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The S-matrix algebra of the AdS2 X S2 superstring

In this paper we find the Yangian algebra responsible for the integrability of the AdS2 X S2 X T^6 superstring in the planar limit. We demonstrate the symmetry of the corresponding exact S-matrix in the massive sector, including the presence of the secret symmetry. We give two alternative presentations of the Hopf algebra, along with related discussions on the issue of evaluation representations. We study the classical r-matrix, and re-discover the need for a secret symmetry also in this context. Finally, taking the simplifying zero-coupling limit of the S-matrix as a generating R-matrix for the Algebraic Bethe Ansatz, we obtain an effective model of free fermions on a periodic spin-chain. This limit should provide hints to the one-loop anomalous dimension of the mysterious superconformal quantum mechanics dual to the superstring theory in this geometry.


Introduction
The remarkable impact integrability has had on the solution of string theory in the AdS 5 × S 5 background [1] motivates trying to apply the same strategy to the other less supersymmetric string backgrounds which are still integrable [2]. One of these is indeed the AdS 2 × S 2 × T 6 background with Ramond-Ramond fluxes in Type II superstring theory, which preserves 8 supersymmetries. One way to generate this background is by taking the near-horizon limit of various intersecting brane configurations in Type IIA/B supergravity, related by T-dualities in the T 6 directions [3]. The dual "field" theory [4] is thought to be either a superconformal quantum mechanics, or a chiral two-dimensional CFT [5]. AdS 2 holography, and AdS 2 × S 2 in particular, is an interesting open problem that can be approached from several distinct directions (see for instance [6] for some recent accounts). Given the reduced dimensionality, it would be tempting to regard it as the simplest example where one can test the AdS/CFT duality, but instead it turns out to be one of the most mysterious.
The AdS 2 ×S 2 (coset) part of the background is conveniently encoded into a Metsaev-Tseytlin [7] type action [8], based on the quotient Although it is not completely clear what representation one should adopt for the massless modes [3,18], in [15] we took the approach advocated in [29] for the AdS 3 case, and assumed that the massless representations and the corresponding S-matrix are the zero-mass / finite h limit of the corresponding massive ones (cf. [30]), at least as far as the psu(1|1) 2 R building block is concerned. We obtained in this way the limiting S-matrices for all the choices of left and right chiralities, and discovered that there exists a canonical Yangian for the massless sector.
In this paper, we obtain several results on the algebraic structure of the exact S-matrix of the system, therefore deepening our understanding of the associated spectral problem. Our aim is to explore the Yangian symmetry for the massive sector. The Yangian relevant to the AdS 5 S-matrix was found in [31], while for AdS 3 it was found in [22] in separate sectors, and a larger version encompassing both left and right algebras was discovered in [32]. The approach we will follow is based on the RTT formulation [33], which was first applied to the AdS 5 S-matrix in [34], and later to the AdS 3 S-matrix in [35].
As discussed above, the crucial new feature in the case of interest is that the representations are long. As a result the canonical realisation of the Yangian, similar in spirit to those in [31,22], results in a representation that is not of evaluation type. This is also a feature for long representations in the AdS 5 case [27]. After investigating this realisation, we find a new alternative realisation of the Yangian that does lead to a representation of evaluation type. This gives more control over the symmetry and its action on one-particle states. Indeed the evaluation representation is the most natural physical manifestation of Yangians in integrable scattering problems. Furthermore, this new realisation and the original one are contained with in a larger family of realisations originating from a symmetry of the restricted Yangian algebra.
Besides the Y(su c (2|2)) Yangian, the S-matrix of the AdS 5 × S 5 superstring admits an additional infinite tower of conserved charges, which constitute the so called secret symmetry of the model [36,37] (see [38] for a review). Such symmetries are present in several other parts of the correspondence, for example in the pure spinor sigma model [39], scattering amplitudes [40] and Wilson loops [41]. Recently, secret symmetries were also found for the AdS 3 superstring [35], providing further evidence of their universal nature in the AdS/CFT framework.
We will begin in section 2 with a brief summary of the key properties of Yangians that will be necessary for the following exposition. In section 3 we use the RTT formulation to construct the Yangian algebra underlying the integrability of the massive sector, including the secret symmetry. Two distinct realisations of this symmetry are given, the first of which is close in spirit to that used in the AdS 5 and AdS 3 cases [34,35] and in general leads to a representation not of evaluation type, while the second is a new realisation leading to a representation that is of evaluation type. We then discuss the issue of evaluation representations in detail, demonstrating a relation to shortening (massless) condition.
In section 4 we explore the strong-and weak-coupling limits. First, we perform a study of the classical r-matrix, and discover that the need for the secret symmetry, based on the residueanalysis at the simple pole in the spectral-parameter plane, is present also in this context. Second, we study an effective Bethe ansatz in the simplifying limit of zero coupling, in which the problem reduces to a standard rational spin-chain in each copy of the symmetry algebra. In this way we obtain the spectrum of free fermions on a periodic chain. This should represent an entry-point to the leading-order (traditionally dubbed one-loop) anomalous dimension for the composite operators of the mysterious superconformal quantum mechanics, supposed to be dual to the superstring theory.

Yangians and integrability
We start by reviewing some of the underlying formalism of Yangians and their various realisations. Here we focus on the details that will be relevant for us when investigating the Yangian of the massive sector of the AdS 2 × S 2 × T 6 superstring.
Yangians: Drinfeld's First Realisation. If g is a Lie superalgebra, its Yangian Y(g) can be obtained via a quantum deformation of the universal enveloping algebra U(g u ), where g u is the polynomial loop algebra of g in the variable u. More precisely, where the vector space Y m (g) corresponds to the m-th level of the Yangian Y(g). In the so-called Drinfeld's first realisation, the generators J I (m) fulfil the graded commutation relations with f AB C being the structure constants of g. Equations (2.2), together with an appropriate set of Serre relations, determine Y(g) uniquely. Notice that Y 0 (g) ≡ g.
Yangians enjoy the translational symmetry which can be clearly seen from (2.1).
A special type of representation is the evaluation representation π E : where u is called the spectral parameter, and π is a chosen representation of g. However, π E might not exist for all representations π. One reason possible reason for this is that inserting π E into the Serre relations actually imposes strong constraints back on π itself, which may or may not be satisfied.
Yangians and integrable systems. Quantum systems exhibiting Yangian symmetry are integrable, as a consequence of the fact that the Yangian allows to construct a solution to the Yang-Baxter equation which controls the inverse scattering problem (and often the S-matrix of the excitations) [42]. Indeed, let R ∈ A ⊗ A be a scattering matrix, with A = Y(g) being the Hopf superalgebra whose coproduct ∆ : A → A ⊗ A defines the action of the conserved charges on two-particles states. If A and R are such that ∆ op (a) R = R ∆(a) ∀a ∈ A , (quasi co-commutativity), (1 ⊗ ∆)(R) = R 13 R 12 , (quasi-triangularity), (2.5) where ∆ op = (σ • ∆) is the opposite coproduct, σ the graded permutation operator and the subscripts 1,2,3 indicate the copy of A in the triple tensor product, then R obeys the quantum Yang-Baxter equation (QYBE) (the hallmark of integrability) and the crossing symmetry equations. This is the content of a famous theorem of Drinfeld, proving the crucial role played by quantum groups in producing solutions to the QYBE with the desired symmetry properties [43].
RTT Realisation of the Yangian. Given a scattering matrix R 12 (u, v) that is a function of two spectral parameters and satisfies the QYBE, one can extract the symmetries of the system using the RTT realisation of the corresponding Yangian [33,34]: The monodromy matrix T (u) plays the role of a generating function for the Yangian charges, which are in turn symmetries of R. We shall now summarise the main features of this construction: see [34] for further details. We will focus on the gl(n|n) case, the one of interest.
Let e A B be the standard basis for the gl(n|n) Lie superalgebra. That is the matrices e A B are such that their only non-vanishing entry is (−) [B] in row A, column B. The symbol [A] stands for the Graßmann grading of the index A. T (u) can then be written as follows: Assuming that T A B (u) is holomorphic in a neighbourhood of u = ∞, the asymptotic expansion is well defined. At this point, one finds that particular combinations of the T A l−1B can be engineered to define Drinfeld's first realisation of Y(gl(n|n)). In particular, defining

10)
J 0 and J 1 span gl(n|n) and Y 1 (gl(n|n)), respectively. The central element U is the braiding factor, and represents a deformation of the co-algebra structure. U = 1 represents the undeformed case.
The key observation is that the R-matrix is a particular representation of T , namely, R(u, v) = (1 ⊗ π v ) T (u), where π v indicates a representation depending on the spectral parameter v. Therefore, the coefficients in the Laurent expansion of R(u, v) can be understood as T A l−1B for the RTT realisation of the underlying symmetry in the representation of interest. This can then be recast in the form of Drinfeld's first realisation using (2.10). The (representationindependent) graded commutation relations for the J A mB are obtained from those for the T A l−1B , by plugging (2.8) and (2.9) into (2.7) and expanding with respect to both u and v.
Finally, the complete Hopf algebra structure of Y(gl(n|n)), in particular, the coproducts and antipodes, can be recovered from the RTT realisation of the Yangian. First, the fusion relation descending from the R-matrix fusion relations, provides the coproducts for the individual generators via the same expansion: indeed, expanding T A B (u) in inverse powers of u gives This can be used to derive the coproducts for J 0,1 : Second, the antipode Σ is a graded linear anti-homomorphism satisfying . (2.14) By expanding: which in turn can be used to determine the antipodes for J 0,1 3 RTT realisation for the Y(gl c (1|1)) Yangian In this section we employ the techniques of [33,34] to construct the RTT realisation for the Y(gl c (1|1)) Yangian. The starting point is the su c (1|1) R-matrix of [15], which describes the scattering of one bosonic state |φ and one fermionic state |ψ transforming in a long representation of the centrally-extended algebra su c (1|1) The explicit form of the representation is given by where the eigenvalues of the central elements P, K and H are given by as a consequence of the algebra relations (3.1). There are no further conditions on the central elements for the algebra to close and hence this 2-dimensional representation is long.
For generic values of P , K and H the tensor product of two of these representations gives a 4-dimensional representation that is fully reducible into two 2-dimensional representations. 1 Therefore, the R-matrix acting on the tensor product is fixed up to two functions by demanding invariance under the symmetry Here ∆ is the coproduct, while ∆ op is the opposite coproduct defined in (2.5). As usual for integrable systems arising in the context of the AdS/CFT correspondence the coproduct is deformed through the introduction of an abelian generator U To admit an R-matrix the coproducts of the central elements should be co-commutatitve, i.e. ∆ op (C) = ∆(C). This relates the central charges P and K to the braiding factor U: where without loss of generality we have taken the constants of proportionality to be equal. In the following we will refer to the algebra (3.1) with these relations (3.6) imposed as the restricted algebra. While the symmetry only constrains the R-matrix up to two functions, demanding that the QYBE is solved (along with imposing various physical requirements such as crossing symmetry and a sensible strong coupling limit) fixes the R-matrix up to a single overall factor.
For reference we will quote the necessary details of the su c (1|1) R-matrix from [15]. The R-matrix in terms of the usual Zhukovsky variables [19,44] x Here is unconstrained as the representation (3.2) is long. The representation parameters a, b, c and d are given in are given in terms of the Zhukovsky variables in [15] -we set the parameter α used there, which controls the normalisation of the bosonic state relative to the fermionic state, to one. The R-matrix is then given by with x ± the kinematic variables associated to the first representation and y ± to the second. The parameterising functions are given by [15] As discussed above, there is an overall factor P 0 that is not fixed by the considerations of symmetry and as such will not be relevant for the following analysis. For concreteness we will fix P 0 such that S 1 = 1 following [34] and so that the expansion of the R-matrix takes the form outlined in section 2.
Finally, let us observe that the AdS 2 ×S 2 worldsheet S-matrix underlying the scattering of the massive modes is built from the tensor product of two copies of this centrally-extended su(1|1) R-matrix.

