The Wald entropy and 6d conformal anomaly

We analyze the Wald entropy for different forms of the conformal anomaly in six dimensions. In particular we focus on the anomaly which arises in a holographic calculation of Henningson and Skenderis. The various presentations of the anomaly differ by some total derivative terms. We calculate the corresponding Wald entropy for surfaces which do not have an Abelian $O(2)$ symmetry in the transverse direction although the extrinsic curvature vanishes. We demonstrate that for this class of surfaces the Wald entropy is different for different forms of the conformal anomaly. The difference is due to the total derivative terms which are present in the anomaly. We analyze the conformal invariance of the Wald entropy for the holographic conformal anomaly and demonstrate that the violation of the invariance is due to the contributions of the total derivative terms in the anomaly. Finally, we make more precise general form of the Hung-Myers-Smolkin discrepancy.


Introduction
More than twenty years ago Wald introduced a very efficient prescription by which the gravitational entropy can be computed for a given gravitational action [1], [2] (see also [3]). The typical gravitational action in question is polynomial in the Riemann curvature and its covariant derivatives. This prescription associates the entropy with a horizon, a co-dimension two surface with very peculiar properties. In particular, it is assumed that there exists a time-like Killing vector which becomes null on this surface. This vector generates an Abelian symmetry in the transverse direction to the surface. As a consequence, the extrinsic curvature of the surface is vanishing.
In a related concept of the entanglement entropy one associates some entropy with arbitrary co-dimension two surface, not necessarily a horizon. The presence of the Abelian symmetry is thus not guaranteed in this case. Nevertheless, there appears to be a certain algorithm [7], [8], [9], [10] to associate some entropy with each individual term, dependent on the curvature, in the quantum effective action. This algorithm certainly deviates from the one proposed by Wald if the surface in question is characterized by a non-trivial extrinsic curvature. However, what happens if the extrinsic curvature is vanishing while the Abelian symmetry is not present? This would be the situation when the Wald algorithm for the entropy might be thought to still work. This was tested by Hung, Myers and Smolkin (HMS) in [4] for a certain class of sixdimensional geometries with the properties as above where they compared the Wald entropy due to the 6d conformal anomaly with the holographic entropy computed by using the holographic prescription by Ruy and Takanayagi [6]. They have found a discrepancy between these two calculations. In the recent works [5] we have made some attempts to explain this discrepancy, for the alternative attempts see [14].
In the present note we deliberately ignore the problem of finding an explanation for the HMS discrepancy. Instead, we make a step back and ask the question which should have been answered first: what is the general form for the discrepancy? The answer to this question is not that trivial as it would seem to be. The reason is as following. In [4] it was considered not the most general form of the conformal anomaly and certainly not the one which actually appears in a holographic calculation as, for instance, in the paper of Henningson and Skenderis [11]. The difference between the two possible forms is due to the total derivatives. In this note we show that what the Wald entropy is concerned these total derivatives are important and can not be disregarded.
This note is organized as follows. In section 2 we present the general form for the trace anomaly in six dimensions. This general form includes the Euler number, three conformal invariants and a set of total derivative terms. In section 2.2 the holographic conformal anomaly calculated by Henningson and Skenderis is presented in this general form and we give the values of the respective conformal charges as well as the charges that correspond to the total derivatives. In section 3 we give the formulas for the Wald entropy due to the 6d conformal anomaly, both in a general form and for the holographic conformal anomaly. In section 4 we consider the HMS geometries and surfaces and compute the Wald entropy for the holographic anomaly presented in two forms, the one in which it appears in the holographic calculation and the other which involves the representation in terms of the conformal invariants, and find a mismatch. In section 5 we explain the mismatch by taking into account the contribution of the total derivatives. The Wald entropy of the latter is non-vanishing in general. In section 6 we discuss the conformal invariance of the corresponding entropy. In section 7 we give a general form for the HMS discrepancy. We conclude in section 8.

2
Conformal anomaly in six dimensions

General form for the anomaly
In a generic conformal field theory in d = 6 the trace anomaly, modulo the total derivatives, is a combination of four different terms where E 6 is the Euler density in d = 6 and, using notations of [12], we have where W αβµν is the Weyl tensor. In (1) the part A C is due to the total derivatives where and the tensor B is defined as Neglecting the contribution of the total derivative terms C k , we give a name to the conformal anomaly (1), the BI -form.

