PLB33881S0370-2693(18)30471-410.1016/j.physletb.2018.06.023The Author(s)TheoryThe exact C-function in integrable λ-deformed theoriesGeorgeGeorgiouabgeorgiou@inp.demokritos.grPantelisPanopoulosappanopoulos@phys.uoa.grEftychiaSagkriotiaesagkrioti@phys.uoa.grKonstantinosSfetsosaksfetsos@phys.uoa.grKonstantinosSiamposc⁎konstantinos.siampos@cern.chaDepartment of Nuclear and Particle Physics, Faculty of Physics, National and Kapodistrian University of Athens, 15784 Athens, GreeceDepartment of Nuclear and Particle PhysicsFaculty of PhysicsNational and Kapodistrian University of AthensAthens15784GreecebInstitute of Nuclear and Particle Physics, National Center for Scientific Research Demokritos, Ag. Paraskevi, GR-15310 Athens, GreeceInstitute of Nuclear and Particle PhysicsNational Center for Scientific Research DemokritosAg. ParaskeviAthensGR-15310GreececTheoretical Physics Department, CERN, 1211 Geneva 23, SwitzerlandTheoretical Physics DepartmentCERNGeneva 231211Switzerland⁎Corresponding author.Editor: N. LambertAbstractBy employing CFT techniques, we show how to compute in the context of λ-deformations of current algebras and coset CFTs the exact in the deformation parameters C-function for a wide class of integrable theories that interpolate between a UV and an IR point. We explicitly consider RG flows for integrable deformations of left–right asymmetric current algebras and coset CFTs. In all cases, the derived exact C-functions obey all the properties asserted by Zamolodchikov's c-theorem in two-dimensions.1Introduction and conclusionsOne of the central objects in two-dimensional field theory, as well as in quantum field theory in general, is the so-called C-function. This is supposed to be a positive and monotonically decreasing function as the theory flows from the UV regime to the IR regime. Another way to state this fact is to say that the flow from the UV to the IR is irreversible. At the UV and IR fixed points the C-function is identified with the central charge of the corresponding conformal field theory (CFT). All these properties of the C-function are encoded in Zamolodchikov's c-theorem [1]. The physical content of the c-theorem is that since the stress-energy tensor couples to all degrees of freedom of a theory, the C-function is a measure of the effective number of degrees of freedom at a certain scale. This is in accordance with the intuition that as we lower the energy scale more and more heavy degrees of freedom decouple from the low-energy dynamics of the theory leading, thus, to a monotonically decreasing C-function.To the best of our knowledge, there is no example in which the C-function has been genuinely computed exactly in the couplings at all points along the RG flow as the theory interpolates between a UV and an IR fixed point. It is the goal of the present paper to provide a calculation of the exact C-function with a clean CFT interpretation for a wide class of (integrable) σ-models constructed in [2–5] representing exact deformations of WZW and gauged WZW CFTs for general groups. These models generalize the construction of [6] (for the SU(2) group case see also [7]). This will be possible since the β-functions, as well as the metrics in the space of couplings are known to all-orders in perturbation theory for the models under consideration. The latter quantities, all two- and three-point correlation functions involving currents and affine primary operators, as well as the associated operator anomalous dimensions, have been computed to all-orders in the deformation parameters λi in a series of papers [8–13]. This was achieved by performing low order perturbative calculations in conjunction with certain non-perturbative symmetries in the space of couplings which these theories possess [14–17,5].11The β-function for these models, that come under the name λ-deformations, have been computed exactly in λ and at large level(s) by either resumming the perturbation series [14,18,19] or by the use of gravitational [16,17] and [10,11,3] or field theoretical methods [20,11,12,5,13].In a generic quantum field theory (QFT) with couplings λi, there exists a function C of the couplings obeying the following relation [1](1.1)dCdlnμ2=βi∂iC=24Gijβiβj. Here, βi=dλidlnμ2 and Gij is the Zamolodchikov metric in the couplings space, of the operators which drive the deformation. The solution of this equation given the appropriate boundary conditions will specify the C-function of our theory. Usually this can be done only up to some finite order in perturbation theory. For our models the β-functions and the metrics Gij have been calculated to all-orders in the perturbative expansion [14,18,19] and [3,5,8,11] enabling us to find exact expressions for the C-functions of the theories under consideration. On general grounds near a fixed point of the RG flow equations the relation of the β-function and the C-function is given by(1.2)βi=124Gij∂jC+⋯,where:Gij=Gij−1, suggesting a gradient flow. This is consistent with (1.1) and becomes an equality for the case of a single coupling. Provided that there is a solution to the system of equations (1.2) as equalities, then (1.1) is certainly solved, leading to a monotonically decreasing behaviour from the UV to the IR as required by the c-theorem.The organization of this work is as follows: In section 2, we will focus on the models constructed in [3], which can be thought of as the deformation of the sum of two WZW models at different levels k1,2, respectively. The operators driving the models away from the UV fixed point are current bilinears with one current obeying a current algebra at level k1 while the other obeys a current algebra at level k2. We examine the RG flows and evaluate the C-function as an exact function of the deformation parameters in the large k1,2 limit and for the case of isotropic couplings λ1,2 (for simplification we place from now on down the indices on the λ's). As discussed in detail in [3], as soon as the perturbation is turned on the model is driven towards another fixed point in the IR. In the case where both couplings are non-zero the IR CFT involves a coset CFT. More precisely, the flow is from the UV point where the symmetry of the theory is Gk1×Gk2 to the IR fixed point where the symmetry of the theory is Gk1×Gk2−k1Gk2×Gk2−k1. In the case where one of the couplings is zero the IR CFT is given by the sum of two WZW models one at level k1 and the other at level k2−k1, namely the flow is from Gk1×Gk2 to Gk1×Gk2−k1. In the same section we also calculate the C-function for the flow from Gk1×Gk2−k1 to Gk1×Gk2−k1Gk2×Gk2−k1 which is realised when one of the couplings, say λ2, is set to λ0=k1k2. In section 3 we consider the flow from the UV coset theory Gk1×Gk2Gk1+k2 to the IR coset theory Gk2−k1×Gk1Gk2 [5]. Finally, in the appendix A, we calculate the C-function for anisotropic couplings. In this theory, which has not been shown to be integrable, each of the two coupling matrices is still diagonal but has different entries in the subgroup H and in the non-symmetric Einstein subspaces. We refer to this case as the four coupling case. In all the examples considered in the present work the derived exact C-functions obey all properties asserted by Zamolodchikov's c-theorem in two-dimensions.2RG flows from group spacesIn this section, we will calculate the exact C-function for the class of theories constructed in [3]. We consider the following action with two coupling matrices (λ1,2)AB(2.1)Sλ1,λ2=Sk1(g1)+Sk2(g2)+k1k2π∫d2σ((λ1)ABJA1+J2−B+(λ2)ABJ2−AJ1+B), where the capital indices run over a semi-simple group G. The effective action for this theory has been constructed in [3], but its explicit form will not be needed for our purposes. Moreover, we will focus on the case of isotropic couplings in which the matrices will be (λi)AB=λiδAB, so that we have just two parameters. A generalization to a case with four couplings will be considered in the appendix A.The Zamolodchikov metric in the coupling space can be calculated to all-orders in λ1,2 and is given in the large k1,2 limit by [14,8](2.2)Gij=δij2dimG(1−λi2)2, where we note that in the large-k expansion, the leading term in the metric presented above depends only on the Abelian part of the current OPEs.22The metric takes the form (2.2) since the current bilinears J1+AJ2−B and J2+AJ1−B in (2.1) do not interact. Therefore, it is a double copy of the single-coupling case [14,8]. If we know the β-function near a CFT fixed point, either at the UV or at the IR, by integrating (1.2) we may know near the same point the behaviour of the C-function, as well. The integration constant is fixed by requiring that the C-function at the CFT point coincides with the central charge of the corresponding CFT. In what follows, the exact expression for C is found by symmetry arguments in conjunction with the perturbative results.Before proceeding, it is convenient to define the combinations of the levels k1,2 (assuming with no loss of generality that k2⩾k1) given byλ0=k1k2⩽1,k=k1k2. The running of the couplings has been computed exactly in the λi's and to leading order for k≫1, by CFT and gravitational methods in [19,10] and [3], respectively(2.3)βi(λi;λ0)=−cG2kλi2(λi−λ0)(λi−λ0−1)(1−λi2)2,i=1,2, where cG is the quadratic Casimir for G in the adjoint representation, see (A.1). This demonstrates that the flows for λ1 and λ2 are decoupled from each other and that there are two fixed points at the UV and at the IR at λi=0 and λi=λ0, respectively. Near the UV fixed point we obtain perturbatively thatβi(λi;λ0)=−cG2kλi2+O(λi3),i=1,2. The central charge at the UV CFT is given by the sum of the central charges of two WZW models at levels k1,2, respectively. Keeping the two leading terms in the 1/k-expansion we obtain that(2.4)cUV=2k1dimG2k1+cG+2k2dimG2k2+cG=2dimG−cGdimG2k(λ0+λ0−1)+⋯. Therefore, using also that at leading order near λi=0, the metric Gij=12δijdimG, we have by solving (1.2) that(2.5)C(λ1,λ2;λ0)=2dimG−cGdimGk(12(λ0+λ0−1)+2λ13+2λ23)+O(λ4). The effective action of the σ-model (2.1), was constructed in [3], and in particular the β-function is explicitly invariant under the symmetry33At the level of the effective action of (2.1), the symmetry (2.6) is accompanied by an inversion of the group elements g1,2→g2,1−1, see Eqs. (2.12) & (2.14) in [3]. This generalizes a similar symmetry found in [16] for the single λ-deformed model [6].