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Holographic fermions in d=2+1 Omid, Hamid 2011

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Holographic Fermions in d = 2+1 by Hamid Omid B.Sc Physics, Isfahan University of Technology, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES (Physics) The University Of British Columbia (Vancouver) August 2011 c© Hamid Omid, 2011 Abstract Recently, a large amount of effort has gone towards using the AdS/CFT conjecture in condensed matter physics . First, we present a review of the conjecture, then we use the conjecture to model 2+ 1-dimensional fermions. We find three kinds of solutions with different kinds of discrete symmetries. We show that Chern-Simons- like electric responses, computed using a holographic model appear with the right quantized coefficients. ii Preface A version of chapter 2 and 3 is submitted to be published. [J. L. Davis, H. Omid, and G. W. Semenoff, Holographic Fermionic Fixed Points in d = 3, [arXiv:1107.4397] [hep-th]] Authors, have the same contributions in the calculations and plots. The idea was developed during several meetings between authors. The calculations in chapter 2 is done by Hamid Omid. The results in chapter 3 are obtained by equal contributions of authors. Most of manuscript was written by G. W. Semenoff and edited by J. L. Davis and Hamid Omid. Hamid Omid is responsible for entirety of the thesis. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Large N Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 AdS Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Motivating Maldacena Conjecture . . . . . . . . . . . . . . . . . 7 1.3.1 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Finite Temperature . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Correlation Functions . . . . . . . . . . . . . . . . . . . 12 1.4 Scalar Field as an Example . . . . . . . . . . . . . . . . . . . . . 13 2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1 D-Branes Configuration . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Dual Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Making it Gauge Invariant . . . . . . . . . . . . . . . . . . . . . 20 2.4 Matching Conditions in the Variational Part . . . . . . . . . . . . 22 2.5 D5-D7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 28 iv 2.7 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Different Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1 Parity and Time Reversal Invariant Solution . . . . . . . . . . . . 32 3.2 Parity and Time Reversal Violating Solution . . . . . . . . . . . . 34 3.3 Solution with a Charge Gap . . . . . . . . . . . . . . . . . . . . . 36 4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Appendix: Legendre Transformation . . . . . . . . . . . . . . . . . . . 45 v List of Figures Figure 1.1 Gluon vertices in QCD . . . . . . . . . . . . . . . . . . . . . 3 Figure 1.2 Gluon vertices in QCD using ’tHooft method . . . . . . . . . 3 Figure 1.3 AdS boundary . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 1.4 Open strings live on the D-branes . . . . . . . . . . . . . . . 8 Figure 1.5 Schematic picture of decoupling in second senario . . . . . . 10 Figure 2.1 D7 with a collapsing sphere . . . . . . . . . . . . . . . . . . 18 Figure 2.2 The spherical symmetry in the D2-brane leaves no tension in the angular directions . . . . . . . . . . . . . . . . . . . . . . 23 Figure 3.1 Numerical solution of (2.54) and (2.55) with f = .8 and Q7 = B = 0. ψ is plotted on the vertical axis, ψ(∞) = pi4 and ψ(0) = 1 2 sin −1 f ≈ 0.46. z is plotted on the rear axis and we see that z∼ 1/r for small r and z∼constant at large r. . . . . . . . . . 35 Figure 3.2 Numerical solution of (2.54) and (2.55) with f = .8, Q7 = B = 0 and pz = 1. ψ is plotted on the vertical axis. It goes smoothly between (1/r,ψ,z) = (0, pi4 ,0.46) and (1/r,ψ,z) = (0,0,−0.46). . . . . . . . . . . . . . . . . . . . . . . . . . . 37 vi Acknowledgments The writing of this dissertation would not have been possible without the help and encouragement of many people. I’m grateful to my supervisor Gordon.W.Semenoff, who let me follow my interests and helped me by his brightness and wide range of knowledge. I would like to thank Joshua Davis, who I learnt a lot from. I would like to thank Ryan McKenzie and Jared Stang for reading the manuscript. I would like to thank my great friends around the world who made life more interesting. I would like to thank all of my great teachers who have spent their life teaching students like me; special thanks goes to Farhang Loran. And last but not least, I highly appreciate my parents for their endless love and devotion to their children. vii Chapter 1 Introduction In this section we provide a very brief introduction to the technology we’re going to use in the next section. This technology enables us to explore areas of physics which were out of reach for a long time. We will be able to look at strongly coupled problems by solving a dual problem in the weakly coupled regime (which is quite fascinating), [1], [2], [3], [4], [5]. 1.1 Large N Limit After the birth of QCD a large amount of effort was devoted to studying the dynam- ics of strongly coupled field theories. Most of our knowledge about quantum field theories comes from perturbative solutions. In contrast to weakly coupled systems, where one can use the coupling as a perturbation parameter, in strongly coupled field theories there is no obvious candidate. Now, we know that gauge theories like QCD play a major role in nature; so, it’s reasonable to develop techniques to study such theories in strongly coupled limits. It was suggested by Gerald t’Hooft [7], that non-Abelian SU(N) gauge theories may get simplified when studied in the limit N→ ∞. In this section we go through his argument and study SU(N) when N→ ∞. The theory is described by a set of Dirac field and gauge fields. The Lagrangian is given by 1 L = ψ̄a(ı∂µ +gAaµbγ µ)ψb−mψ̄aψa− 14F a µνbF µνb a , (1.1) Faµνb = ∂µA a νb+ igA a µcA c νb− (µ → ν). (1.2) Aaνb are the gauge fields where a, b and c runs from 1 to N. The gauge fields, as usual, belong to the adjoint representation of the group so we have Aaνb = A b† νa, (1.3) Aaνa = 0. (1.4) One can rescale the fields to put the Lagrangian in a more convenient form L = N g2 {ψ̄a(ı∂µ +Aaµbγµ)ψb−mψ̄aψa− 1 4 FaµνbF µνb a }, (1.5) Faµνb = ∂µA a νb+ iA a µcA c νb− (µ → ν). (1.6) We have chosen the dependence of the coupling on N so that it’s the most con- venient for our purpose. In the following we will take N → ∞ while keeping g constant. One should be careful in choosing λ because by choosing incorrectly, he ends up with either a trivial theory or one that is not simplified. Now we can look at the dynamics using the usual Feynman Diagrams. The propa- gators are < ψaψ̄b >= δ ab D, (1.7) < AaµbA c νd >= (δ a d δ c b − 1 N δ ab δ c d )Dµν , (1.8) where D is the propagator for a single Dirac field and Dµν the propagator for a single gauge field. The 1N term is here because the gauge group is SU(N); then, the generators are traceless. Actually, for the case where we are interested, for dominant terms we can neglect this one; up to this order, there is no difference between U(N) and SU(N). The Feynman vortices are shown in Fig1.1. Because of the Kronecker Delta, the index at the beginning is the same as the index at the and we can use the following double line notation, as was suggested by t’Hooft Fig 1.2. 2 Figure 1.1: Gluon vertices in QCD Figure 1.2: Gluon vertices in QCD using ’tHooft method We can construct a two dimensional surface by identifying edges which belong to the same double line. Now we can count the power of N. Let the surface have F faces, E edges and V vertices. Every face is a loop and results in a factor of N. Every edge is a propagator which contributes a factor of 1N . Finally, every vertex is a Feynman vertex and carries a factor of N. Thus the graph is proportional to NV+F−E = Nχ , (1.9) 3 where χ is the well known topological invariant, the Euler Characteristic: χ = 2−2g, (1.10) where g is the genus of the surface. This is the important observation one can make. This observation reminds us about the perturbation expansion for amplitudes in String Theory. This is a connection between QCD, which is quantum field theory, and String Theory, which is not a QFT. 1.2 AdS Space In General Relativity the spacetime geometry is given by the Einstein equation Rµν − 12gµνR = 8piGTµν , (1.11) where Rµν is the Riemann Tensor, R is the trace of the Riemann tensor and Tµν is the energy-momentum tensor. At first, Einstein put another part in his equation called the cosmological constant to stabilize his solutions. Later, he called it a big mistake. The equation would be Rµν − 12gµνR+Λgµν = 8piGTµν , (1.12) where Λ is the cosmological constant. One can interpret the cosmological constant as vacuum energy coming from quantum aspects. Considering Einstein universes which have no matter in it one can classify non-flat solutions by sign of Λ. Accord- ing to the sign of Λ one can get topologically different solutions. For mostly plus metric signature, the Λ < 0 solution is called anti de Sitter (AdS), the Λ > 0 case is known as de Sitter(dS) and Λ= 0 is the sphere. It’s useful to consider AdSD as a submanifold embedded in a D+ 1 dimensional pseudo-Euclidian space. Describing the space with coordinates y= (x0, ...,xD) and the metric g = diag(−1,1, ...,1,−1) one can define AdSD as solution of the fol- 4 lowing equation. x20+ x 2 D+1− D−1 ∑ i=1 x2i = L 2, 3 L ∈ℜ. (1.13) The defined hyper-surface has the isometry group of SO(D−1,2). The dimension of SO(D− 1,2) is 12 D(D+ 1), which is the same as the isometry group of Flat Minkowski space in D dimensions. One can parameterize the solution in different ways, one example is using ’global coordinates’, defined as: x0 = Lcoshρ cosτ xD+1 = Lcoshρ sinτ xi = Lsinhρ Ωi 3 Ω2 = 1. (1.14) To cover the whole of the sub-manifold just once, one need to restrict the parame- ters as 0≤ ρ, 0≤ τ ≤ 2pi, (1.15) and Ω as usual are spherical coordinates. Then the induced metric would be ds2 = L2(−cosh2ρ dτ2+dρ2+ sinh2ρ dΩ2). (1.16) Moving to a new coordinate with tanθ = sinhρ helps to understand one of im- portant features of AdS better. AdS, in contrast to the sphere, is an open set with boundary. Later, we see that our dual quantum field theory lives on this boundary Fig 1.3. 5 In thess new coordinates one gets ds2 = L2 cosθ 2 (−dτ2+dθ 2+ sin2θ dΩ2), 3 0≤ θ ≤ 2pi. (1.17) Figure 1.3: AdS boundary The metric is conformally equivalent to S1×SD−1. As one can see, the bound- ary is at θ = pi/2, can be described with the coordinates (τ,Ω) and has the topology of SD−1 (τ spans a S1). Therefore the boundary of the AdSD is equivalent to com- pactified flat Minkowski space in D dimensions. In what follows, we’ll also use Poincaré coordinates, defined as x0 = 1 2y (1+ y2(L2+−→r 2− t2)) xi = Lyri ∀ 1≤ i≤ D−1 xD = 1 2y (1− y2(L2−−→r 2+ t2)) xD+1 = Lyt. (1.18) 6 The domain is y≥ 0 −→r ∈ℜD−1 t ∈ℜ, (1.19) and the metric is ds2 = L2( dy2 y2 + y2(−dt2+−→dr2)). (1.20) 1.3 Motivating Maldacena Conjecture 1.3.1 Zero Temperature AdS/CFT was first proposed by Maldacena [8]; it was developed later by several authors [9], [10]. In this section we briefly go through his argument. There are three types of conjecture one can make. In weakest form this conjecture states a duality between classical gravity on AdS5× S5 and N = 4, SU(N) super Yang- Mills in 4D. In what follows, we consider solitonic solutions of D3 branes in D = 10 type IIB superstring theory. There are two types of strings living in this background. Open strings live on the D3 branes and therefore are four dimensional excitations, while closed strings propagate through the whole of the spacetime. Looking at the low energy limit of both of these strings we find similarities between them and con- clude our conjecture. We start with open strings which live on our D3 branes; see Fig 1.4. Taking the low energy limit of superstring theory for the open string part one can find that the open strings are described by N = 4, SU(N) super Yang-Mills in 4D [13]. Here the rank of gauge group is the same as the number of coincident D3 branes. By taking the low energy limit we mean making the theory close to free ( gs→ 0) 7 and taking the string tension to infinity (α ′→ 0). Doing the calculations explicitly one can find that the relation between the coupling constant in SY M theory and coupling constant in superstring theory is g2SY M = 2pigs. (1.21) Figure 1.4: Open strings live on the D-branes This part of theory gets decoupled from the closed string part which, in the low energy limit describes supergravity. Now, we look at the problem from the closed string perspective. In this per- spective, D3 branes act as sources and cause the spacetime to curve. They are the solitonic solutions of superstring theory. One can check that the following metric is a solution of the equations: ds2 = H(r)− 1 2ηµνdxµdxν +H(r) 1 2 (dr2+ r2dΩ25) µ,ν = 0,1,2,3. (1.22) 8 There will be a background flux given by the following 4-form: C4 = (1−H(r)−1)dx0∧dx1∧dx2∧dx3, (1.23) where the H(r) is given by H(r) = 1+ L4 r4 . (1.24) L which will turn out to be the AdS radius and is given by the string theory as L4 = 4pigsNα ′2. (1.25) Working in the low energy limit that describes the supergravity we ask the curva- ture to be small. It’s simple to check that the the curvature is proportional to 1 (gsN)2 , so, to have a consistent calculation we ask that gsN 1. (1.26) Looking at H(r) one can specify two types of regions. When we get close to horizon, or r→ 0, one gets ds2 = r2 L2 ηµνdxµdxν + L2 r2 (dr2+ r2dΩ25) µ,ν = 0,1,2,3, (1.27) which is just AdS5×S5. In contrast looking far away from the horizon, as one expects, we find the flat spacetime: ds2 = ηµνdxµdxν +(dr2+ r2dΩ25) µ,ν = 0,1,2,3. (1.28) Now let’s take the final step and use the Maldacena limit. In this limit we ask α ′ → 0 while keeping rα ′ constant. By r we mean all of the