One-Loop Holographic Weyl Anomaly in Six Dimensions

We compute $\mathcal O(1)$ corrections to the holographic Weyl anomaly for six-dimensional $\mathcal N=(1,0)$ and $(2,0)$ theories using the functional Schr\"odinger method that is conjectured to work for supersymmetric theories on Ricci-flat backgrounds. We show that these corrections vanish for long representations of the $\mathcal N=(1,0)$ theory, and we obtain an expression for $\delta(c-a)$ for short representations with maximum spin two. We also confirm that the one-loop corrections to the $\mathcal N=(2,0)$ M5-brane theory are equal and opposite to the anomaly for the free tensor multiplet. Finally, we discuss the possibility of extending the results to encompass multiplets with spins greater than two.


I. INTRODUCTION
The trace anomalies for conformal field theories provide an important set of quantities that characterize the theory. In two dimensions, Cardy's formula [1] demonstrates that the central charge c is a reliable measure of the degrees of freedom. Furthermore, its physical implication can be seen from the Zamolodchikov c-theorem [2] which states that an effective c function can be defined that is monotonically decreasing along renormalization group flows to the infrared. While the picture is perhaps the clearest in two dimensions, recent work extending these results to higher dimensional CFTs has further emphasized the importance of trace anomalies in more general situations.
The AdS/CFT correspondence provides an ideal framework for investigating various anomalies, as they may often be reliably computed on both sides of the strong/weak coupling duality. Such calculations can provide a test of the AdS/CFT correspondence and can also yield additional insights on strongly coupled CFTs. Here we focus on the conformal anomaly, which measures the change in the partition function that results from a Weyl scaling of the metric, δg µν = 2δσg µν . In particular, for a partition function given by we define the anomaly A by From the holographic point of view, the leading order Weyl anomaly can be obtained from the regularized classical action [3]. In the AdS 5 /CFT 4 case, this gives the familiar result where IIB supergravity has been compactified on AdS 5 × Σ 5 . Additional corrections to the leading order expression may arise from higher derivative modifications to the supergravity action as well as from quantum (i.e. loop) effects.
Holographically, the log-divergent part of the one-loop effective action provides an O (1) correction to the Weyl anomaly coefficients. This was initially computed for the case of AdS 5 ×S 5 in [4][5][6][7][8], where it was observed that the leading order result (3) is shifted according to N 2 → N 2 −1, in agreement with expectations for SU(N) gauge symmetry. More recently, the one-loop computation in AdS 5 has been extended to holographic field theories with reduced or even no supersymmetry [9][10][11][12].
The one-loop holographic computation is essentially a sum over contributions from all states in the spectrum of single-trace operators. Curiously, when arranged in terms of fourdimensional N = 1 superconformal multiplets, the contribution from long multiplets vanish identically. As a result, only short representations contribute to the O(1) shift in a and c.
This allows for a close connection between the central charges and the superconformal index which encodes knowledge of the shortened spectrum [13,14] (see also [15]).

A. The six-dimensional Weyl anomaly
Here we wish to extend some of the holographic results for the Weyl anomaly coefficients to six dimensions. In general, the anomaly takes the form where E 6 refers to the Euler density, I i are Weyl invariants and the final term is a nonuniversal total derivative. At leading order, Einstein gravity on AdS 7 gives a relation of the form [3] which is the six-dimensional analog of (3). The relation between the c i coefficients arises naturally in the holographic computation, and is further consistent with six-dimensional (2, 0) superconformal invariance.
The most extensively studied (2, 0) theory of relevance is that of N coincident M5-branes, which is dual to supergravity on AdS 7 × S 4 . Here the conjectured expression for the central charges are [16][17][18] where c 1 = 4c 2 = −12c 3 = 96c. information is provided about the c i coefficients.
An alternative approach to the computation of δa and δc was developed in [5][6][7][8] based on a functional Schrödinger approach. In this approach, the contribution of each state to the O(1) shift in the Weyl anomaly takes the form where ∆ is the conformal dimension and b d is the heat kernel coefficient for the corresponding AdS d+1 field when restricted to the d-dimensional boundary. In principle, since the sixdimensional b 6 coefficient may be computed on a general curved background, this allows for a full determination of not just the a coefficient but the c i 's as well.
It has been argued in [12], however, that the expression (7) cannot in general be valid, as the contribution for a single field should have a more complicated dependence on the conformal dimension ∆. This can be seen explicitly in comparison with the expression for δa obtained directly from the one-loop determinant on global AdS. Curiously, however, when (7) is summed over the states of a complete supermultiplet, the resulting expression appears to be valid on Ricci-flat backgrounds as it passes all consistency checks and has the expected connection to the index [12,13].
In this paper, we use (7) to compute the O(1) contribution to the holographic Weyl anomaly of N = (1, 0) theories from maximum spin-2 multiplets in the bulk. Since we consider Ricci-flat backgrounds, we only obtain information on δ(c−a), where c is some linear combination of the Weyl coefficients (and reduces to the c defined above in the N = (2, 0) case). This is similar to the AdS 5 /CFT 4 case, where b 4 ∼ δ(c − a)R 2 µνρσ on Ricci-flat backgrounds. As a consistency check, we find that δ(c − a) vanishes for long representations of N = (1, 0) supersymmetry, as expected.
While we would ideally want an expression for the anomaly contribution from arbitrary higher-spin multiplets, this would require a better understanding of the heat kernel coefficients b 6 of higher-spin operators. This was worked out in [19,20] for general spins in four dimensions. However, a similar expression is lacking in six dimensions. Nevertheless, knowledge of δ(c − a) for spins up to two is sufficient for computing the holographic Weyl anomaly for 11-dimensional supergravity on AdS 7 × S 4 . In this case, we sum the expression for δ(c − a) over the Kaluza-Klein spectrum and find the expected result δ(c − a) = −1/384, in agreement with (6) and the original computation of [17].