(2.6)λi→1λii=1,2,k1→−k2,k2→−k1. Note that, under this transformation the expansion parameter k→−k. The two fixed points of the above symmetry for the parameters λi are at λi=±1. In order to have well behaved expansions of (2.3) around these fixed points we expand(2.7)λi=±1−bik1/3,k→∞,λ0=fixed. Hence, as explained in [8], the C-function to O(1/k) should be of the following form(2.8)C(λ1,λ2;λ0)=2dimG−cGdimG4k(f(λ1;λ0)(1−λ12)3+f(λ2;λ0)(1−λ22)3), with the analytic function f(λ;λ0) obeying, thanks to (2.6), the condition f(λ;λ0)=λ6f(λ−1;λ0−1). Hence, f(λ;λ0) should be a six-degree polynomial in λ with coefficients that will generically depend on λ0. Matching with the perturbative results (2.5) gives(2.9)f(λ;λ0)=(λ0+λ0−1)(1−3λ2−3λ4+λ6)+8λ3. Note that C(λ1−1,λ2−1;λ0−1)=C(λ1,λ2;λ0), implying invariance under (2.6).44A method of deriving what has been called effective central charge is by employing TBA techniques for determining the ground state energy of the system, initially put forward in [25,26].2.1A shortcutUsing (1.1) we obtain that(2.10)∂λC=24Gλλβλ. This relation is exact in λ and should hold for each of the two couplings λ1,2, separately. Then, for our case(2.11)C=cUV−6cGdimGk∫0λ1dxx2(x−λ0)(x−λ0−1)(1−x2)4−(λ1→λ2)x=cUV−cGdimG2kλ134−λ1(3−λ12)(λ0+λ0−1)(1−λ12)3−(λ1→λ2), where cUV is given by (2.4). After some algebra we obtain indeed (2.8) and (2.9). Note that the term arising from the integration is not by itself invariant under (2.6).2.2The equal level caseIn the equal level case, i.e. when λ0=1, the C-function (2.8) becomes (we also take λ1=λ2=λ)(2.12)C(λ)=2dimG−cGdimGk1+2λ+2λ3+λ4(1−λ)(1+λ)3. This is the C-function corresponding to the simplest λ-deformed model of [6]. This becomes strongly coupled at λ=±1. As shown in [6] and in [9] what makes sense near these points are the non-Abelian T-duality and pseudodual model limits (2.7), for λ=1 and λ=−1, respectively. In these limits C(λ) remains indeed finite.Let us mention that the C-function for this particular case, known also as the non-Abelian bosonized Thirring model, was implicitly derived in [14,15], albeit not in the invariant form (2.12). Indeed, the effective potential calculated in [14] satisfies the same differential equation as the C-function. However, unlike this, the effective potential is not invariant under the non-perturbative symmetry λ→1/λ,k→−k.55For certain integrable deformations of the isotropic principal chiral model [21], the Weyl anomaly coefficient for σ-model in the NS–NS sector [22,23] was worked out [24]. This can be interpreted as a “generalized central function”. However, as stressed in [22,23] it can not be in general identified with the C-function of Zamolodchikov since its flow is not generically monotonic due to the indefinite metric in the space of couplings (gμν,Bμν,Φ). It seems worth pursuing investigations in this direction.2.3From Gk1×Gk2 to Gk1×Gk2−k1Gk2×Gk2−k1Expanding (2.8) around the IR fixed point at λ1=λ2=λ0 we obtain that(2.13)C(λ1,λ2;λ0)=2dimG−cGdimGk1+λ042λ0(1−λ02)+O(λ−λ0)2. The leading term should correspond to the large k expansion of the central charge of the IR CFT which was explicitly identified in [3] to be the coset Gk1×Gk2−k1Gk2×Gk2−k1.66It can be easily seen that the central charge of Gk1×Gk2−k1Gk2×Gk2−k1 is invariant under the extension of the symmetry (2.6) which involves the shifts of levelsk1→−k2−2cG,k2→−k1−2cG, valid beyond the large k limit. The transformation of λ is probably not simply an inversion as in (2.6), but it may involve k-corrections as well. Its central charge for k≫1 gives indeed C(λ0,λ0;λ0), i.e. the leading term in (2.13), as it should be. This consists a non-trivial check of our exact formula (2.8).2.4From Gk1×Gk2 to Gk1×Gk2−k1One may consistently set one of the couplings to zero, say λ2=0, as it can be seen from the expressions of the β-functions of the theory (2.3). Renaming λ1 to λ and attending this to the general expression of (2.8), (2.9) we obtain that(2.14)C(λ;λ0)=2dimG−cGdimG2k(λ0+λ0−1)(1−3λ2)+4λ3(1−λ2)3. Expanding around the IR fixed point at λ=λ0 one has that(2.15)C(λ;λ0)=2dimG−cGdimG2k1λ0(1−λ02)+O(λ−λ0)2. The leading term should