All-loop correlators of integrable $\lambda$-deformed $\sigma$-models

We compute the 2- and 3-point functions of currents and primary fields of $\lambda$-deformed integrable $\sigma$-models characterized also by an integer $k$. Our results apply for any semisimple group $G$, for all values of the deformation parameter $\lambda$ and up to order $1/k$. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable $\lambda$-deformed $\sigma$-models. We also present analogous results in two limiting cases of special interest, namely for the non-Abelian T-dual of the PCM and for the pseudodual model.


Introduction and motivation
One of the most intriguing conjectures in modern theoretical physics is the AdS/CFT correspondence [1] which, in its initial form, states the equivalence between type-IIB superstring theory on the AdS 5 × S 5 background and the maximally supersymmetric field theory in four dimensions, i.e. N = 4 SYM. In recent years, a huge progress has been made in calculating physical observables employing both sides of the duality.
These calculations managed to probe the strongly coupled regime of the gauge theory which is practically unaccessible by other means. The key feature that allowed this progress is integrability. N = 4 SYM from one side and the two-dimensional σ-model from the other, are believed to be integrable order by order in perturbation theory. It is clear that one way to construct generalizations of the original AdS/CFT scenario is to try to maintain the key property of integrability.
The aim of this work is to study the structure of a class of two-dimensional σ-models, the so-called λ-deformed models constructed in [2]. For isotropic couplings the deformation is integrable in the group case and in the symmetric and semi-symmetric coset cases [2][3][4][5] (for the su(2) group case integrability is preserved for anisotropic, albeit diagonal couplings [6]). They are also closely related [7][8][9][10][11][12] to the so-called η-deformed models for group and coset spaces introduced in [7,8] and in [13][14][15], respectively. This relation is via Poisson-Lie T-duality and an analytic continuation of coordinates and of the parameters of the σ-models [10][11][12]. There are also embedings of the λ-deformed models as solutions of supergravity [16][17][18].
In particular, we shed light into the structure of the λ-deformed models by computing the two-and three-point functions of all currents and operators exactly in the deformation parameter and up to order 1/k. This work is based and further extends symmetry ideas and techniques originated in our previous work in [19]. The results of this work are summarized in section 7.
Our starting point is the WZW action for a generic semisimple group G, with g ∈ G parametrized by X µ , µ = 1, 2, . . . , dim G.
We will use the representation matrices t a which obey the commutation relations [t a , t b ] = f abc t c and are normalized as Tr(t a t b ) = δ ab . These matrices are taken to be Hermitian and therefore the Lie-algebra structure constants f abc are purely imaginary.
The chiral and anti-chiral currents are defined as The left and right invariant forms L a = L a µ dX µ and R a = R a µ dX µ are related as R a = D ab L b , D ab = Tr(t a gt b g −1 ) . (1.3) We are interested in the non-Abelian Thirring model action (for a general discussion, see [20,21]), namely the WZW two-dimensional conformal field theory (CFT) perturbed by a set of classically marginal operators which are bilinear in the currents where the couplings are denoted by the constants λ ab . An action having the same global symmetries as (1.4), and to which reduces for small values of λ ab has been derived in [2] (see also [22] for the SU(2) case), by gauging a common symmetry subgroup of an action involving the PCM model and the WZW actions. It reads [2] S k,λ (g) = S WZW,k (g) + k 2π where we have assembled in a general real matrix λ the coupling constants λ ab . In addition, this action, as well as (1.4), is invariant under the generalized parity trans- The β-functions for the running of couplings under the Renormanization Group (RG) flow using (1.5) were computed in [23,24] and completely agree with the computation of the same RG-flow equations using CFT techniques based on (1.4) in [25] for a single (isotropic) coupling, i.e. when λ ab = λδ ab and in [26] for symmetric λ ab . Based on that it was conjectured in [23,24] that (1.5) is the effective action for (1.4) valid to all orders in λ and up to order 1/k. In the same works it was realized that (1.5) has the remarkable symmetry These fields are also Virasoro primaries with holomorphic and antiholomorphic dimensions [29] where c R , c R ′ and c G are the quadratic Casimir operators, all non-negative, in the representations R, R ′ and the adjoint representation for which (t a ) bc = f abc . They are defined as In our calculations we will need the basic two-and three-point functions for these fields. For the currents they are given by where we employ the general notation z ij = z i − z j . We will also use the four-point function δ aa 1 δ a 2 a 3 (x 1 − z 1 ) 2 (z 2 − z 3 ) 2 + cyclic in 1, 2, 3 .
(2.6) Similar expressions hold for the antiholomorphic currents as well. Correlators involving both holomorphic and anti-holomorphic currents vanish at the conformal point.
However, as we shall see, this will not be the case in the deformed theory.
The corresponding correlators for the affine primaries are where the superscripts signify the fact that the representations for the different primaries in correlation functions could be, in general, different. However, for the twopoint functions the two representations should in fact be conjugate to each other for the holomorphic and anti-holomorphic sectors separately. As such, they have the same conformal dimensions. Recalling that the matrices t a andt a are Hermitian and after removing the superscripts by relabeling the representation matrices we have that The minus sign in the definition of the conjugate representation is very important for the matrices to obey the same Lie-algebra. It will turn out that, in the deformed theory, for correlation functions involving two primaries to be non-vanishing, their corresponding representations must be conjugate to each other, as well.
Next, consider three affine primaries transforming in the representations (R i , R ′ i ), i = 1, 2, 3. Then the three-point function for them is given by where (2.10) and cyclic permutations of 1, 2 and 3 for the rest. The structure constants C ii ′ ,jj ′ ,kk ′ depend on the representations and implicitly also on k. They obey various properties arising mainly from the global group invariance of the correlation functions, which will be mentioned below in the computation of the three-point functions involving only affine primaries.
Finally, we have the three-point functions with one current and two primaries. They are given by where we have used the fact that, for a non-vanishing result, the representations in which the primaries transform have to be conjugate to each other for the holomorphic and the antiholomorphic sectors, separately. Also I R and I R ′ are the identity elements for the corresponding representations.
Correlators with two currents and one affine primary field are zero at the conformal point and will remain zero in the deformed theory as well.

