Local β-deformations and Yang-Baxter sigma model

Homogeneous Yang-Baxter (YB) deformation of AdS5 × S5 superstring is revisited. We calculate the YB sigma model action up to quadratic order in fermions and show that homogeneous YB deformations are equivalent to β-deformations of the AdS5 ×S5 background when the classical r-matrices consist of bosonic generators. In order to make our discussion clearer, we discuss YB deformations in terms of the double-vielbein formalism of double field theory. We further provide an O(10, 10)-invariant string action that reproduces the Green-Schwarz type II superstring action up to quadratic order in fermions. When an AdS background contains a non-vanishing H-flux, it is not straightforward to perform homogeneous YB deformations. In order to get any hint for such YB deformations, we study β-deformations of H-fluxed AdS backgrounds and obtain various solutions of (generalized) type II supergravity.


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1 Introduction Yang-Baxter (YB) sigma model was originally introduced by Klimčík [1] as a class of Poisson-Lie symmetric sigma models. It is characterized by a classical r-matrix that satisfies the modified classical YB equation (mCYBE). It was later shown to be integrable by constructing the Lax pair [2]. The original YB sigma model can be applied only to sigma models on group manifolds, but it was later generalized to coset sigma models in [3] and to the case of the homogeneous classical YB equation (CYBE) in [4]. An interesting application of YB deformations is an integrable deformation of type IIB superstring theory on the AdS 5 × S 5 background [5][6][7], that has been studied in the context of the AdS/CFT correspondence. Through various examples [8][9][10][11][12][13], it turned out that, when we employ an Abelian classical r-matrix, the YB-deformed AdS 5 ×S 5 superstring can be described as type IIB superstring on a TsT-transformed 1 AdS 5 × S 5 background [14][15][16][17][18][19][20] (see [21] for a clear explanation and generalizations). Namely, Abelian YB deformation was found to be equivalent to a TsT-transformation. For non-Abelian classical r-matrices,

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the deformations of the AdS 5 × S 5 background have not been understood clearly; some deformed backgrounds were obtained through non-commuting TsT-transformations (see for example [22]) and some were obtained through a combination of diffeomorphisms and T -dualities [23], but it is not clear whether an arbitrary YB deformation can be realized as a combination of Abelian T -dualities and gauge symmetries of the supergravity (it was recently shown in [24][25][26][27][28] that YB deformations can be also reproduced from non-Abelian T -dualities [29][30][31][32][33][34][35][36][37][38]). As shown in a seminal paper [22], at least when an r-matrix satisfies a certain criterion called unimodularity, the deformed AdS 5 × S 5 background are solutions of type IIB supergravity. Moreover, for a non-unimodular r-matrix, the deformed AdS 5 ×S 5 background was shown to satisfy the generalized supergravity equations of motion (GSE) [39,40], and a Killing vector I m appearing in the GSE was determined for a general r-matrix. In a recent paper [41] (which appeared a few days after this manuscript was posted to the arXiv), a more tractable expression for I m has been given for r-matrices consisting only of bosonic generators (see appendix A for more details). On the same day, the second version of [42] appeared on the arXiv, which has derived the GSE from the dual sigma model of [35] in a general NS-NS background, and has determined the Killing vector I m in the non-Abelian T -dualized backgrounds.
Recently, in [43][44][45][46][47], the GSE and YB deformations were studied from a viewpoint of a manifestly T -duality covariant formulation of supergravity, called the double field theory (DFT) [48][49][50][51][52][53] and its extensions. Through various examples of non-Abelian YB deformations, it was noticed that YB deformations are equivalent to (local) β-transformations of the AdS 5 × S 5 background [46]. The local β-transformations may be realized as gauge transformations in DFT, known as the generalized diffeomorphism, and for many examples of non-Abelian r-matrices, YB-deformed backgrounds were reproduced by acting generalized diffeomorphisms to the AdS 5 × S 5 background. However, until now the equivalence between YB deformations and local β-transformations has not yet been proven.
In this paper, we show the equivalence for YB deformations of the AdS 5 × S 5 superstring. To be more precise, we show that, for a classical r-matrix consisting of the bosonic generators and satisfying the homogeneous CYBE, the YB deformed AdS 5 ×S 5 superstring action can be regarded as the Green-Schwarz (GS) type IIB superstring action [54] defined in a β-transformed AdS 5 ×S 5 background. During the proof, we perform a suitable identification of the deformed vielbein and make a redefinition of the fermionic variable. These procedures can be clearly explained by using the double-vielbein formalism of DFT [48,[55][56][57][58][59].
We also find a manifestly O(10, 10)-invariant string action that reproduces the conventional GS superstring action up to quadratic order in fermions. In the previous works, T -duality covariant string theories with the worldsheet supersymmetry were studied in [60][61][62]. The GS-type string actions were also constructed in [63][64][65] but the target space was assumed to be flat and have no the R-R fluxes. Our GS-type string action can apply to arbitrary curved backgrounds with the R-R fields and is a generalization of the previous ones (another T -duality manifest GS superstring action in a general background was proposed in [66,67] although the relation to our action is unclear so far).
We expect that the equivalence between YB deformations and β-deformations will hold in more general backgrounds beyond the AdS 5 × S 5 background. As a non-trivial JHEP06(2018)147 example, we study local β-deformations of the AdS 3 × S 3 × T 4 background that contains a non-vanishing H-flux. In this case, due to the presence of H-flux, it is not straightforward to perform YB deformations. 2 Therefore, we do not show the equivalence in this paper. However, thanks to the homogeneous CYBE for the local β-deformations, all examples of the β-deformed backgrounds are shown to satisfy the equations of motion of DFT, or the (generalized) supergravity.
This paper is organized as follows. In section 2, we review the double-vielbein formalism of DFT and find a simple β-transformation rule for the Ramond-Ramond (R-R) fields. We also find the action of the double sigma model for type II superstring that reproduces the conventional GS superstring action. In section 3, we concisely review YB deformations of AdS 5 ×S 5 superstring and show the equivalence of homogeneous YB deformations and local β-transformations. In section 4, we perform β-transformations of the AdS 3 × S 3 × T 4 background and obtain various solutions. We also discuss a more general class of local O (10,10) transformations that are based on the homogeneous CYBE. Section 5 is devoted to conclusions and discussions. Various technical computations are explained in the appendices.

Local β-deformations in DFT
In this section, we review the basics of the type II DFT and find a simple transformation rule for bosonic fields under local β-deformations. We also find a manifestly O(10, 10)-invariant superstring action that reproduces the conventional GS type II superstring action.

DFT fields and their parameterizations
Bosonic fields in DFT are the generalized metric H M N , the T -duality-invariant dilaton d, and the R-R potentialĈ, which is an O(1, D − 1) × O(D − 1, 1) bispinor. In this subsection, we review their definitions and basic properties by turning off fermions (such as gravitino). Here, we employ the double-vielbein formalism developed in [48,[55][56][57][58][59] (see also [73]), which is quite suitable for discussing YB deformations. Following [55][56][57][58][59], we normalize the double vielbeins as By introducing 2D × 2D matrices, where {A} ≡ {a,ā}, the above orthonormal conditions are summarized as (2.9) The matrix V A M is always invertible and the inverse matrix is given by which indeed satisfies (2.11) As long as we raise or lower the indices M, N with η M N and A, B with η AB (namely, a, b andā,b with η ab and ηāb, respectively), there is no difference between V A M and Thus, in the following, we may not show the inverse or the transpose explicitly.

