The first $\alpha'$-correction to homogeneous Yang-Baxter deformations using $O(d,d)$

We use the $O(d,d)$-covariant formulation of supergravity familiar from Double Field Theory to find the first $\alpha'$-correction to (unimodular) homogeneous Yang-Baxter (YB) deformations of the bosonic string. A special case of this result gives the $\alpha'$-correction to TsT transformations. In a suitable scheme the correction comes entirely from an induced anomalous double Lorentz transformation, which is needed to make the two vielbeins obtained upon the YB deformation equal. This should hold more generally, in particular for abelian and non-abelian T-duality, as we discuss.


Introduction
Yang-Baxter deformations were originally constructed as deformations of the Principal Chiral Model and (super)coset sigma models with the interesting property that they preserve integrability [1,2,3,4]. The deformations are built using an R-matrix which solves the classical where c = 0 gives the standard CYBE equation and c = 0 corresponds to the modified CYBE. We will consider only the c = 0 case here, for which the deformed models are often called homogeneous YB models. It was shown in [5,6] that homogeneous deformations can be generated using non-abelian T-duality. One simply adds a closed, non-degenerate, B-field defined on a subalgebra of the isometry algebra and dualizes on that subalgebra. 1 This construction means that these deformations can be defined for a general sigma model as long as it admits isometries that can be dualized. In particular the YB deformation of the Green-Schwarz superstring was constructed in [7]. A special case of this deformation is when the isometries are abelian and in that case the deformed model is simply a T-duality -shift -T-duality (TsT) transformation [8], which are usually called β-shifts or β-transformations in the context of O(d, d).
Just as in non-abelian T-duality [9,10], these models may in principle have a Weyl anomaly. When the anomaly is present the target space fields do not solve the standard supergravity equations but a generalization of these [11,12]. Similar to the non-abelian T-duality case this anomaly is absent if one requires the R-matrix to satisfy a unimodularity condition [13]. This is the case we consider here although unimodularity is not a necessary condition to avoid a Weyl anomaly [14,15,16].
The realization of homogeneous YB models using T-duality makes it natural to try to describe these models using the O(d, d)-covariant language of Double Field Theory (DFT), as was done starting with the work of [17]. In fact the YB deformations take the form of a so-called βtransformation [18,14] in O(d, d) language. This language is particularly useful since, as we will show in this paper, unimodular homogeneous YB deformations leave the generalized fluxes -the basic building blocks in the O(d, d)-covariant formalism -invariant (see also [18,19]). With this observation it becomes very simple to prove that the deformed model solves the low-energy field equations, since those have an O(d, d)-covariant formulation in terms of the generalized fluxes. In fact the same is true for the first α ′ -correction to these equations as shown in [20]. Therefore it is also straightforward to argue that YB-deformed bosonic strings 2 are Weyl invariant at least up to two loops. The fact that all higher derivative corrections should respect the O(d, d) structure suggest that this should even be true to all orders in α ′ , although a complete proof that the string effective action can be written only in terms of generalized fluxes is not known to the authors.
Naively this argument may seem to suggest that the YB-deformed backgrounds should not receive any α ′ -corrections beyond those coming from the intrinsic α ′ -dependence of the original (i.e. undeformed) background. But this is at odds with the results of [21], where non-trivial corrections were found working to second order in the expansion in the deformation parameter. When focusing on the class of TsT transformations, it is also at odds with the fact that abelian T-duality is known to receive α ′ -corrections, as was shown in various works starting from [22,23,24,25,26,27,28], which would be expected to lead to corrections to TsT. As we will explain in more detail in the rest of the paper, the resolution is that while in the doubled formalism there is indeed no correction, corrections appear when one wants to go from the doubled formalism to a standard (super)gravity formulation. In order to do that one has to fix the double Lorentz gaugeinvariance in such a way that the two vielbeins that naturally exist in the doubled formulation are set equal. This requires a certain double Lorentz transformation and -given that the fields of the doubled formulation have an anomalous transformation 3 under double Lorentz transformations [33] -this induces an extra α ′ -correction to the deformed background whose form we determine. A special case of our formula gives the α ′ -correction to TsT transformations.
This discussion naturally connects also to the identification of α ′ -corrections to abelian Tduality transformations, as mentioned above. We will discuss also this and comment on the comparison to the results of [26]. Starting from the corrections to T-duality we will be able to provide an independent way to obtain α ′ -corrections to TsT transformations, which does not make use of the double O(d, d) formulation.
The outline of the paper is as follows. First we give a very brief introduction to the concepts needed from the O(d, d)-covariant formulation as used in DFT. In section 3 we describe what Yang-Baxter deformations are in this language and show that they leave the generalized fluxes invariant. The α ′ -correction to these deformations induced by the compensating anomalous Lorentz transformation is described in section 4. Section 5 focuses on abelian T-duality and TsT transformations and we show that the results agree with those obtained using the O(d, d)-2 Note that while here we consider only the bosonic string for definiteness, very similar results hold for the heterotic string. In fact they can both be treated at the same time by introducing parameters that interpolate between the two as in for example [20]. Note that the relevant equations for the target space fields for the bosonic string are the type II supergravity equations with RR fields set to zero. Therefore we will often loosely refer to them as the (super)gravity equations. 3 The fact that manifest O(d, d) symmetry requires the fields to transform non-covariantly was clarified in the works [29,30,31,32]. covariant formulation. We end with some concluding comments.

O(d, d) covariant formulation of supergravity
We will take inspiration from DFT and use the O(d, d) covariant formulation of (super)gravity. In particular we will work with the so-called frame-like formulation of DFT [34,35,36] where the structure group is taken as two copies of the Lorentz group O(1, d − 1) × O(d − 1, 1). More details and references can be found in the reviews [37,38,39]. However, unlike in DFT, we will always assume that the section condition is solved in the standard way ∂ M = (0, ∂ m ) so that we are really just working with a rewriting of supergravity. Here we will actually consider only the NSNS sector as appropriate for the bosonic string.
In the frame-like formulation one writes the generalized metric in terms of generalized (inverse) vielbeins and flat indices with the metric The generalized metric can be parameterized in the form in terms of the usual metric G and B-field. We take the generalized (inverse) vielbein to be Here e (±) are two sets of vielbeins which transform independently as Λ (±) e (±) under the two Lorentz-group factors. To go to the standard supergravity picture one fixes a gauge e (+) = e (−) = e leaving only one copy of the Lorentz-group.
An important object is the so-called generalized Weitzenböck connection, defined in terms of the generalized vielbeins as From this the generalized fluxes are constructed as whered is the generalized dilaton related to the standard one as e −2d = e −2Φ √ −G. The importance of these objects comes from the fact that the generalized fluxes are scalars under generalized diffeomorphisms. This follows from the fact that a generalized diffeomorphism is implemented by the generalized Lie derivative which acts on a vector field as (2.8) The NSNS sector supergravity equations, or bosonic string low-energy effective equations, can be expressed in terms of the generalized fluxes only. To do this we first introduce the projectors Defining the following projections of the generalized fluxes they take the form where P = P + ,P = P − , (F 2 ) AB = F ACD H CE H DF F BEF and F 2 = H AB (F 2 ) AB . The last line defines the generalized Ricci scalar and these equations of motion can be derived from the action S = dX e −2d R . (2.13) Let us emphasize again that for us this is just a convenient rewriting of the usual bosonic string effective action and equations of motion at lowest order in α ′ .

Yang-Baxter deformations in O(d, d) language
We first need to show how to write YB deformations in O(d, d) language at leading order in α ′ , which will be needed later when discussing their α ′ -corrections. Under a YB deformation we The transformation of the dilaton is such that the generalized dilatond is invariant. The transformation of G and B is equivalent to the following transformation of the generalized metric (2.4) where where k m r are Killing vectors of the undeformed background and R rs is a constant anti-symmetric matrix satisfying (1.1) with c = 0 (r, s are Lie algebra indices). Later we will show that if we just impose that R is constant and anti-symmetric, the additional property of satisfying the CYBE (1.1) will have a natural interpretation. The generalized vielbein (2.5) then transforms as Note that the two sets of vielbeins in (2.5) transform differently, namelỹ This means that if we start from an undeformed background in a gauge such that e (+) = e (−) = e, we will need to accompany the YB deformation by a generalized double Lorentz transformation. We will keepẽ (+) invariant and transformẽ (−) by in order to preserve the gaugeẽ (+) =ẽ (−) . At the (super)gravity level this is of no concern since all objects transform covariantly, but when one considers α ′ -corrections this transformation becomes important due to anomalous transformations of the fields, as we will discuss in the next section.
Note that this is a reformulation of YB deformations in the form of an O(d, d) transformation, in fact h has the form of a so-called β-transformation or β-shift. It is not a standard O(d, d) transformation, such as the ones under which the DFT action is invariant, though. This is first of all because Θ mn is (in general) not constant and second, and more importantly, because Θ mn depends on the background itself since it is constructed using Killing vectors. This is therefore not a symmetry but a map of a background to another background, which is in fact a deformation of the first if we take Θ to be multiplied by a small parameter.
It follows from the transformation of the generalized vielbein that the generalized Weitzenböck connection (2.6) transforms as where we used the fact that any expression with two Θ's contracted (with η M N ) vanishes. Now we use the fact that k r generate isometries, i.e. the generalized Lie derivative of E A M andd along k r vanish 5 Using this fact one finds that the change of the generalized flux F ABC is proportional to the YB equation for R in the form Therefore F ABC is invariant under a YB deformation (see also [18,19]). For F A we find and using (3.8) we find Therefore F A is invariant precisely when the R-matrix is unimodular, since ∇ n Θ mn ∝ f t rs R rs , and f t rs R rs = 0 is the unimodularity condition of [13].
We have therefore shown that the generalized fluxes are invariant under unimodular YB deformations. In fact their derivatives are also invariant since for example Since the (NSNS sector) supergravity equations of motion can be cast in terms of the generalized fluxes and their derivatives, this is enough to conclude that they are invariant under unimodular YB deformations. In other words such YB deformations map SUGRA solutions to SUGRA solutions. Moreover, also the first α ′ -correction to the bosonic string equations can be cast in terms of the generalized fluxes and their derivatives, and therefore our argument shows that in fact the YB deformation preserves Weyl invariance at least to two loops. 6 In fact one would expect that all α ′ -corrections to the equations can be expressed in O(d, d) covariant form, which probably means they can be written only in terms of the generalized fluxes and their derivatives. If this is the case then our argument implies that YB deformations of the bosonic string preserve Weyl-invariance to all loops, i.e. they map a consistent bosonic string to another consistent bosonic string to all orders in α ′ .

The α ′ -correction to YB deformations
Our general argument above has shown that YB deformations preserve two-loop Weyl invariance for the bosonic string. In fact they seem to require no additional α ′ -corrections to the background besides those that are induced from the corrections to the original background. Here we want to understand how this fits with the results of [21] where additional α ′ -corrections were found for YB deformations. The resolution is that the additional α ′ -corrections are indeed absent in the O(d, d) covariant approach, but when one goes down to a standard supergravity formulation one has to fix the double Lorentz symmetry by fixing e (+) = e (−) = e. The double Lorentz transformation required to do this induces, via the anomalous transformation of the generalized vielbein at order α ′ , additional α ′ -corrections to the YB deformed model. Let us now see how this works.
It was shown in [33] that at order α ′ the generalized vielbein acquires an anomalous transformation under (double) Lorentz transformations. The transformation of the vielbein is given by 7 D are parameters of an infinitesimal double Lorentz transformation and the second term is the anomalous piece. Note that we have defined the projected derivatives ∂ where ω (±)cd m = ω m cd ± 1 2 H m cd . This leads to the anomalous infinitesimal transformations 8 Of course, after fixing the gauge e (+) = e (−) = e only the transformations with λ (+) = λ (−) = λ remain and the anomalous Lorentz transformations of the fields becomê We see from these expressions that we can define new fields that transform non-anomalously by 9 The explicit non-covariant terms are constructed such as to cancel the anomalous Lorentz transformations. Notice that the above redefinitions also fix the finite form of the anomalous Lorentz transformations ofḠ,B.

Compensating anomalous transformation
When we are dealing with the YB deformation it is crucial to remember the compensating double Lorentz transformation needed to makeẽ (+) =ẽ (−) given by (3.6). Setting λ (+) = 0 and λ (−) =λ in (4.4) we find that this induces an extra transformation of the fields at order α ′ given by 10δḠ We now need the finite form of the transformation since we are doing a finite transformatioñ Λ = eλ given by (3.6). To find it we use the same strategy as above. We redefine G and B by terms involving the spin connection in such a way that the new fields do not have any anomalous transformation. From this one can then read off the finite form of the transformation.
ForḠ mn this is easily done by noting thatḠ mn ncd is invariant under the above transformation and so the finite transformation forḠ is 11 The bar on the fields is to emphasize that these are the fields coming from the doubled formulation and which have an anomalous Lorentz transformation. Below we will define unbarred fields that transform covariantly. 9 We have included an extra shift of G with H 2 mn to go to the scheme of Metsaev and Tseytlin (MT), see [33] and appendix A.
10 In (4.4) we assumed e (+) = e (−) = e and we were doing a double Lorentz transformation from that starting point. Here we can use the same logic, assuming that we start from the gaugeẽ (+) =ẽ (−) =ẽ for a YB deformation and go back to the situation whereẽ (+) =ẽ andẽ (−) =Λ Tẽ as in (3.5). In this way we construct the inverse of the anomalous transformation we want. We remind that ω (±)cd m = ωm cd ± 1 2 Hm cd , so that the (±) on the torsionful spin-connection should not be confused with the (±) on the two vielbeins coming from DFT. Setting λ (+) = 0 means that for the deformed model we takeẽ =ẽ (+) . 11 Recall that we are computing minus the anomalous transformation we are after. The finite transformation of the CS form is This implies that the transformation of B can be taken to be where B WZW is defined by (4.14) Now that we have found the pieces induced by the compensating double Lorentz transformation we are ready to write the α ′ -correction to the YB-transformed metric and B-field.

The correction to Yang-Baxter deformations
Putting everything together the α ′ -correction to the YB-deformed background in the scheme of Hull and Townsend is 12 The correction to the dilaton follows from the fact that the generalized dilaton e −2d = e −2Φ √ −G is invariant. 13 The term δ ′ (G −B) takes into account the scheme-change of the undeformed background 14 Note that in addition to this, one has the α ′ -corrections to the original background, which will need to be included in (3.1) and will therefore induce a term of the same form -where now δ(G − B) is the correction to the original background.
The spin connection for the YB deformed background entering these expressions is computed using the vielbeinẽ =ẽ (+) defined in (3.5) and is given bỹ 12 See appendix A for the field redefinitions connecting all schemes. Here we set the parameter q of Hull and Townsend to zero. 13 The dilaton in the HT scheme is related to this dilaton as Φ (HT) = Φ + 1 48 α ′ H 2 [21]. 14 For the same reason we have also an additional 1 2ω (−) m cdω(+) ncd term in the correction above, for the schemechange after the deformation.
To see that (4.15) and (4.16) reproduces the results found in [21] one sets B = 0 and expands to order Θ 2 obtaining which, up to a diffeomorphism and B-field gauge transformation, is the same as in [21]. Note that one has to use the fact that the isometry of the vielbein implies that It is worth noting that in the case of a single TsT transformation the correction simplifies. Recall that, given two isometric coordinates y 1 , y 2 , a TsT transformation is implemented by the sequence of T-duality y 1 → T (y 1 ) followed by a shift y 2 → y 2 − ηT (y 1 ) and by another T-duality T (y 1 ) → y 1 . It is understood as a special case of YB with Θ = ηk 1 ∧ k 2 , where k i = ∂ y i are Killing vectors. The above correction simplifies in the TsT case since B WZW vanishes. This follows by noting thatΛ = 1 + 2Θ([1 − (B + G)Θ] −1 ) which means that when Θ has rank 2 the Lorentz transformation is only non-trivial in a 2 × 2 block. In this block it is e λ with λ an anti-symmetric 2 × 2 matrix. Since such a matrix only has one independent component, the RHS of (4.14) vanishes.

Manifestly covariant form of the correction
The expression (4.15) for the α ′ -correction is not manifestly covariant but one can show that it is nevertheless covariant. We start by noting that (4.24) whereω ′(±) =ω (±) − ω. With a bit of algebra one finds where we used the YB equation in the last term of the first expression and also the isometry of B in the next to last term, as well as (4.23). A similar calculation gives Using these expressions we find that (4.15) can be written where we defined (4.28) All terms except B cov−WZW are now manifestly covariant by using (4.23). For the latter the identity where R is the curvature 2-form, implies Therefore also the transformation of B is covariant (up to B-field gauge transformations).

T-duality and TsT transformations
Abelian T-duality transformations are another class of O(d, d) transformations and we can follow exactly the reasoning in section 4 to obtain their α ′ -corrections. When we remain in the noncovariant scheme that comes from DFT, the corrections to the dualized metric and B-field will be given again by the formula (a hat on the field is used to denote the T-dualization) We are assuming that we are dualising along the coordinate y and expressions for the corrections in other schemes will be obtained by implementing the relevant field redefinitions, see appendix A.
In [40] α ′ -corrections to the T-duality rules from the DFT formulation were also discussed. There however instead of writing the generic form of the corrections in terms of the finite form of the Lorentz transformation as above, it was noted thatΛ reduces to a constant 15 when choosing a specific gauge for the vielbein 16 Here we are rewriting the fields in terms of fields of a dimensional reduction
SinceΛ is constant the anomalous Lorentz transformation is trivial in this gauge, and also the α ′ -corrections to T-duality will be trivial. 17 In [40] this observation was used to obtain the α ′ -corrections to the T-duality rules in the scheme of Bergshoeff and de Roo (BR) [41,42]. We use this result as a starting point to write below the T-duality rules to 2 loops in a family of different schemes.
Setting α ′ → 0 they reduce to the Buscher rules that in terms of these fields read simply as σ → −σ and V ↔ W . Here D denotes the covariant derivative with respect to the reduced metric g µν , and w µα β is the reduced spin-connection. We have also defined Apart from the order-α ′ parameters γ ± needed to interpolate between the bosonic and the heterotic strings (see appendix A), the T-duality rules depend on coefficients a i , b i , c i (that are also of order α ′ ) so that they are valid for any scheme related to the one of BR by these field redefinitions By turning on these coefficients we can cover all schemes typically considered in the literature, see appendix A for the field redefinitions relating them. 18 As expected, it is possible to tune the coefficients in order to set to zero all corrections to the T-duality transformations. For generic γ ± it is enough to set and T-duality reduces to the Buscher rules even to 2 loops. We will denote the fields in this (gauge-fixed) scheme by G ′ , B ′ , Φ ′ . When specifying to the bosonic string (γ + = α ′ /2, γ − = 0), they are related to the HT scheme by 19 This matches with the field redefinitions that we would write forḠ,B,Φ as expected. The difference is that here we are also imposing the specific gauge (5.3) and for that reason we denote the fields differently.
The rules above can be compared to the ones first derived by Kaloper and Meissner in [26] for the bosonic string (γ + = α ′ /2, γ − = 0). The scheme used is obtained setting the coefficients to (5.14) and the rest of them equal to zero. To match results, one has to take into account the possibility of transforming the reduced fields by doing diffeomorphisms and gauge transformations. Under such symmetries, the T-dual reduced fields transform as: 17) and the remaining fields transform normally under diffemorphisms. We are restricting to transformations which are first order in α ′ , both for diffeomorphims and gauge transformations. The dw and dβ terms come from gauge transformations of the B field with parameter β µ dx µ + w dy, while v appears when including diffeomorphisms of the form y → y +α ′ v. Choosing the following set of parameters we obtain the following set of ruleŝ and both g µν and φ remain invariant. These match with the rules given by Kaloper and Meissner in [26] up to the sign of the α ′ correction of the b field. 20 use it to obtain an expression for α ′ -corrections to TsT transformations that does not necessarily use all the knowledge of DFT. TsT transformations are a special case of YB deformations, and we will show that the result agrees with that in section 4.
In order to do the TsT transformation we assume that there are two U (1) isometries with corresponding coordinates y 1 and y 2 , and to avoid burdening the notation we will continue labelling by x µ the rest of the coordinates. 22 We will do a T-duality y 1 → T (y 1 ) followed by a shift y 2 → y 2 − ηT (y 1 ) and by another T-duality T (y 1 ) → y 1 . TsT transformations are special cases of YB if we take Θ = ηk 1 ∧ y 2 , where k i = ∂ y i are Killing vectors. Each step will be performed in the scheme that is most convenient. Therefore, when doing T-duality we will prefer to move to the Buscher scheme, while when doing the shift we will prefer to go to a covariant scheme. We will show that the α ′ -corrections to TsT transformations can be understood as arising from these shifts coming from the scheme changes. Because these schemechanging shifts arise at intermediate steps, we will have to look at how they are further modified by the remaining steps in the TsT transformation.
Suppose we start from the HT scheme. In order to do the first T-duality on y 1 we find convenient to first go to the Buscher scheme. This is achieved by implementing the redefinitions (5.13) after taking care of choosing the vielbein as in (5.3). This effectively shifts the fields at order We can immediately account for this contribution in the final result: because we will have to do a TsT transformation including this contribution δ 1 (and we only care about the order α ′ ) we are essentially shifting the original metric and B-field as G − B → G − B + δ 1 (G − B) appearing in the map (3.1). After expanding to first order in α ′ we obtain the first contribution to the α ′ correction of the final result While in the Buscher scheme we can easily do the first T-duality on y 1 because we just need to use the Buscher rules. Notice that under Buscher the gauge choice (5.3) is preserved.
To perform the shift it is more convenient to go back to the HT scheme, which is covariant. That means that we will have to use (5.13) again, although now it will be done using the data of the T-dual background δ 2 (Ĝ mn −B mn ) = + 1 where D = 1 − ηW y 2 (2 − ηW y 2 ) + η 2 e 2σ g y 2 y 2 .
(5.28) At this point one wants to go to the Buscher scheme, in order to perform the last T-duality, which will produce a new correction δ 3 (Ĝ mn −B mn ) = − 1 2ω (−) mabω (+)ab n . Now a double hat is used 22 The reader should be careful, since when doing T-duality along y1 the coordinate y2 should be treated on the same footing as x µ when using the T-duality rules (5.5).
to denote that a T-duality and a shift (followed by the compensating Lorentz transformation) have been implemented. The contribution δ 2 (on which we implement the effect of the shift) and δ 3 can be considered together. In fact all expressions from covariant terms cancel out and we are left with In order to account for the effect of the last T-duality on the above expression one uses: the fact that in the first two terms only (mn) = (y i y j ) contribute, that in the summation of a, b only 1, 2 contribute, the fact that under T-dualitŷ 30) and finally that the last term in (5.29) vanishes if m or n are y i , so that it actually remains the same after T-duality. After taking everything into account the result after the T-duality is simply A tilde denotes the quantities of the TsT-transformed background.
After the last T-duality has been performed, we go back from Buscher to the HT scheme using (5.13) obtaining the final contribution to the α ′ corrections which is Collecting together all contributions we obtain the α ′ correction to the TsT deformed background in the HT scheme  (4.15) in the case of TsT if we remember that B WZW can be taken to be zero, and if we use that we for TsT we can write L −1 dL = dLL −1 because here L is essentially a 2 × 2 anti-symmetric matrix and it commutes with itself.
With a similar reasoning we can obtain the α ′ -corrections to the dilaton of the TsT-transformed background. The simplification in this case is that the dilaton is insensitive to the shift, because by assumption it is isometric and the field redefinitions for the dilaton between the schemes of Buscher and HT are covariant. In HT scheme at generic q we get At q = 1/6 on finds which is in agreement with (4.16), since there the result was written when setting q = 0, and one therefore has the extra H 2 -terms.

Concluding comments
We have seen that the α ′ -correction to YB deformations comes from a compensating Lorentz transformation under which the O(d, d) covariant metric and B-field transform anomalously. It is natural to expect that the same should be true also for T-duality. In fact abelian and non-abelian T-dualities are used to construct the YB deformation and they can also be obtained as a limit (sending the deformation parameter to infinity) of YB deformations. In fact we have already argued that for abelian T-duality the correction is given by precisely the same mechanism. It is therefore very natural to expect the first α ′ -correction to non-abelian Tduality 23 (on a unimodular algebra) to be given by the same expression, with the Lorentz transformation required for NATD substituted forΛ in (4.15) and (4.16).
In [16] YB deformations of strings on AdS 3 × S 3 were studied, and their relation to marginal deformations of WZW models was analyzed. The results of the current paper show that marginal deformations of current algebras include (at least to 2 loops and probably to all loops) also cases which do not solve the "strong version" of the marginality condition of Chaudhuri and Schwartz [46], see [16] for more details. These additional possibilities arise when considering algebras that are not compact. Let us also comment that the deformation generated by the unimodular non-abelian R 9 of [16] must be marginal to all loops, since it can be simply understood as a non-commuting sequence of TsT transformations.
We expect that generalizations of our discussion to a construction in the spirit of the Emodel of Klimčik [47,48,49] will lead to an understanding of the form of α ′ -corrections for the η-deformation [2,3], the λ-deformation [50,51], and to Poisson-Lie T-duality [52].
Another important question we hope to return to is if the structure of the correction found here persists beyond first order in α ′ or whether novel corrections are required at order α ′2 .
Meissner (KP) [55,26], Bergshoeff and de Roo (BR) [41,42]. From [33] (A.1) The symbol ≃ is used when the expressions are simplified by means of the 1-loop equations of motion. We relate the parameters γ ± = ∓(a ± b)/4 to a, b used in [33]. The bosonic string is obtained at γ + = α ′ /2, γ − = 0 and the heterotic string at γ ± = ±α ′ /4. In the following we will specify to the case of the bosonic string. To relate HT and MT schemes we use The parameter q appears in [53], and we normally set q = 0 in the rest of the paper as in [21]. Notice that the sign of the correction to the metric differs from what one would read in [53]. We have checked that this is the correct sign in order to have the correct α ′ -corrections for T-duality and YB deformations. From [55] we read that The fields of the non-covariant scheme that follows from the DFT formulation are denoted simply with a barḠ,B,Φ. They are related to the fields in the HT scheme as