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Thermodynamic and transport properties of a holographic quantum Hall system Hutchinson, Joel 2014

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Thermodynamic and TransportProperties of a Holographic QuantumHall SystembyJoel HutchinsonBSc, University of Alberta, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2014c Joel Hutchinson, 2014AbstractWe apply the AdS/CFT correspondence to study a quantum Hall system at strongcoupling. Fermions at finite density in an external magnetic field are put in via gaugefields living on a stack of D5 branes in Anti-deSitter space. Under the appropriateconditions, the D5 branes blow up to form a D7 brane which is capable of forming acharge-gapped state. We add finite temperature by including a black hole which allowsus to compute the low temperature entropy of the quantum Hall system. Upon includingan external electric field (again as a gauge field on the probe brane), the conductivitytensor is extracted from Ohm’s law.iiPrefaceThis dissertation is an elaboration and summary, with permission, of reference [5], ofwhich I am an author. Figures 8.1, 8.4, 8.2, 8.5, and 8.3 are taken directly from thatpaper. All other figures and tables are my original work. The calculations reported inChapter 8 and Appendix A were originally performed by Charlotte Kristjansen. GordonSemeno↵ supervised the entire project, and was responsible for proposing and advisingthe computations I performed within.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Anti-de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 The boundary of AdS . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 The Poincare´ patch . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Stress-energy and conformal fields . . . . . . . . . . . . . . . . . . 112.3 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Classical string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 The bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Quantum string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.1 A quantum theory of gravity . . . . . . . . . . . . . . . . . . . . . 192.5.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 Chan-Paton factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.4 The superstring spectrum . . . . . . . . . . . . . . . . . . . . . . . 232.5.5 The GSO projection . . . . . . . . . . . . . . . . . . . . . . . . . . 253 D Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 The existence of D branes . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.1 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.2 R-R charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 The D brane action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 D brane gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 The Born-Infeld action . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 D brane dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Dirac-Born-Infeld and super Yang-Mills . . . . . . . . . . . . . . . 343.3.2 Wess-Zumino terms . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.3 The Myers e↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 The AdS/CFT Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 38iv4.1 The geometry of D branes . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.1 Strings in curved space-time . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Supergravity actions . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.3 D3 geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.4 Ramond-Ramond 4-form solution . . . . . . . . . . . . . . . . . . . 414.2 The correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 p =Dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.2 Maldacena’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . 444.2.3 Large N expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.4 Di↵erent forms of the conjecture . . . . . . . . . . . . . . . . . . . 484.2.5 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.6 Excerpts from the dictionary . . . . . . . . . . . . . . . . . . . . . 504.2.7 Tests of AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.8 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.9 Probes, flavours, and defects . . . . . . . . . . . . . . . . . . . . . 545 The Quantum Hall E↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Condensed matter approach . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1.1 Semi-classical description . . . . . . . . . . . . . . . . . . . . . . . 575.1.2 Quantized conductivity . . . . . . . . . . . . . . . . . . . . . . . . 585.1.3 2D electron gas model . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 D3-D5 system at weak coupling . . . . . . . . . . . . . . . . . . . . . . . . 616 A Giant D5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1 D3-D5 becomes D3-D7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1.1 D3-D5 at strong coupling . . . . . . . . . . . . . . . . . . . . . . . 656.1.2 D7 as a giant D5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.1.3 D5-D7 phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . 717 Entropy of the Giant D5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1 Entropy calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.1 Entropy of D5 branes . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.2 Comparison to single quark entropy . . . . . . . . . . . . . . . . . 767.1.3 Entropy of the D7 brane . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Embedding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.3 Weak coupling entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 Conductivity of the Giant D5 Model . . . . . . . . . . . . . . . . . . . . . . . . 868.1 D5 brane case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.1.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.2 D7 brane case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.2.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Appendix A D5 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105vList of Tables2.1 N = 1 supersymmetry states. . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Extended supersymmetry states. . . . . . . . . . . . . . . . . . . . . . . . 154.1 AdS/CFT dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 D5 orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.1 D7 orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70viList of Figures2.1 Hyperboloid of AdSp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Penrose diagram of AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 A graviton mode in the closed string spectrum. . . . . . . . . . . . . . . . 213.1 An example of T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 A double line propagator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 A double line loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.1 D7 brane embedding solutions at r = rh = 0. . . . . . . . . . . . . . . . . 817.2 Embedding solutions at di↵erent temperatures . . . . . . . . . . . . . . . 827.3 Low temperature entropy versus filling fraction . . . . . . . . . . . . . . . 837.4 Zero-temperature entropy at weak coupling. . . . . . . . . . . . . . . . . . 858.1 Deviation of the holographic conductivity from the classical conductivityfor rh = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2 Deviation of the holographic conductivity from the classical conductivityfor rh = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.3 Hall conductivity vs filling fraction . . . . . . . . . . . . . . . . . . . . . . 958.4 Longitudinal resistivity for rh = 0.2 . . . . . . . . . . . . . . . . . . . . . . 968.5 Longitudinal resistivity for rh = 0.4 . . . . . . . . . . . . . . . . . . . . . . 97viiAcknowledgementsI would like to thank Gordon Semeno↵ for guiding this project. I have much gratitudefor Charlotte Kristjansen, who not only was the first to perform the entire conductivitycalculation, but also gave valuable insights for other parts of the work. Lastly, I thankNiko Jokela for sharing his insight into the D7 embedding coordinates, and NSERC forfunding this research.viiiDedicated to Julia Seymour, for her unwavering support.ixChapter 1IntroductionFor a long time, string theory has been hailed for its potential as a grand unifying theoryand a consistent quantum description of gravity; all in all, a reductionist’s dream. In thelast fifteen years, however, some of the most promising string theory research has beendone in more emergent phenomena contained in strongly coupled field theories, withapplication to quantum chromodynamics, and more recently, many condensed mattersystems. This complete turnaround is due to the AdS/CFT correspondence, conjecturedby Juan Maldacena in 1997 [1]. The correspondence establishes an equivalence between ahighly symmetric gauge field theory (N = 4 Super Yang-Mills) and a super-string theory(Type IIB) in a higher dimensional Anti-de Sitter space (AdS). The backbone of thecorrespondence really is a duality in the description of extended p-dimensional objectsin string theory called Dp branes. In a sense, we can think of the gauge field theoryas living on the boundary of AdS. Really, though, these two theories, the boundaryfield theory and interior string theory are two descriptions of the exact same thing.They contain exactly the same information, and as a result, there is an entire dictionaryrelating ingredients on each side. Recently, this correspondence has found applicationto certain condensed matter systems. The string theory model in AdS gives us a so-called holographic model of the physics on the boundary. What makes this dualityincredibly useful is that it is a strong coupling to weak coupling duality. That is, theholographic models probe the strong coupling regime of the boundary field theory usingonly weakly coupled string theory. At low energies, the problem reduces to solvingclassical supergravity.In this thesis, we study a particular condensed matter system in the strong couplingregime, that of a two-dimensional fermion gas in the presence of an external magneticfield. It is well known that the magnetic field causes the fermions to organize themselvesinto Landau levels, which are filled at integer values of the filling fraction, which is the1ratio of charge density to magnetic field. At these values, the system is in a charge-gapped state and the transverse (Hall) conductivity is quantized. This, combined withimpurity e↵ects leads to the so-called Quantum Hall e↵ect, wherein the conductivityforms plateaus at it’s quantized values as a function of the filling fraction.To find the appropriate holographic model to describe this system, we follow reference [2],in which a stack of D5 branes, which are treated as probes, are embedded in AdS to givefermionic flavour fields in the corresponding field theory. Finite density, and externalelectric and magnetic fields can be included by adding gauge fields to the probe brane,while finite temperature can be included by making a black hole in the interior of AdS.With the right background supergravity fields, a dielectric e↵ect occurs in the D5 branes,causing them to blow up into a single D7 brane in the interior of AdS [3]. At integerfilling fraction, this D7 is able to e↵ectively absorb the field lines of the D5s and cap o↵before entering the black hole horizon of AdS. This is a charge gapped quantum Hallstate.The purpose of this thesis is to compute two essential properties of this holographicquantum Hall system. The first is the entropy of the system at low temperatures,and the second is the conductivity tensor. We will find that both of these exhibitdiscontinuities as a function of filling fraction, but the hallmark plateaus of the quantumHall e↵ect will be absent due to the simplifying assumption of translational symmetryin our model. As with all e↵ective models this will not capture all of the physics of itsintended system of study. However, we hope that this perspective will o↵er a new kindof intuition for condensed matter physicists studying quantum Hall systems, whereinthe diculties of understanding strongly correlated electrons is replaced by a ratherelegant gravitational paradigm. Indeed, we will find some indication of the possibilityof Hall plateau formation without impurities to break translational symmetry. This isan exciting alternative for the traditional explanation of conductivity plateaus.The outline of this thesis is as follows. There are several essential ingredients that gointo the AdS/CFT correspondence (and its extension to the whole paradigm of gauge/gravity duality). The first two are in the name: anti de-Sitter space, and conformalfield theories. The third is supersymmetry, since both the gauge field theory on theboundary and the string theory in the interior are supersymmetric. Lastly, of course,is string theory. All of these subjects are covered somewhat briefly in chapter 2. Thereal focus of this list will be on non-perturbative objects of string theory, D branes,since these are really at the core of the AdS/CFT correspondence. Thus, chapter 3 isdevoted to understanding the existence and dynamics of these objects. Chapter 4 willoutline the actual correspondence and how it is used in general. In chapter 5, we explainin detail the quantum Hall system that motivates our study, specifically looking at its2weak coupling behaviour. Chapter 6 is a construction of the giant D5 model, in whichthe stack of D5 branes blows up to form a D7. This chapter is e↵ectively a summaryof references [2] and [4]. Chapters 7 and 8 outline the computation of the entropy andconductivity tensor respectively. The details are relegated to the appendix. Lastly, inchapter 9, we discuss some implications and directions for future work.We should note that throughout this thesis, we will work in natural units unless otherwiseindicated.3Chapter 2PreparationsIn this chapter we discuss four main ingredients that go into the AdS/CFT correspon-dence: Anti-de Sitter space-time, conformal field theories, supersymmetry, and stringtheory.2.1 Anti-de Sitter space2.1.1 The boundary of AdSConsider a flat (p+ 1)-dimensional metric with two time-like directions i.e. a signature(2,p-1) flat metric.ds2 = dX20 + dX21 + . . .+ dX2p1  dX2p . (2.1)Now embed in this space-time a hyperboloid with a radius of curvature L (see figure2.1).X20 +X21 + . . .+X2p1 X2p = L2. (2.2)This hyperboloid defines a p-dimensional Anti-de Sitter space (AdSp), which is a maxi-mally symmetric solution of Einstein’s equations. Note that the two time-like directionsallow for closed time-like curves that wrap around the hyperboloid. It is conventionalto ‘unwrap’ the hyperboloid to avoid this. This amounts to taking the universal cover,allowing 1  t  1. Lastly, we can see from (2.2) that the isometry group of AdSp isSO(p 1, 2), in analogy with the usual 4D Poincare´ isometry group of Minkowski spaceSO(3, 1)4Figure 2.1: The hyperboloid of AdSp embedded in Euclidean space. The red bandillustrates a closed timelike curve.Re-wrting the metric in ‘hyper-polar’ variables gives the global coordinatesds2 = L2( cosh2(⇢)d⌧2 + d⇢2 + sinh2(⇢)d⌦2p2), (2.3)where d⌦p2 is the line element of a (p  2)-sphere. We call the location ⇢ = 0 thePoincare´ horizon.Often, it proves useful to view AdS as a Penrose diagram. To do so, we need to bringthe boundary at x = 1 to a finite coordinate. This can be done with the change ofvariables tan ✓ = sinh ⇢. In this case, the metric becomesds2 =L2cos2 ✓(d⌧2 + d✓2 + sin2 ✓d⌦2p2). (2.4)The Penrose diagram for AdS2 is just a strip, and AdS3 is cylinder (figure 2.2)1. Thiscylinder has boundary R⌧ ⇥ Sp2. The boundary is very important to AdS/CFT, andwe should note that there are many alternative descriptions of it.We can immediately find one very interesting property of this space. If we consider amassless particle at a constant angle on the (p 2)-sphere, its geodesic satisfies ds2 = 0.In global coordinates, this impliesd⇢d⌧= cosh ⇢, (2.5)or in the Penrose coordinates,d✓d⌧= ±1. (2.6)1The cylinder shows the important features of AdS5 as well, we only need to tack on 2-spheres aroundthe circumference.5Figure 2.2: The left diagram shows AdS2 in Penrose coordinates. The right diagramshows AdS3. The shaded regions are the Poincare` patch.In either coordinate, we can integrate this to get the proper time it takes to the particleto travel from the horizon to the boundary⌧ =Z 10d⇢cosh(⇢)=⇡2. (2.7)By symmetry, the particle can travel to the boundary and back again in a proper time⌧ = ⇡. Now consider a stationary observer with d⇢ = 0 located at the Poincare´ horizon.They measure a time interval d⌧ 0 = L cosh(0)d⌧ . So in the interval when the masslessparticle completes its round trip, the observer at the horizon measures a time⌧ 0 = L⇡. (2.8)Thus, even though the boundary is an infinite distance away, signals can be sent toand from the boundary in a finite time. This makes AdS space very unusual and veryinteresting. It means that boundary conditions at infinity can continually a↵ect the dy-namics deep in the interior of the space-time. It also means that Feynman diagrams are6very di↵erent. Usually, in calculating Feynman diagrams, we assume asymptotic statesthat are plane waves at infinity. Now the physics on the boundary of AdS determinesthe asymptotic states.We could also consider massive geodesics. Such particles have four-velocities Uµ thatsatisfy the time-like condition UµUµ = 1, which implies that) L2 cosh2(⇢)✓d⌧d◆2+✓d⇢d◆2= 1, (2.9)where  is the proper time of the particle, and we have considered a constant position onthe p-sphere. Note that the metric is independent of ⌧ and therefore has time-translationsymmetry and a corresponding Killing vector @⌧ (or in components kµ = µ⌧ ). But forany Killing vector, we know that kµpµ = c for some conserved constant c. Thus,kµ =  cosh2 ⇢⌧µ (2.10))12✓d⇢d◆2c22 cosh2(⇢)= 12L2. (2.11)(2.11) is an energy conservation equation. The first term on the left is the kinetic energyof the particle, the second term can be interpreted as a potential energy, and the totalconserved energy is the quantity on the right side, which is always negative. Thus anymassive particle will always be trapped by this potential, and will never be able to escapeto infinity. This makes AdS a convenient box within which to do physics.2.1.2 The Poincare´ patchWhile insightful, the global and Penrose coordinates defined above will not be usedfor calculations. Instead we transform to Poincare` coordinates. To do so, one simplyfollows the analogous transformations that take a Schwarzschild metric into Eddington-Finkelstein coordinates. We definer = Xp +Xp+1, v = Xp1 Xp, Xµ = rLxµ. (2.12)Note that the boundary is now at r = 1. The equation of the hyperboloid (2.2) nowreads L2 = rv +r2L2⌘µ⌫xµx⌫ , (2.13)where ⌘µ⌫ is a (p 1) dimensional Minkowski metric. The induced metric on the hyper-boloid is given by the pullback gij = ⌘ab @Xa@xi @Xb@xj , which simplifies tods2 =r2L2⌘µ⌫dxµdx⌫ + L2r2 dr2. (2.14)7These are the Poincare´ coordinates for AdSp. For this thesis, we will be focusing on thecase of p = 5. These coordinates only a cover half of the hyperboloid, which is referredto as the Poincare´ patch (shown in figure 2.2).There is one last set of coordinates we will need. These are Fe↵erman-Graham coordi-nates given by the transformationz =L2r, (2.15)which gives the transformed metricds2 =L2z2(⌘µ⌫dxµdx⌫ + dz2). (2.16)The boundary in these coordinates is at z = 0. In the opposite limit, z ! 1, thetimelike killing vector becomes null since at a stationary position, d⌧ = L/zdx0 Thus,the Poincare´ horizon is located at z !1.We might also ask if there are any spherically symmetric solutions to Einstein’s equationsthat are asymptotically AdS. Such a solution could describe a black hole embedded inthe space-time. One can check that the following metric in Poincare´ coordinates isexactly such a solutionds2 =r2L2(h(r)dt2 + d~x2) +L2r21h(r)dr2, (2.17)where ~x = (x1, . . . , xp2), h(r) = 1 r4h/r2, and rh is the radial coordinate of the blackhole horizon.2.2 Conformal field theoryConformal symmetry enters the AdS/CFT correspondence in two important but verydistinct ways. The conformal field theory, (CFT) in this correspondence is a quantumtheory with conformal invariance which is used to describe some system in 3+1 dimen-sions. But conformal symmetry enters the AdS side as well, as an invariance on thetwo dimensional string world-sheet. The details of 2D conformal theories involve somesubtleties, and so we will focus here on properties of theories with dimension greaterthan two.82.2.1 Conformal transformationsRecall that conformal transformations are local scale transformations of the metricg0µ⌫ = ⌦(x)gµ⌫ . (2.18)If we let gµ⌫ be the Minkowski metric, then the transformations with ⌦(x) = 1 givesthe isometry group of Minkowski space, which is the Poincare` group. Thus, we seethat the Poincare` transformations are a subset of conformal transformations. Conformaltransformations have the special property that they preserve the angles between vectors.Now if we make an infinitesimal transformation xµ ! xµ + ✏µ(x), then the metrictransforms asgµ⌫ ! gµ⌫  @µ✏⌫  @⌫✏µ +O(✏2). (2.19)Demanding that this be a conformal transformation yields the equation(@µ✏⌫ + @⌫✏µ) = f(x)gµ⌫ (2.20)) f(x) =2d@µ✏µ(x), (2.21)in d dimensions.Let’s work in Euclidean space with metric ⌘µ⌫ for simplicity, then (2.20) implies that@⇢@µ✏⌫(x) + @⇢@⌫✏µ(x) = @⇢f(x)⌘µ⌫ (2.22)) @⌫f(x)⌘⌫⇢ + @µf(x)⌘⌫⇢  @⇢f(x) = 2@µ@⌫✏⇢ (2.23)) 2@2✏⇢(x) = (2 d)@⇢f(x). (2.24)(2.25)Taking a derivative of (2.24) and (2.20) gives) 2@⇢@2✏⇢(x) = (2 d)@2f(x), & 2@µ@2✏µ(x) = @2f(x)d, (2.26)which combine to give us the the condition for f(x) to be conformal(d 1)@2f(x) = 0. (2.27)We see that if d = 1, there is no constraint. Any smooth function f(x) is conformal.However, if d > 2, we havef(x) = A+Bµxµ (2.28)) ✏µ = aµ + bµ⌫x⌫ + cµ⌫⇢x⌫x⇢. (2.29)9aµ is a just a regular translation, and cµ⌫⇢ constitutes so-called special conformal trans-formations. These are essentially an inversion, followed by a translation, followed byanother inversion. To see the e↵ect of bµ⌫ , we plug it into (2.20)[@µ(b⌫⇢x⇢) + @⌫(bµ⇢x⇢)] = 2d@⇢⌘⇢✏⌘µ⌫ (2.30)) b⌫µ + bµ⌫ = 2db⌘µ⌫ . (2.31)If we write bµ⌫ as the sum of a symmetric and anti-symmetric part, bµ⌫ = Sµ⌫ + !