II. THE O(1) CONTRIBUTION TO THE HOLOGRAPHIC WEYL ANOMALY
As indicated above, the six-dimensional Weyl anomaly may be parameterized as where E 6 = ǫ 6 ǫ 6 RRR is the six-dimensional Euler density, and the Weyl invariants are given by Superconformal invariance imposes additional constraints on the anomaly coefficients {a, c i }.
In particular, N = (1, 0) supersymmetry requires c 1 − 2c 2 + 6c 3 = 0, while N = (2, 0) supersymmetry gives an additional constraint c 1 −4c 2 = 0. This suggests the parametrization or equivalently This is designed so that c ′′ vanishes for superconformal theories, and additionally c ′ vanishes when there is extended N = (2, 0) supersymmetry. The coefficient c is chosen to allow for a quantity analogous to c − a in four dimensions, as will become clear below.
The procedure we use to obtain the O(1) shift in the anomaly for N = (1, 0) theories is to sum the expression (7) over complete representations of the corresponding OSp(8 * |2) supergroup. However, we first start with states in the bosonic subgroup OSp(8 * |2) ⊃ SO(2) × SU(4) × SU(2) R labeled by D(∆, j 1 , j 2 , j 3 ) along with R-symmetry representation r. We thus have In the following, we first work out the heat kernel coefficients b 6 (j 1 , j 2 , j 3 ) on a Ricci-flat background, and then perform the sum over complete supermultiplets with maximum spin two.
For an operator ∆ = −∇ 2 − E where E is some endomorphism, the six-dimensional Seeley-DeWitt coefficient b 6 (∆) takes the form [21,22] b 6 (∆) = 1 (4π) 3 Here the A a 's form a basis of curvature invariants [22,23], and the V a 's are built from the endomorphism E and the curvature F ij of the connection [22]. In particular, while the coefficients of the A a 's are universal, the V a terms are specific to the representation.
We follow the conventions spelled out in Appendix A of [22], which also give explicit expressions for the A a 's and V a 's. However, we are concerned with only the combinations which are non-vanishing on Ricci-flat backgrounds. These are The full list of A a 's, and expressions for the V a 's are given in Appendix A.
The invariants E 6 and I 1 , I 2 , and I 3 may be written in terms of the basis A a functions.
On a Ricci-flat background, they become As these quantities are not all independent, we will be unable to determine the individual central charges {a, c i } using only a Ricci-flat background. Note that we may construct two combinations that are total derivatives This allows us to rewrite (15) in terms of the two invariants A 16 and A 17 On a Ricci-flat background, we have the relations E 6 = 32(2I 1 + I 2 ) and I 3 = 4I 1 − I 2 up to a total derivative. As a result, the six-dimensional anomaly, (8), takes the form on Ricci-flat backgrounds. The implication of this expression is that we will only be able to obtain information on the O(1) contribution to c − a and to c ′′ . Since the latter must vanish for superconformal theories, it will serve as a consistency check of our approach. This Ideally, we would like to have an expression for the heat kernel coefficient b 6 (∆) for fields transforming in an arbitrary (j 1 , j 2 , j 3 ) representation of the six-dimensional SU (4) Euclidean rotation group. However, this requires understanding of arbitrary higher-spin Laplacians which currently eludes us. There is also some potential ambiguity in relating 'on-shell' states in AdS 7 to their corresponding boundary Laplacians in the functional Schrodinger approach of [5]. We thus restrict to spins up to two. The relevant b 6 coefficients evaluated on a Ricci-flat background are summarized in Table I. The coefficients for φ, ψ, A µ and B µν were computed in [22], while the remaining ones are worked out in Appendix A.

