis, although very natural in many applications, is not the basis which diagonalizes the adjoint action of the one-dimensional Cartan sub-algebra (H = σ3). In fact σ3|σii ≡ |[σ3, σi]i ,

βi|σii. To diagonalize [σ3, ·] we need to consider a non-hermitian combination J± ∼ σ1 ±iσ2. The

SU(2) latter are the ones which we would naturally identify as {|Tβ i}.

89 Now consider the action of an element of the Cartan Hi on the commutator of two generators labeled by the roots β and β0 respectively

0 sociated to the sum of the eigenvalues β + β and can be thus identiﬁed as |[Tβ, Tβ0 ]i ∼ |Tβ+β0 i.

† Furthemore consider the action of Hi on the hermitian conjugate |Tβi (recall |Tβ i , |Tβi)

† † † † Hi|Tβ i = |[Hi, Tβ ]i = −|[Hi, Tβ] i = −βi|Tβ i. (B.2.6)

† This shows that it is natural to label the hermitian conjugate of |Tβi with the opposite root |Tβ i ≡

|T−βi implying that for any simple Lie group, roots always come in pairs ±βi.

Finally we can derive explicitly the action of a given generator associated to a root Tβ on a generic state |µAi in a representation R

A A A A A A Hi Tβ||µ i = µi Tβ|µ i + [Hi, Tβ]|µ i = (µi + βi)Tβ|µ i. (B.2.7)

A A It follows that Tβ shifts the weight by β, that is Tβ|µ i ≡ |µ + βi.

From the deﬁnition of the Cartan it follows that the vectors corresponding to the Cartan generators, |Hiis, are associated with the “zero” roots.

B.2.3 Simple roots

It is straightforward to count the number of different roots for a given group G of dimension n. In fact there is one root for each generator, with the condition that if β is a root −β is also a root. In addition there are rG generators, namely the generators of the Cartan {Hi}, which are associated to a β = 0 root. This gives us a total of dimG − rG non vanishing roots, grouped in

(dimG − rG)/2 pairs.

A more careful analysis of the group, shows that in order to fully characterize a simple Lie group we don’t need all the roots, only the ones which are linearly independent. Above we

90 have already seen that some of the roots can be written as sums of other roots: the commutator of two generators labeled by β and β0 is associated to a third root β00 = β + β0. The “minimum” number of roots that are needed to characterize a given Lie algebra are called simple roots. They are deﬁned as

Linearly independent. • Positive. •

It can be shown that each Lie group has exactly rG simple groups, where rG is the rank of the group, that is the dimension of the Cartan sub algebra.

The set of simple roots are henceforth indicated with the letter α, by construction {α} ⊂ {β}. Because {α} contain all the important information to characterize the generators of the Lie algebra, we will almost always refer to the simple roots and only very rarely refer to the {β}. Let us stress again that both αs and βs are rG-dimensional vectors.

B.2.4 SU(2)s

Consider a simple root α ∈ {α} of a group G. There are three generators which can be constructed from it (Tα, T−α, α · H). Consider the following convenient normalization

† Tα Tα α · H E = , E− = & O = (B.2.8) α |α| α |α| α |α|2 they satisfy the following commutation relation

[Oα, Eα] = Eα, [Oα, E−α] = −E−α,

[Eα,E−α] = Oα. (B.2.9)

(B.2.9) is isomorphic to an SU(2) algebra. We can construct a triple (B.2.8) for each simple root.

This implies that a simple Lie group of rank rG contains rG independent SU(2)s.

91 With some caveats which is beyond the scope of this short appendix, the normalization of the simple roots is arbitrary. Below we, discussing the SU(N) case, we ﬁx the normalization to

|α| = 1.

