Yang-Baxter deformations of Minkowski spacetime

We study Yang-Baxter deformations of 4D Minkowski spacetime. The Yang-Baxter sigma model description was originally developed for principal chiral models based on a modified classical Yang-Baxter equation. It has been extended to coset curved spaces and models based on the usual classical Yang-Baxter equation. On the other hand, for flat space, there is the obvious problem that the standard bilinear form degenerates if we employ the familiar coset Poincaré group/Lorentz group. Instead we consider a slice of AdS5 by embedding the 4D Poincaré group into the 4D conformal group SO(2, 4) . With this procedure we obtain metrics and B-fields as Yang-Baxter deformations which correspond to well-known configurations such as T-duals of Melvin backgrounds, Hashimoto-Sethi and Spradlin-Takayanagi-Volovich backgrounds, the T-dual of Grant space, pp-waves, and T-duals of dS4 and AdS4. Finally we consider a deformation with a classical r-matrix of Drinfeld-Jimbo type and explicitly derive the associated metric and B-field which we conjecture to correspond to a new integrable system.


Introduction
The integrability of 2D non-linear sigma models such as O(N )-invariant relativistic field theory [1] has been extensively studied in various contexts (for a comprehensive book, see [2]).A prominent example is the AdS/cft correspondence [3].The Green-Schwarz action of type IIB string theory on AdS 5 × S 5 , which is often abbreviated as the AdS 5 ×S 5 superstring, was constructed in [4] and has been shown to be classically integrable in the sense that the Lax pair exists (i.e.kinematical integrability) [5].
For related topics, see the review [6].
It was originally invented for principal chiral models based on the modified classical Yang-Baxter equation (mcybe).It was generalized to coset spaces [14] and then extended to the standard classical Yang-Baxter equation (cybe).In particular, squashed S 3 [7] and 3D Schrödinger spacetime [23] are associated with the mcybe and the cybe, respectively.
An important application of the Yang-Baxter sigma model description are integrable deformations of the AdS 5 × S 5 superstring.A q-deformation with a classical r-matrix satisfying the mcybe has been studied in [24].Jordanian deformations based on the cybe have been proposed in [25].In the latter case, there are a lot of classical r-matrices satisfying the cybe and some of them are associated with well-known gravitational backgrounds, such as Lunin-Maldacena-Frolov backgrounds [26,27], gravity duals for non-commutative gauge theories [28,29], Schrödinger spacetimes [30][31][32] and gravity duals for dipole theories [33][34][35][36][37], as shown in a series of works [38][39][40][41][42]. Very recently, the reality of the classical action has been revisited in [43] and a unified picture of deformed integrable sigma models has been provided in [44].It should be remarked that the Yang-Baxter sigma model approach works well even for non-integrable deformations.The case of AdS 5 × T 1,1 is known to be non-integrable [45].However, deformations of this background [26,46] can be reproduced as Yang-Baxter deformations [47].The above relations between gravity solutions and classical r-matrices may be called the gravity/cybe correspondence (for a brief summary, see [48]).