sentations, given generically by
For maximum spin two, however, we must restrict to j 1 = j 2 = j 3 = 0. In this case, it is a simple exercise to perform the sum (12) over the multiplet using the values of γ 16 and γ 17 given in Table I. Comparison with (18) then allows us to extract δ(c − a) and δc ′′ . The results are summarized in Table II.
As a consistency check, we note that the anomaly coefficient c ′′ vanishes identically after summation over a complete multiplet. This is a requirement of supersymmetry, but is not manifest from the individual b 6 coefficients in Table I. We also see that the anomaly vanishes for the long representation, which agrees with expectations from the AdS 5 case [9,12]. As for the non-vanishing contributions, note that δ(c − a) for the A and D type multiplets are equal and opposite. This must be the case, as A[0, 0, 0; k] and D[0, 0, 0; k + 2] are "mirror shorts" that sum to become a long multiplet.
Finally, recall that the N = (1, 0) theory admits three independent anomaly coefficients, which we have parametrized as a, c and c ′ . Since we only consider Ricci-flat backgrounds, Weyl anomaly coefficients δ(c − a) and δc ′′ . Here k is the SU (2) R Dynkin label (with spin = k/2).
The shortening conditions correspond to those of (19), while the last column is the maximum spin-two long representation.
we have only been able to determine the difference δ(c − a). This may be combined with the holographic δa coefficient obtained in [18] to separate out the contributions to δa and δc. These results are presented in Table III. However, we are unable to determine δc ′ unless we can move away from Ricci-flat backgrounds.
Contribution to the Weyl anomaly coefficients δa and δc from maximum spin two multiplets for the N = (1, 0) theory. Here c is related to the conventional anomaly coefficients c i according to (11). The δa coefficient is computed using the results of [18].
We may perform the same computation for the N = (2, 0) theory, noting however that only the 1/2-BPS multiplets have spins less than or equal to two. In this case, the supercon- The shortening conditions follow the same pattern as (19), however with extended R- Here (k 1 , k 2 ) are Dynkin labels for Sp(4), with (1, 0) denoting the 4 and (0, 1) denoting the 5. For maximum spin two, we restrict to the 1/2-BPS multiplets D[0, 0, 0; 0, k] with ∆ = 2k.
(The case k = 1 is the free tensor multiplet, while k = 2 is the stress tensor multiplet.) The holographic computation of δ(c − a) and δc ′′ for the D[0, 0, 0; 0, k] multiplets are shown in Table IV. The case k ≥ 4 is generic, and we do not include k = 1, which is a supersingleton and would not appear in a holographic computation. The special case k = 3 fits into the generic pattern, and in fact so does k = 2, although it requires separate treatment because of the presence of massless modes. For k = 2, the states in where D(∆; j 1 , j 2 , j 3 ) labels the SO(2, 6) representation and the subscript labels the Sp (4) (Note that the three-form, D(5; 2, 0, 0), is massive, so no subtraction is required.) Taking these null states into account then gives the result δ(c − a) = 13/384 for k = 2 shown in Table IV.
Although k = 2 and k = 3 are special cases, the holographic anomaly coefficient δ(c−a) = (1/384)(6k(k−1)+1) is in fact universal. Combining this with δa = −(7/1152)(6k(k−1)+1) obtained in [18] then allows us to separate out the individual coefficients As an application, consider the N = (2, 0) theory obtained by compactifying 11dimensional supergravity on AdS 7 × S 4 . The Kaluza-Klein spectrum is simply where k = 2 corresponds to the 'massless' supergravity sector. The anomaly coefficients δa and δc may be computed by summing over the Kaluza-Klein levels Following [18], we regulate the sums using a hard cutoff. This amounts to setting ∞ k=1 k n = 0 for any n ≥ 0. This As a result, the regulated anomaly for AdS 7 × S 4 is This is equal and opposite to the result for the conformal anomaly of the free tensor multiplet computed in [16], and agrees with the O(1) contributions in (6).

III. DISCUSSION
While the six-dimensional Weyl anomaly coefficients are conventionally parametrized as a, c 1 , c 2 and c 3 , we have found it convenient to use an alternate linear combination of the c i 's given in (11). Holographically, the leading order anomaly coefficients (assuming Einstein gravity in the bulk) satisfy the relation (While c ′′ = 0 must hold for superconformal theories, this holographic result is independent of whether the theory is supersymmetric or not.) At the one-loop level, we have been able to extend this leading order result by computing δ(c − a) using the expression (7) obtained through a functional Schrödinger method [5][6][7][8].
It is reasonable to question whether the use of (7) is valid, as it disagrees with the direct computation of δa performed in [12,18]. A quick way to see this is to note that δa in Table III is a fourth order polynomial in ∆, while the result of summing (7) over a supermultiplet can be at most quadratic in ∆. (One power comes directly from (7), while another can arise from the dimension of the shortened representation.) If δ(c − a) was expected to be cubic or higher in ∆, then our result, as shown in the last column of Table III, cannot possibly be correct. However, we now demonstrate that c − a can be at most linear in ∆, which is consistent with application of (7).
To see this, recall that, in superconformal field theories, the stress tensor is contained in a multiplet of currents, so that there is a corresponding multiplet of anomalies. For N = (1, 0) theory, the 't Hooft anomalies are characterized by the anomaly polynomial and the relation to the Weyl anomaly coefficients has recently been worked out [29][30][31][32] Since α is the coefficient of the [SU(2) R ] 4 anomaly, it can be at most fifth power in ∆, where the extra power comes from the dimension of the representation. Similarly, β can be at most cubic in ∆, while γ and δ can be at most linear in ∆. This in turn demonstrates that a will be at most fifth power in ∆, c ′ will be at most cubic and c − a will be at most linear. Thus the functional Schrödinger method is indeed compatible with δ(c − a). However, we also see this approach cannot be used to compute either δa alone or δc ′ . Thus, while it would be desirable to compute δc ′ in these theories, we do not expect that it can be done using this approach.
While we have focused on short multiplets with spins ≤ 2, it would be desirable to work more generally with higher-spin multiplets. To do so, we would need knowledge of the b 6 coefficients for arbitrary spin fields. This in turn depends on the form of the higher-spin Laplacian. In general, this depends on the bulk dynamics of the higher-spin field and the further restriction to the boundary following from the procedure of [5][6][7][8]. The Seeley-DeWitt coefficients b n (∆) depend on the field and the form of the second order operator ∆. In four dimensions, the appropriate operators for irreducible fields up to spin two are listed in [20]. Here we write down the analogous operators in six dimensions and compute the contribution of each to the anomaly.
We start with the basis of curvature invariants [22,23] The b 6 coefficient may be computed from the expression (13), where the V a 's are given by Here ∆ = −∇ 2 − E and F ij is the curvature of the connection, [∇ i , ∇ i ] = F ij .

Conformally Coupled Scalar
The conformally coupled scalar has E = − 1 5 R and F ij = 0, so the V -terms are:

Weyl Fermion
The appropriate second order operator for the Dirac fermion may be obtained as the square of the Dirac operator: The endomorphism and curvature of the connection coincide with the result obtained in [22].
Then the V -terms contributing to the anomaly are: The b 6 coefficient is

Vector
The (0, 1, 0) vector representation of SU(4) is a one form, so the correct Laplacian may be obtained by computing the Hodge-deRham operator dδ + δd. We get The endomorphism and curvature of the connection here are: so that (A9)

Self-Dual Three-Form
The field which transforms under the (2, 0, 0) representation is the 10-component selfdual three-form. A three-index antisymmetric tensor has 20 components and the self-duality condition removes half of these. The operator acting on this field is This means that the endomorphism and connection curvature are given by Then we can compute the relevant terms: The self-duality condition reduces each of these terms by a factor of two, reproducing the A 16 and A 17 terms found in table I.

