Yang-Baxter sigma models: Quantum aspects

We study the quantum properties at one-loop of the Yang-Baxter $\sigma$-models introduced by Klim\v{c}\'\ik. The proof of the one-loop renormalizability is given, the one-loop renormalization flow is investigated and the quantum equivalence is studied.


Introduction
The Yang-Baxter σ-models were first introduced by Klimčík [1,2] as a special case, at the classical level, of a non-linear σ-model with Poisson-Lie symmetry [3,4]. Recall that the Poisson-Lie symmetry appears to be the natural generalization of the so-called Abelian T -duality [5] and non-Abelian T -duality [6,7,8] of non-linear σ-models. In particular, two dynamically equivalent σ-models can be obtained at the classical level providing that Poisson-Lie symmetry condition holds. That condition takes a very elegant formulation in the case where the target space is a compact semi-simple Lie group which naturally leads to the concept of the Drinfeld double [9]. The Drinfeld double is the 2n-dimensional linear space where both dynamically equivalent theories live. For the Poisson-Lie σ-models, a proof of the oneloop renormalizability and quantum equivalence was given in [10,11,12,13]. We are interested by a special class of classical Poisson-Lie σ-models, the Yang-Baxter σ-models. Those classical models exhibit the special feature to be both Poisson-Lie symmetric with respect to the right action of the group on itself and left invariant. Thus, using the right Poisson-Lie symmetry or the left group action leads to two different dual theories. Those two dynamically equivalent dual pair of models live in two non-isomorphic Drinfeld doubles, the cotangent bundle of the Lie group for the left action, and the complexified of the Lie group for the right Poisson-Lie symmetry. Classical properties were investigated in the past and it has been showed that Yang-Baxter σ-models are integrable [1]. More recently, based on the previous work of Refs. [14,15,16], authors of Ref. [17] proved that they belong to a more general class of integrable σ-models. In particular, they showed that the ε-deformation parameter of the Poisson-Lie symmetry can be re-interpreted as a classical q-deformation of the Poisson-Hopf algebra.
If classical properties are well investigated, very little is known about the quantum version of the Yang-Baxter σ-models. In the case where the Lie group is SU (2), the Yang-Baxter σ-model coincides with the anisotropic principal model which is known to be one-loop renormalizable. This low dimensional result can let us hope a generalization for any Yang-Baxter σ-models. However, contrary to the anisotropic principal model, the Yang-Baxter σ-models contain a non-vanishing torsion which could potentially gives rise to some difficulties. On the other hand, another generalization of the anisotropic chiral model, the squashed group models are one-loop renormalizable for a special choice of torsion [22]. Furthermore, the one-loop renormalizability of the Poisson-Lie σ-model cannot provide any help here since the proof was established for a theory containing n 2 parameters when the Yang-Baxter σ-models contain only two: the ε deformation and the coupling constant t. At the quantum level, the Yang-Baxter σ-models are no more a special case of the Poisson-Lie σ models. The main result of this article consists in proving the one-loop renormalizability of Yang-Baxter σ-models.
The plan of the article is as follows. In Section 2 we introduced the Yang-Baxter σ-models on a Lie group and all algebraic tools needed. In section 3, the counter-term of the Yang-Baxter σ-models, i.e. the Ricci tensor, is calculated. Section 4 is dedicated to the proof of the one-loop renormalizability, and the computation of the renormalization flow is done in Section 5. In Section 6, we study the quantum equivalence and we express the Yang-Baxter σ-action in terms of the usual one of the Poisson-Lie σ-models. Outlooks take place in Section 7.

