Anomalies, Chern-Simons Terms and Black Hole Entropy

Recent derivations of Cardy-like formulae in higher dimensional field theories have opened up a way of computing, via AdS/CFT, universal contributions to black hole entropy from gravitational Chern-Simons terms. Based on the manifestly covariant formulation of the differential Noether charge for Chern-Simons terms proposed in arXiv:1407.6364, we compute the entropy and asymptotic charges for the rotating charged AdS black holes in higher dimensions at leading order of the fluid/gravity derivative expansion in the Einstein-Maxwell-Chern-Simons system. This gives a result that exactly matches the field theory predictions from Cardy-like formulae.


Introduction
Since the famous demonstration by Bekenstein and Hawking that black hole geometries have entropy, the quest for understanding microscopic origins of that entropy has taken us on a long and fascinating adventure connecting various fields of physics and mathematics. The advent of string theory and AdS/CFT has injected much enthusiasm into this subject, and for a wide class of extremal/near-extremal black holes we can now claim to understand (albeit by somewhat indirect means) where this entropy comes from [2][3][4][5][6][7][8].
Accounting for the entropy of finite-temperature black holes has however been a more difficult endeavor. The major successes in this regard are often linked to anomalies -a paradigmatic result in this direction is the Cardy formula [9] which links the thermodynamic properties of a 2d CFT with the anomaly coefficients (the right and left central charges c R,L ) calculable from the microscopic description. In context of AdS 3 /CFT 2 , this fact has been repeatedly exploited [10] to account for the entropy of finite temperature AdS 3 black holes far from extremality. It is hence an interesting question to ask whether this success can be extended to higher dimensions.
The parity even part of the Cardy formula does not generalize to higher dimensional field theories. 1 However, recently the analogue of the parity odd part of the Cardy formula in higher dimensions has been conjectured [11][12][13][14] and proved [15][16][17]. As we will briefly review below, this generalization -often goes by the name of 'replacement rule'-gives a prescription in which one starts with the anomaly polynomial P CF T of the field theory under question and then, by a series of steps, constructs an expression for the leading parity odd part of the entropy.
It is then natural to enquire whether the replacement rule can be used to account for the leading parity odd part of the black hole entropy. The main aim of this work is to show that this is indeed the case and that successes in AdS 3 /CFT 2 can be extended to the parity odd sector of any even dimensional CFTs.
The anomaly coefficients we are interested in (along with the Weyl anomaly coefficients to whom they are related to via supersymmetry) have been conjectured to determine various terms in different supersymmetric partition functions. Such conjectures have been investigated actively by various authors including [18][19][20][21][22][23][24][25][26]. Although these supersymmetric versions do not yet have a general proof of the type given in [15,17], there is a mounting evidence for their validity. Our arguments in this work about how anomalies show up in gravity computations would hopefully be extended to such supersymmetric versions.
The first step in this direction is to construct a large class of black hole solutions which will play the role of the famous BTZ black hole solution in higher dimensions. In order to have a parity odd part to the entropy associated with anomalies, these black holes should be solutions of a gravitational theory with Chern-Simons terms. The simplest system of this kind is the Einstein-Maxwell-Chern-Simons system with an action (1.1) Here the Chern-Simons part of the Lagrangian is denoted as I CS which is a (d+1)-form. 2 Since Chern-Simons terms are odd forms, this necessarily implies that d = 2n with a positive integer n. The cosmological constant Λ cc is taken to be negative and is given by Λ cc ≡ −d(d − 1)/2 such that G ab is an asymptotically AdS d+1 metric with unit radius, F ab is the Maxwell field strength, G N and g Y M are the Newton and Maxwell couplings respectively. For later use, we also define the normalized Maxwell coupling constant κ q by 16πG N /g 2 Y M = κ q (d − 1)/(d − 2). Large, charged, rotating black hole solutions of this system were constructed in our recent work [27] using fluid/gravity correspondence.
The main aim of this work is to compute the entropy and the asymptotic charges of these solutions. In course of our calculations in this somewhat simplified system, we will exhibit various structural features which we believe would carry over to more complicated examples in string theory. In particular, the bulk of our Appendices are devoted to proving a kind of 'non-renormalization theorem' for anomaly-induced entropy which shows how anomalyinduced entropy does not get corrected in fluid/gravity expansion. We expect this result (along with the various structural cancellations that lead to it) to hold in more complicated examples. In fact, the robustness of anomaly induced entropy in field theory suggests that such a result should hold even when stringy and quantum gravity corrections are taken into account ! The second step is to develop a coherent method to compute the entropy of black hole solutions in the presence of Chern-Simons terms. This involves various subtleties due to the non-covariant nature of Chern-Simons terms in the Lagrangian density. In particular, the original Noether procedure due to Wald [28][29][30] is valid only for the system described by a covariant Lagrangian and thus fails in the case of Chern-Simons terms. Fortunately, this Wald formalism for constructing differential Noether charges was extended to theories with Chern-Simons terms by Tachikawa [31]. As demonstrated by Bonora-Cvitan-Prester-Pallua-Smolic [32], however, this extension suffers from various non-covariance issues for AdS spacetime with dimensions greater than three. In our recent work [1], we identified the root cause of these non-covariance problems to be the choice of a non-covariant pre-symplectic structure in the Tachikawa method.
Further, in that work, we showed that with higher dimensional Chern-Simons terms, one can instead choose a manifestly covariant pre-symplectic structure and implement the Noether procedure without any subtleties. One of the main results of that work was a covariant expression for Chern-Simons contribution ( / δQ Noether ) H to the differential Noether charge co-dimension 2 form / δQ Noether , given as a sum of five terms : where each term on the right hand side is determined by the derivative of the anomaly polynomial P CF T = dI CS (I CS : Chern-Simons terms in the Lagrangian) as Here the spin Hall current (Σ H ) cb a is defined by (Σ H ) cb a ⋆ dx c ≡ −2(∂P CF T /∂R a b ) and (Λ, ξ a ) are the parameters for the U (1) gauge transformation and diffeomorphism, respectively. This ( / δQ Noether ) H is manifestly covariant for any odd dimensional spacetime. This expression with its five terms denoted by T i will play a crucial role in this work. In this paper, we take the third step whereby we evaluate this Noether charge on our black hole solutions and show that this contribution to the black hole entropy is exactly accounted for by the anomaly-induced entropy on the CFT side [13][14][15]17]. In addition, we will also use this covariant differential Noether charge to evaluate the asymptotic charges, that is, the energy-momentum tensor and the charge current of the dual CFT. This completes the holographic and systematic derivation of the replacement rules for these quantities initiated in [27]. Before we proceed to the details of our computation, certain clarifying comments are in order regarding the use of differential Noether charge. For a time-independent black hole solutions with bifurcate Killing horizon, the differential Noether charge form exhibited above can be integrated over the bifurcation surface. Following Wald, we can then derive an integral expression for the total Noether charge of stationary solutions [1] and this gives the correct modification of Wald entropy in the presence of Chern-Simons terms as originally conjectured by Tachikawa [31] : where in the first line, the integrals are over the bifurcation surface with ε ab denoting the binormal at the bifurcation surface. In the second line, we have pulled back these integrals to a spatial slice in the AdS boundary using the ingoing null geodesic prescription for the CFT entropy current J µ S,CFT following [33]. In the above entropy formula, L cov is the covariant part of the gravity Lagrangian which contributes via the famous Wald formula as expected. The CFT anomaly polynomial P CF T = dI CS encodes the information about the Chern-Simons part and we have presented this contribution in terms of the normal bundle connection Γ N on the bifurcation surface and its curvature R N = dΓ N . They are defined using the binormal as (1.5) While we will have much to say about the structure of Tachikawa formula (1.4) especially visà vis the structure of the replacement rule, 3 we will rely directly on ( / δQ Noether ) H for our main results. This is for a computational and a deeper conceptual reason.
The computational reason is this -our solutions are naturally written in ingoing Eddington Finkelstein type coordinates which are unsuited for examining bifurcation surface geometry (especially objects like those defined in Eq. (1.5)). In case of usual Wald formula, this issue does not arise : according to an argument by Jacobson-Kang-Myers(JKM) [34], for time-independent solutions, the first term in (1.4) coming from L cov can be evaluated on an arbitrary spatial slice of the horizon instead of the bifurcation surface. Unfortunately, we have not been able to formulate such a JKM type argument for the second term in (1.4). 4 We will later propose a heuristic expression for the total Noether charge similar to Tachikawa formula which, when evaluated in standard fluid/gravity slicings, does reproduce the answer obtained from the differential Noether charge method (see § § 3.1). It would be interesting to come up with a generalization of [34] to Chern-Simons terms that would justify our proposal. For these reasons, even in the time-independent case, we will rely on ( / δQ Noether ) H to assign entropy and charges .
The conceptual reason is this -our fluid/gravity solutions are in general time-dependent and various steps needed for deriving (1.4) are no more valid. We remind the reader that, unlike the discussion in the previous paragraph, the issue of defining time-dependent entropy current in presence of higher derivative terms is ill-understood even in the absence of CS terms. An implicit assumption here is that the use of differential Noether charge ameliorates these problems, i.e., we advocate that the differential Noether charge is an appropriate way to assign entropy current, energy momentum tensor and charge currents to time dependent black hole solutions at least in the fluid/gravity regime. 5 It would be interesting to see whether this prescription reproduces the specifc subleading 6 time-dependent corrections to anomalous transport predicted by fluid-dynamical considerations (see Sec.12 of [35]).
After this technical aside, let us conclude our introduction by giving the outline of our paper. We will begin in section §2 by briefly reviewing the basic results from previous work that we will need later on. This review naturally falls into two subsections. In § §2.1, we will state the replacement rule derived from CFT considerations while in § §2.2 we present the black hole solutions of interest derived in [27]. This is followed by § §2.3 which contains a description of the manifestly covariant differential Noether charge constructed in [1]. This section ends with § §2.4 which is a summary of new results from this paper for the convenience of the readers.
In the next section §3, we show how the replacement rule for the anomaly-induced entropy current is reproduced from the differential Noether charge for the Chern-Simons terms evaluated at the horizon of our rotating charged AdS black hole solution. This is preceded by a heuristic derivation of the same result using a Tachikawa-like formula on the horizon.
Moving on to section §4, our differential Noether charge is used to derive the CFT stress tensor and current which reproduce field theory expectations. We conclude in section §5 with discussions on future directions.
We relegate various simple examples and many technical details to our Appendices. First, we provide a series of Appendices containing a list of notation (Appendix A) and a collection of useful formulas (Appendix B and C) we use throughout this paper. Rewriting of the Tachikawa entropy formula in terms of the Pontryagin classes is explained in Appendix D. Using our differential Noether charge, we review in Appendix E the well-known AdS 3 derivation of the Cardy formula in the presence of gravitational anomalies. The simple case of Abelian Chern-Simons terms are dealt with in Appendix F. The following Appendix works out in detail the AdS 5 case which shows the essential structures necessary for the computations in arbitrary dimensions. We then describe in Appendix H some structural results regarding the T 3 and T 4 terms appearing in our differential Noether charge Eq. (2.9) on the rotating charged AdS black hole background. Appendice I and J are devoted to the evaluation of T 0 , T 1 and T 2 terms in the differential Noether charge Eq. (2.9) at the horizon. In Appendix K and L we compute the asymptotic charges for our black hole solution. Finally in Appendix M, we compute the Einstein-Maxwell contribution to the entropy.

Review of previous works and summary
In this section, we will review a few relevant recent results which will be useful in the computations of the stress tensor/current and entropy. We start with the recent field-theoretical results on the replacement rule of stress tensor/current and entropy. After this, we move to the dual gravity side and briefly review some important results from our previous papers : the rotating charge-AdS black hole solution dual to charged rotating fluid [27] as well as the manifestly covariant differential Noether charge for Chern-Simons terms and the Tachikawa entropy formula derived from it [1]. In the final part of this section, we summarize our main results in the current paper and compare them with predictions coming from the CFT replacement rule.

Entropy current and stress tensor/current from CFT replacement rule
Through the recent studies on the hydrodynamic description of systems with global anomalies, it has been shown that the leading order anomaly-induced transports are completely captured by the 'the replacement rule' [13][14][15]17]. The statement of the replacement rule is as follows : let us consider an even-dimensional quantum field theory with global anomalies characterized by an anomaly polynomial P CF T . We define the pseudo-vector V µ as ⋆CFT V = u ∧ (du) n−1 ≡ u ∧ (2ω) n−1 where u ≡ u µ dx µ is the fluid velocity one-form and ω ≡ (1/2)ω µν dx µ ∧ dx ν is the vorticity 2-form. Then, the leading anomaly-induced contribution to the Gibbs free energy current G CFT µ is along V µ . If we write G CFT µ = G (V),CFT V µ + . . . where . . . denotes the nonanomalous (and the sub-leading anomalous) contributions, the replacement rule claims that G (V),CFT is completely determined by the anomaly polynomial : Here, µ is the chemical potential for the U (1) charge while T is the temperature. We note that tr[R 2k ] → 2(2πT ) 2k means the replacement of each tr[R 2k ] (k: positive integer) appearing in the anomaly polynomial by 2(2πT ) 2k . Following the standard thermodynamic relations 2) (where G : Gibbs free energy, M : energy, Q : U (1) charge, S : entropy) we also obtain the replacement rule for the anomaly-induced contribution to the stress tensor T anom µν , U (1) current J anom ν and entropy current (J anom S ) ν as We note that this replacement rule was first conjectured in [13,14] based on observations in free theories and was then proved via formal Euclidean methods in [15,17].

Rotating charged AdS black hole solution
Our main interest in the current paper is to consider Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant in (d + 1) dimensions (d = 2n with a positive integer n) and evaluate the Hall contribution to the differential Noether charge constructed in Ref. [1] both at the boundary and horizon of the rotating charged AdS black hole solutions in five and higher dimensions. These black hole solutions are derived via the fluid/gravity derivative expansion in Ref. [27]. For later use, we summarize some key results on these black hole solutions from Ref. [27] and on the differential Noether charge from Ref. [1]. The action of the Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant is given in Eq. (1.1). The rotating charged AdS black hole solution on which we are going to evaluate the Chern-Simons contribution to the differential Noether charge (2.9) takes the following form [27] :

(2.4)
Here P µν ≡ g µν + u µ u ν is the projection operator and We denote the location of the horizon by r = r H (which satisfies f (r H , m, q) = 0). The parameters m and q determine the mass and electric charge of this black hole solution. We also define Ψ(r) ≡ r 2 f (r, m, q)/2 for later use. Throughout this paper, we set the boundary metric to be flat, g µν = η µν and set the velocity vector u µ (normalized such that u µ u µ = −1) to pure rotation, i.e. ∂ (µ u ν) = 0 and u µ ω µν = 0 for the vorticity ω µν = ∂ [µ u ν] . We note that, at the horizon r = r H , we have Φ(r = r H ) = µ and Φ T (r = r H ) = 2πT where µ and T are the U (1) chemical potential and the Hawking temperature, respectively. We will refer the reader to Appendix B.1.2 and our previous paper [27] for a more detailed analysis of these black hole solutions.
The first lines of the metric and gauge field in Eq. (2.4) give the AdS Reissner-Nordstrom solution boosted by a velocity u µ . In particular, we note that, when we take u µ to be a uniformly rotating configuration on a sphere, this first line gives the leading order terms of AdS Kerr-Newman solutions expanded in the fluid/gravity expansion. The second line of the metric and gauge field in Eq. (2.4) is proportional to a pseudo-vector and describes how the AdS Kerr-Newman black hole is dressed by the Chern-Simons contributions.

Manifestly covariant differential Noether charge for Chern-Simons terms
In Ref. [1], we constructed a manifestly covariant differential Noether charge for the Einstein-Maxwell-Chern-Simons system. The charge is split into two contributions where the first term on the right hand side comes from the Einstein-Maxwell part of the Lagrangian while the second one arises from the Chern-Simons terms. The explicit form of the Einstein-Maxwell contribution ( / δQ Noether ) Ein-Max is given by . (2.8) We will call the first term in each line as the Komar contribution, while the second term is referred to as the non-Komar part. The Chern-Simons contribution ( / δQ Noether ) H on the other hand is given by where each term on the right hand side is determined by the derivative of the anomaly polynomial P CF T [F , R] = dI CS (I CS [A, F , Γ, R] : Chern-Simons terms in the Lagrangian) as Here the spin Hall current (Σ H ) cb a is defined by (Σ H ) cb a ⋆ dx c ≡ −2(∂P CF T /∂R a b ). In this paper, we consider the anomaly polynomials of the form 7 (here n = 2k tot +l−1 with k tot ≡ p i=1 k i , and n ≥ 2) or, more generally, a linear combination of it.

Main results
In this subsection, we summarize the main results of this paper : the Chern-Simons contribution to the differential Noether charge evaluated at the boundary and horizon of the rotating charged AdS black hole solution (2.4).

Anomaly-induced currents
Through AdS/CFT correspondence, we can write the differential Noether charge evaluated at the boundary in terms of the stress tensor and current of the CFT living on the boundary : Here, we have used the notation (. . .)| ∞ or simply (. . .) ∞ to denote a quantity (. . .) evaluated at the boundary r → ∞.
In the above equation, the differential Noether charge has been evaluated over a diffeomorphism ξ a and U (1) gauge transformation Λ which, near AdS boundary, asymptote to an arbitrary diffeomorphism and a flavor transformation of the CFT, that is, at the boundary of AdS, we will choose ξ a and Λ such that For simplicity, we choose the vector ξ a to be a boundary vector satisfying ξ r = 0 and ∂ r ξ a = 0 (for example, the Killing vectors corresponding to translations and rotations). We will also take Λ to be independent of r : ∂ r Λ = 0. We note that these {ξ a , Λ} used for computing the stress tensor and currents are state-independent with {δξ a = 0, δΛ = 0}. Our main interest is the anomaly-induced part of the CFT stress tensor, current or charges. These quantities are proportional to ⋆CFT V = ⋆CFT (V µ dx µ ) = ⋆CFT (u ∧ (2ω) n−1 ) and thus are at ω n−1 order in the derivative expansion. Throughout this paper, we in general add a superscript 'anom' to all expressions to denote such types of contribution. For example, the anomaly-induced parts of the stress-energy tensor and current (we simply call them as anomaly-induced currents) are (2.14) To simplify the notation for the differential Noether charge, we will drop the superscript 'anom' and denote the anomaly-induced part by ( / δQ Noether ) in the following part of the paper. Furthermore, due to the splitting in Eq. (2.7), we can also define the two contributions to the anomaly-part of the stress tensor and current : The first terms come from the Einstein-Maxwell part of the Lagrangian, while the second ones are from the Chern-Simons terms. In Ref. [27], the anomaly-induced currents are obtained from the gravity side as 8 is obtained from the anomaly polynomial via the bulk replacement rule [27] : We note that at the horizon, this reduces to the boundary CFT replacement rule in Eq. (2.3) by recalling Φ(r = r H ) = µ and Φ T (r = r H ) = 2πT . As was done in Ref. [27], these results (2.16) match with the ones derived from the CFT Gibbs current by following the discussions in Refs. [13,36,37] (see § §2.1 above), under the assumption In this paper, we have confirmed the correctness of this assumption (2.18) by directly computing these quantities from the differential Noether charge on the gravity side on the rotating charged AdS black hole background.

Entropy current
In order to evaluate the entropy current of our solution, we proceed as follows [29,30]. We consider the differential Noether charge associated with a specific state-dependent diffeomorphism and U (1) transformation on our black hole solution. As before, we choose the diffeomorphism ξ a to be a boundary vector satisfying ξ r = 0 and ∂ r ξ a = 0 and the U (1) transformation Λ to be independent of r : ∂ r Λ = 0. However, the boundary components are now chosen to depend on the particular fluid state under question : we take For convenience, we will use the notation (. . .)| hor or simply (. . .) hor to denote a quantity (. . .) evaluated at the horizon with the substitutions given in Eq. (2.19) and then pulled back to the boundary using ingoing null geodesics [33] . As discussed in [29,30], the differential Noether charge associated with (2.19) evaluated on the horizon corresponds to the entropy of the black hole solution. We can write this entropy by introducing an entropy current as J CFT,anom S µ as This should be thought of as the local version of the formalism developed in [29,30] using the pull-back prescription of [33].
Here, as in (2.15), we can split the entropy current into the contribution coming from the Einstein-Maxwell part of the Lagrangian and the one from the Chern-Simons terms : where the first term vanishes as we have shown in Appendix M For later convenience, it is also useful to define (J (V) S ) l by expanding the entropy current with respect to U (1) chemical potential : (2.23) In this paper, we derive the following expression for the entropy current from the gravity side by computing the entropy of the rotating charged AdS black hole (2.4) : This result is consistent with the CFT prescription for the anomaly-induced entropy current given in Eq. (2.3).

Examples
For the reader's convenience, in Table 1 we present the explicit expressions of G (V) and the anomaly-induced entropy current J for various anomaly polynomials in AdS 3 , AdS 5 and AdS 7 . The results for the more general cases are given in the next section and in the Appendices. Table 1.

CFT replacement rule for entropy current from gravity
In this section, we will show that the CFT entropy current in Eq. (2.24) is reproduced by the black hole entropy coming from the Chern-Simons terms. The parity odd part of the Einstein-Maxwell contribution to the entropy turns out to be zero. The details of the computations are rather standard and straightforward and thus will be presented in Appendix M instead. This section will be devoted to some explicit computations of the Chern-Simons contribution to the entropy. As discussed in the previous section (as well as Introduction), our main derivation here is based on the differential Noether charge at the horizon. Given the length of this computation, however, let us begin instead with a heuristic derivation inspired by Tachikawa entropy formula (1.4). The aim here is to make various assumptions that would directly get us to the heart of how the replacement rule appears in holography. In the next subsection, we will present the results from evaluating the differential Noether charge (2.9) on the rotating charged AdS black hole background (2.4).

Entropy I : Tachikawa-like formula at arbitrary horizon slice
Let us begin by recalling from Ref. [1] that for stationary solutions, at the bifurcation surface, the Chern-Simons terms contribute to black hole entropy through the Tachikawa entropy formula (that is, the second term of (1.4)) for a general anomaly polynomial P CF T . We notice that the rotating charged AdS black hole solution that we constructed in [27] is obtained in the ingoing Eddington-Finkelstein coordinates. This makes the evaluation on the bifurcation surface difficult (since it is at the boundary of such coordinates).
As we described in Introduction, in the usual Wald formula (that is, the first term of (1.4)), these difficulties can be tackled using the arguments of Jacobson-Kang-Myers (JKM) [34]. This JKM type argument ensures that, for stationary solutions, the first term of (1.4) evaluated on an arbitrary horizon slice gives the same answer as when evaluate over the bifurcation surface. In contrast, it is unclear how to convert the Tachikawa formula (3.1) into an expression that can be evaluated over an arbitrary horizon slice. It is especially unclear how to interpret objects like Γ N over an arbitrary slice. This is an indication of broader slice-dependence issues when Wald-like formulae for time-dependent solutions are considered. 9 We will not solve this important issue here. However, we will now make a simple proposal which seems to give the right answers which are consistent with the direct computation based on the differential Noether charge and also the expectations from the CFT side. Let us definẽ We will then assume that with these definitions, the entropy is given by a formula similar to (3.1) where Γ N is replaced byΓ N and R N is replaced byR N . Thus, our goal now is to explicitly evaluateS to obtain the replacement rule for anomaly-induced entropy current. The integrand can then be taken (after pull-back) to be the CFT entropy-current form. Using Eqs. (B.79), (B.64) and (B.68), we find thatΓ N starts at order ω 0 , while F | hor andR N start at ω 1 . In particular, the leading order expressions are given by Finally, using Eq. (B.26), we obtain the leading contribution toS Tachikawa for the rotating charged AdS black hole (which is at order ω n−1 ) : where for any anomaly polynomial P CF T in AdS 2n+1 . Remarkably, this reproduces the CFT replacement rule for the anomaly-induced entropy current, agreeing with the result based on the differential Noether charge without any assumptions in the next subsection. This crude computation inspired by the Tachikawa entropy formula simplifies the treatment of 2nd and higher order terms drastically. That is, since the above derivation uses each building block at its lowest order, obviously if any of them is at 2nd or higher order, the contribution to the entropy will be higher than ω n−1 . This computation also shows that the replacement rule follows from the general structure of black hole solutions and indicates how it might be a robust statement holding beyond the simple model under consideration.
3.2 Entropy II : Evaluation of differential Noether charge at horizon We will now turn to a more honest (but more lengthy) derivation of the replacement rule via the differential Noether charge with no ad-hoc assumptions. To derive the CFT replacement rule for the anomaly-induced entropy current, we start with the manifestly covariant differential Noether charge summarized in Eqs. (2.9) and (2.10). The goal of this subsection is to briefly explain the evaluation of T 0 , T 1 , T 2 , T 3 and T 4 terms in (2.10) on the horizon of the rotating charged AdS black hole solution (2.4). We consider the general anomaly polynomial of the form (2.11). The detail of the computation is provided in the Appendices. For AdS 3 and AdS 5 , it is given in Appendices E and G. For the general cases, the evaluation of T 3 and T 4 (for an arbitrary fixed r) is given in Appendix H, while the rest of the terms, T 0 , T 1 and T 2 on the horizon are calculated in Appendices I and J.
Here is one remark : In the evaluation of the differential Noether charge at the horizon, there are two potential prescriptions depending on whether one does the variation or the evaluation first. In the first prescription, one first sets the radial coordinate to be r = r H and then does the variation with respect to the parameters of the black hole solution, while in the second prescription one first does the variation and set r = r H afterward. In Appendix E, for the rotating BTZ black holes, we have explicitly evaluated the Chern-Simons contribution to the differential Noether charge at the horizon by using these two prescriptions and then obtained the same result. Furthermore, in the case of entropy coming from the Einstein-Maxwell terms, we have also explicitly checked in Appendix M that both prescriptions give the same answer. Although the distinction has not often been discussed in the literature (even in the case of a covariant Lagrangian), it is not known to us if these two prescriptions should also yield the same answer. 10 We take the case of BTZ black hole entropy as well as the Einstein-Maxwell part of the entropy as a hint that it is reasonable to expect that these prescriptions agree in general, and for the rest of this paper we will use the first prescription (due to simplifications in the computations).
Another important comment is that in the evaluation of the differential Noether charge (2.9) with T 0 , T 1 , T 2 , T 3 and T 4 in (2.10), since we evaluate the charge at a given fixed rsurface, the terms proportional to dr do not contribute. Therefore, throughout this paper, we 10 Essentially, the physical difference of these two prescriptions for the evaluation at the horizon traces back to whether one evaluates at the horizon (before the variation) or the new horizon (after the variation). An argument as to why both prescriptions gives the same answer in our computations is that if we include the total contribution to the entropy (including all the terms appearing in the Lagrangian), then on-shell ∂r(/ δQ Noether ) = 0 because of d(/ δQ Noether ) = 0. This implies that / δQ Noether does not depend on r and thus it is the same whether it is evaluated on the original or the new horizon (after the variation).
neglect these terms unless otherwise mentioned (i.e. '=' in the evaluation of the differential Noether charge and T i 's is valid up to these terms). In addition to this, we also set δr = 0.

Terms T 3 and T 4 (from Appendix H)
As a result of the detailed computation in Appendix H, the terms T 3 and T 4 for a general anomaly polynomial (2.11) at an arbitrary fixed r surface of the black hole solution (2.4) are given by where G (V) and H (V) are determined by the corresponding anomaly polynomial P CF T [F , R] via the bulk replacement rule By evaluating these expression for T 3 and T 4 at the horizon r = r H , we obtain We note that the above result can be obtained by assuming that 2nd and higher order terms in the building blocks (that is, R, F etc.) do not contribute to T 3 and T 4 at the leading order of the derivative expansion. We can directly show that the 2nd and higher order terms does not generate the same or lower order contribution by using essentially the same argument for the Einstein source in Appendix D.6 of Ref. [27]. For the readers' convenience, this argument is briefly reviewed in Appendix H.4. For more details of the computations and argument related to T 3 and T 4 , please refer to Appendix H. In Appendix I, we have carried out the evaluation of T 0 , T 1 and T 2 at the horizon, by taking into account the zeroth and 1st order terms in the building blocks (that is, R, F etc.) only. The result for each term at the leading order of the derivative expansion (which turns out to be of order ω n−1 ) is given by

Terms
where we have used (J (V) S ) l defined as in Eq. (2.23). As in the case of T 3 and T 4 , we can obtain this result by assuming that 2nd and higher order terms in the building blocks do not contribute. In Appendix J, we have also taken into account the 2nd and higher order terms in the building blocks and confirmed that these terms do not generate any contribution to T 0 , T 1 and T 2 at the ω n−1 order or lower.

Anomaly-induced contribution to entropy
As summarized in Appendix M, the Einstein-Maxwell part of the differential Noether charge gives no parity odd contribution to black hole entropy. Combined this fact with the results on the Chern-Simons part summarized in the above two subsections, we finally obtain the parity odd contribution to the black hole entropy in the leading order of the derivative expansion as This indeed reproduces the CFT replacement rule for the anomaly-induced entropy current.

CFT current and stress tensor from differential Noether charge
In this section, we will explicitly compute the differential Noether charge evaluated at the boundary ( / δQ Noether )| ∞ of the rotating charged AdS black hole background (2.4) in the leading order of the fluid/gravity derivative expansion. We recall that the differential Noether charge (evaluated at the boundary) splits into the Einstein-Maxwell part and the Chern-Simons part In the following part of this section, we will briefly explain the evaluation of these two terms on the right hand side separately, while the details are provided in the Appendices. In § §4.1, we will confirm Eqs. (2.16) and (2.17) from the evaluation of the first term ( / δQ Noether ) Ein-Max | ∞ . We note that the Einstein-Maxwell terms of the Lagrangian are parity even and thus, in the evaluation of the anomaly-induced contribution (2.16) from the Einstein-Maxwell part at the leading order of the derivative expansion, we drop all parity even contributions with derivatives along the boundary coordinates in § §4.1. In § §4.2, we provide a summary for the evaluation of the second term in Eq. (4.1), which results in and thus justifies the assumptions of Eq. (2.18) (which are used in Ref. [27]). Since the detail of the evaluation of this part involves various technical points and is lengthy, interested readers are kindly referred to Appendix K and L.

