On new proposal for holographic BCFT

Journal of High Energy Physics, Apr 2017

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.

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On new proposal for holographic BCFT

Received: February On new proposal for holographic BCFT Chong-Sun Chu 0 1 2 3 4 Rong-Xin Miao 0 1 2 3 4 Wu-Zhong Guo 0 1 2 3 0 Open Access , c The Authors 1 National Tsing-Hua University , Hsinchu 30013 , Taiwan 2 Hsinchu 30013 , Taiwan 3 Physics Division, National Center for Theoretical Sciences 4 Department of Physics, National Tsing-Hua University This paper is an extended version of our short letter on a new proposal for holographic boundary conformal eld, 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 nd 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 nd 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. ArXiv ePrint: 1701.07202 1Corresponding author. AdS-CFT Correspondence; Anomalies in Field and String Theories; Confor- - mal Field Theory 1 Introduction 2 Holographic boundary Weyl anomaly 2.1 PBH transformation 2.2 Boundary Weyl anomaly 3 Holographic renormalization of BCFT 3.1 3d BCFT 3.2 4d BCFT 4 General boundary condition 5 Holographic entanglement entropy 4.1 4.2 5.1 General boundary Weyl anomaly I General boundary Weyl anomaly II General formula Boundary e ects on entanglement Entanglement entropy for stripe 6 Entanglement wedge 7 Conclusions and discussions A Another derivation of (2.36) B Boundary Weyl anomaly for the proposal of [3] C Derivations of boundary contributions to Weyl anomaly Introduction BCFT is a conformal eld theory de ned on a manifold M with a boundary P 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 d dimensional manifold M to a d + 1 dimensional asymptotically point of view of the UV/IR relation [4] of AdS/CFT correspondence since the presence of boundary in the eld theory introduce an IR cuto and this can be naturally implemented in the bulk with the presence of a boundary. Conformal invariance on M requires that N is part of AdS space. The key point of holographic BCFT is thus to determine the location of boundary Q in the bulk. For interesting developments of BCFT and related topics please see, for example, [5{16]. I = where T is a constant and boundaries M and Q, which makes a well-de ned variational principle on the corner P [17]. Notice that T can be regarded as the holographic dual of boundary conditions of BCFT since it a ects the boundary entropy (and also the boundary central charges, see (2.43), (2.44) below) which are closely related to the boundary conditions of BCFT [3, 5]. Considering the variation of the on-shell action, we have I = For conformal boundary conditions in CFT, Takayanagi [3] proposes to impose Dirichlet boundary condition on M and P , gij jM = abjP = 0, but Neumann boundary condition on Q. And the position of the boundary Q is determined by the Neumann boundary condition = 0: For more general boundary conditions which break boundary conformal invariance even locally, [3, 5] propose to add matter elds on Q and replace eq. (1.3) by 1 T Q ; where we have included 2T h in the matter stress tensor T Q on Q. For geometrical shape of M with high symmetry such as the case of a disk or half plane, (1.3) xes the location of Q and produces many elegant results for BCFT [3, 5, 6]. However since Q is of codimension one and its shape is determined by a single embedding function, (1.3) 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-de ned BCFT with general boundaries. To solve this problem, [5] propose to take into account backreactions of Q. For 3d BCFT, they show that one can indeed nd perturbative solution to (1.3) if one take into account backreactions to the bulk spacetime. In other words although not all the shapes of boundary P are allowed by (1.3) in a given spacetime, by carefully tuning the spacetime (which is a solution to Einstein equations) one can always make (1.3) 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 Q for di erent boundaries of the BCFT. As motivated in [3, 5], the conditions (1.3) and (1.4) 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 elds as long as they satisfy various energy conditions. As a result, it seems one can put the boundary Q at almost any position as one likes. Besides, it is unappealing that the holographic dual depends on the details of matters on another boundary Q. Finally, although eq. (1.4) 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 Q determined by a new condition (1.9). This condition is consistent and provides a uni ed treatment to general shapes of P . Besides, as we will show below, it yields the expected boundary contributions to the Weyl anomaly. Instead of imposing Neumann boundary condition (1.3), we suggest to impose the mixed boundary conditions on Q [18]: 0 0 = 0; 0 0 h 0 0 = 0: 0 0 are the projection operators satisfying 1 1 = 1 1 . Since we could impose at most one condition to x the location of the co-dimension one surface Q, we require trAB = 1 from + + = +. Now the mixed boundary condition (1.5) becomes 0 0 = A = 0; are non-zero tensors to be determined. It is natural to require that eq. (1.7) to be linear in K so that it is a second order di erential equation for the embedding. Thus we propose the choice A = h in [18]. In this paper, we will provide more evidences for this proposal. Besides, we nd that the other choices such as = 1h 1; 2 6= 0; all lead problems. To sum up, we propose to use the traceless condition = 2(1 d)K + 2dT = 0 to determine the boundary Q. Here TBY = 2K is the Brown-York stress tensor on Q. In general, it could also depend on the intrinsic curvatures which we will treat in section 4. A few remarks on (1.9) 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 (1.4) [17]. However, here Q 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 Q has no back-reaction on the geometry just as the boundary M . 3. Eq. (1.9) implies that Q is a constant mean curvature surface, which is also of great interests in both mathematics and physics [19] just as the minimal surface. 4. (1.9) reduces to the proposal by [3] for a disk and halfplane. And it can reproduce all the results in [3, 5, 6]. 