#### One-loop holographic Weyl anomaly in six dimensions

HJE
One-loop holographic Weyl anomaly in six dimensions
James T. Liu 0 1 2
Brian McPeak 0 1 2
0 450 Church St. , Ann Arbor, MI 48109-1040 , U.S.A
1 The University of Michigan , USA
2 Michigan Center for Theoretical Physics, Randall Laboratory of Physics , USA
We compute O(1) corrections to the holographic Weyl anomaly for sixdimensional 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 N = (1, 0) theory, and we obtain an expression for δ(c − a) for short representations with maximum spin two. We also confirm that the one-loop corrections to the 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.
AdS-CFT Correspondence; Anomalies in Field and String Theories; Confor-
1 Introduction
1.1
2 The O(1) contribution to the holographic Weyl anomaly
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
– 1 –
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, δgμν = 2δσgμν . In particular, for a partition function given by
we define the anomaly A by
Z
Z =
Dφ exp (−S[φ]),
Z
δ log Z = −
ddxpdet g δσA.
From the holographic point of view, the leading order Weyl anomaly can be obtained from
the regularized classical action [
3
]. In the AdS5/CFT4 case, this gives the familiar result
HJEP01(28)49
c = a =
N 2
π
3
4 vol(Σ5) ,
where IIB supergravity has been compactified on AdS5 × Σ5. 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 AdS5 × S5 in [4–8], where it was observed that the leading order result (1.3) 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 AdS5 has been extended to holographic field
theories with reduced or even no supersymmetry [9–12].
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]).
1.1
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
(4π)3A = (4π)3hT i = −aE6 + (c1I1 + c2I2 + c3I3) + DμJ μ,
where E6 refers to the Euler density, Ii are Weyl invariants and the final term is a
nonuniversal total derivative. At leading order, Einstein gravity on AdS7 gives a relation of
the form [
3, 16
]
c1 = 4c2 = −12c3 = 96a ∼ O(N 3),
which is the six-dimensional analog of (1.3). The relation between the ci coefficients arises
naturally in the holographic computation, and is further consistent with six-dimensional
(2, 0) superconformal invariance.
– 2 –
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
The most extensively studied (2, 0) theory of relevance is that of N coincident
M5branes, which is dual to supergravity on AdS7 × S4. Here the conjectured expression for
the central charges are [17–19]
1
288
9
4
7
4
1
288
a = −
4N 3 −
N −
,
c = −
(4N 3 − 3N − 1),
(1.6)
where c1 = 4c2 = −12c3 = 96c. The O(N ) terms arise from R4 corrections [17], while
the O(1) terms arise at one-loop [18, 19]. The O(1) shift δa = 7/1152 was computed
in [19] by evaluating the one-loop partition function on global (Euclidean) AdS7 with S6
boundary. However, the conjectured δc = 1/288 has not yet been directly computed, as
the most straightforward computation of one-loop determinants involve highly symmetric
spaces with conformally flat boundaries. In such cases, the Weyl invariants vanish, so no
information is provided about the ci coefficients.
An alternative approach to the computation of δa and δc was developed in [5–8] based
on a functional Schr¨odinger approach. In this approach, the contribution of each state to
the O(1) shift in the Weyl anomaly takes the form
δA = −
Δ −
bd,
1
2
d
2
(1.7)
where Δ is the conformal dimension and bd is the heat kernel coefficient for the
corresponding AdSd+1 field when restricted to the d-dimensional boundary. In principle, since
the six-dimensional b6 coefficient may be computed on a general curved background, this
allows for a full determination of not just the a coefficient but the ci’s as well.
It has been argued in [12], however, that the expression (1.7) cannot in general be
valid, as the contribution for a single field should have a more complicated dependence on
the conformal dimension Δ. 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 (1.7) 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].
Another line of reasoning has been developed to determine the anomaly coefficients
directly from the appropriate conformal higher spin operators on the boundary [20, 21]. In
particular, it is argued that AdS fields with higher dimensions Δ correspond to boundary
fields whose kinetic operators are greater than second order in derivatives. The factorization
of these operators on Ricci-flat backgrounds may serve as a justification of the functional
Schr¨odinger method presented in [5–8].
In this paper, we use (1.7) 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 AdS5/CFT4 case, where b4 ∼ δ(c − a)Rμ2νρσ 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.
