DOCSLIB.ORG

Calabi-Yau Fourfold Compactifications in String Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Calabi-Yau Fourfold Compactifications in String Theory Calabi-Yau Fourfold Compacti¯cations in String Theory Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt der Mathematisch-Naturwissenschaftlich-Technischen FakultÄat der Martin-Luther-UniversitÄatHalle-Wittenberg von Herrn Michael Haack geb. am: 22.5.1973 in: Hamburg Gutachter 1. Prof. Jan Louis 2. Prof. Dieter LÄust 3. Prof. Stefan Theisen Halle (Saale), 2.7.2001 ii iii Danksagung An erster Stelle gilt mein Dank Herrn Prof. J. Louis fÄur den Vorschlag des Themas und die hervorragende Betreuung der Arbeit. Den Mitarbeitern der Arbeitsgruppe Quan- tenfeldtheorie, R. BÄohm, B. Gunara, H. GÄunther, C. Herrmann, M. Klein, M. Mar- quart, A. Micu, T. Mohaupt, H. Singh und M. Zagermann danke ich fÄur eine an- genehme ArbeitsatmosphÄare und anregende GesprÄache. Mein besonderer Dank gilt dabei T. Mohaupt fÄur seine Geduld und Kompetenz, mit der er insbesondere in der An- fangsphase viele meiner Fragen beantwortet hat. Dem Land Sachsen-Anhalt gilt mein Dank fÄur die GewÄahrung eines Landesgraduiertenstipendiums in der Zeit vom April 1998 bis Juni 2000. In der Zeit vom Juli 2000 bis Juni 2001 wurde ich ¯nanziell von der DFG gefÄordert, der ich ebenfalls danken mÄochte. Daneben habe ich von Geldmit- teln des DAAD pro¯tiert, mit deren Hilfe mir drei Auslandsreisen ermÄoglicht wurden. Dem DAAD gilt dabei ebenso mein Dank wie den gastgebenden Arbeitsgruppen von Prof. P. Bin¶etruy (Orsay), Prof. R. Grimm (Marseille) und Prof. S. Thomas (London). Schlie¼lich danke ich meinen Eltern fÄur ihre moralische und ¯nanzielle UnterstÄutzung. Contents 1 Introduction 1 1.1 Why string theory? . ............................. 1 1.2 What is string theory? . ......................... 5 1.3 String theory on Calabi-Yau manifolds . ................. 10 1.4 Dualities and M-theory . ......................... 14 1.5 Topic and organization of the thesis . ................. 17 2 D=3e®ective theories with four supercharges 20 2.1 Heterotic theories in D =3 ......................... 20 2.1.1 D =4;N = 1 heterotic theories . ................. 20 2.1.2 D =4;N = 1 supergravity compacti¯ed on S1 .......... 21 2.1.3 The heterotic D = 3 low energy e®ective Lagrangian . ..... 22 2.2 M-theory compacti¯ed on Calabi-Yau fourfolds . ............. 24 2.3 Duality . ................................. 29 2.3.1 Heterotic { M-theory duality in D =3............... 29 2.3.2 F-theory limit ............................. 33 3 D=2e®ective theories with four supercharges 35 3.1 Heterotic theories in D =2 ......................... 35 3.2 Type IIA theory on Calabi-Yau fourfolds . ................. 38 3.3 Heterotic { type IIA duality in D=2 . ................. 40 4 Inclusion of background fluxes 42 4.1 Compacti¯cations with fluxes . ..................... 42 4.2 Derivation of the potential in D =3 .................... 45 4.3 Some comments about D =2........................ 52 5 Conclusion 54 A Notation and conventions 56 B Calabi-Yau manifolds 57 B.1 (Co)homology primer . ......................... 57 B.2 Complex manifolds . ............................. 59 B.3 Hermitian manifolds ............................. 62 B.4 KÄahler manifolds . ............................. 64 B.5 Holonomy groups . ............................. 66 B.6 Calabi-Yau manifolds . ......................... 67 iv CONTENTS v B.7 Complex structure deformations . .................... 70 B.8 Torus . .................................... 72 B.9K3....................................... 75 B.10 Calabi-Yau threefolds ............................ 77 B.11 Calabi-Yau fourfolds . ............................ 79 C Kaluza Klein reduction 81 C.1 The Kaluza-Klein recipe . ........................ 81 C.2 An example . ................................ 83 C.3 Consistency . ................................ 84 C.4 S1 compacti¯cation of D = 4 supergravity . ................ 87 C.5 D = 11 supergravity on Calabi-Yau fourfolds . ............ 