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University of Cincinnati UNIVERSITY OF CINCINNATI Date:___________________ I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ Topics in supersymmetric gauge theories and the gauge-gravity duality A dissertation submitted to the office of research and advanced studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Ph.D.) in the department of physics of the College of Arts and Sciences B.Sc., Sharif University of Technology, Tehran, Iran, June 2001 Mohammad Edalati Ahmadsaraei June 1, 2007 Dissertation committee chair: Professor Philip C. Argyres Topics in supersymmetric gauge theories and the gauge-gravity duality Mohammad Edalati Ahmadsaraei Department of Physics, University of Cincinnati, P.O. Box 210011, Cincinnati, OH 45221-0011, U.S.A. [email protected] Abstract: In this thesis we use supersymmetry and the gauge-gravity duality to shed light on some dynamical features of strongly-coupled non-Abelian gauge theories. In the first half of the thesis, we consider singular superpotentials of four dimensional = 1 supersymmetric QCD N and show that they must exist. In particular, using some non-trivial consistency checks, we show that these superpotentials, albeit singular, are perfectly sensible, and can be used to reproduce in a simple way both the low energy effective dynamics as well as some special higher-derivative terms. In the second half of the thesis, we investigate the behavior of timelike, spacelike and Euclidean stationary string configurations on a five-dimensional AdS black hole background which, in the context of the gauge-gravity duality, correspond to quark-antiquark pairs steadily moving in an = 4 supersymmetric Yang-Mills thermal N bath. We find that there are many branches of solutions depending on the quark velocity and separation as well as on whether Euclidean or Lorentzian configurations are examined. Such solutions can, in principle, be used to determine some transport properties of strongly- coupled = 4 thermal bath. In particular, using a recently proposed non-perturbative N definition of the jet-quenching parameter, we take the lightlike limit of spacelike solutions to evaluate the jet quenching parameter in the bath. We show that this proposed definition gives zero jet quenching parameter, independent of how the lightlike limit is taken. In fact, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear, rather than quadratic, in the quark separation. To my parents, Reza and Maryam, with gratitude Contents Preface.......................................... vi Citationsandcopyrightnotice. xii Acknowledgements ................................... xiii Notationandabbreviation . xiv 1 Singular effective superpotentials in supersymmetric gauge theories 2 1.1 Effective superpotentials for SU(2) SQCD . ...... 6 1.1.1 Singular superpotentials and the classical constraints ......... 7 1.1.2 Consistency upon integrating out flavors . ..... 10 1.2 Singular superpotentials and the Konishi anomaly equations ......... 10 1.2.1 Directdescription. .. .. 11 1.2.2 Seibergdualdescription . 12 1.3 Singular superpotentials and higher-derivative F-terms ............ 13 1.3.1 Taylor expansion around a vacuum . 15 1.3.2 Feynmanrules............................... 16 1.3.3 Differentflavors.............................. 18 1.4 Singular superpotentials in three dimensions . .......... 24 2 Generalized Konishi anomaly and singular superpotentials of SU(N) SQCD 27 2.1 SU(2) singular superpotentials revisited . .......... 31 iii 2.2 Singular superpotential for Nf = N + 2 SQCD: non-integrability of GKA equations ..................................... 36 2.2.1 ComparisontotheSU(2)solution. .. 40 2.3 Singular superpotential for Nf = N + 2 SQCD: Seiberg dual analysis . 42 2.3.1 Derivation of the effective superpotential . ....... 43 2.3.2 Integrating out the glueball field . .... 46 2.3.3 Comparing to the direct result when Nf = 4 .............. 49 3 Singular superpotentials and higher-derivative F-terms of Sp(N) SQCD 51 3.1 Singular superpotentials of Sp(N) SQCD . ...... 52 3.1.1 Sp(N) SQCD for a small number of flavors . 52 3.1.2 Superpotentials and classical constraints for a large number of flavors 53 3.1.3 Consistency with Konishi anomaly equations: direct description. 55 3.1.4 Consistency with Konishi anomaly equations: Seiberg dual description. 56 3.1.5 Consistency upon integrating out flavors. ...... 58 3.2 Higher-derivative F-terms in Sp(N) SQCD . ...... 58 3.2.1 ThestructureofSp(N)F-terms . 59 3.2.2 Feynmanrules............................... 61 3.2.3 Differentflavors.............................. 