DOCSLIB.ORG

Topology, C*-Algebras, and String Duality

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topology, C*-Algebras, and String Duality Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 111 Topology, C*-Algebras, and String Duality Jonathan Rosenberg American Mathematical Society with support from the National Science Foundation Topology, C*-Algebras, and String Duality http://dx.doi.org/10.1090/cbms/111 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 111 Topology, C*-Algebras, and String Duality Jonathan Rosenberg Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS-Regional Conference in the Mathematical Sciences on Topology, C∗-Algebras, and String Duality held at Texas Christian University, Forth Worth, Texas May 18–22, 2009 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of Mathematical Sciences and NSF Grant #0735233 2000 Mathematics Subject Classification. Primary 81T30; Secondary 81T75, 19K99, 46L80, 58B34, 55R10, 55P65, 55R50, 14J32, 53Z05. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-111 Library of Congress Cataloging-in-Publication Data Rosenberg, J. (Jonathan), 1951– Topology, C∗-algebras, and string duality / Jonathan Rosenberg. p. cm. — (Regional conference series in mathematics ; no. 111) Includes bibliographical references and index. ISBN 978-0-8218-4922-4 (alk. paper) 1. Algebraic topology. 2. C∗-algebras. 3. Duality theory (Mathematics) I. Conference Board of the Mathematical Sciences. II. Title. QA612.R58 2009 514.2—dc22 2009032465 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents The symbol ✰ denotes either a more advanced section, or else a bit of a digression, which can be skipped without interrupting the flow of the rest of the book. Preface vii Chapter 1. Introduction and Motivation 1 1.1. Structure of Physical Theories 1 1.2. Some Basics of String Theory 3 1.3. Dualities Related to String Theory 7 1.4. ✰ More on S-Duality and AdS/CFT Duality 10 Chapter 2. K-Theory and its Relevance to Physics 13 2.1. A Quick Review of Topological K-Theory 13 2.2. K-Theory and D-Brane Charges 21 2.3. K-Homology and D-Brane Charges 22 Chapter 3. A Few Basics of C∗-Algebras and Crossed Products 25 3.1. Basics of C∗-Algebras 25 3.2. K-Theory of C∗-Algebras 29 3.3. Crossed Products 33 Chapter 4. Continuous-Trace Algebras and Twisted K-Theory 37 4.1. Continuous-Trace Algebras and the Brauer Group 37 4.2. Twisted K-Theory 40 4.3. ✰ The Theory of Gerbes 42 Chapter 5. More on Crossed Products and Their K-Theory 47 5.1. A Categorical Framework 47 5.2. Connes’ Thom Isomorphism 51 5.3. The Pimsner-Voiculescu Sequence 52 Chapter 6. The Topology of T-Duality and the Bunke-Schick Construction 55 6.1. Topological T-Duality 55 6.2. The Bunke-Schick Construction 57 Chapter 7. T-Duality via Crossed Products 63 7.1. Group Actions on Continuous Trace-Algebras 63 7.2. The Raeburn-Rosenberg Theorem 68 Chapter 8. Higher-Dimensional T-Duality via Topological Methods 71 8.1. Higher-Dimensional T-Duality 71 8.2. A Higher-Dimensional Bunke-Schick Theorem 72 v vi CONTENTS Chapter 9. Higher-Dimensional T-Duality via C∗-Algebraic Methods 77 9.1. Methodology and a Key Example 77 9.2. Uniqueness of Group Actions 78 9.3. The General Case 81 Chapter 10. Advanced Topics and Open Problems 85 10.1. Mirror Symmetry 85 10.2. Other Solutions to the Missing T-Dual Problem 87 10.3. ✰ Fourier-Mukai Duality 88 10.4. ✰ Refinements of K-Theory 93 10.5. Open Problems 94 Bibliography 95 Notation and Symbols 103 Index 107 Preface This book is the outgrowth of an NSF/CBMS Regional Conference in the Math- ematical Sciences, May 18–22, 2009, organized by Robert Doran and Greg Friedman at Texas Christian University. I am highly indebted to Bob and Greg for their tire- less work in getting together funding for the conference, making all the logistical arrangements, recruiting participants and other speakers, and for keeping me on track during this entire process. The subject of this book, identical to the subject of