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Non-abelian D5 brane dynamics Chaurette, Laurent 2014

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Non-Abelian D5 Brane DynamicsbyLaurent ChauretteB.Sc., Universite´ de Montre´al, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2014c© Laurent Chaurette 2014AbstractThe goal of this thesis is to analyse the non-abelian dual model to the defectprobe D7-brane embedding in AdS5 × S5[1]. The D7-brane picture can bethought of as a large number (N5) of D5-branes growing a transverse fuzzytwo-sphere, called BIon. This non-abelian solution improves our knowledgeof the system by incorporating deviations in 1N25in the number of flavors.Such corrections are important from the point of view of the AdS/CFTcorrespondance as the CFT dual to the probe system is a candidate modelfor graphene, which possesses an emergent SU(4) symmetry. The mainresult of this work is the conductivity for the non-abelian D5 sytem. Wefind that quantum Hall states have a non-integer transverse conductivitythat depends on the number of flavor branes in the model. This deviationscales in 1N25in the number of flavor branes and vanishes in the large N5limit.iiPrefaceChapters 1 and 2 are a description of chosen material done by others andneeded for this work.Chapters 3 and 4 are based on research done during my Master’s de-gree under the supervision of Professor Gordon Semenoff. This work wassupported by funding from the Fonds Que´be´cois de recherche en nature ettechnologies and the Natural Sciences and Engineering Research Council ofCanada.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 BIons and BPS bound . . . . . . . . . . . . . . . . . . . . . . 11.2     Chern-Simons action and Dirac quantization  .  .   .   .   .   .   .   .   . 51.3     Non-abelian brane action     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   . 61.3.1     Non-abelian Born-Infeld action .  .   .   .   .   .   .   .   .   .   .   . 71.3.2 Non-abelian Chern-Simons action   .  .  .  .  .  .  .  .  .  .  .  .  .  . 91.4    Review of the fuzzy sphere  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 92 Abelian Probe Branes . . . . . . . . . . . . . . . . . . . . . . . 122.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Probe brane action . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 D3 probes . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 D5 probes . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 D7 probes . . . . . . . . . . . . . . . . . . . . . . . . 172.3    Near-Horizon limit   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .     192.3.1 D5 probes . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 D7 probes . . . . . . . . . . . . . . . . . . . . . . . . 212.4    Correlators of the defect CFT  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .     212.5    Minkowski vs Clack Iole Fmbedding .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    223 Non-Abelian Probe Brane Action . . . . . . . . . . . . . . . 243.1    D 5-D7  .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .  .   .     253.2    D3-D5 .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    313.2.1     D3   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    31ivTable of Contents3.2.2     D5 with flux   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .     323.3    D3-D7   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .     333.3.1     D3   .   .   .  .   .   .   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .     334 Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Minkowski embeddings . . . . . . . . . . . . . . . . . . . . . 354.2 BH embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 375 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41AppendixA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.1   Trace-log expansion  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   .  .   .    43A.2   Pullback computation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    46A.3   Conformal dimensions   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    48vList of Figures1.1 The brane grows a spike in the transverse direction corre-sponding to a fundamental string attached to the brane. Theendpoint of the string is the Coulomb charge propagating theelectric field on the brane . . . . . . . . . . . . . . . . . . . . 42.1 The defect occupies a 2+1 dimensional volume positioned atz = 0, corresponding to the position of the D-branes, in the3+1 dimensional N = 4 SUSY. . . . . . . . . . . . . . . . . . 202.2 Two different types of embeddings. The circle in the mid-dle represents a black hole with its horizon at r = rh. TheX-direction represents coordinates perpendicular to the D3-branes but parallel to the probe brane. (a) A black holeembedding. The probe brane extends in the radial directionuntil it reaches the horizon of the black hole (b) A Minkowskiembedding. The probe brane pinches off at r0 > rh and doesnot reach the horizon. . . . . . . . . . . . . . . . . . . . . . . 23viChapter 1IntroductionDirichlet-branes have played a tremendous role in the construction of stringtheoretic models since it was suggested in [14] that they are an intrinsicpart of the type IIB model of string theory. Not only they are the extendedobjects on which open strings can end, but they are also BPS objects actingas sources of quantized Ramond-Ramond flux. D-branes now have a life oftheir own as one can write down an action for them recreating the openstring action found from scattering amplitudes.Under the AdS/CFT correspondence D-branes have proven extremelyuseful in describing physical systems at strong coupling from their gravitydual using the Top-Down approach in String Theory. This method consistsin building a specific configuration of D-branes in ten-dimensional spacetimeand using the dictionary of the correspondence to understand its dual fieldtheory. In distinction with the Bottom-Up approach, this method is blindto the four-dimensional dynamics on the gravity side. Even without know-ing the four-dimensional gravity theory, the Top-Down approach remains agreat tool from our good knowledge of brane dynamics. Surely the most fa-mous example of such a D-brane based configuration is the Sakai-Sugimotomodel[16] of AdS/QCD which from probe D8-branes in a D4-branes geome-try reproduces important aspects of QCD such as chiral symmetry breaking.In this chapter, we will review some material we will refer to in thesubsequent sections. We start by introducing the Born-Infeld and Chern-Simons actions for general D-branes as well as some important featuresof branes such as BPS states and Dirac charge quantization. Then, weintroduce the notion of non-abelian brane action[18] [12] and fuzzy sphericalgeometry.1.1 BIons and BPS boundWe describe here the Born-Infeld action which accounts for the coupling ofopen strings to the Neveu-Schwarz background fields Gµν , Bµν and Φ. Forthis discussion, we will work in the type II superstring framework.Let us start with a space-filling D9-brane with gauge field strength Fab in flat11.1. BIons and BPS boundspace. The appropriate action that generalizes non-linear electrodynamicsfor this configuration is simply,S ∼∫d10σ√det (ηab + 2piα′Fab) (1.1)where we remember that α′ is the inverse of the string tension T = 12piα′ andcan be assimilated to the string length α′ = l2s .Upon compactification of the 9th coordinate, T-duality tells us the Dirich-let boundary condition becomes a Neumann boundary condition and theD9-brane is transformed into a D8-brane. The now transverse coordinate isthe compactified component of the gauge field 2piα′A9 = X9. By repeatingthis procedure, we obtain an action for a Dp-brane,S ∼∫dp+1σ√det (ηab + ∂aXm∂bXm + 2piα′Fab) (1.2)We now need a way to implement the background fields in the action. Wenote that the Kalb-Ramond field Bµν is not gauge invariant. Under a space-time gauge transformation δB = dξ, it picks up a term12piα′∫MB + δB =12piα′∫MB +12piα′∫∂Mξ (1.3)The change in B is only affected by the values of ξ on the worldsheet. Wecan see this as a signal that the correct gauge invariant way to implement Bin the action is by mixing with the gauge transformations of the gauge fieldA. As A lives on the worldsheet, its gauge transformations are given by,∫∂MA+ δA (1.4)Therefore, the combination B+2piα′dA preserves spacetime gauge invariancewith the choice of gauge transformation δA = − ξ2piα′ , where ξ is restrictedto the worldsheet. This is good evidence that the Kalb-Ramond field shouldappear in the action with the gauge field strength in the formB + 2piα′F (1.5)The dilaton field must appear in the action as e−Φ to respect the fact thatwe are dealing with open string tree level amplitudes and will be assimilatedto the string coupling. Taking in account an arbitrary background metric,21.1. BIons and BPS boundwe replace ηab + ∂aXm∂bXm by Gab. These changes yield the celebratedBorn-Infeld action for a Dp-brane:SBI = Tp∫dp+1σe−Φ√det (Gab +Bab + 2piα′Fab) (1.6)With Tp being the tension of the brane:Tp =1(2pi)p√α′p+1 (1.7)We will be interested in solutions to the Born-Infeld action. To demonstratean important point, we will simplify the problem by considering a singleDp-brane in Minkowski space with the Dilaton and Kalb-Ramond fields setto zero. We will excite only one transverse direction (X9(r)) and switch onan