5. Eq. (1.9) 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 di culty met in [3, 5]. In fact as we will show in the appendix the proposal (1.4) of [3] Since b1 is expected to satisfy a c-like theorem and describes the degree of freedom on the boundary, thus it is important for b1 to be non-zero. Let us recall that in the presence of boundary, Weyl anomaly of CFT generally pick up a boundary contribution hTaaiP in addition to the usual bulk term i M , i.e. Tii = i M + (x?) hTaaiP , where (x?) is a delta function with support on the boundary P . Our proposal yields the expected boundary Weyl anomaly for 3d and 4d BCFT [20{22]: hTaaiP = c1R + c2Trk2; hTaaiP = Ebdy + b1Trk3 + b2Cac bckba; d = 3; d = 4; CFTs dual to Einstein gravity, R is intrinsic curvature, kab is the traceless part of extrinsic curvature, Cijkl is the Weyl tensor on M and ( Ebdy) is the boundary terms of Euler 4 density E4 used to preserve the topological invariance Since Q is not a minimal surface in our case, our results (2.43), (2.44) are non-trivial generalizations of the Graham-Witten anomaly [23] for the submanifold. The paper is organized as follows. In section 2, we study PBH transformations in the presence of submanifold which is not orthogonal to the AdS boundary M and derive the boundary contributions to holographic Weyl anomaly for 3d and 4d BCFT. In section 3, we investigate the holographic renormalization for BCFT, and reproduce the correct boundary Weyl anomaly obtained in section 2, which provides a non-trivial check of our proposal. In section 4, we consider the general boundary conditions of BCFT by adding intrinsic curvature terms on the bulk boundary Q. In section 5, we study the holographic entanglement entropy and boundary e ects on entanglement. In section 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 section 7. The paper is nished 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 Q. The result agrees with that obtained in section 2 using the PBH transformations. In appendix B, we show that the proposal of [3] always make vanish the central charges c2 and b1 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. and ab are the metrics in N; M; Q and P , respectively. We = (1; : : : ; d + 1), i = (1; : : : ; d), = (1; : : : ; d) and a = (1; : : : ; d 1). The curvatures are de ned by R = @ = R and R = R . The extrinsic pointing outward from N to Q. Note added. Two weeks after [18], there appears a paper [51] which claims that our calculations of boundary Weyl anomaly (2.43), (2.44) are not correct. We nd they have ignored important contributions from the bulk action IN for 3d BCFT and the boundary action IQ for 4d BCFT. After communication with us, they realize the problems and reproduce our results (2.43), (2.44) 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 T as a free parameter rather than to set it zero. Otherwise, the corresponding 2d BCFT becomes trivial since the boundary entropy [3] is by allowing intrinsic curvatures terms on Q, 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 xed, which is completely natural due to the presence of more symmetry. Holographic boundary Weyl anomaly According to [24], the embedding function of the boundary Q 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 nd the leading and subleading terms of the embedding function for Q 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 Q in this approach. So the holographic derivations of boundary Weyl anomaly in this section is very strong. PBH transformation Let us rstly brie y review PBH transformation in the presence of a submanifold [24]. Consider a (p + 1)-dimensional submanifold embedded into the (d + 1)-dimensional bulk N such that it ends on a p-dimensional submanifold @ on the d-dimensional boundary M . Denote the bulk coordinates by X = (xi; ) and the coordinates on = (ya; ) = X ( ). We consider the bulk metric in the FG gauge by the PBH transformation [26]1 , then g ij can be xed ds2 = gij dxidxj g ij = 1Note that in our notation, the sign of curvatures di ers from the one of [24, 26] by a minus sign. hab = @aXi@bXj gij (X; ): = ; ha = 0 To x the reparametrization invariance on , we chose similarly the gauge xing condition Now under a bulk PBH transformation (2.3), (2.4), one needs to make a compensating di eomorphism on [24] such that stay in the gauge (2.7). This gives and ha = @a ~ h + @ ~bhab = 0 in order to As a result, Xi changes under PBH transformation as ~ = (x) and ~a = 2 Xi = ~ @ Xi where ~ is given by (2.8) and ai is given by eq. (2.4). As in the case of the metric, if one expand the embedding function in powers of , Xi( ; ya) = (X0)i(ya) + the rst leading nontrivial term can be xed by its transformation properties [24]. In PBH transformations are a special subgroup of di eomorphism which preserve the xi = ai = 1 Z d 0gij (x; 0)@j (x) + ai0(x): Here (x) is the parameter of Weyl rescalings of the boundary metric, i.e., and ai0(x) is the di eomorphism of the boundary M . To keep the position of @ require that ai0(x)j@ = 0. Next let us include the submanifold. The metric on is given by gi(j0) = 2 gi(j0) on M , we (X0)i = 0; (X2)i = one can solve the second equation of (2.11) by (X2)i = hab = @a(X0)i@b(X0)j gij (X ; ) + (0) (0)cab is the Christo el symbol for the induced metric h ab. = Q and @ = P . Inspired by [3], we relax the assumption of [24] and expand Xi in powers of p in the presence of a boundary: Xi( ; ya) = (X0)i(ya) + p (X1)i(ya) + This means that non-zero X i(ya). Then we have h a = @ Xi@aXj gij (X; ) Imposing the gauge (2.7), we get 2(X2)i@a(X0)j (g0)ij + (X1)i@a(X1)j (g0)ij + (X1)i@a(X0)j (X1)k@k g ij where ki is the trace of the extrinsic curvature of @ ki = (h0)abkaib = (h0)ab (h0)ab is the inverse of h ab which appears in the expansion: (0) (X1)i = j(X1)j ni; hij (X2)j = X i X i, hij := ninj is the zeroth order induced metric on , ki = nik and k = rini. It is worth noting that X i is on longer a vector due to the appearance of the a ne term in eq. (2.18). This is not surprising since we have imposed the gauge (2.7) which xes all the (0)jklnknl is indeed covariant under the residual gauge transformations of the reparametrization of P . Besides, note that coordinates are not vector generally, so there is no need to require X i to be a vector. What must be covariant are the nial results such Weyl anomaly and entanglement entropy. (X0)i = 0; (X1)i = (X2)i = Using the following formulas ni = ki = one can easily check that eqs. (2.17), (2.18) indeed obey the transformations (2.20), (2.21). One may also solve (2.21) directly and obtain for the normal components of X i as: ninj (X2)j = Now let us study the transformations of Xi under PBH. From eq. (2.9), we obtain (0)nnn = ijkninj nk and c1 is a parameter to be determined. Note that a term proportional to ninj @j jX j2 from (2.18) drops out automatically in (2.25) since jX (ya)j is functions of only the transverse coordinates ya, such term vanishes due to the normal derivatives. As we have mentioned, X i is no longer a vector in the normal sense due to the gauge xing (2.7). Instead, X i admit some kinds of deformed covariance under the remaining di eomorphism after xing the FG gauge (2.1) in N and world-volume gauge (2.7) on Q. It is clear that the remaining di eomorphism are the ones on M and P . The key point is that, for every di eomorphism on M , there exists compensating reparametrization on Q in order to stay in the gauge (2.7). As a result, X i is covariant in a certain sense under the combined di eomorphisms on M and Q. As we will illustrate below, the deformed gauge symmetry is useful and it xes the value of the parameter 1 to be zero. x = a1(y)p + a2(y) + ds2M = g ij dxidxj = dx2 + ab + 2xkij + x2qij + To satisfy the gauge (2.7), we should choose the coordinates on Q carefully. For example, the natural one = (y0a; ) with Notice that ni = (1; 0; also that k = niki, we obtain from