NUPHB14511S0550-3213(18)30340-710.1016/j.nuclphysb.2018.11.024High Energy Physics – TheoryWeyl anomaly and the C-function in λ-deformed CFTsEftychiaSagkriotiaesagkrioti@phys.uoa.grKonstantinosSfetsosaksfetsos@phys.uoa.grKonstantinosSiamposb⁎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 AthensAthens15784GreecebTheoretical Physics Department, CERN, 1211 Geneva 23, SwitzerlandTheoretical Physics DepartmentCERNGeneva 231211Switzerland⁎Corresponding author.Editor: Stephan StiebergerAbstractFor a general λ-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 β-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.1IntroductionAccording to Zamolodchikov's c-theorem [1], for a two-dimensional renormalizable quantum field theory (QFT) there is a positive function of the couplings, called the C-function, which monotonically decreases under the renormalization group (RG) flow of the theory from the UV to the IR. At the fixed points of the RG flow the C-function equals the central charges of the corresponding conformal field theories (CFTs). Since the stress–energy tensor couples to all degrees of freedom of a theory, the C-function is associated with the degrees of freedom of the theory at a certain energy scale. Thus, the physical interpretation of the c-theorem is that by flowing towards lower energy scale, progressively more information is lost. Due to the fact that the degrees of freedom are integrated out during the flow, this information loss is irreversible. More intuitively, as the energy scale decreases, heavier degrees of freedom decouple from the low-energy dynamics of the theory, hence leading to a monotonically decreasing C-function.In a generic QFT with couplings λi, the C-function obeys [1](1.1)dCdt=βi∂iC=24Gijβiβj⩾0,βi=dλidt,t=lnμ2, where Gij 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 k1 and k2, mutually interacting via current bilinears(1.2)Sk1,k2λ1,λ2=Sk1(g1)+Sk2(g2)+kπ∫d2σ((λ1)abJ1+aJ2−b+(λ2)abJ2+aJ1−b)+⋯, where k=k1k2 and Ski, i=1,2, are the WZW actions for a group elements gi∈G, of a semi-simple, compact and simply connected Lie group G. The currents are(1.3)Ji+a=−iTr(ta∂+gigi−1),Ji−a=−iTr(tagi−1∂−gi),i=1,2. The ta's are Hermitian matrices, normalized as Tr(tatb)=δab and they obey [ta,tb]=ifabctc, where the structure constants fabc's are taken to be real.The effective action incorporating all-orders in λi'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(1.4)g1→g2−1,g2→g1−1,k1→−k2,k2→−k1,λ1→λ1−1,λ2→λ2−1, 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 β-functions 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, λ2=0, λ1=λ, where the linearized form in λab is also the exact form [5](1.5)Sk1,k2λ=Sk1(g1)+Sk2(g2)+kλabπ∫d2σJ1+aJ2−b. 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(1.6)βab=dλabdt=12kNacd(Nbd(T)c+gbdζc), with(1.7)Nabc=Nabc(λ,λ0−1)=(λaeλbdfedf−λ0−1λeffabe)gfc,Nab(T)c=Nabc(λT,λ0),gab=(I−λTλ)ab,g˜ab=(I−λλT)ab,gab=gab−1,g˜ab=g˜ab−1,ζc=constant,λ0=k1k2. The parameter λ0 is taken to be less than one with no loss of generality and ζa relates to diffeomorphisms. In their absence and for λ0=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 C-function 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 J1+aJ2−b 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 λab=λδab 