DOCSLIB.ORG

University of California San Diego

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
University of California San Diego UNIVERSITY OF CALIFORNIA SAN DIEGO Aspects of Supersymmetric Conformal Field Theories in Various Dimensions A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Emily M. Nardoni Committee in charge: Professor Kenneth Intriligator, Chair Professor Patrick Diamond Professor John McGreevy Professor James McKernan Professor Justin Roberts 2018 Copyright Emily M. Nardoni, 2018 All rights reserved. The dissertation of Emily M. Nardoni is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California San Diego 2018 iii DEDICATION To my family. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . .x List of Tables . xii Acknowledgements . xiii Vita ............................................. xv Abstract of the Dissertation . xvi Chapter 1 Introduction . .1 1.1 Outlook . .1 1.2 Outline . .7 Chapter 2 Technical Introduction . .9 2.1 Supersymmetry: The Basics . .9 2.1.1 The supersymmetry algebra . .9 2.1.2 Irreducible representations of supersymmetry in 4d . 11 2.2 Some Properties of Supersymmetric Gauge Theories . 14 2.2.1 N = 1 supersymmetric actions and the power of holomorphy 14 2.2.2 NSVZ b-function . 16 2.2.3 Moduli space of vacua . 17 2.2.4 A short introduction to anomalies in QFT . 18 2.3 Superconformal Field Theories . 19 2.3.1 A word on conformal symmetry . 19 2.3.2 A tour of SCFTs in various dimensions . 20 2.3.3 Facts about fixed points . 21 v 2.3.4 The a-theorem, and a-maximization . 24 2.3.5 Chiral ring . 26 2.3.6 4d N = 1 superconformal index . 26 Chapter 3 Deformations of WA;D;E SCFTs . 30 3.1 Introduction . 30 3.1.1 The chiral ring of WA;D;E(X;Y) for matrix fields X and Y .. 32 3.1.2 WA;D;E in 4d SQCD with fundamental plus adjoint matter . 34 3.1.3 WA;D;E + DW RG flows . 36 3.1.4 WA;D;E flat direction flows . 38 3.1.5 Outline . 39 3.2 Technical Review . 40 3.2.1 The a-theorem, and a-maximization . 40 3.2.2 Duality for the 4d SCFTs . 41 3.2.3 Moduli spaces of vacua of the theories . 44 3.3 Example and Review: A and Ak One-Adjoint Cases . 46 3.3.1 A Ak flow and Ak duality . 46 ! b 3.3.2 WA + DW deformations and Ak Ak <k RG flows . 48 b k ! 0 3.3.3 Comments on SU(Nc) vs U(Nc) RG flows . 50 3.3.4 SU(Nc) flat direction deformations . 52 3.4 The WDk+2 Fixed Points and Flows . 52 3.4.1 Previously proposed dualities for WDk+2 ........... 53 3.4.2 Matrix-related flat directions at the origin . 56 3.4.3 SU(Nc)-specific (as opposed to U(Nc)) flat directions . 63 3.4.4 Dk+2 RG flows from relevant DW deformations . 67 3.4.5 The SU(Nc) version of the RG flows . 71 3.4.6 The Dodd Deven RG flow and the hypothetical D theory 72 ! even0 3.5 The WE7 Fixed Point and Flows . 74 3.5.1 Previously proposed dualities for WE7 ............ 75 3.5.2 Matrix-related flat directions at the origin . 77 3.5.3 SU(Nc)-specific (as opposed to U(Nc)) flat directions . 79 3.5.4 Case studies in E7 RG flows from DW deformations . 82 vi 3.6 Conclusions, Future Directions, and Open Questions . 89 3.6.1 Recap: puzzles and open questions for the Deven and E7 theories 89 3.6.2 Future directions: aspects of the WE6 and WE8 theories . 90 Chapter 4 Landscape of Simple Superconformal Field Theories in 4d . 