Supergravity one-loop corrections on AdS_7 and AdS_3, higher spins and AdS/CFT

As was shown earlier, one-loop correction in 10d supergravity on AdS_5 x S^5 corresponds to the contributions to the vacuum energy and boundary 4d conformal anomaly which are minus the values for one n=4 Maxwell supermultiplet, thus reproducing the subleading term in their N^2-1 coefficient in the dual SU(N) SYM theory. We perform similar one-loop computations in 11d supergravity on AdS_7 x S^4 and 10d supergravity on AdS_3 x S^3 x T^4. In the AdS_7 case we find that the corrections to the 6d conformal anomaly a-coefficient and the vacuum energy are again minus the ones for one (2,0) tensor multiplet, suggesting that the total a-anomaly coefficient for the dual (2,0) theory is 4 N^3 - 9/4 N - 7/4 and thus vanishes for N=1. In the AdS_3 case the one-loop correction to the vacuum energy or 2d central charge turns out to be equal to that of one free (4,4) scalar multiplet, i.e. is c=+6. This reproduces the subleading term in the central charge c= 6(Q_1 Q_5 +1) of the dual 2d CFT describing decoupling limit of D5-D1 system. We also present the expressions for the 6d anomaly a-coefficient and vacuum energy for a general-symmetry higher spin field in AdS_7 and consider their application to tests of vectorial AdS/CFT with the boundary conformal 6d theory represented by free scalars, spinors or rank 2 antisymmetric tensors.


Introduction
One of the key probes of the AdS/CFT correspondence [1][2][3] is the boundary theory conformal anomaly which is closely related to the simplest correlators of the stress tensor [4][5][6]. In the case of the duality between N = 4 SU (N ) SYM theory and string theory in AdS 5 × S 5 the gauge-theory result for the Weyl anomaly is A 4 = −a E 4 + c W 4 , a = c = (N 2 − 1)k 1 (k 1 = 1 4 is the contribution of a single N = 4 vector multiplet). It is determined by the 2and 3-point correlators of stress tensor and should thus be exact. The N 2 term is indeed reproduced at strong coupling by the classical supergravity action [5].
It was suggested in [7,8] 1 that the -1 term in N 2 − 1 coefficient should come from the one-loop 10d supergravity correction (the contribution of all massive string mode multiplets should vanish). This was recently confirmed in [10] where it was found that the contributions of the massless 5d supergravity modes and the massive S 5 KK modes to the boundary conformal anomaly can be universally described by a simple formula: a p = c p = pk 1 , where p = 1 for a vector multiplet (or boundary doubleton to be omitted), p = 2 for the massless 5d supergravity modes, and p = 3, 4, ... for the massive KK levels. Summing over p using a special regularization prescription ∞ p=1 p = 0 (which is, in fact, required for consistency with the standard ζ-function regularization for the Casimir energy in 10d) gives indeed (a = c) 1−loop sugra = −1.
Below will perform a similar one-loop computation of the boundary a-anomaly in the case of 11d supergravity on AdS 7 × S 4 (correcting an earlier attempt in [11]). This will determine the subleading N 0 term in the a-coefficient of conformal anomaly of the 6d (2,0) theory describing N coincident M5-branes which should be dual to M-theory on AdS 7 × S 4 .
In addition to the duality examples based on AdS 5 × S 5 and AdS 7 × S 4 supergravity backgrounds there is also the duality [1,12] between string theory in AdS 3 × S 3 × T 4 space supported by RR 3-form flux and 2d CFT corresponding to gauge theory describing lowenergy limit D5-D1 system. The central charge of this CFT is c = 6(Q 1 Q 5 + 1) [13,12] (Q i are the number of branes). The leading 6Q 1 Q 5 can be reproduced from the classical action of 10d supergravity on S 3 × T 4 [5,14]. Here we shall demonstrate that the subleading +6 term is reproduced by the one-loop 10d supergravity contribution. This provides a non-trivial test of this AdS 3 /CFT 2 duality.

AdS 7 /CFT 6
The conformal anomaly of a classical Weyl invariant theory in 6d has the following general form [15][16][17] A 6 = a E 6 + W 6 + D 6 , W 6 = c 1 I 1 + c 2 I 2 + c 3 I 3 , where E 6 is the Euler density in six dimensions, W 6 is a combination of three independent Weyl invariants and D 6 is a total derivative term (which can be changed by adding a local counterterm and thus depends on a scheme). Omitting the derivative D 6 term, the conformal anomaly corresponding to a single 6d tensor multiplet [17] and the 6d conformal anomaly contribution coming from the classical 11d supergravity action on S 7 [5] (that should be representing the large N limit of the (2,0) theory result) may be written as The fact that the anomaly in these two cases contains the same Weyl-invariant combination W 6 (so that its Weyl-tensor or B-anomaly part is effectively parametrized by just one overall coefficient c) is related to non-renormalization of the ratio of the 2-and 3-point correlation functions of the corresponding stress tensor [18]. 2 By analogy with a subleading order-N term in the R-symmetry anomaly of (2,0) theory [19] it was suggested in [20] that there should be also order N contributions to a (2,0) and c (2,0) coming from the R 4 term in the M-theory 11d effective action, a (2,0) In [20] the further N 0 corrections a 1 , c 1 were ignored, while the coefficients of order N terms were fixed so that the resulting N 3 + N terms interpolated to N = 1 match the single tensor-multiplet anomalies in (1.3). As in the case of 10d supergravity on S 5 , one may expect that a 1 and c 1 should be determined by the one-loop 11d supergravity correction [11].
Following the example of the D3-brane-based AdS 5 ×S 5 duality where the full anomaly coefficient N 2 − 1 vanishes for N = 1 it is natural to expect that here too the boundary singleton (single M5-brane tensor multiplet) should decouple and thus the full 6d anomaly of the (2,0) theory should vanish for N = 1. This suggests that a 1 and c 1 should be non-zero and given by minus the values for a single tensor multiplet in (1.3) It was noted in [21] that the expression c (2,0) is exactly the same as the central charge of the A N −1 Toda theory at the "symmetric" coupling point (cf. also [22,23]). 3 Here we shall provide support for (1.5) by showing that the one-loop 11d supergravity correction indeed produces the value a 1 = −a tens . Then the expected exact value of a (2,0) Below we shall consider the one-loop 11d supergravity on S 7 supergravity contributions in the case when the 6d boundary of AdS 7 is either S 6 (determining the a-anomaly part of A 6 ) or R × S 5 (finding the vacuum or Casimir energy E c ). We will find that in both cases the result is minus that of a single tensor multiplet We shall use similar methods as in the AdS 5 × S 5 case in [10], i.e. first determine the contributions to a and E c coming from a generic AdS 7 higher spin filed in representation 3 6d CFT with (2, 0) supersymmetry possess a protected sector of operators and observables related to a 2d chiral algebra [21] which is W-algebra labelled by a simply-laced Lie algebra g for a specific value of the central charge. In the g = AN−1 case this leads to c1 = −1. 4 The non-vanishing 1-loop supergravity correction to the conformal anomaly implies that there should be also a similar correction also to the corresponding R-symmetry anomaly (i.e. N → N − 1 in the I8 term in the anomaly [24]) implying its vanishing for N = 1. The chiral anomaly of the boundary theory is accounted for by the Chern-Simons terms in the supergravity action. In the case of AdS5 × S 5 the 1-loop supergravity correction shifts the Chern-Simons coefficient N 2 → N 2 −1 [9]. A similar shift is then expected in the AdS7 × S 7 case where the CS term reproduces the leading N 3 anomaly and also the O(N ) correction [19]. (2,6) and then sum up the contributions of the relevant fields appearing in the supergravity spectrum. We shall also apply our general expressions for a(∆; h 1 , h 2 , h 3 ) and E c (∆; h 1 , h 2 , h 3 ) to provide tests of the vectorial AdS/CFT duality [25][26][27] in the case when the boundary theory is represented by a free scalar, spinor or tensor singleton.
The 2d CFT dual to superstring in AdS 3 × S 3 × T 4 with RR charges Q 5 , Q 1 is described by a coupled system of three (4,4) supersymmetric multiplets (see [13,12] and [28] for a recent review): U (Q 1 ) adjoint vector multiplet, U (Q 1 ) adjoint hypermultiplet, and U (Q 1 )×U (Q 5 ) bi-fundamental hypermultiplet. The contribution to 2d conformal anomaly of a single free (4,4) hypermultiplet (with 4 real scalars and 4 real fermions) is c = 4 + 4 × 1 2 = 6. 5 The 2d vector multiplet has an irrelevant kinetic term and thus contributes to anomaly only through measure (or ghost) factor, with single U (1) vector giving negative contribution c = −1. 6 The U (1) part of the vector multiplet is decoupled (representing the c.o.m. of the bound D5-D1 system) and thus the total central charge count is 7 where the first term is the contribution of bi-fundamental hypers, the second -of adjoint hypers and the third one of the vectors (with the U (1) part subtracted). 8 A peculiarity of the 2d case is that here the subleading (for large Q 5 ) term in the central charge which is responsible for subtraction of the decoupled c.o.m. modes enters with plus rather than minus sign (as was in 4d and 6d examples). Still, we shall demonstrate below that as in the AdS 5 and AdS 7 cases this extra +6 term (which should be protected and thus receive contributions only from the BPS modes) is also reproduced on the dual AdS theory side by the corresponding one-loop correction in 10d supergravity on AdS 3 × S 3 × M 4 with M 4 = T 4 or K3.
More precisely, instead of computing directly the correction to the central c we shall determine the one-loop correction to the AdS 3 vacuum energy or S 1 Casimir energy in 2d; the latter should be directly related to the central charge [31] (1.9) 5 In 2d the conformal anomaly is A2 = 4πb2 = aR, a = 1 6 c, so that c = 1 for one real scalar. 6 The contribution of "non-dynamical" 2d vector gauge field to the central charge is negative (-1) [29] just like that of non-dynamical 2d gravity (-26) [30]. The reason for this -1 contribution can be understood also by giving vector a mass by coupling it to a complex scalar so that it will not contribute to c; then the central charge of the scalar part is reduced by 1 as one scalar component is absorbed by the vector. 7 The same result is found by counting the SU (2) chiral anomaly of the (4,4) superconformal algebra [13,12]. 8 The (4,4) vector multiplet contains one 2d vector Am, 4 scalars φi, 4 real spinors ψ k and 3 auxiliary fields Dr, all having canonical dimensions (i.e. 1 for Am and φi, 1 2 for ψ k and 2 for Dr). With these dimension assignments the corresponding 2d conformal anomaly can be found from the following dimensionless action (same as the standard one but with each kinetic term contianing an extra ∂ −2 factor) d 2 x (A ⊥ m ) 2 + φ 2 i + Dr∂ −2 Dr + ψ k ∂ −1 ψ k . As as a result, the total central charge contribution is c = −1 + 0 + 3 × (−1) + 4 × (− 1 2 ) = −6.
We shall find that the one-loop supergravity contribution gives indeed E c = − 1 2 after summing over the contributions of the KK modes of 10d supergravity on S 3 × M 4 .
The rest of this paper is organized as follows. In section 2 we shall present the expressions for the a-anomaly coefficient and the vacuum energy of a higher-spin field in AdS 7 corresponding to an arbitrary (massive or massless) representation of SO(2, 6), generalising earlier results for symmetric tensors to mixed symmetry case.
In section 3 we shall apply these results to compute the one-loop corrections to the 6d boundary a-anomaly and vacuum energy in 11d supergravity compactified on S 7 obtaining eq. (1.7). As another application, in section 4 we shall perform checks of vectorial AdS 7 /CFT 4 duality in the cases when the boundary 6d theory is represented by free scalars, spinors or (self-dual) rank 2 tensors. We shall find that matching of both a-anomaly and Casimir energy requires particular shifts of the inverse coupling of the AdS 7 higher spin theory.
In section 5 we shall turn to the case of 10d supergravity in AdS 3 × S 3 × M 4 and compute the corresponding one-loop correction to the vacuum energy, demonstrating that it is equal to − 1 2 as in (1.8), thus deriving the subleading term in the central charge (1.9) on the dual string theory side.