Symmetry and correlation functions
In order to compute the correlation functions of currents and of primary fields we will heavily use the symmetry of the effective action for the non-Abelian Thirring model (1.7). First let's consider correlation functions for currents only. At the conformal point when λ = 0 the currents are given in terms of the group element by (1.2) and are, of course, chirally and anti-chirally conserved on shell. Obviously, in the deformed theory these currents will be dressed and will receive λ-corrections. One expects that since their definition contains derivatives there will be operator ambiguities at the quantum level. We propose that these dressed currents are given by These become the correct chiral and anti-chiral currents when λ = 0 (up to a minus sign for J − ). Also, they are components of an on shell conserved current. The attentive reader will notice that the dressed current components in (2.13) are nothing, but, up to a factor of λ, the gauge fields evaluated on-shell in the original construction of (1.5) in [2] by a gauging procedure. Hence, it is natural to consider correlation functions of the J a ± 's as defined above. In addition, we have that Passing to the Euclidean regime we have for the two-point function of the holomorphic component of the currents that with the partition function being where we have used the symmetry of the action (1.7) and the fact that the measure of integration is invariant under g → g −1 , i.e. D[g −1 ] = D[g]. 1 Hence, the partition function of the deformed theory is invariant under the duality-type symmetry. In where we have also employed (2.14). Hence, we obtain that the correlation function should obey the non-trivial identity This identity between current correlators is straightforwardly extendable to higher order correlators involving currents with any type of currents, J a 's orJ a 's.
The overall factors of λ can be absorbed by redefining the currents in (2.13) by a factor of λ. In the following we assume that this is the case which implies also the absence of the factor of λ 2 in the r.h.s. of (2.14).
The above conclusion for the current correlators is in full agreement with [27] who reached the same conclusion using the non-Abelian Thirring model action and certain special properties of the WZW action path integral. The advantage of employing the effective action is that one can employ the duality-type symmetry on correlation functions involving primary fields in the deformed theory which has not been considered before. For these fields we have that, under the inversion of the group element the 1 The measure of integration contains the Haar measure for the semisimple group G which is certainly invariant under g → g −1 , but also the factor det(λ −1 − D T ) arising from integrating out the gauge fields in the path integral [2]. This can be easily seen to transform under g → g −1 and λ → λ −1 as (for a general matrix λ): det(λ −1 − D T ) → (−1) n det λ × det(λ −1 − D T ), with n = dim G and where we have used the property D(g −1 ) = D T (g). This extra constant overall factor cancels out by the same factor arising from the partition function in the denominator in all correlation functions. primary field Φ (1) transforms to its conjugate Φ (2) . Explicitly, we have that (2.20) which means that for the representation matrices we have t (1) ↔t (2) , t (2) ↔t (1) . (2.21) Note that if the inversion of g is followed by the σ → −σ, i.e. the parity transformation (1.6), then 22) and in addition the J a 's andJ a 's are interchanged.