µ⌫ ,then this equation restricts the symmetric part to beSµ⌫ = ↵⌘µ⌫ , (2.32)and the anti-symmetric part is unrestricted. Thus we can write the linear transformationasbµ⌫ = ↵⌘µ⌫ + !µ⌫ . (2.33)The first term is a dilation (sometimes called dilatation) or scale transformation, andthe second is a rotation.So we understand the a↵ect of these transformations on coordinates, but what aboutfields? Recall that we can write an infinitesimal field transformation as  !  iwG,where w is a parameter and G is a generator. For example, the dilation operation scalesthe coordinates and field viax0 = x (2.34)0(x) = (x). (2.35)If the dilation is infinitesimal, we have0(x) ⇡ (x) + @@ (2.36)= (x) + @µxµ, (2.37)giving the generator D ⌘ ixµ@µ.With this in hand, one can work out the conformal algebra, and with a suitable reor-ganizing of the generators, one can show that it satisfies the Lorentz algebra in d + 1dimensions. That is, the conformal group in Rp,q is isomorphic to SO(p + 1, q + 1). Inregular Minkowski space, this means the conformal group is SO(4, 2).10A primary field is one which transforms as0(x0) = @x0@x /d(x), (2.38)and  is referred to as the scaling dimension or conformal dimension. For a scalar fieldunder a dilation, we can use (2.34), (2.35) to get@x0µ@x⌫  = d (2.39)) 0(x0) = @x0@x /d(x), (2.40)so scalar fields are necessarily primary.2.2.2 Stress-energy and conformal fieldsConsider an infinitesimal conformal transformation xµ ! xµ+✏µ. Under this transforma-tion, we can look at the variation of the action using the definition of the Stress-EnergytensorS =12ZddxTµ⌫gµ⌫ . (2.41)Using the conformal transformation of the metric, (2.19), givesS =2dZddxTµµ @⇢✏⇢. (2.42)So if the stress-energy tensor has vanishing trace, then the theory is conformally invariant(at least classically). Of course, the converse is not necessarily true since we could have atransformation ✏µ with vanishing divergence. However, tracelessness of the stress-energytensor turns out to be a useful smoking gun for conformal invariance.We might ask what kind of field theories can be conformally invariant. Actually, we willrephrase this as what kind of field theories can be scale invariant, since it is very unlikelythat a theory is scale invariant but not conformally invariant [6]. Consider a masslessscalar fieldL =12(@⌫)(@⌫), (2.43)and the transformationx0µ = e↵xµ (2.44)0(x0) = (x)e↵ (2.45)= (x) (@µ(x)xµ +(x))↵ +O(↵2). (2.46)11Then the change in the Lagrangian isL = ((@µ)2 + (@µ)2 + (@µ)x⌫@µ@⌫). (2.47)This can only be written as a total divergence if  = 1, in which case,L = @µ✓12(@)2xµ◆. (2.48)Thus conformal invariance restricts  to be 1. The associated conserved current isjµ(x) = @µ + @µ@⌫x⌫ + Lxµ (2.49)⌘ T˜µ⌫ x⌫ , (2.50)where T˜µ⌫ is a modified stress-energy tensor that preserves its properties.Interestingly, if we worked out the change in the Lagranigan due a mass term m22, wewould getLm2 = m22 +m2xµ@µ, (2.51)which can only be written as a total divergence for  = 4. Thus conformal field theoriescannot include massive fields. This fits with our intuition that adding a mass intro-duces a preferred energy scale that breaks scale invariance. We could, however, makethe scalar field interacting by including a 4 term without destroying scale invariance.Furthermore, we can include invariants built out of the space-time coordinates. Poincare`invariance requires that these be composed of separation vectors |xi  xj |. Scale invari-ance requires ratios of these |xixj |/|xkxl|, and special conformal invariance upgradesthese to cross-ratios involving at least four points. |xi  xj ||xk  xl|/(|xi  xk||xj  xl|).Lastly, we should note that conformal symmetry can be (and often is) broken uponquantization of the field theory. Through the so-called operator product expansion [7],one can write products of operators at di↵erent points as (e.g. the stress-energy tensor):T (z)T (w) ⇠c/2(z  w)4+2T (w)(z  w)2+@T (w)(z  w). (2.52)The number c is the conformal anomaly and is set by the model and fields that arepresent. It tells you how the theory reacts to conformal symmetry breaking. If youintroduce finite length scales via boundary conditions (e.g. placing the theory on acylinder), then the free energy, vacuum expectation values, and correlation functionschange according to c.Conformal field theories are very di↵erent from other QFTs. For one thing, two-pointfunctions are completely determined in terms of their scaling dimensions by conformal12symmetry. For another, the S-matrix cannot be defined. Scale invariance preventsus from distinguishing between infinity and the interaction point, so asymptotic statescannot be well defined. We mentioned there is also an issue with using asymptotic statesin AdS, and that is far from the extent of the marriage of these two subjects. Indeed, wecan already see that the conformal group in 3 + 1 dimensions is precisely the isometrygroup of AdS5. This is the basis of what allows us to map degrees of freedom across theAdS/CFT correspondence.2.3 SupersymmetryThe particular field theories used in AdS/CFT are supersymmetric. Supersymmetry(SUSY) exchanges bosons and fermions. For example, if we start with a simple kineticLagrangian for a scalar  and a left-chiral Weyl spinor , we might guess that the super-symmetry transformation is  !  + ⇠ · , where the parameter ⇠ must be Weyl spinorfor Lorentz invariance. Then, dimensionality and invariance of the action determine thelinear transformation of the fermionic field,  !  + (@µ)µ2⇠⇤, where µ are Paulimatrices.Thus the SUSY transformations are parameterized by a two-component spinor ⇠ andits complex conjugate, giving a total of four supercharges denoted by the two left-chiralWeyl spinors Q, Q†. From the corresponding generator, one can compute the SUSYalgebra [8]{Qa, Qb} = 0, {Q†a, Q†b} = 0,{Qa, Q†b} = (µ)abPµ. (2.53)This algebra works when the operators are applied to scalars, but for Weyl spinors, thealgebra requires the Weyl equation in order to close. In other words, the SUSY algebradoes not close o↵-shell. This is inconvenient, and can be worked around by adding anauxiliary field F that vanishes on-shell. The simple kinetic Lagrangian then becomesL = @µ@µ† + †i¯µ⇠ + FF †. (2.54)The linear transformation of F is set by SUSY and Lorentz invariance to be F ! F i⇠†¯µ@µ, and the transformation of  must be modified to  ! +(@µ)µ2⇠⇤+F ⇠.How do these charges act on states? Recall that in the Poincare` group, there areoperators that commute with all generators, called Casimir Operators. These are Pµ,and Wµ ⌘ 1/2✏µ⌫⇢M⇢P⌫ . Wµ shares simultaneous momentum eigenstates with Pµ,13and it’s eigenvalue is the helicity quantum number h which is fixed for a given (massless)field (e.g. h = ±1 for a photon and h = ±2 for a graviton). If we include supercharges,then we find that[Qa,W 0] = 12(3)baQbP3. (2.55)In which case,W0(Q1|p, hi) = E(h+ 1/2)Q1|p, hi (2.56)) Q1|p, hi = |p, h+ 1/2i. (2.57)So supercharges shift the helicity by 1/2, which we would expect since they are respon-sible for exchanging fermions with bosons.The irreducible representations of the SUSY algebra are supermultiplets. From thealgebra (2.53), we see that there must be a state of minimal helicity since (Q2)2 = 0.Then Q†2|p, hmini = |p, hmin+1/2i. But this must be the maximum helicity since (Q†2)2 =0. Including CPT conjugates, we have the following possible states for massless fieldshmin Helicity States Name0 |0i, |0i, |1/2i, | 1/2i Chiral multiplet1/2 |1/2i, |1i, | 1/2i, | 1i Vector/ gauge multiplet1 |1i, | 3/2i, | 1i, | 3/2i Rarita-Schwinger fields (for ±3/2 states)3/2 |3/2i, |2i, | 3/2i, | 2i Graviton (for ±2) and Gravitino (for ±3/2)Table 2.1: N = 1 supersymmetry states.So far, we have only considered one set of supercharges Q1, Q2, Q†1, Q†2, but in principlethere could be N such sets with the charges indexed as QI , I = 1, 2, . . . ,N . The mostgeneral SUSY algebra is then{QIa, QJb } = (i2)abZIJ (2.58){QIa, QJ†b } = (mu)abIJPµ (2.59){QI†a , QJ†b } = (i2)ab(ZIJ)⇤, (2.60)where ZIJ is called the central charge. It is anti-symmetric in I and J , and thus vanishesfor N = 1. For N > 1, the supercharges may be transformed into one another via anSU(N ) rotation [9]. This symmetry is called R-symmetry.Although supergravity theories will be important in this thesis, on the CFT side of thecorrespondence, we will restrict ourselves to field theories without gravity. That means,we cannot have states with helicity 2 (more discussion on this in 2.5.1). In fact, we willfocus on multiplets with maximum helicity equal to one. It turns out that all multiplets14of an N > 4 theory contain helicities higher than one, so N = 4 will be our maximalsupersymmetry. We can then extend our table of massless multiplets as follows:States Multiplet Name Number of Helicity States2|0i, |1/2i, | 1/2i N = 1 Chiral multiplet 4|1/2i, |1i, | 1/2i, | 1i N = 1 Gauge multiplet 44|0i, 2|1/2i, 2| 1/2i N = 2 Hypermultiplet 8|1i, 2|1/2i, 2|0i, 2| 1/2i, | 1i N = 2 Gauge multiplet 8|1i, 4|1/2i, 6|0i, 4| 1/2i, | 1i N = 3 Gauge multiplet 16|1i, 4|1/2i, 6|0i, 4| 1/2i, | 1i N = 4 Gauge multiplet 16Table 2.2: Extended supersymmetry states. The coecient in front of each state isthe number of those states in each multiplet.Of particular importance to us will be the N = 2 hypermultiplet and the N = 4 gaugemultiplet. For the N = 2 hypermultiplet, we can combine the | ± 1/2i states intotwo Weyl fermions, and the four |0i states into two complex scalars, with an overallSU(2) R-symmetry. For the N = 4 gauge multiplet, we have four Weyl fermions ia,i = 1, . . . , 4, six real scalars Xj , j = 1, . . . , 6 and a single gauge field Aµ, with an overallSU(4) R-symmetry.Things look a little di↵erent for massive representations. In that case, one can simplifythe algebra by redefining QI in terms of linear combinations of supercharges to get{QI¯a±, QJ¯†b±} =  I¯¯Jba(M ± Z I¯ J¯). (2.61)From this, one can see that unitarity enforces the so-called BPS boundM  |Z I¯ J¯ |. (2.62)Whenever this bound is saturated, the corresponding multiplet is shortened; the repre-sentation has its dimension decreased by a factor of 1/2 and is called a 1/2 BPS state.From the above fields, we formulate Lagrangians that are invariant under supersymme-try. One particularly special Lagrangian is the N = 4 Super Yang-Mills theory, whichis a supersymmetric extension of a regular Yang-Mills theory to include the fields of theN = 4 gauge multiplet. The Lagrangian is [10]L = Tr✓14g2YMFµ⌫F˜µ⌫  iXi ¯i¯µDµi Xj DµXjDµXj + non abelian terms◆,(2.63)where Dµ is a gauge covariant derivative, and the non-abelian terms are composed ofproducts of field commutators with SU(4) R-symmetry (sometimes denoted SU(4)R)matrices. The couplings g and ✓I have mass dimension zero, which makes this theoryclassically scale invariant. That makes this Lagrangian very special indeed, because it is15has superconformal symmetry. In 3+1 dimensions, this is SO(4, 2)⇥SU(4). Incredibly,this additional conformal symmetry gets passed on to the quantum theory, where it isfound that the beta function vanishes identically i.e. the coupling does not run withscale. It is an excellent example of a CFT.2.4 Classical string theoryTake two tennis balls, tie them together with a string of length l, and spin them around.The tension acting on each ball is a constant, centripetal force:T =pvl/2(2.64)The balls have a combined angular momentum ofJ =2p2vT. (2.65)Now lets make these balls ultra-relativistic, so that v ⇡ c, and p2 = M2. We getJ / M2. (2.66)With this little thought experiment in mind, it’s no surprise that in the early 1960’s,when exactly this relationship turned up for the spectrum of mesons, particle physicistscontemplated the possibility that the quark anti-quark pair were bound together withsome kind of string. Rapid progress was made in understanding the dynamics of thisstring, but the model eventually bowed out to the mathematically elegant formulationof quantum chromodynamics, in which the “string” that tied the quark pair togetherwas seen to be nothing but a flux tube of gluons. For several years string theory laydormant, and when it returned, it was under the new guise of a quantum theory ofgravity. Ironically, string theory’s connection to strongly coupled theories like QCDwould turn out to be one of it’s most powerful features. However, the real connectionlies in the deep duality of the AdS/CFT correspondence.2.4.1 The bosonic stringIn any case, let us examine the dynamics such a one dimensional string. It’s trajec-tory through space-time will fill out a two dimensional world sheet parametrized bycoordinates , ⌧ . For an object in Minkowski space we want to extremize an actionproportional to the length of the object’s world line. So for a string, it’s natural that we16use an action proportional to the area of the world sheet. Indeed, if we give this stringa tensionT =12⇡↵0 , 2 (2.67)and a lengthls = p↵0, (2.68)then the dynamics should minimize the world-sheet area just as water with surfacetension minimizes its surface area. This gives us the Nambu-Goto actionS = TZd2p det , (2.69)where  is the pullback of the metric ⌘µ⌫ in D-dimensional space-time to the world sheet↵ = @Xµ@↵ @X⌫@ ⌘µ⌫ for embedding coordinates Xµ. This action is enticingly simple,containing only one free parameter (not bad for a potential theory of everything!), butwe could write it in a computationally wiser way if we recognize that the D embeddingcoordinates are just scalars living on the world-sheet. So the action should just be theaction for D scalar fields:S =14⇡↵0 Z d2p|g|g↵@↵Xµ@X⌫⌘µ⌫ . (2.70)This is the Polyakov action, and one can check that it gives the same equations of motionas Nambu-Goto. The di↵erence is that now, the metric on the world-sheet g↵ has it’sown dynamics and an equation of motionT↵ ⌘ 1p|g|Sg↵ = 0 (2.71))8<:X˙ ·X 0 = 0X˙2 +X 02 = 0, (2.72)where dots are di↵erentiation with respect to ⌧ and primes are di↵erentiation withrespect to . These can be considered as constraints on the embedding equations ofmotion. The first one simply says that the motion of the string must be orthogonalto , that is, the string supports only transverse oscillations. Note that the Polyakovaction has reparameterization invariance; we have a gauge freedom in how we write theworld-sheet coordinates. In particular, the light-cone gauge, ± = ⌧ ± , is convenientbecause it renders a simple equation of motion for Xµ,@+@Xµ = 0, (2.73)2↵0 is the Regge slope, the slope of the J vs M2 plot you would find in the tennis ball experiment17and simple constraints (2.72)(@+X)2 = 0 = (@X)2, (2.74)where @± ⌘ @± .Moreover, the general solution is a sum of independent + and  functions (left andright moving waves on the string).If we consider a closed string satisfying Xµ(, ⌧) = Xµ( + 2⇡, ⌧), then the solution isjust XµL +XµR withXµL(+) = 1/2xµ + 1/2↵0pµ+ + ip↵0/2Xn 6=0 1n↵µnein+XµR() = 1/2xµ + 1/2↵0pµ + ip↵0/2Xn 6=0 1n ↵˜µnein . (2.75)The first term is just the centre of mass position of the string. The second term is thecentre of mass momentum which we could define to be the zero mode via 1/2↵0pµ =p↵0/2↵µ0 . The remaining sum is just a Fourier sum of oscillation modes. The constraintequations then give a level matching condition Ln = L˜n = 0 whereLn = 1/2Xm ↵nm · ↵m, (2.76)L˜n = 1/2Xm ↵˜nm · ↵˜m. (2.77)The n = 0 constraint equations (2.74) provides the mass spectrum↵02M2 =Xm>0↵m · ↵m. (2.78)If instead we consider an open string, then there are two natural choices of boundaryconditions.x0µ(⌧, 0) = x0µ(⌧,⇡) = 0 (2.79)x˙µ(⌧, 0) = x˙µ(⌧,⇡) = 0 (2.80)The first is a Neumann condition, and the second is a Dirichlet condition. The Dirichletcondition will be very important to us. Choosing it for any direction means that thestring endpoints are at a fixed location in that direction. If we use Dirichlet conditionsfor the directions xi, i = 1, 2, . . . , p, then this means that the open string is fixed to18a p-dimensional hyper-surface. We can similarly find a mode expansion for the openstring spectrum analogous to (2.75).2.5 Quantum string theory2.5.1 A quantum theory of gravityTo quantize the theory, we can compute the conjugate momenta to the embeddingcoordinates Xµ and impose the equal-time Poisson bracket relations. Applying thecanonical quantization to the Fourier expansion for these coordinates (2.75) gives thefollowing commutation relations:[xµ0 , p⌫0 ] = i⌘µ⌫ (2.81)[xµ0 , x⌫0 ] = [pµ0 , p⌫0 ] = [↵µm, ↵˜⌫n] = 0 (2.82)[↵µm,↵⌫n] = [↵˜µm, ↵˜⌫n] = mm+n,0⌘µ⌫ . (2.83)The ↵’s define raising and lowering operators as to be expected since we are just dealingwith free quantum fields. As usual, we can build up the Fock space by demanding that↵µm annihilates the vacuum for m > 0. The full Hilbert space should also be spanned bythe momentum eigenstates so thatpµ0 |k; 0i = kµ|k; 0i (2.84)↵µm|k; 0i = 0. (2.85)So far we just have an infinite set of harmonic oscillators.Focusing now on the open strings, the Hamiltonian can be calculated from the stress-energy tensor to giveH =121Xn=1↵n · ↵n = L0. (2.86)When we promote this to a quantum operator, we have to worry about normal orderingwhich induces an extra term in the Hamiltonian,✏0 ⌘d 221Xn=1n. (2.87)19This is interpreted as the zero-point energy for open strings. Of course this vacuum en-ergy is infinite, but can be renormalized using the analytic continuation of the Riemann-Zeta function⇣(z) =1Xn=1 1nz (2.88)! ⇣(1) =112. (2.89)Then the renormalized zero-point energy becomes✏0 = d 224. (2.90)Now the level matching conditions from the previous section present us with a situationsimilar to that in QED where one imposes a gauge fixing condition to restrict the photonstates to physical ones. This is the Gupta-Bleuler prescription, and it proceeds the sameway for string theory. For a physical state | i, we must have (L0 + ✏0)| i = 0, whichgives the quantum mass spectrumm2 =1↵0 (N + ✏0), (2.91)where N is analogous to the harmonic oscillator number operator N =P1n=1 ↵n · ↵n.Equation (2.91) is peculiar because the ground state, N = 0, has m2 = ✏0↵0 < 0. Anegative mass-squared indicates a Tachyon, but it simple means that we are probingaround a maximum of the potential energy, so this vacuum appears to be unstable.Fortunately, this mode is quite naturally removed by introducing supersymmetry. How-ever, there is an even worse problem with the first excited states, N = 1. The mass-shellcondition m2 = 1↵0 (1 + ✏0) yields massive states of negative norm (giving negative prob-abilities). However, the massless states are perfectly fine. Thus in order to circumventthese negative norms, we are forced to require these states be massless. Thus,(1 + ✏0) = 0 (2.92)) d = 26. (2.93)This is the critical number of dimensions for bosonic string theory. These first excitedstates describe polarizations of a massless spin 1 vector with U(1) gauge symmetry i.e.a photon field [11].Turning now to the closed string, we find the mass spectrumm2 =4↵0 (N  1), (2.94)20which also includes a Tachyon mode for N = 0. For N = 1, m2 = 0, so the first excitedlevel is a massless mode. Like the photon, it has d 2 = 24 polarizations but now mustbe a tensor product of left and right-moving states⇣µ⌫(↵µ1|k; 0i ⌦ ↵˜⌫1|k; 0i), (2.95)where ⇣µ⌫ is the polarization tensor. Each of the states in the tensor product transform asa vector in SO(24). The polarization tensor can then be decomposed into the irreduciblerepresentations of SO(24): a symmetric rank 2 tensor, an anti-symmetric rank 2 tensor,and a trace⇣µ⌫ = gµ⌫ +Bµ⌫ + ⌘µ⌫ . (2.96)The anti-symmetric field Bµ⌫ is called the Kalb-Ramond field. The scalar  is the dilatonfield that turns out to dynamically generate the string coupling gs [12]. gµ⌫ is a masslesssymmetric spin-2 field. This must be a graviton field, which in a coherent state, formsthe space-time metric.3In hindsight, we could have expected that the closed string should contain a gravitonsince there is a mode of oscillation which is exactly the same as the oscillation of space-time points during the passing of a gravitational wave, as shown in figure 2.3.Figure 2.3: A graviton mode in the closed string spectrum.Note that while the first excited states produced a gauge theory for the open string,they produced a gravitational theory for the closed string.3How do we know this field corresponds to the graviton? Suppose we were to set out searching forthe characteristics of a graviton candidate. We could determine its spin by process of elimination [13].The graviton cannot be spin-0 since we cannot couple a scalar to a photon and so light would not bendaround massive objects. It cannot be spin-1/2 or in fact any half-integer spin, since then the emissionof a graviton would alter the internal angular momentum of the gravitating particle, and so could notproduce a static force. Odd integers are eliminated also since they create potentials that are repulsive forlike charges and attractive for unlike charges. Spin-2, however, produces a universally static potential tofit the bill. To make the force long range, the field must be massless. Furthermore, we have to eliminatethe antisymmetric part which would appear in the Lagrangian as a copy of a Maxwell field. So ourbest candidate for the graviton is a massless symmetric spin-2 field. But in fact, the converse of thisis also true [14]. Any massless interacting spin-2 field must have di↵eomorphism invariance as