Integrable S matrix, mirror TBA and spectrum for the stringy $\text{AdS}_{3}\times\text{S}^3\times\text{S}^3\times\text{S}^1$ WZW model

We compute the tree-level bosonic S matrix in light-cone gauge for superstrings on pure-NSNS $\text{AdS}_{3}\times\text{S}^3\times\text{S}^3\times\text{S}^1$. We show that it is proportional to the identity and that it takes the same form as for $\text{AdS}_{3} \times \text{S}^3\times\text{T}^4$ and for flat space. Based on this, we make a conjecture for the exact worldsheet S matrix and derive the mirror thermodynamic Bethe ansatz (TBA) equations describing the spectrum. Despite a non-trivial vacuum energy, they can be solved in closed form and coincide with a simple set of Bethe ansatz equations - again much like $\text{AdS}_{3}\times\text{S}^3\times\text{T}^4$ and flat space. This suggests that the model may have an integrable spin-chain interpretation. Finally, as a check of our proposal, we compute the spectrum from the worldsheet CFT in the case of highest-weight representations of the underlying Ka\v{c}-Moody algebras, and show that the mirror-TBA prediction matches it on the nose.


Introduction
A particularly interesting instance of the correspondence between strings on d-dimensional anti-de Sitter spaces (AdS d ) and conformal field theories in (d−1) dimensions (CFT d−1 ) [1][2][3] is AdS 3 /CFT 2 , see e.g.ref. [4] for a review.Restricting to the most supersymmetric string backgrounds, which preserve 16 Killing spinors, we encounter two classes of geometries: AdS 3 × S 3 × X , with X = T 4 or K3, and AdS 3 × S 3 × S 3 × S 1 .In the former case, supersymmetry requires the radius of AdS 3 and that of S 3 to be equal, R AdS = R S .The supersymmetries arrange themselves into two copies of the psu(1, 1|2) superalgebra, which is enhanced to the "small" N = (4, 4) superconformal algebra in the dual CFT 2 [5].In the case of AdS 3 × S 3 × S 3 × S 1 , the radius R AdS of AdS 3 and the radii of the two three-spheres R 1 and R 2 must obey the curvature constraint so that we have a one-parameter of backgrounds; the relative radii of the spheres are encoded in α, 0 < α < 1, The super-isometries of the background then close into two copies of the exceptional superalgebra d(2, 1; α), and the dual CFT 2 enjoys "large" N = (4, 4) superconformal symmetry [6] first described in ref. [7].
All these backgrounds can be realised from a system of D1, D5, NS5 branes and fundamental strings, yielding a mixture of Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NSNS) fluxes-see ref. [8] for a detailed discussion of the moduli in the AdS 3 × S 3 × T 4 case.The case of NS5 branes and fundamental strings is particularly appealing in that the resulting string theory, in absence of other moduli [8], can be studied relatively easily by CFT techniques on the string worldsheet.In fact, it can be described by a stringy supersymmetric Wess-Zumino-Witten model involving sl (2) and su(2) Kač-Moody algebras [5,6].This can be analysed in detail following the seminal work of Maldacena and Ooguri [9][10][11], see also refs.[12][13][14][15][16]. On the other hand, it is much harder to study the worldsheet CFT when RR fluxes are present [17], see also ref. [18] and references therein for recent work in this direction.More generally, perturbative computations [19][20][21][22][23] seem to suggest that non-protected observables take a substantially more involved form for backgrounds with at least some RR flux.
It is possible to study AdS 3 string backgrounds without relying on conformal symmetry on the worldsheet.As it turns out, the Green-Schwarz action for geometries supported by RR fluxes is classically integrable [24,25], in close analogy with what happens for superstrings on AdS 5 × S 5 .Moreover, backgrounds supported by a mixture of RR and NSNS fluxes are classically integrable too [26].This raised hopes of understanding the (planar) spectrum of strings following the strategy that proved extremely successful in AdS 5 /CFT 4 see e.g.refs.[27,28] (see also ref. [29] for a review of AdS 3 /CFT 2 integrability).A key ingredient is the existence of an integrable worldsheet S matrix for the light-cone gauge-fixed non-linear sigma model (NLSM).Symmetry arguments [30][31][32][33][34][35][36] and explicit computations [19,[37][38][39][40][41][42] resulted in a proposal for such an S matrix up to some overall scalar factors, the so-called dressing factors 1 , valid for almost any value of the background fluxes.Somewhat unexpectedly, the symmetry arguments fail precisely for pure-NSNS fluxes, i.e. when the string theory can be described as a WZW model (see in particular ref.[35] for a discussion on this point).This is a little counterintuive, as we might expect the worldsheet description to be simplest precisely at this point.Moreover, perturbative computations take a qualitatively different form at this point [19], as worldsheet excitations become chiral even in light-cone gauge-as it could be expected from the pp-wave analysis of ref. [47].
Recently, an exact integrable S matrix was proposed for the pure-NSNS AdS 3 ×S 3 ×T 4 theory [48,49] in uniform light-cone gauge [50][51][52] 2 .It takes a remarkably simple form, being proportional to the identity up to a simple dressing factor.For the two-particle S matrix, S(p 1 , p 2 ) = e i Φ(p 1 ,p 2 ) 1 , which trivially satisfies the Yang-Baxter equation.Moreover, the dressing phase Φ(p 1 , p 2 ) takes the Dray-'t Hooft shockwave form [53] and is identical to the one found in ref. [54] for strings in flat space.This suggests that the worldsheet theory in light-cone gauge is an integrable deformation (of the "T T " type [55,56]) of a free theory.The resulting mirror Thermodynamic Bethe Ansatz (TBA) equations [57,58] are so simple that they trivialize and yield back the "asymptotic" Bethe equations-showing that finite-size "wrapping" effects [59] cancel exactly here.Moreover, the equations can be solved in closed form [48] yielding the known stringy WZW result [49], including the sectors corresponding to "spectrally flowed" Kač-Moody representations [9].The fact that the mirror TBA equations trivialize strongly suggests that the underlying integrable model can be interpreted as a quantum-mechanical model-a spin chain.It is worth stressing that it is quite unusual that a set of Bethe equations-let alone TBA equations-can be solved in closed form, even for the simplest integrable models; one notable example which shares many similarities with our case arises in AdS 5 /CFT 4 when restricting to a su(1|1) sector [51,60].
It is a little surprising that the worldsheet S matrix takes the form (1.3) with the same dressing phase Φ(p 1 , p 2 ) both for AdS 3 × S 3 × T 4 and flat space.It is natural to wonder whether this universality holds for more general NSNS backgrounds.The main aim of this paper is to investigate this question for the pure-NSNS AdS 3 × S 3 × S 3 × S 1 background.Despite its relative simplicity, this background is qualitatively different from AdS 3 × S 3 × T 4 : the Killing spinors close into a different superalgebra, light-cone gaugefixing preserves at most one quarter (rather than one half) of the original supersymmetries, and the underlying integrable structure is believed to be related to an alternating, rather then homogeneous, spin-chain [30,61,62].Additionally, it is worth noting that there are ways of fixing light-cone gauge that break all manifest supersymmetry, which nonetheless need to be considered to reproduce the whole spectrum of the model, as it can be seen e.g. from a pp-wave analysis [63].
In this paper we will compute the tree-level bosonic S matrix of light-cone gauge-fixed strings on the pure-NSNS AdS 3 × S 3 × S 3 × S 1 background.This will end up depending on three parameters parameters: k, the level of the (supersymmetric) WZW model, which is the square-radius of AdS 3 in appropriate units, α of eq.(1.2), which encodes the relative curvature of the two spheres, and an additional parameter ϑ which fixes our choice of (non-necessarily supersymmetric) uniform light-cone gauge, cf.ref. [63].We introduce these quantities, as well as fix our notations for the background and string action, in section 2. The computation of the tree-level S-matrix follows a standard route [27,64], and is presented in section 3. We find that indeed the (perturbative) S matrix is proportional to the identity-i.e., compatible with the expansion of (1.3).Moreover, the tree-level phase is the same as the one of flat space [54] and AdS 3 × S 3 × T 4 [48].As a result, we propose that the all-loop S matrix is simply given by the exponentiation of the tree-level result, like in the previous cases.To substantiate our claim, in section 4 we write down the equations describing the specturm.We find again that a simple set of asymptotic Bethe ansatz equations reproduce the WZW spectrum-pointing again to a spin-chain interpretation.A novel subtlety arises when the choice of light-cone gauge does not preserve manifest supersymmetry.In that case, the energy spectrum is shifted by a (negative) ground-state energy, which in the spin chain can be understood as a redefinition of a Fermi sea.We derive this energy shift by working out the Thermodynamic Bethe Ansatz (TBA) equations [57,58] for the mirror model [65].We find that indeed the mirror TBA is not entirely trivial when no supersymmetry is manifestly preserved, but all finite-size corrections boil down to an overall shift of all energy levels, indeed compatibly with a redefinition of the Fermi sea.We comment on our results in section 5 and relegate some technical details to five appendices.

String theory on AdS
We start by briefly recalling some facts about the AdS 3 ×S 3 ×S 3 ×S 1 background supported by pure-NSNS three-form flux.

