A free field perspective of λ -deformed coset CFT's
aa r X i v : . [ h e p - t h ] A p r A free field perspective of λ -deformed coset CFT’s George Georgiou , Konstantinos Sfetsos and
Konstantinos Siampos
Department of Nuclear and Particle Physics,Faculty of Physics, National and Kapodistrian University of Athens,Athens 15784, Greece george.georgiou, ksfetsos, [email protected]
Abstract
We continue our study of λ -deformed σ -models by setting up a / k perturbative expan-sion around the free field point for cosets, in particular for the λ -deformed SU ( ) / U ( ) coset CFT. We construct an interacting field theory in which all deformation effects aremanifestly encoded in the interaction vertices. Using this we reproduce the known β -function and the anomalous dimension of the composite operator perturbing awayfrom the conformal point. We introduce the λ -dressed parafermions which have an es-sential Wilson-like phase in their expressions. Subsequently, we compute their anoma-lous dimension, as well as their four-point functions, as exact functions of the defor-mation and to leading order in the k expansion. Correlation functions with an oddnumber of these parafermions vanish as in the conformal case. ontents β -function 84 Parafermion correlators 11 λ -dressed parafermionic fields . . . . . . . . . . . . . . . . . . . . . . . . 124.2 The anomalous dimension of parafermion bilinear . . . . . . . . . . . . . 174.3 Anomalous dimension of the single parafermion . . . . . . . . . . . . . . 194.4 Four-point correlation functions . . . . . . . . . . . . . . . . . . . . . . . . 214.4.1 Correlation function h Ψ + Ψ − Ψ + Ψ − i . . . . . . . . . . . . . . . . . 224.4.2 Correlation function h Ψ + Ψ + ¯ Ψ − ¯ Ψ − i . . . . . . . . . . . . . . . . . 244.4.3 Correlation function h Ψ + Ψ − ¯ Ψ + ¯ Ψ − i . . . . . . . . . . . . . . . . . 254.4.4 Correlation function h Ψ + Ψ + Ψ + Ψ − i . . . . . . . . . . . . . . . . . 264.4.5 Correlation function h Ψ + Ψ + Ψ + Ψ + i . . . . . . . . . . . . . . . . . 27 ˜ O ±
29B Various integrals 32
A class of integrable two-dimensional field theories having an explicit action realiza-tion, were systematically constructed in recent years [1–10]. They typically representthe effective action corresponding to the deformation of one or more WZW modelcurrent algebra theories at levels k i by current bilinears. They are called λ -deformed1 -models due to the preferred letter used to denote the deformation parameters. Inthis context, a research avenue is the systematic study of various aspects of the cor-responding two-dimensional quantum field theories. Roughly speaking, these fallinto two general categories. In the first, belong studies concerning directly the cou-pling constants in these theories. In particular, the computation of their running un-der the renormalization group flow (RG) ( β -functions) has been exhaustively stud-ied [11, 12, 14–16, 6, 17, 8, 21]. In addition, the geometrical aspects of the space of cou-plings in these theories have been elucidated [18, 19] and the C -function capturingthe number of the degrees of freedom along the RG flow has been evaluated [20, 21].The second category consists of works aiming at discovering how the operators of theCFT respond to the deformation. That includes the computation of the anomalousdimension they acquire [18, 22–25] as well as their dressing induced by the deforma-tion of the original CFT [19]. The above works utilized a combination of CFT andgravitational techniques together with symmetry arguments in the couplings space ofthese models. The results that were obtained are valid to all orders in the deformationparameters and to leading order at the level k .Recently a new purely field theoretic approach to the study of λ -deformed the-ories was initiated in [26]. This method was applied to the isotropic case having asingle coupling λ . The resulting theory is a theory of free fields having certain in-teraction terms with all the coupling constants depending on λ , in a specific manner.Similarly, the dressed operators of the theory, elementary as well composite ones, canbe expressed, building also on results of [19], in terms of the free fields with specificcouplings. In this approach all computations are organized around a free field theoryand not around the conformal point. The advantage is that all information about thedeformation parameter λ is encoded in the coupling constants appearing in the actionand in the various coefficients in the expressions of the operators which are given interms of the free fields. This approach delivered results for the β -function, correla-tion functions and anomalous dimensions in complete agreement with the previousmethods [18, 19], however with much less effort.Most of the studies in this direction were done for the λ -deformed WZW currentalgebra CFTs. However, λ -deformations can be constructed for coset CFTs which havean action realization based on gauged WZW models. In particular, λ -deformed mod-els have been constructed based on the SU ( ) / U ( ) coset CFT in [1], for more general2ymmetric spaces in [2] and the AdS × S superstring in [3]. In the present paperwe will construct and utilize the field theory based on free fields, analog of the con-struction for the group case of [19] for the λ -deformed SU ( ) / U ( ) coset CFT. Withour approach we will be able to compute the anomalous dimension of parafermionicfields which are seemingly very hard to do by means of other methods. In particular,it is much more difficult to apply conformal perturbation theory when the underlyingtheory is a coset CFT instead of a current algebra CFT. Essentially, this is due to thefact that in the former case parafermions are involved instead of currents and theyhave more complicated correlation functions [27].The plan of the paper is as follows: In section 2, the perturbative expansion of theaction around the free field point will be performed. In this way we will construct aninteracting field theory by keeping terms up to sixth order in the fields which sufficesfor our purposes. In the process we will freely use various field redefinitions in orderto simplify the final form of the action. In section 3, we will use this action to com-pute the β -function of the σ -model using standard heat-kernel techniques. In section4, we will introduce the λ -dressed parafermions which contain an important Wilson-like phase in their expressions. Next, we compute the anomalous dimension of thecomposite operator which is bilinear in the parafermions and which drives the modelaway from the conformal point. We also compute the anomalous dimension of a sin-gle parafermion. In this case, the presence of the non-trivial Wilson-like phase playsthe important rôle. In addition, we compute all four-point functions of parafermions.Finally, in section 5 we will present our conclusions and future directions of this work.Last but not least, two appendices follow. In appendix A we calculate the anoma-lous dimension of variations of the composite operator deforming the CFT while inappendix B a list of integrals and the regularization scheme we employ is considered. In this section the interacting field theory corresponding to the λ -deformed coset CFTfor SU ( ) / U ( ) will be constructed. Our approach follows in spirit that of [26]. In thatwork the interacting theory, based on free fields, for the isotropic single λ -deformed σ -model deformation of the WZW model CFT was constructed.3 .1 Expansion around the free point Even though we are interested in the coset case, it is convenient to start with the gen-eral λ -deformed σ -model action for the group case given by [1] S k , λ ( g ) = S WZW,k ( g ) + k π Z d σ R a + ( λ − − D T ) − ab L b − , (2.1)where S WZW, k ( g ) = − k π Z d σ Tr ( g − ∂ + gg − ∂ − g ) + k π Z Tr ( g − d g ) , (2.2)is the WZW action for a group element g of a semi-simple Lie group G at level k and R a + = − i Tr ( t a ∂ + gg − ) , L a − = − i Tr ( t a g − ∂ − g ) , D ab = Tr ( t a gt b g − ) . (2.3)The t a ’s are representation matrices satisfying [ t a , t b ] = i f abc t c , Tr ( t a t b ) = δ ab , a =