88 C.6 Alternative way to D =2 .......................... 91 C.6.1 The heterotic case . ........................ 91 C.6.2 The type IIA case . ........................ 92 Bibliography 94 vi CONTENTS Chapter 1 Introduction 1.1 Why string theory? At the end of the 20th century theoretical physics is in a peculiar situation. Its two corner stones, the standard model of particle physics and the general theory of relativ- ity, are capable to explain almost every experimental piece of data from particle and astrophysics with an impressive accuracy. Nevertheless the physical picture drawn by the standard model and general relativity is not complete. Thus their great success is turned into the main obstacle for progress in theoretical physics today. Extensions of the standard physical picture have to proceed without a clear experimental guidance. However one can make a virtue of necessity because the excellent agreement of the theory with current experiments puts severe constraints on any attempt to go beyond the standard model. There are several good reasons to believe that the standard model of particle physics (SM) and the general theory of relativity (GR) have to be extended. Maybe the most severe one is the fact, that both theories neglect the lessons from the respective other one. GR is a classical theory neglecting all quantum aspects of matter as described ¡ 1 by the SM. A look at Einstein's equation R¹º 2 g¹ºR = T¹º immediately shows that a quantum mechanical treatment of the matter part in T¹º actually requires to treat also space and time quantum mechanically. Such a quantum mechanical treatment of gravity might also solve the problem of singularities encountered quite generically in solutions of GR, see e.g. [1]. Several di®erent suggestions have been made for a quantization of gravity which are reviewed in [2]. None of them is yet completely satisfactory. The covariant approach for example tries to describe quantized gravity as a quantum ¯eld theory of metric fluctuations around a given background metric. This leads however to a non-renormalizable quantum ¯eld theory. The other side of the coin is that the SM is formulated in flat Minkowski space ignoring all gravitational e®ects. Although this is perfectly justi¯ed in earth based experiments there have to be expected new e®ects in `extreme' astrophysical situations as encountered e.g. in the early universe or near a black hole. A ¯rst impression of these new e®ects can be gained through the semi-classical methods of quantum ¯eld theory in curved space [3]. In this approach the quantum ¯elds are de¯ned on a curved background space-time which is however taken to be still a solution of the classical equations of GR. One e®ect predicted via this method is the Hawking radiation of a black hole. A further indication for the incompleteness of the current physical picture is its high 1 2 CHAPTER 1. INTRODUCTION degree of arbitrariness. It has more than 20 free parameters whose values have to be determined by experiment and can not be extracted from the theory itself. As if this was not unsatisfactory enough it turns out that the parameters have to be balanced very accurately in order to lead to a universe containing galaxies, stars and biological life. The arguments supporting the specialty of the actual values of the parameters are nicely reviewed in [4]. If one is not content to blame this to a fortuitous choice of initial conditions one has to ¯nd a more fundamental theory which allows to tackle the question why the parameters have their actual values. It should also shed light on other characteristics of the SM which sofar remain unexplained, e.g. the particle spectrum, the gauge group and the fact that space-time appears to be four dimensional. Several ideas have been proposed to get a handle on the arbitrariness of the SM. The most prominent ones are grand uni¯cation and Kaluza-Klein theories. The idea of grand uni¯cation is the embedding of the SM gauge group into a simple gauge group like