S-Duality and Topological Strings
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Duality and Strings Dieter Lüst, LMU and MPI München
Duality and Strings Dieter Lüst, LMU and MPI München Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Duality of 4 - dimensional string constructions: • Covariant lattices ⇔ (a)symmetric orbifolds (1986/87: W. Lerche, D.L., A. Schellekens ⇔ L. Ibanez, H.P. Nilles, F. Quevedo) • Intersecting D-brane models ☞ SM (?) (2000/01: R. Blumenhagen, B. Körs, L. Görlich, D.L., T. Ott ⇔ G. Aldazabal, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, A. Uranga) Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Duality of 4 - dimensional string constructions: • Covariant lattices ⇔ (a)symmetric orbifolds (1986/87: W. Lerche, D.L., A. Schellekens ⇔ L. Ibanez, H.P. Nilles, F. Quevedo) • Intersecting D-brane models ☞ SM (?) (2000/01: R. Blumenhagen, B. Körs, L. Görlich, D.L., T. Ott ⇔ G. Aldazabal, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, A. Uranga) ➢ Madrid (Spanish) Quiver ! Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. -
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. -
Pitp Lectures
MIFPA-10-34 PiTP Lectures Katrin Becker1 Department of Physics, Texas A&M University, College Station, TX 77843, USA [email protected] Contents 1 Introduction 2 2 String duality 3 2.1 T-duality and closed bosonic strings .................... 3 2.2 T-duality and open strings ......................... 4 2.3 Buscher rules ................................ 5 3 Low-energy effective actions 5 3.1 Type II theories ............................... 5 3.1.1 Massless bosons ........................... 6 3.1.2 Charges of D-branes ........................ 7 3.1.3 T-duality for type II theories .................... 7 3.1.4 Low-energy effective actions .................... 8 3.2 M-theory ................................... 8 3.2.1 2-derivative action ......................... 8 3.2.2 8-derivative action ......................... 9 3.3 Type IIB and F-theory ........................... 9 3.4 Type I .................................... 13 3.5 SO(32) heterotic string ........................... 13 4 Compactification and moduli 14 4.1 The torus .................................. 14 4.2 Calabi-Yau 3-folds ............................. 16 5 M-theory compactified on Calabi-Yau 4-folds 17 5.1 The supersymmetric flux background ................... 18 5.2 The warp factor ............................... 18 5.3 SUSY breaking solutions .......................... 19 1 These are two lectures dealing with supersymmetry (SUSY) for branes and strings. These lectures are mainly based on ref. [1] which the reader should consult for original references and additional discussions. 1 Introduction To make contact between superstring theory and the real world we have to understand the vacua of the theory. Of particular interest for vacuum construction are, on the one hand, D-branes. These are hyper-planes on which open strings can end. On the world-volume of coincident D-branes, non-abelian gauge fields can exist. -
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. -
Introduction to Conformal Field Theory and String
SLAC-PUB-5149 December 1989 m INTRODUCTION TO CONFORMAL FIELD THEORY AND STRING THEORY* Lance J. Dixon Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 ABSTRACT I give an elementary introduction to conformal field theory and its applications to string theory. I. INTRODUCTION: These lectures are meant to provide a brief introduction to conformal field -theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available (or almost available), and most of these go in to much more detail than I will be able to here. Those reviews con- centrating on the CFT side of the subject include refs. 1,2,3,4; those emphasizing string theory include refs. 5,6,7,8,9,10,11,12,13 I will start with a little pre-history of string theory to help motivate the sub- ject. In the 1960’s it was noticed that certain properties of the hadronic spectrum - squared masses for resonances that rose linearly with the angular momentum - resembled the excitations of a massless, relativistic string.14 Such a string is char- *Work supported in by the Department of Energy, contract DE-AC03-76SF00515. Lectures presented at the Theoretical Advanced Study Institute In Elementary Particle Physics, Boulder, Colorado, June 4-30,1989 acterized by just one energy (or length) scale,* namely the square root of the string tension T, which is the energy per unit length of a static, stretched string. For strings to describe the strong interactions fi should be of order 1 GeV. Although strings provided a qualitative understanding of much hadronic physics (and are still useful today for describing hadronic spectra 15 and fragmentation16), some features were hard to reconcile. -