1, 2, . . . , dim G , (2.4)where the f abc ’s are the algebra structure constants which are taken to be real. Thecoupling matrix λ ab parametrizes the deviation from the conformal point at which thecurrents R + and L − are chirally, respectively anti-chirally, conserved.The λ -deformed action for the SU ( ) / U ( ) coset CFT can be obtained from the aboveaction specialized to the SU ( ) case with t a = σ a / √
2, where the σ a ’s are the Paulimatrices, and λ ab = diag ( λ , λ , λ ) where λ corresponds to the U ( ) subgroup of SU ( ) via a limiting procedure [1]. We review this by first parameterizing the groupelement as g = e i ( φ + φ ) σ /2 e i ( π /2 − θ ) σ e i ( φ − φ ) σ /2 , (2.5)where the range of values of the Euler angles are θ ∈ [ π / ] , φ ∈ [
0, 2 π ] , φ ∈ [
0, 2 π ] . (2.6)4nserting the above into (2.1) and taking the limit λ → S k , λ ( g ) = k π Z d σ (cid:18) − λ + λ ( ∂ + θ∂ − θ + tan θ∂ + φ∂ − φ )+ λ − λ ( cos φ∂ + θ − sin φ tan θ∂ + φ )( cos φ∂ − θ − sin φ tan θ∂ − φ ) (cid:19) . (2.7)We note that the coordinate φ has decoupled at the level of the action and the remain-ing two fields are θ and φ . However, this angle will play a very important rôle, as weshall see in the course of the paper. Its presence will be instrumental in determiningthe form of the λ -dressed parafermions and, as a consequence, of their anomalousdimensions.The action (2.7) is invariant under the following two symmetries [13, 14]I : λ → λ − , k → − k ,II : λ → − λ , φ → φ + π θ = θ = ρ √ k (2.9)and defining two new fields as y = r + λ − λ ρ cos φ , y = r − λ + λ ρ sin φ . (2.10)Then, in the large k expansion the action becomes S k , λ = π Z d σ (cid:16) ∂ + y ∂ − y + ∂ + y ∂ − y + g k y ∂ + y ∂ − y + g k y ∂ + y ∂ − y + g k y y ( ∂ + y ∂ − y + ∂ + y ∂ − y )+ y k ( h y + ˜ h y ) ∂ + y ∂ − y + y k ( h y + ˜ h y ) ∂ + y ∂ − y + y y k ( h y + ˜ h y ) ( ∂ + y ∂ − y + ∂ + y ∂ − y ) (cid:17) + · · · , (2.11)5here we have kept terms up to quartic order in the fields. Note that, this is an ex-pansion in the number of fields. Retaining appropriate powers of / k , is just for bookkeeping as we could rescale the fields by a factor of √ k and have k as an overall coef-ficient in the action. The various couplings are given by g =
13 1 + λ − λ , g =
13 1 − λ + λ , g = −
13 1 + λ − λ , h = ( + λ ) ( − λ ) , ˜ h = + λ + λ ( + λ ) , h = ( − λ ) ( + λ ) , ˜ h = − λ + λ ( − λ ) , h = − − λ + λ ( + λ ) , ˜ h = − + λ + λ ( − λ ) . (2.12)Note that (2.11) with the above couplings is invariant under the transformationsI : λ → λ , k → − k ,II : λ → − λ , ( y , y ) → ( − y , y ) . (2.13)which of course correspond to (2.8) when the above zoom-in limit is taken.Next we perform the following field redefinitions y = (cid:16) + a k x + b k x + c k x x (cid:17) x , y = (cid:16) + a k x + b k x + c k x x (cid:17) x . (2.14)Choosing the coefficients as a = − g b = g − h c = g g + g g − g − ˜ h a = − g b = g − h c = g g + g g − g − ˜ h S k , λ = π Z d σ (cid:16) ∂ + x ∂ − x + ∂ + x ∂ − x + ˆ g k x x ( ∂ + x ∂ − x + ∂ + x ∂ − x )+ x x k ( ˆ h x + ˆ˜ h x ) ( ∂ + x ∂ − x + ∂ + x ∂ − x ) (cid:17) + · · · , (2.16)6here the new couplings are denoted by a hat and are given byˆ g = g − g − g = − + λ − λ ,ˆ h = (cid:0) g g − g g + g + h − h − ˜ h (cid:1) = − (cid:18) − λ + λ (cid:19) ,ˆ˜ h = (cid:0) g g − g g + g + h − h − ˜ h (cid:1) = − (cid:18) + λ − λ (cid:19) . (2.17)Obviously, the action (2.16) with the above couplings is invariant under (2.13), wherefor the symmetry II the y a ’s should be replaced accordingly by the x a ’s. We would like to set up a perturbative expansion around the free theory and performquantum computations. Passing to the Euclidean regime we have the following basicpropagators which are consistent with our normalizations h x a ( z , ¯ z ) x b ( z , ¯ z ) i = − δ ab ln | z | , a =
1, 2 , (2.18)where z = z − z . Note that the above propagator implies that h ∂ x a ( z ) ¯ ∂ x b ( ¯ z ) i = π δ ab δ ( ) ( z ) , (2.19)inducing a coupling of the holomorphic and anti-holomorphic sectors which will bevery important in the calculations that follow, in particular in subsubsections 4.4.1 &4.4.5.We will see later that it will be most convenient to define two complex conjugatebosons as x ± = x ± ix . (2.20)In this complex basis, the only non-vanishing two-point function is h x + ( z , ¯ z ) x − ( z , ¯ z ) i = − | z | . (2.21)7inally, for the free theory the holomorphic energy–momentum tensor is given by T = − (cid:16) ( ∂ x ) + ( ∂ x ) (cid:17) = − ∂ x + ∂ x − , (2.22)with a similar expression for the anti-holomorphic one. β -function In this section we use the field theory action (2.16) in order to compute the β -functionfor the deformation parameter λ . The analogous computation for the group case wasperformed in [26]. We will find precisely the result of [14] obtained by gravitationalmethods.In order to obtain the β -function for λ we will employ the background field heat kernelmethod, so that as an initial step we need the equations of motion for the fields x and x derived from the action (2.16). We obtain, up to O ( / k ) , that ∂ + ∂ − x + x x k (cid:18) ˆ g + ˆ h x + ˆ˜ h x k (cid:19) ∂ + ∂ − x + x k (cid:18) ˆ g + ˆ h x + h x k (cid:19) ∂ + x ∂ − x = ∂ + ∂ − x + x x k (cid:18) ˆ g + ˆ h x + ˆ˜ h x k (cid:19) ∂ + ∂ − x + x k (cid:18) ˆ g + h x + ˆ˜ h x k (cid:19) ∂ + x ∂ − x = k we may simplify them by solving for ∂ + ∂ − x a , a =
1, 2. Keeping terms upto O ( / k ) , we find that ∂ + ∂ − x + ˆ g k x ∂ + x ∂ − x + x k (cid:16)(cid:0) ˆ h x + h x (cid:1) ∂ + x ∂ − x − ˆ g x ∂ + x ∂ − x (cid:17) = ∂ + ∂ − x + ˆ g k x ∂ + x ∂ − x + x k (cid:16)(cid:0) ˆ˜ h x + h x (cid:1) ∂ + x ∂ − x − ˆ g x ∂ + x ∂ − x (cid:17) = δ x a of the coordinates x a obey. The fluctuations will be taken around a classical solution which we will stilldenote by x a . They will be casted in the formˆ D ab δ x b = D is a certain second order in the worldsheet derivatives that willalso depend on the classical solution around which we expand. We will present itsexplicit expression after performing the Euclidean analytic continuation and passingto momentum space. In the conventions of [8], we replace ( ∂ + , ∂ − ) by / ( ¯ p , p ) ≡ ( p + , p − ) . Then after dividing by p + p − we obtain for ˆ D the resultˆ D ab = δ ab + k (cid:16) ˆ F + ˆ F ′ + k ˆ F (cid:17) ab , (3.4)where the matrices are given byˆ F = ˆ g p + p − ∂ + x ∂ − x ∂ + x ∂ − x ! ,ˆ F ′ = ˆ g x ∂ + x p + + x ∂ − x p − x ∂ + x p + + x ∂ − x p − ! , (3.5)and ˆ F = p + p − ( ˆ F ) ( ˆ F ) ( ˆ F ) ( ˆ F ) ! , ( ˆ F ) = (cid:0) ˆ h x + ˆ˜ h x (cid:1) ∂ + x ∂ − x − ˆ g x ∂ + x ∂ − x , ( ˆ F ) = (cid:0) ˆ˜ h x + ˆ h x (cid:1) ∂ + x ∂ − x − ˆ g x ∂ + x ∂ − x . (3.6)Note that we have not provided the expressions for ( ˆ F ) and ( ˆ F ) , since their formwill be irrelevant for the discussion that follows. Integrating out the fluctuations, givesthe effective Lagrangian of our model which reads − L eff = L ( ) k , λ + Z µ d p ( π ) ln ( det ˆ D ) − , (3.7)where L ( ) k , λ is the Lagrangian (2.16) on the classical solution. This integral is loga-rithmically divergent with respect to the UV mass scale µ . The logarithmic term is9solated by performing the large momentum expansion of the integrand and keepingterms proportional to / | p | , where | p | = p ¯ p . Next we use the fact thatln ( det ˆ D ) = k Tr ˆ F + k (cid:18) Tr ˆ F −
