Boundary conformal invariants and the conformal anomaly in five dimensions
aa r X i v : . [ h e p - t h ] F e b Boundary conformal invariants and theconformal anomaly in five dimensions
Amin Faraji Astaneh a,b and Sergey N. Solodukhin c a Department of Physics, Sharif University of Technology,P.O.Box 11155-9161, Tehran, Iran b School of Particles and Accelerators,Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395-5531, Tehran, Iran c Institut Denis Poisson UMR 7013, Universit´e de Tours,Parc de Grandmont, 37200 Tours, France
Abstract
In odd dimensions the integrated conformal anomaly is entirely due to the boundaryterms [1]. In this paper we present a detailed analysis of the anomaly in five dimensions.We give the complete list of the boundary conformal invariants that exist in five dimensions.Additionally to 8 invariants known before we find a new conformal invariant that containsthe derivatives of the extrinsic curvature along the boundary. Then, for a conformal scalarfield satisfying either the Dirichlet or the conformal invariant Robin boundary conditionswe use the available general results for the heat kernel coefficient a , compute the conformalanomaly and identify the corresponding values of all boundary conformal charges. e-mails: [email protected] , [email protected] Introduction
The presence of the boundaries brings the new features to the quantum field theory. Oneof the features is the modification of local quantities such as the heat kernel coefficients dueto the boundary terms that results in the boundary terms in the quantum effective action.Those terms were studied for already some time in the literature and the findings, available tothe date, were summarized in a review [3]. The related new feature is a modification of theconformal transformations of the quantum effective action [2] and of the conformal anomaly. Inthe absence of the boundaries the local conformal anomaly is non-trivial only if the dimension d of the space-time is even [4]. It is because it is presented by a combination of the Euler densityand the local conformal anomalies constructed form the Riemann curvature of dimension d . Nosuch invariants exist if d is odd. A related fact is that the Euler number of an odd-dimensionalcompact manifold is zero.In the presence of the boundaries two new things happen. Note that in this case it is moreappropriate to consider the integrated conformal anomaly. Then, in even dimension d = 2 n there appear certain boundary terms in the integrated anomaly. Those boundary terms areconstructed from the Riemann curvature of the bulk metric and the extrinsic curvature K ij ofthe boundary. Since K ij has dimension 1 one can now construct terms of dimension 2 n − d = 4 dimensions wereidentified in [5], [6]. The other new feature is that the anomaly in odd dimension d = 2 n + 1is non-trivial. It is entirely due to the boundary terms [1] that can be of two types: thetopological Euler number of the boundary E n and the local conformal invariants constructedfrom the Riemann tensor and the extrinsic curvature K ij and their derivatives. In dimension d = 3 there is only one such invariant so that the anomaly is determined by two conformalcharges: one for the topological term E and the other for the conformal invariant. In [1] thesecharges were determined for the conformal scalar both for the Dirichlet and conformal Robinboundary conditions and in [7] for the Dirac fermions.The primary goal of the present note is to advance the analysis to the next odd dimension d = 5. Some preliminary list of the conformal invariants in this dimension was given in [1].This list, as it was clear already in the time of writing the paper [1], was rather incomplete sinceit was anticipated that the other, unknown at that time, invariants may exist. The essentialmissing element was one or more invariants that could be constructed from the derivatives ofthe extrinsic curvature along the boundary and that would not vanish if the bulk 5d spacetimewere flat. The differential invariant presented in [1] does not have this last property since, aswe show this in the present paper, it happens to be not an independent invariant as it reducesto a combination of the other invariants constructed from the various contractions of the Weyltensor and the extrinsic curvature tensor. One of the main results of the present paper is thatwe have now found this new invariant, see eq.