Integrability and the AdS(3)/CFT(2) correspondence

We investigate the AdS(3)/CFT(2) correspondence for theories with 16 supercharges using the integrability approach. We construct Green-Schwarz actions for Type IIB strings on AdS(3)xS(3)xM(4) where M(4)=T(4) or S(3)xS(1) using the coset approach. These actions are based on a Z(4) automorphism of the super-coset D(2,1;alpha)xD(2,1;alpha)/SO(1,2)xSO(3)xSO(3). The equations of motion admit a representation in terms of a Lax connection, showing that the system is classically integrable. We present the finite gap equations for these actions. When alpha=0,1/2,1 we propose a set of quantum Bethe equations valid at all values of the coupling. The AdS(3)/CFT(2) duals contain novel massless modes whose role remains to be explored.


Introduction
Several AdS/CFT systems possess integrable structures and are solvable non-perturbatively by Bethe ansatz techniques. Integrability tools such as Bethe ansatz, exact Smatrices, bootstrap and finite-gap integration proved useful in finding the exact spectrum of the AdS 5 /CFT 4 and AdS 4 /CFT 3 systems at the planar/free-string level. We believe that this is not the end of the story and that other integrable AdS/CFT systems should exist. In this paper we extend the integrability approach to the AdS 3 /CFT 2 dual pairs with sixteen supercharges. The AdS 3 backgrounds typically arise from the D1-D5 system in type IIB string theory [1] and are dual to two-dimensional conformal field theories [2].
The AdS 3 /CFT 2 duality may appear simpler than its higher-dimensional counterparts, partly because the conformal symmetry in two dimensions is larger and more constraining, partly because problems with string quantization on Ramond-Ramond backgrounds inevitable in higher dimensions can be circumvented in d = 3. Some AdS 3 × X backgrounds can be supported by pure NSNS flux, which leads to enormous simplification of the worldsheet CFT. The worldsheet sigma-model then admits an NSR description, has extended chiral symmetry, and as a consequence is solvable by representation theory of chiral algebras [3,4,5]. On the contrary, the RR AdS 3 sigma-model [6] does not have useful holomorphicity properties and the usual CFT methods do not work. In this respect, the AdS 3 backgrounds with the RR flux are as complicated as their higherdimensional counterparts such as AdS 5 × S 5 or AdS 4 × CP 3 , which however are exactly solvable due to their integrability. We would like to argue that the appropriate method to attack the problem of string quantization on the RR AdS 3 backgrounds is also the Bethe ansatz. Clear evidence for integrability in AdS 3 /CFT 2 comes from the geometric construction of the Green-Schwarz string action on AdS 3 × S 3 [7,8,9], and from the symmetries of the giant magnons on this background [10].
Within the context of the AdS/CFT correspondence, integrability manifests itself in a number of ways. In this paper we will be mostly concerned with the string (AdS) side of the duality, where the construction is particularly simple. The classical integrability of the string sigma-model follows from the Lax representation of the equations of motion [11]. The monodromy matrix of the Lax connection then generates an infinite set of local or non-local commuting conserved charges. Given the monodromy matrix, one can use the finite-gap methods to solve the equations of motion in terms of a much simpler set of integral equations [12,13]. These same equations arise in the semiclassical limit of the quantum Bethe ansatz [14] that diagonalizes the exact worldsheet S-matrix [15,16]. The experience with AdS 5 /CFT 4 [14] and AdS 4 /CFT 3 [17,18] systems shows that the quantum Bethe equations are very constrained by symmetries [14,19] and can be almost uniquely reconstructed from their semiclassical counterparts [20]. It is not clear to us if the Y-system or the TBA equations [21] can be reconstructed from the classical data. The semiclassical limit is encoded in the Y-system, albeit in a rather non-trivial way [22].
The key property of the AdS 5 × S 5 and AdS 4 × CP 3 backgrounds that guarantees their integrability is the geometric construction of the Green-Schwarz action in terms of the coset superspace [23,24,25,26] 1 . If the coset admits a 4 grading [27], integrability follows automatically, since the equations of motion and Maurer-Cartan equations of any 4 coset can be written as a flatness conditions for a Lax connection [11]. It has been known for a long time that the Green-Schwarz sigma-model on the six-dimensional AdS 3 × S 3 background is the P SU(1, 1|2) × P SU(1, 1|2)/SU(1, 1) × SU(2) supercoset [7] 2 which, as one can check, possesses a 4 grading and is therefore integrable [8] 3 .
We will consider Type IIB strings on AdS 3 ×S 3 ×M 4 where M 4 = T 4 or 4 S 3 ×S 1 . The case of S 3 × S 1 in a certain sense is more general although definitely more complicated. The AdS 3 × S 3 × S 3 × S 1 geometry is a supergravity solution with sixteen supercharges [29,30,31,32,33] and can be supported by either NSNS or RR three-form flux. In the NSNS case the duality is relatively well understood, because one can use the NSR formalism and more or less standard CFT methods to quantize string in this background [4,32,34,33]. On the contrary, the AdS 3 /CFT 2 duality for AdS 3 ×S 3 ×S 3 ×S 1 supported by the RR flux perhaps is the most obscure case among all known AdS/CFT pairs. The AdS/CFT correspondence for AdS 3 × S 3 × S 3 × S 1 was discussed in [32] and at length in [33]. The string in this RR background cannot be quantized by any known method. As far as the dual CFT is concerned, very little is known about it, apart from its rather intricate symmetries. The CFT is probably a resolution of a permutation orbifold [32], and displays a number of unusual features, in particular the non-linear BPS bound [32]. All this makes identification of the spectra and the moduli spaces on the two sides of the duality problematic even at the supergravity level [33].
The radii of the two three-spheres (R ± ) and the AdS radius (l) are not independent in AdS 3 × S 3 × S 3 × S 1 . The supergravity equations of motion require them to satisfy the triangle equality: 1 (1.1) The same triangle equality arises in the invariant bilinear form of the exceptional Lie superalgebra d(2, 1; α) [35,36], eq. (A. 3), and not just by chance -the symmetry of the corresponding AdS/CFT pair is the large N = 4 superconformal algebra [30] whose rigid part is d(2, 1; α). Indeed, the super-isometries of the AdS 3 × S 3 × S 3 background form two copies of d(2, 1; α) [31]. The parameter α, potentially any complex number [35,36], is related to the relative size of the two spheres and takes values between 0 and 1 when a suitable reality condition is imposed on the super-algebra. In view of the triangle equality (1.1) we shall use the trigonometric parameterization: There are two simplifying limits worth mentioning. One is φ = 0. The radii of AdS and of one of the spheres then become equal, while the other sphere blows up to an infinite size. By re-compactifying on T 3 , we get the AdS 3 × S 3 × T 4 background. This limiting case corresponds to the α → 1 degeneration of the d(2, 1; α) algebra, which up to some abelian factors contracts to psu (1, 1|2). The symmetry algebra of AdS 3 × S 3 is indeed P SU(1, 1|2) × P SU (1, 1|2). The other special point is φ = π/4, when the two spheres have equal sizes, √ 2 times smaller than the radius of AdS. The exceptional superalgebra d(2, 1; α) (with α = 1/2) then coincides with the classical osp(4|2) superalgebra from the d(n, m) series.
The plan of the paper is the following. In section 2 we recall the standard classical integrable structure of a general supercoset with Z 4 grading. We then specify the general construction to the AdS 3 × S 3 × S 3 × S 1 case. In section 3 we show that the supercoset action in the flat space limit reproduces the flat space GS action in a particular κ-gauge. In section 4, we demonstrate that, up to terms quadratic in fermions, our supercoset sigma model is indeed a realization of worldsheet theory of GS string on the AdS 3 × S 3 × S 3 × S 1 background with completely fixed κ-symmetry identical to the one used in section 3. In section 5 we investigate the BMN limit of the coset sigma model in light cone gauge and compare it to the supergravity analysis in [37,33]. We make some preliminary steps towards going beyond the strict BMN limit. In section 6 we point out that the type IIB GS string on the purely RR AdS 3 × S 3 × T 4 background can be treated as a limiting case of the AdS 3 × S 3 × S 3 × S 1 super coset action. In section 7 we discuss the general classical integrability scheme for Z 4 symmetric (super)cosets, and derive the finite gap equations entirely in terms of the group-theory data. We then derive classical Bethe equations for AdS 3 × S 3 × S 3 × S 1 and discuss their BMN limit. We propose quantum Bethe equations for two special points, φ = π/4 and φ = 0, the latter case corresponding to AdS 3 × S 3 × T 4 . In section 8 we conclude with a preliminary discussion of the massless modes, which is a novel feature of the AdS 3 /CFT 2 duality compared to AdS 5 /CFT 4 or AdS 4 /CFT 3 . In the appendices we collect the commutation relations of the superalgebra d(2, 1; α), the gamma matrix conventions, the backgroundfield expansion of the general 4 coset action, and some higher order terms of the near-BMN expansion in AdS 3 × S 3 × S 3 × S 1 .

4 cosets
The space-time supersymmetric action of the superstring in flat space [38] can be interpreted as a coset sigma-model [39]. The coset construction readily generalizes to curved space, and in particular allows one to build the Green-Schwarz action for a number of AdS-type backgrounds. The basic example is the Metsaev-Tseytlin action in AdS 5 × S 5 [23] (see [40] for a recent review). As realized in [27], an important feature of the AdS-type cosets is the 4 symmetry 5 , that in particular allows one to construct the Wess-Zumino term necessary for the consistency of the Green-Schwarz action. Integrability arises as a bonus symmetry in all 4 cosets. Indeed, the derivation of the Lax pair for the AdS 5 ×S 5 sigma-model [11] does not really depend on the specifics of the background and relies solely on the existence of the 4 structure. 5 The manifestly 4 invariant form of the Metsaev-Tseytlin action is given in [41].

The action and equations of motion
We start by reviewing the general construction of the sigma-model action for 4 cosets. A coset G/H 0 possesses a 4 symmetry, if the superalgebra g admits a 4 decomposition: consistent with the (anti-)commutation relations: [h n , h m } ⊂ h (n+m) mod 4 . Equivalently the 4 symmetry is associated with an order-four automorphism of the Lie superalgebra g, which is a linear map Ω : g → g that satisfies [Ω(X), Ω(Y )} = Ω([X, Y }) and Ω 4 = id.
The subspace h n then is defined as a subset of generators whose 4 charge is n, in the basis in which Ω is diagonal: The denominator of the coset is the Lie group of the invariant subalgebra h 0 in the 4 decomposition. The fermion number F is the 4 charge mod 2, so that h 0 ⊕ h 2 is the bosonic subalgebra of g, and h 1 , h 3 consist of Grassmann-odd generators. The worldsheet embedding in G/H is parameterized by a coset representative g(x) ∈ G, subject to gauge transformations g( . The action and the equation of motion can be written in terms of the left-invariant current (2. 3) The h 0 component of the current transforms as a connection under the gauge transformations: The other three components transform as matter fields in the adjoint: J a 1,2,3 → h −1 J a 1,2,3 h. The action of the sigma model is 6 Here Str(· ·) denotes the G and 4 invariant bilinear form on g. This action is obviously gauge invariant and 4 -symmetric. The equations of motion for this action and the Bianchi identities for the currents (the Maurer-Cartan equations) read: . These equations admit a Lax representation, they are equivalent to the flatness condition for the connection [11] L a = J a 0 + x 2 + 1 The spectral parameter x is an arbitrary complex number x = ±1. Provided that the currents obey the equations of motion, the Lax connection satisfies And conversely, if the connection L a is flat for any x, the currents satisfy the equations of motion. The Wilson loop of the Lax connection defines an infinite set of conserved charges, which include the global Noether charges of the left group multiplication. These Noether charges are expressed in term of the gauge-invariant right current that can be obtained from the left currents J a n by conjugation with the coset representative g: This current is conserved: as a consequence of the equations of motion.

AdS
The conformal algebra in two dimensions is a two-fold tensor product, with two factors acting independently on the left and right movers. The cosets appropriate for the AdS 3 /CFT 2 correspondence are thus of the form H ×H/H 0 . If H is a superalgebra, such a coset will naturally have a 4 structure. Indeed, one can define a 4 automorphism on g = h ⊕ h by combining the fermion parity with the permutation of the two factors: This map satisfies all necessary requirements: it preserves the (anti)-commutation relations of h ⊕ h and squares to (−1) F such that its forth power is the identity: Ω 4 = id. The 4 grading associated with the automorphism (2.10) is In particular, the invariant subspace is the diagonal bosonic subalgebra.
Thus for any superalgebra H one can construct a 4 invariant coset sigma-model with the global H × H symmetry. The denominator of the coset is the diagonal bosonic subgroup. The bosonic part of the action is the sigma-model with the target space H bos × H bos /H diag isomorphic to 7 H bos . Thus constructed sigma-model will be automatically integrable.
It is not immediately clear if this model is capable of describing the Green-Schwarz superstring on AdS 3 × S 3 × S 3 × S 1 , because of the missing S 1 factor. This factor has to be added by hand. Similar situation occurs in the hybrid sigma-model on AdS 3 ×S 3 ×T 4 [6], where the T 4 factor is completely orthogonal to the non-linear part of the action. The hybrid formalism assumes the conformal gauge from the very beginning [6], and adding an independent CFT is not a problem provided the total central charge vanishes. On the contrary, in the Green-Schwarz action all bosons are coupled to all fermions through the kinetic termθ I ∂ a X M Γ M ∂ b θ J . The desired decoupling of a bosonic direction is essentially equivalent to setting one of the Dirac matrices to zero. This does not sound right at all. In addition, there are 32 fermions in the Green-Schwarz action compared to sixteen (2 × the number of supercharges in d(2, 1; α)) fermion degrees of freedom in the D(2, 1; α) × D(2, 1; α)/SU(1, 1) × SU(2) × SU(2) coset. However, half of the 32 Green-Schwarz fermions are unphysical because of kappa-symmetry. In the next two sections we will demonstrate that by taking a special gauge choice of the kappa-symmetry it is possible to decouple the S 1 factor in the AdS 3 × S 3 × S 3 × S 1 supergeomerty, and that the resulting kappa-fixed GS action coincides with the coset model plus one free boson.
The fact that the S 1 appears as an extra factor in the action without any couplings to the other fields can be anticipated from the structure of the Killing spinors [31]. In particular, the momentum Killing vector for S 1 does not appear on the right-hand-side of the anti-commutator of the Killing spinors for this background and also commutes with all the supercharges [31] 8 .

Flat space limit
In this section we find the flat space limit of the action (2.4) for the D(2, 1; α) × D(2, 1; α)/SU(1, 1) × SU(2) × SU(2) coset, and show that it coincides with the usual flat space Green-Schwarz action [38] in a particular kappa-gauge. 7 In the gauge orbit {(g L h, g R h)|h ∈ H bos }, we can pick a representative by taking h = g −1 L , or in other words impose g L (x) = 1 as a gauge condition. What remains is the group manifold {(1, g)}. 8 We are grateful to Jerome Gauntlett for a detailed explanation of these and related results of [31].
To obtain the flat space limit of the d(2, 1; α) 2 algebra we should rescale the generators as follows and take the R → ∞ limit. In this limit the generators of h 0 become the angular momenta of SO(1, 2) × SO(3) × SO(3) ⊂ SO (1,9), while the elements of h 2 become flat space-time momenta; it is easy to check that, in the R → ∞ limit, the elements of h 2 commute with themselves and the supercharges -as is the case for flat-space momenta.
In the flat space limit the anti-commutators (A.2) become where I = 1, 2, n − R Ṙ n , and the indices L and R distinguish the two copies of d(2, 1; α). Recall that the IIB flat space supersymmetry algebra is where I , J = 1 , 2 counts the amount of 10d supersymmetry,â ,b = 1, . . . , 32 are spinor indices of SO (1,9), M = 0, . . . , 9 is the 10d vector index, C is the charge conjugation matrix, Γ M are 32 × 32 Dirac matrices of SO (1,9) and Γ is the 10d chirality matrix 9 . We would like to identify a sub-algebra of this flat space-time supersymmetry algebra which has the same form as the flat space limit of d(2, 1; α) 2 . In particular, we want to find a subset of 16 fermionic generators which satisfy (3.3). To this end, we define the projection operators Below, we will show that the sixteen supercharges K + q I α satisfy (3.3). This will allow us to show that the flat space limit of the action (2.4) for g = d(2, 1; α) 2 coincides with the usual flat space Green-Schwarz action [38] in a particular kappa-gauge.
First, let us note that as required for a projector. Further, K ± commute with the 10d chirality matrix Γ ≡ Γ 0123456789 and satisfy where t indicates the transpose. As a result, projecting with respect to K ± is compatible with both the 10d Majorana and Weyl conditions. With our choice of gamma matrices, the anticommutator of the supercharges q I projected by K + is The projected supercharges K + (φ)q I have the same commutation relations as the flat space limit of the d(2, 1; α) 2 supercharges (the equations (3.8) and (3.3) differ only by normalization of momenta). As a result the flat-space limit of the 4 coset sigma-model action will match with the flat space Green-Schwarz action in the (fully fixed) kappa gauge One can explicitly check that the form of the actions is indeed equivalent; in doing this it is important to start with the 3d form of the Wess-Zumino term -just as one does for AdS 5 × S 5 [23].

Coset model vs. Green-Schwarz string
In this section we compare the D(2, 1; α)×D(2, 1; α)/SU(1, 1)×SU(2)×SU(2) supercoset with the Green-Schwarz action on AdS 3 × S 3 × S 3 × S 1 , expanded to the second order in fermions 10 . As discussed before, the agreement of the two is not at all obvious. For successful comparison one has to completely fix the kappa-symmetry in the Green-Schwarz action in such a way that the S 1 factor decouples from fermions. In this respect, the background is similar to AdS 4 ×CP 3 , for which the coset description requires partially fixed kappa-symmetry [24,26]. A novel feature is the complete decoupling of a bosonic direction. It is known from the experience with AdS 4 × CP 3 that the coset kappasymmetry gauge may become singular on certain string configurations [26,42]. It would thus be interesting to find the full Green-Schwarz action on AdS 3 ×S 3 ×S 3 ×S 1 , perhaps along the lines of the type IIA AdS 4 ×CP 3 case [26]. For technical reasons we will restrict ourselves to the quadratic part of the action, which is known in a closed form for any type IIB supergravity background [43]. The Green-Schwarz fermions couple to the metric and to the three-form RR flux as [43]: Here E A a is the worldsheet projection of the vierbein: (4.2) 10 We would like to thank A. Tseytlin for suggesting this calculation to us.
For any tangent-space tensor we define The metric coupling in the covariant derivative is the standard spin connection: The fermions are Majorana-Weyl spinors of the same chirality : The explicit form of the Dirac matrices, as given in the appendix C, will be important in our calculation.
In the units where the radius of AdS 3 is set to one, the metric and the RR three-form are where ds 2 (M) and V ol(M) are the standard metricae and volume forms on AdS 3 and S 3 , and U is the periodic coordinate on S 1 . To find an explicit form of the metric, the spin connection and the volume form, we can use the fact that . The metric and the volume form can then be expressed through the Maurer-Cartan forms: where t i are the Lie-algebra generators of sl(2, Ê), su L (2) or su R (2), assumed to be canonically normalized: to η ij = diag(− + +) for sl(2, Ê) and η ij = δ ij for su (2). The metric and the volume form on a group manifold are given by where f i jk are the structure constants. This suggests the following local frame on AdS 3 × S 3 × S 3 × S 1 : (4.9) The factors of cos φ and sin φ take into account that AdS 3 , S 3 + and S 3 − have different radii which satisfy the triangle equality (1.1). For the tangent-space indices we use the Lie-algebra notations from appendix A.
The spin connection on a group manifold is determined by the Maurer-Cartan equations: In our case, Finally, the three-form flux (4.6) has the following tangent-space components: Contracting the RR form with the Dirac matrices we find: This matrix is nilpotent: F / 2 = 0, and is proportional to the projector introduced in the previous section, eq. (3.5): This elucidates the geometric origin of the gauge-fixing condition (3.10).
Assuming that the fermions obey the gauge-fixing condition (3.10), and taking into account the gamma-matrix identities K ±t C = CK ∓ , K ± Γ 012 = Γ 012 K ∓ and D a K ± = K ± D a , we can bring the Green-Schwarz Lagrangian (4.1) to the following form: (4.14) Quite remarkably, this Lagrangian does not depend on the S 1 coordinate U. Before the gauge fixing, the fermions coupled to U via the ∂ a UΓ 9 term in E / a , but in K − E / a K + the ∂ a UΓ 9 term is projected out by virtue of an easily verifiable identity K + Γ 9 K − = 0. The S 1 factor indeed decouples from fermions after the kappa-symmetry fixing condition is imposed. It remains to show that the AdS 3 × S 3 × S 3 couplings in the Green-Schwarz action are the same as in the supercoset model.
Imposing the chiral and the kappa-symmetry conditions and using the explicit form of the projectors, eqs. (C. 15), (C.13), we find that the worldsheet fermions acquire the following form: where the spinors |± φ are defined in (C.14). The remaining tri-spinor indices of θ I aαα are acted upon by the three triplets of gamma matrices γ µ , γ m , γṁ, defined in (A.1).
Plugging the gauge-fixed fermions into the Lagrangian (4.14), we can get rid of the first two spinors in the tensor product with the help of (C.16). Explicitly, 18) and the Green-Schwarz Lagrangian becomes According to (C.12), This Lagrangian should be compared to the quadratic terms in the supercoset action. The expansion of the coset Lagrangian to the second order in fluctuations in an arbitrary bosonic background is given in appendix B. We can specify the general construction to the case of the D(2, 1; α) × D(2, 1; α)/SL(2, Ê) × SU(2) × SU(2) coset. The background bosonic field in (B.1) in this case is parameterized by an element of H bos × H bos /H diag : . Fixing the coset gauge by setting g L = 1 we find: and thus for the background currents (B.2) we get: The fermion part of the fluctuation field in (B.1) can be parameterized as Using the commutation relations of the d(2, 1; α) algebra listed in appendix A, we obtain: where V a is the same combination of the background currents and Dirac matrices as in (4.16), (4.18). Plugging K a and A a , as given in (4.24), into the supercoset Lagrangian (B.9), we find that it exactly agrees with the kappa-fixed Green-Schwarz action (4.19).

The BMN limit
In this section we will quantize the coset sigma-model in the light-cone gauge perturbatively in the sigma-model coupling by expanding the Lagrangian near a light-cone geodesic. Technically, this is the same background-field expansion from appendix B as we used in the previous section to compare the coset sigma-model with the Green-Schwarz string action. The quadratic action describes the BMN limit [44] of a point-like string moving along the light-cone geodesic [45] t = x 0 = ϕ, where t is the global AdS time, x 0 is the worldsheet time direction, and ϕ is an angle on S 3 × S 3 × S 1 . The quadratic terms can be read off the general formula (B.9), with the background taken in the form where D is the dilatation generator of d(2, 1; α) 2 and J is the angular momentum. Higher orders in fluctuations generate the near-BMN expansion [46]. The light-cone frame in AdS 3 × S 3 × S 3 × S 1 is obtained by combining the global AdS time with an angle on S 3 × S 3 × S 1 . The time is conjugate to the dilatation generator 11 For the angular momentum we have more freedom. Potentially one can pick an arbitrary element of su(2) × su(2) × u (1): where P is the generator of the extra U(1) factor. However, there are two conditions to satisfy: J has to be appropriately normalized in order for t ± ϕ to be light-cone directions, and the light-cone gauge should preserve some supersymmetry. We shall see that these two conditions uniquely determine all three coefficients in (5.3). If we relax the supersymmetry condition, there are more solutions which we discuss later in section 8.1.
In the light-cone gauge D + J is fixed, and determines the internal length of the string, and D − J becomes the light-cone Hamiltonian: Both should be null elements of the underlying Lie algebra. This guarantees that the string moves along the light-cone as is required by the Virasoro constraints for the background (5.1). From the explicit form of the metric on d(2, 1; α) in appendix A, we find that C, C ′ , C ′′ must satisfy If we also want to preserve supersymmetry, the Hamiltonian H = D −J should commute with some of the supercharges. The eigenvalues of ad H on the odd generators of the superalgebra are proportional to ±1 ∓ C ∓ C ′ , where the signs take all eight possible values. We thus require that With this choice, the zero norm condition (5.5) becomes The generators of d(2, 1; α) are defined in appendix A and the superscripts L and R denote the generators of the left and right d(2, 1; α) algebras.
The unique solution to this equation is C ′′ = 0, C = cos 2 φ. Then C ′ = sin 2 φ, and The light-cone gauge fixing explicitly breaks part of the original d(2, 1; α) 2 symmetry. By inspecting the symmetries of the giant magnons in AdS 3 × S 3 [47] and drawing some intuition from the spin chain picture in the dual CFT, David and Sahoo argued that the little group of the light-cone gauge in AdS 3 × S 3 × T 4 is P SU(1|1) × P SU(1|1) [10]. The psu(1|1) ⊕ psu(1|1) superalgebra admits a three-parametric central extension [48], and all three central charges appear in the symmetry algebra of the giant magnon [10].
Let us see how the residual symmetries arise in the coset construction of AdS 3 × S 3 × S 3 × S 1 . The global symmetry transformations act on the coset representative (B.1) from the left and in general will change the background. Only those transformations that commute with D and J can be pulled through g B = e Ξ and will act on e . Even these transformation will be non-linearly realized on , with the exception of the transformations from h 0 , for which one can apply a compensating right gauge multiplication such that will transform in the adjoint. Thus the elements of h 0 which commute with D and J act on the transverse field by conjugations and leave the action invariant. There are three U(1) charges that satisfy this condition: However, this is not the end of the story. By analogy with AdS 5 ×S 5 and AdS 4 ×CP 3 , we may expect that the supercharges that commute with D − J will be also preserved, although their algebra, as well as their action on the transverse fields, can be deformed by the gauge fixing. The supercharges that commute with D − J are Q L,R ±±± . They form two copies of the psu(1|1) algebra (which is just the 2d Clifford algebra). We expect that the algebra gets centrally extended by the mechanism described in [16,49].
The quadratic action for fluctuations around the BMN geodesic can be read off from equation (B.9). In particular, the mass-squared operator for the BMN modes is (5.10) Given the tensor product structure of D(2, 1; α) × D(2, 1; α), the mass spectrum can be expressed in terms of the U(1) charges (5.9): The mass takes four possible values 0, 1, cos 2 φ or sin 2 φ, in agreement with the bosonic spectrum found in [33] and the full spectrum at the special value of φ = π/4 [37]. Each mass level contains four states: two bosons and two fermions. This is the dimension of the bifundamental multiplet of psu(1|1) × psu(1|1). The spectrum thus nicely fits into four bifundamental multiplets of the unbroken psu(1|1) × psu(1|1).
To write down the Lagrangian for the fluctuation modes, we can choose the following parameterization of the coset element in (B.1): At the quadratic level the light-cone gauge fixing amounts in replacing the worldsheet metric by η ab = diag(+−), and the light-cone coordinates by their background values: ∂ a t = δ 0 a = ∂ a ϕ. The action can be brought to a nice 2d form by introducing the notations and 16) where I = (2α +α + 5)/2 and m I = (0, sin 2 φ, cos 2 φ, 1). (5.17) In these notations, the light-cone action becomes: For the 2d gamma matrices we take: One can readily expand the action beyond the leading BMN order. For instance, the cubic interaction terms for the bosonic fields are 21) which gives after the gauge fixing: The interactions involving fermions are more complicated and although it is straightforward to include them in the near-BMN expansion, we will not do it here. In appendix D we present the quartic interaction terms for bosons. The heaviest mode (X 4 = X) lies on the threshold of the decay 4 → 2 + 3, since the masses satisfy the sum rule m 2 + m 3 = sin 2 φ + cos 2 φ = 1 = m 4 . A similar sum rule holds in AdS 4 × CP 3 . There, the massive mode disappears from the spectrum, once the quantum corrections are taken into account, by mixing with the continuum of the two-particle states [50]. We expect that the same mechanism is at work here and that the heaviest mode should not be regarded as an elementary excitation.
The appearance of the massless BMN modes is quite unusual in the AdS/CFT context. In the AdS 5 × S 5 and AdS 4 × CP 3 backgrounds, which in other respects are very similar to AdS 3 × S 3 × S 3 × S 1 , the BMN modes are all massive. As we shall see, the massless modes hinder straightforward application of integrability methods to AdS 3 × S 3 × S 3 × S 1 . We will return later to the discussion of their origin and possible implications for the AdS 3 /CFT 2 duality. 6 The AdS 3 × S 3 × T 4 limit The coset action we have constructed in section 2.2 for Type IIB string theory on AdS 3 × S 3 × S 3 × S 1 has the relative radii of the two S 3 's (R ± ) and the radius of S 1 as free parameters (see equation (1.1)). We can take, say, R − → ∞ and decompactify the S 3 . The resulting theory should be Type IIB string theory on AdS 3 × S 3 × T 4 , after periodic identifications in the resulting Ê 3 . In this section we show explicitly how this happens.
A coset action for Type IIB string theory on AdS 3 × S 3 had been written down in [28,7] and basically follows the Metsaev-Tseytlin construction for AdS 5 × S 5 [23]. In the notation of our paper, this action is based on the 4 automorphism (2.10) of the g = psu(1, 1|2) ⊕ psu(1, 1|2) superalgebra, which puts the six-dimensional Type IIB GS action on AdS 3 × S 3 into the general framework of 4 cosets. The coset has 16 fermions, half of which can be removed by fixing kappa-symmetry, leaving 8 physical degrees of freedom: the correct number for a six-dimensional GS action.
We propose that the coset action on AdS 3 × S 3 , when supplemented with four free bosons, in fact describes ten-dimensional Type IIB GS strings on AdS 3 × S 3 × T 4 in a suitable (fully fixed) kappa-symmetry gauge. This may appear puzzling at first, since, as we have just mentioned, the six-dimensional coset action [7] has only 8 physical fermions: a factor of two short of the 16 fermions required in ten dimensions. We are going to argue that the extra T 4 factor in the action changes the number of physical degrees of freedom in the coset sector. At first sight that seems impossible, since one can always go to the conformal gauge, where the T 4 completely decouples. Were this to be true, the coset would lack half of the fermion degrees of freedom. However, the decoupling of the T 4 is not complete, since the four bosons of T 4 do interact with the coset fermions through the 2d metric coupling or, in the conformal gauge, through the Virasoro constraints. The key point here is that the metric transforms non-trivially under kappa symmetry. The metric couplings of the extra bosons violate the kappa symmetry of the action and, in effect, keep all 16 fermions physical. Put differently, the addition of the extra free bosons modifies the Virasoro constraints for the model; a consequence of this modification is that kappa-symmetry of the six-dimensional action is not a symmetry of the ten-dimensional action. As a result the coset + T 4 model has more fermions that just the coset 12 .
We first discuss the R − → ∞ limit for the d(2, 1; α) super-algebra. To reintroduce the dependence on R − we re-scale Rṁ → R − Rṁ. We should also take φ → 0 (c.f. equations (1.2)). The relevant (anti)-commutators then reduce to (6.1) In the limit we also find that the AdS 3 and S 3 radii are equal R + = l (c.f. equation (1.1)). In fact, it is easy to convince oneself that the algebra is now psu(1, 1|2) together with three commuting generators Rṁ. This is a well known property of d(2, 1; α) for α = 1 (see for example [36]). Our coset action is based on a 4 automorphism of d(2, 1; α)⊕d(2, 1; α) constructed in section 2.2. In the R − → ∞ limit this automorphism reduces to the 4 automorphism for psu(1, 1|2) ⊕ psu(1, 1|2) which is used to construct the coset action on AdS 3 × S 3 . Taking into account the extra commuting generators Rṁ, we then find that in the R − → ∞ limit the Type IIB coset action on AdS 3 ×S 3 ×S 3 ×S 1 reduces to the coset action for AdS 3 ×S 3 together with four free bosons for T 4 . Since the original action for AdS 3 ×S 3 ×S 3 ×S 1 has the interpretation of a Green-Schwarz action in the kappa-gauge (3.10), we conclude that the coset action for AdS 3 × S 3 together with four free bosons for T 4 is a Green-Schwarz action for AdS 3 × S 3 × T 4 in the kappa gauge Γ 012345 θ I = θ I . (6.2)

