DOCSLIB.ORG

Why Is the Matrix Model Correct?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Why Is the Matrix Model Correct? View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server hep-th/9710009 IASSNS-HEP-97/108 Why is the Matrix Mo del Correct? 1 Nathan Seib erg School of Natural Sciences Institute for Advanced Study Princeton, NJ 08540, USA We consider the compacti cation of M theory on a light-like circle as a limit of a compacti cation on a small spatial circle b o osted by a large amount. Assuming that the compacti cation on a small spatial circle is weakly cou- pled typ e IIA theory, we derive Susskind's conjecture that M theory com- pacti ed on a light-like circle is given by the nite N version of the Matrix mo del of Banks, Fischler, Shenker and Susskind. This p oint of view pro- vides a uniform derivation of the Matrix mo del for M theory compacti ed p on a transverse torus T for p = 0; :::; 5 and clari es the diculties for larger values of p. 10/97 1 seib [email protected] Ab out a year ago Banks, Fischler, Shenker and Susskind (BFSS) [1] prop osed an amazingly simple conjecture relating M theory in the in nite momentum frame to a certain p quantum mechanical system. The extension to compacti cations on tori T for p =1; :::; 5 was worked out in [1-5]. This prop osal was based on the compacti cation of M theory on N around that circle. In the a spatial circle of radius R in a sector with momentum P = s R s limit of small R M theory b ecomes the typ e IIA string theory and the lowest excitations s in the sector with momentum P are N D0-branes [6]. When the D0-brane velo cities are small and the string interactions are weak the D0-branes are describ ed [7] by the minimal sup ersymmetric Yang-Mills theory with sixteen sup ercharges (SYM). When the velo cities or the string coupling are not small, this minimal sup ersymmetric theory is corrected by higher dimension op erators. The suggestion of BFSS was that M theory in the in nite momentum frame in uncompacti ed space is obtained by considering the minimal SYM p quantum mechanical system in the limit N; R ;P ! 1. For compacti cation on T the s prop osal of [1,2] is to consider Dp-branes, which are describ ed by SYM in p + 1 dimensions, and again to truncate to the minimal theory. This prop osal raised a few questions: 1. Why is this prop osal correct? 2. Why is the theory with small R related to the theory with large R ? s s 3. More sp eci cally, the minimal sup ersymmetric theory is corrected by higher dimension op erators, which are imp ortant when R and the velo cities are not small. Whyisthe s extrap olation from the minimal theory, which is valid at small R , correct for large s R ? s 4. Furthermore, for p 4 the minimal theory is not renormalizable and hence it is ill de ned. Then, higher dimension op erators must b e included in the description. They re ect the fact that the theory must b e emb edded in a larger theory with more degrees of freedom. This theory for p = 4; 5 was found in [3-5]. The pro cedure to nd these extensions of the minimal theories did not app ear systematic. What is then the rule to construct the theory in di erent backgrounds? Susskind noted that the nite N Matrix mo del enjoys some of the prop erties exp ected to hold only in the large N limit and suggested that it is also physically meaningful [8]. He suggested that the matrix mo del describ es M theory compacti ed on a light-like circle N + . Such a compacti cation on a light-like circle with of radius R with momentum P = R nite momentum is known as the discrete light-cone and the quantization of this theory is known as the discrete light-cone quantization (DLCQ). With a light-like circle the value of R can b e changed by a b o ost. Therefore, the uncompacti ed theory cannot b e obtained 1 + by simply taking R to in nity. Instead, it is obtained by taking R; N ! 1 holding P xed. In this note we will relate these two approaches to the Matrix theory. In the pro cess of doing so, we will derive the Matrix mo del and will answer the questions ab ove. We will also present a uniform derivation of the Matrix mo del for M theory on a compacti ed transverse space. We start by reviewing some trivial facts ab out relativistic kinematics. A compacti - cation on a light-like circle corresp onds to the identi cation R p x x 2 ; (1) + R t t p 2 10 where x is a spatial co ordinate, e.g. x . We consider it as the limit of a compacti cation on a space-like circle which is almost light-like q 2 2 R R R s 2 p p + R + x x x s 2 2 2R + + (2) R R t t t p p 2 2 with R R . The light-like circle (1) is obtained from (2) as R ! 