Correlators of massive string states with conserved currents
aa r X i v : . [ h e p - t h ] F e b Correlators of massive string states with conservedcurrents
George Georgiou a , Bum-Hoon Lee b and Chanyong Park ba Demokritos National Research Center, Institute of Nuclear and Particle Physics, Ag.Paraskevi, GR-15310 Athens, Greece, b Center for Quantum Spacetime (CQUeST), Sogang University, Seoul 121-742, Korea [email protected], [email protected], [email protected]
Abstract
We calculate correlation functions of the R -current or the stress-energy tensor T µν with two non-protected operators dual to generic massive string states withrotation in S , in the context of the AdS/CFT correspondence. Field theory Wardidentities make predictions about the all-loop behaviour of these correlators. Inparticular, they restrict the fusion coefficient to be proportional to the R-charge ofthe operators or to their dimension, respectively, with certain coefficients of propor-tionality. We reproduce these predictions, at strong coupling, using string theory.Furthermore, we point out that the recently observed strong coupling factorisationof 4-point correlators is consistent with conformal symmetry and puts constraintson the strong coupling expressions of 4-point correlators involving R-currents or thestress-energy tensor. Introduction
The AdS/CFT correspondence [1] claims a duality between N = 4 Super Yang-Mills(SYM) theory and type-IIB superstring theory on AdS × S background. Remarkably,both theories appear to be integrable, at least at the planar level. The presence ofintegrability allows oneself to hope that one day both theories will be ”solved” and theAdS/CFT correspondence will be proven. On the field theory side this requires two piecesof information. One needs to be able to identify the conformal dimensions of all compositeoperators of the theory, as well as the structure constants that determine the OperatorProduct Expansion (OPE) between two primary operators. Recently, significant progresshas been made in the computation of the planar conformal dimensions of non-protectedoperators for any value of the coupling constant, employing integrability (for a recentreview see [2]). On the other hand, much less is known about the structure constants.Our current knowledge of the structure constants comes from a perturbative expansioneither around λ = 0, or around λ = ∞ where the IIB string theory is approximated bya simpler description. Comparison of the 3-point correlators among half-BPS operatorsin these two different limits led the authors of [3] to conjectured that the correspondingstructure constants are non-renormalised. On the contrary the 3-point correlators amongnon-protected operators receive quantum corrections. On the gauge theory side, the au-thors of [4–7] studied systematically the structure constants and computed the correctionsarising from the planar 1-loop Feynman diagrams. In order to evaluate the correction tothe structure constants which is of order λ one has to take into account intricate mixingeffects related to the fact that the 2-loop eigenstates of the dilatation operator are needed.The importance of this operator mixing was stressed in [8, 5, 9]. Resolution of the mixingmade possible a systematic one-loop study of three-point functions involving single traceconformal primary operators up to length five [6]. On the string theory side it is moredifficult to extract information about non-protected OPE coefficients. This is so because,in the supergravity limit, all non-protected operators acquire large conformal dimensionand decouple. However, one can obtain information about non-BPS structure constantsby studying the BMN limit of type-IIB string theory [10].Recently, another approach to the calculation of n-points correlators involving non-BPS states was developed [11–15]. More precisely, the authors of [12] argued that itshould be possible to obtain the correlation functions of local operators corresponding toclassical spinning string states, at strong coupling, by evaluating the string action on aclassical solution with appropriate boundary conditions after convoluting with the relevantto the classical states wavefunctions. In [13, 16], 2-point and 3-point correlators of vertexoperators representing classical string states with large spins were calculated. Finally, ina series of papers [14, 15, 17] the 3-point function coefficients of a correlator involving a1assive string state, its conjugate and a third ”light” state state were computed. This wasdone by taking advantage of the known classical solutions corresponding to the 2-pointcorrelators of operators dual to massive string states.More recently, an intriguing weak/strong coupling match of correlators involving op-erators in the SU (3) sector was observed [18] . This match was found to hold forcorrelators of two non-protected operators in the Frolov-Tseytlin limit and one short BPSoperator . In [23], this weak/strong coupling match was extended to correlators involvingoperators in the SL (2 , R ) closed subsector of N = 4 SYM theory . Furthermore, by per-forming Pohlmeyer reduction for classical solutions living in AdS but with a prescribednonzero energy-momentum tensor the authors of [25] calculated the AdS contributionto the three-point coefficient of three heavy states rotating purely in S . In the samespirit, the three-point fusion coefficient of three GKP strings was calculated in [27]. Fi-nally, an interesting approach to holographic three-point functions for operators dual toshort string states was developed in [28].The plan for the rest of this paper is as follows. In Section 2, we present the strongcoupling calculation of the 3-point correlator involving an R-current and two scalar oper-ators dual to massive string states with arbitrary charges in S . This particular 3-pointcorrelator obeys a Ward identity which restricts the corresponding structure constant tobe proportional to the R-charge of the non-protected operators to all orders in pertur-bation theory. We verify this expectation at the strong coupling regime by employingthe AdS/CFT correspondence. In Section 3, we present the strong coupling calculationof the 3-point correlator between the stress-energy tensor T µν and the same two scalaroperators of Section 2. In this case, conformal symmetry requires that the structureconstant will be proportional to the conformal dimension of the massive operators witha certain coefficient of proportionality. As in Section 2, the string theory result agreesperfectly with the field theory expectation. Both agreements provide non-trivial tests ofboth the dictionary of the AdS/CFT correspondence and the AdS/CFT correspondenceitself. Finally, in Section 4 we focus on the 4-point correlators between an R-current or astress-energy operator and three scalar operators. We point out that the strong couplingfactorisation of [29] is consistent with conformal symmetry in the case where one of thelight operators is the R-current or the stress-energy tensor. Furthermore, we analyse theconstraints that the aforementioned factorisation imposes on the strong coupling form of The authors of [21] have calculated the one-loop correction to the structure constants of operators inthe SU(2) sub-sector to find that this agreement is spoiled. One can also employ the BMN limit whereagreement at the 1-loop level was found at [22]. For other interesting works on 3-point functions involving operators in the SL (2) sector see [24]. The authors of [26] questioned this result arguing that string solutions with no
AdS charges shouldbe point-like in the AdS space. Correlation functions involving the R-symmetry current have been extensively calculatedin the context of the AdS/CFT duality. In fact one of the first checks of the dualitywas the matching of the parity odd part of the correlator of three R-symmetry operators.The Adler-Bardeen theorem states that this correlator receives contribution only fromone-loop Feynman diagrams. The very same correlation function was calculated at strongcoupling by means of type-IIB supergravity and agreement between the field theory andsupergravity results was found [30, 31]. Another class of correlation functions studiedwas that of an R-symmetry current and two scalar BPS-states [30]. Again the result atstrong coupling, as calculated from supergravity, was found to be in perfect agreementwith the expectation from field theory. In this case, the comparison was possible due tothe fact that the aforementioned correlator satisfies certain Ward identities related to theconservation of the R-current.In all the cases mentioned above one had to restrict himself to BPS operators sinceonly the supergravity approximation was under control. In this section we evaluate thethree-point correlation function of an R-current with two operators dual to semi-classicalstring states. The result of the strong coupling calculation is in perfect agreement withthe conformal Ward identity providing a check of both the dictionary and the AdS/CFTcorrespondence itself.
