BUSSTEPP LECTURES on STRING THEORY Contents
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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. -
Off-Shell Interactions for Closed-String Tachyons
Preprint typeset in JHEP style - PAPER VERSION hep-th/0403238 KIAS-P04017 SLAC-PUB-10384 SU-ITP-04-11 TIFR-04-04 Off-Shell Interactions for Closed-String Tachyons Atish Dabholkarb,c,d, Ashik Iqubald and Joris Raeymaekersa aSchool of Physics, Korea Institute for Advanced Study, 207-43, Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, Korea bStanford Linear Accelerator Center, Stanford University, Stanford, CA 94025, USA cInstitute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA dDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail:[email protected], [email protected], [email protected] Abstract: Off-shell interactions for localized closed-string tachyons in C/ZN super- string backgrounds are analyzed and a conjecture for the effective height of the tachyon potential is elaborated. At large N, some of the relevant tachyons are nearly massless and their interactions can be deduced from the S-matrix. The cubic interactions be- tween these tachyons and the massless fields are computed in a closed form using orbifold CFT techniques. The cubic interaction between nearly-massless tachyons with different charges is shown to vanish and thus condensation of one tachyon does not source the others. It is shown that to leading order in N, the quartic contact in- teraction vanishes and the massless exchanges completely account for the four point scattering amplitude. This indicates that it is necessary to go beyond quartic inter- actions or to include other fields to test the conjecture for the height of the tachyon potential. Keywords: closed-string tachyons, orbifolds. -
UV Behavior of Half-Maximal Supergravity Theories
UV behavior of half-maximal supergravity theories. Piotr Tourkine, Quantum Gravity in Paris 2013, LPT Orsay In collaboration with Pierre Vanhove, based on 1202.3692, 1208.1255 Understand the pertubative structure of supergravity theories. ● Supergravities are theories of gravity with local supersymmetry. ● Those theories naturally arise in the low energy limit of superstring theory. ● String theory is then a UV completion for those and thus provides a good framework to study their UV behavior. → Maximal and half-maximal supergravities. Maximal supergravity ● Maximally extended supergravity: – Low energy limit of type IIA/B theory, – 32 real supercharges, unique (ungauged) – N=8 in d=4 ● Long standing problem to determine if maximal supergravity can be a consistent theory of quantum gravity in d=4. ● Current consensus on the subject : it is not UV finite, the first divergence could occur at the 7-loop order. ● Impressive progresses made during last 5 years in the field of scattering amplitudes computations. [Bern, Carrasco, Dixon, Dunbar, Johansson, Kosower, Perelstein, Rozowsky etc.] Half-maximal supergravity ● Half-maximal supergravity: – Heterotic string, but also type II strings on orbifolds – 16 real supercharges, – N=4 in d=4 ● Richer structure, and still a lot of SUSY so explicit computations are still possible. ● There are UV divergences in d=4, [Fischler 1979] at one loop for external matter states ● UV divergence in gravity amplitudes ? As we will see the divergence is expected to arise at the four loop order. String models that give half-maximal supergravity “(4,0)” susy “(0,4)” susy ● Type IIA/B string = Superstring ⊗ Superstring Torus compactification : preserves full (4,4) supersymmetry. -
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. -
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. -
Pitp Lectures
MIFPA-10-34 PiTP Lectures Katrin Becker1 Department of Physics, Texas A&M University, College Station, TX 77843, USA [email protected] Contents 1 Introduction 2 2 String duality 3 2.1 T-duality and closed bosonic strings .................... 3 2.2 T-duality and open strings ......................... 4 2.3 Buscher rules ................................ 5 3 Low-energy effective actions 5 3.1 Type II theories ............................... 5 3.1.1 Massless bosons ........................... 6 3.1.2 Charges of D-branes ........................ 7 3.1.3 T-duality for type II theories .................... 7 3.1.4 Low-energy effective actions .................... 8 3.2 M-theory ................................... 8 3.2.1 2-derivative action ......................... 8 3.2.2 8-derivative action ......................... 9 3.3 Type IIB and F-theory ........................... 9 3.4 Type I .................................... 13 3.5 SO(32) heterotic string ........................... 13 4 Compactification and moduli 14 4.1 The torus .................................. 14 4.2 Calabi-Yau 3-folds ............................. 16 5 M-theory compactified on Calabi-Yau 4-folds 17 5.1 The supersymmetric flux background ................... 18 5.2 The warp factor ............................... 18 5.3 SUSY breaking solutions .......................... 19 1 These are two lectures dealing with supersymmetry (SUSY) for branes and strings. These lectures are mainly based on ref. [1] which the reader should consult for original references and additional discussions. 1 Introduction To make contact between superstring theory and the real world we have to understand the vacua of the theory. Of particular interest for vacuum construction are, on the one hand, D-branes. These are hyper-planes on which open strings can end. On the world-volume of coincident D-branes, non-abelian gauge fields can exist. -
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. -
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. -
H ( Ff^L°9 . INTERNATIONAL CENTRE for \ | THEORETICAL PHYSICS
IC/87/420 H ( ff^l°9 . INTERNATIONAL CENTRE FOR I 'V. ': \| THEORETICAL PHYSICS HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION OF THE GREEN-SCHWARZ SUPERSTRING E. Nissimov S. Pacheva and INTERNATIONAL ATOMIC ENERGY S. Solomon AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION T T IC/87/420 ABSTRACT International Atomic Energy Agency Based on our recent work, we describe a new generalized model of space- and United Nations Educational Scientific and Cultural Organization time supersymmetric strings - the harmonic superstring. It has the same phys- ical content as the original Green-Schwarz (GS) superstring but it is defined on INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS an enlarged superspace: the D=10 harmonic superspace and incorporates addi- tional fennionic string coordinates. The main feature of the harmonic superstring model is that it contains covariant and irreducible first-class constraints only. This allows for a straight- HARMONIC SUPERSTRING AND COVARIANT QUANTIZATION forward manifestly super-Poincaie'covariant Batalin-Fradkin-Vilkoviski (BFV) OF THE GREEN-SCHWARZ SUPERSTRING * - Becchi-Rouet-Stora-Tyutin (BRST) quantization. The BRST charge has a re- markably simple form and the corresponding BFV hamiltonian exhibits manifest Parisi-Sourlas O5p(l,l|2) symmetry. E. Nissinsov ** In a particular Lorentz-covariant gauge we find the covariant vertex opera- International Centre for Theoretical Physics, Trieste, Italy, tors for the emission of the massless states which closely resemble the covariant "classical" expressions previously guessed by Green and Schwarz. S. Pacheva Institute of Nuclear Research and Nuclear Energy, Boul. Lenin 72, 1784 Sofia, Bulgaria and S. Solomon Physics Department, Weizmann Institute, Hehovot 76100, Israel. MIRAMARE - TRIESTE December 1987 •' To appear in "Perspectives In String Theories" (Proceedings of the Copenhagen Workshop, Denmark, October 1987). -
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. -
The Quantized Fermionic String
Class 2: The quantized fermionic string Canonical quantization Light-cone quantization Spectrum of the fermionic string, GSO projection . The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 . One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring.