Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable σ-model

Journal of High Energy Physics, Oct 2017

Abstract A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter σ-models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model.

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Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable σ-model

JHE Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one F. Delduc 0 2 B. Hoare 0 1 T. Kameyama 0 2 M. Magro 0 2 0 F-69342 Lyon , France 1 Institut fur Theoretische Physik, Eidgenossische Technische Hochschule Zurich 2 Universite Lyon, ENS de Lyon, Universite Claude Bernard, CNRS, Laboratoire de Physique A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter -models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model. Integrable Field Theories; Sigma Models - HJEP10(27) Bi-Yang-Baxter -model plus WZ term 1 Introduction 2 3 4 5 1 2.1 2.2 3.1 3.2 3.3 Conclusion Introduction In [1] Lukyanov constructed a novel four-parameter integrable deformation of the SU(2) principal chiral model (PCM), which preserves a U(1) U(1) subgroup of the original SU(2) SU(2) global symmetry. This four-parameter model generalises [1, 2] a number of previously well-known theories: Fateev's two-parameter deformation of the SU(2) PCM [3]. This identi cation of the Fateev model as a special case of the Lukyanov model resolved the long-standing question of the integrability of the Fateev model. The SU(2) PCM plus the Wess-Zumino (WZ) term with arbitrary coe cient [ 4 ]. For a special value of this arbitrary coe cient one nds the conformal SU(2) Wess Zumino-Witten (WZW) model. The TsT transformation of the SU(2) WZW model, which can also be realised as a gauged WZW model for (SU(2) U(1))=U(1) [5, 6]. Lukyanov's model is de ned by a metric and B- eld. In the undeformed limit, the B- eld vanishes and the metric is the one of the three-sphere. One may then ask if the full fourparameter deformation can be written as an action for a group-valued eld g 2 SU(2), and in turn generalised to arbitrary Lie group G. Our aim in this paper is to answer these questions. To do this we will draw on a number of recent developments, many of which can trace their origins to Klimc k's YangBaxter -model [7, 8], a one-parameter integrable deformation of the PCM for a general group G, whose appellation re ects its dependence on a solution of the modi ed classical Yang-Baxter equation for g = Lie(G). { 1 { mation of the PCM, the bi-Yang-Baxter -model [8, 9], which also incorporates the oneparameter Yang-Baxter deformation of the symmetric space -model [10] for cosets of the type (G G)=Gdiag. Algebraically the two parameters manifest as q-deformations of the G G symmetry, with an independent deformation parameter for each factor of the group G [11] (see also [12, 10, 13]). In [2] it was shown that the bi-Yang-Baxter -model for G = SU(2) is equivalent to Fateev's two-parameter deformation. This model does not have a non-trivial coupling to the B- eld. In contrast the Lukyanov model does have such a coupling. As discussed above, for a certain choice of parameters this B- eld corresponds to a WZ term. In [14] it was pair that encodes its equations of motion, thereby demonstrating the classical integrability of the model. The number of deformation parameters depends on the group G. For G = SU(2) there are four parameters and in this case we explicitly demonstrate equivalence with Lukyanov's model [1]. Therefore, in this sense, the model is the generalisation of Lukyanov's model to arbitrary group G. The construction of the model is split into two stages. In section 2 we consider a three-parameter integrable model: the bi-Yang-Baxter deformation of the PCM plus WZ term, generalising the construction of [14]. Generically this breaks the symmetry of the model from G G to U(1)rank G U(1)rank G, i.e. the Cartan subgroup. We arrive at the Lagrangian and Lax pair for the multi-parameter deformation of the PCM in section 3 by implementing a general TsT transformation that mixes the Cartan generators of the two copies of G, which provides (rank G)2 additional parameters. For G = SU(2) the Cartan subgroup is one-dimensional and therefore there is one additional parameter. In section 4 we demonstrate the equivalence to Lukyanov's model. Finally we conclude in section 5 with comments and open questions. 2 Bi-Yang-Baxter -model plus WZ term In this section we construct a three-parameter integrable deformation of the PCM. Two of these parameters correspond to those of the bi-Yang-Baxter -model while the third is the coupling to the WZ term. To obtain this integrable deformation of the PCM we employ on the following strategy. First of all, we shall view the PCM for a Lie group G as the { 2 { (G G)=Gdiag symmetric space -model, where Gdiag is the diagonal subgroup of G In the framework of integrable deformations, this perspective has been previously used in [25, 11]. Secondly, the Gdiag gauge invariance will be realised by introducing a gauge eld. In subsection 2.1 we start from an ansatz for the action with ve free parameters and derive the corresponding equations of motion. We then determine the conditions for this action to de ne an integrable eld theory in subsection 2.2. We show that a Lax pair exists provided the ve parameters are xed in terms of desired three deformation parameters. 2.1 jR are de ned as ja = ga 1dga (a = L; R). The operators Oab are given by and that its non-trivial kernel is the Cartan subalgebra h of g, i.e. The term SWZ;k in (2.1) denotes the standard Wess-Zumino term, k Z SWZ;k[g] = d 2 d tr g 1 OLL = AdgL1 (1 + L2) 11 + AL2LRR2 ORR = AdgR1 (1 + R2) 11 + AR2RRR2 OLR = ORL = 0; AdgL ; AdgR ; with Adg(x) = gxg 1 for x 2 g. The operator R is a non-split R-matrix on g. It is skewsymmetric and solves the modi ed classical Yang-Baxter equation on g, which means that for x and y in g we have tr(x Ry = tr(Rx y); [Rx; Ry] = R [Rx; y] + [x; Ry] + [x; y]: Furthermore, we take R to be a standard R-matrix, which implies that R3 = R; Rx = 0; 8x 2 h: { 3 { (2.2) (2.3a) (2.3b) (2.4) (2.5) (2.6) The presence of the WZ term indicates that the associated coupling should be quantised in the quantum theory. However, let us note that, in a mild abuse of notation, what we call k is not the standard integer-valued level. The action (2.1) is invariant under Gdiag gauge transformations, gL;R ! gL;Rg0; A (2.7) while jL jR and jL;R A have the homogeneous transformations x ! Adg01x. For the moment the coe cients AL;R are free. The way they depend on L;R and k shall be xed by imposing the existence of a Lax pair. The resulting dependence coincides with the analogous expressions in [15, 16, 14]. Before we proceed to construct the Lax pair let us brie y illustrate the motivation for using a gauge eld. To determine a Lax pair, we will have to explicitly invert operators such as Oab. Without introducing a gauge eld, the Gdiag gauge invariance would be ensured by making use of the projector onto the orthogonal complement of the diagonal subalgebra of g g (see e.g. [25] for the bi-Yang-Baxter case). Such insertions of the projector operator make inverting the relevant operators in a tractable way substantially more di cult. As we shall see in the next subsection, the presence of the gauge eld thus allows the inversion to be done in a simple way. To