– 3 –
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 b6 of higher-spin operators. This was worked out in [22, 23] 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 AdS7 × S4. 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 (1.6) and the original computation of [18].
2
The O(1) contribution to the holographic Weyl anomaly
As indicated above, the six-dimensional Weyl anomaly may be parameterized as
(4π)3A = −aE6 + (c1I1 + c2I2 + c3I3) + DμJ μ,
where E6 = ǫ6ǫ6RRR is the six-dimensional Euler density, and the Weyl invariants are
given by
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
I1 = CamnbCmpqnCpabq,
I2 = CabmnCmnpqCpqab,
I3 = Cmnpq Cmnpq + · · · .
Superconformal invariance imposes additional constraints on the anomaly coefficients
{a, ci}. In particular, N = (1, 0) supersymmetry requires c1 − 2c2 + 6c3 = 0, while
N = (2, 0) supersymmetry gives an additional constraint c1 − 4c2 = 0. This suggests
the parametrization
or equivalently
c1 = 96(c + c′ + c′′),
c2 = 24(c − c′ + c′′),
c3 = −8(c + 3c′ − 3c′′),
c =
c2 − c3
32
,
c′ =
c1 − 4c2
192
,
c′′ =
c1 − 2c2 + 6c3
192
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 (1.7) 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(Δ, j1, j2, j3) along with R-symmetry representation
r. We thus have
δA(rep) = −
2 rep
1 X(Δ − 3)b6(j1, j2, j3).
In the following, we first work out the heat kernel coefficients b6(j1, j2, j3) on a Ricci-flat
background, and then perform the sum over complete supermultiplets with maximum spin
two.
– 4 –
For an operator Δ = −∇2 − E where E is some endomorphism, the six-dimensional
SeeleyDeWitt coefficient b6(Δ) takes the form [16, 24]
b6(Δ) =
Here the Aa’s form a basis of curvature invariants [16, 25], and the Va’s are built from the
endomorphism E and the curvature Fij of the connection [16]. In particular, while the
coefficients of the Aa’s are universal, the Va terms are specific to the representation.
We follow the conventions spelled out in appendix A of [16], which also give explicit
expressions for the Aa’s and Va’s. However, we are concerned with only the combinations
which are non-vanishing on Ricci-flat backgrounds. These are
A5 = (∇iRabcd)2,
A9 = Rabcd∇2Rabcd,
A16 = RabcdRcdef Ref ab,
A17 = Raibj RmanbRimj n.
The full list of Aa’s, and expressions for the Va’s are given in appendix A.
The invariants E6 and I1, I2, and I3 may be written in terms of the basis Aa functions.
On a Ricci-flat background, they become
E6 = 32A16 − 64A17,
I1 = −A17,
I2 = A16,
I3 = 3A5 + 6A9 + 2A16 + 8A17. (2.8)
As these quantities are not all independent, we will be unable to determine the individual
central charges {a, ci} using only a Ricci-flat background. Note that we may construct two
combinations that are total derivatives
D1 = ∇a(Rmnij ∇aRmnij ) = A5 + A9,
D2 = 2∇a(Rmnij ∇mRanij ) = −A5 + A16 + 4A17.
This allows us to rewrite (2.8) in terms of the two invariants A16 and A17
E6 = 32A16 − 64A17,
I1 = −A17,
I2 = A16,
I3 = −A16 − 4A17 + 6D1 − 3D2. (2.10)
On a Ricci-flat background, we have the relations E6 = 32(2I1 + I2) and I3 = 4I1 − I2 up
to a total derivative. As a result, the six-dimensional anomaly, (2.1), takes the form
(2.7)
(2.9)
(4π)3A = 32(c − a)A16 − 64(c − a + 3c′′)A17 + DμJ μ,
(2.11)
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.
– 5 –
Field
φ
ψ
Aμ
Cμ+νρ
Ψμ
Bμν
Gμν
SU(4) Rep
44/9
−202/9
−164/3
−5608/9
3526/9
2992/3
−1388/9
80/9
−436/9
−344/3
26504/9
22012/9
−1616/3
49984/9
17/9
−58/9
−50/3
−8146/9
−1298/9
2269/3
−9236/9
−28/9
140/9
112/3
16352/9
2716/9
−4508/3
18592/9
up to two on a Ricci-flat background. In the last two columns, we tabulate γ16 and γ17, where
(4π)37!b6 = γ16A16 + γ17A17 + DμJ μ.