SU(5), SO(10) or E6. According to this scenario there is only one gauge group at some high energy scale, which can roughly be estimated to be of the order 16 mGUT »O(10 GeV). Through a generalized Higgs mechanism this gauge group is reduced to the SU(3) £ SU(2) £ U(1) of the SM at this scale. Besides the reduction of the number of gauge couplings grand uni¯ed theories achieve a prediction of the electroweak mixing angle and many models also establish a relation between the masses of the b quark and the ¿ lepton, see [5] for an introduction. Another desirable e®ect of grand uni¯cation is that in many models the gauge coupling of the uni¯ed gauge group is asymptotically free, thus removing the possible inconsistencies related to the non-asymptotically free electroweak gauge couplings [6]. It turns out that the idea of grand uni¯cation is not consistent with the particle spectrum of the SM. However there is an extension of the SM in which a uni¯cation of all three couplings at high energies is in fact possible. This is the supersymmetric SM which is reviewed in [7]. It has N = 1 supersymmetry because extended supersymmetries can not accommodate the chiral structure of the SM. As supersymmetry is not observed in nature it must be broken if it is to play any role at all. This leads to a mass split between the bosonic and fermionic partners of the order of the supersymmetry breaking scale. If this scale is roughly the same as the electroweak scale, i.e. mSUSY » 100 GeV, the supersymmetric partners of the SM particles could be heavy enough to escape detection in accelerator experiments so far. In addition such a low supersymmetry breaking scale could save the grand uni¯cation idea. Due to the additional light particles above mSUSY , the renormalization group equations indeed indicate a uni¯cation of all three couplings at a scale » 1016GeV within the minimal supersymmetric extension of the SM, see e.g. [5]. One should bear in mind however that the derivation of this result relies on the assumption that the supersymmetric SM remains a valid description of particle physics up to energies of order O(1016GeV).
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]
  • Arxiv:Hep-Th/0006117V1 16 Jun 2000 Oadmarolf Donald VERSION HYPER - Style
    Preprint typeset in JHEP style. - HYPER VERSION SUGP-00/6-1 hep-th/0006117 Chern-Simons terms and the Three Notions of Charge Donald Marolf Physics Department, Syracuse University, Syracuse, New York 13244 Abstract: In theories with Chern-Simons terms or modified Bianchi identities, it is useful to define three notions of either electric or magnetic charge associated with a given gauge field. A language for discussing these charges is introduced and the proper- ties of each charge are described. ‘Brane source charge’ is gauge invariant and localized but not conserved or quantized, ‘Maxwell charge’ is gauge invariant and conserved but not localized or quantized, while ‘Page charge’ conserved, localized, and quantized but not gauge invariant. This provides a further perspective on the issue of charge quanti- zation recently raised by Bachas, Douglas, and Schweigert. For the Proceedings of the E.S. Fradkin Memorial Conference. Keywords: Supergravity, p–branes, D-branes. arXiv:hep-th/0006117v1 16 Jun 2000 Contents 1. Introduction 1 2. Brane Source Charge and Brane-ending effects 3 3. Maxwell Charge and Asymptotic Conditions 6 4. Page Charge and Kaluza-Klein reduction 7 5. Discussion 9 1. Introduction One of the intriguing properties of supergravity theories is the presence of Abelian Chern-Simons terms and their duals, the modified Bianchi identities, in the dynamics of the gauge fields. Such cases have the unusual feature that the equations of motion for the gauge field are non-linear in the gauge fields even though the associated gauge groups are Abelian. For example, massless type IIA supergravity contains a relation of the form dF˜4 + F2 H3 =0, (1.1) ∧ where F˜4, F2,H3 are gauge invariant field strengths of rank 4, 2, 3 respectively.