It is well-known that conformal symmetry completely fixes the space-time structure of3-point functions of a vector operator V µ , a scalar operator O ∆ and its conjugate ¯ O ∆ asfollows [32] P µ = h V µ ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i = C ( λ ) x − ( x − x ) ( x − x ) E xµ ( x , x ) E xµ ( x , x ) = ( x − x ) µ ( x − x ) − ( x − x ) µ ( x − x ) . (2.1)∆ is the conformal dimension of O ∆ which has to be an eigenstate of the dilatationoperator. If V µ is a conserved current, e.g. one of the components of the R-symmetrycurrent j Rµ , and O ∆ is also an eigenstate of the symmetry generated by j Rµ then (2.1)3beys a certain Ward identity that reads h ∂ µ j Rµ ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i = δ ( x − x ) h δ O ∆ ( x ) ¯ O ∆ ( x ) i + δ ( x − x ) hO ∆ ( x ) δ ¯ O ∆ ( x ) i = J (cid:0) δ ( x − x ) − δ ( x − x ) (cid:1) hO ∆ ( x ) ¯ O ∆ ( x ) i . (2.2)In (2.2) J is R-charge of the operator O ∆ . By differentiating (2.1) one obtains ∂ µ P µ = − C ( λ )2 π (cid:0) δ ( x − x ) − δ ( x − x ) (cid:1) x . (2.3)Comparing (2.2) and (2.3) one gets C ( λ ) = − J π . (2.4)This provides an all-loop prediction for the fusion coefficient C ( λ ). In the next sectionwe will reproduce both the value of (2.4) and the space-time structure of the correlator(2.1) using string theory. h j Rµ ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i at strong coupling In this section we calculate the 3-point function of an R-symmetry current and two op-erators dual to generic classical string states with rotation only in S by employing theAdS/CFT correspondence. In order to perform the calculation one should identify whichsupergravity field is dual to field theory operator j AµB , A, B = 1 , , ,
4. Following [33] weconclude that the dual field is a combination of the metric with one index in
AdS andone index in the internal space S and the 4-form potential with 3 indices in the internalspace j AµB ←→ ( h µα , A µαβγ ) . (2.5)In (2.5) h µα and A µαβγ are the fluctuations of these fields around the AdS × S back-ground. In what follows our string theory computations will be purely at the classicallevel. At this level, A µαβγ will not contribute since it couples to fermions and its contri-bution will be suppressed by 1 / √ λ compared to the one coming from the fluctuations ofthe metric [14]. One can now expand the metric in spherical harmonics of the sphere h µα = X B I µ ( x ) Y I α (Ω) = B [ ij ] µ ( x ) Y [ ij ] α (Ω) + ..., i, j = 1 , ..., j AµB transforms in the SU (4) implies that the sugra field dual tothe R-current is B [ ij ] µ , the first term in the expansion (2.6) (see figure 1 of [33]). The In all equations above we have assumed that O ∆ is normalised to 1. Y [ ij ] α (Ω) is just the component of the [ ij ] Killing vectorof S along the α direction . Without any loss of generality one can take the component ofthe R-current which generates rotations in the [12]-plane. Then the corresponding Killingvector is Y [12] = Y [12] α ∂∂φ α = ∂∂φ . This is just the Killing vector corresponding to the φ -isometry of the sphere . From the last equation it is obvious that the only non-zerocomponent of Y [12] is Y [12] φ = 1 which directly gives Y [12] φ = g φ φ Y [12] φ = sin γ cos ψ .In conclusion we have Y [12] α = Y [12] φ = sin γ cos ψ .We are now in position to evaluate our 3-point correlator. To this end we follow [14]and consider h B i ( ~x ) i = h j [12] i ( ~x ) O ∆ ( x ) ¯ O ∆ ( x ) ihO ∆ ( x ) ¯ O ∆ ( x ) i = (cid:10) B i ( ~x, x = 0) 1 Z string Z DX e − S string [ X, Φ] (cid:11) bulk . (2.7)In (2.7) B i ( ~x, x = 0) = B [12] i ( ~x, x = 0) denotes the boundary value of the sugra field dualto the R-current. By Φ we denote all supergravity fields collectively. Also notice that thestring action depends not only on the string coordinates X ( σ, τ ) but on Φ too. Becausethe sugra field B i ( ~x, x = 0) is