Yang-Baxter invariance of the Nappi-Witten model

We study Yang-Baxter deformations of the Nappi-Witten model with a prescription invented by Delduc, Magro and Vicedo. The deformations are specified by skew-symmetric classical $r$-matrices satisfying (modified) classical Yang-Baxter equations. We show that the sigma-model metric is invariant under arbitrary deformations (while the coefficient of $B$-field is changed) by utilizing the most general classical $r$-matrix. Furthermore, the coefficient of $B$-field is determined to be the original value from the requirement that the one-loop $\beta$-function should vanish. After all, the Nappi-Witten model is the unique conformal theory within the class of the Yang-Baxter deformations preserving the conformal invariance.


Introduction
The Yang-Baxter sigma-model description, which was originally proposed by Klimcik [1], is a systematic way to consider integrable deformations of 2D non-linear sigma models.
According to this procedure, the deformations are specified by skew-symmetric classical r-matrices satisfying the modified classical Yang-Baxter equation (mCYBE) . The original work [1] has been generalized to symmetric spaces [2] and the homogeneous CYBE [3].
In [22], classical r-matrices are identified with exactly-solvable string backgrounds such as Melvin backgrounds and pp-wave backgrounds. In [23], Yang-Baxter deformations of 4D Minkowski spacetime are discussed by using classical r-matrices associated with κdeformations of the Poincaré algebra [24]. Then the resulting deformed geometries include T-duals of (A)dS 4 spaces 1 and a time-dependent pp-wave background. Furthermore, the Lax pair is presented for the general κ-deformations [23,26].
1 T-dual of dS 4 can be derived as a scaling limit of η-deformed AdS 5 as well [25].
As a spin off from this progress, it would be interesting to study Yang-Baxter deformations of the Nappi-Witten model [27]. The target space of this model is given by a centrally extended 2D Poincaré group. Hence the Yang-Baxter deformed Nappi-Witten models can be regarded as toy models of the previous works [22,23], because the structure of the target space is much simpler than that of 4D Minkowski spacetime. This simplification makes it possible to study the most general Yang-Baxter deformation. As a matter of course, it is exceedingly complicated in general, hence such an analysis has not been done yet.
In this article, we investigate Yang-Baxter deformations of the Nappi-Witten model by following a prescription invented by Delduc, Magro and Vicedo [28]. We show that the sigma-model metric is invariant under the deformations (while the coefficient of B-field is changed) by utilizing the most general classical r-matrix. Furthermore, the coefficient of B-field is determined to be the original value from the requirement that the one-loop βfunction should vanish. After all, the Nappi-Witten model is the unique conformal theory within the class of the Yang-Baxter deformations preserving the conformal invariance (i. e., Yang-Baxter invariance).

Nappi-Witten model
In this section, we shall give a concise review of the Nappi-Witten model [27].
The Nappi-Witten model is a Wess-Zumino-Witten (WZW) model whose target space is given by a centrally extend 2D Poincaré group. The associated extended Poincaré algebra g is composed of two translations P i (i = 1, 2) , a rotation J and the center T . The commutation relations of the generators are given by where ǫ ij is an anti-symmetric tensor normalized as ǫ 12 = 1 . It is convenient to introduce a notation of the generators with the group index I like Let us introduce a group element represented by g = exp a 1 P 1 + a 2 P 2 exp u J + v T .
By using this group element g , the left-invariant current A can be evaluated as Here the index α = τ , σ is for the world-sheet coordinates. It is also helpful to introduce the light-cone expression of A on the world-sheet like By using A ± , the classical action of the Nappi-Witten model is given by This action is basically composed of the two parts, 1) the sigma model part and 2) the Wess-Zumino-Witten (WZW) term.
The sigma model part is defined as usual on the world sheet Σ , where we assume that Σ is compact and the periodic boundary condition is imposed for the dynamical variables.
A key ingredient contained in this part is the most general symmetric two-form 2 which satisfies the following condition: Here f IJ K are the structure constants which determine the commutation relations The WZW term in (2.6) also contains Ω IJ , but, apart from this point, it is the same as the usual. The symbol B 3 denotes a 3D space which has Σ as a boundary. Hence the domain of A I is implicitly generalized to B 3 withα = τ , σ and ξ in the WZW term 3 , where the extra direction is labeled by ξ . 2 The overall factor of Ω IJ (i.e., the level of the WZW model) is set to be 1 because it is irrelevant to the deformations we consider later. 3 For the detail of the WZW model, for example, see [29].
A remarkable point is that the action (2.6) can be rewritten into the following form [27]: 4 Here X µ = {u, v, a 1 , a 2 } are the dynamical variables. The metric and anti-symmetric twoform on Σ are described by γ αβ = diag(−1, 1) and ǫ αβ normalized as ǫ τ σ = 1 . Then the space-time metric g µν and two-form field B are given by This is a simple 4D background.
It would be helpful to further rewrite the background (2.10) . By performing the following coordinate transformation with a real constant m [30] the background (2.10) can be rewritten as (2.12) This is nothing but a pp-wave background. Note here that the last two terms of B-field in (2.12) contribute to the Lagrangian as the total derivatives, which can be ignored in the present setup. For this background (2.12), the world-sheet β-function vanishes at the one-loop level [27].