8. Compactification and T-Duality
8. Compactification and T-Duality In this section, we will consider the simplest compactification of the bosonic string: a background spacetime of the form R1,24 S1 (8.1) ⇥ The circle is taken to have radius R, so that the coordinate on S1 has periodicity X25 X25 +2⇡R ⌘ We will initially be interested in the physics at length scales R where motion on the S1 can be ignored. Our goal is to understand what physics looks like to an observer living in the non-compact R1,24 Minkowski space. This general idea goes by the name of Kaluza-Klein compactification. We will view this compactification in two ways: firstly from the perspective of the spacetime low-energy e↵ective action and secondly from the perspective of the string worldsheet. 8.1 The View from Spacetime Let’s start with the low-energy e↵ective action. Looking at length scales R means that we will take all fields to be independent of X25: they are instead functions only on the non-compact R1,24. Consider the metric in Einstein frame. This decomposes into three di↵erent fields 1,24 on R :ametricG˜µ⌫,avectorAµ and a scalar σ which we package into the D =26 dimensional metric as 2 µ ⌫ 2σ 25 µ 2 ds = G˜µ⌫ dX dX + e dX + Aµ dX (8.2) Here all the indices run over the non-compact directions µ, ⌫ =0 ,...24 only. The vector field Aµ is an honest gauge field, with the gauge symmetry descend- ing from di↵eomorphisms in D =26dimensions.Toseethisrecallthatunderthe transformation δXµ = V µ(X), the metric transforms as δG = ⇤ + ⇤ µ⌫ rµ ⌫ r⌫ µ This means that di↵eomorphisms of the compact direction, δX25 =⇤(Xµ), turn into gauge transformations of Aµ, δAµ = @µ⇤ –197– We’d like to know how the fields Gµ⌫, Aµ and σ interact. -
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..................... -
M-Theory Solutions and Intersecting D-Brane Systems
M-Theory Solutions and Intersecting D-Brane Systems A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in the Department of Physics and Engineering Physics University of Saskatchewan Saskatoon By Rahim Oraji ©Rahim Oraji, December/2011. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgrad- uate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Physics and Engineering Physics 116 Science Place University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E2 i Abstract It is believed that fundamental M-theory in the low-energy limit can be described effectively by D=11 supergravity. -
New Worldsheet Formulae for Conformal Supergravity Amplitudes
New Worldsheet Formulae for Conformal Supergravity Amplitudes Joseph A. Farrow and Arthur E. Lipstein Department of Mathematical Sciences Durham University, Durham, DH1 3LE, United Kingdom Abstract We use 4d ambitwistor string theory to derive new worldsheet formulae for tree- level conformal supergravity amplitudes supported on refined scattering equations. Unlike the worldsheet formulae for super-Yang-Mills or supergravity, the scattering equations for conformal supergravity are not in general refined by MHV degree. Nevertheless, we obtain a concise worldsheet formula for any number of scalars and gravitons which we lift to a manifestly supersymmetric formula using four types of vertex operators. The theory also contains states with non-plane wave boundary conditions and we show that the corresponding amplitudes can be obtained from plane-wave amplitudes by applying momentum derivatives. Such derivatives are subtle to define since the formulae are intrinsically four-dimensional and on-shell, so we develop a method for computing momentum derivatives of spinor variables. arXiv:1805.04504v3 [hep-th] 11 Jul 2018 1 Contents 1 Introduction 2 2 Review 4 3 Plane Wave Graviton Multiplet Scattering 6 3.1 Supersymmetric Formula . .8 4 Non-Plane Wave Graviton Multiplet Scattering 9 5 Conclusion 11 A BRST Quantization 13 B Derivation of Berkovits-Witten Formula 16 C Momentum Derivatives 17 D Non-plane Wave Examples 19 D.1 Non-plane Wave Scalar . 19 D.2 Non-plane Wave Graviton . 21 1 Introduction Starting with the discovery of the Parke-Taylor formula for tree-level gluon amplitudes [1], the study of scattering amplitudes has suggested intruiging new ways of formulating quantum field theory. -
What Lattice Theorists Can Do for Superstring/M-Theory
