DOCSLIB.ORG

Abelian Differential, 139 A-D-E Classification, 351 Singularities, 716

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Abelian Differential, 139 A-D-E Classification, 351 Singularities, 716 Index Abelian differential, 139 Dirichlet, 12, 19, 188 A-D-E classification, 351 mixed, 29, 189, 315 singularities, 716 Neumann, 12, 19, 188 ADM tension, 694 Boundary state, 101, 164 AdS/CFT correspondence fermionic, 232 strong, 748 formalism, 93 weak, 748 fractional D-brane, 540 Almost complex structure, 454 BPS, 241 Anomaly bound, 696 gauge, 652 state, 560, 684 gravitational, 250 Brane world scenario, 641 holographic, 754 Breitenlohner-Freedman bound, 765 Anthropic principle, 3, 663 BRST Anti-de-Sitter, 741 charge, 385, 406 global coordinates, 742 current, 113, 387 metric, 735 quantization, 110 Poincare´ coordinates, 742 Bulk, 641 A-roof genus, 633 Bundle Atlas, 450 canonical, 474 Axion, 629 line, 469 stable, 503 vector, 453 Background charge, 395 Buscher rules, 434 Beltrami differential, 129 equation, 129 Betti numbers, 448, 461 Calabi-Yau in Kahler¨ manifolds, 468 four-fold, 478 Bianchi identity, 498, 625, 657 Hodge numbers, 476 Bosonic string holomorphic n-form, 475 map, 561, 567, 568 manifold, 385, 444, 473 propagators, 37 rigid, 532 Bosonization, 341, 391 warped, 662 Boundary changing operator, 98 Calibrated submanifold, 648 Boundary condition Canonical quantization, 35 A-type, 380 Cardy condition, 165, 170 B-type, 380 Cardy formula, 93 R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical 775 Physics, DOI 10.1007/978-3-642-29497-6, © Springer-Verlag Berlin Heidelberg 2013 776 Index Cartan group, 63, 737 matrix, 329 Killing spinor, 182 subalgebra, 328 Killing vector, 18, 129, 182, 737 Cartan-Weyl basis, 328 spin, 66 Casimir, 325 structure, 129 Center of mass position, 24, 29 Ward identity, 84 Central charge, 39, 323 weight, 66 Central extension, 39, 323, 720 Conformal field theory, 63 Chan-Paton factor, 49, 244 boundary, 93 Character, 153, 344 rational, 79, 164, 167, 344 Charge conjugation, 153, 211, 348 super, 355 Chern Conifold, 482 character, 473 Conjugacy class, 331, 392 class, 471 Constraint connection, 457, 470 first class, 9 Chern-Simons primary, 8, 12 3-form, 628 secondary, 9 term, 624 second class, 9, 186, 279 Chiral field, 66 Contorsion, 667 N D 2 superconformal, 377 Coordinates Chirality matrix, 212 homogeneous, 451 Chirality operator, 211 inhomogeneous, 452 Chiral ring, 383, 577 Coset, 331 Christoffel symbols, 509 lattice, 328 Clebsch-Gordan coefficients, 219 Covariant Clifford algebra, 198, 210 derivative, 510 Closed-open string duality, 99 lattice, 416, 578 Closed string, 19 Coxeter element, 524 classical solution, 32 Critical dimension fermionic mode expansion, 189 bosonic string, 47 fermionic spectrum, 207 fermionic string, 203 mode expansion, 23 Critical string, 431 partition function, 148 Crosscap, 124 spectrum, 50 gluing conditions, 167 Cocycle factor, 339 Ishibashi states, 167 Coherent state, 101 state, 166, 248 Cohomology, 461 Crossing symmetry, 83 de Rham, 461 Cross ratio, 82 Dolbeault, 463 Current algebra, 324 Compactification, 264, 440 families, 325 brane world, 641 Cylinder amplitude, 157 flux, 658 heterotic on Calabi-Yau, 497 supersymmetric, 441 type II on Calabi-Yau, 487 Complex D-brane, 30 manifold, 130 fractional, 540 