The complexified double
We considered the case of the Yang-Baxter models studied in Ref. [1]. In that case the Drinfeld double D can be the complexification of a simple compact and simply-connected Lie group G, i.e. D = G C , or the cotangent bundle T * G. Let us consider the case of the complexified Drinfeld double, it turns out that D = G C admits the so-called Iwasawa decomposition In particular, if D = SL(n, C), then the group AN can be identified with the group of upper triangular matrices of determinant 1 and with positive numbers on the diagonal and G = SU (n). Furthermore, the Lie algebra D turns out to be the complex Lie algebra G C , which suggests to use the roots space decomposition of G C : where ∆ is the space of all roots. Consider the Killing-Cartan form κ on G C , and let us take an orthonormal basis H i in the r-dimensional Cartan sub-algebra H C of G C with respect to the bilinear form κ on G C , i.e: This permits to define a canonical bilinear form on H * , and more specifically endows the roots space ∆ ⊂ H * with an Euclidean metric, i.e.
Moreover, the inner product on the roots space part of G C is chosen such as: and to fix the normalization, we impose the following non-linear condition E α = E † −α . With all those conventions, the generators of G C verify: The structure constants N α,β vanish if α + β is not a root.
Since G C is a Lie algebra, the structure constants verify the Jacobi identity which leads on one hand to: and on the other hand to: In the non-vanishing case, the structure constants N α,β can be calculated from the last relation with (n, m) ∈ N such that β + nα and β − mα are the last roots of the chain containing β (see Ref. [21] for more details).
Since H i is an orthonormal basis in H C , we obtain the relations: A basis of the compact Lie real form G of G C can be obtained by the following transformations: with α ∈ ∆ + (positive roots). With our choice of normalization, the vectors of the basis verify κ( and all others are zero.
Let us define now a R-linear operator R : G → G such that: this operator R is the so-called the Yang-Baxter operator [2] which satisfies the following modified Yang-Baxter equation: Let us define the skew-symmetric bracket: which fulfills the Jacobi identity, and defines a new Lie algebra (G, [., .] R ). It turns out that this new algebra is nothing but the Lie algebra of the AN group of the Iwasawa decomposition of G C and will be denoted G R the dual algebra.

The Yang-Baxter action
We shall now consider the action of the Yang-Baxter σ-models [2] expressed on the Lie group G, which takes the expression: where coupling constant and is the deformation parameter.
We can immediately check that the Yang-Baxter models (14) are left action invariant, G acting on himself. Concerning the right Poisson-Lie symmetry, it is well known that such σ-models have to fulfill a zero curvature condition to be Poisson-Lie invariant. Indeed, if we take the following G * -valued Noether current 1-form J(g): we can easily verify that the fields equations of (14) are equivalent to the following zero curvature condition: We remark that if the deformation ε vanishes then the action of the group G is an isometry, since the Noether current are closed 1-forms on the worldsheets and the action (14) coincides with that of the principal chiral σ-model. The operator (1 − εR) −1 on G can be decomposed in a symmetric part interpreted as a metric g on G and a skew-symmetric part interpreted as a torsion potential h on G. An attentive study of the action (14) gives the following expressions for g and h: In order to prove the one-loop renormalizability, we need to calculate the Ricci tensor associated to the manifold (G, g, h).
3 Counter-term of the Yang-Baxter σ-models In this paper, for the calculus of the counter-term, we choose the standard approach [18] based on the Ricci tensor. This choice provides a clear and an elegant expression of the counter-term in terms of the roots of G C . However the calculus could have been done by using our formula of [12] for the counter-term in an equivalent way.

Geometry with torsion on a Lie group G
Let us consider a pseudo-Riemannian manifold (G, g) as the base of its frame bundle, where G is a compact semi-simple Lie group and g a non-degenerated metric. Moreover, we choose the left Maurer-Cartan form g −1 dg, g ∈ G as the basis of 1-forms on G, and in that basis the metric coefficients g ab and the torsion components T abc are all constant.
On that frame bundle we define a metric connection Ω with its covariant derivative D such that Dg = 0. Furthermore, if we define by d D the exterior covariant derivative, the torsion can be written T = d D (g −1 dg). From these definitions we will obtain the expression of the connection Ω.
Metric connection. By using the relation Dg = 0 we obtain: Ω s ac g sb + Ω s bc g as = 0.
With g ab constant and if we denote Ω abc = g as Ω s bc , the previous relation becomes: Thus the two first indices of the connection Ω are skew-symmetric.