(29), that contains derivatives of the extrinsic2urvature tensor and that was absent in [1]. With the new invariant the list of the conformalboundary invariants in five dimensions is now complete.In a wider context the present paper is a part of the on-going study of the various man-ifestations of the boundaries, defects and interfaces in the conformal field theories, [8] - [21].It should be noted that contrary to the situation with the conformal invariants in the bulk ofspace-time, where all such invariants are classified, no such a classification theorem exists ingeneral for the boundary conformal invariants. Discussing the preliminary work we should notethe earlier results in mathematical literature [22] where the conformal invariant boundary struc-tures were studied in various dimensions. The other preliminary work relevant to our presentstudy is the calculation of the heat kernel coefficient a for the various boundary conditionsand the different elliptic operators that was performed in a series of papers [23], [24], [25] thatwas summarized in [26]. It is our other principle goal in the paper to use these earlier results,compute the conformal anomaly in the case of a conformal scalar both for the Dirichlet andconformal Robin boundary conditions and express the anomaly in terms of the complete set ofthe invariants thus determining the exact values of the boundary conformal charges.It should be noted that our findings can be also relevant to the study of the logarithmic termsin the entanglement entropy of a conformal field theory in d = 6. Indeed, the entangling surfacethen has dimension 4 and the logarithmic terms are due to the possible conformal invariantsconstructed from the curvature of spacetime and the extrinsic curvature of the surface. Theprevious study of the logarithmic terms within the holographic paradigm includes [27]. In thepaper we make contact with this previous study.The holographic aspect of the conformal anomaly in five dimensions in the presence of theboundaries is interesting. We, however, leave it for future work. d = 5 We denote by n µ the components of the out-ward normal vector to the boundary ∂ M , n µ n µ =1. The metric induced on the boundary is γ µν = g µν − n µ n ν . In this and the subsequent sectionswe use the notations that the latin indices i , j , k . . . should be understood as projectionsalong the boundary, they take values 1 , , ,
4. The index projected on the normal directionis denoted by n . With these notations for any tensor X µαβ one has that X nij = n µ γ αi γ βj X µαβ .The extrinsic curvature is defined as K ij = γ µi γ νj ∇ ( µ n ν ) , where ∇ µ is the covariant derivativedefined with respect to the 5d metric g µν . The covariant derivative defined with respect to theintrinsic metric γ ij is denoted by ¯ ∇ i and the respective curvature by ¯ R , ¯ R ij and ¯ R ijkl . Therelations between the intrinsic curvature of the boundary and the curvature in the 5d space-timeare given by the Gauss-Codazzi identities presented in Appendix A.3nder the infinitesimal conformal transformations δg µν = 2 σg µν , δn µ = σn µ the Weyl tensortransforms as δW αµβν = 2 σW αµβν . The extrinsic curvature transforms as follows δK ij = σK ij + γ ij ∇ n σ , δK = − σK + 4 ∇ n σ , δ ˆ K ij = σ ˆ K ij , (1)where ∇ n = n µ ∇ µ and ˆ K ij = K ij − γ ij K is the trace free part of the extrinsic curvature.The basic conformal tensors are, thus, the bulk Weyl tensor