distances in the 9 problem. Then, using our formula for L, we find that L4 r4 = 4pigsN α ′2 r4 = 4pigsN α ′4 r4 α ′−2→ ∞, (1.29) showing that in this limit we take the horizon limit and the spacetime would be AdS5×S5. Here, we again got two decoupled theories which can be shown as Fig 1.5. Comparing our story for open strings and closed ones, one finds that two of the decoupled theories are the same. Maldacena proposed that the other two are dual to each other too. One can check this conjecture in several ways. The most trivial one is comparing the symmetry group of them. Figure 1.5: Schematic picture of decoupling in second senario N = 4 SU(N) super Yang-Mills has the symmetry group SO(4,2)×SU(4)R; SO(4,2) is the conformal symmetry group in 4D and SU(4) is the symmetry of the supersymmetry generators in N = 4, which is called R symmetry. On the other hand AdS5×S5 has the symmetry group SO(4,2)×SO(6); considering the home- omorphism SO(6) ' SU(4), we see that the symmetry groups are the same. One 10 can do more non-trivial tests on the conjecture like computing 12 BPS operators and comparing them with field theory results [11], [12]. Maldacena’s conjecture has passed several non-trivial tests making it more reliable and hard to not believe, at least in weakest form. 1.3.2 Finite Temperature When one thinks about introducing temperature in gravity side, a relationship be- tween this temperature and the Hawking temperature seems natural. Following [10] we introduce a horizon to our blackbrane solution and expect it to somehow introduce the temperature consistently in the field theory side. Our new metric will have the more complicated form ds2 = r2 L2 ((1− r 4 h r4 )dτ2+ηi jdxµdxν)+ L2 r2 1 1− r4hr4 dr2+L2dΩ25 i, j = 1,2,3, (1.30) where we have used Euclidean signature. Looking at the near horizon region one gets the following metric ds2 = 4rh L2 (r− rh)dτ2+ L 2 4rh 1 r− rh dr 2+L2dΩ5. (1.31) Moving to the new radial coordinate ρ = √ r− rh, (1.32) one finds the new form of the metric given by ds2 = L2 rh (dρ2+ 4(rhρ)2 L4 dt2)+L2dΩ5. (1.33) 11 This is the metric of flat space in polar coordinates. But to not have a conic singu- larity one needs to have a periodicity in τ , giving τ ∼ τ+β . (1.34) β = piL2 rh . (1.35) It’s well known in field theory that to have temperature one needs to compactify the time; then the temperature is given by β = 1 T . (1.36) This is exactly what happens by introducing horizon in gravity side. On the bound- ary, where the field theory lives, we’ll have the same compactified topology for the time coordinate 1.3.3 Correlation Functions At this level one needs to define the dictionary. In the QFT side one is always in- terested in the correlation functions of different operators living in the theory. So, to have a useful conjecture what one really needs to do is identify the correlation functions with their image in the dual part. Typicaly, correlation functions are calculated using a functional called the genera- tor or the partition function in finite temperature field theory: Z[J] =< e ∫ dxO(x)J(x) >= ∫ Dφe−S[φ ]+ ∫ dxO(x)J(x). (1.37) The AdS/CFT conjecture gives a way to calculate this partition function in terms of the same function in gravity side. In the large N limit, it’s given by the saddle point approximation of the gravity partition functional for the corresponding field in AdS5×S5 side. Looking at the asymptotic behavior of the corresponding field in the AdS5× S5 side, one gets the following behavior φ(z0,−→z ) =< O > z∆+0 + J(−→z )z∆−0 . (1.38) 12 Here, we have identified the coefficient in front of non-normalizable mode as the source in field theory part. The coefficient of the normalizable mode, or z∆+0 , is identified with the VEV or the vacuum expectation value of the operator. In the last equation we are working in the following metric ds2 = 1 z20 (dz20+d −→z 2)+L2dΩ25 µ,ν = 0,1,2,3. (1.39) This is related to usual AdS metric by the following change of coordinates z = L r . (1.40) Then using the common method of the functional derivative one can calculate cor- relation functions of the corresponding operator < T{O(−→z1 )O(−→z2 )}>= δ 2lnZ[J] δJ(−→z1 )J(−→z2 ) |J=0. (1.41) In the following we’ll go through the simplest case which is a scalar of the isometry group in AdS5× S5. We’ll find that the conformal powers are the same as one expects and we find the usual two point function in CFT. 1.4 Scalar Field as an Example In this section we use AdS/CFT conjecture to find the two point function of the operator corresponding the scalar field in gravity side. We consider the low energy action for the scalar field given by the low energy limit of type IIB string theory S = ∫ AdS5×S5 √−ggµν∂µφ∂νφ . (1.42) This results in the equation of motion given by ∂µ(gµν∂νφ) = 0. (1.43) 13 Using the following coordinates ds2 = 1 z20 (dz20+d −→z 2)+L2dΩ25 µ,ν = 0,1,2,3, (1.44) where we have used the Euclidean signature; one gets ∂z0( 1 z30 ∂z0φ)+ 1 z30 ∂ 2µφ = 0. (1.45) Using the Fourier transformation φ(z0,−→z ) = ∫ dk4 2pi ei −→ k .−→z φ(z0,k), (1.46) one gets ( φ ′ z30 )′− k 2 z30 φ = 0, (1.47) which is the Bessel equation for ν = 2. Then the solutions are φ(k,z0) = A(k)z20K2(kz0)+B(k)z 2 0I2(kz0). (1.48) I2(x) blows up at infinity which makes the action badly divergent. We shall ask the solution to be finite at the origin so we demand B(k) to be zero. Around the boundary z0 gets close to zero. The Bessel function has the form of K2(x) = 2 x2 . (1.49) Identifying J as the asymptotic value of φ we find that φ(k,z0) = k2z20 2 K2(kz0)J(k). (1.50) 14 Substituting the solution in the action one gets that S = ∫ AdS5×S5 gµν∂µφ∂νφ (1.51) =− ∫ ∂AdS5 ∂µ( √−ggµνφ∂νφ) = ∫ ∂AdS5 L3 z30 φφ ′ = ∫ dk4 (2pi)4 L3 z30 J(k)J(−k)k 2z20 2 K2(kz0)( k2z20 2 K2(kz0))′. Taking the functional derivative with respect to J(k) twice one finds fk(z0) = k2z20 2 K2(kz0) (1.52) < O(0)O(k)>= kL3 z30 fk(z0) fk(z0)′  z0→0 . Expanding fk(z0) around the boundary and taking the Fourier transformation we find < O(x)O(y)>∼ 1 (x− y)8 , (1.53) which gives the conformal power ∆= 4, as one expects. 