93 4.1 Introduction . 93 4.2 A Landscape of Simple SCFTs . 95 4.3 T0 —Minimal c, Minimal a with U(1) ................ 99 4.4 H0∗—Minimal a ............................ 100 4.5 H1∗—Minimal a with SU(2) ...................... 101 4.6 Discussion . 102 Chapter 5 Interlude: An Introduction to the Theories of Class S ........... 103 5.1 Introduction . 103 5.2 4d SCFTs from 6d . 104 Chapter 6 4d SCFTs from Negative Degree Line Bundles . 109 6.1 Overview and Summary of Results . 109 (m) 6.2 Constructing the BBBW Duals from TN Building Blocks . 111 (m) 6.2.1 TN review . 111 6.2.2 Gluing procedure . 113 6.2.3 Construction of Cg>1;n=0 and computation of a and c .... 114 6.2.4 Operator dimensions and large-N .............. 118 (m) 6.3 Cg=0;n=0 from the Higgsed TN ................... 119 (m) 6.3.1 Constructing Higgsed TN theories . 119 6.3.2 Computation of atrial and ctrial ................. 121 6.3.3 Comments on ruling out g = 0 SCFTs . 122 (mi) 6.4 General Cg;n from TN Building Blocks . 123 6.4.1 Computing atrial and ctrial for g = 1............. 123 6 6.4.2 Comments on operators . 126 6.4.3 Computing atrial and ctrial for the torus . 127 6.5 Future Directions . 128 vii Chapter 7 Structure of Anomalies of 4d SCFTs from M5-branes, and Anomaly Inflow 130 7.1 Introduction . 131 7.2 Structure of Class S Anomalies . 135 7.2.1 Anomalies of the (2;0) theories . 136 7.2.2 Structure of class S anomalies . 138 7.3 Anomalies of (p;q) Punctures in Class S .............. 143 7.3.1 Anomalies of (p;q) punctures . 143 7.3.2 Effective nv;nh for N = 1 theories . 145 7.4 Inflow for Flat M5-branes: A Review . 148 7.4.1 Anomaly inflow . 149 7.4.2 M5-brane inflow . 150 7.5 Class S Anomalies from Inflow . 154 7.5.1 Inflow for curved M5-branes . 155 7.5.2 Sources for connection forms . 160 7.5.3 Variation of the action . 165 7.5.4 Total 4d anomaly from inflow . 172 7.6 Conclusions . 174 7.6.1 Summary . 174 7.6.2 Outlook . 176 Appendix A WA;D;E SCFTs . 178 A.1 ADE Facts . 178 A.2 RG Flows Whose Deformations Seem Irrelevant . 179 A.2.1 E6 A5 ............................ 179 ! A.2.2 E7 D6 ............................ 181 ! A.2.3 E8 D7 ............................ 181 ! Appendix B Relevant Formulae for Gluing Negative-Degree Line Bundles . 184 B.1 Conventions and Main TN Formulae . 184 B.2 Relevant BBBW Results . 185 (m) B.3 ’t Hooft Anomalies for Gluing TN Building Blocks . 187 viii Appendix C Class S Anomaly Conventions . 189 C.1 Anomaly Polynomials and Characteristic Classes . 189 C.2 Anomalies for Regular N = 2 Punctures . 191 C.3 Integral Identities . 193 Bibliography . 194 ix LIST OF FIGURES Figure 1.1: The coupling l as a function of energy scale m plotted for an infrared free theory that breaks down at m LLandau....................5 ∼ Figure 1.2: The coupling l as a function of energy scale m for an asymptotically free theory that develops a dynamical scale at m LQCD.............6 ∼ Figure 1.3: The coupling l as a function of energy scale m for an asymptotically free theory that develops an interacting fixed point at low energies. .6 Figure 3.1: Deforming the WA;D;E 2d N = 2 SCFTs corresponds to adjoint Higgsing of the ADE group, hence cutting the Dynkin diagrams, as illustrated here for the flows in (3.3). This gives the vacua associated with 1d representations of the F-terms. 