Exact results from the geometry of couplings and the effective action

We invent a method that exploits the geometry in the space of couplings and the known all-loop effective action, in order to calculate the exact in the couplings anomalous dimensions of composite operators for a wide class of integrable $\sigma$-models. These involve both self and mutually interacting current algebra theories. We work out the details for important classes of such operators. In particular, we consider the operators built solely from an arbitrary number of currents of the same chirality, the composite operators which factorize into a chiral and an anti-chiral part, as well as those made up of three currents of mixed chirality. Remarkably enough, the anomalous dimensions of the former two sets of operators turn out to vanish. In our approach, loop computations are completely avoided.


Introduction
Two quantities of particular importance exist in any quantum field theory (QFT).
Namely, the β-functions governing the running of the coupling constants with the energy scale and the anomalous dimensions of fundamental and composite operators encoding their scaling properties. Usually, both of them are determined perturbatively order by order.
In this work, we will develop a method which allows one to calculate exact expressions for both the β-functions and the anomalous dimensions of large classes of operators, including composite ones, at one go. The power of our method will be exhibited by considering a certain class of integrable two-dimensional σ-models. These models can be obtained by deforming conformal field theories (CFTs) of the WZW type with bilinear current-current operators. The all-loop, in the deformation parameters but at large WZW levels, effective action of these models was obtained through a gauging procedure initiated in [1] and further developed and exploited in [2][3][4][5] for the cases of equal and unequal levels where mutual and/or self interactions are present. The possibility of using non-trivial automorphisms in the context of single λ-deformations was put forward in [6]. These models are collectively called λ-deformations as they are characterized by a square matrix λ with dimensionality equal to that of the underlying group structure. In addition, exact expressions for three-point correlators of currents and primary fields were calculated for the prototype λ-deformed models [1] possessing a single level in [7] and for the case of two unequal levels of [3] in [8]. The aforementioned models are particularly attractive because they possess certain nonperturbative symmetries in the space of couplings [9][10][11][3][4][5]. As a result, one can exploit these non-perturbative symmetries by combining them with low-order perturbative calculations in order to derive exact expressions for the β-functions and the anomalous dimensions of current and primary operators [12,7,13]. 1 This approach is particularly elegant and effective but it has the drawback that in practice it is difficult to apply for composite operators made up of many fundamental ones since the difficulty of perturbative calculations increases enormously with the length of the operator. A less apparent drawback is that the form of bare operator, i.e. when the deformation is switched off, changes when this is turned on. Both of these drawbacks are rectified in the present work.
Remarkably, in these λ-deformed σ-models the computation of Zamolodchikov's C-function [14], as an exact function of the deformation parameters and to leading order in the large k-expansion, is possible. This was first performed in [15] for the isotropic cases and further generalized for anisotropic λ-deformations in [16].
One of the virtues of the aforementioned constructions [1,[3][4][5] is that the deformed models are integrable for specific forms of the deformation matrix. Besides the case for isotropic such matrices, i.e. when the matrix λ is proportional to the identity, integrability holds for subclasses of anisotropic models as well. In particular, for the λ-deformed SU (2) based models in [17,18], as well as for certain subclasses of those in [4,5]. 2 Integrable deformations based on cosets, symmetric and semi-symmetric spaces were also constructed and studied in [1,27], [28] and [29], respectively. In certain cases, deformed models of low dimensionality were promoted to solutions of type-IIA or type-IIB supergravity [30][31][32][33][34].
Another interesting feature of the λ-deformations is their relation to the so-called η-deformations with the latter being introduced in [35][36][37] for group spaces and in [38][39][40] for coset spaces. This relation is realized by the action of a Poisson-Lie Tduality [41,42] and appropriate analytic continuations which for group and coset spaces was discussed in [43,44] and [18,[45][46][47], respectively. In parallel developments, the dynamics of scalar fields in certain λ-deformed geometries based on coset CFTs has been discussed in [48] while the realization of λ-deformations as theories living on the boundary of Chern-Simons theories was discussed in [49]. Finally, D-branes in 1 The β-functions can also be computed by using a variety of methods. One can either re-sum the series of conformal perturbation theory [19][20][21] or exploit the effective action and use a variant of the background field method [22][23][24]. An independent method is to use the well-known expressions for the β-functions for the metric and the antisymmetric tensors fields in non-linear σ-models [25] which was done in [10,11,3,24]. 2 For the case of the isotropic deformation based on the SU(2) group integrability has been demonstated in [26]. the context of single λ-deformations were studied in [50,6].
The plan of the paper is as follows: In section 2, we introduce the essential features of our method and use it to derive the anomalous dimension of the fundamental current in complete agreement with previous calculations [12]. In section 3, we employ our method to calculate the anomalous dimensions of composite operators build from chiral and/or anti-chiral holomorphic currents. As two explicit examples we consider operators built solely from an arbitrary number of chiral currents as well as operators factorizing into chiral and anti-chiral parts. Surprisingly enough, their anomalous dimensions turn out to be zero to leading order in the large k-expansion. Our third example concerns the fully symmetric operator composed from two chiral and one anti-chiral current. The result for its anomalous dimension is the same as the dimension of the operator J + J − that drives the model off conformality. In all cases the expressions for the anomalous dimensions respect the non-perturbative symmetries of the model [9]. In section 4, we focus on the model constructed in [3]. As in section 3, the anomalous dimensions of purely chiral or anti-chiral current operators build from an arbitrary number of currents are also zero. The same is true for operators that factrorize into chiral and anti-chiral parts. Similarly, the dimension of the mixed operator matches again the dimension of the operator that drives the model off conformality.
In section 5, we consider the model where both self and mutual current-current interactions are present [4]. Using the geometry in the space of couplings and the exact βfunctions of this model, we calculate the exact in the deformation parameters anomalous dimensions of the operators which perturb the CFT, as well as those of the single currents. In section 6, we present a number of perturbative calculations which are in agreement with the exact results obtained in the previous sections. Finally, in section 7 we draw our conclusions. We have also written four appendices containing technical and computational details.

The anomalous dimension of the single current
In this section we will explain the essential features of our method. Focusing on the single λ-deformed σ-models [1] we will show how to compute the anomalous dimensions of the fundamental currents of the model. The method is based on a convenient modification of the gauging procedure of [1] in conjunction with geometrical data defined in the space of couplings of the corresponding two-dimensional field theories. Having the essential and conceptual aspects of the method under control we will extend the construction to include composite operators of currents in the next two sections.
We start with the sum of the WZW model action S k (g) at level k for a group element g in a group G [51], the principal chiral model (PCM) action [52] for the group element g ∈ G with overall coupling κ 2 and a term containing the chiral current of the original WZW model. Specifically this action reads where the last new term has coupling matrix s = s a t a and the overall scaling k has been introduced for later convenience. This extra term is auxiliary and the reason for adding it will become progressively apparent in this section. It will eventually enable us to compute the exact anomalous dimension of the fundamental current in the λdeformed theory of [1] surpassing perturbation theory.
All matrices are expanded using as a basis representation matrices t a obeying [t a , t b ] = i f abc t c and normalized as Tr(t a t b ) = δ ab . As in [1] we gauge the global symmetry acting as g → Λ −1 gΛ andg → Λ −1g . The corresponding gauge invariant action is where D ±g = (∂ ± − A ± )g are the covariant derivatives. The first term is the standard gauged WZW action [53] S k (g, Fixing the gauge in (2.2) by choosingg = 1 one arrives at the following action We are interested in the equations of motion of this action. Varying (2.4) with respect to A ∓ we find the constraints where the covariant derivatives acting on g are now defined as Variation with respect to the group element g gives where the field strength is as usual Substituting (2.6) into (2.7) we obtain that We may use the constraints (2.6) to solve for the gauge fields and upon substitution back into the action (2.4) arrive at a σ-model action. The result is nothing by the λdeformed σ-model action corresponding to the λ-deformed models in the isotropic case, plus a term linear in s. Specifically, we find that where Then the action becomes Obviously, for s = 0 the original λ-deformed theory [1] is recovered. We are interested in computing the anomalous dimensions of the current J a + exactly in λ and to leading order in k at the limit s = 0, that is for the original λ-deformed theory. This limit has to be consistent with the β-function equations, which is the case as we shall see below.
Note that the current J + is "dressed" and replaced by A + as it can be seen from the corresponding last term in (2.12). Obviously for λ → 0 we have that A + ∼ J + .

The RG flow equations
Next we compute the β-function equations for the couplings λ and s. We will follow the background-type method initiated for the λ-deformed models in [22] and applied in full generality in [24]. We choose a particular configuration of the group elements in order to compute the running of couplings. In particular, g = e σ + θ + +σ − θ − , where the matrices θ ± are constant and commuting. Then we have that J ± = −iθ ± and that the matrix D = 1. Then, from (2.10) the background gauge fields are Note that, indeed the above solve the classical equations (2.9). Then the Lagrangian density for the action (2.12) reads The next step is to consider the fluctuations of the gauge fields around (2.13) and let The linearized fluctuations for the equations of motion are given by (2.16) These can be cast in the following form where the operatorD is first order in the worldsheet derivatives. After the Euclidean analytic continuation and in the momentum space we, in the conventions of [4], re- The matrixĈ contains all momentum dependence. Integrating out the fluctuations, gives the effective Lagrangian of our model This integral is logarithmically divergent with respect to the UV mass scale µ which is isolated by performing the large momentum expansion of the integrand and keeping terms proportional to 1 |p| 2 , where |p| 2 = pp. SinceĈ grows with |p| we use the fact that The last tern is the only one contributing to the aforementioned logarithmic divergence which is important for our purposes, thus obtaining (2.21) Next we use the polar coordinates parametrization p = re iφ ,p = re −iφ in which the integration measure is d 2 p = rdrdφ and evaluate Tr(Ĉ −1F ) 2 . The dependence on r is of the form 1/r 2 which upon integration gives the necessary factor of ln µ. Then were we used (2.13) for the background solution for the gauge fields. Also, Tr(Ã + ) a s a , where c G is the eigenvalue of the quadratic Casimir in the adjoint representation defined as f acd f bcd = c G δ ab . In the rest we drop the index in s a since the result for the β-function and later for the anomalous will be independent from it.
As usual in field theory, we demand that the action (2.22) is µ-independent, i.e. ∂ ln µ 2 L eff = 0. For k ≫ 1 this derivative acts only on the coupling constants in L (0) . Then, defining as usual β λ = ∂ ln µ 2 λ and β s = ∂ ln µ 2 s, we obtain that (2.23) For later convenience it is important to find the running of the coupling 3 λ ∼ sλ , (2.24) which will replace s in our considerations. Using the above we obtain that (2.25)

The current anomalous dimension
So far we have kept the couplings λ andλ finite. For small values for these couplings we have from (2.12) that Keeping the discussion general, instead of the single current perturbation, we consider a general perturbation with operators Each one of them has a classical dimension and the β-functions for the couplings are denoted by β i . 3 The proportionality constant below is just an −i. This redefinition is due to the fact that s is purely imaginary in our conventions, i.e. see (2.1). Its inclusion or ommission does not affect the final result for the anomalous dimension.
There is a metric G (0) ij in the space of these couplings defined via the two-point function of the O i 's [19] with line element ds 2 = G (0) ij dλ i dλ j . Renormalizability and the Callan-Symanzik equation give Consider the case of two couplings λ andλ. In general there is a mixing of the two operators even if this is absent at the conformal point. Consequently, their anomalous dimension will be encoded in a matrix γ i j with all four elements nonvanishing. We assume that one of the couplings, sayλ, can be consistently set to zero with the corresponding βλ = 0. In that limit we assume that the mixing vanishes as well. Then, only the entries γ λ λ and γλλ will be non-zero. In our case the operators The latter breaks Lorentz invariance, so that an operator mixing between them may occur. However, in the limitλ → 0 Lorentz invariance is restored and therefore mixing between operators of different chirality is non-existing.
Quite generally, nearλ = 0 we assume for the metric in the coupling space the form (2.29) and that βλ = O(λ). These are a consequence of the decoupling assumption atλ = 0.
Then, in the limitλ → 0 we find using (2.29) that (2.30) and also that Hence, in that limit we have the original λ-deformed theory with just the coupling λ, i.e. (2.12) with the last term absent. However, as a bonus we have the expression for the anomalous dimension of the operator O 2 as well, which was our goal. Note that, only the expressions for these metrics g (0) ii at the strict k → ∞ limit are needed since in (2.30) and in (2.31) the β's are already of O(1/k). In addition, even the overall constant in their specific expressions is immaterial.
Specifically, in our case we have by using (A.8) that Then we first find using (2.30) that which indeed was computed in [7] exploiting this geometric method. In addition, calculating the right hand side of (2.31) we find that (2.34) This result was firstly found in [7] using the symmetries in the coupling space and leading order results from perturbative methods.
Next, we will apply the same method to compute the anomalous dimensions of more complicated composite operators.

Anomalous dimensions of general current composites
In this section, we will extend the formalism explained in the previous section in order to calculate the anomalous dimensions of general operators of the form By construction the overall tensor coefficient should be symmetric in the first m indices, as well as in the last n ones, separately. However, there is no symmetry property relating the a i 's and the b i 's. This tensor can be decomposed into irreducible representations of the group G. As in the case of the single current the above operator will be dressed to an operator O (m,n) λ whose expression will be presented below.
Our starting point is the action (2.2) but with the s-term in the second line replaced by times a factor of (−1) m+n+1 which we introduce so that subsequent expressions are as simple as possible. This action is still gauge invariant and the gauge fixingg = 1 condition leads to The equations of motion for (3.3) with respect to A − and A + are where we have defined the vectors A (m,n ′ ) Hence, the prime indicates that one index is not contracted and is left free in the corresponding tensor coefficient.
Varying the action with respect to the group element g results into the same equation into (2.7) we obtain where the covariant derivatives act as usual, i.e. D + A (m ′ ,n) As a result the equations of motion can be written solely in terms of the gauge fields.
In order to proceed with the calculation one should find a classical solution to (3.5).
Unlike (2.10) for the case of the single current, these are much more difficult to handle due to their nonlinearity. However, since as before we aim at setting eventually the coupling s to zero, we only need the solution valid to O(s). Hence we find that We emphasize that in the second term in each of the above expressions, for the gauge fields entering the definitions (3.6) we should use the leading order expressions given by the first leading terms. The reason is that these terms are already multiplied by s and we only keep terms up to linear order in that parameter. Substitution into the action (3.3) we obtain that where as before in A  We note in passing that (3.9) remains invariant under the generalized symmetry k → −k, λ → λ −1 , g → g −1 , s → sλ m+n , or in terms of the effective coupling This symmetry must be reflected to all physical quantities. For the line element we have which up to linear order inλ must be invariant under (3.11). We are interested for theλ = 0 limit in which the G λλ being linear inλ, does not contribute. Furthermore, at this limit the first term transforms independently and thus is itself invariant under the transformation (3.11). Invariance of Gλλ under (3.11) gives the condition to an overall sign. This is indeed satisfied by our metric component Gλλ in (A.8).

The RG flow equations
As before we choose the group element g = e σ + θ + +σ − θ − for two elements in the Cartan subalgebra of G, so that again J ± = −iθ ± and D = 1. Furthermore, the expressions for the gauge fields on the solution take the form (3.13) The notation A should be self-explanatory, namely in the definition (3.6) we should put for A ± their classical values, in particular the leading sindependent term of (3.13).
Also note that (3.13) should satisfy (3.7) as well. This is warranted if A We are now in the position to write down the expression for the action (3.9) on the classical solution (3.13). To linear order in s we obtain that where we have used the definition θ (m,n) which is analogous to that in (3.4). Note that (3.14) reduces to (2.14) for m = 1, n = 0.
Next we calculate the fluctuations of (3.7) around the classical solution keeping only terms linear in them. The result for the first equation of (3.7) reads In a similar fashion, the fluctuation of the second equation in (3.7) gives We have also defined the following double-primed quantities where as before a prime or two imply that two indices in the tensor S are not contracted. The fluctuations equations (3.16) and (3.17) can be rewritten in the form (2.17) withD =Ĉ +F. In momentum space we havê Each of the entries in the matrices of (3.20) and (3.21) is itself a matrix having two indices. Namely, the matrix components are Given these expressions one can straightforwardly calculate the trace of the matrix (Ĉ −1F ) 2 in (2.21) keeping only the terms that will give rise to non-vanishing contributions upon the angular integration. The latter will contribute an extra factor of 2π. In this way we obtain that Evaluating each of the traces in the right hand side of (3.23) separately one gets that and For the last trace one obtains It can be easily see that in the last term of (3.9) we have that A (m,n) +− ∼ O (m,n) for small s. Hence, the σ-model action (3.9) for small λ and s becomes where we have introduced the effective coupling This is the analog of (2.26) for the single current case. Hence, by taking the limitλ → 0 we will find the anomalous dimension of O (m,n) λ . This will be done by employing (2.31) where βλ now should correspond to this operator and the metric component should be The overall coefficient is irrelevant, but nevertheless it can be found in appendix A, where this metric has been computed. It remains to compute βλ. However, this seems hard in general since for an arbitrary tensor S in the operator (3.1) we expect a mixing of operators under the RG flow even if this tensor corresponds to an irreducible representations of G.
In the following section, we concentrate on important cases where such an operator mixing does not occur and we compute their corresponding anomalous dimensions.
Before we proceed we address two issues. First we note that the background field method in the presence of the, generically irrelevant, operators O (m,n) , may have subtleties. However, in our case we are not interested in obtaining information for the running ofλ to all orders inλ, but only to a linear one for small values of it. In addition, it should be mathematically consistent to setλ = 0. We will see in the examples below involving irrelevant operators that it is indeed consistent to setλ = 0, as in the case for the single current.
The second issue concerns the very form of the ansatz in (3.2). This clearly is not the most general form of a gauge invariant operator one may consider to add. One may add terms with multiple covariant derivatives acting ong and/or the gauge field strength F +− . Upon gauge fixingg = 1 such terms will give rise to descendant-like operators with terms having derivatives on the gauge fields, i.e. ∂ + A ± and the com- For small values of λ these will correspond to derivatives on the currents i.e. ∂ + J ± and the commutator between currents [J + , J − ].
The resulting operators with m(n) in total chiral (anti-chiral) currents and derivatives ∂ + (∂ − ) could in principle mix with the operator in (3.1). Using conformal perturbation theory we have checked that such a mixing does not occur to O(1/k) but it may do so at higher orders for which we are not interested in the present paper. The reason is the following: Quite generally, consider schematically the where the matrices have the form as far as their order in 1/k. The anomalous dimension matrix is given by Clearly, up to O(1/k) there is no mixing since the anti-diagonal entries of γ are of O(1/k 3/2 ) and thus they can be ignored.

Important examples
We are now focused on some basic important examples. These concerns operators which in the UV limit at λ = 0 are a general string of purely chiral current operators as well as mixed current operators. For the former class we generally conclude that the anomalous dimension is zero! For the latter ones there is in general mixing related to the associated representation theory operators. There is not such mixing for the operator having two J + 's and one J − . For this particular operator we find that the anomalous dimension is the same as that of the operator J + J − driving the deformation from the CFT point. We have also checked that the anomalous dimensions of operators factorizing into a chiral and an anti-chiral part also vanishes.
We will mainly concentrate to the case of the SU(N) group for which we have collected some useful formulas in the appendix B.

The chiral operator O (m,0)
The operator whose anomalous dimension we are interested in is at the CFT point of the form ..a m is the completely symmetric rank-m tensor of SU(N). At the CFT point this is a primary field with dimension equal to m [54] where m 3. For m = 2 this field is proportional to the energy momentum tensor. Its λ-dressed version will be given by ( where in the last step we have used (B.11) valid for m = 2, 3, . . . .
Then (2.21) with (3.14) computed with n = 0 is given by where the effective coupling isλ ∼ sλ m . Then, demanding that ∂ ln µ 2 L eff = 0 we obtain to leading order in the 1/k the expression for β λ in (2.23) as well as Using (2.31) with the metric entering given by (3.29) with n = 0, we find that Before commenting on this result we mention that for the case m = 1 (B.11) does not make sense. In that particular case we end up with the result for the anomalous dimensions of the single current in (2.34).
The result (3.36) is robust and has a simple explanation. Recall that the λ-deformed action has two well defined limits involving k → ∞ and λ → ±1 in such a way that k(1 − λ) and k(1 + λ) 3 remain finite. These are the non-Abelian and pseudo-chiral limits, respectively [1,7]. The above suggest that the anomalous dimension of any operator O should have the form where n is a non-negative integer whose value is dictated by the leading order perturbative in λ result, or zero if the operator itself has an 1 k expansion even at the CFT point for λ = 0. The overall function f (λ) is analytic in λ. The symmetry (2.35) should be encoded in the anomalous dimension of the operator which then should remain invariant. That gives the condition An important comment is in order. The operator J a + J a + is at the CFT limit, that is when λ = 0, proportional to the chiral component of the energy momentum tensor. In the λdeformed theory one may readily check that the role of the energy momentum tensor is played by the deformation of J a + J a + , namely O The last proportionality relation to T ++ follows by simply evaluating the energy momentum tensor for the λ-deformed model action (2.12) (with s = 0). As in any σmodel this is classically chirally conserved, i.e. ∂ − T ++ = 0. A less trivial statement is that the following sequence of chiral conservation laws holds, in which the chiral conservation law for T ++ is just the first member. This is a consequence of the fact that the classical equation of motion for the λ-deformed model can be cast as

