Contents
###### Abstract

We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target space is given by squashed S and the isometry is  . It is known that is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the /-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.

KUNS-2467

A deformation of quantum affine algebra

in squashed WZNW models

Io Kawaguchi***E-mail: ioatgauge.scphys.kyoto-u.ac.jp and Kentaroh YoshidaE-mail: kyoshidaatgauge.scphys.kyoto-u.ac.jp

Department of Physics, Kyoto University

Kyoto 606-8502, Japan

## 1 Introduction

The AdS/CFT correspondence [1, 2, 3] has been a fascinating topic in the study of string theory over the decade after Maldacena’s proposal. A tremendous amount of works have been devoted to test and generalize it. Nowadays, the research area covers diverse subjects. A great achievement in the recent progress is the discovery of the integrable structure behind the AdS/CFT correspondence (For a comprehensive review on this subject, see [4]).

In the string-theory side, the integrable structure of two-dimensional string sigma-models with target spacetime AdSS plays an important role [5]. It is closely related to the fact that AdSS is described as a symmetric coset. It leads to an infinite number of the conserved charges constructed, for example, by following the pioneering works [6, 7, 8, 9, 10] (For a comprehensive textbook see [11]). The symmetric cosets that potentially may lead to a holographic interpretation are classified, including spacetime fermions [12].

The next interesting issue is to consider integrable deformations of the AdS/CFT correspondence. There are two approaches. The one is an algebraic approach based on -deformations of the world-sheet S-matrix [13, 14, 15, 16, 17]. The deformed S-matrices are explicitly constructed, while the target-space geometry is unclear. The other is a geometric approach based on deformations of target spaces of the sigma models. It seems likely that the deformed geometries are represented by non-symmetric cosets [18] in comparison to AdSS and hence the prescription for symmetric cosets is not available any more. It is necessary to develop a new method to argue the integrability.

In the latter approach, there is a long history (For classic papers see [19, 20, 21]). Motivated by integrable deformations of AdS/CFT, a series of works have been done on squashed S and warped AdS [22, 24, 25, 28, 26, 27, 29, 23, 30]. Some specific higher-dimensional cases are discussed in [31]. Remarkably, the classical integrable structure of deformed sigma models was recently shown for arbitrary compact Lie groups and the coset cousins by Delduc, Magro and Vicedo [32]. Then, they successively presented a -deformation of the AdSS superstring [33].

Based on the latter approach, we are here concerned with the classical integrable structure of two-dimensional Wess-Zumino-Novikov-Witten (WZNW) models with the target space squashed S . The isometry is given by  . It is partly explained in [24] that there exist a couple of Yangian algebras based on by explicit constructions of non-local conserved charges and direct computations of the Poisson brackets of the charges.

In this paper we consider an infinite-dimensional extension of  . In the case without the Wess-Zumino term, it is just a classical analogue of a quantum affine algebra [26]. When the Wess-Zumino term is added, an additional constant parameter is introduced as its coefficient. A natural question is what happens to the quantum affine algebra. As one may easily guess, a new kind of deformation is induced by the presence of the Wess-Zumino term. The resulting algebra is a classical analogue of the deformed quantum affine algebra. So far, it is not clear what is the mathematical formulation of the deformed quantum affine algebra, though it seems likely to be a two-parameter quantum toroidal algebra [34]. In order to answer the question, it is necessary to see the first realization of the two-parameter quantum toroidal algebra.

This paper is organized as follows. In section 2 the classical action of the squashed WZNW models is introduced. In section 3 we consider the classical integrable structure based on  . This is called the left description. We present a couple of Lax pairs and the associated monodromy matrices. The classical /-matrices are shown to satisfy the extended classical Yang-Baxter equation. The infinite-dimensional extensions of are Yangians. In section 4 we argue the classical integrable structure based on  . This is called the right description. A Lax pair and the associated monodromy matrix are presented. The classical /-matrices satisfy the extended classical Yang-Baxter equation. Remarkably, an infinite-dimensional extension of is shown to be a deformation of quantum affine algebra, where a new deformation parameter is provided by the coefficient of the Wess-Zumino term. In section 5 the gauge equivalence between the left and right descriptions is proven. Under an identification between the spectral parameters, the left Lax pair is related to the right Lax pair via a gauge transformation. In section 6 we argue two degenerate limits in the right description. At some special points in the parameter space, the deformed quantum affine algebra degenerates to a Yangian, according to the enhancement of to  . Section 7 is devoted to conclusion and discussion.

In Appendix A, we explain the computation of the current algebra in detail. Appendix B provides a prescription to treat non-ultra local terms in computing of the Poisson brackets of the conserved charges.

## 2 Preliminary

Let us begin with the setup to fix our notation and convention. The metric of squashed S is first provided in terms of an group element. Then the classical action of the squashed WZNW models is introduced. For later convenience, the equations of motion are explicitly written down.

### 2.1 Squashed S3

The metric of round S can be described as a fibration over S . The squashing is one-parameter deformations of the direction. The metric of squashed S is given by

 ds2=L24[dθ2+sin2θdϕ2+(1+C)(dψ+coshθdϕ)2]. (2.1)

When  , the metric is reduced to that of S with radius and the isometry is  . When  , the isometry is reduced to  .

In order to rewrite the metric (2.1) , let us introduce an group element like

 g = eT3ϕeT2θeT3ψ∈SU(2). (2.2)

Here the generators satisfy the following relations

 [Ta,Tb]=εab  cTc, (2.3)

and normalized as

 Tr(TaTb)=−12δab. (2.4)

Note that is the totally anti-symmetric tensor normalized as  . The indices are raised and lowered by and its inverse, respectively.

It is useful to define as

 T±≡1√2(T1±iT2). (2.5)

Then the commutation relations in (2.3) are rewritten as

 [T±,T∓]=∓iT3,[T±,T3]=±iT±, (2.6)

and the normalization of the generators in (2.4) is given by

 Tr(T±T∓)=Tr(T3T3)=−12. (2.7)

Then the metric (2.1) is rewritten in terms of the group element as

 ds2=−L22[Tr(J2)−2C(Tr[T3J])2], (2.8)

where we have introduced the left-invariant one-form

 J≡g−1dg. (2.9)

Note that can be represented by the angle variables as follows:

 J = T1(sinψdθ−cosψsinθdϕ)+T2(cosψdθ+sinψsinθdϕ)+T3(dψ+cosθdϕ) = T+1√2eiψ(−idθ−sinθdϕ)+T−1√2e−iψ(idθ−sinθdϕ)+T3(dψ+cosθdϕ).

With the metric (2.8) , it is easy to see the invariance under  . The and transformations are the left- and right- multiplications,

 g→eβaTa⋅g⋅e−αT3. (2.10)

Here and are real parameters.

### 2.2 The classical action of the squashed WZNW models

First of all, let us introduce two-dimensional non-linear sigma models whose target space is given by squashed S . The classical action is

 SσM=1λ2∫∞−∞dt∫∞−∞dx ημν[tr(JμJν)−2Ctr(T3Jμ)tr(T3Jν)], (2.11)

where the parameter is the coupling constant and the base space is a two-dimensional Minkowski spacetime with the coordinates (time) and (space) and the metric

 −ηtt=ηxx=+1. (2.12)

Note that the region of the parameter is restricted so that the positivity of the kinetic term is ensured.

The next is to introduce the Wess-Zumino term on squashed S ,

 SWZ=n12π∫10ds∫∞−∞dt∫∞−∞dx ϵ^μ^ν^ρtr(˜J^μ˜J^ν˜J^ρ),˜J≡˜g−1d˜g, (2.13)

where is an integer. Note that the above integral is performed on a three-dimensional base manifold spanned by  . The totally anti-symmetric tensor is normalized as

 ϵtxs=+1. (2.14)

The element is defined on this three-dimensional manifold. It interporates between a constant element at and at  :

 ˜g(t,x,s=0)=g0 :const.,˜g(t,x,s=1)=g(t,x). (2.15)

Note that the Wess-Zumino term (2.13) is the same as in the case of round S and hence it is invariant under the symmetry of the sigma model action (2.11) .

Let us consider the Wess-Zumino-Novikov-Witten models defined on squashed S , which henceforth are called “squashed WZNW models”. The action is given by the sum of in (2.11) and in (2.13) :

 S = SσM+SWZ. (2.16)

The action (2.16) is also -invariant.

From the action (2.16) , the equations of motion are obtained,

 ∂μJμ−2Ctr(T3∂μJμ)T3−2Ctr(T3Jμ)[Jμ,T3]+Kϵμν∂μJν=0, (2.17)

where the new constant is defined as

 K≡nλ28π, (2.18)

and the totally anti-symmetric tensor is normalized as

 ϵtx=+1. (2.19)

The components of the left-invariant one-form are defined as

 Ja≡−2Tr(TaJ), (2.20)

or equivalently

 J=T+J−+T−J++T3J3. (2.21)

In terms of  , the equations of motion are rewritten as

 (1+C)∂μJ3μ+Kϵμν∂μJ3ν=0, (2.22) ∂μJ±μ∓iCJ3μJ±,μ+Kϵμν∂μJ±ν=0.