    [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]
  • Introduction to Quantum Q-Langlands Correspondence
    Introduction to Quantum q-Langlands Correspondence Ming Zhang∗ May 16{19, 2017 Abstract Quantum q-Langlands Correspondence was introduced by Aganagic, Frenkel and Okounkov in their recent work [AFO17]. Roughly speaking, it is a correspondence between q-deformed conformal blocks of the quantum affine algebra and the deformed W-algebra. In this mini-course, I will define these objects and discuss some results in quantum K-theory of the Nakajima quiver varieties which is the key ingredient in the proof of the correspondence (in the simply-laced case). It will be helpful to know the basic concepts in the theory of Lie algebras (roots, weights, etc.). Contents 1 Introduction 2 1.1 The geometric Langlands correspondence.............................2 1.2 A representation-theoretic statement and its generalization in [AFO17].............2 2 Lie algebras 2 2.1 Root systems.............................................3 2.2 The Langlands dual.........................................4 3 Affine Kac{Moody algebras4 3.1 Via generators and relations.....................................5 3.2 Via extensions............................................6 4 Representations of affine Kac{Moody algebras7 4.1 The extended algebra........................................7 4.2 Representations of finite-dimensional Lie algebras.........................8 4.2.1 Weights............................................8 4.2.2 Verma modules........................................8 4.3 Representations of affine Kac{Moody algebras...........................9 4.3.1 The category of representations.............................9 4.3.2 Verma modulesO........................................ 10 4.3.3 The evaluation representation................................ 11 4.3.4 Level.............................................. 11 ∗Notes were taken by Takumi Murayama, who is responsible for any and all errors. Please e-mail [email protected] with any more corrections. Compiled on May 21, 2017. 1 1 Introduction We will follow Aganagic, Frenkel, and Okounkov's paper [AFO17], which came out in January of this year.
    [Show full text]
  • Stringy Instantons, Deformed Geometry and Axions
    Stringy instantons, deformed geometry and axions. Eduardo Garc´ıa-Valdecasas Tenreiro Instituto de F´ısica Teorica´ UAM/CSIC, Madrid Based on 1605.08092 by E.G. & A. Uranga and ongoing work Iberian Strings 2017, Lisbon, 16th January 2017 Axions and 3-Forms Backreacting Instantons Flavor Branes at the conifold Motivation & Outline Motivation Axions have very flat potentials ) Good for Inflation. Axions developing a potential, i.e. through instantons, can always be described as a 3-Form eating up the 2-Form dual to the axion. There is no candidate 3-Form for stringy instantons. We will look for it in the geometry deformed by the instanton. The instanton triggers a geometric transition. Outline 1 Axions and 3-Forms. 2 Backreacting Instantons. 3 Flavor branes at the conifold. Eduardo Garc´ıa-Valdecasas Tenreiro Stringy instantons, deformed geometry and axions. 2 / 22 2 / 22 Axions and 3-Forms Backreacting Instantons Flavor Branes at the conifold Axions and 3-Forms Eduardo Garc´ıa-Valdecasas Tenreiro Stringy instantons, deformed geometry and axions. 3 / 22 3 / 22 Axions and 3-Forms Backreacting Instantons Flavor Branes at the conifold Axions Axions are periodic scalar fields. This shift symmetry can be broken to a discrete shift symmetry by non-perturbative effects ! Very flat potential, good for inflation. However, discrete shift symmetry is not enough to keep the corrections to the potential small. There is a dual description where the naturalness of the flatness is more manifest. For example, a quadratic potential with monodromy can be described by a Kaloper-Sorbo lagrangian (Kaloper & Sorbo, 2009), 2 2 j dφj + nφF4 + jF4j (1) The potential is recovered using E.O.M, 2 V0 ∼ (nφ − q) ; q 2 Z (2) Eduardo Garc´ıa-ValdecasasSo there Tenreiro is a discreteStringy instantons, shift deformed symmetry, geometry and axions.