The mixed operator O (2,1)
The operator whose anomalous dimension we are interested in is of the form where d abc is the completely symmetric tensor of SU(N) of rank three. This operator cannot mix with others and its λ-dressed form is given by (3.10) for m = 2 and n = 1.
Recall that the field Q a = d abc J b + J c + is primary with dimension equal to 2 [54]. Hence, the operator O (2,1) at the CFT point is a primary field with holomorphic and antiholomorphic dimensions (in a Euclidean regime language) dimensions equal to 2 and 1, respectively.   44) where in this case the effective coupling isλ = sλ 3 . Demanding that ∂ ln µ 2 L eff = 0 we obtain to leading order in 1/k the expression for β λ written in (2.23) and that Using (2.31) with the metric entering given by (3.29) again with m = 2 and n = 1 we finally find that which is the same as that in for γ J + J − in (2.33).

λ-deformations with different current algebra levels
In this section, we will use our general formalism in order to calculate the anomalous dimensions of current composite operators in models for which the levels of the chiral and anti-chiral algebras are different. The main motivation is that such models generically have fixed points in the IR corresponding to new CFTs. The first such model was presented in [3] in which one starts with two WZW models at different levels k 1 and k 2 and via a gauging procedure involving two sets of gauge fields A ± and B ± one constructs the all-loop effective action of two mutually interacting WZW models.
The terms driving the models away from the CFT point are J 1+ J 2− and J 2+ J 1− , where the index 1 or 2 indicates that they refer to the first or the second WZW model and the corresponding levels are k 1 and k 2 . We may simplify further the model by consistently setting the coupling of the second of these terms to zero as we will explain below. Then, it turns out that it is consistent to consider operators of a form similar to Our starting point will be eq. (2.6) of [3] but with λ 2 → 0 and λ 1 renamed to λ. It turns out that in this limit, which is consistent quantum mechanically from an RG flow point of view, the leading order term for small remaining coupling λ is J 1+ J 2− which, as mentioned above, is the case we want to concentrate on. Then the last term in the first line of equation (2.6) of [3] remains finite if we first rescale B ± as B ± → √ λ 2 B ± and then take λ 2 → 0. In this limit the effective action analog of (3.3) becomes where as before the expression for A (m,n) +− is given by (3.4). In this procedure the gauge field B ± has decoupled, which is the reason we have not included the term Tr(B + B − ) in the above action, even though its overall coupling constant remains finite.
The equations of motion for (4.2) with respect to A − and A + are given by where we have defined the covariant derivatives D + g 1 = ∂ + g 1 − A + g 1 and D − g 2 = ∂ − g 2 + g 2 A − and the vectors A (m,n ′ ) +− and A (m ′ ,n) +− are given by (3.6). Instead of the levels k 1 and k 2 we will use the parameters Varying the action with respect to the group element g 1 and g 2 results into the following set of equations or equivalently Substituting the constraints (4.3) in (4.5) and (4.6) we obtain that where the covariant derivatives act as explained below (3.7). Hence, the equations of motion have been written in terms of the gauge fields only.

The constraints (4.3) can be easily solved to give
Note that since in the action above in A (m,n) +− only the leading order expressions in s should be used, the λ-dressing of the gauge fields is just a trivial overall constant unlike the case for the gauge fields for the single λ-deformed models in (3.8). Conse-quently, the operator (4.1) does not get change upon the λ-deformation.

The RG flow equations
In order to proceed with the calculation one should find a classical solution to (4.7).
However, as we did in the previous cases we only need to find a solution valid to order O(s). This can be easily obtained if we choose the group elements g i = e σ + θ (i) Furthermore, the expressions for the gauge fields on the solution take the form (4.10) Notice that in the definition (3.6) we should put for A ± their classical values (4.10).
Notice also that (4.10) should also satisfy (4.7). This is guaranteed if A To linear order in s the action (4.2) evaluated on the classical solution (4.10) is where we have used a definition similar to (3.15), i.e. θ (m,n) The linear fluctuations of (4.7) around the classical solution are in order. From the first equation of (4.7) we obtain that In a similar fashion, the second equation in (4.7) gives where the quantities with two primes are defined in (3.18).
These fluctuations can be rewritten in the form (2.17) withD =Ĉ +F. In momentum space we have thatĈ where all matrices appearing in (4.16) and (4.17) are defined as in (3.22).
One can straightforwardly calculate the Tr(Ĉ −1F ) 2 in (2.21). We just need to keep only the terms giving rise to non-vanishing contributions upon the angular integration which will contribute an extra factor of 2π. Evaluating each of the traces in the right hand side of (3.23) separately one gets for the case at hand that and and that (4.20) The various traces appearing in (3.25) and (3.26) should be evaluated case by case since their result depends on the particular form of the operator chosen.
For small values of the parameter s the σ-model action (4.2) becomes where the operator added is defined in (4.1) and the effective coupling as in (3.28).
Hence, by taking the limitλ → 0 we will find the anomalous dimension of O (m,n) .

