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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-Simonslike 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  Scalar Field as an Example . . . . . . . . . . . . . . . . . . . . .  13  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  1.4 2  iv  2.7  Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . .  29  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  Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  43  Appendix: Legendre Transformation . . . . . . . . . . . . . . . . . . .  45  3  4  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 . . . . . . . . . . . . . . . . . . . . . .  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(∞) = 1 2  −1  sin  π 4  and ψ(0) =  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. . . . . . . . . . Figure 3.2  23  35  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, π4 , 0.46) and (1/r, ψ, z) = (0, 0, −0.46). . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  37  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 dynamics 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  1 a L = ψ¯ a (ı∂µ + gAaµb γ µ )ψ b − mψ¯ a ψ a − Fµνb Faµνb , 4  (1.1)  a Fµνb = ∂µ Aaνb + igAaµc Acνb − (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 = Ab† νa ,  (1.3)  Aaνa = 0.  (1.4)  One can rescale the fields to put the Lagrangian in a more convenient form L =  1 a N Faµνb }, {ψ¯ a (ı∂µ + Aaµb γ µ )ψ b − mψ¯ a ψ a − Fµνb 2 g 4 a Fµνb = ∂µ Aaνb + iAaµc Acνb − (µ → ν).  (1.5) (1.6)  We have chosen the dependence of the coupling on N so that it’s the most convenient 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 propagators are < ψ a ψ¯ b >= δba D, < Aaµb Acνd >= (δda δbc −  1 a c δ δ )Dµν , N b d  (1.7) (1.8)  where D is the propagator for a single Dirac field and Dµν the propagator for a single gauge field. The  1 N  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 N1 . Finally, every vertex is a Feynman vertex and carries a factor of N. Thus the graph is proportional to NV +F−E = N χ ,  3  (1.9)  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 1 Rµν − gµν R = 8πGTµν , 2  (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 1 Rµν − gµν R + Λgµν = 8πGTµν , 2  (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. According 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. D−1 2 x02 + xD+1 −  ∑ xi2 = L2 ,  i=1  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 21 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 Ω2 = 1.  (1.14)  To cover the whole of the sub-manifold just once, one need to restrict the parameters as 0 ≤ ρ, 0 ≤ τ ≤ 2π,  (1.15)  and Ω as usual are spherical coordinates. Then the induced metric would be ds2 = L2 (−cosh2 ρ dτ 2 + dρ 2 + sinh2 ρ dΩ2 ).  (1.16)  Moving to a new coordinate with tanθ = sinhρ helps to understand one of important 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 L2 (−dτ 2 + dθ 2 + sin2 θ dΩ2 ), cosθ 2 0 ≤ θ ≤ 2π.  ds2 =  (1.17)  Figure 1.3: AdS boundary The metric is conformally equivalent to S1 × SD−1 . As one can see, the boundary is at θ = π/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 compactified flat Minkowski space in D dimensions. In what follows, we’ll also use Poincar´e coordinates, defined as  1 −r 2 − t 2 )) (1 + y2 (L2 + → 2y xi = Lyri ∀ 1 ≤ i ≤ D−1 1 −r 2 + t 2 )) xD = (1 − y2 (L2 − → 2y xD+1 = Lyt. x0 =  6  (1.18)  The domain is  y≥0 → −r ∈ ℜD−1 t ∈ ℜ, and the metric is ds2 = L2 (  1.3 1.3.1  → − dy2 + y2 (−dt 2 + dr 2 )). y2  (1.19)  (1.20)  Motivating Maldacena Conjecture 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 YangMills 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 conclude 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 = 2πgs .  (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 perspective, 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:  1  1  ds2 = H(r)− 2 ηµν dxµ dxν + H(r) 2 (dr2 + r2 dΩ25 )  8  µ, ν = 0, 1, 2, 3.  (1.22)  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 = 4πgs Nα 2 .  (1.25)  Working in the low energy limit that describes the supergravity we ask the curvature to be small. It’s simple to check that the the curvature is proportional to  1 , (gs N)2  so, to have a consistent calculation we ask that 1.  gs N  (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 µ ν η dx dx + (dr2 + r2 dΩ25 ) µν L2 r2  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 = ηµν dxµ dxν + (dr2 + r2 dΩ25 )  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  r α  constant. By r we mean all of the distances in the  9  problem. Then, using our formula for L, we find that L4 α2 α 4 −2 = 4πg N = 4πg N α → ∞, s s r4 r4 r4  (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 homeomorphism 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  1 2  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 between 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 =  rh4 r2 L2 1 2 µ ν ((1 − )dτ + η dx dx ) + dr2 + L2 dΩ25 i j L2 r4 r2 1 − rh4 r4  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 =  L2 1 4rh (r − rh )dτ 2 + dr2 + L2 dΩ5 . 2 L 4rh r − rh  (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 4(rh ρ)2 2 (dρ 2 + dt ) + L2 dΩ5 . rh L4  11  (1.33)  This is the metric of flat space in polar coordinates. But to not have a conic singularity one needs to have a periodicity in τ, giving τ ∼ τ +β. πL2 β= . rh  (1.34) (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 boundary, 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 interested 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 generator or the partition function in finite temperature field theory: Z[J] =< e  dxO(x)J(x)  >=  Dφ e−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 −z )z∆− . −z ) =< O > z∆+ + J(→ φ (z0 , → 0 0 12  (1.38)  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 −z 2 ) + L2 dΩ2 (dz2 + d → 5 z20 0  µ, ν = 0, 1, 2, 3.  (1.39)  This is related to usual AdS metric by the following change of coordinates L z= . r  (1.40)  Then using the common method of the functional derivative one can calculate correlation functions of the corresponding operator − − < T {O(→ z1 )O(→ z2 )} >=  δ 2 lnZ[J] |J=0 . − − δ J(→ z1 )J(→ z2 )  (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 ∂µ φ ∂ν φ .  (1.42)  This results in the equation of motion given by ∂µ (gµν ∂ν φ ) = 0.  13  (1.43)  Using the following coordinates ds2 =  1 −z 2 ) + L2 dΩ2 (dz2 + d → 5 z20 0  µ, ν = 0, 1, 2, 3,  (1.44)  where we have used the Euclidean signature; one gets ∂z0 (  1 1 ∂z0 φ ) + 3 ∂µ2 φ = 0. 3 z0 z0  (1.45)  Using the Fourier transformation −z ) = φ (z0 , →  − → − dk4 i→ e k . z φ (z0 , k), 2π  (1.46)  one gets (  φ k2 − ) φ = 0, z30 z30  (1.47)  which is the Bessel equation for ν = 2. Then the solutions are φ (k, z0 ) = A(k)z20 K2 (kz0 ) + B(k)z20 I2 (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 ) =  k2 z20 K2 (kz0 )J(k). 2  14  (1.50)  Substituting the solution in the action one gets that S= AdS5  ∂ AdS5 L3 ∂ AdS5  =  gµν ∂µ φ ∂ν φ  (1.51)  √ ∂µ ( −ggµν φ ∂ν φ )  =− =  ×S5  z30  φφ  k2 z20 k2 z20 dk4 L3 J(k)J(−k) K (kz )( K2 (kz0 )) . 2 0 (2π)4 z30 2 2  Taking the functional derivative with respect to J(k) twice one finds k2 z20 K2 (kz0 ) 2  kL3  < O(0)O(k) >= 3 fk (z0 ) fk (z0 )  . z0 →0 z0 fk (z0 ) =  (1.52)  Expanding fk (z0 ) around the boundary and taking the Fourier transformation we find < O(x)O(y) >∼  1 , (x − y)8  which gives the conformal power ∆ = 4, as one expects.  15  (1.53)  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 consider 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 = L4 r4 dt ∧ dx ∧ dy ∧ dz + L4  c(ψ) ¯ 2, dΩ2 ∧ d Ω 2  (2.2)  with ∂ψ c(ψ) = 8 sin2 ψ cos2 ψ = 1 − cos 4ψ.  (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 ¯ 22 , ds2S5 = dψ 2 + sin2 ψdΩ22 + cos2 ψd Ω  (2.4)  ψ ∈ [0, π/2],  (2.5)  and the world-volume of D7 wrapping S2 and S¯2 . This configuration can be stabilized 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 2πα F0 =  L2 L2 ¯ 2 f 1 Ω2 + 2 f 2 Ω2 .  (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 quantization condition we have  17  L2 n1 f1 = , 2πα N7 2 n2 L f2 = . 2πα N7  (2.7)  where n1 and n2 are integers. In [15], Myers and Wapler study the possible embedding 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 potential form. One defines the Field Strength as Fµν = ∂µ Aν − ∂ν Aµ ,  (2.8)  F = dA,  (2.9)  or in closed form  where Aµ is the vector potential with U(1) gauge invariance. The Maxwell equations 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 magnetic 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 forget 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 = Np Tp  dτ p+1 e−φ  ΣqCq ∧ e2πα F ,  −det(g + 2πα F) +  Σ p+1  (2.14)  Σ p+1  where Np is the number of Dp-Branes, Cq is the background and 1  Tp = α  p+ 12  (2π) 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 Ramond fields Cq −→ Cq + dΛq−1 ,  (2.16)  the Chern-Simons term picks up the following extra part dΛ p−1 ∧ e2πα .  20  (2.17)  In the cases that F is closed in the bulk, one can integrate it by part and get a boundary term Λ p−1 ∧ e2πα .  (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 A p −→ A p + ΣΛ p−1 ∧ e2πα ,  (2.19)  and add the following term to the action dA p .  (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 A p 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 2πα = 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 (5)  D5  dA5 =  (5)  ∂ D5  A5 .  (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. (7)  D7  dA5 ∧ 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 (7)  S= D7  (5)  dA5 ∧ F (7) + A5 ∧ δDirac (∂ D5).  (2.24)  Then, by varying A5 , one gets the following terms to vanish (7)  (5)  − δ A5 ∧ dF (7) + δ A5 ∧ δDirac (∂ D5) = 0.  (2.25)  Correlating the variations on the boundary as follows (7)  (5)  δ A5 = Qδ A5 ,  (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 question 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 r0  S= 0  ∞  drL5 (z, z˙) +  r0  drL7 (z, z˙).  (2.28)  Varying with respect to z, keeping the attaching point constant, one gets the equation of motion plus some boundary terms r0  δS =  ∞ ∂ L7 ∂ L7 ∂ L5 ∂ L5 − )δ z + dr( − )δ z + ∂z ∂ z˙ ∂z ∂ z˙ r0 ∂ L5 ∂ L7 ( − )r δ z(r0 ) = 0. ∂ z˙ ∂ z˙ 0  dr( 0  22  (2.29)  The first two terms give the equation of motion, but to satisfy the whole equation one needs to add the following condition (  ∂ L5 ∂ L7 − )r = 0, ∂ z˙ ∂ z˙ 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 independent 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 opposite 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 D2brane 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 = N5 T5  dτ 5+1  −det(g + 2πα F) + 2πα  Σ5+1  C4 ∧ F.  (2.31)  Σ5+1  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)  2πα Ae = −A(r)dt, L2  (2.34)  2πα 2πα Fm = 2 F0 + Bdx ∧ dy, L2 L  (2.35)  one gets the following action for the D5 SDBI = −N5  r0 rh  (r4 + B2 )( f12 + 4cos4 ψ)∆,  (2.36)  with ∆ = 1 + r2 hψ˙ 2 + r4 h˙z2 − A˙ 2 ,  N5 =  2πN5 T5 L6V2,1 , gs  (2.37)  (2.38)  where V2,1 is the volume of the 2-sphere spacetime. The Chern-Simons term would be SCS = −N5 ξ5  r0 rh  drr4 h˙z f2 ,  (2.39)  which gives the final Lagrangian −  L5 = N5  (r4 + B2 )( f22 + 4cos4 ψ)∆ + ξ5 r4 h˙z f2 .  24  (2.40)  Performing the Legendre transformation, as explained in Appendix one, can get the effective Lagrangian or Routhian for ψ −  R5 = N5  ((r4 + B2 )( f22 + 4cos4 ψ) − f22 r4 h + Q25 )(1 + r2 hψ˙ 2 ),  (2.41)  with (r4 + B2 )( f22 + 4cos4 ψ) , 1 + r2 hψ˙ 2 + r2 h˙z2 − A˙ 2  Q5 = A˙ 5  (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 N7 T7  dτ 7+1  =  C4AdS ∧ F ∧ F  −det(g + 2πα F) + 2πα Σ7+1  Σ7+1 5 dC4S  −2πα  ∧ A ∧ F.  (2.43)  Σ7+1  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 )( f22 + 4cos4 ψ)( f12 + 4sin4 ψ)∆, 1 SCS = −N7 ξ7  2 SCS = −2N7 ξ7 B  ∞ r0  ∞ r0  drr4 h˙z f2 f1 ,  (2.45) ∞  drc(ψ)A˙ + 2N7 ξ7 Bc(ψ)A(r) ,  with N7 =  (2.44)  r0  2πN7 T7 L8V2,1 . gs  25  (2.46)  (2.47)  The final Lagrangian would be −  L7 N7  (r4 + B2 )( f22 + 4cos4 ψ)( f12 + 4sin4 ψ)∆  =  (2.48)  ˙ +ξ7 r4 h˙z f1 f2 + 2ξ7 AB. Repeating the procedure as for D5, we find the Routhian for D7 to be  Q7 = A˙ 7  (r4 + B2 )( f12 + 4sin4 ψ)( f22 + 4cos4 ψ) − 2ξ7 Bc(ψ), 1 + r2 hψ˙ 2 + r2 h˙z2 − A˙ 2  −  R7 N7  (1 + r2 hψ˙ 2 )  (2.49)  =  ((r4 + B2 )( f12 + 4sin4 ψ)( f22 + 4cos4 ψ) − ( f1 f2 r4 h − Pz )2 r−4 h−1 + (Q7 + 2ξ7 Bc(ψ))2 . (2.50)  The equation of motion for ψ would then be Γ1 := 1 + r2 hψ 2 ,  (2.51)  Γ2 := (r4 + b2 )g(ψ) − ( f1 f2 )2 r4 h + (Q7 + 2ξ Bc(ψ))2 ),  (2.52)  g :=  ( f12 + 4 cos4 ψ)( f22 + 4 sin4 ψ) − ( f1 f2 )2 ,  ψ r + r−3 2(r + r−3 )ψ −ψ 3 + + Γ1 Γ1 r2 h ψ r2 h(2gr3 − d( f1 f2 r  4 h−P )2 r −4 h−1 ) z  dr  (2.53)  (2.54)  dg dc ) − 21 ((r4 + B2 ) dψ − 2b(Q7 + 2ξ Bc) dψ )  Γ2 r2 h  = 0,  where we’ve put rh = 1. The equation for z would be Pz = r4 h˙z  ((r4 + B2 )( f12 + 4sin4 ψ)( f22 + 4cos4 ψ)(Q7 + 2ξ7 Bc(ψ))2 + ξ7 h 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 d 3 q eiqx ∆ab (q), (2π)3  ja (x) jb (0) =  ∆ab (q) = ∆CS (q) εabc qc + ∆T (q)(q2 δab − qa qb ).  (2.56)  (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 + F˜ 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 ˜ For our purposes, it is also sufficient and consistent to take F˜ of motion for F. 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 = N7 T7 (2πα )2  1 4  d8τ  1 det(g + 2πα F0 ) gµν gλ ρ F˜µλ F˜νρ − i F˜ ∧ F˜ ∧C4 , 2 (2.58)  where we have moved to the Euclidean signature. In addition to this bulk action, in order to maintain invariance under the Ramond form gauge transformation which would shift the function c(ψ) by a constant, it is necessary to add a surface term to this action [14]. The action becomes S = N7 T7 (2πα )2  1 4  d8τ  1 det(g + 2πα F0 ) gµν gλ ρ F˜µλ F˜νρ − i A˜ ∧ F˜ ∧ dC4 . 