By definition, the left-invariant one-form satisfies the flatness condition:

 (2.23)

This condition can also be rewritten in terms of the components as

 ϵμν(∂μJ3ν+iJ+μJ−ν)=0, (2.24) ϵμν(∂μJ±ν±iJ3μJ±ν)=0.

The flatness condition (2.2) enables us to rewrite the equations of motion (2.2) as

 (1+C)∂μJ3μ−iKϵμνJ+μJ−ν=0, (2.25) ∂μJ±μ∓iCJ3μJ±,μ∓iKϵμνJ3μJ±ν=0.

The expressions in (2.25) play an important role in our later discussion.

## 3 The left description

In this section, we discuss the classical integrable structure of squashed WZNW models based on the symmetry. We call it left description. This part contains a short review of the previous work [24].

First of all, we construct an conserved current which satisfies the flatness condition. With the flat and conserved current, we obtain a Lax pair and the corresponding monodromy matrix. Then we compute the classical /-matrices for the Lax pair. Finally, we show that the symmetry is enhanced to the Yangian algebra  .

### 3.1 Lax pairs

The classical action (2.16) has the symmetry and the associated conserved current is given by

 jLμ=gJμg−1−2Ctr(T3Jμ)gT3g−1−KϵμνgJνg−1. (3.1)

The conservation law of this current is equivalent to the equations of motion in (2.17) like

 ∂μjLμ = g[∂μJμ−2Ctr(T3∂μJμ)T3−2Ctr(T3Jμ)[Jμ,T3]+Kϵμν∂μJν]g−1 = 0.

Note that does not satisfy the flatness condition due to the deformation.

One may consider to improve so as to satisfy the flatness condition. This requirement leaves two improved currents [24],

 jL±μ = gJμg−1−2Ctr(T3Jμ)gT3g−1−KϵμνgJνg−1∓Aϵμν∂ν(gT3g−1),

with the coefficient represented by

 A=√C(1−K21+C). (3.3)

The subscripts denote the degeneracy of the improved currents. The improved currents satisfy the following flatness condition,

 ϵμν(∂μjL±ν−jL±μjL±ν)=0. (3.4)

When  , and the improved currents constructed in [22] are reproduced.

With the flat currents, two Lax pairs are constructed as

 LL±t(x;λL±)=12[LL±+(x;λL±)+LL±−(x;λL±)], (3.5) LL±x(x;λL±)=12[LL±+(x;λL±)−LL±−(x;λL±)], LL±+(x;λL±)=11+λL±jL±+,LL±−(x;λL±)=11−λL±jL±−.

Here are spectral parameters and and are defined as

 jL±+≡jL±t+jL±x,jL±−≡jL±t−jL±x. (3.6)

The following commutation relation

 [∂t−LL±t(x;λL±),∂x−LL±x(x;λL±)]=0. (3.7)

gives rise to the conservation law of the flat current (equivalently, the equations of motion) and the flatness condition.

Then the monodromy matrices are defined as

 UL±(λL±)≡Pexp[∫∞−∞dx LL±x(x;λL±)]. (3.8)

The symbol P denotes the path ordering. Due to the flatness of the Lax pairs in (3.7) , the monodromy matrices are conserved,

 ddtUL±(λL±)=0. (3.9)

Thus an infinite set of conserved charges can be obtained by expanding the monodromy matrices with respect to around appropriate points. For example, the monodromy matrices can be expanded around as

 UL±(λL±)=exp[∞∑n=0λ−n−1L±QL±(n)]. (3.10)

In the next subsection, we will discuss the algebra generated by  .

Before closing this subsection, let us discuss the /-matrices computed from the Lax pairs in (3.5) by following the Maillet formalism [35]. One can read off them from the Poisson brackets between the spatial components of the Lax pairs,

 {LL±x(x;λL±)\lx@stackrel⊗,LL±x(y;μL±)}P = [rL±(λL±,μL±),LL±x(x;λL±)⊗1+1⊗LL±x(y;μL±)]δ(x−y) −[sL±(λL±,μL±),LL±x(x;λL±)⊗1−1⊗LL±x(y;μL±)]δ(x−y) −2sL±(λL±,μL±)∂xδ(x−y).