a gaugesymmetry. And so the gµ⌫ field we found above is in fact the graviton field!212.5.2 Perturbation theoryWe could carry on and work out Feynman diagrams and perturbation theory for thebosonic string, but we will be more interested in superstring theory and in particular,its non-perturbative elements. However, there are a few important things to mentionabout perturbative string theory. First, we use an operator formalism to replace excitedstates with vertex operators. This is done by mapping the closed string world-sheetcylinder to the complex plane, and the open string world sheet strip to the upper half-plane. Then incoming states are mapped to vertex operators at the origin, and creationoperators ↵µn are given by residues of the vertex operator @nz xµ(z).The success of string perturbation theory, at least at the few loop level, lies in the factthat it does away with the UV divergences of quantum field theory that arise when theproper time of the internal virtual particle goes to zero, i.e. when an internal loops shrinkto a point. For example, the one loop vacuum diagram is free to contract to a pointin QFT, but the corresponding torus of the closed string diagram has a circumferencebounded below by the size of the string. The path integral for each Feynman diagramreduces to an integral over the moduli space of that diagram’s topology. The modulispace is a space of parameters defining the Riemann surface of the world-sheet modulocoordinate transformations and conformal transformations. In Euclidean signature, itis finite-dimensional [15].2.5.3 Chan-Paton factorsIn the next section we will introduce fermions on the world-sheet, but there is stillsomething missing from our bosonic theory. Ultimately, we hope that string theoryreproduces the standard model, including QCD. To have any chance of describing QCD,we need to include non-abelian fields. The U(1) gauge field in the open string spectrumis not enough. We would like to upgrade this to a U(N) gauge theory. Recall that openstrings with Dirichlet boundary conditions have endpoints that are fixed on hyperplanesin the target space. Suppose there are N such hyperplanes. Since any string can end onany one of N possible hyperplanes, we need to index each string endpoint by assigningit a state i taking on N possible values. Then a general open string state |k; ai canbe decomposed in a basis of N ⇥ N matrices  called Chan-Paton factors. Our pathintegral will then include sums over all possible arrangements of the endpoints, whichcan easily be checked to be invariant under U(N) rotations of  ! UU1. (2.97)22Indeed, we will find that these factors describe part of the gauge symmetry of D branes.2.5.4 The superstring spectrumClearly, we need to introduce fermionic fields if we want to reproduce the standardmodel particles. To do this, we a add Majorana spinors  µ to the world-sheet4. Addinga kinetic Dirac term to the Polyakov action gives:S =14⇡↵0 Z d2(@aXµ@aXµ  i ¯µ⇢a@a µ), (2.98)where ⇢a is a 2 ⇥ 2 Dirac matrix. If we write the two component Majorana spinor as =   +!, then the constraints (2.72) become:(@±X)2 + i2 ± · @± ± = 0. (2.99)In addition, there is a constraint on the conserved current Jµ associated with the super-symmetry of the action:J± ⌘  ± · @±X = 0. (2.100)The equations of motion for  ± yield two consistent fermionic boundary conditions forthe open string: µ+(⌧,⇡) =  µ(⌧,⇡) ! Ramond Sector (2.101) µ+(⌧,⇡) =  µ(⌧,⇡) ! Neveu Schwartz Sector. (2.102)The (anti)periodic condition lets us write  ± as the Fourier series ±(⌧,) = 1p2Xr  µr eir(⌧±), (2.103)where r is an integer for the Ramond (R) sector, and a half-integer for the Neveu-Schwartz (NS) sector.For the closed string, we have to choose boundary conditions for the left and rightmoving modes individually, giving a total of four possible sectors of the theory: NS-NS,4Recall that Majorana spinors are their own charge conjugate [16]23R-R, NS-R, R-NS. The Fourier expansion is µ+(⌧,) = Xr  ˜µr e2ir(⌧+) (2.104) µ(⌧,) = Xr  µr e2ir(⌧). (2.105)Requiring non-negative norms now results in a critical dimension of d = 10, and azero-point energy of:✏0 = 0 (R) (2.106)✏0 = 12(NS). (2.107)The mass-shell constraint is the same as for the bosonic case, only now the numberoperator counts fermionic modes as well 5.Let us look at the open string spectrum first. The ground level is still tachyonic. Thefirst excited level in the NS sector is obtained by acting on the ground state with onecreation operator with the smallest value of r, r = 1/2, which gives  µ1/2|k; 0iNS . Asbefore, the first excited levels are massless, and it turns out this level is actually a space-time boson, one that transforms as a vector under SO(8). We denote its representationby 8v. We have simply rediscovered the transverse polarizations of the original gaugefield Aµ, only now in ten space-time dimensions. In the R sector, the first excited levelis  µ0 |k; 0iR. The  µ0 vectors satisfy (up to an overall coecient) the Cli↵ord algebra{µ,⌫} = 2⌘µ⌫ , (2.110)and so give us spinors transforming in the representation 8s of Spin(8), i.e. masslessspace-time fermions.To produce the closed string spectrum, we take the tensor product of the above represen-tations in each possible combination of left and right-moving modes, and then decompose5The mode operator is nowLn = 12 1Xm=1↵nm · ↵m + 14Xr (2r  n) nr ·  r. (2.108)and the number operator isN =1Xn=1↵m · ↵n +Xr>0 r r ·  r. (2.109)24into the irreducible representations.NSNS : 8v ⌦ 8v ! gµ⌫ , Bµ⌫ ,  (2.111)NS R : 8v ⌦ 8s !  µ,  (2.112)RNS : 8s ⌦ 8v !  0µ, 0 (2.113)R R : 8s ⌦ 8s ! R R forms. (2.114)The NS-NS sector reproduces a ten-dimensional version of the bosonic fields we sawbefore. The  µ, and  0µ are spin-3/2 fields called gravitinos, and the , 0 are spin-1/2dilatinos. The R-R forms will be discussed shortly.2.5.5 The GSO projectionWe still have that annoying Tachyon hanging on in the ground state of the theory. Toget rid of it, we define a projection operatorPGSO =12(1 (1)F ). (2.115)In the NS sector we define(1)F ⌘ (1)Pr>0  r· r , (2.116)while in the R sector,(1)F ⌘ ±01 . . .9 · (1)Pr1  r· r . (2.117)The higher dimensional gamma matrices are discussed in 3.1.2. The important point isthat the operator (1)F , known as the Klein operator, has eigenvalue +1 for bosonicstates and 1 for fermionic states (on the world-sheet). Each fermionic oscillator con-tributes one power of (1), so even numbers of  µ produce world-sheet bosons, andodd numbers produce world-sheet fermions [12]. Applying the projection operator PGSOto all states removes states with an even number of  µ excitations. Then in the NSsector, the Tachyon mode is removed, and as a bonus, the new ground state has anequal number of bosonic and fermionic degrees of freedom. Indeed, it turns out the fullinteracting theory is supersymmetric after this projection.25Chapter 3D BranesThe purpose of this chapter is to illustrate the need for non-perturbative objects, Dbranes in the string theories we have just developed. Then, we will explain how to treatthese things as dynamical objects in their own right.3.1 The existence of D branes3.1.1 T-dualitySuperstring theory has left us with ten space-time dimensions to deal with. Reconcilingthese with the four dimensions we observe requires compactifying the remaining six.Compactification comes with its own uniquely stringy consequences. Suppose we com-pactify one dimension x9 on a circle of radius R. Then the corresponding momenta mustbe quantizedp90 =nR, n 2 Z. (3.1)A closed string will have a second quantum number w indicating the number of timesthe string winds around the compact dimension. This number, the winding number, hasno QFT analog. The oscillator zero-modes then have a component in the x9 directiongiven by↵90 =r↵02,✓nR+wR↵0 ◆, ↵˜90 = r↵02 ✓ nR  wR↵0 ◆. (3.2)We expect the string mass to increase with both momenta and winding number. If thecircle becomes very large, so that the curvature in x9 approaches that of flat space,then the theory should become just like an unbounded QFT. The momentum states willapproach a continuum of plane wave modes, and the winding states will cost too muchenergy to excite. In the opposite limit R ! 0, only very high frequency modes can26satisfy periodicity along x9, which cost too much energy. However, the winding modesare separated by less and less energy, forming a continuum. Explicitly, the spectrum forgeneral R is now given bym2 =n2R2+w2R2↵02 + 2↵0 (N + N˜  2), (3.3)and the level-matching condition is modified tonw +N  N˜ = 0. (3.4)The key insight found in equation (3.3) is that the tower of winding modes in theR ! 0 limit looks just like the tower of momentum modes in the R ! 1 limit. Thiscorrespondence is an exact duality. The spectrum is invariant under the exchangesR $↵0R, n $ w, (3.5)which can be implemented by a space-time parity transformation on just the right-movers.x9R(⌧  ) ! x9R(⌧  ). (3.6)Because of this duality, closed strings do not “feel” circles of radius R <p↵0. Such circlesare mapped onto large circles by the above duality known as T-Duality. Compactifiedgeometries with regions smaller thanp↵0 look very di↵erent to strings.Mathematically, open curves have no winding number, so for open strings, no new towerof winding modes arises upon campactification of x9. However, the interior of the openstring is identical to the interior of a closed string, and so even in the limit R ! 0,the interior of the string is free to vibrate in the full ten dimensions. The endpoints,however, must be constrained to a nine-dimensional hyper-surface, a D brane [17]. Putanother way, the limit R ! 0 causes the spectrum to appear as if another dimension isopening up [18].If we impose Neumann boundary conditions on the string endpoints, then@x9(⌧,)|=0,⇡ = 0 (3.7)) [@+x9L  @x9R]=0,⇡ = 0, (3.8)since @+xR = 0 = @xL. After applying the T-dual transformation (3.6), this becomes[@+x9L + @x9R]=0,⇡ = 0 (3.9)) @⌧x9|=0,⇡ = 0, (3.10)27Figure 3.1: An example of T-duality. Large momentum (purple) strings with smallwinding number on a large cylinder are mapped to small momentum (red) strings withlarge winding number on a small cylinder.which is a Dirichlet boundary condition that fixes the endpoints along one spatial di-rection. So T-duality interchanges Neumann with Dirichlet conditions on the stringendpoints. Even if you start with only Neumann conditions, T-duality makes fixed openstring endpoints truly unavoidable. Of course, we can “T-dualize” in more than onedimension, say 10  p dimensions, forcing Dirichlet conditions in p dimensions whichform a p-dimensional hyper-surface, a Dp-brane.Note that the transformation (3.6) must apply to fermions  as well since the worldsheetis supersymmetric, so that 9(⌧  ) !  9(⌧  ) (3.11)under T-duality. Similarly, the Dirac matrix 9 in the R-sector changes sign underT-duality, so that the whole Klein operator (1)F changes sign.3.1.2 R-R chargesWhat is the relevance of the sign of the Klein operator? Recall that there was a seeminglyinnocuous ± in the definition (2.117) for the Ramond sector. The sign of the Kleinoperator serves to define the space-time chirality of the fields. For the open string, this28is just a matter of convention, but for the closed string, we are free to choose the sign forleft and right-movers independently. The choice leads to two entirely di↵erent theories.Type IIA: We choose opposite chiralities for left and right movers. These are exchangedunder a parity operation, so the resulting theory is non-chiral.Type IIB : We choose the same chirality for left and right-movers. This theory does notrespect parity symmetry and so is chiral.Both of these theories have N = 2 space-time supersymmetry.There are three additional superstring theories we can construct: Type I, which is afurther projection of Type IIB, Heterotic E8⇥E8, and Heterotic SO(32). The Heterotictheories mix bosonic string theory and superstring theory for the closed string [19].However, we will restrict our attention to the Type II theories for this thesis.Now we see that the sign change of the Klein operator under T-duality means that TypeIIA and Type IIB string theories are swapped under T-duality.We are now in a position to fill out 2.114, the R-R sector of the superstring spectrum.Recall that the mass-shell constraint reduced the degrees of freedom of our fields so thatthey transform in Spin(8) which has irreducible representations 16 and 1601.For Type IIA, we choose opposite chiralities for left and right-movers, giving us therepresentation 16⌦160 which decomposes into irreducible representations that are even-rank di↵erential forms (the Cli↵ord algebra guarantees that the representations be an-tisymmetric). We can write the Weyl spinors in a basis of antisymmetric Dirac matrixproducts,( L)a( R)b¯ = Xn evenF (n)µ1...µn([µ1 . . .µn])ab¯ Type(IIA) (3.12)( L)a( R)b = Xn oddF (n)µ1...µn([µ1 . . .µn])ab Type(IIB), (3.13)where the F (n)µ1...µn are n-forms. Now the J = 0 constraint (2.100) on the open stringground state reproduces the massless Dirac equation on the spinors↵0 ·  0| Li ⌦ | Ri = 0. (3.14)1We can build up the Dirac representations of the Cli↵ord algebra 2.110 from tensor products of lowdimensional representations (ie. regular four-dimensional Dirac matrices). In particular, each time weincrease the space-time dimension by two, the size of the Dirac representation goes up by a factor oftwo. Thus in dimension d = 4 + 2n, the Dirac representation has dimension 4(2n) (since in 4-d, Diracspinors have four components). So for d = 10, n = 3, the Dirac representation is 32Dirac, which can bedecomposed into two independent irreducible representation Weyl spinors 16, 160. When n is odd, eachWeyl representation is self-conjugate and when n is even, they are conjugate to one another. d = 10 isspecial in that these Weyl spinors are also Majorana spinors [20]29This equation is more enlightening when written in terms of the “Field strength” F (n)µ1...µn ,@[µF (n)µ1...µn] = 0. (3.15)This is just like the covariant form of the Maxwell equation of motion, i.e. the Bianchiidentity, which defines the electromagnetic field strength in terms of a potential Aµ. Inthis case, there must also be a potential which is an n 1 form, C(n1)µ1...µn1 , satisfyingF (n)µ1...µn = @[µ1C(n1)µ2...µn]. (3.16)We call this n  1 form the R-R potential. This potential must be odd-rank for TypeIIA and even-rank for Type IIB. Furthermore, just as Maxwell’s equations imply thatAµ is invariant under a gauge transformation, equation (3.15) implies that these R-Rpotentials are gauge fields, with the following gauge transformationC(p+1)µ1...µp+1 ! C(p+1)µ1...µp+1 + @[µ1⇤(p)µ2...µp+1], (3.17)for any p-form ⇤(p).Now suppose we have a p + 1-form potential C(p+1). Then it must minimally coupleto a p + 1 dimensional world-volume. Thus, we must have fundamental dynamic p-dimensional objects in the theory. We have only seen one possible candidate, the p-dimensional Dp branes that open strings end on. We have already seen that thesesurfaces are associated with non-abelian degrees of freedom in the Chan-Paton factors.Now we know they also carry R-R charge.3.2 The D brane actionTo summarize, we now have a description of Dp-branes as p-dimensional objects onwhich string endpoints are restricted to move upon, that are charged under a p+1 formR-R potential. For Type IIA, the irreducible representations of 16⌦160 restrict these tobe 0,2,4,6, or 8 dimensional, while for Type IIB, they can be -1,1,3,5,7 or 9 dimensional.The D(-1) brane is called a D-instanton and is localized in both space and time. TheD3 is particularly important. It couples to a 4-form potential and thereby a 5-form fieldstrength F (5), which is self-dual⇤ F (5) = F (5). (3.18)303.2.1 D brane gauge fieldsRecall the T-dual coordinate x9 of the previous section. By computing its value at thestring endpoints, we can determine the length of the string along the T-dual directionx9(⌧,⇡) x9(⌧, 0) =2⇡↵0mR, m 2 Z. (3.19)But this is just an integer number of windings around the T-dual direction, i.e. m isthe open string quantum number analogous to the closed string winding number. If weperiodically identify positions separated by 2⇡↵0R in the x9 direction, then we see thatopen strings start and end on D branes at the same position in x9. Now suppose weadd Chan-Paton factors via the following constant gauge field only along the compactdimensionA9 =12⇡R0BBBBB@1 02. . .0 N1CCCCCA , (3.20)i.e. part of an abelian subgroup of the usual U(N) Chan-Paton factors. The additionof this field will introduce a term iqRAµx˙µ in the action, that will shift the quantizedcanonical momentum along the compact dimension so that a string in state |k; iji willhave momentumpAij = nR + j  i2⇡R . (3.21)Putting this into the mode expansion, we find that the T-dual string length has changed,x9(⌧,⇡) x9(⌧, 0) =2⇡↵0mR+ (j  i)↵0R (3.22)= 2⇡↵0✓mR+ (A9)jj  (A9)ii◆. (3.23)In other words, the string endpoint in state i is now located at 2⇡↵0(A9)ii (up to aconstant winding number in the T-dual direction). The string endpoints are still confinedto D branes, so we see that the introduction of the gauge fields (A9)ii and (A9)jj shiftthe ith and jth D branes from their original positions. Reversing this logic, we findthat setting i = 0 for all 1  i  N restores the U(N) Chan-Paton gauge symmetryand makes the N D branes coincident. This provides a remarkably simple string theoryversion of the Higgs mechanism. Suppose we start with massless open string states, thatis, strings with zero winding number m and momentum number n, then we introduce a31background abelian gauge fieldA9 =12⇡R0BBBBB@0 0i. . .0 01CCCCCA. (3.24)The mass-shell condition gains an extra term from the momentum pAii (3.21), so that itreadsm2ii = 1(2⇡↵0)2 (i)2, (3.25)and depends solely on the separation of the Di brane from the stack. So we startedwith N coincident D branes with U(N) symmetry, and as one of the D branes, Di,separates from the stack, this symmetry is broken U(N) ! U(N  1) ⇥ U(1). Butduring this process a massless open string connecting Di to the stack acquires a massas it is stretched. In section 7.1.2 we will examine this mass in more detail.We could of course generalize A9 to a gauge field Aµ with non-zero components i inall dimensions. Additionally, we could make i position-dependent i ! i(x). Asin the example above, the components of Aµ along the compact dimensions give thespace-time position of the Dp-brane. These form a set of 9  p scalar fields which, forposition-dependent i, describe the shape or embedding of the Dp-brane. This is a smallhint of gauge/gravity duality; a background gauge field describes the embedding of anenergetic, and therefore gravitating, object in space-time2. The remaining componentsAi, i = 0, . . . , p are just gauge fields living on the Dp-brane itself. Interestingly, Ncoincident D branes have embeddings described by matrices in the adjoint representationof U(N), which are understood in terms of non-commutative geometry [21].3.2.2 The Born-Infeld actionWhat is the appropriate e↵ective action to describe these gauge fields Aµ? The aboveanalysis shows us that we can interpret some of these fields in terms of the geometry ofthe D brane. If the D brane has tension, then the simplest action to consider would beone which serves to minimize the world-volume. Consider, for example, a D2-brane inEuclidean space with a constant background abelian gauge fieldAµ = x⌫F⌫µ. (3.26)2The D branes themselves have a tension32Now T-dualize the x2 direction. If we T-dualize one direction along a Dp-brane that isalready in place, then it is mapped to a D(p1)-brane, since the corresponding Dirichletcondition in that direction is replaced with a Neumann condition. Thus in this example,we are left with a D1-brane with positiony2 ⌘ 2⇡↵0x1F12 (3.27)in the T-dual direction, according to the discussion following (3.23). Then for a D1-branetension T1, the action is justS = T1ZD1 ds (3.28)= T1Zdy1p(@1y1)2 + (@1y2)2 (3.29)= T1Zdy1p1 + (2⇡↵0F12)2 (3.30)= T1Zdy1pdet(ab + 2⇡↵0Fab), (3.31)where a, b = 1, 2. Of course, we can generalize this to a Minkowski metric and T-dualizein six directions to getS = Tp Z d4xqdet(⌘µ⌫ + 2⇡↵0Fµ⌫), (3.32)which is the Born-Infeld action [22]. The Born-Infeld action first appeared long beforestring theory, in 1934, as a way to tame the divergence of the self-energy of a point-likecharge in classical electrodynamics [23]. The linearized equations of motion of (3.32) areprecisely Maxwell’s equations, and so it is a very natural generalization of the Maxwellaction to non-linear electrodynamics. The action was developed by introducing a limiton the field strength in the same manner that one introduces a limit on velocity for arelativistic particle. This must have seemed to be an arbitrary maneuver in 1934, but inthe context of D branes, it’s perfectly natural. The abelian background gauge field givesus electrodynamics on the brane. Part of the gauge field strength describes fluctuationsof the brane in space-time (from equation (3.27), v2 ⌘ @1y2 = 2⇡↵0F12). The speed ofthese fluctuations, v2, is bounded by the speed of light, and thus the corresponding fieldstrength is also bounded. It is this bound that eliminates the point-charge divergence.Lastly, we expect the tension Tp to be generated by the strings ending on the D brane.Constructing a quantity with dimensions of mass per unit volume out the basic stringtension parameter ↵0 yieldsTp = 1p↵0(2⇡p↵0)p . (3.33)333.3 D brane dynamics3.3.1 Dirac-Born-Infeld and super Yang-MillsThe Born-Infeld action we have written (3.32) assumes that the D brane is embedded inflat space. We should generalize this to a curved metric, and in fact allow the brane tocouple to all the massless supergravity fields in the low energy limit we are interested in.For the NS-NS sector, (2.111), this