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When D × D matrices, V m a andV mā , are invertible, we can parameterize the double vielbeins as (2.12) where we introduced matrix notations, e ≡ (e m a ) andē ≡ (ē mā ). From (2.9), we find E mn = −E ⊺ mn , g mn ≡ E (mn) = (e η e ⊺ ) mn = −(ēηē ⊺ ) mn . (2.13) By denoting B mn ≡ E [mn] , the parameterizations of the double vielbeins and the generalized metric become The dual parameterization, that can be prescribed when V m a andV mā are invertible, is 16) and it provides the dual parameterization of the generalized metric (2.17) When both parameterizations are possible, comparing (2.15) and (2.17), we obtain In the following, we raise or lower the indices of {e m a ,ē mā ,ẽ m a ,ẽ mā } as and then we obtain relations like (e − ⊺ ) m a = e m a . We can then omit the inverse or the transpose without any confusions as long as the indices are shown explicitly. By using the two metrics, g mn and G mn , we also introduce two parameterizations of the dilaton d, (2.20)

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Ramond-Ramond fields. In order to study the ten-dimensional type II supergravity, let us consider the case D = 10. Associated with the double local Lorentz group O(1, 9) × O(9, 1), we introduce two sets of gamma matrices, (Γ a ) α β and (Γā)ᾱβ, satisfying (2.21) We also introduce the chirality operators 22) and the charge conjugation matrices C αβ andCᾱβ satisfying 3 We raise or lower the spinor indices by using the charge conjugation matrices like We define the R-R potential as a bispinorĈ αβ with a definite chirality where the sign is for type IIA/IIB supergravity. The R-R field strength is defined aŝ

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where D + is a nilpotent operator introduced in [58], and the covariant derivative D M for a bispinor T αβ and the spin connections are defined as [55,56,58] where ∇ M is the (semi-)covariant derivative in DFT [55,75,76] (see also [43] which employs the same convention as this paper). Since D + flips the chirality,F has the opposite chirality toĈ [58] Γ 11FΓ11 = ∓F . (2.30) As it has been shown in [58],F transforms covariantly under the O(1, 9) × O(9, 1) double Lorentz transformations, and transforms as a scalar under generalized diffeomorphisms. Further, from the nilpotency of D + ,F is invariant under gauge transformations of R-R potential 31) and the Bianchi identity is given by As in the case of the democratic formulation [77,78], the self-duality relation for type IIA/IIB supergravity is imposed by hand at the level of the equations of motion.
Section condition and gauge symmetry. In DFT, fields are defined on the doubled spacetime with the generalized coordinates (x M ) = (x m ,x m ), where x m are the standard "physical" D-dimensional coordinates andx m are the dual coordinates. For the consistency of DFT, we require that arbitrary fields or gauge parameters A(x) and B(x) satisfy the so-called section condition [49,51,52], In general, under this condition, fields and gauge parameters can depend on at most D coordinates out of the 2D coordinates x M . We frequently choose the "canonical solution" where all fields and gauge parameters are independent of the dual coordinates;∂ m ≡ ∂ ∂xm = 0. In this case, DFT reduces to the conventional supergravity. Instead, if all fields depend on (D − 1) coordinates x i and only the dilaton d(x) has an additional linear dependence on a dual coordinatesz, DFT reduces to the generalized supergravity as discussed in [43,45].

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When the section condition is satisfied, the gauge symmetry of DFT is generated by the generalized Lie derivative [49,52] This symmetry is interpreted as diffeomorphisms in the doubled spacetime,

Diagonal gauge fixing
In this subsection, we review the diagonal gauge fixing introduced in [55, 58].

NS-NS fields
In order to constrain the redundantly introduced two vielbeins e m a andē mā , we implement the diagonal gauge fixing e m a =ē mā , (2.36) which is important to reproduce the conventional supergravity. Before the diagonal gauge fixing, the double vielbeins transform as and then obtain the following transformation rule: (2.39) At the same time, the dilaton transforms as 40) and the bispinors of R-R fields,Ĉ andF , are invariant.

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As we can see from (2.39), under a (geometric) subgroup (where r mn = 0), e m a andē mā transform in the same manner. However, if we perform a general O (10,10) transformation with r mn = 0, even if we choose the diagonal gauge in the original duality frame (e m a =ē mā ), after the transformation, e m a → e ′ m a andē mā →ē ′ mā , we obtain In order to maintain the diagonal gauge (2.36), we shall simultaneously perform an O(9, 1) local Lorentz transformation for barred tensors that compensates the deviation ofē mā from e m a . Namely, we modify the O(10, 10) transformation as [58] In addition, we relate the two sets of gamma matrices as

R-R fields
According to the diagonal gauge fixing, there is no distinction between the two spinor indices α andᾱ, and we can convert the bispinors into polyforms: From the identity, where ǫ 0···9 = −ǫ 0···9 = 1, the self-duality relation (2.33) can be expressed aŝ Here, we have defined In addition, if we define the components of the spin connections as and compute their explicit forms under the canonical section∂ m = 0 as Originally, the R-R fields were invariant under global O(10, 10) transformations or generalized diffeomorphisms, but after the diagonal gauge fixing, according to the modified transformation rule (2.43), they transform aŝ where Ω is the spinor representation of the local Lorentz transformation (2.42), (2.55) 4 Here, we have used the following identities for type IIA/IIB theory:

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For later convenience, we here introduce several definitions of R-R fields that can be summarized as follows: The quantities at the lower right, polyforms (Ĉ,F) and bispinors (Ĉ,F ), are already defined, which we call (B, Φ)-untwisted fields. There, the curved indices and flat indices are interchanged by using the usual vielbein e m a like (2.50). The quantities at the upper right, which we call the B-untwisted fields, are defined aŝ (2.57) The curved and flat indices are again related aŝ C m 1 ···mn ≡ e m 1 a 1 · · · e mn anĈ a 1 ···an ,F m 1 ···mn ≡ e m 1 a 1 · · · e mn anF a 1 ···an . (2.58) The B-untwisted fields are rather familiar R-R fields satisfyinĝ which can be shown from (2.53). We also define a polyform A and its field strength F as These are utilized in [77,80,81] to define R-R fields as O(D, D) spinors (see also [45]) By using the dual fields (ẽ m a , β mn ,φ), we can also introduce the dual R-R fields, • β-untwisted fields: polyforms (Č,F ) and bispinors (Č,F ), • (β,φ)-untwisted fields: polyforms (Č,F) and bispinors (Č,F ).

Single T -duality
As a simple application of the formula (2.54), let us explain how the R-R fields transform under a single T -duality along the x z -direction, In this case, the vielbein and the dilaton transform as (2.66) and the Lorentz transformation matrix is This can be simplified as and we can easily see that the R-R field transforms under the T -duality as [80] where we have supposed g zz ≥ 0. From the identity (A.20), we obtain By using the B-untwisted R-R potentials,Ĉ = e −ΦĈ andĈ = e −ΦĈ , (2.70) is expressed aŝ This reproduces the famous transformation rule, where we have decomposed the coordinates as {x m } = {x i , x z }.

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It is also noted that, under the single T -duality after taking the diagonal gauge, an arbitrary O (1,9) spinor Ψ α 1 and an O(9, 1) spinor Ψᾱ 2 transform as When we consider a single T -duality connecting type IIA and type IIB superstring, these transformations are applied to the spacetime fermions Θ 1 and Θ 2 introduced later.

β-transformation of R-R fields
In this subsection, we consider local β-transformations This is equivalent to a direct identification of two parameterizations, and consistent with the relation (2.18). If we introduce the flat components of E mn as we obtain (2.81) In terms of E ab , the relation (2.77) can also be expressed as From (2.82), the relation (2.20) between the two dilatons, Φ andφ, can be expressed as

Relation between untwisted R-R fields
From (2.56), the relation between (B, Φ)-untwisted R-R polyforms and the (β,φ)-untwisted R-R polyforms can be expressed aš (2.84) As we show in appendix D by a brute force calculation, if rephrased in terms of bispinors, these relations have quite simple formŝ where AE is an exponential-like function with the gamma matrices totally antisymmetrized [80] In fact, this Ω 0 is a spinor representation of a local Lorentz transformation, 6 as we can show by employing the formula provided below [80] (see appendix E for a proof). In this sense, the (B, Φ)-untwisted fields and the (β,φ)-untwisted fields are related by a local Lorentz transformation.
Formula. For an arbitrary antisymmetric matrix a ab , the spinor representation of a local Lorentz transformation is given by (2.89) 6 Note thatΓ a = Γ 11 Γ a also satisfies the same relation, A key observation is that by using (2.77), (2.79), and (2.82), Λ a b can be decomposed into a product of two Lorentz transformations, where E ′ is defined by Then, we can check the following relations associated with E ′ab : where g ′ mn and B ′ mn are the β-transformed metric and B-field, respectively. From the invariance of d,ẽ m a , andφ under β-transformations, the dilaton Φ in the β-transformed background becomes Corresponding to the decomposition (2.91), we can also decompose Ω as (2.95) where we have defined This gives the desired local Lorentz transformation, The β-transformed R-R field is then expressed aŝ In terms of the differential form, we can express the same transformation rule aŝ

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In terms of the B-untwisted fieldF , the β-untwisted fieldF , and the (β,φ)-untwisted fielď F, we can express the above formula aŝ Namely, the β-or (β,φ)-untwisted field is invariant under β-transformations, which has been shown in [46] (see also [45]) by treating the R-R fields, A and F , as O(D, D) spinors. Specifically, if the B-field and the dilaton Φ are absent before the β-transformation, we have β ab = 0, E a b = δ b a , and β ′ab = −r mnẽ m aẽ n b . Then, (2.95) becomes In section 3, we see that this Ω plays an important role in YB deformations of AdS 5 × S 5 superstring [see eq. (3.64) where 2 η λ ab plays the same role as r ab here].