Gravitino
The gravitino with the gauge condition γ µ ψ µ = 0 corresponds to the (1, 1, 0) representation. In this case the operator O is the square of the Rarita-Schwinger operator: The endomorphism and connection curvature are given by so
This means that the endomorphism and connection curvature are given by so (A18)

Graviton
The symmetric spin-two field is the (0, 2, 0) representation. The appropriate kinetic operator is the Lichnerowicz operator [33]: The endomorphism and connection are given by Then we can compute the relevant terms: Appendix B: Heat kernel for general spins We are interested in a general formula to compute the heat kernel coefficients for spins higher than two, analogous to the algorithm [20] in four dimensions. We consider fields transforming in an irreducible representation of the spacetime symmetry group that are acted on by a generalized second-order operator ∆ = − − E. In four dimensions, the method of computing the heat kernels for general representations assumes that the endomorphism term E for fields transforming as (A, B) of SO(4) ≃ SU(2) L × SU(2) R is given by: for bosonic (A + B = integer) or fermionic (A + B = half-integer, A > B) representations, respectively. Here R abcd + = 1 2 (R abcd + R * abcd ). This prescription is shown to be valid for fields up to spin two in four dimensions, and is conjectured to be the appropriate operator for general spins. In six dimensions, it appears that this prescription is reasonable for bosonic representations, but straightforward generalizations for fermions fail to reproduce the conventional endomorphism terms for the Weyl fermion and gravitino. So it remains unclear what endomorphism term is appropriate for general fermions. Below we use this method for bosonic representations to compute all the V terms, which are built out of the endomorphism E and the connection F ij .

Tracing Over Generators
Computing the heat kernel using this method requires computing the trace of a number of generators; the most we will need is six, as E 3 ∼ Σ 6 . We perform these traces using the algorithm presented in [34], which requires expanding the trace into a sum of symmetric traces, and then writing each symmetric trace in a basis of orthogonal tensors and higher or- Using the fact that T B T A = [T B , T A ] + T A T B and the algebra, we may rewrite this trace as This gives two factors of the non-symmetrized trace plus a term which has a trace over only five generators. Each of the other 718 terms may be dealt with in the same way: commute the generators to put them in the order (ABCDEF ) and keep track of all of the traces over five generators which are picked up along the way. This adds 5·5! terms with five generators.
Using this and rearranging the trace and symmetric trace, we get the schematic relation Each of these five-generator traces may be treated the same way-they each yield a symmetric trace with five generators plus 4 · 4! terms with a trace over four generators. Schematically, the trace may be expanded as and so on, until the result is a sum of symmetric traces of 2, 3, 4, 5, and 6 generators. Clearly this computation is not tractable by hand. Using the XACT package for Mathematica, we calculated all the necessary terms. The symmetric traces over an odd number of generators cancel each other out (which appears to be a sort of generalization of Furry's theorem). The result of this procedure includes a symmetric trace over six generators and a large number of symmetric traces over four generators and two generators.

Dynkin Indices
A representation R with Dynkin labels (a, b, c) has dimension The Weyl character formula may be used to show that The third and fourth order generalization to this index were computed in [35], which finds I 3 (a, b, c) = Dim R 120 (a − c)(a + c + 2)(a + 2b + c + 4) (B11)

Results
As each of the V a coefficients may be reduced to a trace of generators variously contracted with the Riemann tensor, this method will allow each of them to be computed. The entire list of coefficients is presented here: Since these expressions pertain to an endomorphism of the form E = Σ ab R abcd Σ cd , where Σ ab are SU(4) generators in an arbitrary representation specified by Dynkin labels (a, b, c), we refer to this as the "group theory method" for determining the heat kernel coefficients.
Now that the V a 's are known, we may compute the b 6 coefficient using the group theory method. We present the coefficient for a representation R on Ricci-flat backgrounds: b 6 (R) R ab =0 = 1 (4π) 3 In general, the full b 6 coefficients obtained by the group theory method do not match the expressions (A6) and (A15), for the fermion and gravitino, respectively, as the group theory method does not correspond to the square of the Dirac operator when acting on fermions.
This indicates that some modification may be necessary for fermionic representations, as was already noted in the four-dimensional case [20]. Curiously, however, this mismatch disappears when restricted to Ricci-flat backgrounds. This suggests that (B13) may potentially be valid for fermions as well as bosons. If this were true, we could then derive a general expression for δ(c − a) for arbitrary higher spin supermultiplets.
Finally, the expression δA in (12) vanishes in arbitrary backgrounds for long multiplets using the group theory method for the heat kernel. This is in contrast to the conventional method where the fermions are treated by squaring the Dirac operator. There, δA for long multiplets only vanished on Ricci-flat backgrounds, but was otherwise non-vanishing on more general backgrounds.