There are several technical appendices. In appendix A we present the expressions for the Casimir energy, a-anomaly and partition function for the fields of the free (2, 0) multiplet in 6d. In Appendix B we derive the 6d boundary a-anomaly coefficient corresponding to a generic higher spin field on AdS 7 using spectral ζ-function method. Appendix C collects decompositions of tensor products of two SO(2, 6) singleton representations with spin 0, 1 2 , 1 into infinite sums of other representations and the corresponding relations for the characters. These Flato-Fronsdal like relations are used in the discussion of applications to vectorial AdS/CFT duality in section 4. Appendix D contains discussion of some properties of the Casimir energy of spin 0, 1 2 , 1 singletons in AdS d+1 for general d. They are useful in comparing the 6d results to the previously studied 4d case. In appendix E we list the explicit field content of the SU (2, 2|1) × SU (2, 2|1) building blocks appearing in the Kaluza-Klein towers of 6d supergravity compactified on S 3 . Appendix F contains the discussion of the relation between the expression for the 2d Casimir energy in section 5 and the 2d central charge derived [32] using AdS 3 method for short SU (2, 2|1) × SU (2, 2|1) multiplets.
2 Casimir energy and a-anomaly for generic higher spin fields in AdS 7 Given a generic conformal field in 6d we may associate to it a field in AdS 7 correspoonding to the same representation of SO (2,6). That allows to interpret the one-loop contributions for a field in AdS 7 in terms of Casimir energy and conformal anomaly of the boundary field (see [10] and refs. there).
The SO(2, 6) conformal group representations will be denoted as (∆; h) where h = (h 1 , h 2 , h 3 ) are the SO(6) highest weights or Young tableu labels (h i are all integers or all half-integers with h 1 ≥ h 2 ≥ |h 3 |). 9 The unitary irreducible representations of SO(2, 6) have (see, e.g., [33]) (2.1) If ∆ does not saturate the above inequalities then the character of the corresponding massive representation is 10 where d(h) is the multiplicity of the representation If ∆ is at one of the unitarity bounds the corresponding representation is short or massless (i.e. corresponds to a massless field in AdS 7 space) 11 and its character requires a proper subtraction of null states and their descendants. For the ∆ = h 1 + 4 case in (i) in (2.1) we have the following massless representation character (2.5) In the massless case of (iii) with ∆ = h + 2 and h = (h, h, ±h) which corresponds to the singleton representation the character is In particular, it is possible to view the (2, 0) tensor multiplet as supersingleton [35] which is a combination of 6d singletons with h = 0, 1 2 , 1: the one-particle partition functions for a scalar φ, Majorana-Weyl fermion ψ and self-dual tensor T are the characters of the corresponding singleton representations (see also Appendices A and C) 10 The label + indicates that this will represent the partition function of the corresponding AdS7 field with standard (Dirichlet) boundary conditions. Same quantity without + corresponds to associated conformal field in boundary theory (see [10] for details). indicates massive representation character. 11 In general [34], given a field in AdS d+1 (with even d) corresponding to SO(2, d) representation where first k = 0, 1, 2, ... raws of the SO(d) Young tableu may be equal, i.e. h1 = In the case of (2.1) where d = 6 the lower bounds in (i),(ii) and (iii) correspond to k = 0, 1, 2. (2.7) From one-particle partition function Z(q) given by the corresponding SO(2, 6) character one can extract the expression for the Casimir energy E c as [36] For a generic massive representation (∆; h) with the character (2.2) the corresponding Casimir energy is found to be The expression for the a-anomaly can be found from the one-loop partition function on euclidean AdS 7 as explained in appendix B In the case of short representations saturating a unitarity bound one needs to combine the massive representation expression as in (2.4),(2.5),(2.6).
In the special case of the totally symmetric massive spin s tensor representation with h = (s, 0, 0), the expression (2.11) can be written in the following alternative form which is in agreement with the earlier result in [32,26].
3 One-loop correction to vacuum energy and a-anomaly in 11d supergravity on AdS 7 × S 4 Let us now apply the above results (2.10) and (2.11) to compute the corresponding total contribution of the fields in the spectrum of 11d supergravity compactified on S 4 . The corresponding KK spectrum [37,35,38] is given in Table 1 (see also [39]). The massless level p = 2 correspond to the fields of maximal gauged 7d supergravity with AdS 7 vacuum. Contributions of the AdS 7 fields should be summed with multiplicities corresponding to their U Sp(4) = SO(5) representations. 12 Using (2.10) to sum of the vacuum energy contributions at each level p we find The value for the massless multiplet p = 2 is in agreement with [36]. The expressions for the a-anomaly are similar Recalling that for one (2,0) tensor multiplet (see appendix A) we observe that, remarkably, both the vacuum energy and a-anomaly has the following universal expressions for any value of p = 1, 2, 3, ...
