The complete AdS_3 x S^3 x T^4 worldsheet S-matrix

We derive the non-perturbative worldsheet S matrix for fundamental excitations of Type IIB superstring theory on AdS_3 x S^3 x T^4 with Ramond-Ramond flux. To this end, we study the off-shell symmetry algebra of the theory and its representations. We use these to determine the S matrix up to scalar factors and we derive the crossing equations that these scalar factors satisfy. Our treatment automatically includes fundamental massless excitations, removing a long-standing obstacle in using integrability to study the AdS_3/CFT_2 correspondence. The present paper contains a detailed derivation of results first announced in arXiv:1403.4543.


Introduction
The AdS/CFT correspondence is a remarkable equivalence between quantum gauge and gravity theories. In its simplest form it posits a strong/weak duality between superstring theories on AdS d+1 × M 9−d , where M 9−d is a (9 −d)-dimensional compact space, and d-dimensional Conformal Field Theories (CFTs) on the boundary of AdS d+1 [1][2][3]. This conjecture has inspired important advances in our understanding of quantum gravity and Quantum Field Theory (QFT). An intriguing feature of the AdS/CFT duality is the emergence of integrable structures in the 't Hooft, or planar, limit [4] of certain classes of dual theories. The prototypical example is the case of type IIB strings on AdS 5 × S 5 and the dual N = 4 Supersymmetric Yang-Mills (SYM) theory, see [5,6] for a review. Following the discovery of the ABJM Chern-Simons theory [7], integrability was found also to underpin the duality between this CFT and Type IIA string theory on AdS 4 ×CP 3 in the planar limit. 1 The key role of integrability in providing a quantitative handle on both the AdS 5 /CFT 4 and AdS 4 /CFT 3 dualities is rather striking. It hints very strongly that, for certain classes of dual pairs, integrability provides the right set of tools with which to investigate the AdS/CFT correspondence. As a result, identifying other dual pairs where integrable methods may be applicable is an important challenge in developing a detailed understanding of the AdS/CFT correspondence.
Another set of classes where integrability emerges are strings on AdS 3 × M 7 backgrounds with 16 real supersymmetries. The AdS 3 /CFT 2 correspondence is a particularly important example of gauge/string duality. Historically, gravity on AdS 3 gave rise to an early example of holography [9]. The gravity theory was found to have an (infinite-dimensional) conformal symmetry on the boundary whose central charge could be calculated. Further, black hole solutions could be constructed in the gravitational theory [10,11] and their entropy was understood using holography [12]. Moreover, the D1-D5 brane system, whose near-horizon limit gives rise to the AdS 3 /CFT 2 correspondence, has played a central role in the string theory derivation of the black-hole entropy formula [13]. At low energy, such a brane construction gives rise to a 1 + 1 dimensional supersymmetric Yang-Mills theory with matter multiples in the fundamental and adjoint representations, adding new features with respect to N = 4 SYM and ABJM theories.
In the context of string theory, it is natural to first consider AdS 3 backgrounds with maximal supersymmetry. Such backgrounds have 16 real supersymmetries and come in two distinct types. String theory on AdS 3 × S 3 × T 4 gives rise to the small N = (4, 4) superconformal algebra [1,14], 2 while string theory on AdS 3 × S 3 × S 3 × S 1 leads to the large N = (4, 4) superconformal algebra [15]. Both types of backgrounds can be supported by a mixture of Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) fluxes. In the case of pure NS-NS flux, much progress was made by studying the worldsheet theory with two-dimensional CFT techniques [16][17][18][19][20][21][22]. These results can be mapped onto the D1-D5 system via S duality 3 , which however acts in a non-perturbative and non-planar way. It is then natural to ask if backgrounds involving R-R fluxes can be studied more directly [23]. In particular, developing a quantitative understanding of the pure R-R string theory is essential in understanding generic unprotected properties of the D1-D5 system, and a starting point to tackling more general AdS 3 /CFT 2 dualities.
With this motivation in mind, it was realised that the equations of motion of type II string theory on the pure R-R background are integrable [24] 4 and that this extends to mixed fluxes as well [26]. This prompted an extensive investigation of the quantum integrability properties of these backgrounds [24,25,, mainly by means of the Smatrix approach that proved successful in the case of AdS 5 /CFT 4 , see also [49] for a review.
A new feature of the AdS 3 backgrounds is the presence of massless fundamental excitations on the worldsheet. Because massless modes are notoriously difficult to incorporate into integrability constructions [50][51][52], this presented an early challenge to fully understanding the AdS 3 /CFT 2 correspondence using integrable methods. On the other hand, massive S-matrices and Bethe ansatz equations of AdS 3 × S 3 × T 4 [34,35] and AdS 3 ×S 3 ×S 3 ×S 1 [32,33] are relatively well-understood in the pure R-R case; 5 the giant magnon associated to the massive modes was also understood some time ago [56,57]. In [31] massless modes were incorporated in the weakly-coupled spin-chain picture. On the string side, only very recently it has been shown how massless modes can be included in the classical integrability machinery [58]. Both of these results demonstrate that the real intricacies involved in understanding massless modes occur away from the weakly-coupled string and spin-chain regimes.
The aim of this paper is to present in detail how massless excitations can be included in the non-perturbative integrability picture, and how the non-perturbative asymptotic worldsheet S matrix for all fundamental particles can be found in the case of pure R-R AdS 3 × S 3 × T 4 background. These results were first presented in [59].
Our analysis starts from the determination of the off-shell symmetry algebra A of the theory. Before light-cone gauge fixing, the symmetries of AdS 3 × S 3 × T 4 are given by the AdS 3 ×S 3 superisometries 6 psu(1, 1|2) L ⊕psu(1, 1|2) R together with the T 4 isometries. Fixing light-cone gauge breaks some of these symmetries, and in particular halves the supersymmetries. We are interested in the symmetry generators that are linearly realised after gauge fixing, as the S matrix will have to commute with them. Such generators will sit in A, together with some additional central charges which are expected from the case of AdS 5 × S 5 [60][61][62]. We will determine the form of these and find, as it should be, that they have a non-trivial action only on states that do not satisfy the level-matching constraint (i.e. off shell). Once the off-shell algebra of the theory is determined we will use it to constrain the non-perturbative 2 → 2 S matrix, which will then satisfy the Yang-Baxter equation.
Unlike what happened in AdS 5 /CFT 4 [60][61][62], we cannot use the coset action [24,[63][64][65] for our calculations. The coset action requires the use of a particular kappa gauge [24], which does not allow for a straightforward quantization of the massless modes; see [28] for a discussion of the coset kappa gauge. We will therefore work with the Green-Schwarz action [66], in light-cone gauge. Furthermore, we take the decompactification limit, whereby the world-sheet cylinder becomes a plane and the asymptotic states can be defined. It is interesting to note that our results give an example of integrability where the fermionic degrees of freedom do not enter the dynamics through a coset action; similar observations have recently been made in integrable AdS backgrounds which preserve even less supersymmetry [67].
In this way, we are able to establish the off-shell symmetry algebra, including the nonlinear momentum-dependent central extension reminiscent of [61]. As expected, the lightcone-gauge worldsheet theory is non-relativistic. Massive and massless excitations will then have periodic dispersion relations, with the energy of the latter being linear in the momentum for small values of it. Using these results, the two-body S matrix will follow immediately by symmetry arguments, and is fixed up to some dressing factors, for which crossing equations can be written down. As expected, the massive-sector S matrix of [34], including the crossing-invariant dressing factors of [35] can be consistently embedded in the full S matrix of the present paper.
This paper is structured as follows. In section 2 we consider the type IIB superstring action for AdS 3 × S 3 × T 4 in light-cone gauge, and derive its conserved supercurrents. This is done at leading order in the fermions and at subleading order in the bosons. In section 3 we study the symmetry algebra A and the representations that emerge from the supercurrent analysis. We find three short irreducible representations of the centrally extended psu(1|1) 4 ⊕ so(4) algebra 7 : two massive representations of dimension four, and one massless one of dimension eight. In section 4 we deform the representations found perturbatively in order to reproduce the correct non-linear central extension and shortening condition. We also comment on the possibility of quantum corrections to the massless dispersion relation, arguing that they would break part of A. Using those exact representations, in section 5 we construct an invariant S matrix for all of the superstring's excitations, including the massless ones, up to some dressing factors which we constrain by crossing symmetry. We conclude in section 6. We relegate the more technical aspects of our results to the appendices.
2 The off-shell symmetry algebra of superstrings on AdS 3 × S 3 × T 4 In this section we compute the algebra A of off-shell symmetries for classical Type IIB superstring theory on AdS 3 ×S 3 ×T 4 . At first sight it may appear that the natural setting for this would be the coset action [24,[63][64][65], since one can use the algebraic structure of the coset to facilitate the computations. The coset action is obtained from a Green-Schwarz action [66] by fully fixing the kappa symmetry to the so-called coset gauge. While it is useful in the study of the classical integrability of this theory, the coset gauge leads to a kinetic term for the massless fermions which contains no quadratic piece. As a result, computing A using Poisson brackets is not straightforward in the coset gauge. Instead, we will perform the calculations using the Green-Schwarz action in the BMN light-cone kappa gauge. Explicit expressions up to quartic order in fermions have been recently found [68], but we will only work up to quadratic order in fermions and so will use the component action [69]. This section is divided into four parts. In section 2.1 we find the Killing spinors of the background in the metric (2.1). In section 2.2 we write down explicitly the action for Type IIB superstrings on AdS 3 ×S 3 ×T 4 , both before and after imposing the kappa gauge along the BMN light-cone coordinates. In section 2.3 we write down the super-currents for the A charges and in section 2.4 we compute the off-shell algebra A of the classical theory. In appendix A we establish our conventions.

Killing spinors for type IIB supergravity on AdS
In this sub-section we construct the Killing spinors for type IIB supergravity on AdS 3 × S 3 × T 4 . Expressions for these are well-known in the literature [70,71]. We adapt these well-known calculations to the metric and since this metric is well suited for expansion around the BMN ground state. The ten-dimensional Killing spinor equations of Type IIB supergravity on AdS 3 × S 3 × T 4 with R-R flux are where the covariant derivative is given by 5) and the R-R field strength by As is shown in more detail in appendix C, these equations are solved by where ε I 0 are constant 9+1 dimensional Majorana-Weyl spinors 8 , which further satisfy The matricesM andM depend on the AdS 3 × S 3 coordinates and for later convenience we seperate out the dependence on t and φ from the other coordinates by writinĝ and

Type IIB superstring action on AdS
In this sub-section we write down the action for Type IIB superstring action on AdS 3 × S 3 × T 4 . In section 2.2.1 we begin by introducing a set of bosonic vielbeins, particularly adapted to the analysis in the remainder of this section, and expressing the bosonic equations of motion in terms of these. In section 2.2.2 we write down the action to quadratic order in fermions. By picking suitably defined fermionic fields, our action realises the 16 unbroken supersymmetries of the background via linear shifts of the massive fermionic fields. In section 2.2.3 we write down the BMN light-cone kappa gaugefixed action to quadratic order in fermions. Just as was done in [72], we find it useful to redefine the fermions further so that they are neutral under the u(1) charges associated with t and φ translations. The action is then re-expressed in first-order formalism and fully gauge-fixed in the uniform light-cone gauge in section 2.2.4.

A suitable vielbein and bosonic equations of motion
The Lagrangian for the bosonic sigma model is given by where E α A = E m A ∂ α X m denotes the pullback of the vielbein. L B is invariant under SO (1,9) rotations in tangent space. As a result, all vielbeins that describe the same metric will lead to the same bosonic equations of motion, up to field redefinitions. Nevertheless, picking a suitable vielbein may reduce substantially the computational complexity of the analysis. Since we will be working with the metric (2.1), one seemingly natural choice is to pick diagonal vielbeins E m A given in equations (C.1) and (C.13). It turns out that, for the purpose of understanding the realisation of supersymmetry in the Green-Schwarz action, it is instead more conventient to use vielbeinsK m A andǨ m A , which are related to the E m A by orthogonal transformationŝ The matricesM andM are defined in equations (C.25), (C. 16) and (C.5). They follow from considering bilinears formed out of the Killing spinors ε I , (cf. equation (C.24)). As a result, as shown in equations (C.11) and (C. 19),K m A andǨ m A satisfy the Killing vector equation (C.10) and generate the so(2, 2) ⊕ so(4) = sl(2) ⊕ sl(2) ⊕ su(2) ⊕ su(2) isometry algebra of AdS 3 × S 3 .
The bosonic equations of motion that follow from L B are This form of the equations of motion is not only particularly simple, but will prove to be very useful in analysing the supersymmetries of string theory on this background.

Green-Schwarz action before kappa gauge fixing
In this sub-section we write down the Green-Schwarz action for a superstring propagating in AdS 3 ×S 3 ×T 4 up to quadratic order in fermions and construct supercharges preserving the non-gauge-fixed action. The Green-Schwarz action for Type IIB superstrings in a generic supergravity background was constructed in terms of superfields in [66], and explicit expressions in terms of fields are known to quadratic [69] and quartic order [68] in the fermions. We will perform a field redefinition of the conventional fermions [69] so that the 16 real supersymmetries of this background are realised as linear shifts of the massive fermions. 9 The Green-Schwarz Lagrangian can be written as The bosonic Lagrangian L B was discussed in the previous sub-section. We have split the fermionic Lagrangian into two terms: a term dependent on the worldsheet metric, L kin , and the Wess-Zumino term L WZ . In the background we are considering, the former term is [69] L kin = −iγ αβθ (2.20) The definitions of the vielbeins appearing above are given in equations (2.13) and (C.26). The Lagrangian L B + L kin is invariant under the supersymmetry transformations where in the above equation the index A = 0, . . . , 5. By imposing the Majorana condition on the fermions this gives us the expected 16 real supersymmetries of the background. We now consider the Wess-Zumino term 10 After introducing the rotated fermions we find This term is also invariant to quadratic order in the fermions under the supersymmetry transformations (2.21). To see this we can use the identity up to a total derivative. In appendix D we prove (2.24). Together with an obvious extension of the above argument to exressions involvingǨ instead ofK, the Lagrangian L WZ can therefore be written in a form where ϑ − I only appears with a partial derivative acting on it, making the symmetry under shifts of that fermion manifest.

Neutral fermions and the kappa gauge-fixed action
In this sub-section we impose the BMN light-cone kappa gauge on the Lagrangian obtained in the previous sub-section. In addition, we will further redefine the fermions. Recall that the tangent space rotations (2.19) introduced in the previous section were useful for obtaining the supersymmetry transformations before fixing kappa gauge. However,K m A andǨ m A , and therefore also the fermions η K and χ K , transform nontrivially under shifts of the coordinates t and φ. When imposing uniform light-cone gauge it is useful to work with fermions that are uncharged under these shifts [72], which motivates the further re-definition of the fermions. 11 To perform this field redefinition, recall that the rotation matricesM andM can be written in terms of the matrices M 0 and M t (see equation (2.9)), where M 0 is independent of t and φ while M t only depends on those two coordinates. In order to have fermions that are uncharged under shifts of t and φ one needs to multiply the fermions ϑ ± 1 by M −1 t and the fermions ϑ ± 2 with M t . In other words, we define (2.26) We also need to perform the corresponding rotation on the vielbeins defining new viel-beinsÊ andĚ, The components of the inverse vielbeins can easily be read off from equations (C.7) and (C.18) by dropping the first t-and φ-dependent factor. It is useful to introduce light-cone coordinates This leads to The light-cone components of the tangent space metric are given by The bosonic Lagrangian then takes the form We will work in the BMN light-cone kappa gauge The kappa gauge-fixed Lagrangian then takes the form

First-order action and uniform light-cone gauge
To fix the bosonic gauge we will impose uniform light-cone gauge [74]. The simplest way to introduce this gauge is to rewrite the action in a first-order formalism by introducing coordinates From the definition of the light-cone coordinates x ± we then have The isometries generated by shifts in t and φ lead to the conservation of the energy E and angular momentum J For the light-cone momenta we then find The uniform light-cone gauge fixing is now obtained by setting 12 The above gauge condition sets p − to 2. To make the origin of various expressions more clear we generally still write out factors of p − , unless this clutters our formulae excessively. In any case, the correct factors of p − can be restored from dimensional considerations.
To see how this gauge works let us consider the bosonic first-order action, which takes the form where and The equations of motion for the worldsheet metric leads to the Virasoro constraints C 1 = 0 and C 2 = 0. Since ′ x + = 0 we can solve the first constraint by Inserting this into the expression for C 2 we can solve the second constraint for p + . The gauge-fixed action can then be written as 13 with For the transverse fields we impose periodic boundary conditions 14 x i (+r) = x i (−r) and x i (+r) = x i (−r). Since we further assume there is no winding along the angle φ, we find that a physical state should satisfy the level matching condition Here we only consider string states with zero winding number. For more general states the gauge fixing condition becomes x + = τ + 1 2 mσ, where m is the integer winding number along the angle φ. 13 We have omitted the total derivative term p −ẋ − . 14 Here we also ignore possible winding modes along the T 4 directions. See below for a further discussion of these modes.
The gauge-fixed action is invariant under worldsheet translations, which leads to the conservation of the worldsheet momentum From the level matching constraint we then find that a physical string in the zero winding sector has to have vanishing total worldsheet momentum p ws = 0. (2.48) In order to study the worldsheet S matrix we need to be able to create well-defined asymptotic states to scatter. To do this we will from now on work in the decompactification limit by sending the parameter r, which gives the circumference of the worldsheet cylinder, to infinity. Note that after gauge fixing, the light-cone momentum P − is given by Hence, in the large-r limit the light-cone momentum becomes infinite. By imposing periodic boundary conditions on the T 4 coordinates x i we are ignoring winding modes on the torus. This is justified since we study local properties of the field theory on the worldsheet and work in the decompactification limit. If we begin with a string state in the zero winding sector and act on the state with a symmetry generator that acts locally, there is no way to obtain a state with non-zero winding. Similarly, the scattering of two excitations without any winding will not result in non-trivial winding of the out-going states. In the zero-winding sector the u(1) 4 shift isometries of the T 4 are supplemented by an so(4) symmetry, which we will discuss in the next subsection and will play an important role in A when we will use it to constrain the S matrix.
It is furthermore possible to check that, as long as we are in the decompactified theory with P − = ∞, the light-cone Hamiltonian takes the same form in any sector with finite winding on T 4 . This indicates that the S matrix that we will find by this treatment should be valid in any winding sector, and should not depend on the moduli of T 4 . The dependence of the spectrum on winding numbers and torus moduli should then manifest itself only at the level of the Bethe-Yang equations, as it happens in the case of orbifolds, see e.g. [75,76] for a review.

Gauge-fixed action with so(4) 1 ⊕ so(4) 2 bispinor fermions
The fermions appearing in the action (2.33), (2.34) are 32-component 9+1-dimensional spinors. However, these spinors satisfy a number of projections: the 9+1-dimensional Weyl projection, the kappa gauge condition (2.32) as well as equation (2.26). Because of these, writing the fermions as 32 component spinors is rather redundant. In this sub-section we will write down the fully gauge-fixed action in terms of non-redundant physical spinors.
As a result of the above projections, the physical spinors η I and χ I are in fact bispinors of so(4) 1 ⊕ so(4) 2 ⊂ so(8), with so(8) corresponding to rotations transverse to light-cone directions. The algebras so(4) 1 and so(4) 2 correspond to rotations along the non-lightcone AdS 3 × S 3 and T 4 , directions, respectively. 15 While the latter algebra remains unbroken by the background, so(4) 1 is in fact broken to so(2) ⊕ so(2), as can be already seen in the plane-wave limit [77,78]. We will see this breaking in the Lagrangian we write down in this subsection. Nevertheless, it is still convenient to express the fermionic fields that enter the Lagrangian as bispinors of so(4) 1 ⊕ so(4) 2 . We will use the indices a,ȧ (respectively, a,ȧ) to denote the positive and negative chirality so(4) 1 (so(4) 2 ) spinors. Further, we introduce gamma matrices,γ i with i = 1, 2, 3, 4 andτ i , i = 6, 7, 8, 9 for so(4) 1 and so(4) 2 . We write these matrices as 16 with the Clebsch-Gordan coefficients for the decomposition of two so(4) Weyl spinors of opposite chirality given by (2.51) The notation introduced above is purposefully reminiscent of the light-cone gauge in flat space [79] but our exact conventions are slightly different to, for example, those in [80]. The matricesγ i andτ i satisfy the Clifford algebra relations where t = s = σ 3 ⊗ σ 2 . We also introduce (2.53) so that the Lorentz generators take the form Some useful relations involving these gamma matrices are collected in appendix E. In order to obtain compact expressions for the gauge-fixed action we find it necessary to perform a change of basis on the gamma matrices presented in appendix B. These matrices are written as tensor products of five 2 × 2 matrices. Our change of basis takes the form With this change of basis, so(4) 1 and so(4) 2 act non-trivially only on n 2 ⊗n 3 and n 4 ⊗n 5 , respectively, while the 9+1-dimensional Weyl projection acts only on n 1 . The kappa gauge-fixed spinors satisfy Since the action of Γ 1234 and Γ 6789 reduces toγ 1234 andτ 6789 when acting on η I and χ I we see that χ I and η I carry indices (χ I ) ab , (η I )˙a˙b. (2.59) Having introduced this notation we can now re-write L kin in equation (2.33) as 17 (2.60) Above, we have suppressed the spinor indices for compactness and defined Re-writing L WZ in equation (2.34) in terms of so(4) 1 and so(4) 2 bispinors one arrives at a longer expression which we have relegated to appendix G. The above Lagrangian still depends on the worldsheet metric. As discussed above, one way to complete the light-cone gauge fixing is to go to first-order formalism and solve the Virasoro constraints. Alternatively we can impose the condition p − = 2 by solving for the worldsheet metric. Doing this we find that to the relevant order the metric is diagonal with components (2.62) The derivatives of the nondynamic field x − can then be found from the Virasoro constraints.

Supercurrents
In section 2.2.2 we wrote down an action which realised linearly all 16 supersymmetries of our background. However, half of the supervariations (2.21) are incompatible with the BMN light-cone kappa gauge choice (2.32). This is a well known aspect of the light-cone gauge formalism [81]-it implies that such supervariations have to be combined with a compensating kappa transformation in order to preserve the gauge choice (2.32). The eight supercharges that commute with the Hamiltonian and form the fermionic part of A are associated with variations of precisely of this form. Since kappa gauge transformations are known explicitly [66], it is in principal possible to find expression for such compensating kappa gauge transformations. The procedure is however computationally involved. To simplify matters, we will write down the supercurrents corresponding to the A supercharges to first order in fermions and third order in the transverse bosons. When computing the algebra A later in this section we will only need these expressions.
For notational convenience, we split the full supercurrent into parts involving only massless fields, only massive fields and a part involving a mix of massive and massless fields, The supercurrents are given by (2.69) Above, for the massive part of the supercurrents we have only written down the lowestorder-in-bosons expression since it will be all we need later on. For compactness we have also suppressed all spinor indices; re-instating these we have, for example, Using the equations of motion, which are presented in appendix H, we have checked that the currents satisfy the equation and hence are conserved.

The A algebra
In this sub-section we will compute the algebra A. Our computation will be done in a field expansion discussed below. In particular, we will work to leading order in fermions and sub-leading order in bosons. This is the same order to which the corresponding algebra was computed for Type IIB strings on AdS 5 × S 5 [61]. Before describing the details of the computations, let us pause briefly to make two general observations. Firstly, on-shell A reduces to psu(1|1) 4 extended by the Hamiltonian and a central angular momentum and by the torus isometries. This is simply the part of superisometries of the classical string theory on AdS 3 ×S 3 ×T 4 that commutes with the Hamiltonian, and amounts to psu(1|1) 4 ⊕ u(1) 2 ⊕ so(4). The important consistency check then is to see that when going off-shell, by relaxing the level-matching condition, the algebra becomes centrally extended in just the right way. In other words, the Poisson bracket between two different supercharges should result in an expression of the form where the matrix on the right-hand side can be decomposed into the two central charges extending the symmetry algebra.
Secondly, we note that the massive (y i , z i and η I ) and the massless fields (x i and χ I ) each form a consistent closed sector of the equations of motion of the theory. In the classical theory the massive sector is isomorphic to a closed sub-sector of the Type IIB string on AdS 5 × S 5 . 18 As a result, the off-shell computation of A in the massive subsector is identical to the computation performed in [61] and so we will not repeat it here. Instead, we will perform two types of computations that are new to Type IIB strings on AdS 3 × S 3 × T 4 . In section 2.4.1 we restrict to the massless sector of the theory and compute the off-shell algebra A; as anticipated in the previous paragraph, we explicitly see that on-shell the algebra does indeed reduce to psu(1|1) 4 ⊕ u(1) 2 . In section 2.4.3, we compute, off-shell, in the full massive and massless theory the relation (2.72).
We only determine the part of the central charges that depend on the bosonic fields. Since the central charges have to vanish for zero total worldsheet momentum this is enough to reconstruct the full charges. As we will see, the momentum dependence of the central charges comes in through the nonlocal and nondynamic field x − . To capture this dependence we employ a "hybrid" expansion similar to what was used in AdS 5 × S 5 in [61]. This means that we expand the action in the transverse fields to quadratic order in fermions and quartic order in the transverse bosons, but keep any explicit factors of x − unexpanded. This allows us to capture the full momentum dependence of the central charges. It is worth noting at this point that central extensions of the on-shell algebra of this kind had been studied for the plane-wave limit of AdS 3 × S 3 × T 4 in [77].
The fermions in the Lagrangian (2.60) and (G.6) do not have a canonical kinetic term and so will not have a conventional Poisson bracket. It is possible to further redefine the fermionic fields order by order in the field expansion to correct this. However, for our purpose it will be simpler to work with the non-canonical Poisson bracket for the fermions that follows from the Lagrangian (2.60) and (G.6). The Poisson bracket of the fermions η I and χ I is presented below.

The massless sub-sector
As we have noted above, the equations of motion of our system are such that it is consistent to set the transverse massive excitations to zero. In this sub-section we focus on computing A in the purely massless sector. This sector turns out to have a number of simplifying features compared to the full theory and so serves as a good warm-up exercise. We will therefore repeat some of the steps discussed above in more detail. What is more, this sector is not described by a semi-symmetric space coset and so understanding how it enters the integrable machinery is one of the central results of this paper. The massless part of the gauge fixed Lagrangian is given by (2.73) The massless parts of the supercurrents were given in equations (2.64) and (2.67). Notice that to this order in fermions the supercurrents do not contain a term cubic in the bosons, and are in fact the same as they would be in flat space. 19 The non-linear terms in the equations of motion for the fermions are exactly cancelled by the nonlocal exponential part of the supercurrents. We now want to calculate the algebra A obtained by taking Poisson brackets between the supercharges obtained from the currents. To do this we write the action in first-order formalism. The conjugate momenta are of the bosonic fields are given by Inserting this into the Lagrangian we get with the constraints given by (2.76) By solving the constraint C 1 = 0 we obtain Up to quadratic order in the fermions the massless Hamiltonian density H (m) = −p + is given by (2.78) 19 We expect this to be only true because we are working to lowest order in fermions. There does not appear to be an obvious reason for the term quartic in fermions in the action to vanish. Such a term would likely lead to corrections to the supercurrents that are cubic in fermions. Such terms are however absent in the free theory.
From the kinetic term of the action we can read off the (canonical) Poisson brackets Using results from appendix J one can check that the supercharge densities both lead to conserved charges Next we note that up to integrating by parts 20 The last term in the expression in the bracket is exactly the quadratic Hamiltonian in the massless sector. So up to quadratic order in excitations we find (2.84) Similarly we can calculate the commutator between two different supercharges (2.85) At quadratic order we then find (2.86) 20 Here it is useful to note the identities (E.2).
Hence, the central charge takes the form 21 ). This is exactly the form found in [61] for AdS 5 × S 5 , which as we argued coincides with what we must have in our massive sector, as can be seen by an appropriate truncation of the supercoset [49].
To summarize, in this sub-section we have worked in the massless subsector of the full string theory. We have constructed the supercharges and hamiltonian of the theory in the first-order formalism and have shown that they satisfy the commutation relations of A. We have also found that in the off-shell theory the central extension C takes precisely the form expected for A.

Fermionic Poisson brackets
Having found the central charges of A in the massless sector we now want to perform the same calculation again but now including both massive and massless fields. To the order that we will be working in, we only need the dependence of the central charges on the bosonic fields. As explained below, we will in fact only need to consider terms up to quadratic order in both the massless and massive fields, so that the only bosonic quartic terms that we will be interested in contain fields of both masses. Each term in the supercharges contains at least one fermionic field. Since we are only interested in the bosonic field dependence of the central charge, we only need the contribution from the Poisson bracket of two supercharges that comes from the Poisson bracket between two fermions. Any other term will be higher order in fermions.
Since we only need to calculate Poisson brackets between fermionic fields, we do not need to introduce canonical momenta for the bosons. The kinetic terms for the fermions is quite complicated which leads to an involved Poisson structure. We relegate the calculation to appendix I and simply state the non-zero Poisson brackets here The bi-spinor valued matrices A ij are given in equation (I.6) and we have suppressed the bispinor indices in the above so as not to over-clutter the notation. 22 21 The Poisson bracket between the charges Q 1 and Q 2 contains both the charge C and its conjugate C. In section 3 we will write the full algebra A in a more convenient form. Here we report the expression corresponding to the charge C. 22 The reader should note that the epsilon symbols appearing in the Poisson brackets carry different kinds of indices depending on which particular fermions' Poisson bracket one is computing.

Computing the central charge C in the full theory
To establish the off-shell symmetry algebra of Type IIB string theory on AdS 3 ×S 3 ×T 4 we need to check whether the commutation relation (2.72) holds in the full theory. Above, we have demonstrated such a relation in the massless sector of the theory (that is, when massive fields are turned off). Since, as we argued, a similar calculation for the massive sector follows directly from [61], all that we need to worry about now are the mixedmass terms. In this sub-section, we will indeed establish (2.72) by taking into account mixed-mass terms the supercharges expanded to linear order in fermions and cubic order in bosons. To this order we will be showing that such a relation holds with the central charge C taken to zeroth order in fermions and quartic order in bosons.
Using the Poisson brackets given in equation (2.88), we find the Poisson bracket between two supercharges The details of the calculation are given in appendix K. The last line above is a total derivative and can be dropped with a suitable choice of boundary conditions. By partially integrating the second line and again dropping the total derivative we find that the remaining integrand takes the forṁ up to terms that are quartic in the massive fields. The last expression is equal to −p − ′ x − and so we may write Hence, the central charge takes the form with ζ = exp(+2ix − (−∞)), in agreement with the expression found the previous subsection.

Symmetry algebra
In the previous section we have found the off-shell symmetry algebra A for type IIB superstrings on AdS 3 × S 3 × T 4 . We showed that A is given by a central extension of psu(1|1) 4 ⊕ so(4) 2 , where so(4) 2 comes from the torus coordinates. 23 In this section we will first review how this algebra can be constructed by tensoring two copies of su(1|1) 2 c.e. . Then, in subsection 3.2 we will investigate the representations of A in the near-plane wave limit. This can be read-off from the supercurrents obtained in section 2.3 but we collect the results here to set up the notation and conventions that we will use in later sections.
As a preliminary step, it is convenient to rewrite A in components, using the notation introduced in appendix L. We then find the anti-commutation relations take the form where we introduced labels "L" and "R" (left and right) for the supercharges in psu(1|1) 4 . These are inherited from the superisometry algebra su(1, 1|2) L ⊕ su(1, 1|2) R , where they refer to the chirality in the dual CFT 2 . Note that in the leading-order expansion of appendix L, the central charges C, C were linear functions of the worldsheet momentum, C = C = − 1 2 P. This is indeed the leading order term in the expansion of the non-linear relation found in the previous section, cf. (2.92). The so(4) 2 subalgebra arising from the torus directions can be decomposed into su(2) • ⊕ su(2) • , satisfying 3) The supercharges Q j L,R are in the fundamental representation of su(2) • . Indices are therefore raised and lowered by the antisymmetric tensor ǫ˙a˙b and its inverse, so that charges with upper indices transform in the anti-fundamental representation. We then have where Q is any supercharge in the appropriate representation. All of the generators of the centrally extended psu(1|1) 4 commute with su(2) • . The u(1) charges of A are therefore given by the Hamiltonian H, the angular momentum M, two Cartan elements coming from the two su(2)'s and the central elements C, C.
The centrally extended psu(1|1) 4 superalgebra appeared already in the study of the massive sector of the theory in [34], and as discussed there it could be obtained from two copies of the centrally extended su(1|1) 2 . In the next subsection we briefly review that construction.

From
Let us consider su(1|1) L ⊕ su(1|1) R , given by Physically, if we intend to couple these two systems, it is natural to define the positivedefine combination of the two central charges to be the Hamiltonian, while the other one will be an angular momentum: Let us consider a central extension of su(1|1) 2 by setting If we now consider a tensor product of two copies of the above algebra, we have and for the central elements If we now identify the central charges as and consequently drop the indices 1, 2, we are left precisely with (3.1). Constructing psu(1|1) 4 c.e. as a tensor product of two su(1|1) 2 c.e. as described above will be particularly useful in the study of its representations below [34,49].

Representations in the near-plane-wave limit
We expect the fundamental excitations of the GS string to transform in two distinct (not necessarily irreducible) representations of A, for massive and massless particles 24 , with the corresponding modules having the same dimension. This constrains the dimension of the representations of psu(1|1) 4 c.e. ⊂ A that may appear. Since long representations are at least sixteen-dimensional, our representations must instead be short, i.e. they must satisfy the shortening condition [32,34] which can be recast in the form of a dispersion relation in which the eigenvalues of M play the role of a mass term 25 . In section 2.3 we obtained the representation of A in terms of the fields. In order study it more easily, it is useful to go to momentum-space and introduce oscillators. It will be enough to consider the leading-order in the field expansion of the supercharges, which coincides with the leading order in a near-plane-wave [77] or BMN expansion [82].
Let us introduce bosonic creation and annihilation operators, that schematically take the form where ω(p, m) is the dispersion, and fermionic ones where f (p, m), g(p, m) are wavefunction parameters. We will have eight such pairs of operators for bosons and eight for fermions, whose precise form is given in appendix L. We can use them to construct the module of the representation, which is then given by the eight massive states 15) and the eight massless ones Since at this order all excitations are relativistic, we have , (3.17) see also equations (L.15-L. 18). Note that formulae depend on the eigenvalue m of M, which can take value ±1 for massive excitations and 0 for massless ones. It will be convenient to denote ω p = ω(p, ±1),ω p = ω(p, 0), and similarly for f and g. Figure 1: The left and right psu(1|1) 4 c.e. multiplets consists of two bosons Y L,R , Z L,R and of two fermions η L,Ṙ a , corresponding to transverse directions on AdS 3 × S 3 . The fermions carry an indexȧ of the fundamental representation of su(2) • . Note that off-shell any excitation is charged under all supercharges, whereas on-shell left (respectively right) excitations are charged only under left (respectively right) supercharges. We indicate the supercharges whose action corresponds to the outermost arrows of the diagram. The innermost ones follow by Hermitian conjugation.
In terms of the ladder operators the supercharges take a very transparent form, whence their action can be easily read off: where we suppressed the dependence of a † , a and d † , d on the momentum p. Similarly, the Hamiltonian H and the angular momentum M read This sixteen-dimensional module will split in several irreducible ones, which we will describe below one by one. We can label them by the eigenvalue of the angular momentum M.

Left representation
One representation of dimension four (two bosons and two fermions) has eigenvalue +1 under M, and consists only of excitations labelled "L". These correspond to half of the transverse modes on AdS 3 × S 3 , and it can be represented as in the left panel of figure 1. This is a bi-fundamental representation of psu(1|1) 4 c.e. , supplemented by the action of su(2) • on the fermions. In particular, the fermions are in the fundamental representation It is interesting to see what happens if we consider the representation on-shell, i.e. (since we are dealing with a single-particle representation) at zero momentum. Observing that one of the fermion wave-function parameters vanishes then, g p=0 = 0, greatly simplifies the action of the supercharges (3.18). In fact, only left supercharges act nontrivially. For this reason we call this representation left, which explains the name of the corresponding excitations, which then can be thought of as left-movers in the dual CFT 2 . Of course this is only true for an on-shell one-particle state-generally, all states are charged under the whole psu(1|1) 4 c.e. .

Right representation
We have a similar representation with eigenvalue −1 under M, which is depicted in the right panel of figure 1, and consists of "R" excitations. This is again a bi-fundamental representation, closely resembling the left one. We see however that if we take the S 3 excitation |Y L to be the highest weight state in the left-representation, we must take the AdS 3 excitation |Z R to be the highest weight state here. The reason is that we cannot take e.g. Q 1 L and Q R1 to be both lowering operators, if we want the central charge C to be non-vanishing. Instead, we should take e.g. Q 1 L and Q 1 R . This in turn forces the choice of different highest weight states in the two representations. The fermions η Rȧ are in the anti-fundamental of su(2) • .
In the same way as earlier, taking the representation to be on-shell makes it charged under the right supercharges only.