15 Chapter 2 Setup 2.1 D-Branes Configuration We are interested in 2+ 1 dimensional fermions. So, we need to use a system of D-Branes that have a special number of Neumann-Dirichlet boundary conditions for the open strings of interest. This is known to be ND = 6 [13]. Working in low energy limit, because we don’t have any tachyon in spectrum, we only con- sider massless excitations that live in Ramond sectors and therefore are spacetime fermions. In such a configuration, one breaks SUSY completely, and needs to check the stability of the configuration explicitly. We shall do it by checking the BF bound and whether it’s violated or not. We use the following configuration x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 × × × × D7 × × × × × × × × D5 × × × × × × (2.1) We have considered a D5 for a reason that will be clear soon. There are six Neumann-Dirichlet boundary conditions, for the open strings, which are attached to D7−D3, as was promised before. To use the gauge theory-gravity duality, as was discussed before, one should take the large N3 limit, where N3 is the number 16 of D3 branes. Then one gets the usual AdS5×S5 background. we treat D7 and D5 as probes on the background. One should note that there is a separation between D3 and D5 which can be used to introduce a bare mass for the strings attached in between. The metric of the space is, as always, given by AdS5× S5. A Ramond form lives on background given by C4 = L4r4dt ∧dx∧dy∧dz+L4 c(ψ)2 dΩ2∧dΩ̄2, (2.2) with ∂ψc(ψ) = 8sin2ψ cos2ψ = 1− cos4ψ. (2.3) This configuration has been used by other authors [14] to describe the fractional quantum Hall effect. They introduce a topological flux to stabilize the embedding. There was another suggestion by Myers and Wapler [15], where they used instanton bundles to make the embedding stable. In [14], they use the following chart for S5 ds2S5 = dψ 2+ sin2ψdΩ22+ cos 2ψdΩ̄22, (2.4) ψ ∈ [0,pi/2], (2.5) and the world-volume of D7 wrapping S2 and S̄2. This configuration can be sta- bilized by adding world-volume gauge fields with U(1) Dirac monopole fields on one or both of the spheres, turning on a world-volume U(1) flux with 2piα ′F0 = L 2 2 f1Ω2+ L2 2 f2Ω̄2. (2.6) They then stabilize the configuration by introducing a U(1) gauge field on one or both spheres. Since the flux is on a compactified dimension(lives on S2), using the Dirac quanti- zation condition we have 17 L2 2piα ′ f1 = n1 N7 , (2.7) L2 2piα ′ f2 = n2 N7 . where n1 and n2 are integers. In [15], Myers and Wapler study the possible em- bedding of D7. In general, one can have Minkowski or Black Hole embeddings. By Minkowski we mean branes where they don’t enter the horizon formed by D3 geometry; Black Hole embedding is clearly an embedding where branes enter the horizon formed by D3s. There are two ways to get Minkowski embedding: either the brane turns back at a point and goes back to boundary at infinity or the D-brane stops at a point before getting to the horizon. One can make sense of the second scenario by imagining that at this point one of the spheres collapses.(Figure 2.1) Figure 2.1: D7 with a collapsing sphere In [14], to avoid some technical difficulties, they turn the flux on only one of the spheres. This sphere then shrinks to a point at a given distance from the horizon, and the brane stops there. Using these solutions they recover fractional quantum Hall effect results for conductivity. In our model, we are interested in having both of the fluxes on. This will be crucial 18 in constructing a model with the desired symmetries but this causes a problem. Since we have put a finite flux on the sphere with zero volume at a given point, we will get magnetic monopoles at this point living on our D7. From string theory, we know that to create a monopole one needs to attach a D5 to the D7. The other end of D5 can live on two places; either the horizon or the boundary. It cannot live on the D7 because it will source another monopole with the opposite sign, which is not desired. 2.2 Dual Potential Typically, in classical electrodynamics, one uses Maxwell equations in vector po- tential form. One defines the Field Strength as Fµν = ∂µAν −∂νAµ , (2.8) or in closed form F = dA, (2.9) where Aµ is the vector potential with U(1) gauge invariance. The Maxwell equa- tions are then divided into two parts dF = 0, (2.10) which is trivially satisfied since F is exact, and the other parts will come from variation of the Lagrangian, which for Maxwell is d ∗F = J. (2.11) Using the above method is useful when there is no magnetic monopole, as the mag- netic current, is always zero trivially. If there was a magnetic current then F would not be closed anymore and therefore not exact. To deal with this problem, we for- get about the vector potential and treat F as the dynamical variable for variations, and introduce a new vector potential called the Dual Potential. 19 We modify the action as S = ∫ L [F ]+ ∫ dAD−2∧F. (2.12) Here, D is the spacetime dimension and AD−2 is a D−2 form. The Euler equation for F in Maxwell theory would be F = ∗dAD−2, (2.13) and the Euler equation for the Dual potential would recover the Bianchi identity, assuming everything is fine on the boundaries. This method is advantageous because we don’t need to assume F is exact, and we can describe theories including magnetic charges, which is our case of interest. 2.3 Making it Gauge Invariant In general, from string theory for a number of Dp-branes we have the following action − Sp NpTp = ∫ Σp+1 dτ p+1e−φ √ −det(g+2piα ′F)+ ∫ Σp+1 ΣqCq∧ e2piα ′F , (2.14) where Np is the number of Dp-Branes, Cq is the background and Tp = 1 α ′p+ 1 2 (2pi)p . (2.15) The bulk term, or in other words DBI action, simply has gauge invariance; the Chern-Simons part is more difficult. Under the gauge transformation of the Ra- mond fields Cq −→Cq+dΛq−1, (2.16) the Chern-Simons term picks up the following extra part∫ dΛp−1∧ e2piα ′ . (2.17) 20 In the cases that F is closed in the bulk, one can integrate it by part and get a boundary term ∫ Λp−1∧ e2piα ′ . (2.18) If there was no boundary we would not have any worry about gauge invariance, but in our case there is a boundary. To deal with this problem, we introduce a p-form on the Dp-Brane, and couple the gauge transformation of the Ramond-Ramond forms with it Ap −→ Ap+ΣΛp−1∧ e2piα ′ , (2.19) and add the following term to the action∫ dAp. (2.20) Because it’s a total derivative, the Euler equation would remain the same in the bulk. The only thing that one should take into account is that when we do a variation on Ap we can’t ask the variation to vanish on the boundary, as it’s not gauge invariant under the transformation. We couple the variation of the field in two paths, so that they cancel each other on the boundary. Let’s consider the special case of D5−D7 to make things clearer. In this case putting 2piα ′ = 1we’ll get the following Chern-Simons terms on D5 SCS = ∫ C6+C4∧F(5)+C2∧F(5)∧F(5)+C0∧F(5)∧F(5)∧F(5), (2.21) where by F(5) we mean field strength living on D5. To get the gauge invariance back we introduce the following 5-form part to the action ∫ D5 dA(5)5 = ∫ ∂D5 A(5)5 . (2.22) Following our formalism for the cases where there was magnetic charge on the D-brane on the D7 part, we have an extra gauge field which is a 5-form.∫ D7 dA(7)5 ∧F(7). (2.23) Actually, we can write the part of the action containing the 5-form over all of 21 spacetime using the delta function S = ∫ D7 dA(7)5 ∧F(7)+A(5)5 ∧δDirac(∂D5). (2.24) Then, by varying A5, one gets the following terms to vanish −δA(7)5 ∧dF(7)+δA(5)5 ∧δDirac(∂D5) = 0. (2.25) Correlating the variations on the boundary as follows δA(7)5 = QδA (5) 5 , (2.26) one gets the Maxwell equation on the D7 with a monopole as desired dF = QδDirac(∂D5), (2.27) where Q acts like the charge of a D5-brane. 