Weyl anomaly and the $C$-function in $\lambda$-deformed CFTs

For a general $\lambda$-deformation of current algebra CFTs we compute the exact Weyl anomaly coefficient and the corresponding metric in the couplings space geometry. By incorporating the exact $\beta$-function found in previous works we show that the Weyl anomaly is in fact the exact Zamolodchikov's $C$-function interpolating between exact CFTs occurring in the UV and in the IR. We provide explicit examples with the anisotropic $SU(2)$ case presented in detail. The anomalous dimension of the operator driving the deformation is also computed in general. Agreement is found with special cases existing already in the literature.

In a generic QFT with couplings λ i , the C-function obeys [1] where G ij is the Zamolodchikov metric in the space of couplings. For convenience we have used subscript indices in the λ's in order to simplify the expressions and to follow convention used in literature.
Recently, the first examples in literature where the C-function has been computed exactly as a function of the couplings were found [2]. Specifically, this involved the λ-deformed models of [3] and [4,5] based on one or two WZW models, respectively, for the special isotropic cases having one or two deformation parameters. In this research line the main goals we achieve with the present paper are: As a follow-up to [2], we present the exact C-function for the aforementioned (doubly) λ-deformed models but for generic couplings. In addition, we compute the metric in the couplings space geometry, which has potential uses beyond the present paper, as well as the anomalous dimension matrix of the composite operator driving the perturbation away from the conformal point. In the process we show that the C-function is in fact the Weyl anomaly coefficient computed by cleverly utilizing σ-model data corresponding to the λ-deformations.
The action of the doubly deformed models [5] represents the effective action of two WZW models at different Kac-Moody levels k 1 and k 2 , mutually interacting via current bilinears S λ 1 ,λ 2 k 1 ,k 2 = S k 1 (g 1 ) + S k 2 (g 2 ) + k π d 2 σ (λ 1 ) ab J a 1+ J b 2− + (λ 2 ) ab J a 2+ J b 1− + · · · , (1.2) where k = √ k 1 k 2 and S k i , i = 1, 2, are the WZW actions for a group elements g i ∈ G, of a semi-simple, compact and simply connected Lie group G. The currents are The t a 's are Hermitian matrices , normalized as Tr t a t b = δ ab and they obey [t a , t b ] = i f abc t c , where the structure constants f abc 's are taken to be real.
The effective action incorporating all-orders in λ i 's and leading order in 1/k was constructed in [5] and will not be needed for our purposes. It has the remarkable invariance given by which clearly is not a symmetry of its linearized form (1.2).
Due to the fact that the two terms in the perturbation (1.2) have mutually vanishing operator product expansions there is a factorization of the correlation functions which involve current and bilinear current correlators. In particular, the corresponding βfunctions take the form of two copies of the λ-deformed models [6]. This construction has been extended to a multi-matrix deformation of an arbitrary number of mutually interacting WZW models [7]. Due to this factorization property, it is simpler and equivalent to consider the single deformed case, λ 2 = 0, λ 1 = λ, where the linearized form in λ ab is also the exact form [5] S λ For this model the β-functions have been computed to all-orders in the perturbative λ-expansion and up to order 1/k in the large-k expansion in [6]. A slight extension to include diffeomorphisms is worked out in Appendix A where we refer for details.
The end result reads The parameter λ 0 is taken to be less than one with no loss of generality and ζ a relates to diffeomorphisms. In their absence and for λ 0 = 1 the above were derived in [8].
The structure of this work is the following: In subsection 2.1, we compute the Zamolodchikov's metric in the couplings space and in subsection 2.2 the exact Cfunction through the Weyl-anomaly coefficient. As an application, in subsection 2.3 we present the example of the anisotropic SU(2) case. In section 3, we compute the anomalous dimension of the composite operator J a 1+ J b 2− by applying gravitational techniques. Our result for the C-function is compatible with the one in [2] for a diagonal and isotropic matrix and has all the correct properties indicated by Zamolodchikov's c-theorem, while the anomalous dimension matrix at the same limit λ ab = λδ ab reduces to the one found in [9]. Finally, we include two appendices: Appendix A proves the form of the additional (diffeomorphisms) terms in the renormalization group (RG) flows of Eq. (1.6). In Appendix B we derive the general Zamolodchikov metric in the couplings space of the current bilinear operator which drives the perturbation away from the UV fixed point.

The exact C-function
In this section we compute the C-function exactly in (λ 1,2 ) ab and to leading order in the large-k expansion.

