DOCSLIB.ORG

Noncritical String Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Noncritical String Theory Noncritical String Theory Sander Walg Master’s thesis Supervisor: Prof. Dr. Jan de Boer University of Amsterdam Institute for Theoretical Physics Valckenierstraat 65 1018 XE Amsterdam The Netherlands August 26, 2008 ii Abstract In strings theory, a critical dimension, Dc is required to yield consistent theories. For bosonic strings Dc = 26 and for superstrings Dc = 10. These numbers arise naturally from the theory itself. Less familiar are noncrit- ical string theories, theories with D 6= Dc. These theories emerge when background fields are included to the theory, in particular linear dilaton backgrounds. We will study quintessence-driven cosmologies and show an analogy between them and string theories in a timelike linear dilaton theory. We will also present a set of exact solutions for the linear dilaton-tachyon profile system that gives rise to a bubble of nothing. Generalizing this set- ting induces a dimension-changing bubble, which can also be solved exactly at one-loop order. Eventually, we will consider transitions from one theory to another. In this way, noncritical string theories can be connected to the familiar web of critical string theories. Surprisingly, transitions from su- perstring theories can yield pure bosonic theories. Our main focus will be bosonic strings. iv Preface Foreword When I started writing my thesis, I was immediately confronted with a tremendous abundancy of background material on string theory. Even though string theory is relatively new, already a lot of books and an enor- mous amount of articles have been written on the subject, and the level of difficulty varies greatly. Some books were very advanced, others were much more comprehensible but somewhat limited in detail. I found it was a challenging task to restrict my focus only on those subjects relevant to the scope of this thesis, because it is very easy to get lost in all the fascinating features and idea’s that are indissoluble connected to string theory. At first I had some doubt whether I should go into a lot of detail with calculations or not. But gradually, I found that I gained most satisfaction out of explaining most of the intermediate steps needed to complete a cal- culation, whenever I thought they contributed to the clarity of the subject. Whenever a calculation would become to detailed or advanced, I decided to give a reference to the book or article the problem can be found in. I wrote this thesis on the level of graduate students, with preliminary knowledge on quantum field theory, general relativity and string theory. Only chapter 3 involves some knowledge on algebra and representations. It will not be used later on. With this in mind, I still tried to make the thesis as self-contained as possible, reviewing some general facts where needed. Just as many people before me, I too found out that writing a thesis is by far the most challenging and difficult task of completing my studies as a physics student. Nevertheless, I’ve experienced this as a very educative and enjoyable time, and during this period I found out that my interest in doing theoretical research has grown considerably. Acknowledgments First of all, I would like to thank my supervisor Jan de Boer, who has always been an inspiration to me and could explain difficult subject in a very clear way. I always enjoyed his lectures a lot and his interpretations truly made vi Preface complicated subjects completely comprehensible. Secondly, I would like to thank my student friends who were always open to questions and discussions and who helped me with some difficulties regarding LATEX. I also like to thank my other friends, who never lost faith in me and always helped me regain new energy. They always showed a lot of interest in my studies and my thesis, and in turn I enjoyed explaining them the exotic features of particle physics, cosmology, black holes and string theory. I would like to thank my girlfriend Sabine Wirz, who often found me completely worn out after a long day at the university and who has always been there for me. She supported me a great deal and absolutely helped me through this period. And finally I would like to thank my family, who I could rely on all my years at University. In particular I would like to thank my father, Hugo Walg, who somehow always managed to get the best out of me and supported me with his technical backup. And last, I would like to thank my mother, Doroth´eevan Wijhe, who was always able to motivate me, and who made it possible for me to graduate this year. I don’t think I could have succeeded successfully without her support. Contents Preface v Notation and conventions xi Introduction xiii I Theoretical frame-work 1 1 Basic principles on string theory 3 1.1 Why string theory? . 