Holographic BCFT with Dirichlet Boundary Condition

Neumann boundary condition plays an important role in the initial proposal of holographic dual of boundary conformal field theory, which has yield many interesting results and passed several non-trivial tests. In this paper, we show that Dirichlet boundary condition works as well as Neumann boundary condition. For instance, it includes AdS solution and obeys the g-theorem or c-like theorem. Furthermore, it can produce the correct expression of one point function, the boundary Weyl anomaly and the universal relations between them. We also study the relative boundary condition for gauge fields, which is the counterpart of Dirichlet boundary condition for gravitational fields. Interestingly, the four-dimensional Reissner-Nordstrom black hole with magnetic charge is an exact solution to relative boundary condition under some conditions. This holographic model predicts that a constant magnetic field in the bulk can induce a constant current on the boundary in three dimensions. We suggest to measure this interesting boundary current in materials such as the graphene.


Introduction
BCFT is a conformal field theory defined on a manifold M with a boundary P where suitable boundary conditions (BC) are imposed [1,2]. It has important applications in quantum field theory, string theory and condensed matter physics. For interesting developments of BCFT and related topics please see [3,4,5,6,7,8,9,10,11,12,13,14,15]. In the spirit of AdS/CFT [16], Takayanagi [17] proposes to extend the d dimensional manifold M to a d + 1 dimensional asymptotically AdS space N so that ∂N = M ∪Q, where Q is a d dimensional manifold which satisfies ∂Q = ∂M = P . See figure 1 for the geometry. We remark that, unlike M , the bulk boundary Q is not the conformal boundary of the manifold N . In particular, it is located at finite position instead of infinity. A central issue in the construction of the AdS/BCFT is the determination of the location of Q in the bulk. [17] propose to use the Neumann boundary condition to fix the position of Q. The holographic BCFT with Neumann BC has produced many elegant results and passed several non-trivial tests [18,19]. In particular, it obeys the M Q Figure 1: Geometry of holographic BCFT universal relations between Casimir effects and Weyl anomaly [5].
In general, there are more than one consistent BCs for a theory. For example, both Dirichlet BC and Neumann (Robin) BC are well-defined for scalars [20] Dirichlet BC : φ| ∂N = 0, Neumann (Robin) BC : (D n + ψ)φ| ∂N = 0. (1) Similarly, absolute BC works as well as relative BC for Maxwell fields Absolute BC : F na | ∂N = 0, For gravity, there are also several possible choices of BCs such as Dirichlet BC [21,22], Neumann BC and conformal BC [22,23,24]. As we have mentioned above, the Neumann BC plays an important role in the construction of holographic BCFT [17]. As a key characteristic, the BC for holographic BCFT should not only select the solutions to Einstein equations but also determine the location of bulk boundary Q. In other words, the position of bulk boundary Q cannot be freely choosen but determined by the BC of gravity. This is indeed the case for Neumann BC [17].
A natural question is that, "Can Dirichlet BC do the same job for holographic BCFT?". Recall that usually one imposes Dirichlet BC on the AdS boundary M in AdS/CFT. Thus it is natural to impose Dirichlet BC on the bulk boundary Q too. The key problem one needs to clarify is that, could Dirichlet BC fix the bulk boundary Q and yield consistent results for holographic BCFT? In this paper, we give a positive answer to this question. We find that AdS is a vacuum solution to Dirichlet BC and the g-theorem or c-like theorem is obeyed by this theory. What is more, Dirichlet BC can also produce the correct forms of one point funciton, the boundary Weyl anomaly and the universal relations between them. It is remarkable that the central charge for Dirichlet BC is less than or equal to that for Neumann BC. We also study Dirichlet-like BC for gauge fields, the relative BC, and find an exact solution in four dimensions. Interestingly, the holographic theory predicts that there is a constant current on the boundary, when a constant magnetic field is applied in the bulk. And the boundary current gains the maximum value at zero temperature.
