Cosmological Models from String Theory Setups
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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. -
Symmetries of String Theory and Early Universe Cosmology
String Cosmology R. Branden- berger Introduction Symmetries of String Theory and Early T-Duality: Key Symmetry of Universe Cosmology String Theory String Gas Cosmology Structure Robert Brandenberger Perturbations Overview Physics Department, McGill University Analysis DFT Conclusions Yukawa Institute, Kyoto University, Feb. 26, 2018 1 / 61 Outline String Cosmology 1 Introduction R. Branden- berger 2 T-Duality: Key Symmetry of String Theory Introduction T-Duality: Key 3 String Gas Cosmology Symmetry of String Theory String Gas 4 String Gas Cosmology and Structure Formation Cosmology Review of the Theory of Cosmological Perturbations Structure Perturbations Overview Overview Analysis Analysis DFT 5 Double Field Theory as a Background for String Gas Conclusions Cosmology 6 Conclusions 2 / 61 Plan String Cosmology 1 Introduction R. Branden- berger 2 T-Duality: Key Symmetry of String Theory Introduction T-Duality: Key 3 String Gas Cosmology Symmetry of String Theory String Gas 4 String Gas Cosmology and Structure Formation Cosmology Review of the Theory of Cosmological Perturbations Structure Perturbations Overview Overview Analysis Analysis DFT 5 Double Field Theory as a Background for String Gas Conclusions Cosmology 6 Conclusions 3 / 61 Inflation: the Standard Model of Early Universe Cosmology String Cosmology R. Branden- berger Introduction Inflation is the standard paradigm of early universe T-Duality: Key cosmology. Symmetry of String Theory Inflation solves conceptual problems of Standard Big String Gas Bang Cosmology. Cosmology Structure Inflation predicts an almost scale-invariant spectrum of Perturbations Overview primordial cosmological perturbations with a small red Analysis tilt (Chibisov & Mukhanov, 1981). DFT Conclusions Fluctuations are nearly Gaussian and nearly adiabatic. 4 / 61 Map of the Cosmic Microwave Background (CMB) String Cosmology R. -
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. -
Lectures on D-Branes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server CPHT/CL-615-0698 hep-th/9806199 Lectures on D-branes Constantin P. Bachas1 Centre de Physique Th´eorique, Ecole Polytechnique 91128 Palaiseau, FRANCE [email protected] ABSTRACT This is an introduction to the physics of D-branes. Topics cov- ered include Polchinski’s original calculation, a critical assessment of some duality checks, D-brane scattering, and effective worldvol- ume actions. Based on lectures given in 1997 at the Isaac Newton Institute, Cambridge, at the Trieste Spring School on String The- ory, and at the 31rst International Symposium Ahrenshoop in Buckow. June 1998 1Address after Sept. 1: Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris, FRANCE, email : [email protected] Lectures on D-branes Constantin Bachas 1 Foreword Referring in his ‘Republic’ to stereography – the study of solid forms – Plato was saying : ... for even now, neglected and curtailed as it is, not only by the many but even by professed students, who can suggest no use for it, never- theless in the face of all these obstacles it makes progress on account of its elegance, and it would not be astonishing if it were unravelled. 2 Two and a half millenia later, much of this could have been said for string theory. The subject has progressed over the years by leaps and bounds, despite periods of neglect and (understandable) criticism for lack of direct experimental in- put. To be sure, the construction and key ingredients of the theory – gravity, gauge invariance, chirality – have a firm empirical basis, yet what has often catalyzed progress is the power and elegance of the underlying ideas, which look (at least a posteriori) inevitable. -
Gabriele Veneziano
