O(d, d) transformations preserve classical integrability

In this note, we study the action of O(d, d) transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the O(d, d) -transformed model and ﬁnd that they are in general non-local because they depend on the winding modes. We conclude that every O(d, d ; R ) deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as J ¯ J marginal deformations and TsT transformations of the three-sphere with H -ﬂux.


Introduction
This note presents a synthesis of ideas which separately have been subject to intense study. On the one hand, we have non-linear sigma models and their deformations, such as marginal currentcurrent deformations preserving conformality. On the other hand we have the integrability which allows the use of extremely powerful computational techniques and the study of integrable deformations of sigma models which preserve this property. We observe that O(d, d) transformations give rise to examples in both groups of deformations. Using the O(d, d) invariant doubled formalism, we show on general grounds that a sigma model which is classically integrable remains so under any deformation generated by an O(d, d) transformation.
Two-dimensional non-linear sigma models form the basis of the world-sheet description of strings propagating in target space. They encode the massless excitations of the string, namely the metric G μν , the anti-symmetric Kalb-Ramond field B μν , and the dilaton. The field equations which determine the dynamics of these fields are associated to the beta function of the string sigma model: G, B and will solve the Einstein field equations if the corresponding sigma model is conformal. Understanding the moduli space of such two-dimensional conformal field theories is crucial for our understanding of string theory. It is therefore interesting to study the effect of marginal deformations which preserve conformal invariance. One of the traditional approaches is to deform a given theory using a bilinear of left and right-moving conserved currents [1][2][3][4][5][6][7] associated to the isometry group of the target space. Recently, it has been suggested that this type of construction is related to the gravity duals of the TT and JT deformation [8][9][10][11][12][13][14]. Such current-current deformations can be understood as a rotation within the O(d, d; R) group that generalizes the O(d, d; Z) symmetry of string theory [15][16][17][18][19][20][21][22][23][24][25]. For reviews, see [26][27][28]. Also, T-duality, one of the characteristic features of string theory, is part of this O(d, d) group. Its essence lies in the interplay between the momenta and winding modes of closed strings. At the level of the sigma model, T-duality is a consequence of gauging the commuting isometry group on the target space [29,30].
The duality-invariant formulation of string theory has a long history [17,20,31,32]. The idea is based on the introduction of winding coordinates. A sigma model with a T d fiber is naturally extended to an enlarged sigma model endowed with a T 2d fibration containing also the winding coordinates subject to a consistency constraint [33][34][35][36][37]. The resulting double sigma model action is manifestly invariant under O(d, d; Z) transformations. The doubled formalism is particularly effective to describe so-called non-geometric backgrounds [38,39], where the transition functions between local patches involve a T-duality transformation, generalizing the notion of geometry. 1 The relationship between integrability and T-duality is by now well understood. The existence of a Lax pair, giving rise to an infinite number of (non-local) conserved charges, is a sufficient condition for classical integrability. The construction of non-local charges is given in [41][42][43]. In [44] it was shown that starting from a model with a known Lax pair, it is possible to construct the Lax pair for any T-dual system. Related works are [45][46][47][48]. In general, the corresponding conserved charges are non-local. In the context of integrability [49] of the AdS/CFT correspondence [50], the non-local charges of the (T-dual of) AdS 5 × S 5 strings were studied in [51][52][53][54][55][56]. T-duality also plays a role in the study of Yang-Baxter deformations [57][58][59][60][61][62][63][64], a technique which has led to the construction of integrable deformations of AdS backgrounds as well as Minkowski spacetime. Some of these backgrounds can be understood in terms of TsT transformations [65][66][67][68][69][70][71][72][73][74][75][76][77][78], non-Abelian T-dualities [79][80][81][82][83], or generalized T-dualities [84,85]. The latter gives rise to solutions of the generalized supergravity equations [84,86], in which an extra vector field gives rise to non-geometric Q-fluxes [87] (For a physical interpretation of the extra vector, see also [88][89][90]). Generalized supergravity equations have been also studied in the T-duality invariant framework [91][92][93].
From for each element g ∈ O(d, d; R) we obtain a new (generically inequivalent) sigma model S , which in general has only isometry group T d , and not the original group G. Using Noether's procedure we write down the conserved currents J ∈ g for the initial model. If these currents are conserved and flat, they can be used to introduce a Lax pair L that guarantees the classical integrability of the initial model. The O(d, d) transformation g maps the Lax pair into a new pair L for S . These currents do not in general correspond to isometries of S , and so do not stem from Noether's construction. They are non-local. Nonetheless, they are still a one-parameter family of flat currents which is enough to guarantee the classical integrability of the deformed model S . The O(d, d) transformation maps the momenta of S generically into momenta and winding modes of S . The non-locality of the currents is a consequence of the fact that the Lax pair L is constructed using these windings. Having a system that has more currents than isometries, the extra currents being realized in terms of winding modes, should hardly be surprising. The simplest and best-known example of this effect is the compact boson at the self-dual radius that has an SU (2) × SU (2) symmetry as opposed to the geometrical U(1) × U (1). The extra symmetry is naturally interpreted in terms of winding modes [94].
The plan of this paper is as follows. In Section 2 we introduce the basics of the doubled formalism and the group O(d, d). In Section 3 we remind ourselves of the construction of the Lax pairs and conserved charges in classical integrability and how the Lax pair transforms under O(d, d). In Section 4 we put the general formalism into practice using the simple example of O(1, 1)-transforming the sigma model on the two-sphere. In Section 5 we study the Wess-Zumino-Novikov-Witten (WZNW) model on the three-sphere. Specific O(2, 2) elements give rise to JJ deformations, TsT transformation and double T-duality of the original sigma model. For all these cases, we explicitly construct the transformed Lax pairs, thus making the classical integrability of the deformed models manifest. In Section 6 we conclude and present a number of future directions in which our work could be extended.

