Asymmetric $\lambda$-deformed cosets

We study the integrable asymmetric $\lambda$-deformations of the $SO(n+1)/SO(n)$ coset models, following the prescription proposed in \cite{AsyLambda}. We construct all corresponding deformed geometries in an inductive way. Remarkably we find a $Z_2$ transformation which maps the asymmetric $\lambda$--deformed models to the symmetric $\lambda$--deformed models.


Introduction
Integrability plays a key role in obtaining exact results in field theories and string theory. After witnessing a remarkable progress in understanding the integrable string theories it is clearly interesting and important to construct new integrable models. In the last ten years, based on sigma models on group or coset manifolds powerful tools such as η- [2,3,4] and λ-deformations [5,6,7] have been developed to investigate this issue.
The original λ-deformation [5] gives an interpolation between the (gauged) Wess-Zumino-Witten (WZW) model [11]and the non-Abelian T-dual of the Principle Chiral model (PCM) [12]. The deformations of AdS p × S p have been successfully constructed in [6,7]. The novel idea behind the construction of λ-deformation is to combine different integrable models through a gauging procedure. Following the same idea, various of generalizations of λ-deformation have been proposed [8,9,10]. Recently, the authors in [1] introduced a new generalization named asymmetric λ-deformation by modifying the gauging procedure. The key observation in [1] is that different choices of anomaly free gauges can be made in the deformation. It is somehow similar to the gauged WZW model with U(1) symmetry, in which case one can choose either the vector or axial gauge and the two resulted gauged theories are T-dual to each other [13]. Now the deformation breaks the axial-vector duality since the deformation destroys the isometries of the background.
To have a non-trivial asymmetric λ-deformation, the starting model has to possess a Lie algebra with non-trivial outer automorphism group.
In this article, we will study the asymmetric λ-deformation of the SO(n + 1)/SO(n) coset models, and pay special attention to the case that n is odd and the corresponding Lie algebra admits a Z 2 outer automorphism group. A physical motivation to study the deformation of this class of coset models is that the coset SO(n + 1)/SO(n) is isomorphic to S n and by an analytic continuation they can be transformed into AdS n . Then it would be possible to embed the deformed models into supergravity.
The paper has the following organization: In section 2, we review the construction of asymmetric λ-deformation introduced in [1]. To apply the construction, we need to find a suitable coset representative and explicit forms of the outer automorphism in this given representative. So in section 3, we first find the outer automorphism of the SO(n) group following [16] and then develop a gauge fixing scheme similar to [14]. After constructing the deformed models we prove the existence of a Z 2 transformation which maps asymmetric λ-deformed geometry to the symmetric deformed ones. As a by product, we provide a simple recursive method to construct the deformed geometries in our choice of coset representative. Some technical details are presented in the appendices.

Asymmetric λ-deformation
In this section, we briefly review the integrable asymmetric λ-deformation. Here we only focus on symmetric coset models. Similar constructions can also be applied to group and supercoset models. For the details we refer the original article [1]. To introduce the deformation of a coset model G/H, we begin with separating the generators T A of the group G into T a and T α corresponding to the subgroup H and the coset G/H respectively 1 , and then defining the Maurer-Cartan forms (2.1) The asymmetric λ-deformation of coset model G/H is constructed by performing three steps: (i) Combine S P CM (ĝ) on coset G/H with the S W ZW,k (g) on the same group G, where (ii) Gauge the group G whose G L action is given by where g 0 = exp(G A T A ) ∈ G andg 0 = exp(G AT A ) ∈ G have the same parameters G A but they are generated by different embeddings T A andT A of the subalgebra. These two embeddings are related by a linear transformation W , such thatT A = To avoid the gauge symmetry anomaly the transformation W has to be a metric-preserving automorphism of the Lie algebra, i.e., Non-equivalent choice of W is characterized by the outer automorphism group of the Lie algebra. In particular, the choice W = I which is usually called the vector gauge leads to the standard λ-deformation [5]. Therefore only when W is a nontrivial element of the outer automorphism group, following [1] we will call it axial gauge, the deformation can be potentially non-trivial.
(iii) Integrate out the gauge field and fix the gaugeĝ = I.
These procedures give the final action of deformed model: 5) where L, R and D are quantities defined in (2.1) and the operator Ω is given by The background geometry of the target space of this model is given by 2 , With the help of the identity DD T = 1 and 3 W W T = 1, the metric can be simplified as Substituting (2.6) and using the inversion formula of a block matrix, the metric can be cast into the form 2 The Kalb-Ramond field vanishes for a similar argument given in [14,8] 3 W can always be chosen to be diagonal. where Here e are the frames of the gauged WZW model, and the deformation is totally encoded in the matrix P .
In [1], the authors have studied the asymmetric λ-deformations of coset SL(2, R)/U (1) and showed explicitly that the construction leads to a new integrable model. In this paper, we will apply these procedures to the cosets SO(n + 1)/SO(n) for n = 2, 3, . . . .