Strings 2021, June 21 Discussion Session High-Energy Limit of String Theory Gabriele Veneziano Very early days (1967…) • Birth of string theory very much based on the high-energy (Regge) limit: Im A(Regge) ≠ 0! •Duality bootstrap (except for the Pomeron!) => DRM (emphasis shifted on res/res duality, xing) •Need for infinite number of resonances of unlimited mass and spin (linear traj.s => strings?) • Exponential suppression @ high E & fixed θ => colliding objects are extended, soft (=> strings?) •One reason why the NG string lost to QCD. •Q: What’s the Regge limit of (large-N) QCD? 20 years later (1987…) •After 1984, attention in string theory shifted from hadronic physics to Q-gravity. • Thought experiments conceived and efforts made to construct an S-matrix for gravitational scattering @ E >> MP, b < R => “large”-BH formation (RD-3 ~ GE). Questions: • Is quantum information preserved, and how? • What’s the form and role of the short-distance stringy modifications? •N.B. Computations made in flat spacetime: an emergent effective geometry. What about AdS? High-energy vs short-distance Not the same even in QFT ! With gravity it’s even more the case! High-energy, large-distance (b >> R, ls) •Typical grav.al defl. angle is θ ~ R/b = GE/b (D=4) •Scattering at high energy & fixed small θ probes b ~ R/θ > R & growing with energy! •Contradiction w/ exchange of huge Q = θ E? No! • Large classical Q due to exchange of O(Gs/h) soft (q ~ h/b) gravitons: t-channel “fractionation” • Much used in amplitude approach to BH binaries • θ known since ACV90 up to O((R/b)3), universal. -
String-Inspired Running Vacuum—The ``Vacuumon''—And the Swampland Criteria
universe Article String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria Nick E. Mavromatos 1 , Joan Solà Peracaula 2,* and Spyros Basilakos 3,4 1 Theoretical Particle Physics and Cosmology Group, Physics Department, King’s College London, Strand, London WC2R 2LS, UK; [email protected] 2 Departament de Física Quàntica i Astrofísica, and Institute of Cosmos Sciences (ICCUB), Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain 3 Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efessiou 4, 11527 Athens, Greece; [email protected] 4 National Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece * Correspondence: [email protected] Received: 15 October 2020; Accepted: 17 November 2020; Published: 20 November 2020 Abstract: We elaborate further on the compatibility of the “vacuumon potential” that characterises the inflationary phase of the running vacuum model (RVM) with the swampland criteria. The work is motivated by the fact that, as demonstrated recently by the authors, the RVM framework can be derived as an effective gravitational field theory stemming from underlying microscopic (critical) string theory models with gravitational anomalies, involving condensation of primordial gravitational waves. Although believed to be a classical scalar field description, not representing a fully fledged quantum field, we show here that the vacuumon potential satisfies certain swampland criteria for the relevant regime of parameters and field range. We link the criteria to the Gibbons–Hawking entropy that has been argued to characterise the RVM during the de Sitter phase. These results imply that the vacuumon may, after all, admit under certain conditions, a rôle as a quantum field during the inflationary (almost de Sitter) phase of the running vacuum. -
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. -
String Theory and Pre-Big Bang Cosmology
IL NUOVO CIMENTO 38 C (2015) 160 DOI 10.1393/ncc/i2015-15160-8 Colloquia: VILASIFEST String theory and pre-big bang cosmology M. Gasperini(1)andG. Veneziano(2) (1) Dipartimento di Fisica, Universit`a di Bari - Via G. Amendola 173, 70126 Bari, Italy and INFN, Sezione di Bari - Bari, Italy (2) CERN, Theory Unit, Physics Department - CH-1211 Geneva 23, Switzerland and Coll`ege de France - 11 Place M. Berthelot, 75005 Paris, France received 11 January 2016 Summary. — In string theory, the traditional picture of a Universe that emerges from the inflation of a very small and highly curved space-time patch is a