amounts to including gµ⌫ , Bµ⌫ and  in the e↵ectiveaction. At tree level, e = 1/gs, and gauge invariance under the transformationsAµ ! Aµ + ⇤µ2⇡↵0 , (3.34)Bµ⌫ ! Bµ⌫ + @µ⇤⌫  @⌫⇤µ, (3.35)gµ⌫ ! gµ⌫ + @µ⇤⌫ + @⌫⇤µ, (3.36)restricts the appearance of these fields in the action to the following form:S =Tpgs Z dp+1⇠pdet(gab + 2⇡↵0F). (3.37)Here ⇠a are the world-volume coordinates, gab is the pullback of the metric to the world-volume and F = Fab  12⇡↵0Bab. This is the Dirac-Born-Infeld (DBI) action.As with the Born-Infeld action, the linearized DBI action corresponds to the Maxwellaction, i.e. a U(1) Yang-Mills gauge theory3:SDBI = TPgs (2⇡↵0)24 Z dp+1⇠FabF ab +O(F 4) (3.38)SYM =  14g2YM Z dp+1⇠FabF ab. (3.39)To make this correspondence exact, we must setg2YM =gsTP (2⇡↵0)2 . (3.40)Here we see another reason why the D3 brane is special. From (3.33), for p = 3, therelationship between gYM and gs is independent of the energy scale ↵0.The DBI action describes an isolated D brane, but what action should be used for Ncoincident Dp branes? The answer is provided by a full non-abelian action proposedby R. C. Meyers in [3]. It will not be instructive to include it here, but we will discuss3More precisely, this is the dimensional reduction of 10d Yang-Mills to the world-volume of theDp-brane.34a very interesting result of this action, the Myers e↵ect, shortly. The low energy limitof the action naturally describes the low energy limit of the full U(N) N = 1 superYang-Mills theory [24]SYM =12g2YMZdp+1x[Tr(FabF ab) + 2iTr( ¯µDµ )]. (3.41)Recall that both Type II string theories have N = 2 space-time supersymmetry. ThusD branes preserve exactly half the supersymmetries of the bulk. Recall that such a stateis a 1/2 BPS state. These saturate the BPS bound (2.62) and thus have a charge that iscompletely determined by their mass [25]. For D branes, this gives the exact (and verypowerful) relationTp = qp, (3.42)where qp is the R-R charge of the Dp brane [24].Lastly, we should note that if we were to take the dimensional reduction of (3.41) to aflat 6-dimensional torus, we would get 4-dimensional N = 4 super Yang-Mills [9].3.3.2 Wess-Zumino termsWe know that the D brane necessarily couples to the R-R fields. To include this couplingin the e↵ective D brane action, one can repeat the procedure in equations (3.28)-(3.31)for the termRDp Cp+1. Using the T-dual transformations of the R-R potential [26], wearrive at the generic coupling of R-R potential to gauge fields for (possibly multiple) Dpbranes. It is a Chern-Simons (topologically invariant) termSCS =Tp2ZDpXq C(q) ^ Tr(e2⇡↵0F ), (3.43)where q runs over the dimension of all R-R potentials present. For Type IIB stringtheory, q is restricted to be even. In this case, expanding (3.43) to second order in FgivesSCS =NpTp2ZDp C(p+1)+Tp2 (2⇡↵0) ZDp C(p1)^Tr(F)+Tp4 (2⇡↵0)2 ZDp C(p3)^Tr(F^F),(3.44)where Np is the number of Dp-branes. Note that the integral picks out contributionsonly from terms that are p+1-forms to match the dimension of the world-volume. Thismeans the sum over q is limited to q < p + 1. For an abelian configuration, each traceis replaced by a factor of Np.35Later, we will see that the third term above is responsible for incompressible states inour model. The second term, however, deserves comment right now. It represents acoupling between a Dp brane and D(p  2) branes. The interpretation of this couplingis provided by the Myers E↵ect.3.3.3 The Myers e↵ectIn reference [3], Robert Myers proposed a non-abelian extension to the DBI world-volumeaction (3.37) by including non-commuting scalars i, i = p+ 1, . . . 9, that transform inthe adjoint representation of the U(1) gauge group of N coincident D branes (see section3.2.1). In that paper, he considered N D0-branes in a background 4-form field strengthF (4)tijk = ( 2f✏ijk i, j, k 2 {1, 2, 3}0 otherwiseHe then worked out the corresponding potential for the adjoint scalars, and their equa-tion of motion. The abelian configuration [i,j ] = 0 for separated D branes satisfiesthe equation of motion with a potential V () = 0. However, Myers showed that certainnon-commutative N ⇥N matrices i satisfy the equation of motion with a negative andtherefore preferred potential. Thus N separated D branes are unstable to condensationin this background field strength. A remarkable feature of non-commutative geometrylets us interpret this solution as a D2-brane wrapping a 2-sphere with N D0s bound toit. To check this, one simply writes down the metric for a D2 world-volume wrapping a2-sphere of radius R. Then include the same background R-R field (3.3.3), but insteadadd a U(1) field strength F✓ = f02 sin ✓. The single D2-brane is necessarily abelian, soone simply substitutes this solution into the regular DBI + Chern-Simons action. Thecorresponding potential has a unique stable extrema whose value matches that of thenon-commutative solution for the D0 case, provided we identify f0 with the number ofD0 branes, N .The field strength F✓ may look familiar. Indeed its appearance far predates that ofstring theory. In 1931, in an attempt to complete electric-magnetic duality, Dirac con-sidered an electron in the field of a hypothetical magnetic monopole [27]~B = g~rr3. (3.45)But Maxwell’s equations imply that no corresponding vector potential can be foundglobally for R3. If, however, we change to polar coordinates on S2 and define a gaugefieldA± = f02(±1 cos ✓)d (3.46)36which changes by a gauge transformation at the equator, then the potential covers thewhole sphere. But if this is the case, the corresponding wavefunction of the electron mustundergo a gauge transformation  + ! eieg/~  at the equator. The requirement thatthis wavefunction be single-valued yields Dirac’s famous relation between quantizationof electric and magnetic charge [28]. The magnetic monopole has a corresponding fluxf0 on the 2-sphere. If we calculate the field strength from A±, we getF✓ = f02 sin ✓. (3.47)Thus, in the picture above, we can identify the number of D0 branes, N with themagnetic flux on the 2-sphere wrapped by the D2.Note that the D0s we started with cannot carry net C(3) charge, and so neither can theD2. However, the D2 can carry a C(3) dipole and higher multipoles. So we see that theMyers e↵ect is precisely the dielectric e↵ect, but for R-R fields instead of electric fields.A neutral dielectric carries no charge, but when placed in an external electric field gainsan induced polarization. Similarly, a stack of D0s is not charged under C(3), but whenplaced in an external F (4) field, a C(3) polarization is induced. Indeed in both the D0and D2 picture, the leading term in the expansion of the R-R potential is /RdtF (4)t123which corresponds to the dipole of the R-R multipole expansion.This remarkable relation between D0s and a D2 known as the Myers e↵ect has general-izations to higher dimensional branes as well as holographic interpretations to go withthem [29]. In particular, the focus of this thesis is on the holographic realization of D5branes pung out to form a D7 brane.37Chapter 4The AdS/CFT CorrespondenceIt is clear from the previous chapter that string theory contains rich dynamics in thenon-perturbative regime described by D branes. So far, we have approached things fromthe perspective of the string world-sheet (and the brane world-volume), but if we lookfor an e↵ective theory over the whole space-time, we can elucidate the full non-lineardynamics. This should be expected, since at some level, the e↵ective theory should giveus classical general relativity with all of its non-linearity. In general relativity, solutionsof the linearized equations of motion are but a tiny subset of the really interestingsolutions. The others are solitons : localized, static, finite energy solutions like theRiesner-Nordstro¨m metric [30, 31]. These solutions cannot be found be perturbationsabout solutions of the linearized field equations. We will see that the identificationof these gravitational solitons with the gauge theory description of D branes is whatunderpins the entire AdS/CFT correspondence.4.1 The geometry of D branes4.1.1 Strings in curved space-timeWe can easily generalize the Polyakov action (2.70) to strings in a curved backgroundby replacing ⌘µ⌫ with Gµ⌫ to get the so-called Sigma modelS = 14⇡↵0 Z dp|g|g↵@↵Xµ@X⌫Gµ⌫ . (4.1)One can check that the corresponding path integral is that of the Polyakov action withthe insertion of an exponential of a graviton vertex operator i.e. a coherent state ofgravitons. So we have not added any new ingredients by a including a curved backgroundmetric [18].38In addition to reparameterization invariance, the Polyakov action has another symmetry:Weyl Invariance, g↵ ! e(,⌧)g↵ . Weyl invariance is crucial to string theory and mustbe retained at the quantum level. This requires us to ensure that the beta functions foreach of the coupling constants vanish, since otherwise, renormalized quantities wouldhave a scale dependence that would violate Weyl symmetry. Alternatively, we can recallfrom section 2.2.2 that conformal symmetry is linked to tracelessness of the stress-energytensor and imposeT↵↵ = 12↵0µ⌫(G)@Xµ@X⌫ = 0, (4.2)where T↵ = 4⇡p|g| Sg↵ is calculated using a renormalized background metric Gµ⌫ , whichis the renormalized coupling constant for the embedding scalars Xµ. Indeed, one findsthat the coecient µ⌫(G) above is identical to the beta function for this couplingconstant. More exciting is the value of this beta function:µ⌫ = ↵0Rµ⌫ . (4.3)From this we see that Weyl invariance requires Rµ⌫ = 0, which are Einstein’s equationsin vacuum!4.1.2 Supergravity actionsVanishing of the beta functions for all massless fields in the Type II superstring theoriescan be shown to be equivalent to the equations of motion for the following (truncated)e↵ective actions:SIIA = S0  1220 Z d10xp|G| 14(dC(1))2  148(dC(3) + dB(2) ^ C(1))21420ZB(2) ^ dC(3) ^ dC(3) + Fermion terms, (4.4)SIIB = S0 + 1220 Z d10xp|G| 112(dC(1) + C(0) ^ dB(2))2  12(dC(0))21480(dC(4) + dB(2) ^ C(2))2+1420Z(C(4) +12B(2) ^ C(2)) ^ dC(2) ^ dB(2) + Fermion terms. (4.5)Here S0 is the strictly NS-NS part;S0 =1220Zd10xp|G|{e2R + 4(r)2  112(dB(2))2, (4.6)39with R being the Ricci scalar. 0 is related to Newton’s gravitational constant GN via220 = 16⇡GN/g2s . (4.7)However, these actions do not completely encode the dynamics of the R-R fields, andneed to be supplemented with the self-duality of the five-form field strength (3.18). Inthese e↵ective actions, we have restricted ourselves to a low energy ↵0 ! 0 truncation,and are thus left with two classical gravitational theories. The actions (4.4) and (4.5)are the Type II supergravity actions.4.1.3 D3 geometryAs with Einstein’s equations, we can set about looking for non-perturbative solutions tothe equations of motion of the Type II supergravity actions by assuming a high degreeof symmetry. Indeed, using the Riesner-Nordstro¨m metric as a guide, one could look fora similar charged black-hole solution, only with the electric charge replaced by an R-Rcharge. Then one would discover the following particularly nice family of solutions thatare symmetric in p dimensions [32].ds2 =1pHp(r)⇢ 1 ✓rhr ◆7pdt2 + pXi=1 dx2i+qHp(r)⇢1 ✓rhr ◆7p1dr2 + r2d⌦28p◆, (4.8)e2 = g2sqHp(r)(3p), (4.9)C(p+1) = 1gs (Hp(r)1  1)dt ^ dx1 ^ . . . ^ dxp, (4.10)where we have definedr7pp ⌘ (2p⇡)5p✓7 p2 ◆gsN(p↵0)7p, (4.11)↵p ⌘ s1 + ✓ r7ph2r7pp ◆2  r7ph2r7pp . (4.12)The quantity N is the R-R charge 1.N =ZS8p ⇤F (p+2). (4.13)1This is just an integrated form of Gauss’ law: d ⇤ F = ⇤J )RS3 d ⇤ F = qe ) RS2 ⇤F = qe40The time-like killing vector of this metric becomes null at r = rh, so these solutionshave a horizon there. Also, the singularity at r = 0 cannot be removed by a coordinatetransformation; it is real. The sum over dxi elements reveals that these solutions spanthe p + 1 dimensions of Rp+1 and are localized in the remaining 9  p dimensions. Itis tempting to identify these metrics as those of a space-time distorted by a Dp-braneembedded in 9 + 1 dimensions. This temptation is well placed, but for now we will justcall the solutions black p-branes.The case p = 3 is particularly interesting. For that we haver4p = 4⇡gsN↵02= ↵02, (4.14)where we have defined the ’t Hooft coupling  ⌘ 4⇡gsN . Let us examine the nearhorizon r ⇡ rh regime and take the limit as r ! 0. We are going to take this limit ina very particular way. First, we fix the ratio u ⌘ r2pr = p↵0r , and then let u get verylarge, i.e. rp >> r. This corresponds to the so-called “throat” region of the space-time.In this limit,↵3 ! 1, (4.15)H3(r) !✓rpr◆4=↵02r4. (4.16)Before plugging this into the metric (4.8), it is convenient to make the transformationr ! rp↵0 , in which case,H3(r) !1↵02r4 , (4.17)H1/23 (r)r2 ! p↵0, (4.18)yielding the metricds2 =p↵0⇢r2✓1✓rhr◆4◆dt2+3Xi=1 dx2i + 1r2✓1✓rhr ◆4◆1dr2+d⌦25. (4.19)The t, xi and r coordinates appear precisely in the form of the AdS5 black hole metric,(2.17), so the full geometry we have uncovered is that of AdS5 ⇥ S5.4.1.4 Ramond-Ramond 4-form solutionBefore delving into the implications of (4.19), it is worth pausing to write out a morecomplete solution of the R-R 4-form for the choice p = 3. As mentioned before, the41presence of an even-rank R-R form indicates that we are working with a Type IIBsupergravity solution. From (4.10) and (4.17), we getC(4) =1gs (↵02r4  1)dt ^ dx ^ dy ^ dz. (4.20)But recall that we can make any gauge transformation of R-R potentials according to(3.17). This allows us to replace the constant 1 term in (4.20) with a constant ofour choosing. A convenient gauge choice is to make it equal to ↵02r4h (for example byadding a gauge term ⇤µ⌫⇢ = (1+r4h)t✏µ⌫⇢ with µ, ⌫, ⇢ 2 {x, y, z} in (3.17)). The resultingpotential is thenC(4) =↵02gs ✓1 r4hr4◆r4dt ^ dx ^ dy ^ dz. (4.21)If this was all we could do with the 4-form, we would have no hope of seeing anythinglike a quantum hall e↵ect. This is because the term of the D brane action that willultimately be responsible for incompressible states is the last term in (3.44). For thisterm to be non-zero, there needs to be a piece of the R-R 4-form with volume elementsindependent of the AdS5 piece of the metric. That only leaves the coordinates of theS5. For the remainder of this thesis, we will fibrate the S5 with two 2-spheres so thatd⌦5 = d 2 + sin2  (d✓2 + sin2 ✓d2) + cos2  (d✓˜2 + sin2 ✓˜d˜2). (4.22)Then we will postulate a term in the 4-form potential c( )2 d ^ d✓ ^ d^ d✓˜ ^ d˜. Fromthis we can calculate the corresponding field strengthF ✓✓˜˜ = g✓µg⌫g✓˜g˜⇢g Fµ⌫⇢ (4.23)=(p↵0)5sin4  sin2 ✓ cos4  sin2 ✓˜@ c( )2. (4.24)Then from the metric determinantp|g| =q(p↵0)10r6 sin4  cos4  sin2 ✓ sin2 ✓˜, (4.25)we can compute the Hodge dual of this field strength component⇤Frtxyz = 1p!p|g|✏⌫1...⌫prtxyzF ⌫1...⌫p (4.26)=r3sin2  sin ✓ cos2  sin ✓˜@ c( )2. (4.27)42Using the self-duality of F (5), (3.18), we can set this expression to be Frtxyz = 4(p↵0)2r3/gs,resulting in an equation for c( ):@ c( ) = (↵02)gs 8 sin2  cos2  sin ✓ sin ✓˜. (4.28)Integrating this expression givesc( ) =(↵02)gs sin ✓ sin ✓˜✓  14 sin 4  ⇡2◆, (4.29)where again the integration constant is a gauge choice. Thus we can write the full 4-formsolution asC(4) =↵02gs ✓1 r4hr4◆r4dt^dx^dy^dz+12✓  14 sin 4  ⇡2◆d cos ✓^d^d cos ✓˜^d˜.(4.30)4.2 The correspondence4.2.1 p =DpFrom (4.8), and our discussion of the AdS metric, we can identify the radius of curvatureof the p-brane solution metric asL2 =1pHp(r)r2 = p↵0. (4.31)Then for the classical supergravity approximation to be valid, we need the curvature tobe small compared to the scale of quantum corrections;L >> ls (4.32)) 1/4p↵0 >> p↵0 (4.33)) gsN >> 1, (4.34)where we used (2.68) in the second line. Note that this is consistent with the rp >> rapproximation we made earlier. Furthermore, to avoid string loop corrections, we mustkeep the coupling constant small, gs < 1, so that we have 1 << gsN << N , i.e. Nmust be very large. Remember that we defined N as the total R-R charge. But ofcourse, N has another interpretation. Recall from (3.42) and (3.44) that each D branecarries one unit of R-R charge and N coincident D branes carry N units of R-R charge.So N is both the charge of the supergravity solution and the number of coincident Dbranes. The classical supergravity limit then requires a large number of coincident D43branes which carries a large N , U(N) gauge theory. If on the other hand, we wantto do open string perturbation theory, we would want a small coupling constant. ForN coincident D branes, each string gets a Chan-Paton factor N , so that the couplingconstant is gsN , and perturbation theory requires gsN << 1. Thus we have two di↵erentdescriptions valid in two di↵erent regimes. For gsN << 1 (weak coupling), we have openstring perturbation theory with D brane boundary conditions. For gsN >> 1 (strongcoupling), we have classical Type II supergravity.Recall that the U(N) gauge theory in the open string description is broken when coin-cident D branes separate and strings connecting separated branes (which we will call Wbosons) acquire a mass via the Higgs mechanism. In the supergravity description, thereexists a multi-centered solution to the supergravity action, which is a generalization ofthe function Hp(r) [33]:Hp(r) = 1 + ↵p kXi=1 r7pp(i)|~r  ~ri|7p , rp(i) = 4⇡gsNi↵02, (4.35)where now each centre sources an R-R field so that we must integrate ⇤Fp+2 over asphere around each source to get Ni:Ni = ZS8pi ⇤Fp+2. (4.36)Now suppose we place an open string in this metric. Its endpoints must be attached toD branes, but there are none around. However, the strings could end at the singularitieswhich are the centres ~ri. But then we may as well say that there are D branes locatedat these centres. Indeed, one can compare the W-boson mass (from a string stretchedbetween two branes), to the mass of a string suspended between two centres in thismetric and would find they are the same. We will see something similar to the latter ofthese two computations in section 7.1.2, and find that the mass is just Ts|ri rj | (in thestatic gauge r = , t = ⌧). This helps to clarifies the connection between p-branes andDp branes. The Dp brane is located at the centre ~r = ~ri of the supergravity solution, orequivalently, the metric (4.8) gives the geometry near a Dp brane.4.2.2 Maldacena’s conjectureHere is the story so far. We start with Type IIb string theory in ten dimensions. Then weinsert N parallel D3 branes. At low energies, the full theory contains excitations o↵ thebrane, which must be from the closed string spectrum (2.111) - (2.114), and excitationson the brane, which are open strings with endpoints attached to the brane. The lowenergy closed strings are e↵ectively described by the Type IIB supergravity action (4.5),44while the open strings are e↵ectively described by the D brane world-volume actionwhich reduces to N = 4 super Yang-Mills theory for the D3s in the low energy limit(3.39)2.For a linearized metric gµ⌫ = ⌘µ⌫ + 0gshµ⌫ , we can expand the Type IIB action about↵0 = 0 (0 = 0), to getSIIB = 1220g2s Z d10xp|g|R (4.37)=Zd10x[(@h)2 + 0gs(@h)2h+O((0gs)2)], (4.38)which is a free field theory at ↵0 = 0.We can similarly look at the bulk-brane interaction in this limit. The leading term inthis interaction comes from the world-volume coupling to the space-time metric in (3.37)Sint /Tpgs Z dp+1⇠p|g| (4.39)/1gs↵02 Z d4⇠p|g| (4.40)for the D3. But from (4.19), we know thatp|g| / ↵05, and so the leading term in Sintis zero for ↵0 = 0. Thus as ↵0 ! 0, the bulk-brane interactions disappear and we are leftwith N = 4 super Yang-Mills gauge theory for the open strings, and free gravity for theclosed strings.Now what kind of physics does an observer at infinity see deep inside the bulk? Themetric (4.19) is time translation invariant and so has a timelike killing vector ⇠µ =(1, 0, . . . , 0). A static time-like observer with velocity Uµ has a conserved energy E =Uµ⇠µ, and satisfies UµUµ = 1. These two equations fix the observer’s velocity, whichdepends on their radial positionUµ(r) = ✓ 1pgtt , 0, . . . , 0◆. (4.41)The observer will measure the frequency of a signal with wave-vector kµ to be!(r) = kµUµ(r). (4.42)2Recall that upon dimensional reduction, N = 1 supersymmetery can become N = 4. We know theD3 corresponds to N = 4 specifically by comparing fields (see section 2.3). The D3 breaks 10  4 = 6translational symmetries and gains 6 corresponding scalars describing its transverse fluctuations. Onthe brane lives one gauge field Aµ. This matches only with the N = 4 states, which also include fourspin-1/2 fermions. This will all look slightly di↵erent when we introduce a probe D5 brane later.45If we put an emitter at position r, and an observer at infinity, we then get the ratio ofthe frequencies they measure for a signal to be!obs!emiit=pgtt|rpgtt|1 = 1H1/4p (r) , (4.43)where we have set rh = 0 for simplicity. This ratio goes to zero as r ! 0, so as theemitter moves deeper into the bulk, the signal detected at infinity shifts toward theinfrared. So there are two separate sectors in which an observer at infinity detects lowenergy excitations. One is the set of long wavelength massless fields propagating outsidethe throat. The other is the set of red-shifted fields propagating inside the throat, whosegeometry is described by AdS5 ⇥ S5. These two sets do not interact. The fields outsidethe throat have wavelengths too large to probe inside the throat, and the fields insidecannot overcome the gravitational well to get out. Depending on whether we think ofD branes in terms of the open strings that end on them, or in terms of supergravitysolutions, we get two di↵erent pictures in the ↵0 ! 