Integrable S-matrices, massive and massless modes and the AdS_2 x S^2 superstring

We derive the exact S-matrix for the scattering of particular representations of the centrally-extended psu(1|1)^2 Lie superalgebra, conjectured to be related to the massive modes of the light-cone gauge string theory on AdS_2 x S^2 x T^6. The S-matrix consists of two copies of a centrally-extended psu(1|1) invariant S-matrix and is in agreement with the tree-level result following from perturbation theory. Although the overall factor is left unfixed, the constraints following from crossing symmetry and unitarity are given. The scattering involves long representations of the symmetry algebra, and the relevant representation theory is studied in detail. We also discuss Yangian symmetry and find it has a standard form for a particular limit of the aforementioned representations. This has a natural interpretation as the massless limit, and we investigate the corresponding limits of the massive S-matrix. Under the assumption that the massless modes of the light-cone gauge string theory transform in these limiting representations, the resulting S-matrices would provide the building blocks for the full S-matrix. Finally, some brief comments are given on the Bethe ansatz.


Introduction
The remarkable successes of integrability techniques in the study of the AdS 5 × S 5 superstring [1] motivates the application of these methods to other integrable string backgrounds with less supersymmetry [2]. In this work we investigate the AdS 2 × S 2 × T 6 background supported by Ramond-Ramond fluxes in Type II superstring theory, which preserves a quarter of the supersymmetries. These can be found as the near-horizon limit of various intersecting brane solutions of Type IIB supergravity, which are related by T-duality [3]. The dual [4] should be a one-dimensional CFT, and is understood to either be a superconformal quantum-mechanical system or a chiral two-dimensional CFT [5].
The AdS 2 × S 2 part of the background can be written as a Metsaev-Tseytlin [6] type supercoset model [7] for P SU (1, 1|2)/SO(1, 1) × SO (2). The algebra psu(1, 1|2) has a Z 4 automorphism and hence the supercoset model is classically integrable via the same construction as for the AdS 5 × S 5 case [8]. While there exists a classical truncation of the Green-Schwarz action [9] for the AdS 2 × S 2 × T 6 geometry to the supercoset degrees of freedom, there is no κ-symmetry gauge choice which decouples them from the remaining fermions [10]. The integrability of the Green-Schwarz action for the complete background has been demonstrated to quadratic order in fermions [10,11].
The aim of this paper is to use symmetries and integrability to construct exact S-matrices for the scattering of the worldsheet excitations of the decompactified light-cone gauge [12] AdS 2 × S 2 × T 6 superstring. These S-matrices describe the scattering above the BMN vacuum [13], a point-like string moving at the speed of light on a great circle of the two-sphere. The light-cone gauge-fixed Lagrangian [14,15] is in general rather complicated with the interaction terms breaking two-dimensional Lorentz invariance. The quadratic action is however Lorentz invariant and describes 2 + 2 (bosons+fermions) massive modes, the bosons of which are associated to the transverse directions in AdS 2 × S 2 , and 6 + 6 massless modes, associated to the T 6 .
In the AdS 5 × S 5 light-cone gauge-fixed theory all of the excitations have equal non-vanishing mass and furthermore the symmetries completely fix the S-matrix up to an overall phase [16,17]. Here the situation is more similar to AdS 3 ×S 3 ×T 4 for which there are 4+4 massive and 4+4 massless excitations.
In this case the symmetries of the supercoset leaving the BMN string invariant can be used to conjecture an exact S-matrix for the scattering of the massive modes [18,19]. Following a similar approach we observe that the subalgebra of the psu(1, 1|2) symmetry of the AdS 2 × S 2 supercoset preserved by the BMN string is given by psu(1|1) 2 ⋉ R. Relaxing the level-matching condition we extend this algebra by two additional central extensions and conjecture the exact S-matrix for the scattering of massive modes up to an overall factor.
The resulting S-matrix satisfies crossing symmetry [20] and is unitarity so long as the overall factor satisfies the relevant identities. Here the setup is more similar to the AdS 5 × S 5 case as opposed to the AdS 3 × S 3 × T 4 case for which there were multiple phases related by crossing transformations [21]. It was observed in [15] that the one-loop logarithms in the massive S-matrix for AdS 2 × S 2 × T 6 are consistent with part of this overall factor being given by the BES phase [22]. Assuming this to be true we find the remaining rational piece still has to satisfy somewhat complicated relations. Finally, the near-BMN expansion of the exact result is consistent with perturbative computations [14,15,23].
While many features of the construction are similar to the AdS 5 × S 5 and AdS 3 × S 3 × T 4 cases, there are some important differences. In particular, unlike for the AdS 5 × S 5 and AdS 3 × S 3 × T 4 superstrings, the representations we are scattering turn out to be long and hence there is no shortening condition to be interpreted as the dispersion relation. An additional consequence is that the symmetries do not completely fix the S-matrix up to a single overall factor, rather there is an additional undetermined function that can be found by demanding the Yang-Baxter equation is satisfied. These properties are reminiscent of similar features seen for the scattering of long representations of psu(2|2) ⋉ R 3 [24] and also in the Pohlmeyer reduction of strings on AdS 2 × S 2 [25].
The S-matrix has an accidental U (1) symmetry under which the fermions are charged, while the bosons are not. From the perspective of the complete AdS 2 × S 2 × T 6 superstring this U (1) originates from the T 6 compact space [15]. Furthermore, its presence appears to be important to have any hope of applying a Bethe ansatz construction as it allows one to define a pseudovacuum. A conjecture for a set of asymptotic Bethe ansatz equations was given in [10], however, due to the somewhat involved structure of the S-matrix it is not clear how to derive them.
