Universality for Shape Dependence of Casimir Effects from Weyl Anomaly
PPrepared for submission to JHEP
Universality for Shape Dependence of CasimirEffects from Weyl Anomaly
Rong-Xin Miao, a, Chong-Sun Chu, a,b, a Physics Division, National Center for Theoretical Sciences,National Tsing-Hua University, Hsinchu 30013, Taiwan b Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan
E-mail: [email protected] , [email protected] Abstract:
We reveal elegant relations between the shape dependence of the Casimir effectsand Weyl anomaly in boundary conformal field theories (BCFT). We show that for any BCFTwhich has a description in terms of an effective action, the near boundary divergent behaviorof the renormalized stress tensor is completely determined by the central charges of the theory.These relations are verified by free BCFTs. We also test them with holographic models ofBCFT and find exact agreement. We propose that these relations between Casimir coefficientsand central charges hold for any BCFT. With the holographic models, we reproduce not onlythe precise form of the near boundary divergent behavior of the stress tensor, but also thesurface counter term that is needed to make the total energy finite. As they are proportionalto the central charges, the near boundary divergence of the stress tensor must be physical andcannot be dropped by further artificial renormalization. Our results thus provide affirmativesupport on the physical nature of the divergent energy density near the boundary, whosereality has been a long-standing controversy in the literature. Corresponding author. a r X i v : . [ h e p - t h ] M a r ontents A.1 3d BCFT 10A.2 4d BCFT 13
The Casimir effect [1] originates from the effect of boundary on the zero point energy-momentum of quantized fields in a system. As a fundamental property of the quantumvacuum, it has important consequences on the system of concern and has been applied toa wide range of physical problems, such as classic applications in the study of the Casimirforce between conducting plates (and nano devices) [2, 3], dynamical compactification of extradimensions in string theory [4, 5], candidate of cosmological constant and dark energy [6], aswell as dynamical Casimir effect and its applications [7].The near boundary behavior of the stress tensor of a system is crucial to the understandingof the Casimir effect. For a Quantum Field Theory (QFT) on a manifold M of integerdimension d and boundary P , the renormalized stress tensor is divergent near the boundary[8]: (cid:104) T ij (cid:105) = x − d T ( d ) ij ... + x − T (1) ij , x ∼ , (1.1)– 1 –here x is the proper distance from the boundary and T ( n ) ij with n ≥ depend only on theshape of the boundary and the kind of QFT under consideration. For CFT with conformalinvariant boundary condition (BCFT), one further require that divergent parts of renormal-ized stress tensor are traceless in order to get a well-defined finite Weyl anomaly withoutdivergence. It is also natural to impose the conservation condition of energy: lim x → (cid:104) T ii (cid:105) = O (1) , ∇ i (cid:104) T ij (cid:105) = 0 . (1.2)Substituting (1.1) into the above equations, [8] obtains T ( d ) ij = 0 , T ( d − ij = 2 α ¯ k ij , (1.3a) T ( d − ij = − α d − n ( i h lj ) ∇ l k − α d − n ( i h lj ) n p R lp + 2 α d − n i n j − h ij d − k + t ij , (1.3b) t ij := (cid:100) β C ikjl n k n l + β R ij + β kk ij + β k li k lj (cid:101) , (1.4)where n i , h ij and ¯ k ij are respectively the normal vector, induced metric and the traceless partof extrinsic curvature of the boundary P . The tensor t ij is tangential: n i t ij = 0 , (cid:100) (cid:101) denotesthe traceless part, C ijkl is Weyl tensor of M and R ij is the intrinsic Ricci tensor of P . Thecoefficients ( α, β i ) fixes the shape dependence of the leading and subleading Casimir effectsof BCFT. The main goal of this letter is to show that one can fix completely these Casimircoefficients in terms of the bulk and boundary central charges. Consider a BCFT with a well defined effective action. The Weyl anomaly A , defined as thetrace of renormalized stress tensor, can be obtained as the logarithmic UV divergent term ofthe effective action, I = · · · + A log( 1 (cid:15) ) + I finite , (2.1)where · · · denotes terms which are UV divergent in powers of the UV cutoff /(cid:15) , and I finite is the renormalized, UV finite part of the effective action. This part is dependent on thesubtraction scheme. But the dependence is irrelevant for the discussion below and our resultshold for any renormalization scheme.Inspired by [9, 10], let us regulate the effective action by excluding from its volume in-tegration a small strip of geodesic distance (cid:15) from the boundary. Then there is no explicit– 2 –oundary divergences in this form of the effective action, however there are boundary diver-gences implicit in the bulk effective action which is integrated up to distance (cid:15) . The variationof effective action is given by δI = 12 (cid:90) x ≥ (cid:15) √ g ˆ T ij δg ij (2.2)where ˆ T ij = δI √ gδg ij is the non-renormalized bulk stress tensor. The renormalized bulk stresstensor is defined by the difference of the non-renormalized bulk stress tensor against a referenceone [8]: T ij = ˆ T ij − ˆ T ij , (2.3)where ˆ T ij is the