0. This compacti ca- s s tion is related by a large b o ost with 2 R R s p (3) = 1 2 2 2 R R +2R s to a spatial compacti cation on x x R s + : (4) t t 0 A longitudinal b o ost of the light-like circle (1) rescales the value of R . It also rescales the value of the light-cone energy P . Therefore P is prop ortional to R . For small R s the value of P in the system with the almost light-like circle (2) is also prop ortional to R (an exception to that o ccurs when P = 0 for the light-like circle; then P can be non-zero for the almost light-like circle). The b o ost (3) rescales P to be indep endent of R and of order R (if originally P =0, the resulting P after the b o ost can be smaller s than order R ). s Following [9] (as referred to in [10]) wenow consider M theory compacti ed on a light- like circle (1) as the R ! 0 limit of the compacti cation on an almost light-like circle (2) s 2 or as the limit of the b o osted circle (4). This way the DLCQ of M theory discussed in [8] is related to the compacti cation on a small spatial circle as in [1]. For small R the theory s 3 2 compacti ed on (4) is weakly coupled string theory with string coupling g = (R M ) , s s P 2 3 (M is the Planck mass). For xed energies and xed M , and string scale M = R M P P s s P the limit R ! 0 yields a complicated theory with vanishing string scale. s However, as we said ab ove, starting with P of order one, the e ect of the b o ost is to 2 2 is inserted on dimensional grounds). This is exactly (M reduce P to b e of order R M s P P the range of energies in the discussion of [11], which was one of the motivations for the Matrix mo del of BFSS [1]. In order to fo cus on the mo des with such values of P we rescale the parameters of the theory. We do that by replacing the original M theory, which f is compacti ed on a light-like circle of radius R by another M theory, referred to as M f theory with Planck scale M compacti ed on a spatial circle of radius R . The transverse P s f geometry of the original M theory is replaced by that of the M theory. For example, for e a compacti cation on a transverse torus with radii R the other theory has radii R . i i The relations b etween the parameters of these two theories are obtained by combining 2 f f the limit R ! 0 with M !1 holding P R M xed. Therefore we identify s P s P 2 2 f R M = RM ; (5) s P P which is nite in the limit. Since the b o ost do es not a ect the transverse directions, we identify f e M R = M R (6) P i P i and keep it xed. In this limit the energies are nite and we nd string theory with string coupling and string scale 3 3 3 2 4 f 2 4 (RM ) =R eg =(R M ) s s s P P 1 3 3 2 2 2 f f 2 (7) = R M = R ) M (RM s s P s P : For R ! 0 with nite M and R we recover weakly coupled string theory with large s P string tension. This theory is very simple and is at the ro ot of the simpli cation of the Matrix mo del. N + A sector with P = in the original M theory is mapp ed to a sector of momentum R N f P = in the new M theory. In terms of this latter theory it includes N D0-branes. R s 3 Therefore, the original M theory is mapp ed to the theory of D0-branes.
Load more

Recommended publications
  • On the Holographic S–Matrix
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server On the Holographic S{matrix I.Ya.Aref’eva Steklov Mathematical Institute, Russian Academy of Sciences Gubkin St.8, GSP-1, 117966, Moscow, Russia Centro Vito Volterra, Universita di Roma Tor Vergata, Italy [email protected] Abstract The recent proposal by Polchinski and Susskind for the holographic flat space S– matrix is discussed. By using Feynman diagrams we argue that in principle all the information about the S–matrix in the interacting field theory in the bulk of the anti- de Sitter space is encoded into the data on the timelike boundary. The problem of locality of interpolating field is discussed and it is suggested that the interpolating field lives in a quantum Boltzmannian Hilbert space. 1 According to the holographic principle [1, 2] one should describe a field theory on a manifold M which includes gravity by a theory which lives on the boundary of M.Two prominent examples of the holography are the Matrix theory [3] and the AdS/CFT corre- spondence [4, 5, 6]. The relation between quantum gravity in the anti-de Sitter space and the gauge theory on the boundary could be useful for better understanding of both theories. In principle CFT might teach us about quantum gravity in the bulk of AdS. Correlation functions in the Euclidean formulation are the subject of intensive study (see for example [7]-[21]). The AdS/CFT correspondence in the Lorentz formulation is considered in [22]-[29].