The paper is organized as follows. In section 2, to warm up, we study the BCs of gauge fields in holographic BCFT. We show that both the absolute BC and the relative BC work well for gauge fields. In section 3, we generalize our discussions to gravitational fields and show that, similar to Neumann BC, Dirichlet BC can also produce the correct one point function of stress tensor and universal relations between Casimir effects and Weyl anomaly. In section 4, we study the properties of Casimir coefficients and central charges for different BCs. Finally, we conclude with some open questions in section 5.

Boundary Conditions for Maxwell Fields
In this section we study the BCs of Maxwell fields in holographic BCFT. This can be regarded as a warm-up for the gravitational case. We find that both absolute BC and relative BC can yield the expected asymptotic behaviors of one point function of current near the boundary. Furthermore, both BCs can reproduce the universal relations between the current and Weyl anomaly.

Current and Weyl anomaly
For the convenience of readers, let us briefly review the Weyl anomaly induced current for BCFT [25,26]. In general, the renormalized current of BCFT is singular near the boundary and takes the asymptotic form where x is the distance to the boundary, d are the dimensions of spacetime, F ij is a background field strength and n i is inward-pointing normal vector. It is remarkable that β d are determined by the central charges of Weyl anomaly. For example, we have where b 1 , b 2 are defined by the Weyl anomaly of 4d BCFT and 5d BCFT respectively, Notice that, since b 1 is the bulk central charge, it is independent of the choice of BCs. As a result, the near-boundary current (3) is universal in four dimensions. One the other hand, b 2 is the boundary central charge, which implies that the current depends on BCs in five dimensions. Finally, it should be mentioned that there are boundary contributions to the current which can exactly cancel the apparent "divergence" in the bulk current (3) at x = 0 and define a finite total current [25].

Probe limit
In [26], we have proved that the absolute BC can produce the expected form of near-boundary current (3) and the universal relations (4). In this section, we show that this is also the case for relative BC. What is new for relative BC is that there are non-trivial contributions to the current from the bulk boundary Q. Such contributions are important in order to recover the universal relations (4).
We start with the gravitational action for holographic BCFT where F = dA is the bulk field strength which reduces to F = dA on the boundary M , K is the extrinsic curvature on Q and T is a constant parameter which can be regarded as the holographic dual of boundary conditions of BCFT [27,28]. Takayanagi [17] proposes to impose the Neumann BC on Q which can not only fix the location of boundry Q but also the bulk metrics. In other words, not all the solutions to Einstein equations are allowed in holographic BCFT [17]. Instead, they have to satisfy the Neumann BC. Ignoring the bulk Maxwell fields, [17] find AdS is a solution to the Neumann BC (8), provided that the embedding function of Q is given by where we have re-parametrized T = (d − 2) tanh ρ. Now let us turn to discuss BCs of Maxwell fields. In general, there are two consistent BCs on Q for Maxwell fields, i.e., the absolute (electric) BC and relative (magnetic) BC Relative BC : N µ * F µν = 0.
Here N µ is the normal vector on Q and * F is the Hodge dual of F. The absolute (electric) BC is the Neumann BC, since it fixes the current on the boundary. On the other hand, relative (magnetic) BC is the Dirichlet BC in a certain sense, since it fixes the gauge fields on the boundary up to some trivial gauge transformations. To see this, let us recall the fact that N µ * F µν = 0 implies that γ µ ρ γ ν σ F µν = F Qρσ = 0 (γ µ ρ are projector operators from N to Q), which fixes the field strength of induced gauge fields on the bundary.
For simplicity, we focus on the probe limit, i.e., AdS spacetime (9) with the bulk boundary Q located at (10). This is sufficient for the study of the leading order of current. That is because the leading term of current (3) is of order O(F ), while the back-reaction of Maxwell fields to AdS is of order O(F 2 ) [26]. For plane boundary, A µ depends on only the coordinates z and x. And the Maxwell's equations can be solved with A z = A z (z), A x = A x (x) and Similarly, the BCs (11,12) become Relative BC : Inspired by [5], we consider the ansatz for the gauge field where f (0) = 1 and A (i) a are constants. The Maxwell's equations (13) become which has the general solution with α d an integration constant. It should be mentioned that, in order to get regular solutions at x = 0, suitable analytic continuation of the hypergeometric function should be taken when one express the above solutions in terms of the coordinates z and x. Substituting (18) into (14,15), we solve the integration constant for absolute BC and for relative BC. Similarly, suitable analytic continuation of the hypergeometric function should be taken in order to get smooth function at ρ = 0. For example, we have for d = 4, 5, Now we are ready to derive the holographic current for BCFT. Consider the variation of effective action for BCFT, in general, we have [2] where A (1) a x + ..., J i and j (n)a are bulk and boundary operators respectively. In general, both J i and j (n)a are divergent.