construct a Lax pair we follow the method of [14] and start by determining the equations of motion. The equations of motion for the gauge eld read where J L + J R = 0; J L = (OLL + k)(jL J R = (ORR k)(jR A ); A ); J+L = (OLtL k)(j+L J+R = (ORtR + k)(j+R A+); A+): In these expressions, the operators OLtL and ORtR are obtained by taking the transpose of OLL and ORR respectively. This corresponds to ipping the sign of R. The equations of motion for gL and gR are respectively given by D+J L + D J+L 2kF + = 0; D+J R + D J+R + 2kF + = 0: Here we have introduced covariant derivatives D x = @ x + [A ; x] and F + is the eld strength of the gauge eld, 2.2 Lax pair To proceed we treat the equations of motion for the gauge eld (2.8) separately to those the zero curvature condition for the Lax pair. and are, on-shell, just the negative of J L . We shall therefore focus on the currents J L , their equation of motion (2.10a) and the Maurer-Cartan equation, From now on, we explicitly use the relation R3 = R in order to write all operators, such as OLL, as a linear combination of = 1 + R2, R and R2. The operator is the projector on the Cartan subalgebra h. To do this one can use the relations 2 = 1 + R ; 2 = (a + bR + cR2) 1 = a 1 ; + 1 b2 + c2 R = R = 0; ( bR + cR2): jL = a 2 LL + b RLL + d RLL J L + A ; Now expressing the currents jL in terms of J L and A using (2.9a) and (2.12b) leads to where we make use of a general notation for operators dressed by the adjoint action, e.g. LL = AdgL1 AdgL . The coe cients a , b and d are given by a = 1 If we choose AL as in [15, 16, 14], A2L = L2 1 k 2 1 + L2 ; { 5 { (2.11) (2.12a) (2.12b) (2.13) (2.14) (2.16) (2.17) HJEP10(27) Note that the analogous expressions for the right currents are obtained from the left ones by the replacement rule (L; L; AL; k) ! (R; R; AR; k). Let us denote the left-hand side of the Maurer-Cartan equation (2.11) as MCL. Starting from (2.13) we may rewrite MCL as 2 RLL (D J+L D+J L ) then the coe cients a , b and d satisfy the following relations b+(a + d ) = b (a+ + d+); the last line in (2.16) vanishes. The next step is to use (2.18b) and (2.15) to combine the third line of (2.16) with the second one. Finally, we use the equation of motion (2.10a) in the rst line of (2.16). Following these steps we obtain 2 RLL (D J+L D+J L +[J L ; J+L ]): (2.20) The condition (2.17) implies that the operators appearing in the rst and second lines of (2.20) are proportional, with the relative coe cient being equal to (1 + k2 + A2L). Furthermore, these operators are invertible. Therefore, on-shell, the equation MCL = 0 is equivalent to D J+L D+J L + [J L ; J+L ] + (1 + k2 + A2L)F + = 0: Proceeding in the same way for the right currents, choosing in particular one similarly arrives at D J+R D+J R + [J R; J+R] + (1 + k2 + A2R)F + = 0: We now take the sum of (2.21) and (2.23) and use the equations of motion for the gauge eld (2.8) to express the eld strength F + in terms of [J L ; J+L ]. We then use this expression for F + in (2.10a) and (2.21) to obtain with F + D+J L + D J+L 1 2k ( + ( + )[J L ; J+L ] = 0; )[J L ; J+L ] = 0; D+J L D J+L ( + + )[J L ; J+L ] = 0; + = 2(2(1 + k2) + A2L + A2R) ; A2L + AR 2 4k = A2L + A2R + 4k 2(2(1 + k2) + A2L + A2R) : { 6 { (2.18a) (2.18b) (2.21) (2.22) (2.23) (2.24a) (2.24b) (2.24c) (2.25) To construct a Lax pair let us rede ne the gauge eld as The equations (2.24) are then equivalent to A b = A + J L : where Db are covariant derivatives with respect to Ab and The equations (2.27) are equivalent to the atness of the Lax pair Fb + = F + + [J L ; J+L ] = G2[J L ; J+L ]; Db+J L + Db J+L = 0; Db+J L Db J+L = 0; G2 = (4 + (AL + AR)2)(4 + (AL 4(2(1 + k2) + A2L + A2R)2 