structure, 130, 450 intersecting, 310, 646 moduli, 485 magnetized, 317 Conformal R-R charge, 240 bootstrap, 83 SUGRA solution, 701 coordinates, 129 tension, 163, 171, 240 family, 76 Dedekind eta-function, 147 gauge, 16 Dehn twist, 135 Index 777 Descendant metric, 658 field, 69, 741 NS-NS three-form, 657 state, 75 R-R p-form, 658 Dilation, 64, 740 Fractional D-brane, 540 Dimensional reduction, 435, 445 Frenkel-Kac-Segalˇ construction, 327 double, 719 F-theory, 704, 724 Dirac-Born-Infeld action, 436, 630 Fubini-Study metric, 458 Dirac bracket, 186, 279 Fundamental form, 456 Dirac matrix, 210 Fundamental region of torus, 137 Dirichlet boundary condition, 12, 19, 188 Fundamental string Discrete torsion, 300 SUGRA solution, 699 Discriminant, 725 tension, 10, 240 Disk, 94, 101, 145 Fusion rules, 78, 344 Dolbeault cohomology, 384, 463 N D 2 superconformal, 381 Doubling trick, 28, 96, 191 sbu.2/k , 351 Dual Coxeter number, 325 lattice, 327 Gauge hierarchy problem, 439 resonance model, 601 Gauss-Bonnet term, 15 Duality, 688 Gepner model, 567 heterotic-type IIA in 6D, 713 heterotic, 576 S- heterotic-type I in 10D, 710 partition function, 573 strong-weak, 676 type II, 573 S- type IIB in 10D, 705 Ghost, 36, 142 T-, 269, 432 action, 107 target-space, 678 field, 107 U-, 712 number operator, 111 picture Eddington-Finkelstein coordinates, 762 canonical, 404 Effective action, 615 charge, 403 D-brane, 630 system, 394 supergravity, 622 Gluing Effective field theory, 586 automorphism, 164 Einstein frame, 172, 619, 626, 690 conditions, 102, 164, 233 modified, 690 Graviphoton, 489 Einstein spaces, 736 Gravitino, 489, 546 Elliptic curve, 724 two-dimensional, 177 Elliptic fibration, 724 Green-Schwarz mechanism, 499, 629, 651 Energy-momentum tensor, 13, 67 Green-Schwarz superstring, 176 holographic, 757 GSO projection, 203, 230, 551 Euler Beta-function, 607, 636 Euler number, 431, 468, 478 Hagedorn temperature, 93 Harmonic form, 466 Faddeev-Popov ghosts, 53 Hawking temperature, 759 Fefferman-Graham Hermitian expansion, 752 conjugate, 36, 70 gauge, 751 connection, 457 First order system, 393 metric, 456 Fischler-Susskind mechanism, 319 Yang Mills, 470, 503 Fixed point, 528 Heterotic string, 284 Flux, 657 compactification, 497 compactification, 658 covariant lattice, 422, 578 imaginary self-dual, 660 Gepner model, 576 778 Index non-supersymmetric, 425 Kac-Moodyˇ algebra, 321, 323 orbifold, 525 representation, 343 S-duality, 710 SO.2n/, 341 Hirzebruch L-polynomial, 634 sbu.2/k , 349, 381 Hodge Kaluza-Klein decomposition, 384, 462, 463, 467 Ansatz, 435 diamond, 464, 477 modes, 264 numbers, 463 monopole, 720 operator, 465 Kawai-Lewellen-Tye relations, 614 Holography, 749 Killing Holonomy group, 298, 443, 460 group, 132 G2, 445 spinor, 442, 522 SU.2/, 445 equations, 664 SU.3/, 444 vector, 432 Homology conformal, 18, 129, 737 class, 461 Klein bottle, 167, 242 group, 461 amplitude, 168, 242 Horava-Wittenˇ theory, 723 K-theory, 251, 648 Hydrodynamics, 760 K3 surface, 445, 523, 530, 713 Hypermultiplet, 489 universal, 554, 574 Hypersurface, 452 Lagrangian submanifold, 648 Laplacian, 448, 467 Large extra dimension scenario, 642 Large-N expansion, 748 Instanton Lattice, 327 D-, 706 covariant, 416, 578 world-sheet, 487 D5, 392 Yang-Mills, 678 D , 416 Intersecting D-brane 5;1 E5;1, 421 boundary conditions, 