12 Tr ˆ F ‘22 (cid:19) + . . . , (3.8)where we have included only terms potentially contributing to the logarithmicallydivergent term. Then calculating the traces individually one obtainsTr ˆ F = ˆ g p + p − (cid:0) ∂ + x ∂ − x + ∂ + x ∂ − x (cid:1) ,Tr ˆ F = p + p − h(cid:0) ( h − ˆ g ) x + h x (cid:1) ∂ + x ∂ − x + (cid:0) ( h − ˆ g ) x + h x (cid:1) ∂ + x ∂ − x i ,Tr ˆ F ‘22 = g x x p + p − (cid:0) ∂ + x ∂ − x + ∂ + x ∂ − x (cid:1) , (3.9)where again we have not included terms which will vanish upon the angular inte-gration that follows. Using polar coordinates, i.e. p = re i ω , ¯ p = re − i ω , in which theintegration measure is d p = r d r d ω , we evaluate the effective action from (3.7) to be(we return back to the Lorentzian regime) S eff = π Z d σ (cid:20)(cid:16) − ˆ g k ln µ (cid:17)(cid:0) ∂ + x ∂ − x + ∂ + x ∂ − x (cid:1) + ˆ g k x x (cid:16) + ˆ g k ln µ (cid:17)(cid:0) ∂ + x ∂ − x + ∂ + x ∂ − x (cid:1) − ln µ k h(cid:0) h x + ( h − ˆ g ) x (cid:1) ∂ + x ∂ − x + (cid:0) h x + ( h − ˆ g ) x (cid:1) ∂ + x ∂ − x i(cid:21) + · · · . (3.10)The wavefunction renormalization and field redefinition x = (cid:16) + ˆ g k ln µ (cid:17)(cid:18) + ˆ h ˆ x + ( h − ˆ g ) ˆ x k ln µ (cid:19) ˆ x , x = (cid:16) + ˆ g k ln µ (cid:17)(cid:18) + ˆ˜ h ˆ x + ( h − ˆ g ) ˆ x k ln µ (cid:19) ˆ x , (3.11)10uts the kinetic term into a canonical form and (3.10) becomes S eff = π Z d σ (cid:20) ∂ + ˆ x ∂ − ˆ x + ∂ + ˆ x ∂ − ˆ x + k (cid:16) ˆ g + ˆ g + h + h k ln µ (cid:17) ˆ x ˆ x (cid:0) ∂ + ˆ x ∂ − ˆ x + ∂ + ˆ x ∂ − ˆ x (cid:1)(cid:21) + · · · . (3.12)Demanding now that the action (3.10) is µ -independent, i.e. d L eff dln µ =
0, and keepingin mind that we should keep for consistency the leading term in the / k expansion weobtain d ˆ g d ln µ = − k (cid:0) ˆ g + h + h (cid:1) . (3.13)Using the specific expression for the couplings (2.17) we finally get that β λ = d λ d ln µ = − λ k . (3.14)We have, thus, reproduced the result of [14]. The objects naturally arising in the SU ( ) / U ( ) coset CFT are the parafermions [27].These will be denoted by Ψ ± and ¯ Ψ ± for the holomorphic and anti-holomorphic sec-tors, respectively. In this section, we will calculate correlation functions, namely twoand four-point functions, for the λ -deformed versions of the CFT parafermions. Atthe CFT point all correlation functions with an odd number of parafermions vanish.We will show that the same statement is true for the deformed versions of the corre-lators, as well. The first question we address in subsection 4.1 is how the deformationdresses the chirally conserved SU ( ) / U ( ) CFT parafermions which have a classi-cal form in terms of the fields θ and φ [30] . Equipped with their correct form weproceed to calculate in subsection 4.2 the anomalous dimension of the bilinear in theparafermions operator that perturbs the theory away from the free point. Our resultis in complete agreement with the one obtained earlier in [28] by the use of methodsinvolving the geometry in the coupling space. In the next two subsections, we calcu-late the anomalous dimension of the single λ -dressed parafermion and all four-pointcorrelation functions involving parafermionic fields. We will develop methods to deal11ith Wilson-line factors present in the expressions for the parafermion fields. All ourresults will respect the symmetries (2.8) of the action. λ -dressed parafermionic fields In the σ -model (2.1), derived in [1] throughout a certain gauging procedure, the clas-sical equations of motion for the gauge fields assume the following form A + = i ( λ − T − D ) − R + , A − = − i ( λ − T − D T ) − L − . (4.1)As discussed in [22,19], these λ -dependent fields are the counterparts of the chiral andanti-chiral currents of the conformal point to which they reduce, up to overall scales,after taking the limit λ →
0. Hence they provide the correct form of the operator gen-eralizing the chiral and anti-chiral currents in the presence of λ . The above form wasessential for obtaining the correct anomalous dimensions and correlation functions ofcurrents in the free field based approach of [26] and this will be the case here, as well.The aim of this work is to apply the approach of [26] to the case where the de-formed theory is not a deformation of a group based CFT but of a coset CFT, andmore specifically of the SU ( ) / U ( ) coset CFT. In this case the natural chiral ob-jects of the CFT are not currents but parafermions. As it happens for the currentsin the group case, the parafermions are dressed when the deformation parameter λ isturned on. The way the parafermions are dressed should be derived from the samelimiting procedure we used to obtain the action. In order to proceed, recall that wehave parametrized the group element of SU ( ) as in (2.5) and that we have chosenthe deformation matrix to take the form λ ab = diag ( λ , λ , λ ) , where λ correspondsto the Abelian subgroup U ( ) in SU ( ) . In order to obtain the σ -model action (2.7)one should set the parameter λ =
1. In this procedure the Euler angle φ appearingin the group element (2.5) drops out of the σ -model which instead of being three-dimensional becomes two-dimensional (2.7). We should follow the same limiting pro-cedure for the gauge fields (4.1) as well. In doing so, we firstly and most importantlyrealize that the gauge fields A a ± , a =
1, 2, 3 retain an explicit dependence on the an-gle φ which therefore at that level does not decouple. In fact, we will see that thisis a desired feature for the classical description of the parafermions even at the CFTpoint. In particular, we find that, after passing to the Euclidean regime, the gauge12elds projected, via the above limiting procedure, to the coset take the form A + = λ √ (cid:16) Ψ + − Ψ − (cid:17) , A + = − i λ √ (cid:16) Ψ + + Ψ − (cid:17) , A − = − λ √ (cid:16) ¯ Ψ + − ¯ Ψ − (cid:17) , A − = i λ √ (cid:16) ¯ Ψ + + ¯ Ψ − (cid:17) , (4.2)where the Ψ ’s, as we will see, will be the λ -deformed parafermions. They are givenin terms of the angles parametrizing the group element g , as follows after the analyticcontinuation to the Euclidean regime Ψ + = − λ (cid:16) ( e − i φ + λ e i φ ) ∂θ − i ( e − i φ − λ e i φ ) tan θ∂φ (cid:17) e − i φ , Ψ − = − λ (cid:16) ( e i φ + λ e − i φ ) ∂θ + i ( e i φ − λ e − i φ ) tan θ∂φ (cid:17) e i φ ,¯ Ψ + = − λ (cid:16) ( e i φ + λ e − i φ ) ¯ ∂θ + i ( e i φ − λ e − i φ ) tan θ ¯ ∂φ (cid:17) e − i φ ,¯ Ψ − = − λ (cid:16) ( e − i φ + λ e i φ ) ¯ ∂θ − i ( e − i φ − λ e i φ ) tan θ ¯ ∂φ (cid:17) e i φ . (4.3)The components of the gauge fields along the subgroup U ( ) turn out to be A + = i √ ( ∂φ + J ) , A − = i √ (cid:0) ¯ ∂φ − ¯ J (cid:1) , (4.4)where the angle φ has become imaginary after the analytic continuation to the Eu-clidean regime and J and ¯ J are given by J = (cid:0)(cid:0) − λ cos φ + λ (cid:1) tan θ ∂φ − λ sin 2 φ ∂θ (cid:1) tan θ − λ ,¯ J = (cid:0)(cid:0) − λ cos φ + λ (cid:1) tan θ ¯ ∂φ − λ sin 2 φ ¯ ∂θ (cid:1) tan θ − λ . (4.5)Under the non-perturbative symmetries (2.8) the Ψ ’s transform asI : Ψ ± → − λ e ∓ i φ Ψ ∓ , ¯ Ψ ± → − λ e ∓ i φ ¯ Ψ ∓ ,II : Ψ ± → ∓ i Ψ ± , ¯ Ψ ± → ± i ¯ Ψ ± . (4.6)13n addition, the gauge fields satisfy the following equations of motion [2] ∂ A g / h − = − [ A g / h − , A h + ] , ¯ ∂ A g / h + = − [ A g / h + , A h − ] , ∂ A h − − ¯ ∂ A h + = λ − [ A g / h + , A g / h − ] , A h ± = A ± t , A g / h ± = A α ± t α , α =
1, 2 . (4.7)In order to elucidate the expressions for the Ψ ’s we firstly consider the conformallimit λ =
0. Then, these become the standard classical parafermions of the coset SU ( ) / U ( ) CFT that are given by [30] Ψ ± = ( ∂θ ∓ i tan θ∂φ ) e ∓ i ( φ + φ ) , ¯ Ψ ± = (cid:0) ¯ ∂θ ± i tan θ ¯ ∂φ (cid:1) e ± i ( φ − φ ) . (4.8)Using (4.2) and (4.7), we find that the gauge fixing conditions A ± = Ψ ± and ¯ Ψ ± are on-shell chirally and anti-chirally conserved, respec-tively ¯ ∂ Ψ ± = ∂ ¯ Ψ ± = A ± = φ to satisfy the following equations ∂φ = − J , ¯ ∂φ = ¯ J , J = tan θ ∂φ , ¯ J = tan θ ¯ ∂φ . (4.10)Note that J and ¯ J can be obtained from (4.5) after setting λ =