The non-Abelian and pseudodual chiral limits
Besides the small λ ab limit, leading to (1.4), there are two other interesting limits of the action (1.5). They will be instrumental in our computation of correlation functions.
In the first limit [2] one expands the matrix and group elements near the identity as where E is a general dimG square matrix. This leads to Note that our structure constants are purely imaginary so that f ab are indeed real. In this limit the action (1.5) becomes which is the the non-Abelian T-dual with respect to the G L action of the σ-model given by the PCM action with general coupling matrix E ab . We note that in this limit the WZW term in (1.5) does not contribute at all.
To discuss the second new limit, we first recall that the original derivation of the action (1.5) leads for compact groups to the restriction 0 < λ < 1. However, once we have the action we may allow λ to take values beyond this range. For instance, the symmetry (1.7) clearly requires that. Here in order to take a new limit we will extend the range of λ to negative values. We will also need the following equivalent form of the action (1.5) given, after some manipulations needed to combine the quadratic part of the WZW action and the deformation term in (1.5), by where we we remind the reader that our structure constants are purely imaginary.
Then we take the limit where again E is a general dimG square matrix. The various quantities expand as in (2.24) with k replaced by k 1/3 . Then the action (2.26) becomes We see that E can be taken to be symmetric since any antisymmetric piece leads to a total derivative. This action for E ab = δ ab /b 2/3 is nothing by the pseudodual model action [30]. Note that the quadratic part of the WZW action and the deformation term in (1.5) are equally important for the limit (2.27) to exist since each term separately diverges when this limit is taken.
Since the above non-Abelian and pseudodual limits exist at the action level, we expect that physical quantities such as the β-function and the anomalous dimensions of various operators should have a well defined limit as well. This will be an important ingredient in our method of computation.

The regularization method and useful integrals
In the Euclidean path integral the action appears as e −S . The action we will be using is that of the non-Abelian Thrirring model action and will be expanding around the WZW CFT part of it. This is not in contrast with the approach of the last subsection where (1.5) was used, the reason being that the latter is the effective action of the non-Abelian Thrirring model. Hence, it contains all λ-corrections and can be considered as a starting point to find at the quantum level corrections in 1/k. Schematically, to O(λ n ), the correlation function for a number of some generic fields F i , i = 1, 2, . . . , involves the sum of expressions of the type where d 2 z 1...n := d 2 z 1 . . . d 2 z n and for convenience we have dropped k from our notation in the correlation functions · · · k,λ of the deformed theory.
That way one encounters multiple integrals which need to be regularized. Our prescription to do so consists of two steps: • We choose the order of integration from left to right d 2 z 1...n and never permute this order. This is due to the fact that due to the divergences appearing, the various integrations are not necessarily commuting.
• Internal points cannot coincide with external ones. This means that the domain of integration is However, internal points can coincide. Also contact terms, arising from coincident external points will be allowed. The latter is a choice we make and not a part of the regularization scheme. 2 We shall need the very basic integral given by Clearly, if the domain of integration allows, the integral diverges for large distances. 2 All these imply that we will have for the δ-functions arising in performing the various integrations that Note also that in the regularization of [31] no two points, internal or external, can coincide and therefore all δ-functions arising in integrations are set to zero. In contrast in [32] all such δ-functions are kept. The advantage of our regularization is that the symmetry of the correlation functions under k → −k and λ → λ −1 is manifest whereas for the others it is hidden.
The above result is valid provided that the integration is performed in a domain of characteristic size R, e.g. a disc of radius R, with the external points x 1 and x 2 excluded and in addition obeying R ≫ |x 1 |, |x 2 |. The latter conditions are responsible for the translational invariance and the reality of the result. Even then we have to make the replacement |x 12 | 2 → |x 12 | 2 /R 2 on the right hand side of (2.31). However, in our computations there will be integrals of the same kind but with opposite sign and x 1 equal to x 2 and which will have a small distance regulator ε. Hence the factor R will drop out at the end, leaving the ratio ln ε 2 |x 12 | 2 . This means that in practice the domain of integration is R 2 except for the points x 1,2 which are excluded. By appropriately taking derivatives we also have the useful integrals In appendix A we have collected results for some useful to this work integrals. We single out which are valid under the assumptions spelled out below (2.31).