Important examples
In this section we will compute the anomalous dimensions of the same operators as in section 3.

Consider operators of the form
which is similar to (3.32) and at the CFT point are primary fields with dimension m [54] for m 3 and proportional to the energy momentum tensor for m = 2. Certain simplifications occur since for n = 0 most of the matrices in (3.22) vanish. For the seemingly non-vanishing traces appearing in (4.19) and (4.20), we have the same relations as in (3.33).
Then (2.21) summed with (4.11) computed at n = 0 is given by where as beforeλ = sλ m is the effective coupling. Demanding that ∂ ln µ 2 L eff = 0 we obtain to leading order in the 1/k the expression for β λ in the case of unequal levels [3] as well as the β-function for the couplingλ Using (2.31) with the metric entering given by (3.29) again with n = 0 we find that We immediate see that for m 2 the anomalous dimension of the chiral operators vanish due to the group theory identity (B.11), that is as in the case of equal levels in (4.27).
However, for the anomalous dimension of a single chiral current, i.e. when m = 1, this identity does not hold and (4.26) gives that which is in perfect agreement with the expression of the chiral current calculated in equation (2.9) of [8].
Note that, since the equations of motion can be cast in the form (4.29) the classical conservation law (3.40).
Finally, by following the same steps, one can show that the composite operators made from an arbitrary number of anti-chiral currents J 2− have also vanishing anomalous dimensions. This is also, rather trivially, the case for operators built from an arbitrary number of the currents J 2+ or J 1− since these two currents are not present in the operator that deforms the CFT and which is J 1+ J 2− .

The mixed operator O (2,1)
Consider next mixed operators of the form where the effective coupling isλ = sλ 3 . Demanding that ∂ ln µ 2 L eff = 0 we obtain to leading order in the 1/k-expansion the expression for β λ in (4.24) and that Using (2.31) with the metric entering given by (3.29) again with m = 2 and n = 1 we find that which is the same as the anomalous dimension of the operator J a 1+ J a 2− . The latter can be found in equation (2.16) of [8].

λ-deformations of the self-and mutual-type
In this section we consider the λ-deformed model constructed in [4] describing simultaneous interactions of two WZW models of the self-and mutually-interacting type.
At the linearized level the action is S k 1 ,k 2 ,λ,λ (g 1 , g 2 ) = S k 1 (g 1 ) + S k 2 (g 2 ) Various aspects of this model, along with the construction of its all-order in the parameters effective action, can be found in [4] where it was also shown that the RG flow equations of this model have a rich structure.
We will compute the Zamolodchikov's metric for this theory, along with the anomalous dimensions of the composite operators J 1+ J 2− and J 2+ J 1− that drive the pertur-bation away from the CFT point. Then, by taking appropriate limits we will find the anomalous dimensions of currents of the single deformed modes with equal or even unequal levels which have been computed before. Complete agreement will be found.
That gives extra confidence for the validity of the procedure we used in sections 2,3 and 4.
The β−functions for this model can be found in eqs. (4.19) and (4.20) of [4] and read As argued in the end of section 4.1 of [4] it is convenient to rewrite (5.1) after a rescaling so that one may use available results in the literature. Indeed, after the rescaling J i± → J i± / √ k i , i = 1, 2, then (5.1) can be rewritten as where both group indices a, a ′ = 1, 2, . . . , dim G. Notice here that the coupling matrix Λ is non-invertible. However this does not affect our calculations since no inversion operation is needed.

The Zamolochikov metric
We will compute the Zamolochikov metric for (5.1) for finite values of both couplings.
The general form of the Zamolochikov metric was computed in [16]. Recalling the relevant expressions and using a double index notation, we have that where Using the matrix λ in (5.4) we find that Then the explicit form of the metric in the two-dimensional coupling space spanned by λ andλ is found to be Interestingly, this is, at least locally, an AdS 2 space since the corresponding Ricci scalar reads R = −4/ dim G. In addition, it can be shown that (5.8) is invariant under the transformation found in [4] for the full effective action corresponding to (5.1), as are the β-functions (2.23) as well.