B.3 A closer look at SU(N)

Following [75] we can generalize the SU(3) Gell-Mann matrices deﬁning the following normalization

0 1 Tr(T T ) = δ 0 (B.3.1) β β 2 ββ

Explicitly the generators Eβ can be taken as having only one non-zero off-diagonal entry √ with value 1/ 2. Furthermore rSU(N) = N − 1 as there are N − 1 independent traceless diagonal real matrices. The Cartan generators can be chosen as follows   1 Xm   −  − [Hm]i j = √  δi,kδ j,k mδi,m+1δ j,m+1 , m = 1, ..., N 1. (B.3.2) 2m(m + 1) k=1 where δi, j is the standard Kronecker delta. Applying (B.3.2) to the SU(4) we obtain      1 0 0 0   1 0 0 0           0 −1 0 0   0 1 0 0  1   1   H1 =  , H2 = √   , (B.3.3) 2  0 0 0 0  12  0 0 −2 0           0 0 0 0   0 0 0 0     1 0 0 0       0 1 0 0  1   H3 = √   . (B.3.4) 24  0 0 1 0       0 0 0 −3 

We have already seen in (B.2.7) that the roots connect all the different weights. Therefore the set of roots can be obtained from taking all possible combination

βAB = µA − µB (B.3.5)

92 while the N − 1 simple roots are αI = νI − νI+1 (B.3.6)

The vis are deﬁned below. It is straightforward to check that all the βs can be obtained from

AB PB i linear combinations of αs. In fact β = i=A ν .

It is useful to explicitly write the SU(2)s generators in the SU(4) example. Using the expression below for the vis this is readily done      1 0 0 0   0 0 0 0          1  0 −1 0 0  1  0 1 0 0  O ≡ ·   O ≡ ·   1 α1 H =  , 2 α2 H =   , (B.3.7) 2  0 0 0 0  2  0 0 −1 0           0 0 0 0   0 0 0 0     0 0 0 0      1  0 0 0 0  O ≡ ·   3 α3 H =   . (B.3.8) 2  0 0 1 0       0 0 0 −1 

B.3.1 Fundamental

For SU(N) the fundamental representation has N complex dimension and it can be visualized as a column vector with N complex entries    |µ1i       |µ2i      Q ≡  |µ3i  (B.3.9)    .   .       |µNi 

I I I where we labeled the states by their weights, that is Hi|µ i = µi |µ i. Since the weights for the fundamental representation play a special role, we will label them with a different letter, µA =

νA.

93 For completeness, using the deﬁnition of the Cartan operators (B.3.2), we can obtain an explicit expression for the {νA}s ! 1 1 1 1 ν1 = , √ , ..., √ , ..., √ (B.3.10) 2 2 3 2m(m + 1) 2(N − 1)N ! 1 1 1 1 ν2 = − , √ , ..., √ , ..., √ (B.3.11) 2 2 3 2m(m + 1) 2(N − 1)N ! 1 1 1 ν3 = 0, − √ , ..., √ , ..., √ (B.3.12) 3 2m(m + 1) 2(N − 1)N ... (B.3.13) ! m 1 νm+1 = 0, 0, ..., − √ , ..., √ (B.3.14) 2m(m + 1) 2(N − 1)N ... (B.3.15) ! N − 1 νN = 0, 0, ..., 0, ..., − √ (B.3.16) 2(N − 1)N

An important relation is the scalar product of two νis

1 1 νi · ν j = − + δ , (B.3.17) 2N 2 i j from which, using (B.3.6), we can compute the roots’ norm

|αI|2 = αI · αI = 1 (B.3.18)

k We can also write explicit how a generic element of the Cartan sub-algebra ς Hk acts on the I-th state of the fundamental representation:

k I k I I ς Hk|ν i = ς νk|ν i (B.3.19)

In the text we have also chosen a different parametrization which uses N variables related by

k traceless condition, ς Hk ≡ diag(σ1, σ2, ..., σN). This parametrization makes the action evident

k I I I ς Hk|ν i = σ |ν i (B.3.20)

k I I we thus obtain the relation ς νk = σ .

94 B.3.2 Antisymmetric

The anti-symmetric combination of two fundamentals provides an irreducible representation of SU(N), we will refer to this representation as the antisymmetric representation. By construction, the antisymmetric representation is N(N − 1)/2 dimensional. In the SU(4) case we have    0 |µ12i |µ13i |µ14i       0 |µ23i |µ24i  ≡   A   (B.3.21)  0 |µ34i       0  we will see that is more convenient to label the weights with two indices.

Using explicitly the action of the Cartan generators (B.3.2) it is possible to show that the weights µIJ can be written in terms of the {νI}

µIJ = νI + νJ (B.3.22) it then immediately follow the action of a generic element of the Cartan sub-algebra

k IJ k I J IJ I J IJ ς Hk|µ i = ς (νk + νk )|µ i = (σ + σ )|µ i (B.3.23)

The real mass for various elements of the anti-symmetric representation can be readily computed from (B.3.23).

