A Holographic Bound for D3-Brane
Davood Momeni, Mir Faizal, Aizhan Myrzakul, Sebastian Bahamonde, Ratbay Myrzakulov
aa r X i v : . [ h e p - t h ] M a y A Holographic Bound for D3-Brane
Davood Momeni ∗ , Mir Faizal , ,Aizhan Myrzakul ,Sebastian Bahamonde , Ratbay Myrzakulov Department of Physics, College of Science, Sultan Qaboos University ,P.O. Box 36, Alkhod 123, Oman Irving K. Barber School of Arts and Sciences,University of British Columbia - Okanagan, 3333 University Way,Kelowna, British Columbia V1V 1V7, Canada Department of Physics and Astronomy, University of Lethbridge,Lethbridge, Alberta, T1K 3M4, Canada Eurasian International Center for Theoretical Physicsand Department of General Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan Department of Mathematics, University College London,Gower Street, London, WC1E 6BT, UK
Abstract
In this paper, we will regularize the holographic entanglement entropy,holographic complexity and fidelity susceptibility for a configuration ofD3-branes. We will also study the regularization of the holographic com-plexity from action for a configuration of D3-branes. It will be demon-strated that for a spherical shell of D3-branes the regularized holographiccomplexity is always greater than or equal to than the regularized fidelitysusceptibility. Furthermore, we will also demonstrate that the regularizedholographic complexity is related to the regularized holographic entangle-ment entropy for this system. Thus, we will obtain a holographic boundinvolving regularized holographic complexity, regularized holographic en-tanglement entropy and regularized fidelity susceptibility of a configura-tion of D3-brane. We will also discuss a bound for regularized holographiccomplexity from action, for a D3-brane configuration.
In this paper, we will analyse the relation between the holographic complex-ity, holographic entanglement entropy and fidelity susceptibility for a sphericalshell of D3-branes. We shall also analyze the holographic complexity form ac-tion for a configuration of D3-branes. These quantities will be geometricallycalculated using the bulk geometry, and the results thus obtained will be usedto demonstrate the existence of a holographic bound for configurations of D3-branes. It may be noted that there is a close relation between the geometricconfiguration involving D3-branes and quantum informational systems [1]. It is ∗ Corresponding author T provides the microscopic string-theoreticinterpretation of the charges. The most general real three-qubit state can be pa-rameterized by four real numbers and an angle, and that the most general STUblack hole can be described by four D3-branes intersecting at an angle. Thus, itis possible to represent a three-qubit state by D3-branes. A system D3-braneshave been used to holographically analyse quantum Hall effect, as a system ofD3-D7-branes has been used to obtain the Hall conductivity and the topologicalentanglement entropy for quantum Hall effect [3]. The mutual information be-tween two spherical regions in N = 4 super-Yang-Mills theory dual to type IIBstring theory on AdS × S has been analysed using correlators of surface opera-tors [4]. Such a surface operator corresponds to having a D3-brane in AdS × S ending on the boundary along the prescribed surface. This construction relieson the strong analogies between the twist field operators used for the compu-tation of the entanglement entropy, and the disorder-like surface operators ingauge theories. A a configuration of D3-branes and D7-branes with a non-trivialworldvolume gauge field on the D7-branes has also been used to holographicallyanalyse new form of quantum liquid, with certain properties resembling a Fermiliquid [5] The holographic entanglement entropy of an infinite strip subsystemon the asymptotic AdS boundary has been used as a probe to study the ther-modynamic instabilities of planar R-charged black holes and their dual fieldtheories [6]. This was done using a spinning D3-branes with one non-vanishingangular momentum. It was demonstrated that the holographic entanglemententropy exhibits the thermodynamic instability associated with the divergenceof the specific heat. When the width of the strip was large enough, the finitepart of the holographic entanglement entropy as a function of the temperatureresembles the thermal entropy. However as the width became smaller, the twoentropies behave differently. It was also observed that below a critical value forthe width of the strip, the finite part of the holographic entanglement entropyas a function of the temperature develops a self-intersection.Thus, there is a well established relation between different D3-branes con-figurations and information theoretical processes. Thus, it would be interestingto analyze different information theoretical quantities for a configuration of D3-branes. It may be noted that entropy is one of the most important quantitiesin information theoretical processes. This is because entropy measures the loseof information during a process. It may be noted that maximum entropy