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D-Brane Primer

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D-Brane Primer D–Brane Primer Clifford V. Johnson Department of Mathematical Sciences Science Laboratories, South Road Durham DH1 3LE England, U.K. [email protected] Following is a collection of lecture notes on D–branes, which may be used by the reader as preparation for applications to modern research applications such as: the AdS/CFT and other gauge theory/geometry correspondences, Matrix The- ory and stringy non–commutative geometry, etc. In attempting to be reasonably self–contained, the notes start from classical point–particles and develop the sub- ject logically (but selectively) through classical strings, quantisation, D–branes, supergravity, superstrings, string duality, including many detailed applications. Selected focus topics feature D–branes as probes of both spacetime and gauge geometry, highlighting the role of world–volume curvature and gauge couplings, with some non–Abelian cases. Other advanced topics which are discussed are the (presently) novel tools of research such as fractional branes, the enhan¸con mecha- nism, D(ielectric)–branes and the emergence of the fuzzy/non–commutative sphere. (This is an expanded writeup of lectures given at ICTP, TASI, and BUSSTEPP.) Contents 1 Opening Remarks 4 2 String Worldsheet Perspective, Mostly 7 arXiv:hep-th/0007170v3 24 Aug 2000 2.1 ClassicalPointParticles . 7 2.2 ClassicalBosonicStrings. 9 2.3 QuantisedBosonicStrings . 23 2.4 Chan-PatonFactors . .. .. .. .. .. .. .. .. .. 33 2.5 The Closed String Partition Function . 34 2.6 UnorientedStrings .. .. .. .. .. .. .. .. .. .. 39 2.7 StringsinCurvedBackgrounds . 43 3 Target Spacetime Perspective, Mostly 47 3.1 T–DualityforClosedStrings . 47 3.2 TheCirclePartitionFunction . 51 3.3 Self–Duality and Enhanced Gauge Symmetry . 52 1 3.4 T–dualityinBackgroundFields . 54 3.5 AnotherSpecialRadius: Bosonisation . 55 3.6 StringTheoryonanOrbifold . 58 3.7 T–DualityforOpenStrings: D–branes . 61 3.8 D–Brane Dynamics: Collective Co¨ords and Gauge Theory . .. 63 3.9 T–DualityandOrientifolds. 67 3.10 TheD–BraneTension . 70 3.11 TheOrientifoldTension . 75 4 Worldvolume Actions I: Dirac–Born–Infeld 77 4.1 TiltedD–Branes .......................... 78 4.2 TheDirac–Born–InfeldAction . 79 4.3 TheActionofT–Duality. 81 4.4 Non–AbelianExtensions . 81 4.5 Yang–MillsTheory ......................... 82 4.6 BIons, BPS Saturation and Fundamental Strings . 83 5 Superstrings and D–Branes 85 5.1 OpenSuperstrings:FirstLook . 85 5.2 ClosedSuperstrings: TypeII . 90 5.3 Open Superstrings: Second Look — Type I from Type IIB . 93 5.4 The10DimensionalSupergravities . 95 5.5 The K3 Manifold from a Superstring Orbifold . 97 5.6 T–DualityofTypeIISuperstrings . 104 5.7 T–DualityofTypeISuperstrings . 105 5.8 D–BranesasBPSSolitons. 107 5.9 TheD–BraneChargeandTension . 108 5.10 DiracChargeQuantisation . 111 6 Worldvolume Actions II: Curvature Couplings 112 6.1 Tilted D–BranesandBraneswithin Branes . 112 6.2 Branes Within Branes: Anomalous Gauge Couplings . 113 6.3 Branes Within Branes: Anomalous “Curvature” Couplings . 114 6.4 FurtherNon–AbelianExtensions . 116 6.5 EvenMoreCurvatureCouplings . 116 7 The Dp–Dp′ System 118 7.1 TheBPSBound .......................... 121 7.2 FDBoundStates.......................... 122 7.3 TheThree–StringJunction . 124 7.4 0–p BoundStates.......................... 126 2 8 D–Branes, Strong Coupling, and String Duality 129 8.1 D1–BraneCollectiveDynamics . 129 8.2 TypeIIB/TypeIIBDuality . 130 8.3 TypeI/Heterotic . .. .. .. .. .. .. .. .. .. .. 131 8.4 TypeIIA/M–Theory . 133 8.5 E8 E8 HeteroticString/M–TheoryonI . 137 8.6 U–Duality..............................× 138 8.7 U–DualityandBoundStates . 139 9 D–Branes and Geometry I 140 9.1 D–BranesasaProbeofALESpaces . 140 9.2 Fractional D–Branesand Wrapped D–Branes . 147 9.3 Wrapped, Fractional and Stretched Branes. 149 9.4 D–BranesasInstantons . 154 9.5 SeeingtheInstantonwithaProbe . 155 9.6 D–BranesasMonopoles . 159 10 D–Branes and Geometry II 165 10.1 TheGeometryproducedbyD–Branes . 165 10.2 Probing D–Branes’ Geometry with D–Branes: p with Dp .... 169 10.3 TheMetriconModuliSpace . 171 10.4 Probing D–Branes’ Geometry with D–Branes: p with D(p 4). 172 10.5 D2–branes and 6–branes: Kaluza–Klein Monopoles and M–Theory173− 10.6 TheMetriconModuliSpace . 175 10.7 When Supergravity Lies: Repulson Vs. Enhan¸con . 178 10.8 TheMetriconModuliSpace . 183 11 D–Branes and Geometry III: Non–Commutativity 186 11.1 OpenStringswithaBackgroundB–Field . 186 11.2 Non–Commutative Geometry and D–branes . 189 11.3 Yang–Mills Geometry I: D–branes and the Fuzzy Sphere . 191 11.4 Yang–Mills Geometry II: Enhan¸cons and Monopoles . 