Integrability
As we have shown in the previous sections, string theory on the AdS 3 × S 3 × S 3 × S 1 background is described by an integrable sigma-model. Its classical equations of motion can be solved in a quite general form by the finite-gap integration method [51]. The finite-gap method basically performs the separation of variables, always possible in an integrable system. It thus replaces the oscillator expansion in the flat space and serves as a first step towards string quantization via Bethe ansatz. For the AdS 5 × S 5 [12,52,53,54,13,55,56,57] and AdS 4 × CP 3 [17] backgrounds the finite-gap method yields a set of coupled integral equations, which on the one hand parameterize possible classical solutions of the sigma-model and on the other hand can be regarded as the classical limit of the Bethe equations for the quantum spectrum of the string. We first describe the general scheme of finite-gap integration, as applied to the 4 cosets, and then specify the general construction to the case at hand, the sigma model on AdS 3 × S 3 × S 3 × S 1 .

The General Scheme
The general construction mostly follows the derivation of the finite-gap equations for AdS 5 × S 5 [13], but we will use more invariant group-theoretic language, and unlike in all previous studies will not rely on an explicit supermatrix representation of the Lax connection 13 . As usual in integrable systems, instead of solving the non-linear equations of motion one can study the linear problem for the Lax operator (2.6). The fundamental solution of the linear problem is the monodromy matrix of the Lax connection [58]: The contour of integration C x * x * is a closed curve that links the worldsheet 14 , but is otherwise arbitrary. The canonical choice is the equal time section x 0 = x 0 * . Because of the flatness condition (2.7) the monodromy matrix does not change under continuous deformations of the contour.
The monodromy matrix is gauge-dependent and also depends on the base point x * . Under the gauge transformations the monodromy matrix gets conjugated by an element If the base point is shifted to x ′ * , the monodromy matrix is conjugated by the monodromy along the curve connecting x * and Both transformation leave the monodromy matrix in the same conjugacy class. Therefore, the conjugacy class of the monodromy matrix is gauge invariant and independent of the base point (time-independent).
Since the conjugacy class of M(x) is time-independent for arbitrary spectral parameter x, it generates an infinite number of integrals of motion. The set of conjugacy classes is isomorphic to the maximal torus of G modulo Weyl group, and by choosing a Cartan basis H l , the monodromy matrix can be locally represented as The quasi-momenta p l (x) are the gauge-invariant generating functions for the integrals of motion. They are defined up to transformations from the Weyl group and shifts by integer multiples of 2π.
The monodromy matrix M(x) is a meromorphic function of the spectral parameter x whose only possible singularities are located at x = ±1. The nature of these singularities will be discussed later. On the contrary, the quasi-momenta p(x) in general are multivalued functions of x, defined up to Weyl transformations, and can have branch points with the monodromy in the Weyl group. For simplicity, we only consider the case when the monodromies are elementary Weyl reflections (including generalized Weyl reflections specific to supergroups [59]). An arbitrary element of the Weyl group is a product of Weyl reflections. In the Bethe-ansatz language branch points with composite monodromies correspond to stacks of Bethe roots [60] describing composite quantum states of several elementary excitations.
The Weyl reflection with respect to the lth root acts on the lth quasi-momentum as p l (x) → p l (x) − A lm p m (x), where A lm is the Cartan matrix of g. We denote by {a l,i } the set of branch points of the quasi-momentum p l (x) and by {C l,i } the set of cuts that connect these branch points pairwise. The precise nature of the singularity of p(x) at a l,i is determined by the monodromy and depends on whether the lth root of the superalgebra is bosonic or fermionic. For the fermionic root, A ll = 0, and the quasi-momentum shifts by a (known) function which is analytic at x = a l,i : p l → p l + . . .. Consequently, the quasi-momentum p l (x) has a logarithmic singularity. For the bosonic root A ll = 2 and, in addition to the shift by an analytic function, the quasi-momentum changes sign: p l → −p l + . . .. This means that p l (x) has a square root branch point at x = a l,i .
In conclusion, the quasi-momenta are meromorphic functions on the complex plane with cuts C l,i , and can have logarithmic (fermionic) or square root (bosonic) branch points. For the bosonic, square-root cuts, the monodromy condition (7.2) is equivalent to an equation for the continuous part of the quasi-momentum across the cut: where we define: The same equation holds at the end-points of the fermionic cuts, in which case p l (x) drops out of the equation. In addition to the branch cuts, p l has simple poles at x = ±1, where the Lax connection (2.6) itself has a singularity: Parameterization of the residues at x = 1 and x = −1 by their sum and difference is a matter of convenience. The quasi-momenta carry all the information about the conserved quantities in the sigma-model. In particular, their asymptotic behavior at infinity determines the global symmetry charges. Indeed, where k a is the global symmetry current (2.8). Thus, and Further coefficients of the Taylor expansion constitute an infinite set of conserved charges responsible for integrability of the model. Using quasi-momenta, one can also build the canonical set of action variables [56,61]. The information contained in p l (x) is actually redundant because of the 4 symmetry. The symmetry acts on the flat connection according to (2.2), and it is not hard to see that Ω(L a (x)) = L a (1/x). In consequence, the knowledge of the quasi-momenta in the physical region |x| > 1 is sufficient to reconstruct them everywhere in the complex plane. In particular, all the branch points of the quasi-momenta are invariant under inversion. If a l,i is a branch point of p l (x), (possibly) other quasi-momentum will have a branch point at 1/a l,i .
A meromorphic function with the properties listed above can be reconstructed from the discontinuities at its cuts, which for the bosonic quasi-momenta 15 we denote by 2πiρ l (x). For fermionic cuts, the monodromy condition (7.2) completely determines the discontinuity and there is no further freedom. The quasi-momenta thus admit the spectral representation: where C l denotes the collection of cuts in the physical domain |x| > 1 on which p l (x) has a discontinuity. Since the quasi-momenta inside the unit circle can be reconstructed by inversion, we chose to treat separately the cuts at |x| > 1 and their images under The inversion symmetry (7.13) determines the densitiesρ l (x) in terms of ρ k (1/x), and imposes certain constraints on κ l and m l . We find that m l 's must be integers 16 and that κ l and m l satisfy 17 Finally, The integration contours lie entirely outside the unit circle. The condition (7.3) becomes a set of integral equations for the densities: For bosonic nodes of the Dynkin diagram, these equations hold on the cuts of the quasimomenta. For the fermionic nodes, they hold at the positions of singularities. Solutions of these equations describe quasi-periodic solutions of the equations of motion in the sigma-model. The conserved charges for a given solution can be computed by expanding the quasi-momenta at infinity. The classical finite-gap equations have a direct quantum counterpart, the Bethe equations for the quantum spectrum of the sigma-model. 15 Calling the quasi-momenta bosonic or fermionic is a slight abuse of terminology, because all p l (x) are even elements of the Grassmann algebra. We call the quasi-momentum p l fermionic if the lth node of the Dynkin diagram is fermionic. 16 Under x → 1/x, the pole part of p l changes sign and shifts by −2πm l , provided that the equations below are satisfied. If m l 's are integers, the shift has no physical significance, since the quasi-momentum is defined up to an integer multiple of 2π. Typically, the integers m l have the meaning of the winding numbers [52]. 17 Since Ω 2 = (−1) F and H l are Grassmann-even, the matrix S lm squares to one and has eigenvalues one or minus one.