Non-linear sigma models and O(d, d) transformations
Let us first review the basics of the doubled formalism and O(d, d) transformations. We consider a geometric string background, where the D-dimensional target manifold is equipped with a metric G and the closed three-form H -flux. H has locally a two-form potential B. We do not consider the dilaton in this paper. Defining a set of local coordinates Xˆı, î = 1, . . . , D, the string sigma model action is given by where the Hodge duality on the world-sheet satisfies 2 = 1. We assume that the manifold of interest has Euclidean signature. We will use the doubled formalism, which is motivated by the search for a background-independent formulation of string theory and is manifestly invariant under O(d, d) transformations, where d denotes the dimension of the maximal torus T d . From the string sigma model, we read off the metric G ij and the B-field B ij in the isometric directions i, j = 1, . . . , d and package them into the so-called generalized metric It is not hard to see that the generalized metric satisfies where the indefinite matrix L is given by The 2d × 2d generalized metric can be considered as the curved metric of the 2d-dimensional doubled space. Our target space manifold has a group of isometries G. Let us focus on a submanifold M of the target space, on which the maximal torus T d ⊂ G, d ≤ D of the full isometry group acts freely. In other words we separate the coordinates Xˆı into those that describe M (we will call them X i ) and those that describe the base (Y ). In the doubled formalism we extend M to a doubled manifold, whose local patches are formed by a patch of M and a patch of the T-dual M. We denote a set of local coordinates on a patch of the doubled manifold by X I , I = 1, . . . , 2d. It consists of the doublet of local coordinates on M and M: In the following we will always choose the polarization such that the first half of the components of the doubled coordinates are the "physical" ones. The action in the doubled formalism is given by [33][34][35][36][37]

The group O(d, d)
We next present the properties of the group O(d, d) in detail. We write g ∈ O(d, d) as where α, β, γ and δ are d × d matrices. Their index structures are given by The matrix g leaves the indefinite metric (2.4) invariant, where the block matrices satisfy The inverse of g is given by The elements of this group are generated by the following elements.
Diffeomorphisms Given any invertible d × d matrix A, an element of the group of diffeomorphisms is parametrized by corresponding to a general coordinate transformations of the metric and B-field. In terms of the generalized metric, (2.14) B-shift The gauge symmetry of the B-field is also naturally encoded in the doubled formalism.
For an exact two-form d , the matrix for a shift of the B-field is given by β-Transformation The conjugate to the B-shift is the so-called β-transformation. It is encoded by the bi-vector β as The frame after β-transformations is known as a non-geometric frame. The bi-vector corresponds to an antisymmetric tensor obtained by the Seiberg-Witten map [17,95]. The supergravity framework based on this frame is called β-supergravity [96].
T -duality Finally, let us look at the matrix for the Abelian T -duality. We denote by E k the d × d-matrix with 1 in the (k, k)-entry and 0 everywhere else. Then the matrix for T-duality along the k-th direction is given by The metric and B-field transform according to the standard Buscher rules [29,30]. A general O(d, d) element can be decomposed as (2.20)

Deformed metric and B-field
Now we want to express the redefined metric and B-field after a transformation in terms of the fields in the original frame. We use an O(d, d) element g of the form Eq. (2.8) to rotate the generalized metric [97].
The bottom right block gives the inverse of the redefined metric G , The redefined metric is thus written as Next, we look at the top right block of the rotated generalized metric. It reads Multiplying by G from the right, we find

Self-duality constraint
In order to get the right number of (physical) (DOF), we need to impose an extra constraint on the doubled variables.
We start from a set of the pull-backs of dX I which contain both physical and winding momenta: We now impose the self-duality constraint where the Hodge dual on the world-sheet satisfies 2 = 1. As long as this constraint holds, the (EOM) of the doubled sigma model are always satisfied [33][34][35][36][37].
In the following we will use the self-duality constraint to relate the physical coordinates of a model and of its dual. The first component of the constraint can be rewritten as where the J i are the Noether currents associated to the freely acting U(1) d isometries along the Killing vectors k i = ∂ X i . According to Eq. (2.27), the differentials of the winding coordinates X i are interpreted as the Hodge duals of Noether currents J i . The fact that d X i is exact implies the conservation of the currents J i : The differential of the physical coordinates in the original frame is given by Their inverse is given by Using expression (2.27), we can deduce a relation between the physical coordinates of the original and of the dual model:

O(d, d) transformed Lax pairs
We are now in the position to study how the property of classical integrability behaves under O(d, d) transformations. To do so, we first remind ourselves of the construction of the Lax pair.

Lax pairs and conserved charges
If a system has a global symmetry G (which we assume to be a connected Lie group), we can use the Noether procedure to construct the corresponding conserved currents. We denote the Killing vectors associated to the isometries by k i and the conserved (d J = 0) Noether currents by J i .
These currents may additionally fulfill a flatness condition or Maurer-Cartan equation, This flatness is the underlying reason for the classical integrability of a model, as from a flat conserved current J we can always construct the so-called Lax pair: where λ ∈ R is the spectral parameter. In order to preserve flatness, we must set The existence of the Lax pair assures the classical integrability of the model, as each of the L λ gives rise to infinitely many non-local conserved charges.
The construction of these charges is by now standard [42]. Consider the Wilson line W (x, t; λ) defined as the path-ordered exponential of the Lax connection between (x 0 , t 0 ) and (x, t), (3.5) Using W we can now define a one-parameter family of conserved charges (monodromy matrix): Since the Lax pair L λ is flat, if it also vanishes at spatial infinity (L λ (±∞, t) = 0), the oneparameter charge Q(t; λ) is conserved for any λ: Expanding around λ = 0, the condition in Eq. (3.7) is equivalent to the conservation of the infinite set of charges The construction of the infinitely many conserved charges is based only on the existence of the flat of L λ . It can be performed for any model with a one-parameter family of Lax pairs L λ , which is then classically integrable regardless of the nature of the currents. This is in particular also true for non-local currents which do not stem from a Noether construction. We will make use of this fact in the following.

Flat connections and O(d, d) transformations
We have seen how the background data and physical coordinates transform under the action of O(d, d). Suppose that the initial model is integrable in the sense of the existence of Lax pairs which satisfies the zero-curvature condition (3.3 where h ∈ G with G a symmetry group of the initial model. The gauged Lax pair has a vanishing curvature on-shell, as In abstract terms, the flatness condition of the original Lax pair L can be understood as a linear combination of the EOM of the initial model, where again X are the coordinates of the torus and Y the coordinates of the base. The new Lax pair L satisfies by construction where we used (2.31). The flatness condition of the transformed Lax pair can finally be written as a linear combination of the EOM of the deformed model and is hence fulfilled on shell This argument shows that for each Lax pair L in the initial model there is a corresponding flat Lax pair L in the model resulting from the O(d, d) O(1, 1) We start with the simplest model in order to explicitly illustrate the concepts introduced in the last section, reproducing the material in [44] from a different point of view.

Set-up
The sigma model action on the two-sphere reads where , are the angle variables parameterizing the sphere. The SO(3) symmetry of the sphere has three Killing vectors: The corresponding Noether currents are given by As explained above, the J i can be used to construct the Lax pairs L λ , see (3.2). They satisfy the flatness condition Taking the path-ordered exponential of the flat currents, we can now compute infinitely many conserved non-local charges.
Transformations and currents Now we apply an O(1, 1) transformation to the sigma model above. We will see that, while in general the O(3) symmetry is broken, we can still find a set of three conserved flat currents that imply the integrability of the deformed model. It is natural to pick the Killing vector ∂ to define the doubled torus and introduce the coordinate and the corresponding generalized metric (4.8) Under the action of g ∈ O(1, 1) they transform as X → X = g −1 X. (4.10) It is convenient to consider the two connected components of O(1, 1; R) separately, which can be parametrized as (4.11) (4.12) The first connected component includes the identity, so we expect it to describe a continuous deformation of the initial sigma model. We will see that G T describes the T-dual model and its deformations.
In this simple situation, the transformations in the connected component G 0 are rescalings of the initial model. In fact, (4.13) and the deformed action (in the usual polarization where the first component is physical) reads which locally still describes a two-sphere. The situation is more interesting for G T . In this case, e 2t sin 2 ( ) , (4.15) and the deformed sigma model reads It is easy to recognize this as a local rescaling of the T-dual model. The important observation is that this model has only one isometry, corresponding to the Killing vector ∂ , so Noether's construction would only lead to one conserved current. On the other hand we know that this system is related to the original S 2 sigma model by an O(1, 1) transformation. As we have seen in Section 3, we can simply follow the action of this transformation on the three conserved currents in Eq. (4.4). To do so, we need to where a and b were given in Eq. (3.4). • Impose the self-duality condition (4.20) which in our case reads Then we find for the dual Lax pair L 1 (d ) =L 1 (d → d ): where we have first imposed the self-duality condition and then used Eq. (4.15) to relate to the redefined coordinate of the deformed model. Repeating the construction for the other currents we find that the sigma model in Eq. (4.16) admits the following three conserved currents: (4.23) These currents are flat on shell as expected. They are precisely the non-local T-dual currents discussed in [44]. As we have seen, the generalization provided by considering the full O(1, 1) group is limited to local rescalings. The strength of our general formalism is that it can be applied to larger O(d, d) groups, which in general describe non-trivial deformations beyond T-duality, as we will see in the next section.

Example 2: S 3 and O(2, 2)
In this section, we study the group of O(2, 2) transformations for the sigma model on the three-sphere with non-zero H -flux.

WZNW model on the three-sphere
Let us start with the WZNW model on the group SU (2): where κ = 1 at the conformal point and the left/right-invariant Maurer-Cartan one-forms j L/R are defined as for g ∈ SU (2). By construction, the currents satisfy the Mauer-Cartan equations, The variation of the action is given by (5.4) which leads to the EOM: The Noether currents associated to the SU (2) × SU (2) global symmetry are given by We see that they are not only conserved but also flat for any value of κ: The same can be verified for j R . From the conserved and flat currents we can construct the Lax pairs: where a, b contain the spectral parameter λ as before, and are given by (5.9) and satisfy a 2 − b 2 − a = 0. Using the Lax pairs, the infinite number of conserved charges can be constructed in the usual way (see Sec. 3). From now on we will set κ = 1 and look at deformations of the conformal model for ease of notation. It is convenient to pick an explicit parametrization for g ∈ SU (2): where the generators T α , α = 1, 2, 3 are defined in terms of the usual Pauli matrices and satisfy where 123 = 1.
The metric and H -flux are read off from which, for example, leads to the equation of motion for η 14) It is also convenient to decompose the Maurer-Cartan forms on the basis of the T α in the following. Using the SU (2) element (5.10), we explicitly write down (5.2) as where we introduced the coordinates In terms of this decomposition, the flatness condition reads The corresponding Killing vectors k α L/R , α = 1, 2, 3 to the above one-forms are computed as As is clear above, k 2 L and k 2 R are commuting Killing vectors in our choice of coordinates. The associated conserved currents are where J 1 and J 2 are U(1) × U(1) Noether currents for the Killing vectors ∂ ζ 1 and ∂ ζ 2 , respectively: In the following we will pick ζ 1 and ζ 2 to be adapted coordinates in the doubled formalism, so that we can study the corresponding action of O(2, 2) .

Doubled formalism
We start by introducing the doubled coordinates and read from the sigma model the generalized metric The two ingredients of our construction are the choice of an appropriate gauge for the flat currents and the O(2, 2) map. The gauge choice depends only on the torus that we have picked and remains the same for all the transformations in O (2,2).
Gauge choice For L L , we pick so that the gauged Lax pairs decompose as On the other hand, for J R , we choose (5.28) The first components L 1 L/R do not have components in the directions dζ ± , and they will not be affected by O(2, 2) transformations.
Finally let us observe explicitly the connections of the gauged Lax pairs. In the following some example of special interest will be discussed. Using the U(1) 2 conserved currents (5.22) , we compute as well as which obviously vanish under the EOM (5.14) and the conservation of J 1 and J 2 .

Marginal deformations
Current-current deformations of WZNW models are described by O(d, d). In the case of a WZNW model on a compact Lie group, all maximal Abelian subgroups are pairwise conjugated by inner automorphisms so the complete deformation space is D = O(r, r)/(O(r) × O(r)), where r is the rank of the group (see e.g. [7]). In our case, there is only one possible deformation of this kind, which we will call JJ -deformation.
To construct the flat currents of the JJ -deformed model, let us apply the inverse map of the duality automorphism (2.12). It leads to the following relations: We can now explicitly give the deformed Lax pair L =L(dζ 1/2 → dζ 1/2 ): whereas where the inner product of the two-component vectors is used in L 2 L/R . Based on our general arguments in Section 3, we were thus able to explicitly give the Lax pair of the JJ -deformed sigma model, which guarantees its classical integrability.
Given the two-torus generated by (ζ 1 , ζ 2 ), the corresponding TsT transformation consists of 1) a T-duality along ζ 1 direction , 2) a shift ζ 2 by α ζ 1 , 3) a T-duality along ζ 1 direction . (5.47) The T-duality along ζ 1 is realized by the matrix 48) and the doubled coordinates transform as Next, we perform the shift transformation, In order to have a consistent transformation in O (2,2), this shift must be supplemented with an opposite shift of the dual coordinates. Then we have The complete TsT transformation is realized as The equations of motion for ζ 1 and ζ 2 are written as the conservation laws associated with the U(1) 2 isometries along k 1 = ∂ ζ 1 and k 2 = ∂ ζ 2 . The corresponding Noether currents are given by which motivates us to define using J ± = J 1 ± J 2 dζ ± = dζ ± ∓ α J ∓ (α) . (5.60) Now that we have identified the transformation of interest, we find the TsT-deformed Lax pairs as follows: whereas We see that the well-known example of the integrable TsT deformation can be also treated in a systematic manner as an O(d, d) transformation.

Double T-duality
As a final explicit example, let us now consider an O(2, 2) transformation not connected to the identity that describes a one-parameter deformation of the T-dual model. The transformation consists of the combination of 1) an O(2) × O(2) rotation g(α) given by Note that for α = 0 this does not reduce to the identity 2) a diffeomorphism g diff (2.13) characterized by 3) and a B-shift g B (2.15) given by d (α) = 0 s i n (α)(sin(α) + cos(α)) − sin(α)(sin(α) + cos(α)) 0 .
(5.65) All together, we obtain Then the generalized metric transforms as where the metric becomes degenerate and the B-field vanishes. As before, using (5.69), the EOM for ζ 1, , ζ 2 can be expressed as where we define the conserved currents of the dual model to be As a result, we find the deformed flat currents L =L(dζ 1/2 → dζ 1/2 ): whereas Also this example is therefore systematically classified as classically integrable.

Conclusions and outlook
In this paper we showed the classical integrability of models which are obtained from integrable sigma models with a non-Abelian isometry group via an O(d, d) transformation. This class includes T-dual models, JJ -deformed models and TsT-deformed models. To show the classical integrability, we have explicitly constructed the transformed Lax pairs using the doubled formalism and shown that they remain flat.
Our conceptual basis is the string-theoretic picture of the equivalence between momenta and winding modes, showing that integrability goes beyond Noether's construction which is based on the study of isometries only. Locality and non-locality are not physical properties but dependent on the duality frame. Our discussion is another example of a case in which the stringy picture is superior to a purely field-theoretic point of view.
Starting from here, it is possible to develop this line of thought in a variety of directions: • As further concrete examples, one could study WZNW models with a compact group different from SU (2). This case will yield a picture which is qualitatively similar to the explicit examples discussed here. Since the rank is in general larger than one, the possible transformations are richer and will lead to more interesting deformations. • When studying WZNW models on non-compact groups, there are several inequivalent maximal tori that can be considered (because they are not related by inner automorphisms), leading in turn to systems with very different physical properties. • Of course WZNW models are just a natural starting point. We can study in the same way any integrable sigma model with a non-Abelian isometry group of the target space. • O(d, d) transformations capture a number of deformations which can be understood as Yang-Baxter deformations [103,104]. It would be interesting to understand the extent to which these two approaches are related. A recent work exploring this direction is [93]. • It would be interesting to uplift our construction of deformed Lax pairs via the O(d, d)-duality map (2.32) to the cases of other T-dualities such as non-Abelian T-duality [105] and Poisson-Lie T-duality [106,107] based on the so-called E-model [62,64]. In particular, the Poisson-Lie T-duality emerges naturally in Double Field Theory [108] as pointed out in [109]. For a related work, see [110]. should be understood as a solution-generating technique. This difference should appear when studying the algebra satisfied by the charges. In the latter case, we would expect a deformation of the algebra to appear [113,[115][116][117][118][119]. • In the study of integrability, the spectral parameter λ plays an important role. It would be interesting to see whether O(d, d) transformations can be understood in terms of a map acting on the spectral parameter [120].
• One interesting open question that remains is the behavior of integrable models under nonperturbative string dualities such as S-duality. Since the dual system is supposed to describe the same physics as the original one, a reasonable conjecture is that integrability should be preserved. The issue with non-perturbative dualities is that at least some of the perturbative local degrees of freedom in one frame become in general non-local and non-perturbative in the other, so it can be very difficult to identify those that are required to realize integrability from the point of view of the two-dimensional theory. This might explain existing examples in the literature of the non-preservation of integrability in S-dual models [121].
These points go beyond the scope of this work, but would be interesting topics for future research.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.