It is not currently known how the massless modes transform under the symmetry group of the lightcone gauge-fixed theory, and therefore it is not possible to completely determine the corresponding Smatrices. Furthermore, they may depend on the choice of Type II background [3] -in the decompactified light-cone gauge-fixed theory the T 6 formally has an SO(6) symmetry, however, this will be broken by the presence of Ramond-Ramond fluxes. Different choices related by T-duality naively lead to different subgroups. Therefore, we take an alternative approach motivated by the recent explicit computation of the light-cone gauge symmetry algebra for the AdS 3 × S 3 × T 4 superstring [26], the AdS 5 × S 5 version of which was constructed in [27]. Assuming a similar outcome occurs for the AdS 2 × S 2 × T 6 superstring one may expect the massless modes to transform in representations of psu(1|1) 2 ⋉ R 3 , and hence the S-matrices describing their scattering should be built from the massless limits (one massless and one massive or two massless particles) of the massive S-matrix.
The structure of the paper is as follows. In section 2 we describe the near-BMN symmetry algebra and investigate its representation theory. This symmetry is then used in section 3 to determine the exact S-matrix up to an overall phase. We determine the constraints that the phase should satisfy for crossing symmetry and unitarity and compare with perturbation theory. In section 4 we discuss when this symmetry can be extended to a Yangian, finding that it can be done in the standard form for the massless case. Using this Yangian symmetry in section 5 we then construct the massless version of the S-matrix and briefly explore the notions of crossing symmetry and unitarity in this limit. In section 6 we give some initial considerations of the algebraic Bethe ansatz, noting in particular the existence of a pseudovacuum, and we conclude in section 7 with some comments.
2 Symmetry for massive modes of AdS 2 × S 2 The BMN light-cone gauge AdS 2 × S 2 × T 6 superstring action describes 2 + 2 massive and 6 + 6 massless modes. The algebra underlying the scattering of the massive modes is expected to be psu(1|1) 2 ⋉ R 3 , which is found by considering the subalgebra of psu(1, 1|2) that is preserved by the BMN geodesic. The two additional central extensions appear, by analogy with the AdS 5 × S 5 case, in the decompactification limit and relaxing the level-matching condition.
Let us denote the massive boson associated to the transverse direction of S 2 as y and the corresponding boson for AdS 2 as z. The two massive fermions will be represented as two real Grassmann fields ζ and χ. We can then formally define the following tensor product states where φ is bosonic and ψ is fermionic, such that we expect one of the factors of psu(1|1) to act on each of the two entries. Furthermore, as a consequence of the form of the symmetry algebra and the integrability of the theory [10,14] we expect that the S-matrix for y, z, ζ and χ can be constructed as a graded tensor product of an S-matrix for φ and ψ, with each factor S-matrix invariant under the symmetry psu(1|1) ⋉ R 3 .
In this section we will construct the relevant massive representation of psu(1|1) ⋉ R 3 . This representation has an obvious massless limit, and, by analogy with the construction for AdS 3 × S 3 × T 4 [26], one may expect the massless modes to also transform in representations of psu(1|1) ⋉ R 3 in the light-cone gauge-fixed theory. The massless limit is discussed in detail in section 5.
Let us also briefly mention that there is an additional U (1) outer automorphism symmetry [15] of the S-matrix (3.2), under which the psu(1|1) factors transform in the vector representation. The origin of this U (1) symmetry is the T 6 compact space that is required for a consistent 10-d superstring theory.
Under this symmetry (ζ, χ) T also transforms as a vector, while the bosons are uncharged. It is worth noting that taking the tensor product of two copies of any S-matrix for φ and ψ preserving fermion number we find that the U (1) symmetry is present so long as a certain quadratic relation between the parametrizing functions is satisfied (see appendix A). In the case of interest, this quadratic identity turns out to be true just from demanding invariance under the psu(1|1) ⋉ R 3 symmetry and satisfaction of the Yang-Baxter equation. The U (1) does not act in a well-defined way on the individual factor S-matrices and hence for now we will ignore it. We will reconsider it in section 6, where it will play a role in defining a pseudovacuum, an important first step in the algebraic Bethe ansatz.

The gl(1|1) Lie superalgebra and its representations
Let us start by summarizing the relevant information from [28] regarding the Lie superalgebra gl(1|1) and its representations. There are two bosonic generators N and C, with C central, and two fermionic generators Q and S. The commutation relations read The typical (long) irreps are the 2-dimensional Kac modules C, ν , defined by the following non-zero entries on a boson-fermion (|φ , |ψ ) pair of states: As long as C = 0, this module is isomorphic to the anti-Kac module C, ν However, if C = 0, the two modules are not isomorphic and they are no longer irreducible. Rather they become reducible but indecomposable.
To elucidate further we introduce the 1-dimensional modules µ , which form the atypical (short) irreps of gl(1|1). These irreps are characterized by the vanishing of all generators except N, which acts with eigenvalue µ. We then see that for the Kac module, 0, ν , the fermion |ψ spans a sub-representation ν − 1 , and the indecomposable is denoted as The anti-Kac module 0, ν is also reducible but indecomposable and is denoted as with the fermion |ψ once again spanning the sub-representation ν . This indecomposable is not isomorphic to 0, ν . Let us mention that modding out the indecomposable representations by their subrepresentations one obtains the factor representations, which in this case are isomorphic to the short 1-dimensional µ modules and are spanned by the boson |φ .
If we take the tensor product of two typical modules, we get where P ν is the so-called projective module on which C acts identically as zero. The rightmost 1-dimensional short sub-module ν is known as the socle of P ν .
Since N does not appear on the r.h.s. of the commutation relations, the algebra gl(1|1) has a nontrivial ideal generated by Q, S and C. This ideal is the superalgebra sl(1|1). Furthermore, this algebra is also not simple, as C, being central, is a non-trivial ideal. Additionally modding out C gives the algebra psl(1|1), which is still not simple, as the two remaining anti-commuting fermionic generators each form a separate ideal. The fact that psl(1|1) is not simple sets this algebra outside the classification of the possible central extensions of basic classical Lie superalgebras presented in [29].

