Nonperturbative Formulations of Superstring Theory

Total Page:16

File Type:pdf, Size:1020Kb

Nonperturbative Formulations of Superstring Theory NONPERTURBATIVE FORMULATIONS OF SUPERSTRING THEORY BY LUBOSˇ MOTL A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy Written under the direction of Thomas Banks and approved by arXiv:hep-th/0109149v1 19 Sep 2001 New Brunswick, New Jersey October, 2001 ABSTRACT OF THE DISSERTATION Nonperturbative Formulations of Superstring Theory by LuboˇsMotl Dissertation Director: Thomas Banks After a short introduction to Matrix theory, we explain how can one generalize matrix models to describe toroidal compactifications of M-theory and the heterotic vacua with 16 supercharges. This allows us, for the first time in history, to derive the conventional perturbative type IIA string theory known in the 80s within a complete and consistent non- perturbative framework, using the language of orbifold conformal field theory and conformal perturbation methods. A separate chapter is dedicated to the vacua with Hoˇrava-Witten domain walls that carry E8 gauge supermultiplets. Those reduce the gauge symmetry of the matrix model from U(N) to O(N). We also explain why these models contain open membranes. The compactification of M-theory on T 4 involves the so-called (2, 0) supercon- formal field theory in six dimensions, compactified on T 5. A separate chapter describes an interesting topological contribution to the low energy equations of motion on the Coulomb branch of the (2, 0) theory that admits a skyrmionic solution that we call “knitting five- branes”. Then we return to the orbifolds of Matrix theory and construct a formal classical matrix model of the Scherk-Schwarz compactification of M-theory and type IIA string theory as well as type 0 theories. We show some disastrous consequences of the broken supersymmetry. Last two chapters describe a hyperbolic structure of the moduli spaces of one-dimensional M-theory. ii Acknowledgements I am grateful to the many people who helped me along the way, explained many things to me and to those that have shared their friendship and love. For example, I am grateful to Micha Berkooz, Michael Dine, Rami Entin, Michal Fabinger, Tomeu Fiol, Willy Fischler, Ori Ganor, David Gross, Petr Hoˇrava, Marcos Mari˜no, Greg Moore, Arvind Rajaraman, Christian R¨omelsberger, Moshe Rozali, Steve Shenker, Andy Strominger, Lenny Susskind, Herman Verlinde, Edward Witten and many others. A lot of thanks goes to my advisor Tom Banks who not only taught me a lot of things about physics but who was also a significant source of my support in the USA. My work was supported in part by the Department Of Energy under grant number DE- FG02-96ER40559. I would like to thank the ITP in Santa Barbara for hospitality during the final stages of this work. iii Dedication For the victims of 20th century totalitarian regimes, those who have been executed, jailed or dismissed from schools by fascists and communists. iv Table of Contents Abstract .......................................... ii Acknowledgements ................................... iii Dedication ......................................... iv List of Tables ....................................... ix List of Figures ...................................... x 1. Introduction to M(atrix) theory ......................... 1 1.1. BasicsofM(atrix)theory . ... 2 1.1.1. Membranes ................................ 3 2. Matrix string theory ................................ 6 2.1. Reflections on the second quantization and its matrix competitor . 6 2.2. M(atrix) theory in 0 + 0 and 0 + 1 dimensions . ..... 9 2.2.1. An old-fashioned approach to the compactification of one spatial co- ordinate.................................. 11 2.2.2. Limits of the resulting type IIA theory . ..... 13 2.3. Representation of the “long” strings, screwing strings to matrices . 14 2.4. The origin of the level matching conditions and the interactions . 16 2.4.1. The DVV interaction vertex . 17 2.4.2. Compactification to nine dimensions . 19 3. Heterotic matrix models .............................. 21 3.1. Introduction................................... 21 3.2. Heterotic matrix models in ten and eleven dimensions . .......... 23 v 3.2.1. Heterotic string theory as an orientifold of M-theory......... 23 3.2.2. Open membranes in M(atrix) theory . 24 3.2.3. Additional degrees of freedom . 25 3.2.4. Screwing heterotic strings . 29 3.3. Multidimensionalcylinders . ..... 31 3.3.1. TypeIAstrings.............................. 33 3.4. Conclusions .................................... 35 3.4.1. Cosmological constant problem . 36 4. Knitted fivebranes in the (2,0) theory ..................... 38 4.1. Introduction................................... 38 4.2. Reviewofthe(2,0)theory. 39 4.2.1. Realization ................................ 39 4.2.2. Equations of motion for a free tensor multiplet . ....... 