W αµβν and the trace free extrinsiccurvature of the boundary ˆ K ij . The intrinsic Weyl tensor of the boundary metric is expressedin terms of the 5-dimensional Weyl tensor and the extrinsic curvature by means of the Gauss-Codazzi relations. The integrated conformal anomaly in five dimensions can be presented in the form, Z M h T µν i g µν = 15760(4 π ) ( aE + X k =1 c k I k ) , (2)where E is the 4d Euler density integrated over the boundary ∂ M , so that χ [ ∂ M ] = π E is the Euler number of the boundary, and I k , k = 1 , . . . , K ij = K ij − γ ij K . We note that the Euler number of an odd-dimensionalopen manifold is 1/2 of the Euler number of the boundary. This explains why the Euler number E is not an independent quantity and, thus, is absent in the anomaly (2). The numerical pre-factor in (2) is chosen for the further convenience. ( a , c , . . . , c ) are the boundary conformalcharges to be determined below in the paper. The Euler density of the 4-dimensional boundary has the standard expression in terms of theintrinsic curvature of the boundary, E = Z ∂ M ( ¯ R ikjℓ − R ij + ¯ R ) . (3)Applying the Gauss-Codazzi equations, it can be expanded in terms of the curvature tensors ofthe bulk manifold as well as the extrinsic curvature of the four-dimensional boundary, E = Z ∂ M (cid:16) R ijkℓ − R ij + R − R injn + 8 R ij R injn + 4 R nn − RR nn + 4 K ij K kℓ R ikjℓ + 8( K ) ij R ij − KK ij R ij − K R + 2 K R − K ) ij R injn + 8 KK ij R injn + 4 Tr K R nn − K R nn − K + 8 K Tr K + 3( Tr K ) − K Tr K + K (cid:17) , (4)4here we defined ( K ) ij = K ik K kj .Now we list the conformal invariants which can be called algebraic. These invariants do notcontain the derivatives of the extrinsic curvature. The first group of the invariants of this type is constructed from the trace free extrinsic curvatureˆ K ij , I = Z ∂ M ( Tr ˆ K ) = Z ∂ M [( Tr K ) − K Tr K + 116 K ] , (5)and I = Z ∂ M Tr ˆ K = Z ∂ M ( Tr K − K Tr K + 38 K Tr K − K ) . (6) The next set of the invariants is constructed from the 5d Weyl tensor, I = Z ∂ M W ikjℓ = Z ∂ M (cid:18) R ikjℓ − R ij + 518 R + 83 R ij R injn + 49 R nn − RR nn (cid:19) , (7)and I = Z ∂ M W injn = Z ∂ M (cid:18) R ij − R + R injn − R ij R injn − R nn + 29 RR nn (cid:19) . (8) Contractions of the Weyl tensor and the trace free extrinsic curvature give us a new set of theconformal invariants, I = Z ∂ M ˆ K ij ˆ K kℓ W ikjℓ = Z ∂ M (cid:16) K ij K kℓ R ikjℓ + 23 ( K ) ij R ij − KK ij R ij −
112 Tr K R + 18 K R + 12 KK ij R injn − K R nn (cid:17) , (9)5nd I = Z ∂ M ˆ K ik ˆ K kj W injn = Z ∂ M (cid:16) − K ik K kj R ij + 16 KK ij R ij + 112 Tr K R − K RK ik K kj R injn − KK ij R injn −
13 Tr K R nn + 16 K R nn (cid:17) . (10) The invariants of this group will be expressed in terms of the derivatives of the extrinsic curvaturealong the boundary. The first invariant of this kind is I = Z ∂ M W nijk . (11)Using the Gauss-Codazzi equation, W nijk can be expanded in terms of the derivatives of theextrinsic curvature tensor W nijk = 2 ¯ ∇ [ k K j ] i + 23 h i [ j (cid:0) ¯ ∇ ℓ K ℓk ] − ¯ ∇ k ] K (cid:1) . (12)This identity helps us to re-write the invariant I in the form that contains only the extrinsiccurvature tensor and its derivatives along the boundary, I = Z ∂ M (cid:18) ∇ k K ij ¯ ∇ k K ij − ∇ k K ij ¯ ∇ j K ki + 43 ¯ ∇ i K ¯ ∇ j K ij −
23 ¯ ∇ i K ij ¯ ∇ k K kj −
23 ( ¯ ∇ K ) (cid:19) , (13)using (A.10), one re-writes this expression as follows I = Z ∂ M h ∇ k K ij ¯ ∇ k K ij −
83 ¯ ∇ i K ij ¯ ∇ k K kj + 43 ¯ ∇ i K ¯ ∇ j K ij −
23 ( ¯ ∇ K ) − K ij K kℓ R ikjℓ − K ) ij R injn + 2( K ) ij R ij + 2 K Tr K −
2( Tr K ) i . (14)It is instructive to analyse the relation of this invariant to another conformal invariant thatis written in terms of the conformal invariant operator acting on a tensor of rank two. Thefollowing term is introduced in [1] and is known to be a conformal invariant in 5 dimensions, I ( D ) = Z ∂ M Tr ˆ K F ˆ K , (15)where the differential operator F ijkℓ = δ i ( k δ jℓ ) ¯ (cid:3) −
