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An Introduction to Global Supersymmetry ⊘C

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An Introduction to Global Supersymmetry ⊘C An Introduction to Global Supersymmetry Philip C. Argyres Cornell University c ⊘ Contents Preface v 1 N=1 d=4 Supersymmetry 1 1.1 WhySupersymmetry? ........................... 1 1.2 Supersymmetric Quantum Mechanics . 5 1.2.1 Supersymmetry algebra in 0+1 dimensions . 5 1.2.2 Quantum mechanics of a particle with spin . 7 1.2.3 Superspaceandsuperfields. 8 1.3 RepresentationsoftheLorentzAlgebra . .... 15 1.3.1 Tensors ............................... 17 1.3.2 Spinors................................ 19 1.4 Supermultiplets ............................... 26 1.4.1 Poincar´ealgebra and particle states . ... 26 1.4.2 Particle representations of the supersymmetry algebra...... 27 1.4.3 Supersymmetrybreaking. 30 1.5 N=1 Superspace and Chiral Superfields . .. 32 1.5.1 Superspace ............................. 32 1.5.2 GeneralSuperfields . 33 1.5.3 Chiralsuperfields .......................... 34 1.5.4 Chiral superfield action: Kahler potential . .... 36 1.5.5 Chiral superfield action: Superpotential . .... 39 1.6 Classical Field Theory of Chiral Multiplets . ...... 42 1.6.1 Renormalizablecouplings. 42 1.6.2 Generic superpotentials and R symmetries . .. 46 1.6.3 Modulispace ............................ 50 1.6.4 Examples .............................. 51 i 1.7 VectorsuperfieldsandsuperQED . 55 1.7.1 Abelianvectorsuperfield . 55 1.7.2 Coupling to left-chiral superfields: superQED . .... 59 1.7.3 General Abelian gauged sigma model . 62 1.7.4 Higgsingandunitarygauge . 63 1.7.5 Supersymmetry breaking and Fayet-Iliopoulos terms . ...... 64 1.7.6 Solving the D equations ...................... 67 1.8 Non-Abeliansupergaugetheory. 75 1.8.1 Reviewofnon-Abeliangaugetheory . 75 1.8.2 Non-Abelian vector superfields . 82 2 Quantum N=1 Supersymmetry 87 2.1 EffectiveActions .............................. 87 2.2 Non-RenormalizationTheorems . 96 2.2.1 Holomorphyofthesuperpotential . 97 2.2.2 Nonrenormalization theorem for left-chiral superfields...... 99 2.2.3 Kahlertermrenormalization . 102 2.3 Quantumgaugetheories . 106 2.3.1 Gaugecouplings. .. .. 106 2.3.2 ϑ anglesandinstantons . 107 2.3.3 Anomalies .............................. 112 2.3.4 Phasesofgaugetheories . 119 2.4 Non-Renormalization in Super Gauge Theories . ..... 124 2.4.1 Supersymmetric selection rules . 124 2.4.2 Global symmetries and selection rules . 126 2.4.3 IR free gauge theories and Fayet-Iliopoulos terms . ..... 129 2.4.4 Exactbetafunctions . 130 2.4.5 Scale invariance and finiteness . 134 3 The Vacuum Structure of SuperQCD 139 3.1 Semi-classicalsuperQCD . 140 3.1.1 Symmetries and vacuum equations . 140 3.1.2 Classical vacua for Nc > Nf > 0.................. 142 3.1.3 Classical vacua for N N ..................... 144 f ≥ c 3.2 Quantum superQCD: Nf < Nc ....................... 146 iii 3.2.1 N = N 1............................. 148 f c − 3.2.2 N N 1: effectsoftree-levelmasses . 149 f ≤ c − 3.2.3 Integratingoutandin . 153 3.3 Quantum superQCD: N N ....................... 154 f ≥ c 3.3.1 Nf = Nc ............................... 155 3.3.2 Nf = Nc+1 ............................. 158 3.3.3 N N +2 ............................. 161 f ≥ c 3.4 Superconformalinvariance . 163 3.4.1 Representations of the conformal algebra . 164 3.4.2 N=1 superconformal algebra and representations . 169 3.5 N=1duality................................. 170 3.5.1 Checks................................ 173 3.5.2 Matchingflatdirections . 175 4 Extended Supersymmetry 179 4.1 Monopoles.................................. 179 4.2 Electric-magneticduality. 183 4.3 An SU(2)Coulombbranch. 188 4.3.1 Physics near U0 ........................... 191 4.3.2 Monodromies ............................ 193 4.3.3 τ(U)................................. 195 4.3.4 Dual Higgs mechanism and confinement . 197 Bibliography 199 iv Preface These lecture notes provide an introduction to supersymmetry with a focus on the non-perturbative dynamics of supersymmetric field theories. It is meant for students who have had a one-year introductory course in quantum field theory, and assumes a basic knowledge of gauge theories, Feynman diagrams and renormalization on the physics side, and an aquaintance with analysis on the complex plane (holomorphy, analytic continuation) as well as rudimentary group theory (SU(2), Lorentz group) on the math side. More adanced topics—Wilsonian effective actions, Lie groups and algebras, anomalies, instantons, conformal invariance, monopoles—are introduced as part of the course when needed. The emphasis will not be on comrehensive discussions of these techniques, but on their “practical” application. The aims of this course are two-fold. The first is to introduce the technology of global supersymmetry in quantum field theory. The first third of the course introduces the N =1 d = 4 superfields describing classical chiral and vector multiplets, the geometry of their spaces of vacua and the nonrenormalization rules they obey. A excellent text which covers these topics in much greater detail than this course does is S. Weinberg’s The Quantum Theory