32 Figure 3.2: Flows among the fixed points of SQCD with two adjoints. 37 2 Figure 3.3: (aUV aIR)=N plotted for x in the conformal window. The WD theory is − f 4 5 IR-free for x 1. The WD theory requires x > 3:14 for the TrX term in WD ≤ 6 6 to be relevant, while the corresponding term in the Brodie dual is relevant if x < 8:93. The A theory is asymptotically free in both plotted domains. 60 Figure 3.4: Using the Q-flat directions to bypass the strong coupling regime. 63 b Figure 3.5: Flat directions for Dodd, Nc = 2m+kn, integrating out massive fields (denoted by dotted lines) and fields eaten by the Higgs mechanism (not shown). 65 Figure 3.6: Flat directions for Deven, with Nc = 2m + kn. Again we integrate out massive fields (denoted by dotted lines), and those eaten by the Higgs mechanism (not shown). 67 Figure 3.7: We enhance the k odd Dk singularity to a Dk+1 singularity in one of two ways. 73 2 Figure 3.8: (aUV aIR)=N for WE in the UV and A in the IR. The E7 deformation term − f 7 in the UV theory is relevant for x & 4:12, while the corresponding term in b Kutasov-Lin dual is relevant if x . 26:11. The Aˆ theory is UV-free in this whole range. 78 Figure 3.9: Flat directions for E7, Nc = 2m + 3n, integrating out massive fields (denoted by dotted lines) and fields eaten by the Higgs mechanism (not shown). The subscripts label the gauge groups and their matter. 81 x Figure 4.1: The central charges of the 35 “good” theories. The ratios a=c all lie within the range (0:8246;0:9895). The mean value of a=c is 0:8733 with standard deviation 0:03975. 97 Figure 4.2: A subset of the fixed points that can be obtained from SU(2) Nf = 1 ad- joint SQCD with singlets. Note that the graph is not arranged vertically by decreasing a central charge, because the deformations we consider involve coupling in the singlet fields.
Load more

Recommended publications
  • Orthogonal Symmetric Affine Kac-Moody Algebras
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 10, October 2015, Pages 7133–7159 http://dx.doi.org/10.1090/tran/6257 Article electronically published on April 20, 2015 ORTHOGONAL SYMMETRIC AFFINE KAC-MOODY ALGEBRAS WALTER FREYN Abstract. Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification. 1. Introduction Riemannian symmetric spaces are fundamental objects in finite dimensional dif- ferential geometry displaying numerous connections with Lie theory, physics, and analysis. The search for infinite dimensional symmetric spaces associated to affine Kac-Moody algebras has been an open question for 20 years, since it was first asked by C.-L. Terng in [Ter95]. We present a complete solution to this problem in a series of several papers, dealing successively with the functional analytic, the algebraic and the geometric aspects. In this paper, building on work of E. Heintze and C. Groß in [HG12], we introduce and classify orthogonal symmetric affine Kac- Moody algebras (OSAKAs). OSAKAs are the central objects in the classification of affine Kac-Moody symmetric spaces as they provide the crucial link between the geometric and the algebraic side of the theory.