43 ¯ ∇ ( i ¯ ∇ ( k δ j ) ℓ ) −
86 ¯ R i ( k j ℓ ) −
23 ¯ R ( i ( k δ j ) ℓ ) + 16 ¯ Rδ i ( k δ jℓ ) (16)6cts on a tensor with conformal weight +1. It should be noted that there is only one suchdifferential operator that respects the conformal symmetry [28], [29]. The explicit form of thisinvariant reads I ( D ) = Z ∂ M Tr ˆ K F ˆ K = Z ∂ M ˆ K kℓ F ijkℓ ˆ K ij = Z ∂ M h K ij ¯ (cid:3) K ij − K ¯ (cid:3) K + 13 K ij ¯ ∇ i ¯ ∇ j K + 13 K ¯ ∇ i ¯ ∇ j K ij − K ij ¯ ∇ j ¯ ∇ k K ki + 2( K ) ij R injn − KK ij R injn − K ) ij R ij + KK ij R ij −
13 Tr K R nn + 13 K R nn + 16 Tr K R − K R + 2 Tr K − K Tr K −
16 ( Tr K ) + 43 K Tr K − K i . (17)It can be shown using the Gauss-Codazzi relations that in flat bulk spacetime ( R µναβ = 0) allderivatives of the extrinsic curvature in (17) are mutually cancelled.In what follows it will be shown that this term is not an independent invariant. To see this,let us start with the observation that I ( D ) = − I − Z ∂ M ˆ K ij ˆ K kℓ ¯ W ikjℓ (18)Note also that using the Gauss-Codazzi relations one has that Z ∂ M ˆ K ij ˆ K kℓ ¯ W ikjℓ = I + Z ∂ M h − ( R injn − R ij )(( K ) ij − KK ij )+ 13 ( R nn − R )( Tr K − K ) − K + 2 K Tr K + 76 ( Tr K ) − K Tr K + 16 K i . (19)The integral in the right hand side of the above equation can be written in terms of I , I and I . So finally we conclude that I ( D ) = − I − I + I − I + 2 I . (20)This indicates that it is not an independent invariant and, thus, it has to be excluded from thelist. Note that all derivative terms in I ( D ) come just from I , so it is natural that all derivativeterms disappear in the flat spacetime case, where W nijk = 0. To construct the last invariant, we start with the normal derivative of the Weyl tensor, con-tracted with the normal vector and the traceless extrinsic curvature tensor I (1)8 = Z ∂ M ˆ K ij ∇ n W injn , (21)7here by ∇ n W injn we mean n ρ n µ n ν ∇ ρ W iµjν .We can show that δI (1)8 = − Z ∂ M (cid:16) ˆ K ij W injn ∂ n σ + ˆ K ij W nijk ¯ ∇ k σ (cid:17) . (22)The first term in the conformal variation can be removed if we simply add I (2)8 = 12 Z ∂ M K ˆ K ij W injn . (23)Let us focus now on the second term in (22). We can construct an integral with the sametransformation I (3)8 = Z M (cid:18)
19 ¯ ∇ i ˆ K ij ¯ ∇ k ˆ K kj − ( ˆ K ) ij ¯ S ij + 12 Tr ˆ K ¯ S ii (cid:19) , (24)where ¯ S ij = 12 ( ¯ R ij −
16 ¯ Rγ ij ) (25)is the 4d Schouten tensor computed with respect to the intrinsic boundary metric γ ij . Underthe conformal transformations one has δ ( ¯ ∇ i ˆ K ij ¯ ∇ k ˆ K kj ) = − σ ¯ ∇ i ˆ K ij ¯ ∇ k ˆ K kj + 6 ˆ K ij ¯ ∇ k ˆ K kj ¯ ∇ i σ ,δ ¯ S ij = − ¯ ∇ i ¯ ∇ j σ , δ ¯ S ii = − σ ¯ S ii − ¯ (cid:3) σ . (26)Therefore one finds that δI (3)8 = Z ∂ M (cid:20)
23 ˆ K ij ¯ ∇ k ˆ K kj ¯ ∇ i σ + ( ˆ K ) ij ( ¯ ∇ i ¯ ∇ j σ − γ ij ¯ (cid:3) σ ) (cid:21) . (27)Interestingly, we may recast the above expression as follows, δI (3)8 = Z ∂ M ˆ K ij W nijk ¯ ∇ k σ . (28)This is precisely what we need to cancel the second term in the conformal transformation of I (1)(8) . So we can now construct the following conformal invariant, I = I (1)8 + I (2)8 + 2 I (3)8 , I = Z M (cid:18) ˆ K ij ∇ n W injn + 12 K ˆ K ij W injn + 29 ¯ ∇ i ˆ K ij ¯ ∇ k ˆ K kj −
2( ˆ K ) ij ¯ S ij + Tr ˆ K ¯ S ii (cid:19) . (29)Using the Gauss-Codazzi equations and the differential relations (A.9), (A.8) and (A.10) thenew invariant I can be rewritten as follows, I = Z ∂ M h K ij ∇ n R injn − K ∇ n R + 23 K ij K kℓ R ikjℓ − K ik K jk R ij + 13 Tr K R − K R + 53 K ik K kj R injn − KK ij R injn − Tr K R nn + 1124 K R nn + Tr K − K Tr K + 4748 K Tr K − K −
13 ¯ ∇ k K ij ¯ ∇ k K ij + 89 ¯ ∇ i K ij ¯ ∇ k K kj −