40 4.3. Low-energy correction terms – derivation from SUGRA . ......... 42 4.4. Compactification ................................ 45 4.4.1. Dimensional reduction of the correction term . ....... 46 4.4.2. Interpolation between 3+1D and 5+1D . 47 4.4.3. A derivation from 4+1D SYM . 48 4.4.4. Componentform ............................. 49 4.5. Conservedquantities . 51 4.5.1. Traceless stress tensor and supercurrent . ....... 53 4.6. Speculations about the fundamental formulation . .......... 56 4.7. String-like solution of the (2,0) theory . ......... 57 4.7.1. Aroughpicture.............................. 58 4.7.2. TheAnsatz ................................ 58 4.7.3. Numerical solution, tension and speculations . ........ 59 4.8. Discussion..................................... 60 4.9. Appendix A: Formulae for SUSY transformations . ........ 61 vi 4.9.1. SUSYtransformation . .. .. .. .. .. .. .. 61 4.10. Appendix B: Quantization . 62 4.10.1.Commutators............................... 63 4.10.2. Correspondence to Super Yang Mills . 65 4.10.3. Normalization of the current . 67 4.11.AppendixC:Identities . 67 4.11.1. Identities for gamma matrices . 67 4.11.2.Fierzrearrangments . 68 4.11.3. A few notes about spin(5, 1) ...................... 70 5. Nonsupersymmetric matrix models ....................... 72 5.1. The conformal field theory description . ....... 72 5.2. Thematrixmodel ................................ 76 5.2.1. The U(N) U(N)formalism ...................... 78 × 5.2.2. Actions for the local and nonlocal formulations . ........ 79 5.3. Alternative derivation and connection with type 0 theories.......... 81 5.4. Thematrixstringlimits . 83 5.4.1. Rohm compactified type IIA strings . 83 5.4.2. Dependenceonthefluxes . 84 5.4.3. Type0matrixstrings .......................... 87 5.4.4. Breakdown of the cluster property . 88 5.5. Conclusions .................................... 92 6. Asymptotic limits in the moduli space of M-theory ............ 94 6.1. Introduction................................... 94 6.2. Moduli, vacua, quantum cosmology and singularities . ........... 96 6.2.1. Some idiosyncratic views on general wisdom . ...... 96 6.2.2. Quantumcosmology .. .. .. .. .. .. .. .. 98 6.2.3. Kasner singularities and U-duality . 100 6.3. The moduli space of M-theory on rectangular tori . ......... 103 vii 6.3.1. The2/5transformation . 104 6.3.2. Extrememoduli ............................. 105 6.3.3. Coveringthemodulispace. 108 6.4. ModulispaceswithlessSUSY. 112 6.5. Discussion..................................... 116 6.6. Appendix A: The appearance of Γ8 and Γ9,1 .................. 117 7. Hyperbolic structure of the heterotic moduli space ............. 120 7.1. Introduction................................... 120 7.1.1. Thebilinearform............................. 122 7.1.2. Moduli spaces and heterotic S-duality . 124 7.2. CoveringtheSO(32)modulispace . 127 7.2.1. Heterotic strings, type I, type IA and d 9.............. 127 ≥ 7.2.2. Type IA2 and d =8 ........................... 129 7.2.3. Type IA3, M-theory on K3 and d =7 ................. 130 7.2.4. Type IA4,5, type IIA/B on K3 and d = 6, 5 .............. 133 7.2.5. Type IA6 and S-duality in d =4 .................... 134 7.2.6. The infinite groups in d 2....................... 136 ≤ 7.2.7. Thelattices ................................ 137 7.3. Conclusions .................................... 138 References ......................................... 139 Vita ............................................. 146 viii List of Tables 2 2 3.1. Matrix representations of Z2 orbifolds of the sphere S and the torus T : here Ylm are the spherical harmonics and Jr,s are the Fourier modes on the torus. Both are represented by some matrix generators in the regularized case. 25 ix List of Figures 5.1. One-loopdiagrams.. .. .. .. .. .. .. .. .. 90 5.2. Non-vanishing two-loop contributions to the effective potential. Here V ′ 2 ≡ (ǫ(p) 1)V ..................................... 90 − 2 6.1. The structure of the moduli space for T 2. ................... 107 7.1. The limits in d =9................................. 128 x 1 Chapter 1 Introduction to M(atrix) theory Superstring theory has been known to lead to a consistent perturbative S-matrix for a decade or so, since the discoveries of Green and Schwarz in the middle 80s. Superstring the- ory is only consistent in ten spacetime dimensions and it can be approximated by supergrav- ity theories at low energies (or supergravities coupled to supersymmetric Yang-Mills theo- ries). However the most symmetric supergravity theory required eleven spacetime dimen- sions and seemed to be disconnected from superstring theory
Recommended publications
  • M5-Branes, D4-Branes and Quantum 5D Super-Yang-Mills