July 26, 2016 0:28 WSPC/INSTRUCTION FILE review˙2016July23 International Journal of Modern Physics A c World Scientific Publishing Company What lattice theorists can do for superstring/M-theory MASANORI HANADA Yukawa Institute for Theoretical Physics Kyoto University, Kyoto 606-8502, JAPAN The Hakubi Center for Advanced Research Kyoto University, Kyoto 606-8501, JAPAN Stanford Institute for Theoretical Physics Stanford University, Stanford, CA 94305, USA [email protected] The gauge/gravity duality provides us with nonperturbative formulation of superstring/M-theory. Although inputs from gauge theory side are crucial for answering many deep questions associated with quantum gravitational aspects of superstring/M- theory, many of the important problems have evaded analytic approaches. For them, lattice gauge theory is the only hope at this moment. In this review I give a list of such problems, putting emphasis on problems within reach in a five-year span, including both Euclidean and real-time simulations. Keywords: Lattice Gauge Theory; Gauge/Gravity Duality; Quantum Gravity. PACS numbers:11.15.Ha,11.25.Tq,11.25.Yb,11.30.Pb Preprint number:YITP-16-28 1. Introduction During the second superstring revolution, string theorists built a beautiful arXiv:1604.05421v2 [hep-lat] 23 Jul 2016 framework for quantum gravity: the gauge/gravity duality.1, 2 It claims that superstring/M-theory is equivalent to certain gauge theories, at least about certain background spacetimes (e.g. black brane background, asymptotically anti de-Sitter (AdS) spacetime). Here the word ‘equivalence’ would be a little bit misleading, be- cause it is not clear how to define string/M-theory nonperturbatively; superstring theory has been formulated based on perturbative expansions, and M-theory3 is defined as the strong coupling limit of type IIA superstring theory, where the non- perturbative effects should play important roles. -
Chapter 9: the 'Emergence' of Spacetime in String Theory
Chapter 9: The `emergence' of spacetime in string theory Nick Huggett and Christian W¨uthrich∗ May 21, 2020 Contents 1 Deriving general relativity 2 2 Whence spacetime? 9 3 Whence where? 12 3.1 The worldsheet interpretation . 13 3.2 T-duality and scattering . 14 3.3 Scattering and local topology . 18 4 Whence the metric? 20 4.1 `Background independence' . 21 4.2 Is there a Minkowski background? . 24 4.3 Why split the full metric? . 27 4.4 T-duality . 29 5 Quantum field theoretic considerations 29 5.1 The graviton concept . 30 5.2 Graviton coherent states . 32 5.3 GR from QFT . 34 ∗This is a chapter of the planned monograph Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity, co-authored by Nick Huggett and Christian W¨uthrich and under contract with Oxford University Press. More information at www.beyondspacetime.net. The primary author of this chapter is Nick Huggett ([email protected]). This work was sup- ported financially by the ACLS and the John Templeton Foundation (the views expressed are those of the authors not necessarily those of the sponsors). We want to thank Tushar Menon and James Read for exceptionally careful comments on a draft this chapter. We are also grateful to Niels Linnemann for some helpful feedback. 1 6 Conclusions 35 This chapter builds on the results of the previous two to investigate the extent to which spacetime might be said to `emerge' in perturbative string the- ory. Our starting point is the string theoretic derivation of general relativity explained in depth in the previous chapter, and reviewed in x1 below (so that the philosophical conclusions of this chapter can be understood by those who are less concerned with formal detail, and so skip the previous one). -
Black Holes and the Multiverse
Home Search Collections Journals About Contact us My IOPscience Black holes and the multiverse This content has been downloaded from IOPscience. Please scroll down to see the full text. JCAP02(2016)064 (http://iopscience.iop.org/1475-7516/2016/02/064) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 165.123.34.86 This content was downloaded on 01/03/2016 at 21:11 Please note that terms and conditions apply. ournal of Cosmology and Astroparticle Physics JAn IOP and SISSA journal Black holes and the multiverse Jaume Garriga,a;b Alexander Vilenkinb and Jun Zhangb JCAP02(2016)064 aDepartament de Fisica Fonamental i Institut de Ciencies del Cosmos, Universitat de Barcelona, Marti i Franques, 1, Barcelona, 08028 Spain bInstitute of Cosmology, Tufts University, 574 Boston Ave, Medford, MA, 02155 U.S.A. E-mail: [email protected], [email protected], [email protected] Received December 28, 2015 Accepted February 3, 2016 Published February 25, 2016 Abstract. Vacuum bubbles may nucleate and expand during the inflationary epoch in the early universe. After inflation ends, the bubbles quickly dissipate their kinetic energy; they come to rest with respect to the Hubble flow and eventually form black holes. The fate of the bubble itself depends on the resulting black hole mass. If the mass is smaller than a certain critical value, the bubble collapses to a singularity. Otherwise, the bubble interior inflates, forming a baby universe, which is connected to the exterior FRW region by a wormhole.