The supergravity background
Let us write the metric of AdS 3 × S 3 × S 3 × S 1 as Following the notation of [36], we write the metric of AdS 3 and S 3 as where t, φ 5 and φ 8 are isometric coordinates.We consider a background supported by pure NSNS three-form flux: where we have introduced the rescaled Kalb-Ramond field b µν = B µν /R 2 AdS .This background is maximally supersymmetric: the 16 Killing spinors close on two copies of the exceptional super Lie algebra d(2, 1; α) [7].We will refer to these two copies as left and right and denote them as as a bosonic subalgebra, so that so(2, 2) ⊕ so(4) (1) ⊕ so(4) (2) is the global isometry algebra.In the following we will denote with the lowest weight of su(1, 1) L and with j 1 and j 2 the spins of su(2) (1)L ⊕ su(2) (2)L .Similarly ( ˜ , 1 , 2 ) will label the corresponding quantum numbers of d(2, 1; α) R .

String action
The bosonic string action is where the determinant of the worldsheet metric γ αβ has been set to −1 and the 1/R 2 AdS dependence of the string tension canceled against the R 2 AdS dependence of metric and Kalb-Ramond field.Working in first order formalism, define conjugate momenta and rewrite (2.4) as [35,36] where (2.7) Above we have denoted with dot and prime, respectively, derivatives with respect to τ and σ.The Virasoro constraints amount to the equations of motion for the auxiliary field γ αβ .For the bosonic action they are equivalent to setting C 1 = 0 and C 2 = 0.The Virasoro constraints impose [36] γ

Gauge fixing and decompactification limit
In order to streamline the notation let us rewrite equation (1.2) in terms of an angle ϕ Geodesics in AdS 3 × S 3 × S 3 × S 1 can be parametrized by a constant motion along the isometric coordinates, where τ is the affine parameter.Requiring the geodesic to be null amounts to the Pythagorean constraint that can be solved in terms of the angle ϑ as (j 1 + 1 ) cos ϕ = ( + ˜ ) cos ϑ , (j 2 + 2 ) sin ϕ = ( + ˜ ) sin ϑ . (2.12) For ϑ = ϕ, the geodesic is 1 4 -BPS and [24,66,67] We introduce light-cone coordinates x ± and a transverse angle ψ adapted to the geodesic in (2.10), where 0 ≤ a ≤ 1 is a parameter which will be useful to fix a rather general uniform light-cone gauge [50][51][52] x This completely fixes the dynamics of the lightcone coordinates x ± and allows to study the theory in terms of eight transverse worldsheet bosons.The worldsheet Hamiltonian density is related to the lightcone momentum, H = −p + with p + = δS/δ ẋ+ [27], so that we find where we have defined J = cos ϕ cos ϑ (j 1 + 1 ) + sin ϕ sin ϑ (j 2 + 2 ) . (2.17) The second gauge-fixing condition in eq.(2.15) relates the lightcone momentum charge P − to the length of the worldsheet R In the following we will mainly work in the so-called decompactification limit where P − → ∞ and the worldsheet effectively decompactifies into a plane.The change of coordinates in eq.(2.14) implies R = J + aH . (2.19) Gauge dependence (namely, a-dependence) of the worldsheet length is a well known feature of uniform light-cone gauge, which gives a direct link to T T deformations see refs.[48,68] for a discussion of this point).We will discuss in section 4.1 the gauge invariance of the spectrum, see also ref. [69].

Near-BMN expansion
On top of the decompactification limit described in the previous section, one can perform perturbative computations in the k → ∞ limit [64].This is nicely explained in [27] and here we briefly recall the procedure.As the effective string tension k 2π always appears together with the σ derivative of a field we rescale σ as σ → k 2π σ, so that the gauge-fixed action can be written as where the integrand L does no longer depend on k.Rescaling the transverse fields by 2π/k the gauge-fixed action becomes where the subscripts indicate the number of fields in each term.This amounts to a near-BMN expansion [47], where the strict BMN limit corresponds to the quadratic part of the action and higher order terms provide corrections to it.In the following we will frequently omit factors of 2π k , which can be reinstated by counting the interaction order.Moreover, in order to lighten the notation, we will adopt the conventions of [36], where the coordinates of the three-spheres have been rescaled by the radius of the respective sphere and R AdS = 1.In the near-BMN limit, one can solve for the worldsheet metric eq.(2.15) together with the condition detγ = −1.Notice that the worldsheet metric is not constant and is in general a function of the transverse fields.The reader can find the explicit result up to cubic order in appendix A. The Virasoro constraints (2.8) give the derivatives with respect to τ and σ of the lightcone coordinate x − .Again we relegate to appendix A the explicit result.
Once lightcone gauge has been fixed, one can expand (2.4) in the number of transverse fields and compute the bosonic Lagrangian perturbatively.We will express the result in terms of the four complex bosons In terms of AdS 3 × S3 × S 3 × S 1 coordinates introduced above, we see that Z, Z are related to the transverse directions of AdS 3 , Y, Ȳ and X, X to the transverse modes on the two spheres, and W, W are combinations of the flat coordinate w along S 1 and the transverse angle ψ.The complex bosons in eq.(2.22) have a well-defined angular momentum on AdS 3 and on the two spheres, see table 1.It is convenient to introduce the quantum number µ defined as In fact, µ is a central element of the su(1|1) c.e. algebra which is preserved after gauge fixing and it identifies (short) irreducible representations of that algebra [30].In computing the Lagrangian we discard total derivative terms proportional to ẋ− .Then at quadratic order we find: (2.24) Differently from the AdS 3 × S 3 × T 4 case, we find a non-vanishing cubic term The quartic Lagrangian is a sum of terms each containing at most two different fields: where the superscripts indicate the different fields contained in each term and L W W