Current correlators
In this section, we will focus on the two-and three-point functions involving purely currents. These will be computed up to order 1/k and exactly in the deformation parameter λ. To establish our method, employed already in [19], as clearly as possible we first start with the computation of the two-point functions which enables to com-pute the β-function and the anomalous dimensions for the currents known already from using CFT methods in [19,25,26] and from gravitational computations [23,24].
Then we proceed to correlators involving three currents.

Two-point functions
On general grounds the correlator of J a and J b takes the form The result to O(1/k) and O(λ 3 ) was computed in sec. 2 of [19] and reads Comparing with the general form of the two-point function (3.1) we have that and Similarly the correlator of J a andJ b should assume the form At the conformal point this correlator should vanish. We have also allowed for contact terms proportional to the δ-function since these are allowed by symmetry. The coupling functions A and B have to be computed.
After a long computation, all details are given in the appendix B, we found the result which, keeping in mind that we are interested to terms up to O(1/k), is easily seen to be of the form (3.5). Note that this correlator takes the form where γ (J) is the current anomalous dimension given perturbatively by (3.4).

The exact β-function and anomalous dimensions
To compute the wave function renormalization and that for the parameter λ we use the two-point functions J a J b and J aJb . In particular we need the most singular part of these correlation functions. For the purpose of this section let's denote the bare currents by J a 0 andJ a 0 and similarly for the parameter λ 0 .
We need the most singular part of the bare two-point functions up to order 1/k. From where we have kept only the coefficient of the most singular term, i.e. of δ (2) (x 12 ).
The bare quantities and the renormalized ones are related as We make the following ansatz valid to order 1/k in the large k-expansion where the logarithm-independent term in Z −1 has been chosen so that the renormalized two-point function for the J a 's is normalized to one. The pure number coefficients c i are computed so that the renormalized two-point functions are independent of the cutoff ε. We find that the unique choice is given by The β-function is by definition where the bare coupling coupling λ 0 is kept fixed. Next we compute the anomalous dimension of the current (3.15) in agreement of course with (3.4).
The above perturbative expressions are enough to determine the exact in λ dependence of the β-function and of the anomalous dimensions up to order 1/k. As explained, the exact β-function and anomalous dimensions should have a well defined behaviour in the two limiting cases described by the non-Abelian and pseudodual model limits (2.23) and (2.27), respectively. In the isotropic case, which is the case of interest in this work, it implies regularity under the following independent limits where f (λ) and g(λ) are two analytic functions of λ. The assumed pole structure does not exclude the possibility that one of the poles reduces its degree or even ceases to exist. This can happen if the functions in the numerator are zero at λ = 1 or/and λ = −1. In addition, due to the symmetry under (k, λ) → (−k, λ −1 ) we have that All these imply that these functions are in fact polynomials of, at most, degree four Demanding agreement with the perturbative expressions (3.14) and (3.15) we obtain a 0 = a 1 = b 0 = b 1 = 0 and a 2 = b 2 = 1 which completely determines the exact β-function and anomalous dimensions to be and It is also easily seen that the coefficient of the O(λ 3 ) term is in agreement with the perturbative results as well. The above expressions are in full agreement with the results found in [23,24,33] for the β-function and in [19] for the anomalous dimensions.
Note that the β-function and anomalous dimensions of the non-Abelian T-dual limit which are valid for large κ 2 . The anomalous dimensions correspond to which are obtained by taking this limit in (2.13).
The corresponding expressions for the pseudodual model are These are in agreement with the expressions derived in [30] (see above fig. 2) and are valid for small b. The anomalous dimensions correspond to which as before are obtained by taking the appropriate limit in (2.13).