the conference, is inter- disciplinary. Thus it involves a more-or-less equal mixture of topology, operator algebras, and physics. There is also a bit of algebraic geometry, especially in the last chapter. While I assume most readers of this book are probably somewhat fa- miliar with at least one of these subjects, they may not necessarily be knowledgeable about all or even most of them. So I have tried to include some basics on each one. The expert reader can skip over these sections. Roughly speaking, each chapter of this book corresponds to a single lecture from the conference, but “fleshed out” a little more. I have also included some sections, marked with a star, that I didn’t go into in detail in the lectures. These are more advanced, and someone just wanting an overview of the subject can skip these, though they might interest the more advanced reader. Since the book covers a lot of ground, it differs from most con- ventional mathematics books which follow the methodical “theorem-proof” style. There are of course plenty of theorems and proofs, but my main objective has been to show how several seemingly disparate subjects are closely linked with one an- other, and to give readers an overview of some areas of current research. In some places this happens at the expense of trying to cover everything systematically. I would also like to the thank the Conference Board of the Mathematical Sci- ences and the National Science Foundation for their support. NSF Grant DMS- 0735233 supported the conference, NSF Grant DMS-0602750 supported the entire Regional Conference program, and NSF Grants DMS-0504212 and DMS-0805003 supported the research that went into the preparation of the lectures and the writing of this book. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Finally, I would also like to thank the Ruth M. Davis Professorship at the University of Maryland for partial support during the writing of this book, and I would like to thank the many participants at the CBMS conference for their lively participation and penetrating questions at the meeting, and for their valuable vii viii PREFACE feedback on the first draft of this manuscript. They caught many misprints, errors, and omissions, and also made many useful suggestions for improvements, and for this I am very grateful. Jonathan Rosenberg College Park, Maryland Bibliography 1. Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz, Large N field theories, string theory and gravity,Phys.Rep.323 (2000), no. 3-4, 183–386, arxiv.org: hep-th/9905111. MR1743597 (2001c:81134) 2. Clara Luc´ıa Aldana Dom´ınguez, Representation of a gauge group as motions of a Hilbert space, Ann. Math. Blaise Pascal 11 (2004), no. 2, 131–153, available at http://ambp.cedram.org/item?id=AMBP 2004 11 2 131 0. MR2109604 (2005h:58018) 3. E. Alvarez, L. Alvarez-Gaum´e, J. L. F. Barb´on, and Y. Lozano, Some global aspects of duality in string theory,NuclearPhys.B415 (1994), no. 1, 71–100, arxiv.org: hep-th/9309039. MR1265457 (95a:81197) 4. M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), no. 3, 595–687. MR1869850 (2002g:55009) 5. Matthew Ando, Michael J. Hopkins, and Neil P. Strickland, The sigma orientation is an H∞ map,Amer.J.Math.126 (2004), no. 2, 247–334, arxiv.org: math.AT/0204053. MR2045503 (2005d:55009) 6. M. F. Atiyah, K-theory, second ed., Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989, Notes by D. W. Anderson. MR1043170 (90m:18011) 7. M. F. Atiyah and I. M. Singer, The index of elliptic operators. I,Ann.ofMath.(2)87 (1968), 484–530. MR0236950 (38 #5243) 8. Michael Atiyah and Graeme Segal, Twisted K-theory,Ukr.Mat.Visn.1 (2004), no. 3, 287– 330, arxiv.org: math/0407054. MR2172633 9. , Twisted K-theory and cohomology, Inspired by S. S. Chern (Phillip A. Griffiths, ed.), Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, arxiv.org: math/0510674, pp. 5–43. MR2307274 (2008j:55003) 10. Maurice Auslander and Oscar Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.