This leaves us with a holographic determination of δ(c − a), which may be combined with
the result of [19] for the δa coefficient to extract both δc and δa. This, in principle,
provides a complete determination of the O(1) shift in the holographic Weyl anomaly of
N = (2, 0) theories. Unfortunately the additional anomaly coefficient δc′ for N = (1, 0)
theories cannot be determined in this manner on Ricci-flat backgrounds.
Ideally, we would like to have an expression for the heat kernel coefficient b6(Δ) for
fields transforming in an arbitrary (j1, j2, j3) 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 AdS7 to their corresponding boundary Laplacians in the functional
Schrodinger approach of [5]. We thus restrict to spins up to two. The relevant b6 coefficients
evaluated on a Ricci-flat background are summarized in table 1. The coefficients for φ, ψ,
Aμ and Bμν were computed in [16], while the remaining ones are worked out in appendix A.
2.2
N = (1, 0) theory
We now turn to the superconformal theories, starting with the N = (1, 0) theory. We
expect that the anomaly vanishes when summed over long representations, and we will
see that this is indeed the case. The N = (1, 0) superconformal algebra is OSp(8∗|2),
with bosonic subgroup SO(2, 6) × SU(2)R. Here SO(2, 6) is either the isometry group of
AdS7 or the six-dimensional conformal group. We label representations of OSp(8∗|2) ⊃
SO(2, 6) × SU(2)R ⊃ SO(2) × SU(4) × SU(2)R by conformal dimension Δ, SU(4) Dynkin
labels (j1, j2, j3) and SU(2)R Dynkin label k (so that SU(2) ‘spin’ is given by k/2).
Unitary irreducible representations of the N = (1, 0) theory have been studied and
explicitly constructed in [26–30]. The theory has one regular and three isolated short
– 6 –
Δ = 2k + 4
Δ = 2k + 6
k−1,k+1
k−1,k+1
k−1,k+1
k
k−1
k−2
k−3
k−4
1
0
k−2,k
k−1
k−3,k−1
k−2
k−4,k−2
k−3
k
k
0
k
k
k−2,k,k+2
k−1,k+1
k
k−2,k
k−4,k−2,k
k−1
k−3,k−1
k−2
0
k
k
k
k−2,k,k+2
k−1,k+1
k
k−2,k
k−1
−1
0
k−3,k−1,k+1 k−3,k−1,k+1,k+3
k−3,k−1,k+1,k+3
k−2,k,k+2
k−2,k,k+2
representations, given generically by
A[j1, j2, j3; k] :
B[j1, j2, 0; k] :
C[j1, 0, 0; k] :
D[0, 0, 0; k] :
simple exercise to perform the sum (2.5) over the multiplet using the values of γ16 and γ17
given in table 1. Comparison with (2.11) then allows us to extract δ(c − a) and δc′′. The
results are summarized in table 2.
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 b6 coefficients in table 1. We also see that the
anomaly vanishes for the long representation, which agrees with expectations from the
AdS5 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.
– 7 –
Multiplet Δ L[0, 0, 0; k] > 2k + 6 A[0, 0, 0; k]
10Δ2(Δ2 − 2) + 131
according to (2.4). The δa coefficient is computed using the results of [19].
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, we have only been able to determine the difference δ(c − a). This may be
combined with the holographic δa coefficient obtained in [19] to separate out the
contributions to δa and δc. These results are presented in table 3. However, we are unable to
determine δc′ unless we can move away from Ricci-flat backgrounds.
2.3
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
superconformal algebra decomposes as OSp(8∗|4) ⊃ SO(2, 6) × Sp(4)R ⊃ SO(2) × SU(4) × Sp(4)R.