electric field on the brane with no magnetic components. The energydensity in that configuration simplifies to:ε2 = (1 + ~E · ~∇X9)2 + ( ~E − ~∇X9)2=(1+ | ~E · ~∇X9 |)2+ E2 − 2 | ~E · ~∇X9 | + (∇X)2(1.8)This leads to the BPS boundε ≥ 1+ | ~E · ~∇X9 | (1.9)Which is saturated for ~E = ±~∇X9. By solving the equations of motion forX9, one obtains the Laplace equation ∇2X9 = 0, which is solved by theCoulomb solution,X9(r) =cprp−2(1.10)The energy of the Coulomb field is of course infinite and working where thebound is saturated, we impose a cutoff on short distances,E(δ) = TpΩp−1∫ ∞δdr rp−1(~∇X9)2 = TpΩp−1cp(p− 2)X9(δ) (1.11)This result is very interesting as when δ → 0 the BIon grows an infinitetransverse spike with its base being a point charge. The spike has infiniteenergy but a constant energy per unit of length. This behaviour was firstnoticed in [2] and was demonstrated to correspond to an infinitely longfundamental string attached to the brane. This can be seen by setting theconstant cp to cp = 12piα′TpΩp−1(p−2) and the energy of the spike becomes31.1. BIons and BPS boundFigure 1.1: The brane grows a spike in the transverse direction correspondingto a fundamental string attached to the brane. The endpoint of the stringis the Coulomb charge propagating the electric field on the braneexactly the energy of a fundamental string of length X(δ). This value ofcp is not arbitrary but happens to exactly be the value required by chargequantization. This statement is easily verified in the case of a D1-brane andgeneralizable to higher dimensional branes by T-duality.We wish to end this section by verifying that the energy bound we foundabove is truly BPS. This can be done by performing the following super-symmetry analysis. Under a supersymmetry transformation, there are threefermion fields (Gravitino ψM , Dilatino λ and Gaugino χa) whose variationwe must consider. As discussed by Green, Schwarz and Witten [5], thevariation of the dilatino is always zero when the Dilaton and Kalb-Ramondfields are turned off. The question for the gravitino is purely geometricand depends only the background geometry. We will solely consider herethe variation of the gaugino field which is the only non-trivial part. Itsvariation under a supersymmetry transformation is,δχ = ΓµνFµν (1.12)where the matrices Γµν are defined from the 10-dimensional Dirac matricesΓµ byΓµν = [Γµ,Γν ] (1.13)41.2. Chern-Simons Action and Dirac QuantizationA BPS configuration is one for which δχ = 0 for some . As only onetransverse fieldX9 is turned on and the electric field describes a point charge,the non-trivial values for Fµν areF0r = F9r =∂∂rcprp−2(1.14)which leads to the supersymmetry equation,δχ =(Γ0 + Γ9)Γr∂rX (1.15)The configuration is BPS if(Γ0 + Γ9) = 0 (1.16)By going to a Weyl basis for the gamma matrices, it is clear that this equa-tion is satisfied for half of the possible choices of  and the BIon configurationis truly a half BPS state.1.2     Chern-Simons Action and Dirac quantizationTo understand the Dirac quantization on the branes, we recall the usualdevelopment for a magnetic point charge. We first consider a magneticmonopole of charge µm at the origin and integrate the flux over a spheresurrounding the charge. ∫S2F = µm (1.17)The gauge field respects the usual equation F = dA except on a semi-infiniteDirac string ending on the charge. We first take the Dirac string to be onthe positive region of the z-axis. An electric charge µe circling the sphere ona closed loop Γ will interact with the magnetic field and pick up a phaseeiµe∫Γ A (1.18)As we contract Γ around the Dirac string, the phase becomeseiµeµm = e2ipin (1.19)The last equality comes from the fact that the phase needs to be 1 so theDirac string remains invisible even under different orientation of the string.We conclude that the product of the electric and magnetic charges satisfyµeµm = 2pin.51.3. Non-abelian Brane actionIn the previous section, we mentioned that branes are BPS objects. FromNoether’s theorem, they must therefore carry conserved charges. The correctcharges that couples to the branes are the Ramond-Ramond charges. TheDp-brane interacts with the RR potential C(p+1) via the so-called Chern-Simons action, which forms the second piece of the D-brane actionsSCS = −µp∫dp+1σP [∑nC(n)] e2piα′F (1.20)Here, µp is the Dp-brane RR-charge and F is the gauge field strength (Notto be confused with the RR-field strength defined next line). The symbolP [...] denotes the pull-back to the brane worldvolume.The RR potential has a field strength F (p+2) = dC(p+1) which respectsthe Bianchi identityd ∧ F (p+2) = d ∧ ∗F (p+2) = 0 (1.21)The Bianchi identity marks the duality between the field strength and theirHodge dual∗F (p+2) = F (8−p) = dC(7−p) (1.22)This allows us to extend the point charge development by identifying theelectric RR charge to be generated by a Dp-brane while the magnetic chargeis spanned by a D(6-p)-brane transverse to the initial Dp-brane.By following the same argument as for the magnetic point charge, we getthat the integral of the magnetic flux on a sphere surrounding the monopoleis the RR charge ∫S8−p∗F p+2 = µ6−p (1.23)The Dirac string ending on the monopole is now enhanced to a Dirac sheetof 7-p dimensions. A Dp-brane circling the Dirac sheet picks up a phaseeiµpµ6−p and the requirement that the Dirac sheet is invisible leads to thequantization conditionµpµ6−p = 2pin (1.24)1.3      Non-abelian brane actionThe action (S = SBI + SCS) which was presented above for Dp-branes istruly the action of a single brane with U(1) gauge field. Once one considers astack of Nf coincident flavor branes, non-abelian effects are introduced andthe action (1.6)+ (1.20) is not sufficient anymore. These effects can be seen61.3. Non-abelian Brane actionfrom the point of view of strings stretching between the branes. A stringwith its endpoints on different branes possesses a mass proportional to itslength and can be integrated out of the Lagrangian in the low energy limit.But as we bring the branes together, such a string is no longer stretchedand becomes massless. One can not distinguish which brane the endpointis on and the U(1)Nf gauge symmetry is enhanced to a non-abelian U(Nf )symmetry. The gauge field strength is modified with the usual non-abelianstructureFab = ∂aAb − ∂bAa + i[Aa, Ab] (1.25)Non-commutative effects start to appear when we think of a Dp-brane asobtained from a D9-brane by T-duality. T-duality acting on a worldvol-ume coordinate transforms a Dp-brane into a D(p-1)-brane interchangingworldvolume gauge fields into scalars portraying transverse excitations ofthe brane.Ap → Φp (1.26)These transverse coordinates Φi are N ×N matrix-valued scalars with non-trivial commutation relations[Φi,Φj ] = Θij (1.27)The geometry now qualifies as being non-commutative and one can notresolve distances at small wavelengths.In the following section, we wish to review the non-abelian extension ofthe Born-Infeld and the Chern-Simons action which we will later use to findinterresting solutions to various probe-branes configurations.1.3.1      Non-abelian Born-Infeld actionThe extension of the Born-Infeld action to a fully non-commutative theorywas first introduced by Tseytlin [18] but became extensively used in thecontext of Dielectric Branes [12]. In this section we will review the non-abelian formalism to obtain the Born-Infeld action and its main differencewith its abelian counterpart.The way to obtain the action is fairly straightforward. One starts byconsidering a D9-brane, for which there is no non-abelian structure in thegeometry as there are no transverse directions, and starts T-dualizing as be-fore. This time, non-abelian effects of the gauge field are taken into accountduring the procedure and the commutators of the transverse scalars appearin the action. The result of this approach is the non-abelian BI action (NBI)71.3. Non-abelian Brane actiongiven here,SNDBI = −Tp∫dp+1σSTr(e−Φ√−det(P [Eab + Eai(Q−1 − δ)ijEjb] + 2piα′Fab) det(Qij))(1.28)Where we have defined E = G+B and the matrix Qij = δij+i2piα′ [xi, xk]Ekj .The latter is a purely non-commutative addition whose existence comes fromT-duality of the gauge fields turned scalars.We now simply wish to point out the main differences between the non-abelian result (1.28) and the commuting action (1.6).First of all, from the construction of the action, the gauge field possessesobviously a non-abelian symmetry group with field strength given by (1.25).Secondly, the pullback in (1.28) becomes non-abelian. This means thatthe derivatives need to be made covariant appropriately by taking into ac-count the possible commutators:∂axi → Daxi = ∂axi + i[Aa, xi] (1.29a)P [E]ab = EµνDaxµDbxν = Eab + EaiDbxi + EibDaxi + EijDaxiDbxj(1.29b)Thirdly, the bulk supergravity fields are functions of all the spacetimecoordinates and in general are functionals of the non-abelian scalars. Thecorrect way to understand this statement was clarified by Taylor and VanRaamsdonk [17] and consists of expanding the bulk fields in a non-abelianTaylor expansion around the position at which the transverse coordinatesvanish. For example, a general field Tµ1...µk can be written asTµ1...µk(xa, xi) =∞∑n=01n!xi1 ...xin (∂xi1 ...∂xi1 )Tµ1...