310 even, 273, 328 models, 544, 646 Lorentzian, 272 Intersection number, 313, 463 Narain, 273, 293, 715 triple, 496 odd, 328 Ishibashi state, 164, 233, 284 rational, 273 Isometry, 432 self-dual, 287, 328, 423 unimodular, 328 Leech lattice, 335 Jacobi Lefschetz decomposition, 466 theta-function, 227 Level triple identity, 228 Kac-Moodyˇ algebra, 323 Jacobian, 128 matching, 40 j-function, 703 Lichnerowicz operator, 483 Lie algebra, 322 E6, 330 E7, 330 Kahler¨ E8, 330 cone, 484 lattice, 328, 332 complexification, 486 simply laced, 329 form, 457 SO.2n/, 329 manifold, 457 SU.n/, 330 metric, 457 Lie derivative, 432 moduli, 484 Lie group, 321 potential, 457 simply laced, 326 Index 779 Light-cone Modular coordinates group, 136 space-time, 42 invariance, 138, 148 world-sheet, 16 S-transformation, 137, 153 gauge, 42, 200 T-transformation, 137, 153 Line bundle, 469, 726 Moduli, 269, 449 Little group, 46, 178 bundle, 534 Loop group, 321 Calabi-Yau, 483–487 Loop-channel, 159, 165, 236 complex structure, 276, 485 annulus, 245, 539 Kahler,¨ 276, 484 Klein-bottle, 242, 537 space Riemann surface, 140 Mobius¨ strip, 246, 542 stabilization, 643 Loop-channel – tree-channel equivalence, 159, 2-torus, 276, 449 238, 242, 540 Momentum angular, 22 canonical, 12 M-theory, 688, 719 conjugate, 8 M2-brane, 719 space-time, 24, 27 M5-brane, 720, 721 Monodromy charge, 155 Mobius¨ strip, 169, 246 loop-channel, 170, 246 Nambu-Goto action, 10 tree-channel, 169, 247 Narain lattice, 273, 293, 715 transformations, 72 Navier-Stokes equation, 762 Magnetic monopole, 240 Near-horizon Maldacena limit, 745 geometry, 736 Mandelstam variables, 600 limit, 735 Manifold Neumann boundary condition, 12, 19, 188 Calabi-Yau, 473 Neveu-Schwarz sector, 188, 225 complex, 450 Newton’s constant, 434 Hermitian, 457 Niemeier lattices, 424 Kahler,¨ 457 Nijenhuis tensor, 455 Kahler-Einstein,¨ 459 Non-BPS D-branes, 251 orientable, 451 Non-linear sigma model, 315, 428 Ricci-flat, 443 No-scale structure, 661 Mapping class group, 140 NS5-brane, 689 Marginal operator, 560 SUGRA solution, 700 Massless spectrum Gepner model, 574, 577 orientifold, 537 4 T =Z2 orbifold, 529 Open string, 19 4 T =Z2 orientifold, 543 classical solution, 31 6 T =Z3 orbifold, 531 fermionic mode expansion, 191 Membrane, 719 fermionic spectrum, 203 Mirror mode expansion, 27 map, 493 partition function, 157 symmetry, 277, 385, 449, 491, 560 spectrum, 46 transformation, 560 Operator product expansion, 69 Mittag-Leffler theorem, 609 Orbifold, 138 Mode expansion circle S 1, 294 closed string, 24 heterotic, 525 fermionic closed string, 189 supersymmetric, 523 4 fermionic open string, 191 T =Z2, 528 6 open string, 27 T =Z3, 530 780 Index Orientifold, 51, 317 Ramond compactification, 534 ground state, 198, 379 construction, 241 sector, 188, 225 plane, 52, 169, 243, 317 Reflection coefficients, 165 type IIA on CY, 646 Regge 4 type IIB on T =Z2, 535 slope, 11 type IIB on CY, 645 trajectory, 32 Resolution of singularity, 523 Ricci P-transformation, 170, 247, 252 form, 458 Partition function, 80 scalar, 509 torus, 146 tensor, 430, 458, 509 4 T =Z2, 529 Riemann 6 T =Z3, 531 identity, 228, 257 Peccei-Quinn
Load more

Recommended publications
  • JHEP10(2009)051 3 3 2), | S 1 Ads , (1 × 3 Heory P SU August 10, 2009 October 20, 2009 : : September 20, 2009 : Ads Received , Published Unting in D1-D5 Systems