    [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]
  • 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]
  • Lattices and Vertex Operator Algebras
    My collaboration with Geoffrey Schellekens List Three Descriptions of Schellekens List Lattices and Vertex Operator Algebras Gerald H¨ohn Kansas State University Vertex Operator Algebras, Number Theory, and Related Topics A conference in Honor of Geoffrey Mason Sacramento, June 2018 Gerald H¨ohn Kansas State University Lattices and Vertex Operator Algebras My collaboration with Geoffrey Schellekens List Three Descriptions of Schellekens List How all began From gerald Mon Apr 24 18:49:10 1995 To: [email protected] Subject: Santa Cruz / Babymonster twisted sector Dear Prof. Mason, Thank you for your detailed letter from Thursday. It is probably the best choice for me to come to Santa Cruz in the year 95-96. I see some projects which I can do in connection with my thesis. The main project is the classification of all self-dual SVOA`s with rank smaller than 24. The method to do this is developed in my thesis, but not all theorems are proven. So I think it is really a good idea to stay at a place where I can discuss the problems with other people. [...] I must probably first finish my thesis to make an application [to the DFG] and then they need around 4 months to decide the application, so in principle I can come September or October. [...] I have not all of this proven, but probably the methods you have developed are enough. I hope that I have at least really constructed my Babymonster SVOA of rank 23 1/2. (edited for spelling and grammar) Gerald H¨ohn Kansas State University Lattices and Vertex Operator Algebras My collaboration with Geoffrey Schellekens List Three Descriptions of Schellekens List Theorem (H.
    [Show full text]
  • Classification of Holomorphic Framed Vertex Operator Algebras of Central Charge 24
    Classification of holomorphic framed vertex operator algebras of central charge 24 Ching Hung Lam, Hiroki Shimakura American Journal of Mathematics, Volume 137, Number 1, February 2015, pp. 111-137 (Article) Published by Johns Hopkins University Press DOI: https://doi.org/10.1353/ajm.2015.0001 For additional information about this article https://muse.jhu.edu/article/570114/summary Access provided by Tsinghua University Library (22 Nov 2018 10:39 GMT) CLASSIFICATION OF HOLOMORPHIC FRAMED VERTEX OPERATOR ALGEBRAS OF CENTRAL CHARGE 24 By CHING HUNG LAM and HIROKI SHIMAKURA Abstract. This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism. 1. Introduction. The classification of holomorphic vertex operator alge- bras (VOAs) of central charge 24 is one of the fundamental problems in vertex operator algebras and mathematical physics. In 1993 Schellekens [Sc93] obtained a partial classification by determining possible Lie algebra structures for the weight one subspaces of holomorphic VOAs of central charge 24. There are 71 cases in his list but only 39 of the 71 cases were known explicitly at that time. It is also an open question if the Lie algebra structure of the weight one subspace will determine the VOA structure uniquely when the central charge is 24.
    [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]
  • E 8 in N= 8$$\Mathcal {N}= 8$$ Supergravity in Four Dimensions
    Published for SISSA by Springer Received: November 30, 2017 Accepted: December 23, 2017 Published: January 8, 2018 JHEP01(2018)024 E8 in N = 8 supergravity in four dimensions Sudarshan Ananth,a Lars Brinkb,c and Sucheta Majumdara aIndian Institute of Science Education and Research, Pune 411008, India bDepartment of Physics, Chalmers University of Technology, S-41296 G¨oteborg, Sweden cDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore E-mail: [email protected], [email protected], [email protected] Abstract: We argue that = 8 supergravity in four dimensions exhibits an exceptional N E8(8) symmetry, enhanced from the known E7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the = 8 theory to d = 3, a field redefinition to N render the E8(8) invariance manifest, followed by dimensional oxidation back to d = 4. Keywords: Supergravity Models, Superspaces, Field Theories in Lower Dimensions ArXiv ePrint: 1711.09110 Open Access, c The Authors. https://doi.org/10.1007/JHEP01(2018)024 Article funded by SCOAP3. Contents 1 Introduction 1 2 (N = 8, d = 4) supergravity in light-cone superspace 2 2.1 E7(7) symmetry 4 3 Maximal supergravity in d = 3 — version I 5 JHEP01(2018)024 4 Maximal supergravity in d = 3 — version II 5 4.1 E8(8) symmetry 6 5 Relating the two different versions of three-dimensional maximal super- gravity 7 5.1 The field redefinition 7 5.2 SO(16) symmetry revisited 7 6 Oxidation back to d = 4 preserving the E8(8) symmetry 8 7 Conclusions 9 A Verification of field redefinition 10 B SO(16)-invariance of the new Lagrangian 11 C E8 invariance 12 1 Introduction Gravity with maximal supersymmetry in four dimensions, = 8 supergravity, exhibits N both = 8 supersymmetry and the exceptional E symmetry [1].