This is the direct analog to what was found in the case of 10d supergravity on AdS 5 × S 5 in [10] where the role of tensor multiplet was played by N = 4 vector one (or superdoubleton) and instead of the coefficient 6 p 2 − 6 p + 1 we had simply p. 13 To sum over p we shall use the same prescription as in [10], i.e. introducing a sharp cutoff and dropping all divergent terms 14 . 13 For comparison, in the case of 11d supergravity on AdS4 ×S 7 one finds [40,41] that the contributions to the AdS4 vacuum energy sum up to zero at each level p separately, i.e. E + c,p = 0. The boundary conformal anomaly also vanishes as the boundary is 3-dimensional.
This prescription can be justified by using the spectral ζ-function regularization directly in 11d, i.e. before explicitly expanding in modes of S 4 (see below); it is such a regularization that should be consistent with diffeomorphism symmetry of 11d theory. Assuming (3.5) we conclude that if the boundary (2,0) singleton were included in the spectrum of 11d supergravity, the total vacuum energy and a-anomaly would vanish. However, it should be left out representing gauge degrees of freedom. Thus we conclude that the total one-loop supergravity contributions are exactly minus the tensor multiplet ones as claimed in (1.7).
Let us now demonstrate that the prescription (3.5) is indeed equivalent to the use of spectral ζ-function directly in 11d theory. We shall consider the case of the Casimir energy (for a similar discussion on 10d case see [10]). For a massive 7d field in representation (∆; h) the vacuum energy can be extracted from the partition function (2.2) that we may write in the form Then from (2.8),(2.9) we obtain a formal (divergent) expression for E c This sum can be computed using the ζ-function regularization applied to the full effective energy eigenvalue ∆ + n, or, equivalently, by introducing an exponential cutoff via e n → e n e − (∆+n) , doing the sum, expanding in → 0, and finally dropping all singular terms. Keeping finite we may find the contribution to the sum (3.8) from all KK states (taking into account that p = 2 states are massless, cf. (2.4),(2.5)). Denoting the total summand from level p as e n (p; ) and, summing over both n and p = 1, 2, ..., we obtain Thus the finite part of the sum over p ≥ 1 vanishes in agreement with (3.5). Equivalently, 4 Vectorial AdS 7 /CFT 6 duality As in lower dimensions, we may start with a free CFT in 6d described, e.g., by N (complex or real) scalars, spinors or rank 2 antisymmetric tensors and consider the duality between its singlet sector represented by the corresponding bilinear conserved currents and higher spin theory in AdS 7 (see, e.g., [26]). The representation content of the 7d theory is determined from the Flato-Fronsdal type decomposition of the product of 2 singleton representations into sum of higher-spin SO(2, 6) representations described in appendix C (see also [10]). Then using the general expressions for the Casimir energy (2.10) and a-anomaly coefficient (2.11) given in section 2 we may study the matching of these quantities on the two sides of the duality. In what follows we shall denote by K + the two quantities a + and E + c corresponding to AdS 7 field in a generic massless SO(2, 6) representation and also use K = −2 K + for the associated boundary conformal field values. Starting with the case of a free conformal scalar boundary 6d theory, the corresponding fields of the dual AdS 7 theory ("type A" theory) are massless totally symmetric tensors with spin s, for which we find from (2.10),(2.11) The Casimir energy (4.1) is a simple extension of the results in [27]. The a-anomaly expression (4.2) is the same as found in [26]. To sum over spins we shall follow the spectral ζ-function prescription of [26] which is equivalent to introducing the cutoff e − (s+ d−3 2 ) = e − (s+ 3 2 ) and dropping all singular terms in the limit → 0, i.e.
where K φ = (a φ , E c φ ) are the real scalar values from (A.1) and (A.5), As discussed in appendix C, the l.h.s. of (4.4) corresponds to the representation content of the tensor product of two scalar singletons and the associated sum of characters is equal to the partition function of the singlet sector of the 6d U (N ) invariant theory of N free complex scalars, see (C.5). The vanishing to the r.h.s. of (4.5) is consistent with the expectation that the a-anomaly and Casimir energy of the U (N ) 6d CFT which are proportional to N should be exactly reproduced by the classical action of "non-minimal" type A higher spin theory in AdS 7 with the inverse coupling G −1 non−min ∼ N , so that the one-loop HS correction should vanish [25,26].
