Yangian symmetry of string theory on AdS 3 × S 3 × S 3 × S 1 with mixed 3-form ﬂux

We ﬁnd the Yangian symmetry underlying the integrability of type IIB superstrings on AdS 3 × S 3 × S 3 × S 1 with mixed Ramond–Ramond and Neveu–Schwarz–Neveu–Schwarz ﬂux. The abstract commutation relations of the Yangian are formulated via RTT realisation, while its matrix realisation is in an evaluation representation depending on the quantised coefﬁcient of the Wess–Zumino term. The construction naturally encodes a secret symmetry of the worldsheet scattering matrix whose generators map different Yangian levels to each other. We show that in the large effective string tension limit the Yangian becomes a deformation of a unitary loop algebra and we derive its universal classical r-matrix. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP 3 .


Yangian Symmetry and Exact Solvability
There are more symmetries in a theory than those displayed by its Lagrangian. This is particularly evident in four dimensional N = 4 Super Yang-Mills theory, whose planar on-shell scattering amplitudes [1] and Wilson loops [2] are invariant under an infinite tower of non-local conserved charges. The very same symmetry, identified with the Yangian Y[psu(2, 2|4)], was found in type IIB superstrings on AdS 5 × S 5 by studying the coset structure of the background [3]. Generally, if g is a graded Lie algebra, the corresponding Yangian Y[g] is a particular quantum deformation of U (g [u]), the universal enveloping algebra of g-valued polynomials in the spectral parameter u ∈ C.
In terms of Drinfeld first realisation [4][5][6], Y[g] is formulated as where f AB C are the structure constant of g, [X, Y } = XY −(−) |X||Y | Y X is the graded commutator of X and Y and | · | the Graßmann grading. In particular, the Lie algebra g coincides with Yangian symmetry typically appears in integrable quantum field theories: indeed, both N = 4 super Yang-Mills theory [7][8][9] and type IIB superstrings on AdS 5 × S 5 [10][11][12] are dual to an integrable spin chain system. This duality is even more powerful than the AdS/CFT correspondence [13][14][15] as the integrability picture allows for computing the conformal data of N = 4 super Yang-Mills theory at all orders in the gauge coupling constant [16][17][18][19]. Technically, it is therefore very useful to have the control on the dual integrable picture through its symmetries, as the latter are able to strictly constrain or even determine gauge theory or gravity observables. Conceptually, Yangians are fascinating because they link to each other completely different models such as gauge, gravity and condensed matter systems via a rich mathematical structure. Yangians have been found in quite a few instances of the AdS/CFT correspondence [20][21][22][23]. In this paper we will focus on type IIB superstrings on AdS 3 × S 3 × S 3 × S 1 with mixed Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NSNS) flux. We shall now review the main features of the model.
The investigation was generalised by including both RR and NSNS fluxes in [36] directly working on the Green-Schwarz action of the theory, ignoring the coset formulation. In light-cone gauge, the worldsheet scattering is encoded in an su c (1|1) 2 invariant R-matrix, which was derived assuming the integrability of the quantum theory. As in the AdS 5 × S 5 case, to such an integrability shall correspond a Yangian symmetry restricting the observables of the conformal field theory dual, first tackled in [37,38] and deeply studied in [39][40][41]. The goal of this paper is indeed to extract the Yangian of AdS 3 × S 3 × S 3 × S 1 superstrings with mixed flux. Especially, we will employ the scattering matrix found in [36] to define its own symmetry algebra. The procedure we will use is called RTT realisation [42][43][44]: we introduce it in the next subsection.