Anomalous dimensions of the composite operators
In order to compute the anomalous dimension of the bilinear current operators, we will follow the lines of [12,16]. For the general metric (5.5) the Cristoffel symbols were computed to be The anomalous dimension matrix is taken from the work of [19] γ AB where we have used in here our double index notation so that It turns out that the non-zero components of the anomalous dimension tensor are Hence, the anomalous dimension of J 1+ J 1− and J 2+ J 1− are

Two limits and current anomalous dimensions
In theλ = 0 limit only the self-interaction J 1+ J 1− term in (5.1) survives. Then (5.17) simplifies to (5.18) In the above γ J 1+ J 1− coincides with the anomalous dimension of J 1+ J 1− composite operator for the simply deformed model found in [12] (see also (2.33)), while γ J 2+ J 1− is the anomalous dimension of J 1− (and J 1+ due to isotropy) for the same model as in (2.34). This should be the case since at this limit J 2+ is not interacting implying that In the λ = 0 limit, only the self-interaction J 2+ J 1− (5.1) survives. Then, after also the rescalingλ → λ 0λ , we have that As before, γ J 1+ J 1− coincides with the anomalous dimension of J 1− (since J 1+ is noninteracting), while γ J 2+ J 1− is the anomalous dimension of the composite operator found in [8]. The above derivation of anomalous dimensions of single and composite operators via a limiting procedure suggest that one may follow a similar procedure in more complicated models such as the ones in [5].

Verification using perturbation theory
In this section we proceed to verify perturbatively our previous exact results. In order to do this, a set of relations are necessary for our calculation. All our perturbative calculations will be performed in Euclidean signature.
The Callan-Symanzik equation implies that up to O(1/k) the two-point function of an operator takes the form O(1/k)) and the short distance cutoff is ǫ. The indices a, b take values in some generic irreducible representation of the group G. Note that ∆ and∆ may depend on k, but in (6.1) only the k-independent part is kept.
The n-point correlation function for a generic composite field Φ(x) is given by We will be interested in the anomalous dimensions of current composite operators.
Therefore we will need the OPE for two holomorphic currents which reads from which the two-and three-point functions follow. Higher correlators are evaluated using CFT Ward identities. Finally, note that, one should be careful concerning the normal ordering of non-Abelian currents. In that respect we follow the prescription introduced in appendix A of [54]. For two normal ordered product of any two operators operators A and B we will use the notation (AB). For the normal ordered product of more that two operators we will use the nested prescription, i.e. (ABC) = (A(BC)) etc.

Chiral operators
Clearly to O(λ) the anomalous dimension vanishes. The first non-trivial contribution to the anomalous dimension of O (m,0) comes from the two loop-contribution given by Notice that all k-dependence comes entirely from the holomorphic correlation function. Applying the Ward identity for the current at z 1 and suppressing momentarily the d-tensors the holomorphic part of the correlator equals to where m comes from the symmetrization conventions described at appendix B. Note that the contraction with J c 2 (z 2 ) when combined with the anti-holomorphic contribu-tion, either vanishes or gives rise to a bubble diagram. Performing the integral over z 1 for the first and third term, we obtain terms proportional to δ-functions between internal and external points which are set to zero in our regularization description [7] followed throughout this paper. 5 Finally, applying the Ward identity for the current at z 2 and reinstating the d-tensors, the holomorphic part of the correlator multiplied by δ c 1 c 2 (arising form the anti-holomorphic part of correlator) takes the form Having saturated the 1/k power, we only want the Abelian part of the remaining twopoint function given by The parenthesis in the right hand side of (6. The anomalous dimension of O (2,1) is given by (2.33). For small values of λ this be- We would like to verify this against perturbation theory.
For completeness we write down the OPE valid for our purposes We need the tree-level contribution in order to properly normalize our results. We obtain that where we used (6.8), (6.4) and (B.6).
The contribution to O(λ) is given by Evaluating the holomorhic part separately we obtain that (B.7) and (B.6), the integrals appearing are the following ones Having these, is an easy task to gather the terms contributing to the anomalous dimension. Performing the integrations using (6.31), we end up with  (6.33) where the ellipses denote terms contributing to order O(1/k) to the overall normalization or terms higher in the small λ-expansion of the anomalous dimension in (6.1).
We conclude that the anomalous dimension is indeed given by (6.20) up to O(λ 2 ).

Discussion and future directions
We demonstrated how the metric in the space of couplings and the all-loop effective action allows for the calculation of the exact in the deformation parameters anomalous dimensions of composite operators in a wide class of integrable σ-models [1][2][3][4][5].
In our approach loop computations are completely avoided. The method relies on a generalization of the gauging procedure of [1]. Specifically, it consists of adding to the gauged WZW model not only a gauged PCM term, but also an irrelevant, generically Lorentz violating, term involving as many covariant derivatives as the number of currents building the composite operator whose dimension we seek. By fixing the gauge freedom and integrating out the gauge fields, keeping however terms linear in the coupling of the composite irrelevant operator, one can obtain an effective action from which the β-functions of the model can de derived. Then, one uses (2.31) to determine the anomalous dimension of the composite operator in terms of the β-functions and the metric in the space of couplings.
We considered deformations involving self-as well as mutually interacting current algebra theories. We worked out the details for important classes of such operators. In particular, we employed our method in order to calculate the anomalous dimensions of composite operators build from chiral and/or anti-chiral currents. As a first example we considered the operator built solely from an arbitrary number of same chirality currents in (3.32) and in (4.22) for the case of unequal levels. We showed that using the equations of motion, this operator is classically chiral in the λ-deformed theories even though the elementary currents are no longer such away from the CFT point.
Surprisingly enough, their anomalous dimensions turn out to be zero to the leading order in the large k expansion. This result allowed the preservation of the aforementioned chiral conservation laws up to O(1/k). It will be interesting to investigate their fate to O(1/k 2 ) using in particular methods initiated in [55]. In addition, we have also checked that the anomalous dimensions of composite operators which factorize into a chiral and an anti-chiral part is also vanishing to O(1/k). Our last example concerned the fully symmetric operator composed from two chiral currents and one anti-chiral current. In this its anomalous dimension turns out to be the same as that for the operator J a + J a − driving the model off conformality.
As a byproduct of our analyses, we have shown that the anomalous dimension of an operator that does not mix and to O(1/k) in the large k-expansion, vanishes if it does so, up to O(λ 2 ) in the small λ-expansion.
A number of other interesting questions remain to be addressed. One of them is to calculate the anomalous dimensions of generic composite operators comprised of an arbitrary number of chiral and anti-chiral currents or operators involving primary fields. In considering this most general case one will have to deal with the serious problem of operator mixing. Therefore it is imperative to search for a conceivable spin chain description. We expect that this spin chain will most likely be an integrable one at least for the cases where the underlying models are integrable as well. One could also calculate the anomalous dimension of the single currents as well as composite operators made out of them for the most general integrable models constructed in [4] and [5]. It is also possible since we have all the ingredients, albeit technically more difficult, to compute the anomalous dimension of composite operators for the case of anisotropic couplings λ ab . Furthermore, it would certainly be very interesting to find the precise relation, if any, of these general integrable models to those constructed recently in [56,57] and see if our method can be used to derive the anomalous dimensions of composite operators in the latter models too.