    CERN-PH-TH/2010-294 KCL-MTH-10-17 M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills N. Lambert a,∗,† , C. Papageorgakis b,‡ and M. Schmidt-Sommerfeld a,§ aTheory Division, CERN 1211 Geneva 23, Switzerland bDepartment of Mathematics, King’s College London The Strand, London WC2R 2LS, UK Abstract We revisit the relation of the six-dimensional (2, 0) M5-brane Conformal Field Theory compactified on S1 to 5D maximally supersymmetric Yang-Mills Gauge Theory. We show that in the broken phase 5D super-Yang-Mills contains a arXiv:1012.2882v3 [hep-th] 22 Feb 2011 spectrum of soliton states that can be identified with the complete Kaluza-Klein modes of an M2-brane ending on the M5-branes. This provides evidence that the (2, 0) theory on S1 is equivalent to 5D super-Yang-Mills with no additional UV degrees of freedom, suggesting that the latter is in fact a well-defined quantum theory and possibly finite. ∗On leave of absence from King’s College London. †E-mail address: [email protected] ‡E-mail address: [email protected] §E-mail address: [email protected] 1 Introduction Multiple M5-branes are believed to be described at low energies by a novel, interacting, strongly coupled, 6D CFT with (2, 0) supersymmetry. Very little is known about such a theory and it is not expected to have a Lagrangian description. According to the type IIA/M-theory duality it arises as the strong-coupling, UV fixed-point of multiple D4-branes whose dynamics are obtained from open string theory.
    [Show full text]
  • Round Table Talk: Conversation with Nathan Seiberg
    Round Table Talk: Conversation with Nathan Seiberg Nathan Seiberg Professor, the School of Natural Sciences, The Institute for Advanced Study Hirosi Ooguri Kavli IPMU Principal Investigator Yuji Tachikawa Kavli IPMU Professor Ooguri: Over the past few decades, there have been remarkable developments in quantum eld theory and string theory, and you have made signicant contributions to them. There are many ideas and techniques that have been named Hirosi Ooguri Nathan Seiberg Yuji Tachikawa after you, such as the Seiberg duality in 4d N=1 theories, the two of you, the Director, the rest of about supersymmetry. You started Seiberg-Witten solutions to 4d N=2 the faculty and postdocs, and the to work on supersymmetry almost theories, the Seiberg-Witten map administrative staff have gone out immediately or maybe a year after of noncommutative gauge theories, of their way to help me and to make you went to the Institute, is that right? the Seiberg bound in the Liouville the visit successful and productive – Seiberg: Almost immediately. I theory, the Moore-Seiberg equations it is quite amazing. I don’t remember remember studying supersymmetry in conformal eld theory, the Afeck- being treated like this, so I’m very during the 1982/83 Christmas break. Dine-Seiberg superpotential, the thankful and embarrassed. Ooguri: So, you changed the direction Intriligator-Seiberg-Shih metastable Ooguri: Thank you for your kind of your research completely after supersymmetry breaking, and many words. arriving the Institute. I understand more. Each one of them has marked You received your Ph.D. at the that, at the Weizmann, you were important steps in our progress.
    [Show full text]
  • An Introduction to Effective Field Theory
    An Introduction to Effective Field Theory Thinking Effectively About Hierarchies of Scale c C.P. BURGESS i Preface It is an everyday fact of life that Nature comes to us with a variety of scales: from quarks, nuclei and atoms through planets, stars and galaxies up to the overall Universal large-scale structure. Science progresses because we can understand each of these on its own terms, and need not understand all scales at once. This is possible because of a basic fact of Nature: most of the details of small distance physics are irrelevant for the description of longer-distance phenomena. Our description of Nature’s laws use quantum field theories, which share this property that short distances mostly decouple from larger ones. E↵ective Field Theories (EFTs) are the tools developed over the years to show why it does. These tools have immense practical value: knowing which scales are important and why the rest decouple allows hierarchies of scale to be used to simplify the description of many systems. This book provides an introduction to these tools, and to emphasize their great generality illustrates them using applications from many parts of physics: relativistic and nonrelativistic; few- body and many-body. The book is broadly appropriate for an introductory graduate course, though some topics could be done in an upper-level course for advanced undergraduates. It should interest physicists interested in learning these techniques for practical purposes as well as those who enjoy the beauty of the unified picture of physics that emerges. It is to emphasize this unity that a broad selection of applications is examined, although this also means no one topic is explored in as much depth as it deserves.