Massless representation
We expect the remaining eight particles (four bosons and four fermions) to be all massless, at least in this semi-classical analysis. This is indeed the case, and we can check that the massless particles arrange themselves into two irreducible representations of psu(1|1) 4 c.e. , both with fermionic highest weight states, see figure 2. Additionally, note that these representations seem to be left by the argument above. This should be taken with a pinch of salt, because we are considering the situation when the eigenvalue of M precisely vanishes, which as we will see makes the left and right representations practically equivalent.
We have to take the action of so(4) 2 into account. Under this, all of the torus bosons are obviously charged. In our su(2) decomposition we see that the two psu(1|1) 4 c.e. modules are rotated one into another by the action of su(2) • . We then conclude that

Exact representations
In the previous section we constructed representations of A that are valid in the nearplane-wave limit, when C = C = − 1 2 P. On the other hand, we know that instead C = C should be non-linear functions of the world-sheet momentum P given by equation (3.2). In this section, we will show that the representations of section 3.2 can be deformed in such a way so as to satisfy (3.2) together with the shortening condition (3.12) without resorting to any perturbative expansions.
To do this, it will be sufficient to suitably deform the representation parameters ω p , f p and g p . In fact, since the Hamiltonian follows from the central charges, ω p will be fixed in terms of the fermion wave-function parameters. In order to find the central charges C = C, which will now be complex and conjugate to each other, we must introduce complex representation coefficients. Hence, f p will be replaced by a p or its conjugateā p , and g p by b p orb p . These will also suitably depend on the mass m. When expanded in the near-BMN limit, a p andā p reduce to f p (orf p if m = 0) while b p and b p to g p (org p ).
We will proceed as follows. Since the representations of A discussed above must in particular be bi-fundamental representations of psu(1|1) 4 c.e. , and these can be obtained [32,49] from fundamental representations of su(1|1) 2 c.e. , we will focus on these first. In subsection 4.1 we will construct the most general su(1|1) 2 c.e. short fundamental representations. In subsections 4.2 and 4.3 we will show how from these we can find massive and massless psu(1|1) 4 c.e. representations, respectively, and comment on their properties. Then in subsection 4.4 we write down the deformed representation parameters a p , b p for the massive and massless case, and finally in subsection 4.5 we will rule out quantum corrections to the massless dispersion relation by a symmetry argument.
In [32] short representations of the centrally extended psu(1|1) L ⊕ psu(1|1) R algebra were considered. Denoting a boson and fermion excitation of definite momentum p by |φ L p , |ψ L p respectively, the fundamental representation ̺ L is given by Our choice of the representation coefficients ensures that the left-and right-Hamiltonians are positive definite. Furthermore, when we reduce to an one-particle on-shell representation, it must be b p=0 =b p=0 = 0. In this sense, this representation is a left one. We can consider a right representation ̺ R . The module consists of two excitations |φ R p and |ψ R p transforming as Comparing with (4.1), we see that the two representations are related by exchanging the labels L↔R. It we will also be useful to consider a representation ̺ L , ̺ L : This representation differs from ̺ L by the choice of the highest weight state, which is fermionic here rather than bosonic. In a similar way, one can construct a ̺ R representation. It is easy to check that any short representation of psu(1|1) 4 c.e. for which H has real eigenvalues takes the form of one of ̺ L , ̺ R , ̺ L , ̺ R . We will use graded tensor products of these representations to construct short representations of psu(1|1) 4 c.e. .

psu(1|1) 4 c.e. representations for massive excitations
We now want to take the massive near-plane-wave representations discussed in section 3.2 and deform them in such a way as to reproduce the non-linear relation (3.2). We will then see how these can indeed be thought of as coming from the tensor product of suitable pairs of the su(1|1) 2 c.e. representations that we just constructed. If we restrict to on-shell one-particle states, only left (respectively right) supercharges have a non-trivial action on left (respectively right) excitations, of the form where we anticipated the tensor-product structure in the formula label. Off shell, by virtue of the central extension this is supplemented by the action of left supercharges on right-moving states and vice-versa: (4.5)

Bi-fundamental structure
The above left and right psu(1|1) 4 representations can be constructed by tensoring the fundamental representation of (4.1,4.2). To this end, it is sufficient to identify the excitations as where indices have been appropriately raised and lowered. The left psu(1|1) 4 module is then isomorphic to ̺ L ⊗ ̺ L , while the right one to ̺ R ⊗ ̺ R .

Left-right symmetry
It is clear that consistency with the string theory picture requires considering both the left and right representations 26 . On the other hand, "left" and "right" are just labels that can be swapped without affecting the description. This results in a Z 2 left-right symmetry (LR-symmetry) [32,34,49]. In particular, this relates left and right massive excitations as which is compatible with (L.23).

psu(1|1) 4 c.e. representations for massless excitations
Similarly to the previous subsection, we now deform the massless near-plane-wave representations and show how they also enjoy a tensor-product structure. We know that massless bosons transform in the fundamental representation of both su(2) • and su (2) (4.8) Masslessness of the excitations is encoded in the fact that they are annihilated by M, which due to the shortening condition (3.12) plays the role of mass. This results in a constraint on the representation coefficients representations. To see this we note that the massless excitations can be re-written as where the two copies are labelled by an su(2) • index a = 1, 2. Note that we used two modules of the form (̺ L ⊗ ̺ L ) ⊕ (̺ L ⊗ ̺ L ), in agreement with the fact that for massless representations the highest weight state is fermionic.

Equivalent descriptions
It may appear strange that massless excitations come from left representations only. Actually, there are several ways to obtain the massless excitations out of tensor product of su(1|1) 2 modules. Let us introduce rescaled excitations 27 27 As we will see in the next subsection, the ratio ap bp appearing here is essentially the sign of the momentum, due to the massless condition (4.9). which gives the following action of the supercharges where we used (4.9). Thanks to this change of basis, we can identify a different tensor product structure, given by This amounts to considering two modules of the form (̺ R ⊗ ̺ R ) ⊕ (̺ R ⊗ ̺ R ), i.e. to obtain the massless representation out of the massless limit of two right modules, rather than of two left ones. Yet another possibility is to perform the rescaling (4.11) only on one of the psu(1|1) 4 c.e. representations, for instance setting on the fermions. In this way, the massless modules have the form ( where the former term in the direct sum corresponds to a = 1 and the latter to a = 2. The conclusion is that any of the two psu(1|1) 4 modules describing the massless excitations can be equivalently taken to be right or left, up to an inessential change of basis. 28 Therefore, we can think of our representation as coming from any of these choices depending on what is most convenient, as we will discuss in the next subsection.

Left-right symmetry
We have seen that the massive sector should be invariant under exchanging the "left" and "right" labels. This imposes a discrete symmetry on the S-matrix in that sector [32,34]. When we include massless excitations and consider two-particle states as it is necessary for the S-matrix, a consistency condition arises. One should give a prescription on how the massless excitations transform when exchanging L↔R in the massive sector.
Let us start from the case where the massless excitations are in the representation (̺ L ⊗ ̺ L ) ⊕ (̺ R ⊗ ̺ R ). Then, a natural extension of LR symmetry is exchanging the L and R labels in the massless sector too. Following the discussion of the previous subsection, it is clear that those labels are somewhat artificial, because the left and right massless representation are in fact equivalent.
In what follows we will find it more convenient to describe massless excitations using the (̺ L ⊗ ̺ L )⊕(̺ L ⊗ ̺ L ) representation. This is because the action of the su(2) • raising and lowering operators is simpler in this case (in the mixed case, one finds additional factors of a p /b p ). Even in this case, we can consistently implement LR symmetry, by combining the previous prescription with the change of basis (4.14), see also equation (L.24). This gives the prescription where we suppressed the momentum label on the excitations for clarity. Note that the symmetry now depends on the momentum of the particles through the combination ap bp . We will comment on the interpretation of these ubiquitous coefficients in the next section.

Representation coefficients
The representations we constructed can be labelled by the values of the central charges, in terms of which the representation coefficients a p , b p and their conjugates must be determined. There are essentially two independent charges: the angular momentum M, whose absolute value plays the role of the mass, and the central charge C (together with its conjugate C). Their eigenvalues on a one-particle off shell representation are In section 2 we have found their expressions in terms of the momentum p and the coupling constant h to be where we explicitly extracted the dependence on the coupling constant h, and ζ = e 2iξ characterizes the representation. From the shortening condition (3.12) we immediately find the dispersion relation For m 2 = 1, this dispersion relation, strongly reminiscent of the AdS 5 × S 5 case [83], was first found in this context from the study of giant magnons on AdS 3 × S 3 [57]. The parametrisation satisfies equation (4.16) if the Zhukovski parameters x ± satisfy As for the value of ξ, as discussed at length in [62] we can take it to vanish on the oneparticle representation, but it plays an important role in the multi-particle ones. When constructing the two-particle representation out of the tensor product of two one-particle ones, we see that to reproduce the correct value of the central charges we cannot take ξ 1 = ξ 2 = 0 for both of the constituent particles. In fact, imposing that on a two-particle state C gives we see that the parameter ξ must be non-vanishing in one of the representations, resulting in a non-trivial coproduct. [84][85][86] We solve this condition by setting We see that the case m = 1 does not introduce any new feature with respect to the treatment of [34]. However, at m = 0 something new happens already at the level of the representation parameters, in terms of the non-analyticities This is typical in a theory featuring massless excitations, and naturally leads to distinguish left-and right-movers on the worldsheet. Indeed the combination which appears in the formulation of (target-space) left-right symmetry or in the action of the su(2) • raising and lowering operators, shows that excitations have different transformation properties depending on the sign of their momentum. A way to consistently get rid of the non-analyticities is to treat worldsheet left-and right-movers as two genuinely different species of particles, with dispersion If we take both massless psu(1|1) 4 c.e. modules to be in the representation ̺ L ⊗ ̺ L , we find that worldsheet left-movers (respectively right-movers) transform in a definite representation of su(2) • . On the other hand, by virtue of (4.15), the LR symmetry transformation also swaps worldsheet left-and right-moving fermions. In the representation ̺ L ⊗ ̺ R , instead, the situation is reversed as can be seen from (4.11). LR symmetry preserves the worldsheet chirality, while su(2) • rotates worldsheet left-and right-moving fermions into each other.