non-renormalized stress tensor defined for the same CFT without boundary.It is δI = 12 (cid:90) x ≥ (cid:15) √ g ˆ T ij δg ij , (2.4)where I is the effective action of the CFT with the boundary removed, hence the integrationover the region x ≥ (cid:15) . Subtract (2.4) from (2.2) and focus on only the logarithmically divergentterms, we obtain our key formula ( δ A ) ∂M = (cid:18) (cid:90) x ≥ (cid:15) √ gT ij δg ij (cid:19) log(1 /(cid:15) ) , (2.5)where ( δ A ) ∂M is the boundary terms in the variations of Weyl anomaly and T ij is the renor-malized bulk stress tensor. In the above derivations, we have used the fact that I and I havethe same bulk Weyl anomaly so that ( δ A ) ∂M = ( δI − δI ) log(1 /(cid:15) ) . (2.6)We observe that as the right hand side of (2.5) must give an exact variation, this imposesstrong constraints on the possible form of the stress tensor near the boundary since this iswhere one would pick up logarithmic divergent contribution on integration near the boundary.It is this integrability of the variations which helps us to fix the Casimir effects in terms ofthe Weyl anomaly. To proceed, let us start with the metric written in the Gauss normalcoordinates ds = dx + (cid:0) h ab − xk ab + x q ab + · · · (cid:1) dy a dy b , (2.7)where x ∈ [0 , + ∞ ) . The coefficients k ab , q ab , · · · parametrize the derivative expansion (withrespect to both x and y a ) of the metric. Consider variation of the metric with δg xi = 0 and– 3 – g ab = δh ab − xδk ab + · · · . Take first the 3d BCFT as an example. The Weyl anomaly of 3dBCFT is given by [11] A = (cid:90) P √ h ( b R + b Tr ¯ k ) , (2.8)where b , b are boundary central charges which depends on the boundary conditions. Takingthe variation of (2.8), we have b (cid:90) P √ h (cid:104) ( Tr ¯ k h ab − k ac k cb ) δh ab + 2¯ k ab δk ab (cid:105) . (2.9)Now we turn to calculate the variation of Weyl anomaly from the last term of (2.5). Note that C ijkl = (cid:100)R ij (cid:101) = 0 for d = 3 . Note also that ¯ k ij ( x ) = g i (cid:48) i g j (cid:48) j ¯ k i (cid:48) j (cid:48) (0) = ¯ k ij (0) − xk l ( i ¯ k j ) l + O ( x ) ,where g i (cid:48) i is the bivector of parallel transport between x and x = 0 [8]. Taking these facts intoaccount and substitute (1.1) and (1.3) into the last term of (2.5), integrate over x and selectthe logarithmic divergent term, we obtain − α (cid:90) P √ h [( Tr ¯ k h ab − k ac k cb ) δh ab + 2¯ k ab δk ab ]+ (cid:90) P √ h [( β − α ) k ¯ k ab δh ab + β (cid:100) k ac k cb (cid:101) δh ab ] . (2.10)Note that (2.10) is made up of a structure of curvature components different from thoseappearing in (2.9). Integrability of (2.10) gives β = 2 α and β = 0 . Comparing (2.9) with(2.10) gives α = − b . All together, we obtain the relations between the Casimir coefficientsof the stress tensor and the boundary central charges: α = − b , β = − b , β = 0 . (2.11)Similarly for 4d BCFT, we can obtain the shape dependence of Casimir effects from theWeyl anomaly [12, 13] A = (cid:90) M √ g ( c π C ijkl C ijkl − a π E )+ (cid:90) P √ h ( b Tr ¯ k + b C ac bc ¯ k ba ) , (2.12)where a, c are bulk central charges and b , b are boundary central charges. E is the Eulerdensity including the boundary term. To derive t ij , we set δh ij = 0 for simplicity, since itonly affects the third order derivative terms in the stress tensor. Taking variation of (2.12)and comparing the boundary term with the last term of (2.5), we obtain α = b , β = c π + b , β = 0 ,β = 2 b + b , β = − b − b . (2.13)– 4 –t is remarkable that the boundary behavior of the stress tensor is completely determinedby the boundary and bulk central charges However, it is independent of the central chargerelated to Euler density due to the fact that topological invariants do not change under localvariations. We propose that the relations (2.11) and (2.13) between Casimir coefficients andcentral charges hold for general BCFT. Let us verify our general statements with free BCFT. The renormalized stress tensor of 4dfree BCFT has been calculated in [8, 14, 15]. The bulk and boundary central charges for 4dfree BCFTs were obtained in [12]. We summary these results in Table 1 and Table 2. Notethat the results for Maxwell field apply to both absolute and relative B.C. We find these dataobey exactly the relations (2.13). β for Maxwell field is absence in the literature. Here from(2.13), we predict that β = 0 for all 4d free BCFT due to the fact that c = − π b for 4dfree BCFT. As we will show below, this relation is violated by strongly-coupled CFT dual togravity. As a result, β is non-zero in general. Comparing with [15], we note that there is aminus sign typo of β for Maxwell field in [8]. Table 1 . Casimir coefficients for 4d free BCFT α β β β β Scalar, Dirichlet B.C − π − π − π Scalar, Robin B.C − π − π − π (0) 0 − π π Table 2 . Central charges for 4d free BCFT a c b b Scalar, Dirichlet B.C π − π Scalar, Robin B.C π − π Maxwell field π − π Now let us investigate the shape dependence of Casimir effects in holographic modelsof BCFT. Consider a BCFT defined on a manifold M with a boundary P . Takayanagi [16]proposed to extend the d dimensional manifold M to a d + 1 dimensional asymptoticallyAdS space N so that ∂N = M ∪ Q , where Q is a d dimensional manifold which satisfies– 5 – M Q Pz
Figure 1 . BCFT on M and its dual N ∂Q = ∂M = P . The gravitational action for holographic BCFT is [16] ( πG N = 1 ) I = (cid:90) N √ G ( R − (cid:90) Q √ γ ( K − T ) (3.1)plus terms on M and P . Here T is a constant which can be regarded as the holographicdual of boundary conditions of BCFT [17, 18]. A central issue in the construction of theAdS/BCFT is the determination of the location of Q in the bulk. [16] propose to use theNeumann boundary condition K αβ − ( K − T ) γ αβ = 0 (3.2)to fix the position of Q . In [17, 18] we found there is generally no solution to (3.2) for bulkmetric that arose from the FG expansion of a general non-symmetric boundary. The reasonis because Q is of co-dimension one and we only need one condition to determine it’s position,while there are too many extra conditions in (3.2). To resolve this, we suggested in [17, 18] touse the trace of (3.2), (1 − d ) K + dT = 0 , to determine the position of Q . Nonetheless, it isalso possible that one may need to relax the assumption that the bulk metric admits a validFG expansion, as has been attempted in [20] for some non-symmetric boundary in BCFT .In contrast to a FG-expanded metric whose form near the boundary M is completely fixed,a non-FG expanded metric has more degree of freedom. It was suggested in [20] that theembedding equation (3.2) may admit a solution if the bulk metric is also allowed to adjustitself. However in general this is a highly non-trivial problem and there is no systematicmethod available to construct gravity solutions for BCFT in general dimensions d and withan arbitrary non-symmetric boundary ( ¯ k ab (cid:54) = 0 ) that is not FG expanded. Remarkably thisproblem can solved and we will now present the solution.To make progress in this front, we find that one can instead consider an expansion inpowers of small derivatives of the metric and keep both the z and x dependence as exactto construct a perturbative solution to the Einstein equation. For simplicity, we considerthe case of h ab = δ ab here. The more general case of a nontrivial boundary metric can beanalysed. We comment on this in the supplementary information. We find useful to consider– 6 –he following metric ansatz ds = dz + dx + (cid:0) δ ab − x ¯ k ab f (cid:1) dy a dy b z + · · · , (3.3)with f = f ( x, z ) a function such that f ( x,
0) = 1 . To find solution, let us first considerthe region x ≥ and consider the ansatz f = f ( z/x ) . This ansatz plays an importantrole to solve (3.2) for non-symmetric boundary with ¯ k ab (cid:54) = 0 . For simplicity we considera traceless k ab = ¯ k ab extrinsic curvature here. The solution for the general case is givenin the supplementary information. Substituting (3.3) into Einstein equation and writing s := z/x > , we obtain at the order O ( k ) a single equation s (cid:0) s + 1 (cid:1) f (cid:48)(cid:48) ( s ) − ( d − f (cid:48) ( s ) = 0 . (3.4)It has the solution f ( s ) = 1 − α s d F (cid:0) d − , d ; d +22 ; − s (cid:1) d . (3.5)To obtain a solution of the Einstein equation for x < , one may analytic continuate (3.5) tothe region s < . However this solution while continuous at s = 0 , is discontinuous at x = 0 as the region near x = 0 is mapped to widely separated regions s = ±∞ . Another possibilityis to first rewrite the expression (3.5) in terms of x and z , and then analytic continuate theresulting function f ( x, z ) to the region x < . In this way, we obtain a solution of the Einsteinequation that is continuous at x = 0 . For example, for d = 3 , we have f ( x, z ) = 1 − α ( zx − g ( x, z )) , (3.6a) g ( x, z ) = π − − (cid:16) x/ ( z + (cid:112) z + x ) (cid:17) . (3.6b)Let us make some comments. 1. For general d , the perturbation x ¯ k ab f ( x, z ) is finite whichshows that (3.4) is a well-defined metric. 2. Note that formally one can expand f as a powerseries of z and interpret that as a FG expansion of the metric (3.3). However the series doesnot converge whenever x < z . Therefore for the boundary ( x → ) physics we are interestedin, it is necessary to use the exact solution without performing the FG expansion. 3. Theperturbative background (3.3), (3.5) to the Einstein equation is an interesting result whichmay be useful for other studies as well.So far the coefficient α is arbitrary. If we now consider (3.2) in this background, we findthat one can solve the embedding function of Q as x = − sinh( ρ ) z + O ( k ) provided that α isfixed at the same time. Please see the supplementary information for more details. See Table3 for values of α obtained from holography, where we have re-parametrized T = ( d −
1) tanh ρ and θ = π + 2 tan − (cid:0) tanh (cid:0) ρ (cid:1)(cid:1) is the angle between M and the bulk boundary Q . Using(3.3), (3.5), we can derive the holographic stress tensor [21] T ij = lim z → d δg ij z d = 2 α ¯ k ij x d − + O ( k ) , (3.7)– 7 –hich takes the expected form (1.3a). According to [21], T ij (3.7) automatically satisfy thetraceless and divergenceless conditions (1.2). Note that in general the stress tensor (3.7) alsocontains contributions from g ij | z =0 in even dimensions [21]. However, these contributions arefinite, so we can ignore them without loss of generality since we focus on only the divergentparts in this letter.Similarly, we can work out the next order solutions to both the Einstein equation and(3.2), and then derive the stress tensor up to the order O ( k ) by applying the formula (3.7).See the appendix for details. It turns out that the holographic stress tensor takes exactlythe expected expression (1.3) with the coefficients listed in Table 3. These coefficients indeed Table 3 . Casimir coefficients for holographic stress tensor α β β β β − θ − θ − ρ ) tanh ρ tanh ρ +1 ρ − ρ ) tanh ρ tanh ρ +1 satisfy the relations (2.11), (2.13) provided the boundary central charges are given by [22] b = 1 θ , (3.8a) b = 11 + tanh ρ − , b = −
11 + tanh ρ , (3.8b)for 3d and 4d respectively. Since we have many more relations (8) than unknown variables (3),this is a non-trivial check of the universal relations (2.11), (2.13) as well as for the holographicproposal (3.2). In fact, the central charges (3.8a,3.8b) can be independently derived from thelogarithmic divergent term of action by using the perturbation solution of order O ( k d − ) . Onecan consider general boundary conditions by adding intrinsic curvatures on Q [18]. In this casethe boundary central charges change but the relations (2.11), (2.13) remain the same. We canalso reproduce these relations in the holographic model [17, 18]. These are all strong supportsfor the universal relations (2.11), (2.13). The fact that the both the holographic models of[16] and ours [17, 18] verify the universal relations (2.11), (2.13) suggests that both proposalsare consistent holographic models of BCFT. We remark that in general there could be morethan one self-consistent boundary conditions for a theory [19] and so there is no contradictionbetween [16] and [17, 18]. This is supported by the fact that the two holographic models givesdifferent boundary central charges despite the same universal relations are satisfied.From holographic BCFT [16–18], we can also gain some insight into the total energy.Applying the holographic renormalization of BCFT [17, 18], we obtain the total stress tensor: T ij = 2 α ¯ k ij x d − − δ ( x ; P ) 2 α d − k ij (cid:15) d − + O ( k ) , x ∼ (cid:15). (3.9)– 8 –ote that the first term, a local energy density, give rises to a divergence in the total energythat cannot be canceled with any local counterterm in the BCFT, but only with the inclusionof the second term, a surface counterterm as first constructed in [14]. The surface countertermis localized at the boundary surface P , which has been shifted from x = 0 to a position x = (cid:15) .The requirement of finite energy fixes [14] the relative coefficients of the two terms in (3.9).Remarkably the holographic constructions [16–18] reproduce precisely also the surface counterterm with the needed coefficient to make the total energy finite : (cid:82) ∞ (cid:15) dxT ij = O ( k ) < ∞ ,which agrees with the results of [14, 23]. In this letter, we have shown that with the help of an effective action description, the divergentparts of the stress tensor of a BCFT is completely determined by the central charges of thetheory. The found relations between the Casimir coefficients and the central charges areverified by free BCFT as well as holographic models of BCFT. We propose that these relationshold universally for any BCFT. Using the holographic models, we also reproduce remarkablythe precise surface counterterm that is needed to render the total energy of the BCFT finite.Our results are useful for the study of shape dependence of Casimir effects [24–26] andthe theory of BCFT [27, 28]. For Casimir effects where there are spacetime on both sidesof the boundary, it has been argued that the divergent stress tensor originates from theunphysical nature of classical “perfect conductor” boundary conditions [8]. In reality therewould be an effective cut off (cid:15) below which the short wavelength vibrational modes do not“see the boundary”. However for BCFT where there is no spacetime outside the boundary,the divergent one point function of stress tensor is expected and physical. According to [29],one can derive the one point function of an operator in BCFT from the two point functionsof operators in CFT by using the mirror method. Since two point functions are divergentwhen two points are approaching, it is not surprising that the one point function of BCFTdiverge near the boundary. This is due to the interaction with the boundary, or equivalently,the mirror image. Note that although the stress tensor diverges, the total energy is finite.Thus BCFT is self-consistent.Our discussions can be generalized to higher dimensions naturally. Furthermore, ourdiscussions also apply to defect conformal field theory (DCFT) [30] with general codimensions,which is a problem of great interest. For example, the case of codimension 2 DCFT is relatedto the shape dependence of Rényi entropy [9, 10, 31–34]. It is interesting to see whether thespirit of this letter can apply to general QFT. It is also very interesting to generalize andapply the techniques of the holographic models to study the expectation value of current inboundary systems, e.g. edge current of topological materials.– 9 – cknowledgements
We thank John Cardy, WuZhong Guo, Hugh Osborn and Douglas Smith for useful discussionsand comments. This work is supported in part by NCTS and the grant MOST 105-2811-M-007-021 of the Ministry of Science and Technology of Taiwan.