µk(xa, xi) |xi=0 (1.30)Finally, the action depends non-linearly on the non-abelian scalars andwe need a prescription for their precise ordering in the action in order toobtain a gauge invariant quantity. The symmetrized gauge trace is the mostnatural prescription as it specifies that one should take the symmetrizedaverage over all orderings of the matrix-valued quantities [xi, xj ], Daxi, Faband xi in the fundamental representation of U(Nf ).This extension of the BI action is known to correctly predict the interac-tions of low energy superstring theory up to fourth order (F 4,[xi, xj ]4) butneeds corrections at sixth orders.It should be noted that this non-abelian action reproduces the resultsof the abelian system described in 1.6 when all the commutators [xi, xj ]81.4. Review of the Fuzzy Spherevanish. This assures us that an abelian solution to the equations of motionis always a valid solution of the system competing with the non-commutativesolutions.1.3.2 Non-abelian Chern-Simons actionWe will now review the non-abelian extension to the Chern-Simons action.The formalism to approach this problem was discussed in [12]. We will notsketch a proof of its derivation here but only mention that as for the NBIaction, we can verify the consistency of the NCS action by starting from aD9-brane and T-dualizing 8 − p coordinates to get the Dp+1 action. Thefull NCS term can be expressed asSNCS = µp∫dp+1σSTr(P[ei2piα′ıxıx∑nC(n)eB]∧ e2piα′F)(1.31)The symmetrized gauge trace prescription and the dependence of the bulkfields and gauge field strength on the non-abelian scalars follow the treat-ment of the NBI action. The important addition in the non-abelian exten-sion of the CS action arises from the non-abelian interior product ıx. Theinterior product ıx is an operator of degree −1 with its square acting onn-forms as:ıxıxC(n) = xixjCjix3...xn =12Cijx3...xn [xi, xj ] (1.32)We notice that for usual commuting vectors vi, the internal product reducesto the null operator as it is the non-commutativity of the x’s that gives itsstructure. It is thus the exponential of interior products that generate therelevant non-abelian terms in the Chern-Simons action.1.4      Review of the fuzzy sphereIn this section, we review the non-commutative extension of the definitionof the sphere which will be a necessary tool in the following discussion todescribe the action of a stack of D5-branes in Anti-de-Sitter geometry.First of all, we consider the algebra of complex-valued functions on thesphere C(S2). We expand an arbitrary function of this algebra in a polyno-mial expansionA = f0 + fivi + fijvivj + fijkvivjvk + ... (1.33)91.4. Review of the Fuzzy SphereOn the two-sphere, we have three Cartesian coordinates and the index i runsfrom i = 1, 2, 3. We wish to generate the geometry of S2 from a series oftruncations of this algebra. We will call the truncation to order n An.We can start by truncating the algebra to its first term: A1 = C. In thiscase, the algebra collapses to the complex numbers and the geometry of S2is seen as a point.If we truncate the algebra at the linear term, we obtain a four-dimensionalvector space spanned by f0 and fi. Note that A2 is not closed under mul-tiplication for arbitrary vectors vi and does not form an algebra itself. Inorder for the truncation to become an algebra itself, we impose a productrule such that the vectors vi are defined as vi = κσi, where σi are the Paulimatrices and κ2 = r23 . Under this condition, A2 closes and becomes thealgebra of complex 2 × 2 matrices. The geometry can only distinguish thenorth and south poles of the two-sphere.In general, a truncation to an arbitrary level n corresponds to approx-imating the sphere by a n2 dimensional vector space. The truncation Ancloses when the vectors are in the n = 2j + 1 irreducible representation ofSU(2) as vi = κLi. The normalization κ is chosen such that the radius ofthe sphere satisfies the usual definitionr2 =3∑i=1(vi)2(1.34)The right-hand side of equation (1.34) is proportional to the Casimir invari-ant of SU(2):3∑i=1(vi)2= κ23∑i=1(Li)2(1.35a)= Cκ2 (1.35b)= j(j + 1)κ2 (1.35c)The normalization factor κ is thus taken to be κ = r√j(j+1)and the coordi-nates on the sphere arevi =rLi√j(j + 1)(1.36a)[vi, vj ] =ir2j(j + 1)ij kLk (1.36b)101.4. Review of the Fuzzy SphereThe truncation An is described by the algebra of complex n×n matrices Mn.For a small value of n, the commutators are large and the sphere is fuzzy.We can not resolve distances smaller than d ∼ rn . On the other hand, in thelarge n limit, the commutators vanish and the sphere becomes classical.In the space Mn of n × n matrices, we can define a norm such that forany element f ∈Mn,‖f‖2n =1nTr(f∗f) (1.37)The norm can be seen to generalize the notion of integration on the sphere,14pir2∫Ω2f →1nTr(f) (1.38)Where on the left-hand side, f denotes a function in C(S2) while on theright-hand side, f is the associated matrix in Mn.11Chapter 2Abelian Probe BranesIn this chapter, we review the abelian probe brane action of D3, D5 and D7-branes in AdS5 × S5. We will work specifically with probes living on a 2+1dimensional submanifold (defect) of AdS5. We interest ourselves in thesemodels as, when non-abelian effects are included, these systems have thepotential to blow up to a D7-brane. The D7-brane configuration is relevantfrom gauge/gravity duality as, in the dual CFT, the corresponding systemis a 2+1 dimensional gas of electrons exhibiting many interesting featuresas quantum Hall effect and can closely be related to graphene [9] [8].2.1 SetupFrom Maldacena’s gauge/gravity correspondence [10], Anti-de Sitter geome-try is realized as the near-horizon limit of a stack of Nc coincident D3 colourbranes and equivalent to N = 4 Super Yang-Mills theory in 3+1 dimensions.The range of validity of Maldacena’s conjecture will be respected in the limitwhen the string theory is weakly interacting and the number of D3-branesis taken to be large Nc  1. In this background geometry we add Nf flavorprobe branes. Working in the probe limit entails that we can neglect thebackreaction of the probes on the geometry. This corresponds to the limitNc  NfThroughout this work, we will use the orientation of Table (2.1) forthe D-branes. We will also use the following convention: Greek indicesTable 2.1: Orientation of D-branes0 1 2 3 4 5 6 7 8 9D3c X X X XD3f X X X XD5 X X X X X XD7 X X X X X X X Xµ, ν, ... = 0, 1, ..., 9 will refer to spacetime coordinates. Latin indices a, b, ...122.1. Setupwill describe probe branes worldvolume coordinates and transverse direc-tions to the probes will have the Latin indices from the middle of the alpha-bet i, j, .... In Table 2.1 we made the distinction between the backgroundcolour D3-branes(D3c) and the probe flavour branes (D3f ).We will work in static gauge, i.e. the target space coordinate x0 will bedesignated as the time evolution parameter τ . For reasons that will becomeclear later, we will use an S2 × S2 embedding for the S5. This means weparametrize the S5 as the fibration of two S2 over an interval ψ ∈ [0, pi2 ].Both spheres are perpendicular to the probe D3-branes while the first two-sphere (unbarred) is wrapped by the D5-brane and the D7-brane wraps both(barred and unbarred) spheres. The appropriate metric on the S5 is,ds2S5 = dψ2 + sin2 ψ dΩ22 + cos2 ψ dΩ¯22 (2.1)We will study these configurations with non-trivial magnetic field, chargedensity and temperature which, when the non-abelian structure is ignored, isknown to lead to an interesting phase diagram with a BKT phase transition[4].We turn on a gauge field strength as,2piα′Fab =√λα′(−ddra0(r)dt ∧ dr + bdx ∧ dy)(2.2)This accounts for a charge density At(r) =√λa0(r)2pi and aconstant magneticfield B =√λb2pi .The correct way to implement non-zero temperature is via Hawking’sradiation. The metric is then modified following [6] to include an AdSblack-hole.ds2 =√λα′[r2(−h(r)dt2 + dx2 + dy2 + dz2)+dr2h(r)r2+ dψ2 + sin2 ψdΩ22 + cos2 ψdΩ¯22](2.3)The black hole has radius rh with the blackening function defined as h(r) =1−r4hr4 in the case of D3-branes. We remind the reader that α′ is the Reggeslope parameter giving the tension of the string T = 12piα′ and λ = 4pigsN3with the radius of AdS given by L2 =√λα′.In this geometry, the Ramond-Ramond field strength is given by twoparts, one on AdS5 and one on S5.F (5) = 4λα′2 (r3dt ∧ dx ∧ dy ∧ dz ∧ dr + dΩ5)(2.4)132.2. Probe brane actionThe RR potential is defined by F (5) = dC(4) which results once again in twodistinct parts,C(4) = λα′2(r4dt ∧ dx ∧ dy ∧ dz +12(ψ −sin 4ψ4−pi2)dΩ2 ∧ dΩ¯2)(2.5)Where pi2 is a gauge choice corresponding to the value of ψ(r =∞). Through-out this work we will be interested in solutions where ψ is allowed to varywith the radial coordinate ψ = ψ(r), but where z is fixed and does notdepend on r.2.2 Probe brane action2.2.1 D3 probesWe now introduce N3 flavour probe D3-branes to the geometry followingthe orientation of table 2.1 and wish to compute the action for this systemignoring any possible interaction terms between the probes.The Born-Infeld action for this system can be written asSBI3 = −T3gsN3∫d4σ√det (P [G+B]ab + 2piα′Fab) (2.6)Where the tension of the D3-brane is given by T3 = 1(2pi)3α′2 . In this work,the Kalb-Ramond field Bµν will be set to zero. This will be the case for eachtype of embeddings that we will consider. It should also be noticed that theaction is multiplied by an overall factor of N3. Of course, as the probes donot interact, they all contribute the same value to the action, hence thisfactor.The