The shortening conditions follow the same pattern as (2.12), however with extended
Rsymmetry [26–30]
A[j1, j2, j3; k1, k2] :
B[j1, j2, 0; k1, k2] :
C[j1, 0, 0; k1, k2] :
D[0, 0, 0; k1, k2] :
Here (k1, k2) 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 Δ = 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 4. 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 D[0, 0, 0; 0, 2]
– 8 –
D[0, 0, 0; 0, 2]
D[0, 0, 0; 0, 3]
D[0, 0, 0; 0, k ≥ 4]
(0, k)
(1, k − 1)
(2, k − 2)
(0, k − 1)
(1, k − 2)
(3, k − 3)
(0, k − 2)
(2, k − 3)
– 9 –
sponding holographic Weyl anomaly coefficients δ(c−a) and δc′′. Entries are Sp(4)R representations
specified by Dynkin labels (k1, k2).
are
D(4; 0, 0, 0)14 + D
where D(Δ; j1, j2, j3) labels the SO(2, 6) representation and the subscript labels the Sp(4)R
representation. The massless vector, gravitino and graviton representations can be obtained
from the corresponding massive representations by subtracting out null states according to
D(5; 0, 1, 0) = D(5 + ǫ; 0, 1, 0) − D(6; 0, 0, 0),
D
these null states into account then gives the result δ(c − a) = 13/384 for k = 2 shown in
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
δa = −
1
mial in k. As a result, the regulated anomaly for AdS7 × S4 is
Following [19], we regulate the sums using a hard cutoff.
This amounts to setting
Pk∞=1 k
n = 0 for any n ≥ 0. This implies Pk∞=2 f (k) = −f (1), where f (k) is
polynoδa =
1
7
· ,
This is equal and opposite to the result for the conformal anomaly of the free tensor
multiplet computed in [17], and agrees with the O(1) contributions in (1.6).
3
Discussion
While the six-dimensional Weyl anomaly coefficients are conventionally parametrized as a,
c1, c2 and c3, we have found it convenient to use an alternate linear combination of the ci’s
given in (2.4). Holographically, the leading order anomaly coefficients (assuming Einstein
gravity in the bulk) satisfy the relation
1
7
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 [19] then allows us to separate out the individual coefficients
D[0, 0, 0; 0, k ≥ 2] :
δa = −
· (6k(k − 1) + 1) ,
δc = −
(6k(k − 1) + 1) .
1
288
As an application, consider the N = (2, 0) theory obtained by compactifying
11dimensional supergravity on AdS7 × S4. The Kaluza-Klein spectrum is simply
⊕k≥2 D[0, 0, 0; 0, k],
c = a,
c′ = 0,
c′′ = 0.
(3.1)
(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 (1.7) obtained
through a functional Schr¨odinger method [5–8].
It is reasonable to question whether the use of (1.7) is valid, as it disagrees with the
direct computation of δa performed in [12, 19]. A quick way to see this is to note that
δa in table 3 is a fourth order polynomial in Δ, while the result of summing (1.7) over
a supermultiplet can be at most quadratic in Δ. (One power comes directly from (1.7),
while another can arise from the dimension of the shortened representation.) If δ(c − a)
was expected to be cubic or higher in Δ, then our result, as shown in the last column of
(2.16)
(2.17)
table 3, cannot possibly be correct. However, we now demonstrate that c − a can be at
most linear in Δ, which is consistent with application of (1.7).
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
I8 =
[αc2(R)2 + βc2(R)p1(T ) + γp1(T )2 + δp2(T )],
(3.2)
and the relation to the Weyl anomaly coefficients has recently been worked out [20, 31–33]
a = −
α − β + γ +
δ ,
c − a = −
β − 2γ +
δ .
(3.3)
,
c′ =
1
432
1
2
Since α is the coefficient of the [SU(2)R]4 anomaly, it can be at most fifth power in Δ, where
the extra power comes from the dimension of the representation. Similarly, β can be at most
cubic in Δ, while γ and δ can be at most linear in Δ. This in turn demonstrates that a will
be at most fifth power in Δ, c′ will be at most cubic and c−a will be at most linear. Thus the
functional Schr¨odinger 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 b6
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 [5–8]. For higher-spin bosons,
it is natural to take a bulk Laplacian of the form Δ = −
− E with the endomorphism
E = ΣabRabcdΣcd, where Σab are SU(4) generators in the appropriate bosonic higher-spin
representation. However, the situation is less clear for fermions. The natural
generalization would be to simply take Σab to be in a fermionic higher-spin representation. However,
this does not agree with the square of the Dirac operator for ordinary spin-1/2 fermions.