    [Show full text]
  • A Note on Six-Dimensional Gauge Theories
    ILL-(TH)-97-09 hep-th/9712168 A Note on Six-Dimensional Gauge Theories Robert G. Leigh∗† and Moshe Rozali‡ Department of Physics University of Illinois at Urbana-Champaign Urbana, IL 61801 July 9, 2018 Abstract We study the new “gauge” theories in 5+1 dimensions, and their non- commutative generalizations. We argue that the θ-term and the non-commutative torus parameters appear on an equal footing in the non-critical string theories which define the gauge theories. The use of these theories as a Matrix descrip- tion of M-theory on T 5, as well as a closely related realization as 5-branes in type IIB string theory, proves useful in studying some of their properties. arXiv:hep-th/9712168v2 14 Apr 1998 ∗U.S. Department of Energy Outstanding Junior Investigator †e-mail: [email protected] ‡e-mail: [email protected] 0 1 Introduction Matrix theory[1] is an attempt at a non-perturbative formulation of M-theory in the lightcone frame. Compactifying it on a torus T d has been shown to be related to the large N limit of Super-Yang-Mills (SYM) theory in d + 1 dimensions[2]. This description is necessarily only partial for d > 3, since the SYM theory is then not renormalizable. The SYM theory is defined, for d = 4 by the (2,0) superconformal field theory in 5+1 dimensions [3, 4], and for d = 5 by a non-critical string theory in 5+1 dimensions [4, 5]. The situation of compactifications to lower dimensions is unclear for now (for recent reviews see [7, 8]).
    [Show full text]
  • Random Matrix Theory in Physics
    RANDOM MATRIX THEORY IN PHYSICS π π Thomas Guhr, Lunds Universitet, Lund, Sweden p(W)(s) = s exp s2 : (2) 2 − 4 As shown in Figure 2, the Wigner surmise excludes de- Introduction generacies, p(W)(0) = 0, the levels repel each other. This is only possible if they are correlated. Thus, the Poisson We wish to study energy correlations of quantum spec- law and the Wigner surmise reflect the absence or the tra. Suppose the spectrum of a quantum system has presence of energy correlations, respectively. been measured or calculated. All levels in the total spec- trum having the same quantum numbers form one par- ticular subspectrum. Its energy levels are at positions 1.0 xn; n = 1; 2; : : : ; N, say. We assume that N, the num- ber of levels in this subspectrum, is large. With a proper ) s smoothing procedure, we obtain the level density R1(x), ( 0.5 i.e. the probability density of finding a level at the energy p x. As indicated in the top part of Figure 1, the level density R1(x) increases with x for most physics systems. 0.0 In the present context, however, we are not so interested 0 1 2 3 in the level density. We want to measure the spectral cor- s relations independently of it. Hence, we have to remove FIG. 2. Wigner surmise (solid) and Poisson law (dashed). the level density from the subspectrum. This is referred to as unfolding. We introduce a new dimensionless en- Now, the question arises: If these correlation patterns ergy scale ξ such that dξ = R (x)dx.
    [Show full text]
  • JT Gravity and Random Matrix Ensembles
    JT Gravity And Random Matrix Ensembles Edward Witten Strings 2019, Brussels I will describe an extension of this work (Stanford and EW, \JT Gravity And The Ensembles of Random Matrix Theory," arXiv:1907:xxxxx). We generalized the story to include * time-reversal symmetry * fermions * N = 1 supersymmetry This morning Steve Shenker reported on random matrices and JT gravity (Saad, Shenker, and Stanford, \JT Gravity as a Matrix Integral" arXiv:1903.11115 ). * time-reversal symmetry * fermions * N = 1 supersymmetry This morning Steve Shenker reported on random matrices and JT gravity (Saad, Shenker, and Stanford, \JT Gravity as a Matrix Integral" arXiv:1903.11115 ). I will describe an extension of this work (Stanford and EW, \JT Gravity And The Ensembles of Random Matrix Theory," arXiv:1907:xxxxx). We generalized the story to include * fermions * N = 1 supersymmetry This morning Steve Shenker reported on random matrices and JT gravity (Saad, Shenker, and Stanford, \JT Gravity as a Matrix Integral" arXiv:1903.11115 ). I will describe an extension of this work (Stanford and EW, \JT Gravity And The Ensembles of Random Matrix Theory," arXiv:1907:xxxxx). We generalized the story to include * time-reversal symmetry * N = 1 supersymmetry This morning Steve Shenker reported on random matrices and JT gravity (Saad, Shenker, and Stanford, \JT Gravity as a Matrix Integral" arXiv:1903.11115 ). I will describe an extension of this work (Stanford and EW, \JT Gravity And The Ensembles of Random Matrix Theory," arXiv:1907:xxxxx). We generalized the story to include * time-reversal symmetry * fermions This morning Steve Shenker reported on random matrices and JT gravity (Saad, Shenker, and Stanford, \JT Gravity as a Matrix Integral" arXiv:1903.11115 ).