Asymptotic charges I : Einstein-Maxwell contribution
In this subsection, we compute the Einstein-Maxwell contribution to the differential Noether charge, Eq. (2.8), at the boundary of the rotating charged AdS black hole background (2.4). As in Eq. (2.8), we separate this contribution into the Komar part and the non-Komar part, and evaluate them separately. We recall that the Komar part (from the first line in Eq. (2.8)) is given by and the non-Komar part (the second line in Eq. (2.8)) is

(4.5)
We note that the first and third lines respectively are the parity even contribution from the first and second terms of (4.3) in the leading order of the derivative expansion. On the other hand, the second and fourth lines in Eq. (4.5) respectively are the parity odd contribution from the first and second terms of (4.3) in the leading order of the derivative expansion. Now we evaluate this Komar part at the boundary, r → ∞. We look at the contribution from the Einstein part (the first and second lines in Eq. (4.5)) and Maxwell part (the third and fourth lines in Eq. (4.5) separately. We first take the limit r → ∞ of the Einstein part after subtracting the empty AdS contribution (the second term on the left hand side in the following expression) : where we have used On the other hand the Maxwell part of the Komar charge is evaluated at the boundary as (4.8) The two . . . in the above expression denote higher derivative contribution to the parity even and parity odd part, respectively.

Non-Komar variation
We next proceed to the evaluation of the non-Komar part (4.4). We separately evaluate the Einstein part and Maxwell part (the first and the second term in (4.4), respectively).
First, we start with the evaluation of the Einstein part. We note that, by adding Eqs. (C. 19)-(C.22), we obtain (4.9) We pull-back this on a radial slice, contract it with ξ µ using i ξ ⋆CFT 1 = η µν ξ µ ⋆CFT dx ν and then take r → ∞ limit. In the end, we have the following expression : As a next step we evaluate the non-Komar part of the Maxwell contribution at the boundary. We note that this term evaluated for the empty AdS is trivially zero. Near the boundary, since δA| ∞ → δA ∞ + O r −(d−2) and the gauge field fall-offs as in Eq. (B.90), we conclude that Therefore the non-Komar part of the Maxwell contribution vanishes at the boundary.

Einstein-Maxwell contribution to asymptotic charges
Now we combine all the results above to compute the Einstein-Maxwell part of the differential Noether charge (2.8) at the boundary. Subtracting all the non-Komar contributions (see Eq. (4.10) and Eq. (4.11)) from the variation of Komar contribution in Eq. (4.6) and Eq. (4.8), we finally obtain with where we have used which are shown in Eq. (5.9) of [27]. Here again the . . . in the first and third (second and fourth) lines of Eq. (4.13) denote the higher order terms in the parity even (odd) contribution. The expressions in the first lines (i.e. non-V µ parts) in T CFT µν and J CFT µ above are exactly the perfect-fluid ones with pressure p = m/(16πG N ). The anomaly-induced parts (i.e. terms proportional to V µ ) match Eq. (2.16) exactly as claimed.
Of course, when the Lagrangian density contains the Chern-Simons terms, there is potential extra contribution to the CFT stress tensor and current from the Hall part to the differential Noether charge ( / δQ Noether ) H at the boundary. The goal of the next subsection is to confirm that there is no such contribution, that is, to prove Eq. (2.18).

Asymptotic charges II : Chern-Simons contribution
This subsection is devoted to a summary of the results in the evaluation of ( / δQ Noether ) H with the anomaly polynomial (2.11) at the boundary of the rotating charged AdS black hole background. The detail of the computation contains many technical points and thus is provided in Appendix H-L. In the following, we summarize the key results we obtained for the general anomaly polynomial (2.11) in AdS 2n+1 (n ≥ 1). As in Appendix H, for T 3 and T 4 in (2.10), we can obtain their exact expression valid for any fixed r in a relatively simple way. On the other hand, for T 0 , T 1 and T 2 in AdS 7 and higher, we compute them only at the horizon and boundary in Appendices I-J and K-L, respectively. As a comparison, we also briefly comment on the AdS 5 case (see Appendix G for detail).
We again stress that, in the evaluation of T 0 , T 1 , T 2 , T 3 and T 4 , we neglect terms proportional to dr, since these terms do not contribute to ( / δQ Noether ) H at any fixed r.

Terms T 3 and T 4 at boundary (from Appendix H)
As summarized at the beginning of § § §3.2.1, the terms T 3 and T 4 for a general anomaly polynomial at arbitrary fixed r surface are given by (3.7) with (3.9). At the boundary r → ∞, we note that (T 3 + T 4 )| ∞ itself vanishes for AdS 7 and higher. We note that, for AdS 5 , as can be seen in Appendix G, this sum is nonzero but cancels with the rest terms (T 0 + T 1 + T 2 )| ∞ .

boundary (from Appendices K and L)
At the boundary, for AdS 2n+1 (n ≥ 3), all the terms T 0 , T 1 and T 2 at r → ∞ vanish up to the leading order of the derivative expansion. Therefore, for the sum of these three terms, we also have In Appendix K, we confirmed the above statement by taking into account zeroth and first order terms in the building blocks, while in Appendix L, we consider the 2nd and higher order terms and then proved that the statement still holds. We note again that for AdS 5 , the sum (T 0 + T 1 + T 2 )| ∞ is nonzero but cancels with (T 3 + T 4 )| ∞ as can be seen in Appendix G.

Chern-Simons contribution to asymptotic charges
By combining with the results of (T 0 + T 1 + T 2 )| ∞ and (T 3 + T 4 )| ∞ obtained in the previous subsections, we finally confirm that ( / δQ Noether ) H at the boundary vanishes for AdS 5 and higher. Therefore, we have where This verifies one of the main results we claimed in Eq. (2.18).

Discussions and conclusions
The first main result of this paper is that we have reproduced the CFT replacement rule for anomaly-induced entropy current from the dual gravity side. We started with the manifestly covariant differential Noether charge derived in [1] and evaluated it at the horizon of the rotating charged AdS black hole solution constructed by using the fluid/gravity derivative expansion in [27]. In § §3.1, we also showed that the same result can be also obtained by a heuristic Tachikawa-like entropy formula at the horizon. As mentioned in §3, this simpler derivation is based on somewhat ad-hoc proposal about how to lift the bifurcation surface normal bundle connection Γ N onto an arbitrary slice (Eqn. (3.2)). It would be interesting to give a direct derivation of this proposal. The second main result of this paper is to show that at the boundary the Chern-Simons part of our differential Noether charge vanishes in five dimensions and higher (while there is some nontrivial contribution in the case of three dimensions). This completes the holographic and systematic derivation of the replacement rule for the anomaly-induced contribution to the CFT stress tensor and current initiated in [27].
For future works, there are various exciting possibilities generalizing our computations. First of all, it will be useful (in particular, in setups embedded in string theories) to include covariant higher derivative terms in the Lagrangian in the presence of the Chern-Simons terms. 12 Since the field theoretical results do not get corrected, we expect that the extra higher derivative covariant terms would correct the fluid/gravity metric in such a way that the final results for the anomaly-induced currents and stress tensor still agree with the replacement rule. It would present yet another non-trivial check of the replacement rule.
Looking towards a different direction, we recall that for simplicity we have set the magnetic field to be zero and the metric at the boundary to be flat in the current paper. It would be interesting to turn on some nontrivial profiles for them. These generalization will hopefully lead us to a deeper understanding of the replacement rule and, in particular, clarify the role of higher Pontryagin classes. In a similar vein, the various structures of products of Riemann curvatures that we observed (and relied heavily on in our computations) are still somewhat mysterious. It will be meaningful to understand what kind of physical insights or geometric properties of these black hole are encoded in them. Similar objects were also used in [40] in studying the proposal of a local entropy current related to the Wald's construction of black hole entropy as a Noether charge. This enables [40] to study the validity of the second law (i.e. the non-negativity of the divergence of the entropy current). It will be interesting to relate our work to their study as well as extending the analysis of the second law to our setup.
In light of the recent excitement in the area of holographic entanglement entropy [41][42][43][44] (see also [45][46][47]) which potentially sheds light on the emergence of geometry in the context of gauge/gravity dualities [48][49][50][51], a generalization to this line of investigation to include Chern-Simons terms is of great interest. The case of AdS 3 with the gravitational Chern-Simons term was studied in [52] whereby an interesting ribbon-like structure in the bulk encodes the anomaly-induced contribution to the entanglement entropy in CFT 2 with gravitational anomaly. It begs the question of what then generalizes this structure in the higher dimensional holographic entanglement entropy due to Chern-Simons terms. This may well be a concrete arena where we can sharply investigate how the entanglement at the boundary CFT (at least for the chiral degrees of freedom of the field theories) manifests itself geometrically in the bulk. We will report on this aspect in the near future [53]. oretical Physics and in particular to Harvard University for hospitality. T. A. was in part supported by INFN during his stay in the Galileo Galilei Institute for the workshop "Holographic Methods for Strongly Coupled System." T. A. would like to thank the participants of the YITP workshop "Holographic vistas on Gravity and Strings" and "Strings and Fields". T. A. and G. N. are grateful to the participants and organizers of the Solvay Workshop on "Holography for Black Holes and Cosmology". T. A. was financially supported by the LabEx ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. R. L. was supported by Institute for Advanced Study, Princeton. G. N. was supported by DOE grant DE-FG02-91ER40654 and the Fundamental Laws Initiative at Harvard. G. N. was also supported by an NSERC Discovery Grant.

A Notation
Regarding conventions and notations used throughout this paper, we follows our previous papers, Ref. [1,27]. Here we summarize some notations that we frequently use and also define some new useful notation for the purpose of the current paper.

A.1 Summary of notations
• Sometimes, for convenience or to avoid cluttering of indices, we suppress all matrix indices. It is always assumed that objects next to each others are multiplied as matrix multiplications. In particular, we think of (∇ b ξ a ) as a matrix (∇ξ) a b . For example, Another important terminology we will frequently use throughout this paper is the phrase 'building block' which is defined as one of the following objects : These are the basic objects whose products (and appropriate index-contractions) form the basis in the evaluation of T 0 , T 1 , T 2 , T 3 and T 4 .
• When carrying out the fluid/gravity derivative expansion, we need to write down some differential forms at a particular order in the derivative expansion. For this purpose, it is convenient to introduce the following notation : the m-th order terms in the derivative expansion of a form B is denoted as ( B m) a b while the product of k matrix-valued 2forms, ( B m 1 ), · · · and ( B m k ) is written as ( B m 1 . . . B m k ), so that the matrix-valued 2-forms inside the brackets are always multiplied through matrix multiplication. The k-th power of ( B m) is denoted ( B m k ). For example, for the field strength F , we have • In discussing products of curvature two-forms, the cases containing purely ( R 0)'s and ( R 1)'s are of particular significance throughout the computations of the differential Noether charge. Therefore we define χ m (m = 0, 1, 2, · · · ) to denote products of ( R 0)'s and ( R 1)'s with exact m-number of ( R 0)'s wedged with an arbitrary number of ( R 1)'s. For example, χ 0 only includes product of ( R 1)'s, i.e. ( R 1 R 1 . . . R 1 R 1). Another example is χ 1 , which for example contains some of the following possibilities : The classification of χ m has been carried out in Appendix B.2 of Ref. [27] and will be reviewed in Appendix B.1. We also note that we sometimes use χ m to simply denote an element in χ m .
• We define a convenient symbol υ is defined to represent a string made of 2nd or higher order curvature two-form ( R m)'s with m ≥ 2 (for example, ( R 2 2 R 5 R 3)). • We also define (R q (p) ) to denote all possible structures (including those consist of zeroth and first order building blocks only) that can contribute to (R q ) at ω p order. For example, the non-trivial possible structures in (R 3 (2) ) are A complete classification of (R q (p) ) for 0 ≤ p ≤ q + 1 is given in Appendix B.2.2.

A.2 Orientation convention in holography
Let ε µ 1 µ 2 ...µ d be the ε-tensor of the spacetime in which the field theory CFT d lives. Then, there are two possible conventions for the orientation of the dual AdS d+1 .
Let r be the radial coordinate such that r → ∞ is the conformal boundary of the dual AdS d+1 . It is usual in gravity to fix the orientation such that ε rµ 1 µ 2 ...µ d has an opposite sign as compared to ε µ 1 µ 2 ...µ d . For example, if [txyz] formed a right-handed coordinate basis in CFT d then, in this convention [trxyz] forms a right-handed coordinate basis in AdS d+1 .
We use here the opposite convention whereby ε rµ 1 µ 2 ...µ d has the same sign as ε µ 1 µ 2 ...µ d which is more natural from the viewpoint of pullback. Our expressions can easily be adopted to the reverse bulk orientation (we will keep the CFT orientation unchanged) by replacing ⋆ with − ⋆ and ε abcd... with −ε abcd... . Note that we keep ⋆CFT unchanged. In the new convention, the relation between the barred and the unbarred forms becomes V = ⋆ V .

B Useful relations I
This Appendix collects some results needed for the computations in this paper.

B.1 Results from our previous paper
Here we summarize some results from Ref. [27] which are valid for any r.

B.1.1 0th and 1st order terms
Bulk metric and gauge field The bulk metric G ab (in the coordinate x a = {r, x µ }) and its inverse for the rotating charged AdS black hole in d + 1 dimensions (d = 2n with a positive integer n) is given up to the first order of the derivative expansion as while the gauge field A a up to this order is Here f (r), Ψ(r) and Φ (Φ T that we will use later) are defined as The horizon of the black hole is located at r H satisfying f (r H , m, q) = 0. The parameters m and q are related to the mass and electric charge of the black hole solution and κ q is a normalized Maxwell coupling constant defined as 16πG where G N and g Y M are the Newton constant and Maxwell coupling constant, respectively.