Asymmetric λ-deformed SO(n + 1)/SO(n)
The λ-deformation of the cosets SO(n + 1)/SO(n) with λ-deformations for n = 2, 3, 4, 5 in the vector gauge cases have been constructed in [6,7]. These corresponding deformed geometries can be promoted to integrable backgrounds of string theory and their dynamical properties are also analyzed in [15].
In the following discussion we choose the generators of SO(n + 1) to be and embed the subgroup SO(n) as t A = (T n,n+1 ; T n−1,n , T n−1,n+1 ; ..., T 23 , ..., T 2,n+1 a ; T 12 , ..., T 1,n+1 Before performing the deformation procedures we need to solve the outer automorphism group first. Besides that the other preparatory work is to find the coset representative for SO(n + 1)/SO(n) in the axial gauge.

The outer automorphism group
The outer automorphism group of a simply connected group corresponds to the symmetry of its Dynkin diagram. For simple Lie algebras, only the Dynkin diagrams of types A n , D n , and E 6 admit non-trivial symmetries. Therefore for orthogonal groups only the groups SO(2n) with n ≥ 2 have non-trivial outer automorphism groups. The forms of the transformation W are computed in Appendix A. Up to inner automorphism the final results are Notice that W is the diagonal matrix and W = W T = W −1 . This choice of gauge can be viewed as a higher dimensional generalization of the axial gauge used in [17].

The coset representative
An element g n+1 ∈ SO(n + 1) can be decomposed as [18] g n+1 = H n t n (3.4) where and V is a n-vector. The SO(n) gauge rotation R n acts as (2.3) We pick a convenient gauge such that V ′ n = (v n−1 , 0, . . . , 0) such that t ′ n is invariant under the SO(n − 1) gauge rotation R n−1 and H ′ n ≡ g n ∈ SO(n). For g n , a similar decomposition and gauge choice lead to g n−1 and t ′ n−1 . Eventually we can fix all the gauge freedoms and end up with the coset representative The advantage of this coset representative is that it does not depend on the outer automorphsim transformation W . Another possible coset representative can be found in [14].

An Example: SO(4)/SO(3)
Before the generic discussion, let us study the simplest example in details. This example exhibits all the essential features of the general situations. The metrics of the corresponding gauged WZW model (λ = 0) on this coset have been found both with vector and axial gauge in [17]. For the gauged WZW model, it turns out the metrics with different gauges are connected via a coordinate transformation. We will show for λ-deformed model, under a coordinate transformation and an additional transformation of the parameter λ the two metrics with different gauges are related to each other. According to the general discussion (3.3), the non-trivial automorphism W is In order to compare with the results in [17], we parameterize the group SO(4) as

Vector gauge
In this gauge, the coset representative is given by setting θ = 0 in (3.11). Substituting the coset representative into (2.7) leads the metric In the two forms e 2 and e 3 one can recognize that (3.13) are the two frames of the vector gauged SO(3)/SO(2) model [17]. It is more convenient to introduce the new variables x = cos τ cos φ, y = sin τ cos φ. (3.14) With these new variables both e i and J αβ have relatively simpler expressions:

Axial gauge
In this gauge, the coset representative is given by setting τ = 0 in (3.11). Substituting the coset representative into (2.7) leads the metric In e 2 and e 3 again one can recognize that are frames of the axial gauged SO (3)/SO(2) model [17]. Similarly introducing the convenient variables x = cos θ sin φ, y = sin θ sin φ, (3.20) we have Comparing with results (3.12) (3.15), we find when λ = 0 these two metrics are equivalent up to a change of coordinates: The equivalence of vector gauged and axial gauged model is called"self-dual" in [17]. When λ = 0, the metric ds 2 A can be obtained from ds 2 V by performing (3.23) and a Z 2 transformation on the deformation parameter: ( 3.24) This additional Z 2 transformation is non-trivial for this model. One way to show this is to solve the spectrum of scalar on the deformed geometry following [15]. We will give a simple example in the next section. In the next section, we will provide this kind of Z 2 transformation which transforms the deformed geometries in vector gauge into the ones in axial gauge for all the deformed cosets SO(n + 1)/SO(n).

Scalar field
Even though the geometry corresponding to λ-deformed SO(4)/SO (3) coset has no isometries, an algebraic method based on group theory is proposed in [15]. In this appendix, we use their method to give a simple example showing that the Z 2 transformation on the parameter λ is physical. According to [15], the scalar field equation on the deformed geometry reduces to a second order differential equation known as the Heun's equation for polynomial ansatz Q(z): where the parameters (c, h) of the Heun's equation are given by where the matrix elements are defined by Considering the example with (L 2 , L 1 ) = (2, 1), (3.28) leads to which are not equivalent under the transformation λ → −λ.