0 limit. The first gives N = 4 superYang-Mills for open strings and free supergravity for closed strings. The second givesType IIB fields in AdS5 ⇥ S5 for one sector, and free supergravity for the other. Sincein each viewpoint, the free supergravity sector is decoupled from the other sector. It isnatural to identify the other sectors in each viewpoint. This is precisely the conjectureproposed by Juan Maldacena [1]3N = 4 SU(N) super Yang-Mills theory in 3 + 1 dimensions is dual to Type IIBsuperstring theory in AdS5 ⇥ S5As a check, we can compare the symmetries on both sides of the correspondence. Wesaw in section 2.3 that 3+1 dimensional N = 4 super Yang-Mills is scale invariant evenat the quantum level. It has the full conformal symmetry group SO(4, 2). We also sawin 2.1 that this is precisely the isometry group of AdS5. The isometry group of the S5is SO(6) ⇠ SU(4)R, which is also the R-symmetry group of the N = 4 supersymmetryalgebra.Note that if we work in units where the radius of curvature is L = 1, then (4.31) gives↵0 = 1p⇠1pgsN . (4.44)3Recall that we said the gauge theory on the branes was U(N). This can generally be split into aU(1) ⇥ SU(N) theory. The U(1) multiplet describes the center of mass positions of the branes, whichwe can choose to ignore [9]46Thus, we are expanding about a classical gravity solution ↵0 = 0, with quantum grav-ity corrections appearing at O(↵0). On the field theory side, we are doing a large Nexpansion with corrections appearing at O(1/pN).4.2.3 Large N expansionConsider a toy Yang-Mills theory of fields a with adjoint (colour) index 1  a  N21,and flavour index . The SU(N) Yang-Mills Lagrangian contains three and four-pointvertices, and looks likeL() ⇠ Tr[(@)2 + 2 + gYM3 + g2YM4]. (4.45)Under a rescaling ˜ = gYM, we can rewrite this asL() ⇠1g2YM Tr[(@˜)2 + ˜2 + ˜3 + ˜4], (4.46)so that each vertex of a Feynman diagram will carry a factor of1g2YM = 12⇡gs (4.47)⇠N. (4.48)Meanwhile, the propagator will scale ash˜ji ˜lki ⇠ g2YM hjilki (4.49)= g2YM✓lijk  1N ji lk◆, (4.50)which simplifies in the ’t Hooft limit, N !1, to N lijk [34].The index structure suggests a double line notation for the propagator indicating thedirection of colour flow shown in figure 4.1. If we contract over l, k and i, j, we getilj kFigure 4.1the loop shown in figure 4.2, and the trace gives a factor of N in the amplitude. Thusany Feynman diagram in this model with v vertices, p propagators, and l loops will47Figure 4.2: .contribute an overall factor of✓N◆p✓N◆vN l (4.51)= Nvp+lpv (4.52)to the amplitude.Each Feynman diagram can then be associated with the two-dimensional surface thatthey can be plastered onto without any lines rising o↵ the surface. Indeed, the exponentof N is just the Euler characteristic of the corresponding surface  = 2 2g, where g isthe genus of the surface. So now perturbation theory in the large N limit gives a genusexpansion 1Xg=0N22gfg(). (4.53)In this limit, the biggest contribution comes from genus 0 planar diagrams (called planarbecause they can also be plastered onto a flat surface). This expansion looks just like theperturbative expansion for closed strings provided we have gs ⇠ 1N , which is exactly thecase for fixed . Thus, it is important for the ’t Hooft limit that we keep  fixed as wetake N !1. The connection between string theory and gauge theory is ever-prevalent.4.2.4 Di↵erent forms of the conjectureIn the ’t Hooft limit above, we fixed . To make the correspondence more tractable,we will want to make  large, since this means the radius of curvature L ⇠ 1/4 will belarge. In all, we have three forms of the correspondence depending on how bold we wishto be. The statement 4.2.2, is the strongest form. Enforcing the ’t Hooft limit gives usa slightly weaker but more reliable form as it uses a large N field expansion and a smallgs string perturbation expansion. The weakest form of the correspondence is the onewe will use, the large  limit. This yields a classical supergravity theory on the AdSside. On the field theory side, the coupling gYM ⇠q N becomes large so that we probestrong coupling physics in this limit. Thus Maldacena’s conjecture, also known as theAdS/CFT correspondence, is an incredibly powerful conjecture, because in this version,it is a strong/weak coupling duality. It gives us a tool with which we can study strongly48coupled problems using just classical supergravity. Of course, this property also makesit very hard to test.Lastly, note that the decoupling of the two sectors discussed in 4.2.2 only requires thatthe space-time is asymptotically AdS5⇥S5, that is, we can allow for topology-changingobjects deep in the throat. Thus, in its strongest form, the conjecture is a dualitybetween the full Type IIB string theory on any asymptotically AdS5 ⇥ S5 space andN = 4 super Yang-Mills.4.2.5 HolographyOne of the biggest hints as to how the correspondence manifests itself comes from con-sidering black hole information. The Bekenstein bound [35] states that the maximumentropy contained in a space with surface area A is precisely the entropy of a black holewhose event horizon has the same surface areaSmax = SBH =A4GN . (4.54)The second equality above is established by the area theorem, which tells us that thearea of the horizon does not decrease, so this is a natural quantity to associate withentropy [36]. The proof of the first equality is quite simple. Suppose otherwise, thenthe region has an entropy S1 > SBH, so this region is not filled by a black hole. Nowadd matter to the region to form a black hole (or to enlarge any existing black hole).This process must not violate the second law of thermodynamics, so the new entropy isS2  S1. But the new entropy is just that of the black hole formed with surface areaB; S2 = B/4GN . Thus, B  A. But this is a contradiction; we have only increased thedensity of the region, so the black hole must have an area smaller than A.This suggests a “Holographic Principle” wherein the degrees of freedom of a theory ofquantum gravity in some volume should be contained on the boundary of that volume.Indeed if we consider U(N) super Yang-Mills in 3 + 1 dimensions with a UV cuto↵ ,then the number of degrees of freedom should be N2V3/3, since there are N2 degreesof freedom in each pixel of side length . In, AdS5 ⇥ S5, the holographic principle saysthat the number of degrees of freedom is proportional to the area. The area of a surfaceat radius r = 1/ ( should be small for a good UV cuto↵, so we are looking near theboundary at r = 1) is given byA =p|g||r=1/ (4.55)⇠V3N23, (4.56)49In terms of AdS/CFT, this suggests that the CFT “lives” on the boundary. This isnot to say that we supplement the AdS gravity theory with a theory on the boundary.Rather, from the discussion above, we should say the theory lives in the AdS bulk, butthe information is completely encoded on the boundary (or vice versa).4.2.6 Excerpts from the dictionaryIt is time to decide exactly what is meant by duality in the AdS/CFT correspondence.We could just say there is an isomorphism between the Hilbert spaces of each theory,and be done with it, but that is both dicult to prove and not very useful. What wewould like is a one-to-one mapping of fields on each side of the correspondence. However,CFTs like N = 4 super Yang-Mills have no asymptotic states as we saw in section 2.2, soit is preferable to talk about operators on the field theory side. The precise mapping isprovided by the celebrated GKPW rule [37]. It too is a conjecture, though perhaps moredicult to motivate than Maldacena’s. We might try to equate partition functions oneach side of the correspondence, except that there is reason to distinguish between thebulk of AdS and the boundary. This distinction should be manifest in the duality. Thebulk requires data from the boundary in order for the Cauchy problem to be well-posed(recall that fields can propagate to r = 1 and back in finite time). Furthermore, theholographic principle suggests that the CFT lives on the boundary. All of this persuadesus to consider conformal operators on the boundary as sourcing the fields in the bulk(or equivalently, the boundary value of bulk fields source the CFT operators). In thatcase, the GKPW rule is rather intuitiveheR d4x0(~x)O(~x)iSYM = Zstring[(~x, r)|r=1 = 0(~x)], (4.57)where (~x, r) is a Type IIB string field, 0(~x) is its boundary value, O(~x) is its dualCFT operator, and Zstring is the Type IIB partition function which is just eiSSUGRA inthe classical limit. With this, one can compute correlation functions of any operatorO in super Yang-Mills. One just needs to identify the corresponding field , solvethe equations of motion in the supergravity action, compute the on-shell action, andfunctionally di↵erentiate with respect to its boundary value 0. Of course, the hardpart is finding which operator corresponds to which field. We have already seen thatAdS/CFT relates global symmetries of the field theory to local isometries of AdS5⇥S5.It turns out that the entire AdS/CFT dictionary relates global properties in the fieldtheory to local properties on the gravity side. One place to start would be to decomposea field in AdS5 ⇥ S5 into Kaluza-Klein towers, that is, write it in a basis of spherical50harmonics on S5(r, ~y) =1X=0(r)Y(~y), (4.58)where ~y are the coordinates on S5. Then the Klein-Gordon equation fixes the massof the field in terms of the conformal dimension , which in turn fixes the conformaldimension of the CFT operator O. For scalar fields in the bulk, these operators aretraces over scalar fields in N = 4 super Yang-Mills.We will not work out the full dictionary, but simply tabulate some important entriesbelow. Some of these will be justified later.CFT Operator Gravity Fieldm Scalar field  Os Scalar operatorDirac spinor  Of Fermionic operatorGlobal current Jµ Aµ Maxwell fieldEnergy-momentum tensor Tµ⌫ gab MetricSpin SpinTemperature T Hawking temperatureChemical potential µ Aµ(1)Energy scale Radial coordinateTable 4.1: AdS/CFT dictionaryThe last entry in the table warrants a little more explanation. From the field theorypoint of view, an operator O sources a well-defined particle (well-defined in the UV).But it loses its definition as it interacts with the strongly coupled system until it canonly be defined as part of a collective excitation in the IR. In the gravity description, Osources a supergravity field that propagates radially inward. We saw that as it movesdeeper into the throat, its energy as observed at the boundary shifts towards the IR. Ina sense this is just restating the equivalence of symmetries between the CFT and AdS.We can increase the scale of any of the boundary coordinates by a factor of l withouta↵ecting the CFT. We know this must be equivalent to an isometry of the AdS metric.But for the metric (4.8) to be unchanged, r must be scaled by a factor of l1. It is as ifthe extra radial dimension is imprinted on the boundary as the renormalization scale ofthe QFT. Indeed, we already saw that for the holographic principle to work, we had toidentify 1/r with the UV cuto↵.514.2.7 Tests of AdS/CFTThere is an extensive body of work testing various aspects of the correspondence (e.g. [38–40]). Of course, direct tests are dicult, since phenomena on each side of the correspon-dence are worked out in opposite regimes of the coupling. Thus, tests are usually done onquantities that are independent of the coupling. These include certain correlation func-tions and the spectrum of chiral operators. Qualitative tests are more abundant wherethese so-called holographic string theory models are used as a framework for describingstrongly coupled field theories. Some successes include: the discovery of a confiningphase in super Yang-Mills at finite temperature, and the prediction that the entropy ofthis theory changes by a factor of 4/3 between strong and weak coupling. Incredibleadvances have been made in the application of AdS/CFT to super Yang-Mills hydrody-namics where it was discovered that the ratio of viscosity to entropy is ⌘/S = 1/4⇡ [41].Other accomplishments include: the computation of optical conductivity in 2 + 1 di-mensional super Yang-Mills at strong coupling [42], and multiple versions of holographicsuperconductivity [43, 44]. Finally, the confinement-deconfinement transition of superYang-Mills has been shown to be equivalent to a Hawking-Page black hole transition [45].This is but a small sampling of applications to come out of this field.4.2.8 Finite temperatureAdding a finite temperature to the field theory introduces an energy scale which breaksconformal symmetry (and in fact supersymmetry)4, however, as seen above, a slew ofuseful results have come from studying finite temperature AdS/CFT, so we will pushon with a little willing suspension of disbelief.In formulating black hole thermodynamics, it is natural to identify the surface gravity of a black hole with its temperature (in appropriate units). For one, thermal equilibriumoccurs only if  is constant over the horizon. For another, the change in energy of astationary, non-spinning, neutral black hole isdE =8⇡dA, (4.59)4These are some of the issues that plague AdS/CMT (the application of the correspondence tocondensed matter). Perhaps a bigger concern is that condensed matter usually deals with U(1) andSU(2) theories, which are not exactly “large N .” Of course, if the strong form of the duality is true, thisis no issue.52We have already identified the area as A = 4SBH , so this would just be the TdS termin the first law of thermodynamics if we identify = 2⇡T. (4.60)The surface gravity is defined from the time-like killing vector ⇠a = (1, 0, 0, . . . , 0) at thehorizon:ra(⇠b⇠b) = 2⇠a. (4.61)Using, the killing equation, this can be written in a more convenient form: (see [46] fordetails)2 = 12(ra⇠b)(ra⇠b)|r!rh (4.62)= 12gaµgb⌫⌫µtbat|r!rh . (4.63)Plugging in the D3 metric (4.19) and corresponding Christo↵el symbols yields2 =(r4 + r4h)2r6|r!rh (4.64)) T =rh⇡. (4.65)This is the Hawking temperature for an AdS black hole. Since it is the only temperaturearound on the gravity side of the correspondence, it is natural to propose that this is thedual of the field theory temperature. As we turn on temperature in the CFT, a blackhole emerges in the bulk. We can then follow the usual thermal field theory definition ofa grand partition function Z = exp(⌦/T ), so that in the classical limit, we can identify⌦ with the supergravity action. Note that we have used the symbol ⌦ in anticipationof including a finite density and chemical potential in the field theory. In that case, ⌦is the grand potential for the grand canonical ensemble [47]. For our purposes, we willbe interested in the canonical ensemble at fixed charge density, in which case we needto perform a Legendre transform of the supergravity action to get the Helmholtz freeenergy F . It is from the Helmholtz free energy that one can calculate the entropy ofsuper Yang-Mills and get the 4/3 di↵erence between strong and weak coupling mentionedearlier. It is an interesting unresolved question of how exactly the entropy interpolatesbetween the strong and weak coupling values. Is there a discontinuous phase transition?See [48] for a discussion. We too will calculate the holographic entropy in chapter 7, butin a much more dressed up model.534.2.9 Probes, flavours, and defectsNow we have a playground of possibilities to work with. If we start with a stack ofNc coincident D3 branes generating AdS5 ⇥ S5 geometry, and then add Nq coincidentDq branes at radial infinity, we can e↵ectively probe the supergravity solution providedNq << Nc. To see the need for this limit, recall that when embedded in flat space-time(i.e. the boundary of AdS), the Dq branes will curve space-time according to the metric(4.8), with radius of curvature Lq ⇠ H1/4q . Then for Nq << Nc, we haveHq << Hc (4.66)) Lq >> Lc. (4.67)In this case, we can neglect the gravitational backreaction of the Dq branes and treatthem as probes of the AdS geometry. Another nice thing about the probe limit, is that,provided we work to order NqNc, we can ignore quantum loop corrections to the fermionfields in what follows.Strings suspended between the D3 stack and the Dq stack have a gravity description asstrings stretched from the Dq stack to the Poincare` horizon. Though these strings havean infinite proper length, they have a finite energy. One can find a further test of thecorrespondence here if one lets q = 3, Nq = 1. Then the extra D3’s world-volume actionis the full DBI action (3.37) which contains higher derivative terms beyond Yang-Mills,only now the background metric that goes into the DBI action is AdS instead of flatspace. But we have discussed this situation already. It is the Higgs symmetry breakingof U(Nc + 1) ! U(Nc) ⇥ U(1). The DBI action is precisely the e↵ective action for theU(1) fields. The strings connecting the lone brane to the Nc stack have an acquired amass which is why we call them W-bosons. These have been integrated out in the DBIaction, and now just source the F gauge fields living on the single brane. Indeed, if onecalculates the one-loop corrections to the Higgs expectation value in this picture and inthe field theory picture, one would find that they match [49].If instead we keep all the D3s together, then the strings within the stack transform in theadjoint representation of U(N). But we can decompose the adjoint into fundamentaland anti-fundamental representations. For example, for SU(3) this is just the tensorproduct decomposition 3 ⌦ 3¯ = 8  1, where 3, 3¯,8 and 1 are the fundamental, anti-fundamental, adjoint and trivial representations respectively. We can understand thisas each string endpoint being a charge in the fundamental/ anti-fundamental represen-tation. Now again, we can separate some D3s so that a string has one endpoint on twoseparate branes. As viewed from the separated D3 world-volume, this string would forma fundamental representation field. It turns out that it transforms as a vector under54the Lorentz group, so the title of W-boson is indeed very appropriate. But how do wegenerate a fundamental representation spinor field?Suppose instead we introduce a stack of N5 coincident D5-branes. We have some choicein how they are oriented. We can preserve the most supersymmetries by having four oreight dimensions spanned by only one type of D brane [32]. Thus, we are left with thefollowing unique orientation (unique up to relabelling of coordinates):x0 x1 x2 x3 x4 x5 x6 x7 x8 x9D3 X X X XD5 X X X X X XTable 4.2: D5 orientationHere, Xs denote directions along which the branes extend, and blanks are directions inwhich the branes are point like.The addition of D5s breaks half the supersymmetries. If they are separated from theD3 stack, the strings connecting the two groups form the N = 2 hypermultiplet, whichconsists of two complex scalars as well as left and right chiral Weyl spinors. Notethat the two brane-stacks overlap only in 2 + 1 dimensions. This means the masslesshypermultiplet string endpoints are confined to these dimensions. We have uncovereda defect field theory. In the dual description, we still have N = 4 super Yang-Mills in3 + 1 dimensions, but now we also have an N = 2 hyermultiplet confined to a 2 + 1dimensional defect. For the strings connecting the two stacks, one end transforms in thefundamental of U(Nc), and the other transforms in the fundamental of U(N5) (there isno need to distinguish between the fundamental anti-fundamental here). If we considerthe string as a whole, we say that it is a bifundamental field, since it transforms non-trivially under two di↵erent gauge groups [50]. We can consider this set-up either froma D3 world-volume perspective in which U(N5) is a global symmetry, or from the D5perspective in which it is a gauge symmetry. We will take the latter approach.So we now have a supergravity description of fermions propagating along a 2+1 dimen-sional defect. Note, however, that this theory necessarily includes scalar fields. Whatsymmetries remain in such a model? We still have 3D translation and Lorentz invarianceas well as SU(4) R-symmetry. In addition, it is (at least classically) still conformallyinvariant. The D5 world-volume fills AdS4 as well as two of the S5 dimensions. We willalways fibrate S5 into two 2-spheres with an angle  between them. The D5 wraps oneof these 2-spheres and is point-like on the other. Thus, the isometries of its world-volumemust be SO(3, 2) ⇥ SO(3) ⇥ S˜O(3), where we distinguish between the isometry SO(3)of the S2 wrapped by the D5s, and S˜O(3), the isometry of the unwrapped S2.55The full theory is still 4-dimensional, and some work needs to be done to ensure that aset of fields transforms like the 3D hypermultiplet when confined to the defect [51]. Itturns out that these symmetries are enough to completely specify the field theory action.The result has been worked out in [52]. It includes a variety of Yukawa couplings andpotentials, but if we look at the weak coupling, massless regime (actually set the couplingto zero), it reduces to just a kinetic termSweakZd3x[(Dkqm)†Dkqm  i ¯i⇢kDk i], (4.68)where q is the complex scalar,  is a spinor representation of S˜O(3), Dk is a gauge co-variant derivative containing the 4-dimensional gauge field, and ⇢k are gamma matrices.Now, we can ask what would happen if we made these fermions massive. We know thisrequires separating the D5 and D3 stacks from each other along a transverse direction.This would necessarily break the S˜O(3) symmetry because it would single out one of theS˜2 directions. The corresponding fermions arising from strings attached to each stackwould gain a mass that breaks chiral symmetry. Thus, we can associate S˜O(3) withthe chiral symmetry of this model. It turns out that for our purposes, chiral symmetrybreaking will be very important, but we will see how it can arise in a more organic way.For now, we set the fermion mass to zero, yielding a gapless, defect field theory.56Chapter 5The Quantum Hall E↵ectIt is time to ask exactly