Christoffel connection
The components of the Christoffel connection for the bulk metric G ab are given by or in terms of the connection 1-form Γ a b ≡ Γ a bc dx c , by For ∇ a ξ b , since we are considering ξ a with ξ r = 0 and ∂ r ξ a = 0, we obtain (at any fixed r) U (1) field strength By using the notation in Appendix A, the zeroth and first order terms of the field strength F are given by Curvature two-form The curvature 2-form, defined from the Riemann tensor R a bcd as R a b ≡ (1/2)R a bcd dx c ∧ dx d , at zeroth order is given by while the first order terms are

Products of curvature two-form
Let us now consider wedge products of ( R 0)'s and ( R 1)'s. We first consider the products of two curvature two-forms: When two ( R 0)'s are multiplied, we have Therefore, the products of two or more ( R 0)'s vanish identically. For the products of one ( R 0) and one ( R 1), there are following two possibilities : Finally, the following is the products of two ( R 1)'s : Next we consider the products of three curvature two-forms. As a result of ( R 0 R 0) = 0, the following products trivially vanish : Therefore, there is only one nontrivial case with two ( R 0)'s in the product : For the products with only one ( R 0), there three possibilities depending on where ( R 0) is located. The first case is The second case is We also have the case with no ( R 0)'s in the product : We sometimes find it useful to reduce products of ( R 1)'s using Classification of the products of the curvature two-forms As we have shown in Appendix B.2 of Ref. [27], the building block of the wedge product of R's made of ( R 0) and ( R 1) only reduces to the following possibilities : Here by 'reduce' we mean the products of four or more R's (with ( R 0) and ( R 1) only) are zero or written as the one of the above elements wedged by an appropriate power of (2Φ T ω).
In particular, wedge products containing three or more ( R 0)'s (with the rest of R's equal to ( R 1)'s) are zero, i.e. χ m = 0 for m ≥ 3. For more detail, we refer the readers to Appendix B.2 of Ref. [27]. Another useful notation for later purpose isχ 1 , which is defined as all elements in χ 1 excluding ( R 0).
Trace of the products of the curvature two-form Finally, the traces of the wedge products of ( R 0)'s and ( R 1)'s are rather simple : for k ≥ 0.

B.1.2 2nd order terms
Here we summarize some results at the second order of the fluid/gravity derivative expansion which are useful for the purpose of this paper. In Ref. [27], the rotating charged AdS black hole solutions in (d + 1) dimensions is constructed up to the second order in the fluid/gravity derivative expansion (assuming stationary fluid configurations). It is given by where g(r, m, q) and h(r, m, q) are given by For the U (1) field strength, the 2nd order contribution is calculated from the solution as Furthermore, from Appendix B.1 of Ref. [27], the second order curvature 2-form has two types of non-trivial contributions. The one coming from the second order metric and the one coming purely from the zeroth order metric (and its derivatives). To distinguish from the whole 2nd order curvature 2-form ( R 2), we denote the latter contribution by ( R 2 ′ ). They are given by B.2 Structures of (R k ) at arbitrary r In this Appendix, we study the general structures of (R k ) at any fixed r. All equalities are evaluated at a fixed r and terms proportional to dr are ignored.

B.2.1 Some relations
First it is useful to recall from Eq. (B.9) (which is valid at any r) that where G rr = 2Ψ and G rα = u α and we have used that in our setup ∂ r ξ a = 0 and ξ r = 0. We also have On the other hand, where we have ignored dr terms. Thus, for the variation of the Christoffel connection, we have Furthermore, we recall that Moreover, at constant r we have the following relations : Using the above identities, we can show that Consider a product of R of the form (R k ) at constant r (i.e. dr = 0). We will classify all the structures appearing at order ω k−1 , ω k and ω k+1 . The reason why ω k−1 is the lowest non-trivial order is because χ 2 ∝ dr.
Before we start the classifications, we remind the readers of the possible structures in a single product of curvature two-forms denoted as CaseĀ,B,C,D orĒ (as we did in the Einstein sources in Appendix D.6 of Ref. [27]) : We remind the readers of the definition of χ m (see Eq. (B.25)) which denotes a product of (R)'s consisting of m number of ( R 0)'s with the remaining (R)'s being ( R 1)'s. The symbol υ is defined to represent a string made of 2nd or higher order terms ( R m) with m ≥ 2. Now, let us start analyzing CaseĀ to CaseĒ one by one. We further note that whenever we encounter two χ 1 's somewhere in the wedge product, the only non-zero cases are where we defineχ 1 to be the χ 1 containing more than one R's, i.e.

CaseĀ :
Since each υ is at least of order ω 2 and all χ's need to be either χ 0 or χ 1 , the product is in total of order ω k+1 or higher. At ω k+1 -th order, the only nontrivial case is when all υ's are ( R 2)'s and all χ's are Thus, we conclude that caseĀ only has one non-trivial structure CaseB : For caseB, the lowest order starts at ω k for which all υ's are ( R 2)'s and all χ's are . As in the arguments in CaseĀ, this type of the product is non-zero only when ( For the order ω k+1 contributions, there is a few possible structures listed below : where we have used Eq. (B.43) and for any m ≥ 0. Thus, in summary we have

CaseC :
For caseC, the lowest order starts at order ω k−1 for which all υ's are ( R 2)'s and all χ's are Similarly, the order ω k+1 terms are CaseD : We note that in this case, each υ is at least of order ω 2 and hence the total order is at least ω 2k . For k ≥ 1, this means that it could only contribute to order ω k+1 for k = 1 (i.e.
. Thus we conclude that CaseD contains only one structure (B.52)

CaseĒ :
For caseĒ, we can have the following two structures At this point, we would like to summarize the results from the above analysis. Before doing so, we first introduce a useful symbol to denote all possible structures (including those consist of zeroth and first order building blocks only) that can contribute to (R q ) at ω p order. Thus, the summary of the results from above could be stated as : We remind the readers that in the Einstein source computations in Appendix D.6 of Ref. [27], it was proved that 2nd and higher order terms in R do not contribute to (R q ) up to order ω q−1 . This is why 2nd and higher order terms in R do not appear in (R q (q−1) ). Furthermore, up to ω q−2 , all contributions (including zeroth and first order building blocks) to R q vanish.

B.3 Behavior at horizon
In this part, we summarize various quantities evaluated at the horizon and some relation valid at horizon. In the rest of this Appendix, we assume that all equalities are evaluated at the horizon and ignore terms proportional to dr.

B.3.1 0th and 1st order terms
In particular, the metric at the horizon and its inverse simplify as The gauge field and the connection 1-form at the 0th and 1st orders are evaluated at the horizon as for the 0th order part and for the 1st order part.
As for the covariant derivative of ξ a at the horizon, we have at the lowest order of derivative expansion : (B.63) We note that there is no 1st order term in the above equation and higher order terms of the derivative expansion start with the 2nd order. Therefore, we will drop the '(0)' superscript on ∇ξ in the discussions in this subsection.
Let us consider the products of F 's or R's evaluated at the horizon. Since For the product of R's, all products containing more than one ( R 0) vanish or are proportional to dr, i.e. χ m = 0 or proportional to dr for m ≥ 2. We next consider the products of the curvature 2-form containing only one ( R 0), i. e. χ 1 terms. At the horizon, nontrivial components of this kind of the products are (for k ≥ 0) We also note that the existence of more than one χ 1 in a given term implies that such term vanishes, since all the nonzero elements in χ 1 are proportional to u at horizon. For the products with ( R 1)'s only, i.e. χ 0 terms, we have The explicit form of (. . .)'s appearing in the above expressions are not need for our calculation.
The following relations on traces are also useful Here we stress that only 0th and 1st order terms are considered for R, δΓ and ∇ξ, while all terms proportional to dr have been dropped. Furthermore, at the horizon, we can prove the following relations : and in particular it follows that For computing the Chern-Simons contribution to the entropy using Tachikawa-like formula, we also need

B.3.2 2nd and higher order terms
Now we summarize some useful relations relevant to 2nd and higher order terms evaluated at the horizon. First, since ( F 2) ∝ dr, we do not need to deal with it. For product of curvature two-forms, one of the most important relations for our our purpose is tr[ R 2χ 1 ] = 0 which is valid for any r. Moreover, the non-zero components of ( R 0 R 2) and ( R 2 R 0) are Furthermore, in ( R 2χ 1 ) and (χ 1 R 2), the only non-zero objects are (for k ≥ 0) We note that the non-zero possibilities in ( R 2χ 1 ) and (χ 1 R 2) are reducible to ( R 2 R 0) and ( R 0 R 2) (wedged by an appropriate power of (2Φ T ω)).
The following relations are also useful :

B.4 Asymptotic fall-offs at boundary
We summarize the asymptotic behavior of some quantities at r = ∞. We note here again that here we ignore terms proportional to dr.

B.4.1 0th and 1st order terms
In this subsection, all building blocks are considered up to first order and in particular we drop their superscript that we usually use to denote their derivative orders. For Φ, Φ T and Ψ, we have and the gauge field and the connection one-form are Now assuming δr = 0, we have the variation of the gauge field and the connection one-form as follows: For the 0th and 1st order terms in the field strength and curvature two-form, from the explicit form of these terms, we have (note that ( F 0) ∝ dr ∧ u)

B.4.2 Fall-off of 2nd order term at boundary from direct computations
For the field strength, since ( F 2) ∝ dr, we do not need to take this into account when we evaluate the fall-off behaviors of T 1 and T 2 in Appendix. I. For the the second order terms in the curvature two-form ( R 2), we can find their fall-off behaviors from the explicit metric (up to second order) in Eq. (B.27). The contributions to ( R 2) purely from the zeroth order metric can be found in Eq. (B.31) and hence we can just take the r → ∞ limit to see the falloffs : 13 Here, for a general 2nd order building block ( B 2) (or, (2) B) made of the metric (and its derivatives), we introduced the notation ( B 2)| G (0) (or, (2) B| G (0) ) to denote the contribution containing the zeroth order metrics only. The remaining part (which contains a second order metric) is defined as On the other hand, the rest of the contribution to the second order curvature two-form (in which there is a second order metric and the derivatives inside of the definition of the curvature two-form are all ∂ r ) at the boundary is evaluated as follows: We recall the definition of the curvature 2-form (R) a b = dΓ a b + Γ a c ∧ Γ c b . The second order metric can not contribute to ( R 2) a b through the dΓ a b term (at order ω 2 ) since d[( (2) Γ) a b | G (2) ] is proportional to dr or of order ω 3 . Therefore, we only need to compute the 2nd order metric contribution to the term (Γ a c ∧ Γ c b ), which is given by Now, from the explicit second order metric, we deduce the fall-offs of the connection one-form from the second order metric : Combining above and the fall-offs of ( (0) Γ a b ) from Eq. (B.88), we obtain that , (B.95) and thus Therefore, combining the contributions from G (0) and G (2) , we find at infinity. In particular, this indicates that ( R 2) under the rough estimates is We note that in the above derivations, we rely heavily on the explicit form of the second order metric solutions. In general, due to Weyl covariance of the CFT 2n and its extension to the bulk, the fall-off in Eq. (B.98) can be derived without making use of any information about the explicit second order metric solutions. The proof of Eq. (B.98) using Weyl scaling is presented in Appendix L.4. Therefore, the estimate in Eq. (B.98) is in fact valid in more general setups.

B.5 Explicit structures in AdS 5
In Appendix G, we need to deal with traces of products of δΓ a b , ∇ a ξ b and R. Here we summarize some useful results for the computations of these traces. We note that we will ignore all order ω 2 and higher order building blocks in this Appendix.
Let us first consider a product of δΓ a b and R. We have the following results related to this : For a product of ∇ a ξ b and R, we have For a product of δΓ a b and ∇ c ξ d , we obtain

C Useful relations II
This Appendix summarizes some useful relations that are relevant to the computation of the the Einstein-Maxwell part of asymptotic charges and entropy.

C.1 Gauge field, metric and Hodge duals
Here we explicitly compute the Hodge-duals of some quantities on the rotating charged AdS black hole background (2.4). The results summarized here will be useful for the evaluation of the differential Noether charges. For convenience, we start with a summary of the bulk metric and gauge field for the rotating charged AdS black hole solution (2.4) : where we have retained only the leading order contributions to the parity even and parity odd part in the derivative expansion. The gauge field strength is given by It is useful to summarize the Hodge-duals of the following zero forms and one forms : where we have dropped again the sub-leading contributions in the fluid/gravity derivative expansion. Here we also summarize the formulae for Hodge-duals of 2-forms : From this expression, we can obtain By using this, we can also compute the Hodge-dual of the gauge field strength two-form as The definition and detail of the function Q (V) is provided in Ref. [27].
Here we have used ξ µ | hor = u µ /T . Another useful identity about Hodge duals is the following : Let N be a p-form with legs only along the boundary direction and is completely transverse to the velocity u µ . Then, it satisfies

C.2 Christoffel symbols
In the course of the computation of the differential Noether charge, we will also frequently encounter the evaluation of the Christoffel symbols on the rotating charged AdS black hole background. Here we summarize the explicit form of the Christoffel symbols with all the indices lowered on this background : Γ rrr = Γ rrµ = Γ rµr = Γ µrr = Γ µνλ = 0 + . . . , where we have retained only the leading order contributions in the fluid/gravity derivative expansion. This leads to Γ r rr = Γ µ rr = 0 + . . . , In particular, at the horizon, the Christoffel symbols simplify to Γ r rr | hor = Γ µ rr | hor = Γ r µν | hor = 0 + . . . , . We can also write down the connection one-form Γ a b = Γ a bc dx c at the horizon by using this result : (C.14)

C.3 ∇ξ
Let us consider a vector ξ µ along the boundary directions which is only dependent on the boundary coordinates. Then, we have 15) We note that at the horizon, each components of ∇ a ξ b simplifies to where we have used ξ µ | hor = u µ /T .

C.4 Some useful relations for Einstein-Maxwell charges
In the end of this Appendix, we summarize some results useful for the evaluation of the Einstein-Maxwell part of the differential Noether charge on rotating charged AdS black hole back ground. The first useful result is related to ∇ a ξ b . Contracting these with the 2-forms in Eq. (C.6), we obtain 14 Another set of results useful in the computation of the non-Komar part of the Einstein-Maxwell differential Noether charge is 14 This answer can also be derived by directly evaluating (C.17)

D Tachikawa formula in terms of Pontryagin classes
The aim of this Appendix is to rewrite Tachikawa entropy formula in terms of Pontryagin classes of the curvature. This gives a useful expression for the entropy formula, since the Pontryagin classes evaluated on our solution often take a simpler form as compared to traces of the wedge products of the curvature two-form. For mathematical details of the Pontryagin classes, please refer to [54,55]. We start with the Tachikawa entropy formula (the second term of (1.4)) written in terms of the trace of the wedge products of the curvature two-form. The goal of this Appendix is to rewrite this in terms of Pontryagin classes p k (R) which are defined through where t can be thought of formally as a (−2) form. It follows that p k (R) is a 4k form. The first few Pontryagin classes are given by These can be inverted to obtain

(D.3)
In order to rewrite the Tachikawa entropy formula in terms of the Pontryagin classes, we will need to use the following property of the Pontryagin classes : which follows by differentiating Eq.(D.1) with respect to trR 2k : Using Eq.(D.4), we can then use the chain rule to rewrite the Tachikawa entropy formula as The sum above can be simplified by using or in terms of the defining polynomial of the Pontryagin classes This in turn follows from the matrix identity where we have used ε(R + R N ε) = −ε 2 R N + ε 2 R N = 0 on the bifurcation surface. Thus, using Eq. (D.7) in Eq. (D.6), we finally obtain which is the Tachikawa entropy formula written in terms of the Pontryagin classes.

