A Another Derivation of (48)

## Abstract

This paper is an extended version of our short letter on a new proposal for holographic boundary conformal field, i.e., BCFT. By using the Penrose-Brown-Henneaux (PBH) transformation, we successfully obtain the expected boundary Weyl anomaly. The obtained boundary central charges satisfy naturally a c-like theorem holographically. We then develop an approach of holographic renormalization for BCFT, and reproduce the correct boundary Weyl anomaly. This provides a non-trivial check of our proposal. We also investigate the holographic entanglement entropy of BCFT and find that our proposal gives the expected orthogonal condition that the minimal surface must be normal to the spacetime boundaries if they intersect. This is another support for our proposal. We also find that the entanglement entropy depends on the boundary conditions of BCFT and the distance to the boundary; and that the entanglement wedge behaves a phase transition, which is important for the self-consistency of AdS/BCFT. Finally, we show that the proposal of arXiv:1105.5165 is too restrictive that it always make vanishing some of the boundary central charges.

NCTS-TH/1702

On New Proposal for Holographic BCFT

Chong-Sun Chu1 Rong-Xin Miao 2 and Wu-Zhong Guo 3

Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan

Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University, Hsinchu 30013, Taiwan

## 1 Introduction

BCFT is a conformal field theory defined on a manifold with a boundary with suitable boundary conditions. It has important applications in string theory and condensed matter physics near boundary critical behavior [1]. In the spirit of AdS/CFT [2], Takayanagi [3] proposes to extend the dimensional manifold to a dimensional asymptotically AdS space so that , where is a dimensional manifold which satisfies . We mention that the presence of the boundary is very natural from the point of view of the UV/IR relation [4] of AdS/CFT correspondence since the presence of boundary in the field theory introduce an IR cutoff and this can be naturally implemented in the bulk with the presence of a boundary. Conformal invariance on requires that is part of AdS space. The key point of holographic BCFT is thus to determine the location of boundary in the bulk. For interesting developments of BCFT and related topics please see, for example, [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

The gravitational action for holographic BCFT is given by [3, 5] (taking )

 I=∫N√G(R−2Λ)+2∫Q√h(K−T)+2∫M√g K+2∫P√σ θ (1)

where is a constant and is the supplementary angle between the boundaries and , which makes a well-defined variational principle on the corner [17]. Notice that can be regarded as the holographic dual of boundary conditions of BCFT since it affects the boundary entropy (and also the boundary central charges, see (55,56) below) which are closely related to the boundary conditions of BCFT [3, 5]. Considering the variation of the on-shell action, we have

 δI = Missing or unrecognized delimiter for \right (2)

For conformal boundary conditions in CFT, Takayanagi [3] proposes to impose Dirichlet boundary condition on and , , but Neumann boundary condition on . And the position of the boundary is determined by the Neumann boundary condition

 Kαβ−(K−T)hαβ=0. (3)

For more general boundary conditions which break boundary conformal invariance even locally, [3, 5] propose to add matter fields on and replace eq.(3) by

 Kαβ−Khαβ=12TQαβ, (4)

where we have included in the matter stress tensor on . For geometrical shape of with high symmetry such as the case of a disk or half plane, (3) fixes the location of and produces many elegant results for BCFT [3, 5, 6]. However since is of co-dimension one and its shape is determined by a single embedding function, (3) gives too many constraints and in general there is no solution in a given spacetime such as AdS. On the other hand, of course, one expect to have well-defined BCFT with general boundaries.

To solve this problem, [5] propose to take into account backreactions of . For 3d BCFT, they show that one can indeed find perturbative solution to (3) if one take into account backreactions to the bulk spacetime. In other words although not all the shapes of boundary are allowed by (3) in a given spacetime, by carefully tuning the spacetime (which is a solution to Einstein equations) one can always make (3) consistent for any given shape. However, it is still a little restrictive since one has to change both the ambient spacetime and the position of for different boundaries of the BCFT.