The perturbative S matrix
The 2-to-2 perturbative S matrix can be computed using standard QFT techniques, see e.g.[27] for a review.From eqs. (2.24-2.26)one can derive Feynman rules, that we list in appendix C. The 2-to-2 perturbative S matrix can then be defined in terms of scattering amplitudes.We take two particles worldsheet momentum p 1 , p 2 as incoming and obtain particles of momenta p 3 , p 4 as outgoing 3 .Let us define a basis for the S matrix Table 1.We list the spins of the bosonic excitations W, X, Y, Z and of their conjugate, as well as of the eight physical fermions of the Green-Schwarz string, see ref. [63].Spins are quantised in integers and half-integers for bosons and fermions, respectively.The combination µ, which plays the role of the excitations' mass in the dispersion relation (3.4) is an appropriately rescaled combination of the spins, which we write in terms of s = sin ϕ sin ϑ and c = cos ϕ cos ϑ.The total angular momentum of S 3 × S 3 is J , see eq. (2.17); its sign depends on the worldsheet chirality of the particles, as we discuss below.Here we list the values for positive-chirality particles. i where the set of quantum numbers p i and µ i uniquely identifies a particle, according to Table 1.The perturbative S-matrix can be computed in terms of scattering amplitudes, as where iM 12→34 is the scattering amplitude and the tree-level dispersion relation ω(p) can be read off the quadratic Lagrangian (2.24), Notice that this dispersion is chiral, i.e. it distinguishes from left-and right-moving particles on the worldsheet.Given that the dispersion reproduces the energy + ˜ − J of a single excitation, we see that the (quantised) contribution of J , which can be read off at p = 0, comes with a sign depending on the particle's chirality.We are now going to illustrate some examples to clarify the main steps of the computation.

Example: a (vanishing) off-diagonal term
As a first example, we are going to present the computation of the off-diagonal S-matrix element representing a "left" W mode and a "right" X mode scattering into a "right" W mode and a "right" X mode.Notice that here "left" and "right" refer to target space chirality, not to chirality on the worldsheet.That is to say, here left and right are not synonyms of chiral and anti-chiral, but label the two creation/annihilation operators entering the mode expansion of a complex boson.Let us emphasise that this process should vanish by conservation of the u(1) charge relative to particle W .It is still non-trivial to see how this comes about, since the charge of W -particles does not affect the dispersion relation.The amplitude associated to the process in (3.5) is where time flows from left to right and the Feynman diagrams contributing to the amplitude (3.6) can be represented by the following diagrams Exploiting the Feynman rules in appendix C we find where the dispersion relation for p 1 , . . ., p 4 is The energy ω and the momentum p of the virtual particle can be found for each diagram by imposing energy and momentum conservation at each vertex.So, for example, for iM 2 we have In order to further simplify the expressions in (3.8), one needs to resolve the absolute values in (3.9).In order for a scattering event to take place, the two incoming particles (and similarly the two outgoing ones) must have opposite chirality.In particular we will assume p 1 to be anti-chiral and p 2 to be chiral.We hence have the following possibilities: For (i) the delta functions in (3.3) reduce to while for (ii) we find Enforcing energy and momentum conservation, we obtain for (i) and for (ii).Hence, we find and therefore Notice that the vanishing of iM W X→ W X for (ii) is expected from integrability, as the set of initial and final momenta do not coincide.The vanishing of iM 2 and iM 3 in (3.14) comes out in a slightly more subtle way.In fact, once we enforce energy and momentum conservation, the two vertices vanish and at the same time the propagator of the virtual particle is on-shell.Loosely speaking, one may write where in the next-to-the-last equality we have made use of eq.(3.10).In this respect, we see that the iε prescription for regularizing the Feynman propagator plays an essential role.