E Example I : Chern-Simons terms on BTZ black hole
Here we consider U (1) and gravitational Chern-Simons terms in three dimensions and evaluate ( / δQ Noether ) H on the BTZ black hole background. We can obtain the metric, Christoffel symbol and ∇ξ by setting d = 2, κ q = 0 and q = 0 in Eqs. (B.1)-(B.6), (B.7) and (B.9). We also note that g V = 0 for the BTZ black hole since the presence of the Chern-Simons terms does not correct the solution. We note that for the covariant part of the Lagrangian, we only consider the Einstein term without the Maxwell part. In the following, we start with the Hall contribution to the differential Noether charge for an arbitrary fixed r surface of the BTZ black hole solution. We then evaluate it at the boundary and the horizon to obtain the CFT stress tensor, current and entropy. At the horizon, there are two possible prescriptions for the evaluation of the differential Noether charge. One way is to first do the variation with respect to the parameters of the BTZ black hole solution and then set the radial coordinate to be r = r H , while the another way is to first set r = r H and then do the variation. We will comment more on this point in the middle of this Appendix and confirm that we can obtain the same result from these two prescriptions. At the end of this Appendix, we also compare our result with Ref. [31].

E.1 Differential Noether charge
The anomaly polynomial for the three-dimensional Chern-Simons terms is given by (E.1) For later use, we define s ≡ sign(c A ). We note that the BTZ black hole is locally AdS 3 everywhere and thus the Riemann tensor is given by R a bcd = −2δ a [c G d]b and then the spin Hall current in this case becomes (Σ H ) abc = −4c g (1/2!) ε cef R ab ef = 4c g ε abc . By using this, the Hall contribution to the differential Noether charge ( / δQ Noether ) H is written as Let us now explicitly evaluate this ( / δQ Noether ) H on the BTZ black hole solution. The Christoffel symbols in (C.12) simplify by using two facts : in d = 2, κ q = 0 and hence (1/2)(d(r 2 f )/dr) = r. Further, we have g V = 0 since the BTZ black hole solution does not get corrected in the presence of the Chern-Simons terms. Using these, we have G µν = r 2 η µν + mu µ u ν , We also note that Eq. (C.15) simplifies to Using the above results and neglecting the terms proportional to dr, we obtain the following expression for the differential Noether charge ( / δQ Noether ) H on a fixed r slice : Here we have used the relation ε νλ = u ν V λ − V ν u λ which follows from the fact that the two orthogonal vectors u µ and V µ = ε µν u ν form a complete basis. We have also used the fact that in d = 2, η να = V ν V α − u ν u α and thus For later use, here we provide another useful result related to V µ : 15 We note that for a U (1) gauge field in AdS3/CFT2, the boundary conditions depend on the sign of c A . See section. 3.1 of [56].
where we have used the fact that any change in V µ is orthogonal V µ (and thus parallel to u µ ) since V µ V µ = 1, and we also have δu µ ∝ V µ from a similar argument. Then, from Eqs. (E. 6) and (E.7), we obtain

E.2 Asymptotic charge
Now we evaluate the the Hall contribution to the differential Noether charge ( / δQ Noether ) H at the boundary. By using Eqs. (E.5) and (E.8), we have the following expression at the boundary r → ∞ : On the other hand, the Einstein contribution from Eq.
This agrees with the result coming from the replacement rule.
For more details on the construction of the commutator relations and the central extensions such that the charges generate a U (1) Kac-Moody-Virasoro algebra, see [56].

E.3 Entropy
In this part, we compute the differential Noether charge at the horizon to obtain the entropy current. We will carry out this computation by using two different prescriptions which end up with the same final result. We will also comment on the consistency with the original computation on the Chern-Simons contribution to the black hole entropy by Ref. [31].

E.3.1 Entropy: prescription I
As a next step, we evaluate the differential Noether charge at the horizon of the BTZ black hole. Here we carry out the variation (with δr = 0) first and then set r = r H . By expanding out the variation as where the Hall contribution to the entropy current In the above derivation, we have used the fact that any change in V µ is orthogonal V µ and thus parallel to u µ since V µ V µ = 1. In particular it follows that P ν µ δV ν = 0. This result for the entropy current agrees with the CFT replacement rule.

E.3.2 Entropy: prescription II
In the previous section, the computation of ( / δQ Noether ) H was carried out by first taking the variation and then set r = r H of all quantities. Her we consider the second prescription in which we first set r = r H in the expression (E.5) and then carry out the variation. Then ( / δQ Noether ) H at the horizon is evaluated as which agrees with the result from prescription I in Eq. (E.17).

E.3.3 Comparision with Ref. [31]
We now compare our covariant computation with the prescription provided in Ref. [31]. According to Ref. [31], for the gravitational Chern-Simons term in three dimensions, one can write down a non-covariant Komar charge which is twice of what would be expected by the usual Wald formula : where in the last line we have evaluated the Komar charge in our coordinates. Taking r = r H with ξ µ = u µ /T , this gives the same result as our covariant computation :

F Example II : Abelian Chern-Simons terms
For the Abelian Chern-Simons term in (2n + 1) dimensions, the anomaly polynomial is given by where c A is a constant. Here we consider the case with n ≥ 2. The Hall contribution to the differential Noether charge ( / δQ Noether ) H in this case (see [57]) comes only from the first term of T 0 in Eqs. (2.9) and (2.10) : The evaluation of ( / δQ Noether ) H on the background (2.4) goes as follows at the leading order of the derivative expansion. At any fixed r, the leading order contribution to F n−1 is ( F 1 n−1 ) = (2Φω) n−1 since ( F 0) ∝ dr ∧ u. The leading order contribution to ( / δQ Noether ) H therefore is which is of order ω n−1 . We note that the 2nd and higher order terms in A and F etc. do not contribute to this order or lower, since this type of contribution is always accompanied by at least one ( F 0). Now we evaluate ( / δQ Noether ) H in particular at the horizon (r = r H ) and at the boundary (r → ∞). At the horizon, we substitute ξ a | hor = ξ a hor , Λ| hor = Λ hor (see Eq. (2.19)) and Φ(r = r H ) = µ into (F.3), while, at the boundary r → ∞, we use the fall-off behaviors ξ a | ∞ → O(r 0 ) and Φ| ∞ → O(r −(2n−2) ) (see Eq. We note that this is consistent with holographic renormalisation type computation in [58,59].

G Example III: Hall contribution in AdS 5
Before discussing the general anomaly polynomials in general odd dimensional AdS, it is instructive to explain some detail computation of the Hall contribution to the differential Noether charge ( / δQ Noether ) H for AdS 5 with the anomaly polynomial In this Appendix, we analyze this example in detail. We note that, for the anomaly polynomial given in Eq. G.1, each term in the expression Eqs. (2.9) and (2.10) in this case is given by where the spin Hall current is given by (Σ H ) cb a ⋆ dx c = −4c M F ∧ R b a or in components Before starting the computations of each term above, we massage the expression of T 3 and T 4 into the following forms for later use: where ξ ≡ η µν ξ µ dx ν and To derive this, we have used ξ r = 0 and the identities ⋆ dx r = r 3 ( ⋆CFT 1) and i ξ ⋆CFT C = ⋆CFT (C ∧ ξ) valid for any boundary form C as well as the anti-symmetric property of the spin Hall current, (Σ H ) abc = −(Σ H ) acb .
When we evaluate the leading order contribution to ( / δQ Noether ) H , the treatment of 2nd and higher order building blocks is trivial in AdS 5 case, since we are concerned about contributions up to order ω 1 . Thus, in the computation of T 0 , T 1 , T 2 , T 3 and T 4 , we do not need to take into account 2nd and higher order building blocks. In the rest of this Appendix, we therefore deal with 0th and 1st order terms in the building blocks only. We will confirm soon that the leading order contributions to ( / δQ Noether ) H for AdS 5 indeed start with ω 1 . Now we compute each term in (G.2) from the gravity side one by one to evaluate the differential Noether charge at the horizon and boundary.

G.1 Term T 0
For the evaluation of the term T 0 one can compute tr[δΓR] by using Eq. (B.99) and in the end we have the following expression valid at any fixed r : Let us next evaluate this expression at the horizon and boundary. By substituting ξ µ | hor = ξ µ hor = u µ /T , Λ| hor = Λ hor as in Eq. (2.19) at the horizon and the fall-off behaviors of the parameters and fields at infinity summarized in Eqs. (2.13) and (B.86), we can evaluate T 0 there as Therefore we conclude that T 0 vanishes both at the horizon and boundary up to ω 1 order.

G.2 Term T 1
As in the case of the Abelian Chern-Simons terms, we can replace F in T 1 by ( F 1) = (2Φω) since ( F 0) ∝ dr ∧ u . By directly computing the rest part of T 1 by using Eq. (B.101), we obtain the leading order contribution to this term as follows : It is of order ω 1 . By substituting ξ µ | hor = ξ µ hor = u µ /T , Λ| hor = Λ hor as in Eq. (2.19) at the horizon and the fall-off behaviors of the parameters and fields at infinity summarized in Eqs. (2.13) and (B.86), we can evaluate T 1 there as through the expansion in (2.23) as (J . The above result (G.9) will be naturally generalized to higher dimensions, as summarized in § § 4.2 (for the detail of the computation, see Appendices H-J).

G.3 Term T 2
By using Eq. (B.100) we can compute T 2 at the leading order of the derivative expansion as We note that this leading order contribution is of order ω 1 . The second term T 2 at the horizon and the boundary is then calculated as follows. By substituting ξ a | hor = ξ a hor , Λ| hor = Λ hor as in Eq. (2.19) at the horizon and using the fall-off behaviors Eqs. (2.13) and (B.86) at the boundary, we have where in the second equality for the evaluation at the horizon, we have used for any function g(r) of r and a integer m ≥ 0. We also note that we have dropped the terms proportional to dr as well as the total derivative terms d(. . .) in the above expression since they do not contribute the Noether charge after the integration.

G.4 Term T 3
The term T 3 depends on the spin Hall current, but we note that, from the expression of (G.3) and (G.4), we only need to know the specific components of the spin Hall current (Σ H ) aβr to evaluate T 3 . When only zeroth and first order terms in the building blocks are considered, there are three types of terms in (Σ H ) ab c up to ω 1 order :  16) In the following, we first evaluate the contribution to the components (Σ H ) (αβ)r from these three terms, (Σ H ) ′ab c , (Σ H ) ab c , separately. After this, we then combine them to obtain the term T 3 at the leading order.
• From (Σ H ) In this case, using ( F 0) ∝ dr ∧ u and ( R 1) β r ∝ u, we obtain (Σ We first notice that (Σ (2) H ) αβ r = 0 which follows from the fact that ( R 0) β r is proportional to dx β and ( F 1) ∝ ω. Then the components (Σ where G (V) = c M Φ(2Φ 2 T ). Then, by using i ξ [(δu) ∧ u ∧ (2ω)] = V β (δu β ) ⋆CFT ξ we obtain the following expression for T 3 for arbitrary fixed r at the leading order of the derivative expansion: Now we evaluate T 3 at the horizon and boundary. At the horizon, by substituting ξ a | hor = ξ a hor , Λ| hor = Λ hor as in Eq. (2.19) while at the boundary by choosing ξ a as in Eq. (2.13) and using Eq. (B.86), we finally obtain To evaluate the term T 4 at the leading order of the derivative expansion, we first evaluate (T • TermT Here we have used the identity Then, by noticing ⋆CFT V = u ∧ (2ω), we finally have the expression forT Putting the results forT together, we obtain T 4 at the leading order of the derivative expansion valid at arbitrary fixed r as follows : Let us then evaluate the leading order term of T 4 at the horizon and boundary, by substituting Eq. (2.19) at the horizon and by using the fall-off behaviors given in Eqs. (2.13) and (B.86) at the boundary respectively : Here we have also used ⋆CFT (V µ u ν dx ν − u µ V ν dx ν ) = η µν dx ν ∧ (2ω) .

G.6 ( / δQ Noether ) H at horizon and boundary
Let us in the end summarize the above computation of T 0 , T 1 , T 2 , T 3 and T 4 to evaluate ( / δQ Noether ) H both at the horizon and boundary. For the anomaly polynomial P CF T = c M F ∧ tr[R 2 ] in AdS 5 , we find that (T 0 + T 1 + T 2 ) at the horizon and boundary respectively takes the following form : where in the computation at the horizon, we have used Eq. (G.14) and neglected the terms proportional to dr as well as the total derivative term. On the other hand, the terms T 3 and T 4 at arbitrary r are given in (G.18) and (G.24) with G (V) = c M Φ(2Φ 2 T ) and H (V) = c M Φ. In particular, the sum of these two terms (T 3 + T 4 ) at the horizon and boundary is respectively evaluated as H Replacement rule for T 3 and T 4 As in Eq. (2.9), we divide ( / δQ Noether ) H into T 0 , T 1 , T 2 , T 3 and T 4 . In particular, the terms T 3 and T 4 , are related to the spin Hall current (Σ H ) cb a ⋆ dx c ≡ −2(∂P CF T /∂R a b ) and are relatively easy to deal with. The final expression for T 3 and T 4 at a fixed arbitrary r is simple. Moreover, by using essentially the same argument as we used for the Einstein source in Appendix D. 6 Ref. [27], one can prove the statement that the 2nd and higher order building blocks do not contribute to T 3 and T 4 at the leading order of the derivative expansion.
In the following, we will first begin with the general single trace case P CF T = c M F l ∧ tr[R 2k ] and then move on to the purely gravitational multi-trace case ]. Finally, we will consider the most general anomaly polynomial of the form In the first part of this Appendix, we will only consider the zeroth and first order building blocks. At the end, we briefly explain why the 2nd and higher order terms do not contribute to T 3 and T 4 at the leading order based on the argument for the Einstein source in Appendix D.6 of Ref. [27]. We stress that the computation here is valid for any fixed r. In some places, we will specifically evaluate them at infinity r → ∞ and at the horizon r = r H to explicitly display the expressions of T 3 and T 4 .
Before computing T 3 and T 4 for a specific P CF T , similar to the case in AdS 5 (see around Eq. (G.3)), we massage the expression of T 3 and T 4 into the following forms for later use : Let us consider the general single-trace anomaly polynomial P CF T = c M F l ∧tr[R 2k ] admitted by AdS d+1 with d = 2n = 2l + 4k − 2 for d ≥ 6. The spin Hall current in this case is given by As is known from the Einstein source computation carried out in Appendix D.6 of Ref. [27], there are three types of terms in (Σ H ) ab c that can contribute up to ω n−1 order, depending on how ( F 0) and ( R 0) are distributed. For k = 1 (thus n = l + 1), which is just a generalization of Eq.(G.15) in the case of AdS 5 . For k > 1, Here we have used the fact that ( F 0) ∝ dr∧u and ( R 0 2 ) = 0. We note that, for k > 1, the wedge product of odd numbers of the curvature 2-forms (R 2k−1 ) consisting of all ( R 1)'s except exactly one ( R 0) reduces to We also notice that ( F 0) ∧ ( R 0 R 1) = ( F 0) ∧ ( R 1 R 0) = 0, because ( F 0) ∝ dr ∧ u and all the terms in ( R 0 R 1) and ( R 1 R 0) are either proportional to dr or u .
This leads to (Σ In the first equality, we have used the fact that the objects ( R 0) β r , ( R 0 R 1 R 1) β r , ( R 1 R 0 R 1) β r and ( R 1 R 1 R 0) β r are either zero or proportional to dx β , which leads to (Σ (2) H ) (αβ) r = 0. From the above computation, non-trivial components of the spin Hall current relevant to the computation of T 3 is given up to ω n−1 order by where . Therefore, the leading order expression of T 3 for P CF T = c M F l ∧ tr[R 2k ] turns out to be When we evaluate at r = r H (at the horizon) or r → ∞ (at the boundary), substituting ξ a | hor = ξ a hor , Λ| hor = Λ hor as in Eq. (2.19) while at the boundary choosing ξ a as in Eq. (2.13), we have the following expressions : where we have used Eq. (B.86). Hence, T 3 | ∞ = 0 since we are considering n ≥ 2 (where 2k + l ≥ 3 and k ≥ 1).