As motivated in [3, 5], the conditions (3) and (4) are natural from the point of view of braneworld scenario, and so is the backreaction. However from a practical point of view, it is not entirely satisfactory since one has a large freedom to choose the matter fields as long as they satisfy various energy conditions. As a result, it seems one can put the boundary at almost any position as one likes. Besides, it is unappealing that the holographic dual depends on the details of matters on another boundary . Finally, although eq.(4) could have solutions for general shapes by tuning the matters, it is actually too strong since as we will prove in the appendix it always makes vanishing some of the central charges in the boundary Weyl anomaly. In a recent work [18], we propose a new holographic dual of BCFT with determined by a new condition (9). This condition is consistent and provides a unified treatment to general shapes of . Besides, as we will show below, it yields the expected boundary contributions to the Weyl anomaly.

Instead of imposing Neumann boundary condition (3), we suggest to impose the mixed boundary conditions on [18]:

 (Kαβ−(K−T)hαβ)Π   α′β′+αβ=0, (5) Π   α′β′−αβδhα′β′=0. (6)

where and are the projection operators satisfying and . Since we could impose at most one condition to fix the location of the co-dimension one surface , we require and from . Now the mixed boundary condition (5) becomes

 (Kαβ−(K−T)hαβ)Aαβ=0, (7)

where are non-zero tensors to be determined. It is natural to require that eq. (7) to be linear in so that it is a second order differential equation for the embedding. Thus we propose the choice in [18]. In this paper, we will provide more evidences for this proposal. Besides, we find that the other choices such as

 Aαβ=λ1hαβ+λ2Kαβ+λ3Rαβ+⋯,λ1,λ2≠0, (8)

To sum up, we propose to use the traceless condition

 TBYα α=2(1−d)K+2dT=0 (9)

to determine the boundary . Here is the Brown-York stress tensor on . In general, it could also depend on the intrinsic curvatures which we will treat in sect.4. A few remarks on (9) are in order. 1. It is worth noting that the junction condition for a thin shell with spacetime on both sides is also given by (4) [17]. However, here is the boundary of spacetime and not a thin shell, so there is no need to consider the junction condition. 2. For the same reason, it is expected that has no back-reaction on the geometry just as the boundary . 3. Eq. (9) implies that is a constant mean curvature surface, which is also of great interests in both mathematics and physics [19] just as the minimal surface. 4. (9) reduces to the proposal by [3] for a disk and half-plane. And it can reproduce all the results in [3, 5, 6]. 5. Eq. (9) is a purely geometric equation and has solutions for arbitrary shapes of boundaries and arbitrary bulk metrics. 6. Very importantly, our proposal gives non-trivial boundary Weyl anomaly, which solves the difficulty met in [3, 5]. In fact as we will show in the appendix the proposal (4) of [3] is too restrictive and always yields for the central charges in (10,11). Since is expected to satisfy a c-like theorem and describes the degree of freedom on the boundary, thus it is important for to be non-zero.

Let us recall that in the presence of boundary, Weyl anomaly of CFT generally pick up a boundary contribution in addition to the usual bulk term , i.e. , where is a delta function with support on the boundary . Our proposal yields the expected boundary Weyl anomaly for 3d and 4d BCFT [20, 21, 22]:

 ⟨Taa⟩P=c1R+c2Tr¯k2,d=3, (10) ⟨Taa⟩P=a16π2Ebdy4+b1Tr¯k3+b2Cac   bc¯kb a,d=4, (11)

where are boundary central charges, is the bulk central charge for 4d CFTs dual to Einstein gravity, is intrinsic curvature, is the traceless part of extrinsic curvature, is the Weyl tensor on and is the boundary terms of Euler density used to preserve the topological invariance

 Ebdy4=4(2Tr(kR)−kR+23Trk3−kTrk2+13k3). (12)

Since is not a minimal surface in our case, our results (55,56) are non-trivial generalizations of the Graham-Witten anomaly [23] for the submanifold.

The paper is organized as follows. In sect.2, we study PBH transformations in the presence of submanifold which is not orthogonal to the AdS boundary and derive the boundary contributions to holographic Weyl anomaly for 3d and 4d BCFT. In sect. 3, we investigate the holographic renormalization for BCFT, and reproduce the correct boundary Weyl anomaly obtained in sect.2, which provides a non-trivial check of our proposal. In sect. 4, we consider the general boundary conditions of BCFT by adding intrinsic curvature terms on the bulk boundary . In sect.5, we study the holographic entanglement entropy and boundary effects on entanglement. In sect.6, we discuss the phase transition of entanglement wedge, which is important for the self-consistency of AdS/BCFT. Conclusions and discussions are found in sect. 7. The paper is finished with three appendices. In appendix A, we give an independent derivation of the leading and subleading terms of the embedding function by solving directly our proposed boundary condition for . The result agrees with that obtained in sect.2 using the PBH transformations. In appendix B, we show that the proposal of [3] always make vanish the central charges and in the boundary Weyl anomaly for 3d and 4d BCFT. In appendix C, we give the details of calculations for the boundary contributions to Weyl anomaly.

Notations: and are the metrics in and , respectively. We have , , and . The curvatures are defined by , and . The extrinsic curvature on are defined by , where is the unit vector normal to and pointing outward from to .

Note added: Two weeks after [18], there appears a paper [51] which claims that our calculations of boundary Weyl anomaly (55,56) are not correct. We find they have ignored important contributions from the bulk action for 3d BCFT and the boundary action for 4d BCFT. After communication with us, they realize the problems and reproduce our results (55,56) in a new revision of [51]. For the convenience of the reader, we give the details of our calculations in appendix C. We also emphasis here that, from our analysis, it is natural to keep as a free parameter rather than to set it zero. Otherwise, the corresponding 2d BCFT becomes trivial since the boundary entropy [3] is zero when . Besides, we emphasis that, as we previously demonstrated in section 4, by allowing intrinsic curvatures terms on , one can always make the holographic boundary Weyl anomaly matches the predictions of BCFT with general boundary conditions. This may or may not match with the result of free BCFT since so far it is not clear whether and how non-renormalization theorems hold. However in the special case it holds, e.g. in the presence of supersymmetry, it just means the parameters of the intrinsic curvature terms are fixed, which is completely natural due to the presence of more symmetry.

## 2 Holographic Boundary Weyl Anomaly

According to [24], the embedding function of the boundary is highly constrained by the asymptotic symmetry of AdS, and it can be determined by PBH transformations up to some conformal tensors. By using PBH transformations, we find the leading and subleading terms of the embedding function for are universal and can be used to derive the boundary contributions to the Weyl anomaly for 3d and 4d BCFT. It is worth noting that we do not make any assumption about the location of in this approach. So the holographic derivations of boundary Weyl anomaly in this section is very strong.

### 2.1 PBH transformation

Let us firstly briefly review PBH transformation in the presence of a submanifold [24]. Consider a -dimensional submanifold embedded into the -dimensional bulk such that it ends on a -dimensional submanifold on the -dimensional boundary . Denote the bulk coordinates by and the coordinates on by with and . The embedding function is given by .

We consider the bulk metric in the FG gauge

 ds2=dρ24ρ2+gijdxidxjρ. (13)

Here denote the boundary of the metric. It is known that if one assume the metric admits a series expansion in powers of , , then can be fixed by the PBH transformation [26] 4

 (1)gij=−1d−2(R(0)ij−R(0)2(d−1)g(0)ij). (14)

PBH transformations are a special subgroup of diffeomorphism which preserve the FG gauge:

 δρ=−2ρσ(x), (15) δxi=ai=12∫ρ0dρ′gij(x,ρ′)∂jσ(x)+ai0(x). (16)

Here is the parameter of Weyl rescalings of the boundary metric, i.e., and is the diffeomorphism of the boundary . To keep the position of on , we require that .