3 1.2 Relativistic point particle . 5 1.2.1 Point particle action . 5 1.2.2 Auxiliary field . 5 1.3 Relativistic bosonic strings . 6 1.3.1 Polyakov action . 6 1.3.2 World-sheet symmetries . 8 1.4 Solutions of the bosonic string . 9 1.4.1 Choosing a world-sheet gauge . 9 1.4.2 Constraints for embedding coordinates . 9 1.4.3 Solutions for embedding coordinates . 10 1.5 Quantizing the relativistic string . 12 1.5.1 Commutation relations . 12 1.5.2 Mass levels . 13 2 Scale transformations and interactions 15 2.1 Couplings . 15 2.1.1 Interacting theories . 15 2.1.2 Running couplings and renormalization . 16 2.1.3 Renormalization β functions . 16 2.2 Weyl invariance and Weyl anomaly . 17 2.2.1 Weyl invariance . 17 2.2.2 Path integral approach . 18 2.2.3 Critical dimension . 20 viii CONTENTS 2.3 Fields and target space . 21 3 Conformal field theory 23 3.1 Complex coordinates . 23 3.1.1 Wick rotation . 23 3.1.2 Conformal transformation . 24 3.2 Operator-product expansions . 25 3.2.1 Currents and charges . 25 3.2.2 Generators of conformal transformations . 26 3.2.3 Operator-product expansions . 27 3.2.4 Free bosons and OPE’s . 28 3.3 Virasoro operators . 30 3.3.1 Virasoro algebra . 30 3.3.2 Operator-state correspondence . 31 4 Vertex operators and amplitudes 35 4.1 Operator-state correspondence . 35 4.1.1 Applying CFT to string theory . 35 4.1.2 Introducing the vertex operator . 35 4.2 Tachyon tree-diagrams for open strings . 38 4.2.1 Vertex operators for open strings . 38 4.2.2 Emitting one open tachyon state . 39 4.2.3 2-tachyon open string scattering . 40 4.2.4 Vertex operators for excited states . 42 4.3 Tachyon tree-diagrams for closed strings . 43 4.3.1 Closed string tachyon vertex operators . 43 4.3.2 Closed massless string vertex operator . 44 5 Strings with backgrounds 47 5.1 Strings in curved spacetime . 47 5.1.1 Nonlinear sigma model . 47 5.1.2 Coherent background of gravitons . 48 5.2 Other background fields and β functions . 48 5.2.1 β functions up tp first order in background fields . 48 5.2.2 β functions up to first order in α0 . 50 6 Low energy effective action 53 6.1 The string metric . 53 6.1.1 Equations of motion . 53 6.1.2 Spacetime dependent coupling . 55 6.2 Link between β functions and EOM . 55 6.3 The Einstein metric . 56 6.3.1 Effective action in the Einstein frame . 56 6.3.2 Utility of the Einstein frame . 57 CONTENTS ix II Applications on noncritical string theory 59 7 Away from the critical dimension 61 7.1 Constant dilaton . 61 7.1.1 Constant dilaton action . 61 7.1.2 Euler characteristic . 61 7.1.3 UV finite quantum gravity . 62 7.2 Linear dilaton background . 63 7.3 Tachyon profile . 64 7.3.1 On-shell tachyon condition . 64 7.3.2 Liouville field theory . 66 8 Quintessence-driven cosmologies 69 8.1 Quintessent cosmologies . 69 8.1.1 Quintessence . 69 8.1.2 FRW cosmologies in D dimensions . 70 8.1.3 Determining the critical equation of state . 72 8.2 Global structures in quintessent cosmologies . 73 8.2.1 Global structures . 73 8.2.2 Penrose diagrams . 75 9 String theory with cosmological behaviour 77 9.1 Linear dilaton as quintessent cosmologies . 77 9.1.1 Comparing the two theories . 77 9.1.2 Fixing the scale factor . 78 9.2 Stable modes . 80 9.2.1 Stability . 80 9.2.2 Massless modes . 81 9.2.3 Massive modes . 83 10 Exact tachyon-dilaton dynamics 85 10.1 Exact solutions and Feynmann diagrams . 85 10.1.1 Lightcone gauge . 85 10.1.2 Exact solutions . 87 10.2 Bubble of nothing . 88 10.2.1 Bubble of nothing . 88 10.2.2 Particle trajectories . 90 10.3 Tachyon-dilaton low energy effective action . 92 10.3.1 General two-derivative form . 92 10.3.2 Determining the final form . 94 x CONTENTS 11 Dimension-changing solutions 97 11.1 Dimension-change for the bosonic string . 97 11.1.1 Oscillatory dependence in the X2 direction . 97 11.1.2 Classical world-sheet solutions . 99 11.1.3 An energy consideration . 100 11.1.4 Oscillatory dependence on more coordinates . 102 11.2 Quantum corrections . 104 11.2.1 Exact solutions at one-loop order . 104 11.2.2 Dynamical readjustment . 107 11.3 Dimension-change for superstrings . 108 11.3.1 Superstring theories . 108 11.3.2 Transitions among various string theories . 109 12 Summary 111 A Renormalized operators 115 B Curvature 117 B.1 Path length and proper time .