2 (2.59)  Then, with our Ans¨atz for the D7 geometry, and choosing the Ar = 0 gauge, f1 = f2 = f and rh = 0, we obtain S probe =  N3 N7 2π 2  ∞  d3x  dρ 0  1 1 2 (∂ρ Aa )2 + α 2 Fab + i∂ρ c(ψ)εrabc Aa ∂b Ac , 2 4 (2.60) 27  where we have dropped the tildes on F and A, a, b, c are 2+1-dimensional Euclidean indices, α 2 = ( f 2 + 4 sin4 ψ)( f 2 + 4 cos4 ψ) and the radial variable has been transformed to  ∞  ρ(r) = r  d r˜ r˜2  √  ( f 2 +4 sin4 ψ)( f 2 +4 cos4 ψ)  √  1+˜r2 ψ 2 +˜r4 z 2  .  (2.61)  Note that r = 0 and r = ∞ are mapped to ρ = ∞ and ρ = 0, respectively. Using the fourier transform Aa (x) =  d3q (2π)  3 2  eiq·x Aa (q),  (2.62)  the field equation can be written −∂ρ2 + α 2 q2 ± 2iq∂ρ c(ψ) A± (ρ, q) = 0,  (2.63)  and qa Aa = 0. We are considering polarization states which obey [iqb εabc ] Ac± = ±qAa± . The field equation should be solved with the requirement that the solution is regular at the Poincar´e horizon, ρ = ∞. The on-shell action for the gauge field is then  N7 N3 Sˆ = − 4π 2  d 3 q lim Aa (ρ, −q)∂ρ Aa (ρ, q) .  (2.64)  ρ→0  ˆ  The current-current correlator is obtained by taking two derivatives of e−S 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 symmetries 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 = L4 r4 dt ∧ dx ∧ dy ∧ dz + L4  c(ψ) ¯ 2, dΩ2 ∧ d Ω 2  (2.65)  with ∂ψ c(ψ) = 8 sin2 ψ cos2 ψ = 1 − cos 4ψ,  (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 −→  π 2  − 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 )( f12 + 4cos4 ψ)( f22 + 4sin4 ψ) − ( f1 f2 )2 r4 h + (Q + 2ξ c(ψ))2 , (2.67)  the Routhian expands as 2 R7 = −T7 N7 (1 + 21 r2 hφ˙ ) V (r)|ψ0 + ∂∂Vψ |ψ0 φ + 12 ∂∂ ψV2 |ψ0 φ 2 + O(φ 3 ) (2.68) 2 ∼ −T7 N7 V (r)|ψ0 +V (r)|ψ0 12 r2 hφ˙ + ∂∂Vψ |ψ0 φ + 12 ∂∂ ψV2 |ψ0 φ 2 + O(φ 3 ),(2.69)  29  where we have expanded around ψ0 , 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 ∂V ∂ 2V φ− V r2 hφ˙ − 2 dr ∂ψ ∂ψ  = 0.  (2.70)  ψ=ψ0  We now look at the equation at large r. V has the form given by V = r2  ( f12 + 4cos4 ψ)( f22 + 4sin4 ψ) − ( 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)  ∂V = 0, ∂ψ  (2.73)  The leading term is  which results in the following equation cos ψ0 sin ψ0 f22 sin2 ψ0 − f12 cos2 ψ0 + 4cos2 ψ0 sin2 ψ0 (cos2 ψ0 − sin2 ψ0 = 0, (2.74) which has two trivial solutions consisting of 0 and  π 2,  and non-trivial ones that  depend on f1 and f2 . Moving to the next order, one has V  d 4˙ ∂ 2V φ, (r φ ) = dr ∂ ψ2  (2.75)  which leads to the following equation for ∆,  ∆(∆ + 3) =  30  ∂ 2V ∂ ψ2  V  ,  (2.76)  which will give us the following formula for ∆, f12 +16sin6 ψ0 −12sin4 ψ0 f12 +4sin6 ψ0  − 23 ± 12  9 + 16  = − 32 ± 12  9 + 16  ∆± =  f22 +16cos6 ψ0 −12cos4 ψ0 . f22 +4cos6 ψ0  (2.77) (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 = π/4. This restricts f1 and f2 to the following interval | f | = | f1 | = | f2 | ≥ At the bound | f | =  23 50  23 . 50  (2.79)  we get the marginal value given by ∆+ = 32 . At | f | =  1 2  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 Poincar´e patch(zero temperature) 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 conformal power ν=  9 4  2  f + 16 1− − 32 . 