DOCSLIB.ORG

Non-Critical String Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Non-Critical String Theory Extended Conformal Symmetry in Non-Critical String Theory Jan de Boer Extended Conformal Symmetry in Non-Critical String Theory Extended Conformal Symmetry in Non-Critical String Theory Uitgebreide Conforme Symmetrie in Niet-Kritische String Theorie (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Rijksuniversiteit te Utrecht, op gezag van de Rector Magnificus prof. dr. J.A. van Ginkel, ingevolge het besluit van het College van Decanen in het openbaar te verdedigen op woensdag 9 juni 1993 des namiddags te 12.45 uur door Jan de Boer geboren op 29 juni 1967 te Sint Nicolaasga Promotor: Prof. dr. B. Q. P. J. de Wit Dit proefschrift kwam tot stand met de financi¨elesteun van de Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.). Contents 1 Introduction and Outline 1 2 W Algebras 7 2.1 Conformal Field Theory . 7 2.1.1 Why Conformal Field Theory? . 7 2.1.2 Conformal Field Theory . 9 2.1.3 Operator Product Algebras and Field Algebras . 13 2.1.4 The Free Scalar Field . 16 2.1.5 First-Order Systems . 19 2.1.6 The Wess-Zumino-Novikov-Witten Model . 20 2.2 W Algebras . 25 2.2.1 The c = 2 W3 algebra . 25 2.2.2 General W Algebras . 26 2.2.3 Why W Algebras? . 26 2.2.4 W Algebras From sl2 Embeddings . 27 2.2.5 Why sl2 Embeddings? . 29 2.2.6 Gauge Invariant Polynomials and the Miura Transformation . 31 2.2.7 The Structure of the W Algebra . 32 2.2.8 Quantization of the W Algebra . 35 2.2.9 The Stress-Energy Tensor . 40 2.2.10 The Virasoro Algebra . 41 2.2.11 The W3 Algebra . 41 (2) 2.2.12 The W3 Algebra . 42 2.2.13 Free Field Expressions for W Algebras . 43 2.2.14 Some Explicit Results for sln .................... 45 2.2.15 Extensions of W Algebras . 46 3 Chiral W gravity 48 3.1 Gauging the Chiral Algebra . 48 3.1.1 The Free Scalar Field . 50 3.1.2 The Identity Gauge Field . 51 3.1.3 The Chiral Induced Action . 51 3.2 Solution of the Ward Identity . 55 3.2.1 The Virasoro Algebra . 55 3.2.2 The General Case . 56 8 3.3 Quantization . 58 3.3.1 The Induced Action of W3 Gravity . 60 3.3.2 The Effective Action of W3 Gravity . 67 3.3.3 The Effective Action of the WZNW Model . 68 3.3.4 The Effective Action of W3 Gravity, Continued . 71 3.3.5 Remarks . 73 4 Covariant W Gravity 75 4.1 Gauging Non-Chiral Algebras . 75 4.1.1 Example: Free Scalar Field . 75 4.1.2 General Case . 76 4.1.3 Gauging the WZNW Model . 79 4.2 W Gravity . 81 4.2.1 What is W Gravity? . 81 4.2.2 The Classical Covariant Action for W Gravity . 86 4.2.3 The General Structure of the Covariant Action . 89 4.3 Quantum Aspects of the Covariant Action . 90 4.3.1 All-Order Results for the Covariant Action . 90 4.3.2 KPZ Approach . 94 5 The Moduli Space of W Gravity 95 5.1 W Algebras on Riemann Surfaces . 95 5.1.1 General Case . 96 5.2 The Moduli Space Associated to a W Algebra . 98 5.2.1 The Moduli Space Associated to a Chiral Algebra . 98 5.2.2 A Finite Dimensional Model of the Moduli Space for W Algebras 99 5.2.3 The Connection with Higgs Bundles . 103 5.2.4 Reconstruction of the Metric . 108 5.2.5 Relation to Self-Duality Equations . 110 5.3 Generalized Actions . 112 5.3.1 Classical Actions for Chiral and Covariant Gravity on Higher Genus116 5.3.2 Example: Gravity . 117 5.4 Remarks . 118 5.4.1 The Geometrical Description of W Algebras . 118 5.4.2 The Moduli Space for a Surface with Punctures . 120 5.4.3 From Teichm¨ullerspace to the Moduli Space . 121 6 Towards an Exact Solution? 122 6.1 Continuum Approach . 122 6.1.1 Gravity . 122 9 6.1.2 Generalization to W Gravity: BRST Operator . 125 6.1.3 The Spectrum . 127 6.1.4 Correlators . 129 6.2 Topological Approach . 130 6.2.1 Topological W Conformal Field Theory . 130 6.2.2 N = 2 Structure in W Gravity . 132 6.2.3 Relation with G=G models . 134 6.2.4 Topological Gravity . 138 6.2.5 Topological W Gravity . 138 6.2.6 Intersection Theory . 139 6.3 The Discrete Approach . 141 6.3.1 Review of the p-Matrix Model . 143 6.3.2 Genus-Zero Formulation . 146 6.3.3 The Continuum Limit in Genus Zero . 148 6.3.4 The New Continuum Limit . 