A Solutions to holographic BCFT
Here we give details about solutions to the Einstein equations and the boundary conditions(3.2) to the next order in derivative expansion of the boundary metric (i.e. O ( k ) in the caseof a flat boundary metric h ab = δ ab ). Consider the following ansatz for x > , ds = 1 z (cid:104) dz + (cid:0) x X ( zx ) (cid:1) dx + (cid:0) δ ab − x ¯ k ab f ( zx ) − x kd − δ ab + x Q ab ( zx ) (cid:1) dy a dy b (cid:105) + O ( k ) , (A.1)where the functions X ( zx ) and Q ab ( zx ) are of order O ( k ) . We require that f (0) = 1 , X (0) = 0 , Q ab (0) = q ab (A.2)so that the metric of BCFT takes the form (2.7) in Gauss normal coordinates. A.1 3d BCFT
Let us first study the case d = 3 . The generalization to higher dimensions is straightforward.For simplicity, we further set k ab = diag ( k , k ) , q ab = diag ( q , q ) , where k a , q a are constants.Substituting (A.1) into the Einstein equations, and using (A.2) to fix the integral constants,we obtain (3.5) and f ( s ) = 1 − α ( s − g ( s )) Q ( s ) = 18 [4 q (cid:0) s + 2 (cid:1) − α ( k − k ) (cid:0) s − (cid:1) g ( s ) − α ( k − k ) log (cid:0) s + 1 (cid:1) + s (cid:0) α ( k − k ) s + 4 α (cid:1) + s (cid:0) α (cid:0) − k + 8 k k + k (cid:1) − s (cid:0) k − k k − k + q (cid:1)(cid:1) − g ( s ) (cid:0) α k (cid:0) α s + s − (cid:1) + 2 α (cid:0) s + 1 (cid:1)(cid:1) − α g ( s ) (cid:0) k (3 s ( α + s ) + 1) + 2 k k (4 − α s ) (cid:1) ] , – 10 – ( s ) = 18 [4 q (cid:0) s + 2 (cid:1) − α ( k − k ) (cid:0) s − (cid:1) g ( s ) + s (cid:0) α ( k − k ) s − α (cid:1) − α ( k − k ) log (cid:0) s + 1 (cid:1) + s (cid:0) s (cid:0) k + k k − k − q (cid:1) − α (cid:0) k − k k + 7 k (cid:1)(cid:1) +2 g ( s ) (cid:0) α (cid:0) s + 1 (cid:1) − α k (cid:0) α s + s − (cid:1)(cid:1) +2 α g ( s ) (cid:0) k (cid:0) − α s + s + 7 (cid:1) + 2 k k (cid:0) α s + 2 s − (cid:1)(cid:1) ] ,X ( s ) = 14 [ − α ( k − k ) s log (cid:0) s + 1 (cid:1) − α ( k − k ) s + α ( k − k ) g ( s ) (cid:0) α (cid:0) s + 1 (cid:1) g ( s ) + 2 s ( s − α ) + 2 (cid:1) + s (cid:0) α ( k − k ) s − s (cid:0) k + k k + k − q − q (cid:1)(cid:1) ] , (A.3)where s = z/x and g ( s ) = π − − (cid:16) / ( s + √ s + 1) (cid:17) . A continuous solution of theEinstein equations is obtained by first rewriting (A.1) as function of x and z and then analyticcontinutate to the x < region. In this way, we get smooth g ( z, x ) as (3.6). The solution isparametrized by two free parameters α and α .Next we solve (3.2) for the embedding function of Q in the above background. We obtain,for d = 3 , the results x = − sinh( ρ ) z + k cosh ρ d − z + c z + O ( k ) (A.4)with c given by c = − sinh ρ (cid:104) k + 4 k k + 7 k − q + q )+ (cid:0) k + 2 k k + 5 k − q + q ) (cid:1) cosh(2 ρ )+ α ( k − k ) (cid:0) (2 + cosh(2 ρ )) log(coth ρ ) − (cid:1) (cid:105) . (A.5)The boundary conditions (3.2) also restrict solutions (3.6) and fix the integral constants tobe α = − θ , α = − α k , (A.6)where θ = π + 2 tan − (cid:0) tanh (cid:0) ρ (cid:1)(cid:1) is the angle between M and the bulk boundary Q . It shouldbe mentioned that, following our method, the above α is independently obtained in a recentpaper [35]. The derivation of (A.4)-(A.6) is straightforward. For simplicity, let us first focuson the leading order O ( k ) term. From dimensional analysis, the embedding function of Q takes the form x = − sinh( ρ ) z + c kz + O ( k ) with c a dimensionless constant. Substitutingthe metric (A.1) and the embedding function of Q into the conditions (3.2), we get twoindependent equations at order O ( k ) sech ( ρ )( − c + cosh(2 ρ ) + 1) k = 0 , – 11 – α cosh ( ρ ) (cid:16) − (cid:16) tanh ρ (cid:17) + π (cid:17) + 8 c (cid:17) ¯ k ab = 0 . Solving the above equations, we obtain c and α as shown in ( ?? ), (A.6). Similarly, weobtain c and α from (3.2) at order O ( k ) . It is remarkable that the conditions (3.2) fix thebulk metric and embedding function of Q at the same time.Substituting (3.6), (A.1),(A.3), (A.6) into (3.7), we obtain the holographic stress tensor T ij = diag { α ( k − k ) x , α ( k − k ) x − α ( k − k ) x ,α ( k − k ) x − α ( k − k ) x } . (A.7)It is remarkable that all the q a dependence got cancelled away and the stress tensor (A.7)takes exactly the expected form (1.3) with coefficients as listed in Table 3. Recall that k ij in (1.3) is actually a tensor defined at x instead of the boundary x = 0 . It can be obtainedfrom parallel transport of the extrinsic curvature at x = 0 , i.e., ¯ k ij ( x ) = g i (cid:48) i g j (cid:48) j ¯ k i (cid:48) j (cid:48) (0) =¯ k ij (0) − xk l ( i ¯ k j ) l + O ( x ) [8].Further generalization of the our above results is possible. Let us discuss briefly the caseof non-constant metric h ij ( y ) and extrinsic curvature k ij ( y ) . In this case, T ij will include non-diagonal parts generally. These non-diagonal parts obey (1.3b) trivially, since by definition(3.7) T ij automatically satisfy the traceless and divergenceless conditions (1.2), which fixs thenon-diagonal parts of stress tensor as (1.3b) completely.Another generalization is to have more general boundary conditions of holographic BCFTby adding intrinsic curvatures on Q [18]. For example, we consider I = (cid:90) N √ G ( R − (cid:90) Q √ γ ( K − T − λR Q ) , (A.8)with the Neumann boundary condition K αβ − ( K − T − λR Q ) γ αβ − λR Qαβ = 0 . (A.9)Substituting the solutions (3.6) into (A.9), we can solve the embedding function of Q as (A.4)but with different parameter c and different integration constants α = 12 λ sech ρ/ (1 − λ tanh ρ ) − θ ,α = − α k . (A.10)Here T = 2 tanh ρ + 2 λ sech ( ρ ) . From (3.7), we can derive the holographic stress tensorwhich takes exactly the expected form (1.3). It is remarkable that although the centralcharge b = − α changes, the relations (2.11) remain invariant for holographic BCFT withgeneral boundary conditions. The above discussions can be generalized to higher dimensionseasily. The 4d solutions can be used to confirm the universal relations (2.13).– 12 – .2 4d BCFT Now Let us consider the case d = 4 . For simplicity, we also set k ab = diag ( k , k , k ) , q ab = diag ( q , q , q ) , where k a , q a are constants. Substituting (A.1) into the Einstein equations,and using (A.2) to fix the integral constants, we obtain f ( s ) = 1 + 2 α − α (cid:0) s + 2 (cid:1) √ s + 1 , (A.11) X ( s ) = 16 s (2 ( q + q + q ) − k k + k k + k k )) − (cid:0) k + k + k − k k − k k − k k (cid:1) g ( s ) ,Q ( s ) = k g ( s ) + k g ( s ) + k k g ( s )18 ( s + 1) / , + 13 √ s + 1 (cid:104) q (cid:16) s + (cid:112) s + 1 + 2 (cid:17) + q (cid:16) − s + (cid:112) s + 1 − (cid:17) + 3 α (cid:16)(cid:16)(cid:112) s + 1 − (cid:17) s + 2 (cid:16)(cid:112) s + 1 − (cid:17)(cid:17)(cid:105) ,Q ( s ) = k g ( s ) + k g ( s ) + k k g ( s )18 ( s + 1) / + 13 √ s + 1 (cid:104) q (cid:16) s + (cid:112) s + 1 + 2 (cid:17) + q (cid:16) − s + (cid:112) s + 1 − (cid:17) + 3 α (cid:16)(cid:16)(cid:112) s + 1 − (cid:17) s + 2 (cid:16)(cid:112) s + 1 − (cid:17)(cid:17)(cid:105) ,Q ( s ) = ( k + k ) g ( s ) + k k g ( s )18 ( s + 1) / − ( q + q ) (cid:16) s − √ s + 1 + 1 (cid:17) + 3 ( α + α ) (cid:16)(cid:16) √ s + 1 − (cid:17) s + 2 (cid:16) √ s + 1 − (cid:17)(cid:17) √ s + 1 , (A.12)where g i ( s ) are defined by g ( s ) = α (cid:16) α (cid:16) (cid:112) s + 1 + s (cid:0) log (cid:0) s + 1 (cid:1) − (cid:1) − (cid:17) − s + 4 (cid:112) s + 1 − (cid:17) + s g ( s ) = 12 (cid:0) s + 1 (cid:1) ( − s + (cid:112) s + 1 −
