Yang-Baxter Deformation as an O(d,d) Transformation

TL;DR

This paper demonstrates that Yang-Baxter deformation of string sigma models can be understood as an $O(d,d)$ transformation, linking it to duality twists in Gauged Double Field Theory and revealing conditions for fluxes and supergravity solutions.

Contribution

It generalizes YB deformation as an $O(d,d)$ transformation affecting both NS-NS and RR sectors, connecting it to fluxes in Gauged Double Field Theory and supergravity.

Findings

YB deformation acts as an $O(d,d)$ transformation on target space fields.

Deformed fields correspond to duality twisted fields in GDFT.

Non-unimodular R-matrix leads to linear dilaton dependence and generalized supergravity solutions.

Abstract

The rules for Yang-Baxter (YB) deformation for a generic Green-Schwarz string sigma model has been obtained recently. We show that the deformation can be described through the action of a coordinate dependent $O(d,d)$ matrix on the target space fields both in the NS-NS and the RR sectors, generalizing previous results. This enables us to show that the YB deformed fields can be regarded as duality twisted fields in the context of Gauged Double Field Theory (GDFT). We compute the fluxes associated with the twist and show that the conditions on the R-matrix determining the YB deformation give rise to conditions for the fluxes on the GDFT side. More precisely, we show that YB deformation is a process which takes a solution of DFT with geometric flux associated with the isometry group $G$ and deforms it to another solution of DFT with vanishing R-flux and non-vanishing Q-flux given by the structure constants of the dual Lie algebra determined by the R-matrix. We also show that the non-unimodularity of the R-matrix forces the generalized dilaton field to pick up a linear dependence on the winding type coordinates of DFT, implying that the corresponding target space fields satisfy the field equations of DFT in the generalized supergravity frame. This provides a new perspective on the relation between the non-unimodularity of the R-matrix, the trace of the Q-flux and the generalized supergravity equations.

Equations

$$T_{{\rm NATD}}=\left(\begin{array}[]{cc}0&1\\ 1&\theta_{IJ}\end{array}\right).$$

$$G^{\prime }+B^{\prime }=(G+B)[1+\eta R_g(G+B){]}^{-1}.$$

$$R_g=A{d}_{g^{-1}}RA{d}_g,$$

$$R_g(X)=g^{-1}R(gX{g}^{-1})g,g\in G,X\in \mathbf{g},$$

$${\nu }_K{C}_{IJ}^K={\eta }^{-1}{R}_{IJ}^{-1}-{\eta }^{-1}(R_g^{-1}{)}_{IJ}.$$

$$(A{d}_{g^{-1}}{)}_I^J=K_I^J,$$

$$K_I^J=k_I^{\mu }{l}_{\mu }^J.$$

$$k_I=k_I^{\mu }{\partial }_{\mu },$$

$$[k_I,k_J]=-C_{IJ}^K{k}_K.$$

$$r_g^{\ast }\sigma =(A{d}_{g^{-1}})\sigma$$

$$(R_g{)}_L^I=K_J^I{R}_K^J{\tilde{K}}_L^K,$$

$$(R_g{)}^{IM}$$

$${\tilde{K}}_L^K{\kappa }^{LM}=({\tilde{K}}^{-1}{)}_L^M{\kappa }^{LK}.$$

$${\tilde{K}}_L^K{\kappa }^{LM}{\tilde{K}}_M^R={\kappa }^{RK}.$$

$$(R_g{)}^{\mu \nu }=R^{IJ}{k}_I^{\mu }{k}_J^{\nu }.$$

$$G^{\prime }+B^{\prime }=G[1+\eta R_{g}G{]}^{-1}=[G^{-1}+\eta R_g{]}^{-1},$$

$$E\to E^{\prime }=T.E=(aE+b)(cE+d{)}^{-1},$$

$$T=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in O(d,d).$$

$$E^{\prime }$$

$$T_{{\rm YB}}=\left(\begin{array}[]{cc}1&0\\ \Theta&1\end{array}\right).$$

$${\Theta }^{IJ}=\eta (R_g{)}^{IJ}.$$

$$\mathcal{H}⟶{\mathcal{H}}^{\prime }=T\mathcal{H}{T}^T,$$

$${\cal H}=\left(\begin{array}[]{cc}G-BG^{-1}B&BG^{-1}\\ -G^{-1}B&G^{-1}\\ \end{array}\right).$$

$$e^{-2{\varphi }^{\prime }}=e^{-2\varphi }\sqrt{\frac{\det G}{\det G^{\prime }}},$$

$$G^{\prime }=\frac{E^{\prime }+E^{\prime T}}{2}.$$

$$G^{\prime }=\frac{1}{(cE+d{)}^T}G\frac{1}{(cE+d)}.$$

$$\frac{\det G}{\det G^{\prime }}={\left(\det (cE+d)\right)}^2.$$

$${\left(\frac{\det G}{\det G^{\prime }}\right)}^{1/2}=\det (\eta R_g(G+B)+1).$$

$$e^{-2{\varphi }^{\prime }}=e^{-2\varphi }\det (\eta R_g(G+B)+1),$$

$${\varphi }^{\prime }=\varphi -\frac{1}{2}\ln \det (\eta R_g(G+B)+1).$$

Full text

1. 1 Introduction

2. 2 Yang-Baxter Deformation as an $O(d,d)$ transformation

3. 2.1 The YB matrix

4. 2.2 YB deformation as a generalization of the Lunin-Maldacena deformation

5. 3 Transformation of the RR fields

6. 4 Embedding the homogeneous YB model in DFT

7. 4.1 A brief review of DFT

8. 4.2 YB matrix as a twist matrix in GDFT

9. 4.2.1 Properties of the R-matrix and the dressed R-matrix

10. 4.2.2 Fluxes associated with the YB matrix and the CYBE

11. 4.3 The Dilaton Field and the Generalized Supergravity Equations

12. 5 Conclusion

13. A O(D,D) and Its Action on Curved Backgrounds

14. B GDFT and the Fluxes

15. C Index Structure for the Twist Matrices

1 Introduction

Constructing integrable deformations of string backgrounds relevant for AdS/CFT correspondence is an active line of research. An important milestone in this direction was the work of Klimcik [1], where he introduced a particular type of deformation for Principal Chiral Models (PCM) with simple compact Lie group as the target manifold. The resulting deformed sigma model was dubbed Yang-Baxter sigma model, as the deformation is based on solutions of modified classical Yang-Baxter equation (mCYBE). The integrability of the YB sigma model was proved later in [2]. The applicability of YB deformations was extended to symmetric coset spaces in [3], which in turn was applied to $AdS_{5}\times S^{5}$ in [4]. The NS-NS sector of the corresponding deformed supergravity background was found in [5]. Similar deformations for the NS-NS sector of $AdS_{n}\times S^{n}$ supercosets was studied in [6]. The RR sectors of these deformed backgrounds was worked out in [7], and was successfully established for the $n=2$ and $n=3$ cases. The most interesting $n=5$ case could not be understood fully in that work and the full Lagrangian for the deformed $AdS_{5}\times S^{5}$ was found later in [8]. We should note that such deformations of the supergravity backgrounds are usually called $\eta$-deformations.

It is also possible to consider similar deformations based on the classical Yang-Baxter equation (CYBE), as opposed to mCYBE. Following the literature, we call the resulting models homogeneous Yang-Baxter models. Such deformations of the $AdS_{5}\times S^{5}$ string was first studied in [9]. These deformations are particularly interesting, as they provide a natural generalization of the Lunin-Maldacena (LM) deformations [10], obtained by T-duality-shift-T-duality (TsT) transformations [11]. Indeed, when the R-matrix (which is an endomorphism on the Lie algebra of the symmetry group) that solves the CYBE is Abelian, the corresponding deformation is the LM deformation, as was first suggested in [12] and then proved in [13]. The relation between TsT transformations and homogeneous YB deformations was also studied in the papers [14]-[19]. For future reference, we note here that the LM deformations can be generated by the action of a constant $O(d,d)$ matrix on the initial background with commuting isometries, as was first observed in [38]. We will henceforth call this matrix the TsT matrix.

Given the relation between homogeneous YB deformation based on an Abelian R-matrix and the TsT transformation, it is natural to wonder whether a similar relation to T-duality holds for more general R-matrices. Indeed, it was conjectured in [20] that the homogeneous Yang-Baxter model could be obtained by applying Non-Abelian T-duality (NATD) to the original background, with respect to an isometry group determined by the R-matrix. This conjecture was proved by Borsato and Wulff in [21],[22] for the case of Principal Chiral Models (PCM)111See also [23], where they arrive at the same result by utilizing the $O(d,d)$ structure of NATD.. Recently, Borsato and Wulff extended their work to homogeneous YB deformations of more general sigma models than PCMs. In their paper [24], they derived the rules for NATD for a generic Green-Schwarz (GS) string with isometry group $G$, where NATD is applied with respect to a subgroup of $G$. Then, by using the connection between NATD and homogeneous YB models that they had established in [21], [22], they proposed the form of homogeneous YB deformation for a generic GS sigma model and showed that this gave a natural generalization of the YB deformation of PCMs and supercosets.

Recently in [25], the first author showed that the NATD transformation rules obtained in [24] for a generic GS string with isometry group $G$ can be described through the action of a coordinate dependent $O(d,d)$ matrix. The dependence of the NATD matrix on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. Viewing NATD this way made it possible to consider it as a transformation in Double Field Theory (DFT), a formalism which provides an $O(d,d)$ covariant formulation for effective string actions [26]-[30]. This made it possible to prove that the NATD fields (including those in the RR sector) solve supergravity equations when the isometry group is unimodular, whereas in the non-unimodular case they form a solution for generalized supergravity equations (GSE) [31],[32]. NATD (and the related Poisson-Lie T-duality) has been studied in the context of DFT also in the papers [23],[33], [34],[35],[36] and [37].

Given the relation between NATD and homogeneous YB deformations discussed briefly above, the results of [25] implies that it should be possible to describe homogeneous YB deformation also as an $O(d,d)$ transformation. The purpose of this paper is to show that this is indeed the case, for the YB deformation of any Green-Schwarz sigma model. More precisely, we will examine the homogeneous YB deformation proposed in [24] for a generic GS sigma model and show that the deformation in the target space, including the RR fields is generated by a particular non-constant $O(d,d)$ transformation, called the $\beta$-transformation. We call the matrix associated with this transformation the Yang-Baxter (YB) matrix. The R-matrix that determines the YB matrix is constructed by picking up a fixed, constant R-matrix satisfying the CYBE and extending it by the adjoint action to the whole group manifold associated with the isometries of the background. The YB matrix we find in this paper is related to the NATD matrix we studied in [25] by a constant $O(d,d)$ transformation. That the two matrices are related is of course not surprising at all, given the fact that [24] constructed the YB deformed models by utilizing the connection with NATD. We will discuss this relationship in section 2.1. Then, in section 2.2 we will explore the relationship between the YB matrix and the TsT matrix. We will show that, in the special case when the R matrix is Abelian, the YB matrix becomes constant and reduces to the TsT matrix. This directly follows from the fact that the adjoint action is trivial on an Abelian Lie algebra, as we will discuss in subsection 2.2.

Viewing YB deformation as an $O(d,d)$ transformation in the target space is quite useful. First of all, it makes the calculations considerably simpler. Moreover, the transformation rules for the fields in the RR sector almost come for free. Computing the RR fields of the (homogeneous) YB models by the supercoset construction as in [8],[39] is usually cumbersome. An alternative way is to start with the deformed NS-NS fields and determine the deformed RR fields by directly solving the Type IIB supergravity equations as in [7].222In the papers [40],[41], the open-closed string map associated with the YB deformation is extended to the RR sector by utilizing the Page forms. Viewing YB deformation as a $\beta$-transformation gives a more direct and easy to apply formula to determine the RR fields in the deformed background, since the $O(d,d)$ transformation of the NS-NS fields dictates the transformation for the RR fields. As we discuss in section 3, the deformation in the RR sector is completely determined by the $Spin(d,d)$ matrix $S_{{\rm YB}}$ projecting to the YB matrix $T_{{\rm YB}}$ as $\rho(S_{{\rm YB}})=T_{{\rm YB}}$, where $\rho$ is the usual double covering homomorphism between $Pin(d,d)$ and $O(d,d)$. We will discuss this in detail in section 3. Another advantage of viewing YB deformation as an $O(d,d)$ transformation is that it gives a natural embedding of the homogeneous YB model in DFT. As we will discuss in section 4.2, this provides a direct correspondence between the properties of the R-matrix and the deformations of supergravity. We will see that the deformations in the gravity picture, when embedded in DFT, will be encoded in entities called fluxes, which deforms DFT to Gauged Double Field Theory (GDFT), [42],[43],[44],[45]. Then, the conditions to be satisfied by the R-matrix manifest themselves as conditions on these fluxes. More precisely, we will see that the R-matrix satisfies the CYBE if and only if the R-flux vanishes on the DFT side. On the other hand, the structure constants of the dual Lie algebra determined by the R-matrix appears as components of the so called Q-flux on the DFT side, and the R-matrix is unimodular if and only if the Q-flux is traceless. When the R-matrix is non-unimodular, it is known that the YB deformed target space fields constitute a solution to the equations of generalized supergravity [24]. In section 4.3, we look at how Q-flux with non-vanishing trace deforms DFT. We see that the resulting GDFT is consistent only when the generalized dilaton field picks up a linear dependence on the winding type coordinates in a particular way, which does not violate the section condition of DFT. It is known that the field equations of DFT reduce to generalized supergravity equation for this particular solution of the section constraint, [46], [47], [48]. Therefore, our approach provides a new perspective on the connection between the non-unimodularity of the R-matrix, the trace of the Q-flux and generalized supergravity equations.

The $O(d,d)$ structure of YB deformations has already been studied by several groups. Let us review briefly what has been achieved in those works and discuss what has been added in our paper. The first paper, where it was proposed that the homogeneous YB deformation could be obtained by the action of a $\beta$- transformation is [49]. This proposal is based on observation and is supported by working out the example of the $AdS_{5}\times S^{5}$ background. Later in [50], a proof of the proposal of [49] was given for deformations of the $AdS_{5}\times S^{5}$ background, by rewriting the YB deformed action in the form of Green-Schwarz action (up to quadratic order in fermions) and showing that the target space can be obtained by applying $\beta$-transformation on the $AdS_{5}\times S^{5}$ background. It should be noted that their formulas worked only when the B-field of the undeformed background vanished. Indeed, they also considered $\beta$-transformations of the $AdS_{3}\times S^{3}\times T^{4}$ with non-vanishing H-flux and showed that the resulting background solved equations of motion of (generalized) supergravity. However, a proof for the equivalence with YB deformation could not be given for this case, due to complications arising from the presence of the H-flux. Our work extends the work of [50] in two ways. Firstly and most importantly, the equivalence of YB deformations and $\beta$ transformations is not shown merely on a case by case basis and works for the generic case (of which $AdS_{5}\times S^{5}$ is an example). This is because, the YB deformation rules presented in [24] (and which we rewrite here as an $O(d,d)$ transformation) is generic and works for any GS sigma model. Secondly, in our case, we do not need to take the B-field to vanish in the initial background. It is also important to note that our approach in this paper is different from the approach of [49],[50] in that, here we use the framework of GDFT and interpret the $\beta$-transformation associated with the YB matrix as a solution generating transformation, which takes one solution of DFT with geometric flux to another one with dual Q-flux, as we will explain in detail in section 4.2. In contrast, the Q-flux of the YB deformed model cannot be captured in the papers [49],[50]. Later in [51], it is shown that the deformed backgrounds found in [49],[50] belong to a certain class of non-geometric backgrounds, called T-folds [52], and the Q-flux of the YB deformed backgrounds is understood as arising from a non-trivial monodromy in the so called $\beta$-field around closed cycles. Our findings in this paper, which we obtain by computing directly the Q-flux associated with the YB matrix in the framework of GDFT agree with the results of [51] and extends it in the sense that it reveals the role of Q-flux as the dual structure constants to geometric flux. Again, it is worth noting that the monodromy of the $\beta$-field is computed in [51] for the Minkowski and the $AdS_{5}\times S^{5}$ background and for vanishing B-field, whereas our results in this paper applies to any GS sigma model with isometries. The Q-flux of deformed backgrounds and its role as the dual structure constants to geometric flux is also studied in [23], but in the context of Poisson Lie duality of sigma models. However, for the YB deformation, this feature of the deformed models cannot be captured in [23], either.

We should also note that the approach adopted in the papers [40],[41],[53],[54], which regards YB deformation as an open-closed string map [55] also contributes to the understanding of the $O(d,d)$ structure of YB deformations. Indeed, as we will discuss in subsection 2.1, the open-closed string map can be regarded as a special case of $\beta$-transformation. In this picture, the YB deformation is encoded in the non-commutativity parameter $\Theta$ of the open-closed string map333We would like to note that the papers [56, 57] also study the relation between YB deformations and non-commutative deformations within the context of AdS/CFT duality.. In the papers [58]-[60], the open-closed string map was promoted to a solution generating symmetry in generalized supergravity and it was shown that the field equations enforce the CYBE for the antisymmetric bivector $\Theta$. Following this line of work which regards YB deformation as an open-closed string map, the relation between deformations of supergravity backgrounds and solutions of the CYBE was further explored in [61] in the context of $\beta$ supergravity.

The plan of the paper is as follows. In section 2, we find the coordinate dependent $O(d,d)$ matrix $T_{{\rm YB}}$, which we call the YB matrix, that encodes the YB deformation obtained in [24] of a generic GS sigma model. Also in the same section, we discuss how this matrix reduces to the $O(d,d)$ matrix associated with Lunin-Maldacena deformations, when the R-matrix is Abelian. In section 3, we describe the deformation in the RR sector, by identifying the $Spin(d,d)$ matrix that projects to $T_{{\rm YB}}$ under the double covering homomorphism between $O(d,d)$ and $Pin(d,d)$. The purpose of section 4 is to describe the $O(d,d)$ transformation that generates the YB deformation as a solution generating transformation in DFT. After giving a quick review of DFT, we describe in section 4.2 the YB deformed fields as twisted fields in GDFT. In section 4.2.2 we compute the fluxes associated with the twist matrix $T_{{\rm YB}}$ and study the relation between the conditions on the R-matrix and those on fluxes. As a preparation for this computation, we discuss some properties of the R-matrix and the dressed R-matrix in 4.2.1. Then in section 4.3 we discuss how the non-unimodularity of the R-matrix forces the generalized dilaton field to have a linear dependence on winding type coordinates so that the YB deformed fields form a solution of DFT in the generalized supergravity frame. We conclude with discussions and outlook in section 5. The paper is complemented with three appendices.

2 Yang-Baxter Deformation as an $O(d,d)$

transformation

As we discussed in section 1 above, it was shown in [50] that the YB deformation of the string sigma model with target space $AdS_{5}\times S^{5}$ is produced by the action of a coordinate dependent $O(d,d)$ transformation (a particular type of $\beta$-transformation, see (2.20)) on the fields in the target space. Here, we extend their results and show that the correspondence holds for the homogeneous YB deformation of any GS string sigma model, of which $AdS_{5}\times S^{5}$ is a specific example. This correspondence could not be checked in full generality previously, mainly due to complications in formulating the homogeneous YB deformations of type II superstring on backgrounds with non-vanishing H-flux. This was achieved in [24], where they also checked the equivalence to the $\beta$-transformation of [50] for the background $AdS_{3}\times S^{3}\times T^{4}$ with non-vanishing H-flux. Here, we show the equivalence of the YB deformation rules found in [24] with the $\beta$-transformation rules for any GS string sigma model, without having to refer to specifics of the initial supergravity background.

In order to determine the the $O(d,d)$ matrix that produces the YB deformation rules found in [24], analyzing the deformation in the NS-NS sector is sufficient. Once we find it from this analysis, the transformation of the RR fields is determined immediately. In Abelian T-duality, RR fields are packaged in a differential form, which can be a regarded as a spinor field that transforms under $Spin(d,d)$. If the fields in the NS-NS sector transform under $T\in O(d,d)$, then the spinor field that encodes the RR fields transform under $S_{T}\in Spin(d,d)$, which is the element that projects onto $T$ under the double covering homomorphism $\rho$ between $O(d,d)$ and $Spin(d,d)$, that is, $\rho(S)=T$ [62]. Similarly, once we figure out the $O(d,d)$ matrix that produces the YB deformation, the transformation of the RR fields is automatically determined by the corresponding $Spin(d,d)$ matrix, as in Abelian T-duality. We will discuss this in section 3.

2.1 The YB matrix

The rules for NATD for a generic Green-Schwarz string sigma model with isometry group $G$ has recently been obtained in [24]. In [38], we identified the matrix whose action on the target space produces the NATD background presented in [24]. We call this matrix the NATD matrix. It is an $O(10,10)$ matrix and is obtained by embedding (as in (A.3) in Appendix A) the following $O(d,d)$ matrix $T$ in $O(10,10)$:

[TABLE]

Here, $\theta_{IJ}$ is given as $\theta_{IJ}=\nu_{K}C_{IJ}^{\ \ K}$, where $C_{IJ}^{\ \ K},I,J,K=1,\cdots,d={\rm dim}G$ are the structure constants of the Lie algebra of the isometry group $G$ of the initial background, and $\nu_{I}$ are the coordinates of the NATD background. It was shown in [38] that this matrix gives the correct transformation rule for the fields both in the NS-NS and the RR sector.

In the same paper [24], the YB deformation for a generic GS sigma model was proposed by exploiting the relation between NATD and YB deformations [20],[21],[23]. The deformed model they construct is generated by solutions of classical Yang-Baxter equation and it generalizes the construction for PCM and supercosets. The formulas they find for the deformed metric and the B-field are given in equations (4.12)-(4.13) in their paper. As noted in the paper, these transformation rules can be combined in one compact rule, which we present below444Note that due to the convention in [24] there is a difference in the sign in front of the B field term.:

[TABLE]

Here $\eta$ is the deformation parameter, and $R_{g}$ is the dressed R-operator, related to the R-matrix $R$ as

[TABLE]

that is,

[TABLE]

where $\mathbf{g}$ is the Lie algebra of the isometry group $G$. The relation between the NATD and YB parameters is given in [24] as

[TABLE]

Here, the transformation (2.2) acts on the metric $G(x)$ and the B-field $B(x)$ defined in (C.7), that is, it acts on $G_{IJ}$ and $B_{IJ}$ with Lie algebra indices. Note that $l^{I}_{\ i}G_{IJ}l^{J}_{\ j}=G_{ij}$ and similarly for $B_{IJ}$, where $l^{I}_{\ i}$ are the functions on $G$ that determine the left-invariant one-forms $\sigma^{I}=l^{I}_{\ i}d\theta^{i}$. For more details on the index conventions, see Appendix (C).

As discussed in [24], the transformation (2.2) can be seen as a generalization of the open-closed string map of [53],[40],[41]. The best way to see this is to realize that

[TABLE]

where we have defined

[TABLE]

Here, $l^{I}_{\ \mu}$ are as above and $k_{I}^{\ \mu}$ are the components of the right invariant vectors fields $k_{I}$:

[TABLE]

that satisfies

[TABLE]

These are Killing vector fields for the left invariant metric $G_{\alpha\beta}$ in (C.4). The relation (2.6) arises from the well-known relation

[TABLE]

where the left hand side corresponds to the pull-back of the Maurer-Cartan form $\sigma=\sigma^{I}T_{I}=l^{I}_{\ i}d\theta^{i}T_{I}$ under the right translation $r_{g}$, generated by $k_{I}$. Then the dressed R-operator in (2.3) can be written as

[TABLE]

where $\tilde{K}^{I}_{J}=(Ad_{g})^{I}_{J}$. Raising the indices of the left hand side with the Cartan-Killing metric $\kappa^{LM}$ we obtain

[TABLE]

In the last line above we used $\tilde{K}=K^{-1}$, which follows from $Ad_{g^{-1}}=Ad_{g}^{-1}.$ Also, in the second line we used the fact that the Cartan-Killing metric is $Ad$-invariant, that is, $Ad_{g}$ is orthogonal with respect to $\kappa$, which implies that

[TABLE]

Equivalently

[TABLE]

Since the matrix $l^{I}_{\ \mu}$ is invertible, we can use its inverse $l^{\mu}_{\ I}$ to convert all algebra indices $I$ to curved indices $\mu$. Then, acting on the curved background $G_{\mu\nu},B_{\mu\nu}$ in (C.2) the YB transformation rule is again as in (2.2) with

[TABLE]

This is indeed a generalization of the open-closed string map in the papers [58], [59] as it reduces to it for vanishing B-field555It also agrees with the transformation rules given in [61] for non-vanishing B-field.. Indeed for vanishing B-field the transformation rule (2.2) reduces to

[TABLE]

which is the same as equation (1.1) of [59] with $\Theta_{NC}=\eta R_{g}$, where $\Theta_{NC}$ is the noncommutativity parameter666Note that the noncommutativity parameter $\Theta_{NC}$ is identified as $\Theta_{NC}=-2\eta^{\prime}R_{g}$ in the papers [58], [59]. Therefore the deformation parameter $\eta$ in our paper and $\eta^{\prime}$ in these papers are related as $\eta=-2\eta^{\prime}$.. Note that the bi-Killing anzats they introduce in equation (1.2) of [59] is the same as what is given in (2.15).