The l.h.s. of (4.5) corresponds the field content of the "minimal" type A theory in AdS 7 which should be dual to singlet sector of O(N ) invariant free real scalar 6d theory, with the partition function relation given by (C.8) (for similar relations in the case of 3d and 4d cases see [27,10]). Here the non-vanishing r.h.s. may be cancelled against part of the classical contribution of non-minimal type A theory if one assumed that in this case G −1 min ∼ N − 1 [25,27]. Similarly, in the case when the boundary 6d theory is the U (N ) invariant free complex (Weyl) fermion theory or O(N ) invariant free Majorana-Weyl fermion theory (with the dual theory being non-minimal or minimal type B theory in AdS 7 ) we get where the field content corresponds to the one in the r.h.s. of (C.3),(C.6) and (C.9) and K ψ is given in (A.1),(A.5). Here we have also other representations than totally symmetric tensors and thus require general expressions in (2.10),(2.11). As in the scalar case, the non-vanishing r.h.s. of (4.7) may be compensated by assuming that the coupling constant of minimal type B theory is G −1 min ∼ N − 1. When the 6d boundary theory is described by N real or complex self-dual 2-tensors with dual theory being non-minimal or minimal "type C" theory in AdS 7 we find (see Here the non-vanishing result is found in both non-minimal and minimal cases. This is similar to what was found in the case of the AdS 5 /CFT 4 duality with the boundary theory represented by N complex or real Maxwell vectors [10,42]. The (real) vector corresponds to the parity invariant singleton combination {1} c = (2; 1, 0) + (2; 0, 1) in the SO(2, 4) notation. 15 There the r.h.s. of the analogs of eqs.  where the l.h.s. is computed for the representation content appearing in the character relation in (C.11). For example, in the case when the boundary theory is decsribed by N complex (2,0) tensor multiplets we have n φ = 5, n ψ = 4, n T = 1 we get where the tensor multiplet values of K tens. are given in (3.3). This may be compared with the relation found in the case of N = 4 vector multiplet in 4d [10,42]:

One-loop vacuum energy in 10d supergravity on AdS 3 × S 3 × M 4
As discussed in the Introduction, one may also perform a similar one-loop computations in the supergravity sector of type IIB superstring on AdS 3 × S 3 × T 4 to determine the subleading term in the central charge (1.8) or the vacuum energy (1.9). 16 The one-loop AdS 3 vacuum energy can be computed by starting with the spectrum of 6d supergravity on AdS 3 × S 3 as massive KK multiplets on M 4 = T 4 should not contribute due to supersymmetric cancellation. More generally, we may consider in parallel the cases of IIA or IIB supergravities on M 4 = T 4 or K3. The results for the one-loop vacuum energy are expected to be the same. 17 The list of relevant 6d supergravities with N = (n L , n R ) supersymmetry was given in [43], where an algorithm for construction of the corresponding KK spectrum on S 3 was presented. Below we shall consider the following cases: 10d 16 String modes corresponding to massive unprotected multiplets are expected not to contribute to c. 17 For example, type IIB theory on AdS3 × S 3 × T 4 with RR 3-form flux is S-dual to type IIB theory with NSNS flux and as the supergravity theory is S-duality invariant the same should be true for the value of Ec. Since NS-NS sector is common to IIB and IIA theories, the same result should be found also in the corresponding IIA theory.

KK towers of states on S 3
The 6d supergravity fields transform in representations (j 1 , j 2 ) of the 6d little group SO (4) SU (2) × SU (2) (of SO (1,5) in the tangent space). This gives a set Φ of representations of the diagonal subgroup SO(3) SU (2) of SO(4). Considering compactification on S 3 , the above SO(3) can be identified with the factor in S 3 = SO(4)/SO(3). Each representation R ∈ Φ is associated with a tower of KK states with SO(4) representations containing R under restriction to their diagonal SO(3).