Example: a (non-vanishing) diagonal term
We are now going to illustrate how the computation of the non-vanishing diagonal terms works by means of an example: we are going to compute the S-matrix element In principle this may contain two channels: one where the momenta are transmitted along with particles' flavours (the diagonal process), and one where they undergo a reflection.We will see that only the former is non-vanishing.The amplitude associated to the process in (3.18) is where time flows from left to right and the Feynman diagrams contributing to the amplitude (3.19) are Proceeding like in the previous section, we can use the Feynman rules in appendix C to compute the diagrams in eq.(3.20).Again, we can resolve the absolute values entering the dispersion relations in two different ways.For one of these choices the energy and momentum conservation delta functions reduce to while for the other one the set of initial momenta does not coincide with the set of final momenta.Such a process would violate the integrability of the theory, we would hope for it to vanish.Indeed exploiting energy and momentum conservation we find ) so that the amplitude vanishes.Instead, when the sets of initial and final momenta coincide we obtain The vanishing of i Mi for i = 1 . . .4, arises in an particular way: due to energy and momentum conservation, we find p = ω = 0, meaning that the virtual particle has null energy and null momentum.We see that also in this case the iε prescription for the Feynman propagator plays an essential role.We can now exploit (3.3) to compute the S-matrix element in (3.18).We find (3.24)

Tree-level S matrix
The complete computation on the lines of the two previous sections gives us the following result for the S-matrix for a anti-chiral-chiral scattering process: where we defined the (gauge-dependent) combination r i = J i + a ω i .Indeed this combination takes the same form as eq.(2.19) (as well as flat-space strings [54] and pure-NSNS AdS 3 ×S 3 ×T 4 [48]), and compensates the gauge-dependence of the worldsheet volume [69].
It is convenient to re-express the scattering phase in terms of p i , ω i , which emphasises its anti-symmetry: where μ is the non-anomaloues (i.e., p-independent) part of the energy + ˜ − J , and can be defined as μi = µ i sgn p i + µ i .(3.28)This also emphasises that the perturbative S-matrix is consistent with the requirements of crossing symmetry [46].Using that under the crossing transformation p → −p, ω → −ω, so that J → −J and μ → −μ, we get as expected 4 All-loop S matrix and spectrum Building on the simple form of the tree-level bosonic S matrix (3.27), we can now make a proposal for the all loop integrable S matrix by assuming that the tree-level phase-shift simply exponentiates, compatibly with what expected from unitarity.To check whether this is the case, we will compute the finite-volume spectrum of the theory and compare it with the WZW model prediction.