The torsion.
We said that the torsion verifies T = d D (g −1 dg) or in terms of components: Since g −1 dg is the left Maurer-Cartan form on G, we get: on the other hand T a is the 2-form torsion, i.e. we can write it as: Consequently, the components of the torsion are related to the skew-symmetric part of the connection as: with f a bc the structure constants of the Lie algebra G. Note that in the case of the non-linear σ-models the torsion is defined by T = dh where h is the 2-form potential torsion, we will exploit that a little further to express the connection for the Yang-Baxter σ-models.

The connection.
From the relations (20)(22), we can find the components of the connection: with the conventions Ω abc = g as Ω s bc , T abc = g as T s bc and f bca = f s bc g sa . Let us introduce the Levi-Civita connection L which is in fact the second term of the r.h.s in Eq.(23), and rewrite the connection Ω for a totally skew-symmetric torsion: The curvature and the Ricci. By definition the 2-form curvature F fulfills F = d D Ω, i.e.
Moreover, since Ω a b is a 1-form of G, Ω a b = Ω a bc (g −1 dg) c , we obtain the general expression for the curvature: The Ricci tensor is such that Ric ab = F s asb and can be written as: We are now able to decompose the symmetric and skew-symmetric parts of the Ricci tensor in terms of the torsion-less Ricci tensor Ric L and the torsion T as: Ric (ab) = Ric L (ab) + 1 4 T r as T s br (28)

Application to Yang-Baxter
Ricci symmetric part : Recall that in the case of the Yang-Baxter σ-models and with our normalization choice, the metric is given by: Let us introduce the bi-invariant connection Γ on the Lie group G, it corresponds to the Levi-Civita connection in the case of a vanishing deformation, i.e. Γ = L(ε = 0). From the equations (23) we can obtain the Levi-Civita coefficients: where we keep the convention for the indices i ∈ H and α ∈ ∆ + . All others Levi-Civita coefficients are equal to those of the bi-invariant connection Γ.
We can now express the torsion-less Ricci tensor Ric L as a deformation of the usual Ricci tensor Ric Γ of the bi-invariant connection on Lie group, i.e.
It is well-known that for the Riemannian bi-invariant structure the Ricci tensor takes the expression: therefore, the components of Ric L are the following: Concerning the contribution of the Torsion to the symmetric part of the Ricci tensor, we have to express the Torsion in terms of the constant structures of G. For a non-linear σ-model the Torsion 3-form is calculated from the potential torsion 2-form such T = dh, which implies that: Moreover, since the torsion potential involves only root indices the torsion components vanish for the Cartan sub-algebra indices (T ibc = 0). We can now calculate the torsion contribution, and we obtain for the nonvanishing coefficients: In the calculus we used the fact that the Killing κ can be expressed in terms of the root α and the constant structures N α,β such as: Adding both contributions to the Ricci tensor and using our normalization, we obtain the final expression of the symmetric part: We observe that, in the case of the Yang-Baxter model, the torsion induced by the Poisson-Lie symmetry is precisely that which avoids the dependence of the Ricci tensor in the root length (α, α).
Ricci skew-symmetric part : Using the fact the T iab = 0, the only non-vanishing non-diagonal components of the Ricci tensor can be written: The first r.h.s term can be expressed as a function of the structure constants, N α,β such as: The two other terms are nothing but the contribution of the roots space (see Eq.(41)) to the component κᾱᾱ of the Killing form, i.e.: − fᾱ βγ T αβγ κ γγ κ ββ − fᾱβγT αβγ κγγκββ = − ε 2 κᾱᾱ + 2(α, α) .
By summing the Bianchi relations (7) on positive roots, we obtain that: with ρ = 1 2 α∈∆ + α the Weyl vector. Finally, the skew-symmetric part of the Ricci tensor is given by:

One-loop renormalizability
At one-loop the counter-terms for a non-linear σ-model [18] on G are given by: We require, for the renormalizability, that all divergences have to be absorbed by fields-independent deformations of the parameters (t, ε) and a possible non-linear fields renormalization of the fields (g −1 ∂ ± g) a . Thus, if we suppose that all parameters are the independent coupling constants of the theory, the Ricci tensor in our frame has to verify the relations: with u a vector that contributes to the fields renormalization, χ 0 and χ ε are coordinates-independent. Decomposing into symmetric and skew-symmetric parts, the previous relation for the Yang-Baxter σ-models becomes: From the equations (51) and (52), we extract immediately: Since χ 0 and χ ε are now fixed, they have to fulfill in the same time the relation (53), which gives the following constraint: Furthermore, the covariant derivative of u can be easily calculated: Let us define the vector εū = u, and insert (56) in the constraint (55) we obtain: Then, if we imposeū = 4ρ the constraint is fulfilled for any root α since (., .) is the canonical scalar product on R r . We can conclude that the Yang-Baxter σ-models are one-loop renormalizable. We note that it is quite elegant to find a field renormalization given by the Weyl vector.

Renormalization flow
Let us introduce the β-functions of the two parameters (t, ε), they satisfy: where λ = 1 π ln µ, with µ the mass energy scale. We obtain the following system of differential equations: The set of differential equations can be exactly solved, and solutions take the following general expressions: with (A, B, C) ∈ R three integrative constants. We note that divergences occur for ε and t when the energy scaleλ goes to ± 3π 4 +C. On the other hand, forλ → ∞ the parameters ε and t are vanishing, leading to an asymptotic freedom. We can illustrate the situation with the following plot ( Fig.1) of λ as a function of ε where we choose B = 1 and C = 0. Now we will express the Yang-Baxter σ-models in terms of the usual Poisson-Lie σ-models' expression. Recall that general Right symmetric Poisson-Lie σ-models can be written: Here Π R (g) is the so-called Right Poisson-Lie bi-vector and M an n 2 real matrix.
Using the adjoint action of an element g ∈ G we can rewrite the action (14) such as the previous (62), with Let us focus on the dual models, as evoked earlier there exists two nonisomorphic Drinfeld doubles for the action (62). Consequently, we have two different dual theories for one single initial theory on G, and all three are classically equivalent. We will consider each case and argue that they are all quantum-equivalent at one-loop.
We start by considering the Drinfeld double D = G C , in that case we saw that the dual group is the factor AN in the Iwasawa decomposition. The corresponding algebra is the Lie algebra G R generated by the R-linear operator (R − i) on G, whose its group is a non-compact real form of G C (see [2,20] for details). The dual action can be expressed as: K.Sfetsos and K.Siampos proved in [10] that for Right Poisson-Lie symmetric σ-models the quantum equivalence holds providing that the matrix M is invertible. In the Yang-Baxter σ-models this condition is always satisfied and the inverse of M is given by: When we consider the dual model associated to the left action of G, the Drinfeld double is the cotangent bundle T * G = G G * . Then the dual group is the dual linear space G * of G, which is an Abelian group with the addition of vectors as the group law. The corresponding action is that of the non-Abelian T -dual σ-models [6,7,8] and has the well-known expression: It has been showed in [19] that those models are one-loop renormalizable.
Since the action (64) is Left Poisson-Lie symmetric, Sfetsos-Siampos condition [10] still holds (in their Left formulation) and implies again the quantum equivalence at one-loop.

Outlooks
Yang-Baxter σ-models are one case of non-trivial Poisson-Lie symmetric σmodels which keep the renormalizability and the quantum equivalence at the one-loop level, and are known to be classically integrable. Those models appear to be a semi-classical q-deformation of Poisson algebra, and can be a starting point in the quest for a quantum q-deformation fully renormalizable thanks to the relative simplicity of these models containing only two parameters . Furthermore, for low dimensional compact Lie groups G the geometry associated to the Yang-Baxter σ-models can be viewed as a torsionless Einstein-Weyl geometry. We plan in the future to study the Weyl connections with torsion on Einstein manifolds, with the hope to learn more about the geometric aspects of the Poisson-Lie σ-models.
I thank G.Valent for discussions and C.Carbone for proofreading.