In order to recast the transformation rule (2.2) as an $O(d,d)$ transformation, we use the terminology discussed in [25] and write $E=G+B$, which is known as the background matrix [63]. It is known that the action of $O(d,d)$ on the background matrix is by fractional linear transformations:

[TABLE]

where

[TABLE]

Then, (2.2) can be written as

[TABLE]

where

[TABLE]

Here, for simplicity, we have defined777 Note that this means $\Theta$ is in fact equivalent to $\Theta_{NC}$ of [58], [59], see equation (2.16) and the discussion below it. However, we call it $\Theta$ rather than $\Theta_{NC}$ as the focus of this paper is not on the relation to noncommutative deformations.

[TABLE]

This matrix acts only on the isometry directions indexed by $I,J$. The $O(10,10)$ matrix which acts on the full metric and the B-field is obtained by embedding the $O(d,d)$ matrix (2.20) in $O(10,10)$ as in (A.3). When this is done, one gets exactly the deformation rules presented in equations (4.12)-(4.13) in [24] . For simplicity, we will call both matrices (both the $O(d,d)$ and the $O(10,10)$ one) the YB matrix, denoted by $T_{{\rm YB}}$.

It is well known that the transformation of the background matrix $E\rightarrow E^{\prime}=T.E$ under $T\in O(D,D)$ is equivalent to the transformation

[TABLE]

where

[TABLE]

In the context of Double Field Theory (DFT), ${\cal H}$ is called the generalized metric. Taking $T=T_{{\rm YB}}$, one immediately sees that the transformation (2.22) and hence the transformation (2.19) (which is equivalent to (2.2)) is the same as the $\beta$-transformation of [49],[50].

The YB matrix (2.20) also produces the correct transformation rule for the dilaton field. Under $O(d,d)$ the dilaton field $\phi$ transforms as

[TABLE]

where the deformed metric $G^{\prime}$ is read off from the symmetric part of the deformed background matrix $E^{\prime}$, that is

[TABLE]

It is a well known fact that the transformation (2.19) implies for $G^{\prime}$ the following [63]:

[TABLE]

Then,

[TABLE]

When $T$ is as in (2.20) this gives

[TABLE]

Plugging this in (2.24) gives

[TABLE]

which is equivalent to

[TABLE]

This is the same result that has been obtained in [24].

2.2 YB deformation as a generalization of the

Lunin-Maldacena deformation

Before we move on to the transformation for the RR fields, let us explain how the YB matrix reduces to the matrix that generates the TsT transformation, when the R-matrix is Abelian. This matrix, which we will call the TsT matrix, was found in [38] and it generates the $O(10,10)$ transformation associated with single and multi-parameter Lunin-Maldacena deformations of [10],[11]. It is obtained by embedding the following $O(d,d)$ matrix in $O(10,10)$ as in (A.3):

[TABLE]

Here $\Gamma^{IJ}$ is antisymmetric with $I,J=1,\cdots,d$, where $d$ is the dimension of the Abelian isometry group $U(1)^{d}$. For single parameter deformations, the components of $\Gamma^{IJ}$ either vanish or $\Gamma^{IJ}=-\Gamma^{JI}=\lambda$ for non-vanishing components. For multi-parameter deformations, $\Gamma^{IJ}$ can be equated to different deformation parameters. In this way, one can introduce a total of $d(d-1)/2$ independent deformation parameters [64].

Now, assume that the R-matrix $R$ is Abelian. This means that the Lie subalgebra on which $R$ acts as a non-degenerate operator is Abelian888It is known that skew-symmetric solutions for CYBE on a finite dimensional Lie algebra $\mathbf{g}$ are in one-to-one correspondence with quasi-Frobenius subalgebras of $\mathbf{g}$. On such a subalgebra the R-matrix acts as a non-degenerate operator. If this subalgebra is Abelian, the R-matrix is called Abelian; if this subalgebra is unimodular, the R-matrix is called unimodular. The rank of the R-matrix is equal to the dimension of the quasi-Frobenius subalgebra. A quasi-Frobenius algebra (also called symplectic) is a Lie algebra which admits a non-degenerate 2-cocycle. See [65] and the references therein and also the Proposition 22.6 and Proposition 3.1.6 in the book [66].. From the NATD perspective, the subgroup $\tilde{G}$ of $G$ with respect to which the NATD has been performed is Abelian. Restricting to this commutative subgroup and subalgebra, one immediately sees that $R_{g}=R$ in (2.3). This is because for a commutative group (assuming that it is also connected), the adjoint representation is the trivial representation, that is, the adjoint operator $Ad_{A}$ acts as the identity operator on the Abelian Lie algebra of $\tilde{G}$, for all $A\in\tilde{G}$. As a result, one sees that for the deformation parameter (or the non-commutativity parameter in the language of [53], [41]) we have $\Theta^{IJ}=\eta R^{IJ}=$ constant . This can also be seen from (2.5). Since the NATD group $\tilde{G}$ is Abelian, the left hand side of (2.5) will vanish, giving $R_{g}=R$. As a result, for Abelian R-matrices, the antisymmetric deformation parameter $\Theta^{IJ}$ does indeed have constant components and the matrix (2.20) indeed reduces to the TsT matrix (2.31).

3 Transformation of the RR fields

Realizing the deformation of the metric and the B-field as an $O(d,d)$ transformation999To be precise, the transformation is an $O(10,10)$ transformation. However, it is obtained by embedding an $O(d,d)$ matrix in $O(10,10)$ as in (A.3), where $d$ is the dimension of the isometry group. Hence the only non-trivial action is on the isometry directions. For this reason, we usually refer to this action as an $O(d,d)$ transformation, rather than as an $O(10,10)$ transformation. dictates the transformation for the RR fields immediately. In Abelian T-duality, RR fields are packaged in a differential form, which can be a regarded as a spinor field that transforms under $Spin(d,d)$. If the fields in the NS-NS sector transform under $T\in O(d,d)$, then the spinor field that encodes the RR fields transform under $S_{T}\in Spin(d,d)$, which is the element that projects onto $T$ under the double covering homomorphism $\rho$ between $O(d,d)$ and $Pin(d,d)$, that is, $\rho(S)=T$ [62]. This extends to the case of NATD, that is, the transformation of the RR fields is automatically determined by the $Spin(d,d)$ matrix corresponding to the NATD matrix, which had been determined in [25]. So, it is natural to propose that the RR fields in the YB deformed background should be produced by the action of the $Spin(d,d)$ matrix $S_{{\rm YB}}$ projecting to the YB matrix $T_{{\rm YB}}$: $\rho(S_{{\rm YB}})=T_{{\rm YB}}$. Our results should agree with the results found in [50], where the transformation of the RR fields is determined by utilizing the methods in [67]. Here, we adopt the formalism of [62], and the two approaches are known to produce the same results for the case of Abelian T-duality.

Since $T_{{\rm YB}}$ generates a $\beta$-transformation, finding the $Spin(10,10)$ element $S_{{\rm YB}}$ that projects onto it is rather easy. It is given as

[TABLE]

where $\alpha$ is a generic spinor field (regarded as a differential form through the Clifford map as in (3.4) and (3.5) below) and $i_{\Theta}\alpha=\frac{1}{2}\Theta^{IJ}i_{\psi_{I}}(i_{\psi_{J}}\alpha),$ with $i$ being the usual contraction with $i_{\psi_{I}}\psi^{J}=\delta_{I}^{\ J}$. An important feature of $S_{{\rm YB}}$ we should note here is that it is an element of $Spin^{+}(10,10)$, which is the subgroup of $Spin(10,10)$ connected to the identity component. This is in contrast with the fact that $S_{{\rm NATD}}$ is an element of $Spin^{-}(10,10)$.

If $F$ is the spinor field that encodes the RR fluxes of the initial background (assuming that it respects the isometry $G$), the transformed RR fluxes are read off from the spinor field $F^{\prime}$:

[TABLE]

where $B$ is the B-field of the initial background and $B^{\prime}$ is read off from the anti-symmetric part of $E^{\prime}$ in (2.19):

[TABLE]

For more details, we refer the reader to [38],[45] and [68].

An important difference from the Abelian case is the following. Let us write the differential form $F$ as

[TABLE]

where we have decomposed a $p-$form RR flux $G^{(p)}$ according to how many legs it does have along the directions of the isometry group $G$. Since $G$ acts by isometries, the fluxes $F^{(p-a)},\ a=0,1,\cdots,d$ will have no dependence on the isometry coordinates $\theta^{i}$. We map this differential form to a Clifford algebra element in the usual way:

[TABLE]

The difference we have here is that it is $\sigma^{I}$ and not $dx^{i}$ that we identify with the Clifford algebra element $\psi^{I}$, for $I=1,\cdots,d$. On the other hand, for $a=d+1,\cdots,10$, $dx^{a}$ is replaced with $\psi^{a}$, as usual. Here, $\psi^{\alpha},\ \alpha=(I,a)$ are the Clifford algebra elements $\psi^{\alpha}=1/\sqrt{2}\Gamma^{\alpha}$, where $\Gamma^{\alpha}$ are the Gamma matrices. For more details, see [25]. For index conventions, see Appendix (C).