Next let us include the submanifold. The metric on is given by

 hττ=14τ2+1τ∂τXi∂τXjgij(X,τ), (17) hab=1τ∂aXi∂bXjgij(X,τ). (18)

To fix the reparametrization invariance on , we chose similarly the gauge fixing condition

 τ=ρ,  haτ=0 (19)

Now under a bulk PBH transformation (15,16), one needs to make a compensating diffeomorphism on [24] such that and in order to stay in the gauge (19). This gives

 ~ξτ=−2τσ(x)and~ξa=2∫τ0dτ′τ′hττhab∂bσ. (20)

As a result, changes under PBH transformation as

 δXi=~ξα∂αXi−ai, (21)

where is given by (20) and is given by eq.(16). As in the case of the metric, if one expand the embedding function in powers of ,

 Xi(τ,ya)=(0)Xi(ya)+τ(2)Xi(ya)+⋯, (22)

the first leading nontrivial term can be fixed by its transformation properties [24]. In fact, since

 δ(0)Xi=0, δ(2)Xi=−2σ(2)Xi+12(0)hab∂a(0)Xi∂bσ−12(0)gij∂jσ, (23)

one can solve the second equation of (2.1) by

 (2)Xi=12pki, (24)

where is the trace of the extrinsic curvature of

 ki=(0)habkiab=(0)hab(∂a∂b(0)Xi−(0)γcab∂cX(0)i+(0)Γijk∂a(0)Xj∂b(0)Xk), (25)

is the inverse of which appears in the expansion:

 hab=1τ∂a(0)Xi∂b(0)Xjgij((0)X,τ)+⋯:=1τ(0)hab+⋯ (26)

and is the Christoffel symbol for the induced metric .

Now let us focus on our problem with , and . Inspired by [3], we relax the assumption of [24] and expand in powers of in the presence of a boundary:

 Xi(τ,ya)=(0)Xi(ya)+√τ(1)Xi(ya)+τ(2)Xi(ya)+⋯ (27)

This means that is not orthogonal to the AdS boundary generally due to the non-zero . Then we have

 hτa = 1τ∂τXi∂aXjgij(X,τ) (28) = 12τ32(1)Xi∂a(0)Xjg(0)ij+12τ(2(2)Xi∂a(0)Xj(0)gij+(1)Xi∂a(1)Xj(0)gij+(1)Xi∂a(0)Xj(1)Xk∂k(0)gij)+⋯.

Imposing the gauge (19), we get

 (1)Xi=|(1)X| ni, (29) hij(2)Xj=−14hij∂j|(1)X|2−12hij(0)Γjklnknl|(1)X|2, (30)

where is the normal vector pointing inside from to , , is the zeroth order induced metric on , and . It is worth noting that is on longer a vector due to the appearance of the affine term in eq.(30). This is not surprising since we have imposed the gauge (19) which fixes all the reparametrization of except the one acting on [24]. One can easily check that is indeed covariant under the residual gauge transformations of the reparametrization of . Besides, note that coordinates are not vector generally, so there is no need to require to be a vector. What must be covariant are the finial results such Weyl anomaly and entanglement entropy.

Now let us study the transformations of under PBH. From eq.(21), we obtain

 δ(0)Xi=0, (31) δ(1)Xi=−σ(1)Xi, (32) δ(2)Xi=−2σ(2)Xi+12|(1)X|2hij∂jσ−12(1+2|(1)X|2)ninj∂jσ. (33)

Using the following formulas

 δσni=−σni, (34) δσki=−2σki−p ninj∂jσ, (35) δσ(0)Γijk=δij∇kσ+δik∇jσ−(0)gjk∇iσ, (36)

one can easily check that eqs.(29,30) indeed obey the transformations (32,33). One may also solve (33) directly and obtain for the normal components of as:

 ninj(2)Xj=1+|(1)X|22pki−12|(1)X|2(0)Γnnnni+c1(kip+(0)Γnnnni). (37)

Here and is a parameter to be determined. Note that a term proportional to from (30) drops out automatically in (37) since is functions of only the transverse coordinates , such term vanishes due to the normal derivatives.