what strongly coupled system we would like to probe with thegauge/gravity machinery we have developed. Our goal is to investigate the fate of thequantum Hall phenomenon as the coupling becomes strong. First let us look at thephenomenon in detail to see exactly which parts we can capture holographically.5.1 Condensed matter approach5.1.1 Semi-classical descriptionIn the familiar setup of the Hall e↵ect, one puts a current-carrying semi-conductor ina perpendicular magnetic field Bzˆ. The current is deflected by the magnetic field untilcharge builds up on the edge of the semi-conductor and establishes a Hall electric fieldto precisely counteract the Lorentz force~EHall + ~vd ⇥ ~B = 0, (5.1)where ~vd is the drift velocity of the electrons.For an initial current in the yˆ-direction, we have Jy = ⇢vd (where ⇢ is the charge density),and the Hall equation becomesEx  vdB = 0 (5.2)) vd = ExB (5.3)) Jy = ⇢BEx. (5.4)57This is just Ohm’s law, with an o↵-diagonal piece of the conductivity tensorxy = ⇢B . (5.5)For fixed magnetic field, the conductivity just increases linearly with charge density.We could also turn o↵ the current and consider electrons confined to a 2D surface. Thenthe Lorentz deflections will force the electrons into cyclotron motion with a well definedfrequency ! = eBm . We might expect there to be a corresponding harmonic potential sothat at the quantum level, the electron energy is quantized as the harmonic oscillatorlevels. We will see that this is indeed the case. Note that the edges of the sample willexert an impulse that prevents elections there from completing circles. These electronswill travel in semi-circles, inducing a current along the edge.Suppose now we increase the charge density. Then the drift velocity will decrease andso will the cyclotron radius r = mvdeB , but the conductivity will increase. Eventually, thecycles will shrink to the point where impurity e↵ects become important, whereupon theconductivity will plateau until the Fermi energy is raised suciently. To observe thesee↵ects requires small radii and thus strong magnetic fields, as well as low temperatures.This is why it was not observed until 1980 [53].5.1.2 Quantized conductivityTo make more sense of the observed conductivity behaviour, we must move to a quantumpicture. If we define a ratio⌫ = 2⇡⇢B, (5.6)then it is observed that as one tunes a control parameter (⇢ or B), the conductivitybecomes stuck at ⌫2⇡ for integer values of ⌫. When electron interactions are important,there is an additional “fractional quantum Hall e↵ect” for rational values of ⌫ that wewill not discuss here.There are two important ingredients for the traditional integer Hall e↵ect. One is thatthe electron has gapped states in its spectrum. The other is the presence of disorder toensure that such states exist over a range of control parameter (forming a conductivityplateau). The Fermi energy is determined by the control parameters, and disorder givesa finite range of energies over which localized states exist. As we tune the parameter inthe plateau region, the Fermi energy moves through this range. Later we will speculateon an alternative way in which plateaus may appear holographically that has nothingto do with disorder. In the next section, we will see that the presence of the charge58gap and the quantized conductivity are due to the organization of electrons into Landaulevels in the presence of the magnetic field.5.1.3 2D electron gas modelThe Hamiltonian for N electrons in a gauge potential ~A(~r) is given byH =12m⇤ NXi=1(~Pi + e ~A(~ri))2, (5.7)where m⇤ is the electron’s e↵ective mass.With a magnetic field ~B = Byˆ, we are free to choose the Landau gauge for the vectorpotential ~A = Bxyˆ, in which case,H =12m⇤ NXi=1 ✓r2i  2iBxi @@yi + e2B2x2i◆. (5.8)Translational invariance in y gives the Schro¨dinger one-particle ansatz  (t, ~x) =  (x)ei(kyyEt),in which case we can write the Schro¨dinger equation as✓12m⇤ d2dx2 + e2B22m⇤ (x+ kyeB )2◆ (x) = E (x). (5.9)But this is simply the Schro¨dinger equation for a one-dimensional harmonic oscillatorcentred at x0 =kyeB , with frequency ! = eBm⇤ , which is the cyclotron frequency wefound earlier. The spectrum of energies are then just En = !(n + 1/2). Each value ofn = 0, 1, 2, . . . corresponds to a di↵erent Landau level. Let the sample have dimensionsLx⇥Ly. Imposing periodic boundary conditions in y requires ky = 2⇡m/Ly, m 2 Z. Soneighbouring oscillators are separated by x = 2⇡LyB . Then each Landau level containsLxx = LxLyB2⇡ states. So the Landau levels are highly degenerate in strong magneticfields. The number of states per unit area is B2⇡ , and we can define the filling fraction ⌫as the fraction of a Landau level that is occupied.⌫ =electrons/areastates/Landau Level/area= 2⇡⇢B, (5.10)which is what we defined in (5.6).Thus at integer ⌫, a Landau level is completely filled. There is then an energy gap of !to the next Landau level. If the density is increased further, electrons begin to occupythe next Landau level and the chemical potential must jump across the gap. Thus, atinteger filling fraction, the chemical potential has a discontinuity as a function of ⇢. If59we calculate the compressibility  = [⇢2 dµd⇢ ]1, we see that it vanishes at these pointsand so the state is said to be incompressible. Away from these points it only takes aninfinitesimal amount of energy to excite the electrons above the Fermi level, but at thesepoints, it costs ! in energy.The derivation of quantized conductivities is a triumph of a gauge argument due toLaughlin and Halperin [54, 55]. The quantum Hall e↵ect is observed to be robust forall sample geometries, so we can imagine bending the sample to form an annulus. Nowinstead of inducing a current with an applied voltage, we can put a flux  = q0,with 0 = he , through the centre of the annulus. If a wave function extends around theannulus, it will pick up a phase kyLy as y goes from 0 to Ly, as well as an Aharonov-Bohm phase q0. The wave function is single-valued only if q 2 Z. If this is not thecase, we do not get extended wave-functions.Again we are free to change the geometry back to the strip of length Ly. We do so withan electric field along the width of the strip to getH =12m⇤⇢~P + eB✓c+ q0BLy◆yˆ2 + eExx. (5.11)If we slowly turn on one unit of flux, q ! 1, then each electron shifts by x, i.e. oneoscillator site in the x-direction. Of course, only the occupied states move over, so thisprocess is equivalent to moving one electron from each Landau level across the stripwidth. The net charge transferred is exactly ne where n = b⌫c is the number ofoccupied Landau levels. Then using Faraday’s law,Zc ~E · d~l =  Z @ ~B@t · d ~A (5.12))Zc dy⇢yxjx = ddt (5.13)) ⇢yx Z dtJx = 0, (5.14)where c is a contour enclosing the flux, and jx is the current density. But R dtJx is justthe net charge transferred, ne, so⇢yx = 0ne (5.15)) yx = nh/e2 = b⌫c2⇡ , (5.16)where in the last line, we switched to natural units, h/e2 = 2⇡. From here on out, wewill maintain natural units.60For wave-functions that are localized at scales well below the length Ly, there is no specialflux required for single-valuedness, and in fact these states cannot carry current1. Thisis precisely what happens when disorder is introduced, producing random scatteringo↵ impurities. This scattering also broadens the Landau levels into bands (which wewill assume are smaller than the energy gap). In fact, a theorem due to Philip W.Anderson [56], shows that arbitrarily weak impurities in a 2D system force states to belocalized. Of course, there must be some extended state or else no current will flow. Thepresence of the magnetic field helps us out, and it turns out that at the centre of eachLandau level, the localization scale diverges. Thus, as we move the Fermi energy (viaa control parameter) through the localized part of the Landau level, the conductivityremains unchanged until the Fermi energy passes through the extended state energy,whereupon the conductivity jumps to the next plateau.An argument of Lorentz covariance also tells us that disorder is necessary for the for-mation of plateaus at zero temperature. Suppose we did not have impurities. Then thesystem would have translation invariance. If we sit in a partially filled Landau levelwith charge density ⇢ = ⌫2⇡B, then by boosting into a frame with velocity vi, we observea current Ji = ⇢vi. The external magnetic field in this frame comes with a transverseelectric fieldEi = ✏ijvjB (5.17)= ✏ij 2⇡⌫ Jj . (5.18)So the conductivity would just be the classical value. Since this is independent of theboost velocity, it should hold in all frames. Thus we cannot have plateaus in a trans-lationally invariant system at zero temperature. Interestingly, for finite temperature,the system is coupled to a heat bath. This gives a preferred reference frame, destroyingtranslation invariance and opening up the possibility of non-impurity driven plateaus.5.2 D3-D5 system at weak couplingNow we will see how this relates to the holographic D3-D5 system we began to discussin the last chapter. We saw that the corresponding (one-particle) field theory for thissystem had a weak coupling action with a Dirac spinor  and complex scalar qSweak =Zd3[|Dµq|2  i ¯µDµ ], (5.19)1One can show that the current operator is proportional to @H@q . Since the spectrum of a localizedwave-function does not depend on q (there is no Aharonov-Bohm phase), these states cannot carrycurrent.61with Dµ = @µ  iAµ. If we work out the equations of motion for the spinors, we getiµ(@µ + iAµ) ¯ = 0. (5.20)If we want to immerse the fermions in an external B-field, we use (in the Landau gauge)Aµ = (0, 0, Bx). Furthermore, we break up the Dirac spinor into two Weyl spinors =  +  !. The equations of motion are invariant under translations in y, so we canwrite the solution ansatz as  ± =  ±(x)ei(kyyEt). Now the equations of motion can bewritten in the Schro¨dingier form± E ±(x) = ±i1@x ±(x)± i2(@y + iBx) ±(x), (5.21)where i are Pauli matrices. Combining the + and  equations and defining H0 ⌘1@x + 2(@x + iBx), we get,E (x) = iH0 + (5.22)=1EH20 (x) (5.23)) E2 ±(x) = (I@2x + 12(iB) + I(@y + iBx)2) ±(x). (5.24)Completing the square yields(E2 +B) 1±(x) = ✓ d2dx2 B2✓x kyB ◆2◆ 1±(x), (5.25)where we have focused on the first component of each Weyl spinor  1± for simplicity.This is again the same form as the 1D harmonic oscillator equation✓12md2dx2+12m!2x2◆ = E˜ , (5.26)provided we make the following identifications.m = 1/2; !2 = 4B2; E˜ = E2 +B. (5.27)The harmonic oscillator spectrum, E˜ = !(n + 1/2), n = 0, 1, 2, . . ., then gives us theenergy spectrum of these fermionsE2 +B = 2B(n+ 1/2) (5.28)) E =p2Bn. (5.29)62So the fermions in this theory also have Landau levels, but with an energy gapp2B. Itis a promising candidate for quantum Hall states.But we have neglected the scalars! We can repeat the same procedure for them. We getthe equation of motion@⌫(@⌫  iA⌫q) = i(@µq)Aµ +A2q. (5.30)Using the same ansatz, q = q(x)ei(kyyEt), this can also be written in the form of aharmonic oscillator equationE2q(x) =✓d2dx2+B2✓xkyB◆2◆q(x). (5.31)The parameters are the same as for the fermions, except now the energy correspondenceisE2 = E˜ (5.32)) E =pB(2n+ 1). (5.33)We see that the bosons do not have a zero energy mode. If we only look at the firstfermionic Landau level, (n = 0), then we will never excite the bosons since their thresholdenergy ispB.Note that the oscillators are positioned exactly as in the 2D electron gas model, with thesame spacing between centres. So we have the same number of oscillators per Landaulevel. However, in the D3-D5 theory, the spinors have a colour index from SU(Nc), aflavour index from SU(N5) and an isospin index from S˜O(3). Thus, the total Landaulevel degeneracy is 2N5N B2⇡ .So we have a set of Landau levels for each of the N5 flavour states and each of thetwo isospin states (ignore the colour index for now). Suppose we set N5 = 1 and allowa small Coulomb interaction between the fermions. This will make an anti-symmetricspatial wave-function preferable. But the full fermionic wave-function must be anti-symmetric, so that means the fermions will prefer a symmetric isospin configuration.The result is a small gap between the first Landau level of each isospin state. This e↵ectis known as quantum Hall ferromagnetism2. The resulting picture has broken S˜O(3)chiral symmetry, so this e↵ect may also be called magnetic catalysis of chiral symmetrybreaking. Either way, it results in the formation of a chiral condensate and thus weexpect the theory to be confining. So we should not consider the fermions as individual2The sample is ferromagnetic in the sense that the isospins become spontaneously aligned in thepresence of a magnetic field.63quarks, but rather as being bound together in groups of N to form baryons. This iswhy we ignored the colour index above. This also means we have to normalize the fillingfraction. The expression in (5.6) gives the number of quarks in a Landau level dividedby the total number of states in that level. Now we want the number of baryons in eachlevel. Thus,⌫ =2⇡N⇢B. (5.34)This ferromagnetic e↵ect means that the neutral ground state (where the first Landaulevel of one isospin state is full), ⌫ = 0 is gapped. The next gapped state occurs at⌫ = 1, when both isospin states are full. Beyond that, we get the second Landau level(n = 1) with energyp2B. However, this is greater that the energy required to excitebosons. Once we let the bosons loose, there will be no more gapped states since theyhave no Pauli exclusion principle.If we allow the full N5 flavours, we can potentially get 2N5 + 1 Hall states with ⌫ =0,±1, . . . ,±N53 provided the resolution of the degeneracy does not shift any of theseLandau levels above the boson ground state energy. The corresponding quantized con-ductivities are then xy = ± 12⇡ ,± 22⇡ , . . . ,±N52⇡ . In this thesis, we will find that thesegapped states persist at strong coupling. Furthermore, the conductivity remains un-changed at integer filling fraction, but will deviate slightly for non-integer ⌫ when thetemperature is finite.Finally, we should mention that quantum Hall ferromagnetism is observed in graphene,though it has an e↵ective SU(4) symmetry instead of SO(3) [57]. At suciently strongmagnetic fields, and pure samples, the four-fold degeneracy is completely lifted, intro-ducing four new gapped staes and corresponding conductivity plateaus [58]. Of course,in arguing for this e↵ect, we have assumed that the Coulomb interactions are weaky cou-pled so that we could ignore them in the Hamiltonian. But the fine structure constantof graphene is e24⇡~vf ⇡ 300137 , where vf is the speed of light in graphene. This suggeststhat the Coulomb interaction is very strongly coupled. So there must be a frameworkwhich describes how this phenomenon persists at strong coupling. This is of course thegravitational description of the D3-D5 system that we develop in the next chapter.3Particle-hole symmetry dictates that there should be gapped states with negative integer ⌫ as well.64Chapter 6A Giant D5 ModelHere we will outline the holographic giant D5 model developed in [2], that will be usedto explore the fate of the D3-D5 field theory at strong coupling.6.1 D3-D5 becomes D3-D76.1.1 D3-D5 at strong couplingExactly what ingredients do we need for a holographic quantum Hall model? We haveseen that some of the important physics are captured by the D3-D5 model at weakcoupling. Going to the gravity side, we will need to bring with us three important com-ponents. First, to look at thermodynamics, we need a finite temperature and thereforea finite horizon radius rh in the AdS black hole geometry. Second, the magnetic field Bis external to the system, and so will be maintained as is throughout the bulk. We willleave it in the Landau gauge, but change the normalization so thatAy = 1p↵0Bx ⌘ 12⇡↵0 bx. (6.1)Finally, we want to have a finite charge density ⇢ so that we can explore di↵erent valuesof ⌫. From the AdS/CFT dictionary, we know that the global U(1) conserved currentJµ is dual to a local U(1) gauge field Aµ. Thus to have a finite charge density ⇢ = hJ ti,we need to include a temporal component At in this field. For a uniform charge density,this component cannot depend on the x and y coordinates, and there is no obvious needfor it to vary over the 5-sphere, so we will assume it is independent of these coordinates.65It may, however, vary in the IR, so we had better keep a radial dependancyAt = p2⇡ a(r). (6.2)Now for the geometry. We start with the AdS5 ⇥ S5 background metric generated bythe D3 stack (4.19) and pullback to the D5 world-volume which spans AdS4⇥S2. Thatleaves embedding coordinates for the unwrapped sphere S˜2, as well as z, and one moreangular coordinate  . This is the defect orientation on page 70. We want to preservechiral symmetry for now, so we will let the S˜2 embedding be constant. Furthermore,we will keep the SO(3) rotational invariance and translational symmetry, meaning thatno embeddings will depend on x, y or the S2 coordinates. This leaves two dynamicalembedding functions z(r) and  (r). The z(r) embedding will appear in the Lagrangianas a cyclic variable, and the solution to its equation of motion will come with a constantof integration that we are free to set to zero. Doing so will set z(r) to be a constant andthus can be ignored it in the following. The resulting world-volume metric is (from hereon primes will denote di↵erentiation with respect to r):ds2 =p↵0r2(h(r)dt2 + dx2 + dy2) + dr2h(r)r2✓1 + h(r)(r 0(r))2 + h(r)2(r2z0(r))2◆+sin2  (r)(d✓2 + sin2 ✓d2). (6.3)In this setup with no Kalb-Ramond field, the world world-volume field strength is2⇡↵0F = p↵0(a0(r)dr ^ dt+ bdx ^ dy). (6.4)But there is one more ingredient we need: an R-R 4-form C(4) coming from the D3stack, or equivalently the Type IIB supergravity solution. We will use the most generalform that we derived in (4.30). Then the DBI - Wess-Zumino action for N5 of these D5branes isS5 =T5gs N5 Z d6(pdet(g + 2⇡↵0F) + gs2⇡↵0C(4) ^ F), (6.5)where T5 = 1(2⇡)5↵03 , and gs = g2YM4⇡ = 4⇡Nc .For now the last term in (6.5) is totally innocuous, since the volume elements cannotmatch the world-volume element of the D5. Thus we can ignore it. However, for the D7it will become very important.66Evaluating this action, we getS5 = 2⇡T5gs (p↵0)3V2+1N5 Z 1rh dr2 sin2  pb2 + r4p1 + h(r)(r 0(r))2  a0(r)2, (6.6)where V2+1 is the volume of the defect. The temporal gauge field is cyclic, so its equationof motion is simplya0(r) = q5p1 + h(r)(r 0(r))2q4 sin4  (r)(b2 + r4) + q25, (6.7)where q5 is a constant of integration. What is the interpretation of this constant? Weknow that At is dual to J t, and that the precise relation is given by the GKPW rule tocalculate correlation functions. We are told (in the classical supergravity regime) thatfor each J t operator that appears in the correlation function, we di↵erentiate the on-shellaction with respect to the boundary value of the dual field once. ⇢ = hJ ti = 1V2+1 S5At(1) .Variation of the action givesS =Z 1rh dr L(@rAt)@rAt (6.8)= At L(@rAt) 1rh  Z 1rh dr@r L(@rAt)At. (6.9)The last term vanishes by the equation of motion, and we get⇢ = 2⇡T5gs (p↵0)3N5 L(@rAt) 1 (6.10)= q52NcN5(2⇡)2. (6.11)So q5 defines the total charge density of the field theory.As mentioned before, the on-shell action describes the grand potential. For the canonicalensemble, we want to find the Helmholtz free energyF = ⌦+ µN, (6.12)where N = ⇢V2+1 is the total number of baryons. µ is the chemical potential, andhas the holographic dual At(1)1 since this is the term that sources the charge densityoperator on the boundary. Thus we can write the free energy asF5 =2NN5(2⇡)2V2+1✓p2⇡Z 1rh drL+ q5µ◆. (6.13)1This is not necessarily true for non-uniform charge density in the bulk [59]67Now the time-component of the gauge field must vanish at the horizon because thekilling vector @t vanishes there. Thus,µ =p2⇡(a(1) a(rh)) (6.14)=p2⇡Z 1rh a0(r)dr. (6.15)So then,F5 =2NN5(2⇡)2V2+1Z 1rh drp1 + h(r)(r 0(r))2q4 sin4  (b2 + r4) + q25= N˜5Z 1rh drp1 + h(r)(r 0(r))2q4 sin4  f2(1 + r4) + (⇡⌫)2. (6.16)In the last line, we scaled out b usingr ! r/pb (6.17)rh ! rh/pb = ⇡s p2⇡BT (6.18)N˜5 ⌘21/4Nc(2⇡)4V2+1(2⇡B)3/2. (6.19)We have also made use of the parameters f ⌘ 2⇡pN5, and the filling fraction ⌫ = 2⇡⇢NcB .The meat of the problem now is to determine the solution  (r) from the Euler-Lagrangeequationddr@L@ 0(r) = @L@ . (6.20)For this, it is convenient to define a function V5 = 4 sin4  f2(1 + r4) + (⇡⌫)2 so that wecan write the Lagrangian as L =pV5p1 + hr2( 0(r))2. Then the equation of motionreadsddrpV5h(r)r2 0(r)p1 + h(r)r2( 0(r))2 = @ V52pV5p1 + h(r)r2( 0(r))2. (6.21)Consider the series expansion of  (r) at infinity  =  1 + c1r + c2r2 + . . .. Plugging thisinto the equation above gives, to leading order@ V5pV5r!1 = 0, (6.22)This equation is satisfied by  1 = ⇡/2, which is the condition we will use throughout therest of the thesis. The series expansion above indeed satisfies the asymptotic equation ofmotion. By matching conformal dimensions for the constants c1 and c2, one finds thatthey correspond to the bare fermion mass (with conformal dimension equal to 1) and68chiral condensate (with conformal dimension equal to 3) in the field theory. Again, wedo not want to break chiral symmetry explicitly, so we use massless fields c1 = 0, andlet c2 be determined dynamically. Upon solving the equation of motion numerically, itturns out that for the range of parameters we are interested in, one gets the non-constantsolution c2 6= 0, and thus chiral-symmetry is spontaneously broken. This turns out tobe true whenever ⌫ < 1.68f . The full phase transition was investigated in [60]. Thus,we see that quantum Hall ferromagnetism has a strong coupling holographic descriptionas a D5 developing a