Zamolodchikov's metric
Following the discussion in section 1, the metric takes the form of two copies of the single λ-deformed models. Thus, it suffices to focus on the special case with λ 2 = 0, λ 1 = λ whose effective action is given in (1.5). To proceed, we move to the Euclidean worldsheet with complex coordinates z = 1 √ 2 (τ + i σ) andz, yielding the action where we have rescaled the currents as J a i → J a i / √ k i , so that they obey and accordingly for the anti-holomorphic currentsJ a i (z). We will need for our purposes the Abelian (k-independent) part of the Zamolodchikov metric G ab|cd . This computation for the perturbation (2.1) is performed in detail in Appendix B, where we find the result where G ab|cd is given by where g,g were defined in (1.7). This is a positive semi-definite matrix since it is the direct product of such matrices. The inverse metric equals The corresponding line element in the couplings target space is non-negative and moreover it is invariant under the transformation λ → λ −1 , since

The Weyl anomaly coefficient
In order to compute the C-function (1.1) for σ-models corresponding to (1.2) first recall its fundamental property where we have used (2.4) and (2.6). The β ab i with i = 1, 2 are the β-functions corresponding to the two coupling matrices (λ i ) ab . A solution to (2.8) is under the assumption that (β i ) ab d (λ i ) ab is a closed one-form. Integrating (2.9) can still be quite laborious and an alternative method needs to be pursued. We shall demonstrate that for the σ-model (1.2), the C-function is given in terms of the Weyl anomaly coefficient [10,11] and that (2.9) is indeed solved. In the second line we have used for later convenience Generically (2.10) depends explicitly on X µ and it is a constant if and only if where the one-loop β-functions for G µν and B µν are given through [12][13][14] dG µν dt For conformal backgrounds the condition (2.11) is trivially satisfied.
Next, we specialize to the models at hand, whose linearized form was given in (1.2). Following the discussion in section 1, the C-function takes the form of two copies of the single λ-deformed models 1 where C single (λ; k, λ 0 ), corresponds to the single deformed case with action (1.5). We have chosen the dependence on the levels k 1 and k 2 via the parameters 0 < λ 0 < 1 and k ≫ 1. The last term in (2.13) involves the central charge at the UV and has been 1 For a general deformation involving only mutual interactions of the cyclic-type having the form [7] inserted in order to satisfy the conditions (2.14) Explicitly from the standard Sugawara construction Hence the computation boils down to determining C single (λ; k, λ 0 ). This computation heavily depends on several results that can be collectively found in section 2.1.2 of [6]. Here, the corresponding Weyl anomaly coefficient drastically simplifies since the In this case, the Weyl anomaly coefficient simplifies to We clarify that whereas for C double we need to use the action (1.2) in its full nonlinearity, for C single instead, the simple action (1.5) suffices.
Continuing with our computation, the torsion-full Ricci scaler R − can be expressed in terms of the β-function as Note that this is not invariant under the transformation (1.4) Next we evaluate H 2 , using the components of the three-form H in a convenient frame computed in [6]. We find that which similarly to R − is not invariant under the transformation (1.4). Then, plugging the above into (2.17) we find after certain algebraic manipulations that Finally, we should substitute the above into (2.13) and verify, using (1.6) and (2.4), that the system of differential equations (2.9) is indeed obeyed without any diffeomorphisms. This is a formidable task which we did not complete in full generality. We have checked with Mathematica in various examples, involving the groups SU(2), SU (3), SP(4), G 2 and for various couplings (λ i ) ab , that indeed this is the case.
This leaves little doubt that, with the above data, (2.9) is obeyed in general.
For an isotropic coupling λ ab = λδ ab , (2.21) reduces to Eq. (2.14) of [2], corresponding to a flow from G k 1 × G k 2 in the UV point (λ = 0) to G k 1 × G k 2 −k 1 in the IR point (λ = λ 0 ) [5]. For isotropic couplings (λ 1,2 ) ab = λ 1,2 δ ab , (2.13) reduces to Eq. (2.11) of [2], corresponding to a flow from G k 1 × G k 2 in the UV point (λ 1,2 = 0) to Last but not least, C single is invariant under the transformation (1.4), up to a constant Subsequently, one can use the above and (2.13) to prove that C double (λ 1 , λ 2 ; k, λ 0 ), is invariant under the non-perturbative symmetry transformation (1.4). Interestingly, equality of the C-functions under this transformation is achieved only when both couplings are allowed to change under the RG flow so that they both may reach their common fixed value in the IR.
Defining β a = G ab β b , one can prove that β a dλ a is a closed one-form and similarly to (2.9) we find that where c UV is the central charge at the UV fixed point (2.26), namely: λ 1,2 = 0 =λ 1,2 Before closing this section note that the C-function (2.28) is invariant under the trans- and it reproduces the central charges at the UV and the IR 1,2 fixed points (2.26).