T -duality-invariant Green-Schwarz action
In section 3, we study homogeneous YB deformations of the GS type IIB superstring action and show that YB deformations are equivalent to β-deformations of the target space. In order to show the equivalence, it will be useful to manifest the covariance of the GS superstring theory under β-transformations. In this section, we provide a manifestly O(10, 10) T -duality-covariant formulation of the GS type II superstring theory. A manifestly T -duality covariant formulations of string theory, the so-called double sigma model (DSM), has been developed in [60,[82][83][84][85][86][87] for the bosonic string. More recently, the DSM for the GS type II superstring theory was formulated in [64] (see also [60][61][62][63]65] for other approaches to supersymmetric DSMs). The action by Park, in our convention, is given by where γᾱβ is the intrinsic metric on the string worldsheet and and a worldsheet 1-form A M (σ) is defined to satisfy, for arbitrary supergravity fields or gauge parameters T (x). Here, the Dirac conjugates for the spacetime fermions Θ α 1 and Θᾱ 2 are defined respectively as and then we obtainΘ In [64], the target space was assumed to be flat, but here we generalize the action to arbitrary curved backgrounds. In order to consider the superstring action in the presence of fluxes, such as the H-flux and the R-R fluxes, we introduce generalized tensors, (2.109) Then, we add the following term to the DSM action (2.102): By choosing the diagonal gauge, the explicit form of K M N becomes where we defined κ s mn ≡ κ (mn) and κ a mn ≡ κ [mn] and their indices are raised or lowered with the metric g mn . Note that K M N is an O(10, 10) matrix up to quadratic order in Θ I (I = 1, 2).
The modification of the DSM action, S → S + ∆S, is equivalent to the replacement of the generalized metric The non-standard factor −Γ 11 is introduced in the Majorana condition for Θ2 such that the condition becomes the standard Majorana condition after the diagonal gauge fixing;

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The explicit form of M M N is given by Then, we consider an action (2.116) Before we choose the diagonal gauge, spinors Θ I , R-R fields F αβ , and the spin connections Φā bc ,Φ abc are invariant under global O(10, 10) transformations or (finite) generalized diffeomorphisms, while H M N , K M N , Π M , and Σ M transform covariantly and the action is invariant. 8 The global double Lorentz symmetry is manifest but the local one is not manifest because Σ M contains non-covariant quantity dΘ I (or K M N contains the spin connection). The local symmetry becomes manifest only after eliminating the auxiliary fields. On the other hand, if we choose the diagonal gauge fixing, although the global O(10, 10) transformations are manifest, the covariance under generalized diffeomorphisms are lost, because the barred indices are rotated under the compensating local Lorentz transformation. Indeed, the transformation rule of fermionic fields after the diagonal gauge fixing is where Ω is the one given in (2.55), and in general, it is non-constant. Accordingly, dΘ 2 (and thus Σ M also) does not transform covariantly.
It is interesting to note that all information on the curved background is contained in the generalized metric M M N . The usual generalized metric H M N contains only the P -P orP -P components [see (2.6)] while other quantities such as the R-R fluxes are contained in the P -P orP -P components K M N .
In the following, we show that the action (2.116) reproduces the conventional GS superstring action [88] up to quadratic order in fermions Θ I .

Classical equivalence to the type II GS action
In order to reproduce the conventional action, we choose the canonical section∂ m = 0. Then, the condition (2.104) for A M indicates that A M takes the form (A M ) = (0, A m ) and DX M becomes More precisely, as discussed in [64,87], A M does not transform covariantly because dX M does not transform covariantly, and DX M ∧ AM is not invariant under generalized diffeomorphisms. However, the variation is only the total-derivative term and the action is invariant under generalized diffeomorphisms.

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where, for simplicity, we defined P m and treated it as a fundamental variable rather than A m . The action then becomes We can expand the first line as and eliminating the auxiliary fields P m , we obtain By using the explicit expression for Σ M , and neglecting quartic terms in Θ and the topological term, the action becomes (2.123) In order to compare the obtained action with the conventional GS superstring action, let us further expand the Lagrangian as (2.124)

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where we have defined and used the explicit form of the spin connection (2.52), we obtain the type II superstring action where we defined For type IIA superstring, defining Θ ≡ Θ 1 + Θ 2 , we obtain a simple action where we defined On the other hand, for type IIB superstring, using the Pauli matrices σ IJ i (i = 1, 2, 3), we can rewrite the action in a familiar form where we used (A.25) and defined As discussed around (2.74), under a single T -duality along the x z -direction, the fermionic variables transform as Since it flips the chirality of Θ 2 , it maps type IIA and IIB superstring to each other.

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3 YB deformations of AdS 5 × S 5 superstring In this section, we revisit the homogeneous YB deformations of the AdS 5 × S 5 superstring. After a concise review of the supercoset construction of AdS 5 × S 5 superstring, we show that the action of the YB sigma model can be expressed as the GS superstring action in the β-deformed AdS 5 × S 5 background.
3.1 A supercoset construction of AdS 5 × S 5 superstring

Metsaev-Tseytlin action
Before considering YB deformations, we review a supercoset construction of the AdS 5 × S 5 superstring action based on the supercoset .
In order to perform the supercoset construction, we introduce a coset representative g ∈ SU(2, 2|4), 9 and define the left-invariant current A as which satisfies the Maurer-Cartan equation By using the projections , we decompose the left-invariant current as We also define projection operators d ± as Now, we consider the sigma model action [89] where T ≡ R 2 /2πα ′ (R : the radius of AdS 5 and S 5 ) is the dimensionless string tension. In order to relate the supercoset sigma model action to the AdS 5 × S 5 superstring action, let

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us prescribe a concrete parameterization of g and expand the action up the second order in fermions. We first decompose the group element into the bosonic and the fermionic parts, and parameterize the bosonic part g b as (see appendix B for the details) Here, P µ (µ, ν = 0, . . . , 3) and D are the translation and dilatation generators in the conformal algebra so(2, 4), and h i (i = 1, 2, 3) are Cartan generators of the so(6) algebra, given by On the other hand, we parameterize the fermionic part g f as where the supercharges (Q I )αα (I = 1, 2) are labeled by two indices (α ,α = 1, . . . , 4) and θ Iαα (I = 1, 2) are 16-components Majorana-Weyl fermions. Then, we can expand the left-invariant current A as where we have defined and used δ IJθ Iγ a dθ J = 0 and ǫ IJθ I γ ab dθ J = 0. 10 The vielbein e a = e m a dX m takes the form (3.14) 10 We have also used

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and ω ab = ω m ab dX m and R abcd are the corresponding spin connection and the Riemann curvature tensor (see appendix A for our conventions). Using the expansion (3.12), we can straightforwardly obtain Then, the action (3.7) becomes the GS type IIB superstring action (2.132) with the target space given by the familiar AdS 5 × S 5 background: 11 From the supergravity equations of motion (or the Weyl invariance of string theory), the dilaton is determined as Φ = 0.