H.1.2 Term T 4
To calculate T 4 at the leading order of the derivative expansion, we start with the evaluation of (T In deriving the results above, it is helpful to remember which is just a generalization of Eq. (G.22) (in the case of AdS 5 ). We note that in the expression for (Σ (2) ) rα µ , the only difference between k = 1 case and k ≥ 1 case comes from H (V) . Finally, we have the expression for (T (b) 4 ) µ as follows (note that ⋆CFT V = u ∧ (2ω) n−1 ) : Here we consider the case with the purely gravitational anomaly polynomial consisting of multiple traces, Without loss of generality, we can assume that k i = 1 for i ≤ p 0 , i.e. the first p 0 tr[R 2k i ]'s are tr[R 2 ]'s (in particular p 0 = 0 means that the anomaly polynomial P CF T does not contain any tr[R 2 ]). As in the previous case, here we first consider zeroth and first order building blocks and will show at the end of this Appendix that 2nd and higher order building blocks do not contribute to T 3 and T 4 at the leading order.
We first recall that Because of these, by replacing the contributions of F l consisting of ( F 1)'s and at most one ( F 0) by products of traces of (R 2k i ) consisting of all ( R 1)'s and at most one ( R 0) (upon sending Φ to Φ T with some appropriate factors of 2's), we can carry out the same classification of the spin Hall current Σ H as we did for P CF T = c M F l ∧ tr[R 2k ] and define (Σ H ) ′ , (Σ where 2k tot = n + 1. The first two terms on the right hand side are essentially in the same form as (Σ H ) appearing in Appendix H.1, while the third term is the same as (Σ (1) H ) there under the replacement of F l by the traces of the curvature two-forms explained above. Therefore we can straightforwardly compute the nontrivial components of (Σ On the other hand, at r → ∞, we have (H. 40) both of which vanish since k tot ≥ 2 and d ≥ 6.
As a final case, we consider a general mixed anomaly polynomial consisting of multiple traces ] admitted by AdS d+1 with d = 2n = 2l + 4k tot − 2. Again, without loss of generality, we assume that k i = 1 for i ≤ p 0 (p 0 ≥ 0). In the current case, by treating F l and traces of the curvature two-forms as we did in Appendix H.1. We can classify contribution to the spin Hall current up to ω n−1 order into three cases, (Σ H ) ′ , (Σ The expression of (∂P CF T /∂R a b ), up to order ω n−1 , is We can see that the first term on the right hand side of Eq. (H.41) contributes only to (Σ H ) while the rest of the terms contribute only to (Σ (1) H ), and they can be evaluated essentially in the same way as in Appendix H.1. More practically, the contribution to the spin Hall current from the first two terms can be obtained by wedging the result for l = 0 (see Appendix H.2) with (2Φω) l , while the one from the third term can be computed essentially in the same way as (Σ (1) H ) for the case of p = 1 (see Appendix H.1) because the former just contains extra powers of (2Φω) and (2Φ T ω) compared to the latter. In the end, we finally have the expression for T 3 and T 4 at any fixed r as where .
On the other hand, at infinity r → ∞, we have (H. 45) both of which vanish since d ≥ 6 and k tot ≥ 2 .

H.4 On 2nd and higher order building blocks
Up to this point, we only considered the zeroth and first order building blocks. Here we briefly explain why 2nd and higher order building blocks do not contribute to T 3 and T 4 up to ω n−1 order. In Appendix D.6 of Ref. [27], we have proved that these 2nd order and higher order building blocks do not contribute to the Einstein sources. One of the key points of the proof is that the 2nd and higher order building blocks do not contribute to (∂P CF T /∂R a b ) (and thus to the spin Hall current) up to ω n−1 order. Since both T 3 and T 4 are linear in the spin Hall current, this result is sufficient to prove that 2nd and higher order building blocks do not contribute to T 3 and T 4 up to ω n−1 order. I Contributions to T 0 , T 1 and T 2 at horizon: without higher order In this Appendix, we evaluate T 0 , T 1 and T 2 terms for a general anomaly polynomials admitted on AdS 2n+1 with n ≥ 3. We also combine with the results on T 3 and T 4 we obtained in Appendix H to calculate ( / δQ Noether ) H at the horizon. We start with some specific warm-up examples and then consider the general cases.
For the curvature and gauge field strength 2-form at the zeroth and first orders and its products, drastic simplifications occur at the horizon. As we will explain, most of terms appearing in the products vanish or is proportional to dr while the remaining nontrivial terms have simple structures. In particular, for the F l part, since ( F 0) ∝ dr ∧ u, we can replace F l by ( F 1 l ). Therefore, we essentially only need to deal with purely gravitational anomaly polynomials.
Furthermore, due to Eq. (2.19), (Λ + i ξ A)| hor = Λ hor + µ/T = 0 (up to 1st order) and hence T 0 | hor = 0 for a general anomaly polynomial P CF T . Thus, we only need to evaluate T 1 and T 2 at the horizon.
For the rest of this Appendix, we shall first evaluate ( / δQ Noether ) H | hor using zeroth and first order building blocks. Similar to the case of the Einstein sources in Ref. [27], 2nd and higher order building blocks do not contribute to T 1 and T 2 at the horizon up to ω n−1 order of the derivative expansion. As we will show in Appendix J, the restriction at the horizon makes this proof simpler too.
As a first warm-up example, we consider P CF T = c M F l ∧ tr[R 2 ] in AdS 2n+1 with n = l + 1. We notice that all the F 's in T 1 and T 2 can be set to be ( F 1) = (2Φω), since ( F 0) ∝ dr ∧ u. Then the evaluation of T 1 and T 2 is essentially the same as the case of P CF T = c M F ∧tr[R 2 ] in Appendix G (with extra (2Φω)'s). Therefore, using Eq. (G.14) and substituting ξ a | hor = ξ a hor , Λ| hor = Λ hor as in Eq. (2.19), we obtain . This leads to Combining with the results of T 3 and T 4 in Appendix H, Note that T 2 is zero since P CF T does not contain any F 's.
Combining with the result of T 3 + T 4 in Appendix H evaluated at the horizon, we finally have the expression for ( / δQ Noether ) H | hor : As a final warm-up example, we consider P CF T = c g tr[R 4 ] in AdS 7 where n = 3. In this case, the term T 1 is computed as Here we have used δ u ∧ (2ω) n−1 = n(δu) ∧ (2ω) n−1 , (I.10) which comes from Eq. (G.14). As in the previous example, T 2 is obviously zero for purely gravitational anomaly polynomials. For T 3 + T 4 , by setting r = r H in the result obtained in Appendix H, we have Finally, we obtain the following result for ( / δQ Noether ) H evaluated at the horizon : Let us start with the single trace case, P CF T = c M F l ∧tr[R 2k ] in AdS 2n+1 with n = 2k+l−1.
Derivatives of the anomaly polynomial with respect to the curvature two-form and the U (1) field strength are given respectively by As for the U (1) field strength, since ( F 0) ∝ dr ∧ u and ( F 1) = (2Φω), in the evaluation of T 1 and T 2 at the leading order under the assumption that 2nd and higher order terms of F , R etc. do not contribute, we can replace F by (2Φω). Thus, in the evaluation of T 1 , we can just concentrate on the case of the anomaly polynomial of the form (I.14) in AdS 2n+1 with n = 2k − 1, and the result for more general single trace anomaly polynomial P CF T = c M F l ∧ tr[R 2k ] follows from this straightforwardly.

I.4.1 T 1 at horizon
In the evaluation of T 1 , we encounter three kinds of terms which can contribute nontrivially: We first note that (R ∧ . . . ∧ R) with more than one ( R 0) vanishes or is proportional to dr.
Thus, we only have to consider the products with one or no ( R 0). Then for the first kind of possibility, (I.15), we can classify the potential nontrivial contribution as summarized in Table 2. The entries in the second and third column (i.e. the 0's and 1's) under ∇ξ and δΓ indicate which order of derivative expansion in ∇ξ and δΓ are appearing in that structure. For example, when it is 0 under ∇ξ and 1 under δΓ, it means that we are considering the structure containing ( (0) ∇ξ) and ( (1) δΓ). Since we are considering contributions composed of 0th or 1st order terms of ∇ξ, δΓ and R, we can classify the possible nontrivial contributions by the numbers of the 0th order terms. In Table 2, we list up all the possibilities containing two or more zeroth order terms. We note that when the number of the 0th order terms are more than four, then at least one of the two (R ∧ . . . ∧ R)'s contains two or more ( R 0)'s and thus this kind of contribution vanishes. If the number of the zeroth order terms are less than two, then the contributions are of higher order. Note that in the table, we did not specify the explicit contractions of the indices. They should be contracted according to Eq. (I. 15). For the possibilities listed in the table, there are usually two ways that the (R ∧ . . . R) can be contracted, depending if it is next to δΓ or ∇ξ. For example, in the second row, we see that we could have χ 1 in the 3rd column and χ 0 in the 4th column. This means that we can have the following two structures We also note that when both of (R ∧ . . . ∧ R)'s are elements of χ 1 (the first and third case in Table 2), then such a kind of contribution is proportional to u ∧ u and thus vanish (we again stress that we neglect the terms proportional to dr during the evaluation of the first term). As we have shown in Eq. (B.65), the nontrivial elements in χ 1 are ( R 0 R 1 m ) µ r and ( R 1 m R 0) µ ρ only, since ( R 1 R 0 R 1) a b is proportional to dr or zero. # of 0th order terms ∇ξ δΓ (R ∧ . . . ∧ R) (R ∧ . . . ∧ R) Results In a similar way, in the case of the two possibilities given in (I.16), we can summarize potential nontrivial contributions as in Table 3. Table 3. Single trace anomaly polynomial : contributions to T 1 from terms with δΓ and ∇ξ next to each other. # of 0th order terms ∇ξ δΓ R ∧ . . . ∧ R Results In the rest of this subsection, we directly evaluate the possibilities Case A, B, C, D, E and F in the above tables to calculate the leading order contribution to the first term T 1 . Case A (vanish in the end) The followings are four potential nontrivial terms in Case A, but all of them vanish as a result of explicit computations (for 1 ≤ m ≤ 2k − 3) : Therefore, there is no contribution to T 1 from this type of terms.

Case B
There is only one type of contribution in Case B which can be computed as follows (for 1 ≤ m ≤ 2k − 3) : Case C As in Case A, there are four potential nontrivial terms in Case C and only one out of these terms contribute nontrivially (for 1 ≤ m ≤ 2k − 3) : where we have used Eq. (G.14).
Case D (vanish in the end) The following four terms are potentially nontrivial but the direct computation shows that all of them vanish : Therefore there is no nontrivial contribution from Case D.

Case E
There are two non-trivial structures in Case E : Case F There are two potential nontrivial cases one of which vanishes as a result of direct computation : Here we have used Eq. (G.14). Summary for T 1 Now that we have calculated all the contributions in the Table 2 and 3, we sum them up to calculate the leading order contribution to the first term T 1 . The above calculation shows that all the contributions containing more than two 0th order terms vanish. Then the leading order contribution to T 1 contains two 0th order terms and is calculated by summing up the results in Case B, C, E and F. For P CF T = c g tr[R 2k ] with n = 2k − 1, we have where we have used Eq. (I.10) and J (V) As explained before, T 1 in the case of P CF T = c M F l ∧ tr[R 2k ] (with n = 2k + l − 1) is easily obtained by multiplying (2Φω) l to the result for l = 0, which yields 27) where in this case J

I.4.2 T 2 at horizon
Here we evaluate the second term T 2 . For P CF T = c g tr[R 2k ], it is obviously zero and thus we consider P CF T = c M F l ∧ tr[R 2k ] for which T 2 is of the form (I. 28) We note that ∇ b ξ a (R 2k−1 ) b a is either zero or proportional to dr when it contains at least one ( R 0), the U (1) gauge field A at the leading order is Φu, and F can be replaced by the (2µω). Then the leading order contribution to the second term T 2 is given by

I.4.3 ( / δQ Noether ) H at horizon
We recall that for P CF T = c M F l ∧ tr[R 2k ], T 3 + T 4 evaluated at the horizon is given by (see Collecting all these terms, we finally obtain As in the previous case, F can be replaced by (2Φω) to evaluate the leading order contribution. We thus concentrate on the anomaly polynomial of the form ] from which the result for the case with F l is derived straightforwardly. The crucial difference from the single trace case is that when we consider two derivatives of P CF T with respect to R, there are two kinds of terms: the two R-derivatives can act on the same trace or on different traces. As we know that tr[χ 1 ] ∝ dr ∧ u, for the trace tr[R 2k i ] on which the R-derivatives do not act, we can replace it by tr[( R 1 2k i )] = 2(2Φ T ω) 2k i . Therefore, we only need to consider the two-trace case, P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ] with k 1 , k 2 ≥ 1, and the more general case follows from this straightforwardly.
In this two-trace case, we have The results for the terms in the second and third lines can be obtained from the single trace answer by multiplying tr[( R 1 2k 1 )] = 2(2Φ T ω) 2k 1 (or, the one with k 1 replaced by k 2 ). Thus, we will concentrate on the first term. We also restrict ourselves to the case in which the derivatives with respect to R a b and R c d act respectively on tr[R 2k 1 ] and tr[R 2k 2 ] (the other one can be derived by just interchanging k 1 and k 2 ).

I.5.1 T 1 at horizon
In the case of P CF T with a single trace, the possible nontrivial contributions are classified as in Table 2 and 3. In the double trace case, there is the following extra contribution related to the first line of (I.32) : whose potential nontrivial contribution at low orders are classified as in Table 2, though the contraction structures are different. For clarify, we shall present such classification in Table  4. Table 4. Multi-trace anomaly polynomial : contributions to T 1 from terms with δΓ and ∇ξ in different traces.
# of 0th order terms ∇ξ δΓ (R ∧ . . . ∧ R) (R ∧ . . . ∧ R) Results For Case A, there are four potential nontrivial contributions but all of them vanish (or proportional to dr) as follows : due to Eq. (B.73)-(B.74). We note that, strictly speaking, there are other four cases where k 1 and k 2 are interchanged, but they are obviously zero (or proportional to dr) for the same reason.