Load more

Recommended publications
  • Geometric Analysis of Shapes and Its Application to Medical Image Analysis
    Geometric Analysis of Shapes and Its application to Medical Image Analysis by Anirban Mukhopadhyay (Under the DIRECTION of Suchendra M. Bhandarkar) Abstract Geometric analysis of shapes plays an important role in the way the visual world is per- ceived by modern computers. To this end, low-level geometric features provide most obvious and important cues towards understanding the visual scene. A novel intrinsic geometric sur- face descriptor, termed as the Geodesic Field Estimate (GFE) is proposed. Also proposed is a parallel algorithm, well suited for implementation on Graphics Processing Units, for efficient computation of the shortest geodesic paths. Another low level geometric descriptor, termed as the Biharmonic Density Estimate, is proposed to provide an intrinsic geometric scale space signature for multiscale surface feature-based representation of deformable 3D shapes. The computer vision and graphics communities rely on mid-level geometric understanding as well to analyze a scene. Symmetry detection and partial shape matching play an important role as mid-level cues. A comprehensive framework for detection and characterization of partial intrinsic symmetry over 3D shapes is proposed. To identify prominent overlapping symmetric regions, the proposed framework is decoupled into Correspondence Space Voting followed by Transformation Space Mapping procedure. Moreover, a novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric embedding is also proposed. The ultimate goal of geometric shape analysis is to resolve high level applications of modern world. This dissertation has focused on three different application scenarios. In the first scenario, a novel approach for the analysis of the non-rigid Left Ventricular (LV) endocardial surface from Multi-Detector CT images, using a generalized isometry-invariant Bag-of-Features (BoF) descriptor, is proposed and implemented.
    [Show full text]
  • On the Holographic S–Matrix
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server On the Holographic S{matrix I.Ya.Aref’eva Steklov Mathematical Institute, Russian Academy of Sciences Gubkin St.8, GSP-1, 117966, Moscow, Russia Centro Vito Volterra, Universita di Roma Tor Vergata, Italy [email protected] Abstract The recent proposal by Polchinski and Susskind for the holographic flat space S– matrix is discussed. By using Feynman diagrams we argue that in principle all the information about the S–matrix in the interacting field theory in the bulk of the anti- de Sitter space is encoded into the data on the timelike boundary. The problem of locality of interpolating field is discussed and it is suggested that the interpolating field lives in a quantum Boltzmannian Hilbert space. 1 According to the holographic principle [1, 2] one should describe a field theory on a manifold M which includes gravity by a theory which lives on the boundary of M.Two prominent examples of the holography are the Matrix theory [3] and the AdS/CFT corre- spondence [4, 5, 6]. The relation between quantum gravity in the anti-de Sitter space and the gauge theory on the boundary could be useful for better understanding of both theories. In principle CFT might teach us about quantum gravity in the bulk of AdS. Correlation functions in the Euclidean formulation are the subject of intensive study (see for example [7]-[21]). The AdS/CFT correspondence in the Lorentz formulation is considered in [22]-[29].
    [Show full text]
  • Stephen Hawking (1942–2018) World-Renowned Physicist Who Defied the Odds
    COMMENT OBITUARY Stephen Hawking (1942–2018) World-renowned physicist who defied the odds. hen Stephen Hawking was speech synthesizer processed his words and diagnosed with motor-neuron generated the androidal accent that became disease at the age of 21, it wasn’t his trademark. In this way, he completed his Wclear that he would finish his PhD. Against best-selling book A Brief History of Time all expectations, he lived on for 55 years, (Bantam, 1988), which propelled him to becoming one of the world’s most celebrated celebrity status. IAN BERRY/MAGNUM scientists. Had Hawking achieved equal distinction Hawking, who died on 14 March 2018, was in any other branch of science besides cos- born in Oxford, UK, in 1942 to a medical- mology, it probably would not have had the researcher father and a philosophy-graduate same resonance with a worldwide public. As mother. After attending St Albans School I put it in The Telegraph newspaper in 2007, near London, he earned a first-class degree “the concept of an imprisoned mind roaming in physics from the University of Oxford. He the cosmos” grabbed people’s imagination. began his research career in 1962, enrolling In 1965, Stephen married Jane Wilde. as a graduate student in a group at the Uni- After 25 years of marriage, and three versity of Cambridge led by one of the fathers children, the strain of Stephen’s illness of modern cosmology, Dennis Sciama. and of sharing their home with a team of The general theory of relativity was at that nurses became too much and they sepa- time undergoing a renaissance, initiated in rated, divorcing in 1995.