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.2The exact C-functionIn this section we compute the C-function exactly in (λ1,2)ab and to leading order in the large-k expansion.2.1Zamolodchikov's metricFollowing 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 λ2=0, λ1=λ whose effective action is given in (1.5). To proceed, we move to the Euclidean worldsheet with complex coordinates z=12(τ+iσ) and z¯, yielding the action(2.1)Sk1,k2λ=Sk1(g1)+Sk2(g2)−λabπ∫d2zOab(z,z¯),Oab(z,z¯)=J1a(z)J¯2b(z¯), where we have rescaled the currents as Jia→Jia/ki, so that they obey(2.2)Jia(z1)Jib(z2)=δabz122+ifabckiJic(z2)z12+⋯,z12=z1−z2,i=1,2 and accordingly for the anti-holomorphic currents J¯ia(z¯).We will need for our purposes the Abelian (k-independent) part of the Zamolodchikov metric Gab|cd. This computation for the perturbation (2.1) is performed in detail in Appendix B, where we find the result(2.3)〈Oab(x1,x¯1)Ocd(x2,x¯2)〉λ=Gab|cd|x12|4, where Gab|cd is given by(2.4)Gab|cd=12(g˜−1⊗g−1)ab|cd=12g˜acgbd, 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(2.5)Gab|cd=(G−1)ab|cd=2(g˜⊗g)ab|cd=2g˜acgbd,Gab|mnGmn|cd=δacδbd. The corresponding line element in the couplings target space is non-negative(2.6)dℓ2=Gab|cddλabdλcd⩾0 and moreover it is invariant under the transformation λ→λ−1, since(2.7)g−1→−λg−1λT,g˜−1→−λTg˜−1λ.2.2The Weyl anomaly coefficientIn order to compute the C-function (1.1) for σ-models corresponding to (1.2) first recall its fundamental property(2.8)dCdt=∑i=12βiab∂C∂(λi)ab=24∑i=12Gab|cdiβiabβicd=12∑i=12Tr(βiTg˜i−1βigi−1)⩾0, where we have used (2.4) and (2.6). The βiab with i=1,2 are the β-functions corresponding to the two coupling matrices (λi)ab. A solution to (2.8) is(2.9)(βi)ab=124∂C∂(λi)ab,where:(βi)ab=Gab|cdiβicd, under the assumption that (βi)abd(λi)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](2.10)Cdouble=2dimG−3(R−112H2+4∇2Φ−4(∂Φ)2)xxxxx=2dimG−3(R−+16H2+4∇2Φ−4(∂Φ)2) and that (2.9) is indeed solved. In the second line we have used for later convenience the torsion-full Ricci scalar R−=R−14H2.Generically (2.10) depends explicitly on Xμ and it is a constant if and only if(2.11)4dGμνdt∂νΦ+dBνρdtHμνρ=2∇ν(dGμνdt), where the one-loop β-functions for Gμν and Bμν are given through [12–14](2.12)dGμνdt=Rμν−14H2μν+2∇μ∂νΦ,dBμνdt=−12∇ρ(e−2ΦHρμν). 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 models11For a general deformation involving only mutual interactions of the cyclic-type having the form [7] Lpert=kπ∑i=1nλi+1abJ(i+1)+aJi−b, with J(n+1)±a=J1±a, the expression (2.13) generalizes toCn(λi,;ki)=∑i=1nCsingle(λi;kiki+1,kiki+1)−(n−1)cUV,kn+1=k1,cUV=∑i=1n2kidimG2ki+cG.(2.13)Cdouble(λ1,λ2;k,λ0)=Csingle(λ1;k,λ0)+Csingle(λ2;k,λ0−1)−cUV, where Csingle(λ;k,λ0), corresponds to the single deformed case with action (1.5). We have chosen the dependence on the levels k1 and k2 via the parameters 0<λ0<1 and k≫1. The last term in (2.13) involves the central charge at the UV and has been inserted in order to satisfy the conditions(2.14)Cdouble(0,0;k,λ0)=Csingle(0;k,λ0)=Csingle(0;k,λ0−1)=cUV. Explicitly from the standard Sugawara construction(2.15)cUV=2k1dimG2k1+cG+2k2dimG2k2+cG=2dimG−cGdimG2k(λ0+λ0−1)+O(1k2). Hence the computation boils down to determining Csingle(λ;k,λ0). 