150 Bibliography 153 Samenvatting 170 Dankwoord 173 Curriculum Vitae 174 1 Introduction and Outline The ingenious step that led Einstein from the special to the general theory of rela- tivity, was to extend the invariance under Poincar´etransformations by invariance under general co-ordinate transformations. Later, similar ideas have been successfully applied to other theories. In Maxwell theory, we have the freedom to add the divergence of an arbitrary space-time dependent phase factor to the vector potential. If we replace the phase factor by an arbitrary group, this leads to Yang-Mills theory with a local non-abelian gauge invariance. Any matter theory with a rigid invariance can be made into a gauge theory, by replacing the rigid invariance by a local one. A somewhat more involved example is supergravity, which results from the passage from global to local supersymmetry. As an example, consider the action for a free complex scalar field, Z 4 µ ∗ S ∼ d x @µφ∂ φ (1.1) The action is invariant under the rigid U(1) transformation φ ! eiθφ. To make it invariant under local U(1) transformations, i.e. with θ an arbitrary function of xµ, one can use the principle of `minimal coupling'. Introduce a real gauge field Aµ(x), and modify the action to Z 4 µ ∗ S ∼ d x (Dµφ)(D φ) ; (1.2) where Dµ = @µ + iAµ is the covariant derivative. If the gauge field transforms under a local U(1) transformation as Aµ ! Aµ − @µθ, the modified action has a local U(1) in- variance. Associated to every rigid invariance is a conserved current or Noether current. The integral of this conserved current is a conserved quantity, that can be identified with the generator of the rigid symmetry. An important feature of the extended action (1.2) is that the term linear in Aµ couples to the Noether current of the rigid symmetry. Indeed, (1.2) contains the term Z Z 4 µ ∗ ∗ µ 4 µ i d x Aµ(φ∂ φ − φ @ φ) ≡ d x AµJ ; (1.3) where J µ is the conserved current associated to the rigid U(1) invariance. 2 Introduction and Outline The action (1.1) has another invariance, namely Poincar´einvariance. To make this invariance local, we also have to introduce a gauge field, but now minimal coupling does not work. The role of the gauge field is played by a metric on space-time, and the gauged action reads Z 4 p µν ∗ S ∼ d x −g g @µφ∂νφ (1.4) If we write the metric as gµν = ηµν +hµν, where ηµν is the usual flat space metric, we can expand the action in powers of h. The term linear in h is proportional to the Noether current associated to the global translational invariance, which is the energy-momentum tensor of (1.1). Thus, there is a close analogy between the coupling to a metric and the coupling to a gauge field. In both (1.2) and (1.4) the gauge fields are non-dynamical. To provide them with some dynamics, a special Lagrangian has to be chosen, that does not break the gauge invariance, and gives a kinetic term for them. For the U(1) gauge field this is the usual Maxwell action Z 4 µ ν ν µ SMW ∼ d x (@µAν − @νAµ)(@ A − @ A ); (1.5) whereas for gravity it is the Einstein-Hilbert action Z 4 p SEH ∼ d x −gR: (1.6) The Maxwell action is renormalizable, and the quantization of the U(1) gauge field can proceed without problems. In contrast, the Einstein-Hilbert action is non-renormalizable, and it does not define a consistent quantum theory of gravity. In this thesis we investigate the transition from rigid to local invariance in two- dimensional field theory, treating all symmetries, including gauge symmetries and gen- eral co-ordinate transformations, on an equal footing. The class of two-dimensional field theories we will discuss have the special property that they are invariant under local rescalings or Weyl transformations of the metric. These theories are called conformal field theories. An example is the two-dimensional version of (1.4), Z 2 p µν ∗ S ∼ d x −g g @µφ∂νφ : (1.7) Usually conformal field theories are not presented in a covariant form, but in a form where the symmetries (general co-ordinate invariance and Weyl invariance) have been used to bring the metric in some fixed form. On a flat two-dimensional space, the metric can always be brought in the form ds2 = dt2 − dx2. On a non-trivial curved two-dimensional manifold, the space of metrics modulo general co-ordinate and Weyl transformations is a finite dimensional space, called the moduli space of the manifold.