The Z 2 transformation
In the section, we will not restrict ourselves to SO(2n)/SO(2n − 1) but consider the general coset SO(n + 1)/SO(n) model with the automorphism transformation even though for the case when n is even the transformation is just a gauge transformation.
The deformed geometries (2.7) depend on W in a rather complicate way. However we prove the deformed geometries can be transformed back to the symmetric λ-deformed (W = I) ones by the Z 2 transformation 4 : where the coordinates θ i of the target space are defined in (3.9). The proofs are presented in the Appendix B. One prediction of our results is that the spectrum of scalar field on the λ-deformed SO(2n + 1)/SO(2n) is invariant under λ → −λ since W is a gauge transformation in this case.
For the frames e α , since the expression of L has been shown in (B.11), we only need to consider (d 1 | n+1 − 1) −1 d 2 | n+1 . The block matrix inversion formula gives where * stands for the irrelevant elements. Substituting the identity (B.5) we get with the first case Combining the two facts we get 3.46) which complete our recursion relations.

Summary
In this article we explored the asymmetric λ-deformation introduced in [1]. For the SO(n + 1)/SO(n) coset model we found a Z 2 transformation (3.32) which maps the asymmetric λ-deformation to the symmetric λ-deformations . When the deformation parameter λ vanishes, these two different deformed models reduce to the gauged WZW models in the axial and vector gauge, respectively. The gauged WZW models in different gauges are dual to each other. We gave evidences to show that the deformation break this duality so that the asymmetric λ-deformation leads to new integrable models. Furthermore, we construct the resulting geometries for arbitrary n recursively (3.38), (3.41) and (B.11). It would be interesting to extend the current results to the cases of supergroups so that the integrable deformation of string theory on AdS p × S p can be constructed and studied.

Acknowledgments
The work was in part supported by NSFC Grant No. 11335012, No. 11325522 and No. 11735001.

A The outer automorphism of so(n)
For the Lie algebra, W is called the outer automorphism transformation if there is no where T ij are denoted by (3.1). According to the symmetry of the Dynkin diagrams of D n , the outer automorphism W are [16] W Notice that W is a linear transformation. Therefore, By (A.4) and commutators of SO(n) group we can get an outer automorphism transformation, As for so (4), which is isomorphism to so(3) ⊕ so (3). The generators are The commutators are The adjoint representations of the generators are where σ i are Pauli matrices in the adjoint representation, that is, (A.10) Consider an automorphism W , Under the basis J = (J + , J − ), However, by the definition of outer automorphism, there is no solution about W = e x A ad J A , so W is an outer automorphism. Under the basis T ij , .14) B Proof of the Z 2 transformation In this appendix we will prove that under the combination W and (3.32) the geometry (2.7) is invariant. Our main method is the mathematical induction. We begin with proving some import properties of matrix D AB defined in (2.1) under the coset representative (3.8): The matrix D can be decomposed as (B.6) Using the matrix (B.5), we can expression off-diagonal blocks with the diagonal blocks as For the diagonal blocks d 1 and d 4 we provide recursion relationships: cos θ n−1 I n−1 , R(θ) = cos θ − sin θ sin θ cos θ (B.8) and The frames e α Let us study how the frames The expression means that the element of the matrix in the (i,j) position is the element of L, whose position in L is the same as the position of T i,j in t A . So under the transformation (3.32) the forms transform as Next we focus on the term W a d 1 in the combination in which we already have tan θ n−1 → − tan θ n−1 under Z 2 . Using the recursion relation (B.8) and (B.9) , one can get cosθ n−2 −sinθ n−2 cosθ n−1 sinθ n−2 cosθ n−1 cosθ n−2 I n−2 cosθ n−2    cos θ n−1 d 2 | n = d 1 | n C n−2 tan θ n−2 cos θ n−1 , cos θ n−1 d 3 | n = − tan θ n−2 cos θ n−1 where a is the index of so(n), a ′ is the index of so(n − 1) and α ′ is the index of so(n) − so(n − 1). Observing that the (3.32) has the same form for a generic n, below we will use an inductive proof to show that W a d 1 is invariant under (3.32). Assume that W a d 1 | n and W α d 4 | n cosθ n−1 are invariant one can find that W a d 1 | n+1 and W α d 4 | n+1 cosθ n are also invariant from their expressions. For n = 2, the result is simply proved by a direct substitution. Therefore, combining the transformations (B.12) we conclude under the Z 2 transformation, the frames transform as The matrix P P T Recall the definition of the matrix P Using the identities DD T = W W T = I, one can show Therefore, we solve Substituting the relation (B.9), the first term in the bracket can be rewritten as We conclude that under the Z 2 transformation, the matrix P P T is invariant.

The dilaton
The deformed dilaton is given by The determinant det(W ) = ±1 can be absorbed into Φ 0 so we focus on det(N). To compute this determinant we rewrite matrix N as The identity (B.17) implies that the eigenvalues of P −1 are ±1 − W α λ −1 so that det[P −T ] is a constant and can be absorbed into Φ 0 . Therefore after absorbing all the constants we end up with Using the previous results in the appendix, we conclude that the dilaton does not change under Z 2 .
To summarize, in this appendix by calculating the transformations of the frames e α , the deformation matrix P P T and det(N) we have proved that deformed geometry is invariant under (3.32).

C Property of matrix Q
In this appendix we first derive the recursion relation for Q, and then prove Q = Q T by induction.