Three-point functions
We consider the J J J and J JJ correlators. The remaining correlators JJJ and JJ J can be easily obtained by applying the parity transformation to the first two. The results of this subsection match those obtained in [32], where current-current perturbations of the WZW model on supergroups were studied with a different regularization scheme. Before moving to our analysis, let us note that analogue perturbations of the WZW models on supergroups were studied in [34], but the perturbation consists of the term J a + D ab J b − added to the action; effectively the non-critical WZW model.

The J J J correlator
From appendix C we have that the, up to O(λ 3 ), correlator reads The ansatz for the all-loop expression takes the form where f (λ) is everywhere analytic and obviously f (0) = 1 to agree with the CFT result. As before this form takes into account that under the limit (3.16) the correlator is well behaved. Invariance of the above expression under the duality-type symmetry Consistency with the perturbative expression up to O(λ) (3.27) gives c = 1. Therefore, the all-loop correlator reads As a check we see that this expression reproduces the O(λ 2 ) and O(λ 3 ) terms in the perturbative expression (3.26).

The J JJ correlator
The perturbative calculation of this correlator is performed in appendix D. The result We now make a similar to (3.27) ansatz for the all-loop expression , (3.31) where f (λ) is everywhere analytic and f (0) = 1. Invariance of the above expression under the duality-type symmetry yields Hence, we find the all-loop expression (3.33) whose expansion around λ = 0 agrees with (3.30).
Note that implementing the non-Abelian and pseudodual limits both lead to finite (non-zero) expressions for all of the above three-point functions. In these limiting cases the results are valid for large κ 2 and small b where we refer to (3.16) for the definition of these parameters. We mention also that, our results for these correlators agree with those done for supergroups in [32] after an appropriate rescaling of the currents that presumably takes into account the different regularization schemes used in that work.

Primary field correlators
The purpose of this section is to compute two-and three-point functions of arbitrary primary fields. This will allow us to extract their anomalous dimensions and the deformed structure constants in the OPEs.

Two-point functions
After a long computation, all details of which are given in the appendix E, we found that a perturbative computation up to O(λ 3 ) and to order 1/k, gives for the two-point function of primary fields the result We see that due to the deformation there is an operator mixing so that one should proceed by choosing an appropriate basis in which the dimension matrix is diagonal.
For convenience we will adopt the double index notation I = (ii ′ ). Then there is a matrix U chosen such that where N I are the eigenvalues of the matrix t a ⊗ t * a . Note also that U is λ-independent as well as k-independent. Then in the rotated basis Φ (1) the correlator (4.1) becomes diagonal, i.e.

Φ
(1) where perturbatively To determine the exact anomalous dimension of the general primary field we first realize that we should include in the above expression the k-dependent part coming from the CFT dimensions of ∆ R and∆ R ′ in (2.3) up to order 1/k. Hence the anomalous dimension is given by As in the case of currents we make the following ansatz for the exact anomalous di- where the yet unknown function should be analytic in λ. Using the symmetry (1.7) and the transformation of the primary fields under this symmetry (2.20), we have that which implies the following relations between the various unknown functions Hence, these functions should be fourth order polynomials in λ with related coefficients. It turns out that comparing with the perturbative expression (4.6) up to O(λ 2 ) we determine all these functions to be Therefore, the exact in λ anomalous dimension is It is easily checked that this expression is in agreement with the O(λ 3 /k) term in (4.6).
Note also that in the non-Abelian limit the above anomalous dimensions have a well defined and different than zero limit. In contrast the limit is zero in the pseudodual limit. This expression also applies for current current perturbations of the WZW model on supergroups with vanishing Killing form [31].
Finally, the two point functions take the form (4.12)