It is useful to explicitly write down how the various components of the antisymmetric tensor transform under the various SU(2)’s. Using the formalism thus far introduced we obtain

1 OI|µJKi = αI · (νJ + νK)|µJKi = δIJ − δI+1,J + δIK − δI+1,K |µJKi. (B.3.24) 2 it follows that only the states such that J, K , I, I + 1 are charged under the I-th SU(2) form- ing doublets. We can then fully write down the decomposition of the elements of the antisymmetric under the three SU(2)s contained in SU(4). Each SU(2) embedding will be denoted

95 by the corresponding simple root αI:      |µ13i   |µ14i  1 →     12 34 α   ,   , |µ i & |µ i ,  |µ23i   |µ24i       |µ12i   |µ24i  2 →     14 23 α   ,   , |µ i & |µ i , (B.3.25)  |µ13i   |µ34i       |µ13i   |µ23i  3 →     12 34 α   ,   , |µ i & |µ i .  |µ14i   |µ24i 

96 APPENDIX C ZERO MODES AND CALLIAS INDEX THEOREM

Counting fermionic zero modes in an instanton/monopole background is a very useful tool to study the Coulomb branch of 3D N = 2 SUSY gauge theories. In this appendix we detail how to count these zero modes and derive (6.1.2).

C.1 Callias Index Theorem

We may now state the Callias index theorem. For a fermion in representation R with weights wi, the number of zero modes in an instanton/monopole background, (4.3.10), is

1 X N = sign(w · σ )(w · g ). (C.1.1) 2 i 0 i 0 i

For a careful proof see the original paper [76] or [19] for a more physical derivation. From the sign function one can see that conjugate representations have the same number of zero modes.

To emphasize this 4D intuition, we refer to these 3D instantons as instanton/monopole solutions. See AppendixC or the original literature [53, 54] for details. We thus expect a 3D instanton solution associated with each SU(2) subgroup. In particular, SU(N) has (N − 1) independent solutions, one for each SU(2) subgroup. These are labeled by topological indices

(n1, n2, ..., nN−1) which are the charges of the independent 4D BPS monopoles.

C.2 Fundamental representation

Let us computer the number zero modes of a fermion in the fundamental representation. We indicate the weights of the fundamental as {νi}, where i = 1, ..., N. In this case we have

νi · σ0 = σi, νi · g0 = g0 i = ni − ni−1 with n0 = nN = 0. (C.2.1)

97 The number of zero modes is

N 1 X N = sign(σ ) n − n − , (C.2.2) 2 i i i 1 i=1 which is the standard result that each fundamental fermion has ni zero modes in the region σi >

0 > σi+1 [19,20,60]. In each region only the i-th fundamental instanton/monopole conﬁguration contributes to the number of zero modes. Thus, in (6.1.2), each and provide n1 zero modes in region I, n2 zero modes in region II and III, and n3 in region IV.

C.3 Adjoint representation

Now consider the number of gaugino zero modes. The weights of the adjoint representation are the roots βi j ≡ νi − ν j, which we have expressed in terms of the fundamental weights, νi. The number of zero modes is

N 1 X N = sign(σ − σ ) (n − n − ) − (n − n − ) (C.3.1) adj 2 i j i i 1 j j 1 i, j=1 XN−1 XN−1 h i = ni sign(σi − σ j) − sign(σi+1 − σ j+1) (C.3.2) i=1 j=1 XN−1 = 2n j. (C.3.3) j=1

Therefore the number of gaugino zero modes is independent of the values of the σi. This result matches with [19, 20, 60].