Integrable Scattering matrices from Hopf Algebras
Integrable quantum field theories enjoy an infinite number of conserved charges. Such a symmetry, which we denote by A, is so constraining that it completely fixes the scattering matrix of the system. This situation is rigorously described if A has got a Hopf algebra structure. Indeed, the coproduct map ∆ : A → A ⊗ A naturally provides a multiparticle representation of the conserved charges: conventionally, J ∈ A acts on in-states via ∆(J) and on out-states via ∆ op (J) = P • ∆(J), where P(X ⊗ Y ) = (−1) |X||Y | Y ⊗ X is the graded permutation operator. The symmetry acts on antiparticle states through the antipode Σ : A → A, which is a C-linear anti-homomorphism.
This provided, the scattering matrix S acting on the Hilbert space H can be expressed as The R-matrix depends on two spectral parameters u 1 , u 2 ∈ C related to the momenta of the scattering excitations and is fully determined by the quasi co-commutativity condition as well as by the crossing equations Importantly, (1.5) implies that, if C ∈ A is central, it must be co-commutative: ∆ op (C) = ∆(C).
We shall also require that the underlying Hopf algebra A is almost quasi triangular, meaning that the fusion relations 3 hold. This is not restrictive as (1.7) and (1.5) imply the quantum Yang-Baxter equation (QYBE) [50] which is a necessary and characterising condition for R-matrices of integrable systems.

From R-matrices to Hopf Algebras
The above discussion can be read backwards; indeed, an R-matrix satisfying the QYBE defines its own Hopf algebra structure by the RTT relations where the monodromy matrix T : C → End(H ) ⊗ A generates the conserved charges. If we focus on the AdS 3 ×S 3 ×S 3 ×S 1 worldsheet scattering, End(H ) = su c (1|1) 2 ⊂ gl(2|2) and we can express the R-matrix and the monodromy matrix in the standard basis 4 {e a b } of the gl(2|2) superalgebra.
This embedding will be particularly convenient as it allows for dealing with both copies of su c (1|1) at the same time. Then, assuming that T (u) is holomorphic in a neighbourhood of u = ∞, with T a (n) b being the abstract (representation independent) generators of A. Similarly, the R-matrix takes the form (1.11) Substituting (1.10) and (1.11) in (1.9) yields The abstract commutation relations of A are therefore recovered by expanding this constraint where U is a central element. Furthermore, by comparing the RTT relations (1.9) with the QYBE (1.8), one observes that R and T (u) can be connected via a representation map ρ u depending on the spectral parameter u: gives access to the representations T a (n) b = ρ u T a (n) b . As for the Hopf algebra structure, the fusion relations (1.7) induce the coproducts while the antipodes descend from (1.6): If the Hopf algebra of the system is a Yangian over a graded Lie algebra g, Y[g], it is useful to define the objects where the constants h, m appear for future convenience. These Js satisfies (1.18) A great advantage of the RTT realisation is that it automatically provides the consistency conditions of the Hopf algebra, for instance [51,52] µ where µ and ǫ are the Hopf algebra multiplication and counit respectively.

Outline
To make the paper self-contained, in Section 2 we re-derive the AdS 3 × S 3 × S 3 × S 1 superstring worldsheet scattering matrix, originally found in [36]. In Section 3 we use the RTT realisation to extract its Yangian symmetry. We also derive the Yangian evaluation representation and show its dependence on the quantised coefficient of the Wess-Zumino term appearing in the AdS 3 × S 3 × S 3 × S 1 superstring action. In Section 4 we perform the large effective string tension limit and demonstrate that the resulting classical r-matrix can be written in a universal, representation independent form as a tensor product of u(1|1)[u, u −1 ] 2 loop algebra generators.

Note
While completing this paper I became aware of [53], which has some overlap with this work. I am very grateful to the authors for sharing their draft before publication.