of aregion of space scales with its area, and this observation has been motivatedfrom the physics of black holes. This observation has led to the development ofthe holographic principle [7, 8]. The holographic principle equates the degreesof freedom in a region of space to the degrees of freedom on the boundary sur-rounding that region of space. The AdS/CFT correspondence is a concrete real-izations of the holographic principle [9], and it relates the string theory in AdSto a superconformal field theory on the boundary of that AdS. The AdS/CFTcorrespondence in turn can be used to holographically obtain the entanglemententropy of a boundary field theory. The holographic entanglement entropy of aconformal field theory on the boundary of an AdS solution is dual to the areaof a minimal surface defined in the bulk. Thus, for a subsystem as A , we candefine γ A as the ( d − ∂A . Now using this subsystem, the holographic entanglement entropy2an be expressed as [12, 13] S A = A ( γ A )4 G d +1 (1)where G is the gravitational constant for the bulk AdS and A ( γ A ) is the areaof the minimal surface. Even though this quantity is divergence, it can be reg-ularized [14, 15]. The holographic entanglement entropy can be regularized bysubtracting the contribution of the background AdS spacetime from the defor-mation of the AdS spacetime. Thus, for the system studied in this paper, let A [ D γ A )] be the contribution of a D3-brane shell and A [ AdS ( γ A )] be con-tribution of the background AdS spacetime, then the regularized holographicentanglement entropy will be given by∆ S A = A [ D γ A )] − A [ AdS ( γ A )]4 G d +1 . (2)In this paper, we will use this regularized holographic entanglement entropy.The entropy measures the loss of information during a process. However, itis also important to know how easy is it for observer to extract this information.The complexity quantified this idea relating to difficulty to extract information.It is expected that complexity is another fundamental physical quantify, asit is an important quantity in information theory, and law of physics can berepresented in terms of informational theoretical processes. In fact, complexityhas been used in condensed matter systems [16, 17] and molecular physics [18,19]. Complexity is also important in the black hole physics, as it has beenproposed that even thought the information may not be ideally lost during theevaporation of a black hole, it would be effectively lost during the evaporationof a black hole. This is because it would become impossible to reconstruct itfrom the Hawking radiation [20]. It has been proposed that the complexity canbe obtained holographically as a quantity dual to a volume of codimension onetime slice in anti-de Sitter (AdS) [21, 22, 23, 24], Complexity = V πRG d +1 , (3)where R and V are the radius of the curvature and the volume in the AdS bulk.As it is possible to define the volume in different ways in the AdS, differentproposals for the complexity have been made. If this volume is defined to be themaximum volume in AdS which ends on the time slice at the AdS boundary, V = V (Σ max ), then the complexity corresponded to fidelity susceptibility χ F ofthe boundary conformal field theory [25]. This quantity diverges [26]. However,we will regularize it by subtracting the contribution of the background AdSspacetime from the contribution of the deformation of AdS spacetime. So,let V [ D max )] be the contribution of a D3-brane shell and V [ AdS (Σ max )] becontribution of the background AdS spacetime, then we can write the regularizedfidelity susceptibility as∆ χ F = V [ D max )] − V [ AdS (Σ max )]4 G d +1 . (4)It is also possible use a subsystem A (with its complement), to define a volumein AdS as V = V ( γ A ). This is the volume which is enclosed by the minimal3urface used to calculate the holographic entanglement entropy [27]. Thus,using V = V ( γ A ), we obtain the holographic complexity as C A . As we want todifferentiate between these two cases, we shall call this the quantity define by V = V (Σ max ) as fidelity susceptibility, and the quantity denied by V = V ( γ A )as holographic complexity. The holographic complexity diverges [26]. We willregularized it by subtracting the contributions of the background AdS from thedeformation of the AdS spacetime. Now if V [ D γ A )] is the contribution ofa D3-brane shell and V [ AdS ( γ A )] is the contribution of the background AdSspacetime, then we can write the regularized holographic complexity as∆ C A = V [ D γ A )] − V [ AdS ( γ A )]4 G d +1 . (5)It may be noted that there is a different proposal for calculating the holo-graphic complexity of a system using the action [28, 29]. According to thisproposal the holographic complexity of a system can be related to the bulkaction evaluated on the Wheeler-deWitt patch, C W = A ( W ) π ¯ h , (6)where A ( W ) is the action evaluated on the Wheeler-DeWitt patch W , with asuitable boundary time. To differentiate it from the holographic complexity cal-culated from volume C , we shall call this quantity ”holographic