AR)2) : L ( ) = Ab + G integrable model with AL;R given by (2.17) and (2.22). TsT transformation 3 G The three-parameter deformation of the PCM constructed in section 2 breaks the global G symmetry of the action. As a consequence of the property (2.5) the symmetry that remains is the Cartan subgroup speci ed by the kernel of the operator R. By implementing TsT transformations [ 26, 5, 6 ] on the corresponding shift isometries we are able to introduce additional deformation parameters while preserving integrability [27{29]. In this section we perform a general TsT transformation with each of the two shift isometries coming from a di erent copy of G. 3.1 On the action Our starting point is the action (2.1). As shown in subsection 2.2 the equations of motion for gL and gR and the Maurer-Cartan equations follow from a Lax pair if AL and AR are xed in terms of L , R and k as A2L = L2 1 k 2 { 7 { which are invariant under the original Gdiag gauge transformations (2.7), and the rescaled projections of la onto the Cartan subalgebra (recall that = 1+R2 is the projector onto h) LL = (1 + L2 k) lL ; LR = (1 + R2 k) lR ; we use (2.4) and (2.5) to rewrite the action (2.1) in the form S[g~L;R; xL;R; A] = d d + SWZ;k[g~L] SWZ;k[g~R] k d 2 tr A (|~L+ |~R ) A+(|~L + |~R ) ; (3.7) where the operators OL;R are given by S![g~L;R; x^L;R; A] = d 2 tr l+L OL lL (1 + R2)LL+!tOe 1!LL OL;R = 1 + AL;RR + L2;R : To implement the TsT transformation we rst T-dualise xL ! x~L, then perform the shift xR = x^R + !x~L, where ! is a constant linear operator on the Cartan subalgebra h containing (rank G)2 additional parameters, and nally implement the reverse T-duality x~L ! x^L. Eventually we arrive at the action where |~a is the left-invariant one-form associated with g~a, i.e. |~a = g~a 1dg~a. It is important to note that the parameterisation (3.2) introduces a new left-acting Cartan gauge symmetry xa ! xa + a; g~a ! exp( a)g~a: As we will see this symmetry survives the TsT transformation (up to potential total derivatives). Therefore for now we leave it un xed, using the xa coordinates to implement the deformation, and x it only at the end. De ning the combinations HJEP10(27) such that Z Z Z Z Z 1 1 Z Z { 8 { (3.3) (3.4) (3.5) (3.6) (3.8) d 2 tr l+R OR lR (1 + L2)LR+!O 1!tLR + LL+O + LR+Oe 1!(LL + (1 + L2)@ x^L) + SWZ;k[g~L] SWZ;k[g~R] k d 2 tr A (|~L+ |~R ) A+(|~L + |~R ) ; (3.9) Note that O and Oe are related as follows !tOe 1 ! = (!tOe 1!)t = O 1!t!; !O 1!t = (!O 1!t)t = Oe 1!!t: In order to recast the action (3.9) in a form generalising (2.1) we parameterise g~a = exp(ya)g^a; g^a 2 G; ya 2 h: with Setting we nd that the x^a dependence drops out of the action up to the total derivative k k2!t!) 1 t which we also drop. We expect to be able to remove the dependence on x^a in this way as a consequence of the left-acting Cartan gauge invariance (3.4). As foreseen this symmetry survives the TsT transformation up to potential total derivatives that we ignored in the T-dualisations. S![gL;R; A] = Renaming g^a as ga, we are nally left with the action Z d 2 X where the dressed operators are now given by OLL;! = AdgL1 1 + ALR + ORR;! = AdgR1 1 + ARR + 2 L 2 R (1 + R2)(1 + L2 + k)(1 + L2 (1 + L2)(1 + R2 + k)(1 + R2 OLR;! = AdgL1 (1 + L2 + k)(1 + R2 + k)(O 1 t ! ) ORL;! = AdgR1 (1 + L2 k)(1 + R2 k)(Oe 1!) with AL;R de ned in terms of L;R and k in (3.1), O and Oe given in (3.10) and we recall that ! is an arbitrary constant linear operator on h. As we will shortly demonstrate via the existence of a Lax pair this multi-parameter deformation of the PCM is integrable. Before we do so, let us brie y consider various limits of (3.14) in order to gain a better