All-loop S matrix and asymptotic Bethe ansatz
We propose the two-particle S-matrix of this model takes the form with, in the a-gauge of section 2.3 where for a single excitation we have4 while J i is the contribution to the angular momentum on S 3 ×S 3 , i.e. the charge under J of the i-th particle, cf.eq.(2.17).Here we are assuming that H(p) does not receive corrections in 1/k with respect to its one-loop counterpart ω(p)-i.e., minding the factor of k/2π in front of p, H(p) = ω( k 2π p).In the supersymmetric case ϕ = ϑ, the equality between H(p) and ω(p) is the consequence of a shortening condition [30,36], while more generally this is an assumption.As for the value of the masses, recalling that µ i is a combination of spins, which are quantised, we take the masses of bosons and fermions to be the same as in the pp-wave limit [63], see table 1.Let us emphasise once more that, when ϑ = ϕ, the masses of the bosons and those of the fermions are different.
With this input, it is immediate to write down the asymptotic Bethe ansatz (ABA) equations for an M -particle state, which are which we supplement by the definition of the worldsheet size R in the a-gauge-cf.eq.(2.19)and by the level matching condition, where J 0 is the vacuum R-charge and the last equality is the level-matching condition 5 .
Taking the logarithm of eq.(4.5) we get6 where we used the level matching condition.These equations can be solved much like in ref. [49].Let us briefly recall how this goes in the simplest case, corresponding to the highest weight representations of the WZW model; this will be sufficient to illustrate some novel features with respect to the AdS 3 × S 3 × T 4 case.We have Plugging this in the dispersion relation, and assuming that we can take the positive (resp.negative) branch of the dispersion for p i > 0 (resp.p i < 0) 7 we find which gives We have arrived at a closed-form expression for the energy, like in ref. [49].As remarked, note that this is a rather unique occurrence even in the theory of integrable models.Interestingly, a strikingly similar result was found in the context of AdS 5 /CFT 4 [51,60], and it would be interesting to investigate the similarities of these two set-ups further.Following again ref. [49], we expect this to match with the WZW result where we identify the Green-Schwarz vacuum, labelled by J , with a specific (physical) state arising from the su(2) (1) ⊕su(2) (2) Kač-Moody modules with highest weights (j 0,1 , j 0,2 ).For AdS 3 × S 3 × T 4 we observed that the matching required taking as a reference state the lightest physical states in the Ramond-Ramond sector of the theory.Here there is one such state, and it has R-charge J 0 = sin ϕ sin ϑ (2j 0,1 + 1) + cos ϕ cos ϑ (2j 0,2 + 1) . (4.11) The expression for the vacuum R-charge can be further simplified by recalling that ϑ encodes the ratio of R-charges of the light-cone-gauge vacuum along the two spheres as in eq.(2.12), so that specialising to our case and Using this expression we reproduce the WZW energy formula of appendix D, seemingly for any value of ϕ and ϑ: not only we reproduce the mode-number-dependence in the square root, but we can also match the μtot charge with the charges of the WZW description.More in detail, in the supersymmetric case when ϑ = ϕ (that is, when the highestweights are j 0,1 = j 0,2 ≡ j 0 ), the vacuum (the reference state in the Ramond-Ramond sector) is BPS and it has R-charge J 0 = 2j 0 + 1.Note that for each given j 0 = j 0,1 = j 0,2 there are four BPS states: the one which we considered in the R-R sector, and three more in the NS-R, R-NS and NS-NS sectors; in the Green-Schwarz language, they are related to each other by acting with a zero-mode of a massless fermion [66,67].Things are more involved in the non supersymmetric case, when j 0,1 = j 0,2 .The energy H tot = ∆ − J of the R-R reference state is still zero-as correctly reproduced by the asymptotic Bethe ansatz-but this reference state is no longer the lowest-energy state.In fact, of the four "would-be-BPS states" described above, the NS-R, R-NS, and NS-NS states acquire a negative energy, as we schematically represent in the diamond below Such a shift is to be expected for a non-supersymmetric theory, and arises from the polarisation of the vacuum, see ref. [63] for its explicit computation.On the other hand, the energy shifts are proportional to the energies of the lightest fermions.In view of this, we can treat the state with no excitations in the asymptotic Bethe ansatz (the R-R state) as a pseudo-vacuum, and postulate that the true vacuum (the NS-NS state) is obtained by acting with on the pseudo-vacuum with two fermion zero-modes with negative light-cone energy.This redefinition of the Fermi sea is harmless, and allows us to interpret this model as one where the asymptotic Bethe ansatz is exact, i.e.where no particle production occurs, even at the microscopic level. 8hus, we have found a set of asymptotic Bethe ansatz equations that match the (exact) known WZW spectrum-suggesting the existence of an underlying integrable spin chain.The main novelty with respect to AdS 3 × S 3 × S 3 × S 1 is the necessity to treat the ABA vacuum as a pseudo-vacuum, and the appearance of states with negative light-cone energy in the sectors with ϑ = ϕ.At this stage, the analysis of the non-supersymmetric sectors relied on the known WZW result and on an ad-hoc prescription.We know however that, if our proposal for the S matrix is correct, the exact spectrum including the true ground state should emerge from the mirror thermodynamic Bethe ansatz (TBA).We shall see below that this is indeed the case, and highlight as we progress the physical reasons for the drastic simplification of the mirror TBA equations with respect to a generic integrable model.

The mirror theory and TBA equations
The use of the thermodynamic Bethe ansatz [57] to compute the finite-volume spectrum of two-dimensional integrable models is well-established [58].The basic idea is that the finitevolume properties of the model of interest are encoded in the finite-temperature properties of a mirror model, where the roles of time and space have been interchanged.When the integrable theory is not Poincaré invariant, as it is the case for light-cone gauge-fixed string NLSM and indeed for our model, the mirror model is a genuinely new theory [65], see also ref. [70] for a pedagogical introduction to the (mirror) thermodynamic Bethe ansatz.
As explained in detail in refs.[65,70], we can think of the mirror model as being related to the original one by an analytic continuation which sends the momentum of the original model p into the energy of the mirror one, H, and similarly energy in the mirror momentum: Here and below we denote with a "bar" all quantities computed in the mirror model.In this way, from the original dispersion relation we find the mirror dispersion, where we have explicitly highlighted the dispersion of chiral and anti-chiral particles by means of Heaviside's Theta function.We have that p > 0 for chiral particles, and p < 0 for anti-chiral ones.In the mirror theory, the "mass" µ plays the role of a purely imaginary twist of the Bethe equations.These take the form where we made it explicit that the fermions of the mirror model are anti-periodic (while the ones arising from the Green-Schwarz string were periodic) [65], and the "volume" β is of course the inverse temperature of the original model.The mirror Bethe equations are written in terms of the "mirror-mirror" scattering matrix S(p i , pj ), which is the analytic continuation of the original S matrix.We have where9 where we highlighted split into chiral-anti-chiral and anti-chiral-chiral scattering.
The only subtle step remaining before obtaining the mirror TBA equations is to formulate the string hypothesis, i.e. describing all solutions of the equations (4.19) in the thermodynamic limit.In general these include particles, bound states and "Bethe strings", see e.g.refs.[65,71].Here, like for pure-NSNS AdS 3 × S 3 × T 4 , due to the linearity of the equations (4.19), we assume that only the (8|8) fundamental particles appear in the thermodynamic limit.The mirror TBA equations for the ground state then are written in terms of (8|8) pseudo-energies a (u), where u is some appropriate parametrisation, and take the familiar form10 where γ is an appropriate contour which depends on the state we consider and and finally The total energy and total momentum are then expressed in terms of the pseudo-energies:

Ground-state energy
The ground-state mirror TBA equations follow from eq. (4.21) by taking the contour γ to run on the mirror-momentum line.It is convenient to take the parameter v to be the mirror momentum p itself.Then Let us explicitly split the pseudo-energies between chiral and anti-chiral particles, which we denote with ± a .Then where The ground state energy and momentum (4.24) are given by We start by imposing level-matching condition for the ground state, P 0 = 0, which gives K + + κ + = K − + κ − so that from the expressions of (4.27) we find Using the linearity of the pseudo-energies, the integral can be computed analytically, as we do in appendix E. This computation relies on essentially two physical features of the model: first, crossing symmetry requires that the masses {µ i } come in pairs of different signs.This enforces reality of the TBA integrals (despite H(p) being complex for real p) and greatly simplifies their analytic form.Secondly, supersymmetry of the original model dictates that-even when the choice of the light-cone geodesic does not preserve supersymmetry manifestly-the sum of the square masses is the same for the bosons as it is for the fermions.We refer the reader to appendix E for details of how affects the computation of the integrals.Eventually we find that both κ ± and K ± are proportional to the differences of the bosonic and fermionic masses, so that perfectly matching eq.(4.15).Moreover, in the case where the theory is manifestly supersymmetric, and ϑ = ϕ, we find that the ground-state energy vanishes, as expected.
The simplicity of the solution is quite striking, though not entirely surprising in light of the simplicity of the stringy WZW spectrum.Still, it might be quite interesting to further investigate this structure by computing the finite-size corrections [52] to giant-magnon solutions [39].

Excited states from contour deformation
Following ref. [73], we take the equation for the excited states to have the same forms of eq.(4.21) up to modifying the integration contour γ.Different states are given by different and inequivalent choices of γ; for two contours to be inequivalent, they should encompass different singularities of the integral.These can arise only from singularities of the logarithm in Λ(v), namely when the rapidity v takes a value v * such that To make the equations more transparent, we can deform the contour γ back to the real mirror-momentum line (like we had for the ground-state equations), picking up appropriate residues in the process, so that for a state where the contour encompassed M singularities of the form (4.33) we have The equations for the total energy and momentum (4.24) are similarly modified: Using the form of the scattering phase and of the kernel we see that yet again ± a (v) ∝ Ha (v) up to a fermion sign ψ a .A necessary condition for eq.(4.33) to have a solution is that ± a (v * ) is purely imaginary, which as expected happens precisely when v * is on the real-momentum (not mirror momentum) line and Ha (v * ) = ip * .In other words, such a singularity correspond to a particle in the original (string) theory.Using this information we can write H(p * j ), where we indicated the sum of all momenta of (anti-)chiral excitations arising from the residues at {v * j } j=1,...M as P * ± .Once again, by solving the level-matching constraint we find that K + = K − and κ + = κ − is a solution.Moreover, the integrals give the exact same values as in eq.(4.31), see appendix E. Using these expressions we find simply Imposing the quantisation condition (4.33) we have that where we already picked the sign of ν j .As before, we restrict to the "unflowed" sector, see ref. [49] for a discussion of the mode-number identification in the spectrally-flowed sectors.
Bearing in mind the level-matching condition we hence have which corrects the asymptotic Bethe ansatz result (4.10) simply by shifting the vacuum energy to the lowest-energy state, yielding a perfect agreement with the WZW result-see appendix D.