possibility, not a necessity: quite different initial conditions are possible, and not necessarily unlikely. In particular, the duality symmetries of string theory suggest scenarios in which the Universe starts inflating from an initial state characterized by very small curvature and interactions. Such a state, being gravitationally unstable, will evolve towards higher curvature and coupling, until string-size effects and loop corrections make the Universe “bounce” into a standard, decreasing-curvature regime. In such a context, the hot big bang of conventional cosmology is replaced by a “hot big bounce” in which the bouncing and heating mechanisms originate from the quan- tum production of particles in the high-curvature, large-coupling pre-bounce phase. Here we briefly summarize the main features of this inflationary scenario, proposed a quarter century ago. In its simplest version (where it represents an alternative and not a complement to standard slow-roll inflation) it can produce a viable spectrum of density perturbations, together with a tensor component characterized by a “blue” spectral index with a peak in the GHz frequency range. -
Extreme Gravity and Fundamental Physics
Extreme Gravity and Fundamental Physics Astro2020 Science White Paper EXTREME GRAVITY AND FUNDAMENTAL PHYSICS Thematic Areas: • Cosmology and Fundamental Physics • Multi-Messenger Astronomy and Astrophysics Principal Author: Name: B.S. Sathyaprakash Institution: The Pennsylvania State University Email: [email protected] Phone: +1-814-865-3062 Lead Co-authors: Alessandra Buonanno (Max Planck Institute for Gravitational Physics, Potsdam and University of Maryland), Luis Lehner (Perimeter Institute), Chris Van Den Broeck (NIKHEF) P. Ajith (International Centre for Theoretical Sciences), Archisman Ghosh (NIKHEF), Katerina Chatziioannou (Flatiron Institute), Paolo Pani (Sapienza University of Rome), Michael Pürrer (Max Planck Institute for Gravitational Physics, Potsdam), Thomas Sotiriou (The University of Notting- ham), Salvatore Vitale (MIT), Nicolas Yunes (Montana State University), K.G. Arun (Chennai Mathematical Institute), Enrico Barausse (Institut d’Astrophysique de Paris), Masha Baryakhtar (Perimeter Institute), Richard Brito (Max Planck Institute for Gravitational Physics, Potsdam), Andrea Maselli (Sapienza University of Rome), Tim Dietrich (NIKHEF), William East (Perimeter Institute), Ian Harry (Max Planck Institute for Gravitational Physics, Potsdam and University of Portsmouth), Tanja Hinderer (University of Amsterdam), Geraint Pratten (University of Balearic Islands and University of Birmingham), Lijing Shao (Kavli Institute for Astronomy and Astro- physics, Peking University), Maaretn van de Meent (Max Planck Institute for Gravitational -
8. Compactification and T-Duality
8. Compactification and T-Duality In this section, we will consider the simplest compactification of the bosonic string: a background spacetime of the form R1,24 S1 (8.1) ⇥ The circle is taken to have radius R, so that the coordinate on S1 has periodicity X25 X25 +2⇡R ⌘ We will initially be interested in the physics at length scales R where motion on the S1 can be ignored. Our goal is to understand what physics looks like to an observer living in the non-compact R1,24 Minkowski space. This general idea goes by the name of Kaluza-Klein compactification. We will view this compactification in two ways: firstly from the perspective of the spacetime low-energy e↵ective action and secondly from the perspective of the string worldsheet. 8.1 The View from Spacetime Let’s start with the low-energy e↵ective action. Looking at length scales R means that we will take all fields to be independent of X25: they are instead functions only on the non-compact R1,24. Consider the metric in Einstein frame. This decomposes into three di↵erent fields 1,24 on R :ametricG˜µ⌫,avectorAµ and a scalar σ which we package into the D =26 dimensional metric as 2 µ ⌫ 2σ 25 µ 2 ds = G˜µ⌫ dX dX + e dX + Aµ dX (8.2) Here all the indices run over the non-compact directions µ, ⌫ =0 ,...24 only. The vector field Aµ is an honest gauge field, with the gauge symmetry descend- ing from di↵eomorphisms in D =26dimensions.Toseethisrecallthatunderthe transformation δXµ = V µ(X), the metric transforms as δG = ⇤ + ⇤ µ⌫ rµ ⌫ r⌫ µ This means that di↵eomorphisms of the compact direction, δX25 =⇤(Xµ), turn into gauge transformations of Aµ, δAµ = @µ⇤ –197– We’d like to know how the fields Gµ⌫, Aµ and σ interact. -