0. Moreover, using(4.10) one can show that φ satisfies on-shell the compatibility condition (cid:0) ∂ ¯ ∂ − ¯ ∂∂ (cid:1) φ = ∂ ¯ J + ¯ ∂ J = U ( ) isometry of the SU ( ) / U ( ) cosetCFT. Given (4.11), one can solve (4.10) to express the angle φ as a line integral φ ( z , z ) = φ ( z ) − φ ( z ) = Z C (cid:0) ∂φ d z + ¯ ∂φ d ¯ z (cid:1) = Z C ( − J d z + ¯ J d ¯ z ) , (4.12) The second of (4.7) seems singular in the λ = A g / h ± ∼ λ ,so that λ − [ A g / h + , A g / h − ] ∼ λ , as λ → In a closed curve it would identically vanish due to Stokes theorem and (4.11) i I C ( − d z J + d ¯ z ¯ J ) = Z S d z (cid:0) ∂ ¯ J + ¯ ∂ J (cid:1) = C = ∂ S . C is a curve connecting the arbitrary base point ( z , ¯ z ) with the end point ( z , ¯ z ) . Note that, given the above, the phase φ is independent from the choice of C but depends only the base and end points. However, correlation functions should notdepend on the arbitrary base point. In the following sections, we will see that this isindeed the case.However, in the case of non-zero deformation λ = A ± = A + = − i √ F , A − = i √ F , (4.13)where F is at the moment an arbitrary function which will be later specified and ¯ F isits complex conjugate. Equivalently, using (4.4), we find ∂φ = − J F , ¯ ∂φ = ¯ J F , (4.14)with J F = J + F . The equations (4.14) uniquely determines φ on-shell provided thatthe following consistency condition is satisfied (cid:0) ∂ ¯ ∂ − ¯ ∂∂ (cid:1) φ = ∂ ¯ J F + ¯ ∂ J F = F and ¯ F should satisfy the relation ∂ ¯ F + ¯ ∂ F = λ − λ (cid:16) ( ∂θ ¯ ∂θ − ∂φ ¯ ∂φ tan θ ) sin 2 φ + ( ∂θ ¯ ∂φ + ∂φ ¯ ∂θ ) tan θ cos 2 φ (cid:17) . (4.16)Similarly to (4.12), we can solve (4.14) and (4.15) and express φ as a line integralthrough φ ( z , z ) = φ ( z ) − φ ( z ) = Z C ( − J F d z + ¯ J F d ¯ z ) . (4.17)In what follows, we shall expand the parafermions (4.3), as well as the phase (4.14),for k ≫ / k . This is a straightforward calculation whichcan be summarized in the following steps. Firstly, we zoom θ around zero as in (2.9).Then we express the variables ( ρ , φ ) in terms of ( y , y ) by using (2.10). Subsequently,we perform the field redefinition of (2.14) keeping terms up to order / k . Finally, werewrite all expressions in terms of chiral coordinates, namely x ± = x ± ix . The15nd result is given by the following expressions (cid:16) we dismiss an overall factor of √ k ( − λ ) (cid:17) Ψ + = (cid:16) ∂ x − + F + k ( − λ ) (cid:17) e − i φ , Ψ − = (cid:16) ∂ x + − F − k ( − λ ) (cid:17) e i φ ,¯ Ψ + = (cid:16) ¯ ∂ x + − ¯ F + k ( − λ ) (cid:17) e − i φ , ¯ Ψ − = (cid:16) ¯ ∂ x − + ¯ F − k ( − λ ) (cid:17) e i φ , (4.18)where F + = ( x + − x − ) (cid:16) ( + λ ) ∂ x + − λ∂ x − ) (cid:17) , F − = ( x + − x − ) (cid:16) ( + λ ) ∂ x − − λ∂ x + ) (cid:17) ,¯ F + = ( x + − x − ) (cid:16) ( + λ ) ¯ ∂ x − − λ ¯ ∂ x + ) (cid:17) ,¯ F − = ( x + − x − ) (cid:16) ( + λ ) ¯ ∂ x + − λ ¯ ∂ x − ) (cid:17) . (4.19)Furthermore, the currents (4.5) have the following large k expansion J = i k ( − λ ) (cid:16) (( + λ ) x + − λ x − ) ∂ x − − (( + λ ) x − − λ x + ) ∂ x + (cid:17) ,¯ J = i k ( − λ ) (cid:16) (( + λ ) x + − λ x − ) ¯ ∂ x − − (( + λ ) x − − λ x + ) ¯ ∂ x + (cid:17) . (4.20)Note that the condition (4.14) determines φ provided that F satisfies on-shell the con-dition ∂ ¯ F + ¯ ∂ F = − i λ k ( − λ ) ( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) . (4.21)Using the equation of motion (3.2), we find that to O ( / k ) (4.21) is solved by F = − i λ k ( − λ ) ( x + ∂ x + − x − ∂ x − ) , (4.22)Employing the latter and (4.20), we find that J F = i k + λ − λ ( x + ∂ x − − x − ∂ x + ) ,¯ J F = i k + λ − λ (cid:0) x + ¯ ∂ x − − x − ¯ ∂ x + (cid:1) , (4.23) To O ( / k ) the equations of motion (3.2) read ∂ ¯ ∂ x ± = φ through (4.17). Let us note that the parafermions in (4.18), withthe phase factor φ of (4.17) with (4.23), are not on-shell chirally and anti-chirally con-served for λ =
0, since¯ ∂ Ψ ± = ∓ i ∂ x ∓ ¯ F e ∓ i φ , ∂ ¯ Ψ ± = ± i ¯ ∂ x ± F e ∓ i φ . (4.24)We close this subsection by noticing that in this set of variables the two symmetries of(2.8) are mapped respectively toI : λ → λ − , k → − k ,II : λ → − λ , x ± → ± ix ± , (4.25)under which the parafermions (4.18) transform asI : Ψ ± → Ψ ± , ¯ Ψ ± → ¯ Ψ ± ,II : Ψ ± → ∓ i Ψ ± , ¯ Ψ ± → ± i ¯ Ψ ± , (4.26)while (4.20) and (4.23) remain intact. In the computations which follow we shall makeuse of (4.18), with the phase factor φ of (4.17), (4.23), keeping terms up to order O ( / k ) . The aim of this subsection is to calculate the anomalous dimension of the operatorthat perturbs the SU ( ) / U ( ) CFT. This operator is bilinear in the parafermions andits form is given by the classical parafermions bilinear [1] O = Ψ + ¯ Ψ − + Ψ − ¯ Ψ + , (4.27)where we note that the phase factors cancel, as one may readily verify using (4.8).Taking the large k -limit we obtain that the leading term of O is given by 2 ( ∂ x ¯ ∂ x − ∂ x ¯ ∂ x ) . Had we kept the leading correction, a bilinear in the parafermions operatorwould have been of the generic form (we ignore the irrelevant factor of 2) O = ∂ x ¯ ∂ x − ∂ x ¯ ∂ x + k ∑ i = c i O i + . . . , (4.28)17or some coefficients c i and a basis of operators with engineering dimension (
1, 1 ) O i . However, notice that the overlap of the leading and subleading term in (4.28) is zero.This means that one can safely ignore the subleading term in (4.28), at least up to O ( / k ) which is the order we are working in the present paper.In order to proceed, we will also need the interaction terms of the action (2.16) up toorder O ( / k ) . There is a single such term which in the Euclidean regime reads S int = ˆ g π k Z d z x x ( ∂ x ¯ ∂ x + ∂ x ¯ ∂ x )= − ˆ g π k Z d z ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) . (4.29)The two-point function then reads hO ( z , ¯ z ) O ( z , ¯ z ) i = | z | − ˆ g π k Z d z h (cid:0) ∂ x ( z ) ¯ ∂ x ( ¯ z ) − ∂ x ( z ) ¯ ∂ x ( ¯ z ) (cid:1) × (cid:0) ∂ x ( z ) ¯ ∂ x ( z ) − ∂ x ( z ) ¯ ∂ x ( ¯ z ) (cid:1)(cid:0) x x ( ∂ x ¯ ∂ x + ∂ x ¯ ∂ x ) (cid:1) ( z , ¯ z ) i = | z | + g π k ( I + I ) , (4.30)where the minus sign in the second term above, corresponding to a single insertion ofthe interaction term, is due to the fact that in the Euclidean regime we have the term e − S int in the correlators. By inspecting (4.30) it is straightforward to see that we needtwo kinds of contractions, namely I = Z d z h ∂ x ( z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) ih ∂ x ( z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) i = Z d z ( z − z ) ( z − z )( ¯ z − ¯ z )( ¯ z − ¯ z ) (4.31) Such a basis is O = x x ∂ x ¯ ∂ x , O = x x ∂ x ¯ ∂ x , O = x ∂ x ¯ ∂ x , O = x ∂ x ¯ ∂ x . I = Z d z h ∂ x ( z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) ih ∂ x ( z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) i = Z d z ( z − z )( z − z ) ( ¯ z − ¯ z ) ( ¯ z − ¯ z ) . (4.32)Using (B.4) twice we find that hO ( z , ¯ z ) O ( z , ¯ z ) i = | z | (cid:18) + g k (cid:18) + ln ε | z | (cid:19)(cid:19) , (4.33)from which we read the anomalous dimension of the parafermion bilinear to be γ O = g k = − k + λ − λ . (4.34)This expression is in perfect agreement with equation (4.16) of [28], where the anoma-lous dimension of the perturbing operator was found by using the geometry in thespace of couplings. Notice that in the conformal point the anomalous dimension ofthe composite operator is twice the anomalous dimension of the holomorphic (anti-holomorphic ) parafermion which equals − / k [27].The reader may wonder if there are other operators with the same engineeringdimension as O which may mix with it. The operators of such equal engineeringdimension are˜ O = ∂ x ¯ ∂ x + ∂ x ¯ ∂ x + O ( / k ) , ˜ O ± = ∂ x ¯ ∂ x ± ∂ x ¯ ∂ x + O ( / k ) , (4.35)where the corrections are of same form with those for O in (4.28). It turns out that theseoperator does not mix