BPS spectrum on AdS$_3\times $S$^3 \times $S$^3 \times $S$^1$

The BPS spectrum of string theory on AdS$_3\times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ is determined using a world-sheet description in terms of WZW models. It is found that the theory only has BPS states with $j^+ = j^-$ where $j^{\pm}$ refer to the spins of the $\mathfrak{su}(2)$ algebras of the large ${\cal N}=4$ superconformal algebra. We then re-examine the BPS spectrum of the corresponding supergravity and find that, in contradistinction to previous claims in the literature, also in supergravity only the states with $j^+=j^-$ are BPS. This resolves a number of long-standing puzzles regarding the BPS spectrum of string theory and supergravity in this background.


Introduction
For the case of AdS 3 , the AdS/CFT correspondence has been understood in quite some detail. In particular, there is very convincing evidence that the CFT dual of string theory on AdS 3 × S 3 × M 4 is (on the moduli space of) the symmetric orbifold of M 4 , where M 4 = T 4 or M 4 = K3, see e.g. [1] for a review. For example, the BPS spectrum of the two descriptions matches perfectly, and their correlation functions agree [2][3][4]. More recently, it was also found that the symmetric orbifold is a natural extension of the CFT dual of a supersymmetric higher spin theory on AdS 3 [5]. For either choice of M 4 , the dual CFT has the (small) N = 4 superconformal symmetry [6][7][8].
On the other hand, the situation is much less clear [9] for the other maximally supersymmetric AdS 3 background AdS 3 × S 3 × S 3 × S 1 , and no convincing proposal for what precisely the dual CFT should be exists to date, see however [10] for a recent attempt. This is a bit surprising since the corresponding dual CFT has an even larger symmetry algebra, the so-called large N = 4 superconformal algebra A γ [11], see also [12][13][14][15]. One of the reasons why the case with large N = 4 superconformal symmetry is more difficult is related to the fact that the BPS bound for A γ is in general stronger than the corresponding BPS bound [16][17][18] of the supergravity symmetry algebra D(2, 1|α) [19]. Both algebras contain an su(2) ⊕ su(2) subalgebra, and the BPS bounds for the two algebras only agree if the spins with respect to these two algebras (j + , j − ) coincide, j + = j − . In particular, this leads to the somewhat mysterious phenomenon that any supergravity BPS state with j + = j − has to acquire nontrivial quantum corrections upon quantisation in order to satisfy (let alone saturate) the BPS bound of A γ [9,19]. This is not just a theoretical possibility since, according to the analysis of [19], such BPS states exist in supergravity.
In this paper we revisit this somewhat unsatisfactory situation. We begin by studying string theory on AdS 3 ×S 3 ×S 3 ×S 1 from a world-sheet prespective, following the analysis of [20]. The relevant background has pure NS-NS flux, and the AdS 3 factor is described by a WZW model associated to sl(2, R) as in [21], while for the S 3 factors we have the familiar su(2) WZW models. It was shown in [20] that the spacetime CFT (i.e., the dual CFT) has large N = 4 superconformal symmetry A γ . Thus we can analyse which states of the spacetime spectrum saturate the A γ BPS bound, and we find that the only states with this property have j + = j − . Furthermore, since string theory reduces to supergravity in the limit of vanishing string size, our analysis also leads to a prediction for what the supergravity BPS spectrum should be. This suggests that all the BPS states of supergravity also have j + = j − .
Given that this conclusion is in contradiction to the claim of [19], we perform then a first principle supergravity analysis, following the strategy of [22] suitably adjusted to the current setting. We should stress that this somewhat tedious analysis was not done in [19] where it was simply assumed that for each harmonic there would be a BPS state -as was indeed the case for AdS 3 × S 3 × M 4 with M 4 = T 4 or K3. Using this assumption the BPS states were then organised into supermultiplets, using group theoretic methods [19].
In our analysis, we start with 9-dimensional supergravity (with a pure NS-NS background) and compactify it on S 3 × S 3 , making an expansion in terms of spherical harmonics on the two spheres. We will concentrate on the scalar fields coming from the NS-NS fields in 9 dimensions; this is sufficient for the analysis of the BPS spectrum since every BPS multiplet in the list of [19] contains at least one such field. The resulting field equations are then Klein-Gordon equations from the viewpoint of AdS 3 , and hence we can easily read off their masses as a function of the spins along the two S 3 's. The analysis is however quite tedious, but with the help of Mathematica, 1 we have managed to find all eigenfunctions and identify their corresponding masses. The result turns out to be exactly as predicted from the world-sheet analysis: the only BPS states of supergravity appear for j + = j − . This is the main result of this paper. As was alluded to before, it resolves the puzzle about the mysterious quantum corrections of supergravity BPS states: the only BPS states that would have to behave in this manner arise for j + = j − -and the resolution is simply that no such supergravity BPS states exist! In fact, our analysis also shows that the actual masses of the supergravity states with j + = j − do not just obey the supergravity BPS bound, but in fact also the stronger A γ BPS bound (and saturate neither).
As a consequence of our analysis, also the question about the CFT dual to string theory on AdS 3 × S 3 × S 3 × S 1 needs to be re-examined. In particular, the most natural candidate theory at least for the case when the two S 3 's have the same size, the symmetric orbifold of the so-called S 0 theory [9,20], was largely discarded in [9] because of its failure to reproduce the BPS spectrum of supergravity -but since the latter was wrongly identified, this conclusion does not hold any more. In fact, it now seems that this symmetric orbifold is a viable candidate, and we are in the process of exploring this possibility further [23].
The paper is organised as follows. In Section 2 we review the world-sheet description of this background and then analyse its spacetime BPS spectrum. The main result of this section is eq. (2.9) which gives a lower bound on the conformal dimension of any spacetime state as a function of its spins. It follows from this bound that the only BPS states arise for j + = j − . We furthermore speculate in Section 2.2 what the supergravity incarnation of this result should be. In Section 3 we then perform the supergravity analysis from first principles: we first determine the 9-dimensional vacuum solution (Section 3.1), and deduce the equations of motion for the fluctuations around this vacuum solution (Section 3.2). The various components are then expanded in terms of spherical harmonics (Section 3.3), and the different modes are diagonalised into eigenfunctions of the Laplace operator on AdS 3 ; for three of the scalar fields this is done explicitly in Section 3.4, while the analysis of the remaining seven scalars is quite complicated and has been relegated to an appendix (Appendix B), with only the results being given in Section 3.5. The full scalar spectrum is then analysed in Section 3.6, and the above statements about the supergravity BPS bound are derived. Our analysis culminates in the description of the full supergravity spectrum in eq. (3.80). Finally, Section 4 contains our conclusions and outlines future directions of research. There are three appendices: in Appendix A we review the D(2, 1|α) and the large N = 4 superconformal algebra A γ and describe their BPS bounds. Appendix B contains part of the general supergravity analysis, while Appendix C deals with the special features that arise for the harmonics with small spin.