    [Show full text]
  • D-Brane Actions and N = 2 Supergravity Solutions
    D-Brane Actions and N =2Supergravity Solutions Thesis by Calin Ciocarlie In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Submitted May 20 , 2004) ii c 2004 Calin Ciocarlie All Rights Reserved iii Acknowledgements I would like to express my gratitude to the people who have taught me Physics throughout my education. My thesis advisor John Schwarz has given me insightful guidance and invaluable advice. I benefited a lot by collaborating with outstanding colleagues: Iosif Bena, Iouri Chepelev, Peter Lee, Jongwon Park. I have also bene- fited from interesting discussions with Vadim Borokhov, Jaume Gomis, Prof. Anton Kapustin, Tristan McLoughlin, Yuji Okawa, and Arkadas Ozakin. I am also thankful to my Physics teacher Violeta Grigorie who’s enthusiasm for Physics is contagious and to my family for constant support and encouragement in my academic pursuits. iv Abstract Among the most remarkable recent developments in string theory are the AdS/CFT duality, as proposed by Maldacena, and the emergence of noncommutative geometry. It has been known for some time that for a system of almost coincident D-branes the transverse displacements that represent the collective coordinates of the system become matrix-valued transforming in the adjoint representation of U(N). From a geometrical point of view this is rather surprising but, as we will see in Chapter 2, it is closely related to the noncommutative descriptions of D-branes. A consequence of the collective coordinates becoming matrix-valued is the ap- pearance of a “dielectric” effect in which D-branes can become polarized into higher- dimensional fuzzy D-branes.
    [Show full text]
  • Three Approaches to M-Theory
    Three Approaches to M-theory Luca Carlevaro PhD. thesis submitted to obtain the degree of \docteur `es sciences de l'Universit´e de Neuch^atel" on monday, the 25th of september 2006 PhD. advisors: Prof. Jean-Pierre Derendinger Prof. Adel Bilal Thesis comittee: Prof. Matthias Blau Prof. Matthias R. Gaberdiel Universit´e de Neuch^atel Facult´e des Sciences Institut de Physique Rue A.-L. Breguet 1 CH-2001 Neuch^atel Suisse This thesis is based on researches supported the Swiss National Science Foundation and the Commission of the European Communities under contract MRTN-CT-2004-005104. i Mots-cl´es: Th´eorie des cordes, Th´eorie de M(atrice) , condensation de gaugino, brisure de supersym´etrie en th´eorie M effective, effets d'instantons de membrane, sym´etries cach´ees de la supergravit´e, conjecture sur E10 Keywords: String theory, M(atrix) theory, gaugino condensation, supersymmetry breaking in effective M-theory, membrane instanton effects, hidden symmetries in supergravity, E10 con- jecture ii Summary: Since the early days of its discovery, when the term was used to characterise the quantum com- pletion of eleven-dimensional supergravity and to determine the strong-coupling limit of type IIA superstring theory, the idea of M-theory has developed with time into a unifying framework for all known superstring the- ories. This evolution in conception has followed the progressive discovery of a large web of dualities relating seemingly dissimilar string theories, such as theories of closed and oriented strings (type II theories), of closed and open unoriented strings (type I theory), and theories with built in gauge groups (heterotic theories).
    [Show full text]
  • Notes on Matrix Models (Matrix Musings)
    Notes on Matrix Models (Matrix Musings) Dionysios Anninos1 and Beatrix M¨uhlmann2 1 Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK 2 Institute for Theoretical Physics and ∆ Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands [email protected] & [email protected] Abstract In these notes we explore a variety of models comprising a large number of constituents. An emphasis is placed on integrals over large Hermitian matrices, as well as quantum me- chanical models whose degrees of freedom are organised in a matrix-like fashion. We discuss the relation of matrix models to two-dimensional quantum gravity coupled to conformal mat- ter. We provide a brief overview of a variety of more general quantum mechanical matrix models and their putative worldsheet interpretation, as well as other recent developments on large N systems. We end with a discussion of open questions and future directions. arXiv:2004.01171v3 [hep-th] 12 Apr 2021 1 Contents 1 IntroductionA;B 4 2 Large N vector integrals6 2.1 Saddle point approximation . .7 2.2 Vector integrals . .8 2.3 A perturbative expansion: the Cactus diagrams . 10 2.4 Diagrams resummed . 11 3 Large N integrals over a single matrix IC 14 3.1 Eigenvalue distribution . 14 3.2 Saddle point approximation & the resolvent . 17 3.3 General polynomial & multicritical models . 22 3.4 Large N factorisation & loop equations . 23 3.5 A perturbative expansion: the Riemann surfaces . 26 4 Large N integrals over a single matrix IIC;D 31 4.1 Orthogonal polynomials .