of Fields III: Supersymmetry [1]. These notes try to follow the notation and conventions of Weinberg’s book; they also try to provide alternative (often more qualitative) explanations for overlapping topics, instead of reproducing the exposition in Weinberg’s book. Also, the student is directed to Weinberg’s book for important topics not covered in these lectures (supersymmetric models of physics beyond the standard model and supersymmetry breaking), as well as to the original references. The second second aim of these notes is to use our understanding of non-perturbative aspects of particular supersymmetric models as a window on strongly coupled quan- tum field theory. From this point of view, supersymmetric field theories are just es- pecially symmetric versions of ordinary field theories, and in many cases this extra symmetry allows the exact determination of some non-perturbative properties of these theories. This gives us another context (besides lattice gauge theory and semi-classical expansions) in which to think concretely about non-perturbative quantum field theory in more than two dimensions. To this end, the last two-thirds of the course ana- v vi PREFACE lyzes examples of strongly coupled supersymmetric gauge theories. We first discuss non-perturbative SU(n) N=1 supersymmetric versions of QCD, covering cases with completely Higgsed, Coulomb, confining, and interacting conformal vacua. Next we describe d=4 theories with N=2 and 4 extended supersymmetry, central charges and Seiberg-Witten theory. Finally we end with a brief look at supersymmetry in other dimensions, describing spinors and supersymmetry algebras in various dimensions, 5- dimensional N=1 and 2 theories, and 6-dimensional N=(2, 0) and (1, 1) theories. These notes owe a large intellectual debt to Nathan Seiberg: not only is his work the main focus of much of the course, but also parts of this course are modelled on two series of lectures he gave at the Institute for Advanced Study in Princeton in the fall of 1994 and at Rutgers University in the fall of 1995. These notes grew, more immediately, out of a graduate course on supersymmetry I taught at Cornell University in the fall semesters of 1996 and 2000. It is a pleasure to thank the students in these courses, and especially Zorawar Bassi, Alex Buchel, Ron Maimon, K. Narayan, Sophie Pelland, and Gary Shiu, for their many comments and questions. It is also a pleasure to thank my colleagues at Cornell—Eanna Flanagan, Kurt Gottfried, Tom Kinoshita, Andr´e Leclair, Peter Lepage, Henry Tye, Tung-Mow Yan, and Piljin Yi—for many helpful discussions. I’d also like to thank Mark Alford, Daniel Freedman, Chris Kolda, John March-Russell, Ronen Plesser, Al Shapere, Peter West, and especially Keith Dienes for comments on an earlier version of these notes. Thanks also to the physics department at the University of Cincinnati for their kind hospitality. Finally, this work was supported in part by NSF grant XXXX. Philip Argyres Ithaca, New York January, 2001 Chapter 1 N=1 d=4 Supersymmetry 1.1 Why Supersymmetry? Though originally introduced in early 1970’s we still don’t know how or if supersym- metry plays a role in nature. Why, then, have a considerable number of people been working on this theory for the last 25 years? The answer lies in the Coleman-Mandula theorem [2], which singles-out supersymmetry as the “unique” extension of Poincar´ein- variance in quantum field theory in more than two space-time dimensions (under some important but reasonable assumptions). Below I will give a qualitative description of the Coleman-Mandula theorem following a discussion in [3]. The Coleman-Mandula theorem states that in a theory with non-trivial scattering in more than 1+1 dimensions, the only possible conserved quantities that transform as tensors under the Lorentz group (i.e. without spinor indices) are the usual energy- momentum vector Pµ, the generators of Lorentz transformations Jµν , as well as possible scalar “internal” symmetry charges Zi which commute with Pµ and Jµν. (There is an extension of this result for massless particles which allows the generators of conformal transformations.) The basic idea behind this result is that conservation of Pµ and Jµν leaves only the scattering angle unknown in (say) a 2-body collision. Additional “exotic” conservation laws would determine the scattering angle, leaving only a discrete set of possible angles. Since the scattering amplitude is an analytic function of angle (assumption # 1) it then vanishes for all angles. We illustrate this with a simple example. Consider a theory of 2 free real bose fields φ1 and φ2: 1 1 = ∂ φ ∂µφ ∂ φ ∂µφ . (1.1) L −2 µ 1 1 − 2 µ 2 2 Such a free field theory has infinitely many conserved currents. For example, it follows 1 2 CHAPTER 1. N=1 D=4 SUPERSYMMETRY µ µ immediately from the equations of motion ∂µ∂ φ1 = ∂µ∂ φ2 = 0 that the series of currents J = (∂ φ )φ φ ∂ φ , µ µ 2 1 − 2 µ 1 J = (∂ φ )∂ φ φ ∂ ∂ φ , µρ µ 2 ρ 1 − 2 µ ρ 1 J = (∂ φ )∂ ∂ φ φ ∂ ∂ ∂ φ , etc. (1.2) µρσ µ 2 ρ σ 1 − 2 µ ρ σ 1 are conserved, µ µ µ ∂ Jµ = ∂ Jµρ = ∂ Jµρσ =0, (1.3) leading to the conserved charges d−1 Qρσ = d x J0ρσ, etc.
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