    [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]
  • CERN Celebrates Discoveries
    INTERNATIONAL JOURNAL OF HIGH-ENERGY PHYSICS CERN COURIER VOLUME 43 NUMBER 10 DECEMBER 2003 CERN celebrates discoveries NEW PARTICLES NETWORKS SPAIN Protons make pentaquarks p5 Measuring the digital divide pl7 Particle physics thrives p30 16 KPH impact 113 KPH impact series VISyN High Voltage Power Supplies When the objective is to measure the almost immeasurable, the VISyN-Series is the detector power supply of choice. These multi-output, card based high voltage power supplies are stable, predictable, and versatile. VISyN is now manufactured by Universal High Voltage, a world leader in high voltage power supplies, whose products are in use in every national laboratory. For worldwide sales and service, contact the VISyN product group at Universal High Voltage. Universal High Voltage Your High Voltage Power Partner 57 Commerce Drive, Brookfield CT 06804 USA « (203) 740-8555 • Fax (203) 740-9555 www.universalhv.com Covering current developments in high- energy physics and related fields worldwide CERN Courier (ISSN 0304-288X) is distributed to member state governments, institutes and laboratories affiliated with CERN, and to their personnel. It is published monthly, except for January and August, in English and French editions. The views expressed are CERN not necessarily those of the CERN management. Editor Christine Sutton CERN, 1211 Geneva 23, Switzerland E-mail: [email protected] Fax:+41 (22) 782 1906 Web: cerncourier.com COURIER Advisory Board R Landua (Chairman), P Sphicas, K Potter, E Lillest0l, C Detraz, H Hoffmann, R Bailey
    [Show full text]
  • Prvních Deset Abelových Cen Za Matematiku
    Prvních deset Abelových cen za matematiku The first ten Abel Prizes for mathematics [English summary] In: Michal Křížek (author); Lawrence Somer (author); Martin Markl (author); Oldřich Kowalski (author); Pavel Pudlák (author); Ivo Vrkoč (author); Hana Bílková (other): Prvních deset Abelových cen za matematiku. (English). Praha: Jednota českých matematiků a fyziků, 2013. pp. 87–88. Persistent URL: http://dml.cz/dmlcz/402234 Terms of use: © M. Křížek © L. Somer © M. Markl © O. Kowalski © P. Pudlák © I. Vrkoč Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Summary The First Ten Abel Prizes for Mathematics Michal Křížek, Lawrence Somer, Martin Markl, Oldřich Kowalski, Pavel Pudlák, Ivo Vrkoč The Abel Prize for mathematics is an international prize presented by the King of Norway for outstanding results in mathematics. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) who found that there is no explicit formula for the roots of a general polynomial of degree five. The financial support of the Abel Prize is comparable with the Nobel Prize, i.e., about one million American dollars. Niels Henrik Abel (1802–1829) M. Křížek a kol.: Prvních deset Abelových cen za matematiku, JČMF, Praha, 2013 87 Already in 1899, another famous Norwegian mathematician Sophus Lie proposed to establish an Abel Prize, when he learned that Alfred Nobel would not include a prize in mathematics among his five proposed Nobel Prizes.