4 Embedding the homogeneous YB model in DFT

In the previous section, we saw that the fields in the target space of the YB deformed GS sigma model can be obtained by applying a coordinate dependent $O(d,d)$ matrix on the initial target space fields. In this section we describe this $O(d,d)$ transformation as a solution generating transformation in Double Field Theory (DFT). For this purpose, we start with a quick review of DFT.

4.1 A brief review of DFT

DFT is a field theory defined on a doubled space, which implements the $O(d,d)$ T-duality symmetry of string theory as a manifest symmetry. In addition to the standard space-time coordinates, the doubled space also includes dual coordinates, which are associated with the winding excitations of closed strings on backgrounds with non-trivial cycles. The space-time and dual coordinates $X^{M}=(\tilde{x}_{\mu},x^{\mu})$ transform as a vector under the T-duality group $O(d,d)$.

In DFT, the semi-Riemannian metric and the B-field are encoded in the generalized metric ${\cal H}$, which is an element in $SO^{-}(d,d)$. In order to write down the Lagrangian for the RR sector, one needs the spinor field $\mathbb{S}\in Spin^{-}(d,d)$ that projects onto ${\cal H}$ under the double covering homomorphism between $Spin(d,d)$ and $SO(d,d)$, that is $\rho(\mathbb{S})={\cal H}$. To describe the p-form fields in the RR sector, one starts with the democratic formulation of Type II supergravity. In this formulation, the p-form fields are packaged in a polyform as in (3.4), which is then mapped to a spinor field $\chi$ as in (3.5). In DFT, this spinor field is allowed to have dependence both on the space-time coordinates and the winding type dual coordinates. Another dynamical field in DFT is the generalized dilaton field $d$, which is a generalized scalar field. A generalized scalar field and more generally generalized tensor fields are defined according to how they transform under generalized diffeomorphisms of DFT. Infinitesimal transformations under generalized diffeomorphisms are generated by the generalized Lie derivative, which defines the D-bracket that generalizes the Lie bracket. Anti-symmetrization of the D-bracket gives the C-bracket. The gauge algebra of infinitesimal transformations under generalized diffeomorphisms closes under the C-bracket. For more details see [45], [68].

In its current formulation, DFT is a consistent field theory only when a certain constraint, called the strong constraint is satisfied. When the strong constraint is satisfied generalized tensor fields become sections of the direct sum of the tangent and the cotangent bundle101010Here, the tangent bundle is the union of all ‘physical tangent spaces’ determined at each point of the doubled manifold by the strong constraint as a maximally isotropic subspace of the doubled tangent space. Although the structure of a doubled manifold and its tangent space is not very well understood, this choice of a physical tangent space can at least be done in a local chart., generalized diffeomorphisms reduce to the semi-direct product of space-time diffeomorphisms and B-field gauge transformations, and the C-bracket becomes the Courant bracket. This is the framework of generalized geometry program of Nigel Hitchin [69],[70]. A trivial solution of the strong constraint occurs when all the fields and gauge parameters in the theory are independent of the winding type coordinates. Such fields are said to belong to the supergravity frame and it can be shown that the DFT action reduces to the standard supergravity action in the supergravity frame. Another well known solution of the strong constraint is provided when all the fields except for the dilaton field depend on all but one of the space-time coordinates and have no dependence on the winding type coordinates, whereas the dilaton field does also have a linear dependence on the winding type coordinate dual to the remaining space-time coordinate. In such a frame, DFT equations are known to reduce to generalized supergravity equations of [31],[32]111111Generalized supergravity equations (GSE) is a deformation of supergravity equations determined by a Killing vector field. In an adopted coordinate system this Killing vector field generates translations along the space-time coordinate on which the DFT fields have no dependence.. Henceforth, we will call this frame the generalized supergravity frame.

4.2 YB matrix as a twist matrix in GDFT

From the way they are constructed the target space fields of the YB deformed GS sigma model form a solution of supergravity equations, if the corresponding R-matrix is unimodular and of GSE if not unimodular [24]. Then, according to the discussion above, one can construct DFT fields corresponding to these target space fields which solve the field equations for DFT either in the supergravity frame or in the generalized supergravity frame, depending on the unimodularity of the R-matrix. On the other hand, the initial target space fields before the deformation also form a solution for the field equations of DFT in the supergravity frame, as they belong to the target space of a GS string sigma model. We showed in 2.1 that the homogeneous YB model is obtained through the action of the $O(d,d)$ matrix $T_{{\rm YB}}$. Hence, the YB deformation of the GS sigma model is indeed a solution generating $O(d,d)$ transformation for the DFT fields corresponding to the target space fields. A natural question to ask at this point is as to how the condition of unimodularity and the classical Yang-Baxter equation arises on the DFT side. For addressing this question, we find it useful to utilize the methods developed in [25], where we considered DFT fields whose dependence on the doubled DFT coordinates are separated as $\phi(x,Y)=U(Y).\phi(x)$. Here, the $O(d,d)$ matrix $U(Y)$ is called the twist matrix, $\phi(x,Y)$ denotes a generic DFT field, $(x,Y)$ are DFT coordinates and the action of $U(Y)\in O(d,d)$ is determined by how $\phi$ transforms under the $O(d,d)$ duality group of DFT (or more generally under $Spin(d,d)$ if $\phi$ is a spinor field in DFT). Using terminology from Scherk-Schwarz compactifications, we call the fields $\phi(x,Y)$ twisted fields, whereas the fields $\phi(x)$ are called untwisted fields. We showed in [25] that the twisted fields satisfy the field equations of DFT, if and only if the untwisted fields satisfy the field equations of Gauged Double Field Theory (GDFT), provided that $U(Y)$ satisfies a certain set of conditions.121212GDFT is a deformation of DFT, obtained from a Scherk-Schwarz reduction and the deformation is determined entirely by the so called fluxes associated with the twist matrix $U(Y)$. For more details, see Appendix B. This discussion is relevant for analyzing the homogeneous YB model, as the fields here are also twisted fields with the twist matrix being $T_{{\rm YB}}$, as was shown in section 2.1. This perspective will allow us to read off the CYBE and the condition of unimodularity in terms of fluxes on the GDFT side.

From the rules (2.22) and (3.2) we presented in the section 2.1 and section 3, we know that the fields in the target space of the homogeneous YB model are of the following form131313For now, we assume that the R-matrix is unimodular. As will be discused in section 4.3, if the R-matrix is not unimodular, the generalized dilaton field is forced to have a linear dependence on the winding type coordinates.:

[TABLE]

where the fields on the right hand side without prime are the DFT fields constructed out of the target space fields before the deformation. The coordinates $\nu=\{\nu^{I}\},I=1,\cdots,d$ are regarded as coordinates on the group manifold $G$ by using the relation (2.5) derived in [24]. The fields before and after the deformation have no dependence on the winding type coordinates, that is, they belong to the supergravity frame. Also, the fields both before and after the deformation have flat indices ${}^{A}=(_{\alpha},\ ^{\alpha})$ with $\alpha=(I,a)$. For details on the index conventions, see Appendix (C). In (4.3), $\rho(S_{{\rm YB}})=T_{{\rm YB}}$ and $B^{\prime}(x,\nu)$ is read off from the antisymmetric part of ${\cal H}^{\prime}(x,\nu)$ in (4.1). The spinor field ${\cal K}$ in (4.2) is defined as ${\cal K}=C^{-1}\mathbb{S}$ and its transformation rule is determined by that of ${\cal H}$ since $\rho(\mathbb{S})={\cal H}$. For more details, see [45]. Note that the fields ${\cal H}^{AB}(x),{\cal K}(x),d(x),F(x)$ satisfy the field equations of GDFT with geometric fluxes $f_{IJ}^{\ \ K}=C_{IJ}^{\ \ K}$ associated with the isometry, as discussed in Appendix (C).

Hence, the DFT fields constructed from the fields in the target space of the homogeneous YB model are indeed twisted fields, where the twist matrix is the YB matrix $T_{{\rm YB}}$ in (2.20). The twist matrix $T_{{\rm YB}}$ does satisfy the constraints (B.8, B.9) presented in Appendix (B). Firstly, it is obtained by embedding an $O(d,d)$ matrix in $O(10,10)$ and the untwisted fields do not depend on the corresponding $d$ coordinates. This ensures that (B.9) is satisfied. Also, $T_{{\rm YB}}$ does not depend on the winding type coordinates, which then implies that (B.8) is also satisfied. Another condition that needs to be imposed on the twist matrices for a consistent reduction is that the associated fluxes defined in (B.5) should be constant. In subsection (4.2.2) we will compute these fluxes and show that they are indeed constant. Moreover, we will show that the CYBE and the unimodularity condition for the R-matrix become conditions on the fluxes. As a preparation for this computation, we give below a brief review of R-matrices and the related algebraic structures.

4.2.1 Properties of the R-matrix and the dressed

R-matrix

Let $G$ be a semisimple Lie group with Lie algebra $\mathbf{g}$. Let $R$ be an endomorphism on $\mathbf{g}$ with $R^{I}_{\ J}$ being the components of the corresponding matrix with respect to a fixed basis $T_{I}$ of $\mathbf{g}$:

[TABLE]

Let $R^{IJ}=\kappa^{IL}R^{J}_{\ L}$, where $\kappa$ is the non-degenerate Cartan-Killing form on $\mathbf{g}$. We assume that $R$ is skew-symmetric with respect to $\kappa$ so that $R^{IJ}=-R^{JI}$. The CYBE for the operator R is the following:

[TABLE]

With respect to the fixed basis $T_{I}$ this is equivalent to the following equation

[TABLE]

where $C_{IJ}^{\ \ K}$ are the structure constants of the Lie algebra $\mathbf{g}$ with respect to the basis $T_{I}$ and all indices are raised and lowered by the Killing form $\kappa$. The equation (4.6) is a sufficient condition for the bracket $[,]_{R}$ to satisfy the Jacobi identity (see, for example, the book [71]), where

[TABLE]

Then, this yields a new Lie algebra $\mathbf{g}_{R}$ with the same underlying vector space as that of $\mathbf{g}$ and the new bracket $[,]_{R}$. The structure constants $\tilde{C}_{IJ}^{\ \ K}$ of $\mathbf{g}_{R}$ with respect to the basis $T_{I}$ can be computed to be

[TABLE]

Lowering the index $L$ and using the fact that $C_{IJK}$ are totally anti-symmetric in all of its indices we obtain

[TABLE]

Now we raise the indices $I$ and $J$ to obtain

[TABLE]

Note that

[TABLE]

where

[TABLE]

are elements of the dual Lie algebra $\mathbf{g}_{R}^{*}$, where we have identified $\mathbf{g}_{R}$ and its dual $\mathbf{g}_{R}^{*}$ by the bilinear from $\kappa$ in the usual way.

The dressed matrix is defined as in (2.3), which we rewrite here for convenience:

[TABLE]

An important property that we will use in computing the fluxes is that the dressed R-matrix is also an R-matrix that satisfies the CYBE (whenever $R$ does) and as such it defines a Lie bracket $[,]_{R_{g}}$, which is in fact equivalent to $[,]_{R}$141414This, of course, is natural as $R_{g}$ is constructed by taking an R-matrix $R$ at the identity element of the group manifold $G$ and by extending it to the whole manifold by the adjoint action, which is an automorphism of the Lie bracket.. Then the equations (4.7) and (4.9) are still true, when one replaces $R$ with $R_{g}$. These facts can be shown easily by using the fact that $Ad_{g}$ is an automorphism of the Lie bracket.

We showed in (2.12) that the following holds for the dressed R-operator

[TABLE]

where $K^{I}_{J}$ is as in (2.7). The coordinate dependence of the dressed R-matrix comes from the functions $K^{I}_{J}$. So, in computation of the fluxes, we will encounter terms of the form $\partial_{L}K^{I}_{J}$, where $\partial_{L}=l_{L}^{\ i}\partial_{i},$151515 The reason why we have derivatives $\partial_{I}$ rather that $\partial_{i}$ becomes clear in the computation (4.21). where $l^{I}_{\ i}l_{L}^{\ i}=\delta^{I}_{L}.$ In order to calculate this, first note that the functions $K^{I}_{J}$ are just

[TABLE]

where $k_{I}$ are the right-invariant vector fields in (2.8) and $\sigma^{I}$ are the left-invariant one-forms. Now, using the fact

[TABLE]

we immediately obtain

[TABLE]

4.2.2 Fluxes associated with the YB matrix and the

CYBE

We are now ready to compute the fluxes associated with the twist matrix (2.20). We have

[TABLE]

with $T_{a}^{\ b}=\delta_{a}^{\ b},\ T^{a}_{\ b}=\delta^{a}_{\ b},\ T_{a}^{\ J}=T^{a}_{\ J}=T_{I}^{\ b}=T^{I}_{\ b}=0$. For more details on the index structure, see Appendix C. The embedded $d\times d$ matrix, which we have also called the YB matrix has components $T^{I}_{\ J}=\delta^{I}_{\ J},\ T_{I}^{\ J}=\delta_{I}^{\ J},\ T_{IJ}=0$ and $T^{IJ}=\Theta^{IJ}=\eta R_{g}^{IJ}$. Then, the formula in (B.5) for the fluxes becomes

[TABLE]

We expand this as

[TABLE]

In the second line we used the fact that the twist matrix $T_{{\rm YB}}$ has no dependence on the winding type coordinates and in the third line we used the fact that the twist matrix depends only on the coordinates along the isometry directions. (Recall that the doubled coordinate $A$ decompose as ${}^{A}=(_{\alpha},\ ^{\alpha})$, where $\alpha=(I,a)$ with the indices $I$ labelling the isometry directions.)

Using the properties of the $O(d,d)$ matrices in general (see Appendix A), and the structure of the YB matrix in particular, we find that the only non-vanishing fluxes are

[TABLE]

and

[TABLE]

Note that the second equality in (4.23) above holds because the components $\Omega^{JK}_{\ \ I}$ and $\Omega^{K\ J}_{\ I}$ are zero due to the structure of the YB matrix (2.20). Now, using equations (4.18) and (2.12) we find

[TABLE]

which gives

[TABLE]

We can also compute the Q-flux by using (4.18) and (2.12):

[TABLE]

Now recall that $R^{IJ}$ are the components of an R-matrix that satisfies the CYBE (4.7) and the relation (4.11). As a result, the dressed R-matrix $R_{g}$ (2.3) also satisfies (4.7) and (4.11). Using these in (4.25) and (4.26) respectively we obtain

[TABLE]

Therefore, we conclude that YB deformation is a process which takes a solution of DFT with geometric flux associated with the isometry group $G$ and deforms it to another solution of DFT with vanishing R-flux and non-vanishing Q-flux. The non-vanishing Q-flux is given by the structure constants of the dual Lie algebra $\mathbf{g}_{R}^{*}$ of $\mathbf{g}_{R}$ determined by the R-matrix $R$. This is in agreement with the results of [51], which finds that the YB deformation of the Minkowski space and the background $AdS_{5}\times S^{5}$ induces Q-flux on the resulting backgrounds, arising from a non-trivial monodromy in the so called $\beta$-field around closed cycles. What we find here by using the framework of GDFT agrees with their result and extends it for YB deformation of any GS sigma model. Also, using the framework of GDFT reveals the role of the Q-flux as the structure constants of a Lie algebra dual to the isometry algebra. The Q-flux of deformed backgrounds and its role as the dual structure constants to geometric flux is also studied in [23], but in the context of Poisson Lie duality of sigma models. However, for the YB deformation, this feature of the deformed models cannot be captured in [23].

Before we move on to the discussion of the non-unimodularity condition for the R-matrix in the next subsection, we would like to make a comment. In obtaining the equation (4.27), we used the fact that if the CYBE (4.7) is satisfied by the R-matrix (and hence by the dressed R-matrix), then the R-flux computed in (4.25) vanishes. In fact, the converse statement is also true: if we demand the R-flux to vanish161616Note that this is in fact a reasonable thing to demand, as the non-vanishing R-flux is usually a sign of non-geometry, even locally., then the CYBE must be satisfied by the R-matrix that determines the YB deformation. Indeed, using the fact that the structure constants $C_{IJ}^{\ \ K}$ are antisymmetric in their lower indices we see that the equation $R^{IJK}=0$ for the R-flux is equivalent to the CYBE (4.7) for the dressed R-matrix.

4.3 The Dilaton Field and the Generalized Supergravity

Equations

In this section, we analyze the case when the R-matrix that determines the YB deformation is non-unimodular. As we discussed in the introduction, unimodularity of the R-matrix is required for the homogeneous YB model to be a solution of supergravity equations. On the other hand, when it is non-unimodular, the model is known to be a solution of GSE [65].

The condition for unimodularity presented in [65] is171717This also means that the quasi-Fobenius subalgebra of $\mathbf{g}$ determined by $R$ is unimodular, see [65].:

[TABLE]

Since the Lie algebra $\mathbf{g}$ we start with is unimodular so that $C_{IJ}^{\ \ I}=0$ in any basis, a careful examination of (4.11) shows that the condition (4.29) is satisfied if and only if the structure constants $\tilde{C}^{IJ}_{\ \ K}$ are traceless. This then implies that the R-matrix is unimodular if and only if the Q-flux generated by the YB matrix is traceless, see (4.28). This is in agreement with the results of [51], which also finds that the non-unimodularity of the R-matrix is measured by the trace of the Q-flux.