Three-point functions
To leading order in the λ-expansion after a straightforward computation this correlator is found to be Even for dimensional reasons we should be able to cast the above expression in a form in which all space dependence is in terms of ratios ε 2 /|x ij | 2 . In order to do that we first recall that the structure constants C ii ′ ,jj ′ ,kk ′ are factorized according to their holomorphic and antiholomorphic content as (4.14) An important constraint, arises by making use of the global Ward identity. It reads From (4.15) it is straightforward to obtain the following relations Using the above relations we can rewrite the three-point function as The next step is to pass to the rotated basis. By using the double index notation we introduced before we have that where in the new basis the structure constants read while the result for the correlator at O(λ) is given by From this result we can write down the exact expression in λ for the three-point function. It is given by where γ 12;3 is given by (4.20) The other differences of dimensions γ 23;1 ,γ 23;1 and γ 13;2 ,γ 13;2 are obtained by performing cyclic permutations in the indices 1, 2 and 3.
We now turn our attention to the three-point function coefficientsC I JK (k, λ). At λ = 0 these coefficients are considered as known since they are in principle fully determined from the WZW CFT data. On general grounds the following perturbative expansion holdsC where note the leading coefficientC (0) I JK in 1/k expansion does not depend on λ. This is so because such a term being k-independent and simultaneously having possible poles only at λ = ±1 and preserving the symmetry k → −k, λ → λ −1 cannot be finite either in the non-Abelian T-dual or in the pseudodual limit. Using the same line of reasoning as in the rest of this paper we conclude that the first correction to the three-point function should be of the form 3 with We saw from the O(λ) calculation that a (1) I JK = 0. Furthermore, it is not difficult to see that a (2) I JK = 0 too. Indeed, by inspecting the O(λ 2 ) calculation one can see that in order to remain to order 1/k either the two holomorphic or the two anti-holomorphic currents should be contracted through the Abelian part of their OPE. Then the resulting integrals will be of the form which can only produce logarithms. But the logarithms have to be combined and exponentiated to give the differences of the anomalous dimensions. Thus, no finite part will be present at this order and as a result a (2) I JK = 0, as well. Thus, we conclude that where as explained, the constantC (1) I JK (0) is fully determined from the WZW CFT initial data. As a result we have determined the exact in λ three-point function coefficient of three-primary fields up to order 1/k. 3 Notice that here we are using the duality (1.7) followed by parity. Under this combined symmetry

Mixed JΦΦ and J ΦΦ correlators
In this section we focus on the mixed correlators involving two primary fields and one current. From appendix F one can read off the O(λ 3 ) result which is given by The similar expression for the correlatorJ a reads Getting inspired by the previous computations and by the expression in (5.1) we conclude that the all-loop mixed correlators should assume the following form where the functions f 1 (λ) and f 2 (λ) are everywhere analytic and f 1 (0) = f 2 (0) = 1.
As usual, the denominator of (5.3) is written in such a way that the correlator has well-defined non-Abelian and pseusodual limits.
Applying the duality (1.7), as well as the corresponding transformation rules for the currents (2.14) and primary fields (2.20) we obtain that where on the right hand side of the last equation we have changed the order of the indices for convenience. Subsequently, the left hand side of the above can be rewritten using appropriately (5.3). We have that Hence, comparing (5.4) with (5.5) we have the two equivalent conditions Plugging the latter into (5.3) we find after some rearrangement that Similar reasoning leads to