3.1
The Y(su c (1|1)) and Y(gl c (1|1)) Yangians and their RTT realisation The centrally extended Y(su c (1|1)) Yangian is defined by the graded commutation relations which extend the algebra (3.1). Here, m, n ≥ 0 indicate the level of the corresponding generator, with Q 0 , S 0 , P 0 , K 0 and H 0 playing the role of the original generators in (3.1).
It is worth noting that, in contrast to ordinary situations, the infinite-dimensional algebra (3.12) contains infinitely many finite-dimensional su c (1|1) subalgebras. Indeed, the set of generators Qm, Sn, P 2m , K 2n and Hm +n forms a subalgebra for allm,n ≥ 0. This is due to the fact that the generators on the right-hand side of the relations (3.12) are central.
Another consequence of this, again in contrast to usual, is that for an arbitrary representation one cannot generate the whole infinite dimensional algebra by (anti) commuting a finite set of generators. However, considering the natural lift of the 2-dimensional representation (3.2) it is relatively easy to see that the relations which follow from the graded commutation relations (3.12), can be used to solve recursively for the higher-level representation parameters given their values at level 0 and 1.
As we will see, one property that does carry down from the higher-dimensional cases is that the su c (1|1) R-matrix (3.9), (3.10), (3.11) exhibits an additional family of symmetries, B n , n ≥ 1, known as bonus or secret. These symmetries enhance the Y(su c (1|1)) Yangian to some indented Yangian-like quantum group we call Y(gl c (1|1)), which contains all the generators in (3.12) along with B n (not including B 0 ).
To construct the RTT realisation of the Yangian we introduce a spectral parameter u defined in terms of x ± , and similarly v for y ± . We shall use two different definitions for the spectral parameter, but in both cases the expansion of x ± in powers of u −1 will take the form Once this expansion has been specified, we follow the construction outlined in section 2. That is we expand the R-matrix R(x ± , y ± ) in inverse powers of one of the two spectral parameters, say u, 2 and from the resulting Laurent coefficients extract a series of generators J m whose graded commutation relations reproduce those of the underlying infinite-dimensional symmetry algebra. We can then define abstract generators J m , of which J m are a representation. At this point we can construct generators satisfying (3.12) along with their coproducts and antipodes. It should be noted that this construction automatically leads to the restricted Yangian, for which, in addition to (3.6), the higher-level central charges P m and K m are defined in terms of lower-level central elements. We refer the reader to [34,35] for a more complete discussion.

Canonical representation
Let us start by considering the canonical spectral parameter and a Hopf algebra structure that is close in spirit to that used in the AdS 5 and AdS 3 cases [34,35]. The spectral parameter is given by (3.17) Following the procedure outlined above we then identify the following combinations 3 which satisfy the defining commutation relations (3.12). Evaluated in the representation arising from the expansion of R-matrix, the level-0 generators coincide with those used in [15].
Coproducts. The level-0 coproducts are given in (3.5), while the level-1 coproducts can be constructed from (2.13) and read where P 0 and K 0 are defined in terms of lower-level central elements in (3.18). Indeed inserting these definitions, the coproduct for H 1 becomes manifestly co-commutative as expected.
The coproducts for P 1 and K 1 can be obtained from the graded commutation relations If P 1 and K 1 are defined in terms of lower-level central elements as in (3.18) we find that these coproducts are also co-commutative as required. Moreover, we can compute the coproducts for the level-2 central charges (3.21) Doing so, we find that for co-commutativity of ∆(P 2 ) and ∆(K 2 ) we require where the normalisation is fixed by matching with the expansion of the R-matrix, 4 while the coproduct ∆(H 2 ) is automatically co-commutative upon using the definitions of P 0,1 and K 0,1 in (3.18).
With these definitions of the generators, one can show that the representation of the Yangian arising from the expansion of the R-matrix is in general not evaluation. However, if the eigenvalues of the central elements satisfy which has the interpretation as a (massless) shortening condition, we find that the representation does become of evaluation type, in agreement with [15]. We will return to this issue in the following sections.
Crossing. Using the general expression for the antipodes of (2.16) we can derive the antipode for the generators of interest i.e. it is involutive for the generators of Y 0,1 (su c (1|1)).
Secret symmetry. While J 1 01 + J 2 02 is central, the combination The coproduct reads while the antipode for the secret symmetry is As in the AdS 5 and AdS 3 case, the antipode is not an involution when acting upon the secret symmetry.