2.4 Matching Conditions in the Variational Part Consider two D-branes that are attached to each other. One starts from the horizon and goes to r0, and at this point attaches to another D-brane, in our case D7. A ques- tion of interest is the matching condition one should use at this point. Describing the whole system with a single Lagrangian, one needs to have a very constrained relation between momentums at the attaching point. Consider the following action S = ∫ r0 0 drL5(z, ż)+ ∫ ∞ r0 drL7(z, ż). (2.28) Varying with respect to z, keeping the attaching point constant, one gets the equa- tion of motion plus some boundary terms δS = ∫ r0 0 dr( ∂L5 ∂ z − ∂L5 ∂ ż )δ z+ ∫ ∞ r0 dr( ∂L7 ∂ z − ∂L7 ∂ ż )δ z+ (2.29) ( ∂L5 ∂ ż − ∂L7 ∂ ż )r0δ z(r0) = 0. 22 The first two terms give the equation of motion, but to satisfy the whole equation one needs to add the following condition ( ∂L5 ∂ ż − ∂L7 ∂ ż )r0 = 0, (2.30) which is nothing but continuity of the momentum. For angular coordinates, we should be more careful. Depending on the symmetry of the boundary on which the attaching happens, the tension on special angular directions may get canceled automatically. This stems from the fact that although the Lagrangian may be in- dependent for some angular variables, when we use the partial integration on the boundary, the symmetry of thess variables cause pairs of forces which act in oppo- site directions. For example, consider the following example illustrated in Figure 2.2. Here we have a fundamental string attached to a D2-brane. Because the D2- brane has spherical symmetry at the attachment point, the angular tension gets canceled automatically. Figure 2.2: The spherical symmetry in the D2-brane leaves no tension in the angular directions 2.5 D5-D7 In this section we derive the Lagrangian for our D7 and D5. 23 D5 We start from the D5 action. Since it will end on the D7, it has a boundary thus we get a gauge invariance issue which will be solved by introducing new terms as was suggested in section 2.3 − S5 N5T5 = ∫ Σ5+1 dτ5+1 √ −det(g+2piα ′F)+2piα ′ ∫ Σ5+1 C4∧F. (2.31) Using the Ramond form (2.2) and the special form of F (2.6) as the background field, plus the magnetic and electric field that we are interested in F = Fe+Fm, (2.32) Fe = dAe, (2.33) 2piα ′ L2 Ae =−A(r)dt, (2.34) 2piα ′ L2 Fm = 2piα ′ L2 F0+Bdx∧dy, (2.35) one gets the following action for the D5 SDBI =−N5 ∫ r0 rh √ (r4+B2)( f 21 +4cos4ψ)∆, (2.36) with ∆= 1+ r2hψ̇2+ r4hż2− Ȧ2, (2.37) N5 = 2piN5T5L6V2,1 gs , (2.38) where V2,1 is the volume of the 2-sphere spacetime. The Chern-Simons term would be SCS =−N5ξ5 ∫ r0 rh drr4hż f2, (2.39) which gives the final Lagrangian − L5 N5 = √ (r4+B2)( f 22 +4cos4ψ)∆+ξ5r 4hż f2. (2.40) 24 Performing the Legendre transformation, as explained in Appendix one, can get the effective Lagrangian or Routhian for ψ − R5 N5 = √ ((r4+B2)( f 22 +4cos4ψ)− f 22 r4h+Q25)(1+ r2hψ̇2), (2.41) with Q5 = Ȧ5 √ (r4+B2)( f 22 +4cos4ψ) 1+ r2hψ̇2+ r2hż2− Ȧ2 , (2.42) as the conserved current of the cyclic variable A. One should note that we have used the fact that D5 enters the horizon to set the z momentum equal to zero. D7 In this case, we get more terms simply because of the higher dimension of the brane. − S7 N7T7 = ∫ Σ7+1 dτ7+1 √ −det(g+2piα ′F)+2piα ′ ∫ Σ7+1 CAdS4 ∧F ∧F −2piα ′ ∫ Σ7+1 dCS 5 4 ∧A∧F. (2.43) We have divided the Chern-Simons term in two parts to make it gauge invariant. The difference between this shape and original one is a boundary term that does not contribute to the equation of motion. One gets SDBI =−N7 ∫ ∞ r0 √ (r4+B2)( f 22 +4cos4ψ)( f 21 +4sin4ψ)∆, (2.44) S1CS =−N7ξ7 ∫ ∞ r0 drr4hż f2 f1, (2.45) S2CS =−2N7ξ7B ∫ ∞ r0 drc(ψ)Ȧ+2N7ξ7Bc(ψ)A(r) ∣∣∣∞ r0 , (2.46) with N7 = 2piN7T7L8V2,1 gs . (2.47) 25 The final Lagrangian would be − L7 N7 = √ (r4+B2)( f 22 +4cos4ψ)( f 21 +4sin4ψ)∆ (2.48) +ξ7r4hż f1 f2+2ξ7ȦB. Repeating the procedure as for D5, we find the Routhian for D7 to be Q7 = Ȧ7 √ (r4+B2)( f 21 +4sin4ψ)( f 22 +4cos4ψ) 1+ r2hψ̇2+ r2hż2− Ȧ2 −2ξ7Bc(ψ), (2.49) − R7 N7 √ (1+ r2hψ̇2) = √ ((r4+B2)( f 21 +4sin4ψ)( f 2 2 +4cos4ψ)− ( f1 f2r4h−Pz)2r−4h−1+(Q7+2ξ7Bc(ψ))2. (2.50) The equation of motion for ψ would then be Γ1 := 1+ r2hψ ′2, (2.51) Γ2 := (r4+b2)g(ψ)− ( f1 f2)2r4h+(Q7+2ξBc(ψ))2), (2.52) g := √ ( f 21 +4cos4ψ)( f 22 +4sin 4ψ)− ( f1 f2)2, (2.53) ψ ′′ Γ1 −ψ ′3 r+ r −3 Γ1 + 2(r+ r−3)ψ ′ r2h + (2.54) ψ ′r2h(2gr3− d( f1 f2r4h−Pz)2r−4h−1)dr )− 12((r4+B2) dgdψ −2b(Q7+2ξBc) dcdψ ) Γ2r2h = 0, where we’ve put rh = 1. The equation for z would be Pz = r4hż √ ((r4+B2)( f 21 +4sin4ψ)( f 22 +4cos4ψ)(Q7+2ξ7Bc(ψ))2 ∆ +ξ7h f1 f2. (2.55) 26 In addition, we are interested in finding the current-current correlation in the dual field theory, and deriving ∆CS, which is defined below. In the field theory side, just by asking for Lorentz symmetry, we get a very restricted form for the current-current correlator given by 〈 ja(x) jb(0)〉= ∫ d3q eiqx (2pi)3 ∆ab(q), (2.56) ∆ab(q) = ∆CS(q) εabcqc+∆T (q)(q2δab−qaqb). (2.57) The U(1) current is the field theory dual to the U(1) gauge field whose curvature is F . To get the 2-point function of this current, it is sufficient to study fluctuations of the field strength F = F0 + F̃ about F0 up to second order. To that order, it is consistent to simply insert the solution for the F = F0 geometry into the equation of motion for F̃ . For our purposes, it is also sufficient and consistent to take F̃ to have non-zero components only on the AdS5 space and to only depend on the coordinates (r, t,x,y). The action up to our desired order is S = N7T7(2piα ′)2 ∫ d8τ [ 1 4 √ det(g+2piα ′F0) gµνgλρ F̃µλ F̃νρ − i 1 2 F̃ ∧ F̃ ∧C4 ] , (2.58) where we have moved to the Euclidean signature. In addition to this bulk action, in order to maintain invariance under the Ra- mond form gauge transformation which would shift the function c(ψ) by a con- stant, it is necessary to add a surface term to this action [14]. The action becomes S = N7T7(2piα ′)2 ∫ d8τ [ 1 4 √ det(g+2piα ′F0) gµνgλρ F̃µλ F̃νρ − i 1 2 Ã∧ F̃ ∧dC4 ] . (2.59) Then, with our Ansätz for the D7 geometry, and choosing the Ar = 0 gauge, f1 = f2 = f and rh = 0, we obtain Sprobe = N3N7 2pi2 ∫ d3x ∫ ∞ 0 dρ ( 1 2 (∂ρAa)2+ 1 4 α2F2ab+ i∂ρc(ψ)εrabcAa∂bAc ) , (2.60) 27 where we have dropped the tildes on F and A, a,b,c are 2+1-dimensional Euclidean indices, α2 = ( f 2+4sin4ψ)( f 2+4cos4ψ) and the radial variable has been trans- formed to ρ(r) = ∫ ∞ r dr̃ r̃2 √ ( f 2+4sin4ψ)( f 2+4cos4ψ)√ 1+r̃2ψ ′2+r̃4z′2 . (2.61) Note that r = 0 and r =∞ are mapped to ρ =∞ and ρ = 0, respectively. Using the fourier transform Aa(x) = ∫ d3q (2pi) 32 eiq·xAa(q), (2.62) the field equation can be written[ −∂ 2ρ +α2q2±2iq∂ρc(ψ) ] A±(ρ,q) = 0, (2.63) and qaAa = 0. We are considering polarization states which obey [iqbεabc]Ac± = ±qAa±. The field equation should be solved with the requirement that the solution is regular at the Poincaré horizon, ρ =∞. The on-shell action for the gauge field is then Ŝ = − N7N3 4pi2 ∫ d3q lim ρ→0 [ Aa(ρ,−q)∂ρAa(ρ,q) ] . (2.64) The current-current correlator is obtained by taking two derivatives of e−Ŝ by the boundary value of the gauge field Aa(0,q) and then setting Aa(0,q) = 0. 