Near the boundary, there is no meaning to distinguish the divergent parts of the bulk current J i and boundary operators j i . Only the combination of the bulk current and boundary operators have physical meaning. According to [29,30], one can always regulate the effective action by excluding from its volume integration a small strip of geodesic distance from the boundary. Then there is no explicit boundary divergences in this form of the effective action, however there are boundary divergences implicit in the bulk effective action which is integrated up to distance . Now the variation of effective action becomes where M denotes x ≥ . The boundary operators are absorbed into the total bulk current J i in (24). One can recover the boundary operators after the integral up to x = . Let us consider a simple example with g 0 = h = 1 andĴ i = l i /x m . We have (25) which recovers the divergant parts of boundary operators.
Let us go on to consider the holographic theory. Consider the variation the on-shell action (7) with respect to gauge fields, we have To make consistent (26) and (24), we should transform the integral on Q into the integral on M . There are two ways to do so. First, one can do the integral along the bulk direction z on Q and get integrals on P . Then we choose a suitable renormalization as above to transform the divergent terms on P into the integral on M . Second, one can choose the same coordinates (x, y a ) on Q as on M and rewrite δA i | Q in terms of δA i . Then integral on Q becomes integral on M . These two methods give the same results for the divergent parts of current. Below we take the second method for simplicity. The key point is that, on-shell, both A µ on M and Q are functions of background gauge fields A i of BCFT. For example, from (16) we have on M Similarly, from (10,16) we have on Q Substituting (27) and (28) into (26), we can derive the current via the definition where we have ignored the hat for the bulk current, g 0 is determinant of the metric of BCFT. For our case, we have We discuss the current for absolute BC and relative BC separately. For absolute BC (11), the current on Q vanishes and (26) is simplified as from which one can read off the current [26] In the above derivations, we have used (16,18) and The near-boundary current (31) takes the expected form (3) with β d = −α Ad . Furthermore, it is proved in [26] that the coefficients α A4 , α A5 (21) obey the universal relations (4). Now let us go on to discuss the relative BC, which is one of the new results of this paper. For relative BC, there are non-trivial cotributions to the current from the near-boundary region of Q. Substituting (15,16,28) where f (s), α Rd are given by (18,20). Following the methods for absolute BC, one can derive the first line of (32). As for the second line, to obtain the contribution from Q to the leading term of current, we need to work out . This means that one needs to consider back-reactions to AdS for d ≥ 6, which are quite complicated. Fortunately, we do not need to do so. Since we focus on the solutions depending on only z and x in the bulk, the induced gauge field A| Q depends on only x: where γ m are constant vectors to be determined. Recall that the relative BC (12) is equivalent to fixing the field strength of induced gauge fields, i.e., F Q = d(A| Q ) = 0. Thus the relative BC (12) gives γ m = 0. As a result, we have The cases for d = 4 and d > 4 are a little different. We discuss them respectively below. For d = 4, there are non-trivial contributions from Q to the current. We have on leading term which yields the current Note that (36) is exactly the same as the current with absolute BC (21,31). This is consistent with the fact that the current is independent of BCs in four dimensions. As we have mentioned above, the current with absolute BC obey the universal law (4) [26]. So does the current with relative BC (36). This is a non-trivial test of the validity of relative BC (Dirichlet BC) in holographic BCFT.
As for d ≥ 5, it is remarkable that the contributions on Q cancel out, As a result only the usual terms on M contribute to the current. We thus have which takes very similar expression as the current with absolute BC (31). In [26], we have proved for d = 5 that the current with absolute BC (31) satisfies the universal relation (4).