induced metric on the probe worldvolume isds24 =√λα′(r2(−hdt2 + dx2 + dy2)+ dr2(1hr2+ ψ˙2))(2.7)In our convention, the dot defines a derivative with respect to r. The BIaction for the probe D3-brane isSBI3 = −N3N3∫dr√(1 + hr2ψ˙2 − a˙2)(r4 + b2) (2.8)Here we have defined the constantN3 =T3gsλα′2V2+1 (2.9)142.2. Probe brane actionThe second part of the action is given by the Chern-Simons (CS) term,SCS3 = −µ3N3∫d4σP [C4] (2.10)Where the D3 RR charge is µ3 =T3gs. The pullback picks up a term fromthe derivative of the z-direction,SCS3 = −N3′N3′∫dr r4z˙ (2.11)but as we mentioned above, we only interest ourselves in solutions wherez does not depend on r which allows us to neglect the CS action for thisconfiguration. The only relevant term is the BI action from which we findthe equations of motion for ψ(r). To do so, we start by removing the cyclicvariable a from the Lagrangian via a Legendre transformation that takes usto the Routhian,R3 = S3 −∫dr a˙∂S3∂a˙(2.12)The result is,R3 = −N3N3∫dr√(r4 + b2) + q23√1 + hr2ψ˙2 (2.13)Here, q3 is the constant of the equation of motion for the cyclic variable a˙.q3 =1N3N3∂S3∂a˙=a˙√r4 + b2√1 + hr2ψ˙2 − a˙2(2.14)This constant can be related to the total charge density viaρ =1V2+1∂S3∂ ddrAt=1V2+12pi√λ∂S3∂a˙=q3NcN3pi√λ(2.15)Where in the last equality, we have used the fact that gs = λ4piNc . We nowhave the freedom to rescale the value of r in such a way that the magneticfield is taken outside of the integral. This is done while remembering thatthe Landau level filling fraction is the charge density per units of magneticfield,ν =2piρN3B(2.16a)q3b=νλ4piN3=piνf3(2.16b)152.2. Probe brane actionwhere f3 was defined as f3 =4pi2N3λ .Rescaling r to get rid of b in the integral, we get the Routhian,R3 = −N3N3f3(2piB√λ)3/2 ∫dr√f23 (r4 + 1) + (piν)2√1 + hr2ψ˙2 (2.17)In this particular case, it appears that ψ is also a cyclic variable and theaction is trivial. The corresponding constant of the equations of motion is0 =ddrhr2ψ˙√f23 (r4 + 1) + (piν)2√1 + hr2ψ˙2 (2.18)Even though this system has no dynamical degrees of freedom, it becomesnon-trivial once the full non-abelian action is considered, which we will in-vestigate later.2.2.2 D5 probesOnce we have done the D3 problem, embedding a stack of commuting D5-branes in this geometry is not a lot of work. Here, the probes live on a 2+1dimensional defect on AdS5 and wrap a two-sphere on the S5.The Born-Infeld action is written asSBI5 = −T5gsN5∫d6σ√det (P [G+B]ab + 2piα′Fab) (2.19)with the D5-brane tension T5 = 1(2pi)5α′3 . This time, the induced metricreceives a contribution on the wrapped sphere,ds26 =√λα′(r2(−hdt2 + dx2 + dy2)+ dr2(1hr2+ ψ˙2)+ sin2 ψdθ2 + sin2 ψ sin θdφ2)(2.20)The BI action for this configuration is thus written as,SBI5 = −N5N5∫dr√4 sin4 ψ (r4 + b2)√1 + hr2ψ˙2 − a˙2 (2.21)Where N5 is defined in a similar fashion as N3 asN5 = 2piT5gs(√λα′)3V2+1 (2.22)162.2. Probe brane actionThe CS action for the probe D5-branes receives its only contributionfrom the termSCS5 = −µ5N5∫d6σP [C(4)] ∧ 2piα′F (2.23)As we did not turn on a flux for the gauge field strength on the two-sphere,this term cannot have a contribution on the D5 worldvolume and the CSaction vanishes once again.The full action is given only by the BI term to which we can apply thesame procedure as before to remove the dependence on a˙ and rescale themagnetic field appropriately. The corresponding Routhian is,R5 = −N5N5f5(2piB√λ)3/2 ∫dr√4 sin4 ψf25 (r4 + 1) + (piν)2√1 + hr2ψ˙2(2.24)Here, f5 was obtained by the same procedure as before, defining the constantof the equation of motion for a˙ as q5b =piνf5. In this case, we get f5 =2piN5√λ.Applying the Euler-Lagrange equations on the Routhian give us theequations of motion for the radius of the two-sphere ψ(r),0 =h(r ddr)2ψ1 + hr2ψ˙2+ 2r4hr3(1 +1hr2ψ˙2)+ hrψ˙(1 +8f2 sin4 ψr44 sin2 ψf2(1 + r4) + (piν)2)−8 sin3 ψ cosψf2(1 + r4)4 sin2 ψf2(1 + r4) + (piν)2(2.25)This time, the dynamics for ψ are not trivial and the system possessesvery interesting solutions for different values of the parameters f, ν, rh whichwe will look at closely in the next chapters.2.2.3 D7 probesFinally, we look at the embedding of a single probe D7-brane in theAdS5×S5geometry. For this particular configuration, we will consider a worldvolumegauge field with flux on both two-spheres. Later on, the fluxes will be seenby the D7-branes as units of Dirac monopole generated by lower-dimensionalbranes. Adding a flux, the worldvolume field strength becomes,2piα′Fab =√λα′(−ddra(r)dt ∧ dr + bdx ∧ dy +f72dΩ2 +f¯72dΩ¯2)(2.26)172.2. Probe brane actionThe induced metric on the D7-branes is now,ds26 =√λα′(r2(−hdt2 + dx2 + dy2)+ dr2(1hr2+ ψ˙2)+ sin2 ψdΩ22 + cos2 ψdΩ¯22)(2.27)The BI action for the D7-branes is,SBI7 = −N7∫dr√(f27 + 4 sin4 ψ) (f¯27 + 4 cos4 ψ)(r4 + b2)√1 + hr2ψ˙2 − a˙2(2.28)with the definition N7 =T7gs(√λα′)4(2pi)2 V2+1. It is important to pointout thatN7 =N3N3f3=N5N5f5(2.29)which will allow us to consider a single D7-brane as a stack of D3-branesgrowing two transverse fuzzy two-spheres (or one fuzzy four-sphere) or as astack of D5-branes growing a transverse fuzzy two-sphere.The Chern-Simons action receives a contribution from the second Cherncharacter,SCS7 = −µ7∫d8σ(2piα′)2P [C(4)] ∧ F ∧ F (2.30)which this time is non-zero as the RR potential C(4)θφθ¯φ¯lies on the D7-braneworldvolume. The result for the CS action isSCS7 = −2b N7b∫dr a˙ c(ψ) (2.31)The full action is still cyclic in a˙ and after the now usual computationswe get the following Routhian,R7 = −N7(2piB√λ)3/2 ∫dr√(f27 + 4 sin4 ψ) (f¯27 + 4 cos4 ψ)(r4 + 1) + [piν + 2c(ψ)]2√1 + hr2ψ˙2(2.32)182.3. Near-Horizon LimitFrom the Routhian, we find the equations of motion for ψ0 =h(r ddr)2ψ1 + hr2ψ˙2+ 2r4hr3(1 +1hr2ψ˙2)+ hrψ˙(1 +2r4(f27 + 4 sin4 ψ) (f¯27 + 4 cos4 ψ)(1 + r4)(f27 + 4 sin2 ψ) (f¯27 + 4 cos4 ψ)+ [piν + 2c(ψ)]2)−8 sinψ cosψ(1 + r4)(sin2 ψf¯27 − cos2 ψf27)+ 4 sin3 2ψ cos 2ψ(1 + r4) + 2 sin2 2ψ[piν + 2c(ψ)](1 + r4)(f27 + 4 sin2 ψ) (f¯27 + 4 cos4 ψ)+ [piν + 2c(ψ)]2(2.33)2.3      Near-horizon limit2.3.1 D5 probesIn the near-horizon geometry generated by the D3-branes 2.3, we remarkthat the worldvolume of the probe D5-branes is an AdS4 × S2 submanifoldof AdS5 × S5[7]. This can be seen as the D5-branes sit at z = 0, ψ = pi2 .The world-volume metric as seen by the probe D5-branes isds26 =√λα′(r2(−hdt2 + dx2 + dy2)+dr2hr2+ sin2 ψdΩ22)(2.34)There are two types of excitations present in this limit: the closed type IIBstrings propagating in AdS5 × S5 and the stretched 3-5 open strings livingin AdS4 × S2. The most interesting feature of this construction is that thegauge/gravity duality acts twice. In the dual picture, we get anN = 4 SYMtheory in 3+1 dimensions at the boundary of AdS5 × S5 while the probesgenerate a 2+1 (z = 0) defect CFT living at the boundary of AdS4 × S2.3-5 strings provide an interaction between the 3+1 N = 4 SYM and thedefect CFT.The probe D5-branes break the usual SO(2, 4) symmetry of AdS5 toSO(2, 3), the symmetry of AdS4. It also breaks the SO(6) symmetry of theS5 to SO(3) × SO(3¯) as the D5-branes wrap an S2 leaving the transverseS¯2 intact.From the field theory perspective, the SO(2, 3) symmetry generated bythe probes are associated to conformal symmetry in 2+1 dimensions. Thedefect is situated at z = 0 from the 3+1 dimensional perspective. TheSO(3) × SO(3¯) ∼= SU(2)V ector × SU(2)Hyper is the unbroken R-symmetry192.3. Near-Horizon LimitFigure 2.1: The defect occupies a 2+1 dimensional volume positioned atz = 0, corresponding to the position of the D-branes, in the 3+1 dimensionalN = 4 SUSY.with 8 conserved supercharges. The original sixteen supercharge vector mul-tiplet of 3-3 and 5-5 open strings transform respectively as a vector multi-plet and a hypermultiplet under the eight preserved supercharges. The 3-5strings transform as a bifundamental, the tensor product of the fundamentalrepresentation of each gauge group.At weak coupling, the defect action in the field theory contains onlyfermions ψ and their scalar superpartners φ.S =∫d3xN3∑α=1N5∑λ=1[ψ¯λαiγµDµψλα +Dµφ¯λαDµφλα] (2.35)Under R-symmetry, the fermions, assimilated to electrons, transform un-der the spinor representation of S¯O(3) while the scalars are in the spinorrepresentation of SO(3). Both fermions and scalars transform under thefundamental representation of the gauge groups U(N5) and SU(Nc).Once we study the system under non-zero magnetic fields and density 2.2,the probe branes worldvolume is not exactly AdS4×S2 anymore as the par-allel two-sphere is not maximal. Also, in general there can be a momentumin the z-direction. The line element on the D5 world-volume becomes the202.4. Correlators of the Defect CFTfollowing,ds26 =√λα′(r2(−hdt2 + dx2 + dy2)+ dr2(1hr2+(dψdr)2+ hr4(dzdr)2)+ sin2 ψdΩ22)(2.36)2.3.2 D7 probesFrom the point of view of the D7-branes, the story is almost the same. Theworldvolume symmetry of the probes is AdS4×S2× S¯2 instead of AdS4×S2as the second sphere is also on the worldvolume of the D7 but the discussionabout the near-horizon limit remains similar.A way to achieve a different model is by considering probe D7-branesthat wrap an S4 in S5 instead of two two-spheres. The probe worldvolumegeometry becomes AdS4×S4 and the main difference arising from this setupis that the original SO(6) R-symmetry gets broken to SO(5). The fermionsare in the spinor representation of SO(5) while the scalars remain unchangedunder