2.6 Discrete Symmetries We are interested in applying our holographic description to model systems which have charge conjugation (C) ,parity (P) and time reversal (T ) as discrete symme- tries which may break dynamically, but we don’t want to break them explicitly. One can observe such anomalous behaviors in nature in systems such as graphene [6]. Our Lagrangian breaks all of the symmetries. To restore them, we augment their most naive definitions with extra isometries. Let’s start with C, which is the most simple one. C is defined as F −→ −F for all worksheet gauge fields that do not leave the DBI term invariant. It can be augmented with a change of orientation of both of the spheres we used for the S5 chart, so that the DBI term remains invariant. 28 Parity in the 2+1 dimensions, where our QFT lives, is more tricky; it’s defined as the transformation (t,x,y) −→ (t,−x,y). The Ramond 4-form was given by C4 = L4r4dt ∧dx∧dy∧dz+L4 c(ψ)2 dΩ2∧dΩ̄2, (2.65) with ∂ψc(ψ) = 8sin2ψ cos2ψ = 1− cos4ψ, (2.66) where the AdS5 part changes sign under the reversing isometry we mentioned. The AdS5 part of F ∧F changes sign under this transformation as well. To make the Chern-Simons term invariant, we augment the reversing isometry of AdS5 by the interchanging the two spheres with each other. This would change the sign of S5 part of the Ramond 4-form and F ∧F . To get this exchange isometry we do the following transformation,ψ −→ pi2 −ψ . The exchange isometry will be a symmetry of the background field F0 if and only if f1 = f2. T in our case, which is 2+ 1 dimensional, is easy. It’s defined as (t,x,y) −→ (−t,x,y) which can be restored the same way as P. 2.7 Asymptotic Behavior We linearize the action around a constant value of ψ . We are interested in the asymptotic behavior of the ψ coordinate because in dual conformal field theory it represents an operator O, which is the mass operator. The constant value which we expand around is not necessarily a solution which we should be careful about. Defining V (ψ) = √ (r4+B2)( f 21 +4cos4ψ)( f 22 +4sin4ψ)− ( f1 f2)2r4h+(Q+2ξc(ψ))2, (2.67) the Routhian expands as R7 =−T7N7(1+ 12 r2hφ̇) ( V (r)|ψ0 + ∂V∂ψ |ψ0φ + 12 ∂ 2V ∂ψ2 |ψ0φ 2 ) +O(φ 3) (2.68) ∼−T7N7 ( V (r)|ψ0 +V (r)|ψ0 12 r2hφ̇ + ∂V∂ψ |ψ0φ + 12 ∂ 2V ∂ψ2 |ψ0φ 2 ) +O(φ 3),(2.69) 29 where we have expanded aroundψ0, and have ignored cubic and higher order terms and assumed that in the asymptotic limit the derivative of ψ goes to zero. The equation of motion is then d dr ( V r2hφ̇ ) − ∂ 2V ∂ψ2 φ − ∂V ∂ψ ∣∣∣ ψ=ψ0 = 0. (2.70) We now look at the equation at large r. V has the form given by V = r2 √ ( f 21 +4cos4ψ)( f 22 +4sin4ψ)− ( f1 f2)2+O(r−2). (2.71) Assuming a power law behavior at the boundary as one expects, we get φ = r∆. (2.72) The leading term is ∂V ∂ψ = 0, (2.73) which results in the following equation cos ψ0sin ψ0 ( f 22 sin 2ψ0− f 21 cos2ψ0+4cos2ψ0 sin2ψ0(cos2ψ0− sin2ψ0 ) = 0, (2.74) which has two trivial solutions consisting of 0 and pi2 , and non-trivial ones that depend on f1 and f2. Moving to the next order, one has V d dr (r4φ̇) = ∂ 2V ∂ψ2 φ , (2.75) which leads to the following equation for ∆, ∆(∆+3) = ∂ 2V ∂ψ2 V , (2.76) 30 which will give us the following formula for ∆, ∆± = −32 ± 12 √ 9+16 f 2 1+16sin 6ψ0−12sin4ψ0 f 21+4sin 6ψ0 (2.77) =−32 ± 12 √ 9+16 f 2 2+16cos 6ψ0−12cos4ψ0 f 22+4cos 6ψ0 . (2.78) To have a stable embedding, one needs the conformal powers to be positive. In what follows, we are interested in the special case of equal fluxes, f1 = f2. In this case, the non-trivial solution where we are interested is ψ0 = pi/4. This restricts f1 and f2 to the following interval | f |= | f1|= | f2| ≥ √ 23 50 . (2.79) At the bound | f | = √ 23 50 we get the marginal value given by ∆+ = 3 2 . At | f | = 12 we get the classical conformal dimension which is ∆+ = 2. When | f | = 1, we get ∆+ = 3. Now let’s look at the solution near the origin for the Poincaré patch(zero tempera- ture) and zero magnetic field and charge. In this case, D7 enters the horizon, so the only smooth solution has Pz = 0. Expanding around r = 0, one gets equation 2.70 back. The homogenous part has different solutions, but the stable solution goes to the minimum of V (ψ), which happens at sin(2ψ) = f assuming f1 = f2 = f . Using the same equation for the conformal powers as ∆±, one gets the following confor- mal power ν = √ 9 4 +16 1− f 2 4− f 2 − 32 . (2.80) The other family of solutions, which diverge at the origin, we drop. So, around origin one gets ψ = arcsin f2 + cr ν + . . . . (2.81) 31 Chapter 3 Different Scenarios 3.1 Parity and Time Reversal Invariant Solution The parity invariant solution is given by the solution which, under ψ −→ pi4 −ψ , is invariant. Clearly, the only possible solution is the constant solution given by ψ = pi 4 , (3.1) z = z0− f 2√ 1+2 f 2 1 r , (3.2) pz = 0. (3.3) where we get a black hole embedding of D7. D7 starts from the boundary at infin- ity, and keeping the angle ψ , fixed enters the horizon. Its geometry will be given by AdS4×S2×S2 In non-trivial cases, we will have a solution in the following form around the boundary ψ = pi4 + ψ1 r∆− + ψ2 r∆+ + . . . , (3.4) 32 where ∆+ = 32 + √ 9 4 −8 1− f 2 2 f 2+1 , (3.5) ∆− = 32 − √ 9 4 −8 1− f 2 2 f 2+1 , (3.6) where the ∆+ is the conformal power of the related dual operator on the CFT side, which here is the Fermionic mass operator at weak coupling.This interpretation is not valid at strong coupling, due to operator mixing. Then the correlation function for this operator is given by 〈O(x)O(y)〉= const.|x− y|2∆+ . (3.7) ψ1 is usually interpreted as the source, and ψ2 as the condensate. There are situa- tions where they may change role, see [14]. When ψ = pi4 , α and c(ψ) are independent of ρ , and the Maxwell equation (2.63) can be solved exactly. The current-current correlator may then be extracted by taking two functional derivatives of the on-shell action (2.64) by boundary data Aa(ρ = 0,q). The result is [16] ∆T = N3N7 2pi2 f 2+1 q , ∆CS = 0. (3.8) ∆CS vanishes due to parity symmetry. The function ∆T(q), whose dependence on q is determined by conformal symmetry together with the fact that ja is a conserved current, now determines the current-current correlator at strong coupling. It is qualitatively similar to the weak coupling result for massless fermions, ∆T = N3N716piq but with a coefficient that depends on f . Recall that f > √ 23 50 , and is typically of order one. 33 3.2 Parity and Time Reversal Violating Solution At small r, a solution of (3.4) must have the form ψ = arcsin f2 + cr ν + . . . , ν = √ 9 4 +16 1− f 2 4− f 2 − 32 . (3.9) If we search for a solution where ψ depends on r with boundary behavior (3.4) and (3.9), once one of the three constants (ψ1,ψ2,c) is fixed, the other two are determined by requiring that the solution is nonsingular. (Solutions at r ∼ 0 exist only when pz = 0.) Setting ψ1 to some fixed value corresponds to turning on a source for the operator O with conformal dimension ∆+, and whose expectation value 〈O〉 is then