String Phenomenology
Introduction Model Building Moduli Stabilisation Flavour Problem String Phenomenology Joseph Conlon (Oxford University) EPS HEP Conference, Krakow July 17, 2009 Joseph Conlon (Oxford University) String Phenomenology Introduction Model Building Moduli Stabilisation Flavour Problem Chalk and Cheese? Figure: String theory and phenomenology? Joseph Conlon (Oxford University) String Phenomenology Introduction Model Building Moduli Stabilisation Flavour Problem String Theory ◮ String theory is where one is led by studying quantised relativistic strings. ◮ It encompasses lots of areas ( black holes, quantum field theory, quantum gravity, mathematics, particle physics....) and is studied by lots of different people. ◮ This talk is on string phenomenology - the part of string theory that aims at connecting to the Standard Model and its extensions. ◮ It aims to provide a (brief) overview of this area. (cf Angel Uranga’s plenary talk) Joseph Conlon (Oxford University) String Phenomenology Introduction Model Building Moduli Stabilisation Flavour Problem String Theory ◮ For technical reasons string theory is consistent in ten dimensions. ◮ Six dimensions must be compactified. ◮ Ten = (Four) + (Six) ◮ Spacetime = (M4) + Calabi-Yau Space ◮ All scales, matter, particle spectra and couplings come from the geometry of the extra dimensions. ◮ All scales, matter, particle spectra and couplings come from the geometry of the extra dimensions. Joseph Conlon (Oxford University) String Phenomenology Introduction Model Building Moduli Stabilisation Flavour Problem Model Building Various approaches in string theory to realising Standard Model-like spectra: ◮ E8 E8 heterotic string with gauge bundles × ◮ Type I string with gauge bundles ◮ IIA/IIB D-brane constructions ◮ Heterotic M-Theory ◮ M-Theory on G2 manifolds Joseph Conlon (Oxford University) String Phenomenology Introduction Model Building Moduli Stabilisation Flavour Problem Heterotic String Start with an E8 vis E8 hid gauge group in ten dimensions. -
Phenomenological Aspects of Type IIB Flux Compactifications
Phenomenological Aspects of Type IIB Flux Compactifications Enrico Pajer Munchen¨ 2008 Phenomenological Aspects of Type IIB Flux Compactifications Enrico Pajer Dissertation an der Fakult¨at fur¨ Physik der Ludwig–Maximilians–Universit¨at Munchen¨ vorgelegt von Enrico Pajer aus Venedig, Italien Munchen,¨ den 01. Juni 2008 Erstgutachter: Dr. Michael Haack Zweitgutachter: Prof. Dr. Dieter Lust¨ Tag der mundlichen¨ Prufung:¨ 18.07.2008 Contents 1 Introduction and conclusions 1 2 Type IIB flux compactifications 9 2.1 Superstring theory in a nutshell . 10 2.2 How many string theories are there? . 13 2.3 Type IIB . 15 2.4 D-branes . 17 2.5 Flux compactifications . 18 2.5.1 The GKP setup . 19 2.6 Towards de Sitter vacua in string theory . 21 3 Inflation in string theory 27 3.1 Observations . 27 3.2 Inflation . 29 3.2.1 Slow-roll models . 32 3.3 String theory models of inflation . 35 3.3.1 Open string models . 35 3.3.2 Closed string models . 36 3.4 Discussion . 37 4 Radial brane inflation 39 4.1 Preliminaries . 40 4.2 The superpotential . 42 4.3 Warped Brane inflation . 44 4.3.1 The η-problem from volume stabilization . 44 4.3.2 F-term potential for the conifold . 46 4.4 Critical points of the potential . 47 4.4.1 Axion stabilization . 48 4.4.2 Volume stabilization . 48 4.4.3 Angular moduli stabilization . 50 4.4.4 Potential with fixed moduli . 51 4.5 Explicit examples: Ouyang vs Kuperstein embedding . 52 4.5.1 Ouyang embedding . 52 4.5.2 Kuperstein embedding .