    [Show full text]
  • Electric-Magnetic Duality, Matrices, & Emergent Spacetime
    Electric-Magnetic Duality, Matrices, & Emergent Spacetime Shyamoli Chaudhuri1 1312 Oak Drive Blacksburg, VA 24060 Abstract This is a rough transcript of talks given at the Workshop on Groups & Algebras in M Theory at Rutgers University, May 31–Jun 04, 2005. We review the basic motivation for a pre-geometric formulation of nonperturbative String/M theory, and for an underlying eleven- dimensional electric-magnetic duality, based on our current understanding of the String/M Duality Web. We explain the concept of an emerging spacetime geometry in the large N limit of a U(N) flavor matrix Lagrangian, distinguishing our proposal from generic proposals for quantum geometry, and explaining why it can incorporate curved spacetime backgrounds. We assess the significance of the extended symmetry algebra of the matrix Lagrangian, raising the question of whether our goal should be a duality covariant, or merely duality invariant, Lagrangian. We explain the conjectured isomorphism between the O(1/N) corrections in any given large N scaling limit of the matrix Lagrangian, and the corresponding α′ corrections in a string effective Lagrangian describing some weak-coupling limit of the String/M Duality Web. arXiv:hep-th/0507116v4 23 Aug 2005 1Based on talks given at the Workshop on Groups & Algebras in M Theory, Rutgers University, May 31–Jun 04, 2005. Email: [email protected] 1 Introduction Understanding the symmetry principles and the fundamental degrees of freedom in terms of which nonperturbative String/M theory is formulated is a problem of outstanding importance in theoretical high energy physics. The Rutgers Mathematics workshop on Groups & Algebras in M Theory this summer devoted part of its schedule to an assessment of the significance of Lorentzian Kac-Moody algebras to recent conjectures for the symmetry algebra of String/M theory.
    [Show full text]
  • Soft Gravitons and the Flat Space Limit of Anti-Desitter Space
    RUNHETC-2016-15, UTTG-18-16 Soft Gravitons and the Flat Space Limit of Anti-deSitter Space Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Willy Fischler Department of Physics and Texas Cosmology Center University of Texas, Austin, TX 78712 E-mail: fi[email protected] Abstract We argue that flat space amplitudes for the process 2 n gravitons at center of mass → energies √s much less than the Planck scale, will coincide approximately with amplitudes calculated from correlators of a boundary CFT for AdS space of radius R LP , only ≫ when n < R/LP . For larger values of n in AdS space, insisting that all the incoming energy enters “the arena”[20] , implies the production of black holes, whereas there is no black hole production in the flat space amplitudes. We also argue, from unitarity, that flat space amplitudes for all n are necessary to reconstruct the complicated singularity arXiv:1611.05906v3 [hep-th] 8 Apr 2017 at zero momentum in the 2 2 amplitude, which can therefore not be reproduced as → the limit of an AdS calculation. Applying similar reasoning to amplitudes for real black hole production in flat space, we argue that unitarity of the flat space S-matrix cannot be assessed or inferred from properties of CFT correlators. 1 Introduction The study of the flat space limit of correlation functions in AdS/CFT was initiated by the work of Polchinski and Susskind[20] and continued in a host of other papers. Most of those papers agree with the contention, that the limit of Witten diagrams for CFT correlators smeared with appropriate test functions, for operators carrying vanishing angular momentum on the compact 1 space,1 converge to flat space S-matrix elements between states in Fock space.