    [Show full text]
  • 1 Getting a Knighthood
    Getting a knighthood (Bristol, June 1996) It's unreal, isn't it? I've always had a strong sense of the ridiculous, of the absurd, and this is working overtime now. Many of you must have been thinking, ‘Why Berry?’ Well, since I am the least knightly person I know, I have been asking the same question, and although I haven't come up with an answer I can offer a few thoughts. How does a scientist get this sort of national recognition? One way is to do something useful to the nation, that everyone agrees is important to the national well- being or even survival. Our home-grown example is of course Sir Charles Frank's war work. Or, one can sacrifice several years of one's life doing high-level scientific administration, not always received with gratitude by the scientific hoi polloi but important to the smooth running of the enterprise. Sir John Kingman is in this category. Another way – at least according to vulgar mythology – is to have the right family background. Well, my father drove a cab in London and my mother ruined her eyes as a dressmaker. Another way is to win the Nobel Prize or one of the other huge awards like the Wolf or King Faisal prizes, like Sir George Porter as he then was, or Sir Michael Atiyah. The principle here I suppose is, to those that have, more shall be given. I don't qualify on any of these grounds. The nearest I come is to have have won an fairly large number of smaller awards.
    [Show full text]
  • Towards a String Dual of SYK Arxiv:2103.03187V1 [Hep-Th] 4 Mar
    Towards A String Dual of SYK Akash Goel and Herman Verlinde Department of Physics, Princeton University, Princeton, NJ 08544, USA Abstract: We propose a paradigm for realizing the SYK model within string theory. Using the large N matrix description of c < 1 string theory, we show that the effective theory on a large number Q of FZZT D-branes in (p; 1) minimal string theory takes the form of the disorder averaged SYK model with J p interaction. The SYK fermions represent open strings between the FZZT branes and the ZZ branes that underly the matrix model. The continuum SYK dynamics arises upon taking the large Q limit. We observe several qualitative and quantitative links between the SYK model and (p; q) minimal string theory and propose that the two describe different phases of a single system. We comment on the dual string interpretation of double scaled SYK and on the relevance of our results to the recent discussion of the role of ensemble averaging in holography. arXiv:2103.03187v2 [hep-th] 24 Aug 2021 Contents 1 Introduction2 2 SYK from the Two Matrix Model4 2.1 FZZT-branes in the two matrix model4 2.2 Kontsevich matrix model from FZZT branes6 2.3 SYK matrix model from FZZT branes7 3 Towards the Continuum SYK model8 3.1 Non-commutative SYK8 4 From Minimal Strings to SYK 10 4.1 SYK and the (p; q) spectral curve 11 4.2 FZZT brane correlation function 12 4.3 Minimal String-SYK phase diagram 13 5 SYK as a Non-Critical String 14 6 Conclusion 16 A D-branes in Minimal String Theory 18 A.1 Matrices and the non-commutative torus 23 A.2 Non-commutative SYK 24 A.3 Matrix SYK 26 A.4 Mapping between Matrix SYK and Non-Commutative SYK 27 { 1 { 1 Introduction The SYK model is the prototype of a maximally chaotic quantum system with low energy dynamics given by near-AdS2 gravity.
    [Show full text]
  • Karen Keskulla Uhlenbeck
    2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems.
    [Show full text]
  • Introduction
    Proceedings of Symposia in Pure Mathematics Volume 81, 2010 Introduction Jonathan Rosenberg Abstract. The papers in this volume are the outgrowth of an NSF-CBMS Regional Conference in the Mathematical Sciences, May 18–22, 2009, orga- nized by Robert Doran and Greg Friedman at Texas Christian University. This introduction explains the scientific rationale for the conference and some of the common themes in the papers. During the week of May 18–22, 2009, Robert Doran and Greg Friedman orga- nized a wonderfully successful NSF-CBMS Regional Research Conference at Texas Christian University. I was the primary lecturer, and my lectures have now been published in [29]. However, Doran and Friedman also invited many other mathe- maticians and physicists to speak on topics related to my lectures. The papers in this volume are the outgrowth of their talks. The subject of my lectures, and the general theme of the conference, was highly interdisciplinary, and had to do with the confluence of superstring theory, algebraic topology, and C∗-algebras. While with “20/20 hindsight” it seems clear that these subjects fit together in a natural way, the connections between them developed almost by accident. Part of the history of these connections is explained in the introductions to [11] and [17]. The authors of [11] begin as follows: Until recently the interplay between physics and mathematics fol- lowed a familiar pattern: physics provides problems and mathe- matics provides solutions to these problems. Of course at times this relationship has led to the development of new mathematics. But physicists did not traditionally attack problems of pure mathematics.