Load more

Recommended publications
  • Report of the Supersymmetry Theory Subgroup
    Report of the Supersymmetry Theory Subgroup J. Amundson (Wisconsin), G. Anderson (FNAL), H. Baer (FSU), J. Bagger (Johns Hopkins), R.M. Barnett (LBNL), C.H. Chen (UC Davis), G. Cleaver (OSU), B. Dobrescu (BU), M. Drees (Wisconsin), J.F. Gunion (UC Davis), G.L. Kane (Michigan), B. Kayser (NSF), C. Kolda (IAS), J. Lykken (FNAL), S.P. Martin (Michigan), T. Moroi (LBNL), S. Mrenna (Argonne), M. Nojiri (KEK), D. Pierce (SLAC), X. Tata (Hawaii), S. Thomas (SLAC), J.D. Wells (SLAC), B. Wright (North Carolina), Y. Yamada (Wisconsin) ABSTRACT Spacetime supersymmetry appears to be a fundamental in- gredient of superstring theory. We provide a mini-guide to some of the possible manifesta- tions of weak-scale supersymmetry. For each of six scenarios These motivations say nothing about the scale at which nature we provide might be supersymmetric. Indeed, there are additional motiva- tions for weak-scale supersymmetry. a brief description of the theoretical underpinnings, Incorporation of supersymmetry into the SM leads to a so- the adjustable parameters, lution of the gauge hierarchy problem. Namely, quadratic divergences in loop corrections to the Higgs boson mass a qualitative description of the associated phenomenology at future colliders, will cancel between fermionic and bosonic loops. This mechanism works only if the superpartner particle masses comments on how to simulate each scenario with existing are roughly of order or less than the weak scale. event generators. There exists an experimental hint: the three gauge cou- plings can unify at the Grand Uni®cation scale if there ex- I. INTRODUCTION ist weak-scale supersymmetric particles, with a desert be- The Standard Model (SM) is a theory of spin- 1 matter tween the weak scale and the GUT scale.
    [Show full text]
  • String Theory. Volume 1, Introduction to the Bosonic String
    This page intentionally left blank String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries.
    [Show full text]
  • Spectra of Conformal Sigma Models
    Spectra of Conformal Sigma Models Dissertation zur Erlangung des Doktorgrades an der Fakultät für Mathematik, Informatik und Naturwissenschaften Fachbereich Physik der Universität Hamburg vorgelegt von Václav Tlapák aus Berlin Hamburg 2014 Tag der Disputation: 30. Januar 2015 Folgende Gutachter empfehlen die Annahme der Dissertation: Prof. Dr. Volker Schomerus Prof. Dr. Gleb Arutyunov Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit den Spektren von konformen Sigma Modellen, die auf (verallgemeinerten) symmetrischen Räumen defi- niert sind. Die Räume, auf denen Sigma Modelle ohne Wess-Zumino-Term konform sind, sind Supermannigfaltigkeiten, also Mannigfaltigkeiten, die fermionische Richtungen aufweisen. Wir stellen die Konstruktion von Ver- tex Operatoren vor, gefolgt von der Hintergrundfeld-Entwicklung. Für semi- symmetrische Räume berechnen wir anschließend die diagonalen Terme der anomalen Dimensionen dieser Operatoren in führender Ordnung. Das Resul- tat stimmt mit dem für symmetrische Räume überein, jedoch treten nicht- diagonale Terme auf, die hier nicht weiter betrachtet werden. Anschließend präsentieren wir eine detaillierte Analyse des Spectrums des Supersphären S3j2 Sigma Modells. Dies ist eins der einfachsten Beispie- le für konforme Sigma Modelle auf symmetrischen Räumen und dient als Illustration für die Mächtigkeit der vorgestellten Methoden. Wir verwenden die erhaltenen Daten, um eine Dualität mit dem OSP(4j2) Gross-Neveu Modell zu untersuchen, die von Candu und Saleur vorgeschlagen wurde. Wir verwenden dazu ein Resultat, welches die anomalen Dimensionen von 1 2 BPS Operatoren zu allen Ordnungen berechnet. Wir finden das gesamte Grundzustandsspektrum des Sigma Modells. Darüber hinaus legen wir dar, dass sowohl die Zwangsbedingungen als auch die Bewegungsgleichungen des Sigma Modells korrekt vom Gross-Neveu Modell implementiert werden. Die Dualität wird weiterhin durch ein neues exaktes Resultat für die anomalen Dimensionen der Grundzustände unterstützt.
