Renormalization Group Flows and Continual Lie Algebras
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Renormalization Group Flows from Holography; Supersymmetry and a C-Theorem
© 1999 International Press Adv. Theor. Math. Phys. 3 (1999) 363-417 Renormalization Group Flows from Holography; Supersymmetry and a c-Theorem D. Z. Freedmana, S. S. Gubser6, K. Pilchc and N. P. Warnerd aDepartment of Mathematics and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 b Department of Physics, Harvard University, Cambridge, MA 02138, USA c Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA d Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract We obtain first order equations that determine a super symmetric kink solution in five-dimensional Af = 8 gauged supergravity. The kink interpo- lates between an exterior anti-de Sitter region with maximal supersymme- try and an interior anti-de Sitter region with one quarter of the maximal supersymmetry. One eighth of supersymmetry is preserved by the kink as a whole. We interpret it as describing the renormalization group flow in J\f = 4 super-Yang-Mills theory broken to an Af = 1 theory by the addition of a mass term for one of the three adjoint chiral superfields. A detailed cor- respondence is obtained between fields of bulk supergravity in the interior anti-de Sitter region and composite operators of the infrared field theory. We also point out that the truncation used to find the reduced symmetry critical point can be extended to obtain a new Af = 4 gauged supergravity theory holographically dual to a sector of Af = 2 gauge theories based on quiver diagrams. e-print archive: http.y/xxx.lanl.gov/abs/hep-th/9904017 *On leave from Department of Physics and Astronomy, USC, Los Angeles, CA 90089 364 RENORMALIZATION GROUP FLOWS We consider more general kink geometries and construct a c-function that is positive and monotonic if a weak energy condition holds in the bulk gravity theory. -
Quantum Groups and Algebraic Geometry in Conformal Field Theory
QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford -
Conformal Symmetry in Field Theory and in Quantum Gravity
universe Review Conformal Symmetry in Field Theory and in Quantum Gravity Lesław Rachwał Instituto de Física, Universidade de Brasília, Brasília DF 70910-900, Brazil; [email protected] Received: 29 August 2018; Accepted: 9 November 2018; Published: 15 November 2018 Abstract: Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Keywords: quantum gravity; conformal gravity; quantum field theory; non-local gravity; super- renormalizable gravity; UV-finite gravity; conformal anomaly; scattering amplitudes; conformal symmetry; conformal supergravity 1. Introduction From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and also due to relativity changes of periods of clocks to measure time intervals). -
Introduction to Supersymmetry(1)
Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N. -
Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum -
Inönü–Wigner Contraction and D = 2 + 1 Supergravity
Eur. Phys. J. C (2017) 77:48 DOI 10.1140/epjc/s10052-017-4615-1 Regular Article - Theoretical Physics Inönü–Wigner contraction and D = 2 + 1 supergravity P. K. Concha1,2,a, O. Fierro3,b, E. K. Rodríguez1,2,c 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile 3 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile Received: 25 November 2016 / Accepted: 5 January 2017 / Published online: 25 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract We present a generalization of the standard cannot always be obtained by rescaling the gauge fields and Inönü–Wigner contraction by rescaling not only the gener- considering some limit as in the (anti)commutation relations. ators of a Lie superalgebra but also the arbitrary constants In particular it is well known that, in the presence of the exotic appearing in the components of the invariant tensor. The Lagrangian, the Poincaré limit cannot be applied to a (p, q) procedure presented here allows one to obtain explicitly the AdS CS supergravity [7]. This difficulty can be overcome Chern–Simons supergravity action of a contracted superal- extending the osp (2, p) ⊗ osp (2, q) superalgebra by intro- gebra. In particular we show that the Poincaré limit can be ducing the automorphism generators so (p) and so (q) [9]. performed to a D = 2 + 1 (p, q) AdS Chern–Simons super- In such a case, the IW contraction can be applied and repro- gravity in presence of the exotic form. -
Renormalization Group Flows and Continual Lie Algebras
hep–th/0307154 July 2003 Renormalization group flows and continual Lie algebras Ioannis Bakas Department of Physics, University of Patras GR-26500 Patras, Greece [email protected] Abstract We study the renormalization group flows of two-dimensional metrics in sigma models us- ing the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by (d/dt; 1), with anti-symmetric G Cartan kernel K(t, t′) = δ′(t t′); as such, it coincides with the Cartan matrix of the − superalgebra sl(N N + 1) in the large N limit. The resulting Toda field equation is a | non-linear generalization of the heat equation, which is integrable in target space and arXiv:hep-th/0307154v1 17 Jul 2003 shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via B¨acklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten’s two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormal- ization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. -
POLCHINSKI* L.Vman L,Ahoratorv ~!/"Phvs,W