Case B
For Case B, we have only one possibility (and the case with k 1 and k 2 interchanged) : Case C In this case, there are four potential non-trivial cases but one out of the four cases gives nontrivial contribution (as in the case of Case A and B, there is another nontrivial contribution from k and l interchanged) : where we have used Eq. (G.14). Summary for T 1 Now we collect all the terms to calculate the first term T 1 . For P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ] (with n = 2k 1 + 2k 2 − 1), by using Eq. (G.14), the first term T 1 at the leading order is given by (do not forget the second and third terms in (I.32)) where we have used Eq. (I.10) and the fact that in this case J For more general multi-trace terms in the anomaly polynomial, ] (with n = 2k tot + l − 1 and k tot = p i=1 k i ), by recalling that F can be replaced by (2Φω) and tr[R 2k i ] by 2(2Φ T ω) k i when the derivatives with respect to R do not act there, the first term T 1 at the leading order is given by 38) where in this case J

I.5.2 T 2 at horizon
For the anomaly polynomial without F 's, this term is obviously zero and thus we consider more general cases including the multiple traces, ]. The evaluation of T 2 at the leading order is the same as the single trace case except that we need to replace the traces tr[R 2k i ] without the derivative acted on by tr[( R 1 k i )] = 2(2Φ T ω) k i . Then the final result is given by We recall that T 3 + T 4 evaluated at the horizon is given by (see Appendix H) Therefore, collecting all the terms, we finally obtain J Contributions to T 0 , T 1 and T 2 at horizon: with higher order In Appendix I, we have assumed that the 2nd and higher order building blocks do not generate any non-trivial contribution to ( / δQ Noether ) H | hor up to ω n−1 order. We confirm this statement here. As we have proved this statement for T 3 and T 4 in Appendix H.4, here we will prove that 2nd and higher order building blocks do not contribute to T 0 , T 1 and T 2 (at the horizon) at the leading order. In the following, we check this statement term by term. Since T 0 | hor = 0 due to (Λ + i ξ A)| hor = 0 to all order, we do not need to deal with T 0 | hor .
Here is one important remark that will be useful : As we have seen, when 2nd and higher order building blocks are neglected, the leading order contribution to T 1 and T 2 is of order ω n−1 .
We emphasize that in this Appendix, we are considering all contributions which contain at least one 2nd or higher order building block.

J.1 T 1 at horizon
Let us consider the anomaly polynomial of the form ] with n = 2k tot − 1. We will take into account F l later. Without loss of generality, we assume k i = 1 for i ≤ p 0 (p 0 ≥ 0). Furthermore, let us denote the total number of the derivative in ∇ξ and δΓ byÑ . One important remark is that at the horizon, ( R 0) is either proportional to dr or u and thus the number of ( R 0)'s needs to be less than two to have nontrivial contribution to T 1 .
When we evaluate the derivative (∂ 2 P CF T /∂R a b ∂R c d ), there are four possibilities : • Case 1 :Ñ ≥ 1.
• Case 2 :Ñ = 0 and both of the derivatives act on the same trace tr[R 2k i ] with i ≤ p 0 , i.e.
where without loss of generality, we have assumed that i = 1.
• Case 3 :Ñ = 0 and both of the derivatives act on the same trace tr[R 2k i ] with i > p 0 , i.e.
• Case 4 :Ñ = 0 and the two derivatives act on different traces Since we are considering the contributions containing at least one 2nd or higher order term in T 1 , at least two R's must be ( R 0) to contribute to T 1 up to ω n−1 order. Therefore, this gives vanishing contribution.

J.1.2 Case 2
In this case, to circumvent the appearance of two or more ( R 0)'s, the R's need to contain one ( R 0), one ( R 2) and the rest are set to ( R 1). This indicates that tr[R 2k 2 ] ∧ · · · ∧ tr[R 2kp ] contains tr[ R 2χ 1 ] or tr[χ 1 ], but both of them are zero or proportional to dr. We thus conclude that this case give vanishing contribution to T 1 .

J.1.3 Case 3 and Case 4
For Case 3 and Case 4, we note that the derivative (∂ 2 P CF T /∂R a b ∂R c d ) contains two (R ∧ R ∧ . . . ∧ R)'s (we note that one of the (R ∧ R ∧ . . . ∧ R)'s can be δ a b which means we are in Case 3) and a product of traces of the form T = T 1 ∧ T 2 ∧ T 3 , where T 1 (T 2 , T 3 , respectively) is the wedge product of the trace of the form T R (1) (T R (2) , T R (3) , respectively) only, with the T R (i) 's are defined Ref. [27] as : Here, the symbol χ represents one of the elements in χ 0 ∪ χ 1 ∪ χ 2 , that is, it is a string made of ( R 0)'s and ( R 1)'s only. On the other hand, the symbol υ is defined to represent a string made of 2nd or higher order terms ( R m) with m ≥ 2. We note that the existence of ( R 0) in T 1 is not allowed since tr[χ 1 ] ∝ dr and thus we set all the trace in T 1 to be tr[χ 0 ]. Therefore, we might as well neglect the T 1 in the following discussion. where (I ≥ 1). We notice that we regard δ a b as an element of CaseĒ with χ 0 .

Case 3 :
When we consider the contribution of order ω n−1 or lower, to avoid appearance of more than one or they are proportional to : Here, we have used Eqs. (B.76), (B.78) and (B.85).

Case 4 :
As in Case 1, up to ω n−1 order, to avoid appearance of more than one ( R 0)'s in two derivatives of the anomaly polynomial, i.e. (∂ 2 P CF T /∂R a b ∂R c d ), we need to require that one of the (R ∧ . . . ∧ R)'s is of the form in CaseĒ while the other (R ∧ . . . ∧ R)'s is either of the form in CaseB,C orĒ and that there is no terms in T 3 . There are three possibilities as before : • When both of (R ∧ R ∧ . . . ∧ R)'s are χ 0 and T 2 contains only one trace of the form tr[ R 2χ 1 ], this type of terms vanishes since tr[ R 2χ 1 ] = 0. To summarize up to this point, for P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ] ∧ · · · ∧ tr[R 2kp ], we have shown that 2nd and higher order building blocks do not give any nontrivial contribution to T 1 up to ω n−1 order.
Let us next consider the case with ]. Since ( F 0) ∝ dr ∧ u, F l starts at order ω l . As for the rest part of T 1 , the argument goes in the same way as we did for P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ] ∧ · · · ∧ tr[R 2kp ] (that is, we know that for this anomaly polynomial, T 1 at the leading order is ω n−1 and this does not contain 2nd and higher order terms).
From the above arguments, we conclude that there is no contribution to the first term T 1 up to ω n−1 order from second or higher order building blocks. estimate is enough for AdS 11 and higher), while more detailed analysis is still required for some low-dimensional examples (that is, AdS 7 and AdS 9 ). We thus analyze the latter cases more carefully after the rough estimate.
Here, we define more precisely what we mean by the rough estimate and introduce some notation we will employ in this Appendix. For the building blocks, we define their fall-offs by the slowest damping component and denote it by '∼' : For example, for ∇ξ, the component with the slowest fall-off is ∇ µ ξ r which behaves as r 3 at the boundary, and thus in the rough estimate we have ∇ξ ∼ r 3 . Here we remind the readers that we are ignoring terms proportional to dr in this Appendix. The rough estimate of the fall-off behavior for the products of R's goes as follows : we just forget all their index-contractions and replace each building block by its fall-off from the rough estimate. For example, Of course, we can consider the case with more ( R 0)'s, but the above possibilities are enough for our purpose, as we will see later.
The refined estimate takes into account the contraction structure of each building block. To distinguish the refined estimate from the rough estimate, we will denote the refined estimate with ' '. Here is a simple example: for the rough estimate, we have while for the refined estimate, the fall-offs are much faster (due to explicit contractions of the indices) : 16 In the course of the rough estimate, since we do not take into account the index contraction, there is no distinction between the anomaly polynomial with a single trace or multiple traces. For example, for the rough estimate, we do not distinguish P CF T = c g tr[R 2(k 1 +k 2 ) ] and P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ]. In the following part, we therefore carry out the rough estimate for the following two cases: (1) purely gravitational anomaly polynomial (including both single-and multi-trace cases) (2) mixed anomaly polynomial (including both single-and multi-trace cases).

K.2 Rough estimate for purely gravitational anomaly polynomials
Let us consider a general purely gravitational anomaly polynomial containing 2k curvature two-forms R (k ≥ 2) on AdS 2n+1 with n = 2k − 1 (note that n ≥ 3 and n is an odd number in this case). In this case, since both T 0 and T 2 are zero from the definition of these terms, we only need to consider the first term T 1 .
Before the estimate, we remind the readers that when there are three or more ( R 0)'s in T 1 , then T 1 contains at least one χ 2 or tr[χ 1 ] both of which are proportional to dr ∧ u. The symbol χ m is defined as products of (R)'s consist of m number of ( R 0)'s with the remaining (R)'s being ( R 1)'s (see Appendix B.1.1 for useful facts about χ m ). Therefore, we only need to consider the cases with two or less ( R 0)'s.
Through the rough estimate, the first term T 1 behaves as Thus, depending on the number of ( R 0)'s in T 1 , there are three possibilities with the following fall-off behaviors: Since n ≥ 3, the first case vanishes at infinity while the second and third cases vanish for n ≥ 5 (note that n is an odd integer). Therefore, for AdS 11 and higher, the rough estimate is sufficient to imply that T 1 | r→∞ ∼ 0. Thus, the only nontrivial case is when n = 3 which corresponds to the anomaly polynomial P CF T = c g tr[R 4 ] and P CF T = c g tr[R 2 ] ∧ tr[R 2 ] on AdS 7 . We will carry out the refined estimate for these cases later in Appendix K.4.

K.3 Rough estimate for mixed anomaly polynomials
Now we consider the anomaly polynomial containing F l and 2k curvature two-forms (where l ≥ 1 and k ≥ 1). This anomaly polynomial is admitted on AdS 2n+1 with n = 2k + l − 1.
Here we consider n ≥ 3 only and thus 2k + l ≥ 4 needs to be satisfied. In the following, we evaluate the three terms T 0 , T 1 , T 2 one by one.

K.3.1 Zeroth term T 0
There are two terms in T 0 . From Eq. (2.10), we see that the first and the second term each contains (∂ 2 P CF T /∂F ∂F ) and (∂ 2 P CF T /∂R a b ∂F ) respectively. Here we assume (Λ + i ξ A) fall-offs as r 0 at most. For the first term, since ( F 0) ∝ dr ∧ u, all the F 's need to be replaced by ( F 1). We also note that all the R's need to be replaced by ( R 1) since tr[χ 1 ] ∝ dr ∧ u and R's only appear in the form of tr[R 2k i ] (since there is no derivative with respect to R a b ). Then the fall-off behavior is roughly estimated as which vanishes at infinity. Similarly, for the second term in T 0 , we can replace all the F 's by ( F 1)'s, thus the second term in T 0 behaves as Since the second term in T 0 contains only one derivative with respect to R a b , there is only one (R ∧ · · · ∧ R) (without trace) wedged by tr[R 2k i ]'s. Since χ 2 ∝ dr ∧ u and tr[χ 1 ] ∝ dr ∧ u, we only need to consider the case with one or less ( R 0). In particular, the ( R 0) must be located in the (R ∧ · · · ∧ R) part, while all the tr[R 2k i ]'s must be replaced by tr[ R 1 2k i ]. Therefore, there are two potential nontrivial terms which fall-off as both of which vanish at infinity (note that 2k + l ≥ 4 for n ≥ 3).
To summarize, the zeroth term T 0 vanishes at infinity for n ≥ 3.

K.3.2 First term T 1
Similar to T 0 , all the F 's need to be replaced by ( F 1) to have a non-trivial result. Then the fall-off behavior of this term is (K.10) Then, as in the case of a purely gravitational anomaly polynomial, there are three possibilities depending on the number of ( R 0)'s : These three terms vanish when n ≥ 3, n ≥ 4 and n ≥ 5, respectively. We thus need to carry out the refined estimate for AdS 7 (n = 3) and AdS 9 (n = 4).

K.3.3 Second term T 2
As a final part for the rough estimate, we evaluate T 2 . Again, by noting that ( F 0) ∝ dr ∧ u, we replace all the F 's by ( F 1). Then the fall-off behavior of this term is As in the case of the second term in T 0 , this term contains only one derivative with respect to the curvature two-form and thus at most one ( R 0) is allowed to have a nontrivial contribution. Therefore, the followings are two non-trivial possibilities : both of which vanish for n ≥ 3 (note again that 2k + l ≥ 4 for n ≥ 3).
To summarize, we conclude that: • T 0 , T 1 and T 2 vanish at infinity for AdS 2n+1 with n ≥ 5.
• For AdS 7 and AdS 9 the refined estimate needs to be done for T 1 only.
• In the case of AdS 7 , from (K.11) (which is valid for l = 0 case too), the subtle case is when one or two R's are ( R 0) and T 1 in the rough estimate behaves as r 3−l and r 8−l , respectively.
• On the other hand, for AdS 9 , the only subtle case is when there exist two ( R 0)'s where T 1 in the rough estimate behaves as r 3−l .

K.4 Refined estimate of T 1 for AdS 7
In this part, we carry out the refined estimate of T 1 for the anomaly polynomials admitted on AdS 7 . For AdS 7 , there are three types of the anomaly polynomials, P CF T = c M F 2 ∧ tr[R 2 ], c g tr[R 2 ] ∧ tr[R 2 ] and c g tr[R 4 ]. We will evaluate how they behave at infinity one by one. We stress again that here we still neglect 2nd and higher order building blocks, but we will take them into account at the end of this Appendix. We also recall that we have confirmed through the rough estimate that T 1 vanishes at infinity when it contains no ( R 0)'s. We thus consider the case with one or two ( R 0)'s.
In this case, the first term T 1 is given by This case does not contain any ( R 0) and thus we conclude that T 1 vanishes at infinity.
The first term T 1 for this anomaly polynomial is composed of two types of terms, depending on where the two derivatives with respect to R act : For the first term in (K.15), since ( R 0 2 ) = 0 and tr[( R 0 R 1)] ∝ dr ∧ u, this term does not contribute at infinity. For the second term in (K.15), direct computations show that To summarize, we have shown that T 1 vanishes at infinity for the anomaly polynomial As a final example for AdS 7 , we carry out the refined estimate of T 1 at infinity for the anomaly polynomial P CF T = c g tr [R 4 ]. The first term T 1 in this case is given by (K.18) The first and the second terms in (K.18) have three potential contributions, but all of them vanish at infinity in the following way : For the third term, there are three potential contributions: For the first line, we have used which behaves as r −3 for AdS 7 . For the other two terms, we evaluated them directly to show the fall-off behavior. In the end, we confirmed that T 1 → 0 at infinity.