The centrally-extended psu(1|1) Lie superalgebra
We are now ready to introduce the centrally-extended version of the algebra we discussed in the previous section, which is relevant for the scattering of the massive modes of the AdS 2 × S 2 × T 6 superstring.
The algebra psu(1|1) ⋉ R 3 is defined by the commutation relations The states |φ and |ψ , introduced in (2.1), then transform in the following representation: Here a, b, c, d, C, P and K are the representation parameters that will eventually be functions of the energy and momentum of the states. For the supersymmetry algebra to close the following conditions should be satisfied ab = P , cd = K , ad + bc = 2C . (2.11) This representation corresponds to the typical (long) Kac module C, ν discussed in the previous section.
We will be interested in a particular real form of the algebra (2.9), which is given by These relations further constrain the representation parameters as follows The closure conditions (2.11) imply that (2.14) Unlike the AdS 5 × S 5 case, with the larger symmetry algebra psu(2|2) 2 ⋉ R 3 , here we are scattering long representations and hence there is no shortening condition -that is, ad − bc is free to take any value, which we will denote as m. From the reality conditions (2.13) we see that m is real. Furthermore, we will choose the branch of the square root such that m is positive m = ad − bc = 2 C 2 − P K , C 2 > P K .
Later it will be useful to solve the set of equations (2.11) for a, b, c and d in terms of m, C, P and K a = α e − iπ 4 C + m 2 Here α is a phase parametrizing the normalization of the fermionic states with respect to the bosonic states and can be a function of the central extensions.
To define the action of this symmetry on the two-particle states we need to introduce the coproduct 17) and the opposite coproduct, defined as where J is an arbitrary abstract generator (prior to considering a representation), and P defines the graded permutation of the tensor product.
The coproduct differs from the trivial one by the introduction of a new abelian generator U, with . This is done according to a Z-grading of the algebra, whereby the charges −2, −1, 1 and 2 are associated to the generators K, S, Q and P respectively, while C remains uncharged. The action of U on the single-particle states is given by This braiding allows for the existence of a non-trivial S-matrix.
One important consequence of the non-trivial braiding (2.17) is that it leads to a constraint between U and the eigenvalues of the central charges. This follows from the requirement that, to admit an S-matrix, the coproduct of any central element should be equal to its opposite. 1 This implies We fix the normalization of P relative to K by taking both constants of proportionality to be equal to 1 2 h where the reality conditions (2.13) require that h is real. 2 The parameter h is a coupling constant and eventually should be fixed in terms of the string tension, which we will return to in section 3.2. Acting on the single-particle states then gives us the relations where U should satisfy, as a consequence of (2.13), the following reality condition which, for an invertible R-matrix, necessarily implies ∆ op (c) = ∆(c). This is expressed by saying that the coproduct of c is co-commutative. 2 The reality conditions (2.13) do allow for the introduction of an additional phase into the constants of proportionality, i.e. 1 2 he iϕ and 1 2 he −iϕ . However, this phase does not appear in the S-matrix and thus we set ϕ = 0.
The relation (2.14) in terms of C, U and m is then given by While this is a single equation for three undetermined parameters, we will later still attempt to interpret it as a dispersion relation with C, U and m defined in terms of just two kinematic variables, the energy and momentum. These precise definitions are not fixed by symmetry considerations, and hence should be found from direct string computations.
It is now useful to introduce the Zhukovsky variables x ± , in terms of which we will write the S-matrix, in place of the central extensions, C and U . These are defined as [16,31] In these variables the dispersion relation (2.23) takes the following familiar form The representation parameters a, b, c and d in (2.16) and (2.31) are then given by Here we clearly see that the advantage of these variables is that the parameters a, b, c and d do not depend on m and hence, written as a function of x ± and m, neither will the S-matrix. Finally, let us note that for the reality conditions (2.13) we have the usual (x ± ) * = x ∓ .
We could also eliminate the central extensions, C and U , in terms of two variables that will later be identified with the energy and momentum. Motivated by the AdS 5 × S 5 case we write where e is the energy and p is the spatial momentum. Solving for x ± in terms of e and p we find (2.29) Using (2.21) and (2.28) we can substitute in for C, P and K in terms of the energy and the momentum in (2.23) to find the following familiar dispersion relation It is important to emphasize that here m is algebraically a free parameter. However, for (2.30) to really be interpreted as a dispersion relation m should be fixed by the spectral analysis of the theory. In terms of the energy and the momentum the representation parameters a, b, c and d (2.16) are given by In the AdS 5 × S 5 and AdS 3 × S 3 × M 4 models, the choice of the phase factor α that is appropriate for the light-cone gauge-fixed string theory is As we will see, this is also a natural choice for α in the AdS 2 × S 2 theory.

Tensor product of irreps and scattering theory
In this section we consider the tensor product of two of the irreps we discussed in the previous section, with the aim of constructing the relevant scattering theory. In particular, we want to investigate the persistence of the phenomenon observed for gl(1|1) modules in section 2.1, namely complete reducibility of the tensor product of two 2-dimensional irreps, for generic values of the momenta, into two 2-dimensional irreps of the same type.