R-Symmetry. This configuration happens to be unstable as it wasshown [15] to contain a tachyon which violates the Breitenlohner-Freedmanbound. This is a good motivation for us to focus on S2×S2 embeddings forthe rest of this work.2.4      Correlators of the defect CFTAs it was pointed out in the previous section, the introduction of probebranes corresponds as the insertion of a 2+1 dimensional CFT located atz = 0. It is thus crucial to understand how the behavior of correlationfunctions in the CFT will be affected by the insertion of the defect. Therewill be two types of primary operators: those living in the 3+1 N = 4SYM denoted O4(~x, z) and the defect primary operators O3(~x) situated atz = 0. We use the notation ∆4, ∆3 for the conformal dimension of primaryoperators O4 and O3 respectively. An interesting feature of the dCFT isthat the defect introduced at z = 0 creates a ”length scale” generatingnon-vanishing one-point function for the operator O4,〈O4(~x, z)〉 =C4z∆4(2.37)while the operator O3, of course does not see that length scale and its one-point function vanishes.212.5. Minkowski vs Black Hole EmbeddingWe can also obtain non-trivial values for the product of two ambientoperators of different conformal dimensions ∆4, ∆′4,〈O4(~x1, z1)O4 (~x2, z2)′〉 =1z∆41 z∆′42f(ξ) (2.38)which is the usual behavior for a two point correlation function at the ex-ception that the result is multiplied by an arbitrary function f(ξ) withξ = (x1−x2)24z1z2. It is easily verified that the parameter ξ is invariant underthe 2 dimensional translations and O(2, 1) rotations that leave the bound-ary invariant allowing the introduction of the arbitrary function f(ξ) in thetwo-point function.We can also compute the two-point function between one ambient oper-ator O4 and one defect operator O3. To do so we use the operator productexpansion[11] of the ambient operator at the boundary,O4(~x, z) =∑nC∆4∆nz∆4−∆nO¯∆n(~x) (2.39)with the bar on the summed operators to make explicit the fact that theseoperators live at the boundary z = 0. The sum runs over all such possi-ble operators and we write their conformal dimension as ∆n. Using thisboundary OPE, the two-point function is found from the contraction withan operator O3(~x)〈O4(~x, z)O3(~x′)〉 =C∆4∆3z∆4−∆3d2∆3(2.40)Here d2 is the square of the distance between the position of the insertedoperators d2 =| ~x− ~x′ |2 +z22.5      Minkowski vs black hole embeddingIn the AdS-Schwarzschild background, there are two types of possible em-beddings for the branes. The first one is the Minkowski embedding andcorresponds to branes that end outside of the horizon. Such a scenario ispossible if the D-brane worldvolume on the S5 collapses to zero before reach-ing the black hole. When electric gauge fields are considered in a Minkowskiembedding solution, at the point where the branes pinch-off there is nowherefor the field lines to go. However, we can include point charges in the formof strings stretching from the probe branes to the horizon which palliate222.5. Minkowski vs Black Hole Embeddingthe problem. In the probe D5-brane system discussed above Minkowski em-beddings arise when the filling fraction is set to zero. Then, the Routhian2.24 vanishes when the angle ψ(r) dynamically goes to zero. The probeD7-brane system also can have a Minkowski embedding but only when oneof the sphere has no magnetic flux. When f7 is zero, the same argumentapplies and the brane worldvolume vanishes at ψ(r) = 0 for filling fraction1 in this case. This solution corresponds to an integer Quantum Hall statewhere the conductivity recreates the famous plateaus of graphene.The second type of embedding is the Black Hole embedding which occurswhen the worldvolume does not pinch-off and the branes reach the blackhole. This is achieved in the D5 and in the D7 picture when the fillingfraction is not an integer. Then, there is a residual charge preventing theenergy to vanish and the brane has to reach the black hole horizon.Figure 2.2: Two different types of embeddings. The circle in the middlerepresents a black hole with its horizon at r = rh. The X-direction representscoordinates perpendicular to the D3-branes but parallel to the probe brane.(a) A black hole embedding. The probe brane extends in the radial directionuntil it reaches the horizon of the black hole (b) A Minkowski embedding.The probe brane pinches off at r0 > rh and does not reach the horizon.23Chapter 3Non-Abelian Probe BraneActionThe purpose of this chapter is to consider the full non-abelian action for D3and D5 probe branes in AdS5 × S5. In the previous chapter, we computeda simplified embedding of probe branes where the interactions between thebranes were neglected, hence, the symmetries of the stack of probe braneswas given by U(1)Nf . But the full picture is somewhat different. It can berealized by bringing together a number of parallel non-coincident branes.An open string with both ends on the same brane describes a U(1) masslessgauge field which accounts for the U(1)Nf symmetry discussed previously.On the other hand, an open string with both ends on different branes isstretched and thus massive. As we bring the branes closer to one another,the stretched strings become massless and the gauge symmetry is enhancedto U(Nf ).The first non-commutative BIon solution was discovered in [12] and com-puted the action of N0 flavour D0-branes in a colour D2-branes background.It was shown that by letting three transverse directions to be excited theD0-branes could grow a two-dimensional transverse fuzzy sphere. This con-figuration, when the number of flavours is infinite, is dual to a single flavourD2-brane wrapping an S2 with N0 units of Dirac flux. The two configura-tions are dual in the sense that their energies are the same as well as theirequilibrium radii. Once one examines the D0-branes configuration at finitevalues of N0, 1N20corrections arise when compared to the D2-brane treat-ment. The non-commutative solution describes well the core of the BIonwhile the D2 configuration depicts the spike of the BIon precisely.In the following sections, we wish to apply a similar treatment to dif-ferent embeddings. This will allow us to improve our knowledge of somecommutative setups of probe-branes by providing 1N2fcorrections for theseembeddings. First, we look at non-commutative D5-branes growing a trans-verse fuzzy two-sphere. This should be dual to the D7-probe configurationmentioned in section 2.2.3 when only a flux on the sphere transverse to the24Chapter 3. Non-Abelian Probe Brane ActionD5 is considered (f7 = 0). Secondly, we look at non-commutative flavourD3-branes blowing up on a D5-brane or a D7-brane by exciting whetherone or two transverse two-spheres. We dismiss the solution of D3-branesgrowing a fuzzy S4 that would be dual to the S4 embedding of a D7-branebecause of the tachyon instability mentioned earlier.3.1     D 5-D7Here, we enhance the solution of Section 2.2.2 to the full non-abelian solutionwhere the stack of D5-branes will grow a transverse BIon spike in the formof a fuzzy sphere. An infinite number of terms will appear in the actionand it will be necessary to make an expansion where the number of probeD5-branes is large. We will truncate the action to second order in 1N25. Thiswill allow us to find corrections to the D7-brane solution which we knowshould be the large N5 limit of the non-abelian solution.Non-abelian BI actionLet us start by remembering the non-abelian extension of the Born-Infeldaction (NBI):SNDBI = −Tp∫dp+1σSTr(e−Φ√−det(P [Eab + Eai(Q−1 − δ)ijEjb] + 2piα′Fab) det(Qij))(3.1)There are four transverse directions to the probe D5-branes and thusfour coordinates become fuzzy and are included in the matrix Qij = δij +i2piα′ [Xi, Xk]Ekj . The solution we are looking for is a stack of D5-branesgrowing a transverse two-sphere inside the S5 geometry. To achieve this, wewill make the ansatz where the coordinate z is proportional to the identitymatrix and does not possess a non-abelian structure. To exhibit the fuzzytwo-sphere geometry we go to Euclidean coordinates on the transverse S¯2X1 = cosψ sin θ˜ cos φ˜ (3.2a)X2 = cosψ sin θ˜ sin φ˜ (3.2b)X3 = cosψ cos θ˜ (3.2c)This provides us with a better understanding of the fuzzy coordinates. Thescalars are now seen as fuzzy distances interpreted as the transverse excita-tions of the brane. As mentioned above, the scalars are enhanced to matrices25Chapter 3. Non-Abelian Probe Brane Actionin the N5 = 2j + 1 irreducible representation1 of SU(2).X1 =2 cosψ√N25 − 1Lx (3.3a)X2 =2 cosψ√N25 − 1Ly (3.3b)X3 =2 cosψ√N25 − 1Lz (3.3c)cos2 ψ = X21 +X22 +X23 (3.3d)Here, the transverse directions have been normalized following the develop-ment of 1.4 to span a fuzzy sphere of radius cosψ. To preserve the SO(3)invariance on S¯2, we consider for ψ only functions of the radius coordinate,ψ = ψ(r). In that coordinate system, following 1.30 the non-abelian Taylorexpansion of the metric in the transverse directions becomes the unit matrix,Gij =√λα′δij (3.4)while the matrix Qij involving the commutators of Φi takes the formQij = Iij −4√λ cos2 ψ2pi(N25 − 1)ijkLk (3.5)This is a 3N5 × 3N5 matrix and the determinant is taken over the indiceson the sphere i = 1, 2, 3. Taking the determinant of this matrix results inan N5 ×N5 matrix which will later be traced on its U(N5) indices. To findthis determinant we use the identitydet(1 +A) = exp{Tr3×3(log(1 +A)} (3.6)This computation is detailed in the Appendix. The result is given as,detQij =(1 +16λ cos4 ψ4pi2(N25 − 1)2−i16λ3/2 cos6 ψ3(2pi)3(N25 − 1)2)IN×N (3.7)where the last term on the right-hand side is imaginary. This might seembothering at a first glance but this term comes from a product of three Limatrices and all odd powers of L matrices vanish under the symmetrized1A reducible representation could be decomposed as a sum of irreducible representa-tions and would correspond to a solution with parallel D7-branes where the number ofD7-branes would be one-to-one with the number of irreps in the decomposition26Chapter 3. Non-Abelian Probe Brane Actiontrace prescription. Upon tracing the U(N5) indices, the last term of 3.7 willdrop off and only the first two terms will contribute to the action.We also need the inverse matrix Q−1Q−1 − δ =4√λ cos2 ψ2pi(N25 − 1)ijkLk +16λ cos4 ψ(2pi)2(N25 − 1)2ilkljmLkLm + ... (3.8)from which we can evaluate the second term in the pullback of equation3.1. It appears that once again the odd powers of L matrices vanish underthe symmetrized trace prescription. The computation is detailed in theappendix and up to second order in 1N5 , we getP [Gai(Q−1 − δ)ijGjb] = −16λ sin2 ψ cos4 ψ ψ˙2(2pi)2(N25 − 1)2(3.9)We can finally write the term in the first determinant of 3.1,P [Gab +Gai(Q−1 − δ)ijGjb] + 2piα′Fab =√λα′−hr2 0 0 −a˙ 0 00 r2 b 0 0 00 −b r2 0 0 0a˙ 0 0 1hr2 + ψ˙2(1− 16λ sin2 ψ cos4 ψ(2pi)2(N25−1)2)0 00 0 0 0 sin2 ψ 00 0 0 0 0 sin2 ψ sin2 θ(3.10)Under the symmetrized trace prescription, the Lagrangian is propor-tional to the unit matrix and the full equation is multiplied by N5 = 2j+ 1.We are now in possession of the complete non-abelian Born-Infeld actionincluding corrections up to 1N25.SNBI5 = −2N5N5∫dr sin2 ψ√(1 +4λ cos4 ψ4pi2(N25 − 1))(r2 + b2)√(1 + hr2ψ˙2(1−16λ sin2 ψ cos4 ψ(2pi)2(N25 − 1)2)− a˙2)(3.11)We wish to remove the λ dependence and rewrite the action in terms of the27Chapter 3. Non-Abelian Probe Brane Actionparameter f5 =2piN5√λ.SNBI5 = −2N5N5f5∫dr sin2 ψ√(f25 + 4 cos4 ψ +4 cos4 ψN25 − 1)(r2 + b2)√(1 + hr2ψ˙2(1−16 sin2 ψ cos4(N25 − 1)f25)− a˙2)(3.12)By comparing this result to the D7-brane picture 2.28 we notice that bothactions agree at order 1N5 when we do not consider a flux on the first sphereS2, corresponding to f7 = 0. The non-abelian treatment provides a 1N25correction to the BI action as expected.Non-abelian CS actionFor the NCS term, the relevant RR potential is once again the C(4) (2.5)generated by the background D3-branes. In the abelian treatment, we foundno Chern-Simons term as the only possible term P [C(4)] ∧ F could not bepulled-back to the worldvolume indices of the D5-branes. The non-abeliandescription is more general and allows new interactions through the internalproduct introduced in 1.3.2 by stripping off pairs of indices of the RR po-tential. For the D5-branes in a C(4) background, there are generally threeterms that can arise from this procedure,SNCS5 = (2piα′)Tpgs∫dp+1σSTr(P[−12ı4xC(4)]∧ F ∧ F ∧ F+ iP[ı2xC(4)]∧ F ∧ F + P[C(4)]∧ F )(3.13)From our ansatz, F 3 = 0 as we did not put any gauge field on the wrappedsphere. Also, as the integral is taken on the D5-branes worldvolume, thelast term of 3.13 vanishes as for the abelian CS action. The only resultingcontribution comes from the second term of 3.13 by stripping off the twotransverse components from the RR potential via the internal product.To expose the SO(3) symmetry, we once again go to Cartesian coordinateson S¯2 where the S5 part of the Ramond-Ramond potential now takes theformC(4)ijθφ = λα′2 c(ψ)2 cos3 ψijk2xkdxi ∧ dxj ∧ dΩ2 (3.14)28Chapter 3. Non-Abelian Probe Brane ActionAs the RR potential is a function of the transverse fields, we follow the argu-ment discussed 1.30 and make a non-abelian Taylor expansion to computeits interior product. In this particular case, the expansion is trivial as thepotential depends linearly on the fields:ıxıxC(4)ijθφ = xjxixk∂kC(4)ijθφ(r)= λα′2 ic(ψ)√N25 − 1dΩ2(3.15)The gauge trace simply gives a factor of N5 as the interior product of theRR potential is proportional to the identity. The full Chern-Simons actionfor the configuration we are studying is:SNCS5 = −N5N5f52b√1− 1N25∫dr a˙ c(ψ) (3.16)It should be noted that in a large j expansion of the Chern-Simons actionthe prefactor depending on j is 2√1− 1N25∼ 2 + 1N25. When compared to theD7-brane model, the NCS action receives no corrections at first order in a1N5expansion.Routhian and equations of motionNow that we are in possession of a non-abelian action for the D5-branesconfiguration we wish to obtain the equations of motion for the dynamicalvariable ψ. To do so, we once again find the Routhian for this action througha Legendre transformation to remove the cyclic variable aRN5 = −N5N5f5(2piB√λ)3/2 ∫dr√1 + hr2ψ˙2(1−16 sin2 ψ cos4 ψ(N25 − 1)f25)√√√√4 sin2 ψ(1 + r4)(f25 + 4 cos4 ψ +4 cos4 ψ(N25 − 1)) + [piν +2√1− 1N25c(ψ)]2(3.17)29Chapter 3. Non-Abelian Probe Brane ActionApplying the Euler-Lagrange equations on the Routhian gives the equationsof motion for ψ0 =hA(ψ)(r ddr)2ψ1 + hr2A(ψ)ψ˙2+hr2 dA(ψ)dψ ψ˙22(1 + hr2A(ψ)ψ˙2+ hrA(ψ)ψ˙1 +8 sin4 ψr4(f25 + 4 cos4 ψ + 4 cos4 ψ(N25−1))4 sin2 ψ(1 + r4)(f25 + 4 cos4 ψ + 4 cos4 ψ(N25−1)) + [piν + 2√1− 1N25c(ψ)]2−8 sin3 ψ cosψf25 + 4 sin3 2ψ cos 2ψ(1 + r4) + 4√1− 1N25sin2 2ψ[piν + 2√1− 1N25c(ψ)]4 sin2 ψ(1 + r4)(f25 + 4 cos4 ψ + 4 cos4 ψ(N25−1)) + [piν + 2√1− 1N25c(ψ)]2(3.18)Where we have defined,A(ψ) = 1−16 sin2 ψ cos4 ψ(N25 − 1))f25dA(ψ)dψ=32 sinψ cosψ3(3 sin2−1)(N25 − 1)f25(3.19)It is also useful to compute the Hamiltonian for this system as this willgive us a tool to compare the energy of two different solutions. This can beobtained via a Legendre transformation:EN5 =N5N5f5(2piB√λ)3/2√4 sin2 ψ(1 + r4)(f25 + 4 cos4 ψ + 4 cos4 ψ(N25−1)) + [piν + 2N5√(N25−1)c(ψ)]2√1 + hr2(1− 16 sin2 ψ cos4 ψ(N25−1)f25)ψ˙2(3.20)Let us note that the action and Routhian we found from the non-abelianD5 picture is in agreement with the D7 equations found in section 2.2.3where only f¯7 is turned on and f7 = 0 as we did not include a flux on theS2 wrapped by the D5-branes. Turning on the flux on the sphere has veryimportant physical differences with the present case. When a flux is turnedon, the energy of the probes does not vanish anymore for the angle ψ = 0which prevents the sphere from pinching off inside the bulk and prohibitsMinkowski embeddings. This type of solutions can instead be portrayed by303.1. D3-D5D3-probe branes blowing up to a higher dimensional (D5 or D7) brane withflux. We will investigate the action for this phenomenon in the followingsection.3.2     D3-D5We now investigate the non-abelian action for a stack of coincident D3 branesblowing up to a D5-brane on a fuzzy two-sphere. In the dual picture, theD5-brane contains N3 units of Dirac monopole flux on the two-sphere. Theidea is similar to the D5-D7 system and we shall quickly go over the detailsto avoid redundancy.3.2.1     D3The NBI action is computed in a similar way. In this case, the transversetwo-sphere which becomes fuzzy is taken to be the one with radius sinψ asit is the one wrapped by the D5-brane. Following the same derivation, wefind,SNBI3 = −N3N3∫dr√(r4 + b2)(1 +4λ sin4 ψ4pi2(N23 − 1))√1 + hr2ψ˙2(1−16λ sin4 ψ cos2 ψ(2pi)2(N23 − 1)2)− a˙2(3.21)while the Chern-Simons action vanishes. Indeed, the RR potential C(4)θφθ¯φ¯commutes on its two indices θ¯,φ¯ and one cannot strip off all of its indiceswith the internal product.The full action is given by the NBI term only and following the usualprocedure we find the Routhian and equations of motion for ψ.R3 = −N3N3(2piB√λ)3/2 ∫dr√1 + hr2ψ˙2(1−16 sin4 ψ cos2 ψ(N23 − 1)f2)√(1 + r4)(1 +4λ sin4 ψ4pi2(N23 − 1))+(q3b)2(3.22)where as usual, q3 is the conserved quantity related to the cyclic variable a˙,q3 =1N3N3∂S3∂a˙=a˙√(1 + r4)(1 + 4λ sin4 ψ4pi2(N23−1))√1 + hr2ψ˙2(1− 16λ sin4 ψ cos2 ψ(2pi)2(N23−1)2)− a˙2(3.23)313.1. D3-D53.2.2      D5 with fluxWe now look at the commuting D5-brane system dual to the non-commutativeD3-brane configuration above. Adding a Dirac monopole flux on the wrappedsphere, the worldvolume gauge field strength on the sphere is written as,2piα′Fθφ =f2sin θdθ ∧ dφ (3.24)where the factor of 2 is for later convenience. The action is fully determinedby the BI part,R5 = −N5(2piB√λ)3/2 ∫dr√(f2 + 4 sin4 ψ)(1 + r4) +(q5b)2√1 + hr2ψ˙2(3.25)To make the contact between the two systems, we notice thatN3N3N5=2piN3√λ(3.26)and that the values of the charges can be rewritten in terms of the fillingfraction,q3b=piνf3, f3 =4pi2N3λ(3.27a)q5b=piνf5, f5 =2pi√λ(3.27b)and both actions can be written is a similar way,R3 = −N3N3f3(2piB√λ)3/2 ∫dr√1 + hr2ψ˙2(1−16 sin4 ψ cos2 ψ(N23 − 1)f2)√4pi2λ(1 + r4)(λf234pi2+ 4 sin4 ψ +4 sin4 ψ(N23 − 1))+ (piν)2(3.28)R5 = −N3N3f3(2piB√λ)3/2 ∫dr√4pi2λ(f2 + 4 sin4 ψ)(1 + r4) + (piν)2√1 + hr2ψ˙2(3.29)Our results agree up to order 1N23as long as the flux on the sphere isnormalized appropriately,f =√λ2pif3 (3.30)323.2. D3-D73.3      D3-D7This time we will look at a system where instead of growing a transversefuzzy sphere, the D3-branes are excited in four transverse directions andthe Myers effect happens on both fuzzy two-spheres at once, one of radiussinψ and the other of radius cosψ. In the general case, the number ofbranes blowing up on each transverse two-sphere can be different. This willbe accounted for by introducing a quantity N3 