79 ¯ ∇ i K ij ¯ ∇ j K + 2572 ( ¯ ∇ K ) i . (30)8o the best of our knowledge the invariant I is a new invariant that was not available in theliterature before. It has been for a long time an important missing element in the discussionof the conformal anomalies in five dimensions. Having said that we should notice that in someparticular case this invariant reduces to the one that already appeared in the literature onthe holographic computation of the entanglement entropy in d = 6. Indeed, provided the 5dmanifold is flat and using the Gauss-Codazzi equations, one has that R nijk = 0 → ¯ ∇ k K ij = ¯ ∇ j K ik . (31)In this case I takes a simpler form, I flat = Z ∂ M (cid:20)
18 ( ¯ ∇ K ) + Tr K − K Tr K −
13 ( Tr K ) + 4748 K Tr K − K (cid:21) . (32)Interestingly, in this form, it is related to invariant T found by Safdi [27] in a holographiccalculation of the entanglement entropy on a 6 dimensional flat manifold, I flat = 18 ( T + 103 I − I ) . (33)It should be noted that no conformal invariant form of T was given in [27]. Thus, our result(29), (30), among other things, offers a conformal invariant form, valid in a curved spacetime,for this holographic calculation. The free conformal scalar field in five dimensions is described by a conformal Laplace operator, D = − ( ∇ + E ) , E = − R . (34)In the space-time with boundaries it should be supplemented by a boundary condition. Thereare two possible boundary conditions consistent with the conformal symmetry: the Dirichletcondition and the conformal Robin condition,Dirichlet b.c. : φ | ∂ M = 0 , Robin b.c. : ( ∂ n − S ) φ | ∂ M = 0 , S = − K . (35)The important object that encodes the main information about the quantum field theory is theheat kernel K ( D, s ) = e − sD and its small s expansion,Tr K ( D, s ) = X p =0 a p ( D ) s ( p − / , s → a p ( D ) are the heat kernel coefficients that are represented by integrals over the manifoldand its boundary of the local invariants constructed from the curvature of the bulk metric, theextrinsic curvature of the boundary and the quantities E and S .The integrated conformal anomaly in dimension d = 5 is determined by the coefficient a , Z M h T µν i g µν = a . (37)As was explained above, there is no a bulk term in a and the entire contribution comes onlyfrom the boundary ∂ M . The exact form of the coefficient a is available in the literature fora rather general elliptic operator with the boundary condition being a mixture of the Dirichletand the Robin conditions, see [23], [24], [25], [26]. In what follows we use the general form of a presented in [23].For the conformal operator D and the Dirichlet boundary condition we find that a ( D )5 = − π ) Z ∂ M (cid:16) R µανβ − R µν + 516 R − R injn + 16 R ij R injn − R nn + 52 RR nn + 48 (cid:3) R −
45 ¯ (cid:3) R − ∇ n R + 24 ¯ (cid:3) R nn + 15 ∇ n R nn − K ∇ n R + 32 K ij K kℓ R ikjℓ + 16 K ik K kj R ij + 14 KK ij R ij + 258 Tr K R − K R − K ik K kj R injn + 494 KK ij R injn − K R nn − K R nn − K + 172 K Tr K + 77732 ( Tr K ) − K Tr K − K + 3558 ¯ ∇ k K ij ¯ ∇ k K ij −
114 ¯ ∇ i K ij ¯ ∇ k K jk −
294 ¯ ∇ k K ji ¯ ∇ j K ik + 58 ¯ ∇ i K ij ¯ ∇ j K − K ¯ ∇ i ¯ ∇ j K ij − K jk ¯ ∇ k ¯ ∇ i K ij + 2854 K ij ¯ ∇ i ¯ ∇ j K + 54 K ij ¯ (cid:3) K ij + 6 K ¯ (cid:3) K − ∇ i K ¯ ∇ i K (cid:17) , (38)while for the conformal Robin boundary condition we have that a ( R )5 = 15760(4 π ) Z ∂ M (cid:16) R µανβ − R µν + 516 R − R injn + 16 R ij R injn − R nn + 52 RR nn + 48 (cid:3) R −
45 ¯ (cid:3) R − ∇ n R + 24 ¯ (cid:3) R nn + 15 ∇ n R nn − K ij ∇ n R injn − K ∇ n R + 32 K ij K kℓ R ikjℓ + 16 K ik K kj R ij + 432 KK ij R ij −