The world-sheet analysis
In this section we analyse the string spectrum on AdS 3 × S 3 × S 3 × S 1 for the case of pure NS-NS flux. This background may be described by a WZW model as in [20]. More specifically, the AdS 3 factor is captured by an N = 1 superconformal sl(2, R) WZW model at level k, while for the two S 3 factors we have N = 1 superconformal WZW models based on su(2) at levels k ± . Finally, the S 1 factor is just described by a free boson and a free fermion. After decoupling the respective fermions, the bosonic currents (that commute with the fermions) then have levels k +2, and k ± −2, respectively. The requirement that the string theory is critical, i.e., has total central charge c = 15, implies a relation between the levels, see e.g., [20] Geometrically, k ± describe the sizes of the two S 3 's, and (2.1) implies that the size of the AdS 3 space (that is described by k) is fixed in terms of these two.
To be more specific, we denote the bosonic modes of sl(2, R) at level k + 2 by J a n , while K ±,a n are the modes for su(2) at level k ± − 2. (In either case a = 3, ±.) The corresponding fermions will be denoted by ψ a r and χ ±,a r , respectively, while for the S 1 factor we have the bosonic and fermionic modes α n and b r , respectively. We also denote the corresponding supersymmetric currents by J a n and K ±,a n ; their levels are then k, and k ± , respectively. The N = 1 superconformal generators can be constructed in the usual manner [24].
In the following we shall mainly concentrate on the NS sector of the theory; we shall come back to the R sector at the end of this section. There the physical states Φ are characterised by the condition with a similar relation for the right-movers. We shall furthermore mainly consider the unflowed sector of the sl(2, R) WZW model [21]; we will later comment on the situation in the flowed sectors. The affine representations that appear in the spectrum are (in the unflowed sector) conventional highest weight representations, and can be characterised by , where j 0 is the spin of the sl(2, R) highest weight representation, while j ± 0 are the spins of the two su(2) highest weight representations. In our conventions, j 0 ≥ 0 labels the 'highest weight state' that is characterised by 3) and the Casimirs of the sl(2, R) and su(2) representations are Let us denote by N the excitation number of the physical state; then the mass-shell condition -the last equation in (2.2) -implies that We have analysed the other two constraints of (2.2) on the low-lying (N ≤ 3 2 ) physical states following the analysis of [25]. Modulo spurious null-states, we have found that the resulting spectrum has the standard form one expects from a light-cone gauge approach, where the two light-cone directions (whose oscillators do not create physical states) are the Cartan direction of sl(2, R), as well as the circle direction associated to the S 1 .

The spacetime BPS spectrum
It was shown in [20], building on [26], that the spacetime theory has a large N = 4 superconformal symmetry. The large N = 4 superconformal algebra is generated by the Virasoro algebra -the asymptotic symmetry algebra arising from AdS 3as well as two affine su(2) Kac-Moody algebras that arise from the two S 3 factors. Furthermore there is a u(1) algebra corresponding to the S 1 . 2 In the unflowed sector, the levels of the two su(2) algebras of the spactime N = 4 superconformal algebra can be identified with the levels of the two su(2) algebras on the world-sheet, while the central charge is equal to c = 6k, with k the level of the sl(2, R) WZW model [20,26]. The BPS bound of the large N = 4 superconformal algebra has the form [16][17][18], see also [19] h where h is the conformal dimension, while j ± are the spins with respect to the two su(2) algebras. Furthermore, u is the u(1) charge. In the following we shall only consider the neutral sector u = 0 -we have checked that there are no BPS states for u = 0.
In terms of world-sheet parameters, we can identify h with the eigenvalue j of J 3 0 , while j ± are the spins of K ±,a 0 . We want to ask which physical states of the world-sheet theory saturate the BPS bound (2.6). In order to identify the potential BPS states we will fix j ± , and look for the physical state with the lowest value of j. The eigenvalues j ± differ from those of the ground states j ± 0 by the charges that are carried by the oscillators, and we define If ∆ ± = 0, then we can use all the oscillators to lower j, leading to [Note that N has to be half-integer (because of the GSO projection); furthermore, for N > 1 2 we can use the fermionic mode ψ − −1/2 once, but then it becomes more efficient to use the bosonic J − −1 modes.] On the other hand, if ∆ + = 0, we need |∆ + | − 1 2 oscillators to obtain the correct j + spin and similarly for j − , and then only the remaining oscillators can be used to reduce the eigenvalue of j. Thus it seems plausible -and it is not hard to show rigorously, although the argument is a bit tedious -that the BPS states can only occur for ∆ + = ∆ − = 0 and N = 1 2 . (Note that this is also what one would have guessed on general grounds since N = 1 2 characterises the 'supergravity' states.) Then j ± = j ± 0 , and determining j 0 from the mass-shell condition and plugging it into (2.8) leads to where we have used that k is given by, see eq. (2.1) (2.10) One can convince oneself that (2.9) satisfies the BPS bound (A.25) provided that j 0 ≤ k+1 2 ; this latter condition is a consequence of the no-ghost theorem [21,25] and guarantees unitarity. More specifically, after squaring (j + 1 2 ) from (2.9) and comparing to (2.11), the BPS bound becomes (2.12) The first factor is clearly non-negative, while the square bracket defines an ellipse in the (j + , j − )-plane. One can see that this lies outside the variety -this is just the mass-shell condition for the maximal choice of j 0 = k+1 2 - 13) but they touch at the point (2.14) Thus the BPS bound is always satisfied, but it can (and is) only saturated for the case This is the main result of our world-sheet analysis. We have also performed a similar analysis for the spectrally flowed sectors in the NS sector. They give rise to further BPS states, as explained in detail in [33]. However, since supergravity corresponds to the regime k ± → ∞, these additional states are not important for the present analysis. Thus from now on we will only talk about the unflowed sector of string theory.
As regards the situation in the R sector, the analysis is similar to the above, and the only BPS states exist for j + = j − . The corresponding values of j + = j − are shifted by 1 2 relative to the NS analysis. These BPS states are therefore the states associated to the j + 1 2 term in (A.26), and they are required in order to get the complete multiplet of the large N = 4 superconformal algebra. Thus our analysis predicts that the entire BPS spectrum of string theory on AdS 3 ×S 3 ×S 3 ×S 1 consists of the representations BPS spectrum of string theory:

Supergravity interpretation
Given the general relation between string theory and supergravity, one should expect that (2.9) also has a direct supergravity interpretation. Let us consider a scalar field on AdS 3 that arises from a massless scalar field in 10 dimensions upon KK reduction on S 3 × S 3 × S 1 . If we take the KK momentum along the S 1 to be trivial, then its mass (in AdS 3 units, i.e, relative to the size k), is where the two terms on the right-hand-side are the eigenvalues of the Laplacian on the sphere for the spherical harmonic labelled by (j + , j − ). Again these eigenvalues are evaluated in appropriate units, i.e., relative to the sizes k ± of the corresponding S 3 's. We can convert this expression into the conformal dimension of the dual CFT, using the general relation between the mass of a scalar field and the left-moving conformal dimension ∆ = h +h (with h =h), This differs by a shift of 1 from (2.9) -as we shall see in the next section, the analysis is a bit more subtle, and there is in fact one scalar component for which (2.9) is precisely reproduced -but the dependence on the spins is exactly as in (2.9). For the supergravity analysis the relevant symmetry algebra is D(2, 1|α), whose BPS bound takes the form (A.12) (see Appendix A.1) (2.20) Provided that the actual supergravity analysis leads to (2.19) (or rather to h − 1), it follows by similar arguments as above that it can only be saturated if j + = j − . Indeed the supergravity bound (2.20) is weaker than the stringy bound (2.11), and hence at most those states that saturate the stringy bound can saturate the supergravity bound. Happily, the only states that saturate the stringy bound occur for j + = j − where the two bounds coincide.
If this somewhat sketchy line of reasoning is correct it would suggests that the BPS spectrum of supergravity on AdS 3 × S 3 × S 3 × S 1 consists only of representations of D(2, 1|α) with j + = j − . This is contrary to what was claimed or assumed in the literature before, see in particular [19]. We shall therefore, in the next section, perform a careful and detailed supergravity computation to confirm this claim explicitly.

Supergravity approach
Let us start with 10-dimensional IIB supergravity. The bosonic part of the action is, in the string frame, given by Here, Φ is the 10d dilaton field, g is the 10d metric with R its associated curvature scalar, and B is the Kalb-Ramond field. C 0 , C 2 and C 4 are the R-R-fields, whose fields strengths are defined as Finally, we have to impose self-duality onF 5 , We will look at solutions on AdS 3 × S 3 + × S 3 − × S 1 where we have pure NS-flux on the AdS 3 factor and through the two S 3 ± . It is consistent to set all R-R fields (as well as all the fermions) to zero. Then we are only left with the NS-NS fields, i.e., the metric, the dilaton and H, whose action isS

The 9d vacuum solution
We now compactify the theory on a circle to get an effective 9d theory. In the process, we have to dimensionally reduce the fields, see, e.g., [27]. The 10d metric g gives rise to a 9d-metric, which we again call g, a vector A µ = g µ,10 and a scalar k = g 10, 10 .
The Kalb-Ramond field B leads to a vectorÂ µ = B µ,10 and a 9d two-form B. We call the field strengths associated to A andÂ, F andF , respectively. The resulting action is then -this is eq. (21.25) of [27], after changing the signature, rescaling the fields and setting Ψ = log k, The result is invariant under T-duality, which interchanges A andÂ. The field equations for Ψ and the one-forms read It is hence consistent to set them all to zero. The Φ-equation of motion is We set Φ = 0 (or Φ at least constant), which imposes the condition The g-equation of motion is the usual Einstein equation, where we have used (3.10). Taking the trace gives i.e. H is closed and coclosed and hence harmonic. In order to find the vacuum solution corresponding to AdS 3 × S 3 + × S 3 − , let us denote their radii by r (for AdS 3 ), and r ± (for S 3 ± ). All factors are maximally symmetric, and thus the Riemann tensor takes the form if all indices are in one factor. We thus have where we view g AdS 3 etc. as 10d fields, i.e., g AdS 3 M N = 0 except when 0 ≤ M, N ≤ 2, etc. Contracting indices gives Following [9], we now make the ansatz for H where ω AdS 3 are the volume forms on AdS 3 , etc. This form is trivially closed. The volume forms are harmonic and hence also coclosed, so H obeys the equation of motion. The normalisations are chosen as and similarly for AdS 3 , which can be obtained by analytic continuation. We have |ω AdS 3 | 2 = −1 and |ω S 3 ± | 2 = 1, i.e. in total: The Einstein equations exhibit a connection between the λ's and the radii, namely and furthermore the vanishing of R implies The fluxes through S 3 ± are given by and are always integers. As a consequence, (3.22) coincides with the string theory criticality condition on the levels, see eq. (2.1) where we defined in analogy k as k = 4π 2 r 2 .

Equations of motion for quadratic perturbations
We now consider the fluctuations around this background geometry. To second order in the variations the above action becomes where L is explicitly given as Here we wrote δg M N = h M N , and we treat h M N as a tensor, i.e., The first terms are kinetic terms, but there are also mass terms. The connection is still the Levi-Civita connection of the background metric, and we raise and lower indices with the background metric. We used that AdS 3 × S 3 + × S 3 − has vanishing Ricci scalar and vanishing |H| 2 ; we have also already inserted the other background values of the fields and used the Einstein equation of the background solution. Note that there are no interference terms between Z,Ẑ, ψ, and the other fields, and hence we can treat the fluctuations of F ,F and Ψ separately from the rest. The resulting equations of motion for the remaining quadratic fluctuations are where δB M N = X M N .

Expanding the Fields
Following [22], we parametrise the metric fluctuations as Here and from now on, greek indices refer to AdS 3 , latin indices from the beginning of the alphabet a, b, . . . to S 3 + , while the latin indices from the middle of the alphabet i, j, . . . refer to S 3 − . We will use capital latin letters to indicate 9d-indices. For the antisymmetric tensor field, we have H M N P = 3∂ [M B N P ] , and we parametrise the fluctuations as We expand the fluctuations in harmonic functions on the two S 3 's as Here i ∈ N 0 , and the relation to the previously used j variables is The space of 1-forms on S 3 + is spanned by and the space of traceless symmetric 2-tensors is spanned by As in [22], we choose the 'Lorentz gauge' This gauge removes 9 degrees of freedom of the metric and 8 degrees of freedom of X, which are the right numbers. Hence locally, this gauge is admissible, and one can also confirm this more carefully. We should note that this gauge choice breaks the manifest symmetry between the two spheres (since we impose the divergence condition only with respect to the a-coordinates on S 3 + ). However, as we shall see later, the resulting spectrum will be symmetrical with respect to exchanging the two spheres. We should note that eq. (3.49) implies in particular that Additionally, there is a gauge freedom in the decoupled subsystem which will be fixed in the next section.