Corrections to the massless dispersion relation
As we have seen in the previous section (see for example equation (4.26)), the massless modes have rather different properties to the massive ones. One may wonder whether the massless modes could receive corrections which would give them masses in the quantum theory. Even in integrable theories, it may happen that quantum corrections dress the particle with a mass, 29 and it is interesting to investigate whether this can be the case here. As we are about to show, this is impossible unless part of the so(4) 2 invariance is also broken.
We have seen that the massless module is constructed out of two bi-fundamental psu(1|1) 4 c.e. ones. Each of them can be equivalently left (L) or right (R), giving the possibilities However, only when the representation coefficients satisfy |a 2 p | = |b p | 2 , all of these cases are equivalent, up to a rescaling of the basis vectors. If we think of deforming the representations to switch on a mass ε > 0 (coming from e.g. quantum corrections), these four choices will no longer be equivalent, and will lead to different deformations, which we will consider separately. The non-vanishing masses will appear in particular in the matrix representation of M. This will read either where we used a block form and the ± signs correspond to LL/RR, or (4.29) In this notation each block corresponds to a bi-fundamental representation, with the two upper blocks being the massive left and right representations of [34]. We can immediately establish that the choice of (4.28) is incompatible with crossing invariance. In fact, charge conjugation will flip the sign of all the u(1) charges, see also equation (5.42) below. Therefore, a necessary condition for crossing invariance is that each representation appears together with its conjugate, which is indeed the case in the massive sector, as one can easily read-off from the first two diagonal entries of M above. This is not the case when we choose to deform the LL (or RR) representation for ε > 0: both representations will have positive (or negative) eigenvalues of modulus ε. On the other hand, the choice of equation (4.29) is compatible with crossing symmetry, but does not respect the so(4) invariance. In fact, all of the supercharges should commute with su(2) • , and therefore so should M. But su(2) • rotates the two mass-ε representations into each other, so that its raising and lowering operators cannot commute with M when ε = 0. This also clearly shows how both obstructions disappear at ε = 0. 29 A prototypical example being the Gross-Neveu model [87].

S matrix
In the previous section we have constructed the representations of A valid for arbitrary values of the momentum and of the coupling constant h, as long as we are in the decompactification limit. The mere fact that the two-particle S matrix should commute with the off-shell symmetry algebra will allow us to fix it almost completely, up to a small number of functions-the dressing factors.
In subsection 5.1 we recall the construction of the su(1|1) 2 c.e. invariant S matrices of [32]. By tensoring these in a suitable way, in subsection 5.2 we obtain the A-invariant S matrix up to the dressing factors. In subsection 5.3 we show that these dressing factors can in turn be constrained by unitarity. In subsection 5.4 we show that the S matrix satisfies the Yang-Baxter equation, and therefore can be used to define an integrable theory. Finally, in subsection 5.5, we constrain the dressing factors by crossing symmetry.
The S-matrix scattering two fundamental particles is an operator-in fact, a finitedimensional matrix-that relates in-and out-states. Schematically, where outgoing momenta are permuted. Due to the presence of massless excitations, defining a scattering matrix may appear problematic. In the familiar relativistic case, the equivalent of the dispersion relation (4.26) is linear in p. Therefore, the group velocity of a wave-packet is i.e., massless relativistic particles move at a constant velocity-the speed of light. Particles with the same worldsheet chirality then cannot scatter, regardless of the value of their momentum. 30 In the case of interest to the present paper however, factorised massless scattering appears to be simpler than in the relativistic case. In fact, from equation (4.26) we find the group velocity v non-rel = ± h cos p 2 .
We see then that massless excitations of different momenta have different velocity, so that we can expect them to scatter in the usual way. It could be interesting to investigate the near-plane-wave limit of our S matrix, in which the theory becomes approximately relativistic.
The two-particle S-matrix has to satisfy a number of consistency conditions. The first requirement, is that S commutes with all the generators of A acting on two-particle excitations S (12) (p, q) Q (12) (p, q) = Q (12) (q, p) S (12) (p, q).

(5.4)
Additionally, braiding unitarity is the requirement that acting twice with the S-matrix is equivalent to acting with the identity operator and physical unitarity demands that S is unitary as a matrix Furthermore, as we will see our S-matrix will satisfy the Yang-Baxter equation, meaning that it can consistently be used to define multi-particle scattering. In what follows, it will be useful to define, by means of the permutation matrix Π, which then satisfies Conservation of energy and of the other central charges implies that the scattering can be broken down in several sectors, consistently with our discussion of the representations. We naturally find purely massive scattering, which is the one that was already discussed in [34], purely massless scattering, and finally mixed massive-massless scattering. In each of these parts we will construct the S matrix as one that describes the scattering of suitable su(1|1) 4 c.e. modules. In fact, in view of our discussion of these modules, it is convenient to first consider suitable su(1|1) 2 c.e. invariant S matrices that will serve as building blocks of the full S matrix.

The su(1|1) 2 c.e. invariant S matrices
In section 4.1 we constructed the short representations of psu(1|1) 2 , namely ̺ L , ̺ R , ̺ L , ̺ R . The scattering of these leads to distinct S matrices. We start by discussing the case in which both excitations are left ones, which allows for four different S matrices: (5.12) As indicated by their labels, the S matrices above scatter particles in the representations  4). However, that linear relation allows for an arbitrary prefactor in each S matrix. Such prefactors, until we take non-linear constraints such as unitarity and crossing symmetry into account, are merely a matter of convention. The explicit parametrisation of the S-matrix element is given in appendix M, with a convention that slightly differs from the original one of [32].
When the two excitations do not have the same LR flavor we find S-matrices such as The S matrices above scatter particles in the representations ̺ L ⊗ ̺ R , ̺ R ⊗ ̺ L and ̺ L ⊗ ̺ R , ̺ R ⊗ ̺ L . The former pair was one of the main results of [32], while the latter corresponds to the so-called second central extension in appendix D in the same reference. Note that we could write down four more S matrices corresponding to scattering processes where one of the excitations is in the representation ̺ R . We will not be needing their explicit form, which in any case follows from similar changes of bases. The S-matrix elements obey a constraint due to left-right symmetry, which simply reads The explicit expression of these S-matrix elements is also given in appendix M. The case where we scatter particles in the representations ̺ R ⊗ ̺ R , ̺ R ⊗ ̺ R , etc. follows from equations (5.9)-(5.12) by LR symmetry.

The S-matrix from a tensor product
As discussed in section 4, the excitations of the AdS 3 × S 3 × T 4 superstring transform in four bi-fundamental representations of psu(1|1) 4 c.e. , two massive (̺ L ⊗ ̺ L and ̺ R ⊗ ̺ R ) and two massless ones (both of the form ̺ L ⊗ ̺ L ). The corresponding S matrices can be obtained from graded tensor products of the ones introduced in the previous section, which takes the general form up to a prefactor. The tensor product⊗ is graded, i.e. it takes into account the signs arising from fermion permutations where the symbol ǫ is one for fermions and zero for bosons. We collect explicit expressions for the graded tensor products used in the construction of the S-matrix blocks in appendix N. Each of the psu(1|1) 4 c.e. -invariant blocks that we obtain in this way should be multiplied by a (dressing) scalar factor. A priori these may all differ, but as we will see some of those are in fact related by additional symmetries, namely left-right symmetry and su(2) • . To describe these blocks it is convenient to split the S-matrix in three sectors: massive, massless, and mixed-mass. Schematically where S •• scatters two massive particles, S •• scatters two massless ones, and the remaining blocks describe mixed-mass scattering. 31 Below, we will construct the S matrix block by block, having particular care to keep track of the number of independent scalar factor that we should allow for. We will come back to those factors at the end of the discussion.

Massive sector (••)
In the massive sector, the psu(1|1) 4 c.e. modules are ̺ L ⊗̺ L and ̺ R ⊗̺ R , which we labelled "left" and "right" depending on their eigenvalues under M. The S matrix that scatters two left-modules is then left -left: S LL⊗ S LL . These two matrices can in principle be multiplied by two arbitrary scalar factors containing the dressing factors, which we call σ •• and σ •• respectively. The two remaining blocks (RR and RL) can be obtained in a similar way, and moreover are related to the LL and LR blocks by left-right symmetry. In particular, this symmetry constrains the RR and RL scalar factors in terms of σ •• and σ •• . The massive fermions are also charged under su(2) • . It is easy to see that invariance under such transformations is guaranteed by the tensor product structure, so that no additional requirement should be imposed on S •• . Therefore, the massive sector of our S-matrix is given precisely by the matrix proposed in [34], and schematically reads

Mixed-mass sector (•• and ••)
In the mixed-mass sector we scatter one massive particle with one massless one, or vice versa. Let us focus on the former possibility. The relative coefficient between the blocks is fixed by su(2) • action. If instead we started from a right massive particle, we would have found two blocks of the form massive (right) -massless: S RL⊗ S RL , (5.25) which again form a doublet of su(2) • . The two pairs of blocks of (5.24) and (5.25) are related to each other by left-right symmetry. Therefore, we are left only with a single undetermined scalar factor for the scattering of a massive particle with a massless one, i.e. for the whole S •• . We denote the corresponding dressing phase as σ •• , and we have If we now consider the scattering of a massless particle with a massive one, analogous considerations yield the S •• up to a dressing factor σ •• , (5.27)

Massless sector (••)
We are left with the scattering of massless particles, each transforming in two copies of ̺ L ⊗ ̺ L . Constructing the S-matrix for these psu(1|1) 4 c.e. modules would lead to 16 seemingly unrelated blocks. However, each of the modules is part of a su(2) • doublet. As a result, the blocks must arrange themselves in an su(2) • covariant expression. In other words we may decompose S •• as S su(2) ⊗ S LL⊗ SLL , (5.28) and the su(2) invariant S-matrix takes the familiar form 33 S su(2) (p, q) = 1 1 + ς pq Π + ς pq 1 , (5.29) in terms of the permutation operator Π and of an undetermined function ς pq which we will constrain later. The action of S su (2) can be represented in a block matrix form, whereby the whole S •• takes the form  up to an overall coefficient containing the dressing factor σ •• .