58 Tr K R − K R + 1332 K ik K kj R injn − KK ij R injn − K R nn − K R nn + 2318 Tr K − K Tr K + 37532 ( Tr K ) + 14732 K Tr K − K + 5358 ¯ ∇ k K ij ¯ ∇ k K ij + 494 ¯ ∇ i K ij ¯ ∇ k K jk + 1514 ¯ ∇ k K ji ¯ ∇ j K ik + 58 ¯ ∇ i K ij ¯ ∇ j K − K ¯ ∇ i ¯ ∇ j K ij − K jk ¯ ∇ k ¯ ∇ i K ij + 3154 ¯ ∇ i ¯ ∇ j K + 114 K ij ¯ (cid:3) K ij − K ¯ (cid:3) K − ∇ i K ¯ ∇ i K (cid:17) . (39)10e shall insert here R µανβ = R ikjℓ + 4 R nijk + 4 R injn , R µν = R ij + 2 R in + R nn . (40)The other important point is that the general expression for the coefficient a given in [23]contains some sort of redundancy since the differential relation (A.11), which together with(A.8) generalises the Gauss-Codazzi identities, was not taken into account. In fact, a similarredundancy is present in the coefficient a given in [3], where the relation (A.8) was not takeninto account. Since these relations contain the higher order normal derivatives one might worrywhether one would have to specify an extension for the normal vector our-side the boundary tomake these relations valid. We, however, have checked that the relations (A.7) - (A.11) do notdepend on the way the normal vector is extended. Thus, using the Gauss-Codazzi equationsand the identities (A.7), (A.11) we finally arrive at the expression a ( D )5 = 15760(4 π ) Z ∂ M (cid:16) − R ikjℓ + 8 R ij − R − R injn − R ij R injn + 10 R nn − RR nn − K ij ∇ n R injn + 158 K ∇ n R + 2774 K ij K kℓ R ikjℓ − K ik K kj R ij + KK ij R ij −
258 Tr K R + 3516 K R + 4994 K ik K kj R injn − KK ij R injn −
258 Tr K R nn − K R nn + 3278 Tr K − K Tr K + 198332 ( Tr K ) + 14132 K Tr K + 65128 K − ∇ k K ij ¯ ∇ k K ij + 1652 ¯ ∇ i K ij ¯ ∇ k K jk − ∇ i K ij ¯ ∇ j K + 28516 ¯ ∇ i K ¯ ∇ i K (cid:17) (41)for the conformal scalar operator with the Dirichlet boundary condition and a ( R )5 = 15760(4 π ) Z ∂ M (cid:16) R ikjℓ − R ij + 516 R + 22 R injn + R ij R injn − R nn + 52 RR nn − K ij ∇ n R injn + 158 K ∇ n R − K ij K kℓ R ikjℓ + 1094 K ik K kj R ij + 132 KK ij R ij −
58 Tr K R − K R + 414 K ik K kj R injn − KK ij R injn −
358 Tr K R nn + 5516 K R nn + 2318 Tr K + 10 K Tr K − K ) + 14732 K Tr K − K + 2558 ¯ ∇ k K ij ¯ ∇ k K ij − ∇ i K ij ¯ ∇ k K jk + 1354 ¯ ∇ i K ij ¯ ∇ j K − ∇ i K ¯ ∇ i K (cid:17) (42)for the conformal scalar operator with the conformal Robin boundary condition (35). The relation (A.8) has appeared earlier in [30] and [5]. a − c − c c −
818 818 c −
272 272 c −
98 1898 c c − c − − Following the earlier works [23] - [26], we adopted in this paper the convention that thecomponents of the normal vector n µ are always outside the covariant derivatives, for instance ∇ n R nn ≡ n α n µ n ν ∇ α R µν , ∇ n R ninj ≡ n α n µ n ν ∇ α R µiνj etc. This guarantees that these terms donot depend on how the components of the normal vector are extended outside the boundary.It should be noted that the elliptic operators in question do not require any such extension sothat the respective heat kernel coefficients contain only terms that are independent of the waythe vectors normal to the boundary are extended to the nearest vicinity of the boundary. This,in particular, explains why in the heat kernel coefficients there do not appear any terms thatcontain the normal derivative of the extrinsic curvature, ∇ n K ij , since for these terms one wouldneed to specify the extension of the normal vector outside the boundary .The expressions for a , obtained above, should be now compared with the general form (2)of the anomaly decomposed over the possible conformal invariants. We have checked that thedecomposition is indeed possible and unique both for a ( D )5 and a ( R )5 so that the list of conformalinvariants E , I , . . . , I is indeed complete. The details of the analysis are given in AppendixB. The decomposition allows us to determine the values of the boundary conformal charges inthe case of a conformal scalar field. For each boundary condition, one arrives at 27 equationson 9 charges. That solution of this highly overdetermined system of algebraic equations existsis a nice check on our formulas. The result is presented in Table 1. The values found for theEuler charge a agree with the earlier computations [14], [31]. We thank D. Vassilevich for correspondence on this point. Conclusions