eq. (2.25) = ; x = a1(y0)p ya = y0a + a2(y0) + a2(y0) = x = x0 + cx02 + O(x03) Now let us use the remaining di eomorphism to x the parameter c1. Consider a remaining which keeps the position of P and the gauge eqs. (2.1), (2.7). From eqs. (2.29), (2.31), the embedding functions given by di eomorphism (2.32), we have x0 = x cx2 + O(x3) = a1(y0)p Since the new coordinate x0 satis es the gauge (2.1), (2.7), it must take the form (2.25) eq. (2.25), we get a02(y0) = c a12(y0) + 2cc1 for the new coordinate x0. Comparing eq. (2.34) with the coe cients of in eq. (2.33), we As a summary, by using the PBH transformations and the covariance under remaining di eomorphism, we nd the leading and subleading terms of embedding functions are universal and take the following form (X1)i = j(X1)j ni; (X2)i = In the Gauss normal coordinates (2.26), the embedding function has very elegant expression x = a1(y)p These are the main results of this section. One may still doubt eq. (2.36) due to the noncovariance. Actually, we can derive it from the covariant equation (1.9) together with the gauge (2.7). So it must be covariant under the remaining di eomorphism. This is a nontrivial check of our results. Please see the appendix for the details. Besides, we have checked other choices of boundary conditions such as eq. (1.7) with A = h They all yield the same results eqs. (2.35), (2.36), (2.37). This is a strong support for the universality. 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 P below. Let us quickly recall our main setup. Consider the asymptotically AdS metric ds2 = dz2 + gij dxidxj where z = p , gij = gi(j0) + z2gi(j1) + , gi(j0) is the metric of BCFT on M and gi(j1), xed uniquely by the PBH transformation, is given by (2.2). Without loss of generality, we choose Gauss normal coordinates for the metric on M ds2M = gi(j0)dxidxj = dx2 + ab + 2xkij + x2qab + x3lab + Expanding it in z, we have x = a1z + a2z2 + + (bd+1 ln z + ad+1)zd+1 + where ai and bd+1 are functions of ya. By using the PBH transformation, we know that a2 is universal and can be expressed in terms of a1 and the extrinsic curvature k through eq. (2.37). a1 can be determined by the boundary condition on Q. Noting that K + O(z), we get the leading term of eq. (1.7) as = 0; denotes higher order terms in z. It is remarkable that we can solve a1 from eq. (2.41) without any assumption of Aij except its trace is nonzero. In other words, we can solve a1 from the universal part of the boundary conditions. From eqs. (2.37), (2.41), we nally obtain T = (d a1 = sinh a2 = where we have re-parameterized the constant T in terms of , which can be regarded as the holographic dual of boundary conditions for BCFT. That is because, as will be clear a ects the boundary central charges as the boundary conditions do. It should be mentioned that one can also obtain a1; a2 by directly solving the boundary condition eq. (1.9) or eq. (1.7) with A = h for (T; a1; a2) but di erent results for (a3; a4; . They yield the same results 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. (2.38){(2.42) into the action (1.1) and selecting the logarithmic divergent terms after the integral along x and z, we can obtain the boundary Weyl anomaly. We note that IM and IP do not contribute to the logarithmic divergent term in the action since they have at most singularities in powers of z 1 but there is no integration alone z, thus there is no way for them to produce log z terms. We also note that only a2 appears in the nal results. The terms including a3 and a4 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 hTaaiP = sinh hTaaiP = 1 Ebdy + cosh(2 ) cosh(2 )Cac bckba; for 4d BCFT. for 3d BCFT; which takes the expected conformal invariant form [20{22]. It is remarkable that the coe cient of Ebdy takes the correct value to preserve the topological invariance of E4. This 4 is a non-trivial check of our results. Besides, the boundary charges c1; b1 in (1.10), (1.11) are expected to satisfy a c-like theorem [5, 7, 29]. As was shown in [3, 6], null energy condition on Q implies decreases along RG ow. It is also true for us. As a result, eqs. (2.43), (2.44) indeed obey the c-theorem for boundary charges. This is also a support for our results. Most importantly, our con dence is based on the above universal derivations, i.e., we do not make any assumption except the universal part of the boundary conditions on Q. Last but not least, we notice that our results (2.43), (2.44) are non-trivial generalizations of the Graham-Witten anomaly [23] for the submanifold, i.e., we nd 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 boundary central charge related to the Weyl invariant p has been suggested that the coe cient of Ck in (2.44) 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 in [29], the proposal of [22] is suspicious. It means that there could be no independent bckba. 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{33] for examples. To summarize, by using the universal term in the embedding functions eq. (2.37) and the universal part of the boundary condition eq. (1.7), 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 Q, the holographic derivations of boundary Weyl anomaly here is very strong. On the other hand, since the terms including a3 and a4 automatically cancel each other out in the above calculations, so far we cannot distinguish our proposal (1.9) from the other possibilities such as eq. (1.7) with A = h . We will solve this problem in the next section. Holographic renormalization of BCFT In this section, we develop the holographic renormalization for BCFT. We nd that one should add new kinds of counterterms on boundary P in order to get nite action. Using 3d BCFT Iren = this scheme, we reproduce the correct boundary Weyl anomaly eqs. (2.43), (2.44), which provides a strong support for our proposal eq. (1.9). Let us use the regularized stress tensor [34] to study the boundary Weyl anomaly. This method requires the knowledge of (a3; a4; ) and thus can help us to distinguish the proposal (1.9) from the other choices. we will focus on the case of 3d BCFT in this The rst step is to nd a nite action by adding suitable covariant counterterms [34]. We to Weyl anomaly. For example, we may add p there is no freedom to add other counterterms, except some nite terms which are irrelevant KM2 to IP . However, these terms are invariant under constant Weyl transformations. Thus they do not contribute to the boundary Weyl Anomaly. In conclusion, the regularized action (3.1) is unique up to some irrelevant nite counterterms. From the renormalized action, it is straightly to derive the Brown-York stress tensor In sprint of [5, 34, 35], the boundary Weyl anomaly is given by Bab = 2(KMab KM ab) + 2( hTaaiP = lim 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 P . Instead, we can play a trick. From the constant Weyl transformations ab ! e we can read o the boundary Weyl normally as 0 hTaaiP = which agrees with eq. (3.3) exactly. Substituting eqs. (2.38){(2.42) into eq. (3.3), we obtain hTaaiP = sech2( )[48a3 +sinh( ) 2R+6q 3k2 6Trk2 +sinh(3 ) 2q k Comparing eq. (3.5) with eq. (2.43), we nd that they match if and only if a3 = sinh( ) cosh(2 )( 2R 4q + k2 + 10Trk2) 8q + 3k2 + 12Trk2 ; (3.6) which is exactly the solution to our proposed boundary condition (1.9). One can check that eq. (1.7) with the other choices A = h gives di erent a3 and thus can be ruled out. Following the same approach, we can also derive boundary Weyl anomaly for 4d BCFT, which agrees with eq. (2.44) if and only if a3 and a4 are given by the solutions to condition (1.9). This is a very strong support to the boundary condition (1.9) we proposed. To end this section, let us talk more about the stress tensors on P . In general, since the Brown-York stress tensor on Q is non-vanishing, we have Iren = 1 Z 2 P 2 Q From the viewpoint of BCFT, the variations of e ective action should takes the form e = 1 Z 2 P where J and O are the currents and operators on P , respectively. After the integration the fact that the integration on Q, i.e. dzzm, cannot produce terms of order O(z0). An = 0 and e 2 advantage of Teab is that it is always nite by de nition T ab = p 0 The integration of the other components of h on Q give the new operator O on P . It is worth noting that since h is related to gij on-shell, the new operator O coming from h is also related to geometric quantity derived from gi(j0). According to [40], such geometric quantity appears naturally as the new operator on the boundary of BCFT. 4d BCFT make the action Now we study the holographic renormalization for 4d BCFT, which is more subtle. We nd that one has to add squared extrinsic curvature terms on the corner P in order to We propose the following renormalized action Iren = Similar to the case of 3d BCFT, IM includes the usual counterterms in holographic renorterm for RM on M . It is worth noting that the induced metric on Q is AdS-like, i.e., it can be rewritten into the form of eq. (2.38) except that now gij are in powers of z instead of z2. In spirit of the holographic renormalization for asymptotically AdS, one can add a constant term and an intrinsic curvature term RP into IP . However, they are not enough to make the action nite. Instead, we have to add the extrinsic curvature terms TrKQ2 on P . This maybe due to the presence of the singular corner P and the non-AdS metric on Q. Note that RP O(z2) are designed to delete the O( z1 ) divergence in the action.2 It should be mentioned that KQ ab can be regarded as new boundary operator from the viewpoints of BCFT, since it is de ned by the embedding from P to Q rather than to the spacetime where BCFT lives. On the other hand, KM ab is not an independent operator, since it is de ned by the derivatives of the metric for BCFT. As a result, if we denotes the location of P . This means there is energy owing outside P , which is not a well-de ned BCFT. For these reasons, we propose eq. (3.9) as the renormalized action. Substituting eqs. (2.38){(2.42) into the action (3.9), we can solve , a nite action. It is remarkable that a3 and a4 disappear in the divergent terms of the action (3.9) once we impose the universal relations (2.42). Thus the solutions to , are irrelevant to a3 and a4. After some calculations, we get = 0: A quick way to derive eq. (3.10) is to consider AdS5 in the bulk and choose spherical coordinates and cylindrical coordinates on M for and , respectively. Note that the new O( ) vanish for the trivial boundary condition = 0. Now we are ready to calculate the boundary contributions to Weyl anomaly. Similar to the 3d case, from the constant Weyl transformations ab ! e KM , RP ! e 2 RP and TrKQ2 ! e 2 TrKQ2, we can read o the boundary Weyl KM + (d 3)( RP + anomaly as hTaaiP = 0 hTaaiP = 2 To make eq. (3.11) nite, we solve a3 = ; (3.12) which is exactly the solution to our proposed boundary condition (1.9). Substituting the above a3 into eq. (3.11), we get 27sech2( )(48a4 +qk 6l 2kTrk2 6Trk3 +Tr(kR)+7Tr(kq)) 3 cosh(2 ) k 12q +k2 +27Trk2 +27l+90Trk3 9Tr(kR) 63Tr(kq) q +9kR+9kq 81l 13k3 +45kTrk2 54Tr(kR)+54Tr(kq) 2We have KQba O(1) and KQba O(z). Thus only the combination TrKQ2 is of order O(z2). Comparing eq. (3.14) with eq. (2.44), we nd that they match if and only if a4 = +4 cosh(2 ) 6kR +cosh(4 ) 14k3 21kq +90kTrk2 +135l+108Trk3 90Tr(kR) 144Tr(kq) 4k3 +6kq 9 3l+6Trk3 +Tr(kR) 5Tr(kq) 2k3 +9kq 18kTrk2 9 3l+12Trk3 2Tr(kR) 8TR(kq) which is again the solution to the boundary condition (1.9) we proposed. The other choices of boundary conditions give di erent a3 and a4 and thus can be excluded. In the above calculations, we have used the following formulas Trk3 = Cac bckba = kTrk2 + Trk3; Tr(kR) + in Gauss normal coordinates (2.39). Since the calculations are quite complicated, the non-patient readers can study some simple examples instead. For example, AdS in spherical coordinates and cylindrical coordinates are good enough to reproduce most of the results above. To sum up, we have developed a scheme of holographic renormalization for BCFT. We nd that it reproduces the correct boundary Weyl anomaly eqs. (2.43), (2.44) only when Q is determined by eq. (1.9). This is a non-trivial check of our proposal for holographic BCFT. General boundary condition In this section, we consider more general boundary conditions for BCFT. As we have mentioned before, the constant T in the gravitational action eq. (1.1) can be regarded as the holographic dual of boundary conditions for BCFT, since it is closely related to boundary central charges. Naturally, we propose to add intrinsic curvature terms on Q to mimic general boundary conditions. For simplicity, we focus on the case of Ricci scalar. Now the gravitational action for holographic BCFT becomes I = is a constant. Similarly, we suggest to impose the mixed boundary conditions on Q with the non-trivial one given by = 2(1 d)K + 2dT + 2 (d 2)RQ = 0: Below we will apply the methods of section 3 and section 4 to investigate the boundary contributions to Weyl anomaly. As it is expected, we nd the boundary central charges depend on the new parameter . And again, these two methods give the same results only if the bulk boundary Q is determined by the traceless-stress-tensor condition eq. (4.2). Now let us use the method of section 2 to derive the boundary Weyl anomaly. For simplicity, we focus on AdS4 with spherical coordinates and cylindrical coordinates below. The generalization to higher dimensions and other metrics is straightforward. ds2 = ds2 = dz2 + dr2 + r2d 2 + r2 sin2 d 2 dz2 + dr2 + r2d 2 + dy2 spherical coordinates cylindrical coordinates: r = r0 + sinh where k is r20 for sphere and r10 for cylinder. From the leading term of eq. (4.2), we can re-express T in terms of and . In general, we get T = (d Substituting eqs. (4.3){(4.6) into the action (4.1) and selecting the logarithmic divergent term after the integral along r and z, we can obtain the boundary Weyl anomaly. Similarly, one can check that IM and IP in the action (4.1) and a3; a4 in the embedding function (4.5) are irrelevant in the above derivations. Rewriting the