Yang-Baxter deformed Nappi-Witten model
In this section, let us consider Yang-Baxter deformations of the Nappi-Witten model.

A Yang-Baxter deformed classical action
As explained in the previous section, the Nappi-Witten model contains the WZW term.
Hence it is not straightforward to study Yang-Baxter deformations of this model. Our 4 In this derivation, we have used the identities (12) and (13) in [27].
strategy here is to follow a prescription invented by Delduc, Magro and Vicedo [28]. This is basically a two-parameter deformation. It is an easy task to extend their prescription to the Nappi-Witten model.
A deformed action we propose is the following: 5 Here the deformed current J is defined as First of all, the classical action (3.1) includes three constant parameters η ,Ã and k . The deformation is measured by η andÃ . The last parameter k is regarded as the level. When η =Ã = 0 and k = 1 , the action (3.1) is reduced to the original Nappi-Witten model.
A key ingredient contained in J is a linear operator R: g → g . In the context of Yang-Baxter deformations, it is supposed that R should be skew-symmetric and satisfy the (modified) Yang-Baxter equation [31] [ The constant parameter ω can be normalized by rescaling R , hence it is enough to consider the following three cases: ω = ±1 and 0 . In particular, the case with ω = 0 is the homogeneous CYBE.

The general solution of the (m)CYBE
In this subsection, we derive the general solution of the (m)CYBE.
Let us start from the most general expression of a linear R-operator: Here M IJ is an anti-symmetric 4 × 4 matrix which is parametrized as can be rewritten into the following form: Note that we define Ω IJ as the inverse matrix of Ω IJ . Then, by putting the expression (3.5) into (3.6) , the most general solution can be determined like Here the condition (3.6) has led to the following constraints: Then we have also supposed that ω ≥ 0 in order to preserve the reality of the background 6 .
After all, m 3 , m 5 and m 6 have survived as free parameters of the R-operator as well as ω .

The general deformed background
Let us consider a deformation of the Nappi-Witten model with the general solution (3.7) .
The resulting background is given by Here we have ignored the total derivative terms that appeared in the B-field part. This background (3.8) can be simplified by performing a coordinate transformation a 1 → a 1 cos(m u) + a 2 sin(m u) + C 1 cos (C 3 u) + C 2 sin (C 3 u) , a 2 → − a 2 cos(m u) + a 1 sin(m u) − C 2 cos (C 3 u) + C 1 sin (C 3 u) , where we have introduced the following quantities: (3.10) After performing the transformation (3.9) , the resulting background is given by the following pp-wave background equipped with a B-field: (3.11) Here we have ignored the total derivative terms again.
Note that the B-field in (3.11) can be rewritten as (up to total derivative terms)

Yang-Baxter deformations and conformal invariance
Finally, let us show that the original Nappi-Witten model is the unique conformal theory within the class of the Yang-Baxter deformations preserving the conformal invariance.
Due to the requirement of the vanishing β-function at the one-loop level, the two B-fields should be identical as follows: 2m (the original) = k C 3 − 2mÃ m 6 (the deformed) . (3.13) This condition indicates two interesting results. The first one is the Yang-Baxter invariance of the Nappi-Witten model. If we start from the case with k = 1 , then the original system is invariant under the Yang-Baxter deformations preserving the conformal invariance, which are specified by the parameters satisfying the condition . In other words, the Yang-Baxter invariance follows from the conformal invariance.
The second is that the Yang-Baxter deformation may map a non-conformal theory to the conformal Nappi-Witten model. Suppose that we start from the case with k = 1 . Then, by performing a Yang-Baxter deformation with parameters satisfying the condition (3.14) the resulting system becomes the Nappi-Witten model. In other words, the coefficient of B-field can be set to the conformal fixed point by an appropriate Yang-Baxter deformation.

Conclusion and discussion
In this article, we have studied Yang-Baxter deformations of the Nappi-Witten model. By There are many future directions. It would be interesting to consider a supersymmetric extension of our analysis by following [33]. The number of the remaining supersymmetries should depend on Yang-Baxter deformations because the coefficient of B-field is changed. It is also nice to investigate higher-dimensional cases (e.g., the maximally supersymmetric ppwave background [34]). As another direction, one may consider non-relativistic backgrounds such as Schrödinger spacetimes [35] and Lifshitz spacetimes [36]. Although there is a problem of the degenerate Killing form similarly, it can be resolved by adopting the most general symmetric two-form [37] as in the Nappi-Witten model. It would be straightforward to apply the techniques presented in [37] to Yang-Baxter deformations by following our present analysis.
It should be remarked that the most interesting indication of this work is the universal aspect of the dual gauge-theory side. According to our work, pp-wave backgrounds would have a kind of rigidity against Yang-Baxter deformations. This result may indicate that the ground state and lower-lying excited states of the spin chain associated with the N = 4 super Yang-Mills theory are invariant. It is quite significant to extract such a universal characteristic after classifying various examples. This is the standard strategy in theoretical physics and would be much more important than identifying the associated dual gauge theory for each of the deformations.
We hope that our result could shed light on a universal aspect of Yang-Baxter deformations from the viewpoint of the invariant pp-wave geometry.