Nuclear Physics B231 (lq84) 269-295 ' Norlh-tlolland Publishing Company RENORMALIZATION AND EFFECTIVE LAGRANGIANS Joseph POLCHINSKI* l.vman l,ahoratorv ~!/"Phvs,w. Ilarcard Unn'er.~itv. ('amhrtdge. .'Was,~adntsett,s 0213b¢. USA Received 27 April 1983 There is a strong intuitive understanding of renormalization, due to Wilson, in terms of the scaling of effective lagrangians. We show that this can be made the basis for a proof of perturbative renormalization. We first study rcnormalizabilit,.: in the language of renormalization group flows for a toy renormalization group equation. We then derive an exact renormalization group equation for a four-dimensional X4,4 theorv with a momentum cutoff. Wc organize the cutoff dependence of the effective lagrangian into relevant and irrelevant parts, and derive a linear equation for the irrelevant part. A length} but straightforward argument establishes that the piece identified as irrelevant actually is so in perturbation theory. This implies renormalizabilitv The method extends immediately to an}' system in which a momentum-space cutoff can bc used. but the principle is more general and should apply for any physical cutoff. Neither Weinberg's theorem nor arguments based on the topology of graphs are needed. I. Introduction The understanding of renormalization has advanced greatly in the past two decades. Originally it was just a means of removing infinities from perturbative calculations. The question of why nature should be described by a renormalizable theory was not addressed. These were simply the only theories in which calculations could be done. A great improvement comes when one takes seriously the idea of a physical cutoff at a very large energy scale A. -
Notes on Statistical Field Theory
Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual).......................... -
Functional Renormalization Group Approach and Gauge Dependence
Functional renormalization group approach and gauge dependence in gravity theories V´ıtor F. Barraa1, Peter M. Lavrovb,c2, Eduardo Antonio dos Reisa3, Tib´erio de Paula Nettod4, Ilya L. Shapiroa,b,c5 a Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil b Department of Theoretical Physics, Tomsk State Pedagogical University, 634061 Tomsk, Russia c National Research Tomsk State University, 634050 Tomsk, Russia d Departament of Physics, Southern University of Science and Technology, 518055 Shenzhen, China Abstract We investigate the gauge symmetry and gauge fixing dependence properties of the effective average action for quantum gravity models of general form. Using the background field formalism and the standard BRST-based arguments, one can establish the special class of regulator functions that preserves the back- ground field symmetry of the effective average action. Unfortunately, regardless the gauge symmetry is preserved at the quantum level, the non-invariance of the regulator action under the global BRST transformations leads to the gauge fixing dependence even under the use of the on-shell conditions. Keywords: quantum gravity, background field method, functional renormal- ization group approach, gauge dependence arXiv:1910.06068v3 [hep-th] 5 Mar 2020 1 Introduction The interest to the non-perturbative formulation in quantum gravity has two strong motivations. First, there are a long-standing expectations that even the perturbatively non- renormalizable models such as the simplest quantum gravity based on general relativity 1E-mail address: [email protected] 2E-mail address: [email protected] 3E-mail address: eareis@fisica.ufjf.br 4 E-mail address: [email protected] 5E-mail address: shapiro@fisica.ufjf.br may be quantum mechanically consistent due to the asymptotic safety scenario [1] (see [2, 3] for comprehensive reviews). -
Supergravity Backgrounds and Symmetry Superalgebras
Turkish Journal of Physics Turk J Phys (2016) 40: 113 { 126 http://journals.tubitak.gov.tr/physics/ ⃝c TUB¨ ITAK_ Review Article doi:10.3906/fiz-1510-8 Supergravity backgrounds and symmetry superalgebras Umit¨ ERTEM∗ School of Mathematics, College of Science and Engineering, The University of Edinburgh, Edinburgh, United Kingdom Received: 12.10.2015 • Accepted/Published Online: 08.12.2015 • Final Version: 27.04.2016 Abstract: We consider the bosonic sectors of supergravity theories in ten and eleven dimensions corresponding to the low energy limits of string theories and M-theory. The solutions of supergravity field equations are known as supergravity backgrounds and the number of preserved supersymmetries in those backgrounds are determined by Killing spinors. We provide some examples of supergravity backgrounds that preserve different fractions of supersymmetry. An important invariant for the characterization of supergravity backgrounds is their Killing superalgebras, which are constructed out of Killing vectors and Killing spinors of the background. After constructing Killing superalgebras of some special supergravity backgrounds, we discuss the possibilities of the extensions of these superalgebras to include the higher degree hidden symmetries of the background. Key words: Supergravity backgrounds, Killing spinors, Killing superalgebras 1. Introduction The unification of fundamental forces of nature is one of the biggest aims in modern theoretical physics. The most promising approaches for that aim include the ten-dimensional supersymmetric string theories and their eleven-dimensional unification called M-theory. There are five different string theories in ten dimensions: type I, type IIA and IIB, and heterotic E8 × E8 and SO(32) theories. However, some dualities called T-duality, S-duality, and U-duality between strong coupling and weak coupling limits of these theories can be defined and these dualities can give rise to one unified M-theory in eleven dimensions [1, 2, 3, 4]. -
Remarks on A2 Toda Field Theory 1 Introduction
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Remarks on A2 Toda Field Theory S. A. Apikyan1 Department of Theoretical Physics Yerevan State University Al. Manoogian 1, Yerevan 375049, Armenia C. Efthimiou2 Department of Physics University of Central Florida Orlando, FL 32816, USA Abstract We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, af- ter integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differ- ential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A2 Toda field theory. 1 Introduction The Toda field theory (TFT) provides an extremely useful description of a large class of two-dimensional integrable quantum field theories. For this reason these models have attracted a considerable interest in recent years and many outstanding results in various directions have been established. TFTs are divided in three broad categories: finite Toda theories (FTFTs) for which the underlying Kac-Moody algebra [1, 2] is a finite Lie algebra, affine Toda theories (ATFTs) for which the underlying Kac-Moody algebra [email protected] [email protected] 1 is an affine algebra and indefinite Toda theories (ITFTs) for which the under- lying Kac-Moody algebra is an indefinite Kac-Moody algebra.