    [Show full text]
  • Durham Research Online
    Durham Research Online Deposited in DRO: 30 January 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Bhattacharya, Jyotirmoy and Lipstein, Arthur E. (2016) '6d dual conformal symmetry and minimal volumes in AdS.', Journal of high energy physics., 2016 (12). p. 105. Further information on publisher's website: https://doi.org/10.1007/JHEP12(2016)105 Publisher's copyright statement: Open Access, c The Authors. Article funded by SCOAP3. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk Published for SISSA by Springer Received: November 16, 2016 Revised: December 8, 2016 Accepted: December 12, 2016 Published: December 20, 2016 6d dual conformal symmetry and minimal volumes JHEP12(2016)105 in AdS Jyotirmoy Bhattacharya and Arthur E.
    [Show full text]
  • Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
    Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings.
    [Show full text]
  • 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).
    [Show full text]
  • Off-Shell Interactions for Closed-String Tachyons
    Preprint typeset in JHEP style - PAPER VERSION hep-th/0403238 KIAS-P04017 SLAC-PUB-10384 SU-ITP-04-11 TIFR-04-04 Off-Shell Interactions for Closed-String Tachyons Atish Dabholkarb,c,d, Ashik Iqubald and Joris Raeymaekersa aSchool of Physics, Korea Institute for Advanced Study, 207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea bStanford Linear Accelerator Center, Stanford University, Stanford, CA 94025, USA cInstitute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA dDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail:[email protected], [email protected], [email protected] Abstract: Off-shell interactions for localized closed-string tachyons in C/ZN super- string backgrounds are analyzed and a conjecture for the effective height of the tachyon potential is elaborated. At large N, some of the relevant tachyons are nearly massless and their interactions can be deduced from the S-matrix. The cubic interactions be- tween these tachyons and the massless fields are computed in a closed form using orbifold CFT techniques. The cubic interaction between nearly-massless tachyons with different charges is shown to vanish and thus condensation of one tachyon does not source the others. It is shown that to leading order in N, the quartic contact in- teraction vanishes and the massless exchanges completely account for the four point scattering amplitude. This indicates that it is necessary to go beyond quartic inter- actions or to include other fields to test the conjecture for the height of the tachyon potential. Keywords: closed-string tachyons, orbifolds.
    [Show full text]
  • TASI Lectures on String Compactification, Model Building
    CERN-PH-TH/2005-205 IFT-UAM/CSIC-05-044 TASI lectures on String Compactification, Model Building, and Fluxes Angel M. Uranga TH Unit, CERN, CH-1211 Geneve 23, Switzerland Instituto de F´ısica Te´orica, C-XVI Universidad Aut´onoma de Madrid Cantoblanco, 28049 Madrid, Spain angel.uranga@cern,ch We review the construction of chiral four-dimensional compactifications of string the- ory with different systems of D-branes, including type IIA intersecting D6-branes and type IIB magnetised D-branes. Such models lead to four-dimensional theories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Stan- dard Model, and review their main phenomenological properties. We finally describe how to implement the tecniques to construct these models in flux compactifications, leading to models with realistic gauge sectors, moduli stabilization and supersymmetry breaking soft terms. Lecture 1. Model building in IIA: Intersecting brane worlds 1 Introduction String theory has the remarkable property that it provides a description of gauge and gravitational interactions in a unified framework consistently at the quantum level. It is this general feature (beyond other beautiful properties of particular string models) that makes this theory interesting as a possible candidate to unify our description of the different particles and interactions in Nature. Now if string theory is indeed realized in Nature, it should be able to lead not just to `gauge interactions' in general, but rather to gauge sectors as rich and intricate as the gauge theory we know as the Standard Model of Particle Physics.