proportional to the other constant ψ2. Since ∆+ is positive, the operator is relevant and the dual field theory is no longer a conformal field the- ory. However, it will flow to another conformal field theory in the infrared, small momentum limit. The equations (2.54) and (2.55) can be solved numerically. An example of a solution is depicted in figure 3.1. For the solution in figure 3.1, where ψ is not constant, we cannot obtain an exact solution of the Maxwell equation (2.63). However, it is easy to solve in the limits where q is large or small compared to other dimensional parameters. Here, the only dimensional parameters in the problem come from the boundary behavior and we can use, for example, ψ1/∆−1 to compare with q. When q is large, as is shown in the [16], the solution is identical to (3.8). The theory has an ultraviolet fixed point which is identical to the parity and time reversal invariant conformal field theory that corresponds to the constant ψ = pi4 solution. On the other hand, when q is small, we find ∆T = N3N7 2pi2 2 f q + . . . , ∆CS = N3N7 pi2 ( f √ 1− f 2− arccos f )+ . . . , (3.10) where corrections are of higher order in q. These functions characterize the elec- tromagnetic properties of the infrared fixed point of the field theory dual of the non-constant ψ solution. That theory is apparently gapless and parity and time reversal violating. We note that ∆CS(0) differs from the one-loop result. By doubling the degrees of freedom, we could find a parity invariant config- uration. This could be, for example, an exotic time reversal invariant phase of 34 Figure 3.1: Numerical solution of (2.54) and (2.55) with f = .8 and Q7 =B= 0. ψ is plotted on the vertical axis, ψ(∞) = pi4 and ψ(0) = 1 2 sin −1 f ≈ 0.46. z is plotted on the rear axis and we see that z∼ 1/r for small r and z∼constant at large r. graphene. In that case, two D7 branes behave as we have described above and two D7-branes are their parity mirror, with ψ(r) replaced by pi2 −ψ(r). Since the pairs of branes behave differently, the global U(4) symmetry is broken to U(2)×U(2). In this case, this is not spontaneous breaking, instead it is explicit breaking by turning on the boundary condition (3.4) for two of the D7-branes and a similar boundary condition but with ψ1,ψ2 replaced by −ψ1,−ψ2 for the other two D7- branes. At weak coupling, this would correspond to turning on the parity invariant mass that corresponds to a charge density wave in graphene which was discussed in reference [6]. At weak coupling, this would seem to gap the spectrum, but for this particular solution, at strong coupling gapless charged excitations seem to sur- vive. The parity even part of the current-current correlator ∆T would be as we computed above with N7 = 4, however ∆CS would cancel, and would therefore be zero, consistent with parity and time reversal invariance. On the other hand, it 35 would re-appear in a flavor current correlation function, and it would give the so- called “valley Hall effect”, originally described in reference [6]. The value of ∆CS in (3.10) with N7 set to 4 describes the valley Hall effect at strong coupling. 3.3 Solution with a Charge Gap It is interesting to ask whether we can find a theory with a charge gap. Charged particles are the low energy modes of strings which are suspended between the Poincaré horizon and the D7-brane and the mass of a string state is roughly pro- portional to its length. In the previous two examples, the charged particles were massless, corresponding to the fact that the D7-brane comes arbitrarily close to the Poincaré horizon and the 3-7 strings could be arbitrarily short. To get a gapped so- lution, we need a configuration of D7-brane which does not approach the Poincaré horizon, something similar to a “Minkowski embedding” where the D7 brane pinches off before it reaches the Poincaré horizon at r = 0. The D7-brane can pinch off smoothly when one of the S2’s collapses, for exam- ple, when ψ→ 0. However, this would not be compatible with the existence of the magnetic flux f on S2 unless there is a magnetic source. Here, a magnetic source would be supplied by nD D5-branes where nD is the number of units of monopole flux. The flat space configuration of the D5-branes is given in the table in (2.1). On AdS5× S5, each D5 brane wraps S̃2 ⊂ S5 and has nD units of Dirac monopole charge on S̃2. However, it can be argued that such a D7-brane with a shrinking S2 attaching to a suspended D5-brane is not stable, the D7 always has smaller tension than the D5. If the D5 where connected to the horizon, it would simply pull the D7 brane to the horizon. This would seem to rule out the possibility of finding gapped solutions using D5-branes suspended between the D7-brane and the Poincaré horizon. The only alternative is that the D5 brane is connected to r→ ∞ and that the imbalance of tensions pulls the D7-brane back to infinity. We can indeed find numerical solutions of the D7 equation which behave in this way. An example is depicted in figure 3.2. The D7-brane begins at (r,ψ,z) = (∞, pi4 ,−0.46). As ψ decreases, r decreases until it reaches a minimum. Then it starts increasing and returns to the asymptotic region at (r,ψ,z) = (∞,0,0.46). During this interval, 36 Figure 3.2: Numerical solution of (2.54) and (2.55) with f = .8, Q7 = B = 0 and pz = 1. ψ is plotted on the vertical axis. It goes smoothly between (1/r,ψ,z) = (0, pi4 ,0.46) and (1/r,ψ,z) = (0,0,−0.46). the coordinate z increases steadily over a finite range. At the latter endpoint, S2 has collapsed to a point, leaving a source with nD Dirac monopoles on the D7 world-volume. We can think of the flux on the collapsing S2 as being sourced by a D5-brane sitting at r = ∞. The solution depicted in figure 3.2 is fundamentally different from the solutions that we have found in the previous two Sections since it goes to the boundary at two different locations. On the AdS5, these locations are separated in the coordinate z, so they nominally correspond to two different dual quantum field theories. The operator corresponding to fluctuations ofψ , for example, in the asymptotic regime where ψ → pi4 behaves as in equation (3.9), whereas in the regime where ψ→ 0 it has ∆− = 1,∆+ = 2, independent of f and identical to the quantities for a 37 D5-brane with flat space configuration depicted in (2.1). In addition, to solve the Maxwell equation on the world-volume requires two sets of asymptotic data. For example, if we require that the world-volume gauge field goes to Aa(x) at the ψ→ pi4 asymptote and Ãa(x) at the ψ→ 0 asymptote, it is straightforward to solve the large momentum limit where we obtain two decoupled currents [16] 〈 ja jb〉= N3N72pi2 f 2+1 q ( q2δab−qaqb ) , 〈 ja j̃b 〉 = 0 〈 j̃a j̃b 〉 = N3N7 2pi2 f √ f 2+4 q ( q2δab−qaqb ) . (3.11) The j− j correlator reproduces the high energy limit (3.8) of our previous solu- tions. The j̃− j̃ correlator produces what would be expected for the solution of the D5-brane geometry with the constant angle ψ = 0. The