Let us proceed by constructing a 4-dimensional representation of the algebra (2.9). To do this we start with the bosonic state Let us assume that the action of the central elements on this state is given by This assumption will be justified by the concrete example we will consider later in our treatment of the scattering theory. We can then construct two more states by considering the action of Q and S The action of the central elements on these new states is then easily seen to be given by (P, K, C)|w 1 = (P, K, C)|w 1 , (P, K, C)|w 1 = (P, K, C)|w 1 . (2.36) We can then look at the action of Q and S on |w 1 and |w 1 Here we see that we have generated one additional new state where we have chosen a normalization depending on Given the real form we are interested in, see eq. (2.12), and the assumption that C 2 > P K, or equivalently that M is real and non-zero (we will briefly discuss the case when M vanishes at the end of this section), the above normalization implies that |w 0 has the same norm as |w 0 . Therefore, the action of Q and S on |w 1 and |w 1 is given by Again it is clear that the action of the central elements on |w 0 is given by Finally, the action of Q and S on |w 0 is given by Therefore, in summary, we have constructed the following 4-dimensional representation: However, using the fact that we see that defining the linear combinations Furthermore, Using the definition of M (2.39) one can easily see that and hence the 4-dimensional representation we constructed is actually reducible and is formed of two To conclude, let us briefly mention orthogonality. Here we will make use of the real form of the algebra given in eq. (2.12), and the assumption that M is real. We then have Using the conjugation relations we find that Therefore, the two representations are orthogonal.
We will now apply the above construction to the tensor product of two of the 2-dimensional representations of section 2.2. This will be relevant to the scattering theory discussed in section 3. In this case we have four states that are acted on as follows by the generators of the algebra (C, P, K)|φφ = (C, P, K)|φφ (C, P, K)|φψ = (C, P, K)|φψ where the labels 1, 2 refer to the first and second entry in the tensor product and we recall that the action on the tensor product is given by the coproduct (2.17), so that These relations imply It is then clear that the bound-state points occur when either ℓ ac = 0 or ℓ bd = 0. Furthermore, for the scattering of two physical states, i.e. when the following reality conditions are satisfied To explicitly find the decomposition into two irreps, let us start by taking It then follows that Alternatively we could have started by taking in which case we end up with It is easy to see that these states are proportional to each other from the identity This same identity, along with the reality conditions, can be used to see that Working with the state (2.60) we can apply the fermionic generators to find One can then check 3 that so that using (2.63) it is clear that QS|Φ ± ∝ |Φ ± and SQ|Φ ± ∝ |Φ ± and hence this shows explicitly that {|Φ ± , |Ψ ± } form two 2-dimensional irreps.
At the bound-state points ℓ ac = 0 or ℓ bd = 0 one of the irreducible blocks contains |φφ , as either |Φ + or |Φ − aligns to this state. Therefore, one can focus on the φ φ → φ φ entry of the S-matrix (supplemented by the appropriate dressing phase) to ascertain whether this corresponds to a pole in the s-channel in the physical region.
3 This is seen explicitly from the following algebra: The explicit derivation is Let us finally make the important observation that the arguments of this section cannot be applied for the M = 0 case (such as, for instance, the scattering of two massless particles with the momenta taken at the bound-state point 5 ).
In this case what we find is the analog of the projective indecomposable representation of section 2.1. In particular, one can check that, at M = 0, the state |w where we have used M 2 = 4(C 2 − P K) = 0 to derive the last proportionality statement. However, this is the only state which satisfies these properties, meaning we do not have two solutions to these conditions (as we did in the M = 0 case above). Therefore, there is only one irreducible 2-dimensional block, containing the states {|w

S-matrix for massive modes of AdS 2 × S 2
In this section we study the S-matrix for the massive modes of the light-cone gauge AdS 2 × S 2 × T 6 superstring. As mentioned in section 2 from the structure of the symmetry algebra and the integrability of the theory we expect the S-matrix for the massive fields y, z, ζ and χ to be constructed from the graded tensor product of two copies of an S-matrix describing the scattering of 1 + 1 massive modes, φ and ψ. The former are defined in terms of the latter in (2.1).
The excitations φ and ψ should transform in the massive representation of psu(1|1) ⋉ R 3 discussed in section 2.2. Their S-matrix is then fixed by demanding invariance under this symmetry Accounting for conservation of fermion number, the most general form for the S-matrix is where x ± , m are the kinematic variables associated to the first particle and x ′± , m ′ to the second particle, that is As a consequence of the discussion in section 2.3 this symmetry will only fix the S-matrix up to two arbitrary functions. One of these functions can be found by requiring the S-matrix also satisfies the given by 6,7 P 0 is an overall factor that sits outside the matrix structure and is not fixed by symmetries or the Yang-Baxter equation. Let us emphasize that, as discussed beneath eq. (2.26), when written in these variables the S-matrix is independent of m and m ′ , which can take any value. The limits m → 0 and m ′ → 0 are subtle however, and will be discussed in detail in section 5. Let us also note that if we take α to be given by (2.32), which is the choice suitable for string theory, then Q 1 = Q 2 and R 1 = R 2 . From now on we will take α to be given by this value.
This S-matrix can be thought of as a 4 × 4 block diagonal matrix One can then check that each of the two 2 × 2 blocks have equal trace and determinant, The second of these equations is particularly important as it implies the tensor product of two copies of the S-matrix possesses an additional U (1) symmetry, which will be discussed further in section 6 and appendix A. 6 Note that here we are choosing the branch so that x − 1 4 and similarly for x ′± . For p ∈ [−π, π] this corresponds to taking the branch cut on the negative real axis. 7 The solutions that violate crossing symmetry are given by f = 0 and f → ∞. (For the latter one should first rescalẽ P 0 by f −1 and then take f → ∞.) As φ and ψ are real, the two processes should be related by a crossing transformation. However, if f vanishes then so does the amplitude for the first of these processes, but not for the second. Similarly, if f → ∞ then the amplitude for the second process vanishes, but not for the first. Consequently, in both cases the two processes cannot be related by a crossing transformation and hence there is a violation of crossing symmetry as claimed. It is interesting to note that taking f = 0 and f → ∞ we recover the massive S-matrices of the AdS 3 × S 3 × T 4 light-cone gauge superstring [18,19]. The symmetry is enhanced accordingly Finally, let us remark that a significant difference with respect to the AdS 5 × S 5 S-matrix of [16] is the presence of scattering processes sending two bosons into two fermions (and vice versa) in (3.2). This makes the formal embedding of (3.2) into the AdS 5 × S 5 S-matrix hard to implement. Embedding the algebra psu(1|1) ⋉ R 3 into psu(2|2) ⋉ R 3 is possible. 8 To embed the S-matrix, one possibility would be that two bosons of the same type scatter to two fermions of the same type in AdS 5 × S 5 (for instance, φ 1 φ ′1 → ψ 3 ψ ′3 , in the notation of [16]). A scattering process like this one is not present in the AdS 5 ×S 5 S-matrix. It is prohibited by the extra su(2) 2 symmetry which is present in psu(2|2) ⋉ R 3 but not in An alternative would be to allow different AdS 5 × S 5 states for the two particles being scattered, for instance (φ 1 , ψ 3 ) and (φ ′2 , ψ ′4 ). Reducing the S-matrix of [16] according to this choice would leave us with many terms, but not the one involving This process is not present before reduction, yet it would be necessary to reproduce the AdS 2 S-matrix.