    [Show full text]
  • Heterotic String Compactification with a View Towards Cosmology
    Heterotic String Compactification with a View Towards Cosmology by Jørgen Olsen Lye Thesis for the degree Master of Science (Master i fysikk) Department of Physics Faculty of Mathematics and Natural Sciences University of Oslo May 2014 Abstract The goal is to look at what constraints there are for the internal manifold in phe- nomenologically viable Heterotic string compactification. Basic string theory, cosmology, and string compactification is sketched. I go through the require- ments imposed on the internal manifold in Heterotic string compactification when assuming vanishing 3-form flux, no warping, and maximally symmetric 4-dimensional spacetime with unbroken N = 1 supersymmetry. I review the current state of affairs in Heterotic moduli stabilisation and discuss merging cosmology and particle physics in this setup. In particular I ask what additional requirements this leads to for the internal manifold. I conclude that realistic manifolds on which to compactify in this setup are severely constrained. An extensive mathematics appendix is provided in an attempt to make the thesis more self-contained. Acknowledgements I would like to start by thanking my supervier Øyvind Grøn for condoning my hubris and for giving me free rein to delve into string theory as I saw fit. It has lead to a period of intense study and immense pleasure. Next up is my brother Kjetil, who has always been a good friend and who has been constantly looking out for me. It is a source of comfort knowing that I can always turn to him for help. Mentioning friends in such an acknowledgement is nearly mandatory. At least they try to give me that impression.
    [Show full text]
  • Supergravity at 40: Reflections and Perspectives(∗)
    RIVISTA DEL NUOVO CIMENTO Vol. 40, N. 6 2017 DOI 10.1393/ncr/i2017-10136-6 ∗ Supergravity at 40: Reflections and perspectives( ) S. Ferrara(1)(2)(3)andA. Sagnotti(4) (1) Theoretical Physics Department, CERN CH - 1211 Geneva 23, Switzerland (2) INFN - Laboratori Nazionali di Frascati - Via Enrico Fermi 40 I-00044 Frascati (RM), Italy (3) Department of Physics and Astronomy, Mani L. Bhaumik Institute for Theoretical Physics U.C.L.A., Los Angeles CA 90095-1547, USA (4) Scuola Normale Superiore e INFN - Piazza dei Cavalieri 7, I-56126 Pisa, Italy received 15 February 2017 Dedicated to John H. Schwarz on the occasion of his 75th birthday Summary. — The fortieth anniversary of the original construction of Supergravity provides an opportunity to combine some reminiscences of its early days with an assessment of its impact on the quest for a quantum theory of gravity. 280 1. Introduction 280 2. The early times 282 3. The golden age 283 4. Supergravity and particle physics 284 5. Supergravity and string theory 286 6. Branes and M-theory 287 7. Supergravity and the AdS/CFT correspondence 288 8. Conclusions and perspectives ∗ ( ) Based in part on the talk delivered by S.F. at the “Special Session of the DISCRETE2016 Symposium and the Leopold Infeld Colloquium”, in Warsaw, on December 1 2016, and on a joint CERN Courier article. c Societ`a Italiana di Fisica 279 280 S. FERRARA and A. SAGNOTTI 1. – Introduction The year 2016 marked the fortieth anniversary of the discovery of Supergravity (SGR) [1], an extension of Einstein’s General Relativity [2] (GR) where Supersymme- try, promoted to a gauge symmetry, accompanies general coordinate transformations.
    [Show full text]