Holographic conformal anomaly
In this paper, we are mostly interested in the holographic conformal anomaly computed in [11]. It is derived by a standard holographic procedure from the 7-dimensional gravitational action This anomaly takes the form It can be represented in the form (1) as follows So that we find for the charges b k and a, in agreement with [4] and [13]. We also find that and all other d k vanish.

General formula
Suppose that the gravitational action W is a function of the Riemann curvature and its covariant derivatives (and contains two derivatives at most). Then the corresponding Wald entropy is computed according to the formula [2] where ǫ µν = n a µ n b ν ǫ ab , n a µ , a = 1, 2 is a pair vectors normal to surface Σ, so that we have that We notice the symmetrization in the last term in (11). It will be important when we discuss the Wald entropy due to the total derivatives.

Entropy due to the holographic conformal anomaly
Using the above definition we compute the corresponding Wald entropy. In the case of the holographic conformal anomaly (7) we find where the indexes a and b correspond to projections on the normal vectors n µ a , a = 1, 2 and we have introduced the notations ∇ a ≡ n µ a ∇ µ , R aa ≡ n µ a n ν a R µν .

Entropy due to conformal anomaly in the BI -form
On the other hand, for a generic anomaly in the form A BI (1) we have that as was first derived in [4]. Notice that the contribution of the total derivative terms C k is neglected in (14). This contribution is indeed supposed to vanish when considered in the standard situation of a Killing horizon.  [4] have considered the following six-dimensional geometries and the four-dimensional surfaces: All these geometries are characterized by same properties: i) they are products of constant curvature spaces; that is why, in the Wald entropy (13) or (15) all terms with covariant derivatives vanish; ii) the time coordinate x 1 = t lies in the flat sub-space, so that components of the Riemann tensor R µναβ or the Ricci tensor R µν vanish if one of the indexes is 1; iii) the surface Σ has two normal vectors, one of which, n 1 , is time-like. The corresponding extrinsic curvature vanishes because it is the Killing vector. The surface Σ has a component which is the minimal surface embedded in a sphere. The corresponding normal vector, n 2 , lies along this sphere. The corresponding extrinsic curvature vanishes since it is the minimal surface; vi) in particular, we have that R 2 aa = R 2 ab and R abcb = 0 for all these geometries; v) in all these cases the Abelian O(2) symmetry in the transverse direction to Σ is absent although the components of the extrinsic curvature of Σ vanish.

Entropy due to holographic conformal anomaly
First, we compute the entropy (13) due to the holographic conformal anomaly: c). S h W = 27π 400 where V Σ is volume of Σ.

Entropy due to holographic conformal anomaly in BI -form
Now we compute the Wald entropy for the same holographic anomaly but represented in the BI form (8). We find a). S h BI = π 400 These results for the Wald entropy agree with the entropy computed in [4] when we choose the particular values (9) for the conformal charges b k .

The mismatch
Clearly, we have a mismatch, ∆S h = S h W − S h BI , between these two calculations: a ′ ). ∆S h = π 24 5 Contribution of the total derivative terms

Some general formulas
Trying to understand the mismatch which we have just observed we have to consider carefully the total derivative terms which are in general present in the conformal anomaly. As wee see from (8) only terms C k , k = 3, 4, 5, 6, 7 contribute to the holographic conformal anomaly. Before using the Wald formula (11) and computing the entropy it is required to first transform each C k to a form which would contain terms symmetric in second covariant derivatives of the curvature. The terms C 3 and C 5 already take this form. The corresponding Wald entropy calculated using (11) gives zero entropy, For the other terms the situation is less trivial. Commuting the covariant derivatives one finds where I 1 and I 2 are defined in (12) and we have introduced Now we apply the general formula (11) and compute the Wald entropy, where we define R abab = n µ a n ν a n α b n β b R µανβ and R aa = n µ a n ν a R µν , and the terms s 1 and s 2 have been defined in (15). We used the Bianchi identities and, in particular, that when derived (31). This identity, in particular, shows that the right hand side of (32), and its projections on the transverse subspace, vanishes for a product of constant curvature spaces. We then use that if the extrinsic curvature of surface Σ vanishes we have the following relation where ∆ Σ is the Laplace operator defined on surface Σ and R Σ is the intrinsic curvature of Σ. The integration of (33) over a closed surface Σ gives zero and the expressions (31) are simplified We note that these formulas are valid for arbitrary geometry and any entangling surface for which the extrinsic curvature vanishes. If the geometry in question is a product of constant curvature spaces, as in the examples considered in section 4.1, then the terms with derivatives in (34) vanish. Each entropy in (34) is a priori non-vanishing and we shall demonstrate this below for the concrete examples. Accidentally, we observe that the Wald entropy for invariants C 4 and C 6 is the same, up to a sign.