4− f 2  (2.80)  The other family of solutions, which diverge at the origin, we drop. So, around origin one gets ψ=  arcsin f 2  + crν + . . . .  31  (2.81)  Chapter 3  Different Scenarios 3.1  Parity and Time Reversal Invariant Solution  The parity invariant solution is given by the solution which, under ψ −→  π 4  − ψ, is  invariant. Clearly, the only possible solution is the constant solution given by ψ=  π , 4  z = z0 −  (3.1) f2 1+2f2  1 , r  pz = 0.  (3.2) (3.3)  where we get a black hole embedding of D7. D7 starts from the boundary at infinity, 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=  π 4  + rψ∆−1 + rψ∆+2 + . . . ,  32  (3.4)  where 2  ∆+ = 32 +  9 4  f − 8 21− , f 2 +1  ∆− = 23 −  9 4  f , − 8 21− f 2 +1  (3.5)  2  (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 situations where they may change role, see [14]. When ψ = π4 , 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 =  N3 N7 f 2 + 1 , ∆CS = 0. 2π 2 q  (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 = but with a coefficient that depends on f . Recall that f > order one.  33  23 50 ,  N3 N7 16πq  and is typically of  3.2  Parity and Time Reversal Violating Solution  At small r, a solution of (3.4) must have the form ψ=  arcsin f 2  9 4  + crν + . . . , ν =  2  1− f + 16 4− − 32 . f2  (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 theory. 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 1/∆−  and we can use, for example, ψ1  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 =  π 4  solution.  On the other hand, when q is small, we find ∆T =  N3 N7 2 f N3 N7 + . . . , ∆CS = 2 ( f 2 2π q π  1 − f 2 − arccos f ) + . . . ,  (3.10)  where corrections are of higher order in q. These functions characterize the electromagnetic 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 configuration. 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(∞) = π4 and ψ(0) = 12 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  π 2  − 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 D7branes. 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 survive. 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 socalled “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 Poincar´e horizon and the D7-brane and the mass of a string state is roughly proportional 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 Poincar´e horizon and the 3-7 strings could be arbitrarily short. To get a gapped solution, we need a configuration of D7-brane which does not approach the Poincar´e horizon, something similar to a “Minkowski embedding” where the D7 brane pinches off before it reaches the Poincar´e horizon at r = 0. The D7-brane can pinch off smoothly when one of the S2 ’s collapses, for example, 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 S˜2 ⊂ S5 and has nD units of Dirac monopole charge on S˜2 . 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 Poincar´e 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) = (∞, π4 , −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, π4 , 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 of ψ, for example, in the asymptotic regime where ψ →  π 4  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 → π asymptote and A˜ a (x) at the ψ → 0 asymptote, it is 4  straightforward to solve the large momentum limit where we obtain two decoupled currents [16] ja jb =  N3 N7 f 2 + 1 2 q δab − qa qb , 2π 2 q  ja j˜b = 0  f2 +4 2 q