Now, assume that condition (4.29) is not met, that is, the R-matrix is not unimodular. As a result, the Q-flux is trace-full. Looking at (B.7), one immediately sees that the trace of the Q-flux is just the fluxes $f^{I}$ defined there. Indeed,

[TABLE]

since all other components of $\Omega$ are zero due to the structure of the YB matrix. Note that this means, by using (4.28) that

[TABLE]

As a result, when the R-matrix is not unimodular, the fluxes $f^{I}$ are non-vanishing and this contributes to $\eta^{I}$, whose definition is given in (B.5). However, it is well-known that the GDFT action with non-vanishing $\eta_{A}$ is not consistent. This follows from the requirement of gauge invariance of the GDFT action. In the case of the NS-NS sector, this analysis was made in [44]. For the RR sector, it was shown in [45] that the RR sector of the GDFT action is gauge invariant, only when the fluxes $\eta^{A}$ vanish, see equation (4.66) there. In order to satisfy this condition, the $f^{I}$ part in (B.5) should be compensated by a non-trivial dilaton anzats. A similar situation was considered in [25] and [73]. Rewriting (B.5) in components, we see that we need to have

[TABLE]

When the twist matrix is equal to the YB matrix $U^{-1}=T_{{\rm YB}}$, we have $(U^{-1})^{IJ}=\eta\Theta^{IJ}$, and $(U^{-1})^{I}_{\ J}=\delta^{I}_{\ J},\ (U^{-1})_{I}^{\ J}=\delta_{I}^{\ J},\ (U^{-1})_{IJ}=0$. Plugging this in (4.32) we get

[TABLE]

Combining (4.35) and (4.34) we obtain

[TABLE]

In other words, $\sigma$ is linear in the winding type coordinates and does not depend on the standard coordinates. Then, the generalized dilaton field is of the form:

[TABLE]

Due to the linear dependence of the generalized dilaton field on the winding type coordinates181818Note that, due to the appearance of the term $e^{-\sigma(Y)}$ in the anzats (B.3), the spinor field $F=e^{-B}\not{\partial}\chi$ also has a dependence on the winding type coordinates. The other DFT fields ${\cal H}$ and $\mathbb{S}$ have no dependence on winding type coordinates. For more details on a similar situation, see [73]. we are now in the generalized supergravity frame discussed at the end of section 4. It was shown in [46] and [48] that the field equations of DFT in this frame reduce to GSE, where the components $I^{m}$ of the Killing vector field $I$ that defines the deformation in GSE is identified with $d^{J}$, where

[TABLE]

see equation (3.51) of [48]. When we compare (4.36) with (4.37), we see that the components of the Killing vector field $I$ are given by $-f^{I}/2$ in our formalism. Note that this is in agreement with the results of [34] (see equation (3.63) in that paper), but disagrees with a factor of 2 with the result $I^{i}=f^{j}_{ij}$ of [72] obtained for NATD. The source of this discrepancy was discussed and resolved in [35], as will discuss shortly.

Let us also discuss the consistency of our results with those of [41] and [53]. Looking at (4.23), one sees that the trace of the Q-flux is equivalent to the divergence of $\Theta$:

[TABLE]

where $\Theta^{IK}=\eta(R_{g})^{IK}$ is given in (2.12). Recall we discussed in section 2.1 that $\Theta^{IK}$ is in fact equivalent to the noncommutativity parameter $\Theta_{NC}$ of the papers [58], [59] (see footnote (7)). In the papers [41], [53] the divergence of $\Theta_{NC}$ is identified with the Killing vector field $I$ that defines the GSE, see equation (3) in [53]. In our approach based on the framework of GDFT, (4.38) equates the divergence of $\Theta^{IK}$ to the trace of the Q-flux, which arises as a measure of the non-unimodularity of the R-matrix. On the other hand, as we discussed above, the trace of the Q-flux is equivalent to $f^{I}$, which are the type of fluxes that force the YB deformed fields to lie in the generalized supergravity frame in which the field equations of DFT reduce to GSE with $I^{I}=-\frac{1}{2}f^{I}$. To summarize, we have the following relations in our framework:

[TABLE]

Comparing (4.39) with $I^{I}=\nabla_{K}\Theta_{NC}^{KI}$ presented as equation (3) of [53], we see that there is a mismatch with a factor of 2. This discrepancy is in fact caused by the factor 2 mismatch mentioned at the end of the previous paragraph between the results for the components of the Killing vector field $I$ found in [34] (and here) and in [72]. A resolution for this discrepancy is introduced in [35]191919We thank the anonymous referee for informing us on the discussion in [35] on how to resolve the discrepancy between the results of [34] and of [72]., which involves a field redefinition for the vector field $I$, as described in equation (6.67) of [35]. They redefine the vector field $I$ so that $I^{J}=d^{J}\rightarrow I^{J}=\frac{1}{2}\eta\tilde{C}^{IJ}_{\ \ I}+d^{J}$ , where $d^{J}$ is as in (4.37)202020We have rewritten equation (6.67) of [35] in our conventions. We expand the Killing vector field $I$ with respect to left invariant vector fields so that $I=I^{J}\partial_{J}=I^{J}l^{i}_{\ J}\partial_{i}$. What we call $l^{i}_{\ J}$ here is called $v_{a}^{m}$ in [35]. Also, we need the deformation parameter $\eta$ in our transformation, whereas it is not needed in [35], where they work in the setting of Poisson Lie T-duality.. Then, $d^{J}=I^{J}-\frac{1}{2}\eta\tilde{C}^{IJ}_{\ \ I}=I^{J}+\frac{1}{2}f^{J},$ where we have used (4.31). Comparing it with equation (4.36) above, we see that this gives $-\frac{1}{2}f^{J}=d^{J}=I^{J}+\frac{1}{2}f^{J}$ which gives $I^{J}=-f^{J}$ as in equation (6.71) of [35]. Plugging this in (4.39), we obtain

[TABLE]

which is exactly the same as equation (3) of [53]212121Recall that $\partial_{K}=l^{i}_{\ K}\partial_{i}$, which is indeed a covariant derivative on a group manifold..

Recall that the fields in the homogeneous YB model are known to form a solution of GSE, when the R-matrix is not unimodular. The discussion above shows that for this case, the deformed fields are forced to lie in the frame in which these equations can be embedded in field equations of DFT. Hence, viewing YB deformation as a solution generating transformation in DFT provides a nice framework in which the unimodular and non-unimodular cases can be considered simultaneously.

5 Conclusion

In this paper, we studied the homogeneous YB deformation proposed in [24] for a generic GS sigma model. We showed that the deformation in the target space fields, including the RR fields is generated by a non-constant $O(d,d)$ transformation, generalizing previous results obtained by [49] and [50]. The associated matrix, which we call the YB matrix, is determined by an R-matrix satisfying the CYBE. The coordinate dependence comes from extending this fixed R-matrix to a dressed R-matrix on the whole group manifold (of the isometry group) by the adjoint action. Since the adjoint action is an automorphism of the Lie bracket, the resulting dressed R-matrix also satisfies the CYBE, whenever the initial R-matrix does. Relating this to the open-closed string map approach of [53],[40],[41], we see that their bi-Killing anzats for the noncommutativity parameter arises naturally from this construction with the adjoint action.

The YB matrix is related by a constant $O(d,d)$ transformation to the matrix found in [25] that generates NATD backgrounds, solidifying the relation between NATD and YB deformations. It also provides a nice framework to see how the homogeneous YB deformation generalizes the Lunin-Maldacena deformation, generated by TsT transformations. Note that the TsT transformation was generalized in [64] to a solution generating transformation for 11 dimensional supergravity. Using the formalism there and the results we obtain here, we expect it to be possible to also generalize the YB deformation to a solution generating transformation in 11 dimensions. We would like to note that generalized YB deformations in 11 dimensions has been studied very recently [74] within the framework of Exceptional Field Theory, a formalism which provides a U-duality covariant formulation for 11 dimensional supergravity.