To do so, we take into account the back-reactions to AdS from the bulk Maxwell fields up to order O(F 2 ) and then calculate the holographic Weyl anomaly. Finally, we compare the central charge b 2 in holographic Weyl anomaly with the current coefficient β 5 = −α R5 and verify (4). Following the same approach, we can prove that the current with relative BC also obey the universal relation (4). Since the proof is almost the same as that for absolute BC, we do not repeat it here. In fact, one only needs to replace α 5 by α R5 in the proof of [26].
Now we have proved that both absolute BC and relative BC can derive the correct one point function of current and universal relations between the current and Weyl anomaly. This is a strong support that both absolute BC (Neumann BC) and relative BC (Dirichlet BC) are consistent in holographic BCFT.

Exact solutions
In the above subsection, we have investigated perturbation solutions in holographic BCFT. In general, it is a non-trivial problem to find exact solutions to Einstein-Maxwell equations after imposing BCs on Q. Here we notice that the four-dimensional Reissner-Nordström black holes are exact solutions to holographic BCFT, provided that the bulk boundary Q is perpendicular to the AdS boundary M 1 . Interestingly, the magnetic charged Reissner-Nordström black holes satisfy the relative (magnetic) BC and predict that there is a constant boundary current on P , when a constant magnetic field is applied on M for 3d BCFT. Now let us list the main results. We find the electric charged Reissner-Nordström black holes satisfy the Neumann BC (8) and absolute BC (11) imposed on Q.
metric : Q : and recall that we have T = ρ = 0. In the dual picture, we have zero background gauge field but non-zero charge density on M The above picture is a little trivial, since it does not tell us new stories compared with the case without boundary. Now let us consider magnetic charged Reissner-Nordström black holes, which is more interesting. One can check that, for T = ρ = 0, the magnetic charged Reissner-Nordström black holes satisfy the Neumann BC (8) for metrics and the relative BC (12) for gauge fields metric : Q : where f (z) = 1 − M z 3 + B 2 z 4 /4 and the constant a y is the background gauge field on the boudnary P . In the dual picture, we have a constant magnetic field on M and non-zero current on P where z h is the location of outer horizon, i.e., f ( To derive the boundary current (44), let us consider the variation of action. Substituting (42) into (26), we have which gives the boundary current (44). Notice that since the boundary current is finite, there is no need to perform the approach (24) of sect.2.2, which is developed for the divergent parts of current.
Let us make some comments on the boundary current (44). First of all, the holographic BCFTs with relative BCs predict that there is a constant boundary current on P , when a constant magnetic field is applied on M in three dimensions. Secondy, the current depends on both the magnetic field and the temperature where we have M 4 ≥ (4/27)B 6 in order to avoid the naked singularity, and the temperature vanishes when the lower bound of mass is saturated. Thirdly, the absolute value of current decreases as the temperature increases. In particular, the current gains its maximum absolute value at zero temperature while vanishes in high temperature limit Fourthly, it should be mentioned that T = ρ = 0 is the dual BC preserving the maximum (half) supersymmetry [33]. And it seems that the non-renormalization theorem holds for such BCs. As a result, the dual strongly-coupled BCFTs can be mimicked by free BCFTs. Thus the above prediction would also apply to some kinds of free BCFTs with suitable BCs. For general BCs, there would be both bulk current and boundary current. Last but not least, it is expected that a constant magnetic field in the bulk M can induce constant boundary current on the boundary P for general boundary quantum field theory (BQFT). The boundary current increases as the magnetic field is enhanced, while decreases as the temperature and the mass of charged particles increase. In the zero temperture and mass limit, from the dimension analysis and isotropy, the boundary current takes the general form where we have recoverd the units, e is the charge, c is the speed of light, is the Planck constant and λ 0 is a dimensionless constant determined by the theory. It is remarkable that this current is detectable. For B = 1T and γ 0 = 1, we have It is interesting to measure such boundary current in some (1+2) dimensional systems with boundaries, such as graphene [32]. To enhance this effect, one should try to decrease the temperture and the effective mass of charged particles in materials.
To end this section, we remark that the holographic BCFT (7) has electromagnetic duality in four dimensions. Thus the above electric/magnetic charged black holes, and the absolute/relative BCs are dual to each other, respectively.

Holographic BCFT with Dirichlet BC
In this section, we study the Dirichlet BC for holographic BCFT. We find that, similar to Neumann BC, Dirichlet BC can also yield the expected one point function of stress tensors and the universal realtions between stress tensors and Weyl anomaly.
Consider the variations of action (7) with respect to the metric, we have To have a well-defined action principle, one can impose either Neumann BC (8) or the Dirichlet BC Dirichlet BC fixes the induced metric on the boundary. To proceed, we must specify what kinds of metrics we choose. The most simple and natural one is the AdS metric, which satisfes with ρ a free parameter of the model. The Dirichlet BC (53) is the central assumption of this paper. Of course, one can choose the other metrics such as those of asymptotically AdS, black holes and so on. For simplicity, we focus on AdS and leave the study of other choices to future work. Another reason to choose the AdS metric is that it can be easily written into covariant form (53). We remark that (53) is the gravitational counterpart of the relative BC (12) for gauge fields, since they both fix the curvatures of induced fields on the boundary.
The good BCs for holographic BCFT should satisfy the following requirements: (1) It should be neither too strong nor too weak. In particular, it allows AdS to be a solution.
If the BC allows no solutions or only a limited number of solutions, it is too strong. On the other hand,it is too weak if all the solutions to EOM are allowed. In general, there is a narrow window of consistent BCs. We also hope AdS is a solution so that we can apply AdS/CFT.
(2) It can fix the location of bulk boundary Q.
This is the central problem in the construction of AdS/BCFT.
(3) It can produce the expected one point fucntion of stress tensor (current), Weyl anomaly and universal relations between them.
In this section, we will show that Dirichlet BC satisfies all the above requirements. So the holographic BCFT with Dirichlet BC is well-defined. As a quick check, we notice that the Poincare AdS (9) together with the embedding funciton of Q (10) are indeed solutions to the Dirichlet BC (53). As a result, holographic BCFT with Dirichlet BC share most of the advantages of holographic BCFT with Neumann BC [17]. In particular, it obeys the holographic g-theorem, or c-like theorem due to the fact that the two kinds of holographic BCFT have the same g-functions and boundary central charges related to Euler densities. To see this, let us show some details below. We require that the Poincare AdS (9) is a solution to Dirichlet BC (53) for some suitable embedding function of Q. For simplicity, we assume the embedding function is independent of y a x = −F (z). (54) Substituting (9,54) into (53), we get for d ≥ 3. We choose the boundary at x = 0 which fixes the integration constant F (0) = 0. We make the choice that F (z) = sinh ρ in order to get same solution as Neumann BC. Note that, the trace of extrinsic curvature is positive for this choice for ρ > 0. Finally we obtain the embedding function (10). The case for d = 2 is a little special, since the curvatures (53) have only one independent component in two dimensions. For d = 2, we solve where c 1 is a integration constant. Note that the left hand side of (56) is positive, while the right hand side of (56) could be negative for sufficiently large z. To avoid inconsistency, we must have c 1 = ∞ and (56) reduces to (55). Following the above method, we get the embedding function (10) for d = 2.
Applying the conformal map [17], we can obtain the holographic dual of BCFT on a round disk from the one on half space (9,10). In this way, we get the metric and the embedding function of Q One can check that the above solutions indeed satisfy the Dirichlet BC (53). Following [17,18,19], one can further derive the partition function and then read off the boundary entropy and boundary central charges related to Euler densities . For example, for d = 2 we get the boundary entropy which is a good candidate for the g-function. Applying the null energy condition on Q, i.e., , we get the holographic g-theorem which shows that the boundary degree of freedom is a monotonically decreasing function under the RG flow. Notice that if we choose K < 0 for the solutions, we would get S bdy = − ρ 4 , which disagrees with g-theorem.
As a summary, the Dirichlet BC (53) indeed allows AdS to be a solution. It is remarkable that, in addition to the Dirichlet BC (53), we impose an extra condition that the extrinsic curvature is positive definite K ≥ 0 for ρ ≥ 0 2 . This requirement is reasonable. That is because, according to [24], the extrinsic curvature should be either positive definite or negative definite in order to have a well-defined quantization for Dirichlet BC. We make the physical choice K ≥ 0 in order to satisfy the holographic g-theorem.

Casimir effects and Weyl anomaly
Let us quickly review the Casimir effects and Weyl anomaly for BCFTs. It is found in [34] that the renormalized stress tensor of BCFT is divergent near the boundary, where x is the proper distance from the boundary,k ij is the traceless part of extrinsic curvature and α is a constant which depends only on the kind of BCFT under consideration. The coefficient α fixes the shape dependence of the leading Casimir effects of BCFT.
Interestingly, the authors of [5] observe that the Casimir coefficient is related to the central charge of Weyl anomaly. For example, there are universal relations where b i are boundary central charges of Weyl anomaly of 3d BCFT and 4d BCFT [35,3,4], respectively Under some assumptions, the authors of [5] further check that the holographic BCFT with Neumann BC [17] is consistent with the universal relations (62) between Casimir effects and Weyl anomaly. In this paper, we give a solid proof of the universal relations (62) for both the holographic BCFTs with Dirichlet BC (53) and Neumann BC [17].

Solutions and Stress Tensors
To study the one point functon of stress tensor, one needs to derive the perturbation solutions around (9) and (10). Following [5], we take the following ansatz for the metric and the embeding function of Q where k denotes the trace of the extrinsic curvature, f (s) is a function and λ 2 is a constant to be determined. We require that f (0) = 1 (67) so that the metric of BCFT takes the form in Gauss normal coordinates Note that we expand the solutions in extrinsic curvatures, or equivalently, the derivatives O(k) ∼ O(∂). Besides, we focus on the case without y dependent, which is sufficient to obtain the Casimir coefficients and central charges of Weyl anomaly.
Substituting (65) into Einstein equations, we obtain at the order O(k) a single equation which can be solved as Imposing either Neumann BC [17] or Dirichlet BC (53), we can fix the location of Q (66) and derive In fact, λ 2 can be fixed by the symmetry of asymptotically AdS [28]. Thus it is universal and independent of BCs. Now the BCs on Q (66,71) become Dirichlet BC : f (−cschρ) = 0.
Substituting (70) into the above BCs, we obtain the integration constant for Neumann BC and Dirichlet BC respectivelȳ We remark that the gravitational solutions (70,74,75) in D = (d + 1) dimensions are exactly the same as the solutions of gauge fields (18,19,20) in (D + 2) dimensions. As we have mentioned before, suitable analytic continuation of the hypergeometric function should be taken in order to get smooth function at ρ = 0. For example, we have for d = 4, 5, Now let us go on to derive the holographic stress tensors. We firstly discuss the case of Neumann BC. Using (65), (70), we can derive the holographic stress tensor [36] T ij = dh which takes the expected form (61). Here h (d) ij is defined in the Fefferman-Graham expansion of the asymptotic AdS metric Now let us turn to the case of Dirichlet BC. Similar to the current of gauge fields, in general, there are potential contributions from the bulk boundary Q to the stress tensors. Taking the metric variation of the action (7) with T = (d − 2) tanh ρ and using (66,71,70,73), we have at leading order where γ ij is the AdS metric due to Dirichlet BC (53). Recall that we focus on the case that γ ij is a function of only x. Then the most general form of γ ij takes where f 1 = 1−x f 2 f 2 , h ab = δ ab and (h ab , k ab , q ab , ...) are defined in the Gauss normal coordinate To derive the divergent parts of stress tensor via ab , ab .
In this way, we derive Note that δγ xx is not important for our purpose, since the stress tensors on Q (80) have no xx−components at the leading order. Substituting (84) into (80), we derive Similar to the case of gauge fields, the contributions on Q vanish. Note that the stress tensor with d = 2 is the counterpart of the current with d = 4 of sect.2.2. Similar to the case of current, there would be non-trivial contributions on Q to the stress tensor with d = 2. However, due tok ab = 0 for d = 2, such potential contributions vanish. From (85) we finally obtain the stress tensor for Dirichlet BC which takes the same form as Neumann BC (78). In the next section, we shall test the universal relations (62) for both Dirichlet BC and Neumann BC.

Holographic Weyl anomaly
In this section, we derive the holographic Weyl anomaly and verify the universal relations (62) between Casimir effects and Weyl anomaly for 3d BCFT and 4d BCFT. To do so, we need to work out the perturbation solutions up to order O(k 2 ) ∼ O(∂ 2 ) for 3d BCFT and O(k 3 , kq) ∼ O(∂ 3 ) for 4d BCFT, respectively. Since the calculations are quite complicated, below we take 3d BCFT as an example to illustrate the key points of derivations and only claim the final result for 4d BCFT.
Following [5], we take the following ansatz for metrics where the functions X 2 ( z x ) and Q ab ( z x ) are of order O(∂ 2 ), X 3 ( z x ) and H ab ( z x ) are of order O(∂ 3 ) and ... denotes higher orders. We set that so that the metric of BCFT takes the form in Gauss normal coordinates For simplicity, we focus on the solutions without y a dependence. We further set k ab = diag(k 1 , ..., k d−1 ), q ab = diag(q 1 , ..., q d−1 ), where k a , q a are constants. Then the embedding function of Q takes the form where the second term is fixed by the symmetry of asymptotically AdS [28], c i are constants of order O(∂ i−1 ) which can be determined by BCs. Now let us focus on the second order solutions for 3d BCFT. Substituting (87) into the Einstein equations, we solve [5] f (s) = 1 − α 1 (s − g(s)) where s = z/x and g(s) = π 2 − 2 tan −1 1/(s + √ s 2 + 1) . To rewrite the above solutions in functions of x and z, we should consider suitable analytic continuation in order to get smooth functions at x = 0. In this way, we get smooth g(z, x) as Imposing NBC (8) or DBC (53), we can solve the integration constants whereᾱ 3 are given by (76) and (77) for NBC and DBC, respectively. The BCs can not only fix the bulk solutions but also the location of Q (90). We obtain for both NBC and DBC where f 4 (ρ) = π 2 + 2 tan −1 tanh ρ 2 . It is interesting that NBC and DBC yield almost the same solutions except a different parameter α 1 . Now we are ready to derive the holographic Weyl anomaly for 3d BCFT. The following approach applies to both NBC and DBC. By using Einstein equations, we can rewrite the on-shell gravitational action (7) without gauge fields as To get the holographic Weyl anomaly, we need to do the integration along x and z, and then select the UV logarithmic divergent terms. We divide the integration region into two parts: region I is defined by (z ≥ 0, x ≥ 0) and region II is defined by the complement of region I. Let us first study the integration in region I, where only the bulk action in (95) contributes.
Integrating along z, expanding the result in small x and selecting the 1/x term, we obtain Next let us consider the integration in region II. In this case, both the bulk action and boundary action in (95) contribute. For the bulk action, we first do the integral along x, which yields a boundary term on Q. Note that since only the UV logarithmic divergent terms are related to Weyl anomaly, we keep only the lower limit of the integral of x. Adding the boundary term from bulk integral to the boundary action in (95), we obtain where we have used α 1 = −α N 3 (76) for NBC and α 1 = −α D3 (77) for DBC above. Combining together (96) and (97) and noting that α 1 = −ᾱ 3 , we finally obtain the Weyl anomaly (63) with the boundary central charge b 1 =ᾱ 3 . Hence we obtain the universal relation (62) for 3d BCFT. It should be mentioned that since we focus on solutions without y a dependence, the first term R of Weyl anomaly (63) vanishes. Actually, it is easy to recover this term.