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where we introduced a notation g T i g −1 ≡ [Ad g ] i j T j . By using our parameterization (3.9), the Killing vectors on the AdS 5 background are given bŷ (3.20) The Lie brackets of these vector fields satisfy the same commutation relations (B.25) as the conformal algebra so(2, 4) (with negative sign, (3.21)

YB deformed AdS 5 × S 5 backgrounds
Let us now consider (homogeneous) YB deformations of the AdS 5 × S 5 superstring action.
A key ingredient that characterizes the YB deformation is an R-operator. It is a linear operator R : g → g, that solves the homogeneous CYBE where X, Y ∈ g. We also define the dressed R-operator R g as which also satisfies the homogeneous CYBE (3.22), as long as R satisfies the homogeneous CYBE. Then, the action of YB-deformed AdS 5 × S 5 superstring is given by where we defined linear operators O ± as and η ∈ R is a deformation parameter. This action reduces to the undeformed AdS 5 × S 5 action (3.7) by taking η = 0.
In this paper, we consider a class of R-operators that can be specified by using a skew-symmetric classical r-matrix. By introducing an r-matrix r ∈ g ⊗ g of the form

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the R-operator is defined as Then, the YB deformations are characterized only by the r-matrix. In terms of the rmatrix, the homogeneous CYBE (3.22) can be expressed as

Preparations
In the following, we rewrite the YB-deformed action in the form of the conventional GS action, and show that the target space is a β-deformed AdS 5 × S 5 background. In order to determine the deformed background, it is sufficient to expand the action up to quadratic order in fermions, This kind of analysis has been performed in [74] for the q-deformation of AdS 5 × S 5 and in [13] for homogeneous YB deformations. However, the obtained deformed actions in the previous works are very complicated and it is not easy to read off the deformed background explicitly. In this paper, we provide a general formula for the deformed background for an arbitrary r-matrix satisfying homogeneous CYBE, though our analysis is limited to the cases where the r-matrices are composed only of the bosonic generators of the superalgebra g. 12 In order to expand the YB sigma model action (3.25), let us introduce some notations. Since we are supposing that the r-matrices are composed of the bosonic generators, the dressed R-operator R g b acts as According to the definition (3.28), the (dressed) R-operator is skew-symmetric and by choosing X and Y as P a or J ab , we obtain the following relations: For later convenience, we introduce the deformed currents as

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By using the results of appendix F, it can be expanded as As it turns out, e a ± and W ab ± , respectively, play the roles of the two vielbeins and the torsionful spin connections ω ± (2.126) in the deformed background. 13

NS-NS sector
Let us first consider the NS-NS part of the YB sigma model action, From (F.4), we can easily see that it takes the form By comparing this with the NS-NS part of the GS action [i.e. the first line of (2.132)], we can regard (3.39) as the NS-NS part of the string sigma model on a deformed background Then, we obtain In the original AdS 5 × S 5 background, the B-field is absent and we have Therefore, the deformation can be summarized as By comparing this with the β-transformation rule (2.76), we can regard the YB deformation as β-deformation with the parameter where ω [±] represent the spin connections associated with the vielbeins e± and H ′ 3 represents the H-flux in the deformed background.

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If we compute the dual field in the deformed background, we obtain The dual metric is invariant under the deformation G mn → G ′ mn = G mn , while the β-field, which is absent in the undeformed background, is shifted as β mn = 0 → β ′mn = −r mn .
In addition, the YB-deformed dilaton Φ ′ that is consistent with the kappa invariance (or supergravity equations of motion) has been proposed in [13,22] as (3.46) In order to compare this with the β-transformation law of the dilation, we consider the two vielbeins e ±m a = e b k ±b a introduced in (3.36). Here, we can rewrite k ±a b as by using r mn of (3.44) and B mn = 0 in the undeformed background. Then, e ±m a becomes Recalling Φ = 0 in the undeformed background, the transformation rule (3.46) can be understood as the β-transformation. Therefore, NS-NS fields are precisely β-deformed under the homogeneous YB deformation. For later convenience, let us rewrite r mn of (3.44) by using the r-matrix instead of λ ab . From the definition, λ ab can be expressed as (3.52) By using the r-matrix r = 1 2 r ij T i ∧ T j , this can be expressed as

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and we obtain (3.54) By using the Killing vectors (3.19) we obtain a very simple expression The β-field after the deformation takes the form and we can calculate the associated non-geometric R-flux By using the Lie bracket for the Killing vector fields upon using the homogeneous CYBE (3.29). This shows the absence of the R-flux in homogeneous YB-deformed backgrounds as noted in [46].

R-R sector
Next, we determine the R-R fields from the quadratic part of the YB sigma model action S (2) , and show that the R-R fields are also β-deformed with the r mn given in (3.55).
As noticed in [13,74], the deformed action naively does not have the canonical form of the GS action (2.132), and we need to choose the diagonal gauge and perform a suitable redefinition of the bosonic fields X m . Since the analysis is considerably complicated, we relegate the details to appendix H, and here we explain only the outline.
The quadratic part of the deformed action S (2) can be decomposed into two parts For a while, we focus only on the first part S c (2) since the second part δS (2) is completely cancelled after some field redefinitions. The explicit expression of S c (2) is given by where D ±ᾱ Θ I ≡ ∂ᾱ+ 1 4 W ±ᾱ ab Γ ab Θ I andF 5 is the undeformed R-R 5-form field strengtĥ Next, we eliminate the barred vielbein e +m a by using which follows from (3.36). By further using the identity [recall the formula (2.89)] (3.65) We then perform a redefinition of the fermionic variables Θ I , which corresponds to the transformation rule of fermions (2.117) under the diagonal gauge fixing. As the result of the redefinition, we obtain where the derivatives D ′ ± are defined as

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As we show in appendix G, the spin connection ω ′ab associated with the deformed vielbein e ′a and the deformed H-flux H ′ abc satisfy and D ′ ± can be expressed as Then, the deformed action (3.67) becomes the conventional GS action at order O(θ 2 ) by identifying the deformed R-R field strengths aŝ The transformation rule (3.71) has originally been given in [22] but our expression of Ω (3.64) may be more useful to recognize the homogeneous YB deformations as β-twists. Indeed, (3.71) is precisely the β-transformation rule of the R-R field strengths (2.98).
Another evidence for the equivalence between YB deformations and local β-deformations, based on the κ-symmetry variations, is given in appendix I. Finally, let us consider the remaining part δS (2) . This is completely canceled by redefining the bosonic fields X m [13,74], as long as the r-matrix satisfies the homogeneous CYBE. Indeed, this redefinition gives a shift S (0) → S (0) + δS (0) , and as explained in appendix H.2, the deviation δS (0) satisfies a quite simple expression (the shift of S (2) is higher order in θ) where CYBE  In the previous section, we have shown that the YB sigma model on the AdS 5 × S 5 background associated with an r-matrix r = 1 2 r ij T i ∧ T j can be regarded as the GS superstring theory defined on a β-deformed AdS 5 × S 5 background with the β-deformation parameter r mn = −2 η r ijT m iT n j . The same conclusion will hold also for other backgrounds in string theory.

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In this section, we study deformations of an AdS background with H-flux. In the presence of H-flux, it is not straightforward to define the YB sigma model, and we shall concentrate only on β-deformations. As an example, we here consider the AdS 3 × S 3 × T 4 solution which contains the non-vanishing H-flux Using the Killing vectorsT i of the AdS 3 × S 3 × T 4 background, we consider local βdeformations with deformation parameters of the form, r mn = −2 η r ijT m iT n j . We consider several r-matrices r ij satisfying the homogeneous CYBE, and show that all of the βdeformed backgrounds satisfy the equations of motion of (generalized) supergravity.
In order to find the Killing vectors explicitly, we introduce a group parameterization for AdS 3 × S 3 (for simplicity, we do not consider the trivial T 4 directions) Here, similar to the AdS 5 × S 5 case (see appendix B), we have introduced the so(2, 2) × so(4) generators (Pǎ , Pâ , Jǎb , Jâb) (ǎ,b = 0, 1, 2;â,b = 3, 4, 5) as the following 8 × 8 supermatrices: where 2 × 2 gamma matrices γǎ and γâ are defined as We have also defined the conformal basis {P µ , M µν , D , K µ } as The generators of su(2) L × su(2) R ≃ so(4) are defined as JHEP06 (2018)147 which satisfy the commutation relations, By computing the Maurer-Cartan 1-form A = g −1 dg, and using the supertrace formula we can reproduce the above metric (4.1). Then, we can find the Killing vectorsT i of this background associated with the generator T i by using the formula (3.19), orT m i = Str g −1 T i gP a e am . The result is summarized aŝ (4.10) We note that among the AdS isometries,P µ ,M 01 , andD are symmetry of the B-field, while the special conformal generatorsK µ change the B-field by closed forms, In the following, we first study β-deformations by using Killing vectorsP µ ,M 01 ,D, and T R 4 . Then, non-trivial cases using the Killing vectorsK µ are studied in section 4.3.

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4.1.1 r = 1 2 P 0 ∧ P 1 Let us first consider an Abelian r-matrix . After the β-transformation, we obtain the background (4.14) As we have mentioned above, we can also obtain the background by a TsT transformation from the background (4.1); (1) T-dualize along the x 1 -direction, (2) (active) shift x 0 → x 0 + η x 1 , (3) T-dualize along the x 1 -direction. This background is of course a solution of supergravity.
As a side remark, noted that this background interpolates a linear dilaton background in the UV region (z ∼ 0) and the undeformed AdS 3 × S 3 × T 4 background in the IR region (z → ∞). Indeed, by performing a coordinate transformation the deformed background becomes In the asymptotic region e −2ρ ≫ η −1 (i.e. z ∼ 0), the background approaches to a solution that is independent of the deformation parameter η where we ignored the constant part of the dilaton and rescaled light-cone coordinates x ± as The AdS 3 part of the background (4.16) is precisely the geometry obtained via a null deformation of SL(2, R) WZW model [90] (see also [91]), which is an exactly marginal deformation of the WZW model (see [92][93][94][95] for recent studies). Note also that, under a formal T -duality along the ρ-direction, the solution (4.17) becomes the following solution in DFT:

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where the dilaton depends linearly on the dual coordinateρ. This background can be also interpreted as the following solution of GSE: As the second example, let us consider an Abelian r-matrix For convenience, let us change the coordinates such that the background (4.1) becomes In this coordinate system, the Killing vectors take the form,P + = ∂ + andT R 4 = ∂ ψ . Then, the associated β-deformed (or TsT-transformed) background is given by This background has been studied in [96], where the twist was interpreted as a spectral flow transformation of the original model in the context of the NS-R formalism.
4.1.3 r = 1 2 D ∧ M 01 Let us also consider a slightly non-trivial example r = 1 2 D ∧ M 01 , which is also an Abelian r-matrix. The associated β-deformed background is given by We can easily check that this is a solution of the supergravity. In order to obtain the same background by performing a TsT transformation, we should first change the coordinates such that the Killing vectorsD andM 01 become constant, and perform a TsT transformation, and then go back to the original coordinates. The β-transformation is much easier in this case.

Non-unimodular deformations
Let us next consider β-deformations associated with non-Abelian r-matrices. In particular, we consider non-unimodular r-matrices, namely non-Abelian r-matrices satisfying In general, as was shown in [22], YB deformations associated with non-unimodular rmatrices give backgrounds that do not satisfy the usual supergravity equations but rather the GSE [39,40,[43][44][45], which include non-dynamical Killing vector I m (see appendix A).
As it was observed experimentally [47,[98][99][100], the extra vector I m typically takes the form (see appendix A for a derivation in the case of the AdS 5 × S 5 superstring) (4.31) Using r mn = −2 η r ijT m iT n j obtained in (3.55) and the Killing property ofT i , we can also express the experimental formula as [46,47,[98][99][100] I m = D n r nm , (4.32)

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where D n is the usual covariant derivative associated with the undeformed AdS 3 × S 3 × T 4 background.
Interestingly, as we explain in section 4.2.2, in some examples, even for non-unimodular r-matrices, the β-deformed backgrounds satisfy the usual supergravity equations of motion. Such example has not been observed in the case of the AdS 5 × S 5 background, 14 and this is due to a particular property of the AdS 3 × S 3 × T 4 background as explained below.
Let us consider the simplest non-unimodular r-matrix r = 1 2c µ D ∧ P µ , satisfying The β-deformed background becomes where c µ ≡ η µν c ν . Although this is not a solution of the usual supergravity, by introducing a Killing vector, it becomes a solution of the GSE.

r
The next example is a non-unimodular r-matrix r = 1 2c µ M 01 ∧ P µ , satisfying The β-deformed background becomes where c µ ≡ η µν c ν . As usual, by introducing this background satisfies the GSE.
14 See a recent paper [41] for a general analysis of such backgrounds, called the "trivial solutions" of GSE.

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Here, note that the defining properties of I m , require that the parameters should satisfy c 0 = ±c 1 . In terms of DFT, the above deformed background can be expressed as This solves the equations of motion of DFT for arbitrary parameters c µ , but they satisfy the strong constraint only when c 0 = ±c 1 . Therefore, we have to choose c 0 = ±c 1 . In fact, this background has a distinctive feature that has not been observed before. According to the classification of [22], the condition for a YB-deformed background to be a standard supergravity background is the unimodularity condition. However, in this example, the background (4.37) satisfies the GSE even if we perform a rescaling I m → λ I m with arbitrary λ ∈ R. In particular, by choosing λ = 0, the background (4.37) without I m satisfies the usual supergravity equations of motion. As we explain below, the reason for the unusual behavior is closely related to the degeneracy of (g ± B) mn .
According to [43], the condition for a solution of the GSE to be a standard supergravity background is given by£ where ∇ M is the (semi-)covariant derivative in DFT and (4.43) In our example with c 0 = ±c 1 , (g ±B) mn I n = 0 is satisfied, and this leads to Y M = ±X M . Then, from the null and generalized Killing properties of X M the condition (4.42) is automatically satisfied, and our GSE solution is also a solution of the standard supergravity. If we regard the background (4.37) as a solution of supergravity, the strong constraint is satisfied for an arbitrary c µ and it is not necessary to require c 0 = ±c 1 .

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The homogeneous CYBE requires a 0b1 −ā 1b0 = 0 , −ā 0b0 + a 1b1 = 0 , (4.46) and we consider a non-trivial solution The non-unimodularity becomes The corresponding β-deformed background is given by (4.49) By introducing a Killing vector, this background becomes a solution of the GSE. In particular, when c = d, this becomes a supergravity background. Similar to the previous example, the Killing vector again satisfies (g±B) mn I n = 0, and even if we rescale the Killing vector as I m → λ I m , this is still a solution of the GSE. As a particular case λ = 0, the background (4.49) becomes a solution of the usual supergravity.
Let us also consider the case, In this case, the β-deformed background becomes and this is a solution of the GSE for arbitrary a µ ≡ ηā µ . In this case, we can freely rescale the Killing vector I m only when a 0 = ±a 1 .

"β-deformations" with generalized isometries
In the previous subsections, we have not considered the special conformal generatorsK µ . As in the case of AdS 5 × S 5 background, if there is no B-field, we can obtain various solutions from β-deformations usingK µ . However, in the AdS 3 × S 3 × T 4 background, we cannot naively useK µ according to £K µ B 2 = 0. Indeed, even for a simple Abelian r-matrix, such as r = 1 2 K 0 ∧ K 1 or r = 1 2 K + ∧ P + , the β-deformed background does not satisfy the supergravity equations of motion. In this subsection, we explain how to utilize the special conformal generators, and obtain several solutions from (generalization of) β-deformations.
In the canonical section∂ m = 0, if there exists a pair (v m ,ṽ m ) satisfying it means that the background admits a generalized Killing vector (V M ) = (v m ,ṽ m ) satisfying as well as the conformal algebra so(2, 2) by means of the C-bracket

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If we could find generatorsT ′ i which satisfy they are on a common D-dimensional section, and we can find a duality frame where the generalized Killing vectors take the form (T ′ M i ) = (T ′ m i , 0). If it is possible, the generalized Killing vectors reduces to the usual Killing vector and we can consider the usual β-deformations in such duality frame. However, it seems unlikely to be the case in the AdS 3 ×S 3 ×T 4 background, and in the following, we employ the above set of generalized Killing vectors.
4.3.1 r = 1 8 K + ∧ P + Let us first consider an Abelian r-matrix r = 1 8 K + ∧ P + associated with the Abelian generalized isometries; [K + ,P + ] C = 0. SinceK + has the dual components, it is not clear how to perform a "β-deformation." We thus change the generalized coordinates such that the dual components disappear.
We here employ the simple coordinate transformation law by Hohm and Zwiebach [101]. Namely, under a generalized coordinate transformation x M → x ′M , the generalized tensors are transformed as (4.60) We can easily check that a generalized coordinate transformatioñ indeed removes the dual components; (K ′ M + ) = (K ′ m + , 0) and (P ′ M + ) = (P ′ m + , 0). In fact, is precisely a B-field gauge transformation, In the transformed background, the B-field is shifted

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we can check the isometries Then, we can perform the usual β-deformation associated with r = 1 8 K + ∧ P + , This is a new solution of the usual supergravity.

General procedure
In general, it is not easy to find a generalized coordinate transformation like (4.61), which removes the dual components of the generalized Killing vectors. However, in fact, it is not necessary to find such a coordinate transformation. As it is clear from the above procedure, for an r-matrix, r = 1 2 r ij T i ∧ T j , associated with the generalized Killing vectors, the previous deformation is simply a transformation When we consider a non-unimodular r-matrix, we suppose that the formula (4.30) will be correct in a duality frame whereT i take the form (T M i ) = (T m i , 0). Then, the deformed background will be a solution of modified DFT (mDFT) [43] with In terms of the GSE, it is a solution with I m =Î m and Z m = ∂ m Φ + I n B nm +Î m .

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4.3.3 r = 1 2 K + ∧ K − For an Abelian r-matrix r = 1 2 K + ∧ K − , we do not find a generalized coordinate system where dual components of bothK M + andK M − vanish. However, from the general procedure (4.68), we can easily obtain the deformed background We can easily see that this is a solution of the usual supergravity.
4.3.4 r = 1 2 M +− ∧ K + or r = 1 2 D ∧ K + Let us next consider a non-unimodular r-matrix r = 1 2 M +− ∧ K + , satisfying In this case, the deformed background satisfies the equations of motion of mDFT with X M = ηK M + . Similar to the example studied in section 4.2.2, we can freely rescale X M as X M → λ X M (λ ∈ R), and in a particular case λ = 0, (4.72) can be regarded as a solution of the usual supergravity.
Interestingly, we can obtain the same background also by considering an r-matrix r = 1 2 D∧K + , satisfying I = η [D, K + ] = −η K + . This also may be related to the degeneracy of (g ± B) mn in the AdS 3 × S 3 × T 4 background. It is interesting to find a set of generalized Killing vectorsT M i and an r-matrix (satisfying CYBE) that satisfy the above condition with c ij = 0, but here we simply require c ij = 0. Note that the same the null condition c ij = 0 is known in the context of the non-Abelian T -duality. As it has been studied in [102,103], the null condition played an important role in gauging non-Abelian isometries.

Short summary
Let us summarize this subsection. Usually, we prepare a bi-vectorr = 1 2 r ijT i ∧T j satisfying the homogeneous CYBE (or the Poisson condition)

Conclusions
In this paper, we have shown that, after suitable field redefinitions, a homogeneous YBdeformed AdS 5 × S 5 superstring action associated with bosonic isometries can be always express as the usual GS superstring action up to quadratic order in fermions. The deformations were made only in the supergravity backgrounds and they were identified as local β-deformations. We have also found a DSM action that reproduces the GS type II superstring action up to quadratic order in fermions. After taking the diagonal gauge, the spacetime fermion is transformed as Θ 2 → Ω Θ 2 under β-transformations. We found an explicit form of Ω in terms of the β-transformation parameter and the NS-NS fields. Moreover, β-deformations of H-fluxed AdS backgrounds were also studied and various solutions are obtained. In several examples of non-unimodular deformations, we unexpectedly obtained solutions of the usual supergravity.
In this paper, we have mainly focused on the YB deformations of the AdS 5 × S 5 superstring, but the equivalence between homogeneous YB deformations and local βdeformations will be shown also for other backgrounds. 15 Indeed, if we observe the NS-NS part of the deformed action discussed in section 3.2.2, it is clear that we have not used specific properties of the psu(2, 2|4) algebra. At least when the B-field is absent, and the algebra g admits a projection P to the bosonic coset generators P a and κ ab ≡ Str(P a P b ) is non-degenerate, the bosonic part of a coset YB sigma model associated with a skewsymmetric R-operator becomes where P (Aᾱ) ≡ eᾱ a P a , R g (P a ) ≡ λ a b P b , and k +ab ≡ [(κ −1 + η κ −1 λ) −1 ] ab . According to the skew-symmetry λ ab = −λ ba [λ ab ≡ (κ −1 λ) ab ] , the YB deformation can be regarded as a (local) β-deformation. The YB deformations of R-R fields are rather non-trivial, but we expect that the equivalence between YB deformations and local βdeformation will be shown in more general cases, as long as the r-matrix consists of bosonic generators. In a general analysis performed in [22], the YB sigma model action associated with a general r-matrix satisfying the homogeneous (or modified) CYBE has been expressed in the standard form of the GS superstring to all order in fermions. When the r-matrix contains fermionic generators, we expect the deformation of the target space can be no longer regarded as a local β-deformation. It will be an interesting future direction to clarify what kind of deformations are made in such cases. As we showed in section 4, all of the β-deformed AdS 3 × S 3 × T 4 backgrounds are solutions of (generalized) supergravity and string theory is well-defined in such backgrounds. In this sense, the local β-deformations are certain duality transformations in string theory. It will be interesting future work to check the integrability of string sigma model on the 15 See a recent paper [104] for a generalization of the "YB deformations" (or the open-closed string map) beyond the coset spaces.

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β-deformed AdS 3 × S 3 × T 4 backgrounds obtained in this paper. It is also important to formulate the homogeneous YB deformations of type II superstring in H-fluxed background and show the equivalence to β-transformations.
In this paper, we obtained the O(10, 10)-invariant DSM action for type II superstring. Although we considered only up to the quadratic order in fermionic variables, it is important to obtain the complete action. In fact, a T -duality manifest GS superstring action has been proposed also in [66,67]. In our approach, the R-R field strengths are contained in the P -P orP -P components of the generalized metric M M N , but also in the approach of [66,67], they will appear in the "left-right" mixing terms. It is interesting future work to make the connection between the double-vielbein formalism and the approach of [66,67] clearer, and obtain a T -duality manifest GS superstring action in arbitrary curved backgrounds. In the conventional GS superstring, from the requirement of the kappa invariance, the (generalized) supergravity equations of motion has been obtained [40]. By generalizing this analysis, it will be important to derive the type II DFT equations of motion from the T -duality manifest GS superstring action.
It is also important to formulate the DSM that manifests the symmetry of non-Abelian T -dualities. When the target space is a group manifold, such DSM has been formulated in [105,106] and its symmetry is clearly discussed in [107] (see also [28,108] for relevant recent works). Its generalization to coset space will be important for a clearer understanding of YB deformations. In the usual DSM, the generalized vector DX M has the form As discussed in [109], in the case of β-transformations (s m n = δ m n and r mn = −r nm ), by requiring the set of generalized Killing vectors V (m) to form a closed algebra by means of the C-bracket, the homogeneous CYBE for the bi-vector [r,r] S = 0 (r ≡ 1 2 r mn ∂ m ∧ ∂ n ) is required. In the case of an Abelian r-matrix, we can find a coordinate system where all of the relevant Killing vectorsT i are constant vector, and the bi-vector r mn = −2 η r ijT m iT n j automatically satisfies the requirement (5.5) and we can perform the β-deformation without breaking the integrability. This is the usual constant β-shift in the presence of Abelian isometries, that can also realized as a TsT-transformation. In the case of non-Abelian r-matrices, since we cannot find a coordinate system where all of the Killing vectorsT i are constant vectors, the requirement (5.5) is too restroctive. It will be interesting future work to relax the requirement (5.4) by reformulating the DSM that manifests the symmetry of non-Abelian isometries.