nal results into covariant form, where RQ = RQ hTaaiP = 1 Ebdy + cosh(2 )(1 hTaaiP = sinh Interestingly, now the central charges with respect to R and Trk2 become independent, which implies that there are two independent boundary central charges for 3d BCFT generally. This is the expected result, since every independent Weyl invariant should correspond to an independent central charge. The above discussions can be easily generalized to higher dimensions and general metrics. For 4d BCFT, we obtain hTaaiP = 1 Ebdy + cosh(2 )(1 cosh(2 )(1 Now the central charges related to Trk3 and Cac bckba are still not independent. One can check that, by adding more general curvatures in IQ, the boundary central charges can indeed become independent. For example, let us consider the action I = 2RQ RQ RQ )+2 . Following the above approach, we derive cosh(2 )(1 We remark that in obtaining the results (4.8) and (4.10), it is necessary to consider non General boundary Weyl anomaly II In this section, we take the method of section 3 to study the boundary Weyl anomaly for general boundary conditions. Due to the Ricci scalar in IQ (4.1), we should add new a Gibbons-Hawking-York term KQ in IP . Recall that the induced metric on Q is AdS-like, i.e., it can be rewritten into the form of eq. (2.38) except that now gij are in powers of z instead of z2. In spirit of the holographic renormalization for asymptotically AdS, one can add a constant term and intrinsic curvature terms on P . Besides, from the experience of section 3, one has to add extrinsic curvature terms in order to make the action generally. This is may because of the presence of the corner P and the non-AdS metric on Q. Based on the above discussions, we propose the following renormalized action for 3d 2 KQ + RP + and 4d BCFT Ire = RM where ; ; are parameters and will be determined below. For 3d BCFT, we have 2 sech , and ; are free parameters since they are related to nite counterterms. Below, we focus on 4d BCFT. For simplicity, we focus on AdS with spherical coordinates and cylindrical coordinates. ds2 = ds2 = ds2 = dz2 + dr2 + r2d 2 dz2 + dr2 + r2d 2 + sin2 d 2 + dy22 dz2 + dr2 + r2d 2 + dy12 + dy22 spherical coordinates; cylindrical coordinates I; cylindrical coordinates II; r = r0 + sinh z2 + a3z3 + a4z4 + where k take values ( r30 ; r20 ; r10 ) for the metrics (4.13), (4.14), (4.13), respectively. From the traceless-stress-tensor condition eq. (4.2), we can solve the above embedding function. For the spherical metric (4.13), we can get exact solution r = For the rst kind of cylindrical metric eq. (4.14), we obtain a3 = a4 = a3 = a4 = As for the second second of cylindrical metric eq. (4.15), we have (9 sinh(2 ) 4 (9 cosh(2 ) ( 84 sinh + 44 sinh(3 ) + cosh 11 cosh(3 )) (9 sinh(2 ) 36 cosh(2 ) + 28 ) (4 (47 sinh(3 ) 81 sinh( )) + 19 cosh( ) 47 cosh(3 )) Substituting eqs. (4.13){(4.16) into the action (4.11) and requiring the action nite, = 4 sech : Again, a3 and a4 do not appear in the divergent terms of the action (4.11). Actually, we can use only two of the three examples in eqs. (4.13), (4.14), (4.15) to derive eq. (4.22). The third one provides a double check of our calculations. Now we are ready to calculate the boundary contributions to Weyl anomaly. Similar to the cases of section 3, with the help of constant Weyl transformations, we can read o the boundary Weyl anomaly as = 2 d +(d 1)( 2 (d 2)KQ +(d 3)( RP + TrKQ2) ; (4.23) Substituting eqs. (4.13){(4.22) into the above formula, we can derive the boundary Weyl anomaly for the three examples in eqs. (4.13), (4.14), (4.15), which exactly agrees with the result eq. (4.8) of last subsection. This is a strong support to our proposal of holographic Holographic entanglement entropy General formula Let us go on to discuss the holographic entanglement entropy. Following [36, 37], it is not di cult to derive the holographic entanglement entropy for a d-dimensional BCFT, which is also given by the area of minimal surface where A is a (d 1)-dimensional subsystem on M , and A denotes the minimal surface which ends on @A. What is new for BCFT is that the minimal surface could also end on SA = the bulk boundary Q, when the subsystem A is close to the boundary P . See gure 2 for example. We could keep the endpoints of extreme surfaces A0 freely on Q, and select the one with minimal area as A. It follows that A is orthogonal to the boundary Q when they intersect nQj A\Q = 0: Here nQ is the normal vector of Q and naA are the two independent normal vectors of A. It is easy to see that if A0 is not normal to Q, one can always deform A0 to decrease the area until it is normal to Q.3 Let us take an example in gure 1 to illustrate this. For simplicity, we focus on static spacetime and constant time slice. Then the normal vector of A alone time is orthogonal to nQ trivially. It is worth keeping in mind that the induced metric on constant time slice is Euclidean and positive de nite. Below we focus on the case extreme surfaces OA in gure 1, where O is a xed point in the bulk, and OA is not normal to the boundary Q. Then select an arbitrary point B alone OA as long as it is near enough to the boundary. Starting from B, we can construct a minimal surface BA0 that is normal to the boundary and ending on the boundary at A0 . Since the metric is positive de nite and B is near enough to the boundary, we have BA > BA0 and thus OBA > OBA0. Next we construct a minimal surface OA0 linking A0 and O. By de nition, it is smaller than OBA0. As a result, we have OBA > OBA0 > OA0. If OA0 is not orthogonal to Q either, we can repeat the above approach again and again until the extreme surface is normal to Q. Now it is clear that the minimal area condition leads to the orthogonal condition (5.2). Another way to obtain the orthogonal condition is that, otherwise there will arise problems in the holographic derivations of entanglement entropy by using the replica trick. In the replica method, one considers the n-fold cover Mn of M and then extends it to the bulk as Nn. It is important that Nn is a smooth bulk solution. As a result, Einstein 3We thank Dong for emphasizing this point to us. equation should be smooth on surface A. Now the metric near A is given by [37] ds2 = " @zf @zf exactly the orthogonal condition (5.2). It should be mentioned that the smooth requirement of the general boundary conditions (1.7) may yield more constraints in addition to the orthogonal condition (5.2). includes higher curvature terms, sometimes the smooth requirement even leads to contradictions. This can also help us to exclude a large class of A in the boundary condition (1.7). Since our boundary condition (1.9) yields the expected orthogonal condition (5.2), this is also a support to our proposal. In summary, the holographic entanglement entropy for BCFT is given by RT formula (5.1) together with the orthogonal condition (5.2). As we will show below, there appear many new interesting properties for entanglement due to the presence of boundaries. Boundary e ects on entanglement Let us take an