field theory dual to this the D5-brane geometry is well known [15]. On the other hand, a small momentum expansion is diagonalized by linear combinations of the currents with 〈 j+a j+b〉= N3N74pi εacbqc+ . . . , (3.12) 〈 j−a j−b〉= N3N7pi2ρm (δab− qaqb q2 )+ εacbqc∆ (−) CS (0)+ . . . , . (3.13) where ρm = ∫ ∞ rmin dr̃ r̃2 √ ( f 2+4sin4ψ)( f 2+4cos4ψ)√ 1+r̃2ψ ′2+r̃4z′2 , (3.14) ∆(−)CS (0) = N3N7 pi2 ∫ pi/4 0 dψ(1− cos4ψ) ( 1− ρ(ψ) ρm )2 , (3.15) where rmin in (3.14) is the minimum value of r that the solution reaches. Correc- tions in (3.12) and (3.13) are of order q2. The currents j± are linear combinations of j, j̃ which are normalized so that the charges obtained by integrating the time components of j± are integers, as were the charges of j, j̃. We then observe that the j+ current has the value of ∆CS(0) that would be expected for a system of N3N7 fermions with a parity violating mass gap. 38 This is identical to the non-interacting one-loop result for massive fermions. Note that j− has a superfluid-like pole in its 2-point function, indicating spon- taneous breaking of a phase symmetry. 39 Chapter 4 Remarks A few remarks about the solutions that we have found are in order. Solutions that enter the horizon have the eqution of motion given by r d dr ln [ ( f 2+4cos4ψ)( f 2+4sin4ψ)− f 4 1+ ( r ddrψ )2 ] = 6 ( r d dr ψ )2 . (4.1) The right-hand-side is positive and we can conclude that the left-hand-side is a monotonically increasing function of r. If we assume that the logarithmic derivative of ψ vanishes at both limits, r→∞ and r→ 0, integrating this equation yields the sum rule ( f 2+4cos4ψ(∞))( f 2+4sin4ψ(∞))− f 4 ( f 2+4cos4ψ(0))( f 2+4sin4ψ(0))− f 4 = exp ( 6 ∫ ∞ 0 dr r ( r d dr ψ )2) . (4.2) This indicates that any solution of the equation of motion will necessarily have smaller V = ( f 2 + 4sin4ψ)( f 2 + 4cos4ψ)− f 2 at r = 0, that is in the infrared limit, than at r=∞, the ultraviolet limit. If we interpret the evolution from large r to small r as a renormalization group flow, we have found a quantity which definitely 40 decreases. What is more, it is directly related to the Routhian1 R7 = N3N7V2+1 2pi2 ∫ ∞ 0 dr r2 √ ( f 2+4sin4ψ)( f 2+4cos4ψ)− f 2 √ 1+ ( r d dr ψ )2 , (4.3) whose variation gives the equation of motion. It is the density in this integral which must decrease, an analog of an H-theorem which has recently been discussed for nonsupersymmetric 2+1-dimensional field theories [17]. If we evaluate the left-hand-side of (4.4) with the solution that we have found, it reads 1+2 f 2 f 2(4− f 2) = exp ( 6 ∫ ∞ 0 dr r ( r d dr ψ )2) . (4.4) The left-hand-side is greater than one for all values of f 2 < 1 and it equals one when f 2 = 1. When f → 1, the operator that we have perturbed the conformal field theory by when we switched on ψ1 approaches a marginal operator. It should be possible to develop a perturbative approach where this flow can be analyzed explicitly. As another observation, in the same solution, the scale symmetry of the em- bedding equation for the D7-brane is broken only by one of the parameters in the boundary condition, say ψ1. Then the other parameters in the boundary conditions are determined by ψ1 and the relationship is governed by scale symmetry, ψ2 = g( f )ψ ∆+/∆− 1 . It would be interesting to compute the function g( f ) to see if it has any special behavior. Finally, the gapped solution that we have found in section 3.3 seems to be the unique example of a simple solution with a mass gap. We have outlined why it is unlikely that a solution could exist with the D5-brane reaching the Poincaré hori- zon, rather than being located at the large r boundary. We believe that the argument is robust. We cannot rule out more complicated complexes of branes with multiple 1This is obtained by the Legendre transform R(ψ(r),ψ ′(r), pz) = L(ψ(r),ψ ′(r),z′(r))− z′(r)pz and setting pz = 0 to focus on embeddings entering the Poincaré horizon. 41 intersections. The solution that we did find had an additional asymptotic region which essentially doubles the degrees of freedom. It also has a spontaneously bro- ken U(1) symmetry which we find explicitly when we study the current-current correlation functions. This breaking could proceed with a ψ̄ψ̃-condensate which breaks the U(1)×U(1) gauge symmetry to a diagonal U(1). This would be suffi- cient to gap the spectrum of fermions which live on both the D7 and D5 branes. 42 Bibliography [1] S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, Phys. Rev. Lett. 101, 031601 (2008) [arXiv:0803.3295 [hep-th]]. [2] S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, JHEP 0812, 015 (2008) [arXiv:0810.1563 [hep-th]]. [3] S. Hartnoll, Class. Quant. Grav. 26, 224002 (2009) arXiv:0903.3246. [4] M. Rangamani, Class. Quant. Grav. 26, 224003 (2009) arXiv:0905.4352. [5] J. McGreevy, arXiv:0909:0518. [6] G. W. Semenoff, Phys. Rev. Lett. 53 2449 (1984). [7] G. ’t Hooft, Nucl. Phys. B72 (1974) 461. [8] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). [9] S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Phys. Lett. B428, 105 (1998). [10] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)253291. [11] E. D’Hoker, D. Z. Freedman and W. Skiba, Phys.Rev. D59, 045008 (1999), arXiv:9807098 [hep-th]. [12] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, arXiv:9908160 [hep-th]. [13] J. Polchinski, “String theory. Vol. 2: Superstring Theory and Beyond,” Cam- bridge, UK: Univ. Pr. (1998) 402 p. 43 [14] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, arXiv:1003.4965 [hep- th]. [15] R. C. Myers and M. C. Wapler, JHEP 0812, 115 (2008) [arXiv:0811.0480 [hep-th]]. [16] J. L. Davis, H. Omid, and G. W. Semenoff, arXiv:1107.4397 [hep-th]. [17] I. R. Klebanov, S. S. Pufu and B. R. Safdi, arXiv:1105.4598 [hep-th]. 44 Appendix: Legendre Transformation In this section, we derive the Legendre transformation of a particular form of La- grangian densities which appears very often in D-brane dynamics. Working in AdS5 × S5 has the advantage of having highly symmetric Lagrangian densities which result in the following form of Lagrangian L = √ f (ψ, ψ̇,r)+gi j((ψ, ψ̇,r))żiż j +hi(ψ, ψ̇,r)żi, (4.5) where ψ is the noncyclic coordinate we want to find the Routhian for and f ,gi j,hi are arbitrary functions ofψ, ψ̇ and r but not żi. zi are the cyclic coordinates we want to Legendre transform. In this Lagrangian, the role of time is played by the radial coordinate r which is the only coordinate one integrates over to get the action. Each coordinate has a conjugate momentum given by Pi = gi j żi√ f +gi j żiż j +hi, (4.6) or equivalently √ f +gi j żiż j = gi j żi Pi−hi . (4.7) Substituting into 4.5 gives L = żi( gi j Pi−hi +hi). (4.8) 45 Now performing the Legendre transformation on all of the cyclic coordinates gives R = żi( gi j Pj−h j +h j−Pj). (4.9) Substituting żi into this expression, one gets R = √ f (1−g−1i j (P−h)i(P−h) j), (4.10) which can be used to derive the equation of motion for ψ . 46

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