understanding of the model. First we note that, as expected, upon setting ! = 0 we recover the three-parameter deformation of section 2, i.e. the bi-Yang-Baxter deformation of the PCM plus WZ term. Additionally setting either L or R to zero we expect to nd the one-parameter Yang-Baxter deformation of the PCM plus WZ term constructed in [14]. The model of [14] depends on a single eld g 2 G and hence to explicitly check this { 9 { relation we integrate out the gauge eld. This is done in section 3.3 for the multi-parameter deformation (3.14), with the resulting action given in (3.36). The latter only depends on gL and gR through the combination g = gLgR 1 as a consequence of the gauge symmetry (2.7), and indeed setting ! = R = 0 we recover the model of [14]. It is also interesting to consider the limit k = 0, that is when the WZ term is no longer present. In this case we can rewrite the deformed action in a form familiar in the context of Yang-Baxter deformations where the operator O is given by which in turn is de ned in terms of a linear operator R acting on g g ! ! 1 R O = p1 + L2 0 0 p1 + R2 1 RgL;R p1 + L2 0 p1 + R2 ! ; RgL;R = R = AdgL1 0 0 AdgR1 LR p(1 + L2)(1 + R2)! AdgL 0 0 AdgR p(1 + L2)(1 + R2)!t ! : 0 ! ; RR For all X = (xL; xR)t and Y = (yL; yR)t in g g the operator R satis es the modi ed classical Yang-Baxter equation [RX; RY ] Note that the right-hand side of (3.19) is independent of ! and hence if we additionally set L = R = 0 the operator R satis es the classical Yang-Baxter equation. In this case we are left with the homogeneous Yang-Baxter deformation of the PCM with an abelian R-matrix, which is equivalent to a series of TsT transformations [18{24]. Alternatively we may set ! = 0, in which case we recover the bi-Yang-Baxter sigma model of [8, 9, 25, 11]. 2 and ( + 1)2!!t = ( + 1)2!t! = 1 then the operator R 2R2 = 1: Therefore, 1R de nes a complex structure on G G. Yang-Baxter deformations based on complex structures have been explored in [30] and typically give rise to particularly The Lax pair (2.29) for the three-parameter model described by the action (2.1) is given by L ( ) = A + J L + G (3.17) (3.18) (3.19) (3.20) (3.21) with the parameters , G given in (2.25) and (2.28) respectively. The zero-curvature equation for L ( ) implies the equations of motion (2.10) and Maurer-Cartan equations, (2.21) and (2.23). Furthermore, it should be supplemented with the equations of motion for the gauge eld (2.8), which are constraint equations xing the gauge eld in terms of the group elds. The Lax pair (3.21) and the constraint equations (2.8) are written in terms of the currents J a and the gauge eld A , where the dependence on ga is contained within the former. Therefore, to determine the Lax pair and constraint equations for the TsT transformed model we just implement the transformation on J a , which gives the TsT transformed currents. It then follows that the Lax pair for the model described by the TsT transformed action (3.14) has the same form (3.21) as it had before transformation, only now with J a given by TsT transformed expressions for the currents. The same holds for the constraint equations (2.8). To construct the currents of the TsT transformed model we start from those of the three-parameter model de ned by (2.9). Using the parameterisation (3.2) these can be written as J L = J~L + Adg~L1(1 J R = J~R + Adg~R1(1 (3.22) where the J~a are simply obtained from J a by the replacement ga ! g~a. The currents J a , and thus the Lax pair, equations of motion and Maurer-Cartan equations, only depend on derivatives of the Cartan subalgebra valued elds xa. Then, following, for example, [28], we track the fate of the