As we have mentioned, is no longer a vector in the normal sense due to the gauge fixing (19). Instead, admit some kinds of deformed covariance under the remaining diffeomorphism after fixing the FG gauge (13) in and world-volume gauge (19) on . It is clear that the remaining diffeomorphism are the ones on and . The key point is that, for every diffeomorphism on , there exists compensating reparametrization on in order to stay in the gauge (19). As a result, is covariant in a certain sense under the combined diffeomorphisms on and . As we will illustrate below, the deformed gauge symmetry is useful and it fixes the value of the parameter to be zero.

Without loss of generality, we consider the Gauss normal coordinates on

where is located at , and is determined by

 x=a1(y)√τ+a2(y)τ+⋯ (39)

To satisfy the gauge (19), we should choose the coordinates on carefully. For example, the natural one does not work. Instead, we should choose with the embedding functions given by

 ρ=τ, (40) x=a1(y′)√τ+a2(y′)τ+⋯ (41) ya=y′a−14σab∂ba21(y′)τ+⋯ (42)

Notice that and for the Gauss normal coordinates (38). Recall also that , we obtain from eq.(37)

 a2(y′)=−1+a21(y′)2pk−c1kp. (43)

Now let us use the remaining diffeomorphism to fix the parameter . Consider a remaining diffeomorphism

 x=x′+cx′2+O(x′3) (44)

which keeps the position of and the gauge eqs.(13,19). From eqs.(41,43,44), we have

 x′=x−cx2+O(x3)=a1(y′)√τ−(1+a21(y′)2pk+c1kp+c a21(y′))τ+⋯ (45)

Since the new coordinate satisfies the gauge (13,19), it must take the form (37) because of PBH transformations. Substituting and into eq.(37), we get

 a′2(y′)=−1+a21(y′)2pk−c1kp−c a21(y′)+2cc1 (46)

for the new coordinate . Comparing eq.(46) with the coefficients of in eq.(45), we find that they match if and only if . Hence our claim.

As a summary, by using the PBH transformations and the covariance under remaining diffeomorphism, we find the leading and subleading terms of embedding functions are universal and take the following form

 (1)Xi=|(1)X| ni, (47) (2)Xi=1+|(1)X|22pki−14hij∂j|X(1)|2−12(0)Γinn|(1)X|2 (48)

In the Gauss normal coordinates (38), the embedding function has very elegant expression

 x=a1(y)√τ−1+a21(y)2pk τ+⋯ (49)

These are the main results of this section. One may still doubt eq.(48) due to the non-covariance. Actually, we can derive it from the covariant equation (9) together with the gauge (19). So it must be covariant under the remaining diffeomorphism. This is a non-trivial check of our results. Please see the appendix for the details. Besides, we have checked other choices of boundary conditions such as eq.(7) with . They all yield the same results eqs.(47,48,49). This is a strong support for the universality.

### 2.2 Boundary Weyl anomaly

In this section, we apply the method of [25] to derive the Weyl anomaly (including the boundary contributions to Weyl anomaly [5]) as the logarithmic divergent term of the gravitational action. For our purpose, we focus only on the boundary Weyl anomaly on below.