C.4 Antisymmetric representation

For antisymmetric matter, the weights can be labelled with two indices (wanti)i j = νi + ν j with i , j. The number of zero modes is thus

N 1 X N = sign(σ + σ ) (n − n − ) + (n − n − ) . (C.4.1) anti 2 i j i i 1 j j 1 i, j

98 This depends on the sign of σi + σ j. Referring to (6.1.2), in regions I and II (σ1 + σ2; σ1 +

σ3; σ1 + σ4) > 0 and (σ2 + σ3; σ2 + σ4; σ3 + σ4) < 0. Therefore each antisymmetric fermion

I, II has Nanti = 2n1 zero modes. In regions III and IV both σ2 + σ3 and σ1 + σ4 ﬂip sign. Inserting III, IV this into the above formula yields Nanti = 2n3. The zero mode counting for the fundamental and the adjoint representations matches the results in the literature. On the other hand, our results for antisymmetrics are obtained in a generic point on the Coulomb branch and are thus expected to differ from the conclusions of [77] where the zero mode counting is performed at a center-symmetric point.

99 BIBLIOGRAPHY

[1] Csaba Cski, Mario Martone, Yuri Shirman, Philip Tanedo, and John Terning. Dynamics of 3D SUSY Gauge Theories with Antisymmetric Matter. 2014.

[2] N. Seiberg. Exact results on the space of vacua of four-dimensional SUSY gauge theories. Phys. Rev. D, 49:6857–6863, 1994.

[3] N. Seiberg. Electric–magnetic duality in supersymmetric non-Abelian gauge theories. Nucl. Phys., B435:129–146, 1995.

[4] Kenneth A. Intriligator and N. Seiberg. Lectures on supersymmetric gauge theories and electric- magnetic duality. Nucl. Phys. Proc. Suppl., 45BC:1–28, 1996.

[5] Nathan Seiberg and Edward Witten. Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. Nucl. Phys., B431:484, 1994.

[6] Nathan Seiberg and Edward Witten. Electric - magnetic duality, monopole condensation, and conﬁnement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys., B426:19–52, 1994.

[7] Alexander M. Polyakov. Quark Conﬁnement and Topology of Gauge Groups. Nucl.Phys., B120:429–458, 1977.

[8] Richard P. Feynman. The Qualitative Behavior of Yang-Mills Theory in (2+1)-Dimensions. Nucl.Phys., B188:479, 1981.

[9] Mithat Unsal. Magnetic Bion Condensation: a New Mechanism of Conﬁnement and Mass Gap in Four Dimensions. Phys.Rev., D80:065001, 2009.

[10] Mithat Unsal. Abelian Duality, Conﬁnement, and Chiral Symmetry Breaking in QCD(Adj). Phys.Rev.Lett., 100:032005, 2008.

[11] Philip C. Argyres and Mithat Unsal. The Semi-Classical Expansion and Resurgence in Gauge Theories: New Perturbative, Instanton, Bion, and Renormalon Effects. JHEP, 1208:063, 2012.

[12] Erich Poppitz and Mithat Unsal. Seiberg-Witten and ‘Polyakov-Like’ Magnetic Bion Con- ﬁnements are Continuously Connected. JHEP, 1107:082, 2011.

[13] Erich Poppitz and Mithat Unsal. Conformality Or Conﬁnement: (Ir)Relevance of Topolog- ical Excitations. JHEP, 0909:050, 2009.

[14] M. Shifman and M. Unsal. Multiﬂavor QCD* on R(3) × S 1: Studying Transition from Abelian to Non-Abelian Conﬁnement. Phys.Lett., B681:491–494, 2009.

100 [15] Sidney R. Coleman. The Quantum Sine-Gordon Equation as the Massive Thirring Model. Phys.Rev., D11:2088, 1975.

[16] S. Mandelstam. Soliton Operators for the Quantized Sine-Gordon Equation. Phys.Rev., D11:3026, 1975.

[17] N. Seiberg and E. Witten. Gauge dynamics and compactiﬁcation to three-dimensions. In Saclay 1996, The mathematical beauty of physics, pages 333–366.

[18] Kenneth A. Intriligator and N. Seiberg. Mirror symmetry in three-dimensional gauge theories. Phys. Lett. B, 387:513, 1996.

[19] Jan de Boer, Kentaro Hori, and Yaron Oz. Dynamics of N=2 supersymmetric gauge theories in three-dimensions. Nucl.Phys., B500:163–191, 1997.

[20] Ofer Aharony, Amihay Hanany, Kenneth A. Intriligator, N. Seiberg, and M. J. Strassler. Aspects of N = 2 supersymmetric gauge theories in three dimensions. Nucl. Phys., B499:67– 99, 1997.

[21] Andreas Karch. Seiberg duality in three-dimensions. Phys.Lett., B405:79–84, 1997.