δab − qa qb . q  N3 N7 f j˜a j˜b = 2π 2  (3.11)  The j − j correlator reproduces the high energy limit (3.8) of our previous solutions. The j˜ − j˜ 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 N3 N7 εacb qc + . . . , 4π N3 N7 (−) = 2 (δab − qqa q2 b ) + εacb qc ∆CS (0) + . . . , . π ρm  j+a j+b =  (3.12)  j−a j−b  (3.13)  where ∞  ρm = rmin (−) ∆CS (0)  d r˜ r˜2  √  ( f 2 +4 sin4 ψ)( f 2 +4 cos4 ψ)  N3 N7 = 2 π  √  1+˜r2 ψ 2 +˜r4 z 2  π/4 0  ,  ρ(ψ) dψ(1 − cos 4ψ) 1 − ρm  (3.14) 2  ,  (3.15)  where rmin in (3.14) is the minimum value of r that the solution reaches. Corrections in (3.12) and (3.13) are of order q2 . The currents j± are linear combinations of j, j˜ which are normalized so that the charges obtained by integrating the time components of j± are integers, as were the charges of j, j˜. We then observe that the j+ current has the value of ∆CS (0) that would be expected for a system of N3 N7 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 spontaneous 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  ( f 2 + 4 cos4 ψ)( f 2 + 4 sin4 ψ) − f 4 d d ln =6 r ψ 2 dr dr 1+ r d ψ  2  .  (4.1)  dr  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 + 4 cos4 ψ(∞))( f 2 + 4 sin4 ψ(∞)) − f 4 = exp 6 ( f 2 + 4 cos4 ψ(0))( f 2 + 4 sin4 ψ(0)) − f 4  ∞ 0  d dr r ψ r dr  2  . (4.2)  This indicates that any solution of the equation of motion will necessarily have smaller V = ( f 2 + 4 sin4 ψ)( f 2 + 4 cos4 ψ) − 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 =  N3 N7V2+1 2π 2  ∞  dr r2  ( f 2 + 4 sin4 ψ)( f 2 + 4 cos4 ψ) − f 2  0  1+ r  d ψ dr  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+2f2 = exp 6 f 2 (4 − f 2 )  ∞ 0  d dr r ψ r dr  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 embedding 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 )ψ1 +  .  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 Poincar´e horizon, 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 1 This  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 Poincar´e 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 broken U(1) symmetry which we find explicitly when we study the current-current ˜ correlation functions. This breaking could proceed with a ψ¯ ψ-condensate which breaks the U(1) ×U(1) gauge symmetry to a diagonal U(1). This would be sufficient 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,” Cambridge, UK: Univ. Pr. (1998) 402 p. 43  [14] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, arXiv:1003.4965 [hepth]. [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 Lagrangian 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=  ˙ r) + gi j ((ψ, ψ, ˙ r))˙zi z˙ j + hi (ψ, ψ, ˙ r)˙zi , f (ψ, ψ,  (4.5)  where ψ is the noncyclic coordinate we want to find the Routhian for and f , gi j , hi are arbitrary functions of ψ, ψ˙ and r but not z˙i . 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 z˙i + hi , f + gi j z˙i z˙ j  (4.6)  gi j z˙i . Pi − hi  (4.7)  or equivalently f + gi j z˙i z˙ j = Substituting into 4.5 gives L = z˙i (  gi j + hi ). Pi − hi  45  (4.8)  Now performing the Legendre transformation on all of the cyclic coordinates gives R = z˙i (  gi j + h j − Pj ). Pj − h j  (4.9)  Substituting z˙i into this expression, one gets R=  f (1 − g−1 i j (P − h)i (P − h) j ),  which can be used to derive the equation of motion for ψ.  46  (4.10)  

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