198 12 Closing Remarks 201 A Collection of (Hopefully) Useful Formulae 202 B List of Inserts 207 3 1 Opening Remarks These lecture notes are supposed to represent, at least in part, the introductory lectures on D–branes which I have given at a few schools of one sort or another. At some point last year, while preparing to write some lecture notes in a publishable form, it occurred to me that nobody really wanted another set of introductory “D–notes”, (as I like to call them). After all, there have been many excellent ones, dating as far back as 1996, not to mention at least two excellent text books. 1,2 Another problem which occurs is that well–meaning organisers want me to give introductory lectures on D–branes, and assume that the audience is going to “pick up” string theory along the way during the lectures, but still seem quite keen that I get to the “cool stuff” —and usually in four or five lectures. Not being able to bear the thought that I might be losing some of my audience, I thought I’d write some notes for myself which try to take the “informed beginner” from the very start (classical point particles), all the way to the modern applications (AdS/CFT, building a crystal set, whatever), but making sure to stop to smell the flowers along the way. It’ll be a bit more than four lectures, unfortunately, but one should be able to pick and choose from the material. This clearly calls for something somewhere between a serious text (for which there is no need just yet) and another set of short lectures, and here they are. I was hoping for them to be at least in part a sort of useful toolbox, and not just a tour of what’s happened or happening. So as a result it reaches much further back than other D–notes, and also necessarily a bit further forward, so as to make contact with (and serve as a launching pad for) the other lecturers’ material at the school. Due to the remarkable activity in the field, there is not a complete list of every paper written on each topic. I am trying instead to supply a collection of notes that can be worked through as preparatory material, so I list some of the papers I found useful to this task, with an occasional partial attempt at historical context. In terms of the later choice of topics, the notes hopefully fill in some of the holes that other lecture notes on D–branes have left. There is a rather detailed table of contents for aid in searching for topics, and a list of some of the useful formulae that I (for one) like to have to hand. (It’s probably best just to tear those off and throw the rest away.) There are lots of figures and (hopefully) helpful insert boxes to help the reader, especially in the earlier parts. I should mention that I think of this as a natural offspring of the 1996 project with Joe Polchinski and Shyamoli Chaudhuri, 3 and have inevitably borrowed many bits straight from there, and also from Joe’s excellent TASI 4 notes from the same year. 4 Those form a sort of core which I have chipped away at, twisted, stretched, embellished, and any other verb you can think of except “improved”, simply because those lecture notes were trying to do something quite different and still serve their purpose very well. Perhaps in conflict with some students’ memory of the actual lectures, the reader will not find any Star Wars references or jokes scattered within. This is partly because I ran out of good Episode names, but mostly because I actually saw last year’s film. Quite seriously, I hope that the reader can use this collection of notes as a means of pulling together lots of concepts that are in and around string theory in a way that prepares them for actually doing research in this wonderful and exciting subject. Since I wrote no major problems to be done along with the material, let me end with a few suggestions for some daily calisthenic exercises, if you will, of a type which I have heard that all of the top researchers in the field employ on a regular basis. They are listed on the next page. They should be done first thing in the morning, or at least about the same time each day. As experience is gained, you will find it fulfilling to make up some of your own. I hope that you enjoy the notes. Look out for a fully hyperlinked web version. —cvj 5 Daily Calisthenic Exercises a Wind an F–string around as large a circle as you can manage (or have • room for in your office). Tip: If it is an open string, make sure that you firmly fix the ends on a handy D–brane before letting go, or you’ll risk getting a nasty cut as it snaps back. Stretch a D–string between two parallel D–branes. In the old days, we • used to do this with F–strings, but that is now considered to be well beneath most young researchers’ abilities. Do not be tempted to do this at strong coupling; all benefits of the exercise will be lost!. Try lifting a Coulomb branch from time to time, but be careful! This • is one of the more advanced operations, and you should lift steadily to avoid any long term back pain.
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