RG flows for $\lambda$-deformed CFTs

We study the renormalization group equations of the fully anisotropic $\lambda$-deformed CFTs involving the direct product of two current algebras at different levels $k_{1,2}$ for general semi-simple groups. The exact, in the deformation parameters, $\beta$-function is found via the effective action of the quantum fluctuations around a classical background as well as from gravitational techniques. Furthermore, agreement with known results for symmetric couplings and/or for equal levels, is demonstrated. We study in detail the two coupling case arising by splitting the group into a subgroup and the corresponding coset manifold which consistency requires to be either a symmetric-space one or a non-symmetric Einstein-space.


Introduction
A class of integrable theories smoothly interpolating between exact CFTs in the UV and in the IR was constructed in [1]. These models are based on current bilinear deformations of two independent WZW models at different positive levels k 1,2 and at the linear level they are of the form S λ 1 ,λ 2 = S k 1 (g 1 ) + S k 2 (g 2 ) + 1 Notice that the above models are driven away from the CFT point by mutual interaction of the currents of the two independent WZW actions via current bilinears. This is different than the original λ-deformations introduced in [2] in which the currents belongs to the same WZW action.
Following the lines of discussion in [4], factorization of correlators involving current operators as well as composite current operators, implies that the β-function for the couplings λ 1 and λ 2 are the same as in the single λ-deformed theory, since the correlation functions from which they are derived involve only currents and as such they take the form of two copies of λ-deformed models. In particular, in the case of isotropic couplings, i.e. (λ i ) ab = λ i δ ab , these read [1] where t = ln µ 2 and µ is the energy scale and c G is the second Casimir in the adjoint representation defined from the relation f acd f bcd = c G δ ab .
The goal of this work is to obtain the RG flows for generic couplings matrices (λ 1,2 ) ab in the action (1.2) and to show that the result takes the form of (2.30) below with the definition (2.9). The plan of this work is as follows: In section 2, we tackle initially the single coupling matrix case using three independent methods, that is the one-loop effective theory for quantum fluctuations around a classical background, gravitational techniques and a CFT approach. Then, we work out the two coupling matrices case. In section 3, we focus on an example based on a class of non-symmetric coset Einstein spaces. We conclude with section 4, where we summarize our results and we give an outlook on future directions.

The single coupling matrix
In this section we will consider RG flows of the action (1.2) when λ 2 = 0, while the other coupling matrix λ 1 , renamed as λ, remains general. Then (1.2) simplifies to We shall compute its RG flows using three completely independent methods, the oneloop effective action for quantum fluctuations, gravitational techniques and CFT results, all in agreement. Obviously, with this action, as compared to (1.2), the first two computational methods simplify considerably, especially the one involving gravitation techniques. In contrast, the CFT method is based on the form of the perturbation being bilinear in the currents and as such is insensitive to the details of the action for finite values of the couplings.

The one-loop effective action
To compute the β-function we need to specify a classical background solution and compute the quantum fluctuations around it. The discussion of this section goes along the lines of [5,4]. In these works the method is described and applied for the isotropic case, i.e. when the λ's are proportional to the identity which also correspond to integrable σ-models. However, until the present work it wasn't clear whether or not the method could be extendable to other cases beyond integrable ones, let alone for general deformation matrices.
The equations of motion of (2.1) are given by [1] where At first we assume a background solution of (2.2) for which the Lagrangian density is of course deduced from (2.1) and reads Next we vary the equations of motion (2.2) obtaining the first order matrix equation Then, the one-loop effective action in momentum space, after Wick rotating to Euclidean space and integrating out the fluctuations in the Gaussian path integral, reads − L eff = L (0) + µ d 2 p (2π) 2 ln det D −1/2 , d 2 p = dp 1 dp 2 , (2.5) where µ is the energy scale cutoff and the matrix D is given by with p ± = 1 2 (p 1 ± ip 2 ) . To extract the β-function we need to compute using (2.5) the logarithmic contribution in µ. To do so we first rewrite the determinant as ln det D = ln det E + Tr ln I 2d where Then we compute the inverse of the matrix E which we write as where the various entries are given by To proceed we expand the field dependent term of (2.7) which yields the only non-vanishing logarithmic contribution in (2.5). It is simply a matter of algebra to prove that Putting everything together in (2.5) we find that where the dots denote the two WZW actions which are λ-independent. The one-loop β-function is derived by demanding that the effective action is independent of the cutoff scale µ, yielding The levels k 1,2 retain their topological nature at one-loop in the large k 1,2 expansion as they do not run with the energy scale.
The system of RG flows (2.11) contains as a subclass the symmetric case k 1 = k 2 [6,4]. For an isotropic coupling matrix λ ab = λδ ab it is in agreement with (1.3). Furthermore (2.11) is invariant under the transformation 1 and also retains its form under the transformation In fact a symmetric coupling matrix λ remains symmetric under the RG flows, as it can be readily checked from (2.11). For this class of coupling matrices there exists an analogue formula in the literature [7]. We show in section 2.1.3 that it is equivalent with (2.11).