It would be very interesting to generalize a correspondence of this type to the case of flat space, instead of AdS 5 × S 5 .In particular, one expects a relationship between certain classical r-matrices and Melvin twists [49].Therefore, in this paper, we consider Yang-Baxter deformations of 4D Minkowski spacetime.For flat space, we however immediately encounter a problem if we employ the familiar coset Poincaré group/Lorentz group, namely the degeneracy of the bilinear form.A possible resolution1 is to consider instead a slice of AdS 5 by embedding the 4D Poincaré group into the 4D conformal group SO (2,4) .This embedding appears to work well, as we can reproduce the deformed metric and B-field associated to well-known backgrounds such as T-duals of Melvin backgrounds [52][53][54], Hashimoto-Sethi backgrounds [55], time-dependent backgrounds of Spradlin-Takayanagi-Volovich [56], the T-dual of Grant space [57], pp-wave backgrounds, and T-duals of dS 4 and AdS 4 , as Yang-Baxter deformations.Finally we consider a deformation based on a classical r-matrix of Drinfeld-Jimbo type [58][59][60] and explicitly derive the associated metric and B-field.
In the following, we embed most of these 4D σ-models into string theory.In doing so, one can hope to understand the integrability of said models from a string theory perspective.We start with the simplest ansatz, in which we assume the extra six dimensions to be flat (as resulting from a T 6 -compactification).Here, we will be looking for solutions which contain only a dilaton and no Ramond-Ramond-fields.It turns out that for a large class of the models studied, such a string theory embedding can be found.They all turn out to be T sT transformations of flat space.As we are starting from an integrable background (i.e.flat space with identifications) and T-duality preserves integrability [61][62][63][64][65], this fact explains the integrability of these models from a string theory point of view.
For other classes of r-matrices, this is not the case, but they can instead be related to known integrable models with dS 4 or AdS 4 target space via T-duality.In order to complete them to full string theory solutions, the introduction of RR-fields is necessary (in the case of dS 4 , even imaginary RR-form fields are needed), so these models cannot be captured with the simplest possible ansatz.A similar approach in the case of the AdS 2 × S 2 geometry has been discussed in [66], where type II theories are reduced to 4D theories plus a T 6 .The plan of this paper is as follows.In Section 2, we introduce Yang-Baxter deformations of 4D Minkowski spacetime.In Sections 3-5, we will provide some examples of classical r-matrices and the associated metrics and two-form B-fields.
There is in particular the class of models that correspond to T sT -transformations of flat space (Sec.3).In Section 4, we consider non-twist cases including the S-dual of the pp-wave background, T-duals of dS 4 and AdS 4 , and two-parameter deformations.
In Section 5, we extend the formulation from the cybe to the mcybe and then study a deformation with a classical r-matrix of Drinfeld-Jimbo type.Section 6 is devoted to conclusion and discussion.