Besides making calculations easier, our approach gives a natural embedding of the homogeneous YB model in DFT. This enabled us to show that the YB deformed fields can be regarded as duality twisted fields in the context of GDFT. We computed the fluxes associated with the YB matrix and showed that the conditions on the R-matrix determining the YB deformation manifested themselves as conditions on the fluxes on the GDFT side. More precisely, we showed that YB deformation is a process which takes a solution of DFT with geometric flux associated with the isometry group $G$ and deforms it to another solution of DFT with vanishing R-flux and non-vanishing Q-flux given by the structure constants of the dual Lie algebra determined by the R-matrix. We also showed that for YB deformations based on a non-unimodular R-matrix, the deformed fields are forced to lie in a frame in which the generalized supergravity equations can be embedded in DFT. Hence, viewing YB deformation as a solution generating transformation in DFT provides a nice framework in which the unimodular and non-unimodular cases can be considered simultaneously. Our approach based on GDFT also enabled us to discuss the relation between the non-unimodularity of the R-matrix, the trace of the Q-flux and the generalized supergravity equations. We hope that the approach we pursue here contributes towards a better understanding of the exciting relation between the CYBE and (generalized) supergravity.

Acknowledgments

We would like to thank E. O Colgain, N. S. Deger and M. M. Sheikh-Jabbari for comments and discussions. This work has been partially supported by the Turkish Council of Research and Technology (TÜBİTAK) through the ARDEB 1001 project with grant number 114F321. We also acknowledge the financial support of Scientific Research Coordination Unit of Istanbul Technical University (ITU BAP) under the project TGA-2018-41742.

Appendix A O(D,D) and Its Action on Curved Backgrounds

Let $T$ be a matrix in $O(d,d,R)$. Then

[TABLE]

where $\eta$ is the $O(d,d)$ invariant metric with

[TABLE]

where $I_{d}$ is the $d\times d$ identity matrix. Writing the equations above in indices for the YB matrix and its inverse we get:

[TABLE]

In indices $\eta$ has the following form:

[TABLE]

If we write $T$ as

[TABLE]

then, (A.1) implies

[TABLE]

T can be embedded in $O(D,D,R)$ as follows [63]:

[TABLE]

where $\hat{a},\hat{b},\hat{c},\hat{d}$ are $D\times D$ matrices defined below:

[TABLE]

The action of the $O(D,D)$ matrix $\hat{T}$ on the background matrix $E$ is defined as below [63]:

[TABLE]

The transformed metric and the transformed B-field are read off from $E^{\prime}$ as

[TABLE]

Appendix B GDFT and the Fluxes

GDFT is obtained from a Scherk-Schwarz reduction of DFT with the following duality twisted reduction anzats: [45]:

[TABLE]

Here, $X$ denote collectively the coordinates of the reduced theory. The $Y$ coordinates are the internal coordinates, which will be integrated out eventually. One can further decompose these coordinates into dual and standard coordinates as $Y=(\tilde{y},y)$ and $X=(\tilde{x},x)$. The twist matrix $U(Y)$ is an element of $O(D,D)$ and $\rho(S)=U^{-1}$, where $\rho$ is the double covering homomorphism: $\rho:Pin(D,D)\rightarrow O(D,D)$. $B(X,Y)$ in (B.3) is read off from the antisymmetric part of ${\cal H}(X,Y)$ in (B.1).

The reduced theory is determined by the so called fluxes $f_{ABC},\ \eta_{A}$, which are defined as below:

[TABLE]

[TABLE]

Note that $\Omega_{ABC}$ are antisymmetric in the last two indices: $\Omega_{ABC}=-\Omega_{ACB}$. We also make the following definition

[TABLE]

For a consistent reduction, constraints that should be obeyed by the twist matrices are as follows [42]-[44]:

[TABLE]

where $g$ is any of the DFT fields (${\cal H},\mathbb{S},\chi$). In addition to the above, the weak and the strong constraint has to be imposed on the external space so that

[TABLE]

for any fields or gauge parameters $V,W$ that has dependence on the coordinates of the external space only. Also, all fluxes must be constant for the consistency of the reduced theory, which ensures that the $Y$ dependence is completely integrated out in the reduced theory.

Appendix C Index Structure for the Twist Matrices

Our index conventions are as follows:

$M,N,\cdots$: Doubled coordinates; ${}^{M}=(_{\mu},\ ^{\mu})$,

$A,B,\cdots$: Doubled coordinates; ${}^{A}=(_{\alpha},\ ^{\alpha})$,

$\mu=(i,m),\ \ \ \mu=1,\cdots,10;\ \ i=1,\cdots,d,\ \ d={\rm dim}G$,

$\alpha=(I,a),\ \ \ \alpha=1,\cdots,10;\ \ I=1,\cdots,d$.

According to the embedding rules in (A.3), a twist matrix $T\in O(D,D,R)$, which only twists the $d$ isometry directions is of the following form:

[TABLE]

with $T_{m}^{\ a}=\delta_{m}^{\ a},\ T^{m}_{\ a}=\delta^{m}_{\ a},\ T_{m}^{\ I}=T^{m}_{\ I}=T_{i}^{\ a}=T^{i}_{\ a}=0$.

There are two types of twist matrices relevant for this paper: $L$ and $T_{{\rm YB}}$. Let us discuss the index structure of these matrices.

The twist matrix $L$ arises when the Lie group $G$ acts on the background by isometries. In this case we can write

[TABLE]

where $\theta^{i},i=1,\cdots,d$ are coordinates for $G$ and $\sigma^{a}=\delta^{a}_{\ m}dx^{m}$ and $\sigma^{I},I=1,\cdots,dimG$ are the left invariant 1-forms on $G$; $\sigma^{I}=l^{I}_{\ i}d\theta^{i}$. Similarly,

[TABLE]

Since the group $G$ acts on the background by isometries, all the $\theta$ dependence of the fields are encoded in $l^{I}_{\ i}$. We define the matrices $G(x,\theta),G(x),B(x,\theta)$ and $B(x)$ from

[TABLE]

where $\wedge$ denote the obvious wedge product of matrices and $d\textbf{x}$ and $\sigma$ denote the 10-vectors with components $(dx^{1},\cdots,dx^{10})$ and $(\sigma^{1},\cdots,\sigma^{d},dx^{d+1},\cdots,dx^{10})$, respectively. Then the background matrix $E=G+B$ has the following form:

[TABLE]

where $l$ is the $GL(10)$ matrix obtained by embedding in the $GL(d)$ matrix $l_{d}$ with components $(l_{d})^{I}_{\ i}=l^{I}_{\ i}$. The embedding is as described above so that $(l_{d})^{I}_{\ m}=l^{a}_{\ i}=0$ and $(l_{d})^{a}_{\ m}=\delta^{a}_{\ m}$. This is equivalent to the following $O(10,10)$ action :

[TABLE]

where $L$ is the following $O(10,10)$ matrix:

[TABLE]

One can show that (C.9) is equivalent to [75]:

[TABLE]

Since the twist matrix $L$ operates between curved and flat indices, the index structure of it is as follows:

[TABLE]

where we have identified

[TABLE]

From (C.10) we read off $L_{m}^{\ a}=\delta_{m}^{\ a},\ L^{m}_{\ a}=\delta^{m}_{\ a},\ L_{m}^{\ I}=L^{m}_{\ I}=L_{i}^{\ a}=L^{i}_{\ a}=0$ and $L_{i}^{\ I}=(L_{d})_{i}^{\ I}=l_{i}^{\ I},\ L^{i}_{\ I}=(L_{d})^{i}_{\ I}=l^{i}_{\ I}$, where $l^{i}_{\ I}l^{I}_{\ j}=\delta^{i}_{\ j}$.

One can show that (see [38]) (C.11) implies the following for the other DFT fields:

[TABLE]

where $\rho(S_{L})=L$.

From the formulas (B.5) with $(U^{-1})^{M}_{\ A}=L^{M}_{\ A}$, one finds the the only non-vanishing flux is the geometric flux

[TABLE]

This follows from the fact that $\sigma^{I}=l^{I}_{\ i}d\theta^{i}$ are left-invariant one-forms and as such they satisfy

[TABLE]

Now, let us discuss the index structure of the twist matrix $T_{{\rm YB}}$ (2.20). First, recall that $\Theta^{IJ}=l^{I}_{\mu}\Theta^{\mu\nu}l^{J}_{\nu}$ where $\Theta^{\mu\nu}=\eta k_{J}^{\mu}R^{JL}k_{L}^{\nu}$. Then we can write

[TABLE]

Hence, the index structure is as follows

[TABLE]

Then,

[TABLE]

with $T_{a}^{\ b}=\delta_{a}^{\ b},\ T^{a}_{\ b}=\delta^{a}_{\ b},\ T_{a}^{\ J}=T^{a}_{\ J}=T_{I}^{\ b}=T^{I}_{\ b}=0$. The embedded $d\times d$ matrix, which we have also called the YB matrix has components $T^{I}_{\ J}=\delta^{I}_{\ J},\ T_{I}^{\ J}=\delta_{I}^{\ J},\ T_{IJ}=0$ and $T^{IJ}=\Theta^{IJ}$.