[22] O. Aharony. IR duality in d= 3 N= 2 supersymmetric USP(2Nc) and U(Nc) gauge theories. Phys. Lett. B, 404:71, 1997.

[23] Anton Kapustin and Matthew J. Strassler. On mirror symmetry in three-dimensional Abelian gauge theories. JHEP, 9904:021, 1999.

[24] M. J. Strassler. Conﬁning phase of three-dimensional supersymmetric quantum electrody- namics. In *Shifman, M. A. (ed.): The many faces of the superworld, pages 262–279.

[25] N. Dorey and D. Tong. Mirror symmetry and toric geometry in three-dimensional gauge theories. JHEP, 0005:018, 2000.

[26] D. Tong. Dynamics of N=2 supersymmetric Chern-Simons theories. JHEP, 0007:019, 2000.

[27] A. Giveon and D. Kutasov. Seiberg Duality in Chern-Simons Theory. Nucl. Phys., B812:1, 2009.

[28] V. Niarchos. Seiberg Duality in Chern-Simons Theories with Fundamental and Adjoint Matter. JHEP, 0811:001, 2008.

[29] Anton Kapustin. Seiberg-like duality in three dimensions for orthogonal gauge groups. 2011.

[30] T. Dimofte, D. Gaiotto, and S. Gukov. Gauge Theories Labelled by Three-Manifolds.

101 [31] Francesco Benini, Cyril Closset, and Stefano Cremonesi. Comments on 3d Seiberg-like dualities. JHEP, 1110:075, 2011.

[32] Ofer Aharony and Itamar Shamir. On O(Nc) d = 3 N = 2 supersymmetric QCD Theories. JHEP, 1112:043, 2011.

[33] C. Hwang, K. J. Park, and J. Park. Evidence for Aharony duality for orthogonal gauge groups. JHEP, 1111:011, 2011.

[34] Anton Kapustin, Hyungchul Kim, and Jaemo Park. Dualities for 3d Theories with Tensor Matter. JHEP, 1112:087, 2011.

[35] H. Kim and J. Park. Aharony Dualities for 3d Theories with Adjoint Matter.

[36] Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg, and Brian Willett. 3d dualities from 4d dualities. 2013.

[37] Ofer Aharony, Nathan Seiberg, and Yuji Tachikawa. Reading between the lines of four- dimensional gauge theories. 2013.

[38] J. Park and K. J. Park. Seiberg-like Dualities for 3d N=2 Theories with SU(N) gauge group.

[39] V. Niarchos. Seiberg dualities and the 3d/4d connection. JHEP, 1207:075, 2012.

[40] Csaba Csaki, Martin Schmaltz, and Witold Skiba. Conﬁnement in N = 1 SUSY gauge theories and model building tools. Phys. Rev., D55:7840–7858, 1997.

[41] Csaba Csaki, Martin Schmaltz, and Witold Skiba. A Systematic approach to conﬁnement in N=1 supersymmetric gauge theories. Phys.Rev.Lett., 78:799–802, 1997.

[42] Csaba Csaki. The Conﬁning N=1 supersymmetric gauge theories: A Review. pages 489– 499, 1998.

[43] Ian Afﬂeck, Jeffrey A. Harvey, and Edward Witten. Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions. Nucl.Phys., B206:413, 1982.

[44] Kenneth Intriligator and Nathan Seiberg. Aspects of 3d N=2 Chern-Simons-Matter Theo- ries. 2013.

[45] S. Weinberg. The Quantum Theory of Fields, Volume III Supersymmetry. Cambridge Univer- sity Press, 1 edition.

[46] A.N. Redlich. Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions. Phys.Rev., D29:2366–2374, 1984.

102 [47] A.N. Redlich. Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions. Phys.Rev.Lett., 52:18, 1984.

[48] Amihay Hanany and Edward Witten. Type IIB superstrings, BPS monopoles, and three- dimensional gauge dynamics. Nucl.Phys., B492:152–190, 1997.

[49] Jan de Boer, Kentaro Hori, Hirosi Ooguri, and Yaron Oz. Mirror symmetry in three- dimensional gauge theories, quivers and D-branes. Nucl.Phys., B493:101–147, 1997.