Let us quickly recall our main setup. Consider the asymptotically AdS metric

 ds2=dz2+gijdxidxjz2 (50)

where , , is the metric of BCFT on and , fixed uniquely by the PBH transformation, is given by (14). Without loss of generality, we choose Gauss normal coordinates for the metric on

where the boundary is located at . The bulk boundary is given by . Expanding it in , we have

where and are functions of . By using the PBH transformation, we know that is universal and can be expressed in terms of and the extrinsic curvature through eq.(49). can be determined by the boundary condition on . Noting that , we get the leading term of eq.(7) as

 (Kαβ−(K−T)δαβ)Aβα=⎛⎜ ⎜⎝(1−d)a1√1+a21+T⎞⎟ ⎟⎠Aαα+⋯=0, (53)

where denotes higher order terms in . It is remarkable that we can solve from eq.(53) without any assumption of except its trace is nonzero. In other words, we can solve from the universal part of the boundary conditions. From eqs.(49,53), we finally obtain

 T=(d−1)tanhρ∗,a1=sinhρ∗,a2=−Trk2(d−1)cosh2ρ∗,⋯, (54)

where we have re-parameterized the constant in terms of , which can be regarded as the holographic dual of boundary conditions for BCFT. That is because, as will be clear soon, affects the boundary central charges as the boundary conditions do. It should be mentioned that one can also obtain by directly solving the boundary condition eq.(9) or eq.(7) with . They yield the same results for but different results for .

Now we are ready to derive the boundary Weyl anomaly. For simplicity, we focus on the case of 3d BCFT and 4d BCFT. Substituting eqs.(50-54) into the action (1) and selecting the logarithmic divergent terms after the integral along and , we can obtain the boundary Weyl anomaly. We note that and do not contribute to the logarithmic divergent term in the action since they have at most singularities in powers of but there is no integration alone , thus there is no way for them to produce terms. We also note that only appears in the final results. The terms including and automatically cancel each other out. This is also the case for the holographic Weyl anomaly and universal terms of entanglement entropy for 4d and 6d CFTs [27, 28]. After some calculations, we obtain the boundary Weyl anomaly for 3d and 4d BCFT as

 ⟨Taa⟩P=sinhρ∗ R−sinhρ∗ Tr¯k2,                                    for 3d BCFT, (55) ⟨Taa⟩P=18Ebdy4+(cosh(2ρ∗)−13)Tr¯k3−cosh(2ρ∗)Cac   bc¯kb a,  for 4d BCFT. (56)

which takes the expected conformal invariant form [20, 21, 22]. It is remarkable that the coefficient of takes the correct value to preserve the topological invariance of . This is a non-trivial check of our results. Besides, the boundary charges in (10, 11) are expected to satisfy a c-like theorem [5, 7, 29]. As was shown in [3, 6], null energy condition on implies decreases along RG flow. It is also true for us. As a result, eqs.(55, 56) indeed obey the c-theorem for boundary charges. This is also a support for our results. Most importantly, our confidence is based on the above universal derivations, i.e., we do not make any assumption except the universal part of the boundary conditions on . Last but not least, we notice that our results (55,56) are non-trivial generalizations of the Graham-Witten anomaly [23] for the submanifold, i.e., we find there exists conformal invariant boundary Weyl anomaly for non-minimal surfaces.

We remark that based on the results of free CFTs [21] and the variational principle, it has been suggested that the coefficient of in (56) is universal for all 4d BCFTs [22]. Here we provide evidence, based on holography, against this suggestion: our results agree with the suggestion of [22] for the trivial case , while disagree generally. As argued in [29], the proposal of [22] is suspicious. It means that there could be no independent boundary central charge related to the Weyl invariant . However, in general, every Weyl invariant should correspond to an independent central charge, such as the case for 2d, 4d and 6d CFTs. Besides, we notice that the law obeyed by free CFTs usually does not apply to strongly coupled CFTs. See [30, 31, 32, 33] for examples.