Once again the embedding of S-matrices seems not to be straightforward.

The overall factor and crossing symmetry
As currently written the factorP 0 is neither a phase factor or antisymmetric. Indeed, given the reality conditions (x ± ) * = x ∓ and (x ′± ) * = x ∓ , the functions f , s 1,2 and t 1,2 satisfy the following relations 9 and hence, as a consequence of braiding and QFT unitarity the overall factor should satisfy 10 . To isolate an antisymmetric phase factor, we can define P 0 as follows: The second equality follows from eq. (3.8). As claimed the unitarity conditions for P 0 are then (3.14) 8 One may for instance use Q 3 1 + Q 4 2 and S 1 3 + S 2 4 of AdS 5 × S 5 as Q and S of the smaller algebra, see section 2.2. This would be consistent with the central extensions. We thank B. Stefanski for discussions regarding this point. 9 If we consider the case for which m = m ′ , then on-shell, that is when the dispersion relations (3.3) are satisfied, we have that t 1 ≈ t 2 , which given the reality conditions is real. 10 Note that the AdS 5 × S 5 S-matrix contains copies of the 2 × 2 block: Taking into account the factor of 2, when t i = 0 (3.12) simplifies toP 0P * 0 =P 0 (x, x ′ )P 0 (x ′ , x) = 1 so thatP 0 is an antisymmetric phase factor. This is the familiar AdS 5 × S 5 story.
Crossing symmetry provides an additional constraint on the overall factorP 0 , which takes the form Here the label c denotes that the corresponding arguments are taken as (x ′± , x ± ) instead of the original (x ± , x ′± ) where the "crossed" Zhukovsky variablesx ± are, as usual, given bȳ corresponding toē = −e andp = −p. It is useful to note that we have the following identities The relation (3.15) translates to the following constraint for P 0 and hence it appears that we either have a simple crossing relation or simple unitarity relations.
Using Hopf algebra arguments, we have checked that crossing symmetry is present for the representation of interest for any value of m and m ′ . Denoting the symmetry algebra as A, the antipode Σ is found from the defining rule where µ is the multiplication map, η : C → A is the unit and ǫ : A → C is the counit, which annihilates all generators apart from 1 and e ip (acting on which, it returns 1). The antipode being a Lie algebra anti-homomorphism, we simply need to derive This map is idempotent and therefore equal to its inverse. We impose where C is the charge conjugation matrix and the label str denotes supertransposition. The fundamental crossing relation for an R-matrix 11 is then given by (cf. [20]) which projects into representations as and an analogous equation for the second factor. Here str i denotes the supertranspose for factor i. 11 For our purposes, S-matrices will be representations of abstract R-matrices.
To conclude this discussion of the overall factor, we compare with the unitarity and crossing relations of the AdS 5 × S 5 light-cone gauge-fixed theory. The amplitude for the scattering process Y 11 Y 11 → Y 11 Y 11 , otherwise known as the SU (2) sector, is given by [16,31,17,33] x where Here θ bes is the BES phase [22]. S 0 satisfies the following unitarity and crossing relations [20] S 0 S 0 while σ bes = exp(iθ bes ) satisfies the following crossing relation Therefore, comparing with equations (3.12), (3.14), (3.15) and (3.18) we see that if the BES phase is part of the overall factor P 0 , or equivalentlyP 0 , the crossing and unitarity relations for the remaining (rational) piece will still be non-trivial.

Comparison with perturbation theory
Defining the effective string tension the tree-level S-matrix for the scattering of massive modes in the light-cone gauge AdS 2 × S 2 × T 6 superstring following from near-BMN perturbation theory can be be found by suitably truncating the corresponding result for AdS 5 × S 5 or AdS 3 × S 3 × T 4 [23] (various components were also computed in [15]). This gives where the functions l i are defined as The parameter a is the standard gauge-fixing parameter of the uniform light-cone gauge [12]. In [15] it was shown that to one-loop the near-BMN dispersion relation is given by The one-loop near-BMN result can be constructed via unitarity methods following [34]. Doing so we find that the S-matrix takes the following form where the expansion of the phase factor σ AdS2 is given by is fixed by the requirement of unitarity. As observed in [15] the logarithms are consistent with the overall phase being related to the BES phase [22].
We define the near-BMN expansion of the exact result as follows , 35) and similarly for e ′ , p ′ and m ′ . Here for generality we have allowed for various rescalings, however, for simplicity we will assume that the ρ i are constants. 12 Expanding the exact dispersion relation (2.30) in the near-BMN regime, we recover (3.31) if we take Further expanding the exact S-matrix (3.4) in the near-BMN regime, taking α given by (2.32), and fixing the overall factorP 0 such that any one of the eight amplitudes agrees with perturbation theory, we find that, so long as the remaining seven also agree with perturbation theory, (3.30) and (3.32). 12 To be completely general, one could in principle let e, m and p be arbitrary functions of e and p. However, naively truncating the classical/tree-level results for AdS 5 × S 5 and AdS 3 × S 3 × T 4 , for example [27,26], to the massive sector of AdS 2 × S 2 × T 6 the ansatz (3.35) seems reasonable. Of course to check this claim one should construct the light-cone gauge symmetry algebra explicitly.