Co-commutativity and shortening condition
Let us now investigate what happens if we try to impose evaluation representation onto the Hopf algebra structure described in section (3.1.1). In particular, we will show that this demonstrates the existence of representations for which one of the higher-level central charges is not cocommutative and hence does not admit an R-matrix.
In evaluation representation we have J n = u n J 0 ∀ J ∈ {Q, S, H, P, K} , (3.29) which manifestly satisfies the algebra relations (3.12). As discussed above, in order to have a co-commutative coproduct for all the level-0 central charges one can impose where we recall that the constant h is independent on the representation space of the coproduct. Similarly, for the level-1 central charges for co-commutativity one can impose It then follows that for a representation of evaluation type the spectral parameter is given by where we use H m , P m , K m and U to denote both the generator in evaluation representation and its eigenvalue, as it is always clear from context which is meant.
At this point, we compute the coproduct of the level-2 central charge P 2 using The expression one obtains is rather lengthy, however it simplifies considerably if one takes the antisymmetric combination which is precisely the quantity that determines whether the coproduct is co-commutative or not.
There are two notable contributions to δP 2 , coming from two separate pieces of the coproduct of P 2 . The first contribution comes from the part of ∆(P 2 ) arising when the S 0 generators in ∆(Q 1 ) meet among themselves in the anti-commutator (3.33). Upon antisymmetrisation these terms contribute to δP 2 . In fact this would be the only surviving term had we set H 0 = u = 0 in both representation spaces. In principle this could already be enough to conclude that there exist representations with non co-commutative P 2 . Nevertheless, it is instructive to continue.
The remaining terms reduce to and, combining all contributions together, we obtain This means that we can achieve co-commutativity if we demand that where the constant does not depend on the representation space. The relation (3.38) is nothing else than the known shortening condition, which is in this way reinterpreted as the condition that makes the central charges' coproduct co-commutative at higher levels (similarly to what the Serre relations do for the canonical part of the Yangian).

Evaluation representation
The Hopf algebra structure discussed in section 3.1.1, which was motivated by similar constructions in the AdS 5 and AdS 3 cases, turned out to give a representation of the Yangian that was not of evaluation type. It turns out that there is an alternative Hopf algebra structure we can put on the same infinite-dimensional algebra, such that the representation arising from the expansion of the R-matrix is evaluation.
To do this we introduce a new spectral parameter such that the expansions of x ± (again assuming that M (x + , x − ) = 2) are given by Let us now define the following (as an alternative to (3.18)) combinations of generators which also satisfy the defining commutation relations (3.12). Again, evaluated in the representation arising from the expansion of R-matrix, the level-0 generators coincide with those used in [15].
Coproducts. The level-0 coproducts are given in (3.5), while the level-1 coproducts can be constructed from (2.13) and read