simple example to illustrate the boundary e ects on entanglement entropy. Consider Poincare metric of AdS3 ds2 = dz2 + dx2 presence of boundary, now there are two kinds of minimal surfaces, one ends on Q and the other one does not. It depends on the distance d that which one has smaller area. From eqs. (5.1), (5.2), we obtain time, yi are coordinates along the surface, x n1 , r is coordinate normal to the surface, + 2 n is the Euclidean Einstein equations Rzz = extrinsic curvature of Q as where dc = l e 2 + 1 l is the critical distance. The parameter can be regarded as the holographic dual of the boundary condition of BCFT, since it a ects the boundary entropy [3] and the boundary central charges (2.43), (2.44) as the boundary condition does. It is remarkable that entanglement entropy (5.6) depends on the distance d and boundary when it is close enough to the boundary. This is the expected property from the viewpoint of BCFT, where the correlation functions depend on the distance to the boundary [40]. IA = SACF T where SCF T is the entanglement entropy when the boundary disappears or is at in nity. A For simplicity, we focus on the case 0. In the holographic language, SCF T is given by A SBCF T and IA is always non-negative. It is expected that boundary does not a ect the A eq. (5.7). As a result, IA is not only non-negative but also nite. For the example discussed above, we have IA = which is indeed both non-negative and nite. Actually in this simple example, IA is just one half of the mutual information between A and its mirror image, so it must be non-negative nite. See gure 2 for example. For this simple case, the metric at the mirror image O0 of a point O is given by the metric at the point O. One should keep in mind that the mirror image is only an auxiliary tool, there is no real spacetime outside the boundary Q. Entanglement entropy for stripe In this subsection, we study the entanglement entropy of stripe in general dimensions. Consider a BCFT de ned on M the half space x x1 < 0, and consider a subsystem A given by the constant time slice l < x1 < 0, L < x2; x3; = arctan(sinh( )), where is the parameters that we used in previous section. is the angle between Q and M . See gure 3. Let the minimal surface dary condition x(0) = l: The induced metric on the minimal surface and gives the equation of motion ds2 = (1 + x0(z))dz2 + Pid=21 dxi2 = C; C = constant; x = (x0; z0) be the coordinates of the point P 0 where Q and intersect. It is x0 z0 tan . Denote the unit normal vectors of Q by nQ, the unit normal vector of x(z0) = Now we solve (5.11) together with the boundary conditions (5.9), (5.12). Using the condition (5.12) we have x0(z0) = d 1 = zd 1 cos ; x0(z) = We also have the relation This allow us to solve for z , z = x0 + l = x0(z)dz + F ( ) = tan (cos ) d 1 tan (cos ) d 1 ; A2 = A0 where B(x; a; b) is the incomplete beta function. When d 3, there always exist some can also show that c is a monotone decreasing function of d. In particular in the limit In the limit x = have two minimal surface solutions, the desired solution is the one with a smaller area. Consider rst the surface x = l. It is easy to obtain its area where A0 is the area of the entangling surface. The area of the other surface is A1 = =z xd 1p 2 2 (d 1) 2 (d 1) Here is the cuto and 2F1(a; b; c; z) is the hypergeometric functions. In the limit z ! +1, as a result, A2 ! A1. One could also show A2 is a monotone decreasing function < c. Therefore in the region < c, A2 < A1, the entanglement entropy is We remark that our holographic calculation suggests that there is a phase transition at the critical value stripe l. But it is probably related to the shape of the entangling surface in general. As the is expected to be dual to the boundary condition of BCFT, it is interesting to explore what is the nature of the boundary condition in the eld theory that would lead to this phase transition in the BCFT. Entanglement wedge According to [38, 39], a sub-region A on the AdS boundary is dual to an entanglement wedge EA in the bulk where all the bulk operators within EA can be reconstructed by using only the operators of A. The entanglement wedge is de ned as the bulk domain of dependence of any achronal bulk surface between the minimal surface A and the subsystem A. Apparently, it seems to con ict with the holographic proposal of BCFT by [3] and us, where the holographic dual of A is given by N , which is larger than EA generally. Of course, there is no contradiction. That is because CFT and BCFT are completely di erent theories. For CFT, although we do not know the information outside, there still exists spacetime outside A. As for BCFT, there is no spacetime outside A at all. Besides, we should impose suitable boundary conditions for BCFT, while there is no need to set boundary condition on the entangling surface for CFT. It is interesting to study the entanglement wedge in the framework of AdS/BCFT. For simplicity, we focus on the static spacetime and constant time slice. Recall that the entanglement wedge is given by the region between the minimal surface A and the subsystem A on M . A key observation is that entanglement wedge behaves a phase transition and becomes much larger than that within AdS/CFT, when A is increasing and approaching to the boundary. See gure 5 for example. This phase transition is important for the selfconsistency of holographic BCFT. If there is no phase transition, then the entanglement wedge is always given by the rst kind (left hand side of gure 5). When A lls with the whole boundary M and P , there are still large space left outside the entanglement wedge, which means there are operators in the bulk cannot be reconstructed by all the operators on the boundary. Thanks to the phase transition, the entanglement wedge for large A is given by the second kind (right hand side of gure 5). As a result all the bulk operators can be reconstructed by using the operators on the boundary. Conclusions and discussions In this letter, we have proposed a new holographic dual of BCFT, which can accommodate all possible shapes of the boundary P with a uni ed prescription. The key idea is to impose the mixed boundary condition (1.9) so that there is only one constraint for the co-dimension one boundary Q. In general there could be more than one self-consistent boundary conditions for a theory [41], so the proposals of [3] and ours have no contradiction in principle. However, the proposal of [3] is too restrictive to include the general BCFT. The main advantage of our proposal is that we can deal with all shapes of the boundary P easily and that it can accommodate nontrivial boundary Weyl anomaly as is needed in a general BCFT. It is appealing that the bulk boundary Q is given by a constant mean curvature surface, which is a natural generalization of the minimal surface. Applying the new AdS/BCFT, we obtain the expected boundary Weyl anomaly for 3d and 4d BCFT and the obtained boundary central charges satisfy naturally a c-like theorem holographically. As a