To compute the above Poisson bracket, we have to use the current algebra for  ,

 {jL±,at(x),jL±,bt(y)}P = εab  cjL±,ct(x)δ(x−y)−2Kδab∂xδ(x−y), (3.12) {jL±,at(x),jL±,bx(y)}P = εab  cjL±,cx(x)δ(x−y)+(1+K2+A2)δab∂xδ(x−y), {jL±,ax(x),jL±,bx(y)}P = −(K2+A2)εab  cjL±,ct(x)δ(x−y) −2Kεab  cjL±,cx(x)δ(x−y)−2Kδab∂xδ(x−y).

The explicit expressions of /-matrices are given by

 rL±(λ±,μ±) = hLC,K(λ±)+hLC,K(μ±)2(λ±−μ±)[T−⊗T++T+⊗T−+T3⊗T3], (3.13) sL±(λ±,μ±) = hLC,K(λ±)−hLC,K(μ±)2(λ±−μ±)[T−⊗T++T+⊗T−+T3⊗T3],

where a scalar function is defined as

 hLC,K(λ)≡A2+(λ+K)21−λ2.

The /-matrices satisfy the extended classical Yang-Baxter equation*** The /-matrices depend on and individually (not only  ) and they satisfy the extended classical Yang-Baxter equation. Thus the classification of the /-matrice are subtle.,

 [(r−s)L±12(λL±,μL±),(r+s)L±13(λL±,νL±)] + [(r+s)L±12(λL±,μL±),(r+s)L±23(μL±,νL±)] + [(r+s)L±13(λL±,νL±),(r+s)L±23(μL±,νL±)]=0.

It should be noted that, when  , the function is reduced to

 hLC,0(λ)≡C+λ21−λ2.

Thus the /-matrices in (3.13) reproduce the results without the Wess-Zumino term [25] .

### 3.2 Yangians

So far, the monodromy matrices have been introduced and an infinite number of the conserved charges are obtained by expanding them with respect to  .

For the first three levels, the explicit expressions of are given by

 QL±,a(0) = ∫∞−∞dx jL±,at(x), (3.15) QL±,a(1) = 14∫∞−∞dx∫∞−∞dy ϵ(x−y)εa bcjL±,bt(x)jL±,ct(y)−∫∞−∞ jL±,ax(x), QL±,a(2) = 112∫∞−∞dx∫∞−∞dy∫∞−∞dz ϵ(x−y)ϵ(x−z)δbc ×[jL±,bt(x)jL±,at(y)jL±,ct(z)−jL±,at(x)jL±,bt(y)jL±,ct(z)] −12∫∞−∞dx∫∞−∞dy ϵ(x−y)εa bcjL±,bt(x)jL±,cx(y)+∫∞−∞dx jL±,at(x).

Note that these charges can be directly constructed from the flat currents recursively by following the BIZZ construction [8] .

The next task is to show that the conserved charges satisfy the defining relations of Yangian  . The Poisson brackets of the first two levels are given by [24]

 {QL±,a(0),QL±,b(0)}P = εab  cQL±,c(0), (3.16) {QL±,a(1),QL±,b(0)}P = εab  cQL±,c(1), {QL±,a(1),QL±,b(1)}P = εab  c[QL±,c(2)+112(QL±(0))2QL±,c(0)+2KQL±,c(1)].

The Serre relations are shown as

 {QL±,3(1),{QL±,+(1),QL±,−(1)}P}P=14QL±,3(0)(QL±,+(0)QL±,−(1)−QL±,−(0)QL±,+(1)), (3.17) =14QL±,±(0)(QL±,±(0)QL±,∓(1)−QL±,∓(0)QL±,±(1))−14QL±,3(0)(QL±,±(0)QL±,3(1)−QL±,3(0)QL±,±(1)), {QL±,±(1),{QL±,±(1),QL±,3(1)}P}P=14QL±,±(0)(QL±,±(0)QL±,3(1)−QL±,3(0)QL±,±(1)).

Thus the defining relations of Yangian at the classical level are satisfied in the sense of Drinfeld’s first realization [36, 37].

Here we should comment on the treatment of non-ultra local terms contained in the current algebra of  . They develop ambiguities in computing the Poisson brackets of the conserved charges and there might be the possibility that the defining relations of are spoiled. In the present case, the presence of the Wess-Zumino term make the situation worse. It develops non-ultra local terms even in the Poisson brackets of and hence cause ambiguities in computing the usual Lie algebra part. The treatment of the non-ultra local terms is argued in Appendix B in detail.