In this note we have presented an exhaustive discussion of the boundary conformal invariantsin five dimensions. In particular, and this is one of our main results, we have found a newconformal invariant I that contains the derivatives of the extrinsic curvature tensor along theboundary. In a flat space limit this invariant is related to the one found holographically by Safdi.We note that the invariants discussed in this paper do not depend on the way the normal vectoris extended outside the boundary. Only invariants of this type may appear in the heat kernel ofan elliptic operator that knows nothing about such an extension. We, however, note here for therecord that there exists a family of the conformal tensors and the respective conformal invariantsthat can be defined provided an extension of the normal vector is given. These invariants havenot been discussed in this paper.We use the available in the literature general results for the heat kernel coefficient a andcompute the integrated conformal anomaly for a conformal scalar satisfying either the Dirichletor the conformal Robin boundary conditions. The anomaly is then uniquely decomposed overthe set of conformal invariants we have found. We then compute the respective conformalcharges. This is our second main result. It would be interesting to extend our results andcompute the conformal anomaly and the respective boundary charges for the conformal fieldsof higher spin. Although we do not anticipate any principle difficulties we do not present thisanalysis here leaving it to the future.Among other possible applications, our results will be useful in classifying the possibleconformal invariants that appear in the logarithmic terms of entanglement entropy of a d = 6conformal field theory. The holographic derivation of the conformal anomaly in five dimensionsis yet another interesting subject to explore. We leave these directions for future work. Acknowledgements
We thank Cl´ement Berthiere for collaboration at the beginning of this project. We are gratefulto Dmitri Vassilevich for a useful correspondence. This project was started when both of us werevisiting the Theory Division at CERN in the spring of 2018. We thank CERN for hospitality.13
Gauss-Codazzi relations
The Gauss-Codazzi identities give us relations between the bulk curvature R , the intrinsiccurvature on the boundary ¯ R and the extrinsic curvature K ij , R ikjl = γ µi γ αk γ νj γ βℓ R µανβ = ¯ R ikjℓ − ( K ij K kℓ − K iℓ K jk ) , (A.1) R nijk = γ µi γ νj γ ρk n α R αµνρ = ( ¯ ∇ k K ij − ¯ ∇ j K ik ) , (A.2)where γ µi represents the projection operation. Contracted Equations R in = R ni = ( ¯ ∇ j K ji − ¯ ∇ i K ) . (A.3) R ij = ¯ R ij + R injn + ( K ij − KK ij ) . (A.4) Doubly Contracted Equations R = ¯ R + 2 R nn + ( Tr K − K ) . (A.5)Thus, one finds for G nn = G µν n µ n ν , G µν = R µν − g µν R , G nn = −
12 ¯ R −
12 ( Tr K − K ) . (A.6) Differential Equations (cid:3) R = ¯ (cid:3) R + ∇ n R + K ∇ n R , (A.7) ∇ n G nn = ∇ n ( R nn − R ) = K ij R ij − KR nn − ¯ ∇ i ¯ ∇ j K ij + ¯ (cid:3) K , (A.8) ∇ n R ij = ∇ n R injn − K kℓ R ijkℓ − K ki R jnkn − K kj R inkn + KR injn + K ij R nn − Tr K K ij + KK ik K kj + ¯ ∇ i ¯ ∇ k K kj + ¯ ∇ j ¯ ∇ k K ki − ¯ (cid:3) K ij − ¯ ∇ i ¯ ∇ j K , (A.9)¯ ∇ k K ij ¯ ∇ j K ik = ¯ ∇ i K ij ¯ ∇ k K kj + K ij K kℓ R ikjℓ − K ik K jk R ij + K ik K jk R injn − K Tr K + ( Tr K ) + T.D. , (A.10) ∇ n G nn = − R ij R injn + R nn − K ik K kj R ij + Tr K R nn + K ij ∇ n R ij − K ∇ n R nn − ¯ ∇ i K ij ¯ ∇ j K + ( ¯ ∇ K ) + T.D. , (A.11)where we defined ∇ n G nn = n α n µ n ν ∇ α G µν and ∇ n G nn = n α n β n µ n ν ∇ α ∇ β G µν .14 Algebraic equations for the conformal charges
Matching the coefficients of the various terms in a ( D,R )5 with the corresponding factors in thedecomposition over the basis of the conformal invariants one arrives at the following algebraicequations for the conformal charges: Tr K : − a + c + c = ( 3278 , ,K Tr K : 8 a − c + 2 c − c = ( − , , ( Tr K ) : 3 a + c − c = ( 198332 , − ,K Tr K : − a − c + 38 c + 4748 c = ( 14132 , ,K : a + 116 c − c − c = ( 65128 , − ,R ikjℓ : a + c = ( − , ,R ij : − a − c + 19 c = (8 , − ,R : a + 518 c − c = ( − ,