Classical Bethe equations for D(2, 1; α) × D(2, 1; α)
The symmetrized Cartan matrix of 18 The second factor in the tensor product acts on the ± indices. The 4 symmetry operator acts on the Cartan generators simply by permutation of the two D(2, 1; α) factors: The vector κ l can be found by evaluating the monodromy matrix on the vacuum solution (5.1) and calculating the residue of the quasi-momenta at x = ±1: Here, E = D/ √ λ is the ratio of the energy of the string to its tension which we denote by 19 √ λ/(2π). These data leads to the following set of classical Bethe equations: The integer m is the winding number of the string. The equations can be summarized in a diagram shown in fig. 1. The nodes of the diagram correspond to the densities ρ ± n . The left hand side of the equations is determined by the Dynkin labels, and the right hand side by the links. The original Dynkin links from the Cartan matrix (7.18) determine coefficients in the first term in (7.17). The second term, which is associated with the inversion symmetry, produces additional links on the Dynkin diagram, shown in fig. 1 by broken lines. 18 In this section, we denote quantities related to the left (right) d(2, 1; α) by +(−). 19 In the AdS 5 /CFT 4 case, λ is also the 't Hooft coupling of the dual super-Yang-Mills theory. The precise nature of the parameter λ in the CFT 2 dual of the AdS 3 × S 3 × S 3 × S 1 background is not clear to us.
Physical states should in addition satisfy the level-matching condition P ∈ 2π .

BMN limit
The trivial solution of the finite-gap equations, with zero densities, describes the BMN vacuum. The BMN modes correspond to the vanishingly small cuts whose position is determined by the no-force condition, the vanishing of the left-hand side the Bethe equations. For instance, the left hand side of (7.21) vanishes at where we have set the winding number m to zero and also neglected the difference between E, which enters the classical Bethe equations, and J , which plays the role of the length of the string in the light-cone gauge. This is justified for small deviations from the BMN vacuum. The set of points {x n } determines the locus at which short cuts with infinitesimal filling fractions can emerge. The solution with infinitesimal cuts corresponds to exciting a number of BMN modes. Their occupation numbers N n are proportional to the filling fractions of the cuts: The precise relationship between the occupation numbers are the filling fractions is derived in [12]: The energy then is Because the length of the string in the light-cone gauge is 2πJ , the combination n/J plays the role of the worldsheet momentum. The spectrum of small fluctuations thus describes particles with the dispersion relation ε(p) = p 2 + sin 4 φ . (7.28) Similarly, the densities ρ ± 3 (x) describe particles with mass cos 2 φ. The heavy modes are more tricky. They correspond to stacks [60] that cross from node 1 to node 3 through node 2. The stack, roughly speaking, is a set of overlapping densities on different nodes. In this particular case it is a simultaneous solution of a pair of equations 2 sin 2 φ J x The solution is only possible in the thermodynamic limit n ∼ m ∼ J → ∞, since n and m must satisfy n/m = tan 2 φ, in which case the stack corresponds to a particle of mass 1. The stacks are also responsible for the correct four-fold degeneracy at each mass level. For instance, the bosonic members of the P SU(1|1) × P SU(1|1) multiplet of mass sin 2 φ are the single-node solutions for the densities ρ + 1 and ρ − 1 . The fermions in the same multiplet are the 1 + − 2 + and 1 − − 2 − stacks.
We have correctly reproduced the massive part of the BMN spectrum. However, the massless modes, that we also found in section 5, are completely missing. As we explain below in section 8, the finite-gap equations do not capture massless modes and describe only those solutions of the sigma-model in which the massless modes are not excited. Although this makes our analysis incomplete, we will proceed with quantization of the classical Bethe equations obtained above.