Deformations of 4D Minkowski spacetime
In this section we consider Yang-Baxter deformations of 2D sigma-model actions whose target space is given by 4D Minkowski spacetime.

Coset construction of AdS 5 revisited
Let us remind ourselves of the coset construction of AdS 5 with Poincaré coordinates.
It is well known that AdS 5 can be represented by a symmetric coset as and its metric can be computed via a coset construction.To express the generators of the Lie algebras so(2, 4) and so (1,4) it is necessary to first introduce some quantities.
We first introduce the gamma matrices γ µ and γ 5 defined as One can introduce n µν and n µ5 defined as Then the Lie algebra so(2, 4) is spanned by the above quantities, and the subalgebra so(1, 4) is generated by Now we can compute the metric of AdS 5 with Poincaré coordinates by using a coset representative, where the p µ (µ = 0, 1, 2, 3) are defined as The left-invariant one-form A = g −1 dg can now be evaluated easily.The metric is defined as where P is a coset projector from so(2, 4) to so(2, 4)/so (1,4) and is defined as (2.9) The resulting metric is given by (2.10) Note that the AdS radius is set to 1 .

A conformal embedding of 4D Minkowski spacetime
Here we are interested in a coset construction of 4D Minkowski spacetime.However, there is the obvious problem that the standard bilinear form degenerates if we naively employ the usual coset ISO(1, 3)/SO (1,3) .To avoid this, it is convenient to represent 4D Minkowski spacetime instead as a slice of AdS 5 in Poincaré coordinates.
A possible representation of the group element g is the following: In comparison to (2.6), the radial coordinate z is not contained.In other words, to express 4D Minkowski spacetime, a section at z = 1 of AdS 5 has been taken at this stage: 4D Minkowski spacetime = AdS 5 z=1 . (2.12) Note here that the 4D Poincaré algebra iso(1, 3) and the 4D Lorentz algebra so(1, 3) are generated by the following generators, respectively, Thus it makes sense to use the generators p µ to parameterize the coset representative of ISO(1, 3)/SO(1, 3) as (2.11) .Eventually the left-invariant one-form is written as a linear combination of p µ .
By dropping γ 5 of P in (2.9) , we introduce the projector for 4D Minkowski spacetime by Then, it is straightforward to compute the metric, This result is the starting point of our argument in the following.
The definition of the projector (2.14) is justified as follows.From an algebraic point of view, we are able to realize 4D Minkowski spacetime as an embedded coset by considering the quotient of so(2, 4)/so(1, 4) by γ 5 .That is where the bilinear form of so(2, 4) is not degenerate.On the right-hand side, the appropriate coset projector turns out to be This is the reason why the 4D Minkowski projector P is given by (2.14) instead of the AdS 5 coset projector P in (2.9)2 .
The point is that we consider a conformal embedding of 4D Minkowski spacetime as in Eq. (2.16) and use the coset projector P in (2.14) to avoid the degeneracy of the bilinear form of the 4D Poincaré group ISO(1, 3) .After that, the projected one-form P (A) is expanded in terms of γ µ (µ = 0, 1, 2, 3) and the trace operation in the action leads to non-vanishing quantities as we will see in the next subsection.

Yang-Baxter sigma model for 4D Minkowski spacetime
Yang-Baxter deformations have only been discussed for curved backgrounds so far.
However, it is possible to generalize the formulation to Minkowski spacetime.
The deformed action is given by where A α = g −1 ∂ α g and g is given in Eq. (2.11) .Here η is a constant parameter and the action (2.18) is reduced to the undeformed one for η = 0 .The base space is 2D Minkowski spacetime with the metric γ αβ = diag(−1, 1) .The anti-symmetric tensor ǫ αβ is normalized as ǫ τ σ = 1 .The operator R g is defined as where a linear operator R : so(2, 4) → so(2, 4) is a solution of the cybe , The R-operator is related to the skew-symmetric classical r-matrix in the tensorial notation through where the classical r-matrix is given by satisfying the cybe, The generators a i , b i are elements of so (2,4).This means that the Yang-Baxter deformations are investigated within so (2,4) .
It is easy to extend this formulation to the mcybe, which will be discussed in Section 5.
It is worth remarking on the order of the operations of taking a slice at z = 1 and performing a Yang-Baxter deformation.Naively, it seems likely that the two operations are commutative.This is however not the case.The deformation may depend on the radial z-direction.For example, if the classical r-matrix contains γ 5 , which deforms the z-direction, then the two operations do not commutate.On the other hand, they commute for all the examples listed in Sec.3.6.The magnetic case for example corresponds to a slice of the gravity dual for non-commutative gauge theory [28,29] and in fact the classical r-matrix is identical to the one found in [39].

Embedding into string theory
Starting from a 2D integrable model, it is natural to ask whether it can be embedded into string theory.This requires in general supplementing the fields we can read off from the action by other fields in order to solve the one-loop beta function of string theory.As a result of Klimcik's procedure, we always get a 4D metric and B-field.
For the missing internal directions, we make the simplest ansatz of assuming them to be flat, as resulting from an internal T 6 with vanishing B-field.Adding only a dilaton will be sufficient to produce a solution for a large class of deformations, which all turn out to be related via T-duality to flat space.The minimal ansatz for the dilaton is that it preserves the same symmetries as the known fields.When the r-matrix only contains p µ or n µν , this is sufficient to find a solution.More general rmatrices which contain also γ 5 require also Ramond-Ramond fields in order to solve the equations.We have not actually been able to find the string theory embedding for all of these cases.