with O and among themselves at the free field point and also at O ( / k ) . Moreover, one may easily show that ˜ O has the opposite anomalous dimensionas that in (4.34). The anomalous dimension of ˜ O ± is computed for completeness inappendix A. The goal of this subsection is to compute, using the free field expansion, the two-pointfunctions of the parafermions to order / k from which one can read the anomalous19imension of the deformed parafermion.We first consider the correlator h Ψ + Ψ + i . This vanishes at the conformal point since itis not neutral. Employing the expressions above we find that h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) i =+ ˆ g π k Z d z h ∂ x − ( z ) ∂ x − ( z ) (cid:0) ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) (cid:1) ( z , ¯ z ) i + k ( − λ ) ( h F + ( z , ¯ z ) ∂ x − ( z ) i + h ∂ x − ( z ) F + ( z , ¯ z ) i ) − i h ∂ x − ( z ) ∂ x − ( z ) φ ( z , z ) i − i h ∂ x − ( z ) ∂ x − ( z ) φ ( z , z ) i = φ we will use the leading O ( / k ) of (4.17) with (4.23) so thatthe corresponding terms in (4.36) are indeed of O ( / k ) . The second of (4.36) involvesthe interaction term of the action and vanishes due to the fact that the interactionterm is normal ordered. The third and fourth line of (4.36) originate from the O ( / k ) corrections to the parafermion operators and from the phase φ , respectively. Theyboth vanish either because in the process one necessarily encounters propagators ofthe form h x + x + i or h x − x − i which are identically zero (terms in the fourth line) orbecause of normal ordering (terms in the third line).We now turn to the neutral two point function h Ψ + Ψ − i . To O ( / k ) this correlatorequals to h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) i = − z + k ( − λ ) ( h F + ( z , ¯ z ) ∂ x + ( z ) i − h ∂ x − ( z ) F − ( z , ¯ z ) i )+ ˆ g π k Z d z h ∂ x − ( z ) ∂ x + ( z ) (cid:0) ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) (cid:1) ( z , ¯ z ) i− i h ∂ x − ( z ) ∂ x + ( z ) φ ( z , z ) i + i h ∂ x − ( z ) ∂ x + ( z ) φ ( z , z ) i . (4.37)In the above expression, apart from the free part, only the last two terms related to thephase factor in the parafermions are non-vanishing and equal to z k + λ − λ (cid:18) − Z z z d z + Z z z d z (cid:19) ( z − z ) ( z − z ) = z k + λ − λ (cid:18) + ln ε z (cid:19) + k + λ − λ ε z . (4.38)20ll other terms in (4.38) vanish due to the fact that both the interaction term and F ± arenormal ordered. In evaluating the line integrals of (4.38) we have introduced a smalldistance cut-off ε so that the integration point never coincides with the end points ofthe integration. Therefore, we find that h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) i = − z − k + λ − λ (cid:16) + ln ε z (cid:17)! + ε k + λ − λ z . (4.39)The / ε pole can be absorbed by a field redefinition of the Ψ ± ’s, namely Ψ ± → Ψ ± + δ Ψ ± , where δ Ψ ± = − f ± ε k x ∓ , f + − f − = + λ − λ . (4.40)Note that we do not need to specify both parameters f ± , just their difference as aboveand in addition this relation is invariant under the symmetries (4.26). At λ =
0, thecorrelator (4.38) is in agreement with the CFT result to leading order in / k after onemakes the the following rescaling Ψ = i √ (cid:18) − k (cid:19) Ψ CFT . (4.41)Finally, from (4.39) one can read the anomalous dimension of Ψ which is given by γ Ψ = − k + λ − λ . (4.42)Notice that γ Ψ is half the anomalous dimension of the composite operator (4.34). Atthe conformal point it is indeed, the deviation from unity of the holomorphic dimen-sion (equal to the anti-holomorphic one) of the parafermion which, as already noted,equals 1 − / k . For completeness, we note that the conjugate correlator h ¯ Ψ + ¯ Ψ − i takesthe form (4.39) as well. In this section, we calculate four-point correlators of parafermions. To begin with,note that at the CFT point correlation functions with an odd number of parafermionsvanish. In the following we will argue that the same is true for the deformed modelas well. Indeed, to O ( / k ) the number of fields involved in the correlator is odd sincethe phase may contribute two and the insertion from the interacting part of the action21our fields. Any correlator with an odd number of free fields vanishes. This argu-ment should hold to all orders in the large k expansion since every extra power of / k contributes two more fields in the correlators.At the CFT point correlators with odd number of holomorphic (or antiholomorphic)parafermions also vanish identically due to charge conservation. However, CFT pointcorrelators with an even number of holomorphic (or antiholomorphic) parafermionsand neutral do not vanish. Away from the CFT point correlators with an even numberof fields receive corrections. Using conformal perturbation one can easily convinceoneself that the corrections come to even order in the λ expansion. h Ψ + Ψ − Ψ + Ψ − i Consider, as our first example, the four-point function h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) i = z z + z z + k ( − λ ) ( h F + ( z , ¯ z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i + h F + ( z , ¯ z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i−h F − ( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x + ( z ) i − h F − ( z , ¯ z ) ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) i ) (4.43) − i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i − i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) i + ˆ g π k Z d z h ∂ x − ( z ) ∂ x + ( z ) ∂ x − ( z ) ∂ x + ( z ) (cid:0) ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) (cid:1) ( z , ¯ z ) i .To calculate this is a rather tedious but straightforward task. To start, let us brieflydescribe the various contributions. The second and third line of (4.43), originate fromthe / k corrections of Ψ ’s (4.19) and involve only free contractions, straightforwardlyyielding 2 k + λ − λ z z z + z z z − z z z − z z z ! . (4.44)22he fourth and fifth line, which originate from the phase φ , give after performingordinary integrations over z the following contribution − k + λ − λ z z + z z ! + k + λ − λ z z z z + k + λ − λ z z + z z ! ln z z z z z z . (4.45)Finally, from the last line we find2 ˆ g k z z z + z z z − z z z − z z z ! . (4.46)This result is easily obtained since the integration over z is immediately performedsince the free contractions necessarily give contact terms of the form δ ( ) ( z − z i ) .Adding the various contributions we finally find h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) i = − k + λ − λ z z + − k + λ − λ z z + k + λ − λ z z z z + k + λ − λ z z + z z ! ln z z z z z z . (4.47)At λ =
0, this is in agreement with the CFT result [27] after the rescaling (4.41) and ofcourse in the large k expansion up to O ( / k ) . Furthermore, this result is in agreementwith perturbation which predicts that corrections to the correlator under consider-ation should start at order λ . This is so because the contribution linear in λ willinvolve a single anti-holomorphic parafermion and will, thus, be vanishing. Lastly,we would like to comment on the efficiency of the method initiated in [26] and fur-ther developed here. Our method provides the exact in the deformation parameter λ correlators in contradistinction to the usual conformal perturbation theory which onlygives results order by order in the λ expansion. The power of our approach resides onthe fact that the exact in λ effective action (see (2.7)) is at hand.23 .4.2 Correlation function h Ψ + Ψ + ¯ Ψ − ¯ Ψ − i Our second four-point correlation function has the following large k expansion h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) i = k ( − λ ) (cid:0) h F + ( z , ¯ z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + h F + ( z , ¯ z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + h ¯ F − ( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) i + h ¯ F − ( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) i (cid:1) (4.48) − i h φ ( z , z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i − i h φ ( z , z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + ˆ g π k Z d z h ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) (cid:0) ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) (cid:1) ( z , ¯ z ) i .A non-vanishing correlator requires equal number of x + and x − ’s. Clearly, only thelast line may contribute, giving h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) i = ˆ g π k Z d z h ∂ x − ( z ) ∂ x − ( z ) ¯ ∂ x − ( ¯ z ) ¯ ∂ x − ( ¯ z )( x + ∂ x + ¯ ∂ x + )( z , ¯ z ) i = ˆ g π k Z d z (cid:18) ( z − z )( z − z ) ( ¯ z − ¯ z )( ¯ z − ¯ z ) + ( z − z ) ( z − z )( ¯ z − ¯ z )( ¯ z − ¯ z ) + ( z − z )( z − z ) ( ¯ z − ¯ z ) ( ¯ z − ¯ z ) + ( z − z ) ( z − z )( ¯ z − ¯ z ) ( ¯ z − ¯ z ) (cid:19) . (4.49)To evaluate the above integrals, we use (B.3) four times with appropriate renaming ofthe z i ’s. Adding them up we find a vanishing result h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) i = .4.3 Correlation function h Ψ + Ψ − ¯ Ψ + ¯ Ψ − i The third four-point corellator that we will consider is the following h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) ¯ Ψ + ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) i = z ¯ z + k ( − λ ) (cid:0) h F + ( z , ¯ z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i − h F − ( z , ¯ z ) ∂ x − ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i−h ¯ F + ( z , ¯ z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x − ( ¯ z ) i + h ¯ F − ( z , ¯ z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) i (cid:1) (4.51) − i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i− i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + i h φ ( z , z ) ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) i + ˆ g π k Z d z h ∂ x − ( z ) ∂ x + ( z ) ¯ ∂ x + ( ¯ z ) ¯ ∂ x − ( ¯ z ) (cid:0) ( x + − x − )( ∂ x + ¯ ∂ x + − ∂ x − ¯ ∂ x − ) (cid:1) ( z , ¯ z ) i .To evaluate (4.51) is a rather cumbersome but straightforward computation. Let usbriefly describe the various contributions. The second and third line, related to the / k corrections of the parafermions Ψ ’s (4.19), trivially vanish once we ignore contactterms involving only external points. The fourth and fifth lines, that are related to thephase φ of Ψ ’s, yield after performing ordinary integrations the following result1 k + λ − λ z ¯ z − − ln ε z − ln ε ¯ z + ln | z | | z | | z | | z | + (cid:18) z z z z + z z z z + ¯ z ¯ z ¯ z ¯ z + ¯ z ¯ z ¯ z ¯ z (cid:19)(cid:19) . (4.52)Finally, the last line of (4.51) originating from the interaction insertion equals to4 k ˆ g z ¯ z (cid:18) ln | z | | z | | z | | z | + (cid:18) z z z z + z z z z + ¯ z ¯ z ¯ z ¯ z + ¯ z ¯ z ¯ z ¯ z (cid:19)(cid:19) . (4.53)In order to obtain the last result, we have used twice (B.3) with an appropriate renam-ing of the z i ’s. 25dding all the above contributions we find that h Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) ¯ Ψ + ( z , ¯ z ) ¯ Ψ − ( z , ¯ z ) i = z ¯ z − k + λ − λ − k + λ − λ (cid:18) ln ε z + ln ε ¯ z (cid:19)! . (4.54)At λ =
0, this result is under the rescaling (4.41) in agreement with the CFT resultwhich enforces factorization of the four-point function into two-point functions holo-morphic and anti-holomorphic, respectively. h Ψ + Ψ + Ψ + Ψ − i The only contribution for this correlator comes from the free contractions involvingthe / k terms in the expression for the parafermions Ψ ’s (4.18) and (4.19), yieldingfinally h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ − ( z , ¯ z ) i = λ k ( − λ ) z z z − z z z + z z z − z z z − z z z − z z z ! = / k expan-sion since the correlator at hand is not a neutral one.26 .4.5 Correlation function h Ψ + Ψ + Ψ + Ψ + i There are two contributions for this correlator. The first one comes from the / k termsof the Ψ ’s (4.18) and (4.19), it involves only free contractions and reads1 + λ k ( − λ ) (cid:16) h ( x + ∂ x + )( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z ) i + h ( x + ∂ x + )( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z ) i + h ( x + ∂ x + )( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z ) i + h ( x + ∂ x + )( z , ¯ z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z ) i (cid:17) . (4.56)The second contribution comes from the interaction insertion and is given byˆ g π k Z d z h ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z ) ∂ x − ( z )( x + ∂ x + ¯ ∂ x + )( z , ¯ z ) i . (4.57)To evaluate this integral we contract ¯ ∂ x + with the ( ∂ x − ) i ’s yielding contact terms pro-portional to δ ( ) ( z − z i ) . Evaluating then the integral we obtain an expression with freecontractions among external point only, which is, after substituting ˆ g from (2.17), isprecisely the opposite of that in (4.56). Combining these two contributions we find thevanishing result h Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) Ψ + ( z , ¯ z ) i = / k expansion.A couple of important comments are in order. The first one concerns the / ε polesin the calculation of the four-point functions. These infinite contributions have beensuppressed in the presentation because on can absorb them by employing preciselythe same redefinition (4.40) used in the two-point function (4.37). The second com-ment concerns the form of the two and four-point correlators. By just inspecting theresults one can see that they behave as if the theory was conformal albeit with a re-defined level ˜ k = k − λ + λ . In particular, this implies that the λ -dressed parafermions ofthe deformed theory will satisfy the same Poisson bracket algebra as the one satisfiedby the parafermions at the conformal point but with ˜ k in the place of k .27 Discussion and future directions
We studied the λ -deformed SU ( ) / U ( ) σ -model by using a free field expansion in-stead of conformal perturbation and the underlying parafermionic algebras. Expand-ing along the lines of [26] the perturbation is organized as a series expansion for largevalues of k . This approach has the advantage that all deformation effects are fullyencoded in the coupling constant coefficients and in the form of the operators. Thisprescription allowed for the computation the RG flow of the deformation parame-ter λ and the anomalous dimension of the parafermion bilinear perturbation, drivingthe theory away from the conformal point, as exact functions of λ and up to O ( / k ) .Then, we introduced the corresponding λ -dressed parafermions containing a non-local phase in their expressions – which ensures on-shell chirality at the conformalpoint. Subsequently, we evaluate their anomalous dimensions and their four-pointfunctions, odd-point correlation functions identically vanish, again to all-orders in λ and up to O ( / k ) . The derived results are in agreement with CFT expectations at λ = SU ( ) / U ( ) is the simplest possible symmetric coset space. It should be possibleto study other λ -deformed coset CFT based on symmetric spaces using free fields asa basis. Furthermore, it was argued in [28] that the λ -deformed SU ( ) × SU ( ) / U ( ) and SU ( ) / U ( ) CFTs are closely related since they share the same symmetries and β -function to all-orders in λ and k . It would be interesting to check and understandthis relation further using the techniques of the present work.Another interesting direction is to extend the current set for deformations of non-symmetric coset CFTs. A class of such integrable models is the λ -deformed G k × G k / G k + k coset CFTs, which, assuming that k > k , flows in the infrared to the G k × G k − k / G k coset CFT [6]. These models have a richer parameter space and asa result a very interesting generalization of the duality-type symmetries (2.8) (see eq.