    [Show full text]
  • Contemporary Mathematics 442
    CONTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors http://dx.doi.org/10.1090/conm/442 Lie Algebras, Vertex Operator Algebras and Their Applications In honor of James Lepowsky and Robert Wilson on their sixtieth birthdays CoNTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 17810, 17837, 17850, 17865, 17867, 17868, 17869, 81T40, 82823. Photograph of James Lepowsky and Robert Wilson is courtesy of Yi-Zhi Huang. Library of Congress Cataloging-in-Publication Data Lie algebras, vertex operator algebras and their applications : an international conference in honor of James Lepowsky and Robert L. Wilson on their sixtieth birthdays, May 17-21, 2005, North Carolina State University, Raleigh, North Carolina / Yi-Zhi Huang, Kailash Misra, editors. p. em. ~(Contemporary mathematics, ISSN 0271-4132: v. 442) Includes bibliographical references. ISBN-13: 978-0-8218-3986-7 (alk. paper) ISBN-10: 0-8218-3986-1 (alk. paper) 1. Lie algebras~Congresses. 2. Vertex operator algebras. 3. Representations of algebras~ Congresses. I. Leposwky, J. (James). II. Wilson, Robert L., 1946- III. Huang, Yi-Zhi, 1959- IV.
    [Show full text]
  • Groups and Group Functors Attached to Kac-Moody Data
    GROUPS AND GROUP FUNCTORS ATTACHED TO KAC-MOODY DATA Jacques Tits CollAge de France 11PI Marcelin Berthelot 75231 Paris Cedex 05 1. The finite-dimenSional complex semi-simple Lie algebras. To start with, let us recall the classification, due to W. Killing and E. Cartan, of all complex semi-simple Lie algebras. (The presen- tation we adopt, for later purpose, is of course not that of those authors.) The isomorphism classes of such algebras are in one-to-one correspondence with the systems (1.1) H , (ei)1~iSZ , (hi) 1~i$ Z , where H is a finite-dimensional complex vector space (a Cartan subalgebra of a representative G of the isomorphism class in question), (~i) i~i$~ is a basis of the dual H* of H (a basis of the root system of @ relative to H ) and (hi) i$i~ ~ is a basis of H indexed by the same set {I ..... i} (h i is the coroot associated with di )' such that the matrix ~ = (Aij) = (~j(hi)) is a Cartan matrix, which means that the following conditions are satisfied: the A.. are integers ; (C1) 1] (C2) A.. = 2 or ~ 0 according as i = or ~ j ; 13 (C3) A. ~ = 0 if and only if A.. = 0 ; 13 31 (C4) is the product of a positive definite symmetric matrix and a diagonal matrix (by abuse of language, we shall simply say that ~ is positive definite). More correctly: two such data correspond to the same isomorphism class of algebras if and only if they differ only by a permutation of the indices I,...,~ .
    [Show full text]
  • Lectures on Conformal Field Theory and Kac-Moody Algebras
    hep-th/9702194 February 1997 LECTURES ON CONFORMAL FIELD THEORY AND KAC-MOODY ALGEBRAS J¨urgen Fuchs X DESY Notkestraße 85 D – 22603 Hamburg Abstract. This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras. These lectures were held at the Graduate Course on Conformal Field Theory and Integrable Models (Budapest, August 1996). They will appear in a volume of the Springer Lecture Notes in Physics edited by Z. Horvath and L. Palla. —————— X Heisenberg fellow 1 Contents Lecture 1 : Conformal Field Theory 3 1 Conformal Quantum Field Theory 3 2 Observables: The Chiral Symmetry Algebra 4 3 Physical States: Highest Weight Modules 7 4 Sectors: The Spectrum 9 5 Conformal Fields 11 6 The Operator Product Algebra 14 7 Correlation Functions and Chiral Blocks 16 Lecture 2 : Fusion Rules, Duality and Modular Transformations 19 8 Fusion Rules 19 9 Duality 21 10 Counting States: Characters 23 11 Modularity 24 12 Free Bosons 26 13 Simple Currents 28 14 Operator Product Algebra from Fusion Rules 30 Lecture 3 : Kac--Moody Algebras 32 15 Cartan Matrices 32 16 Symmetrizable Kac-Moody Algebras 35 17 Affine Lie Algebras as Centrally Extended Loop Algebras 36 18 The Triangular Decomposition of Affine Lie Algebras 38 19 Representation Theory 39 20 Characters 40 Lecture 4 : WZW Theories and Coset Theories 42 21 WZW Theories 42 22 WZW Primaries 43 23 Modularity, Fusion Rules and WZW Simple Currents 44 24 The Knizhnik-Zamolodchikov Equation 46 25 Coset Conformal Field Theories 47 26 Field Identification 48 27 Fixed Points 49 28 Omissions 51 29 Outlook 52 30 Glossary 53 References 55 2 Lecture 1 : Conformal Field Theory 1 Conformal Quantum Field Theory Over the years, quantum field theory has enjoyed a great number of successes.