(3.42)
Here ∆(H 1 ) is written in a manifestly co-commutative form. To check that these coproducts, along with those in (3.5), obey the graded commutation relations (3.12) one needs to use the definitions of P 0 and K 0 in (3.41).
The coproducts for P 1 and K 1 can be obtained from the graded commutation relations (3.20). If P 1 and K 1 are defined in terms of lower-level central elements as in (3.41) we find that these coproducts are also co-commutative as required. Furthermore, we can compute the coproducts for the level-2 central charges using (3.21). Co-commutativity of ∆(P 2 ) and ∆(K 2 ) then requires where normalisations are fixed by matching with the expansion of the R-matrix, while the coproduct ∆(H 2 ) is automatically co-commutative by the definitions of P 0,1 and K 0,1 in (3.41).
From (3.41) we find that the representation of the Yangian arising from the R-matrix expansion is indeed of evaluation type with spectral parameter (3.39) Crossing. Using the general expression for the antipodes of (2.16) we can derive the antipode for the generators of interest. These turn out to be the same as for the canonical case, i.e. (3.24).
Secret symmetry. While J 1 01 + J 2 02 is central, the combination The coproduct reads while the antipode for the secret symmetry is As before, the antipode is not an involution when acting upon the secret symmetry.
It is worth highlighting that the modification of the combinations in (3.41) has altered the commutation relations involving B 1 , changing the tail.

Freedom in the realisation of the Yangian
The two different Hopf algebra structures described in sections 3.1.1 and 3.1.3 are indicative of a larger possible freedom, which we will now describe. Let us consider the defining graded commutation relations (3.12), but in particular focus on the restricted form in which P m and K m are defined in terms of lower-level central charges.
Motivated by the definitions of P 0,1,2 and K 0,1,2 in (3.18), (3.22), (3.41), (3.43), we postulate that the following relation is true for all levels: If we now consider the following redefinitions: where y q,s and z q,s are functions of the braiding factor U. We are interested in finding a set of these functions such that the algebra relations (3.12) and (3.49) are still satisfied by the new generators. Indeed such a solution exists and is given by I.e. the freedom is parameterised by two functions, y and z, of the braiding factor U. The freedom parameterised by y is a generalisation of the symmetry (2.3). For a representation of evaluation type, its effect is to shift the spectral parameter by y(U ).
The redefinitions (3.50) will modify many of the relations underlying the Hopf algebra structure, including the relations between P m and K m and the lower-level central charges, the coproducts of the generators and the commutation relations involving the secret symmetry B 1 . If we demand that the antipode structure (3.24) is preserved, we find Σ(y(U)) = y(U) , Σ(z(U)) = z(U) . Observing that mapping between the Hopf algebra structures in sections 3.1.1 and 3.1.3 precisely takes the form given above in (3.50) and (3.51), we investigate what happens if we take a more general ansatz for the level-1 Yangian supercharges to preserve the antipode structure. By anti-commuting Q 1 and S 1 we obtain the central charges We can now ask for what choices of the function z(U) we can have an evaluation type representation of the Yangian. In particular, this would imply the following two relations Combining (3.55) with the conditions just above reveals that a necessary requirement for an evaluation type representation is There are two cases solutions of interest to this condition. The first is Finally, let us observe that the generator we see that P 1 and K 1 in (3.55), along with the tails in (3.61) vanish. It is therefore natural to ask if the existence of this choice (3.62) is related to the existence of the secret symmetry.

Strong and weak coupling expansions 4.1 Strong coupling expansion and the classical r-matrix
As was done in the AdS 5 case [45,37] it is instructive to study the so-called classical r-matrix of the system. This can be obtained by expanding the quantum R-matrix at strong coupling. In standard quantum group theory, the knowledge of the classical r-matrix and of its Lie bi-algebra structure allows one to reconstruct the quantum group underlying the exact problem. This is still an open problem for AdS superstrings, nevertheless much can be learnt from this exercise.
where C is the Casimir operator for the gl(1|1) tensor algebra.
The classical r-matrix admits the following expression in terms of the generators of the gl(1|1) algebra The same matrix can then be rewritten as an element in the tensor product of two copies of the loop algebra gl(1|1) u,u −1 where The peculiarity of (4.11) is that it is representation independent, and can therefore be taken as a candidate for the universal classical r-matrix in the AdS 2 case.