Normalisation of the sectors
Before moving on to discuss the non-linear constraints on the S matrix it is convenient to fix its normalisation. There is some arbitrariness in doing so because, as we mentioned, any of the blocks discussed above can be multiplied by an arbitrary prefactor.
Here we chose these prefactors in such a way as to reproduce scattering elements that are compatible with perturbative results (in the massive sector, where there exists a proposal for the dressing factors [35]) and make their symmetry properties as manifest as possible. In particular, we dictate the form of the following boson-boson processes In the massive sector, this is the same normalisation used in [34,35]. The necessary prefactors by which the formulae of appendix M must be multiplied are given in appendix O.

Physical and braiding unitarity
We have already mentioned that braiding and physical unitarity are necessary properties for the self-consistency of our construction. It is easy to see that they result in (mild) constrains on the five scalar factors, i.e.
together with a constrain on the undetermined function ς pq appearing in the su(2) Smatrix, where * denotes complex conjugation.

The Yang-Baxter equation
A necessary condition for our S-matrix to describe an integrable system is that the Yang-Baxter equation (YBE) holds. In a matrix language, this is a cubic equation on a three-particle vector space Since our S-matrix decomposes into a tensor-product structure, so does the Yang-Baxter equation, which therefore can be checked directly for the fundamental S-matrices of section 5.1 together with the su(2) S-matrix of section 5.2. While the former automatically satisfy the YBE, the latter in general does not. In fact, it is well known from the study of the Heisenberg model that, for the su(2)-invariant S-matrix (5.29) to be integrable, the relative coefficient between between the identity and the permutation operator cannot be arbitrary. In our normalisation, the YBE imposes ς(p, q) − ς(p, r) + ς(q, r) = 0 .
(5.37) Therefore, ς pq must be the difference of two appropriately defined rapidities, which together with equation (5.33) implies that w(p) is real.

Crossing invariance
Another natural requirement on our S-martrix is crossing invariance [89]. The crossing transformation involves analytic continuation of the S-matrix to an unphysical channel, so that momentum and energy flip signs In string-theory discussions of crossing it is useful to introduce a complex variable z that takes values in a "rapidity torus", 34 so that the dispersion relations are uniformised [35,49,89,90]. The analytic continuation amounts to z → z + ω 2 , where ω 2 is half of the imaginary period of the torus. In terms of the Zhukovski parameters x ± and of the function η this gives In order to impose crossing symmetry on the S-matrix, one needs to find the matrix C p that implements the crossing transformation on the one-particle states. In general C p is momentum dependent and it turns out that we will need this dependence for massless fermions. The charge conjugation matrix C p acts on the su(2) charges as and flips the sign of the central charges, giving in particular while the supercharges one has Here st denotes supertransposition, defined as Q st = Q t Σ. The diagonal matrix Σ is the fermion-sign matrix, taking values +1, −1 on bosons and fermions respectively. If we work in the basis then the matrix for the crossing transformation can be written as 35 The crossing equations can be derived in a standard way [5,49] and are most simply expressed in terms of the matrix S, and read In fact, taking the symmetry properties of the scalar factors into account, it will be sufficient to consider the crossing equation in either variable, e.g. the first. Such a matrix equation automatically yields a left-hand side which is proportional to the identity matrix, and constrains the normalisation of certain products of S-matrix elements to be one. In appendix P we write down such constraints in components. They are equivalent to the following equations for the scalar factors where we indicate the crossed momenta by a bar, The equations in (5.47) are the crossing equations for the scalar factors in the massive sector. They were already derived in [34] and a solution to them was proposed in [35]. Equation (5.49) contraints the scalar factor of the massless sector, while (5.48) gives the crossing equations for massive-massless and massless-massive scalar factors. Finally, if we use the fact that the scalar factor ς pq is given by the difference of two rapidities as in equation (5.38), we have that its crossing symmetry simply amounts to the well-known equation w(p) = w(p) + i .

Discussion and outlook
In this paper we have given a detailed exposition of the results announced in [59]. We have found the symmetries of type IIB AdS 3 × S 3 × T 4 superstrings with R-R flux from the light-cone gauge-fixed action, and illustrated how they can be used to fix the exact non-perturbative worldsheet S matrix up to five crossing-symmetric dressing factors. We have also written down the crossing relations that these dressing factors satisfy.
Our results provide a comprehensive framework for investigating the AdS 3 /CFT 2 correspondence using integrability tools. In particular, we have shown how to include the massless modes into the non-perturbative worldsheet S matrix of the theory, thus solving this long-standing obstacle.
The next natural step in this investigation is to determine the form of such factors: so far, a proposal [35] exists only for the ones related to scattering processes in the massive sector, i.e. σ •• and σ •• . Finding the remaining factors will likely require new insights into the analytic structure of the rapidity curve for massless excitations, as well as guidance from perturbative calculations [40].
Another interesting direction would be to write down the Bethe-Yang equations describing the asymptotic spectrum and thereby extending the results presented for the massive modes in [34] to the complete theory. While diagonalising the S matrix is a relatively straightforward task, it would be very interesting to see how the asymptotic N = (4, 4) symmetry is realised on the spectrum. It will be particularly interesting to see how the AFS phase [91] generalises to the massless modes setting and how the semi-classical features such as finite gap equations [92,93] or Landau-Lifshitz equations [94][95][96][97][98][99] emerge in this setting. The former would then need to be compared to the recent proposal for the finite-gap equations of [58].
Since the massless dispersion relation is reminiscent of the ones of giant magnons [83], it is natural to look for similar classical solitonic solution in the massless sector. It appears however that such solutions cannot be straightforwardly constructed only out of bosonic fields, as the related equations of motions are essentially free.
Additionally, note that our description is valid even in the presence of non-trivial winding on the torus because we are in the strict decompactification limit. At the level of the Bethe-Yang equations we should instead be able to distinguish the winding sectors and torus moduli. It would also be very interesting if a similar set of Bethe ansatz equations could be extracted from the dual CFT, perhaps by techniques similar to the ones described in [29]. Further, given the advances in the thermodynamical Bethe Ansatz/quantum spectral curve program [100][101][102][103][104][105][106][107], it would be interesting to see how massless modes will appear in that setting.
Let us remark that the methods presented here should be applicable to more general cases. One is the AdS 3 × S 3 × S 3 × S 1 background, which is also classically integrable [24] and whose massive-sector S matrix and Bethe-Yang equations were found in refs. [32,33]. While this case is somewhat more complicated that the AdS 3 × S 3 × T 4 one-the dual CFT is still to be precisely identified [108,109]-the presence of so-called large N = (4, 4) symmetry yields a rich algebraic structure, which was fruitfully employed in the study of higher spin theories [110].
It would also be interesting to consider the case where AdS 3 backgrounds are sup-ported by a mixture of R-R and NS-NS fluxes. These are also classically integrable [26], and interpolate between the pure R-R superstrings described here and supersymmetric WZW models [20][21][22]. Recently, considerable effort was put into studying the massive sector of the mixed-flux AdS 3 × S 3 × T 4 background, both in terms of their world-sheet S matrix [42,43,48] and by semi-classical techniques [44,47]. The study of their off-shell symmetries and representations could shed new light on the structure of the massive sector and of the massless one, which at this time remains quite obscure. Finally, the prospect of studying deformations and orbifolds of these backgroundsas it was done for AdS 5 × S 5 , see e.g. [75,76] for a review-is extremely appealing, as they include BTZ-like black-hole backgrounds [10], and also appear to be classically integrable [111,112]. This may allow to put, for the first time, an integrability handle on string theory in black-hole backgrounds.
We are confident that we will witness significant progress in these directions in the near future, and we hope to report on some of these topics soon.
in terms of this algebra. so(4) 2 corresponds to rotations along T 4 . Underlined indices will always refer to so(4) 1 . Indices a, b, . . . = 1, 2 andȧ,ḃ, . . . = 1, 2 are used for the two Weyl spinors of so(4) 1 , while i, j, . . . = 1, . . . , 4 are used for the vector of so(4) 1 . Further, throughout the paper i, j, . . . = 1, 2 will denote the two directions of AdS 3 transverse to t; the corresponding coordinates will be denoted as z i , with the understanding that z 3 ≡ z 4 ≡ 0. Similarly, i, j, . . . = 3, 4 will denote the two directions of S 3 transverse to φ; the corresponding coordinates will be denoted as y i , with y 1 ≡ y 2 ≡ 0. By a slight abuse of notation we will sometimes write expressions like ǫ ij z i ∂ α z j or ǫ ij y i ∂ α y j with the understanding that

B Spinor and Gamma matrix conventions
For AdS 3 and S 3 we consider the three-dimensional gamma matrices 36 We further define The ten-dimensional gamma matrices are then given by We then have 36 Our conventions are the same as those of [24], except for the definition of γ 0 and γ 2 .
The gamma matrices satisfy It is useful to note the relations The Majorana spinors satisfy the conditions (B.8)

C Killing spinors and a preferred choice of vielbeins
In this appendix we collect some of the computational details that are useful for the calculation of Killing spinors done in section 2.1. We begin by presenting solutions of the Killing spinor equation on S 3 and AdS 3 and, using these, we construct Killing spinors in the full AdS 3 × S 3 × T 4 geometry. In parallel with this construction, we also introduce a particular choice of vielbeins for the geometries in question. Such a choice is of course in some sense arbitrary and can be gauged away. However, many of the detailed computations performed in this paper simplify significantly in this frame.
Killing spinors on S 3 . The S 3 metric (2.2) can be written in terms of a diagonal dreibein 37 (C. 2) The S 3 Killing spinors satisfy [71] ∂ m η I These equations are solved bỹ where η I 0 are constant spinors with two complex components. Let us consider the first of these solutions. We note that whereM AB andM AB are orthogonal matrices. Using these matrices we can introduce new dreibeins, obtained by a rotation in tangent spacê The components of the inverse dreibein are given by a fairly compact expression and can be written in the factorized form Like all vielbeins, theK n A are covariantly constant Here,ω mAB is the spin connection in the rotated tangent space.K m A andω mAB further satisfy the relationω which is not true for generic vielbeins and spin-connections. Equations (C.8) and (C.9) together give the relation This is the Killing vector equation; similar equations hold forǨ n A . The vectorsK m A andǨ m A together generate the so(4) = su(2) ⊕ su(2) isometry algebra of S 3 , Killing spinors on AdS 3 . We consider the AdS 3 metric with the diagonal dreibein 38 (C.13) The AdS 3 Killing spinors satisfy [70,71] ∂ α ǫ I These equations have the solutions where ǫ I 0 are constant spinors. 38 For AdS 3 we denote the tangent space directions A = 0, 1, 2.