These KK fields carry also representation of the AdS 3 isometry group SO(2, 2) (or global part of 2d conformal group) which are are labelled by scaling dimension and spin (∆, s), with ∆ ≥ |s|. The values of (∆, s) can be determined by re-organizing the KK towers in short supermultiplets of SU (2, 2 |1) × SU (2, 2 |1) since its generators include the dilatation (Virasoro L 0 ) and spin operators. The relevant short representations (J) s of SU (2, 2 |1) have the following content (J) s : where |0 is the lowest weight of the representation in the usual oscillator construction [44], Q ± are the supercharges, and j is SU (2) spin. Thus, in general, each short (J) s representation contains four SO(2, 2) representations. Using (5.2) and that ∆ = L 0 + L 0 , s = L 0 − L 0 one obtains the quantum numbers of representations in the tensor products (J, J) s . Let us now list the KK towers that appear in the theories in (5.1). For (2, 0) 6d supergravity, or IIB theory dimensionally reduced on K3 the field content is a graviton, five self-dual two-forms, four gravitinos, and n T = 21 tensor multiplet of one anti self-dual two-form, four fermions and five scalars (see also [45,46] Reorganising KK towers in short multiplets of SU (2, 2 |1) × SU (2, 2 |1), we find where The towers in the first and second sums are called spin-2 and spin-1 towers because of the maximum spin of their bottom floor = 0. The explicit field content is collected in Appendix E and their 6d origin is discussed in [45]. 18 We shall keep nT generic because this will be useful in comparing with IIA case.

Vacuum energy
The AdS 3 vacuum energy contributions of the above KK towers can be computed using the expressions for the characters or one-particle partition functions of the corresponding SO(2, 2) representations which we shall first recall. SO(2, 2) viewed as global conformal group in 2d is generated by the L 0 , L ±1 and L 0 , L ±1 Virasoro generators. Unitary irreducible representations of SO(2, 2) are massive for ∆ > |s| and massless for ∆ = |s|. A massive representation is built on a ground state |h, h with hh > 0. Thus, both L −1 and L −1 give a non zero result and the resulting character is (see, e.g., [44]) 20 (5.11) 19 Here we combine representations related by conjugation (j1, j2) → (j2, j1) since they give same contribution to KK spectrum. 20 The double factor of 1/(1 − q) takes into account multiple applications of both L−1 and L−1.
A massless representation with ∆ = |s| > 0 has conformal weights (h, 0) or (0, h). Acting with the lowering operators L −1 and L −1 on |h, h only one of them gives a non-zero result. As a consequence, here Finally, for ∆ = s = 0, we have only the ground state |0, 0 and Z + (0; 0) = 1. The expressions (5.11) and (5.12) can be used to prove that SU (1, 1 | 2) short multiplets obey the important relation E c = − 1 12 c, see (1.9). We discuss this in details in Appendix F. The contribution from a particular SO(2, 2) representation to the AdS 3 vacuum or S 1 2d Casimir energy E c can then be computed using (2.8),(2.9). Explicitly, for a massive field in AdS 3 , we may write the partition function (5.11) as We then obtain a formal (divergent) expression for the corresponding E c as (cf. In addition, we then need to sum over the KK states. There will be divergences coming from the sum over n, but also from the sum over the KK level . Like in AdS 5 × S 5 case [10] and AdS 7 × S 4 case in section 3 the total sum may be again computed using the ζ-function regularization applied to the full effective 6d energy eigenvalue ∆ + n, or, equivalently, by introducing the cutoff e n → e n e − (∆+n) , doing the sum, expanding in → 0, and dropping all singular terms. Applying this procedure to theKK towers appearing in (5.4),(5.7) and (5.9), we obtain where E c,extra is the contribution from the ( 1 2 , 1 2 ) s representation appearing in (5.4), (5.7), and (5.9) in the bottom part of the KK towers.