(2.17) in that work) which dictates their behavior and which reduces to the first onein (2.8) for equal levels. In addition, in these models the perturbation is driven bynon-Abelian parafermion bilinears [31] which are much less understood than theirAbelian counterparts [30] we utilized in the present work. We believe that our freefield techniques will be instrumental in understanding these type of non-symmetriccoset CFTs. 28he present setup may be applied in the study of the two-parameter integrabledeformations of symmetric coset spaces constructed in [7]. Such type of deforma-tions are applicable to a restricted class of symmetric spaces provided that they sat-isfy a certain gauge invariance condition involving a non-trivial solution of the Yang–Baxter equation (see Eq.(5.19) in that work). Examples include the SU ( ) / U ( ) [7], the SO ( N + ) / SO ( N ) [32] and the recently studied C P n models [33]. In the aforemen-tioned examples, this second parameter can be set to zero via parameter redefinitionsand/or diffeomorphisms. Such redefinitions come with caveats due to the topologi-cal nature of k and the non-trivial monodromies that the associated parafermions willpresumably have. Therefore, a proper study of this class of models, in the presencecontext in terms of free fields, should include both deformation parameters. Acknowledgments
The work of G. Georgiou and K. Siampos on this project has received funding fromthe Hellenic Foundation for Research and Innovation (H.F.R.I.) and the General Sec-retariat for Research and Technology (G.S.R.T.), under grant agreement No 15425.The research work of K. Sfetsos was supported by the Hellenic Foundation for Re-search and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects tosupport Faculty members and Researchers and the procurement of high-cost researchequipment grant” (Project Number: 16519).
A The anomalous dimension of ˜ O ± In this appendix we compute the anomalous dimension of the operator ˜ O ± definedin (4.35). The operator ˜ O + arises, in the large k -limit, if we had we considered a per-turbation of the form (4.27) but with a negative relevant sign. At the CFT point thisshould have anomalous dimension − / k since it corresponds to just the difference oftwo parafermion bilinears. 29onsider the two-point function h ˜ O c ( z , ¯ z ) ˜ O c ( z , ¯ z ) i = + c | z | − ˆ g π k Z d z h (cid:0) ∂ x ( z ) ¯ ∂ x ( ¯ z ) + c ∂ x ( z ) ¯ ∂ x ( ¯ z ) (cid:1) × ( ∂ x ( z ) ¯ ∂ x ( ¯ z ) + c ∂ x ( z ) ¯ ∂ x ( ¯ z )) (cid:0) x x ( ∂ x ¯ ∂ x + ∂ x ¯ ∂ x ) (cid:1) ( z , ¯ z ) i = + c | z | − ˆ g π k ∑ i , j , k = c i + j − I ijk , (A.1)where for convenience we have slightly generalize ˜ O ± to ˜ O c ( z , ¯ z ) = ∂ x ( z ) ¯ ∂ x ( ¯ z ) + c ∂ x ( z ) ¯ ∂ x ( ¯ z ) by introducing an arbitrary relative constant c . In the various integrals I ijk the index structure implies that it arises from the i -th, the j -th and the k -th term inthe first, second and third factors in the integrand in (A.1). They are given by I = Z d z h ∂ x ( z ) ∂ x ( z ) ∂ x ( z ) x ( z , ¯ z ) ih ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) i = Z d z (cid:12)(cid:12)(cid:12)(cid:12) ( z − z ) ( z − z ) + ( z − z ) ( z − z ) (cid:12)(cid:12)(cid:12)(cid:12) . (A.2)To evaluate the integral we first expand the absolute value I = Z d z | z − z | | z − z | + Z d z | z − z | | z − z | (A.3) + Z d z ( z − z ) ( z − z )( ¯ z − ¯ z )( ¯ z − ¯ z ) + Z d z ( z − z ) ( z − z )( ¯ z − ¯ z )( ¯ z − ¯ z ) .Then we use twice (B.5) and (B.4) for the first and second line respectively, yielding I = − π | z | (cid:18) + ln ε | z | (cid:19) + π | z | (cid:18) + ln ε | z | (cid:19) = O ± as explainedbelow (B.5). It is easily seen that adding to ˜ O ± in (4.35) a term proportional to ε k x x does the job without giving rise to a mixing with the other operators of engineeringdimension two, i.e. with O and ˜ O . 30n addition, we have that I = Z d z h ∂ x ( z ) ∂ x ( z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) ih ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) i , (A.5)has no contribution since it gives rise to contact terms on external points. In addition, I = Z d z h ¯ ∂ x ( ¯ z ) ∂ x ( z ) ∂ x ( z ) x ( z , ¯ z ) ih ∂ x ( z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) i = Z d z | z − z | | z − z | , (A.6)and I = Z d z h ¯ ∂ x ( ¯ z ) ∂ x ( z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) ih ∂ x ( z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) i = Z d z | z − z | | z − z | , (A.7)Adding the last two integrals and using twice (B.5) we find I + I = − π | z | (cid:16) + ln ε | z | (cid:17) , (A.8)where the divergent pieces have been dismissed (see discussion below (A.4)). Simi-larly, I = Z d z h ∂ x ( z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) ih ¯ ∂ x ( ¯ z ) ∂ x ( z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) i = I (A.9)and I = Z d z h ∂ x ( z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) ih ¯ ∂ x ( ¯ z ) ∂ x ( z ) ∂ x ( z ) x ( z , ¯ z ) i = I . (A.10)Finally, I = Z d z h ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) ∂ x ( z ) x ( z , ¯ z ) ih ∂ x ( z ) ∂ x ( z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) i , = I (A.11)31nd I = Z d z h ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) ¯ ∂ x ( ¯ z ) x ( z , ¯ z ) ih ∂ x ( z ) ∂ x ( z ) ∂ x ( z ) x ( z , ¯ z ) i = I . (A.12)Putting all together in (A.1) and using (A.8), we find: h ˜ O c ( z , ¯ z ) ˜ O c ( z , ¯ z ) i = + c | z | (cid:18) + c + c ˆ g k (cid:16) + ln ε | z | (cid:17)(cid:19) . (A.13)Hence, the anomalous dimension of ˜ O ± reads γ ˜ O ± = ± g k = ∓ k + λ − λ . (A.14)Hence, the operator ˜ O + has an anomalous dimension that matches with (4.34) asexpected and now has been explicitly verified. The operator ˜ O − corresponds to amarginally irrelevant operator. B Various integrals
In this appendix we shall provide the various (single) integrals which appear in thepresent work. Our regularization scheme is as follows: Internal points in integralscan coincide with external ones but coincident external points are not allowed. Inaddition, to regularize infinities we introduce a very small disc of radius ε aroundexternal points as well.We shall need the basic integrals Z d z ( z − z )( ¯ z − ¯ z ) = π ln | z | , Z d z ( z − z )( ¯ z − ¯ z ) = π ln ε , Z d z ( z − z ) ( ¯ z − ¯ z ) = π z , Z d z ( z − z )( ¯ z − ¯ z ) = π ¯ z , (B.1)along with the identity 1 ( z − z )( z − z ) = z (cid:18) z − z − z − z (cid:19) . (B.2)32sing the above regularization scheme and (B.1), (B.2) we find that Z d z ( z − z )( z − z ) ( ¯ z − ¯ z )( ¯ z − ¯ z ) = − π z ¯ z (cid:18) ln | z | | z | | z | | z | + z z z z + ¯ z ¯ z ¯ z ¯ z (cid:19) .(B.3) Z d z ( z − z ) ( z − z )( ¯ z − ¯ z )( ¯ z − ¯ z ) = π | z | (cid:18) + ln ε | z | (cid:19) + π | z | Z d z ¯ z − ( z − z ) ( ¯ z − ¯ z ) − z − ( z − z )( ¯ z − ¯ z ) ! = π | z | (cid:18) + ln ε | z | (cid:19) ,(B.4)since it can be easily seen that in our regularization scheme the integral in the last lineof (B.4) is zero. For instance, R d z ( z − z )( ¯ z − ¯ z ) = i H C ε d z | z − z | , where C ε is a small circleof radius ε surrounding z = z . Then, the last integral indeed vanishes. In addition,using (B.1), (B.2) we find Z d z | z − z | | z − z | = − π | z | (cid:18) + ln ε | z | (cid:19) + | z | Z d z | z − z | . (B.5)Finally, the integral in the last line of (B.5) equals π / ε and is clearly divergent as ε → O ± in appendix A. In order to absorb this divergent piece the operators require aredefinition similar to that for the single parafermion in subsection 4.3. With this inmind we may safely ignore this term all together and set it to zero.Finally, using (B.1), (B.2) we find Z d z | z − z | | z − z | = − π | z | ln ε | z | . (B.6) References [1] K. Sfetsos,
Integrable interpolations: From exact CFTs to non-Abelian T-duals ,Nucl. Phys.