Load more

Recommended publications
  • M2-Branes Ending on M5-Branes
    M2-branes ending on M5-branes Vasilis Niarchos Crete Center for Theoretical Physics, University of Crete 7th Crete Regional Meeting on String Theory, 27/06/2013 based on recent work with K. Siampos 1302.0854, ``The black M2-M5 ring intersection spins’‘ Proceedings Corfu Summer School, 2012 1206.2935, ``Entropy of the self-dual string soliton’’, JHEP 1207 (2012) 134 1205.1535, ``M2-M5 blackfold funnels’’, JHEP 1206 (2012) 175 and older work with R. Emparan, T. Harmark and N. A. Obers ➣ blackfold theory 1106.4428, ``Blackfolds in Supergravity and String Theory’’, JHEP 1108 (2011) 154 0912.2352, ``New Horizons for Black Holes and Branes’’, JHEP 1004 (2010) 046 0910.1601, ``Essentials of Blackfold Dynamics’’, JHEP 1003 (2010) 063 0902.0427, ``World-Volume Effective Theory for Higher-Dimensional Black Holes’’, PRL 102 (2009)191301 0708.2181, ``The Phase Structure of Higher-Dimensional Black Rings and Black Holes’‘ + M.J. Rodriguez JHEP 0710 (2007) 110 2 Important lessons about the fundamentals of string/M-theory (and QFT) are obtained by studying the low-energy theories on D-branes and M-branes. Most notably in M-theory, recent progress has clarified the low-energy QFT on N M2-brane and the N3/2 dof that it exhibits. Bagger-Lambert ’06, Gustavsson ’07, ABJM ’08 Drukker-Marino-Putrov ’10 Our understanding of the M5-brane theory is more rudimentary, but efforts to identify analogous properties, e.g. the N3 scaling of the massless dof, is underway. Douglas ’10 Lambert,Papageorgakis,Schmidt-Sommerfeld ’10 Hosomichi-Seong-Terashima ’12 Kim-Kim ’12 Kallen-Minahan-Nedelin-Zabzine ’12 ..
    [Show full text]
  • 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]
  • 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]
  • S-Duality, the 4D Superconformal Index and 2D Topological QFT
    Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N S-duality, the 4d Superconformal Index and 2d Topological QFT Leonardo Rastelli with A. Gadde, E. Pomoni, S. Razamat and W. Yan arXiv:0910.2225,arXiv:1003.4244, and in progress Yang Institute for Theoretical Physics, Stony Brook Rutgers, November 9 2010 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. Moore-Seiberg groupoid of Σ = (generalized) 4d S-duality • Vast generalization of “ =4 S-duality as modular group of T 2”. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory.
    [Show full text]
  • UV Behavior of Half-Maximal Supergravity Theories
    UV behavior of half-maximal supergravity theories. Piotr Tourkine, Quantum Gravity in Paris 2013, LPT Orsay In collaboration with Pierre Vanhove, based on 1202.3692, 1208.1255 Understand the pertubative structure of supergravity theories. ● Supergravities are theories of gravity with local supersymmetry. ● Those theories naturally arise in the low energy limit of superstring theory. ● String theory is then a UV completion for those and thus provides a good framework to study their UV behavior. → Maximal and half-maximal supergravities. Maximal supergravity ● Maximally extended supergravity: – Low energy limit of type IIA/B theory, – 32 real supercharges, unique (ungauged) – N=8 in d=4 ● Long standing problem to determine if maximal supergravity can be a consistent theory of quantum gravity in d=4. ● Current consensus on the subject : it is not UV finite, the first divergence could occur at the 7-loop order. ● Impressive progresses made during last 5 years in the field of scattering amplitudes computations. [Bern, Carrasco, Dixon, Dunbar, Johansson, Kosower, Perelstein, Rozowsky etc.] Half-maximal supergravity ● Half-maximal supergravity: – Heterotic string, but also type II strings on orbifolds – 16 real supercharges, – N=4 in d=4 ● Richer structure, and still a lot of SUSY so explicit computations are still possible. ● There are UV divergences in d=4, [Fischler 1979] at one loop for external matter states ● UV divergence in gravity amplitudes ? As we will see the divergence is expected to arise at the four loop order. String models that give half-maximal supergravity “(4,0)” susy “(0,4)” susy ● Type IIA/B string = Superstring ⊗ Superstring Torus compactification : preserves full (4,4) supersymmetry.