T sT -duals of flat space
In this section, we have collected a class of models for which the simplest ansatz for a string theory embedding can be used, in which we assume the extra six dimensions to be flat (as resulting from a T 6 -compactification) and we introduce only a dilaton field.They all turn out to be T sT -transformations of flat space with identifications.
In cases in which the r-matrix contains a p µ , one or both T-dualities act trivially.
We will present the examples in the following order.We start with a rather

T-dual of Melvin Background
Here we will consider an explicit example of a Yang-Baxter deformation.Our first example is the classical r-matrix The associated geometry is a Melvin twist of 4D Minkowski spacetime.
Let us first compute the explicit form of A α and a deformed current J α defined as First of all, P (A α ) is evaluated as Note here that P (A α ) can also be expressed as Then, by plugging (3.3) into (3.4) and solving the four equations, the explicit form of P (J α ) can be determined to be where the components of j µ α are given by ) . (3.6) Thus the classical action can be rewritten as where we have performed a coordinate transformation, From this action, one can read off the metric and B-field, It should be remarked that the Yang-Baxter deformations cannot reproduce the associated dilaton, although it may be possible to perform a supercoset construction in principle.However we can embed the background in string theory observing that the one-loop beta function vanishes by adding a dilaton [52][53][54] The background obtained in this way is a T sT transformation of flat space on the torus generated by x 3 and θ, i.e. the result of the following chain of transformations: 1. T -duality in x 3 ; 2. shift θ → ηx 3 + θ; 3. T -duality in x3 .Consistency requires x3 to be periodic with period x3 ≃ x3 + 2π/η and x 3 with period x 3 ≃ x 3 + α ′ η/(2π).

T-dual of Grant space
Let us consider the classical r-matrix The derivation is almost the same as in the previous subsection, hence we will not repeat the detailed explanation but simply present the deformed metric and B-field.
The associated metric and B-field are given by Here the light-cone coordinates are given by and we have introduced new coordinates given by The metric and B-field in (3.12) agree with (2.7) and (2.8) in [57] for x 2 = 0 .
The background can be embedded in string theory by adding a dilaton which solves the beta function equations.In fact this is an exact string theory, resulting from a T sT transformation of flat space on the torus generated by x 1 (which has periodicity α ′ η/(2π)) and φ.

Time-like Melvin circle
Let us consider the classical r-matrix This r-matrix is Abelian and the associated metric and B-field are given by The Ricci scalar curvature is negative, and this background has a curvature singularity at r = 1/η.This background is related to the previous one by analytic continuation By supplementing the fields by a dilaton which solves the beta function equations, we obtain a T sT -dual of flat space on the torus x 0 , θ .Note that the Melvin circle x 0 has periodicity x 0 ≃ x 0 + α ′ η/(2π).

PP-wave background
Let us consider the classical r-matrix The associated deformed background is where we have introduced the polar coordinate system for x 1 and x 2 given by This is a pp-wave background which can also be understood as a generalization of a T sT transformation obtained as 1. a T-duality from θ to θ, followed by 2. the shifts x 0 → η θ + x 0 , x 3 → −η θ + x 3 and the final 3. T-duality from θ to θ.

Hashimoto-Sethi background
We next consider the classical r-matrix r = 1 2 √ 2 p 2 ∧ (n 01 + n 13 ) . (3.24) The resulting metric and B-field are given by (3.25) Note that this background depends on the light-cone time x + explicitly.The associated dilaton to complete the string embedding is taken to be The metric and B-field (3.25) agree with those of the Hashimoto-Sethi background.To show this agreement, one has to introduce new coordinates, This reproduces the expression in Eq. ( 25) of [55] where the background is shown to be the result of a T sT transformation.

Spradlin-Takayanagi-Volovich background
Let us here consider the classical r-matrix r = 1 2 n 12 ∧ n 03 . (3.28) Then the associated metric and B-field are given by (3.29) The light-cone coordinates are given in (3.13) .This background (3.29) is really time-dependent and has a curvature singularity.
By using the coordinates in (3.14), one can rewrite the expressions in (3.29) as Note that the coordinates in (3.14) do not cover the whole x 0 -x 3 plane and the background (3.30) contains no singularity.Then the metric and B-field in (3.30) agree with those of (6.1) in [56].This is a time-dependent background realized by a T sT transformation of Minkowski spacetime on the torus generated by φ and θ.
The associated dilaton is (3.31)

Locally flat spaces
In this section, we consider classical r-matrices of the type In these cases the associated geometries are locally flat and the B-field is closed.The backgrounds are nevertheless non-trivial and are associated to non-commutative field theories [67].The following is a list of possible r-matrices.
(1) Magnetic flux r = 1 2 p i ∧p j (i = j, i, j = 1, 2, 3) (Melvin Shift Twist) With this r-matrix, the following metric and B-field are obtained: This background is obtained via a T sT transformation on the torus x i , x j .
A multi-parameter generalization of this type is obtained using the r-matrix where a 1 , a 2 , a 3 ∈ R are deformation parameters.The resulting metric and B-field are given by This r-matrix leads to the metric and B-field This corresponds to turning on an electric flux and the range of η is restricted to −1 < η < 1 .Once more we can obtain this background via a T sT transformation on the torus x 0 , x i .The background is related to the non-commutative open string theory (ncos) [68].
A multi-parameter generalization of this type is obtained by the r-matrix r = 1 2 p 0 ∧ (a 1 p 1 + a 2 p 2 + a 3 p 3 ) , (3.37) where a 1 , a 2 , a 3 ∈ R are deformation parameters.The resulting metric and B-field are given by In this case, we obtain the following metric and B-field (3.39) This corresponds to turning on a light-like flux resulting from a T -duality in x 2 , opposite shifts in x 0 and x 3 and a final T -duality in x2 .The background is related to the light-like non-commutativity argued in [69].

Non-Twist cases
So far, we have discussed the cases associated with T sT transformations.In this section, we shall consider the other cases, for instance, including a single T-duality, or a chain of T sT -transformations followed by an S-duality.We will refer to the cases as non-twist cases.
As the first example, we revisit the pp-wave case and argue its relation to S- Let us remark that the presence of γ 5 in the all of the r-matrices appearing in this section plays a crucial role.It can intuitively be interpreted as a deformation of the radial direction of AdS 5 and indicates that the location of the slice is moved.This is also the reason why the integrability of the T sT -cases of the previous section is obvious, while in general it is not automatic in the non-twist cases, except for special cases such as the T-duals of dS 4 and AdS 4 .

PP-wave revisited -S-duality
In the previous section, we have considered the pp-wave background (3.22).We will show that the same metric can be reproduced by considering the r-matrix The resulting metric and B-field are given by As noted above, the metric is the same as in (3.22), but the B-field is different.In particular, the B-field carries an index for the radial direction r , which is not a U (1)direction.Hence it does not seem likely that the background (4.2) can be obtained via a simple twisting procedure, as opposed to (3.22).This difference comes from the fact that the second r-matrix (4.1) is non-Abelian, while the first one (3.21) is Abelian.This non-Abelian r-matrix appears to be related to a chain of dualities.In fact, the r-matrix (4.1) is identical to the one in (2.17) of [41] that has been employed to study a deformation of AdS 5 × S 5 .It has been shown in [42] that this r-matrix corresponds to a duality chain of three TsT transformations and an S-duality3 .Thus non-Abelian r-matrices appear to be related to intricate duality chains.
As another possibility, non-Abelian r-matrices may indicate deformations beyond the standard twists or dualities.To elaborate this statement, it would be interesting to study more non-trivial examples of classical r-matrices, as we will show in the next subsection.