1) + 36 α (cid:0) s + 1 (cid:1) ( − s + 2 (cid:112) s + 1 − − α (cid:16) − (cid:16)(cid:112) s + 1 − (cid:17) + s (cid:16) s − (cid:112) s + 1 + 108 (cid:17) + 6 (cid:0) s + 1 (cid:1) / log (cid:0) s + 1 (cid:1)(cid:17) g ( s ) = − (cid:0) s + 1 (cid:1) (cid:16) − s + (cid:112) s + 1 − (cid:17) + 6 α (cid:0) s + 1 (cid:1) (cid:16) − s + 2 (cid:112) s + 1 − (cid:17) + α (cid:16) (cid:16)(cid:112) s + 1 − (cid:17) + s (cid:16) s + 11 (cid:112) s + 1 − (cid:17) − (cid:0) s + 1 (cid:1) / log (cid:0) s + 1 (cid:1)(cid:17) g ( s ) = − (cid:0) s + 1 (cid:1) (cid:16) − s + (cid:112) s + 1 − (cid:17) − α (cid:0) s + 1 (cid:1) (cid:16) − s + 2 (cid:112) s + 1 − (cid:17) + α (cid:16) s − (cid:16)(cid:112) s + 1 − (cid:17) + s (cid:16) − (cid:112) s + 1 (cid:17) + 6 (cid:0) s + 1 (cid:1) / log (cid:0) s + 1 (cid:1)(cid:17) g ( s ) = − (cid:0) s + 1 (cid:1) (cid:16) − s + (cid:112) s + 1 − (cid:17) − α (cid:0) s + 1 (cid:1) (cid:16)(cid:16) (cid:112) s + 1 − (cid:17) s + 2 (cid:16)(cid:112) s + 1 − (cid:17)(cid:17) + α (cid:16) (cid:16)(cid:112) s + 1 − (cid:17) + 8 s (cid:16) (cid:112) s + 1 − (cid:17) − (cid:0) s + 1 (cid:1) / log (cid:0) s + 1 (cid:1) + s (cid:16) (cid:112) s + 1 − (cid:17)(cid:17) – 13 – ( s ) = 3 (cid:0) s + 1 (cid:1) (cid:16)(cid:16) (cid:112) s + 1 − (cid:17) s + 8 (cid:16)(cid:112) s + 1 − (cid:17)(cid:17) +12 α (cid:0) s + 1 (cid:1) (cid:16)(cid:16)(cid:112) s + 1 − (cid:17) s + 4 (cid:16)(cid:112) s + 1 − (cid:17)(cid:17) + α (cid:16) s (cid:16)(cid:112) s + 1 − (cid:17) + 28 (cid:16)(cid:112) s + 1 − (cid:17) + 6 (cid:0) s + 1 (cid:1) / log (cid:0) s + 1 (cid:1) + s (cid:16) − (cid:112) s + 1 (cid:17)(cid:17) . (A.13)Note that since the full expressions of Q ab are too complicated, we only list the results with k = q = 0 for Q ab in (A.12). We want to stress that we focus on the general case withnonzero k and q , we just do not list the full expressions for simplicity.The above solutions work well for x > . A continuous solution of the Einstein equations isobtained by first rewriting (A.1),(A.11),(A.12),(A.13) as functions of x and z and then analyticcontinutate to the x < region. In fact, we only need to replace all √ s = (cid:113) z x in(A.11),(A.12),(A.13) by √ x + z /x . One can check that after the analytic continutation,the metric (A.1) are solutions to Einstein equations for x ∈ ( −∞ , ∞ ) . What is more, now itbecomes continuous at x = 0 (see xf ( s ) as an example). The above solution is parametrizedby three free parameters α , α and α .Next we solve (3.2) for the embedding function of Q in the above background. We obtain,for d = 4 , the results x = − sinh( ρ ) z + k cosh ρ d − z + c z + O ( k ) (A.14)with c given by c = − e − ρ sinh( ρ ) (cid:2) t ( k + k + k ) + t ( k k + k k + k k ) + t ( q + q + q ) (cid:3) , (A.15)with t i given by t = 8 + 48 sinh(2 ρ ) + 20 sinh(4 ρ ) + 5 log (cid:0) coth ( ρ ) (cid:1) + cosh(4 ρ ) (cid:0) log (cid:0) coth ( ρ ) (cid:1) + 20 (cid:1) + cosh(2 ρ ) (cid:0) (cid:0) coth ( ρ ) (cid:1) + 44 (cid:1) t = 16 + 24 sinh(2 ρ ) + 4 sinh(4 ρ ) − (cid:0) coth ( ρ ) (cid:1) − cosh(4 ρ ) (cid:0) log (cid:0) coth ( ρ ) (cid:1) − (cid:1) + cosh(2 ρ ) (cid:0) − (cid:0) coth ( ρ ) (cid:1)(cid:1) t = − e ρ (cosh(2 ρ ) + 2) . (A.16)The boundary