OPEs and equal-time commutators
In this section we use the two-point and three-point functions of the currents and primary fields to find their OPE algebra up to order 1/k reads and exact in the defor-mation parameter λ. The result is Having the OPEs at our disposal, we can easily compute the equal-time commutator of the currents and primaries through a time-ordered limiting procedure and the following representations of Dirac delta-function Employing Eqs. (6.1), (6.3) and (6.4), we find to order 1 These equal-time commutators turn out to be isomorphic to two commuting copies of current algebras with opposite levels where S a = 1 2π The parameter λ does not appear in this algebra but it does in the time evolution of the system due to the fact that, as it turns out, the Hamiltonian in terms of S a andS a is λ-dependent (cf. eq. (2.11) of [2]). Also, the λ-dependence still appears in the OPEs of the S a andS a among them. The reasons is that the OPEs, unlike the commutators (6.7), are not computed at equal times.
Finally we take the classical limit of (6.5) and appropriately rescaling the currents, we find Rajeev's deformation of the canonical structure of the isotropic PCM [35] (recall that, in our conventions the group structure constants f abc are taken to be imaginary) realized through the action (1.5) of [2] e = 2d(λ) = 1 That the deformed brackets (6.9) follow as the classical limit of the OPEs provides actually, for the isotropic case, the mathematical proof that the action (1.5) is in fact the effective action of the non-Abelian Thirring model action (1.4). The reason is that (1.5) provides, as was shown in [2], a realization of (6.9) which in turn was derived by using (1.4) as the starting point.

Conclusions
In this work we have computed all possible two-and three-point functions of current and primary field operators for the λ-deformed integrable σ-models. One direction for extending our work would be to consider cases beyond isotropy, i.e. when the matrix λ is not proportional to the identity. In particular, we believe that the equal-time commutators of the currents and primaries will take the form of (6.7), under an analogue to (6.8) relation. Another direction would be to calculate the subleading, in the 1/k expansion, terms of all physical quantities such as the β-function, the anomalous dimension matrix and the fusion coefficients. These line of research would, hopefully, be culminated by finding the exact in both λ and k expressions for these physical quantities as well as the underlying effective action.

Acknowledgments
The authors would like to thank each others home institutes for hospitality. The research of K. Siampos is partially supported by the Germaine de Stael France-Swiss bilateral program (project no 32753SG). K. Sfetsos and K. Siampos would like to thank the TH-Unit at CERN for hospitality and financial support during the final stages of this project. K. Siampos would like also to thank the ICTP, Trieste for hospitality during the final stages of this project.

A Various integrals
In this appendix we assemble all the integrals that will be needed in our perturbative calculations. In all integrals considered below the integration domain is a disc of radius R in which the various external points labeled by x's are excluded. This can be done by encircling them with circles having arbitrarily vanishing radius. One way to prove the expressions below is to use Stokes' theorem in two-dimensions for appropriately chosen vectors and contours.
The first set of integrals is the exact version of the integrals in (2.31) and (2.33) where the R > |x 1,2 |. By taking derivatives we may compute the exact analog of the integrals in (2.32).
A generalization of the first of the above integrals is given by . This relation can be proved by first performing a partial fraction decomposition and then use (A.1). A special case of this is when the denominators are cubic polynomials. Namely, which is the exact analogs of the integrals in (2.34).
Another important integral is given by The large R limit of this is given in (2.35) and is necessary for the derivation of the two-loop contribution in (3.6).

B Perturbative computation of the JJ correlator
In this appendix we present the perturbative calculation of the JJ two-point function. At the conformal point it vanishes.
Order O(λ): To that order we have that Order O(λ 2 ): To that order we have that where we have broken the integral J into the four integrals J ij , resulting from multiplying out the terms in the parenthesis, in a rather self-explanatory notation. We have that where we have used (2.35). Also Note that for J 12 we have first performed the z 2 -integration which is not in accordance with our regularization prescription. However, we now show that the same result follows if we do first the z 1 according to our regularization. We easily find that where we have used the fact that the second integral in the second line above can be obtained from (B.4) (with ε = 1). A simple algebra gives the same expression as in (B.5). Finally Therefore collecting all contributions we find that (B.8) To that order we have that The four-point function is given by (2.6) and is multiplied by the analogous four-point function for antiholomorphic currents. Keeping terms up to O(1/k), disregarding terms giving rise to bubbles and taking into account the above permutation symmetry we arrive at Performing the integrations first over z 1 and then over z 2 we obtain that In conclusion (B.1), (B.2) and (B.10) combine to (3.6) in the main text.