Like we did for the S 3 we introduce matricesM
These matrices are orthogonal with respect to a metric of signature (−1, +1, +1). The rotated dreibeins are defined bŷ are their inverses can be written in components aŝ Like in the S 3 case these dreibeins are Killing vectors satisfying the so(2, 2) = sl(2)⊕sl(2) algebra Killing spinors on AdS 3 × S 3 × T 4 . The above constructions of Killing spinors on AdS 3 and S 3 can readily be used to show that the spinors (2.7) satisfy the Killing spinor equations (2.4). Writing the Killing spinors in the penta-spinor notation used to define the gamma matrices in appendix B, we have where ǫ I 0 , η I 0 and ψ I 0 are constant two-component spinors. In this basis, the matricesM andM are given bŷ and satisfŷ (C.26) of the various vielbeins in the two subspaces.
After performing the tangent space rotations we find new spin connectionsω mAB anď ω mAB , and corresponding covariant derivativesD m andĎ m satisfyinĝ and similar forĎ m . The spin connections can be written as which leads to (C.29)

D Proof of the identity (2.24)
To prove the identity (2.24) we write The expression in the square brackets is antisymmetrized in m and n. By multiplying it from the left and the right byM andM −1 we geť Using the relations The first term on the right-hand side can be shown to be zero using the covariant constancy of the vielbein. Using the expressions for the spin connection in section C we can check that the second term is proportional to the projector (1 + Γ 0123456 ). Hence the expression appearing in (2.24) vanishes.

E Useful identities for so(4) gamma matrices
The following identities involving blocks of the so(4) gamma matricesγ i andτ i are useful γ iγj γ kγl = δ jk δ il − δ jl δ ik + δ kl δ ij + ǫ ijkl − ǫ jklm γ i m + δ jk γ il − δ jl γ ik + δ kl γ ij , γ i γ jγk γ l = δ jk δ il − δ jl δ ik + δ kl δ ij − ǫ ijkl + ǫ jklmγi m + δ jkγil − δ jlγik + δ klγij , τ k τ ij = ǫ kijlτ l − δ kiτ j + δ kjτ i . We also use the relations F Relations between Γ A and so(4) 1 ⊕ so(4) 2 gamma matrices Let us relate the action on the fermions of the ten dimensional gamma matrices of appendix B with the action of the so(4) 1 ⊕ so(4) 2 gamma matrices introduced in section 2.2.5. The chiral spinors η I and χ I have eigenvalue +1 under Γ 0123456789 . Here we write the action of the gamma matrices that preserve this chirality. Performing the change of basis (2.55) we find the decomposition where the arrows indicate a restriction to the upper left 16 × 16 block. We further find and It is also useful to note that Γ 12 → +γ 12 ⊗ 1, Γ ij → +γ ij ⊗τ 6789 , for i = 1, 2, j = 3, 4, G L WZ in so(4) 1 ⊕ so(4) 2 components In this appendix we write down the expression for L WZ in terms of so(4) 1 ⊕ so(4) 2 bispinors introduced in section 2.2.5. In order to do this notice first that the matrix M 0 , when written after the basis change (2.55), is of the form where m 0 is a 4 × 4 matrix. In other words it acts non-trivially only on so(4) 1 . Explicitly we find and it is also useful to note The above matrices are written in block form, with the blocks having the same index structure as that given in equation (2.50). Using this block structure we define It is straightforward to check that Equipped with these observations, it is now straightforward though laborious to ex-press L WZ given in equation (2.34), in terms of so(4) 1 ⊕ so(4) 2 bispinor fermions, As expected, many of the terms above break the so(4) 1 explicitly.

H Equations of motion
In this appendix we write down the bosonic and fermionic equations of motion for the physical fields of the fully gauge-fixed theory to leading order in fermionic fields and sub-leading order in bosonic fields. The bosonic equations of motion arë The fermionic equations of motion are (H.7)

I Poisson bracket for η I and χ I
In this appendix we compute the Poisson brackets for η I and χ I . To do this let us write all the terms in the action that have a τ -derivative acting on the fermions up to quadratic order in transverse bosons. Keeping only the terms up to the order in the number of fields that we require, L kin gives rise to two such terms From L WZ we get additional terms as well as terms that mix massless and massive fields, These terms can be diagonalised by a field redefinition. Let us introduce a new set of spinors ψ a , a = 1, . . . , 4 by The kinetic term then take the canonical form 2iψ a ǫǫψ a , (I.5) provided the coefficients A ab are chosen as It is useful to note that the coefficients satisfy The fermions ψ a have canonical Poisson brackets The Poisson brackets for the η i and χ I then follow immediately and are given in equation (2.88).

J Some Poisson brackets used in section 2.4.1
To calculate the Poisson bracket between the exponential factor in Q and H we need the relations (J.1) Using these relations we find For the Poisson bracket between the Hamiltonian and the quadratic supercharge we note

(J.4)
We also collect here some formulas that are useful in deriving equation (2.82). First, note that for a constant α e αx − (σ) [Q 2 (σ), H] and Putting this together we find, up to total derivatives of expressions that vanish at ±∞, Hence, if we set α = γ 34 (J.8) the expression e αx − Q 2 gives a conserved charge.

K Derivation of equation (2.89)
In this appendix we give the details of the how equation (

L Oscillator algebra
As a preliminary step to facilitate the study of non-perturbative representations, let us rewrite the supercurrents in components and expand them in terms of oscillators in a momentum-basis. In this way, it will be easier to read off the form of the representations, and to deform their dependence on the momentum. In order to elucidate the structure of the representations it will be sufficient to consider their leading order in a field expansion. We introduce complex coordinates (L.6) On the right-hand side of the above expressions for η I (respectively χ I ) the indices 1, 2 correspond to the unbroken su(2) • (respectively su(2) • ), and complex conjugation is indicated by a bar. We write superscript su(2) • indices for left massive fermions and subscript indices for right massive fermions (the opposite for their conjugates) because they transform in the fundamental and anti-fundamental representations, respectively. The canonical anti-commutation relations take the form {η Lȧ (σ 1 ), η˙b L (σ 2 )} = {η˙b R (σ 1 ), η Rȧ (σ 2 )} = δ˙ḃ a δ(σ 1 − σ 2 ), Using these expressions, we can write the leading order expression of the charges, recalling that we take ǫ 12 = −ǫ 12 = +1,  (L.15) The corresponding creation operators are found by taking the complex conjugate of the above expressions, and raising and lowering the su(2) indices by the tensor ǫ, yielding We define states by acting with the creation operators on the vacuum. We have eight massive excitations |Z L,R = a † L,R z |0 , |Y L,R = a † L,R y |0 , |η˙a L = d˙a † L |0 , |η Rȧ = d † Rȧ |0 , (L. 20) and eight massless ones |T˙a a = a˙a a † |0 , |χ a = d a † |0 , | χ a =d a † |0 . (L.21) The same notation is used in the main text for the states on which the non-perturbative S matrix acts. Finally, the supercharges in terms of ladder operators take the form Q˙a L = dp (d˙a † L a Ly + ǫ˙a˙b a † Lz d Lḃ )f p + (a † Ry d˙a R + ǫ˙a˙b d † Rḃ a Rz )g p + ǫ˙a˙bd a † a˙b a + a˙a a † d a f p , Q Rȧ = dp (d † Rȧ a Ry − ǫ˙a˙b a † Rz d˙b R )f p + (a † Ly d Lȧ − ǫ˙a˙b d˙b † L a Lz )g p + d a † aȧ a − ǫ˙a˙b a˙b a †d a g p . (L.22) Note that these supercharges are manifestly covariant under so(4) 2 = su(2) • ⊕ su(2) • , and furthermore enjoy a discrete "left/right" symmetry under while in the left-right sector Note that for convenience we normalise these S-matrices in a different way from what was done in [32]. In particular, in order to satisfy unitarity one would need to multiply them by an appropriate scalar factor.

N Explicit form of the S-matrix elements
For the reader's convenience we write the explicit form of the S-matrix in the different sectors. To this end, we introduce the graded tensor product⊗ for the matrices S so that up to prefactors S psu(1|1) 4

N.1 The mixed-mass sector
In the following we write the S-matrix in the case of a massive excitation that scatters with a massless excitation. We write only the matrix part of it. These elements need to be multiplied by the appropriate scalar factors introduced in (O.1). In the case of left massive excitations that scatter with massless excitations transforming in the ̺ L ⊗ ̺ L representation of psu(1|1) 4

N.2 The massless sector
•• : where we defined

P Crossing equations for the S-matrix elements
For completeness we present the crossing equations also in terms of the S-matrix elements.
To obtain simpler expressions we choose to focus on processes involving the highest weight states of the left, right and massless modules, which are enough to constrain all of the dressing factors. We then rewrite the crossing equations in terms of these scattering processes and the ones with the conjugates of the highest weight states. For example, in the massive sector we can consider