To summarize, by using the universal term in the embedding functions eq.(49) and the universal part of the boundary condition eq.(7), we succeed to derive the boundary contributions to Weyl anomaly for 3d and 4d BCFTs. Since we do not need to assume the exact position of , the holographic derivations of boundary Weyl anomaly here is very strong. On the other hand, since the terms including and automatically cancel each other out in the above calculations, so far we cannot distinguish our proposal (9) from the other possibilities such as eq.(7) with . We will solve this problem in the next section.

## 3 Holographic Renormalization of BCFT

In this section, we develop the holographic renormalization for BCFT. We find that one should add new kinds of counterterms on boundary in order to get finite action. Using this scheme, we reproduce the correct boundary Weyl anomaly eqs.(55,56), which provides a strong support for our proposal eq.(9).

### 3.1 3d BCFT

Let us use the regularized stress tensor [34] to study the boundary Weyl anomaly. This method requires the knowledge of and thus can help us to distinguish the proposal (9) from the other choices. we will focus on the case of 3d BCFT in this subsection.
The first step is to find a finite action by adding suitable covariant counterterms[34]. We obtain

 Iren = ∫Ndx4√G(R−2Λ)+2∫Qdx3√h(K−T)+2∫Mdx3√g(K−2−12RM) (57) +2∫Pdy2√σ(θ−θ0−KM),

where includes the usual counterterms in holographic renormalization [34, 35], is a constant [5], is the Gibbons-Hawking-York term for on . Notice that there is no freedom to add other counterterms, except some finite terms which are irrelevant to Weyl anomaly. For example, we may add and to . However, these terms are invariant under constant Weyl transformations. Thus they do not contribute to the boundary Weyl Anomaly. In conclusion, the regularized action (57) is unique up to some irrelevant finite counterterms.

From the renormalized action, it is straightly to derive the Brown-York stress tensor on

 Bab=2(KMab−KMσab)+2(θ−θ0)σab (58)

In sprint of [5, 34, 35], the boundary Weyl anomaly is given by

 ⟨Taa⟩P=limz→0Baaz2=limz→04(θ−θ0)−2KMz2, (59)

where , and . Actually since we are interested only in boundary Weyl anomaly, we do not need to calculate all the components of Brown-York stress tensors on . Instead, we can play a trick. From the constant Weyl transformations , , and , we can read off the boundary Weyl normally as

 ∫Pdy2√σ0⟨Taa⟩P=∫Pdy2√σ(4(θ−θ0)−2KM), (60)

which agrees with eq.(59) exactly.

Substituting eqs.(50-54) into eq.(59), we obtain

 ⟨Taa⟩P=−14sech2(ρ)[48a3+sinh(ρ)(2R+6q−3k2−6Trk2)+sinh(3ρ)(2q−k2−4Trk2)] (61)

Comparing eq.(61) with eq.(55), we find that they match if and only if

 a3=148sinh(ρ)(cosh(2ρ)(−2R−4q+k2+10Trk2)−4R−8q+3k2+12Trk2), (62)

which is exactly the solution to our proposed boundary condition (9). One can check that eq.(7) with the other choices gives different and thus can be ruled out. Following the same approach, we can also derive boundary Weyl anomaly for 4d BCFT, which agrees with eq.(56) if and only if and are given by the solutions to condition (9). This is a very strong support to the boundary condition (9) we proposed.

To end this section, let us talk more about the stress tensors on . In general, since the Brown-York stress tensor on is non-vanishing, we have

 δIren=12∫M√g0TijMδg(0)ij+12∫P√σ0Tabδσ0ab+12∫Q√hTαβQδhαβ (63)

From the viewpoint of BCFT, the variations of effective action should takes the form

 δIeff=12∫M√g0TijMδg(0)ij+12∫P√σ0(Tabeδσ0ab+JδO) (64)

where and are the currents and operators on , respectively. After the integration along on , we can identify with . Since , integration of on can also contribute to the stress tensor on . So and are different generally. Interestingly, they always yield the same Weyl anomaly due to and the fact that the integration on , i.e. , cannot produce terms of order . An advantage of