[50] Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, and Zheng Yin. Mirror symmetry in three-dimensional theories, SL(2,Z) and D-brane moduli spaces. Nucl.Phys., B493:148–176, 1997.

[51] C. Csaki, M. Martone, Y. Shirman, F. Tanedo, and J. Terning. to appear.

[52] M. Shifman. Advanced Topics in Quantum Field Theory, A lecture course. Cambridge Univer- sity Press, 1 edition.

[53] E. Weinberg. Fundamental monopoles and multi-monopole solutions for arbitrary simple gauge groups. Nucl. Phys., B167:500, 1980.

[54] E. Weinberg. Fundamental monopoles in theories with arbitrary symmetry breaking. Nucl. Phys., B203:445, 1982.

[55] Anton Kapustin. Wilson-’t Hooft operators in four-dimensional gauge theories and S- duality. Phys.Rev., D74:025005, 2006.

[56] Vadim Borokhov, Anton Kapustin, and Xin-kai Wu. Monopole operators and mirror symmetry in three-dimensions. JHEP, 0212:044, 2002.

[57] Vadim Borokhov, Anton Kapustin, and Xin-kai Wu. Topological disorder operators in three-dimensional conformal ﬁeld theory. JHEP, 0211:049, 2002.

[58] Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg, and Brian Willett. 3d dualities from 4d dualities for orthogonal groups. 2013.

[59] Kenneth Intriligator. Matching 3d N=2 Vortices and Monopole Operators. 2014.

[60] E. Poppitz and M. Unsal. Index theorem for topological excitations on R3 × S 1. JHEP, 0903:027, 2009.

[61] E. Poppitz and T. Sulejmanpasic. (S)QCD on R3 × S1: Screening of Polyakov loop by fundamental quarks and the demise of semi-classics. JHEP, 1309:128, 2013.

[62] John Preskill. Magnetic Monopoles. Ann.Rev.Nucl.Part.Sci., 34:461–530, 1984.

103 [63] K. Lee and P. Yi. Monopoles and Instantons on Partially Compactiﬁed D-Branes. Phys. Rev., D56:3711, 1997.

[64] K. Lee. Instantons and Magnetic Monopoles on R3 × S1 with Arbitrary Simple Gauge Groups. Phys. Lett., B426:323, 1998.

[65] K. Lee and C. Lu. SU(2) Calorons and Magnetic Monopoles. Phys. Rev., D58:025011, 1998.

[66] N. Michael Davies, Timothy J. Hollowood, and Valentin V. Khoze. Monopoles, afﬁne algebras and the gluino condensate. J. Math Phys., 44:3640, 2003.

[67] T. M. W. Nye and M. A. Singer. An L2-index theorem for Dirac operators on S 1 × R3.

[68] T. C. Kraan and P. van Baal. Periodic instantons with nontrivial holonomy. Nucl. Phys., B533:627–659, 1998.

[69] T. C. Kraan and P. van Baal. Monopole constituents inside SU(N) caldrons. Phys. Lett. B, 435:389–395, 1998.

[70] T. C. Kraan and P. van Baal. Constituent monopoles without gauge ﬁxing. Nucl. Phys. Proc. Suppl., 73:554–556, 1999.

[71] A.J. Niemi and G.W. Semenoff. Axial Anomaly Induced Fermion Fractionization and Ef- fective Gauge Theory Actions in Odd Dimensional Space-Times. Phys.Rev.Lett., 51:2077, 1983.

[72] Edward Witten. Supersymmetric index of three-dimensional gauge theory. Shifman M. A. (ed.): The many faces of the superworld, page 156, 1999.

[73] E. Witten. Constraints on Supersymmetry Breaking. Nucl. Phys., B202:253, 1982.

[74] A. V. Smilga. Witten index in N = 1 and N = 2 SYMCS theories with matter. 2013.

[75] H. Georgi. Lie Algebras In Particle Physics: from Isospin To Uniﬁed Theories. Westview press, 2 edition.

[76] C. J. Callias. Index theorems on open spaces. Comm. Math. Phys., 63:213, 1978.

[77] M. Shifman and M. Unsal. QCD-like Theories on R3 × S 1: A Smooth Journey from Small to Large r(S 1) with Double-Trace Deformations. Phys. Rev. D, 78:065004, 2008.

104