516 ) ,R injn : − a + c = ( − , ,R ij R injn : 8 a + 83 c − c = ( − , ,R nn : 4 a + 49 c − c = (10 , − ,RR nn : − a − c + 29 c = ( − ,
52 ) ,K ij K kℓ R ikjℓ : 4 a + c − c + 23 c = ( 2774 , −
974 ) ,K ik K kj R ij : 8 a + 23 c − c + 2 c − c = ( − , ,KK ij R ij : − a − c + 16 c = (1 ,
132 ) , Tr K R : − a − c + 112 c + 13 c = ( − , −
58 ) ,K R : 2 a + 18 c − c − c = ( 3516 , −
54 ) ,K ik K kj R injn : − a + c − c + 53 c = ( 4994 ,
414 ) ,KK ij R injn : 8 a + 12 c − c − c = ( − , − , Tr K R nn : 4 a − c − c = ( − , −
358 ) ,K R nn : − a − c + 16 c + 1124 c = ( − , , ¯ ∇ k K ij ¯ ∇ k K ij : 2 c − c = ( − , , ¯ ∇ i K ij ¯ ∇ k K kj : − c + 89 c = ( 1652 , − , ¯ ∇ i K ij ¯ ∇ j K : 43 c − c = ( − , , ( ¯ ∇ K ) : − c + 2572 c = ( 28516 , − ,K ij ∇ n R injn : 23 c = ( − , − ,K ∇ n R : − c = ( 158 ,
158 ) . eferences [1] S. N. Solodukhin, Phys. Lett. B , 131 (2016) [arXiv:1510.04566 [hep-th]].[2] J. S. Dowker and J. P. Schofield, J. Math. Phys. , 808 (1990).[3] D. V. Vassilevich, Phys. Rept. , 279 (2003) [hep-th/0306138].[4] D. M. Capper and M. J. Duff, Nuovo Cim. A , 173 (1974).S. Deser, M. J. Duff and C. J. Isham, Nucl. Phys. B , 45 (1976).M. J. Duff, Class. Quant. Grav. , 1387 (1994).[5] D. Fursaev, JHEP , 112 (2015) [arXiv:1510.01427 [hep-th]].[6] C. P. Herzog, K. W. Huang and K. Jensen, JHEP , 162 (2016) [arXiv:1510.00021[hep-th]].[7] D. V. Fursaev and S. N. Solodukhin, Phys. Rev. D , no. 8, 084021 (2016)[arXiv:1601.06418 [hep-th]].[8] C. P. Herzog and K. W. Huang, JHEP (2017), 189 [arXiv:1707.06224 [hep-th]].[9] C. Berthiere and S. N. Solodukhin, Nucl. Phys. B (2016), 823-841 [arXiv:1604.07571[hep-th]].[10] C. Berthiere, Phys. Rev. B (2019), no.16, 165113 [arXiv:1811.12875 [cond-mat.str-el]].[11] A. F. Astaneh and S. N. Solodukhin, Phys. Lett. B (2017), 25-33 [arXiv:1702.00566[hep-th]].[12] A. F. Astaneh, C. Berthiere, D. Fursaev and S. N. Solodukhin, Phys. Rev. D (2017)no.10, 106013 [arXiv:1703.04186 [hep-th]].[13] C. P. Herzog, K. W. Huang and D. V. Vassilevich, JHEP (2020), 132 [arXiv:2005.01689[hep-th]].[14] D. Rodriguez-Gomez and J. G. Russo, JHEP (2017), 066 [arXiv:1710.09327 [hep-th]].[15] C. Herzog, K. W. Huang and K. Jensen, Phys. Rev. Lett. (2018) no.2, 021601[arXiv:1709.07431 [hep-th]].[16] K. W. Huang, JHEP (2016), 013 [arXiv:1604.02138 [hep-th]].[17] K. Jensen and A. O’Bannon, Phys. Rev. Lett. (2016) no.9, 091601 [arXiv:1509.02160[hep-th]]. 1718] N. Drukker, M. Probst and M. Tr´epanier, “Defect CFT techniques in the 6d N = (2 , (2017), 034 [arXiv:1612.06386 [hep-th]].[20] D. Seminara, J. Sisti and E. Tonni, JHEP (2017), 076 [arXiv:1708.05080 [hep-th]].[21] D. Seminara, J. Sisti and E. Tonni, JHEP (2018), 164 [arXiv:1805.11551 [hep-th]].[22] M. Glaros, A. R. Gover, M. Halbasch and A. Waldron, “Singular Yamabe ProblemWillmore Energies,” arXiv:1508.01838 [math.DG]. A. R. Gover and A. Waldron, “Con-formal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem,”arXiv:1506.02723 [math.DG]. Yu. Vyatkin, “Manufacturing Conformal Invariants of Hy-persurfaces”, PhD Thesis, Auckland, 2013.[23] T. P. Branson, P. B. Gilkey, K. Kirsten and D. V. Vassilevich, Nucl. Phys. B , 603-626(1999) [arXiv:hep-th/9906144 [hep-th]].[24] T. P. Branson, P. B. Gilkey and D. V. Vassilevich, Boll. Union. Mat. Ital. B , 39-67(1997) [arXiv:hep-th/9504029 [hep-th]].[25] K. Kirsten, Class. Quant. Grav. , L5-L12 (1998) [arXiv:hep-th/9708081 [hep-th]].[26] K. Kirsten, “Spectral functions in mathematics and physics,” [arXiv:hep-th/0007251 [hep-th]].[27] B. R. Safdi, JHEP , 005 (2012) [arXiv:1206.5025 [hep-th]].[28] V. P. Gusynin and V. V. Romankov, “Conformally Covariant Operators and EffectiveAction in External Gravitational Field,” Sov. J. Nucl. Phys. (1987), 1097-1099.[29] J. Ben Achour, E. Huguet and J. Renaud, Phys. Rev. D , 064041 (2014)[arXiv:1311.3124 [gr-qc]].[30] D. M. McAvity and H. Osborn, Class. Quant. Grav. (1991), 603-638[31] Y. Wang, “Defect a -Theorem and aa