Anomalous dimension of the bilinear current
In this section we compute the anomalous dimension matrix for the bilinear current operator. To do so, we recall results of [16] O ab (x 1 ,x 1 )O cd (x 2 ,x 2 ) λ,k = 1 |x 12 | 4 G ab|cd + γ ab|cd ln where γ ab cd = ∇ ab β cd + ∇ cd β ab = ∇ ab β cd + G ab|mn G cd|pq ∇ pq β mn , with ∇ ab β cd = ∂ ab β cd + Γ cd ab|mn β mn . The Γ cd ab|mn are the standard Christoffel symbols and can be computed throughout the Zamolodchikov metric (2.4) where we denoted ∂ m 1 m 2 = After some algebra we find the anomalous dimension matrix (3.2) This expression is quite involved and we could not further simplify it. Still, it transforms as a mixed tensor under the duality-type symmetry as expected. Specializing to an isotropic coupling λ ab = λδ ab , we obtain The corresponding anomalous dimension is found from the eigenvalue problem γ ab cd δ cd = γ δ ab , (3.8) which coincides with that in Eq. (2.16) of [9] Other checks for equal level include the SU(2) case with anisotropic coupling and the two coupling case using a symmetric coset, see Eq.(3.11) and the equation after (3.15) of [17], respectively. Finally, we note that when the current bilinear is restricted to the Cartan subgroup then (3.2) for the corresponding anomalous dimension vanishes, in accordance with the fact that the perturbation is then exactly marginal.

Outlook
In this paper we presented the exact C-function for the doubly λ-deformed models for generic couplings. This was done by computing the general metric in the space of couplings and subsequently, incorporating the exact β-function for these models. We demonstrated that the Weyl anomaly is indeed Zamolodchikov's C-function. In addition, we have computed the anomalous dimension matrix of the composite current bilinear operator driving the perturbation away from conformality.
Our results also provide C-functions for the so-called η-deformations for group and coset spaces introduced in [18][19][20][21][22]. The reason is that these models are related to symmetric λ-deformations ( that is when the levels of the CFTs are equal) via Poisson- Lie T-duality and appropriate analytic continuations [23][24][25][26]. In particular, the background fields, the β-functions, the C-functions, etc map to each other. However, the analytic transformation spoils the UV behavior of the η-deformed models, as compared to that of the λ-models. In particular, there is no UV fixed point and they generically possess cyclic RG-flows [27].
Finally, we note that the all loop effective action representing, for small couplings, simultaneously self and mutually interacting current algebra CFTs realized by two different WZW models were constructed in [28]. It will be very interesting to extend the results of the present paper in this most general case as well.

Acknowledgments
K. Sfetsos would like to thank the Theoretical Physics Department of CERN for hospitality and financial support during part of this research.

A Renormalization and diffeomorphisms
The scope of this appendix is to work out the presence of diffeomorphisms ξ's for the RG flows (1.6), of the σ-model (1.5), which were explicitly worked out in [6]. Consider the generic one-loop RG flow [12][13][14] d dt where µ is the RG scale, R − MN is the torsion-full Ricci and (ξ M , ζ M ) correspond to diffeomorphisms and gauge transformations respectively. For the scope of this appendix, it suffices to only consider ξ M . The above expression can be rewritten equivalently in The term ξ A involves two contributions [6] where the first one is vanishing through (2.16) and the second one incorporates additional diffeomorphisms that might be needed for ensuring consistency of the RG flow in cases with a reduced λ ab . Next we rewrite theξ A term

B Computation of Zamolodchikov's metric
In this appendix we compute the Zamolodchikov metric (2.4) for the composite operator O ab in (2.1). The metric in the couplings space can be found through the two-point function [1], given in (2.3). Following the lines of appendix A.2 in [17], we can write the two-point function as a series expansion with the two-point function of O ab evaluated at the conformal point Next we work out (B.2) by performing the appropriate contractions avoiding bubble and disconnected diagrams and keeping only the Abelian part, we find the recursive relation aa 3 a 4 ···a 2n c|b 2 b 3 ···b 2n d .

(B.4)
This is solved by a fact that can be proven by induction as follows: • It is obvious that for n = 1 (B.5) holds, since from (B.4) and (B.3) we have that • We assume that (B.5) holds for any order up to n − 1 • We prove that (B.5) holds for n. By multiplying (B.4) with the λ's we find For the last two terms we can easily substitute (B.6) for G (2n−2) . However, since the contracted indices of the first term do not follow the pattern of (B.6), a bit more work in needed. In the first line, we substitute G (2n−2) by its recursive relation (B.4). We have ..a 2n c|bb 5 b 6 ...b 2n d where in the second and third line we can use (B.6) for G (2n−4) . We end up with (λ T λ) m bd = (I − λλ T ) −1 ac (I − λ T λ) −1 bd =g ac g bd = g −1 ⊗ g −1 ab|cd .

(B.8)
A comment is in order related to the additional scaling factor 1/2 in (2.4) versus (B.8) which contains no such factor. To understand its appearance we consider the doubled deformed action (1.2) with λ 1 = λ 2 = λ. Analytically continuing to a Euclidean worldsheet and rescaling the currents as J a i → J a i / √ k i (as in Eq. (2.1)), we obtain S λ k 1 ,k 2 = S k 1 (g 1 ) + S k 2 (g 2 ) − λ ab π d 2 z O ab (z,z) + · · · , We are going to normalize the two-point function of O ab to one, so that it matches with the conventions used in the proof of the c-theorem (1.1) in [1]. This normalization introduces the additional scaling factor 1/2 in (2.4). Note that for the single λ-deformed model [3], the analogue scaling factor is one [16,17].