Yangian symmetry 4.1 Massive case
In this section we would like to discuss the issue of Yangian symmetry. The first observation is that, in the massive case (we can fix m = m ′ = 1 for the purposes of this section), we could not apply the same standard Yangian symmetry of the R-matrix which works for the massless case (see section 4.2). The massive representation is a long one (cf. section 2.2), and a similar result was found for long representations of psu(2|2) ⋉ R 3 [24]. The long representations studied in [24] bear a strong resemblance to the ones in this paper, up to the different dimensionality.
We proceed by postulating the commutation relations of the standard sl(1|1) Yangian in Drinfeld's second realization [35,36] (with central extensions) One can check that the coproducts obtained from and their opposites satisfy the defining relations (4.1) and hence provide homomorphisms of the Yangian.
The antipode Σ can be easily found from (4.2) using the defining property where ǫ annihilates all level 1 generators. Combined, this defines the Hopf algebra structure of the standard Yangian.
One can construct a family of representations of the Yangian (4.2) starting from a slightly simpler level-zero (Lie algebra) representation compared to the one we use in section 2.2. Determining the level 1 generators in this representation, we can obtain all the central elements up to and including level 2, together with their coproducts and opposite coproducts. 13 Following the strategy of [24], one can check whether all the central coproducts are co-commutative, as this is a necessary condition for the existence of an R-matrix scattering two such representations (see footnote 1). We found that for all members of the family of representations. This implies that at least one representation of the standard Yangian does not admit an R-matrix, excluding the existence of a universal R-matrix.
However, it is likely that the massive R-matrix may admit a coproduct which is not precisely the same as for massless representations, but still of the type found in [37]. Moreover, considerations as in footnote 3 of [24] are likely to apply. We leave this investigation for future work.

Massless case
The situation is different for the massless limit m = m ′ = 0. In this case, in the absence of the central extensions (b = c = 0, i.e. considering again the gl(1|1) algebra), the representation would 13 In the absence of non-central Cartan elements, we cannot mechanically generate the level 2 and higher supercharges and they would have to be guessed. However we do not need them for the sake of this argument. become one of the reducible but indecomposable modules of section 2.1. In fact, in that case the condition m = ad − bc = ad = 0 would force one of the fermionic generators to be identically zero. The indecomposable would then be made up of short 1-dimensional gl(1|1) irreps. This suggests that the Yangian might now be straightforwardly derived from the standard one.
Indeed, this time we construct an evaluation representation of the Yangian (4.2) does not directly transfer to long ones as it stands [37,38].
The crossing symmetry transformation reveals an interesting property, related to what was observed in [18] for the case of AdS 3 × S 3 × T 4 , namely the existence of two different Yangian spectral (evaluation) parameters for the particle and the anti-particle representations. Here, the difference is superficial, as the massless condition makes the two spectral parameters coincide. In fact, the antipode obtained from applying (4.3) reads This effectively amounts to a shift in the spectral parameter u by one of the central elements. When plugging this into the relation and postulating that the anti-particle representation is also of evaluation type, that is we see that the conditions (4.7) and (4.6) reduce to the same equation that holds true for the level 0 charges, i.e. (3.21), provided that the anti-particle spectral parameter is chosen to be For massless particles, (4.10)

Derivation from Yangian invariance
The S-matrix describing the scattering of two massless excitations can be directly obtained by imposing Lie algebra and Yangian invariance for two m = 0 representations of section 2.2, or as an m, m ′ → 0 limit of the massive S-matrix. In the latter case, one has to treat various 0 0 limiting expressions, which come from the function f in eq. (3.5). 14 Taking care when resolving these singular limits we find agreement with the result from imposing Yangian invariance. In the massless limit the dispersion relation in terms of the Zhukovsky variables takes the form [26] 15 In terms of the energy and momenta this translates to and hence there are two branches of the dispersion relation depending on the sign of sin p 2 [26] 16 In the following we will use the convention that σ = +1 corresponds to a particle moving from left spatial infinity to right spatial infinity, i.e. right-moving, while σ = −1 corresponds to a left-moving particle.
For σ = σ ′ = +1, the Yangian invariance fixes the S-matrix up to two undetermined functions χ ++ 1,2 : We have checked that the Yangian representation with the coproducts taken in the appropriate branches -and away from the bound-state point (see footnote 5) -is fully reducible simultaneously at level zero and one, which is consistent with the appearance of two undetermined functions in the scattering matrix.
In order to match the limit from the massive S-matrix, the functions χ ++ 1,2 should be chosen as follows: where f ++ is the limit of f . The limit of f is not fixed by the comparison with the Yangian S-matrix.
However, imposing the Yang-Baxter equation There is a second solution x + = x − , however, this corresponds to p = 0 and therefore is not physically sensible. 16 Although the doubly-branched dispersion relation e = 2h| sin p 2 | is non-relativistic, there are some similarities with the kinematics of massless relativistic scattering. Following [39], in the relativistic case one has A boost sends the rapidity u → u+λ, with λ ∈ R, hence the two branches can never be connected by such a transformation.