and N¯3 which respectivelycorrespond to the number of D3-branes blowing up on the fuzzy S2 and S¯2.In the dual D7 picture, taking N3 6= N¯3 corresponds to having two differentvalues of flux on each sphere which is translated as f7 6= f¯73.3.1     D3The non-abelian Born-Infeld action is only modified from the previous sys-tems through the Qij matrix. Once both spheres are written in Cartesiancoordinates,X1 = sinψ sin θ cosφ X¯4 = cosψ sin θ˜ cos φ˜X2 = sinψ sin θ sinφ X¯5 = cosψ sin θ˜ sin φ˜X3 = sinψ cos θ X¯6 = cosψ cos θ˜(3.31)Qij becomes a (3N3 + 3N¯3)× (3N3 + 3N¯3) block-diagonal matrix as the Xicommute with the X¯i while each group has non-trivial commutation relationwithin each themselves.This time, the symmetrized trace picks up two factors, N3 and N¯3 ac-counting for all possible orderings of the matrices X and X¯. The non-abelianBI action then take the form,SNBI3 = −N3N3N¯3∫dr√(r4 + b2)(1 +4λ sin4 ψ4pi2(N23 − 1))(1 +4λ cos4 ψ4pi2(N¯23 − 1))√1 + hr2ψ˙2(1−16λ sin4 ψ cos2 ψ4pi2(N23 − 1)2−16λ sin2 ψ cos4 ψ4pi2(N¯23 − 1)2)− a˙2(3.32)In this system, there is also a Chern-Simons contribution to the actionwhich arises by taking off all indices of C(4) via the internal product.SNCS3 = −T3gs∫d4σ Tr (2piα′)2P [ı4C(4)] ∧ F ∧ F (3.33)333.2. D3-D7Here, the internal product is easily taken as there are two distinct contrac-tions,ı4C(4) = [Φi,Φj ][Φ¯k¯, Φ¯l¯]C(4)ijk¯l¯(3.34)finally, the CS action is written as followsSNCS3 = −N3N3N¯3f32b2√(1− 1N23)(1− 1N¯23)∫dr a˙ c(ψ) (3.35)with f3 picking up the new term N¯3,f3 =4pi2N3N¯3λ(3.36)Following the same procedure as usual, we find the Routhian,R3 = −N3N3N¯3f3′(2piB√λ)3/2 ∫dr√1 + hr2ψ˙2(1−16λ sin4 ψ cos2 ψ4pi2(N23 − 1)2−16λ sin2 ψ cos4 ψ4pi2(N¯23 − 1)2)√(1 + r4)(λf234pi2N¯23+ 4(1 +1N23 − 1)sin4 ψ)(λf234pi2N23+ 4(1 +1N¯23 − 1)cos4 ψ)+ ξ233¯(3.37)withξ33¯ = piν +2√(1− 1N23)(1− 1N¯23)c(ψ) (3.38)The Routhian agrees up to 1N23or 1N¯23corrections with the D7 probe braneaction introduced in section 2.2.3 as long as the flux on the spheres in theD7 picture satisfiesf27 =λf234pi2N¯23, f¯27 =λf234pi2N23(3.39)We notice that in the special case f7 = f¯7, we getf3 =λf234pi2N¯23=λf234pi2N23(3.40a)f3 = f27 = f¯27 (3.40b)Let us finish this section by noting that the non-abelian action for ourconfiguration of D3-branes is not trivial as was its commuting counterpart.While the abelian action 2.2.1 had no dynamical degrees of freedom, thenon-abelian action possesses interesting dynamics for the variable ψ(r). Thenon-commutative solutions we analysed correspond to a single D5-brane orto a D7-brane depending if we allowed one or two two-spheres to be excited.34Chapter 4ConductivitiesHaving found an action and equations of motion for various non-abelianprobe branes systems, we are now able to compute interesting quantities forthese configurations. In this section, we find the longitudinal and transverseconductivity for the non-abelian D3-D5 system which will be the correctedvalues of the D3-D7 system for a non-infinite number of flavors.4.1 Minkowski embeddingsWe advertised the D3-D5 configuration possesses Minkowski embeddingswhich are Quantum Hall states for particular values of the filling fractionν. Such a claim needs to be supported by a computation of longitudinaland transverse conductivity as a QH state is one of charged matter in 2+1dimensions with null longitudinal conductivity and quantized transverse con-ductivity. The correct procedure was elaborated for the D3-D7 scenario in[1] and we will use the same method here to find the conductivities.Our goal is to find the associated conserved currents to the gauge fields〈J i〉 = σijEj (4.1)in the presence of a constant background electric field. We thus need addsome extra gauge fields which will yield the needed conserved currents,Ax(r, t) =√λ2pi(et+ ax(r)) , Ay(r, x) =√λ2pi(bx+ ay(r)) (4.2)These additional gauge fields introduce modifications to the action thatare readily computed. For the D3-D5 system, these are,SBI5 = −N5N5f5∫dr√4 sin4 ψ(f25 + 4 cos4 ψ +4 cos4 ψN25 − 1)√(r4 + b2 −e2h)(1 + hr2ψ˙2)+ (hr4 − e2) a˙2y + hr4a˙2x − (r4 + b2) a˙20 − 2bea˙0a˙y(4.3)354.1. Minkowski embeddingsSCS5 = −N5N5f52√1− 1N25∫dr (ba˙0 + ea˙y) c(ψ) (4.4)from which we find the equations of motion for the gauge fields,d = γ((r4 + b2)a˙0 + bea˙y)−2√1− 1N25bc(ψ) (4.5a)jx = −γhr4a˙x (4.5b)jy = −γ((hr4 − e2)a˙y − bea˙0)−2√1− 1N25bc(ψ) (4.5c)Where γ is defined for convenience as,γ =√√√√√4 sin4 ψ(f25 + 4 cos4 ψ + 4 cos4 ψN25−1)(r4 + b2 − e2h)(1 + hr2ψ˙2)+ (hr4 − e2) a˙2y + hr4a˙2x − (r4 + b2) a˙20 − 2bea˙0a˙y(4.6)The equations for a˙0 and a˙y are coupled but can be solved for each variableindependently,a˙0 =d˜(hr4 − e2)+ bej˜yγr4 [h (r4 + b2)− e2](4.7a)a˙y =bed˜−(r4 + b2)j˜yγr4 [h (r4 + b2)− e2](4.7b)where a tilde means that we added the term 2√1− 1N25bc(ψ). For example,d˜ = γ((r4 + b2)a˙0 + bea˙y). We can now put these back in our equation forγ and get,γ =√h(1 + b2r4 −e2hr4)(hr4Γ + hd˜2 − j2x − j˜2y)−(hbd˜− ej˜y)2(1 + b2r4 −e2hr4)√1 + hr2ψ˙2(4.8)where the factor of Γ was defined to beΓ = 4 sin4 ψ(f25 + 4 cos4 ψ +4 cos4 ψN25 − 1)(4.9)364.2. BH embeddingsFor the solution to be regular, when the branes pinch off the currentsneed to vanish: d˜(r0) = j˜y(r0) = jx = 0. From this, the longitudinal currentvanishes directly,σxx =JxEx= 0 (4.10)while the transverse conductivity yields,σxy =JyEx=ν2pi(4.11)Let us also remember that these Minkowski embeddings are found when thefilling fraction ν is ν = 1√1− 1N25≈ 1 + 12N25. In this case, the conductivityis quantized in units of 1√1− 1N25instead of being integer. This is indeed avery surprising result as it postulates that for any finite number of probeD5-branes, the plateaus for the conductivity at strong coupling are slightlyseparated from their weak coupling counterpart which are found for eachinteger values of ν.4.2 BH embeddingsWhen in the presence of a Black Hole Embedding, the probe branes fallinside the horizon and there is no gap between them and the backgroundD3-branes. This represents a metallic state and we should expect that bothlongitudinal and transverse conductivity be non-zero.Asking that the action should be real, the radicand in the numerator of(4.8) needs to be positive. At the specific value r = r∗,h(r∗)(r4∗ + b2) = e2 (4.12)the first term of the radicand vanishes. For the radicand to be positive, thelast term needs to cancel out at that point. This corresponds to the relation,j˜y(r∗) =1eh(r∗)bd˜(r∗) (4.13)To find a second relation is a bit trickier. An argument detailed in [13]tells us that the second piece of the first term of the radicand also vanishesat r = r∗. This can be seen from the fact that the second term is negativeat the horizon and positive at the boundary. Therefore it possesses a zero.But if that zero does not coincide with the zero of the first term, then in the374.2. BH embeddingsregion between the two zeros, the radicand is negative and the action doesnot satisfy the reality condition. This allows us to write our second relation,j2x + j˜y(r∗)2 = h(r∗)r4∗Γ + h(r∗)d˜(r∗)2 (4.14)These equations would be very hard to solve exactly but the conductivi-ties are only given by the approximation of the linear response. This meansthat we can consider the electric field e as a small perturbation and expandthe solution to first order. When e = 0, equation (4.12) tells us that r∗ = rh.Taking e to be non-zero, we are looking for a solution in the vicinity of theblack hole radius of the type r∗ = rh(1 +x). Solving the quadratic equationfor r4∗ by neglecting any e4 term, yields the solution,r4∗ = r4h(1 +e2r4h + b2)(4.15)Replacing this value of r∗ in our equations for the conductivities (4.13),(4.14),σxy =JyEx=N32pi2br4h + b2d˜(rh) +2√1− 1N25bc(rh) (4.16a)σxx =JxEx=N32pi2r2hb2 + r4h√d˜(rh)2 + (r4h + b2)Γ(rh) (4.16b)Both longitudinal and transverse conductivities are non-zero for theblack-hole embedding as expected for a metallic state. Once again, thedifference between the conductivities for the non-abelian D5-branes systemonly differs from the D7-brane system by a factor of 1√1− 1N25term.38Chapter 5ConclusionIn this work, we evaluated the action for a stack of coincident probe flavorD5-branes embedded in AdS5×S5 geometry. We chose an S2×S2 embeddingon the S5 and took into account the non-abelian U(Nf ) effects. We founda spherically symmetric solution to the system where the D5-branes growa transverse fuzzy two-sphere spike. This solution, in the large number offlavors limit, is dual via Myers effect to a D7-brane formulation where asingle D7-brane wraps the two-sphere transverse to the D5-branes with NFunits of RR flux on that sphere. The non-abelian D5-branes picture allowsus to look for 1Nf corrections to the D7 setup. We find that the initial probeD7-brane brane action does not receive any correction to order N−1f . Theleading contribution to the difference between both setups is introduced atorder N−2f .We computed conductivities for the non-abelian solution and found thatthey deviate from the usual integer values by a factor of Nf√C= 1√1− 1N2fwhich depends strictly on the number of flavor branes probing the geome-try. This result is the direct consequence of the change in the non-abelianChern-Simons action. We wish to emphasize that this result is particularlysurprising as CS actions are always quantized in integer units. We can pointout that a similar result for the non-abelian CS action was obtained in [3]but from the point of view of D1-branes probing the geometry. One canargue that such a result is not physical. Indeed, in the weak coupling limit,the value of the transverse conductivity is integer and a no-renormalizationtheorem states that this results holds at any level in perturbation theory.Thus, the conductivity at strong coupling should also take integer valueswhich disagrees with the non-abelian probe branes calculation.It was suggested to us2 that the change in the CS action might be can-celled by including fermions to the model. The Born-Infeld and Chern-Simons actions only take bosonic degrees of freedom into account and itwould be possible that a fermionic one loop computation gives a factor that2We are grateful to Rob Myers for pointing out this idea to us39Chapter 5. Conclusionwould correct the CS action back to an integer value. We will leave theresolution of this question for a further work and remember that the resultsfound here need to be taken with a grain of salt.40Bibliography[1] Oren Bergman, Niko Jokela, Gilad Lifschytz, and Matthew Lippert.Quantum Hall Effect in a Holographic Model. JHEP, 1010:063, 2010.