Centrally Extended su c (1|1) 2 Algebra and its Representations
As we already mentioned, the fundamental symmetry of the worldsheet theory in light-cone gauge is su c (1|1) 2 , which is made of two copies of su(1|1) and two central charges entangling them, where we labelled each copy by left (L) and right (R). Each su(1|1) A is a three-dimensional superalgebra generated by two supercharges Q A , S A and a central charge quently, the algebra su c (1|1) 2 has got 4 fermionic generators Q L , S L , Q R , S R and 4 central charges H L , H R , P, K whose commutation rules are The elementary excitations of the AdS 3 × S 3 × S 3 × S 1 worldsheet scattering are represented by two bosonic states |φ A and two fermionic ones while the value of the central charges on ν L is The action on the right module ν R = {|φ R , |ψ L } follows from LR symmetry; namely, it is obtained by substituting L with R in (2.3) and (2.4): for instance, Such a representation satisfies the shortening condition (H L + H R ) 2 = (H L − H R ) 2 + 4 P K and is conveniently written in terms of Zhukovski variables x ± L , x ± R . These are kinematical variables depending on the momentum p and the mass m of the corresponding scattering excitation, as well as on the effective string tension h and the quantised coefficient κ of the Wess-Zumino term in the L , x ± R are defined by the constraint where κ L = −κ R = |κ| and we chose the branch cut such that log e 2πi = 0. The link with the particle momentum is given by To simplify the R-matrix entries, we define whose inverse relations read In this setting, the representation coefficients assume the compact form The Lie algebra su c (1|1) 2 is enhanced to a Hopf algebra by introducing 5 which are the coproducts of the supercharges in Drinfeld first realization. Acting with P provides the opposite coproducts (2.10) The coproduct is an algebra homomorphism, therefore ∆(H A ), ∆(P ), ∆(K) are obtained by anticommuting (2.9): for example, (2.11) Using (2.4) it is straightforward to check that the central charges are co-commutative.
The coproducts provide the two-particle representation of su c (1|1) 2 , which is sufficient to solve the scattering problem as the system is assumed integrable and n-body processes factorise into 2-body ones, as allowed by the QYBE.

AdS
Using the basis (φ L , φ R , ψ L , ψ R ), the R-matrix for the worldsheet scattering of strings on AdS 3 × S 3 × S 3 × S 1 with mixed flux assumes the form in the LL sector and in the LR sector. According to LR symmetry, the RL and RR sectors are found by just swapping L and R in the above expressions. Then, (1.5) fixes the LL and LR scattering elements: where x ± A j are the kinematic variables referring to the state |Φ A j , with Φ = φ, ψ; A = L, R and j = 1, 2. For simplicity, we will consider scattering particles with equal masses m, leaving m unspecified. Accordingly, α RL , . . . , ρ RR are obtained by exchanging L and R in (2.14). As a result, the R-matrix is fixed up to four scalar factors σ LL , σ LR , σ RL , σ RR : these are the dressing phases determined by the crossing equations (1.6). Such scalars encode important physical informations (e.g. the bound states of the system); however, they do not affect the Hopf algebra structure and specifying their exact form will be unnecessary.

Representations of the RTT Generators
In the form (1.11), the R-matrix entries (2.14) are distributed as In a neighbourhood of u j = ∞, the x ± A j given by satisfy (2.5). The spectral parameters u j expressed in terms of x ± A j are then Thanks to (3.1) one can expand the R-matrix and read off the algebra commutation relations from (1.12) and the representations of their generators from (1.13). To begin with, we find The expressions of the higher generators are rather lenghty, e.g. (3.5)

Abstract Commutation Relations
Expanding (1.9) around ∞ with respect to both u 1 and u 2 gives the abstract commutation relations for the Ts. A sample of the latter is Notice that m, h and κ combine to structure constants. The commutation rules of Ts provide Js': for instance, the relations (3.6) imply

Lie Superalgebra from RTT
We now reconstruct the whole Hopf algebra A. By choosing we exactly reproduce the su c (1|1) 2 graded commutation relations: The coproducts (2.9) are recovered by applying ∆ to (3.8) by means of (1.17). As for the antipodes, the equations (1.18) provide which preserve the algebra commutation relations. Notice that the antipode is involutive when evaluated on su c (1|1) 2 .