The above are the contributions from the massive SO(2, 2) representations. As discussed in [45,43], the resolution of the missing states puzzle raised in [48] amounts to the re-introduction of the massless representations ( = −1 states in the spin 2 and 3 2 towers). These are massless multiplets in AdS 3 that do not carry propagating degrees of freedom. Their structure is presented in Appendix E. For these multiplets we find

(5.18)
This is also the result for IIA theory on K3, as follows from (5.7). From (5.9) we also get exactly the same result for IIA or IIB theory on T 4 , 19) in agreement with the claim in (1.9).
B a-anomaly from spectral ζ-function in AdS 7 The a-coefficient of the boundary conformal anomaly can be determined from the logarithmic IR singular part of the one-loop partition function in Euclidean AdS 7 with boundary S 6 , i.e. hyperboloid H 7 (see, e.g., [55,32]) Here ζ(z) is the spectral zeta function found by evaluating the trace of the H 7 heat kernel [56] associated with the 7d operator O and R is an IR cutoff regularising the volume of H 7 . 22 Here the two additional terms are related to gauge freedom in the rank 2 tensor potential. 23 In the tensor case, the factor 1 2 is absent due to the self-duality condition.
Below we shall consider the operator O corresponding to a generic massive (or massless) higher spin field in representation (∆; h) generalizing the expression in [32] found in the totally symmetric tensor case h 1 = s, h 2 = h 3 = 0 24 Here D 2 is the standard Laplacian in AdS 7 defined on transverse field. The discussion will be parallel to the one in AdS 5 case in [10]. The spectral ζ-function of the operator O can be expressed in terms of the heat kernel Since H 7 is homogeneous, the trace over the position x gives a factor of (regularized) volume, i.e.
where tr is the trace over the representation indices of the operator and ζ(z; x) does not actually depend on x.
One can use the results for the heat kernel of the Laplacian in AdS 2n+1 with even n derived in [56,59] applying them to the case of n = 3. It is convenient to start with heat-kernel for the sphere S 7 and then analytically continue to AdS 7 . Let us consider a field on S 7 transforming under the tangent space rotations in a representation G of SO(7). Since S 7 = SO(8)/SO (7), the heat kernel receives contributions from each representation R of SO(8) that contains G when restricted to SO(7). Let us denote R and G by the corresponding weights as were all labels are integer or half integer. The branching condition on the representation R is with the additional requirement that i − g i ∈ Z. The heat kernel at the coincident points, traced over representation indices, can be written as where E

(H)
R are the eigenvalues of the Laplacian −D 2 on S 7 expressed in terms of the second Casimir values for the two representations and d R is the dimension of R C 2 (G) = g 2 3 + g 3 + g 1 (g 1 + 5) + g 2 (g 2 + 3) , (B.10) 24 For the general form of X see [57,34,58].
The analytic continuation from S 7 to AdS 7 amounts to [56,59] with the sum over 1 becoming an integral over λ ≥ 0. Finally, considering states saturating the inequalities (B.6) and identifying ( 2 , 3 , 4 ) = h = (h 1 , h 2 , h 3 ), we find that the eigenvalues of the operator (B.2) for the representation (∆; h) are The regularised volume may be written as Vol(H 7 ) = 1 3 π 3 log R + ... where the IR cutoff R is the radius of S 6 measured in 7d metric dρ 2 + sinh 2 ρ dΩ 2 6 at large ρ. Doing the analytic continuation (B.12) in the dimension d R in (B.11) we finally obtain (B.14) Integrating over λ and taking the z-derivative at z = 0 we may then use (B.1) to find the expression forâ + in (2.11).
Here the l.h.s. may be interpreted as the one-particle partition functions corresponding to the single sector of the U (N ) boundary theory, with The case of the real O(N ) invariant theory is found by an appropriate Z 2 projection. The corresponding sums then represent the partition functions of the singlet sector of O(N ) invariant free real scalar, Majorana fermion, and real self-dual tensor theories in 6d: These relations (C.5)-(C.7) can be generalized by considering a tensor product of the linear combination of singletons: . This gives for the corresponding characters Also, the analog of (C.10) is It is useful to derive the general expressions for the Casimir energy for spin j = 0, 1 2 , 1 SO(2, d) singletons in the general case of even dimension d = 2, 4, 6, ... of the boundary.