64 4 String configurations for moving quark-antiquark pairs in a thermal bath 70 4.1 Equationsofmotion ............................... 74 4.2 Timelike Lorentzian solutions . ..... 76 4.2.1 Timelike Lorentzian: perpendicular velocity . ........ 77 4.2.2 Timelike Lorentzian: parallel velocity . ....... 79 4.3 Euclidean strings and their energetics . ........ 81 4.3.1 Euclidean: perpendicular “velocity” . ...... 81 4.3.2 Euclidean: parallel “velocity” . .... 85 iv 5 Spacelike string configurations and jet-quenching from a Wilson loop 86 5.1 String embeddings and equations of motion . ...... 89 5.2 Spacelikesolutions .............................. .. 93 5.2.1 Perpendicular velocity . 93 5.2.2 Parallelvelocity.............................. 99 5.3 Application to jet-quenching . ..... 100 5.3.1 EuclideanWilsonloop . .. .. 101 5.3.2 SpacelikeWilsonloop .......................... 103 5.4 Appendix ..................................... 106 5.4.1 Euclideanaction ............................. 106 5.4.2 Spacelikeaction.............................. 110 Bibliography ...................................... 117 v Preface Quantum field theory which combines the principles of quantum mechanics and special rel- ativity is the language by which we describe modern particle physics. Using this language one can calculate the typical observable quantities in particle physics such as scattering am- plitudes and particle lifetimes. In fact, the Standard Model of particle physics is a special kind of a quantum field theory: a gauge theory with a particular gauge group. A (quantum) gauge theory is a quantum field theory with a symmetry (or more accurately, a redundancy) called gauge symmetry. Understanding the dynamics of gauge theories is of utmost impor- tance for at least two reasons. Firstly, gauge theories give a fairly complete description of the physics of elementary particles at energies that we can reach experimentally these days in particle accelerators. Secondly, over the past thirty years or so, it has become clear that we have to think of such gauge theories as effective theories: a low energy approximation to a deeper theory that may fundamentally be different from a gauge theory (for example, string theory). Therefore, the better we understand the dynamics of gauge theories, the more we can constrain the ultimate theory whose effective descriptions are supposed to be the gauge theories we know of at low energies. The weak-coupling regime of gauge theories, though very important, is relatively easy to understand. One just needs to apply the well-studied tools of quantum field theory such as perturbation theory and Feynman diagrams to the problem at our disposal. The difficult part is to understand the strong-coupling behavior of such theories where perturbation theory is no longer applicable as a result of the coupling being strong. For example, crucial dynamical features such as quark confinement, dynamical generation of a mass gap and chiral symmetry breaking which are all so characteristic of the strongly-coupled regime of asymptotically-free gauge theories can not be analytically addressed using conventional tools of quantum field theory (except for lattice simulations which are numerical techniques). Thesis in a nutshell Thus, the question is: how can one obtain a better understanding of the strongly-coupled dynamics of gauge theories such as quantum chromodynamics (QCD)? One approach which comprises the first half of the thesis is to enlarge the symmetries of the theory to make it supersymmetric. (See [1] for a general introduction to supersymmetry.) The more symmetry available in a system, the easier it is to be solved exactly. A theory with a certain amount vi of supersymmetry is more constrained than an ordinary non-supersymmetric one, hence generically more amenable to exact analysis. Another interesting approach which is the subject of the second half of the thesis is to use the celebrated gauge-gravity duality , also known as Maldacena’s conjecture, or AdS/CFT correspondence. (For a detailed review of the subject, see [2].) This duality maps the strong-coupling regime of gauge theories to the weakly-coupled supergravity. Although constructing a supergravity solution dual to QCD is extremely difficult, one can nevertheless come up with solutions dual to QCD-like theories, and learn non-trivial lessons. In what follows, along with explaining these two approaches, we present new results on the strongly-coupled dynamics of supersymmetric gauge theories, as well as on certain features of finite-temperature QCD-like plasmas at strong coupling. Part I: singular effective
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