derivatives @ xa through the TsT transformation. In the rst step, that is under the T-duality xL ! x~L, one has 1 1 + L2 (LL where La are de ned in (3.6). The second step is a translation of xR and implies x^R = xR !x~L ) Finally, the second T-duality, x~L ! x^L, gives 1 LL (1 + L2)(!t(LR + (1 + R2)@ x^R) where O is de ned in (3.10). Once the TsT transformation is performed, we x the gauge x^L = x^R = 0 using the gauge symmetry (3.4). Recall that, as discussed in subsection 3.1, this symmetry survives the TsT transformation up to total derivatives, which do not contribute to the equations of motion. For this gauge choice the expressions for @ xa in (3.23) and (3.24) become (1 + R2)O 1!t!LL (1 + L2)O~ 1!!tLR 1!tLR ; O O ~ 1!LL : Substituting into (3.22) we nd expressions for TsT transformed currents J a as a function of the eld g~a 2 G. Finally, to match with the action (3.14) we rename g~a as ga, after (3.23) (3.24) (3.25) (3.26) j j L L + A = QLJ L ; A+ = PLtJ+L ; j j R R + A = A+ = QRJ L ; PRtJ+L ; where QL, QR, PL and PR are the following operators OLR;!(ORR;! k) 1ORL;! 1 1 + OLR;!(ORR;! k) 1 ; k ORL;!(OLL;! + k) 1OLR;! 1 1 + ORL;!(OLL;! + k) 1 ; PL = 1 + (ORR;! + k) 1ORL;! OLL;! k OLR;!(ORR;! + k) 1ORL;! k) 1OLR;! ORR;! + k ORL;!(OLL;! k) 1OLR;! 1 1 ; : which these currents are expressed as follows J L = (OLL;! + k)(jL J+L = (OLtL;! J R = (ORR;! where the various operators are de ned in (3.15). Therefore, the Lax pair of the TsT transformed model (3.14) takes the form (3.21) with J a now given by (3.27). As before, this Lax pair should be supplemented by constraint equations of the form (2.8), again with J a given by (3.27). Note that these results also follow from direct computation, in the spirit of subsection 2.2, starting from the action (3.14). Let us now eliminate the gauge eld from the action (3.14). The resulting action will be the starting point in the next section for the comparison with Lukyanov's model. The equations of motion for the gauge eld (2.8) and the de nitions of J a given in (3.27) can be used to write the left-invariant currents ja as QL = OLL;! + k QR = ORR;! PR = 1 + (OLL;! 1 2 and the currents J L as where (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) Inverting these relations it is then possible to express the gauge eld as A = jL + jR (QL QR) J L ; A+ = j+L + j+R t PL PRt J+L : 1 2 J L = (QL + QR) 1 j ; J+L = PLt + PR t 1 j+; These results enable us to rewrite the rst term in the Lagrangian for the action (3.14) as j = jL jR: tr h(j+a = 1 2 h tr j+ (QL + QR) 1 j + j+ (PL + PR) 1 j i : The last term of (3.14) is proportional to the gauge eld. We can therefore use the relation (3.30) to obtain k tr A (j+L jR ) + A+(jL jR ) = k tr j+LjR j+R jL 1 2 1 2 h h + k tr j+ (QL QR) (QL + QR) 1 j i k tr j+ (PL + PR) 1 (PL PR) j i The rst term in (3.34) may be combined with the WZ terms associated with gL and gR using the Polyakov-Wiegmann formula [31] where the operators QL, QR, PL and PR are de ned in (3.29). It is straightforward to check that, as indicated, this action only depends on gL and gR through the combination g = gLgR 1. This is expected as a consequence of the Gdiag gauge symmetry (2.7). 