    [Show full text]
  • Type IIA D-Branes, K-Theory and Matrix Theory Petr Hofava A
    © 1998 International Press Adv. Theor. Math. Phys. 2 (1998) 1373-1404 Type IIA D-Branes, K-Theory and Matrix Theory Petr Hofava a a California Institute of Technology, Pasadena, CA 91125, USA horava@theory. caltech. edu Abstract We show that all supersymmetric Type IIA D-branes can be con- structed as bound states of a certain number of unstable non-supersym- metric Type IIA D9-branes. This string-theoretical construction demon- strates that D-brane charges in Type IIA theory on spacetime manifold X are classified by the higher K-theory group K_1(X), as suggested recently by Witten. In particular, the system of N DO-branes can be obtained, for any iV, in terms of sixteen Type IIA D9-branes. This sug- gests that the dynamics of Matrix theory is contained in the physics of magnetic vortices on the worldvolume of sixteen unstable D9-branes, described at low energies by a £7(16) gauge theory. e-print archive: http://xxx.lanl.gov/abs/hep-th/9812135 1374 TYPE 11 A D-BRANES, K-THEORY, AND MATRIX THEORY 1 Introduction When we consider individual D-branes in Type IIA or Type IIB string theory on R10, we usually require that the branes preserve half of the original su- persymmetry, and that they carry one unit of the corresponding RR charge. These requirements limit the D-brane spectrum to p-branes with all even values of p in Type IIA theory, and odd values of p in Type IIB theory. Once we relax these requirements, however, we can consider Dp-branes with all values of p.
    [Show full text]
  • Strings, Links Between Conformal Field Theory, Gauge Theory and Gravity Jan Troost
    Strings, links between conformal field theory, gauge theory and gravity Jan Troost To cite this version: Jan Troost. Strings, links between conformal field theory, gauge theory and gravity. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2009. tel-00410720 HAL Id: tel-00410720 https://tel.archives-ouvertes.fr/tel-00410720 Submitted on 24 Aug 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ecole´ Normale Sup´erieure Universit´ePierre et Marie Curie Paris 6 Centre National de la Recherche Scientifique Th`esed’Habilitation Universit´ePierre et Marie Curie Paris 6 Sp´ecialit´e: Physique Th´eorique Strings Links between conformal field theory, gauge theory and gravity pr´esent´eepar Jan TROOST pour obtenir l’habilitation `adiriger des recherches Soutenance le 20 mai 2009 devant le jury compos´ede : MM. Adel Bilal Rapporteurs : Bernard Julia Costas Kounnas Dieter L¨ust Matthias Gaberdiel Marios Petropoulos Marios Petropoulos Jean-Bernard Zuber Eliezer Rabinovici Contents 1 Before entering the labyrinth 7 2 Strings as links 15 2.1 Down to earth . 15 2.2 Lofty framework . 16 3 Continuous conformal field theory 23 3.1 One motivation .
    [Show full text]
  • Nonrelativistic String Theory and T-Duality Arxiv:1806.06071V3 [Hep-Th]
    Nonrelativistic String Theory and T-Duality Eric Bergshoeff a, Jaume Gomis b, and Ziqi Yan b aVan Swinderen Institute, University of Groningen Nijenborgh 4, 9747 AG Groningen, The Netherlands bPerimeter Institute for Theoretical Physics 31 Caroline St N, Waterloo, ON N2L 6B9, Canada E-mail: [email protected], [email protected], [email protected] Abstract: Nonrelativistic string theory in flat spacetime is described by a two- dimensional quantum field theory with a nonrelativistic global symmetry acting on the worldsheet fields. Nonrelativistic string theory is unitary, ultraviolet complete and has a string spectrum and spacetime S-matrix enjoying nonrelativistic symmetry. The worldsheet theory of nonrelativistic string theory is coupled to a curved space- time background and to a Kalb-Ramond two-form and dilaton field. The appropriate spacetime geometry for nonrelativistic string theory is dubbed string Newton-Cartan geometry, which is distinct from Riemannian geometry. This defines the sigma model of nonrelativistic string theory describing strings propagating and interacting in curved background fields. We also implement T-duality transformations in the path integral of this sigma model and uncover the spacetime interpretation of T-duality. We show that T-duality along the longitudinal direction of the string Newton-Cartan geometry describes relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, which is otherwise only defined by a subtle infinite boost limit. This relation arXiv:1806.06071v3 [hep-th] 21 Nov 2018 provides a first principles definition of string theory in the discrete light cone quan- tization (DLCQ) in an arbitrary background, a quantization that appears in nonper- turbative approaches to quantum field theory and string/M-theory, such as in Matrix theory.