    [Show full text]
  • Twenty Years of the Weyl Anomaly
    CTP-TAMU-06/93 Twenty Years of the Weyl Anomaly † M. J. Duff ‡ Center for Theoretical Physics Physics Department Texas A & M University College Station, Texas 77843 ABSTRACT In 1973 two Salam prot´eg´es (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor 2 gµν(x) Ω (x)gµν (x) displayed by classical massless field systems in interac- tion with→ gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic re- view. arXiv:hep-th/9308075v1 16 Aug 1993 CTP/TAMU-06/93 July 1993 †Talk given at the Salamfest, ICTP, Trieste, March 1993. ‡ Research supported in part by NSF Grant PHY-9106593. When all else fails, you can always tell the truth. Abdus Salam 1 Trieste and Oxford Twenty years ago, Derek Capper and I had embarked on our very first post- docs here in Trieste. We were two Salam students fresh from Imperial College filled with ideas about quantizing the gravitational field: a subject which at the time was pursued only by mad dogs and Englishmen. (My thesis title: Problems in the Classical and Quantum Theories of Gravitation was greeted with hoots of derision when I announced it at the Cargese Summer School en route to Trieste. The work originated with a bet between Abdus Salam and Hermann Bondi about whether you could generate the Schwarzschild solution using Feynman diagrams. You can (and I did) but I never found out if Bondi ever paid up.) Inspired by Salam, Capper and I decided to use the recently discovered dimensional regularization1 to calculate corrections to the graviton propaga- tor from closed loops of massless particles: vectors [1] and spinors [2], the former in collaboration with Leopold Halpern.
    [Show full text]
  • From Vibrating Strings to a Unified Theory of All Interactions
    Barton Zwiebach From Vibrating Strings to a Unified Theory of All Interactions or the last twenty years, physicists have investigated F String Theory rather vigorously. The theory has revealed an unusual depth. As a result, despite much progress in our under- standing of its remarkable properties, basic features of the theory remain a mystery. This extended period of activity is, in fact, the second period of activity in string theory. When it was first discov- ered in the late 1960s, string theory attempted to describe strongly interacting particles. Along came Quantum Chromodynamics— a theoryof quarks and gluons—and despite their early promise, strings faded away. This time string theory is a credible candidate for a theoryof all interactions—a unified theoryof all forces and matter. The greatest complication that frustrated the search for such a unified theorywas the incompatibility between two pillars of twen- tieth century physics: Einstein’s General Theoryof Relativity and the principles of Quantum Mechanics. String theory appears to be 30 ) zwiebach mit physics annual 2004 the long-sought quantum mechani- cal theory of gravity and other interactions. It is almost certain that string theory is a consistent theory. It is less certain that it describes our real world. Nevertheless, intense work has demonstrated that string theory incorporates many features of the physical universe. It is reasonable to be very optimistic about the prospects of string theory. Perhaps one of the most impressive features of string theory is the appearance of gravity as one of the fluctuation modes of a closed string. Although it was not discov- ered exactly in this way, we can describe a logical path that leads to the discovery of gravity in string theory.
    [Show full text]
  • 8.821 String Theory Fall 2008
    MIT OpenCourseWare http://ocw.mit.edu 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.821 F2008 Lecture 11: CFT continued; geometry of AdS Lecturer: McGreevy October 17, 2008 In this session, we are going to talk about the following topics. 1. We are making a few comments about CFT. 2. We are discussing spheres and hyperboloids. 3. Finally we are focusing on Lorentzian AdS and its boundary. 1 Conformal Symmetry 1.1 Weyl anomaly Quantumly, conformal symmetry in a curved space (with even number of dimensions) could be anomalous, that is ds2 → Ω(x)ds2 could be no longer a symmetry of the full quantum theory. This µ anomaly can be evaluated from the following diagram with operator Tµ inserted at the left vertex. Figure 1: A contribution to the Weyl anomaly. The conformal anomaly signals a nonzero value for the trace of the energy-momentum tensor. In a curved spacetime, it is related to the curvature: µ D/2 Tµ ∼ R 1 where R denotes some scalar contractions of curvature tensors and D is the number of spacetime dimensions; the power is determined by dimensional analysis. For the special case of D = 2, this is c T µ = − R(2) (1) µ 12 where R(2) is the Ricci scalar in two dimensions and c is the central charge of the Virasoro algebra µ of the 2d CFT. Also in D = 4, the anomaly is given by Tµ = aW + cGB where W and B are defined as 1 W = (Weyl tensor)2 = R....R − 2R..R + R2 (2) ...
    [Show full text]