complexity fromaction”, and denote it by C W (as it has been calculated on a Wheeler-DeWittpatch). This quantity also diverges [30]. We shall regularize it by subtractingthe contributions of AdS spacetime from the contributions of the deformationof AdS spacetime. So, if A [ D W )] is the contribution of a D3-brane shell and A [ AdS ( W )] is the contribution of the background AdS spacetime, then we canwrite the regularized holographic complexity from action as∆ C W = A [ D W )] − A [ AdS ( W )] π ¯ h . (7)It may be noted that this proposal is very different from the other proposals tocalculate complexity of a boundary theory. This difference occurs as there aredifferences in the definition of complexity for a boundary field theory. So, thisproposal cannot be directly related to the proposals where the complexity canbe calculated from the volume of a geometry. In fact, it is possible to have thesame volume for two theories with different field content. In this paper, we willfirst use calculate a bound for the D3-brane geometries using the volume of ashell of D3-branes. Then we shall calculate a different holographic bound for aconfiguration of D3-branes using the action of this system.In this paper, we will analyze a specific configuration of D3-branes, anddiscuss the behavior of these regularized information theoretical quantities forit. It is possible to use static gauge, and write the bosonic part of the action forsuch a system in AdS × S background as [36] A = 12 πg s k Z (cid:18) √− h − q − det ( G µν + kF µν ) (cid:19) d x + χ π Z F ∧ F, (8)where k = p g s N/π and G µν = h µν + k ∂ µ φ I ∂ ν φ I φ . (9)4ere h µν = φ η µν , h = det h µν , with η µν being the four dimensional Minkowskimetric. Thus, we can write √− h = φ , where φ = P ( φ I ) , and φ I are six scalarfields corresponding to the six dimensions transverse to the D3-brane geometry.It may be noted that R F ∧ F term only contributes to the magnetically chargedconfigurations. The D3-brane can be placed at a fixed position on S , suchthat the five scalars fields corresponding to the S geometry will not have anycontribution. We shall consider the spherically symmetrical static solutions,centered at r = 0, for this geometry. So, the electric field ~E and the magneticfields ~B will only have radial components, which we shall denote by E and B .So, all fields of this system are only functions of the radial coordinate r , E ( r ), B ( r ), φ ( r ). Thus, we can write det( − G µν ) = φ G rr = φ [ φ + γ ( φ ′ /φ ) ] , and − det( G µν + γF µν ) = φ (cid:18) G rr − γ E φ (cid:19) (cid:18) γ B φ (cid:19) . (10)So, the Lagrangian density for this system can be written as L = 1 γ φ − s(cid:18) γ [( φ ′ ) − E ] φ (cid:19) (cid:18) γ B φ (cid:19) ! + g s χBE. (11)where γ = q N π = R √ T D , T D is D3-brane tension. There are two BPSsolutions for this geometry, φ ± = µ ± Q/r.
The probe D3-brane solution dis-cussed here describes a BIon like spike (either up to the AdS boundary ordown to the Poincare horizon, depending on the sign in φ ± ). This solutionalso breaks the translational symmetry in the field theory, and preserves therotational invariance.It is also possible to analyze a probe D3-brane with Q = 0 , E = 0 , and B = 0. Now we will analyze such a specific solution representing a D3-braneconfiguration, and analyze these quantities for that specific geometric configu-ration. It is possible to study such a D3-brane shell. The metric for the nearhorizon geometry of D3-brane shell is given by [31] ds = R z h ( z ) (cid:16) X µ =0 dx µ dx µ (cid:17) + R h ( z ) (cid:16) dz z + d Ω (cid:17) (12)where the function h ( z ) is defined as h ( z ) = (cid:26) , z ≤ z ( z z ) , z ≥ z . (13)For this geometry, the entangled region is a strip with width ℓ in the D3-braneshell defined by the embedding A = { x = x ( z ) , t = 0 } . The area functional canbe expressed as A ( γ A ) = 2 π R L Z z ∗ h ( z ) p x ′ ( z ) + h ( z ) z dz , (14)where x ′ ( z ∗ ) = ∞ . The Euler-Lagrange equation for x ( z ) has the following form x ′ ( z ) p x ′ ( z ) + h ( z ) = h ( z ∗ ) h ( z ) (cid:16) zz ∗ (cid:17) (15)5he total length can be obtained by ℓ = 2 Z z ∗ dzh ( z ) h h ( z ∗ ) h ( z ) (cid:0) zz ∗ (cid:1) q − (cid:0) h ( z ∗ ) h ( z ) (cid:0) zz ∗ (cid:1) (cid:1) i / . (16)We can also write the volume V ( γ A ) as V ( γ A ) = 2 π R L Z z ∗ h ( z ) / z x ( z ) dz . (17)We can solve Eq. (15) exactly, and obtain x ( z ) = C + R h ( z ∗ ) z √ − h ( z ∗ ) z + z dz , z ≤ z C + R h ( z ∗ ) z z √ − z h ( z ∗ ) + z dz , z ≥ z , (18)where, C and C are integration constants. The maximal volume, which isrelated to the fidelity susceptibility, is given by V (Σ max ) = 2 π R L Z z ∞ h ( z ) / z .dz (19)Now we will use h ( z ), and split the integral into two parts: R z ∞ = R z + R z ∞ z ,to obtain V (Σ max ) = − π R L z + z ∞ z ∞ z . (20)It may be noted that by setting C = C = L , the difference of the volumes(17) and (20), is given by V ( γ A ) − V (Σ max ) = R z h ( z ∗ ) z √ − h ( z ∗ ) z + z dz , z ≤ z R z ∞ z h ( z ∗ ) z z √ − z h ( z ∗ ) + z dz , z ≥ z . (21)Since h ( z ∗ ) >