T-dual of dS 4
Now we consider the classical r-matrix Note that this r-matrix contains γ 5 and leads to the metric and B-field given by where we have introduced new coordinates r , θ and φ through Note here that the above B-field can be rewritten in the form of a total derivative.
The deformed background (4.4) is already simple.However, in order to understand the background well, we can perform a T-duality along the x 0 -direction 4 .The resulting background is given by Note that the B-field has disappeared now.Then, by performing a coordinate transformation from x 0 to t , one can reproduce the well-known metric of dS 4 in static coordinates, with a cosmological horizon at r = 1/η .Thus we have shown that the background (4.4) is nothing but a T-dual of dS 4 .The dS 4 geometry cannot be realized as a twist of 4D Minkowski spacetime, hence it is quite remarkable that the T-dual of dS 4 has been obtained as a Yang-Baxter deformation.

T-dual of AdS 4
As another example, let us consider the classical r-matrix This r-matrix also contains γ 5 .The associated metric and B-field are given by where we have introduced new coordinates t , θ and φ through Note here that the B-field can be recast as a total derivative.
As in the previous case, it is nice to perform a T-duality along the x 1 -direction.
Then the resulting background is given by Now the B-field has disappeared.Let us perform a coordinate transformation, Then the resulting metric is given by By replacing the coordinates (with a double Wick rotation) by one can obtain the standard metric of AdS 4 with the global coordinates Note that η 2 measures the curvature.

More complicated backgrounds
Here, we give a list of classical r-matrices for which the corresponding backgrounds have not been identified yet.All of the classical r-matrices we consider here satisfy the cybe given in (2.23) .Nevertheless, this does not guarantee integrability without explicitly constructing the Lax pair.In the previous cases, the string embedding assured integrability via T-duality to integrable backgrounds.We conjecture that also the following examples are new integrable models, but at this point have no explicit proof.
In the following, we assume that the light-cone coordinates are defined in (3.13) and x 1 , x 2 are rewritten with the polar coordinates given in (3.23).

One-parameter deformations
Let us consider more complicated one-parameter deformations.
The first example is and the resulting metric and B-field are given by and the metric and B-field are given by This is an Abelian r-matrix in which γ 5 is not contained but n 05 and n 35 carry the index 5.Note that the Ricci scalar and Riemann square vanish and x + = ±|η| −1/2 are just coordinate singularities.

Two-parameter deformations
It may be interesting to consider two-parameter deformations.
The first example is given by where s 1 and s 2 are arbitrary constant parameters.This r-matrix has been used to study a two-parameter deformation of AdS 5 in [41].The associated metric and B-field are given by When s 1 = −s 2 = s , the metric simplifies to A T-duality along the x 0 -direction leads to the geometry of dS 3 × R .
The second example is given by where a, b ∈ R are deformation parameters and E ≡ 4 i=1 E ii .The resulting metric and B-field are given by The third example is given by where a 1 , a 2 ∈ R are deformation parameters.The resulting metric and B-field are given by which satisfies the modified Yang-Baxter equation, where M, N ∈ so (2,4).In comparison to the cybe in (2.20) , the right-hand side of (5.2) is modified 6 .The r-matrix (5.1) was utilized to study an integrable deformation of the AdS 5 × S 5 superstring [24].The deformed metric and B-field were explicitly computed in [71].
The computation scheme is essentially identical because we do not mention the the associated Lax pair and the kappa transformation related to the Green-Schwarz string action.
The resulting metric and B-field are given by (5.3) Here we have performed a coordinate transformation, x 0 = r sin θ sinh t , x 1 = r cos θ cos φ , x 2 = r cos θ sin φ , x 3 = r sin θ cosh t , (5.4) and rescaled η → η/2 .It is worth noting that the metric in (5.3) would be regular as opposed to the case of the deformation of AdS 5 .The scalar curvature has no singularity.We are now working on a slice of AdS 5 at z = 1 which is why the singular surface of the metric identified in [71] has been avoided.

