DOCSLIB.ORG

Supergravity in D = 11 and D = 10 1 P-Forms 2 Spinors in D Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Supergravity in D = 11 and D = 10 1 P-Forms 2 Spinors in D Dimensions Supergravity in d = 11 and d = 10 The first equality shows that the integral does not depend on the metric. The second one Felix Br¨ummer keeps general covariance manifest (here the metric and Hodge ∗ are those of M, so ∗Ω is a scalar). 1 p-forms A familiar example where p-form notation can sometimes be useful is electromagnetism in d = 4. These are geometric animals which will frequently There is a 1-form A with components Aµ (the gauge appear, so we start by giving a (somewhat sloppy) potential), which is locally determined only up to overview of p-form notation. A p-form in d ≥ p a U(1) gauge transformation dimensions is an antisymmetric tensor field with p indices. In particular, 0-forms are scalars and 1- A → A + d Λ (6) forms are vectors. All operations on a p-form Ω with Λ any 0-form. It gives rise to a 2-form F = can be defined in terms of its components Ω µ1...µp d A with components F (the field strength). F is in some local coordinates. What we need is µν invariant under gauge transformations of Eq. (6) by Eq. (2). The kinetic action in some 4d space-time • Addition: Two p-forms are added by adding S is their components. 1 Z 1 Z √ • Exterior differentiation: The d operator turns S = − F ∧∗F = − d4x −g F F µν . (7) 2 4 µν a p-form into a (p + 1)-form. It acts as S One may couple an external particle of charge q (d Ω) = (p + 1) ∂ Ω . (1) νµ1...µp [ν µ1...µp] (an “electron”) to the gauge field. This is done Here (as always) [ ] denotes weighted antisym- by integrating the 1-form A over the particle’s 1- metrization. Note that dimensional worldline C ⊂ S: 1 Z Z d d (anything) = 0 by symmetry . (2) S = − F ∧ ∗F + q A + ... (8) 2 S C • The wedge product: Given a p-form Ω and a Alternatively one could group all charged particles q-form Ω0, one may form a (p + q)-form Ω ∧ Ω0 into a current density 1-form J: whose components are 1 Z Z 0 (p + q)! 0 S = − F ∧ ∗F + A ∧ ∗J + ... (9) (Ω ∧ Ω )µ1...µp ν1...νq = Ω[µ ...µp Ω . 2 p! q! 1 ν1...νq] S S (3) In this formulation Maxwell’s equations are very • Hodge dualization: The ∗ operator turns a p- short: form into a (d−p)-form in d-dimensional space: d F = 0, d ∗F = ∗J. (10) √ −g The homogeneous equation, or “Bianchi identity” (∗Ω) = ν1...νp Ω . µ1...µd−p p! µ1...µd−p ν1...νp is as usual a direct consequence of working with a (4) gauge potential. In p-form language it follows from Here g = det(gµν), and the symbol with all F = d A and Eq. (2). lower indices is defined to be ±1 or 0 as usual, with indices raised by the inverse metric gµν. 2 Spinors in d dimensions • Integration: A p-form can be integrated over a p-dimensional manifold. In local coordinates, The d-dimensional Clifford algebra can be defined Z Z p in analogy to the four-dimensional case. It is repre- Ω = d x Ω0 1 ··· (p−1) 0 d−1 M sented by d Dirac matrices Γ ,..., Γ such that Z (5) p √ = d x −g (∗Ω) . {Γµ, Γν} = 2 ηµν 1 . (11) 1 A representation of the Lorentz algebra so(d−1, 1) It turns out that one may consistently impose is then furnished by this condition on Dirac spinors if d ≡ 0, 1, 2, 3, 4 mod 8. One may impose it on Weyl spinors only if i Σµν = − [Γµ, Γν] . (12) d ≡ 2 mod 8. 4 From these considerations we obtain the num- In d = 2, a possible choice of Dirac matrices is ber of real degrees of freedom f in the minimal spinor representation in d dimensions: 0 1 0 1 Γ0 = , Γ1 = . (13) d 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 0 f 1 2 4 8 8 16 16 16 16 32 64 From this one can iteratively construct the Γµ for even d: 3 Why d = 11 is special −1 0 Γµ = Γeµ ⊗ , (µ = 0, . , d − 3), 0 1 Using the above table we can deduce that the 0 1 0 −i maximal dimension for supersymmetry is d = 11. Γd−2 = 1 ⊗ , Γd−1 = 1 ⊗ , 1 0 i 0 The argument goes as follows: For arbitrary d we d−4 (14) can always compactify on a torus T to d = 4, which preserves all supersymmetries. Supersym- where Γeµ are the Dirac matrices in d−2 dimensions, metry transformations are generated by spinorial and 1 is the 2d/2−1 × 2d/2−1 unit matrix. The di- supercharges. A d = 4 