Nevertheless, it is possible that the use of a universal endomorphism term for bosons and
fermions would be appropriate when tracing over supermultiplets. Along these lines, we
have computed the b6 coefficient for general higher-spin representations in appendix B.
Finally, part of our motivation for exploring the O(1) contributions to the holographic
Weyl anomaly is to make a connection to the N = (1, 0) superconformal index. As in
the AdS5/CFT4 case, the bulk one-loop corrections to the Weyl anomaly vanish for long
representations, so it is natural to expect that these corrections can be obtained from the
index. More generally, we anticipate that the one-loop matching between δ(c − a) and the
index can be extended to the full set of anomaly coefficients a and ci, even in theories
without holographic duals. This would then open up a new path towards characterizing
superconformal field theories in six dimensions.
Acknowledgments
We wish to thank A. Arabi Ardehali, F. Larsen and P. Szepietowski for stimulating
discussions. This work was supported in part by the US Department of Energy under Grant
No. DE-SC0007859. The work of B. McPeak was partially supported by a Rackham 2016
Spring/Summer Research Grant.
A
Heat kernel for spins up to two
The Seeley-DeWitt coefficients bn(Δ) depend on the field and the form of the second order
operator Δ. In four dimensions, the appropriate operators for irreducible fields up to spin
two are listed in [23]. 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 [16, 25]
A1 = 2R, A2 = (∇aR)2 , A3 = (∇aRmn)2 , A4 = ∇aRbm∇bRam, A5 = (∇aRmnij)2 ,
A6 = R R, A7 = Rab Rab, A8 = Rab∇m∇bRam, A9 = Rabmn Rabmn, A10 = R3
A11 = RRa2b,
A12 = RRa2bmn, A13 = RamRmiRia, A14 = RabRmnRambn,
A15 = RabRamnlRbmnl, A16 = RabcdRcdef Ref ab, A17 = RaibjRmanbRimjn.
The b6 coefficient may be computed from the expression (2.6), where the Va’s are given by
V1 = ∇kFij∇kF ij, V2 = ∇jFij∇kF ik, V3 = Fij F ij, V4 = FijF jkFki,
V5 = RmnijF mnF ij, V6 = RjkF jnF kn, V7 = RFijF ij, V8 =
2E, V9 = E E,
V10 = ∇kE∇kE, V11 = E3, V12 = EFi2j, V13 = R E, V14 = Rij∇i∇jE,
V15 = ∇kR∇kE, V16 = E2R, V17 = E R, V18 = ER2,
V19 = ERi2j, V20 = ERi2jkl.
(A.1)
(A.2)
Here Δ = −∇2 − E and Fij is the curvature of the connection, [∇i, ∇j] = Fij. Below we
present the V -terms for each field after tracing over the representation.
A.1
Conformally coupled scalar
The conformally coupled scalar has E = − 1 R and Fij = 0, so the V -terms are:
5
V4
0
V14
V5
0
V15
V6
0
V16
− 15 A1 215 A6
V8
V18
V9
V19
V10
The b6 coefficient is
V7
0
V17
8
16
3
of the Dirac operator:
The appropriate second order operator for the Dirac fermion may be obtained as the square
The endomorphism and curvature of the connection coincide with the result obtained
Then the V -terms contributing to the anomaly are (after tracing):
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:
OAμ = − Aμ + RνμAν.
Eab = −Rab,
(Fij)ab = Rabij,
V7
V17
V8
V18
V9
V19
V10
V20
(A.4)
(A.5)
(A.7)
(A.8)
Self-dual three-form
The field which transforms under the (2, 0, 0) representation is the 10-component self-dual
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
OCμνρ = − Cμνρ + RμλCλνρ + RνλCμλρ + RρλCμνλ
− RμνλσCλσρ − RνρλσCμλσ − RρμλσCλνσ.
This means that the endomorphism and connection curvature are given by
Eabc
def = −3R[a[dδbe δcf]] + 3R[ab
Then we can compute the relevant terms:
(A.10)
(A.11)
HJEP01(28)49
−6A5
−6A15
where V11 = −A10 +6A11 −3A12 −6A13 −12A14 +12A15 −2A16 +8A17. (Again, all V -terms
are given after tracing over the representation.) So the b6 coefficient is given by
The self-duality condition reduces each of these terms by a factor of two, reproducing the
A16 and A17 terms found in table 1.
A.5
Gravitino
The gravitino with the gauge condition γμψμ = 0 corresponds to the (1, 1, 0) representation.
In this case the operator O is the square of the Rarita-Schwinger operator:
Oψμ = − ψμ +
1
4
2
Rψμ − 1 γργσRρσμνψν.
(A.13)
The endomorphism and connection curvature are given by
and
The adjoint representation (1, 0, 1) corresponds to the two-form computed in [16].
OBμν = − Bμν + RμλBλν − RνλBλμ − Rμν ρσBρσ.
This means that the endomorphism and connection curvature are given by
Eabcd = −2R[[acδbd]] + Rabcd,
(Fij)abcd = 2Rij[a
V8
−4A1
V13
V18
−4A10
(A.15)
(A.16)
(A.17)
V5
−4A16
V10
V15
V20
−4A12
4A17
V4
V9
V14
V19
−4A11
A6 + A9
A2 + A5
The symmetric spin-two field is the (0, 2, 0) representation. The appropriate kinetic
operator is the Lichnerowicz operator [34]:
Ohμν = − hμν + Rμλhλν + Rνλhλμ − 2Rμρνσhρσ.
The endomorphism and connection are given by
Then we can compute the relevant terms:
Eμρσν = −2R{{μρδνσ}} + Rμ ρν σ + Rμ σν ρ,
(Fab)ρμσν = 2Rab{μ{ρδνσ}}.
and
where V11 = −3A11 − 16A13 − 6A14 − 18A15 − A16 + 8A17, and these terms are given after
tracing over the representation. The b6 coefficient is
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 [23] 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 Δ = −
− 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:
1
A
E = ΣabRabcdΣcd
or
E =
ΣabR+abcdΣcd,
(B.1)
for bosonic (A + B = integer) or fermionic (A + B = half-integer, A > B) representations,
respectively. Here R+abcd = 12 (Rabcd + 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 Fij .
B.1
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 E3 ∼ Σ6. We perform these traces using the
algorithm presented in [35], 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 order
Dynkin indices. For example, the trace of two generators of an irreducible representation is
Tr[TRATRB] = I2(R)gAB.
(B.2)
Here R refers to the representation, and the capital Roman letters A, B, . . . = 1, 2, . . . , 15
label the generators of SU(4).
Each SU(4) index is interchangeable with a pair of
antisymmetrized six-dimensional spacetime indices {µ, ν }.
If the number of generators is greater than two, we will first need to break the trace
into a sum of symmetrized traces. For a trace of n generators, this is accomplished by
writing out each of the n! terms in the symmetrized trace, and then using commutation
relations to return each term to the original order, plus a number of traces of lower numbers
of generators. For example, we may look at the trace of six generators. First consider the
symmetrized trace
STr[TATBTC TDTETF ]
=
1
6!
STr[TATBTC TDTE TF ] =
(Tr[TATBTC TDTE TF ]
1
6!
(Tr[TATBTC TDTE TF ] + Tr[TBTATC TDTETF ] + 718 more terms) .
(B.3)
Using the fact that TBTA = [TB, TA] + TATB and the algebra, we may rewrite this trace as
+ Tr[TATBTC TDTE TF ] + Tr[fBAX T X TC TDTE TF ] + 718 more terms).
(B.4)
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
Tr[TATBTC TDTETF ] = STr[TATBTC TDTETF ] −
· 600 Tr[T T T T T ].
(B.5)
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
1
6!
Tr[TATBTC TDTE TF ]
(B.6)
HJEP01(28)49
= STr[TATBTC TDTE TF ] −
600 STr[T T T T T ] −
· 96 Tr[T T T T ]
,
1
6!
1
5!
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.