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 diffeomorphisms ξA vanish22For a reduced λab, an additional diffeomorphism might be needed for ensuring consistency of the RG flow, see Eq. (1.6) and its derivation performed in Appendix A. In that case, the dilaton contribution has to be included as in Eq. (2.10).(2.16)ξA=ω−CA|C=0, corresponding to a constant dilaton since ξA=2∂AΦ. In this case, the Weyl anomaly coefficient simplifies to(2.17)Csingle(λ;k,λ0)=2dimG−3(R−+16H2). We clarify that whereas for Cdouble we need to use the action (1.2) in its full non-linearity, for Csingle 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(2.18)R−=−2Tr(dλdtλTg˜−1)=ddtlndetg˜. Note that this is not invariant under the transformation (1.4)(2.19)R−→−2Tr(dλdtλ−1g˜−1). Next we evaluate H2, using the components of the three-form H in a convenient frame computed in [6]. We find that(2.20)H2=λ0k(IabcIpqrg˜apg˜bqg˜cr+3NbcdNqregde2g˜bqg˜cr+cGdimG),Iabc=λ0−1fabdg˜cd+Nbcd(λTg˜)da+Ncad(λTg˜)db, 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(2.21)Csingle(λ;k,λ0)=(2−cGλ02k)dimG+6Tr(βλTg˜−1)−λ02k(IabcIpqrg˜apg˜bqg˜cr+3NbcdNqregde2g˜bqg˜cr). 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),G2 and for various couplings (λi)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 λab=λδab, (2.21) reduces to Eq. (2.14) of [2], corresponding to a flow from Gk1×Gk2 in the UV point (λ=0) to Gk1×Gk2−k1 in the IR point (λ=λ0) [5]. For isotropic couplings (λ1,2)ab=λ1,2δab, (2.13) reduces to Eq. (2.11) of [2], corresponding to a flow from Gk1×Gk2 in the UV point (λ1,2=0) to Gk1×Gk2−k1Gk2×Gk2−k1 in the IR point (λ1,2=λ0) [5].Last but not least, Csingle is invariant under the transformation (1.4), up to a constant(2.22)Csingle(λ−1;−k,λ0−1)−Csingle(λ;k,λ0)=cGdimG2k(λ0+λ0−1). Subsequently, one can use the above and (2.13) to prove that Cdouble(λ1,λ2;k,λ0), 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.2.3The anisotropic SU(2) exampleIn the anisotropic SU(2) case we have six couplings, parameterized as(2.23)(λ1)ab=diag(λ1,λ2,λ3),and(λ2)ab=diag(λ˜1,λ˜2,λ˜3), with the metrics of the composite operator given by(2.24)Gab=δab2(1−λa2)2,G˜ab=δab2(1−λ˜a2)2,a=1,2,3. To compute the exact in λ's and leading order in k, β-functions of this model, we employ the results of [6,15]. We find that [8](2.25)β1=dλ1dt=−2(1+λ12)λ2λ3−(λ0+λ0−1)λ1(λ22+λ32)k(1−λ22)(1−λ32) and cyclic in λ1,2,3. The β-functions for the λ˜a, are obtained by simply relabeling λa→λ˜a. The fixed points of the β-functions and the corresponding CFTs read [5](2.26)UV:λ1,2=0=λ˜1,2,SU(2)k1×SU(2)k2,IR1:λa=λ0=λ˜a,SU(2)k1×SU(2)k2−k1SU(2)k2×SU(2)k2−k1,IR2:λa=λ0,λ˜a=0,SU(2)k1×SU(2)k2−k1. A comment is in order related to the UV fixed point. At first the choice of λ1,2=0=λ˜1,2 is just a matter of convention as other pairs of λ's could have been chosen. One can show that this point corresponds to an exact CFT, as the parameters (λ3,λ˜3) can be absorbed by an O(4,4) duality transformation on the exact SU(2)k1×SU(2)k2 string background. This is consistent with the perturbation being Lpert=kπ(λ3J1+3J2−3+λ˜3J2+3J1−3), i.e. in the Cartan subalgebra of SU(2)×SU(2), and hence exactly marginal.Defining βa=Gabβb, one can prove that βadλa is a closed one-form and similarly to (2.9) we find that(2.27)βa=124∂Cdouble∂λa,β˜a=124∂Cdouble∂λ˜a, with(2.28)Cdouble(λa,λ˜a;k,λ0)=cUV−6k(f(λa;λ0)+f(λ˜a;λ0)),f(λa;λ0)=4λ1λ2λ3−(λ0+λ0−1)(λ12λ22+λ22λ32+λ12λ32−λ12λ22λ32)(1−λ12)(1−λ22)(1−λ32), where