The scalars from the decoupled subsystem
As was mentioned before, we can analyse the fluctuations of Ψ, F andF separately from the rest; since this part of the analysis is simpler, let us first explain that. The equation of motion of Ψ is simply Expanding Ψ in terms of harmonics, and using (B.1), we get where To fix the gauge, we require the equivalent of the last equation of eq. (3.49) The equations of motion, split into the different components, read We shall in the following always focus on scalar quantities, i.e., on modes that are scalar with respect to AdS 3 as well as the two S 3 ± . In the present context, there are two of those, namely ∇ µ Π µ and ∇ i Ω i . Thus, we have an overconstrained system, since the scalar parts of (3.58) -(3.60) constitute three equations for these two fields. Let us extract the scalar parts of these three equations by applying ∇ µ to the first equation, ∇ a to the second one and ∇ i to the third one; the result is For + > 0, the second equation implies that there is actually only one scalar field, which we may take to be ∇ µ Π µ ; for + = 0, the situation is more complicated and requires fixing the residual gauge, as explained in Appendix C. Assuming + > 0, the first and third equation are equivalent to (3.64) (Later, we will also have a similar situation for the dilaton, the metric and the Kalb-Ramond field, i.e., the overconstrained system of scalar equations will imply some algebraic relationships among the fields, and once we have found these, there are some linear dependencies among the remaining equations.) We hence get again one set of scalar fields with mass Thus in total we find three sets of scalar fields, one from Ψ, one from A M and one fromÂ M , all having equal mass.

The scalars from the remaining fields
The analysis of the remaining fluctuations is more complicated, and the details are spelled out in Appendix B (and have been largely performed with the help of Mathematica, see also the ancillary workbook of the arXiv submission). It follows from eqs. (B.70) -(B.76) that all the remaining fields mix, but one can diagonalize the corresponding matrix. The eigenvectors are complicated, but the eigenvaluesthey correspond to the masses of the particles, since the above system of equation is simply the Klein-Gordon equation of seven coupled scalar particles on AdS 3 -are quite simple. Five of the eigenvalues of the matrix are directly of the above form where m 2 + , − is defined in (3.55), see also (3.65). The remaining two eigenvalues are more complicated and take the form (3.71) Hence their associated conformal dimensions, which are obtained via r 2 λ 2 = h(h−1), are where h + , − is the conformal dimension associated to m 2 + , − , In the final step we have used eq. (3.46) to convert the integer valued labels ± into the spin labels j ± , and eq. (3.23) to change from the radii to the levels we used earlier.

The full scalar spectrum
Taking these results together, we thus conclude that the supergravity spectrum contains the ten scalar fields Note that we have suppressedh andj ± , since they agree with h and j ± , respectively, as we are only considering scalars, see the comment after eq. (3.60). We have furthermore assumed that j ± ≥ 1; for small values of j ± (or rather ± ) the above analysis has to be adjusted, and this is explained in detail in Appendix C. (In particular, it follows that if j ± = 1 2 , the fields with spin j ± − 1 are absent; in addition, there are further restrictions if either (or both) spins vanish, j ± = 0.) The above spectrum accounts correctly for the scalar modes of the NS-NS fields; there are also scalar fields arising from the R-R fields.
We note that h j + ,j − agrees precisely with our naive prediction from above, see eq. (2.19). As was explained there, this differs by 1 from (2.9); thus the only state that can directly satisfy the BPS bound is the one corresponding to the eigenvalue h j + ,j − − 1, and it actually only saturates the bound if j + = j − .
We can now compare this to the supergravity spectrum spelled out in [19], where it was claimed that it takes the form Bosonic: except that the multiplet [0, 0; 0, 0] s does not appear in the sum. Fermionic: where the first and second factor corresponds to the left-and right-movers, respectively (and our conventions for the D(2, 1|α) representations are described in Appendix A.1.1). Since we have only analysed the scalar fields from the NS-NS sector, we cannot see all states from our above analysis; however, each multiplet (with the exception of the supergravity multiplet [ 1 2 , 1 2 ; 0, 0] s and [0, 0; 1 2 , 1 2 ] s ) contains at least one NS-NS scalar field, and thus we can deduce the conformal dimensions of the relevant multiplets. In particular, in the first sum in (3.75) the ground states are NS-NS scalars, and they contain the four modes On the other hand, the ground states of the second sum in (3.75) are R-R states, and their NS-NS contributions are then Similarly, the two fermionic multiplets contain one NS-NS scalar each In particular, it therefore follows that the first multiplet (3.77) is only BPS if j + = j − , since only in this case does h j+,j − −1 saturate the BPS bound; if j + = j − , it is actually not BPS and combines with the second term in (3.75) to form a long multiplet, see eq. (A.14). The analysis in the other sectors works similarly (as does the various shortenings for small spin), and we therefore conclude that the correct supergravity spectrum takes the form except that, as before, see eq.