    Published by IOP Publishing for SISSA Received: August 10, 2009 Accepted: September 20, 2009 Published: October 20, 2009 3 Asymptotic symmetries of string theory on AdS3 × S JHEP10(2009)051 with Ramond-Ramond fluxes Sujay K. Ashok,a,b Raphael Benichouc and Jan Troostc aInstitute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai, 600113 India bPerimeter Institute for Theoretical Physics, Waterloo, Ontario, ON N2L2Y5, Canada cLaboratoire de Physique Th´eorique, Unit´eMixte du CRNS et de l’Ecole´ Normale Sup´erieure associ´ee `al’Universit´ePierre et Marie Curie 6, UMR 8549, 24 Rue Lhomond, Paris 75005, France E-mail: [email protected], [email protected], [email protected] Abstract: String theory on AdS3 space-times with boundary conditions that allow for black hole states has global asymptotic symmetries which include an infinite dimensional conformal algebra. Using the conformal current algebra for sigma-models on P SU(1, 1 2), | we explicitly construct the R-symmetry and Virasoro charges in the worldsheet theory describing string theory on AdS S3 with Ramond-Ramond fluxes. We also indicate how 3 × to construct the full boundary superconformal algebra. The boundary superconformal algebra plays an important role in classifying the full spectrum of string theory on AdS3 with Ramond-Ramond fluxes, and in the microscopic entropy counting in D1-D5 systems. Keywords: AdS-CFT Correspondence, Conformal Field Models in String Theory ArXiv ePrint: 0907.1242 c SISSA 2009 doi:10.1088/1126-6708/2009/10/051 Contents 1 Introduction 1 3
    [Show full text]
  • Exercise 3 — Supersymmetry for Superstrings
    Arash Ardehali String Theory | Prof. Sheikh-Jabbari Exercise 3 | Supersymmetry for Superstrings 1. SUSY generalities (suggested reading : Vafa et. al., Mirror symmetry, Ch 10) Supersymmetry often leads to: I) stability of quantum systems, and II) control over the non-perturbative regime of quantum systems. As for item I, in string theory supersymmetry allows removing the bosonic string tachyon in Minkowski background. (Question: do you remember the masses of the open and closed bosonic string tachyons in Minkowski background?) In other words, the superstring spectrum, unlike the bosonic spectrum, does not signal a perturbative instability in the system. The fact that supersymmetry (SUSY) removes the tachyon (after GSO, of course), is in accord with the general wisdom that supersymmetry often stabilizes quantum systems. • Write down the N = (1; 1) SUSY algebra (in terms of Hilbert-space operators) in 1+0, 1+1, and 1+3 dimensional quantum field theories. [Hint: ask Google.] Make sure you understand the role of y. Show that in all cases the SUSY algebra implies a positive semi-definite energy. [Hint: write the expectation value of the Hamiltonian in terms of the supercharge operators.] • Write down the SUSY algebra in 1+1 dimensional conformal field theory (CFT). This CFT could be the one living on the worldsheet of superstrings. Does the algebra imply a positivity constraint on the spectrum? Task: read about Witten's SUSY-inspired proof of the positive energy theorem in clas- sical GR. As for item II, in string theory supersymmetry allows understanding various strong- weak dualities between the five superstring theories and M-theory.