supercharge has f = 4 de- mension of the Dirac representation is thus 2d/2, grees of freedom; a d = 11 supercharge has f = 8·4 and there are 2d/2+1 real degrees of freedom con- degrees of freedom, i.e. it corresponds to 8 distinct A tained in a 2d/2-dimensional complex Dirac spinor. supercharges {Q }, or N = 8 SUSY, after torus compactification to d = 4. A hypothetical d ≥ 12 5 The analogue of γ in d = 4 is supercharge would correspond to at least N = 16 SUSY in d = 4. We will show that such a large Γ = id/2−1Γ0Γ1 ··· Γd−1 . (15) number of d = 4 supercharges would lead to 4d Γ has the properties fields with spin > 2, and comment on why this is unacceptable. 2 1 µ µν (Γ) = , {Γ, Γ } = 0, [Γ, Σ ] = 0 . (16) In d = 4 the supercharges are written as two- component complex Weyl spinors which satisfy the The eigenvalues of Γ are ±1. Dirac spinors in even algebra (in the absence of central charges) d are reducible since they admit a decomposition into two Weyl spinors of opposite Γ eigenvalues, or A µ A {Qα , QβB˙ } = 2σ ˙ Pµ δB, chiralities: The Dirac representation splits into two αβ Weyl representations of dimension 2d/2−1. {Q, Q} = {Q, Q} = [P, Q] = [P, Q] = [P, P ] = 0 (19) If d is odd, one obtains the Clifford algebra by taking the Dirac matrices of the (d−1)-dimensional Here Pµ is the four-momentum, α and β˙ are Weyl d Clifford algebra and adding Γ = Γ. The Dirac rep- spinor indices, A and B label the supercharges, and (d−1)/2 resentation of the Lorentz algebra is then 2 - we have suppressed all indices in the second line. dimensional and irreducible. Consider now a massless supermultiplet, P 2 = In certain dimensions it is possible to fur- 0. In a reference frame where P = (−E, 0, 0,E), ther halve the independent degrees of freedom of the first line of Eq. (19) becomes a spinor by imposing the Majorana condition 4 E 0 {QA, Q } = δA . (20) ζ∗ = Bζ (17) α βB˙ 0 0 B A where B is some product of Dirac matrices satisfy- The operators Q2 and Q2˙A are represented by zero ing in this frame since they anticommute with every- µν −1 µν∗ A B Σ B = −Σ . (18) thing. The operators Q1 lower the helicity of any 2 1 state by 2 .A N = 8 massless multiplet contains a Here F4 = d A3 is the U(1) field strength 4-form. spin-2 field (a graviton); its two polarisation states The Chern-Simons term is gauge invariant up to |h+2 i and |h−2 i with helicities ±2 are related by a total derivative, since under a gauge transforma- A repeatedly acting with the Q1 : tion Eq. (22) it transforms as 1 8 Z |h−2 i ∼ Q1 ··· Q1 |h+2 i . (21) A3 ∧ F4 ∧ F4 A ficticious massless multiplet of N = 16 Z Z → A ∧ F ∧ F + d Λ ∧ F ∧ F (24) SUSY in d = 4 would have to contain a spin- 3 4 4 2 4 4 4 field. It is often stated that interacting theo- Z Z = A ∧ F ∧ F + d (Λ ∧ F ∧ F ) . ries with fields of spin > 2 cannot be consistently 3 4 4 2 4 4 coupled to gravity, although recently some excep- The last equality follows from the product rule and tions may have been discovered (involving an infi- from d F = 0 nite number of fields, and on curved backgrounds). 4 Barring such exotic constructions, the maximal su- The terms in the action involving Ψ are some- persymmetry in d = 4 would be N = 8, and the what more complicated. They are however also maximal dimension for supersymmetry d = 11. fully fixed by local supersymmetry and gauge in- variance. The above arguments can also be applied, with slight modifications, to massive multiplets, and generalized to the case non-vanishing central 5 Type IIA in d = 10 charges. When compactifying d = 11 SUGRA on S1 and 4 d = 11 supergravity truncating to the massless spectrum, one obtains type IIA SUGRA in d = 10. The number of real supercharges is still 32, contained in two Majorana- In d = 11 there is a unique supergravity theory; it Weyl spinors of opposite chiralities. The field con- contains 32 real supercharges, or equivalently a d = tent is 11 Majorana spinor supercharge. The gravitational multiplet is the unique supersymmetry multiplet. • the graviton and its gravitino superpartner, It contains • a real scalar dilaton φ and its dilatino super- • the graviton, partner ψ, • a gravitino Ψ, • a 2-form gauge potential B2 with field strength H = d B , • and a 3-form A3 .