light with respect to the massive string states one cantreat the string in first quantised string theory while the sugra field in the supergravityapproximation. One can then expand the string action in powers of the sugra fields. Therelevant field for us is h µα . S str = √ λ π Z d σ √ gg αβ ∂ α X M ∂ α X N G MN + ( f ermions ) ⇒ δS str δh µβ ( X ) = √ λ π Z d σ ( ∂ τ X µ ∂ τ X β + ∂ σ X µ ∂ σ X β ) (2.8)Plugging (2.8) in (2.7) and keeping only the linear in h µβ term we get h B i ( ~x ) i = − (cid:10) B i ( ~x, x = 0) δS str [ X, Φ = 0] δh µβ ( Z ) h µβ ( Z ) (cid:11) bulk = − √ λ π Z d σ ( ∂ τ Z µ ∂ τ Z β + ∂ σ Z µ ∂ σ Z β ) (cid:10) B i ( ~x, x = 0) B µ ( z ) (cid:11) bulk Y [12] β (Ω) , (2.9)where we have substituted the relevant spherical harmonic. Here we should mention thatin the leading approximation the string path integral of (2.7) is dominated by the classical Here we enumerate the 15 Killing vectors of S by two antisymmetric indices [ ij ] , i, j = 1 , ..., We consider a parametrisation of the 5-sphere in terms of the angles ( γ, ψ, φ , φ , φ ) with ds S = dγ + cos γdφ + sin γ ( dψ + cos ψdφ + sin ψdφ ), where x + ix = sin γ cos ψe iφ , x + ix =sin γ sin ψe iφ , x + ix = cos γe iφ with P i =1 x i = 1. . This bulk-to-boundary propagator reads [30] G µi ( z ; ~x, x = 0) = Γ(4)2 π Γ(2) z z + ( ~z ) − ~z ) (cid:0) δ µi − z − x ) µ ( z − x ) i z + ( ~z − ~x ) (cid:1) (2.10)In (2.10) Greek letters µ = 0 , , , , i = 0 , , , ∂AdS . In order to simplify thecalculation we take the limit where the BPS operator insertion on the boundary is very faraway from the insertions of the stringy states, that is x i → ∞ . Then the bulk-to-boundarypropagator simplifies to G µi ( z ; ~x, x = 0) = 3 π ( z ( ~x ) ( δ µi − x µ x i ~x ) if µ = 0 , , , z ( ~x ) ( − z ( z − x ) i ~x ) ≈ µ = 4 since δ i = 0 . • String solution
At this point we need to write down the AdS part of the classical string solution propa-gating from the boundary to the boundary of the AdS space. Since we want string stateswhich are dual to scalar operators we need to consider only strings with rotation purely in S . Consequently the solution is pointlike in AdS. In the Poincare patch with Euclideansignature it reads [12] z = A cosh( kτ ) , z = A tanh( kτ ) , γ, ψ, φ , φ , φ = γ, ψ, φ , φ , φ ( σ, τ ) . (2.11)In this solution the spatial separation of the heavy operators living on the boundary isalong the Euclidean time direction and reads | x | = 2 A . We should stress that ouranalysis is valid for a generic sting solution with arbitrary charges in S .Plugging the expressions for the bulk-to-boundary propagator, the relevant sphericalharmonic (see discussion above (2.7)) and for the string solution in (2.9) we obtain afterperforming the τ integral h B i ( ~x ) i = − √ λ π ~x ) ( δ µi − x µ x i ~x ) I µ ,I µ =0 = Z d σ ∂ τ Z µ ∂ τ Z β Y β z = Z d σ ∂ τ z ∂ τ φ sin γ cos ψ A cosh ( kτ ) =4 A Z dσ sin γ cos ψ ∂φ ∂τ = 4 A J π √ λ , (2.12) We should mention that the capital symbols like Z correspond 10-dinensional coordinates while thesmall ones like z ~x live in the 4-dimensionalCFT. φ angle, J = √ λ π R dσ sin γ cos ψ ∂φ ∂τ . We should mention that because our stringsolution extends only in the 0 and 4-directions µ of (2.12) can be 0 or 4. But as we sawbelow (2.10) in our limit x i → ∞ the component of the bulk-to-boundary propagator G i = 0. Thus the only value µ can take is 0. So overall one gets h B i ( ~x ) i = − J A π ~x ) ( δ i − x x i ~x ) (2.13)To be able to compare with field theory one should take the same limit x i → ∞ to thecorresponding expression (2.1) for the 3-point correlator in field theory . After half pageof algebra the leading term is h j Ri ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i = C ( λ ) x µ x − ~x ) ( δ µi − x µ x i ~x ) . (2.14)In order to make contact with (2.13) we have to take into account that the separationof the heavy operators is | x | = 2 A and divide (2.14) by the 2-point function of thenon-BPS operators to obtain h j Ri ( x ) O ∆ ( x ) ¯ O ∆ ( x ) ihO ∆ ( x ) ¯ O ∆ ( x ) i = C ( λ )(2 A ) ~x ) ( δ i − x x i ~x ) . (2.15)Direct comparison of (2.15) and (2.13) gives the string theory prediction for the fusioncoeeficient C ( λ >>