    [Show full text]
  • Some Suggestions for the Term Paper for Physics 618: Applied Group Theory
    Preprint typeset in JHEP style - HYPER VERSION Some Suggestions for the Term Paper for Physics 618: Applied Group Theory Gregory W. Moore Abstract: Here are some ideas for topics for the term paper. The paper is due MONDAY MAY 13, 2019 LATE PAPERS WILL NOT BE ACCEPTED. I would appreciate it if you send me a pdf copy of the paper at [email protected] in addition to putting a hardcopy in my mailbox. The topics range from easy to challenging. For each topic I give some indication of a point of entry into the literature on the subject. These are not meant to be reference lists and part of the project is to do literature searches to see what is known about the subject. This is an opportunity for you to explore some topic you find interesting. You can write a short (≥ 10pp:) expository account of what you have read, or go for a more in depth investigation in some direction. You do not need to do original research, although if you can do something original that would be great. Feel free to choose your own topic (if it is not on this list please discuss it with me first). The only boundary condition is that the topic should have something to do with applications of group theory to physics. Version February 19, 2019. Contents 1. Wigner's Theorem 5 2. The Spectral Theorem 5 3. C∗-algebras And Quantum Mechanics 6 4. Jordan Algebras 6 5. Stabilizer Codes In Quantum Information Theory 6 6. Crystallographic groups and Bravais lattices 6 7.
    [Show full text]
  • Field Theory Dynamics from Branes in String Theory
    February 1, 2008 hep-th/9812072 Field Theory Dynamics from Branes in String Theory Andreas Karch Institut f¨ur Physik, Humboldt Universit¨at Berlin Invalidenstr. 110 10115 Berlin, Germany Abstract: I review certain aspects of Hanany-Witten setups and other approaches used to embed (and solve) gauge theories in string theory. Applications covered include dualities in 4 and 3 dimensions, fixed points in 6 dimensions, phase transitions between different geometric backgrounds and dualities between branes and geometry. arXiv:hep-th/9812072v1 8 Dec 1998 Contents 1 Motivation and Introduction 4 1.1 Motivation.................................... 4 1.2 Introduction................................... 5 2 D-branes and non-perturbative effects in string theory 7 2.1 Thebreakdownofperturbationtheory . .... 7 2.2 A string theoretic description of D-branes . ....... 9 2.3 D-branesandgaugetheory. 11 2.3.1 Gaugetheoryontheworldvolume . 11 2.3.2 CompactificationsandD-branes . 13 2.4 EngineeringGaugetheories . .. 13 2.4.1 GeometricEngineering . 13 2.4.2 BranesasProbes ............................ 15 2.4.3 Hanany-Wittensetups . 15 2.5 D-branesanddualities . .. .. 16 2.5.1 StringDualitiesandM-theory . 16 2.5.2 MatrixTheory ............................. 18 2.5.3 WorldvolumetheoryoftheNS5brane . 20 3 Exploiting the SYM D-brane correspondence 22 3.1 Classical Hanany-Witten setups . .... 22 3.1.1 Branegineering ............................. 22 3.1.2 There’ssomuchmoreonecando . 26 3.2 Solvingthequantumtheory . 29 3.2.1 Bendingandquantumeffects . 29 3.2.2 LiftingtoM-theory........................... 31 4 Applications of the brane construction 35 4.1 Dualitiesinthebranepicture . ... 35 4.1.1 ExactS-duality ............................. 35 4.1.2 Seiberg duality and Mirror symmetry . .. 37 2 4.1.3 S-dual =1pairsrevisited . .. .. 44 N 4.2 Non-trivialRGfixedpoints .
    [Show full text]