4 Equivalence with the Lukyanov model for G = SU(2) In this section we prove that the action (3.14) corresponds to the Lukyanov model [1] for G = SU(2). Let us start by noting that SU(2) has rank one. Therefore in this case the operator !, introduced in section 3, contains just a single parameter. In a slight abuse of notation we will also call this parameter !, with the operator given by multiplying by the identity (acting on the Cartan subalgebra). For G = SU(2), the action (3.14) thus de nes a four-parameter integrable deformation of the SU(2) PCM. As a rst order check of equivalence we observe that this is the same number of deformation parameters as in the Lukyanov model. To demonstrate the full equivalence we shall start with the action (3.36), obtained after eliminating the gauge eld. Partial identi cation of this four-parameter deformation with the Lukyanov model, that is to say with some deformation parameters set to zero, has already been shown in [2]. For this reason we use the same parameterisation of g 2 SU(2) as in [2]. We then compute the corresponding metric and B- eld and show that there exists a coordinate transformation, and a map between the parameters AL, AR, k, ! and Lukyanov's parameters , p, h and h, such that this metric and B- eld coincides with those of [1]. Let us take the SU(2) group element g(r; ; ) = e T 3( + ) r 1l p Here T i are the generators of su(2) satisfying [T i; T j ] = ij kT k; tr(T iT j ) = i; j; k = 1; 2; 3; where the totally anti-symmetric tensor ijk is normalised as 123 = +1 and the su(2) indices are raised and lowered by ij and its inverse. The R-matrix acts on the generators as R(T +) = iT +; R(T ) = iT ; R(T 3) = 0; where T = p12 (T 1 The computation of the metric and B- eld is rather lengthy but ultimately straightforward. In order to see the equivalence with the metric and B- eld of Lukyanov's model one needs to perform the following coordinate transformations for the angle variables fL+ + fR+ f L f R 4(1 k!)(1 + L2)(1 + R2)ALAR 4(1 + k!)(1 + L2)(1 + R2)ALAR log log 4 + (AL AR) 2 4 + (AL AR)2 + 4r2ALAR 4 + (AL AR) 2 4 + (AL AR)2 + 4r2ALAR ; ; (4.4) and = = 1 + 2 where fL;R are given by f f L = AL(1 + L2)(k R2 R = AR(1 + R2)(k L2 (1 + R2)(1 + R2 (1 + L2)(1 + L2 k2)!); k2)!): The resulting metric becomes block diagonal, i.e. gr 1 = gr 2 = 0. We also introduce a new radial coordinate z related to r by (4.2) (4.3) (4.5) (4.6) (4.7) (4.8) (4.9) r = s (1 2(1 )(1 + z) z) ; and de ne Lukyanov's parameters ( ; p; h; h) as = p4 + (AL + AR)2 p4 + (AL p4 + (AL + AR)2 + p4 + (AL AR)2 AR)2 ; p2 = h = h h = 4H (p4 + (AL + AR)2 + p4 + (AL AR)2)H0 L2(1 + R2)AR ; R2(1 + L2)AL ; where the quantities H0, H+ and H are given by H0 = p(1 + L2)(1 + R2)(1 k2!2); H+ = kp L2 R2 + !2(1 + L2)(1 + R2) (4 + A2L + A2R + !2(1 + L2 k2)(1 + R2 k2)); H = !(k2 + (1 + L2)(1 + R2)): With these identi cations we indeed recover the metric and B- eld of [1]. In particular, up to a total derivative, the Lagrangian corresponding to (3.36) is [1, 2] h 2 2 + [C(z) i 1 ; where we have rewritten the Lukyanov background using new angle variables ( 1; 2), related to the original ones (v; w) through overall factor T is equal to 1 = 12 R 1(v w), 2 = 12 (v + w) [2]. The T = 2((1 + L2)(1 + R2) + k2) 2 + L2 + R2 ; while the components of (4.9) are U (z) = 4(1 m2 z2)(1 2z2) ; 2R m c + c D(z) = R2(1 + z) 2 + (p2 + p 2) (2 + p2 + p 2)z Q(z); D^ (z) = (1 z) 2 + (p2 + p 2) + (2 + p2 + p 2)z Q(z); C(z) = (p2 p 2)R(1 z2)Q(z); B(z) = h(c2 1)(c z) h(c2 1)(c + z) Q(z); where Q(z) is given by Finally, we recall the de nitions of c, c, m and R, Q(z) = 4(1 (c + 1)(c 1) 2)(c + z)(c z) : c = r 1 + h2 2 + h2 ; m = p( + p2)( + p 2); c = R = s s 1 + h2 2 + h2 ; (c 1)(c + 1) (c + 1)(c 1) : The expressions of c, c and m in terms of the parameters AL, AR, k and ! are cumbersome. We shall therefore not reproduce them here. However, let us point out that the relations (4.7) and (4.13) lead to a simple expression for R, This expression is interesting because it has simple limits. Indeed, we have R = 1 when ! = 0 or k = 0. This result is consistent with those obtained previously in [2]. 