[2] Curtis G. Callan and Juan Martin Maldacena. Brane death and dy-namics from the Born-Infeld action. Nucl.Phys., B513:198–212, 1998.[3] Neil R. Constable, Robert C. Myers, and Oyvind Tafjord. The Non-commutative bion core. Phys.Rev., D61:106009, 2000.[4] Nick Evans, Astrid Gebauer, Keun-Young Kim, and Maria Magou.Phase diagram of the D3/D5 system in a magnetic field and a BKTtransition. Phys.Lett., B698:91–95, 2011.[5] Michael B. Green, J.H. Schwarz, and Edward Witten. SUPERSTRINGTHEORY. VOL. 2: LOOP AMPLITUDES, ANOMALIES AND PHE-NOMENOLOGY. 1987.[6] Gary T. Horowitz and Andrew Strominger. Black strings and P-branes.Nucl.Phys., B360:197–209, 1991.[7] Andreas Karch and Lisa Randall. Open and closed string interpretationof SUSY CFT’s on branes with boundaries. JHEP, 0106:063, 2001.[8] C. Kristjansen, R. Pourhasan, and G.W. Semenoff. A HolographicQuantum Hall Ferromagnet. JHEP, 1402:097, 2014.[9] Charlotte Kristjansen and Gordon W. Semenoff. Giant D5 Brane Holo-graphic Hall State. JHEP, 1306:048, 2013.[10] Juan Martin Maldacena. The Large N limit of superconformal fieldtheories and supergravity. Adv.Theor.Math.Phys., 2:231–252, 1998.[11] D.M. McAvity and H. Osborn. Conformal field theories near a boundaryin general dimensions. Nucl.Phys., B455:522–576, 1995.[12] Robert C. Myers. Dielectric branes. JHEP, 9912:022, 1999.41[13] Andy O’Bannon. Hall Conductivity of Flavor Fields from AdS/CFT.Phys.Rev., D76:086007, 2007.[14] Joseph Polchinski. Dirichlet Branes and Ramond-Ramond charges.Phys.Rev.Lett., 75:4724–4727, 1995.[15] Soo-Jong Rey. String theory on thin semiconductors: Holographic real-ization of Fermi points and surfaces. Prog.Theor.Phys.Suppl., 177:128–142, 2009.[16] Tadakatsu Sakai and Shigeki Sugimoto. Low energy hadron physics inholographic QCD. Prog.Theor.Phys., 113:843–882, 2005.[17] Washington Taylor and Mark Van Raamsdonk. Multiple D0-branes inweakly curved backgrounds. Nucl.Phys., B558:63–95, 1999.[18] Arkady A. Tseytlin. Born-Infeld action, supersymmetry and stringtheory. 1999.42Appendix AA.1     Trace-log expansionIn this section we will compute the determinant of the matrix Qij mentionedin 3.7.Let us start withQij = 1−4√λ cos2 ψ2pi(N25 − 1)ijkLk (A.1)Which we will write for later convenience asQ = 1 +A (A.2)We use the identity:det(1 +A) = expTr log(1 +A) (A.3)And replace the log by its Taylor expansionTr log(1 +A) =∞∑k=1(−1)kTrAkk(A.4)The problem is now simplified to finding the trace of powers of the matrixAij = −4√λ cos2 ψ2pi(N25 − 1)ijkLk = βijkLk (A.5)Where we absorbed all the coefficients into the constant β. The matrix Ais traceless so the first term will come from the square of AA2 = β2iklkjmLlLm= β2(−δijδlm + δimδlj)LlLm= β2(−δijL2 + LjLi)(A.6)With traceTrA2 = −2β2(N25 − 1)4= −4λ cos4 ψ2pi2(N25 − 1)(A.7)43A.1. Trace-Log ExpansionThe third power of A is computed in the same fashionA3 = β3iklkmnmjoLlLnLo= β3(−δimδln + δinδlm)mjoLlLnLo= β3(−ijoL2Lo + ljoLlLiLo)= β3(−ijoL2Lo + ljoLiLlLo + ljo[Ll, Li]Lo)= β3(−ijoL2Lo +ljo2Li[Ll, Lo] + iljolikLkLo)= β3(−ijoL2Lo +i2ljolokLiLk + i(δijδok − δoiδjkLkLo))= β3(−ijoL2Lo − iLiLj + iδijL2 − iLjLi)= β3(ijk(1− L2)Lk + iδijL2 − 2iLjLi)(A.8)The cube of A is now expressed in a simple form to take its trace. By doingso, the first term on the rhs vanishes as it is antisymmetric in i, j. We areleft withTrA3 =iβ3(N25 − 1)4= −i16λ3/2 cos6 ψ(2pi)3(N25 − 1)2(A.9)The fact that the result is imaginary could seem disturbing at first but weremind the reader that the action is defined by the symmetrized trace of thedeterminants. Under the symmetrisation prescription, all odd powers of Acancel out in the expansion leaving us only with even powers of A which areall real valued. This can easily be seen by inverting, for example, L1 and L2in A.8 providing an overall minus sign from the commutators.There is no need in computing the expansion of the logarithm any furtheras the determinant is taken only over the three indices i = 1, 2, 3. Therefore,the result can be non-zero only up to third power in A. Let us put the termstogetherTr log(1 +A) =β2(N25 − 1)4+iβ3(N25 − 1)12+ ... (A.10)The determinant we are looking for is found by exponentiating the result ofA.10det(1 +A) = 1 +β2(N25 − 1)4+iβ3(N25 − 1)12(A.11)After taking the symmetrized trace, the only relevant terms aredet(1 +A) = 1 +β2(N25 − 1)4(A.12)44A.1. Trace-Log ExpansionAnd using the definition f = 2piN5√λ∼ 1det(Q) = 1 +4 cos4 ψf2+4 cos4 ψ(N25 − 1)f2(A.13)45A.2. Pullback ComputationA.2      Pullback computationWe present here the details of the computation of the quantity P [Gai(Q−1 − δ)ijGjb]needed in the non-abelian Born-Infeld action. First of all, we recall that(Q−1 − δ)ij = β ijkLk + β2 ilkljmLkLm + ... (A.14)where β = 4√λ cos2 ψ2pi(N25−1).As the metric is diagonal and the transverse directions depend only on theradial coordinate r, the lone term contributing in the pullback isP [Gri(Q−1 − δ)ijGjr] = ∂rXkGkiGmj (Q−1 − δ)im ∂rXlGjl=4√λα′ sin2 ψ ψ˙2N25 − 1Li(Q−1 − δ)ij Lj(A.15)where in the second line we have used Gij =√λα′δij and Xi =cosψ√j(j+1)Li.The first term in the expansion is a product of three matrices,βijkLiLkLj = βijk2[Li, Lk]Lj= iβijkikl2LlLj= −iβL2(A.16)Once again, the same argument allows us to dismiss this term as it vanishesunder the symmetrized trace prescription.The next term is even in powers of L and will therefore contribute tothe action,β2ilkljmLiLkLmLj = β2 (−δijδkm + δimδjk)LiLkLmLj= β2(LiLjLiLj − LiLjLjLi)= β2LiLj [Li, Lj ]= iβ2ijkLiLjLk= iβ2ijk2Li[Lj , Lk]= −β2ijkjkl2LiLl= −β2L2 = −β2(N25 − 1)4(A.17)46A.2. Pullback ComputationThis term is the only relevant term up to order 1N25. To verify this assertion,let us compute the following term in the expansion. The next even term inpowers of L is a product of six matrices,β4iklkmnmopojqLiLlLnLpLqLj = β4 (−δimδln + δinδlm) (−δmjδpq + δmqδpj)LiLlLnLpLqLj= β4(L2L2L2 − 2L2LiLjLiLj + LiLkLiLjLkLj)= β4(−L2Li[Lj , Li]Lj + Li[Lk, Li]LjLkLj)= iβ4(−jikL2LiLkLj + kilLiLlLjLkLj)= iβ4(−jik2L2Li[Lk, Lj ] +kil2[Li, Ll]LjLkLj)= β4(jikkjl2L2LiLl −kililm2LmLjLkLj)= β4(L2L2 − LkLjLkLj)= −β4 Lk[Lj , Lk]Lj= −iβ4 jkiLkLiLj= β4jkikil2LlLj= β4 L2 =β4(N25 − 1)4(A.18)We notice that the factor β scales as 1N5 and therefore the first term A.17in the expansion is already a 1N25correction to the action while the secondterm A.18 scales as 1N45which we can neglect.Putting everything together, we obtain,P [Gri(Q−1 − δ)ijGjr] = −√λα′ sin2 ψ ψ˙216λ2 cos4 ψ(2pi)2(N25 − 1)2∼ −16√λα′ sin2 ψ cos4 ψ ψ˙2f2(N25 − 1)(A.19)47A.3. Conformal DimensionsA.3      Conformal dimensionsTo understand the asymptotic behavior of ψ(r), we expand the action tolinear order around the boundary value of ψ. We remind the reader thatthe potential for the non-abelian D3-D5 system is,V (ψ) =√√√√√4 sin4 ψ (r4 + b2)(f25 + 4 cos4 ψ +4 cos4 ψN25 − 1)+piν −2√1− 1N25bc(ψ)2(A.20)Calling ψ(r) = ψ∞ + φ(r) with φ(r) a small deviation from the value atinfinity scaling as φ(r) ∼ r∆, we expand the Routhian in the near-boundaryregion,R5 ∼ −N5N5f5((1 +12hr2φ˙2(r))V (r) |ψ∞ +φ(r)∂V (r)∂ψ|ψ∞ +φ2(r)2∂2V (r)∂ψ2|ψ∞)(A.21)Applying the Euler-Lagrange equation to the Routhian yields the equationof motionddr(hr2φ˙V (r))−∂V (r)∂ψ|ψ∞ −φ(r)∂2V (r)∂ψ2|ψ∞= 0 (A.22)Considering only the leading term, this is−∂V (r)∂ψ|ψ∞= 0 (A.23)which gives the condition,4 sin3 ψ∞ cosψ∞(f25 + 4 cos4 ψ∞ +4 cos4 ψN25 − 1− 4 sin2 ψ∞ cos2 ψ∞(1 +1N25 − 1))= 0(A.24)This equation possesses trivial solutions (ψ∞ = 0, pi2 ) and one complicatedsolution that depends on values of f5 and N5. We are interested in thesolution ψ∞ = pi2 for which the D5-branes wrap a maximal two-sphere atthe boundary.The first correction term to the equations of motion yields a quadraticequation for the power ∆,∆(∆ + 3)V (r) |ψ∞=∂2V (r)∂ψ2|ψ∞ (A.25)48A.3. Conformal DimensionsEvaluating the potential and its second derivative at ψ = pi2 gives usV (r) |ψ=pi2 = 2r2f5 ,∂2V (r)∂ψ2|ψ=pi2 = −4r2f5 (A.26)which gives the following relation for the conformal dimension,∆2 + 3∆ + 2 = 0 (A.27)We find two solutions,∆+ = −1 , ∆− = −2 (A.28)as claimed above. The conformal dimensions indicate the behavior of thesolution near the boundary. We can thus expand ψ for large values of r inthe following way,ψ(rmax) ∼pi2+mr+cr2(A.29)49


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