A Conventions and formulas
Differential form and curvature. The antisymmeterization is defined as For conventions of differential forms, we use on string worldsheet while in the spacetime, we define The spin connection is defined as where e a ≡ e m a dx m and ω a b ≡ ω m a b dx m . The Riemann curvature tensor is defined as (Generalized) supergravity. Our conventions for the type II GSE [39,40,[43][44][45] are as follows: where D m is the usual covariant derivative associated with g mn and we have defined The Killing vector I = I m ∂ m is defined to satisfy When I = 0, the GSE reduce to the usual supergravity equations of motion.
When we describe the GSE as a special case of the modified DFT [43], the Killing vector I m and U m are packaged into a null generalized Killing vector In a particular gauge U m = I n B nm (see [45] for the details), the dual components of X M vanish. We also define a generalized null vector As shown in [22], a YB deformed AdS 5 × S 5 background associated with a nonunimodular r-matrix is a solution of the GSE. From various examples, we experimentally know that the Killing vector I m in the deformed background takes the form (4.31) Here, we derive the experimental formula for YB deformed AdS 5 × S 5 backgrounds. In the case of the AdS 5 × S 5 backgrounds, a general formula for I was obtained in [22], and by neglecting contributions from fermionic generators, a simple expression

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was given in a quite recent paper [41]. This expression becomes where we have used Then, the curved components become Formulas for gamma matrices and spinors. Products of antisymmetrized 32 × 32 gamma matrices satisfy where the under-barred indices are totally antisymmetrized and the integer r takes values r = |p − q| , |p − q| + 2 , . . . , p + q − 2 , p + q , (A. 19) and u, v and w are non-negative integers. As particular cases, we obtain (A.21) 16 Here, indices i, j, k are restricted to a subalgebra of so(2, 4) × so (6) such that the r-matrix is invertible (see [22] for more details). Accordingly, the indices in (A.17) are also limited to the subalgebra. 17 As shown in [24][25][26][27][28], YB deformations can be also realized as non-Abelian T -dualities. In this context, a relation between the trace of the structure constant f ki k and the Killing vector I m was noted in [24] (see also [26]). This relation was further clarified in [27] and a simple expression I i = f ki k was obtained in [42].

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The conformal basis, {P µ , M µν , D, K µ }, of a bosonic subalgebra su(2, 2) ∼ = so(2, 4) that corresponds to the AdS isometries, can be constructed from Pǎ and Jǎb as where P µ , M µν , D, and K µ represent the translation generators, the Lorentz generators, the dilatation generator, and the special conformal generators, respectively. On the other hand, a bosonic subalgebra su(4) ∼ = so(6) that corresponds to the isometries of S 5 are generated by Pâ and Jâb. We choose the Cartan generators of su(4) as follows (B.14) For later convenience, let us also define 16 × 16 matrices γ a ,γ a , and γ ab as We can easily see γǎb = γ [ǎ γb ] and γâb = γ [â γb ] . If we also defineγǎb ≡γ [ǎγb ] and γâb ≡γ [âγb ] , they satisfyγ where R ab cd are the tangent components of Riemann tensor in AdS 5 × S 5 , whose nonvanishing components are . (B.18) Fermionic generators. The fermionic generators (Q I )αα (α,α = 1, . . . , 4) are given by As discussed in [74], these matrices do not satisfy the reality condition (B.1) but rather their redefinitions Q I do. The choice, Q I or Q I , is a matter of convention, and we here employ Q I by following [74]. We also introduce Grassmann-odd coordinates θ I ≡ (θαα) I

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Since the matrices Q I satisfy the condition (B.20) is equivalent to the Majorana condition or more explicitly,θαα Commutation relations. The generators of su(2, 2|4) algebra, P a , J ab , Q I , and Z satisfy the following commutation relations: and the psu(2, 2|4) algebra is obtained by dropping the last term proportional to Z.
is spanned by the following generators: Then, from the definition of d ± (3.5), we obtain

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B.2 Connection to ten-dimensional quantities By using the 16 × 16 matrices γ a defined in (B.15), the 32 × 32 gamma matrices (Γ a ) α β are realized as We can also realize the charge conjugation matrix as The 32-component Majorana-Weyl fermions Θ I expressed as which satisfies the chiral conditions The Majorana condition is given bȳ This decomposition leads to the following relations between 32-and 8-component fermions: The second relation plays an important role for a supercoset construction of the AdS 5 × S 5 background since the R-R bispinor in the AdS 5 × S 5 background takes the form Indeed, we obtainθ We can also show the following relations: 18

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C Geometry of reductive homogeneous space In this appendix, we review geometry of reductive homogeneous spaces (see for example [111,112] for more details).

C.1 Generalities
Let us consider a homogeneous space G/H and decompose the Lie algebra as a direct sum of vector spaces, , we need to simultaneously perform a local right multiplication, Then, if we expand the left-invariant Maurer-Cartan 1-form as we obtain the following transformation laws under the left multiplication (C.1): where we have defined h P a h −1 = Λ b a P b and h J i h −1 ≡ Λ j i P j . This shows that Ω i behaves as a connection of H. From the decomposition (C.2), the Maurer-Cartan equations become If we regard e a as the vielbein on G/H and suppose the absence of torsion the Maurer-Cartan equations show that the spin connection can be expressed as Moreover, the associated Riemann curvature tensor is expressed as (C.7) -54 -

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In order to obtain the Killing vectors on G/H, let us consider an infinitesimal left multiplication under which the coordinates are supposed to transform as We obtain and this leads to We thus obtain the following expression: Under the same variation, we obtain If we define the metric on G/H as We can check that the variation is the same as the Lie derivative, δ ǫ e m a = ǫ i £T i e m a = ǫ i T n i ∂ n e m a + ∂ mT n i e n a , (C. 18) and the invariance of the metric indicates thatT m i are Killing vectors associated with the generator T i . From On the other hand, from [T i , T j ] = f ij k T k , we can also express the left-hand side as and by comparing these, we obtain In the case of the (bosonic) coset the two sets of generators are given by and it is a symmetric coset space (f ab c = 0). The normalization 1 √ 2! is introduced to prevent overcounting coming from the summation of antisymmetrized indices. Quantities with the index i always contains the factor 1 √ 2! and, for example, the Maurer-Cartan 1-form (C.2) is expressed as From (C.6) and f ab c = 0, the spin connection becomes where we used D Equivalence of (2.84) and (2.85) In this appendix, we prove that the relation (2.84), is equivalent to the relation (2.85), Since e Φ−φ = (det E a b ) − 1 2 from (2.83), we here show the equivalence of two relations, whereF ≡ e −ΦF andF ≡ e −φF .

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D.1 Evaluation ofF = e −B 2 ∧ e −β∨F Let us first evaluate e −B 2 ∧ e −β∨F . This can be expanded as where the square bracket [n] denotes the integral part of n, and in the second equality, we have used relations (2.63) and (2.81). Then,F with flat indices becomeŝ Next, by using the definitions, let us expand the right-hand side ofF =FΩ −1 0 aŝ

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where we used the formula (A.18) and defined r ≡ 2ℓ−k+s 2 . Then, the R-R field strengtĥ F a 1 ···a k with flat indices becomeŝ where the under-barred indices are totally antisymmetrized and non-negative integers ℓ and r run over the region where the following relations are satisfied: We can further expand the right-hand side of (D.8) as 19 where s and t are defined as and non-negative integers ℓ, r, and u run over the region where are satisfied. If we change the variables, we obtain a more explicit expression, This precisely matches with (D.5) and the equivalence has been proven. 19 We used the identity for arbitrary totally antisymmetric tensors C ab , Aa 1 ···ar , and Ba r+1 ···a 2ℓ , where 0 ≤ r ≤ 2ℓ, 2s ≡ r − u, and 2t ≡ 2ℓ − r − u. The range of the summation over u is as follows: u = 0 , 2 , . . . , min(r, 2ℓ − r) for r even , 1 , 3 , . . . , min(r, 2ℓ − r) for r odd .