B880 (2014) 225, arXiv:1312.4560 [hep-th].[2] T. J. Hollowood, J. L. Miramontes and D.M. Schmidtt,
Integrable Deformations ofStrings on Symmetric Spaces , JHEP (2014) 009, arXiv:1407.2840 [hep-th].[3] T. J. Hollowood, J. L. Miramontes and D. Schmidtt,
An Integrable Deformation of theAdS × S Superstring , J. Phys.
A47 (2014) 49, 495402, arXiv:1409.1538 [hep-th].334] G. Georgiou and K. Sfetsos,
A new class of integrable deformations of CFTs ,JHEP (2017) 083, arXiv:1612.05012 [hep-th].[5] G. Georgiou and K. Sfetsos,
Integrable flows between exact CFTs ,JHEP (2017) 078, arXiv:1707.05149 [hep-th].[6] K. Sfetsos and K. Siampos,
Integrable deformations of the G k × G k / G k + k coset CFTs ,Nucl. Phys. B927 (2018) 124, arXiv:1710.02515 [hep-th].[7] K. Sfetsos, K. Siampos and D. C. Thompson,
Generalised integrable λ - and η -deformations and their relation ,Nucl. Phys. B899 (2015) 489, arXiv:1506.05784 [hep-th].[8] G. Georgiou and K. Sfetsos,
Novel all loop actions of interacting CFTs: Construction,integrability and RG flows , Nucl. Phys.
B937 (2018) 371, arXiv:1809.03522 [hep-th]].[9] G. Georgiou and K. Sfetsos,
The most general λ -deformation of CFTs and integrability ,JHEP (2019) 094, arXiv:1812.04033 [hep-th].[10] S. Driezen, A. Sevrin and D. C. Thompson, Integrable asymmetric λ -deformations ,JHEP , 094 (2019), arXiv:1902.04142 [hep-th].[11] D. Kutasov, String Theory and the Nonabelian Thirring Model ,Phys. Lett.
B227 (1989) 68.[12] B. Gerganov, A. LeClair and M. Moriconi,
On the beta function for anisotropic cur-rent interactions in 2-D , Phys. Rev. Lett. (2001) 4753, hep-th/0011189.[13] D. Kutasov, Duality Off the Critical Point in Two-dimensional Systems With Non-abelian Symmetries , Phys. Lett.
B233 (1989) 369.[14] G. Itsios, K. Sfetsos and K. Siampos,
The all-loop non-Abelian Thirring model and itsRG flow , Phys. Lett.
B733 (2014) 265, arXiv:1404.3748 [hep-th].[15] K. Sfetsos and K. Siampos,
Gauged WZW-type theories and the all-loop anisotropicnon-Abelian Thirring model , Nucl. Phys.
B885 (2014) 583, arXiv:1405.7803 [hep-th].[16] C. Appadu and T.J. Hollowood,
Beta function of k deformed AdS × S string theory ,JHEP (2015) 095, arXiv:1507.05420 [hep-th].3417] E. Sagkrioti, K. Sfetsos and K. Siampos, RG flows for λ -deformed CFTs ,Nucl. Phys. B930 (2018) 499, arXiv:1801.10174 [hep-th].[18] G. Georgiou, K. Sfetsos and K. Siampos,
All-loop anomalous dimensions in integrable λ -deformed σ -models , Nucl. Phys. B901 (2015) 40, arXiv:1509.02946 [hep-th].[19] G. Georgiou, P. Panopoulos, E. Sagkrioti and K. Sfetsos,
Exact results from the ge-ometry of couplings and the effective action ,Nucl. Phys.
B948 (2019), 114779, arXiv:1906.00984 [hep-th].[20] G. Georgiou, P. Panopoulos, E. Sagkrioti, K. Sfetsos and K. Siampos,
The exact C-function in integrable λ -deformed theories ,Phys. Lett. B782 (2018), 613, arXiv:1805.03731 [hep-th].[21] E. Sagkrioti, K. Sfetsos and K. Siampos,
Weyl anomaly and the C-function in λ -deformed CFTs , Nucl. Phys. B938 (2019) 426, arXiv:1810.04189 [hep-th].[22] G. Georgiou, K. Sfetsos and K. Siampos,
All-loop correlators of integrable λ -deformed σ -models , Nucl. Phys. B909 (2016) 360, 1604.08212 [hep-th].[23] G. Georgiou, K. Sfetsos and K. Siampos, λ -deformations of left-right asymmetricCFTs , Nucl. Phys. B914 (2017) 623, arXiv:1610.05314 [hep-th].[24] G. Georgiou, K. Sfetsos and K. Siampos,
Double and cyclic λ -deformations and theircanonical equivalents , Phys. Lett. B771 (2017) 576, arXiv:1704.07834 [hep-th].[25] G. Georgiou, E. Sagkrioti, K. Sfetsos and K. Siampos,
Quantum aspects of doublydeformed CFTs , Nucl. Phys.
B919 (2017) 504, arXiv:1703.00462 [hep-th].[26] G. Georgiou and K. Sfetsos,
Field theory and λ -deformations: Expanding around theidentity , Nucl. Phys. B950 (2020) 114855, arXiv:1910.01056 [hep-th].[27] V.A. Fateev and A.B. Zamolodchikov,
Parafermionic Currents in the Two-Dimensional Conformal Quantum Field Theory and Selfdual Critical Points in Z(n) In-variant Statistical Systems ,Sov. Phys. JETP (1985) 215, [Zh. Eksp. Teor. Fiz. (1985) 380].[28] G. Georgiou, E. Sagkrioti, K. Sfetsos and K. Siampos, An exact symmetry in λ -deformed CFTs , JHEP (2020) 083, arXiv:1911.02027 [hep-th].3529] P. Bowcock, Canonical Quantization of the Gauged Wess–Zumino Model ,Nucl. Phys.
B316 (1989) 80.[30] K. Bardakci, M. J. Crescimanno and E. Rabinovici,
Parafermions From Coset Models ,Nucl. Phys.
B344 (1990) 344.[31] K. Bardakci, M. J. Crescimanno and S. Hotes,
Parafermions from nonabelian cosetmodels , Nucl. Phys.
B349 (1991) 439.[32] O. Lunin and W. Tian,
Analytical structure of the generalized λ -deformation ,Nucl. Phys. B929 (2018), 330, arXiv:1711.02735 [hep-th].[33] S. Demulder, F. Hassler, G. Piccinini and D. C. Thompson,
Integrable deformationof CP n and generalised Kähler geometryand generalised Kähler geometry