B.2
Orthogonal tensors
The symmetrized traces may be expanded in a set of orthogonal symmetric tensors. The
two needed for this calculation are
STr[T AT BT C T D] = I4(R)dA⊥BCD + I2,2(R)(δABδCD + δAC δBD + δADδBC )/3,
(B.7)
and
STr[T AT BT C T DT E T F ] = I6(R)dA⊥BCDEF +I4,2(R)(dA⊥BCDδEF +dA⊥BCEδDF +···)/15
+I3,3(R)(dA⊥BC d⊥DEF +dA⊥BDdC⊥EF +···)/10+I2,2,2(R)(δABδCDδEF +···)/15. (B.8)
⊥
Note that I6 = 0 for all representations of SU(4). The tensors dABCD and dABC are
⊥ ⊥
fixed by the condition of orthogonality; dABC is the six-dimensional epsilon tensor
(recalling that A = {µ 1ν1}, etc.) The fourth order dABCD
⊥
six-dimensional metric — its terms include gμ1ν4 gμ2ν3 gμ3ν2 gμ4ν1 and the other 47 ways of
arranging the indices. The indices I4,2, I3,3, and I2,2,2 are not unique; imposing
orthogonality and other group-theoretic relations yields the system of equations (158)–(160) in [35].
Solving these allows I4,2, I3,3, and I2,2,2 to be expressed in terms of the Dynkin indices I4,
may be expressed in terms of the
I3, and I2.
B.3
Dynkin indices
A representation R with Dynkin labels (a, b, c) has dimension
DimR(a, b, c) =
(a + 1)(b + 1)(c + 1)(a + b + 2)(b + c + 2)(a + b + c + 3).
(B.9)
1
12
I2(a, b, c) =
60
I3(a,b,c) =
I4(a,b,c) =
120
3360
B.4
Results
As the trace of each of the Va 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 traced coefficients is presented here:
DimR (a−c)(a+c+2)(a+2b+c+4)
DimR 3a4 +8a3b+4a3c+24a3 +2a2b2 +2a2bc+30a2b
−4a2c2 +6a2c+54a2 −12ab3 −18ab2c−50ab2 +2abc2 −28abc
−34ab+4ac3 +6ac2 −2ac+24a−6b4 −12b3c−48b3 +2b2c2
−50b2c−122b2 +8bc3 +30bc2 −34bc−104b+3c4 +24c3 +54c2 +24c .
(B.11)
HJEP01(28)49
DimR 3a2 + 2a(2b + c + 6) + 4b2 + 4b(c + 4) + 3c(c + 4) .
(B.10)
I
2
2
DimR
I
3
2
Dim2R
I
2
3
DimR
DimR
I
2
2
I
2
2
DimR
34
+
+
3
3
51
51
11A6 − 4A7 + 5A9 I4,
11A2 −
4A3 + 5A5 I4,
51
51
The Weyl character formula may be used to show that
The third and fourth order generalization to this index were computed in [36], which finds
V10 = − A2 +
A3 − 25A5 I2 +
V1 = − A25 I2,
V5 = − A16 I2,
2
V2 = (A4 −A3)I2,
V6 = − A15 I2,
2
V3 = − A29 I2,
V7 = − A12 I2,
2
V4 =
7A10 + 209A11 − 183A12 − 437A13 − 437A14 + 57A15 − 101A16 + 437A17
V13 = − A26 I2,
V14 = −(A8 −A13 +A14)I2,
V15 = − A22 I2,
476
84
952
56
136
8
51
17
17
15A12 +
68
15A16
34
I
2
2
DimR
51
+
476
8A17 I4
153
952
476
84
38A17
51
51
952
168
I2I4 ,
DimR
11A12 −
4A15 +
5A16 I4,
3
51
51
6
204
11A10 −
4A11 +
3
51
5A12 I4,
Since these expressions pertain to an endomorphism of the form E = ΣabRabcdΣcd, where
Σab 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 Va’s are known, we may compute the b6 coefficient using the group theory
method. We present the coefficient for a representation R on Ricci-flat backgrounds:
b6(R)
Rab=0 (4π)37!
1
A5 17DimR
3150I22 +9DimR −
In general, the full b6 coefficients obtained by the group theory method do not match the
expressions (A.6) and (A.15), 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 [23]. Curiously, however, this
mismatch disappears when restricted to Ricci-flat backgrounds. This suggests that (B.13)
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, we find that the expression δA in (2.5) vanishes on arbitrary (ie not just
Ricciflat) 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. This complete vanishing of
δA for long multiplets is consistent with expectations from AdS5/CFT4 [9, 12], and lends
credibility to the idea that the group theory method may yield the correct expression for
δA for general spins.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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