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]
  • TASI Lectures on String Compactification, Model Building
    CERN-PH-TH/2005-205 IFT-UAM/CSIC-05-044 TASI lectures on String Compactification, Model Building, and Fluxes Angel M. Uranga TH Unit, CERN, CH-1211 Geneve 23, Switzerland Instituto de F´ısica Te´orica, C-XVI Universidad Aut´onoma de Madrid Cantoblanco, 28049 Madrid, Spain angel.uranga@cern,ch We review the construction of chiral four-dimensional compactifications of string the- ory with different systems of D-branes, including type IIA intersecting D6-branes and type IIB magnetised D-branes. Such models lead to four-dimensional theories with non-abelian gauge interactions and charged chiral fermions. We discuss the application of these techniques to building of models with spectrum as close as possible to the Stan- dard Model, and review their main phenomenological properties. We finally describe how to implement the tecniques to construct these models in flux compactifications, leading to models with realistic gauge sectors, moduli stabilization and supersymmetry breaking soft terms. Lecture 1. Model building in IIA: Intersecting brane worlds 1 Introduction String theory has the remarkable property that it provides a description of gauge and gravitational interactions in a unified framework consistently at the quantum level. It is this general feature (beyond other beautiful properties of particular string models) that makes this theory interesting as a possible candidate to unify our description of the different particles and interactions in Nature. Now if string theory is indeed realized in Nature, it should be able to lead not just to `gauge interactions' in general, but rather to gauge sectors as rich and intricate as the gauge theory we know as the Standard Model of Particle Physics.
    [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]
  • 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.
    [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]
  • 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.
    [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]
  • Holography and Compactification
    PUPT-1872 ITFA-99-14 Holography and Compactification Herman Verlinde Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 and Institute for Theoretical Physics, University of Amsterdam Valckenierstraat 65, 1018 XE Amsterdam Abstract arXiv:hep-th/9906182v1 24 Jun 1999 Following a recent suggestion by Randall and Sundrum, we consider string compactification scenarios in which a compact slice of AdS-space arises as a subspace of the compactifica- tion manifold. A specific example is provided by the type II orientifold equivalent to type I theory on (orbifolds of) T 6, upon taking into account the gravitational backreaction of the D3-branes localized inside the T 6. The conformal factor of the four-dimensional metric depends exponentially on one of the compact directions, which, via the holographic corre- spondence, becomes identified with the renormalization group scale in the uncompactified world. This set-up can be viewed as a generalization of the AdS/CFT correspondence to boundary theories that include gravitational dynamics. A striking consequence is that, in this scenario, the fundamental Planck size string and the large N QCD string appear as (two different wavefunctions of) one and the same object. 1 Introduction In string theory, when considered as a framework for unifying gravity and quantum mechan- ics, the fundamental strings are naturally thought of as Planck size objects. At much lower energies, such as the typical weak or strong interaction scales, the strings have lost all their internal structure and behave just as ordinary point-particles. The physics in this regime is therefore accurately described in terms of ordinary local quantum field theory, decoupled from the planckian realm of all string and quantum gravitational physics.
    [Show full text]