Discussion and Outlook
In this paper we have analysed the BPS spectrum of supergravity and string theory on AdS 3 × S 3 × S 3 × S 1 (with pure NS-NS flux). We have found that both spectra only contain BPS states in representations for which j + = j − . Furthermore, the BPS spectra of both descriptions agree since the BPS part of supergravity -the first sum in (3.80) -agrees exactly with (2.16). Here we have used (A.26), namely that each BPS representation of the large N = 4 superconformal algebra contains two BPS representations of the supergravity symmetry algebra D(2, 1|α). Our finding resolves a number of long-standing puzzles. In particular, the mysterious fact that the BPS bound for D(2, 1|α) is strictly weaker than that of the large N = 4 superconformal algebra -compare (A.12) and (A.25) -seemed to imply that the supergravity BPS states have to acquire miraculous quantum corrections [9,19] in order to even satisfy the stringy BPS bound. This problem has now disappeared since the bounds only differ for j + = j − -but for that case, there simply aren't any supergravity BPS states (and the supergravity states that do exist already satisfy the stringy BPS bound without any correction).
The other important consequence is that our result changes significantly the expectations for what the CFT dual of string theory on AdS 3 × S 3 × S 3 × S 1 should be. In particular, the symmetric orbifold of the S 0 theory that was first proposed for the case of k + = k − in [20] and essentially ruled out because of its failure to reproduce the alleged BPS spectrum of supergravity [9] now appears to be a viable candidate after all; we will come back to analysing this possibility in more detail elsewhere [23]. We should also mention that, using integrability techniques, the BPS spectrum of string theory was analysed in [28], and that also from that perspective only BPS states with j + = j − were found. This suggests that the BPS spectrum agrees for all (generic) points in moduli space (as is also the case for AdS 3 × S 3 × M 4 with M 4 = T 4 or M = K3); thus one should be able to identify the dual CFT directly based on the BPS spectrum, without any need to resort to the index techniques of [29].

Acknowledgements
This paper is largely based on the Master thesis of one of us (LE). We thank Alessandro Sfondrini for comments on the draft and for sharing with us [28] prior to publication. We also thank Jan de Boer and Greg Moore for email correspondences, and Kevin Ferreira and Juan Jottar for discussions. The work of MRG is partly supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. He also thanks ITP of the Chinese Academy of Science for hospitality during the final stages of this work.
while [L m , A ±,i 0 ] = 0. Furthermore, the expressions α ± i ab are the 4 × 4 matrices that satisfy the relations The parameter γ that appears in these commutation relations is expressed in terms of α as Note that the algebra is isomorphic under γ ↔ (1 − γ); in terms of α this is the transformation α ↔ α −1 .

A.1.1 BPS representations
The highest weight representations of D(2, 1|α) are labelled by j + , j − , h, where j ± are the spins of the two su(2) algebras generated by A ± i 0 , while h is the eigenvalue of L 0 . (The highest weight states are annihilated by the positive modes, G a 1/2 and L 1 .) A generic (long) representation has the form (A.11) where the different lines correspond to states with conformal dimension h = h 0 , h = h 0 + 1 2 , h = h 0 + 1, h = h 0 + 3 2 and h = h 0 + 2, respectively, whose su(2) ⊕ su(2) representation is given. The BPS bound takes the form, see e.g., [9,19] The corresponding BPS representation then consists of the subset of states, see [19] eq. (4.2) Here h 0 = ( 1 1+α j − + α 1+α j + ) saturates the BPS bound. We shall denote the long representation (A.11) as [j + , j − ], and the short representation (A.13) as [j + , j − ] s . Note that each long representation contains the set of states corresponding to two short representations The above description is only correct if j ± ≥ 1; for small values of j ± the representations are further shortened; explicit formulae for these representations are given in [19, eq. (4.3)].