non-trivial embedding.6.1.2 D7 as a giant D5So far we have a model of N5 D5 branes with world-volume gauge fields coupled to anexternal R-R 4-form. But we know from our discussion of the Myers e↵ect, that Dpbranes in an external R-R potential can pu↵ out to form a D(p + 2) brane. Thus wemight expect that under certain conditions, the D5 stack will pu↵ out from the boundaryof AdS to form a D7 brane that wraps the extra 2-sphere2. We know that in the Myerse↵ect, the lower dimensional brane is interpreted in the higher dimensional brane world-volume as a unit of magnetic flux. From (3.47), we know that this will appear in thegauge field strength asF✓ = N52 sin ✓, (6.23)in which case he full D7 field strength will appear as2⇡↵0F = p↵0✓ ddra(r)dr ^ dt+ bdx ^ dy +f2sin ✓˜d✓˜ ^ d˜◆. (6.24)The DBI/ Wess-Zumino action is nowS7 =T7gs Z d8✓pdet(g + 2⇡↵0F) + gs (2⇡↵0)22 C(4) ^ F ^ F◆. (6.25)Note that unlike the D5 case, the Wess-Zumino term for the D7 has a non-zero contri-butionC(4) ^ F ^ F = a0(r)bc( )d✓ ^ d ^ d✓˜ ^ d˜ ^ dr ^ dt ^ dx ^ dy, (6.26)where c( ) is defined in (4.29). The non-radial directions are integrated over to give afactor of 4V2+1(2⇡)4.We orient the D7-brane as shown in the table below, where x3 is the z direction and x9is the  direction.2Really this 2-sphere is an approximation to the non-abelian fuzzy sphere configuration of the D5branes.69x0 x1 x2 x3 x4 x5 x6 x7 x8 x9D3 X X X XD7 X X X X X X X XTable 6.1: D7 orientationThe z(r) embedding is again satisfied by a constant, and the world-volume metric isthends2 =p↵0r2(h(r)dt2 + dx2 + dy2) + dr2h(r)r2✓1 + h(r)(r 0(r))2 + h(r)2(r2z0(r))2◆+sin2  (r)(d✓2 + sin2 ✓d2) + cos2  (d✓˜2 + sin2 ✓˜d˜2)). (6.27)Note that the radius of the second 2-sphere is cos2  . The full sphere coordinates mustsatisfyx1 + x2 + x3 + x4 + x5 = 1. (6.28)By choosing the radius of the first 2-sphere to be sin2  , we have x1 + x2 + x3 = sin2  .So in order to satisfy (6.28), we must have x4 + x5 + x6 = cos2  .Now we can repeat the same analysis as above, for the D7. The action becomesS7 = 2Nc(2⇡)4V2+1Z 1rh dr✓2 sin2  p(f2 + 4 cos4  )(b2 + r4)p1 + h(r 0)2  a0(r)2+2a0(r)bc( )◆. (6.29)The a(r) equation of motion now reads:2 sin2  p(f2 + 4 cos4  )(b2 + r4)a0(r)p1 + h(r 0(r))2  2bc( ) = q7. (6.30)Again, we Legendre transform the on-shell action to get the free energy and scale out b.F7 = N˜7Z 1rh drp1 + h(r 0(r))2 (6.31)⇥q4 sin4  (f2 + 4 cos4  )(1 + r4) + (⇡(⌫  1) + 2  1/2 sin(4 ))2,where N˜7 ⌘ 2Nc(2⇡)4V2+12⇡Bp 3/2 = N˜5.The equation of motion for  (r) is identical to the D5 case, (6.21), only with V5 replacedbyV7 = 4 sin4  (f2 + 4 cos4  )(1 + r4) +✓⇡(⌫  1) + 2 12sin 4 ◆2. (6.32)70At large r, the equation of motion reduces to@ V7pV7r!1 = 0. (6.33) 1 = ⇡/2 is also a solution to this equation, though it has another given byf2 + 4 cos4  = 4 sin2  cos2  (6.34))  =12arccos✓12p1 2f2  1◆. (6.35)Since this solution only works for small values of f , (f < 1/p2), we will use the firstsolution which does not restrict our parameters. The latter solution has been investi-gated in [61]. More importantly, if we look at the the free energy, (6.31), the integrandbecomes identical to that of the D5 stack for  ! ⇡/2. So with this choice of boundarycondition at infinity, the D7 brane reduces to N5 D5s at the boundary of AdS. It reallyis like a giant D5 brane.There is another remarkable thing to notice about (6.31), and that is ⌫ = 1 is a specialvalue. It is the only value at which the D7 can cap o↵ before entering the horizon atsome radius r0, in a so-called Minkowski embedding. To see this, note that for such anembedding, the profile d dr will diverge at r0. Then the only way to keep the free energyfinite, is if ⌫ = 1 and  (r0) = 0. The latter condition is indeed verified by numericallysolving the equation of motion for  (r). The Minkowski embedding enforces a minimumlength for strings stretching from the probe brane to the horizon, and thus forms a chargegapped state. So we see that this system recovers the essential property of a quantumhall system that we were looking for. A gapped state occurs only at integer fillingfraction. Note that this argument only works for ⌫ = 1. To get the higher integer Hallstates one must consider multiple D7 branes with the D5 flux split between them. Thisis investigated in [2, 4].6.1.3 D5-D7 phase diagramTo determine if the blow up to a D7 really occurs, we must know if the D7 systemoutlined above is energetically preferable to the D5 system outlined in section 6.1.1.This requires computing the on-shell free energy in each case. First one must computethe embedding solution  (r) numerically. This can essentially be done with a shootingmethod. Some results are given in section 7.2. The integral over r in the free energies isformally divergent, but a comparison between F5 and F7 only requires their di↵erence,which is finite. This di↵erence was computed at zero temperature in [2] for a variety ofparameters (⌫, f). There it was found that as one tunes ⌫ from 0 to 1, there is a critical71value ⌫c, dependent on f , where the blown up D7 configuration does have lower energythan the D5s. It continues to be favourable for ⌫ > ⌫c, and at ⌫ = 1, the favoured D7gets a Minkowski embedding. In that paper, the authors confirmed that the D5 chiralsymmetry breaking solution is stable below the bound ⌫/f < 1.68. They also found thatthe D7 chiral symmetry breaking solution is not only stable beyond this regime, but isin fact, energetically preferable.The D5-D7 phase diagram was extended to finite temperature in [4], where it was foundthat for large f , the D5-D7 transition occurs at ⌫ = 0.5.To summarize, if we start at ⌫ = 0 and zero temperature, we have a D5 system withSU(N5) symmetry. As soon as the magnetic field is turned on, the S˜O(3) symmetry isbroken via magnetic catalysis of chiral symmetry breaking. Then if we increase ⌫ to ⌫c,the D7 takes over, breaking the SU(N5) symmetry. This continues to ⌫ = 1, where theD7 forms a gapped state. If, however, we turn up the temperature, we will eventuallyrestore the S˜O(3) symmetry. This chiral symmetry restoration phase was found to occuratrh ⇡ 0.4 (6.36)T ⇡ 0.32pB1/4 . (6.37)72Chapter 7Entropy of the Giant D5 ModelIn this chapter we investigate some thermodynamics of the model setup in the previouschapter. Specifically, we will find an analytic solution for the entropy of this system atvery low temperatures.7.1 Entropy calculation7.1.1 Entropy of D5 branesThe entropy can be found from the first derivative of the free energy with respect totemperature. Starting with the setup outlined in section 6.1.1, we had the free energygiven by (6.16).Now, the entropy is justS5 = @F5@rh @rh@T (7.1)= ⇡1/4p2⇡B@F5@rh . (7.2)And,@F5@rh = N˜5✓L(rh, rh) Z 1rh dr@rhL(rh, r)◆, (7.3)where L(rh, r) refers to the integrand of (6.16) (i.e. the transformed Lagrangian).The first term in the bracket is this integrand evaluated at r = rh,L(rh, rh) = q4 sin4( (rh))f2(1 + r4h) + (⇡⌫)2. (7.4)73The second term is:Z 1rh dr@rhL(rh, r) =  Z 1rh dr✓@L@h @rhh+ @L@ 0(r)@rh 0(r) + @L@ @rh ◆= Z 1rh dr✓@L@h @rhh+ @L@ 0(r)@rh 0(r) + ddr✓ @L@ 0◆@rh ◆= @L@ 0(r)@rh 1rh  Z 1rh dr@L@h @rhh. (7.5)In the second line above, we used the equation of motion for  (r), and in the third, weintegrated by parts.Until now, we have ignored regularization, and the Lagrangian is divergent as r ! 1.Often in AdS/CFT, this problem requires “holographic renormalization” through theaddition of counter-terms to the boundary action [62], but here we can address theproblem simply by subtracting a background Lagrangian that matches our Lagrangianasymptotically, so as to cancel the divergence. All of our energies will then be in referenceto this background system. In this case, the obvious choice of background system to useis the same D5 model, but at zero charge density. LBG = limq5!0 L. Looking at (6.16),we do indeed get LBG|r!1 = L, so our free energy and entropy will be finite. With thisbackground subtraction, the first term in (7.5) becomes✓@L@ 0(r)  @LBG@ 0(r)◆@rh 1rh . (7.6)Near the boundary, we maintain the asymptotic form of the  (r) solution with zeromass for the fundamental representation fields,  (r) = ⇡2 + c2r2 + . . .. We see thatlimr!1 @rh (r) = 0, and so (7.6) is completely determined at the horizon where:@L@ 0(r) r=rh = hr2 0(r)p1 + h(r 0(r))2q4 sin4  (r)f2(1 + r4) + (⇡⌫)2r=rh = 0 = @LBG@ 0(r) r=rh ,(7.7)since L and LBG have the same dependency on  0(r).All that remains to evaluate is the second term in (7.5). Here we appeal to a low tem-perature calculation. In particular, we assume that terms of order T 4 can be neglectedin the free energy (or T 3 can be neglected in the entropy). In this case, we can expand74the second term to second order in temperature (or equivalently rh):Z 1rh dr@L@h @rhh= Z 1rh dr@L@h @rhhrh=0+ rh @L@h @rhhr!rh| {z }L1  Z 1rh ✓ ddrh✓@L@h◆@rhh+ @L@h @2rhh◆| {z }L2 drrh=0+r2h2!dL1drh + L2|r!rh  Z 1rh ✓ d2dr2h✓@L@h◆@rhh+ 2 ddrh✓@L@h◆@2rhh+ @L@h @3rhh◆drrh=0+ O(r3h). (7.8)Now,@rhh = 4r3h/r4@L@h=12(r 0(r))2s4 sin4  f2(1 + r4) + (⇡⌫)21 + h(r 0(r))2 (7.9)L1 = 2rhq(⇡⌫)2 + 4f2(1 + r4h) sin4( (rh)) 0(rh)2 (7.10)L2 =2r2hp(⇡⌫)2 + 4f2(1 + r4) sin4( (r)) 0(r)2(3r2 + (3r4  r4h) 0(r)2)r(r2 + (r4  r4h) 0(r)2)3/2 .(7.11)After regularization, the integrals on the right hand side will be finite and the presenceof the h derivatives will cause them to vanish in the limit rh ! 0.So the first two lines on the right hand side of (7.8) are zero and only the quadraticterm contributes.Z 1rh dr@L@h @rhh = 4 0(0)2q4 sin4  (0)f2 + (⇡⌫)2r2h +O(r3h). (7.12)One can easily see that the embedding equation of motion, (6.21), requires  (0) = 0 forthe D5, but the value  0(0) will be determined later in section 7.2.Thus we finally have all the terms in equation (7.3), and so the D5 entropy to secondorder in temperature (or equivalently horizon radius) is:S5 ⇡N˜5⇡2⌫1/4p2⇡B(1 4 0(0)2r2h)=pNcN5V2+1q5(2⇡)2(1 4 0(0)2r2h), (7.13)75where we have truncated the L(rh, rh) term, (7.4), to second order, assuming  0(0) tobe finite. There are two things to note about this expression. First, it is linear in ⌫even at finite low temperature. Second, it scales asp at T = 0. This is strangebecause we are working in the regime of large . Such a large non-zero entropy at zerotemperature is often a sign of degeneracy or an unstable solution. It may also be anartifact of the large N limit. Stability can be tested by looking at the equations ofmotion of fluctuations about the background fields, but we will not report on that here.This kind ofp behaviour of the entropy is often encountered in holographic models,and we will discuss its interpretation below. In any case, we can evaluate the bracket ifwe know the embedding solution (or rather it’s derivative) at the horizon. However, wewill postpone this until after discussing the D7 brane.7.1.2 Comparison to single quark entropyThe above result has a nice interpretation from string theory following a similar analysisdone by Herzog et al [63]. What is the free energy of a single string connecting a probebrane in AdS to the horizon? Consider the AdS5 part of the black-brane supergravitysolutionds2 =✓ hr2dt2 + r2dx2 + r2dy2 + r2dz2 +dr2hr2◆. (7.14)The Nambu-Goto action,S = Ts Z dd⌧pdetgab, (7.15)requires the pullback of the AdS4 metric to the string world-sheet : gab = @Xµ@xa @X⌫@xb Gµ⌫ .Here Xµ are the space-time coordinates and xa are the world-sheet coordinates , ⌧ .Recall that the action is reparameterization invariant, so we are free to choose the staticgauge  = r, ⌧ = t. In this case,g = 1hr2 + r2x02 + r2y02 + r2z02g⌧⌧ = hr2 + r2x˙2 + r2y˙2 + r2z˙2g⌧ = r2x˙x0 + r2y˙y0 + r2z˙z0, (7.16)where dots are derivatives with respect to ⌧ and primes are derivatives with respect to. Then,detgab = g2⌧  gg⌧⌧ (7.17)= r4(x˙x0 + y˙y0 + z˙z0)2  ✓ 1hr2+ r2x02 + r2y02 + r2z02◆r2(h+ x˙2 + y˙2 + z˙2).76We can then compute the quantity@pdetgab@X˙↵ = Tspdetgab [(X˙ ·X 0)X 0⌫ X 02X˙⌫ ]G↵⌫ , (7.18)which allows us to find the canonical energy density:⇡⌧t ⌘ @L@X˙t = Tspdetgabhr2✓ 1hr2 + r2x02 + r2y02 + r2z02◆. (7.19)The free energy is then just the integral of this along the stringF = Zd⇡⌧t . (7.20)We want one end of the string to act as a source for the gauge fields on the probe Dbrane. The other end cannot just end anywhere, and so must stretch to the black holehorizon (or the Poincare´ horizon if there was no black hole). So the endpoints are atrh and rb, where rb lies somewhere on the probe brane. One can easily check that theequations of motion for the string are satisfied by the constant solutionx(r, t) = x0, y(r, t) = y0, z(r, t) = z0. (7.21)In which case,pdetgab = 1. (7.22)So in the static gauge,F = Ts Z rbrh dr = Ts(rb  rh). (7.23)In the zero temperature limit, the free energy must recover the Lagrangian quark massm, so Tsrb = m. The string tension is related to the ’t Hooft coupling via Ts = p2⇡ .Thus,F = m12pT, (7.24)and the entropy of a single quark isS =p2, (7.25)which is of course temperature independent. At zero temperature, the entropy we foundfor the D5 brane was (from (7.13)):S5 =pNcN5V2+1q5(2⇡)2=pV2+1⇢2. (7.26)So we see that, at least at zero temperature, the probe D5 stack entropy is precisely theentropy for a single quark times the total charge of the system.777.1.3 Entropy of the D7 braneWe repeat the analysis of section 7.1.1 for the case of the blown up D7 brane outlinedin 6.1.2.To get the entropy, we again evaluate equation (7.3), only using the Lagrangian fromF7 (equation (6.31)). This time we have for the first term:L(rh, rh) = q4 sin4  (rh)(f2 + 4 cos4  (rh))(1 + r4h) + (⇡(⌫  1) + 2 (rh) 1/2 sin 4 (rh))2.(7.27)The second term is evaluated using (7.5) where again the zero charge density backgroundLagrangian has the same  0(r) dependancy, so that@L@ 0(r)@rh 1rh = @LBG@ 0(r)@rh 1rh = 0. (7.28)The final term is evaluated using the rh expansion, equation (7.8). For the D7,@L@h=12(r 0)2p4 sin4  (f2 + 4 cos4  )(1 + r4) + (⇡(⌫  1) + 2  1/2 sin(4 ))2p1 + h(r 0)2L1 = 2rh✓4(1 + r4h)(f2 + 4 cos4( (rh))) sin4( (rh)) + (⇡(⌫  1) + 2 (rh)1/2 sin(4 (rh)))2◆1/2 0(rh)2 (7.29)L2 =2r2h 0(r)2(3r2 + (3r4  r4h) 02)r(r2 + (r4  r4h) 02)3/2 ✓4(1 + r4)(f2 + 4 cos4  ) sin4  + (⇡(⌫  1)+2  1/2 sin(4 ))2◆1/2. (7.30)So the order T 0 and T 1 terms vanish using the same arguments as in the D5 case. Thenon-zero quadratic terms are:dL1drh |rh!0 = 2 0(0)✓4(f2 + 4 cos4( (0))) sin4( (0)) + (⇡(⌫  1) + 2 (0)1/2 sin(4 (0)))2◆1/2(7.31)L2|r!rh |rh!0 = 6 0(0)✓4(f2 + 4 cos4( (0))) sin4( (0)) + (⇡(⌫  1) + 2 (0)1/2 sin(4 (0)))2◆1/2. (7.32)78We can now write out the entropy for the D7 brane.S7 ⇡ N˜7⇡1/4p2⇡B✓L(rh, rh) 4r2h 0(0)2L(0, 0)◆= pNcV2+1B(2⇡)2✓L(rh, rh) 4r2h 0(0)2L(0, 0)◆. (7.33)We need to expand L(rh, rh) to second order in rh:L(rh, rh) ⇡ L(0, 0) + dL(rh, rh)drh rh=0rh + d2L(rh, rh)dr2h rh=0 r2h2= L(0, 0) +✓@L(rh, rh)@  0(rh) + @L(rh, rh)@rh ◆rh=0rh+✓@2L(rh, rh)@ 2 0(rh)2 + @L(rh, rh)@  00(rh) + @2L(rh, rh)@r2h ◆rh=0 r2h2(7.34)But from (7.27) we can see that@L(rh, rh)@rh rh=0 = 0 = @2L(rh, rh)@r2h rh=0. (7.35)We can use a trick to rewrite @L(rh,rh)@ . Using the equation of motion for  , we get@L(rh, rh)@ =✓ddr@L(r, rh)@ 0 ◆r=rh=✓ddrp4 sin4  (f2 + 4 cos4  )(1 + r4) + (⇡(⌫  1) + 2  1/2 sin(4 ))2p1 + h(r 0(r))2⇥h(r2 0(r))◆r=rh=q4 sin4  (f2 + 4 cos4  )(1 + r4) + (⇡(⌫  1) + 2  1/2 sin(4 ))2⇥r2 0(r)@rhr=rh= L(rh, rh)(4rh 0(rh)) (7.36)And therefore,@2L(rh, rh)@ 2= L(rh, rh)(4rh 0(rh))2. (7.37)So the second order expansion of L(rh, rh) becomesL(rh, rh) ⇡ L(0, 0)14rh✓rh 0(rh)2◆rh=0+2r2h✓rh 0(rh) 00(rh)+4r2h 0(rh)4◆rh=0.(7.38)79Thus, the D7 entropy to second order in temperature is:S7 ⇡pNcV2+1B(2⇡)2L(0, 0)⇢1 4rh✓rh 0(rh)2◆rh=0+ 2r2h✓ 2 0(rh)2  rh 0(rh) 00(rh) + 4r2h 0(rh)4◆rh=0. (7.39)To evaluate this requires knowing  (0),  0(0), and  00(0), or at least whether theyblow up at rh ! 0, and this must be determined numerically. However, we can getan analytic expression if we consider the regime of large f . This is shown in the nextsection. Meanwhile, the numerical embedding solutions reveal thatdd ln r (r)|r=rh=0 = 0) rh 0(rh)|rh=0 = 0. (7.40)So only one of the quadratic terms in rh above contribute, and the D7 entropy takes thesame form as the D5 entropyS7 ⇡pNcV2+1B(2⇡)2L(0, 0)⇢1 4 0(0)2r2h . (7.41)7.2 Embedding solutionsAt the horizon, the equation of motion that  must satisfy is@ VVrh = 0, (7.42)for the potentialsV5 = 4 sin4  f2(1 + r4h) + (⇡⌫)2 (7.43)V7 = 4 sin4  (f2 + 4 cos4  )(1 + r4h) + (⇡(⌫  1) + 2  1/2 sin 4 )2. (7.44)The minimum of V5 is simply  = 0 for which the entropy is given in equation (7.13).At finite temperature, the derivative  0(rh) is set by the equation of motion 0(rh) = 2 sin3  (rh) cos (rh)f2(1 + r4h)rh(4 sin4  (rh)f2(1 + r4h) + (⇡⌫)2) . (7.45)Since we know  (0) = 0, we can expand  in a Taylor series around rh = 0,  (rh) =a1rh + a2r2h + . . ., and taking the limit of the above equation as rh goes to zero gives 0(0) = 0. Thus, the second term in (7.13) vanishes, and the D5 entropy is constant up80to third order in the temperature:S5 =pNcV2+1B(2⇡)2⌫ +O(r3h) (7.46)Turning now to the D7 brane, the minimum of V7 depends on the values of ⌫, f andmust be solved numerically. We solve this in the range 0 < ⌫ < 1 for di↵erent valuesof f ensuring that the equation of motion is also satisfied1. The embedding angle  (0)drops continuously as ⌫ is increased from 0 to 1, and as f is increased (see figure 7.1).Out[2999]=0.2 0.4 0.6 0.8 1.0n0.51.01.5yH0Lf=0.0f=0.5f=1f=5Figure 7.1: D7 brane embedding solutions at r = rh = 0 for di↵erent values of f , ⌫.The full solution  (r) is found by iterating over a grid of initial values of  (0). Fora given initial value, the derivative  0(0) is set by the equation of motion. We thenintegrate the solution up to a large value of r, and fit the result to the asymptotic form = ⇡2 + mr + c2r2 . The solution with parameters closest to  1 = ⇡/2 and m = 0 is thentaken to be the embedding solution. In this way, c2 is determined dynamically. Theresult is plotted for di↵erent values of rh, in figure 7.2.From these solutions, we confirm that limrh!0 rh 0(rh) = 0, so that the D7 entropy tosecond order in temperature isS7 ⇡pNcV2+1B(2⇡)2q4 sin4  (0)(f2 + 4 cos4  (0)) + (⇡(⌫  1) + 2 (0) 1/2 sin 4 (0))2⇥✓1 4 0(0)2r2h◆. (7.47)1This is only really a concern at ⌫ = 1 where the minimum of V7 is also a root and the left hand sideof the equation of motion above has a discontinuity.81Out[735]=0.0 0.5 1.0 1.5 2.0 2.5r0.51.01.5yHrLrh=0.10rh=0.20rh=0.23pê2Figure 7.2: Embedding solutions at di↵erent temperatures for ⌫ = 0.6, f = 1 Allsolutions asymptote to ⇡/2.We can solve this analytically by considering the large f regime of the potential V7.Then we haveV7 ⇡ 4f2 sin4  + (⇡(⌫  1) + 2  1/2 sin(4 ))2, (7.48)which is always non-negative and also temperature independent. To minimize at largef , sin4  must be small and so we are forced to consider  near 0 or ⇡. Then in thebracket, the linear term in  guarantees that  is near 0. For small  , the potentialsimplifies greatly,V7 ⇡ (⇡(⌫  1))2 +O( 2). (7.49)But V7|rh=0 is exactly what is under the square root of equation (7.47). So for large f ,the zero temperature entropy becomesS7 ⇡pNcV2+1B(2⇡)2⇡|⌫  1|. (7.50)Comparing this to the D5 entropy, we see that the entropy is just a simple sawtoothfunction of ⌫ with a transition between D5 and D7 occurring at ⌫ = 0.5. This is the lastplot in figure 7.3. The other plots show the overall entropy for di↵erent values of f thathave been found by plugging the numerical solutions for  (0) into equation (7.47) atrh = 02. Note that the crossover between solutions takes place at lower filling fractionsas f is decreased, and discontinuities appear.2Using a small non-zero value of rh would not visibly a↵ect these plots since  0(0)rh is negligible820.2 0.4 0.6 0.8 1.0n0.20.40.60.81.01.21.4Sf=1.00.2 0.4 0.6 0.8 1.0n0.51.01.5Sf=2.00.2 0.4 0.6 0.8 1.0n0.51.01.5Sf=10.0Figure 7.3: Low temperature entropy versus filling fraction for di↵erent values of f .The crossover is the D5-D7 transition ⌫c where their Free energies are equal. The unitsof entropy arepNcV2+1B(2⇡)2 .83We could also ask about the heat capacity, which is found from the entropy viacv = T @S@T . (7.51)For the D5, we can guarantee that the leading order of the low temperature heat capacityis T 3 or higher. For the D7, the heat capacity goes as order T 2 or higher.7.3 Weak coupling entropyWe may make a qualitative comparison to the zero-temperature entropy of the theoryat weak coupling. The degeneracy of the Landau levels is d ⌘ 2N5Nc B2⇡ , and the totalfraction of levels filled is ⌫N5 . Thus, the number of micro-states available is⌦ =✓d⌫N5d◆, (7.52)and the entropy isS = ln✓d!( ⌫N5d)!(d ⌫N5d)!