In the non-relativistic case we have the two branches with x + a pure phase for real momentum and energy. As the S-matrix is not of difference form there is a priori no notion of boosts and hence it is not clear if the presence of two branches represents an obstruction to interpreting the σ = σ ′ = ±1 scattering. However, as pointed out in [26], while the small momentum dispersion relation is relativistic, for the exact non-relativistic dispersion relation, the group velocity v = ∂e ∂p is a non-trivial function of p and hence one may hope to give a physical interpretation to the σ = σ ′ = ±1 scattering. requires that The Yang-Baxter equation for σ = σ ′ = +1 scattering (5.7) does not allow for non-constant limits of the function f . In particular, the condition it imposes reads (we denote lim m, If f ++ 13 f ++ 23 = 1, we immediately get f ++ = ±1. If f ++ 13 f ++ 23 = 1, we find However, the l.h.s. of (5.10) does not depend on p 3 , and hence we should impose that the derivative of the r.h.s. with respect to p 3 is zero. Doing so, we find that either once again f ++ = ±1, or, if f ++ = ±1, then should be independent of p 1 . Let us call this function ω(p 3 ). This implies that Plugging this expression back into (5.10) we find that eitherω(p) = 0, in which case f ++ = 1 and we are done, orω(p) =ω −1 (p). Finally, substituting into (5.9) we find thatω(p) is a constant and hence f ++ = 0. This then demonstrates that the solutions of (5.9) are f ++ = ±1, 0.
As in the relativistic case [39], a different situation applies for σ = +1, σ ′ = −1. The Yangian invariance again fixes the S-matrix up to two undetermined functions χ +− 1,2 : Again one can check that the Yangian representation with the coproducts taken in the appropriate branches -and away from the bound-state point (see footnote 5) -is fully reducible simultaneously at level zero and one, which is as before consistent with the appearance of two undetermined functions in the scattering matrix. In order to match the limit from the massive S-matrix, the functions χ +− 1,2 should be chosen as follows: where f +− is the limit of f . For this mixed case the limit of f is also not fixed by the comparison with the Yangian S-matrix. Once again, the Yang-Baxter equation fixes this limiting value. In order to write down the Yang-Baxter equation for the mixed case, we need to first calculate the S-matrix for σ = σ ′ = −1, as schematically it is given by (5.14) we find the following possibilities for the limits of f : where µ 1 and µ 2 are arbitrary constants.
The various choices for f ++ , f +− , f −+ , f −− can be further restricted by considering crossing symmetry. Although in the massless case there is no clear physical interpretation of crossing, see, for example, [39], one may nevertheless demand that it is still present. Let us recall that the crossing transformation simultaneously changes the sign of the energy and momentum, therefore the crossing of a + (−) particle is still a + (−) particle. Consequently in the crossing relation (3.24) we should consider two massless Smatrices of the same type. Considering the various possible limits of f , we find that the choices f ++ = 0 and f −− = 0 are incompatible with crossing. Indeed, before taking the massless limit, the function f satisfies the following crossing transformation with respect to the first particle: which is is clearly problematic for f → 0. We are then left with the following choices for the limits of f It is worth noting that for the crossing relation to be satisfied for these choices we should not only consider two massless S-matrices of the same type, but also with the same limit of f . Now that we are left with the choices in eq. (5.21), let us recall that in the massive case the sign of f is not determined by symmetry or the Yang-Baxter equation, rather from comparing with perturbation theory. This is consistent with the residual ambiguity we are finding in this limit.
If we look at the BMN limit (see section 3.2) for the σ = σ ′ = ±1 S-matrices, we don't necessarily expect to (and indeed we do not) find the identity. This expectation comes from the fact that the quadratic Lagrangian of the light-cone gauge-fixed theory is relativistic and it is not clear how one should perform a perturbative computation for the scattering of two massless relativistic particles on the same branch, or if there should be a perturbative expansion at all.
For the σ = −σ ′ = ±1 S-matrices one may expect the limit to be better behaved as perturbative computations can be carried out. Indeed, assuming that the phase goes like one plus corrections, then for the σ = −σ ′ = +1 case we find that if f +− = 1 the S-matrix is the identity at leading order, while for the σ = −σ ′ = −1 case the same is true, but with f −+ = −1. Therefore, we end up with the following choices for the limits of f We may attribute some physical meaning to this result by considering the group velocities For a physically realizable scattering process with σ = −σ ′ = +1 the group velocities satisfy v > v ′ , while for a scattering process with σ = −σ ′ = −1 we have v ′ > v. Therefore, we may associate lim m,m ′ →0 f → 1 with v > v ′ and lim m,m ′ →0 f → −1 with v < v ′ . This is consistent with the crossing symmetry discussed above as the group velocity is invariant under the crossing transformation. Furthermore, one may expect the σ = −σ ′ = +1 and σ = −σ ′ = −1 S-matrices to be related upon interchanging the arguments.
Indeed, so long as f +− = −f −+ , the following equation is satisfied for real momenta 18 The corresponding relation for the σ = σ ′ = ±1 S-matrices is given by To conclude, let us briefly comment on unitarity. Motivated by the physical interpretation outlined above, one may expect that braiding unitarity for the massless S-matrix will involve one S-matrix with f → 1 and one with f → −1, and indeed, one can explicitly check that braiding unitarity relations can be constructed in this way. They are given by These relations can also be found by taking the massless limit of the braiding unitarity relation for the massive S-matrix. Finally, one can see that by combining (5.24), (5.25) and (5.26), all the four massless S-matrices are also QFT unitary so long as the overall factors satisfy appropriate constraints.

Massless limits and symmetry enhancement
Let us now consider taking the various massless limits of the parametrizing functions of the massive Smatrix, i.e. one massless and one massive or two massive particles. Here we work in terms of the variables x ± , x ′± as it allows us to consider the four cases of section 5.1 at the same time. For convenience we introduce the following notation for the massless Zhukovsky variables The parametrizing functions are then given by on the individual factor S-matrices. This symmetry is expected from string theory as a consequence of the additional compact space T 6 required for a consistent 10-d superstring theory [15]. 19 Under this symmetry the bosons y and z are uncharged, while the fermions (ζ, χ) T form an SO(2) vector.