    [Show full text]
  • Modular Constraints on Superconformal Field Theories
    KIAS-P18096 Modular Constraints on Superconformal Field Theories Jin-Beom Bae, Sungjay Lee, and Jaewon Song Korea Institute for Advanced Study 85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Korea Abstract We constrain the spectrum of N = (1; 1) and N = (2; 2) superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the Γθ congruence subgroup of the full modular group SL(2; Z). We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without U(1) charges. Especially, the arXiv:1811.00976v1 [hep-th] 2 Nov 2018 upper bounds on the twist gap for the non-current primaries exhibit interest- ing peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to Γθ. Some of the candidate theories have been discussed by H¨ohnin the context of self-dual ex- tremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS3. Contents 1 Introduction 1 2 Preliminaries 7 2.1 Representations of N = 1 Superconformal Algebra . .7 2.2 Representations of N = 2 Superconformal Algebra . .9 2.3 Semi-Definite Programming . 12 3 Charge Independent Bounds 14 3.1 Spin-independent bounds on the gap . 14 3.2 Spin-dependent Bounds for N = (1; 1) SCFT . 16 3.3 Spin-dependent Bounds for N = (2; 2) SCFT . 26 4 Charge Dependent Bounds 29 4.1 HFK-type Bound for the Charged States .
    [Show full text]
  • Comments on String Theory on Adss
    © 1998 International Press Adv. Theor. Math. Phys. 2 (1998) 733-782 Comments on String Theory on AdSs Amit Giveona, David Kutasovb, Nathan Seibergc aRacah Institute of Physics The Hebrew University Jerusalem 91904, Israel b Department of Physics University of Chicago 5640 S. Ellis Av. Chicago, IL 60637, USA ^Department of Physics of Elementary Particles Weizmann Institute of Science Rehovot, Israel cSchool of Natural Sciences Institute for Advanced Study Olden Lane Princeton, NJ, USA Abstract We study string propagation on AdSs times a compact space from an "old fashioned" worldsheet point of view of perturbative string theory. We derive the spacetime CFT and its Virasoro and current algebras, thus establishing the conjectured AdS/CFT correspondence for this case in the full string theory. Our results have implications for the extreme IR limit of the Dl - D5 system, as well as to 2+1 dimensional BTZ black holes and their Bekenstein-Hawking entropy. e-print archive: http://xxx.lanl.gov/abs/hep-th/9806194 734 COMMENTS ON STRING THEORY ON AdSz 1 Introduction The purpose of this paper is to study string propagation on curved spacetime manifolds that include AdS^. We will mostly discuss the Euclidean version also known as H^ = 51/(2, C)/SU(2) (in Appendix A we will comment on the Lorentzian signature version of AdS^, which is the SX(2, R) group man- ifold) . At low energies the theory reduces to 2 + 1 dimensional gravity with a negative cosmological constant coupled (in general) to a large collection of matter fields. The low energy action is but we will go beyond this low energy approximation.