Conclusion and Discussion
In this paper, we have discussed Yang-Baxter deformations of 4D Minkowski spacetime using a conformal embedding of the spacetime into AdS The backgrounds that we have identified are summarized in Table 1.
There are many open questions.The undeformed case is trivially integrable and hence the integrability should be preserved under Yang-Baxter deformations.It would be interesting to consider what happens to the Lax pairs and to study the associated algebras.
It would be most important to generalize our argument to 10D Minkowski spacetime in order to extend our argument to a consistent string theory.For this purpose, we have to consider the 10D conformal group SO(2, 10) and realize 10D Minkowski spacetime as a slice of 11D AdS space.We expect that in this case 10D supersymmetric configurations like the fluxtrap backgrounds [72] could be reproduced as Yang-Baxter deformations.
As opposed to deformations of AdS 5 × S 5 , our computations are very simple and hence it is easier to generalize the Yang-Baxter deformations.A possible direction is a supercoset construction including spacetime fermions.Furthermore, it would be very interesting to investigate some quantum aspects such as the relation between string spectra and Yang-Baxter deformations.
general example containing both p and n in the r-matrix, which corresponds to the T-dual of Melvin background.The r-matrices of the next two examples presented in Sec.3.2.1 and Sec.3.2.2have the same structure of p ∧ n and correspond to the T-dual of Grant space and T-duals of Minkowski spacetime.The next example, the pp-wave background in Sec.3.3, has a more complicated r-matrix with the structure r ∼ (p 0 − p 3 ) ∧ n.Next we discuss the Hashimoto-Sethi background which has two ns appearing in the r-matrix, r ∼ p µ ∧ (n νρ + n σγ ).The next two examples have again a simpler structure of the r-matrix, with the Spradlin-Takayanagi-Volovich background, where only two n's appear in Sec.3.5, and with models with only p's in the r-matrix appearing in Sec.3.6.All the models considered in this section can be embedded in string theory as T sT -duals of flat space.Let r = a ∧ b and let α and β be the dual coordinates to a and b in the sense of the Lie algebra (concretely, if a = p µ then α = x µ and if a = n µν then α is the angle in the plane x µ , x ν ).Impose periodic boundary conditions on α and β and consider the torus with parameter τ generated by α, β .The effect of the deformation is to transform τ into τ η = τ /(1 + ητ ).In string theory this is realized via the following sequence of transformations (T sT ): 1. T -dualize α → α; 2. shift β → η α + β; 3. T -dualize α → α.[26].In the special case of a = p µ and b = n ρσ , this corresponds to the Melvin twist where x µ is the Melvin circle.
duality.Then we will reproduce T-duals of dS 4 and AdS 4 respectively, as Yang-Baxter deformations.These examples may be regarded as non-trivial examples of our procedure.Finally we provide more complicated examples including two-parameter deformations.

( 4 . 27 )
When a 1 = −a 2 , the background in(3.22)  is reproduced.Thus this background can be regarded as a deformation of the pp-wave background.5A deformation of Drinfeld-Jimbo typeSo far, we have considered classical r-matrices satisfying the classical Yang-Baxter equation (2.23) (or (2.20)).Here, as an exceptional case, let us consider a classical r-matrix of Drinfeld-Jimbo (DJ) type,

5 .
Via this procedure we have succeeded in reproducing the metric and B-field of well-known backgrounds such as T-duals of Melvin backgrounds, Hashimoto-Sethi backgrounds, time-dependent backgrounds of Spradlin-Takayanagi-Volovich, the T-dual of Grant space, pp-wave backgrounds, and T-duals of dS 4 and AdS 4 .For a large class of them, an embedding into string theory exists as a T sT -transformation of flat space with identifications, where the extra six dimensions are flat and only a dilaton is introduced.Finally we have studied a deformation by using a classical r-matrix of Drinfeld-Jimbo type and have derived the associated metric and B-field explicitly.

Table 1 :
A catalog of classical r-matrices and the associated backgrounds.