    [Show full text]
  • 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)..........................
    [Show full text]
  • S-Duality in N=1 Orientifold Scfts
    S-duality in = 1 orientifold SCFTs N Iñaki García Etxebarria 1210.7799, 1307.1701 with B. Heidenreich and T. Wrase and 1506.03090, 1612.00853, 18xx.xxxxx with B. Heidenreich 3 Intro C =Z3 C(dP 1) General Conclusions Montonen-Olive = 4 (S-)duality N Given a 4d = 4 field theory with gauge group G and gauge N coupling τ = θ + i=g2 (*), there is a completely equivalent description with gauge group G and coupling 1/τ (for θ = 0 this _ − is g 1=g). Examples: $ G G_ U(1) U(1) U(N) U(N) SU(N) SU(N)=ZN SO(2N) SO(2N) SO(2N + 1) Sp(2N) Very non-perturbative duality, exchanges electrically charged operators with magnetically charged ones. (*) I will not describe global structure or line operators here. 3 Intro C =Z3 C(dP 1) General Conclusions S-duality u = 1+iτ j(τ) i+τ j j In this representation τ 1/τ is u u. ! − ! − 3 Intro C =Z3 C(dP 1) General Conclusions S-duality in < 4 N Three main ways of generalizing = 4 S-duality: N Trace what relevant susy-breaking deformations do in different duality frames. [Argyres, Intriligator, Leigh, Strassler ’99]... Make a guess [Seiberg ’94]. Identify some higher principle behind = 4 S-duality, and find backgrounds with less susy that followN the same principle. (2; 0) 6d theory on T 2 Riemann surfaces. [Gaiotto ’09], ..., [Gaiotto, Razamat! ’15], [Hanany, Maruyoshi ’15],... Field theory S-duality from IIB S-duality. 3 Intro C =Z3 C(dP 1) General Conclusions Field theories from solitons One way to construct four dimensional field theories from string theory is to build solitons with a four dimensional core.
    [Show full text]
  • Scale Without Conformal Invariance in Membrane Theory
    Scale without conformal invariance in membrane theory Achille Mauria,∗, Mikhail I. Katsnelsona aRadboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Abstract We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flex- ible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elas- ticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to D − 1, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a sym- metry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures.
    [Show full text]
  • Lectures on D-Branes
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server CPHT/CL-615-0698 hep-th/9806199 Lectures on D-branes Constantin P. Bachas1 Centre de Physique Th´eorique, Ecole Polytechnique 91128 Palaiseau, FRANCE [email protected] ABSTRACT This is an introduction to the physics of D-branes. Topics cov- ered include Polchinski’s original calculation, a critical assessment of some duality checks, D-brane scattering, and effective worldvol- ume actions. Based on lectures given in 1997 at the Isaac Newton Institute, Cambridge, at the Trieste Spring School on String The- ory, and at the 31rst International Symposium Ahrenshoop in Buckow. June 1998 1Address after Sept. 1: Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris, FRANCE, email : [email protected] Lectures on D-branes Constantin Bachas 1 Foreword Referring in his ‘Republic’ to stereography – the study of solid forms – Plato was saying : ... for even now, neglected and curtailed as it is, not only by the many but even by professed students, who can suggest no use for it, never- theless in the face of all these obstacles it makes progress on account of its elegance, and it would not be astonishing if it were unravelled. 2 Two and a half millenia later, much of this could have been said for string theory. The subject has progressed over the years by leaps and bounds, despite periods of neglect and (understandable) criticism for lack of direct experimental in- put. To be sure, the construction and key ingredients of the theory – gravity, gauge invariance, chirality – have a firm empirical basis, yet what has often catalyzed progress is the power and elegance of the underlying ideas, which look (at least a posteriori) inevitable.
    [Show full text]