Explaining the mismatch
Now let us compute the contribution of these to total derivative terms to the entropy, for various geometries discussed before. Our aim is to demonstrate that The values for the charges d k are given in (10).
We find in the following cases: and hence and hence and hence and hence We see that in all these cases we obtain a complete agreement with (24)-(27) and confirm (35). The mismatch thus is indeed due to the contributions of the total derivative terms which appear in the conformal anomaly when we pass from the form (7) to the BI -form (8).

Conformal invariance
In this section we would like to check whether the Wald entropy in question is conformal invariant. We restrict ourselves to the infinitesimal conformal transformations, δ σ g µν = g µν σ .
Then we find in six dimensions that First, we analyze the conformal invariance of the Wald entropy (13) due to the holographic conformal anomaly. In order to simplify the transformations we consider the case of a geometry with constant curvature, i.e. the condition ∇ ρ R αβµν = 0 is assumed to be valid. Then we find for the variations of the quantities, integrated over the surface Σ, Since the extrinsic curvature of Σ vanishes we have σ = ∇ 2 a σ + ∆ Σ σ . Finally, we use the commutation relation where ∇ a = n µ a ∇ µ , a = 1, 2 are derivatives in the transverse sub-space and we neglected all terms with derivatives along the surface Σ since, after integration over Σ, these terms give zero. Putting everything together we find for the conformal transformation of the Wald entropy (13) due to the holographic conformal anomaly. This conformal transformation vanishes in many particular cases. For instance, it vanishes if the six-dimensional space-time is Einstein, i.e. R µν = Λg µν . In the presence of O(2) symmetry in the transverse sub-space we have that R ab = λδ ab , a = 1, 2 and the tensor R ab − 1 2 R cc δ ab = 0 so that (47) vanishes, as expected. However, for the geometries considered in section 4.1 we have that R 11 = 0 and R 22 = 0. In this case the right hand side of (47) is non-zero. It is proportional to R 22 (∇ 2 2 − ∇ 2 1 )σ . This quantity is non-vanishing if σ is a generic function of coordinates x 1 and x 2 orthogonal to surface Σ.
The terms (15) in the Wald entropy that are due to the conformal invariants I 1 , I 2 and I 3 are conformal invariant by construction. That the total Wald entropy (13) is not conformal invariant is due to the presence of the terms (31) which correspond to the total derivative terms. Indeed, we find for the conformal transformation of these terms where in the last line we used (45) and that W acbc = R acbc + δ ab ( 1 20 R − 1 4 R cc ). Thus, to this order the entropy S (C 7 ) W is conformal invariant while the non-invariance of the Wald entropy (13) is entirely due to the terms S

The HMS discrepancy for a general 6d anomaly
The discrepancy discovered in [4] is a mismatch between the holographic calculation of the universal term in the entropy and the Wald entropy computed for the holographic conformal anomaly. However, in [4] it was considered only the BI -form of the conformal anomaly and the total derivative terms were neglected. As we explained in this note the actual holographic anomaly differs from the BI -form by certain total derivative terms. For the geometries considered in [4] the Wald entropy of these total derivatives is non-vanishing, as we have just demonstrated. Thus, the finding of [4] should be completed by adding the corresponding contributions of the total derivative terms.
For an arbitrary conformal anomaly taking the form (1) with the conformal charges b k and the charges d k the discrepancy takes the following form where in the first line we used the result of [4]. This is a general form for the discrepancy. For the geometries more general than those considered in section 4.1 there may appear some terms with derivatives of the curvature in the first line of (49). The analysis of [4] does not allow us to identify those terms. We also notice that d 4 − d 6 = b 2 for the holographic anomaly obtained from the d = 7 Einstein action.

Conclusions
In this note we have analyzed the general form for the conformal anomaly in six dimensions and the corresponding Wald entropy. We have demonstrated that while the Wald entropy is concerned the total derivative terms which generically appear in the conformal anomaly can not be neglected. Their contribution is essential for the consistency when the entropy of the holographic conformal anomaly is compared to the entropy computed for the anomaly expressed in terms of the conformal invariants. The other important role of the total derivative terms is that their presence in the anomaly breaks the conformal invariance of the corresponding Wald entropy. Finally, we use our findings and give a general form for the discrepancy found in [4].