    [Show full text]
  • Effective Conformal Descriptions of Black Hole Entropy
    Entropy 2011, 13, 1355-1379; doi:10.3390/e13071355 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Effective Conformal Descriptions of Black Hole Entropy Steven Carlip Department of Physics, University of California at Davis, Davis, CA 95616, USA; E-Mail: [email protected]. Received: 1 July 2011; in revised form: 12 July 2011 / Accepted: 19 July 2011 / Published: 20 July 2011 Abstract: It is no longer considered surprising that black holes have temperatures and entropies. What remains surprising, though, is the universality of these thermodynamic properties: their exceptionally simple and general form, and the fact that they can be derived from many very different descriptions of the underlying microscopic degrees of freedom. I review the proposal that this universality arises from an approximate conformal symmetry, which permits an effective “conformal dual” description that is largely independent of the microscopic details. Keywords: black holes; Bekenstein–Hawking entropy; conformal dual 1. Introduction Since the seminal work of Bekenstein [1] and Hawking [2], we have understood that black holes behave as thermodynamic systems, with characteristic temperatures and entropies. A central task for quantum gravity is finding a statistical mechanical description of this behavior in terms of microscopic states. One important clue may come from the surprising “universality” of these thermodynamic properties, which manifests itself in two ways: – The Bekenstein–Hawking entropy A S = (1) 4G takes the same simple form for all black holes, independent of the charges, spin, and even the number of spacetime dimensions. If the action differs from that of general relativity, the entropy is modified, but again in a simple and universal way [3].
    [Show full text]
  • The Duality Between F-Theory and the Heterotic String in with Two Wilson
    Letters in Mathematical Physics (2020) 110:3081–3104 https://doi.org/10.1007/s11005-020-01323-8 The duality between F-theory and the heterotic string in D = 8 with two Wilson lines Adrian Clingher1 · Thomas Hill2 · Andreas Malmendier2 Received: 2 June 2020 / Revised: 20 July 2020 / Accepted: 30 July 2020 / Published online: 7 August 2020 © Springer Nature B.V. 2020 Abstract We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank 16. Weconstruct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial Higgsing of the heterotic string gauge algebra using two Wilson line parameters. Keywords F-theory · String duality · K3 surfaces · Jacobian elliptic fibrations Mathematics Subject Classification 14J28 · 14J81 · 81T30 1 Introduction In a standard compactification of the type IIB string theory, the axio-dilaton field τ is constant and no D7-branes are present. Vafa’s idea in proposing F-theory [51] was to simultaneously allow a variable axio-dilaton field τ and D7-brane sources, defining at a new class of models in which the string coupling is never weak. These compactifications of the type IIB string in which the axio-dilaton field varies over a base are referred to as F-theory models.