◆. (7.53)Since the degeneracy is very large, we may use Stirling’s approximation to write this as:S ⇡ d ln✓dd ⌫N5d◆+ ⌫N5d ln✓d ⌫N5d⌫N5d ◆= d⌫N5ln✓⌫N5◆+✓⌫N5 1◆ln✓1⌫N5◆, (7.54)which is shown in figure 7.4.Note the symmetry about ⌫ = 0.5 in both the weak coupling and f = 10 strong couplingcase (last plot in figure 7.3). This is a reflection of particle-hole symmetry, since we candetermine the entropy by calculating the number of occupied or unoccupied sites. Thus,the first two plots in 7.3 indicate that particle-hole symmetry is somehow broken as welower f , and thereby lower the number of flavours in the strongly coupled field theory.84Out[1730]=0.2 0.4 0.6 0.8 1.0nêN50.10.20.30.40.50.60.7SêdFigure 7.4: Zero-temperature entropy at weak coupling.85Chapter 8Conductivity of the Giant D5ModelIn this chapter we compute the conductivity tensor of giant D5 model, and compare theHall conductivity to the classical value.8.1 D5 brane case8.1.1 SetupIt turns out to be much more convenient to compute the conductivity in Fe↵erman-Graham coordinates. Recall from section 2.1 that the AdS metric in Fe↵erman-Grahamcoordinates was defined byds2 =L2z2(⌘µ⌫dxµdx⌫ + dz2), (8.1)where we know that L2 =p↵02. Comparing to our usual Poincare´ AdS black holecoordinates, we see that the two coordinates should be related viadz2z2=dr2r2h(8.2)) dr = rphzdz, (8.3)86where we have chosen the sign so that z goes to zero at the AdS boundary. Integratingthis result, with the appropriate choice of integration constant givesz2 =2r2 +qr4  r4h , zh = p2rh = p2⇡T . (8.4)These can be inverted to getr2 =(z4 + z4h)z2z4h , h(z) = 1 4z4z4h(z4 + z4h)2 . (8.5)The world-volume metric of the D5 branes can now be written asds2 =p↵0[gttdt2 + gxx(dx2 + dy2) + gzzdz2 + sin2  (d✓2 + sin2 ✓d2)]. (8.6)With the above transformations, we can find the metric components:gtt = r2h = 1z2 (1 z4/z4h)21 + z4/z4h (8.7)gxx = gyy = r2 = 1z2 (1 + z4/z4h) (8.8)gzz = 1z2 (1 + z2 0(z)2). (8.9)The 2-sphere metric remains unchanged, though now  is a function of z.To find the conductivity we need to include an external electric field ~E =p2⇡ exˆ. Fur-thermore, we expect a non-zero current in the field theory hJxi, hJyi. These are dualto the gauge field components Ax, Ay in exactly the same way that hJti was dual to At(see section 6.1.1). This a gives a more elaborate gauge field on the probe branes,2⇡↵0F = p↵0[a0(z)dz ^ dt+ f 0x(z)dz ^ dx+ f 0y(z)dz ^ dy + bdx ^ dyedt ^ dx]. (8.10)Again, Wess-Zumino terms will not contribute to the D5 action. Evaluating the remain-ing DBI action,S5 = T5gs N5 Z d6p det(g + 2⇡↵0F), (8.11)requires the determinant det(g + 2⇡↵0F) = detA detB, whereA =0BBBBB@gtt e 0 a0(z)e gxx b f 0x(z)0 b gxx f 0y(z)a0(z) f 0x(z) f 0y(z) gzz 1CCCCCA , B =  sin2  00 sin2  sin2 ✓! . (8.12)87This yields,S5 =2pNcN5(2⇡)3V2+1Z zh0dzpS, (8.13)whereS ⌘ 4 sin4  ✓2a0(z)bef 0y(z) a0(z)2(b2 + g2xx) e2(f2y + gxxgzz)+gtt[(f 0x(z)2 + f 0y(z)2)gxx + (b2 + g2xx)gzz]◆. (8.14)Note that the limits e, f 0y(z), f 0x(z) ! 0, gives back the action we had earlier for theD5s.Now just as a(z) was cyclic in the case with no electric field, fx(z) and fy(z) are cyclicnow, and their equations of motion each yield an integration constant which is a con-served charge q, qx, qy. These are related to the components of the current densities inthe same way as beforeq =(2⇡)22NcN5 ⇢, (qx, qy) = (2⇡)22NcN5 (Jx, Jy), (8.15)where(Jx, Jy) =  1V2+1✓ S5f 0x(z) , S5f 0y(z)◆. (8.16)The gauge field equations of motion then readq =1pS4 sin4  [bef 0y(z) a0(z)(b2 + g2xx)], (8.17)qx = 1pS4 sin4  [gttgxxf 0x(z)], (8.18)qy = 1pS4 sin4  [a0(z)be+ f 0y(z)(gxxgtt  e2)]. (8.19)8.1.2 ConductivityIn analogy with the Legendre transform we used earlier,F5 = S5 +At(1)⇢V2+1, (8.20)we similarly do a Legendre transform for the other two gauge components to get theRouthianR5 = S5 +At(1)⇢V2+1 +Ax(1)JxV2+1 +Ay(1)JyV2+1 (8.21)=2NcN5(2⇡)2V2+1p2⇡Z zh0dz[L5 + qa0(z) + qxf 0x(z) + qyf 0y(z)], (8.22)88where again we assumed that the gauge fields vanish at the horizon (this will be consis-tent with their equations of motion). We get,R5 = 2NcN5(2⇡)2V2+1p2⇡Z zh0dz4 sin4  pS[gzz(gttb2 + gttg2xx  gxxe2)]. (8.23)The details of how we extract the conductivity from this Routhian are relegated to theappendix. However, the outline goes as follows. By solving for the gauge fields in termsof their conserved charges, we can write the Routhian in the schematic formR5 =2NcN5(2⇡)2V2+1p2⇡Z zh0dzrgzzgtt 1gxxpBC A2, (8.24)where we have definedA ⌘ qbgtt  qyegxx (8.25)B ⌘ gttg2xx + gttb2  gxxe2 (8.26)C ⌘ 4 sin4  gttg2xx + gttq2  gxx(q2x + q2y). (8.27)Now we follow a beautiful little argument due to O’Bannon [64], by looking at theroots of A, B, and C. It turns out that reality of the Routhian requires that these allshare a single common zero at some position z⇤. That gives three equations B(z⇤) = 0,A(z⇤) = 0, and C(z⇤) = 0 with three unknowns z⇤, qx, and qy. Solving for the latter twogivesqy = qbe(gxx(z⇤)2 + b2) . (8.28)andqx = egxx(z⇤)(gxx(z⇤)2 + b2)q4 sin4  (z⇤)(gxx(z⇤)2 + b2) + q2. (8.29)This two equations allow us to directly read o↵ the conductivity tensor.hJii = xiE (8.30)) xi = 2NcN5(2⇡)pqie|e=0, (8.31)where i = x, y. Written in terms of the rh, ⌫ and f variables from before we have thefinal expressions for the conductivity tensorxx = 2Nc(2⇡)2 f✓ r2h1 + r4h◆q4 sin4  (zh)(1 + r4h) + (⇡⌫/f)2 , (8.32)89andboxedxy = Nc⌫2⇡ ✓1 r4h1 + r4h◆. (8.33)Here we set e = 0 to look at the linear response, and we may simply insert the horizonembedding  (rh), that we found in the entropy calculation, into xx.8.2 D7 brane case8.2.1 SetupBy looking at the actions for the D5 and D7 (6.6), (6.29), we see that we can get the D7action from the D5 action by adding a Wess-Zumino term and using the replacementrule4 sin4  ! 4 sin4  (f2 + 4 cos4  ) (8.34)in the integrand S.The Wess-Zumino term with the new gauge fields isT7(2⇡↵0)22C(4) ^ F ^ F =Nc(2⇡)3✓c( )2(ba0(z) ef 0y(z)V2+1)◆. (8.35)At this point, for convenience, we have redefined c( ) !Rd✓˜d˜c( ).The action is now,S7 =2Nc(2⇡)4V2+1Z 10dzpS(f2 + 4 cos4  ) +12c( )(a0(z)b ef 0y(z)). (8.36)In fact, our replacement rule allows us to compute the conserved charges with ease.From,q =(2⇡)42NcN5V2+1 S7a0(z) , (8.37)we note that the derivative with respect to a0(z) gives a Wess-Zumino contribution of1/2bc( ), so thatq =4 sin4  pSpf2 + 4 cos4  (bef 0y(z) a0(z)(b2 + g2xx)) 12bc( ). (8.38)90Similarly, qx receives no Wess-Zumino contribution since the functional derivative is withrespect to f 0x(z), and qy receives a contribution of 1/2ec( ), yieldingqx = 4 sin4  pSpf2 + 4 cos4  (gttgxxf 0x(z)) (8.39)qy = sin4  pS[a0(z)be+ f 0y(z)(gxxgtt  e2)] 12ec( ). (8.40)8.2.2 ConductivityWe can apply our replacement rule to find the Routhian as well. The Wess-Zuminoterms (WZ) contribute to the Legendre transform viaZdza0(z)(WZ)a0(z) Z dzf 0x(z)(WZ)f 0x(z)Z dzf 0y(z)(WZ)f 0y(z) = 12 Z dzc( )(a0(z)bef 0y(z)).(8.41)So the Legendre transform serves to precisely to remove the Wess-Zumino term fromthe action (of course it still a↵ects the conserved charges), so we can write the RouthianasR7 = 2Nc(2⇡)4V2+1Zdz4 sin4  pSpf2 + 4 cos4  [gzz(gttb2 + gttg2xx  gxxe2)]. (8.42)This is exactly the same as the D5 Routhian integrand (modulo thepf2 + 4 cos4  factor), so we can write it asR7 = 2Nc(2⇡)4V2+1Zdzrgzzgtt 1gxxpBC A2, (8.43)only we need to adjust the charges sinceqD5 = qD7 + 12bc( ), & qD5y = qD7y + 12ec( ), (8.44)so that the functions A and C change toA ! (q + 1/2bc( ))bgtt  (qy + 2ec( ))egxx (8.45)C ! 4 sin4( )(f2 + 4 cos4  )gttg2xx + gtt(q + 2bc( )) gxx[q2x + (qy + 2ec( ))2].However, the function B is unchanged, and thus so is its zero z⇤. The same argumentapplies as before; the equations A(z⇤) = 0 and c(z⇤) = 0, determine the charges to beqy = ✓bq  12c( (z⇤))gxx(z⇤)b2 + g2xx(z⇤) ◆e (8.46)qx = gxxb2 + g2xx(z⇤)q4 sin4  (z⇤)(f2 + 4 cos4  (z⇤))(b2 + g2xx(z⇤)) + (q + 1/2bc( (z⇤)))2e.91Again, we can read o↵ the conductivities at e = 0,xx = Nc2⇡2✓ r2h1 + r4h◆q4 sin4  (zh)(f2 + 4 cos4  (zh))(b2 + g2xx(zh)) + (q + 1/2bc( (zh)))2(8.47)xy = N2⇡⌫ + ✓(1 ⌫) + sin 4 (zh)2⇡  2⇡ (zh)◆✓ r4h1 + r4h◆ , (8.48)where we have used lime!0 z⇤ = zh.8.3 ResultsComparing our results to the classical linear Hall conductivity xy = Nc⌫2⇡ , we see thatthe D5 hall conductivity lies below this linear curve at finite temperature, and matchesthe classical case at zero temperature.For the D7 Hall conductivity, note that figure 7.1 shows that  (rh) ! 0 at ⌫ = 1 for allvalues of f . In this case, we get D7xy = Nc⌫2⇡ . So the D7 Hall conductivity always returnsto the classical value at ⌫ = 1. Below ⌫ = 1, D7xy always lies above the classical curve.The amount of deviation of each conductivity tensor component from the linear curveis shown in figure 8.1 for rh = 0.2 and in figure 8.2 for rh = 0.4.These deviations from the classical value hint at a plateau. Indeed, if we could take thelarge rh limit, we would find that D5xy ! 0 throughout the range of ⌫ over which theD5 dominates. Furthermore, in the limit of large f ,  (rh) ! 0 (see 7.1) and soD7xy ! N⌫2⇡ + N(1 ⌫)2⇡ (8.49)=N2⇡. (8.50)In this case, the conductivity would exhibit perfect integer plateaus with a transitionat the D5-D7 transition. Of course we cannot actually take this limit, because thetemperature is numerically confined to rh . 0.4. At higher temperatures, the chiralsymmetry is restored. Pushing rh to the edge of this limit, we see that the jump inconductivity is rather small. This is plotted with the classical Hall conductivity forcomparison at large f in figure 8.3.Beyond ⌫ = 1, we know that a composite system takes over and we will not discussthis here. Indications are that the conductivity tensor in the interval 1  ⌫  2 isqualitatively the same as the interval 0  ⌫  1.920.2 0.4 0.6 0.8 1.0n-0.3-0.2-0.10.10.20.3sxyf=1.00.2 0.4 0.6 0.8 1.0n0.20.40.60.8sxxf=1.00.2 0.4 0.6 0.8n51015rxxf=1.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=2.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=2.00.2 0.4 0.6 0.8n51015rxxf=2.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=10.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=10.00.2 0.4 0.6 0.8n2468101214rxxf=10.0Figure 8.1: The first column shows the deviation of the holographic Hall conductivityfrom the classical one. The second column shows the holographic longitudinal conduc-tivity. Each row corresponds to a di↵erent value of f . As we move along ⌫ in the x-axis,a transition occurs between the D5 (blue) and the D7 (red). The units of the y-axesare Nc2⇡ r4h/(1 + r4h) for the first column andNc2⇡ r2h/(1 + r4h) for the second column. Thetemperature for all of these plots corresponds to rh = 0.2.The ohmic conductivity xx is just a temperature factor times the Routhian evaluatedat the horizon, and thus is similar to the low-temperature entropy. Indeed, at lowtemperature and large f , the plot of xx vs ⌫ becomes identical to the entropy plot inthis regime with the appropriate unit conversion (see the second column, third row of8.1).For convenience, we also compute the resistivity,⇢xx = xx2xx + 2xy . (8.51)930.2 0.4 0.6 0.8 1.0n-0.10.10.20.3sxyf=1.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.50.6sxxf=1.00.2 0.4 0.6 0.8n5101520rxxf=1.00.2 0.4 0.6 0.8 1.0n-0.3-0.2-0.10.10.20.30.4sxyf=2.00.2 0.4 0.6 0.8 1.0n0.20.40.60.8sxxf=2.00.2 0.4 0.6 0.8n51015rxxf=2.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=10.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=10.00.2 0.4 0.6 0.8n246810rxxf=10.0Figure 8.2: The first column shows the deviation of the holographic Hall conductivityfrom the classical one. The second column shows the holographic longitudinal conduc-tivity. Each row corresponds to a di↵erent value of f . As we move along ⌫ in the x-axis,a transition occurs between the D5 (blue) and the D7 (red). The units of the y-axesare Nc2⇡ r4h/(1 + r4h) for the first column andNc2⇡ r2h/(1 + r4h) for the second column. Thetemperature for all of these plots corresponds to rh = 0.4.In the large f limit, we haveD5xx = r2h1 + r4h Nc⌫2⇡ (8.52)D5xy = ✓1 r4h1 + r4h◆Nc⌫2⇡ (8.53)D7xx = r2h1 + r4h Nc2⇡ (1 ⌫) (8.54)D7xy = Nc2⇡✓⌫ + (1 ⌫) r4h1 + r4h◆, (8.55)940.2 0.4 0.6 0.8 1.0n0.20.40.60.81.0sxyFigure 8.3: The Hall conductivity vs filling fraction for the D5 brane (red), the D7brane (blue), and the classical case (green), for f = 10. The temperature correspondsto rh = 0.4.which gives,⇢D5xx = 2⇡N⌫ r2h (8.56)⇢D7xx =  2⇡Nc r2h(⌫  1)r4h + ⌫2 . (8.57)For smaller values of f , ⇢xx is shown in 8.4 and 8.5.950.2 0.4 0.6 0.8 1.0n-0.3-0.2-0.10.10.20.3sxyf=1.00.2 0.4 0.6 0.8 1.0n0.20.40.60.8sxxf=1.00.2 0.4 0.6 0.8n51015rxxf=1.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=2.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=2.00.2 0.4 0.6 0.8n51015rxxf=2.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=10.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=10.00.2 0.4 0.6 0.8n2468101214rxxf=10.0Figure 8.4: Longitudinal resistivity as a function of filling fraction. Each row cor-responds to a di↵erent value of f . As we move along ⌫ in the x-axis, a transitionoccurs between the D5 (blue) and the D7 (red). The units of the y-axes are 2⇡Nc r2h. Thetemperature for all of these plots corresponds to rh = 0.2.960.2 0.4 0.6 0.8 1.0n-0.10.10.20.3sxyf=1.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.50.6sxxf=1.00.2 0.4 0.6 0.8n5101520rxxf=1.00.2 0.4 0.6 0.8 1.0n-0.3-0.2-0.10.10.20.30.4sxyf=2.00.2 0.4 0.6 0.8 1.0n0.20.40.60.8sxxf=2.00.2 0.4 0.6 0.8n51015rxxf=2.00.2 0.4 0.6 0.8 1.0n-0.4-0.20.20.4sxyf=10.00.2 0.4 0.6 0.8 1.0n0.10.20.30.40.5sxxf=10.00.2 0.4 0.6 0.8n246810rxxf=10.0Figure 8.5: Longitudinal resistivity as a function of filling fraction. Each row cor-responds to a di↵erent value of f . As we move along ⌫ in the x-axis, a transitionoccurs between the D5 (blue) and the D7 (red). The units of the y-axes are 2⇡Nc r2h.Thetemperature for all of these plots corresponds to rh = 0.4.97Chapter 9ConclusionWe have seen a lot in this thesis, so let us review the salient points. We used gauge/grav-ity duality to investigate the charge transport, and thermodynamic properties a 2 + 1dimensional field theory of fermions in a magnetic field. At strong coupling, we wereable to describe this field theory as a classical supergravity theory with additional ingre-dients, such as finite density, and external electric and magnetic fields, comprising gaugefields on a stack of probe D5 branes. At a certain filling fraction, the Ramond-Ramondsupergravity solution facilitates a Myers e↵ect in which the D5 branes blow up to forma D7 brane. At integer filling fraction, that D7 is then able to essentially dissolve thefield lines from the D5 system, allowing it to cap o↵ before entering the horizon. Thisyields a charge-gapped state in the dual theory, corresponding to an integer Hall state.We studied the low temperature entropy of this system and found that below the D5-D7 transition, the entropy has a simple interpretation in terms of individual quarks i.e.strings stretched between the probe brane and the horizon.We argued for the D5-D7 transition based on a comparison of their energies. This doesnot guarantee that the D7 is stable when it takes over. To check this, one would have tolook at the dynamics of fluctuations in the probe brane geometry and gauge fields. Thecorresponding equations of motion are highly complicated and non-linear, and require asophisticated numerical determinant method to solve [65, 66]. Furthermore, this cannotbe done with the coordinates used in this thesis. When the D7 reaches a Minkowskiembedding, it ends at some particular radial coordinate. For a complete fluctuationanalysis, this radial coordinate must be allowed to fluctuate, and this cannot be doneby simply allowing the embedding angle  to fluctuate. However, the equations ofmotion have been solved for a slightly di↵erent D7 system in transformed coordinatesthat account for this problem [44]. There it was found that for a careful choice ofboundary conditions for the fluctuations, a superfluid mode results that has a field98theory interpretation as an anyonic superfluid. There is a tantalizing possibility thatsuch a mode exists for our model as well.Lastly, we calculated the conductivity tensor for this system. As expected, transla-tion invariance prevented any integer Hall plateaus from forming. However, hints of aplateau occurring at the D5-D7 transition appeared. This opens up the possibility ofnon-impurity driven plateaus forming in a strongly coupled quantum Hall system. Re-moving the homogeneity condition in the field theory dimensions of our model would bea good direction for future work. It is entirely possible that this system is unstable toinhomogenous condensates. Such an instability was found in another D7 system in [67],where it was observed that above a critical density, a striped phase formed. Perhaps asimilar phenomenon in our system could yield integer Hall plateaus.In any case, it is clear that holography has a exceedingly high potential as a tool forprobing strongly-coupled condensed matter. For now, quantitative holographic predic-tions for the quantum Hall ferromagnet remain in the future, but the insight gainedfrom viewing this system in terms of supergravity is tremendously illuminating.99Bibliography[1] Juan Maldacena. The large n limit of superconformal field theories and supergravity.International journal of theoretical physics, 38(4):1113–1133, 1999.[2] Charlotte Kristjansen and Gordon W Semeno↵. Giant d5 brane holographic hallstate. Journal of High Energy Physics, 2013(6):1–28, 2013.[3] Robert C Myers. Dielectric-branes. Journal of High Energy Physics, 1999(12):022,1999.[4] C Kristjansen, R Pourhasan, and GW Semeno↵. A holographic quantum hall fer-romagnet. Journal of High Energy Physics, 2014(2):1–36, 2014.[5] Joel Hutchinson, Charlotte Kristjansen, and Gordon W Semeno↵. Conductivitytensor in a holographic quantum hall ferromagnet. Physics Letters B, 2014.[6] Philippe Di Francesco. Conformal Field Theory. 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Energy loss of a heavy quark moving through script n= 4 supersymmetricyang-mills plasma. Journal of High Energy Physics, 2006(07):013, 2006.[64] Andy O’Bannon. Holographic thermodynamics and transport of flavor fields. arXivpreprint arXiv:0808.1115, 2008.[65] Irene Amado, Matthias Kaminski, and Karl Landsteiner. Hydrodynamics of holo-graphic superconductors. Journal of High Energy Physics, 2009(05):021, 2009.[66] Matthias Kaminski, Karl Landsteiner, Javier Mas, Jonathan P Shock, and JavierTarrio. Holographic operator mixing and quasinormal modes on the brane. Journalof High Energy Physics, 2010(2):1–37, 2010.[67] Oren Bergman, Niko Jokela, Gilad Lifschytz, and Matthew Lippert. Striped insta-bility of a holographic fermi-like liquid. Journal of High Energy Physics, 2011(10):1–14, 2011.104Appendix AD5 ConductivityWe start with D5 Routhian in Fe↵erman-Graham coordinates (8.23)R5 = 2NcN5(2⇡)2V2+1p2⇡Z zh0dz4 sin4  pS[gzz(gttb2 + gttg2xx  gxxe2)]. (A.1)Now we can solve the q equations for the gauge fields. For f 0y(z), this gives1 =qybef 0y(z) + qya0(z)(b2 + g2xx)qbea0(z) + qf 0y(z)(gxxgtt  e2) (A.2)) f 0y(z) = b2qy  qyg2xx + qbeqgxxgtt  qe2 + qybe a0(z). (A.3)Solving for f 0x(z) gives,f 0x(z) = [gttgxx(q2x + 4gttgxx sin4  )]1/2 ✓(b2gtt + gxx(gttgxx  e2)◆q2x✓gzz(e2q+gttgxxq + beqy)2 + a0(z)2gxx(e2q2  gttgxxq2  2beqqy + (b2 + g2xx)q2y)◆(e2q+gttgxxq + beqy)21/2 (A.4)And lastly, a0(z) is,a0(z) = pgttpgzzpb2gtt + gxx(gttgxx  e2)(beqy  e2q + gttgxxq) gxx(b2gtt+gxx(gttgxx  e2))(4gttgxx sin4  (b2gtt + gxx(gttgxx  e2)) + gtt(gttgxxq2(b2 + g2xx)(q2x + q2y)) + 2begttqqy + e2(gxxq2x  gttq2))1/2 (A.5)105Inserting these identities into (8.23) givesR5 =2NcN5(2⇡)2V2+1p2⇡Z zh0dzrgzzgtt 1gxxpBC A2, (A.6)where we have definedA ⌘ qbgtt  qyegxx (A.7)B ⌘ gttg2xx + gttb2  gxxe2 (A.8)C ⌘ 4 sin4  gttg2xx + gttq2  gxx(q2x + q2y). (A.9)Now we follow the argument of [64]. First, we look to see of there are any roots of thefunction B at some value z = z⇤. The equation B(z⇤) = 0 has 16 solutions, four of whichare real. Of those four, only one of those lies in the range z 2 (0, zh). The solution isfound as follows:B(z⇤) = 0) e2z4⇤(z4⇤ + z4h) = (z4⇤  z4h)2(z8⇤ + z8h + z4hz4⇤(2 + b2z4h))z8h(z4 + z4h)) 4✓z4⇤zh◆e˜2✓z4⇤z4h + 1◆2 = ✓z4⇤z4h  1◆2✓✓z4⇤z4h + 1◆2 + 4b˜2 z4⇤z4h◆. (A.10)In the last line, we introduced the scaled fields e˜ ⌘ z2h2 e and b˜ ⌘ z2h2 b. If we also letZ ⌘ z4⇤/z4h, and introduce the coecients a2 ⌘ 2(4b˜2+4e˜21) and a3 ⌘ 4(e˜2 b˜2), thenthis now amounts to finding the roots of a quarticZ4 + a3Z3 + a2Z2 + a3Z + 1 = 0. (A.11)Dividing by Z2 and changing variables to x = Z + 1/Z, we getx2 + a3x+ a2 = 0)Z2 + 1Z= 2(e˜2  b˜2) +q4(e˜2  b˜2)2  2(4b˜2 + 4e˜2  1)) Z = e˜2  b˜2 +q1 + (e˜2  b˜2)2 + 2(e˜2 + b˜2) + 1r[(e˜2  b˜2) +q(e˜2  b˜2)2 + 2(e˜2 + b˜2) + 1]2  1, (A.12)where we have selected the only root in (0, zh).Now lets turn our attention to the function C. At the horizon, gtt ! 0 and gxx ! 2/z2h,so thatC(zh) =  2z2h (q2x + q2y) < 0. (A.13)106Meanwhile at the AdS boundary, we have gttg2xx ! 1/z6, gtt ! 1/z2, and gxx ! 1/z2,so that near z = 0,C(z) ⇡4z6q2z2(q2x + q2y)z2, (A.14)which is positive as z ! 0. Thus C must cross the axis at least once in the interval(0, zh). Suppose it had a zero at some zc 6= z⇤ (without loss of generality let zc > z⇤).Then for z 2 (z⇤, zc), the product BC would be negative. Looking at (A.6), we see thiswould cause the Routhian to become imaginary in that interval. Furthermore, A mustalso share the same zero at z⇤ since otherwise, the Routhian would be imaginary at thatpoint. Thus demanding reality of the Routhian forces A, B, and C to share a zero atz⇤, giving us three equations:B(z⇤) = 0 ) gtt(z⇤) = gxx(z⇤)e2(gxx(z⇤)2 + b2) (A.15)A(z⇤) = 0 ) qbgtt(z⇤) = qyegxx(z⇤) (A.16)C(z⇤) = 0 ) 4 sin4  (z⇤)g2xx(z⇤)e2 + q2e2 gxx(z⇤)2(gxx(z⇤)2+b2)(gxx(z⇤)2 + b2) . (A.17)Combining (A.15) and (A.16) givesqy = qbe(gxx(z⇤)2 + b2) . (A.18)While combining (A.17) and (A.18) givesqx = egxx(z⇤)(gxx(z⇤)2 + b2)q4 sin4  (z⇤)(gxx(z⇤)2 + b2) + q2. (A.19)Now note thatz4⇤z4h e˜!0 = b˜2 +qb˜4 + 2b˜2 + 1r(b˜2 +qb˜4 + 2b˜2 + 1)2  1= 1 (A.20)) gxx(z⇤)2|e˜!0 = 4z4h . (A.21)Equations (A.18), (A.19), and (8.15) directly give us the conductivity tensor.hJxi = xxE (A.22)xx = 2NcN5(2⇡)pqxe|e=0. (A.23)107Here we take e ! 0 to eliminate non-linear terms in e, since we are interested in thelinear response. Evaluating this, we getxx = 2NcN5(2⇡)p✓2/z2h4/z4h + b2◆q4 sin (zh)(4/z4h + b2) + q2. (A.24)To be consistent with the entropy results, let us convert back to rh = q2b 1zh , f = 2⇡pN5,and q = ⇡b⌫f . Then we get the longitudinal conductivity in it’s final form,xx = 2Nc(2⇡)2 f✓ r2h1 + r4h◆q4 sin4  (zh)(1 + r4h) + (⇡⌫/f)2 . (A.25)Similarly, the Hall conductivity is found fromhJyi = xyEx (A.26)) xy = 2NcN5(2⇡)pqb(4/z4h + b2) (A.27)) xy = Nc⌫2⇡ ✓1 r4h1 + r4h◆ . (A.28)Note that upon setting e = 0, qx and qy vanish and thus so do the gauge fields f 0x(z),and f 0y(z). Therefore, we may simply insert the horizon embedding  (rh), that we foundin the entropy calculation, into xx.108

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