Conclusions
We saw that the bosonic tree-level worldsheet S matrix for pure-NSNS AdS 3 ×S 3 ×S 3 ×S 1 is proportional to the identity, and it takes the "shockwave" form [53] already found for flatspace strings [54] and for pure-NSNS AdS 3 × S 3 × T 4 [32].Based on this (and on a previous analysis of the Green-Schwarz fermion contributions [63]) we made a proposal for the exact worldsheet S matrix.We took it to be diagonal, and simply given by the exponentiation of the tree-level result.To check our proposal, we wrote down the mirror TBA equations and solved them analytically in the sector corresponding to highest-weight representations of the worldsheet WZW model, cf.ref. [67,74], finding a perfect match.In the process, we noted that the mirror TBA boils down to a set of equations which can be though of as spin-chain Bethe ansatz-like for AdS 3 × S 3 × T 4 [49].There is an important subtlety here, with respect to that case: generically, the spin-chain vacuum is only a pseudo-vacuum, i.e. it is not the lowest-energy state of the theory.Lower-energy states can by acting with two negative-energy fermion zero-modes.Only in the case where the spin-chain pseudo-vacuum describes a BPS state we find that this is a true (albeit degenerate) vacuum.One immediate question is whether our conjectured S matrix, and therefore the mirror TBA, reproduces the whole spectrum of the theory as predicted by the worldsheet WZW model-not just the highest-weight representations.In particular, new representations can be obtained from the highest-weight ones by means of spectral flow [9].We expect these to appear when we relax some assumptions that we have made in solving the mirror TBA equations.In ref. [49] we discussed at some length how, when the effective size of the worldsheet R eff is comparable with the WZW level k, spectrally-flowed sectors emerge.More specifically, the m-th spectrally flowed sector appears when m ≤ R eff /k < (m+1).In the case at hand, we expect a similar qualitative structure, with the important caveat that different bounds arise depending on the mass of each particle, m ≤ (µR eff )/k < (m + 1).This is due to the fact that there are three WZW levels, one for each Kač-Moody algebra; indeed the structure of the spectral flow for the AdS 3 × S 3 × S 3 × S 1 WZW model is rather intricate, see e.g.ref. [74] for a detailed discussion.
An additional family of states comes from continuous representations [9], which seem to be qualitatively different from the discrete one (spectrally flowed or not).Indeed these representations have recently attracted much attention, both for AdS 3 × S 3 × T 4 [75][76][77] and AdS 3 × S 3 × S 3 × S 1 [76].The role of these representations in describing the theory's spectrum seems especially crucial when the level of the WZW model is small; then the dual appears to be a symmetric-product orbifold CFT [75][76][77].At this stage it is not clear how these states emerge from the worldsheet S matrix in light-cone gauge, even in the simpler case of AdS 3 × S 3 × T 4 .This is certainly something that should be understood, and we hope to return to this question in the near future.
Another interesting feature of stringy WZW models is that not only their spectrum, but also their correlation functions are substantially simpler than for RR string backgrounds [11,15,[78][79][80][81]. Integrability also allows the computation of stringy correlation functions, at least in the planar limit: Basso, Komatsu, and Vieira recently proposed an "hexagon" formalism [82] for the computation of three-point functions, which was later extended to four-point [83][84][85] and well as higher-point [86] correlation functions and, at least in some special cases, to non-planar corrections [87][88][89].That proposal has been developed in the context of the AdS 5 /CFT 4 duality.Despite many impressive advances, a major stumbling block remains the systematic incorporation of finite-size ("wrapping") corrections.For three-point functions, this can be done order-by-order [90][91][92]-by formulae similar to those developed by Lüscher [93,94]-while for higher-point correlators many puzzles remain [95].In any case, nothing like a "mirror TBA" for the hexagon program has been developed yet.Perhaps the pure-NSNS backgrounds-with their almost-trivial wrapping structure-may prove to be the ideal playground for advancing the hexagon program.
Finally, an important ingredient in recognising the integrable structure of pure-NSNS backgrounds was their relation to T T deformations [55,56] of free theories [48].It would be interesting to study in greater detail this relation, and explore recently-proposed ideas concerning the worldsheet description of the T T deformation of the dual CFT 2 [96,97].
where using the short-hand notation c ≡ cos ϕ cos ϑ and s ≡ sin ϕ sin ϑ

E TBA integrals
Here we will compute the integral K and the sum κ defined as where we assumed that c ≥ 0 (which will turn out to be the case).This is immediately zero in presence of manifest supersymmetry.More generally, recall that the masses {µ a } come in pairs of opposite signs, as a consequence of crossing invariance.Note the identity Li 2 e it + Li 2 e −it = π As already observed in ref. [63], the second sum vanishes; this is due to so(8) triality 11 .Hence the integral K does not depends on c at all and takes the value (E.9)