    [Show full text]
  • Arxiv:1805.06347V1 [Hep-Th] 16 May 2018 Sa Rte O H Rvt Eerhfudto 08Awar 2018 Foundation Research Gravity the for Written Essay Result
    Maximal supergravity and the quest for finiteness 1 2 3 Sudarshan Ananth† , Lars Brink∗ and Sucheta Majumdar† † Indian Institute of Science Education and Research Pune 411008, India ∗ Department of Physics, Chalmers University of Technology S-41296 G¨oteborg, Sweden and Division of Physics and Applied Physics, School of Physical and Mathematical Sciences Nanyang Technological University, Singapore 637371 March 28, 2018 Abstract We show that N = 8 supergravity may possess an even larger symmetry than previously believed. Such an enhanced symmetry is needed to explain why this theory of gravity exhibits ultraviolet behavior reminiscent of the finite N = 4 Yang-Mills theory. We describe a series of three steps that leads us to this result. arXiv:1805.06347v1 [hep-th] 16 May 2018 Essay written for the Gravity Research Foundation 2018 Awards for Essays on Gravitation 1Corresponding author, [email protected] [email protected] [email protected] Quantum Field Theory describes three of the four fundamental forces in Nature with great precision. However, when attempts have been made to use it to describe the force of gravity, the resulting field theories, are without exception, ultraviolet divergent and non- renormalizable. One striking aspect of supersymmetry is that it greatly reduces the diver- gent nature of quantum field theories. Accordingly, supergravity theories have less severe ultraviolet divergences. Maximal supergravity in four dimensions, N = 8 supergravity [1], has the best ultraviolet properties of any field theory of gravity with two derivative couplings. Much of this can be traced back to its three symmetries: Poincar´esymmetry, maximal supersymmetry and an exceptional E7(7) symmetry.
    [Show full text]
  • Introduction to String Theory A.N
    Introduction to String Theory A.N. Schellekens Based on lectures given at the Radboud Universiteit, Nijmegen Last update 6 July 2016 [Word cloud by www.worldle.net] Contents 1 Current Problems in Particle Physics7 1.1 Problems of Quantum Gravity.........................9 1.2 String Diagrams................................. 11 2 Bosonic String Action 15 2.1 The Relativistic Point Particle......................... 15 2.2 The Nambu-Goto action............................ 16 2.3 The Free Boson Action............................. 16 2.4 World sheet versus Space-time......................... 18 2.5 Symmetries................................... 19 2.6 Conformal Gauge................................ 20 2.7 The Equations of Motion............................ 21 2.8 Conformal Invariance.............................. 22 3 String Spectra 24 3.1 Mode Expansion................................ 24 3.1.1 Closed Strings.............................. 24 3.1.2 Open String Boundary Conditions................... 25 3.1.3 Open String Mode Expansion..................... 26 3.1.4 Open versus Closed........................... 26 3.2 Quantization.................................. 26 3.3 Negative Norm States............................. 27 3.4 Constraints................................... 28 3.5 Mode Expansion of the Constraints...................... 28 3.6 The Virasoro Constraints............................ 29 3.7 Operator Ordering............................... 30 3.8 Commutators of Constraints.......................... 31 3.9 Computation of the Central Charge.....................
    [Show full text]
  • S-Duality-And-M5-Mit
    N . More on = 2 S-dualities and M5-branes .. Yuji Tachikawa based on works in collaboration with L. F. Alday, B. Wecht, F. Benini, S. Benvenuti, D. Gaiotto November 2009 Yuji Tachikawa (IAS) November 2009 1 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 2 / 47 Montonen-Olive duality • N = 4 SU(N) SYM at coupling τ = θ=(2π) + (4πi)=g2 equivalent to the same theory coupling τ 0 = −1/τ • One way to ‘understand’ it: start from 6d N = (2; 0) theory, i.e. the theory on N M5-branes, put on a torus −1/τ τ 0 1 0 1 • Low energy physics depends only on the complex structure S-duality! Yuji Tachikawa (IAS) November 2009 3 / 47 S-dualities in N = 2 theories • You can wrap N M5-branes on a more general Riemann surface, possibly with punctures, to get N = 2 superconformal field theories • Different limits of the shape of the Riemann surface gives different weakly-coupled descriptions, giving S-dualities among them • Anticipated by [Witten,9703166], but not well-appreciated until [Gaiotto,0904.2715] Yuji Tachikawa (IAS) November 2009 4 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 5 / 47 Contents 1. Introduction 2. S-dualities 3. A few words on TN 4. 4d CFT vs 2d CFT Yuji Tachikawa (IAS) November 2009 6 / 47 S-duality in N = 2 . SU(2) with Nf = 4 .
    [Show full text]