Furthermore (Q 2 , Q 1 ) T and (S 2 , S 1 ) T are also charged as SO (2) vectors under the symmetry. 20 Here we will summarize the relevant details of this symmetry. Explicit details (including the expansion of the tensor product) are given in appendix A. Defining and their conjugates, we have the following actions of the U (1) generator, J U (1) , To proceed with the algebraic Bethe ansatz (ABA) technique one constructs the monodromy matrix as a string of R-matrices acting on an auxiliary space a and on N physical spaces where · denotes multiplication in the auxiliary space. A(λ), B(λ), C(λ) and D(λ) are operators on N -particle physical space, while the 2 × 2 matrix acts on the auxiliary space. As a consequence of the Yang-Baxter equation one has Taking the trace tr a1 ⊗ tr a2 on both sides of (6.4), one finds that the transfer matrix T (λ) ≡ tr T a (λ) = A(λ) + D(λ) satisfies: [T (λ), T (λ ′ )] = 0 . (6.5) As T (λ) is an N th order polynomial in λ (with the highest-power coefficient chosen equal to 1), we see that (6.5) implies that T (λ) generates N non-trivial independent commuting operators.
To find the simultaneous eigenvectors of all the commuting charges (which include the Hamiltonian), one assumes that B(λ) is a creation operator acting on a pseudo-vacuum |vac , which is annihilated by C(λ): The pseudo-vacuum should be a highest-weight T (λ)-eigenstate, whether or not that is the true ground state of the Hamiltonian. The vectors (6.6) are not immediately eigenstates of T (λ) because of unwanted terms obtained when acting with T (λ). These unwanted terms are cancelled by imposing the Bethe equations, providing the quantization condition for the momenta of excitations.
Let us now give some initial observations on applying the ABA procedure to the S-matrix for the light-cone gauge AdS 2 × S 2 × T 6 superstring. We can immediately remark that a single copy of the 19 We are grateful to O. Ohlsson Sax and P. Sundin for pointing out to us the existence of this symmetry in the superstring theory. 20 Here the subscripts on the supercharges Q and S refer to the two copies of psu(1|1) in the full symmetry algebra. In particular the charges with the label 1 act on the first entry in the tensor product (2.1), while the charges with the label 2 on the second entry.
centrally-extended S-matrix does not seem to admit a pseudovacuum on which to construct the ABA procedure. However, when we take the tensor product of two copies there is a pseudovacuum. This is given by a uniform sequence of either all |θ + states or, alternatively, |θ − . In fact, thanks to the conservation of the additional U (1) charge discussed above and in appendix A, these states are the only ones with maximal (minimal) such charge, and therefore have to be eigenvalues of the transfer matrix. By a similar logic they are also annihilated by some of the lower-corner entries of the (now 4-dimensional) transfer matrix. This in principle could allow the ABA procedure to be applied. However, this still remains technically challenging given the complexity of the parametrizing functions of the S-matrix.

Comments
In this paper we have constructed the S-matrix describing the scattering of massive modes of the AdS 2 × S 2 × T 6 light-cone gauge superstring. A significant difference with the AdS 5 × S 5 and AdS 3 × S 3 × T 4 light-cone gauge superstrings is that the massive excitations transform in long representations of the symmetry algebra psu(1|1) 2 ⋉ R 3 . Consequently there is no shortening condition and the dispersion relation is not fixed by symmetry. Furthermore, the symmetry only fixes the S-matrix up to an overall phase, for which we have given the crossing and unitarity relations, which appear to be more complicated than those in the AdS 5 × S 5 case. The exact form of both the dispersion relation and the phase remain to be determined.
The massless limits (one massive and one massless or two massless particles) of the massive S-matrix have been studied in detail. The resulting expressions should play the role of building blocks for the S-matrices of the massless modes of the AdS 2 × S 2 × T 6 superstring. As for the AdS 3 × S 3 × T 4 case [26], the precise nature of this construction requires the knowledge of how all the states transform under the full light-cone gauge symmetry algebra including any additional bosonic symmetries originating from the T 6 compact directions.
In the massless limit the light-cone gauge symmetry psu(1|1) 2 ⋉ R 3 can be extended to a Yangian of the standard form. However this does not generalize in an obvious way to the massive S-matrix. It would be interesting to see if there exists a non-standard Yangian in this case. We are also currently investigating the presence of the secret symmetry [40,41] and the RTT realization of the symmetry algebra [42,43]. Finally, we gave some initial considerations regarding the Bethe ansatz for the massive S-matrix, in particular highlighting the existence of a pseudovacuum. Due to the complexity of the parametrizing functions of the S-matrix and the fact that we are considering long representations of the symmetry algebra the completion of the algebraic Bethe ansatz remains an open problem.
Appendix A: Expansion of tensor product and U(1) symmetry In this appendix we will write explicitly the full expression for the tensor product of two copies of the S-matrix given in (3.2). This will allow us to demonstrate the existence of the U (1) symmetry that was important in section 6 for the Bethe ansatz.

Fermion-Boson
S|θ ± y ′ = S 1 T 2 |θ ± y ′ ∓ iQ 1 R 2 |θ ± z ′ + S 1 R 2 |yθ ′ ± ± iT 1 Q 2 |zθ ′ ± S|θ ± z ′ = S 2 T 1 |θ ± z ′ ± iQ 2 R 1 |θ ± y ′ − S 2 R 1 |zθ ′ ± ± iT 2 Q 1 |yθ ′ ± Fermion-Fermion Provided that which was indeed the case for the S-matrix under consideration in the main text (3.8), it is clear that this S-matrix commutes with a U (1) symmetry acting on the states as follows Finally for completeness we give the commutation relations of the full algebra under which the Smatrix is invariant. First let us define where the subscripts on the supercharges Q and S refer to the two copies of psu(1|1) in the full symmetry algebra. In particular the charges with the label 1 act on the first entry in the tensor product (2.1), while the charges with the label 2 on the second entry.