Gravitational techniques
We are going to re-derive (2.11), using gravitational methods along the lines of [6,4].
The line element of (2.1) reads Hence, the unhatted and hatted indices denote the Maurer-Cartan forms of g 1 and g 2 respectively. By introducing the vielbeins as well as the double index notation A = (a,â), the line element can be written as whereG ab = λ 2 0g ab . Note that in the previous calculations, as well as in the following ones an overall factor of k 1 2π is not included. We may now proceed with the computation for the spin connection ω AB . Since the tangent metric G AB is constant, ω AB is antisymmetric. A practical way to compute it is by first define the quantities Then simply from which we can also extract the useful quantity ω AB|C . Employing the above along with (2.13) we find that In order to find the torsion-full spin connections we need the two-form of (2.1) which is given by where B 0 is the two-form corresponding to the two WZW models with The field strength of the two-form B is (2.16) The torsion-full spin connections are defined as Using the above along with (2.15) and (2.16) we find that and that We are now in position to compute the torsion-full Ricci tensor by a rewriting The one-loop RG flow equations read [8][9][10] 19) or equivalently in the tangent frame e A = e A M dX M they take the form where the second term corresponds to diffeomorphisms along ξ M . This term can be absorbed by choosing the vector ξ A = ω −C A|C. The left-hand side of the above equa-tion equals Employing the above in (2.20) and reinserting the overall k 1 , which does not flow, leads to the one-loop β-functions of (2.11).

CFT approach
Another approach to the β-functions (2.11) is to employ CFT techniques. Let us review the results of [7]. One considers a perturbation of the form where J a + , J b − satisfy currents algebras at levels k 1 , k 2 respectively and d A ab 's are pure numbers that define the perturbation. The d A ab 's were taken to be symmetric in the lower indices a, b = 1, 2, . . . , dim(G). The upper index A takes as many values as the number of independent coupling constants h A . Making contact with our notation, we have that and so it applies only for symmetric matrices λ ab . The following three conditions ensure closeness of this algebra and renormalizability at all orders as well as the consistency relations Finally one defines the quantities Then the β-functions are given by [7] dh In fact (2.11) is equivalent to (2.24). Indeed, first note the relation where we have used (2.22) and the second of (2.23). Similarly we find that where the matrix g = I d − λ 2 was defined in (2.8). 2 Using the above expressions we can easily prove that (2.11) is equivalent with (2.24), after we contract the latter with d A ab . The terms appearing in (2.24) are mapped in order to the quadratic, quartic and cubic in λ ab 's of (2.11).

The two coupling matrices
Before closing this section, we tackle the general case for the two coupling matrices λ 1,2 for which the action is (1.2). To compute the one-loop RG flows, one may follow the one-loop effective action approach or employ gravitational techniques, as in sections 2.1.1 and 2.1.2 respectively. However, it is apparent that the one-loop effective action approach is much simpler. First we consider the action [1] which after solving for the gauge fields gives (1.2). 2 For symmetric coupling matrices λ ab , there is no distinction between the g andg defined in (2.8).
The equations of motion of (2.25) are simply two copies of (2.2) for the coupling matrices λ 1 and λ 2 respectively [1] (2.27) The expressions of the gauge fields in terms of the J a 1± and the group elements is much more complicated than that for the single coupling case in (2.3). These can be found in [1] and will not be needed for our purposes. Therefore the non-vanishing logarithmic divergent piece, analogs of (2.5) and (2.7) factorizes and upon integrating over d 2 p, the end result simply reads where the N 's were given in (2.9). Then one can prove on the nose that (2.25) satisfies Demanding that the effective action is independent of the cutoff scale µ, leads to As in the single coupling case, the system is invariant under, implemented as in footnote 1 where the levels k 1,2 retain their topological nature at one-loop in the large k 1,2 expansion. This set of RG flows is invariant under the interchange of the indices 1 and 2.