Outer Generators
As in [20,22], the sum is an outer central generator of su c (1|1) 2 . Since A never appears on either the left or the right hand side of the non-trivial A's commutation relations, it can be modded out of the algebra. In other words, if B L and B R are the bosonic generators whose action on the supercharges reads 6

Y[su c (1|1) 2 ] Yangian from RTT
we obtain the commutation relations for the level one of the Y[su c (1|1) 2 ] Yangian: The corresponding coproducts are 17) and ∆ (J R ) = ∆ (J L ) L→R for the R generators. Finally, ∆ acts on the Yangian central charges as Ultimately, the antipodes are The antipode Σ acts as an involution on the level one partners of the su c (1|1) 2 generators. This will no longer be true for the secret symmetry, as we shall see in the next subsection.

Secret Symmetry
The outer generator is not central and cannot be quotiented out as it was previously done for A. As a result, the bosonic are independent. Their commutation relations read where the others are given by the swapping L ↔ R. The coproducts are and ∆(ß R ) = ∆(ß L ) L→R , while the antipodes read The existence of these additional generators implies that the symmetry of the theory is not just of the secret symmetry first found in the AdS 5 × S 5 case [54][55][56] and then discovered also in the worldsheet scattering on AdS 3 × S 3 × M 4 with RR flux [21] as well as in the massive sector of the AdS 2 × S 2 superstring [22]. On the field theory side, the secret symmetry corresponds to an helicity operator acting on scattering amplitudes [57] as well as on Wilson loops [58][59][60]. On the string theory side, the existence of secret symmetries was demonstrated by means of the pure spinor formalism [61]. As anticipated, the antipode is not involutive on ß A : The order u −2 deviation from Z A (u) = ½ is proportional to the shift in Σ 2 (ß A ), while higher orders provide Σ 2 (J a (+2) a ), Σ 2 (J a (+3) a ) and so forth.

Evaluation Representation
Acting with the function ρ u on the Yangian charges one finds that the latter are in evaluation holds. In general, ifĴ is the level one counterpart of J, one obtainŝ Then, the quantised coefficient of the Wess-Zumino term κ splits the evaluation representation in two branches: the left one, with effective spectral parameter u L = u + (2 κ/h); and the right one, with u R = u−(2 κ/h). In the limit κ → 0, corresponding to RR flux only, the two branches collapse and become one, with a single spectral parameter u. This was the situation analysed in [21,62].

Loop Algebra and Universal Classical r-matrix 4.1 Loop Algebra
In this section we study the classical limit of the AdS 3 × S 3 × S 3 × S 1 R-matrix and its Yangian by taking the effective string tension h to be very large, namely 7 h → ∞. To this aim, we parametrise the Zhukovsky variables x ± L,R in terms of the constants h, m and of a new variable z [63]: (4.1) Consequently, we expand with respect to h −1 . Such x ± L,R satisfy the constraint (2.5) because κcorrections are of order h −2 . In this regime, the spectral parameter u reads u = z + z −1 and the representations of the Yangian generators become Let G be one of the generators reported above. Then, we can uplift G to its level n counterpart 7 By AdS/CFT, this limit corresponds to the strong coupling limit on the gauge theory side.

Classical r-matrix
Using (4.1), the R-matrix expands as where r is the classical r-matrix of the system satisfying the classical Yang-Baxter equation Therefore, the right-hand side of (4.7), also known as the cobracket of J, has to match the commutator [∆ 0 (J), r]. The classical r-matrix obtained from (2.14) reads The generators Q L , S L , . . . , ß L , ß R are genuine symmetries of the r-matrix: for instance, which agrees with (4.7). In the region |u 2 | < |u 1 |, the r-matrix can be rewritten as 8 where the redefinitions , were used. The object in (4.10) is representation independent; therefore, it is a well-posed candidate for the universal r-matrix of the loop algebra (4.3). In the light of (4.10), (4.8) is in fact in evaluation representation.
Finally, notice that the identifications map the r-matrix (4.10) to the one found in [53] up to central shifts.

Discussion
In this work we have investigated the integrability of type IIB superstrings on AdS 3 × S 3 × S 3 × S 1 background with mixed RR and NSNS flux from an algebraic viewpoint. Our analysis has shown that the worldsheet scattering of the theory enjoys an infinite dimensional symmetry A endowed with a Hopf algebra structure and spanned by an endless tower of generators. Specifically, we have used the R-matrix to write down the RTT relations and expanded these with respect to the spectral parameter, obtaining the abstract commutation relations of A. Furthermore, expanding the R-matrix itself has provided the representations of the A generators. We have observed that the presence of both RR and NSNS fluxes deforms not only the algebra representations but also the abstract commutation relations of the RTT generators.
The infinite dimensional algebra A is organised in levels: the bottom level, A −1 , only contains the identity and a braiding factor U. The latter is a central element deforming commutation relations, coproducts, antipodes and so forth. The next level, A 0 , coincides with the building block 8 The r-matrix for |u1| < |u2| is obtained by making the replacement G in (4.10).
of the light-cone off-shell symmetry algebra of the model, su c (1|1) 2 , whose central charges depend on the braiding factor U and vanish in the limit U → ½. The level zero of A also includes two outer generators of su c (1|1) 2 , B L , B R . These add up to a central generator A that never appears in the commutation relations of A, implying that B L , B R actually represent one single generator. We have summarised this situation by referring to A 0 as gl c (1|1) 2 /A. Subsequently, we have demonstrated that the level one of A, A 1 , contains level one charges of the Yangian Y[su c (1|1) 2 ]. Such charges are in evaluation representation, where each copy of su c (1|1) presents its own spectral parameter depending on the quantised coefficient of the Wess-Zumino term. We have also found that at level one there are two bonus generators, ß L , ß R , mapping each level of A to the next one. In contrast to B L and B R , ß L and ß R are algebraically independent and span the AdS 3 × S 3 × S 3 × S 1 counterpart of the secret symmetry found in AdS 5 and AdS 2 , as well as in another AdS 3 case.
We have checked that the antipode fails to be involutive when evaluated on ß L , ß R , and displayed that the failure is quantitatively related to the non-trivial center of A. By taking into account this secret symmetry, we have concluded that the worldsheet scattering matrix of superstrings on Finally, taking the large effective string tension limit, we have showed that Y[gl c (1|1) 2 /A] reduces to a deformation of the u(1|1)[u, u −1 ] 2 loop algebra. Then, we have formulated a candidate for the universal, representation independent u(1|1)[u, u −1 ] 2 r-matrix, linked to the classical limit of the R-matrix by means of the evaluation representation. We have explicitly verified that such a universal r-matrix, in analogy to those found in [22,64,65], is classically co-commutative with respect to the u(1|1)[u, u −1 ] 2 generators regardless the representation.

Outlook
It would be very interesting to find a universal R-matrix for the worldsheet scattering on AdS 3 ×S 3 × S 3 × S 1 with both RR and NSNS fluxes in terms of a Drinfeld quantum double construction [66,67].
Furthermore, it would be fascinating to see how q-Poincaré symmetry [53,68,69] behaves in presence of mixed flux. Another intriguing direction would be translating the Yangian symmetry derived in this paper into constraints applicable on observables of the conformal field theory dual [38][39][40][41], reverse-engineering what it was done in the context of N = 4 super Yang-Mills theory [2,70,71].
Finally, it would be exciting to extend the analysis of this paper to other supergravity backgrounds; for instance, by extracting the Yangian corresponding to the AdS 2 ×S 2 ×S 2 ×T 4 superstring [72,73].

Acknowledgements
I would very much like to thank Riccardo Borsato, Joakim Strömwall and Alessandro Torrielli for valuable comments on the draft and for sharing a copy of [53] whilst in preparation. I am extremely grateful to Lorenz Eberhardt, Guido Festuccia, Olaf Lechtenfeld and Marius de Leeuw for illuminating discussions. This work was supported by the Riemann Fellowship and by the ERC STG Grant 639220.