A The Zamolodchikov's metric for O (m,n)
The purpose of this Appendix is to calculate the leading term in the large k-expansion of the Zamolodchikov's metric for operators of the form (3.1). We will do this in the strict k → ∞ limit in which the current algebra becomes Abelian. The action arising from deforming a WZW action is (3.27) which we reproduce here for convenience in the Euclidean regime (A.1) We will now follow the lines of appendix A.2 in [12] in order to compute the exact in λ and zeroth order inλ Abelian part of the Zamolodchikov's metric in the coupling space of λ andλ. In what follows we will consider m n. The resulting metric will be the same for m < n as well.
For the G λλ = |x 12 | 4 O(x 1 )O(x 2 ) part of the metric and to O(λ 0 ), the result is the same as in the simply deformed case computed in [12]. The off-diagonal term G λλ originating from the correlator O(x 1 )O (m,n) (x 2 ) is zero to O(λ 0 ) in accordance with the assumption in (2.29). Thus, we only need to compute the Abelian part of Gλλ = We can now find a recursive relation for G (r) by first recalling that the points z 1 , . . . , z k are internal, while x 1,2 are external ones and we must avoid disconnected and bubble diagrams. Picking up the currents J e 1 (z 1 ) andJ e 1 (z 1 ) and keeping the above in mind we have contractions of the internal-internal and internal-external type. The holomorphic J e 1 can be contracted with any of the other (r − 1) internal currents J e i 's (with j = 1). This should be combined with the contraction of the anti-holomorphic current J e 1 with one of the (r − 2) other internalJ e j 's (with j = 1, so that we avoid bubbles and disconnected diagrams) or with any of the 2n externalJ b i andJ d i . In addition, the holomorphic J e 1 can be contracted with any of the 2m external currents J a i and J c i and the result should be combined with the contraction ofJ e 1 with anyone of the (r − 1) other internalJ e i 's (with i = 1). Thus, we have for G (r) the recursive relation G (r) = π 2 [(r − 1)(r − 2) + 2n(r − 1) + 2m(r − 1)] G (r−2) = π 2 (r − 1) r + 2(m + n − 1) G (r−2) , where we used that d 2 z (z − x) 2 (z −ȳ) 2 = π 2 δ (2) (x − y). Solving it, we find that G (r) = π 2k (r − 1)!! r + 2(m + n − 1) !! 2(m + n) − 2 !! G Summarizing the above, the Abelian (for k → ∞), exact in λ butλ-independent com-ponents of the Zamolodchikov's metric are given by Notice that in the special case of m = 2, n = 0 we should use that S ab;0 = δ ab .

B Elements of SU(N) group theory
Here we use as references [58][59][60]. Consider a basis of N × N traceless matrices {t a }, a = 1, 2, . . . , N 2 − 1 and the normalization condition The multiplication law of two of them can be decomposed as where f abc is totally antisymmetric and d abc is symmetric and traceless. The coefficient of the first term is dictated by the normalization condition. In our normalization the eigenvalue of the quadratic Casimir is In a given irreducible representation R with elements (t a ) αβ , α, β = 1, 2, . . . , dim R the completeness relation reads  Note that m = 2, we use the convention that d (2) ab = δ ab .

C Supplement to the perturbative calculations
We present the detailed calculation of the perturbative contributions to the anomalous dimensions discussed in the main text. For this, we first write down a list of integrals needed to be evaluated during our computation and use them when applicable.
The basic technique for our computation is the use of Stokes theorem in two dimen- where M is the two-dimensional region and ∂M the contour with positive rotational index when circles counterclockwise. Our integrating functions are not holomorphic so we cannot apply Cauchy's theorem but it can be easily shown that the only parts contributing to the integrals appearing below are the contours around the poles. In most cases we apply partial integration treating z andz as independent variables, with a non-vanishing contribution coming from the superficial terms in general. In what follows x i denote external points while z i internal, with i = 1, 2. The first set of integrals is the standard ones (C.5)