K.5 Refined estimate of T 1 for AdS 9
Here we carry out the refined estimate for AdS 9 . We recall that, as a result of the rough estimate, the only potential nontrivial case is when T 1 contains two ( R 0)'s. For this Appendix, we will focus on such possibilities. In this case, T 1 behaves as T 1 ∼ r 3−l . We also notice the relation (K.21) in this case is δΓ a b ( R 0) b c r −(d−3) = r −5 , while the rough estimate gives δΓ a b ( R 0) b c ∼ r 3 .
In this case, T 1 is given by which does not contain any R. Therefore, this term vanishes at infinity as a result of the rough estimate.
The first term T 1 for this anomaly polynomial is which vanishes since ( R 0 2 ) = 0. As for the second term, as a result of (K.21) (for AdS 9 ), the refined estimate changes the fall-off behavior of T 1 by at least a factor r −8 compared to the rough estimate This is enough to confirm that T 1 , which fall-off as r 3−l = r 2 under the rough estimate, vanishes at infinity.
In this case, T 1 is given by (K.24) The first and second terms vanish as a result of ( R 0 2 ) = 0. On the other hand, for the third term, Eq. (K.21) (for AdS 9 ) improves the fall-off behavior at infinity by at least a factor r −8 compared to the rough estimate. As in the previous case, this is enough to prove that T 1 in this case vanishes at infinity.
L Why ( / δQ Noether ) H | ∞ = 0 in AdS d+1>7 ? : with higher order In this Appendix, we will focus on contributions containing at least a 2nd or higher order building block and prove that the 2nd and higher order building blocks do not contribute to ( / δQ Noether ) H at r → ∞. Before we begin the general argument, it is instructive to study in details all cases in AdS 7 and a particular case in AdS 15 .

L.1 Example : AdS 7
In AdS 7 , we consider contributions up to ω 2 order. Hence, we only need to consider the case containing exactly one 2nd order building block (which contributes at ω 2 ) with the rest of the building blocks set to zeroth order. We will show that such contributions are zero at any fixed r. In particular, they vanish at r → ∞.
The terms T 0 , T 1 and T 2 for this anomaly polynomial are given by The first term in T 0 does not contain ω 2 order contribution because of tr[( R 0) 2 ] = 0 and tr[( R 0 R 2)] = 0. For T 1 , T 2 and the second term in T 0 , since ( F 0) ∝ dr ∧ u, one needs to set one F to be ( F 2). In that case, since F , tr[δΓR] and tr[∇ξR] start at ω 1 order, these terms all become ω 3 or higher order.
In this case, T 0 and T 2 are trivially zero, while T 1 is The first term does not contribute to ω order because of tr[( R 0) 2 ] = 0 and tr[( R 2 R 0)] = 0. For the second term, let us first consider the case when the second order term is located in tr [δΓR] . Then, since tr[∇ξR] starts at ω 1 order, the second term in T 1 starts at ω 3 or higher order. The argument is the same when tr[δΓR] and tr[∇ξR] are interchanged.
This is the final case admitted in AdS 7 , in which a little bit detailed analysis is required. For this anomaly polynomial, T 1 is given by (T 0 and T 2 are trivially zero as in the previous example) These terms all turn out to be zero due to Eq. (B.39).

L.2 Example : AdS 15
Now we consider one specific example in AdS 15 which will be useful when we deal with the general cases. In AdS 15 , we consider up to ω 6 order terms. Here we discuss the first and second line separately. The first line in T 1 is easy to deal with. There are two points to notice: first of all, tr[R 4 ] already starts at order ω 4 and it is equal to tr[( R 1 4 )]. Secondly, the terms in the curly bracket starts at order ω 2 and 2nd (and higher order) building blocks do not contribute to the curly bracket terms up to ω 2 order. Thus, the lowest order contributions therefore start at order ω 6 , and this order ω 6 contribution does not contain any 2nd or higher order building blocks.
For the second line in T 1 , as we have shown in Appendix B.2.3, both tr[∇ξR 3 ] and tr[δΓR 3 ] start at order ω 3 and such contributions do not contain 2nd or higher order building blocks.
To summarize, we conclude that 2nd and higher order building blocks do not contributing to ( / δQ Noether ) H at infinity up to order ω 6 .

L.3 General arguments on 2nd and higher order terms
In Appendix. K, we used only zeroth and first order building blocks in the estimates of ( / δQ Noether ) H at r → ∞. Here we will deal with contributions containing at least one higher order building blocks, i.e. at least one (m) δΓ, (m) ∇ξ, ( F m) or ( R m) for m ≥ 2. We will show that all such contributions to ( / δQ Noether ) H vanish up to ω n−1 order.
Before the proof, we here summarize some important facts. Let us consider tr[R 2k ], tr[δΓR 2k−1 ], and tr[∇ξR 2k−1 ]. As we have seen in the computation for the Maxwell sources in Appendix C.3 of Ref. [27] and Appendix B.2.3, these traces start at ω 2k , ω 2k−1 and ω 2k−1 order respectively, and these leading order terms contain zeroth and first order building blocks only. For the wedge product of the gauge field strength F l , the leading order contribution is of the form ( F 1) l and thus is of order ω l . Second and higher order terms in F can only start contributing to F l at ω l+1 -order or higher. These results will become important in the later computations.
We stress that in this Appendix, by 'all contributions', we mean all contributions containing at least a second or higher order building block.

Strategy
Let us first explain our strategy. Our arguments rely on the following two facts (which will be shown next): 2. Fact 2: At ω n−1 -order, all contributions vanish at any fixed r with one exception : terms having exactly one ( R 2), ( (0) δΓ), ( (0) ∇ξ) along with products of ( R 0)'s and ( R 1)'s.
Once these two facts are confirmed, we only need to deal with the exceptional case in Fact 2 which contains exactly one ( R 2). Moreover, to realize ω n−1 order contribution, the derivative (∂ 2 P CF T /∂R∂R) needs to contain one ( R 2), one ( R 0) and the rest of R's are ( R 1). Now we consider r → ∞. From Appendix B.4.2 (in particular Eq. (B.98)), we note that the fall-offs of ( R 2) in the rough estimate are ( R 2) ∼ r 2 ∼ ( R 0) .
(L.6) Thus, we can replace ( R 2) by ( R 0) under the rough estimate. Therefore, what we need to evaluate is This term, however, has been estimated in Appendix K.3.2 to fall-off sufficiently fast and vanish in AdS 11 and higher. We note that, for AdS 7 (n = 3), in Appendix. L.1, we have already shown explicitly that at any fixed r, there is no lower contributions than ω n−1 and that the nontrivial ω n−1 order contributions do not contain any 2nd and higher order building blocks.
In the case of AdS 9 (n = 4), we note that the anomaly polynomials are just the ones in AdS 7 wedged with an extra F . Since F starts at ω 1 (since ( F 0) ∝ dr ∧ u), essentially the same argument as the AdS 7 case lead to the proof that there is no lower contributions than ω n−1 and that the nontrivial ω n−1 order contributions do not contain any 2nd and higher order building blocks.
In the rest of this part, we will prove Fact 1 and Fact 2 case by case. Before doing so, we remind the readers that χ m denotes all possible products of ( R 0)'s and ( R 1)'s containing exactly m-number of ( R 0)'s wedged with an arbitrary number of ( R 1)'s (see Appendix B.1.1). We sometimes use χ m to simply denote an element in χ m . A useful symbol (R q (p) ) (L.8) is defined to denote all possible structures (including those consist of zeroth and first order building blocks only) that can contribute to (R q ) at ω p order. Here is a summary of the results from Appendix B.2 regarding classifications of (R q (p) ): We note that, when Einstein sources were evaluated in Appendix D.6 of Ref. [27], it was shown that that 2nd and higher order terms in R do not contribute to (R q ) at order ω q−1 (and lower). This is why 2nd and higher order terms in R do not appear in (R q (q−1) ). Furthermore, up to ω q−2 , all contributions (including zeroth and first order building blocks) to R q vanish.
L.3.1 Single-trace case 1: P CF T = c g tr[R 2k ] We begin by first dealing with the case of P CF T = c g tr[R 2k ] for k ≥ 2 in AdS 2n+1 with n = 2k − 1. In this case, T 0 and T 2 are trivially zero and thus we consider T 1 only.
Single-product terms: These two terms are in fact zero due to for arbitrary ( R m) (m ≥ 0). Thus, for the single-product, we proved Fact 1 and Fact 2 .
Double-product terms: The contributions of this type up to ω n−1 are : where we have used Eq. (B.41) andχ 1 is defined in Eq. (B.42). First, the ω n−2 order terms all vanish as a result of for arbitrary ( R m) with m ≥ 0. Thus, Fact 1 is confirmed. For the ω n−1 order contributions, one can show that all ω n−1 order contributions vanish except for the case containing ( (0) δΓ), ( (0) ∇ξ), one ( R 0), one ( R 2) (and the rest of R's are all ( R 1)'s). To show that, we use Eq. (L.15) and the identities 16) for any m ≥ 0 and any δΓ of any order. We could prove such identities by making use of the first Bianchi identity for ( R m) and the fact that δΓ a [bc] = 0 together with the following : Now we include a U (1) gauge field and consider the most general anomaly polynomial in the single-trace form, P CF T = c M F l ∧ tr[R 2k ], admitted in AdS 2n+1 with n = 2k + l − 1. In this case, the gauge field part F l at the lowest order is ( F 1 l ) and is of order ω l . Combining with the result for the previous case, for T 1 , we confirmed Fact 1 and Fact 2.
For T 2 , we encounter the term of the form From Appendix. B.2.3, we know that tr[δΓR 2k−1 ] starts at ω 2k−1 and consists of purely zeroth and first order building blocks. Hence, the lowest order contribution to this term is of order ω l−1+2k−1 = ω n−1 , which does not contain any 2nd and higher order terms in the building blocks Finally, let us consider T 0 . It contains the following two types of terms : (L. 20) We note that the gauge field parts at the leading order are (0) δA (0) (Λ + i ξ A) ∧ ( F 1 l−2 ) (thus ω l−2 -order) and (0) (Λ + i ξ A) ∧ ( F 1 l−1 ) (thus ω l−1 -order) only. From the Maxwell sources computations (see Appendix C.3 of Ref. [27]) and Appendix B.2.3, the terms tr[R 2k ] and tr[(δΓ)R 2k−1 ] starts at ω 2k and ω 2k−1 order respectively, and these leading order terms do not contain 2nd and higher order building blocks. Therefore, the leading order contribution to T 0 is of order ω n−1 and does not contain any 2nd and higher order terms of the building blocks.
To summarize, we confirmed Fact 1 and Fact 2. Let us begin with the purely gravitational anomaly case. Since the second order and higher terms in the building blocks do not contribute to tr[R 2q ] at ω 2q order, without loss of generality, we only need to consider the two-trace case, P CF T = c g tr[R 2k 1 ] ∧ tr[R 2k 2 ] (in AdS 2n+1 where n = 2k 1 + 2k 2 − 1 and k 1 , k 2 ≥ 1). Depending on how the two R-derivatives act on this anomaly polynomial, there are two type of contribution: single-trace case and double-trace case. For the former, by noticing again the fact that the leading order contribution to tr[R 2k ] is ω 2k and does not contain any 2nd order building blocks, we can deal with this type of term in the same way as the single-trace anomaly polynomial case. We therefore concentrate on the double-trace terms appearing in T 1 : We note that there is another case where k 1 and k 2 are interchanged, but since we can treat it in the same way, we consider the above case only. The rest of the arguments follow along the same line of arguments as the AdS 15 case in Appendix. L.2. As we have summarized, the traces in this expression start at ω 2k 1 −1 and ω 2k 2 −1 order, respectively, and these leading order contributions do not contain any 2nd order building blocks. More precisely, they are of the form To summarize, we have proven even stronger statements than Fact 1 and Fact 2 that we used in the single-trace case for the terms in Eq. (L.21). That is, here we have shown that for such type of terms, the first non-trivial contributions start at ω n−1 and that 2nd (or higher) order building blocks do not contribute at order ω n−1 for any fixed r-surface. As in the previous case, we only need to consider the double-trace case, P CF T = c M F l ∧ tr[R 2k 1 ] ∧ tr[R 2k 2 ] (in AdS 2n+1 where n = 2k 1 + 2k 2 + l − 1, and k 1 , k 2 , l ≥ 1), and more general cases follow from this immediately.
Let us start with the first term T 1 . We recall that the lowest order contribution from F l is ( F 1) l only and thus is at order ω l . Then the above results for the purely gravitational anomaly polynomial cases (double-trace and single-trace cases) are enough to prove Fact 1 and Fact 2.
For T 0 and T 2 , since they contain only single R-derivative, the proof of Fact 1 and Fact 2 essentially follows from the one for the case with the mixed single-trace anomaly polynomial we have investigated above (by recalling that the leading order term of tr[R 2k ] is tr[( R 1 2k )] only and is of order ω 2k ).
To summarize, we have confirmed Fact 1 and Fact 2 for general anomaly polynomials. We therefore see that in T 0 and T 2 the second and higher order building blocks do not contribute up to order ω n−1 for any fixed r-surface. For T 1 , we have a weaker statement that these contributions do not contribute at r → ∞.

L.4 Weyl covariance in AdS 2n+1 /CFT 2n
In this paper, since we are studying AdS 2n+1 with a CFT 2n dual, it is useful to understand how fields in the boundary CFT 2n transform under the Weyl rescaling. We follow closely the discussions in Ref. [59][60][61]. We first review the Weyl scaling of boundary fields and then extend it to bulk fields. We then apply this analysis to estimate the fall-off of the higher-order terms in the curvature two-forms at the boundary.
behaviors of the curvature two-form at ω m order is estimated as In particular, for m = 2, this gives the estimate (L.38) We see that this reproduces all the fall-offs in Eq. (B.97) except that we can improve the estimate of the fall-off for ( R 2) µ r by one power of 1/r. We note that in the above estimates, when we construct the two-forms appearing in a particular component of ( R m), we have started with the boundary two-form discussed above and then compensated the Weyl weight (difference between the Weyl weights of a particular component of ( R m) and the boundary two-form) by just multiplying an appropriate power of r. This provides the estimates when q = m = 0 only. With q, m = 0, we are also allowed to use Φ T and Φ (both are with the Weyl weight +1) to compensate the Weyl weight. Since both Φ T and Φ have faster fall-offs than r, they will not affect the modest fall-off estimates above.

M Entropy : Einstein-Maxwell contribution
This Appendix is devoted to the evaluation of the Einstein-Maxwell contribution to the black hole entropy. The Einstein-Maxwell contribution to the differential Noether charge is given in Eq. (2.8) of § § §2.3. We first evaluate the Einstein part (the first line of Eq. (2.8)) to confirm that this part reproduces the non-anomalous CFT result. After this, we will confirm that the Maxwell part (the second line of Eq. (2.8)) does not give any nontrivial contribution at the horizon. In the evaluation, we use the two prescriptions for the differential Noether charge introduced in § §E.3 and confirm the matching of the final result explicitly.

M.1 Entropy current from Komar charge
We first evaluate the Komar charge at the horizon by taking ξ µ = u µ /T , the appropriately normalized Horizon generator. We note that this is the Killing vector which preserves the horizon. To the order we are working in, the horizon is given by the surface r = r H with f (r H ) = 0 and 4πT = r 2 H f ′ (r H ). We obtain from the Komar part that We note that this is essentially the prescription to compute CFT entropy current given in Eq. (2.3).