cUV is the central charge at the UV fixed point (2.26), namely: λ1,2=0=λ˜1,2(2.29)cUV=6−6k(λ0+λ0−1)+O(1k2). Before closing this section note that the C-function (2.28) is invariant under the transformation(2.30)λa→λa−1,λ˜a→λ˜a−1,k→−k,λ0→λ0−1, and it reproduces the central charges at the UV and the IR1,2 fixed points (2.26).3Anomalous dimension of the bilinear currentIn this section we compute the anomalous dimension matrix for the bilinear current operator. To do so, we recall results of [16](3.1)〈Oab(x1,x¯1)Ocd(x2,x¯2)〉λ,k=1|x12|4(Gab|cd+γab|cdlnε2|x12|2), where(3.2)γabcd=∇abβcd+∇cdβab=∇abβcd+Gab|mnGcd|pq∇pqβmn, with ∇abβcd=∂abβcd+Γab|mncdβmn. The Γab|mncd are the standard Christoffel symbols and can be computed throughout the Zamolodchikov metric (2.4)(3.3)Γm1m2|n1n2p1p2=12Gp1p2|q1q2(∂m1m2Gq1q2|n1n2+∂n1n2Gq1q2|m1m2−∂q1q2Gm1m2|n1n2), where we denoted ∂m1m2=∂∂λm1m2. With the help of the identity λg−1=g˜−1λ, the result can be brought into the form(3.4)Γm1m2|n1n2p1p2=δn1p1δm2p2(λg−1)m1n2+δm1p1δn2p2(λg−1)n1m2. After some algebra we find the anomalous dimension matrix (3.2)(3.5)γabcd=12k(δapδbqδcmδdn+Gab|mnGcd|pq){N(T)kni[(δmpλks−δpkλms)fqsfgfi−λ0−1fmkpgqi+λpl(Nmklgqi+Nmkqgli)]+Nmki[(δnqλsi−δiqλsn)fpsfg˜fk−λ0fniqg˜pk+λlq(Nni(T)pg˜lk+Nni(T)lg˜pk)]+δpmNlrsNns(T)r(λg−1)lq+δqnNmrsN(T)rls(λg−1)pl}. This expression is quite involved and we could not further simplify it. Still, it transforms as a mixed tensor under the duality-type symmetry(3.6)γabcd→λeaλbfλcg−1λhd−1γefgh, as expected. Specializing to an isotropic coupling λab=λδab, we obtain(3.7)γabcd=cGλ2(1+λ2)(λ0+λ0−1)−4λ(1−λ2)3δacδbd+λ2(λ0+λ0−1)−2λ(1−λ2)2facefbde+λ2(1+3λ2)(λ0+λ0−1)−2λ(3+λ2)(1−λ2)3fadefbce. The corresponding anomalous dimension is found from the eigenvalue problem(3.8)γabcdδcd=γδab, which coincides with that in Eq. (2.16) of [9](3.9)γ=cGλ3(λ0+λ0−1)λ(1+λ2)−2(1+4λ2+λ4)k(1−λ2)3. 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.4OutlookIn 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 [18–22]. 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 [23–26]. 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.AcknowledgementsK. Sfetsos would like to thank the Theoretical Physics Department of CERN for hospitality and financial support during part of this research.Appendix ARenormalization and diffeomorphismsThe 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 [12–14](A.1)ddt(GMN+BMN)=RMN−+∇N+ξM+∇[MζN],t=lnμ2, where μ is the RG scale, RMN− is the torsion-full Ricci and (ξM,ζM) correspond to diffeomorphisms and gauge transformations respectively. For the scope of this appendix, it suffices to only consider ξM. The above expression can be rewritten equivalently in the tangent frame eA=eAMdXM(A.2)ddt(GAB+BAB)=RAB−+∇B−ξA. The term ξA involves two contributions [6](A.3)ξA=ω−CA|C+ξˆA, 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 λab. Next we rewrite the ξˆA term(A.4)∇B−ξˆA=eBM(∂MξˆA+ωA−C|MξˆC), where eAM is the inverse of eAM, i.e. eAMeAN=δMN. Plugging (A.4) into (A.2) along with the results of section 2.1.2 of [6], leads to the consistent set of RG flows(A.5)dλabdt=12kNacd(Nbd(T)c−λ0gbdg˜ceξˆe),ξˆe=constant, which take the form of (1.6) where ζc=−λ0g˜ceξˆe.Appendix BComputation of