conditions (3.2) also fix all the integral constants of the solutions (A.1),(A.11),(A.12),(A.13) α = − ρ ) + 1) , – 14 – = − ρ ) + cosh( ρ )) (cid:2) k + 25 k k + 25 k k − k − k − k k − q + 12 q + 12 q +4 (cid:0) k − k + k ) k − k − k + 2 k k − q + 3 q + 3 q (cid:1) sinh(2 ρ )+3 (cid:0) k − k + k ) k − k − k − k k − q + 4 q + 4 q (cid:1) cosh(2 ρ ) (cid:3) α = α [ k ↔ k , q ↔ q ] . (A.17)It should be mentioned that, following our method, the above α is independently obtainedin a recent paper [35], which exactly agrees with our results when using our notations. Thederivation of (A.14)-(A.17) is straightforward. For simplicity, let us first focus on the leadingorder O ( k ) term. From dimensional analysis, the embedding function of Q takes the form x = − sinh( ρ ) z + c kz + O ( k ) with c a dimensionless constant. Substituting the metric(A.1) and the embedding function of Q into the conditions (3.2), we get two independentequations at order O ( k ) sech ( ρ )( − c + cosh(2 ρ ) + 1) k = 0 , k ( α (sinh(2 ρ ) + cosh(2 ρ ) + 1) + 6 c ) − ( k + k ) (2 α (sinh(2 ρ ) + cosh(2 ρ ) + 1) + 3( − c + cosh(2 ρ ) + 1)) = 0 . (A.18)Solving the above equations, we obtain c and α as shown in (A.14), (A.17) c = cosh ( ρ )6 , α = − ρ ) + 1) . (A.19)Similarly, we can obtain c , α , α from Neumann boundary conditions (3.2) at the next order O ( k , q ) . It is remarkable that the conditions (3.2) fix the bulk metric and embedding functionof Q at the same time.Substituting the solutions (A.1),(A.11),(A.12),(A.13),(A.17) into the formula T ij = lim z → d δg ij z d , (A.20)and noting BCFT is defined in x ∈ [0 , ∞ ) , we obtain the holographic stress tensor withnon-zero components given by T xx = − k + k + k − k k − k k + k k x (tanh( ρ ) + 1) ,T = − (2 k − k − k )3(1 + tanh ρ ) x + cosh( ρ ) − sinh( ρ )18 x (cid:2) k (5 sinh( ρ ) + 8 cosh( ρ )) + (cid:0) k + k (cid:1) (7 cosh( ρ ) − ρ ))+2 k k (sinh( ρ ) + 4 cosh( ρ )) − k ( k + k ) (sinh( ρ ) + 19 cosh( ρ )) + 3 ( q + q ) sinh( ρ ) − q sinh( ρ ) (cid:3) ,T = T [ k ↔ k , q ↔ q ] , – 15 – = T [ k ↔ k , q ↔ q ] . (A.21)We can rewrite the above holographic stress tensor into convariant form: T ij = 2 α (¯ k ij − xk l ( i ¯ k j ) l ) x + α ( n i n j − h ij ) Tr ¯ k x + p C ikjl n k n l + p k ¯ k ij + p ( k il k lj − h ij Tr k ) x (A.22)where ¯ means traceless parts, C ikjl n k n l = − ¯ q ij + k ¯ k ij , α is given by (A.17), n i =( − , , , , h ij = diag (0 , , , and p i are given by p = p = tanh( ρ )tanh( ρ ) + 1 , p = − ρ ) − ρ ) + 1) . (A.23)Now let us turn to the field theoretical result of BCFT stress tensor (1.3a,1.3b,1.4), whichtakes the form T ij = 2 α (¯ k ij − xk l ( i ¯ k j ) l ) x + α ( n i n j − h ij ) Tr ¯ k x + β C ikjl n k n l + β k ¯ k ij + β ( k il k lj − h ij Tr k ) x (A.24)Recall that k ij in eqs.(1.3a,1.3b,1.4) is actually a tensor defined at x instead of the boundary x = 0 . It can be obtained from parallel transport of the extrinsic curvature at x = 0 , i.e., ¯ k ij ( x ) = g i (cid:48) i g j (cid:48) j ¯ k i (cid:48) j (cid:48) (0) = ¯ k ij (0) − xk l ( i ¯ k j ) l + O ( x ) [8].Comparing the holographic stress tensor (A.22) with the field theoretical result (A.24),we get β = p = tanh( ρ )tanh( ρ ) + 1 , β = p = − ρ ) − ρ ) + 1) , β = p = − ρ ) − ρ ) + 1) . (A.25)Now it is easy to check that the Casimir coefficients α , β i indeed satisfy the universal relations α = b , β = c π + b , β = 0 ,β = 2 b + b , β = − b − b . (A.26)provided the boundary central charges are given by b = 11 + tanh ρ − , b = −
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