Applying the conformal map [17], we can obtain the holographic dual of BCFT on a round disk from the one on half space (9,10). Following the above approach, we can derive the R term in Weyl anomaly and read off the central charge b 1 = sinh ρ. We remark that holographic BCFT with NBC and DBC have the same boundary central charges related to Euler densities but different boundary central charges related to extrinsic curvatures. Since the c-like theorem on the boundary, or generalized g-theorem, apply to the central charges related to Euler densities. The holographic BCFT with NBC and DBC both obey the c-like theorem. Following the same approach, we can derive the holographic Weyl anomaly for 4d BCFT and verify the universal relation (62). Since the calculations are similar to the 3d case but quite complicated, we do not repeat it here.

Casimir Coefficients and Central Charges
In the above sections, we have obtained the Casimir coefficients and central charges for holographic BCFT with NBC and DBC, respectively. In this section we study the interesting characteristics of these charges. For simplicity we only list the main results and discuss them briefly.
(1) The Casimir coefficients for NBC are greater than or equal to those for DBC.
From (74) and (75), it is easy to check that where the equality is saturated when the bulk boundary Q is pulled back into the AdS boundary, i.e., ρ → ∞. Interestingly,ᾱ N d gains its lower bound whileᾱ Dd gets its upper bound at ρ → ∞.
According to [37], we haveᾱ d = λ d c nn , where λ d is a positive factor and c nn is the central charge defined by the two point funciton of the displacement operator < D n (x)D n (0) >= c nn x 2d .
From (100), we have c nn ≥ 0 and thus It should be mentioned that [6] find that b 2 = π 2 8 c nn for 3d BCFT and b 4 = 2π 2 15 c nn for 4d BCFT. Using the univeral realtion (62), we getᾱ d ≥ 0 for d = 3, 4. This can be regarded as an independent test of our claim above.
Interestingly, the positive bound is automatically satisfied by NBC. As for DBC, the bound yields T = (d − 1) tanh ρ ≥ 0, which is reasonable if we take T as the tension of the brane Q [17]. (3) The Casimir coefficients for NBC reduce to the ones of free BCFT, while the Casimir coefficients for DBC vanish, when bulk boundary Q is perpendicular to the AdS boundary M .
From (74) and (75), we have where C T is the central charge defined in the two point funciton of stress tensor far away from the boundary < T µν (x)T λρ (0) >= C T |x| 2d I µν,λρ (x) with I µν,λρ a dimensionless tensor fixed by symmetry. The central charge C T for CFTs dual to Einstein gravity is given by Using the results of [34,35,3], we notice that (102) is satisfied by 3d and 4d free BCFTs. It is expected that (102) applies to free BCFTs in general dimensions.
To end this section, let us draw the figures of Casimir coefficients for NBC and DBC in three and four dimensions, respectively. See figure 2 and figure 3.

Conclusions and Discussions
In this paper, we have investigated the holographic BCFT with Dirichlet BC and find it works as well as the one with Neumann BC. For example, AdS is a vacuum solution to Dirichlet BC and the g-theorem or c-like theorem is obeyed by this theory. Furthermore, it gives the expected one point funciton of the stress tensor (current) and the universal relations between the stress tensor (current) and Weyl anomaly. It is remarkable that the boundary central charge related to B-type Weyl anomaly, or equivalently, the Casimir coefficient for Dirichlet BC is less than or equal to the one for Neumann BC. We have also studied the relative BC for gauge fields, which is the counterpart of Dirichlet BC for gravitational fields. We find an exact solution to this BC, which implies that a constant magnetic field in the bulk can induce a constant current on the boundary in three dimensions. And the boundary current gets the maximum value at zero temperature. It is interesting to measure this boundary effect in laboratory. In this paper, we discuss only the tip of the iceberg for Dirichlet BC, i.e., we fix the boundary metric to be that of AdS. It is natural to study more general metrics on Q, such as the ones describing gravitational waves and black holes. Besides, one can generalize the discussions of this paper to other fields, such as scalars and higher spin fields. We leave this exercise to the interesting readers. Finally, it is also interesting to study other kinds of BCs for holographic BCFT, such as the conformal BC or mixed BC. We notice that the conformal BC is more subtle, which is less restrictive than Dirichlet BC and Neumann BC. We leave a careful study of this problem to future work.