    [Show full text]
  • Construction and Classification of Holomorphic Vertex Operator Algebras
    Construction and classification of holomorphic vertex operator algebras Jethro van Ekeren1, Sven Möller2, Nils R. Scheithauer2 1Universidade Federal Fluminense, Niterói, RJ, Brazil 2Technische Universität Darmstadt, Darmstadt, Germany We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens’ classification of V1-structures of meromorphic confor- mal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds. 1 Introduction Vertex algebras give a mathematically rigorous description of 2-dimensional quantum field the- ories. They were introduced into mathematics by R. E. Borcherds [B1]. The most prominent example is the moonshine module V \ constructed by Frenkel, Lepowsky and Meurman [FLM]. This vertex algebra is Z-graded and carries an action of the largest sporadic simple group, the monster. Borcherds showed that the corresponding twisted traces are hauptmoduls for genus 0 groups [B2]. The moonshine module was the first example of an orbifold in the theory of vertex algebras. The (−1)-involution of the Leech lattice Λ lifts to an involution g of the associated lattice vertex g algebra VΛ. The fixed points VΛ of VΛ under g form a simple subalgebra of VΛ. Let VΛ(g) be the unique irreducible g-twisted module of VΛ and VΛ(g)Z the subspace of vectors with integral g g L0-weight. Then the vertex algebra structure on VΛ can be extended to VΛ ⊕ VΛ(g)Z. This sum \ g is the moonshine module V .
    [Show full text]
  • Exact Thresholds and Instanton Effects in D = 3 String Theories
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Preprint typeset in JHEP style. - HYPER VERSION hep-th/0006088 NBI-HE-00-29 NORDITA-2000/54 HE HUTP-00/A022 LPTHE-00-22 Exact thresholds and instanton effects in D =3string theories Niels A. Obers∗ Nordita and Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen, DENMARK E-mail: [email protected] Boris Pioline† Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138, USA E-mail: [email protected] Abstract: Three-dimensional string theories with 16 supersymmetries are believed to possess a non-perturbative U-duality symmetry SO(8; 24; Z). By covariantizing the heterotic one-loop amplitude under U-duality, we propose an exact expression for the (∂φ)4 amplitude, that reproduces known perturbative limits. The weak-coupling ex- pansion in either of the heterotic, type II or type I descriptions exhibits the well-known instanton effects, plus new contributions peculiar to three-dimensional theories, includ- ing the Kaluza-Klein 5-brane, for which we extract the summation measure. This letter is a post-scriptum to hep-th/0001083. Keywords: Nonperturbative Effects, String Duality, p-branes. ∗Work supported in part by TMR network ERBFMRX-CT96-0045. yOn leave of absence from LPTHE, Universit´e Pierre et Marie Curie, PARIS VI and Universit´e Denis Diderot, PARIS VII, Bo^ıte 126, Tour 16, 1er ´etage, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE Contents 1. Introduction In a recent article [1], we have submitted the conjectured heterotic-type II duality to its most detailed test to date, by showing the identity of the heterotic one-loop–exact four–gauge-boson F 4 threshold in six dimensions with the tree-level amplitude in type IIA compactified on K3.