Quantum Bethe equations
In the previously studied cases of AdS 5 × S 5 and AdS 4 × CP 3 the relationship between classical and quantum Bethe equations follows a regular pattern dictated by the structure of the Dynkin diagram. At the quantum level, the densities ρ l (x) describe macroscopic distributions of the Bethe roots, the solutions of the quantum Bethe equations. When the number of roots is very large and the sums over the roots can be replaced by the integrals over their densities, the quantum Bethe equations reduce to the finite-gap integral equations for the densities. By way of observation, one can notice that the quantum Bethe equations for AdS 5 × S 5 and AdS 4 × CP 3 can be inverse engineered from the finite gap equations by applying a simple set of regular rules for each element in the Dynkin diagram. The key observation is that only three distinct structures appear in the finite-gap equations, namely: (i) the normal Dynkin links; (ii) the inversion symmetry links between a momentum-carrying node and "wrong" fermionic nodes (all possible links of type appear in the equations); (iii) the inversions symmetry links that connect the momentum-carrying nodes pairwise. It is quite remarkable that the classical Bethe equations in our case also contain only these three types of structural elements. We can thus apply the same set of rules as in AdS 5 /CFT 4 and AdS 4 /CFT 3 to discretize the classical Bethe equations derived above. There is one subtlety though. The discretization is straightforward only when the elements of the Cartan matrix are integers, which is not the case for the d(2, 1; α) algebra in general. The two exceptions are φ = π/4 (considered here) and φ = 0 (discussed below), and we will restrict our attention to these two special cases.
When φ = π/4, the d(2, 1; α) superalgebra coincides with osp(4|2) whose Cartan matrix has integer entries. In quantum theory, the cuts of the classical spectral curve get discretized and become the arrays of Bethe roots. The asymptotic Bethe ansatz determines the positions of the roots in the spectral plane, x l,i , through a system of discrete functional equations (the Bethe equations). The coupling constant of the sigmamodel (playing the role of ) 2π/ √ λ does not appear in the equations explicitly and only enters through the quantum parameters x ± defined by the Jukovsky map: x ± + 1 The function h(λ) cannot determined by integrability alone, but at strong coupling should behave as in order to reproduce the correct dispersion relation ε(p) = p 2 + 1/4. The Bethe equations are constructed according to the Dynkin diagram in fig. 2, which is obtained from the classical Dynkin diagram 1 by (i) assigning x−x ± type interactions to the normal links; (ii) assigning the x−1/x ± type interactions to the anomalous fermionic links; (iii) and finally associating the BES/BHL phase with the anomalous bosonic links.
The conjectured set of Bethe equations thus reads 20 The energy and momentum are given by The elementary excitation associated to single Bethe roots then have the dispersion 20 We denote by 1, 2, and 3 the upper nodes of OSp + (4|2) and by1,2, and3 the lower nodes of OSp − (4|2). The nodes 1 and 3 (1 and3) carry momentum 1 (−1).
which reproduces the dispersion relation of the BMN modes provided that the function h(λ) behaves at strong coupling as (7.31).
The BES/BHL phase [62] admits the following integral representation [63]: 34) where the integration contour is the unit circle: |z| = 1, |w| = 1. From this representation one can readily infer the analytic structure and the symmetry properties of σ(x, y) [64]. At the lowest order in the strong-coupling expansion, h → ∞, the dressing factor reduces to the AFS phase [20]: and this is all one needs to reproduce the classical Bethe equations, which indeed follow from the quantum equations above in the h → ∞ limit, upon the identification As we discussed above, the classical finite-gap equations describe only massive modes of the string. The sole reason to quantize this incomplete set of equations is their striking similarity to the finite-gap equations in AdS 5 × S 5 and AdS 4 × CP 3 . In those cases, the resulting quantum Bethe equations describe the full asymptotic spectrum at any coupling. For sure, this cannot be true here, because the massless modes are missing already at the classical level. Our best hope is that the equations above represent a truncation of the hypothetical complete Bethe ansatz system that also takes into account the massless modes of the string. In an integrable system, in which action-angle variables separate, such truncations are in many cases possible. For instance, the Bethe equations for the su(2) sector of AdS 5 /CFT 4 [20] can be reconstructed from the finite-gap equations on S 3 ×R 1 [12], which ignores most part of the string modes in AdS 5 ×S 5 . At the moment we cannot justify that ignoring the massless modes we get a consistent truncation of the Bethe equations in AdS 3 /CFT 2 . It is also possible that the inclusion of massless modes modifies the equations in a more essential way than just adding extra nodes into the Dynkin diagram.

AdS
We can apply the same general formalism to the coset part of the AdS 3 × S 3 × T 4 background, which is the sigma-model on P SU(1, 1|2) × P SU(1, 1|2)/SU(1, 1) × SU (2). The Cartan matrix of P SU(1, 1|2) × P SU(1, 1|2) is The 4 symmetry operator again is the permutation: From these data we infer the classical Bethe equations ( fig. 3): Ex + m The quantum Bethe equations can be reconstructed by applying the same set of rules as before ( fig. 4): Outlook: Some comments on the massless modes A new feature of the AdS 3 /CFT 2 dual theories, as compared with the AdS 5 /CFT 4 and AdS 4 /CFT 3 cases, is the presence of massless modes. We have encountered these massless modes as appearing either from the free boson(s) on S 1 or T 4 , or from an extra massless mode in the BMN limit of the AdS 3 × S 3 × S 3 × S 1 theory; there are also fermionic massless modes related to the bosonic modes by supersymmetry. The integrability techniques we have used throughout this paper have little to say about the massless degrees of freedom. It is not difficult to see why the finite-gap methods fail to capture the massless modes, just by looking at the bosonic string on the S 3 ×S 3 ×Ê 1 subspace of AdS 3 ×S 3 ×S 3 ×S 1 .
The Ê 1 factor is the global time direction in AdS 3 . At the classical level, all three factors are independent and related only by the Virasoro constraints. It is known that the Lax connection, in effect, imposes the constraints automatically [65]. In the case at hand the Lax connection actually over-imposes the Virasoro by setting the energy momentum tensors T 1,2 ±± for each of the three-spheres to a constant independently, and thus kills two degrees of freedom instead of one -we want only the sum T 1 ±± + T 2 ±± to be constrained. The massless excitation arises from the combination of the longitudinal modes on the two spheres such that their Virasoro-violating contributions mutually cancel. The finite-gap method eliminates this mode from the very beginning.
By applying the finite-gap method to the bosonic AdS 5 × S 5 or AdS 4 × CP 3 backgrounds one imposes two constraints and thus eliminates just the right amount of unphysical, longitudinal modes of the string. In AdS 3 × S 3 × S 3 , we eliminate three degrees of freedom, one of which is actually physical. Refs. [66] discuss how longitudinal modes can be incorporated in the classical/quantum Bethe ansatz. In the case of the S 3 sigma-model discussed there, the longitudinal degree of freedom corresponds to the extra rapidity variables in the Bethe equations. Perhaps our equations from section 7 could also be augmented by on or more extra rapidity node(s) in order to describe the missing massless degrees of freedom.
In this section we collect together some first steps we have taken towards understanding the zero-modes. We hope to return to these issues in the future.