correspond to the large k expansion of the central charge of the IR CFT, which consists of a sum of two WZW models at levels k1 and k2−k1 [3]. For k≫1 we get indeed C(λ0;λ0), i.e. the leading term in (2.15), as it should be. Note also that (2.14) is not invariant under λ→1/λ and k→−k. This is not a surprise in the sense that at the end point of the flow the central charge cIR is not invariant under this symmetry.2.5From Gk1×Gk2−k1 to Gk1×Gk2−k1Gk2×Gk2−k1There is another consistent truncation in the coupling space. This is to set one of the couplings, say λ2 to λ0 and rename λ1 as λ. Plugging these to the general expression for C(λ1,λ2;λ0) (2.8) we obtain that(2.16)C(λ;λ0)=2dimG−cGdimG2k1−3λ2+λ04λ4(3−λ2)+4λ0(1−λ02)λ3λ0(1−λ02)(1−λ2)3. Near the UV fixed point at λ=0 we obtain the leading term in (2.15), as expected since the IR fixed point of the flow described in the previous section is the UV fixed point of the present flow. Similarly, near the IR fixed point at λ=λ0 we retrieve (2.13), as one should expect. This C-function is invariant under λ→1/λ and k→−k.3RG flows from coset spacesIn this section, we evaluate the exact C-function for the flow from the coset theory with symmetry Gk1×Gk2Gk1+k2 (UV fixed point) to the coset theory with symmetry Gk2−k1×Gk1Gk2 (IR fixed point).77Other related works studying aspects of RG flows of coset theories include [27–31]. These theories were constructed first time in [4] for G=SU(2) where an explicit action was given. The effective action for this, taken into account all loop effects in the λi's integrability properties, as well as the running of the coupling were subsequently examined for a general group G in [5], where the corresponding RG flow was also identified.We start by defining the parameterssi=kik1+k2,i=1,2 andλ1−1=s2−3s1,λ2−1=s1−3s2,λ3−1=(s1−s2)2=1−4s1s2,λf−1=1−8s1s2. The one-loop β-function exact in λ and to leading order for k≫1 is given [5] by88For equal levels k2=k2=k we have that λ1=λ2=λf=−1 and λ3→∞. Then, βλ=−cG4kλ, that is the perturbative example is also exact (for k≫1) as found in [16,20] for all symmetric spaces.(3.1)βλ=−cGλ(1−λ1−1λ)(1−λ2−1λ)(1−λ3−1λ)2(k1+k2)(1−λf−1λ)2. The central charge at the UV CFT Gk1×Gk2Gk1+k2 is easily obtained to be(3.2)cUV=dimG−cGdimGk1+λ02+λ042λ0(1+λ02)+⋯. The central charge at the IR CFT Gk2−k1×Gk1Gk2 is when λ=λ2<0 and is given by(3.3)cIR=dimG−cGdimGk1−λ02+λ042λ0(1−λ02)+⋯. The action of the theory is invariant under the following remarkable symmetry transformation [5](3.4)λ→1−(s1−s2)2λ(s1−s2)2−(1−8s1s2)λ,ki→−ki,i=1,2 and the same is true for the β-function. This symmetry has two fixed points for the parameter λ given by λ=1 and λ=λf. Near these points the limitsλ=1−bk,k→∞, andλ=λf−bk,λ0=1−n2k,k→∞, of all quantities should be well defined (as they are for the β-function).The metric in the space of couplings is given by(3.5)Gλλ=8s12s22dimG(1−λ)2(1−λf−1λ)2. It has the correct behaviour being singular at the fixed points of the symmetry transformation and also for k1=k2 we get λf=−1 reducing, thus, to the known one, namely Gλλ=dimG2(1−λ2)2. Furthermore, the symmetry (3.4) leaves invariant the line element with metric (3.5). Finally, the overall coefficient is chosen so that the correct central charge in the IR is reproduced. It will be interesting to derive (3.5) along the lines of [14,8].To obtain the exact in λ and to O(1/k) one solves (2.10)(3.6)C(λ;λ0)=dimG−cGdimGk1+λ02+λ042λ0(1+λ02)x−16cGdimGkλ05λ2(1+λ02)7x×3(1+λ02)2+2(1+10λ02+λ04)λ−(5−22λ02+5λ04)λ2(1−λ)(1−λf−1λ)3. At the IR fixed point situated at λ=λ2 we find (3.3), as it should be. In addition, C(λ;λ0) is invariant under the symmetry (3.4).99We note that in the equal level case, cf. footnote 8, the C-function drastically simplifies toC(λ)=dimG−3cGdimG4k1+λ21−λ2.Appendix AThe case of four couplingsA.1PreliminariesIn this appendix, we consider again the class of models presented in [3], but with the couplings matrices being more general. Namely, let's consider again the action (2.1) and split the semi-simple group G into one of its subgroups H and the corresponding coset space G/H. In the split index A=(a,α), Latin (Greek) letters denote subgroup (coset) indices. Then for non-symmetric Einstein-spaces G/H [32,33] we have that(A.1)fACDfBCD=cGδAB,facdfbcd=cHδab,fαγδfβγδ=cG/Hδαβ,faγδfbγδ=(cG−cH)δab,fαγcfβγc=12(cG−cG/H)δαβ. Among the above identities, the non-trivial one that essentially defines a non-symmetric Einstein space, is the one involving purely Greek indices.We will work for diagonal deformation matrices (λi)AB=λHiδab+λiδαβ, where λHi and λi denote the subgroup and coset deformation parameters respectively. The β-functions for each (λi)AB are given by [13]βλH=−(λH−λ0)(λH−λ0−1)2k×(cHλH2(1−λH2)2+(cG−cH)λ2(1−λ2)2),βλ=−12k(cG/Hλ2(λ−λ0)(λ−λ0−1)(1−λ2)2+cG−cG/H2λ(1−λ2)(1−λH2)×((λ0−1−λH)(λ0λH−λ2)+(λ0−λH)(λ0−1λH−λ2))). These β-functions are invariant underλ→λ−1,λH→λH−1,k1,2→−k2,1, and have as fixed points(A.2)(λH,λ)={(0,0),(λ0,λ0),(λ0,0)}. In the case at hand, the Zamolodchikov metric in the space of couplings Gij reads(A.3)Gij=12(Gij(1)00Gij(2)),Gij(i)=(dimG/H(1−λi2)200dimH(1−λHi2)2),i=1,2.A.2The C-functionWe will compute the general C-function corresponding to the action (2.1). From (1.2) with the use of (A.3) we end up with 4 differential equations. However, from the form of the two β-functions we know that the couplings (λ1)AB and (λ2)AB are independent to each other. Hence, the occurring differential equations are pairwise decoupled.The general solution of these equations is given by(A.4)C(λ1,λH1,λ2,λH2;λ0)=2dimG−14k(C˜(λ1,λH1,λ0)+C˜(λ2,λH2;λ0)), whereC˜(λ,λH;λ0)=1(1−λ2)3(cGdimGg1+cGdimHg2+cHdimHg3), where the functions gi(λ,λH;λ0), i=1,2,3 are given byg1=(λ0+λ0−1)(1−3λ2−3λ4+λ6)+8λ3,g2=12λ2(λ0+λ0−1−2λ)−12λ2(1−λ2)(1−λH2)(λ0+λ0−1−2λH),g3=−2((λ0+λ0−1)(1+3λ2)−8λ3)+2(1−λ2)(1−λH2)3[(λ0+λ0−1)(1+4λ2+λ4−3(1+λ2)2λH2+6λ2λH4)+4(1+4λ2+λ4)λH3−12λ2λH(1+λH4)]. In order to find (A.4), we used the consistency relationcG/H=cG−2dimH(cG−cH)dimG/H, which can be easily proved by appropriately tracing the free indices of the second line in (A.1). The function (A.4) indeed gives the correct UV and IR values for the central charges (2.4) and (2.13). It remains invariant under the symmetry (λi→λi−1,λHi→λHi−1,λ0→λ0−1,k→−k). The C-function we have derived, satisfies (1.1), has the right behaviour when λ=λH or λ=0 and monotonically decreases as one transverses from the UV to the IR.A.2.1Other important RG flows1.If we take λ2=λH2=0 and λ1=λH1=λ, (A.4) takes the formC(λ,λ,0,0;λ0)=2dimG−cGdimG2k(λ0+λ0−1)(1−3λ2)+4λ3(1−λ2)3 which is invariant up to a non-vanishing constant under the duality-type symmetry and has the correct IR central charge at λ=λ0, corresponding to the flow from Gk1×Gk2 to Gk1×Gk2−k1.2.By taking λ2=λH2=λ0 and λ1=λH1=λ, (A.4) reduces toC(λ,λ,λ0,λ0;λ0)=2dimG−cGdimG2k1−3λ2+4λ04λ4(3−λ2)+λ0(1−λ02)λ3λ0(1−λ02)(1−λ2)3 representing the flow from Gk1×Gk2−k1 to Gk1×Gk2−k1Gk2×Gk2−k1, with the correct IR central charge at λ=λ0.3.Taking λ1,2=0 and λH1=λH2=λ, (A.4) reduces toC(0,λ,0,λ;λ0)=2dimG−cGdimG2k(λ0+λ0−1)−cHdimHkλ3(4+λ(λ0+λ0−1)(λ2−3)(1−λ2)3.4.Finally, considering λ1,2=0=λH2 and λH1=λ, (A.4) reduces toC(0,λ,0,λ;λ0)=2dimG−cGdimG2k(λ0+λ0−1)−cHdimH2kλ3(4+λ(λ0+λ0−1)(λ2−3)(1−λ2)3. 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