Weak coupling limit and Bethe equations
In this section we study the leading-order weak-coupling (h → 0) term in the R-matrix, extracting from it a set of Bethe equations. These equations should relate to the leading-order first-level nested Bethe equations one would in principle obtain from the spin-chain Hamiltonian of the putative dual superconformal quantum mechanics that is meant to live on the boundary of AdS 2 . In AdS 5 parlance, this would be called the one-loop nearest-neighbour spin-chain Hamiltonian [47].
The advantage of restricting to this limit is that we can avoid one crucial complication present when dealing with the full R-matrix. To admit a pseudo-vacuum we need to take the tensor product of two copies of the centrally-extended su(1|1) R-matrix. (This tensor product is the one relevant for building up the AdS 2 ×S 2 worldsheet S-matrix [15]). The corresponding pseudovacuum is a specific fermionic linear combination of the states in the two copies, with a definite charge under a certain U (1) quantum number. The corresponding U (1) symmetry does not act in a well-defined way on the individual copies. Performing the algebraic Bethe-ansatz procedure starting from the full pseudo-vacuum is at the moment an open issue.
Dealing with the individual copies, which do not admit a pseudo-vacuum in their own right, could in principle be approached by adapting alternative methods (such as, for instance, the one of Baxter operators). However, the limit h → 0 switches off the most unconventional entries of the R-matrix and allows for the existence of a pseudo-vacuum separately in each copy. Moreover, it drastically simplifies all the remaining entries, allowing for an almost straightforward treatment.

Weak coupling R-matrix
The R-matrix up to order h has the form where with the parameterising functions given by The maps between x ± , u and p are The factors of e ip k in (4.14) and (4.15) are the result of a Drinfeld twist [48]. The presence of such a twist was already observed in the same scaling limit of the AdS 5 R-matrix [1]. Indeed, R 12 can be written as where T 12 = diag e α(p 1 +p 2 ) , e − i 4 p 2 +β(p 1 +p 2 ) , e (β− i with α and β arbitrary coefficients. The entries ofR 12 are given bỹ with the same associations of letters to entries as in (4.13) and (4.14).
Taking h = 0, the entries ofR 12 involvingÃ 12 drop out, and we recover the gl(1|1) R-matrix, written down in its canonical rational form.

Bethe ansatz and twist
As we have seen, at leading order the R-matrix reduces to the canonical rational (Yangian) R-matrix R can of gl(1|1), decorated by a Drinfeld twist. Let us rewrite the twist as where E ij are unit matrices, i.e. 1 in row i, column j and zero everywhere else, and we denote by i the numerical coefficient multiplying the momentum in the respective spaces of the T 12 matrix. Similarly, we write the canonical R-matrix as (4.20) such that at leading order the R-matrix reads We are now ready to write down the monodromy matrix. Denoting the auxiliary space with the label 0, we have Plugging the explicit expressions (4.19) and (4.20) into (4.22) we obtain (4.23) Now we insert the dependence of the twist on the momentum. Most of the factors appearing in the auxiliary space cancel, such that we are left with We can therefore define a new set of states and matrices Using these new vectors and unit matrices, the expression for M becomes indistinguishable from the canonical one. Consequently the Bethe ansatz reduces to the standard one, except with the vectors |a now appearing in the wave functions: in particular, the Bethe equations for M magnons will be In the effective model obtained from this limit, excitations propagate as free fermions on a periodic one-dimensional lattice.

Bethe equations via the algebraic Bethe ansatz
To conclude this discussion of the weak coupling limit let us recall how the algebraic Bethe ansatz procedure works for the standard gl(1|1) rational R-matrix, which, as we have just shown, is relevant in the h → 0 limit. As in section 2, we define T (u) as in [34] T (u) = with The RTT equations determine the commutation relations for A, B, C and D. In particular, we will need where f (µ, λ) = R 11 (µ, λ)/R 33 (µ, λ) , The standard rational monodromy matrix admits a pseudo-vacuum state Ω, such that  is an eigenstate of t(λ). For instance, for M = 2 one gets Substituting in the entries of the rational (weak coupling) R-matrix, we see that (4.35) is the same as (4.27). This is the expected result for the gl(1|1) R-matrix 6 (see for instance [44,49]).