    [Show full text]
  • String Theory 101 Lectures at the International School of Strings and Fundamental Physics
    String Theory 101 Lectures at the International School of Strings and Fundamental Physics Munich July 26 - August 6 2010 Neil Lambert Theory Division CERN 1211 Geneva 23 Switzerland Email: [email protected] 1 Contents 1 Introduction: Why String Theory? 3 2 Classical and Quantum Dynamics of Point Particles 4 2.1 Classical Action . 4 2.2 Electromagnetic field . 7 2.3 Quantization . 8 3 Classical and Quantum Dynamics of Strings 10 3.1 Classical Action . 10 3.2 Spacetime Symmetries and Conserved Charges . 13 3.3 Quantization . 14 3.4 Open Strings . 23 3.5 Closed Strings . 27 4 Light-cone gauge 31 4.1 D = 26, a =1................................. 33 4.2 Partition Function . 34 5 Curved Spacetime and an Effective Action 36 5.1 Strings in Curved Spacetime . 36 5.2 A Spacetime Effective Action . 38 6 Superstrings 39 6.1 Type II strings . 39 6.2 Type I and Heterotic String . 44 6.3 The Spacetime Effective Action . 46 2 1 Introduction: Why String Theory? The so-called Standard Model of Particle Physics is the most successful scientific theory of Nature in the sense that no other theory has such a high level of accuracy over such a complete range of physical phenomena using such a modest number of assumptions and parameters. It is unreasonably good and was never intended to be so successful. Since its formulation around 1970 there has not been a single experimental result that has produced even the slightest disagreement. Nothing, despite an enormous amount of effort. But there are skeletons in the closet.
    [Show full text]
  • The Early Years of String Theory: a Personal Perspective
    CALT-68-2657 THE EARLY YEARS OF STRING THEORY: A PERSONAL PERSPECTIVE John H. Schwarz California Institute of Technology Pasadena, CA 91125, USA Abstract arXiv:0708.1917v3 [hep-th] 3 Apr 2009 This article surveys some of the highlights in the development of string theory through the first superstring revolution in 1984. The emphasis is on topics in which the author was involved, especially the observation that critical string theories provide consistent quantum theories of gravity and the proposal to use string theory to construct a unified theory of all fundamental particles and forces. Based on a lecture presented on June 20, 2007 at the Galileo Galilei Institute 1 Introduction I am happy to have this opportunity to reminisce about the origins and development of string theory from 1962 (when I entered graduate school) through the first superstring revolution in 1984. Some of the topics were discussed previously in three papers that were written for various special events in 2000 [1, 2, 3]. Also, some of this material was reviewed in the 1985 reprint volumes [4], as well as the string theory textbooks [5, 6]. In presenting my experiences and impressions of this period, it is inevitable that my own contributions are emphasized. Some of the other early contributors to string theory have presented their recollections at the Galileo Galilei Institute meeting on “The Birth of String Theory” in May 2007. Since I was unable to attend that meeting, my talk was given at the GGI one month later. Taken together, the papers in this collection should
    [Show full text]
  • Introduction to Superstring Theory
    Introduction to Superstring theory Carmen Nunez Instituto de Astronomia y Física del Espacio C.C. 67 - Sue. 28, 1428 Buenos Aires, Argentina and Physics Department, University of Buenos Aires [email protected] Abstract 1 Overview The bosonic string theory, despite all its beautiful features, has a number of short­ comings. The most obvious of these are the absence of fermions and the presence of tachyons in spacetime. The tachyon is not an actual physical inconsistency; it indicates at least that the calculations are being performed in an unstable vacuum state. More­ over, tachyon exchange contributes infrarred divergences in loop diagrams and these divergences make it hard to isolate the ultraviolet behaviour of the "unified quan­ tum theory" the bosonic string theory gives rise to and determine whether it is really satisfactory. Historically, the solution to the tachyon problem appeared with the solution to the other problem, the absence of fermions. The addition of a new ingredient, supersym- metry on the world-sheet, improves substantially the general picture. In 1977 Gliozzi, Scherk and Olive showed that it was possible to get a model with no tachyons and with equal masses and multiplicities for bosons and fermions. In 1980, Green and Schwarz proved that this model had spacetime supersymmetry. In the completely consisten- t tachyon free form of the superstring theory it was then possible to show that the one-loop diagrams were completely finite and free of ultraviolet divergences. While most workers on the subject believe that the finiteness will also hold to all orders of perturbation theory, complete and universally accepted proofs have not appeared so far.