by-product, we give a holographic disproof of the proposal [22] and is sensitively dependent on the choices of boundary conditions of non-free BCFT. Besides, we nd the holographic entanglement entropy is given by the RT formula together with the condition that the minimal surface must be orthogonal to Q if they intersect. The presence of boundaries lead to many interesting e ects, e.g. phase transition of the entanglement wedge. Of course, many things are left to be explored, for instance, the holographic Renyi entropy [43, 44], the edge modes [45, 46], the shape dependence of entanglement [47, 48], the applications to condensed matter and the relation between BCFT and quantum information [49]. Finally, it is straightforward to generalize our work to Lovelock gravity, higher dimensions and general boundary conditions. Acknowledgments We would like to thank X. Dong, L.Y. Hung, F.L. Lin for useful discussions and comments. This work is supported in part by the National Center of Theoretical Science (NCTS) and the grant MOST 105-2811-M-007-021 of the Ministry of Science and Technology of Taiwan. Another derivation of (2.36) In section 2.1, we have obtained the key result (2.36) from the PBH transformation together with the explicit requirement of covariance under the residual di eomorphism of the gauge xing condition (2.7). In this appendix, we derive eq. (2.36) directly from the covariant equation (1.9) and the gauge xing (2.7). The analysis is manifestly covariant with respect to (2.7) and provides an independent derivation of the (2.36). on Q. The components of K small , we have K = Kab = Ki = 1+(Xi)2 4 3=2 2 1 XiXi+ 12(0)ki XiXkXm (1) (2) (1) (1) (1) m 1+(Xi)2 1+(Xi)2 To compute K, we note that the extrinsic curvature K on Q is 1 XiXi+ 12(0)ki XiXkXm X(1)i+ 1+(Xi)2 K = nQK ; K = @ @ X is the Christo el symbol for the induced metric h and nQ is the unit normal vector can be worked out easily. Expanded in powers of p 1 (0)kmi X(1)iX(1)kX(1)m+O(1); (A.3) p XiXi+ 12(0)ki XiXkXm (1) (2) (1) (1) (1) m (1+(Xi)2)2 4 Xikaib+O( ); (A.4) Kaib = p 1+(Xi)2 1+(Xi)2 1+(Xi)2 (1+(Xi)2)2 Since nQ @X = 0, we have @ The trace K is nQ = n K = (p+1)p niQX(1)i 2 (p + 1)(Xi)2 + pnjQX(2)j + niQki+ (1) Generally n and ni can be expanded as nQ = p1 (n0)Q + (n1)Q + ; niQ = p1 (n0)iQ + (n1)iQ + Taking them to (A.7) we have 1 + (Xi)2 (n0)Q = (n1)Q = 1 + (Xi)2 Using the relation h~ i = 0. Hence where ds2Q = h d d and as a result boundary submanifold P . Using n @aX to P . We have the following relations X(1)i = (n0)iQ = = G (n0)iQX(1)i = (n1)iQX(1)i = X(1)iX(2)i + 12 (0)kmi X(1)iX(1)kX(1)m 2 q(X(1)i)2(1 + (X(1)i)2)3=2 where ni is the unit normal vector of P . Taking (A.9)(A.14)(A.16) into (A.8) we have K = (p+1)(n0)iQX(1)i +p (n0)iQki p (1+(X(1)i)2)3=2q(X(1)i)2 niX(1)i = niX(2)i = eqs. (2.17), (2.18). Combining eqs. (A.18), (A.19) and eqs. (2.17), (2.18) together, we recover exactly eqs. (2.35), (2.36). Boundary Weyl anomaly for the proposal of [3] In this appendix, we show that the BC (1.4) proposed by [3] always make vanish the central charges c2 and b1 in the boundary Weyl anomaly (1.10), (1.11) for 3d and 4d BCFT. Since b1 is expected to satisfy a c-like theorem and describes the degree of freedom on the boundary, thus it is important for b1 to be non-zero. We emphasis that this holds for any energy-momentum tensor T Q on Q as long as the BC (1.4) holds. In this sense, the proposal of [3] is too restrictive to include the general BCFT, in particular, the non-trivial Let us rst start with a simple example to see explicitly how c2 and b1 vanish in the proposal of [3]. Consider AdS with cylindrical coordinates on M eqs. (4.4), (4.15) so that only the Trkd 1 terms are non-vanishing in the Weyl anomaly (1.10), (1.11). We note that in the present case, the equation (1.4) does not admit a solution with a constant T term and one needs to include on Q either nontrivial matter elds or higher derivatives gravitational action terms. For simplicity, let us consider the addition of an intrinsic Ricci scalar RQ on Q. In other words, we focus on the action (4.1). Requiring all the components of stress tensors on Q vanishing, we get the following exact solutions ; T = (d 1) coth(2 ); r = r0 + sinh Substituting eqs. (4.4), (4.15), (B.1) into the action (4.1) and selecting the logarithmic divergent term after integration alone r and z, we nd hTaaiP = 0 generalized to include general higher curvature terms, i.e., we replace RQ by L(RQ action (4.1). Using the trick of [27], we expand L(RQ ) around a `background-curvature' h ). Then we nd only the rst a few terms up to d 1 contribute to the boundary Weyl anomaly for d-dimensional BCFT. We have worked out the cases for 3d and 4d BCFT on cylinders and nd they all yield eq. (B.2). So the boundary Weyl anomaly c2; b1 indeed vanish for 3d and 4d BCFT in the proposal of [3]. We have also constructed a model with only matter on Q (non-minimally Now let us present the general proof. Consider the following action I = T + Lm( )) + 2 where Lm( ) is the Lagrangian of matter elds on Q. According to [25], we can derive the Weyl anomaly as the logarithmic divergent term of the gravitational action. Recall that IM and IP do not contribute the logarithmic divergent term.4 Considering the variation of the on-shell action, we have I = 1 T Q 4Instead of ln z, IM and IP may contribute terms such as zn ln z with n > 1, which vanish in the limit denotes E.O.M for matter elds on Q, P is the conjugate momentum of along the direction nP , which is the normal vector pointing from Q to P . If one impose the BC (1.4), one obtain for arbitrary boundary variations gij , where we have used the EOM E due to the BC (1.4). This is the main reason why the proposal of [3] yields trivial boundary central charges c2 and b1 in eqs. (1.10), (1.11). In fact as we will show below, the integration on M and P in eq. (B.5) are not su cient to produce the full structures of the boundary I = Weyl anomaly. of the Weyl anomaly A To proceed, we note that the logarithmic divergent term of I is equal to the variation Ijln = A = ( I)P jln = i M ). Then from eqs. (B.5), (B.6), we get jln = 0 M ( gij ) (B.10) Since there is no integration alone z on M and P , the only way to produce ln z in I is that the integral element includes ln z. There are two possible sources for ln z: one is the expansion of gij and the other one is the expansion of the embedding function (2.40) gij = gi(j0) + z2gi(j1) + x = a1z + a2z2 + + zd(gi(jd=2) + hij (d=2) ln z) + + (bd+1 ln z + ad+1)zd+1 + ; for even d Note that there is no ln z term in gij when d is odd. As a result, there is no bulk Weyl anomaly T i same order O(kd) where k is the trace of the extrinsic curvature of P . In general, E.O.M for matter elds E = 0 will also give ln z terms in . However, such terms are expected to yield new contributions to Weyl anomaly in addition to the geometric Weyl invariant such as eqs. (1.10), (1.11). See [35, 50] for some examples. Since here we are interested only in the geometric Weyl invariant which de nes c2 and b1, we will ignore these ln z terms of = 0) in this appendix. Of course, can inherit ln z terms from gij (B.7) and Let us rstly consider the case without the boundary P , i.e., the standard case of AdS/CFT. From the above discussions, we must have ( I)M jln = Khij ) gij = When d is odd, we have T i in gij and thus in ( I)M . When d is even, one can check eq. (B.9) by straightforward In the presence boundary P , the formulas of ( I)M and (pg0 T i any change. So eq. (B.9) is still satis ed up to a possible boundary term ) do not have M on P from Notice that only the terms linear in hij contribute to hTaaiP at all. Actually, the terms linear in hij z ln z, which vanish in the limit z ! 