Conclusions
In this paper, we have performed a series of studies on the conjectured exact S-matrix for the massive excitations of the AdS 2 × S 2 × T 6 superstring. This S-matrix encodes the integrability of the quantum problem, and is supposed to be the first step towards the complete solution of the theory in the "planar" limit (no joining or splitting of strings). This in turn is expected to provide information on the spectrum of the elusive superconformal quantum mechanics, which should be holographically related to the superstring in this background.
Our main results are as follows: • By employing the technique of the RTT realisation, we have found the presence of Yangian symmetry for the massive sector, and given two alternative presentations -both in the spirit of Drinfeld's second realisation [50] -along with the map relating them. We have studied the Yangian coproduct, and found the conditions under which we can have a consistent evaluation representation. In order to ascertain these requirements, we studied the co-commutativity of the higher central charges, which is a necessary condition for the existence of an R-matrix. We discovered that shortening is one way to have a consistent evaluation representation, exactly as it was noticed in AdS 5 [27]. However, we demonstrated explicitly that there is a second route, which crucially for the AdS 2 × S 2 superstring holds for long representations.
• We also found, as in the higher dimensional cases, a secret symmetry, which is present only at level 1 of the Yangian and higher. This confirms the ubiquitous presence of this symmetry in all the known manifestations of integrability in AdS/CFT.
• We have studied the classical r-matrix of the problem, and rediscovered from its analytic structure the need for the extra gl type (secret) generator. This is also similar to the situation in the AdS 5 case.
• We have taken the first steps towards a derivation of the Bethe equations starting from our S-matrix (inverse scattering method). At zero coupling, we discovered that our S-matrix becomes (up to a twist which is easily dealt with) two copies of the standard rational gl(1|1) R-matrix. This allows one to define a pseudo-vacuum in each copy individually, and enormously simplifies the problem. Taking the zero-coupling limit of the S-matrix as a generating R-matrix for the Algebraic Bethe Ansatz, we obtain an effective model of free fermions on a periodic spin-chain. Let us note that this should relate to the would-be one-loop result of the first nested level in the Bethe ansatz of the putative spin-chain, describing the superconformal quantum mechanics dual to the superstring.
There are a number of future directions which we plan to explore: • It seems that we are re-discovering many of the features of the AdS 5 (and, to a certain extent, the AdS 3 ) Yangian. In the AdS 2 case, however, the algebra is small enough that we are able to say a more, especially in terms of alternative presentations. This means that we may hope to find the complete Drinfeld second realisation and derive the universal R-matrix through a suitable ansatz. This would also help to understand the higher dimensional cases. In turn, we would then be able to finally construct the much sought after exotic quantum group, which should quantise the classical r-matrix algebra, and would prove the algebraic integrability of the system.
• The most urgent challenge is probably to derive the full set of Bethe (Beisert-Staudacher) equations for the spectral problem. They encode the information of the planar anomalous dimensions in the dual theory, and would hence provide vital information regarding the nature of the holographic dual to the AdS 2 superstring theory. The simplifying assumption of zero coupling of course eliminates those entries which are responsible for the full pseudovacuum being a mixed state in the two copies. Therefore, a more sophisticated technique, rather than the simple algebraic Bethe ansatz computation we have performed here, might be required. This should tie in with a thorough off-shell worldsheet analysis in the spirit of [51,29].
• Further directions include studying D-branes in this background, and performing a boundary integrability analysis as recently done in [52]. Also, it would be illuminating to continue the perturbative and unitarity analyses of [18,26,53,54], obtaining further information on the dressing phase and the dispersion relation.