    [Show full text]
  • The Conformal Bootstrap and Super W-Algebras
    THE CONFORMAL BOOTSTRAP AND SUPER W -ALGEBRAS Jose´ M. Figueroa-O’Farrill\ and Stany Schrans] Instituut voor Theoretische Fysica, Universiteit Leuven Celestijnenlaan 200 D, B–3001 Heverlee, BELGIUM ABSTRACT We undertake a systematic study of the possible extensions of the N = 1 super Vi- 1 7 rasoro algebra by a superprimary field of spin 2 ≤ ∆ ≤ 2 . Besides new extensions which exist only for specific values of the central charge, we find a new non-linear algebra (super W2) generated by a spin 2 superprimary which is associative for all values of the central charge. Furthermore, the spin 3 extension is argued to be the symmetry algebra of the m = 6 super Virasoro unitary minimal model, by exhibiting the (A7,D4)-type modular invariant as diagonal in terms of extended characters. \ e-mail: [email protected] ] Onderzoeker I.I.K.W., Belgium; e-mail: [email protected] §1 Introduction Extended conformal and superconformal algebras have received a great deal of attention lately[1],[2],[3]. Their study is relevant for the classification problem of rational conformal field theories[4] (RCFTs) since every RCFT is by definition a minimal model of its chiral algebra (the operator subalgebra generated by its holomorphic fields) which, since it contains the Virasoro algebra as a subalgebra, extends it. As shown by Cardy[5], a conformal field theory (CFT) which is ra- tional relative to the Virasoro algebra (i.e. , which contains a finite number of Virasoro primaries) must necessarily have c < 1. Similar arguments show that a superconformal field theory (SCFT) which is rational relative to the N = 1 super 3 Virasoro algebra must have c < 2 .
    [Show full text]
  • Introduction to Superstring Theory
    Introduction to Superstring Theory Carmen N´u~nez IAFE (UBA-CONICET) & PHYSICS DEPT. (University of Buenos Aires) VI ICTP LASS 2015 Mexico, 26 October- 6 November 2015 . Programme Class 1: The classical fermionic string Class 2: The quantized fermionic string Class 3: Partition Function Class 4: Interactions . Outline Class 1: The classical fermionic string The action and its symmetries Gauge fixing and constraints Equations of motion and boundary conditions Oscillator expansions . The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? . The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. If there is some other stable vacuum, this is not an actual inconsistency. Why superstrings? The spectra of bosonic strings contain a tachyon ! it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 @2V (T ) j − 4 ) action: M = @T 2 T =0 = α0 = we are expanding around a maximum of V .
    [Show full text]
  • Extended Kac-Moody Algebras '• • and Applications
    \j LAPP-TH-335/91 Avril 9l I 1 Extended Kac-Moody algebras '• • and Applications E. Ragoucy and P. Sorba Laboratoire Physique Théorique ENSLAPP" - Groupe d'Annecy Chemin de Bellevue BP 110, F-74941 Annecy-le- Vieux Cedex, France Abstract : * We extend the notion of a Kac-Moody algebra defined on the S1 circle to super Kac-Moody algebras defined on M x GN, M being a smooth closed compact manifold of dimension greater than one, and G/v the Grassman algebra with N generators. We compute all the central extensions V-, »! of these algebras. Then, we determine for each such algebra the derivation algebra constructed : from the M x <7/v diffeomorphisms. The twists of such super Kac-Moody algebras as well as , the generalization to non-compact surfaces are partially studied. Finally, we apply our general construction to the study of conformai and superconformai algebras, as well as area-preserving j diffeomorphisms algebra and its supersymmetric extension. Wf A review to be published in Int. Jour, of Mod. Phys. A "URA 14-36 du CNRS, associée à l'Ecole Normale Supérieure de Lyon, et au Laboratoire d'Annecy-le-Vieux de Physique des Particules. ••• \ ; Contents 8 Appli< 8.1 (, 1 Introduction. 8.2 I- 8.3 h-. FUNDAMENTALS 6 O Cone U The usual Kac-Moody algebras. G 2.1 Loop algebra 6 A Deter •****< 2.2 Twists of loop algebras 7 A.I h 2.3 Central extension of a loop algebra 7 A.2 C, 2.4 Kac-Moody algebras and Hochschilq (cyclic) cohomology 8 A.3 C 2.5 Derivation algebra S B KM a 2.6 The Sugawaraand GKO constructions 10 B.I K B.2 h Central extensions for generalized loop algebraB.
    [Show full text]
  • Comments on String Theory on $ Ads 3$
    EFI-98-22, RI-4-98, IASSNS-HEP-98-52 hep-th/9806194 Comments on String Theory on AdS3 Amit Giveon1, David Kutasov2,3, Nathan Seiberg4 1Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel 2Department of Physics, University of Chicago, 5640 S. Ellis Av., Chicago, IL 60637, USA 3Department of Physics of Elementary Particles, Weizmann Institute of Science, Rehovot, Israel 4School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ, USA We study string propagation on AdS3 times a compact space from an “old fashioned” worldsheet point of view of perturbative string theory. We derive the spacetime CFT and its Virasoro and current algebras, thus establishing the conjectured AdS/CFT correspon- dence for this case in the full string theory. Our results have implications for the extreme IR limit of the D1 D5 system, as well as to 2+1 dimensional BTZ black holes and their − arXiv:hep-th/9806194v2 1 Jul 1998 Bekenstein-Hawking entropy. 6/98 1. Introduction The purpose of this paper is to study string propagation on curved spacetime manifolds + that include AdS3. We will mostly discuss the Euclidean version also known as H3 = SL(2, C)/SU(2) (in Appendix A we will comment on the Lorentzian signature version of AdS3, which is the SL(2,R) group manifold). At low energies the theory reduces to 2 + 1 dimensional gravity with a negative cosmological constant coupled (in general) to a large collection of matter fields. The low energy action is 1 2 = d3x√g(R + ) + ... (1.1) S 16πl l2 p Z but we will go beyond this low energy approximation.
    [Show full text]
  • Embedding LFHQCD in Worldsheet String Theory
    October 1, 2019 Embedding LFHQCD in Worldsheet String Theory Harun Omer UMass Dartmouth 285 Old Westport Rd Dartmouth MA 02747 [email protected] Abstract Light-front holographic quantum chromodynamics (LFHQCD) has been proposed as an approximation to QCD which is non-perturbative and at the same time an- alytically tractable. It can be derived from the holographic light-cone Hamiltonian and effectively corresponds to a one-dimensional superconformal quantum theory. It is proposed in this paper to extend the underlying quantum group to N = 2 SCFT. This provides an avenue for bottom-up string theory model-building which is firmly grounded in phenomenology. Moreover, the peculiar properties of the model carry over to provide novel solutions to string theory problems, including supersymmetry without predicting superpartners as well as the breaking of the conformal symmetry and the introduction of a low-energy scale such that the massive Virasoro modes correspond to hadronic states reminiscent of what was once envisioned in the Veneziano model. 1. Introduction In the last few years, a series of papers has been published about Light-Front Holographic QCD (LFHQCD) [1,2,3,4,5,6,7,8,9,10,11,12,13]. Helpful reviews are [14,15]. LFHQCD is arXiv:1909.12866v1 [hep-th] 27 Sep 2019 able to reproduce a number of spectroscopic features of hadrons within the framework of a rather simple, semi-classical non-perturbative model. One starts with anti-de Sitter space and uses the holographic principle to obtain a field theory in one lower dimension in the limit of Nc →∞ colors. The equations of motion can be matched with the bound-state equations for two massless constituents in light front (LF) quantization.
    [Show full text]