    [Show full text]
  • Supersymmetry: Lecture 2: the Supersymmetrized Standard Model
    Supersymmetry: Lecture 2: The Supersymmetrized Standard Model Yael Shadmi Technion June 2014 Yael Shadmi (Technion) ESHEP June 2014 1 / 91 Part I: The Supersymmetrized SM: motivation and structure before 2012, all fundamental particles we knew had spin 1 or spin 1/2 but we now have the Higgs: it's spin 0 of course spin-0 is the simplest possibility spin-1 is intuitive too (we all understand vectors) the world and (QM courses) would have been very different if the particle we know best, the electron, were spin-0 supersymmetry: boson $ fermion so from a purely theoretical standpoint, supersymmetry would provide an explanation for why we have particles of different spins Yael Shadmi (Technion) ESHEP June 2014 2 / 91 The Higgs and fine tuning: because the Higgs is spin-0, its mass is quadratically divergent 2 2 δm / ΛUV (1) unlike fermions (protected by chiral symmetry) gauge bosons (protected by gauge symmetry) Yael Shadmi (Technion) ESHEP June 2014 3 / 91 Higgs Yukawa coupling: quark (top) contribution also, gauge boson, Higgs loops Yael Shadmi (Technion) ESHEP June 2014 4 / 91 practically: we don't care (can calculate anything in QFT, just put in a counter term) theoretically: believe ΛUV is a concrete physical scale, eg: mass of new fields, scale of new strong interactions then 2 2 2 m (µ) = m (ΛUV ) + # ΛUV (2) 2 m (ΛUV ) determined by the full UV theory # determined by SM we know LHS: m2 ∼ 1002 GeV2 18 if ΛUV = 10 GeV 2 36 2 we need m (ΛUV ) ∼ 10 GeV and the 2 terms on RHS tuned to 32 orders of magnitude.
    [Show full text]
  • All Flat Manifolds Are Cusps of Hyperbolic Orbifolds 1 Introduction
    ISSN 1472-2739 (on-line) 1472-2747 (printed) 285 Algebraic & Geometric Topology Volume 2 (2002) 285–296 ATG Published: 18 April 2002 All flat manifolds are cusps of hyperbolic orbifolds D.D. Long A.W. Reid Abstract We show that all closed flat n-manifolds are diffeomorphic to a cusp cross- section in a finite volume hyperbolic n + 1-orbifold. AMS Classification 57M50; 57R99 Keywords Flat manifolds, hyperbolic orbifold, cusp cross-sections 1 Introduction By a flat n-manifold (resp. flat n-orbifold) we mean a manifold (resp. orbifold) En/Γ where En is Euclidean n-space and Γ a discrete, cocompact torsion-free subgroup of Isom(En) (resp. Γ has elements of finite order). In [7], Ham- rick and Royster resolved a longstanding conjecture by showing that every flat n-manifold bounds an (n + 1)-dimensional manifold, in the sense that each diffeomorphism class has a representative that bounds. Flat manifolds and orbifolds are connected with hyperbolic orbifolds via the structure of the cusp ends of finite volume hyperbolic orbifolds—if M n+1 is a non-compact finite volume hyperbolic orbifold, then by a standard analysis of the thin parts, a cusp cross-section is a flat n-orbifold (see below and [14]). In the early eighties, motivated by this and work of Gromov [6], Farrell and Zdravkovska [5] posed the following geometric version: Question Does the diffeomorphism class of every flat n-manifold have a rep- resentative W which arises as the cusp cross-section of a finite volume 1-cusped hyperbolic (n + 1)-manifold? We note that it makes sense to ask only for bounding up to diffeomorphism type, since it is a well-known consequence of Mostow Rigidity that there are algebraic restrictions on the isometry type of the flat n-manifolds that can arise as cusp cross sections of finite volume hyperbolic (n + 1)-manifolds.
    [Show full text]