5 Conclusion In this paper we have presented a new multi-parameter integrable deformation of the PCM for a general group G. The rst step of its construction was the derivation of the integrable bi-Yang-Baxter deformation of the PCM plus WZ term. The second was the implementation of a general TsT transformation mixing the Cartan generators of the two copies of G. R = 1 k! 1 + k! : (4.10) (4.11) (4.12) (4.13) (4.14) This multi-parameter integrable model generalises Lukyanov's four-parameter deformation of the SU(2) PCM [1] to arbitrary group G. Therefore the construction con rms the proposal of [2] on the algebraic origin of the four parameters: two correspond to the bi-Yang-Baxter deformation, one parameterises the coupling to the WZ term, and the nal parameter is generated by a TsT transformation. There are a number of possible open questions whose investigation would further probe the properties of this integrable -model. One of the most important is the study of its classical Poisson structure, Hamiltonian integrability and twist function in the spirit of [10, 11, 13, 32]. To determine the twist function, it is enough to consider the three parameter case. Indeed, the twist function is not changed under a TsT transformation [32]. One particular aim is to understand the q-deformed algebra of hidden charges. Furthermore, the extension to the a ne algebra as considered in [33] for the Yang-Baxter -model would be interesting to investigate (see also [34, 35, 16] for the SU(2) case). Finally, studying this -model at the Hamiltonian level would indicate if it is also possible to reinterpret it as a dihedral a ne Gaudin model [36]. In [37] the Yang-Baxter deformation of the PCM plus WZ term of [14] was recast in the framework of E -models [38, 39] (a rst-order action de ned on the Drinfel'd double). Understanding how to formulate the bi-Yang-Baxter deformation of the PCM plus WZ term (and TsT transformations thereof) presented in sections 2 (and 3) in this language may prove useful in gaining a deeper understanding of the underlying algebraic structure of the model. Setting L = R = ! = 0 and k = 1, the deformed action simpli es to the WZW action for the group G. This model is conformal, as are its deformations associated with TsT transformations. It would be interesting to investigate which other points in parameter space correspond to conformal sigma models at the quantum level and hence de ne string backgrounds. In particular, this would involve generalising the one-loop renormalisation analysis, including UV and IR xed points, of [1] beyond the SU(2) case. Finally, there are a class of superstring backgrounds for which the Green-Schwarz worldsheet action takes the form (at least in part) of an integrable supercoset -model [40{ 44]. For the maximally symmetric AdS5 S5 background the PSU(2; 2j4)=(SO(1; 4) SO(5)) supercoset model of [40] captures the full Green-Schwarz string. Generalising the bosonic construction of [10], the Yang-Baxter deformation of this model was constructed in [45, 46]. Particularly relevant to the constructions of this paper are string backgrounds for which the superisometry takes the form of a product group. For example, the AdS3 background is related to the supercoset PSU(1; 1j2)2=(SU(1; 1) SU(2))diag, and the AdS3 S1 background to the supercoset D(2; 1; )2=(SU(1; 1) SU(2) SU(2))diag [47]. In these cases one can construct a bi-Yang-Baxter deformation of the supercoset -model [25], S3 T 4 or alternatively introduce a WZ term [48]. 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F. Delduc, B. Hoare, T. Kameyama, M. Magro. Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable σ-model, Journal of High Energy Physics, 2017, 212, DOI: 10.1007/JHEP10(2017)212