1) = − J π . (2.16)This strong coupling result is in perfect agreement with the all-loop prediction based onthe field theoretic Ward identity (2.4). Furthermore, the spacetime structure of the cor-relator computed by means of AdS/CFT is the one specified by the conformal invarianceof field theory. T ij and two massive string states In this section we calculate the 3-point function of two operators dual to semi-classicalstring states and the stress-snergy tensor. 2- and 3-point functions involving the stress-energy tensor have been calculated in the past, particularly in relation to the conformal Without taking the x i → ∞ limit, after some tedious calculations, we have also reproduced the exactresult (2.1) obtained in the CFT. V ij andtwo scalar eigenstates of the dilatation operator is constrained by conformal symmetryup to a scalar coefficient C ( λ ) h V ij ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i = C ( λ ) x − ( x − x ) ( x − x ) F ij ( x, x , x ) F ij ( x, x , x ) = E xi ( x , x ) E xj ( x , x ) − δ ij x ( x − x ) ( x − x ) , (3.1)where E xi ( x , x ) is a vector defined in (2.1). In the case where V ij = T ij conservation of T ij implies that the fusion coefficient is proportional to the conformal dimension ∆( λ ) ofthe heavy operator C ( λ ) = − π ∆( λ ) . (3.2)This is an all-loop statement which we will verify at strong coupling in the next section.As in the previous section we will need the leading term of the correlator (3.1) in the limit x i → ∞ . A simple calculation shows that h V ij ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i x i →∞ = C ( λ ) x − ~x ) (cid:0) x k x l j ki ( x ) j lj ( x ) − δ ij x (cid:1) ,j ij ( x ) = δ ij − x i x j ~x . (3.3)By dividing with the 2-point function and putting our heavy operators along the 0-direction at a distance of x = 2 A as we did with our string solution we get h V ij ( x ) O ∆ ( x ) ¯ O ∆ ( x ) ihO ∆ ( x ) ¯ O ∆ ( x ) i x i →∞ = C ( λ )( ~x ) (2 A ) (cid:0) j i ( x ) j j ( x ) − δ ij (cid:1) . (3.4) h T ij ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i at strong coupling The calculation for the case in hand proceeds in a similar way to that of section 2.2.The field dual to the stress-energy tensor T ij is the fluctuations of the metric g µν = g AdSµν + h µν , µ, ν = 0 , ..., T ij ←→ h µν . It is apparent that in whatfollows we will need the bulk-to-boundary propagator for the graviton. This can be readof the solution to the linearised equations of motion in the covariant gauge of de Dondertype ∇ µ ( h µν − δ µν h ) [37, 35]. h µν ( x , ~x ) = k G Z d d yK ( x, ~y ) j iµ ( x − ~y ) j νj ( x − ~y ) E ij,kl ˆ h kl ( ~y ) , (3.5)8here ˆ h kl ( ~y ) is the value of the graviton on the boundary, k G = d +1 d − d ) π d/ Γ( d/ and E ij,kl = ( δ ik δ jl − δ il δ jk ) − d δ ij δ kl . Finally, the scalar propagator K ( x, ~y ) and the inversion tensorare given by K ( x, ~y ) = x d (cid:0) x + ( ~x − ~y ) (cid:1) d , j iµ ( x ) = δ iµ − x µ x i x . (3.6)In the equations above d is the dimension of the AdS boundary which in our case is d = 4.We are now in position to evaluate our 3-point correlator at strong coupling. Byfollowing the same steps as in section 2.2 we obtain h ˆ h ij ( ~x ) i = − (cid:10) ˆ h ij ( ~x, x = 0) δS str [ X, Φ = 0] δh νµ ( Z ) h νµ ( Z ) (cid:11) bulk = − √ λ π Z d σ ( ∂ τ Z µ ∂ τ Z ν + ∂ σ Z µ ∂ σ Z ν ) (cid:10) ˆ h ij ( ~x, x = 0) h νµ ( z ) (cid:11) bulk . (3.7)Taking into account that the AdS part of the string solution (2.11) does not depend on σ and keeping only the leading terms in our usual limit x i → ∞ we get h ˆ h ij ( ~x ) i = − k G √ λ π Z d σ z ( ~x ) ∂ τ Z µ ∂ τ Z ν j kµ ( z − ~x ) j νl ( z − ~x ) E kl,ij = − k G √ λ π ~x ) ( δ k − x x k ~x )( δ l − x x l ~x ) E kl,ij Z d σ z ∂ τ z ∂ τ z (3.8)In order to proceed we need to calculate the integral and simplify the tensorial structurein (3.8). This is easily done Z d σ z ∂ τ z ∂ τ z = Z d σ z ( ∂ τ z ) = A Z dσdτ k cosh k τ = A πk δ k − x x k ~x )( δ l − x x l ~x ) E kl,ij = ( δ i − x x i ~x )( δ j − x x j ~x ) − δ ij (3.9)In (3.9) i, j = 0 , , , h which are parallelto the boundary. Inserting (3.9) in (3.8) one obtains h ˆ h ij ( ~x ) i = − k G √ λ ~x ) A k (cid:0) ( δ i − x x i ~x )( δ j − x x j ~x ) − δ ij (cid:1) (3.10)Direct comparison of (3.10) and (3.4) shows that they have the same spacetime structure and that the strong coupling value of the fusion coefficient is C ( λ >>
1) = − π √ λk = − π √ λk = − π E, (3.11) Similarly to the vector operator, (3.1) can be exactly reproduced in string calculation without takingthe x i → ∞ limit. E = √ λk is the energy of the string (2.11) with arbitrary charges in S . Thus thestring theory result (3.11) is in complete agreement with the all-loop field theory expec-tation (3.2) based on Ward identities since according to the AdS/CFT correspondence∆( λ ) = E ( λ ).An important comment is in order. According to the AdS/CFT correspondence, thefield theory correlators we are considering, h j Rµ O ∆ ¯ O ∆ i and h T µν O ∆ ¯ O ∆ i , should beequal to correlators of the corresponding string vertex operators which schematically areof the form h V µ V heavy V heavy i and h V µν V heavy V heavy i [13, 29]. Here V µ and V µν are thevertex operators for the massless string modes that correspond to j µ and T µν respectively,while V heavy are the vertex operators for the massive string states which are dual toprimary operators in field theory. The semiclassical value of the aforementioned stringcorrelators takes the form of a ”light” vertex operator integrated over the classical stringsolution sourced by V heavy (see (2.9) and (3.7)) [29]. One could have determined these”light” vertex operators explicitly. Then integrating the ”light” vertex operators over theclassical string solution would have given results identical to ours. In fact, one shouldunderstand the analysis following (2.7) and (3.7) as an effective way to determine thecontribution coming from these supergravity vertex operators. Recently, it was observed that the 4-point function of two operators dual to massive stringstates and two light BPS operators dual to sugra states factorises as a product of two3-point correlators for very large values of the coupling constant λ >>
1. More precisely,in [29] it was argued that hO L ( x ) O L ( x ) O ∆ ( x ) ¯ O ∆ ( x ) i = hO L ( x ) O ∆ ( x ) ¯ O ∆ ( x ) ihO L ( x ) O ∆ ( x ) ¯ O ∆ ( x ) ihO ∆ ( x ) ¯ O ∆ ( x ) i + O ( √ λ ) . (4.1)One can imagine that in the place of the massive string states one takes two BPS operatorswith a very large angular momentum in S such that these operators can have a descriptionin terms of a classical point like string state. One can then use (4.1) to calculate thestrong coupling limit of the 4-point correlator. The result obtained this way disagreeswith the result calculated in the supergravity approximation [38] when the charges of thetwo ”heavy” BPS states are extrapolated to very large values. This is so because thesupergravity approximation is not a priori valid if the charges of the supergravity states By light we mean operators with conformal dimensions ∆ << √ λ . √ λ . Thus correlators involving ”heavy” string modes cannot be compared tosupergravity correlators. It might happen that in particular cases one can extrapolate thesupergravity result to very large values of the charges ( J ∼ √ λ ) but that should not beexpected in general .In this section we point out that this strong coupling factorisation