## 4 The right description

The classical integrable structure of the squashed WZNW models can also be described based on as another description. This description is called right description. A Lax pair and the associated monodromy matrix are presented. Then an infinite-dimensional extension of is argued. The resulting algebra is a deformation of the standard quantum affine algebra. In the right description, the /-matrices are deformed by an additional term, in comparison to the case without the Wess-Zumino term.

### 4.1 Lax pair

A Lax pair which respects is given byThe anisotropic Lax pair with is constructed originally by Cherednik [19]. See also [20].

 LRt(x;λR)=12[LR+(x;λR)+LR−(x;λR)], (4.1) LRx(x;λR)=12[LR+(x;λR)−LR−(x;λR)], LR±(x;λR)=−sinh(α±β)sinh[α±(β+λR)][T+J−±+T−J+±+cosh(α±λR)coshαT3J3±].

New constants and are related to and like

 tanhα=iCA,tanhβ=iCKA(1+C). (4.2)

Note that and have the periodicities,

 α∼α+πi,β∼β+πi. (4.3)

In the Lax pair (4.1), a spectral parameter has been introduced. This is seemingly independent of at this stage, but eventually there is a relation between them as we will see later. The Lax pair (4.1) is referred to as the right Lax pair hereafter.

The relations given in (4.2) imply the inequalities for  :

 −tanh2α=C2A2=C(1+C)1+C−K2>−1, (4.4) −tanh2β=C2K2A2(1+C)2=CK2(1+C)(1+C−K2)>−1.

With the kinematic restriction  , these inequalities are equivalent to

 (C−K+1)(C+K+1)(C−K2+1)>0.

Due to the relations in (4.2) and the reality of and  , and must be real or purely imaginary. When  , and are real (up to the shift of with ) . On the other hand, when  , and are purely imaginary. Thus the allowed region of and can be expressed on the -plane as depicted in Fig. 1.

The relations given in (4.2) can be solved for and  , and hence  , (and  ) are written in terms of and  ,

 K=sinh2βsinh2α,C=−sinh(α+β)sinh(α−β)cosh2α, (4.5) A=−2isinh(α+β)sinh(α−β)sinh2α.

Note that are invariant under the shift of and by  .

The equations of motion in (2.17) and the flatness condition of are reproduced from the commutation relation,

 [∂t−LRt(x;λR),∂x−LRx(x;λR)]=0. (4.6)

#### Monodromy matrix

With the right Lax pair given in (4.1) , the associated monodromy matrix is defined as

 UR(λR)≡Pexp[∫∞−∞dx LRx(x;λR)]. (4.7)

The flatness of the Lax pair (4.6) ensures that the monodromy matrix is a conserved quantity,

 ddtUR(λR)=0. (4.8)

Thus, by expanding , an infinite number of conserved charges are constructed.

### 4.2 q-deformation of su(2)R

In the squashed WZNW models, the symmetry, which is preserved by round S , is broken to  , due to the deformation term. The Noether current for is given by

 jR,3μ = −[(1+C)J3μ+KϵμνJ3,ν] = −coshβcoshα(coshβcoshαJ3μ+sinhβsinhαϵμνJ3,ν).

For later convenience, the expressions have been given in terms of and  , as well as in terms of and  .

It is known that there exist non-local conserved currents which correspond to the broken components in the case without the Wess-Zumino term [25]. Let us show that this is the case even in the squashed WZNW models.

For this purpose, it is helpful to introduce a non-local function  ,

 χ(x)=−12∫∞−∞dy ϵ(x−y)jR,3t(y). (4.10)

This function satisfies the differential equation,

 ∂μχ=−ϵμνjR,3,ν=(1+C)ϵμνJ3,ν+KJ3μ. (4.11)

This relation ensures the conservation law of the non-local currents, as we will see later.

With  , the conserved, non-local currents are constructed as

 jR,±μ = eγ±χiR,±μ, (4.12) iR,±μ = −(J±μ+(K±iA)ϵμνJ±,ν) =

Here are defined as

 γ±≡(1+CCA∓iK)−1=−ie±βsinh2α2coshβ. (4.13)

The non-local currents give rise to the conserved charges

 QR,±=∫∞−∞dx jR,±t(x), (4.14)

as well as the standard Noether charge of  ,

 QR,3=∫∞−∞dx jR,3