An application
In this section we analyze the β-function (2.11) of the action (2.1) in a simple example, that is a two coupling case using a splitting of the group indices into subgroup and corresponding non-symmetric coset space having special properties.   and in addition we assume that In general c G > c H , c G/H but there is conclusion for the relation between c H and c G/H . 3 The Ricci tensor for a non-symmetric space with Killing metric δ αβ , reads

It turns out that the truncation (3.1) is a consistent if and only if (3.4) is satisfied
Therefore demanding to be an Einstein space, yields (3.4) or (3.5).  Table 1 with λ 0 = 0.8. Clearly the point (λ 0 , λ 0 ) is an IR attractor.
which is invariant under This symmetry if inherited by corresponding symmetry for the σ-models backgrounds.
The above system of RG flow equations has the fixed points We may assume without loss of generality that 0 < λ 0 < 1. From this and the fact that the levels are positive, we deduce that the physical ones are the first three. The RG flow using those fixed points are depicted in the Fig. 1.
Setting λ H = λ 0 is consistent with the corresponding equation of motion in (3.6), provided that λ 0 = 1. Then, for the remaining coupling we find that This β-function has the following properties: 1. It is invariant under the symmetry λ → λ −1 and k 1,2 → −k 2,1 , which is the left over symmetry (3.7) after setting λ H = λ 0 .
2. It has a fixed point at λ = 0, near which the β-function (3.8) reads Hence, the operator that drives the perturbation is relevant and has dimension It is not clear to us whether or not this fixed point corresponds to an exact CFT.
3. It has an IR fixed point at λ = λ 0 , provided that λ 0 = 1. At that point the CFT corresponding to the action (2.1) has been identified in G k 1 × G k 2 −k 1 CFTs [1].
The β-function (3.8) for λ near λ 0 reads and so the operator that drives the perturbations has dimension 4. For equal levels λ 0 = 1, the β-function (3.8) reads (3.9) Note however, that setting λ H = 1 is not a fixed point of the flow (3.6), except if the subgroup H is an Abelian one. Nevertheless the expression (3.9) remains valid as we prove in Section 3.2 through a direct computation in the coset. This β-function does not admit a new fixed point in the IR. Instead, the theory is driven at strong coupling.

Coset computation
We will compute the β-function (3.9) using the one-loop effective action projected at the coset G/H. We need to determine a specific background solution and evaluate its quantum fluctuations, following as before the techniques presented in [5,4]. Specializing the equations of motion for the subgroup and coset (A h ± = A a ± t a , A g/h ± = A α ± t α ) and the coupling matrix λ AB λ ab = δ ab , λ αβ = λδ αβ , we find that [3] (3.10) Moreover we fix the residual gauge through the covariant gauge fixing condition At first we specify a background solution of (3.10) and (3.11) where we set A h ± = 0, so that we project to the coset G/H. The Lagrangian density for this background reads Next we vary the equations of motion (3.10) and the covariant gauge fixing condition (3.11) obtaining for the fluctuations (δA To evaluate the one-loop effective Lagrangian, we Wick rotate to Euclidean space and then we integrate out the fluctuations in the Gaussian path integral. The result in momentum space reads − L eff E = L (0) + µ dp 1 dp 2 (2π) 2 ln det D −1/2 , (3.14) After some algebra we find that where the dots denote as before λ-independent terms. Again, the one-loop RG flows can be found by demanding that (3.16) is independent of the cutoff scale µ. Using

Outlook
We studied quantum properties of the actions (1.2) and (2.1) describing smooth interpolations between exact CFTs [1]. We proved that are one-loop renormalizable and we derived its renormalization group flows for general coupling matrices in (2.30) and (2.11). The derivation was achieved by computing the one-loop effective action of fluctuations around a background solution and from gravitational techniques. Our results are in agreement with limit cases existing in the literature. Namely, for symmetric couplings and different levels in [7] and for general couplings but equal levels in [6].
We elucidated our results by studying the two coupling case arising form splitting the group into a subgroup and the corresponding coset manifold. This is consistent if the latter is either a symmetric-space or a non-symmetric Einstein-space. It is interesting to compute correlation functions and anomalous dimensions of operators in these two-coupling theories. This would generalize analogous computations for the isotropic case in [15][16][17]. Finally it would interesting to apply the one-loop effective action techniques for the closely related η-deformed models which were introduced for semi-simple groups and symmetric cosets in [18][19][20] and [21,22] respectively. The goal would be to derive the general one-loop RG flow equations as has been performed only for isolated cases [23][24][25].