Semi-classical lightlike geodesics
Throughout this section we ignore the spacetime S 1 , as it plays no role in the classical analysis -its equation of motion is decoupled from the others, apart from the Virasoro constraints, and so can be set to zero for simplicity. We use complex embedding coordinates X i , Y i and Z i for AdS 3 and the two S 3 's The Virasoro constraints are The Cartan generators of the Noether charges are where above there is no summation on the index i = 1, 2. Using the global symmetry of the background a point-particle world-line can always be rotated to for some, as yet undetermined, κ, ω i functions of the world-line parameter t. Simple solutions of the equations of motion 21 can be found for for constant κ, ω i . In this case the Lagrange multipliers are constant 13) and the Virasoro constraints reduce to which can be solved in terms of an angle variable θ ω 1 = κ cos θ , ω 2 = κ sin θ . (8.15) The angle θ denotes the relative angle between the great circles on the two S 3 factors. In the coset language the fact that such a one-parameter family of geodesics exists corresponds to the fact that (for C ′′ = 0) equation (5.5) is solved by The Noether charges of these solutions are (8.17) 21 A particular case of such solutions was analyzed in [67].
These solutions correspond to a different choice of the light-like geodesic, compared to the supersymmetric case studied in section 5. One picks a different linear combination of the Noether charges to define the angular momentum: where the φ dependence is chosen for later convenience. Since J has to be suitably normalised (in the measure given by equation (A.3)), we must take in other words we should choose them as for a free parameter λ. The choice of j + = j − = 1, or λ = φ corresponds to the BPS geodesic of section 5. Evaluating E − J on solutions (8.11)-(8.12) we find The BPS groundstate corresponds to In particular, for the choice j ± = 1 made in section 5, we find ω 1 = κ cos 2 φ and ω 2 = κ sin 2 φ. If we define J ⊥ as the operator 'orthogonal' to J in the coset metric J ⊥ = sin λ cos φJ + 1 − cos λ sin φJ − 1 , (8.23) then our light-like geodesic solutions carry the following J ⊥ charges κl 2 α ′ (sin λ cos θ sec φ − cos λ sin θ csc φ) . (8.24) In particular, the BPS groundstate has J ⊥ = 0. The existence of a continuous family of light-like geodesics is a semi-classical manifestation of the massless modes we have encountered; it is a new feature of the AdS 3 background.

Large charge limit
The finite gap equations do not capture the above semi-classical solutions. One way to see this is to consider the large charge limit of the string action [68,69,70]. For simplicity, we restrict ourselves initially to a bosonic subsector of our action which is the analogue of the SU(2) subsector originally considered in [68]. To take the large charge limit one redefines the global coordinates as follows After taking t = κτ (κ is the large-spin parameter here), redefining v = cos 2 φ κτ +ṽ(τ , σ) , (8.28) w = sin 2 φ κτ +w(τ , σ) , (8.29) and rescaling τ → κτ , to leading order in κ, the Virasoro constraint (8.6) reduces to while the other Virasoro constraint (8.5) determines ∂ τ (ṽ +w) in terms of the other fields. The coordinates conjugate to the charges J and J ⊥ are and we can eliminate the dependence of the Lagrangian on k through the Virasoro constraints. The Lagrangian then reduces to Notice that in this limit the field k ⊥ has no time derivative in the action. In the finite gap equations one simply ignores thew dependent term above; the other terms reduce to standard SU(2)/U(1) Landau-Lifshitz sigma models with a normalisation of the kinetic term that reflects the relative size of the two CP 1 factors. We can also construct the Landau-Lifshitz sigma model for the full theory. To do this we need to identify the sub-group H which preserve the vacuum of the model.
using the definition given in [70] for a Landau-Lifshitz sigma model on a general coset G/H. This sigma model will have 6 + 6 degrees of freedom as is evident from counting the super-dimension of the coset. 23

Discussion
It would be very interesting to understand how the Bethe equations we postulate arise in the dual CFT. The CFT dual for M 4 = T 4 is well understood (see for example [2,71]). It is a deformation of the symmetric product orbifold Sym Q 1 Q 5 (M 4 ), where Q 1 , Q 5 are the numbers of coincident D1-and D5-branes correspondingly, such that the latter are wrapping M 4 and the former are transverse to M 4 . These CFTs possess "small" N = (4, 4) super conformal symmetry with four supercurrents 24 . The BMN limit of this dual pair was investigated in [72]. It should be possible to construct a suitable spin-chain which could presumably match our Bethe equations at weak coupling and perhaps shed some light on the problem of massless modes. Indeed some first steps in this direction have been taken in [10]. By contrast the CFT dual for M 4 = S 3 ×S 1 is probably one of the most obscure amongst AdS/CFT pairs [29,30,31,4,32,34,33]. The symmetries dictate that this CFT has a "large" N = (4, 4) super-conformal symmetry, which appears to be more difficult to treat. As a result, to date a suitable dual CFT candidate has not been identified. In fact, a number of potential duals do not appear to satisfy the requirements of the duality [33]. In this context, we hope that unraveling the integrability structures of superstring theory on this background may be helpful for further understanding of correct CFT dual for this background.
We have derived the classical Bethe equations for the supercoset model on the AdS 3 × S 3 × S 3 × S 1 background. We also made a guess for the quantum Bethe equations. But both of these equations miss the massless string modes. In addition, we only see the rigid, D(2, 1; α) × D(2, 1; α) part of the target-space Virasoro symmetry. We believe that these two problems are related, and that understanding the target space symmetry enhancement may also shed light on the massless modes. The appearance of the targetspace Virasoro algebra is understood at the classical supergravity level [73] and at the quantum level in the NSNS AdS 3 backgrounds [3,74], including AdS 3 × S 3 × S 3 × S 1 [4]. In an interesting recent development, the target space Virasoro generators were constructed for the string on AdS 3 ×S 3 ×T 4 with the RR flux [75]. Perhaps incorporating these results into the integrability approach can provide the missing information on the massless degrees of freedom on the worldsheet.
Note added: When we were preparing this paper for publication, ref. [76] appeared on the ArXiv, in which the spin chain for the symmetric orbifold CFT on T 4 was constructed. It would be extremely interesting to see if the λ → 0 limit of the Bethe equations (7.42) is capable of capturing a part of the spectrum of this spin chain. center for support. B.S. is supported by an ARF from EPSRC. The work of K.Z. was supported in part by the BQR ENS, in part by the Swedish Research Council under the contract 621-2007-4177, in part by the RFFI grant 09-02-00253, and in part by the grant for support of scientific schools NSH-3036.2008.2.
To describe the action of the sl(2) generators on the supercharges we introduce three sets of Pauli matrices: and is consistent with the (− + . . . +) signature in the target space. We will also use another basis of supercharges: The commutators with the sl(2) generators then take the form with the gamma-matrices rotated to a different basis:γ µ = (σ 3 , iσ 2 , −iσ 1 ). The normalization also changes: StrQ aααQ bββ = −ε ab ε αβ εαβ . (A.6)

B Background-field expansion for 4 cosets
In this appendix we expand the sigma-model action (2.4) to the second order in fluctuations around an arbitrary bosonic background g B (x). This is necessary for comparison of the coset model to the Green-Schwarz action, and also for fixing the light-cone gauge in the BMN limit. The expansion can be done quite generally, starting with the coset representative in the form g = g B e , (B.1) where (x) = X A (x)T A ∈ h 1 ⊕ h 2 ⊕ h 3 is the fluctuation field. For the background currents we introduce the following notations: According to the discussion in the main text, A a ∈ h 0 is the background gauge field. By D a we will denote the corresponding covariant derivative: Unlike the covariant derivative D a , which commutes with the 4 grading, the long derivative D a does not have definite 4 charge. Thus, (D a ) n = D a n for any n, also (D a ) 2 = D a 2 , but (D a ) 1,3 = D a 1,3 + [K a , 3,1 ]. We can now expand the action (2.4) in powers of . To simplify the result, one should use the identity: (B.9) The second term contains the covariantized derivative ∇ a K b = D a K b + Γ b ac K c and vanishes on-shell, when K a satisfies the equations of motion. Let us stress that we have not used the equations of motion for K a in deriving (B.9). The equations (B.4) that we used are identities valid for any K a and A a of the form (B.2).

D Quartic terms in the near-BMN Lagrangian
The quartic near-BMN Lagrangian is already rather complicated. We list here its bosonic part before the gauge fixing: