$T$-folds from Yang-Baxter deformations

Yang-Baxter (YB) deformations of type IIB string theory have been well studied from the viewpoint of classical integrability. Most of the works, however, are focused upon the local structure of the deformed geometries and the global structure still remains unclear. In this work, we reveal a non-geometric aspect of YB-deformed backgrounds as $T$-fold by explicitly showing the associated $\mathrm{O}(D,D;\mathbb{Z})$ $T$-duality monodromy. In particular, the appearance of an extra vector field in the generalized supergravity equations (GSE) leads to the non-geometric $Q$-flux. In addition, we study a particular solution of GSE that is obtained by a non-Abelian $T$-duality but cannot be expressed as a homogeneous YB deformation, and show that it can also be regarded as a $T$-fold. This result indicates that solutions of GSE should be non-geometric quite in general beyond the YB deformation.


Introduction
A prototypical example of the AdS/CFT correspondence [1] is a conjectured equivalence between a type IIB superstring theory on AdS 5 × S 5 and the four-dimensional N = 4 SU(N) super Yang-Mills theory in the large N limit. Nowadays, it is well recognized that an integrable structure underlies this correspondence (for a comprehensive review, see [2]). In particular, straints of the Green-Schwarz type IIB string theory on an arbitrary background. Therefore, the GSE has now been established on the fairly fundamental ground.
For the homogeneous CYBE case [12], there exists a significant criterion to identify whether a YB-deformed background is a solution of type IIB supergravity or GSE, before deriving the concrete expression of the resulting deformed background, namely, at the level of classical r-matrix. It is called the unimodularity condition [55]. When this condition is satisfied, the deformed background is a solution of type IIB supergravity, but if not, the background is a solution of the GSE. Various solutions of GSE have been obtained from YB deformations with non-unimodular classical r-matrices [52,54]. As it has been shown in [23][24][25][26], classical r-matrices, which characterize homogeneous YB deformations of AdS 5 geometry, are closely related to non-commutative parameters in the dual open-string description and, as pointed out in [33], they are nothing but β-fields [66]. In terms of the β-field, the non-unimodularity is measured as [25,26,33] 1 where G mn is the so-called the open-string metric that will be defined later. The quantity on the left-hand side is basically the trace of a non-geometric Q-flux, and this result indicates that YB deformations with non-unimodular r-matrices lead to non-geometric backgrounds.
In this paper, we will concentrate on YB deformations of Minkowski and AdS 5 × S 5 backgrounds, and find that the deformed backgrounds we consider here belong to a specific class of non-geometric backgrounds, called T -folds [67]. As far as we know, the YB-deformed backgrounds have not been recognized as T -folds so far, hence this is the first work that clearly states the relation between YB-deformed backgrounds and T -folds. Moreover, it is worth noting that our examples have an intriguing feature that the R-R fields are also twisted by the T -duality monodromy, in comparison to the well-known T -folds which include no R-R fields.
This paper is organized as follows. Section 2 provides a brief review of T -folds, including two examples that are well-known in the literature. In Section 3, we consider GSE solutions which can be realized as YB deformations of Minkowski spacetime and AdS 5 × S 5 , and argue that these deformed backgrounds are regarded as T -folds. In addition, we study a solution of GSE that is obtained by a non-Abelian T -duality but not as a YB deformation, and show that this can also be regarded as a T -fold. Section 4 is devoted to conclusions and discussions. In Appendix A, we discuss how to generalize the Penrose limit [68,69] so as to produce various GSE solutions. To be pedagogical, Appendix A.1 is devoted to a review of Penrose limit of Poincaré AdS 5 . In Appendix A.2, we discuss the modified Penrose limit with a rescaling of the deformation parameter. Then, we apply it to YB-deformed background and reproduce the deformed Minkowski backgrounds discussed in Section 3.4.

A brief review of T -folds
In this section, let us explain what is T -fold. A T -fold is supposed to be a generalization of the usual manifold. It locally looks like a Riemannian manifold, but which is glued together not just by diffeomorphisms but also by T -duality. It plays a significant role in studying non-geometric fluxes beyond the effective supergravity description.
As illustrative examples, we revisit two well-known cases in the literature, corresponding to a chain of duality transformations [70,71] and to the codimension-1 5 2 2 -brane solution [72].
It is conjectured that string theories are related by some discrete dualities. One thing that can occur is that, by duality transformations, a flux configuration transforms into a nongeometric flux configuration, which means that it cannot be realized in terms of the usual fields in 10/11-dimensional supergravities. Therefore, dualities suggest that we need to go beyond the usual geometric isometries to fully understand the arena of flux compactifications.
For the case of T -duality, one proposal to address this problem is the so-called doubled formalism. This construction consists of a manifold in which all the local patches are geometric. However, the transition functions that are needed to glue these patches not only include usual diffeomorphisms and gauge transformations, but also T -duality transformations.
T -fold backgrounds are formulated in an enlarged space with a T n ×T n fibration. The tangent space is the doubled torus T n ×T n and is described by a set of coordinates Y M = (y m , y m ) which transforms in the fundamental representation of O(n, n). The physical internal space arises as a particular choice of a subspace of the double torus, T n phys ⊂ T n ×T n . Then T -duality transformations O(n, n; Z) act by changing the physical subspace T n phys to a different subspace of the enlarged T n ×T n . For a geometric background, we have a spacetime which is a geometric bundle, T n phys = T n . 2 Nevertheless non-geometric backgrounds do not fit together to form a conventional manifold. That is to say, despite of they are locally well-defined, their global description is not valid. Instead, they are globally well-defined as T -folds.
This formulation is manifestly invariant under the T -duality group O(n, n; Z). However, to make contact with the conventional formulation, one needs to choose a polarization, i.e., 2 We can also have T n phys =T n , which corresponds to a dual geometric description.
a particular choice of T n phys ⊂ T n ×T n . This means that we have to break the O(n, n; Z) and pick n coordinates out of the 2n coordinates (y m ,ỹ m ). Then, T -duality transformations allow to identify the backgrounds that belong to the same physical configuration or duality orbit and just differ on a choice of polarization 3 .
Due to the O(n, n) symmetry, it is convenient to introduce the generalized metric H M N on the double torus,

A toy example
We start by reviewing a toy model example that involves several duality transformations of a given background. This example has been discussed in [70,71]. To be pedagogical and provide simple exercises, this subsection presents geometric cases like a twisted torus and a torus with H-flux before introducing a T -fold example.

Twisted torus
Let us consider the metric of a twisted torus, Note that this is not a supergravity solution for m = 0, but still is a useful example to reveal a non-geometric global property. As this background has isometries along y and z directions, these directions can be compactified with certain boundary conditions. For example, let us take (x, y, z) ∼ (x, y + 1, z) , (x, y, z) ∼ (x, y, z + 1) .

(2.4)
Apparently, there is no isometry along the x direction, but there actually exists a deformed Killing vector, Thus, this isometry direction can be compactified as (x, y, z) ∼ e k (x, y, z) = (x + 1, y, z + m y) . (2.6) According to this identification, a 1-form e z ≡ dz − m x dy is globally well-defined [70], and the metric (2.3) is also globally well-defined.
When this background is regarded as a 2-torus T 2 y,z fibered over a base S 1 x , the metric of the 2-torus takes the form Then, as one moves around the base S 1 x , the metric is transformed by a GL(2) rotation. That is to say, for x → x + 1, the metric is given by This monodromy twist can be compensated by a coordinate transformation Thus the metric is single-valued up to the above coordinate transformation. Then this background can be understood to be geometric because general coordinate transformations belong to the gauge group of supergravity.

Torus with H-flux
When a T -duality is formally performed on the twisted torus (2.3) along the x direction, we obtain the following background If we consider the generalized metric (2.1) on the doubled torus (y, z,ỹ,z) associated to this background, then we can easily identify the induced monodromy when x → x + 1. In this case, the monodromy matrix is given by Then, the induced monodromy can be compensated by a constant shift in the B-field, This shift transformation, which makes the background single-valued, belongs to the gauge transformations of supergravity. Hence we conclude that the background is geometric.

T -fold
Finally, let us perform another T -duality transformation along the y-direction on the twisted torus (2.3). Then we obtain the following background [70]: (2.14) In this case, neither general coordinate transformations nor B-field gauge transformations are enough to remove the multi-valuedness of the background. This can also be seen by calculating the monodromy matrix. The associated generalized metric is given by Then, we find that, upon the transformation x → x + 1, the induced monodromy is In summary, we conclude that a non-geometric background with a non-trivial O(n, n; Z) monodromy transformation, such as a β-transformation, is a T -fold. The background (2.14) is a simple example.
From a viewpoint of DFT, by choosing a suitable solution of the section condition, the β-transformations can be realized as the gauge symmetries. Indeed, the above O(2, 2; Z) monodromy matrix Ω can be canceled by a generalized coordinate transformation on the double torus coordinates (y, z,ỹ,z), In this sense, the twisted doubled torus is globally well-defined in DFT.
In addition, it is also possible to make the single-valuedness manifest by introducing the dual fields G mn and β mn [66,[74][75][76][77] defined by 18) or equivalently, The dual metric G mn is precisely the same as the open-string metric [78], and the original metric g mn may be called the closed-string metric. In terms of these fields, the generalized metric can be parameterized as (see for example [79]) 20) which is referred to as a non-geometric parameterization of the generalized metric. At the same time, the parameterization of the DFT dilaton is also changed by introducing the dual dilatonφ, In the non-geometric parameterization, the background (2.15) becomes ds 2 dual ≡ G mn dx m dx n = dx 2 + dy 2 + dz 2 , β yz = m x , (2.22) and the O(2, 2; Z) monodromy matrix (2.16) corresponds to a constant shift in the β field; β yz → β yz + m . Namely, up to a constant β-shift, which is a gauge symmetry (2.17) of DFT, the background becomes single-valued.
In this paper, we define a non-geometric Q-flux as [80] Q p mn ≡ ∂ p β mn . (2.23) Then, upon a transformation x → x + 1, the induced monodromy on the β-field is measured by an integral of the Q-flux, This expression plays the central role in our argument.
After this illustrative example we conclude that Q-flux backgrounds are globally welldefined as T -folds. In the next subsection, let us explain a codimension-1 example of the exotic 5 2 2 -brane by using the above Q-flux.

Codimension-1 5 2 2 -brane background
The second example is a supergravity solution studied in [72]. It is obtained by smearing the codimension-2 exotic 5 2 2 -brane solution [81,82], which is related to the NS5-brane solution by two T -duality transformations. It is also referred to as a Q-brane, as it is a source of Q-flux, as we are going to check. The codimension-1 version of this solution is given by .

(2.25)
With the non-geometric parameterization (2.20), this solution is simplified as (2.26) Assuming that the y direction is compactified with y ∼ y +1, the monodromy under y → y +1 is given by a constant β-shift; β zw → β zw + m . By employing the knowledge on T -folds introduced in this section, we will elaborate on a non-geometric aspect of YB-deformed backgrounds as T -folds.

Non-geometric aspects of YB deformations
Let us show that various YB-deformed backgrounds can be regarded as T -folds.
Subsec. 3.1 is devoted to a brief review of the generalized supergravity to fix our convention and notation. In Subsec. 3.2, we explain how the homogeneous Yang-Baxter deformations are interpreted as β-twists and how a YB-deformed background can be derived from a given classical r-matrix. In Subsec. 3.3, the general structure of T -duality monodromy is revealed for the YB-deformed backgrounds studied in this paper. In Subsec. 3.4, various T -folds are obtained as YB-deformations of Minkowski spacetime. In Subsec. 3.5, we study a certain background which is obtained by a non-Abelian T -duality but is not described as a Yang-Baxter deformation. It is shown that this background is a solution of GSE and can also be regarded as a T -fold. In Sec. 3.6, in order to study a more non-trivial class of T -folds with R-R fields, we consider some backgrounds obtained as YB-deformations of AdS 5 × S 5 .

Generalized supergravity
The generalized type IIB supergravity equations of motion were originally derived in [64,65].
Just for later convenience, we will follow the convention utilized in [32] hereafter.
Then the generalized type II supergravity equations of motion are given by where I = I m ∂ m is a Killing vector satisfying Here D m is the covariant derivative associated with the metric g mn , * is the Hodge star operator, and ι I is the interior product with the vector I. In addition, we have introduced the following quantities: Here, 0 ≤ p ≤ 9 takes an even/odd number for type IIA/IIB theory, respectively. The R-R field strengths should satisfy the self-duality relation, Given the R-R field strengths, the R-R potentials can be determined through the relation, Note that when I = 0, the above expressions reduce to those of the usual supergravity.
It is also convenient to define the R-R fields (F, A) and (F,Č) as satisfying Here, for a bi-vector β mn and a p-form α p , we have defined In order to distinguish three definitions of R-R fields, we call (F,Ĉ) B-untwisted R-R fields while (F,Č) β-untwisted R-R fields, Following the same terminology, we call the dual metric and the dual dilaton (G mn ,φ) the β-untwisted fields, important role, as we will discuss later.
Before closing this subsection, it is worth noting the divergence formula observed in [25,26,33]. For the solutions of GSE obtained as YB deformations and a non-Abelian T -duality discussed in this paper, the Killing vector I can always be found from the following formula: whereD is associated with the β-untwisted metric g mn . The general proof of this expression for the general YB deformations based on the mCYBE and the homogeneous CYBE will be reported in the coming paper [83].

YB deformations as β-deformations
YB deformations of type IIB string theory on AdS 5 ×S 5 have been presented in [10,12]. It used to be quite a difficult problem to read off the full expression of YB deformed background, because it is necessary to perform supercoset construction but it is really complicated and the computation becomes messy.
In the pioneering work [42], the supercoset construction was done for the q-deformed AdS 5 ×S 5 . Then the technique was generalized to the homogeneous CYBE case in [52]. After these developments, this technique was refined in [55] based on κ-symmetry. In the recent paper [33] 4 , a much simpler way has been proposed. This is a direct formula between the fields in GSE and classical r-matrices satisfying the homogeneous CYBE and relies on the divergence formula (3.11). In the following, we will give a brief review of this simple formula and explain how to use it by taking a simple example.
As we introduced in Sec. 2, the β-deformations (or the β-transformations) belong to a particular class of O(D, D) transformations under which the β-field is shifted as while the β-untwisted fields remain invariant, which satisfies the homogeneous classical Yang-Baxter equation (CYBE), Here r ij is a constant skew-symmetric matrix and T i 's are the elements of the Lie algebra g associated with the bosonic isometry group G, satisfying commutation relations An important observation made in [33] is that a YB-deformed background associated with the classical r-matrix (3.14) can also be generated by a β-deformation, Here, a real constant η is a deformation parameter andT i are Killing vector fields on the original background satisfying the same commutation relations (3.16). Since β mn = 0 in the undeformed background, we obtain the following expression for the YB-deformed background.
In terms of the usual supergravity fields (g mn , B mn , Φ,F,Ĉ), the YB-deformed background can be expressed as where the β-untwisted fields (G mn = g mn ,φ = Φ,F =F,Č =Ĉ) are the original undeformed background with B 2 = 0. The deformed background solves the (generalized) supergravity equations of motion (3.1). In this way, we can generate YB-deformed backgrounds by using Furthermore, it is interesting to note that the homogeneous CYBE (3.15) can also be expressed as where [ , ] S denotes the Schouten bracket and the tri-vector R is known as the non-geometric R-flux. The Schouten bracket is defined for a p-vector and a q-vector as where the checkǎ i denotes the omission of a i . This fact implies that the non-geometric R-flux vanishes for the homogeneous YB-deformed backgrounds (as far as the undeformed background has vanishing B-field).

Minkowski and AdS 5 × S 5 backgrounds
In the following subsections, we will consider YB deformations of 10D Minkowski spacetime and the AdS 5 × S 5 background. Before presenting various examples, we will introduce the coordinate systems and show the explicit form of the Killing vector fieldsT i in 10D Minkowski spacetime and the AdS 5 × S 5 background.
For a 10D Minkowski spacetime, we take the standard Minkowski metric, where η mn = diag(−1, +1, . . . , +1) . In this coordinate system, the Killing vector fields These vector fields realize the following Poincaré algebra: Here P m and M mn are the translation and Lorentz generators of the Poincaré group ISO (1,9) .
When we consider the AdS 5 × S 5 background as the original background, we choose the following coordinate system: ds 2 S 5 = dr 2 + sin 2 r dξ 2 + cos 2 ξ sin 2 r dφ 2 1 + sin 2 r sin 2 ξ dφ 2 2 + cos 2 r dφ 2 3 . (3.25) The R-R 5-form field strength in the AdS 5 × S 5 background is given bŷ where the volume forms ω AdS 5 and ω S 5 are defined as, respectively, and * is the Hodge star operator associated with the undeformed AdS 5 × S 5 background, It is also convenient to define ω 4 as Note that dω 4 = 4 ω S 5 .
The non-vanishing commutation relations for the isometry group SO(2, 4) of AdS 5 are (3.31) We will use the above Killing vector fields (3.24), (3.31) to obtain an explicit expression of the β-field β (r) = η r ijT i ∧T j associated with a given r-matrix r = 1 2 r ij T i ∧ T j . In the following, we will omit the superscript (r) for the YB-deformed backgrounds.

An example: the Maldacena-Russo background
To demonstrate how to use the formula (3.19), let us consider a YB-deformed AdS 5 × S 5 background associated with a classical r-matrix [23], This r-matrix is Abelian and satisfies the homogeneous CYBE (3.15). The associated YB deformed background is derived in [23,52].
The classical r-matrix (3.32) leads to the associated β-field, Then, the AdS 5 part of a 10 × 10 matrix (G −1 − β) is where we have ordered the coordinates as (z , x 0 , x 1 , x 2 , x 3 ) . By using the inverse of the matrix (3.34) and the formula (3.19), we obtain the NS-NS fields of the YB-deformed background, The next task is to derive the R-R fields of the deformed background. From the undeformed R-R 5-form field strength (3.26) of the AdS 5 × S 5 background, the R-R fields F are given by (3.36) This is nothing but a linear combination of the deformed R-R field strengths with different rank. Hence we can readily read off the following expressions: Furthermore, the B-untwisted R-R fieldsF can be computed aŝ (3.38) Namely, we obtainF The full deformed background, given by (3.35) and (3.39), is a solution of the standard type IIB supergravity. This background is nothing but a gravity dual of non-commutative gauge theory [59,60].
Thus, nowadays, we do not have to perform supercoset construction to obtain the full expression of YB-deformed background. Just by using a simple formula (3.19), given a classical r-matrix, the full background can easily be derived.

T -duality monodromy of YB-deformed background
As we explained in the previous subsection, the YB-deformed background described by (H, d, F) always has the following structure: At this stage, we know only the local property of the YB-deformed background.
In the examples considered in this paper, the bi-vector r mn (or the β-field in the YBdeformed background) always has a linear-coordinate dependence. Suppose that r mn depends on a coordinate y linearly like, r mn = r mn y +r mn (r mn : constant,r mn : independent of y) , (3.42) and the β-untwisted fields are independent of y. Then, from the Abelian property, e r 1 +r 2 = e r 1 e r 2 = e r 2 e r 1 , e −(r 1 +r 2 )∨ = e −r 1 ∨ e −r 2 ∨ = e −r 2 ∨ e −r 1 ∨ , is a gauge symmetry of String Theory and the background can be identified up to the gauge transformation. In this example of T -fold, the monodromy matrices for the generalized metric and R-R fields are Ω a 0 and e −ωa 0 ∨ , respectively, while the dilaton d is single-valued. Note that the R-R potential A has the same monodromy as F .

YB-deformed Minkowski backgrounds
In this subsection, we study YB-deformations of Minkowski spacetime [85,86]. We begin by

Abelian example
Let us consider a simple Abelian r-matrix [85] The corresponding YB-deformed background becomes (3.46) It seems very messy, but after moving to an appropriate polar coordinate system (see Sec. 3.1 of [85]), this background (3.46) is found to be the well-known Melvin background [87][88][89].
In [85], it was reproduced as a Yang-Baxter deformation with the classical r-matrix (3.45).
For later convenience, we will keep the expression in (3.46).
The dual parameterization of this background is given by Hence, under a shift x 2 → x 2 + η −1 , the background receives the β-transformation,

Non-unimodular example
Let us consider a non-unimodular classical r-matrix 5 The corresponding YB-deformed background becomes (3.52) Apparently, this background has a coordinate singularity at x 0 + x 1 = ±1/η. But when the dual parameterization (2.20) is employed, the dual fields are given by and they are regular everywhere. 6 By introducing a Killing vector I with the help of the divergence formula (3.11) as the background (3.52) with this I solves GSE. 5 As far as we know, this example has not been discussed anywhere so far. 6 A similar resolution of singularities in the dual parameterization has been argued in [90,91] in the context of the exceptional field theory.
Since the β-field depends on x 1 linearly, as one moves along the x 1 direction, the background is twisted by the β-transformation. In particular, when the x 1 direction is identified with period 1/η, this background becomes a T -fold with an O(10, 10; Z) monodromy, (3.55) Note that an arbitrary solution of GSE can be regarded as a solution of DFT [32]. Indeed, by introducing the light-cone coordinates and a rescaled deformation parameter as the present YB-deformed background can be regarded as the following solution of DFT: where only (x + , x − ,x + ,x − )-components of H M N are displayed. Note here that the dilaton has an explicit dual-coordinate dependence because we are now considering a non-standard solution of the section condition which makes this background a solution of GSE rather than the usual supergravity.
Before perfoming this YB deformation (i.e.η = 0), there is a Killing vector χ ≡ ∂ + , but the associated isometry is broken for non-zeroη . However, even after deforming the geometry, there exists a generalized Killing vector which goes back to the original Killing vector in the undeformed limit,η → 0 . In order to make the generalized isometry manifest, let us consider a generalized coordinate transformation, By employing Hohm and Zwiebach's finite transformation matrix [92], the generalized Killing vector in the primed coordinates becomes constant, χ = ∂ ′ + . We can also check that the generalized metric in the primed coordinate system is precisely the undeformed background. Namely, at least locally, the YB deformation can be undone by the generalized coordinate transformation 7 . This fact is consistent with the fact that YB deformations can be realized as the generalized diffeomorphism [33].
Non-Riemannian background: Since the above background has a linear coordinate dependence onx − , let us rotate the solution to the canonical section (i.e. a section in which all of the fields are independent of the dual coordinates). By performing a T -duality along the The resulting background is indeed a solution of DFT defined on the canonical section. However, this solution cannot be parameterized in terms of (g mn , B mn ) and is called a non-Riemannian background in the terminology of [93]. This background does not even allow the dual parameterization (2.20) in terms of (G mn , β mn ) 8 .

Non-unimodular example
The next example is the classical r-matrix [86], This classical r-matrix is a higher dimensional generalization of the light-cone κ-Poincaré r-matrix in the four dimensional one.
By using the light-cone coordinates, In the study of YB deformations of AdS 5 , the similar phenomenon has already been observed in [54]. 8 For another example of non-Riemannian backgrounds, see [93]. A classification of non-Riemannian backgrounds in DFT has been made in [94]. In the context of the exceptional field theory, non-Riemannian backgrounds have been found in [91] even before [93]. There, the type IV generalized metrics do not allow both the conventional and dual parameterizations similar to our solution (3.61).
the corresponding YB-deformed background becomes (3. 64) In terms of the dual parameterization, this background becomes This background can also be regarded as the following solution of DFT: When one of the (x 2 , x 3 , x 4 )-coordinates, say x 2 , is compactified with the period x 2 ∼ x 2 + η −1 , the monodromy matrix is given by ∈ O(10, 10; Z) , (3.68) and in this sense the compactified background is a T -fold. In terms of the non-geometric Q-flux, this background has the following components of it: (3.69)

A non-geometric background from non-Abelian T -duality
Before considering YB-deformations of AdS 5 × S 5 , let us consider another example of purely NS-NS background, which was found in [16] via a non-Abelian T -duality.
The background takes the form, , where ds 2 T 6 is the flat metric on a 6-torus. In terms of the dual parameterization, this background takes a Friedmann-Robertson-Walker-type form, (3.71) Note here that this background cannot be represented by a coset or a Lie group itself. This is because the background (3.70) contains a curvature singularity and is not homogeneous.
Hence the background (3.70) cannot be realized as a Yang-Baxter deformation and is not included in the discussion of [27][28][29].
It is easy to see that the associated Q-flux is constant on this background (3.71), As stated in [16], this background is not a solution of the usual supergravity. However, by using the divergence formula I m =D n β mn again and introducing a vector field as 75) we can see that the background (3.70) together with this vector field I satisfies GSE. Thus, this background can also be regarded as a T -fold solution of DFT.
In this paper, we have considered just one example of non-Abelian T -duality, but it would be interesting to study a lot of examples as a new technique to generate GSE solutions. In fact, it is well-known that non-Abelian T -duality is a systematic method to construct T -fold solutions in DFT.

YB-deformed AdS 5 × S 5 backgrounds
We show that various YB deformations of the AdS 5 × S 5 background are T -folds. We consider here examples associated with the following five classical r-matrices: The classical r-matrices other than the first one are non-unimodular. Note here that the S 5 part remains undeformed and only the AdS 5 part is deformed. As shown in App. A, through the (modified) Penrose limit, the second and third examples are reduced to the two examples discussed in the previous subsection.

Non-Abelian unimodular r-matrix
Let us consider a non-Abelian unimodular r-matrix (see R 5 in Tab. 1 of [55]), where, for simplicity, it is written in terms of the light-cone coordinates, 9 The corresponding YB-deformed background is given by In terms of the dual fields, we obtain the following expression: (3.79) It is straightforward to check that the R-R field strengths are given bŷ This background has the following components of Q-flux: Accordingly, for example, when the x 3 direction is compactified with a period x 3 ∼ x 3 + η −1 1 , this background becomes a T -fold with the monodromy, The R-R fields F are also twisted by the same monodromy, Note that the R-R potentials are twisted by the same monodromy as well, though their explicit forms are not written down here.
3.6.2 r = 1 2 P 0 ∧ D Let us next consider a classical r-matrix [50,54], Because [P 0 , D] = 0 , this classical r-matrix does not satisfy the unimodularity condition. By introducing the polar coordinates, the deformed background can be rewritten as [54] 10 This background is not a solution of the usual type IIB supergravity, but that of GSE [64].
By setting η = 0, this background reduces to the original AdS 5 × S 5 .
In the dual parameterization, the dual metric, the β field and the dual dilaton are given by (3.87) The Killing vector I m satisfies the divergence formula, The Q-flux has the following non-vanishing components: Thus, when at least one of the (x 1 , x 2 , x 3 ) directions is compactified, the background can be interpreted as a T -fold. For example, when the x 1 direction is compactified, the monodromy is given by (3.90) From (3.86), the R-R potentials can be found as follows: Providing the B-twist, we obtain (3.92) We can further compute the β-untwisted fields, As expected, the β-untwisted R-R fields are precisely the R-R fields in the undeformed background, and they are single-valued. In terms of the twisted R-R fields, (F, A), the R-R fields have the same monodromy as (3.90), (3.94)

A scaling limit of the Drinfeld-Jimbo r-matrix
Let us consider a classical r-matrix [53,54], which can be obtained as a scaling limit of the classical r-matrix of Drinfeld-Jimbo type [39,40].
By using the polar coordinates (ρ, θ), the YB-deformed background, which satisfies GSE, is given by [53,54] , The R-R potentials can be found as follows: (3.98) Then the corresponding dual fields in the NS-NS sector are given by and the Killing vector I m again satisfies the divergence formula, Providing the B-twist to the R-R field strengths, we obtain Furthermore, the β-untwist leads to the following expressions: These are the same as the undeformed R-R potentials.
Then the non-zero component of Q-flux are given by When the x 2 -direction is compactified as x 2 ∼ x 2 + η −1 , this background becomes a T -fold with the monodromy, (3.103) Let us consider a non-unimodular r-matrix 11 , Here we have introduced the light-cone coordinates and polar coordinates as The YB-deformed background is given by The R-R potentials are also given bŷ (3.107) The dual fields are given by (3.108) and the Q-flux has the following non-vanishing components: In a similar manner as the previous examples, by compactifying one of the x 1 , x 2 , and x 3 directions with a certain period, this background can also be regarded as a T -fold. For example, if we make the identification, x 3 ∼ x 3 + η −1 1 , the associated monodromy becomes 3 .

(3.110)
A solution of Generalized Type IIA Supergravity Equations: In the background (3.97), by performing a T -duality along the x 1 -direction (see [32] for the duality transformation rule), we obtain the following solution of the generalized type IIA equations of motion: Here the R-R potentials are given bŷ This background cannot be regarded as a T -fold, but it is the first example of the solution for the generalized type IIA supergravity equations.
The final example is associated with the r-matrix [54] This r-matrix is called the light-cone κ-Poincaré. Again, by introducing the coordinates, The R-R potentials can be found as follows: The corresponding dual fields are given by and it is easy to check that the divergence formula is satisfied: We can calculate other types of the R-R field fields as The non-geometric Q-flux has the non-vanishing components,  Then, we have elucidated that the simple formula (3.19) proposed in [33] and the divergence formula (3.11) reproduce various YB-deformed backgrounds. This means that the YB deformation with a classical r-matrix r = 1 2 r ij T i ∧ T j satisfying the (homogeneous) CYBE, is equivalent to the β-deformation with the deformation parameter r mn = 2 η r ijT m iT n j . (4.1) We also considered a known background obtained by the non-Abelian T -duality and showed that the extra vector I determined by the divergence formula (3.11) makes the background a solution of GSE.
We have then computed monodromy matrices for various YB-deformed backgrounds and a non-Abelian T -dual background. In order to clarify the general pattern, let us consider a YB deformation associated with a classical r-matrix, where b µ = 0 for YB-deformations of Minkowski spacetime, and a µνρ and b µ should be chosen such that r satisfies the homogeneous CYBE. In this case, the β-field in the YB-deformed background becomes and this provides the constant Q-flux, By compactifying some of x µ directions, the background becomes a T -fold. Importantly, as long as the r-matrix solves the homogeneous CYBE, the deformed background is a solution of DFT. Therefore, the YB deformation is a very systematic procedure to obtain solutions with Q-fluxes in DFT. Although we have considered YB deformations of Minkowski and AdS 5 × S 5 backgrounds, it is applicable to more general cases such as AdS 3 × S 3 × S 3 × S 1 solutions.
On the other hand, let us remember that the GSE exhibits one isometry direction. This may suggest that they are effectively a 9-dimensional theory. In this respect, as it was denoted in [31], it is still an open problem what is the explicit relation, if any, between the GSE and the 9-dimensional gauged supergravities that involve the gauging of the trombone symmetry of type IIB supergravity [95,96] 12 . If this were the case, then an additional question is in order. As the trombone symmetry is considered an accidental symmetry which is broken at higher order α ′ -corrections [95], it would be interesting to seek for the relation between type IIB supergravity and GSE, including α ′ -corrections.
This relation has been further generalized by intriguing works [103,104]. Hence by generalizing our work to include the modified CYBE case, it should be possible to study the PL T-duality in our context. In fact, it is remarkable that the PL T-duality in DFT has been discussed in the recent work [105] from another angle, the global structure of DFT 13 , independently of a series of our works. As a matter of course, these directions meet up at some point.
In summary, we have shed light on a non-geometric aspect of YB deformation. Namely, using the formulas (3.19) and (3.11), we have established a mapping between YB deformations and non-geometric backgrounds involving Q-fluxes. We hope that our result could be the starting point to delve into the relation between integrable deformations and non-geometric backgrounds.
AdS 5 × S 5 backgrounds may disappear under the Penrose limit. In that case, the resulting backgrounds become purely NS-NS solutions of GSE.
Penrose limit [68,69] is formulated for the standard supergravity. But, at least so far, there is no general argument on Penrose limit for the GSE case. Hence, it is quite non-trivial whether it can be extended to GSE or not. Here, we will not discuss a general theory of Penrose limit for GSE, but explain how to take a scaling of the extra vector I. The point here is that a YB-deformed background contains a deformations parameter and I is proportional to it. Hence, there is a freedom to scale the deformation parameter in taking a Penrose limit.
Without scaling the deformation parameter, 5D Minkowski spacetime is obtained as in the undeformed case. On the other hand, by taking an appropriate scaling of the deformation parameter, one can obtain a non-trivial solution of GSE with non-vanishing extra vector fields. We refer to the latter manner as the modified Penrose limit. As a result, this modified Penrose limit may be regarded as a technique to generate solutions of GSE 14 .

A.1 Penrose limit of Poincaré AdS 5
Let us first recall how to take a Penrose limit of the Poincaré metric of AdS 5 .
The first task is to determine a null geodesic. Here we are interested in a radial null Here τ is an affine parameter and the symbol "·" denotes a derivative in terms of τ . From the energy conservation, we obtain that Hereafter, we will set E = 1 by rescaling τ . Then the equation (A.2) can be rewritten aṡ 14 Without any general argument, it is not ensured that the resulting background should satisfy the GSE.
However, this point can be overcome by directly checking the GSE for the resulting background. As far as we have checked, it seems likely that this procedure works well.
Hence, we will take a solution as r = −τ , (A.5) by adjusting an integration constant to be zero. Then t can also be determined as follows: As a result, the radial null geodesic is described as Let us take a Penrose limit by employing the radial null geodesic (A.7). The first step is to introduce a new variablet as a fluctuation around the null geodesic as Then, the metric of Poincaré AdS 5 is rewritten into the pp-wave form: Next, by further transforming the coordinates as Finally, by taking the R → ∞ limit, the metric of 5D Minkowski spacetime is obtained.

A.2 Penrose limits of YB-deformed AdS 5 × S 5
Our aim here is to consider the modified Penrose limit of YB-deformed AdS 5 × S 5 with classical r-matrices satisfying the homogeneous CYBE. In the following, we will focus upon two examples of non-unimodular classical r-matrices.

−→ [solution of section 3.4.2]
The first example is a YB-deformed background associated with r = 1 2 P 0 ∧ D, which was studied in section 3.6.2. To take a Penrose limit of the background (3.86), let us rescale the fields as follows: (A.12) After performing a coordinate transformation for the radial direction, the radial null geodesic is given by (A.14) This expression coincides with the one (A.7) even after performing the deformation.
As in the case of Poincaré AdS 5 , a new variablet is introduced as a fluctuation around the null geodesic (A.14): If the R → ∞ limit is taken naively, one can perform the usual Penrose limit, but it again leads to 5D Minkowski spacetime as in the case of the Poincaré AdS 5 .
It is interesting to add a modification to the usual process. That is to rescale the deformation parameter η as well, We refer to this modification as the modified Penrose limit.
By taking the R → ∞ limit and also the flat limit of the S 5 part, we obtain the YBdeformed Minkowski background (3.52) with the following identifications: {x + , x − , x, y, z, η} ←→ {r, v, ρ sin θ cos φ, ρ sin θ sin φ, z, ξ} . Let us next consider another YB-deformed background studied in Sec. 3.6.3. To consider a Penrose limit of the background (3.97), let us rescale the fields as follows: After performing a coordinate transformation, we obtain a radial null geodesic, which again takes the form, Let us next introduce a new variablet as a fluctuation around the null geodesic (A.21): Then, we perform a further coordinate transformation As in the previous case, the deformation parameter is rescaled as After taking the R → ∞ limit, the resulting background is given by (3.64) with the following replacements: {r, v, p, θ, ξ} → {x + , x − , x 2 + y 2 , arctan(y/x) , η} .
(A. 25) Note that all of the R-R fluxes have vanished again as in the previous example (3.52).