Zamolodchikov's metricIn this appendix we compute the Zamolodchikov metric (2.4) for the composite operator Oab 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(B.1)Gab|cd=|x12|4〈Oab(x1,x¯1)Ocd(x2,x¯2)〉λ=Gab|cd(0)+∑n=1∞(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)π2n(2n)!, where(B.2)Gaa1...a2nc|bb1⋯b2nd(2n)|x12|4=∫d2z1…2n〈J1a(x1)J1a1(z1)⋯J1a2n(z2n)J1c(x2)〉〈J¯2b(x¯1)J¯2b1(z¯1)⋯J¯2b2n(z¯2n)J¯2d(x¯2)〉, with the two-point function of Oab evaluated at the conformal point(B.3)Gab|cd(0)=|x12|4〈Oab(x1,x¯1)Ocd(x2,x¯2)〉CFT=δacδbd=(I⊗I)ab|cd.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(B.4)1π2Gaa1⋯a2nc|bb1⋯b2nd(2n)=(2n−1)(2n−2)δa1a2δb1b3Gaa3a4⋯a2nc|bb2b4⋯b2nd(2n−2)+2(2n−1)δaa1δb1b2Ga2a3..a2nc|bb3b4⋯b2nd(2n−2)+2(2n−1)δa1a2δbb1Gaa3a4⋯a2nc|b2b3⋯b2nd(2n−2). This is solved by(B.5)(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)=π2n(2n)!∑m=0n(λλT)acm(λTλ)bdn−m,n⩾1, 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 thatλa1b1λa2b2Gaa1a2c|bb1b2d(2)=2π2(λa1b1λa2b2δaa1δb1b2Ga2c|bd(0)+λa1b1λa2b2δa1a2δbb1G(0)ac|b2d)=2π2((λλT)acδbd+δac(λTλ)bd).•We assume that (B.5) holds for any order up to n−1(B.6)(∏i=12n−2λaibi)Gaa1⋯a2n−2c|bb1⋯b2n−2d(2n−2)=π2n−2(2n−2)!∑m=0n−1(λλT)acm(λTλ)bdn−m−1.•We prove that (B.5) holds for n. By multiplying (B.4) with the λ's we find1π2(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)==(2n−1)(2n−2)(λλTλ)a3b2(∏i=42nλaibi)Gaa3a4⋯a2nc|bb2b4⋯b2nd(2n−2)+2(2n−1)(λλT)aa2(∏i=32nλaibi)Ga2a3⋯a2nc|bb3b4⋯b2nd(2n−2)+2(2n−1)(λTλ)bb1(∏i=32nλaibi)Gaa3⋯a2nc|b1b3b4⋯b2nd(2n−2). 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 have1π4(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)==(2n−1)⋯(2n−4)(λλTλ)a3b2δa3a4δb2b5(∏i=42nλaibi)Gaa5a6⋯a2nc|bb4b6⋯b2nd(2n−4)+2(2n−1)⋯(2n−3)(λλTλ)a3b2δaa3δb2b4(∏i=42nλaibi)Ga4a5…a2nc|bb5b6…b2nd(2n−4)+2(2n−1)⋯(2n−3)(λλTλ)a3b2δa3a4δbb2(∏i=42nλaibi)Gaa5⋯a2nc|b4b5⋯b2nd(2n−4)+2π2n−4(2n−1)!(λλT)aa2∑m=0n−1(λλT)a2cm(λTλ)bdn−m−1+2π2n−4(2n−1)!(λTλ)bb1∑m=0n−1(λλT)acm(λTλ)b1dn−m−1, where in the second and third line we can use (B.6) for G(2n−4). We end up with1π4(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)==(2n−1)⋯(2n−4)(λTλλTλλT)b4a5(∏i=62nλaibi)Gaa5a6⋯a2nc|bb4b6⋯b2nd(2n−4)+2π2n−4(2n−1)!(λλTλλT)aa4∑m=0n−2(λλT)a4cm(λTλ)bdn−m−2+2π2n−4(2n−1)!(λTλλTλ)bb4∑m=0n−2(λλT)acm(λTλ)b4dn−m−2+2π2n−4(2n−1)!(λλT)aa2∑m=0n−1(λλT)a2cm(λTλ)bdn−m−1+2π2n−4(2n−1)!(λTλ)bb1∑m=0n−1(λλT)acm(λTλ)b1dn−m−1. By continuing the recursion of the first line down to G(0) (where the term containing internal-internal contractions has a vanishing coefficient) only terms of internal-external contractions survive, giving(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)==2(2n−1)!π2n∑p=1n∑m=0n−p((λλT)acm+p(λTλ)bdn−m−p+(λλT)acm(λTλ)bdn−m), where the double sum rewrites to∑p=1n∑m=0n−p((λλT)acm+p(λTλ)bdn−m−p+(λλT)acm(λTλ)bdn−m)=n∑m=0n(λλT)acm(λTλ)bdn−m. Assembling all these together we obtain(B.7)(∏i=12nλaibi)Gaa1⋯a2nc|bb1⋯b2nd(2n)=π2n(2n)!∑m=0n(λλT)acm(λTλ)bdn−m. Using the latter into (B.1) we have(B.8)Gab|cd=∑n=0∞∑m=0n(λλT)mac(λTλ)bdn−m=∑n=0∞(λλT)acn×∑m=0∞(λTλ)mbd=(I−λλT)ac−1(I−λTλ)bd−1=g˜acgbd=(g˜−1⊗g−1)ab|cd.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 λ1=λ2=λ. Analytically continuing to a Euclidean worldsheet and rescaling the currents as Jia→Jia/ki (as in Eq. 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