0. Thus, we have (d=2) and bd+1 take the form 0 M ( gij ) = 0 for arbitrary boundary variations. For 3d BCFT, ary Weyl anomaly (1.10). For 4d BCFT, T i M ( gij ) are non-zero. Note that M ( gij ) is proportional to the Weyl tensor C and its derivatives. Therefore for the simple implies that R p 0 hTaaiP is a topological invariant. So b1 related to Trk3 must vanish in the boundary Weyl anomaly (1.11). Notice that in this argument we only require Cijkl to vanish at the boundary P . It can be nontrivial inside M . For instance, the following M ( gij ) = 0. This together with eq. (B.11) metric gij (0) with a free parameter c works well for our purpose: = 0 and M ( gij ) disappear. Eq. (B.11) implies that the action (B.3) and the proof proceeds exactly the same way. Therefore we independent of the form of the matter or gravitational action, the proposal of [3] always components of stress tensors on Q vanish T require only the trace of the stress tensor to vanish as in our proposal then the integral on Q in eq. (B.4) is no longer zero and one can indeed obtain non-trivial boundary central charges c2 and b1 in eqs. (1.10), (1.11). Finally we remark that, as the careful readers may notice also, the solution (B.1) with surprising since the parameter does not lie in the \physical range". In fact the solutions to our proposal TBY stress tensors. Generally as long as the parameters of the higher curvature terms lie in some \physical" region, there is an unique solution which satis es the universal law for a2 and give the non-trivial boundary central charges. We select this kind of solution as the physical one. However when one set the parameters of higher curvature terms to the critical value as in eq. (B.1), the physical solution is replaced by a di erent solution which 2(d 1) Trk as in eq. (2.42). This is not violate the universal law of a2 = 2(d 1) Trk. Actually, the same situation already appears in [24]: for higher curvature gravity such as Lovelock gravity, the bulk entangling surfaces obtained by minimizing the entropy functional are not unique. One usually select the one which can be continuously reduced to the minimal surface when the parameters of higher curvature terms are all turned o . This kind of surface always satisfy an universal relation for a2 [24]. However if one set the parameters of higher curvature terms to the critical value as in eq. (B.1), there exist solutions which violate the universal relation [24]. Thus, the universality of a2 in our proposal has the same meaning as the one in [24]: it holds as long as the parameters of higher curvature terms lie in the physical ranges. Curiously, the proposal of [3] has solution only if the parameter of higher curvature terms takes the critical value central charges. = 2(d1 2) coth and this prevents the realization of non-trivial boundary Derivations of boundary contributions to Weyl anomaly In section 2.2, we have shown the key steps of holographic derivations of boundary contributions to Weyl anomaly. Here we provide more details. We work in Gaussian normal coordinate and nd the following formulas useful: T = (d x = sinh z2 + a3z3 + a4z4 + Since we want to consider the general boundary condition (1.7), we keep a3 and a4 o metric (2.39), we have G and the BCFT boundary G = h = 1 q(g0)p1+gxxx02 1+ 1 z2(g1)aa + 2(d 2)(d 1) g = k2 +q 2Trk2 x k3 +3kq 6kTrk2 +3l+8Trk3 6Tr(kq) x3 + (g1)ii = 2 2q +3Trk2 2kTrk2 +kq +3l+6Trk3 +Tr(kR) 5Tr(kq) denotes terms of order O(k4) which do not contribute to boundary Weyl anomaly for 3d BCFT and 4d BCFT. For the boundary action IQ = 2 RQ T ), we need 3dl 4dTrk3 +4dTr(kq)+kq 2kTrk2 +6l+10Trk3 9Tr(kq)+Tr(kR) z2x+ 2(d 2)(d 1) (g1)aa = R k2 +Trk2 kq 2kTrk2 +2Trk3 nu = pgxx +x02 where x0 = @zx and anomaly for 3d BCFT and 4d BCFT. Tr(kq)+Tr(kR) x+ 3dl 4dTrk3 +4dTr(kq)+kq 2kTrk2 +6l+10Trk3 9Tr(kq)+Tr(kR) z2x+ denotes higher order terms irrelevant to the boundary Weyl 1 Z q(0)h48a3 3 sinh 2R+k2 2q+2Trk2 +sinh(3 ) Now we are ready to derive the boundary Weyl anomaly. Substituting the above formulas into the action (1.1) and selecting the logarithmic divergent terms after the integral along x and z, we can obtain the boundary Weyl anomaly. For 3d BCFT, we have IN = IQ = ln 1 Z q(0)h48a3 +sinh( ) 2R+k2 +6q 14Trk2 +sinh(3 ) where we have ignored terms without ln 1 above. Combining IN and IQ together, we get Ijln 1 = Trk2) sinh ; which exactly gives the boundary Weyl anomaly (2.43). It is remarkable that a3 and all non-conformal invariant terms automatically cancel each other out. Similarly, for 4d BCFT we have IN = ln 1 Z q(0) 1 h 576a3k sinh 576a4 2k3 +23kq 16kR 30kTrk2 +45l+72Trk3 66Tr(kq)+24Tr(kR) 4 cosh(2 )(kq 2kR+9l+12Trk3 12Tr(kq)+6Tr(kR)) +cosh(4 ) 2k3 3kq+6kTrk2 9l 24Trk3 +18Tr(kq) i IQ = ln 1 Z q(0) 1 h1728a3k sinh +1728a4 +26k3 135l 69kq 60kR Combining the above IN and IQ together, we obtain Ijln 1 = Z q(0) 1 h5k3 9k(3R+Trk2)+54Tr(kR) +54kTrk2 216Trk3 +198Tr(kq)+144Tr(kR) +12 cosh(2 ) k3 +4kq+kR 9kTrk2 +9l+30Trk3 21Tr(kq) 3Tr(kR) +3 cosh(4 ) 2k3 6kTrk2 +24Trk3 18Tr(kq)+3kq+9l +3 cosh(2 ) k3 +3k(q+R 3Trk2)+18Trk3 9Tr(kq) 9Tr(kR) i which yields exactly the boundary Weyl anomaly (2.44). In the above calculations, we have used eqs. (3.15), (3.16), (3.17). Similar to the 3d case, a3, a4 and all of the non-conformal invariant terms automatically cancel each other out in the nal results. To end this appendix, let us discuss the physical meaning of the parameter we have mentioned, can be regarded as the holographic dual of boundary conditions of BCFT since it a ects the boundary entropy [3] and also the boundary central charges (2.43), (2.44) which are closely related to the boundary conditions of BCFT. To cover the general boundary condition, it is natural to keep free rather than to set it zero. If we set BCFT. Furthermore, it is expected that the boundary central charges related to di erent conformal invariants are independent in general. As a result we must keep free. Of course, as discussed in section 4 one could add intrinsic curvature terms on Q in order to make all the boundary central charges independent. Finally, we notice that for 4d BCFT, the case special choice of boundary conditions that preserve half of supersymmetry [51]. 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On new proposal for holographic BCFT, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)089