    [Show full text]
  • A View from the Bridge Natalie Paquette
    INFERENCE / Vol. 3, No. 4 A View from the Bridge Natalie Paquette tring theory is a quantum theory of gravity.1 Albert example, supersymmetric theories require particles to Einstein’s theory of general relativity emerges natu- come in pairs. For every bosonic particle there is a fermi- rally from its equations.2 The result is consistent in onic superpartner. Sthe sense that its calculations do not diverge to infinity. Supersymmetric field theory has a disheartening String theory may well be the only consistent quantum impediment. Suppose that a supersymmetric quantum theory of gravity. If true, this would be a considerable field theory is defined on a generic curved manifold. The virtue. Whether it is true or not, string theory is indis- Euclidean metric of Newtonian physics and the Lorentz putably the source of profound ideas in mathematics.3 metric of special relativity are replaced by the manifold’s This is distinctly odd. A line of influence has always run own metric. Supercharges correspond to conserved Killing from mathematics to physics. When Einstein struggled spinors. Solutions to the Killing spinor equations are plen- to express general relativity, he found the tools that he tiful in a flat space, but the equations become extremely needed had been created sixty years before by Bernhard restrictive on curved manifolds. They are so restrictive Riemann. The example is typical. Mathematicians discov- that they have, in general, no solutions. Promoting a flat ered group theory long before physicists began using it. In supersymmetric field theory to a generic curved mani- the case of string theory, it is often the other way around.
    [Show full text]
  • Collective Coordinates for D-Branes 1 Introduction
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server UTTG-01-96 Collective Co ordinates for D-branes Willy Fischler, Sonia Paban and Moshe Rozali Theory Group, Department of Physics University of Texas, Austin, Texas 78712 Abstract We develop a formalism for the scattering o D-branes that includes collective co- ordinates. This allows a systematic expansion in the string coupling constant for such pro cesses, including a worldsheet calculation for the D-brane's mass. 1 Intro duction In the recent developments of string theory, solitons play a central role. For example, in the case of duality among various string theories, solitons of a weakly coupled string theory b ecome elementary excitations of the dual theory [1]. Another example, among many others, is the role of solitons in resolving singularities in compacti ed geometries, as they b ecome light[2]. In order to gain more insight in the various asp ects of these recent developments, it is imp ortant to understand the dynamics of solitons in the context of string theory.In this quest one of the to ols at our disp osal is the use of scattering involving these solitons. This includes the scattering of elementary string states o these solitons as well as the scattering among solitons. An imp ortant class of solitons that have emerged are D-branes [4]. In an in uential pap er [3]itwas shown that these D-branes carry R R charges, and are therefore a central ingredient in the aforementioned dualities.
    [Show full text]
  • King's Research Portal
    King’s Research Portal DOI: 10.1088/1742-6596/631/1/012089 Document Version Publisher's PDF, also known as Version of record Link to publication record in King's Research Portal Citation for published version (APA): Sarkar, S. (2015). String theory backgrounds with torsion in the presence of fermions and implications for leptogenesis. Journal of Physics: Conference Series, 631(1), [012089]. 10.1088/1742-6596/631/1/012089 Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. •Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. •You may not further distribute the material or use it for any profit-making activity or commercial gain •You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Geometry, Quantum
    Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2010 DOI: 10.4171/OWR/2010/25 Geometry, Quantum Fields, and Strings: Categorial Aspects Organised by Peter Bouwknegt, Canberra Dan Freed, Austin Christoph Schweigert, Hamburg June 6th – June 12th, 2010 Abstract. Currently, in the interaction between string theory, quantum field theory and topology, there is an increased use of category-theoretic methods. Independent developments (e.g. the categorificiation of knot invariants, bun- dle gerbes and topological field theories on extended cobordism categories) have put higher categories in the focus. The workshop has brought together researchers working on diverse prob- lems in which categorical ideas play a significant role. Mathematics Subject Classification (2000): 81T, in particular 81T45 and 81T13. Introduction by the Organisers The workshop Geometry, Quantum Fields, and Strings: Categorial Aspects, organ- ised by Peter Bouwknegt (Australian National University, Canberra), Dan Freed (University of Texas, Austin), and Christoph Schweigert (University of Hamburg) was held June 6th–June 12th, 2010. The meeting was attended by 52 participants from all continents. 18 talks of one hour each were contributed to the workshop. Moreover, young researchers were offered the possibility to present short contributions. On Monday and Wednesday evening a total of 11 short talks were delivered. We would like to stress the high quality and level of interest of these contributions. The two sessions have received much attention and have led to much additional scientific discussion about the work of younger participants. For this reason, these contributions are covered in these proceedings as well. Another special event was a panel discussion on Tuesday evening on the topic “Whither the interaction of Geometry-QFT-String?”.