A.2 The large N = 4 superconformal algebra
The large N = 4 superconformal algebra A γ whose wedge algebra is D(2, 1|α) is defined by (we follow the conventions of [30]), In terms of the levels of the two su(2) algebras, we have

A.2.1 The BPS Bound
The highest weight representations of the large superconformal N = 4 algebra A γ are characterised by (h, j ± , u), where h is the conformal dimension of the highest weight states, while j ± are the spins of the two affine su(2) algebras, and u denotes the u(1)-charge, i.e. the eigenvalue under U 0 . If we require unitarity, we need that j ± ≤ k ± /2. However, as explained in [16], unitarity actually requires that The BPS bound takes the form [16][17][18] h ≥ 1 (A. 25) Note that this bound differs from the the corresponding BPS bound of the wedge algebra D(2, 1|α), see (A.12); apart from the additional u 2 term there is in particular also the (j + − j − ) 2 term. If we denote the corresponding representation by [j + , j − , u] then it only satisfies the BPS bound of D(2, 1|α) if u = 0 and j + = j − . On the other hand, if this is the case, the BPS representation [j + , j − , u] of the linear A γ algebra contains actually two BPS representations of D(2, 1|α) This is basically a consequence of the fact that in addition to the four supercharges (that also appear in D(2, 1|α)), A γ also contains four free fermions.

B The supergravity analysis
In this section we give some more details of the supergravity analysis of section 3.
To start with, let us collect various identities that describe the action of differential operators on the spherical harmonics: i.e. for each ± ∈ N 0 we have one set of harmonics.

B.3 Scalar Part of the Equations of Motion
From now on, we discuss the generic case where + ≥ 2 and − ≥ 2. For low ± , there are additional issues to be taken care of; these cases are discussed in Appendix C. We now extract the scalar part of these equations. We have the following scalars: Since in the following, only Ω ( + 0)( − 0) for some quantity Ω appears and we will consider fixed + and − , we will use the shorthand notation Let us first discuss the equations. (B.37) gives immediately D = 0, and this then also implies immediately the scalar parts of (B.38) and (B.39). On (B.28), we will apply ∇ a ∇ b , then we will get an algebraic equation relating Φ, M, N and P. Extracting the scalar part of (B.33) by applying ∇ a ∇ i will yield a further algebraic equation. A last algebraic equation will come from a combination of (B.25) and (B.27). We then have four algebraic equations, cutting down the number of scalar fields to seven. These seven fields combine with the three scalar fields we found earlier to yield a total of ten scalar fields in the compactification. Let us begin with this outlined program. From (B.28), we can upon application of ∇ a ∇ b directly deduce that 3M + N + 3P − 4Φ = 0 .
C Special cases at low ±

C.1 Residual gauge transformations
Before discussing the various special cases, we have to find all residual gauge transformations, since those gauge away modes with low + . There are two of those, modifying scalar fields. AdS 3 ×S 3 -reparametrisations: They induce the following transformations on the fields (only the non-trivial transformations are displayed): Stückelberg shift symmetries: Transforming the coordinate system with this vector field preserves the gauge (3.49) and we find for the metric We will not need the transformation properties of the antisymmetric tensor field.

C.2 Metric, Dilaton and Kalb-Ramond Field
Let us first describe how the analysis of Appendix B gets modified for small ± ; the corresponding analysis for the decoupled subsystem of Section 3.4 will be described below in Section C.3. where λ 6/7 are as in (3.71). Since the equations of motion of the residual gauge transformation (C.22) and the eigenvector corresponding to 3r −2 coincide, we can gauge that eigenvector away. Thus, we find again the same states as in the previous subsection, except that + and − are interchanged. Again, using a residual gauge transformation, we can gauge the eigenvector corresponding to 3r −2 away.