C Perturbative computation of the J J J correlator
In this appendix we present the perturbative calculation of the J J J three-point correlator. The O(λ) contribution to this correlators vanishes since J = 0. Proceeding to higher orders in the λ-expansion we have: Using the identity Employing again an analogue of the identity (C.6) we get that where we have included the contact term involving external points. This will be neglected in the main text.

E Perturbative computation of the ΦΦ correlator
In this appendix we present the perturbative calculation of the ΦΦ correlator.
Order O(λ): To that order we have that where we have used (2.11) and (2.12) and wrote the result as a direct product of matrices. The representations involved are in fact conjugate to each other. Therefore, using (2.8) we find that Order O(λ 2 ): To that order we find that where the four different terms are computed below and arise by using the current Ward identity with respect to the current J a (z 1 ).
• The first term is where we have kept only contributions which will give terms of O(1/k) to the final result. In addition we used the integral d 2 z 12 (z 1 − x 1 )(z 2 − x 2 )z 2 12 = π d 2 z 2 (x 1 −z 2 )(z 2 − x 2 ) = π 2 ln |x 12 | 2 , (E.5) as well as the same with x 2 → x 1 in which case |x 12 | 2 → ε 2 in the result above.
• The second term is where as before we have kept only contributions providing at most O(1/k) terms in the final result.
• The fourth term is more involved to compute. The result is (E.8) Note that this is expected since it is just the sum of the other two non-vanishing terms with the representations R and R ′ exchanged.
• The first term originates from the contraction of the current J a (z 1 ) with the primary field Φ (1) and leads to The next step is to contract one of the remaining holomorphic currents, lets say J b (z 2 ).
This current can not be contracted with any of the external primaries because in that case the last holomorphic current should also be contracted with an external field too and as a result this contribution will be of order 1/k 3/2 . Since in this calculation we keep terms of order O(1/k) this contribution can be ignored. For the same reason the holomorphic currents J b (z 2 ) and J c (z 3 ) can not be contracted through the non-Abelian part of their OPE but only via the Abelian part. Once we have contracted all the holomorphic currents we start treating the anti-holomorphic ones by choosingJ a (z 1 ) Next we employ the Ward identity for the current at the pointz 1 . This current can not be contracted with the other anti-holomorphic currents through a δ-Kronecker term because in such a case this term will be proportional either to f abc δ ab = 0 or to f abc δ ac = 0. AlsoJ a (z 1 ) can not be contracted with the external primary fields because in such a case the corresponding diagram will disconnected, thus it will be the product of a bubble involving the points z 2 and z 3 times the rest of the diagram. Consequently, the only contribution that remains comes from the non-abelian contraction of either J a (z 1 ) with eitherJ b (z 2 ) orJ c (z 3 ). In both cases the resulting diagrams will be disconnected, i.e. they will be the product of the tree-level ΦΦ correlator times a bubble involving all interactions points z i , i = 1, 2, 3. Therefore, we conclude that J (2) ii ′ ,jj ′ = 0 . (E.14) • The third term originates from the contraction of the current J a (z 1 ) with J b (z 2 ) through the Abelian term of their OPE The last holomorphic current, i.e. the one at point z 3 should necessarily be contracted with each of the external primaries giving a factor of 1/ √ k and leaving us with a sum of two correlators involving two primaries and the three anti-holomorphic currents.
Choosing the current at z 1 to be the one for which we will apply the Ward identity we obtain the following terms: i) the term whenJ a (z 1 ) is contracted withJ a (z 2 ). This diagram will have a factor of 1 z 2 12z 2 12 indicating that it is disconnected and should, thus be ignored.
ii) the term arising from the contraction ofJ a (z 1 ) withJ c (z 3 ) through the non-Abelian term of their OPE will also be zero since we have saturated the powers of 1/k and all remaining contractions should be Abelian resulting to the factor of f acd δ ad = 0.
iii) the term arising form the contraction ofJ a (z 1 ) withJ c (z 3 ) through the Abelian term