    [Show full text]
  • Collective Coordinates for D-Branes
    UTTG-01-96 Collective Coordinates for D-branes Willy Fischler, Sonia Paban and Moshe Rozali∗ Theory Group, Department of Physics University of Texas, Austin, Texas 78712 Abstract We develop a formalism for the scattering off D-branes that includes collective co- ordinates. This allows a systematic expansion in the string coupling constant for such processes, including a worldsheet calculation for the D-brane’s mass. 1 Introduction In the recent developments of string theory, solitons play a central role. For example, in the case of duality among various string theories, solitons of a weakly coupled string theory become elementary excitations of the dual theory [1]. Another example, among many others, is the role of solitons in resolving singularities in compactified geometries, as they become light [2]. In order to gain more insight in the various aspects of these recent developments, it is important to understand the dynamics of solitons in the context of string theory. In this quest one of the tools at our disposal is the use of scattering involving these solitons. This includes the scattering of elementary string states off these solitons as well as the arXiv:hep-th/9604014v2 11 Apr 1996 scattering among solitons. An important class of solitons that have emerged are D-branes [4]. In an influential paper [3] it was shown that these D-branes carry R R charges, − and are therefore a central ingredient in the aforementioned dualities. These objects are described by simple conformal field theories, which makes them particularly suited for explicit calculations. In this paper we use some of the preliminary ideas about collective coordinates de- veloped in a previous paper [5], to describe the scattering of elementary string states off D-branes.
    [Show full text]
  • WKB and Wall-Crossing a “Combinatorial” Picture of Moduli Spaces of Higgs Bundles
    WKB and Wall-Crossing A “combinatorial” picture of moduli spaces of Higgs bundles Andrew Neitzke, Harvard University (work in progress with Davide Gaiotto, Greg Moore) Simons Center for Geometry and Physics, January 2009 Preface Greg’s talk described the physical meaning of the Kontsevich-Soibelman WCF in terms of the interplay between four-dimensional and three-dimensional field theories. One begins with a four-dimensional field theory with 8 supercharges, and reduces it on S1 of radius R. The resulting three-dimensional theory is (on one branch) a sigma model into a hyperk¨ahler manifold M, with metric depending on R. Preface To describe the geometry of M we introduced a family of Darboux coordinates × × Xγ : M × C → C To prove the Kontsevich-Soibelman WCF the key property of Xγ is I Jumps: The collection {Xγ} jumps by the symplectomorphism Ω(γ;u) Kγ along the ray `γ = {ζ : Z(γ; u)/ζ ∈ R−}. To construct the metric on M one also needs I Asymptotics: limζ→0 Xγ exp(−πRZ(γ; u)) exists. Preface How should we think about the Xγ? We have a definition in terms of quantum field theory, but how to make it mathematically intelligible? The approach of Greg’s talk was to declare that the Xγ are defined by these two desired properties. Drawback: it’s not obvious that such functions really exist! To get them you have to solve a Riemann-Hilbert problem, which might have solution only at large R. But we think they should exist for all R. Preface For quantum field theories that come from branes in string theory the situation becomes more geometric and we can understand Xγ more intrinsically.
    [Show full text]
  • 3 Sep 2015 Permutation Orbifolds in the Large N Limit
    RUNHETC-2015-09 Permutation Orbifolds in the large N Limit ⋆ Alexandre Belin , Christoph A. Keller♠, Alexander Maloney September 7, 2015 Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA 94305, USA ♠ NHETC, Rutgers, The State University of New Jersey, Piscataway, USA ⋆ Department of Physics, McGill University, Montr´eal, Canada [email protected], [email protected], [email protected] Abstract The space of permutation orbifolds is a simple landscape of two dimensional CFTs, arXiv:1509.01256v1 [hep-th] 3 Sep 2015 generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions factorize at large N and thus reduce to those of a generalized free field in AdS. We show that this happens not just for symmetric orbifolds, but also for permutation groups which act “democratically” in a sense which we define. 1 Contents 1 Introduction 3 1.1 AdS/CFT and the Space of CFT2’s.................... 3 1.2 HolographicCFTs ............................. 4 1.3 PermutationOrbifolds ........................... 5 2 Permutation Orbifolds and their spectrum 6 2.1 Oligomorphic families GN ......................... 6 2.1.1 Untwistedstates .......................... 7 2.1.2 Twistedstates............................ 8 2.2 Oligomorphicgroups ............................ 10 2.3 Examples of oligomorphic families . 11 2.4 Growthintheuntwistedsector .
    [Show full text]