JHEP06(2018)147 E The spinor rotation Ω
In this appendix, we prove the formula (2.89) in an arbitrary even dimension D. Namely, we prove that the spinor representation of a local Lorentz transformation Λ a b ≡ (O −1 If we define matrices we can easily show the identity, and this leads to Choosing the lower sign, we obtain the desired relation, In the following, we rescale Ω − and define Ω (a) such that Ω −1 (a) = Ω (−a) . The relation (E.4) implies that Ω −1 ± is proportional to Ω ∓ , and we denote their relation as , and then we find that Ω (a) ≡ |Ω| −1 Ω − satisfies the relation Ω −1 (a) = Ω (−a) . We can compute |Ω| 2 = Ω − Ω + as

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where S 2p is the symmetric group on a set of 2p indices, sgn(σ) is the sign of a permutation σ ∈ S 2p and ε b 1 ≡ η b 1 b 1 is −1 for b 1 = 0 and +1 for b 1 ≥ 1. The Pfaffian is the polynomial in matrix elements of the antisymmetric matrix A(b 1 , . . . , b 2p ) which is defined by As it is well known, the square of the Pfaffian Pf its Taylor series around x = 0 is where the coefficients c 2p (p = 0 , 1 , . . . , D/2) are given by From this, we finally obtain In this appendix, we expand the operators To this end, we first use the parameterization g = g b · g f , and expand R g (X) as ) g b and χ ≡ Q I θ I . We can then expand O ± as , The inverses can be also expanded as Order O(θ 0 ). The leading order part O −1 ±(0) of the inverse operators act as where we have used (3.31) and defined k ±a b as Note that k ±a b satisfies k ±ab ≡ k ±a c η cb = k ∓ba due to the antisymmetry of λ ab given in (3.33).
Order O(θ 1 ). At the next order, we obtain where "(J-term)" represents terms proportional to J ab that are not relevant to our computation. Then, the operations of the inverse operators are (F.7)

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Order O(θ 2 ). Finally, the operators at the quadratic order are given by The inverses are Operators of O −1 ±(2) on other generators are not necessary for the computation of the action.

G Deformed torsionful spin connections
In this appendix, we show that two torsionful spin connections W ab ± introduced in (3.36) satisfy the following relations (we basically follow the discussion of [22]): which are assumed in (3.69).

G.1 Two expressions of the deformed H-flux
In order to show (G.1) and (G.2), we here obtain two expressions for the deformed Hflux. Let us begin by considering two expressions of the deformed B-field [recall (3.40) and (3.36)] where J ± are defined in (3.34) and J (n) ± ≡ P (n) J ± . Since we are only interested in the B-field at order O(θ 0 ), in the following computation, we ignore terms involving the grade-1 and 3 components of A and J ± (where we have d ± ∼ 2 P (2) ).

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The exterior derivatives of the two expressions in eq. (G.3) become Here, in the third line, we have used a relation for g-valued 1-forms B and C, and in the last equality, we have used the relation It is easy to see that the last term in (G.4) vanishes by using the cyclic property of the supertrace and the homogeneous CYBE Now, we utilize the deformed structure equation [22] dJ where we have repeatedly used A = O ± (J ± ), and in the last equality, we have used the homogeneous CYBE. In the following computation, since terms involving J ± are irrelevant, we have ± } , (G.9) and then (G.4) is simplified as ± ) θ=0 , (G.10)

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where, in the last equality, we have used relations 20 We can further rewrite the expression (G.10) by using the operator we can rewrite (G.10) as Finally, by introducing a notation and using e a + = Λ b a e ′b , we obtain two expressions for the deformed H-flux

G.2 Deformed torsionful spin connections
By considering the leading order part O(θ 0 ) of (G.9), we obtain where the spin connections (ω [±] ) ab associated with the deformed vielbeins e a ± have the form In particular, for the spin connection ω ′ab ≡ ω ab [−] associated with the deformed vielbeins e ′a = e a − , using the formula H ′ abc = −3 M −1 [c,ab] in (G.16), we obtain the first relation (G.1) as

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where in the last equality we have used 21 On the other hand, if we take the upper sign in (G.17), from e a + = Λ b a e ′b , we obtain From the upper sign of (G. 18), ef ] in (G.15), and the identity This together with (G.21) shows the second relation (G.2),

H Deformed Lagrangian at order O(θ 2 )
In this appendix, we show that the YB-deformed sigma model action can be rewritten as the conventional GS superstring action (up to quadratic order in fermions) by performing suitable field redefinitions. where we have defined and Aᾱ (a) (a = 0, 1, 2) have the following form as we can see from (3.12): (H.3) 21 In order to show the relation W ab + = W ab − − e ′c (M −1 )c ab , it will be easier to observe the bosonic part of the relation J

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The result is In the second equality, we have used where the projection operators π IJ ± are defined by π IJ ± = δ IJ ± σ IJ 3 2 . (H.14) Now, we decompose the deformed Lagrangian into two parts L (2) = L c (2) + δL (2) , (H. 15) where L c (2) takes the form of the canonical GS Lagrangian after taking the diagonal gauge (see section 3.2.3) while δL (2) is the remaining part. The explicit form of L c (2) is given by (H. 17) This was originally considered in [13,74] such that the unwanted terms involving As we show below, in fact, δL (2) is completely canceled out under the redefinition (H.20) when the r-matrix satisfies the homogeneous CYBE, which has been checked for specific examples in the previous works. For simplicity, we introduce a shorthand notation with combinatoric factors discussed around (C.25). In this notation, the commutation relations of bosonic generators {P a , J i } and matrices {γ a , γ i } become

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Then, δL (2) in (H.18) can be expressed as A computation of δL (0) . A straightforward computation shows where we have used e ±m a = k ±b a e m b and δW ±m k = ±2 η e ±m c λ c k . From this, we can easily see that the terms involving σ IJ 1θ I γ k ∂ᾱθ J in δL (0) and δL (2) cancel each other out. We can further compute ∂ p k +ab and ∂ m λ a k as follows by recalling their original definitions. The ∂ m λ a k can be obtained from where we have used ∂ m R g ( · ) = −[A m , R g (·) ] + R g ([A m , · ]) , A m | θ=0 = e m a P a − ω m k J k , Similarly, we obtain In fact, the YB sigma model action defined in (3.25) is invariant under the following κsymmetry variations (see [7] for the details): −β + Pᾱβ + Q 2 κ 2ᾱ , J where κ Iᾱ (I = 1, 2) are local fermionic parameters and we have defined Υ ≡ diag(1 4 , −1 4 ).
Here and hereafter, worldsheet vectors with (±) are projected with the projection operator P ±ᾱβ , like Jᾱ ±(±) ≡ Pᾱβ ± J ±β . In this appendix, following the procedure of [74], we rewrite the κ-variations (I.1) and (I.2) as the standard κ-variations in the GS type IIB superstring [113], where the detailed notations are explained below. In the course of the rewriting, we need to identify the supergravity background (e ′ m a , B ′ mn ,F ′ ) as the β-deformed AdS 5 × S 5 background. In this sense, the following computation serves as a non-trivial check of the equivalence between YB deformations and local β-deformations.
Bosonic fields. We first consider the κ-symmetry transformation of the bosonic fields X m , which can be extracted from the grade-2 component of (I.1). From (I.1), we can easily see The left-hand side can be expanded as Now, by performing the redefinition (3.72) of X m , the term proportional to σ IJ 1 disappears and we obtain 0 = e mb δ κ X m + i 2 δ IJ δ c b + 2 η σ IJ 3 λ b c θ Iγc δ κ θ J k ba − P a + O(θ 3 ) . (I.8)

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Then, solving the equation (I.8) for δ κ X m , we obtain where it is note that the inverse of e ±m a = e m b k ±b a is e ±a m = (k −1 ± ) a b e b m . Finally, by using Λ a b Γ b = Ω −1 Γ a Ω and the redefined fermions Θ ′ I given in (3.66), (I.9) becomes which is the usual κ-variation (I.3) of X m .
Fermionic fields. Next, let us consider the κ-variations of fermionic variables. These can be found from +β } . (I.11) Indeed, the left-hand side gives In order to evaluate the right-hand side, we use the following relations [74]: (I.14) By using relations (B.33) and (B.35), these can be rewritten as Pᾱβ + e +β a Γ a K 2ᾱ + O(Θ 2 ) , where we have used commutation relations of su(2, 2|4) algebra It is noted that the redefinition (3.72) of X m does not affect the variation of the worldsheet metric at the leading order in θ.
Then, by using the 32 × 32 gamma matrices, the variation can be expressed as In this way, we have obtained the standard κ-variation of the worldsheet metric (I.5).
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