of 4-point correlatorsis consistent with conformal symmetry in the case where one of the light operators is theR-current or the stress-energy tensor . Furthermore, we will analyse the constraints onthe form of 4-point correlators that are imposed from the aforementioned factorisationat strong coupling. We start with the 4-point correlation function among a conservedcurrent J µ and three scalar operators one of which O δ is not charged under the symmetrygenerated by J µ while the other two O ∆ and ¯ O ∆ have charges J and − J respectively. Inthat case the 4-point correlator can be expressed as follows [40] hO ∆ ( x ) ¯ O ∆ ( x ) O δ ( x ) j µ ( x ) i = (4.2) (cid:16) ξ h − K x µ ( x , x ) F ( ξ, η ; λ ) − η h − K x µ ( x , x ) F ( η, ξ ; λ ) (cid:17) hO ∆ ( x ) ¯ O ∆ ( x ) O δ ( x ) i , where h = D/ D is the dimension of the spacetime while ξ and η are the conformalcross-ratios and K µ is given by ξ = x x x x , η = x x x x K x µ ( x , x ) = Γ( h ) J π h x x x ( ( x ) µ x − ( x ) µ x ) . (4.3)Note that conformal symmetry is not enough to fully constrain the spacetime structureof the 4-point correlator as was the case with the 3-point function. Instead there is anarbitrary function of the cross-ratios F ( η, ξ ; λ ) which has to be determined by directcalculation.However, at strong coupling the factorisation (4.1) implies that (4.2) leads to ξ h − K x µ ( x , x ) F ( ξ, η ; λ ) − η h − K x µ ( x , x ) F ( η, ξ ; λ ) = hO ∆ ( x ) ¯ O ∆ ( x ) j µ ( x ) i x (4.4)This equation can be satisfied only if its left hand side is independent of x . This canonly happen if ξ h − x x x F ( ξ, η ; λ ) = η h − x x x F ( η, ξ ; λ ) (4.5) We thank Arkady Tseytlin for bringing this point to our attention. For weak/strong matching of 4-point correlation functions see [39]. x disapperas from the parenthesis of (4.2). Plugging the value h = 2in (4.5) we obtain F ( ξ, η ; λ ) = F ( η, ξ ; λ ) , for λ >> . (4.6)Thus we conclude that at strong coupling the function F ( ξ, η ; λ ) should be symmetricunder the exchange ξ ↔ η . In the case where j µ is the R-current direct comparisonof (4.6) and (2.1), (2.4) leads to the conclusion that F is not only symmetric but alsocompletely independent of the cross-ratios ξ, η . F ( ξ, η ; λ ) = F ( η, ξ ; λ ) = − J π + O ( ξ, η ; ( √ λ ) ) . (4.7)We now turn to the last correlation function to be considered. This is the similarto (4.2) with the stress-energy tensor in the place of the R-current. As above conformalsymmetry constrains to some extent this 4-point function [40] hO ∆ ( x ) ¯ O ∆ ( x ) O δ ( x ) T µν ( x ) i = (cid:16) K x µν ( x , x ) ˆ F ( ξ, η ; λ ) + K x µν ( x , x ) φ ( ξ, η ; λ ) + K x µν ( x , x ) φ ( η, ξ ; λ ) (cid:17) hO ∆ ( x ) ¯ O ∆ ( x ) O δ ( x ) i , (4.8)where K x µν ( x , x ) = x x x F µν ( x , x , x ) , (4.9)where F µν is defined in (3.1). Strong coupling factorisation results to K x µν ( x , x ) ˆ F ( ξ, η ; λ ) + K x µν ( x , x ) φ ( ξ, η ; λ ) + K x µν ( x , x ) φ ( η, ξ ; λ ) = hO ∆ ( x ) ¯ O ∆ ( x ) T µν ( x ) i x . (4.10)Comparing the last equation to (3.1)and (3.2) we conclude that φ ( ξ, η ; λ ) = φ ( η, ξ ; λ ) = 0 √ λ + O ( ξ, η ; ( √ λ ) )ˆ F ( ξ, η ; λ ) = − λ >> π + O ( ξ, η ; ( √ λ ) ) . (4.11)As in the case of the R-current correlators the leading in large λ expressions for theundetermined by the conformal symmetry functions ˆ F ( ξ, η ; λ ) , φ ( ξ, η ; λ ) are independentfrom the cross-ratios ξ and η . We do not expect that this property will hold when α ′ corrections are taken into account or at the weak coupling expansion. Acknowledgments
12e wish to thank George Savvidy and Dimitrios Zoakos and especially Arkady Tseytlinfor useful discussions and comments. This work was partly supported by the GeneralSecretariat for Research and Technology of Greece and from the European Regional De-velopment Fund (NSRF 2007-13 ACTION,
KP H Π I Σ).This work was also partly supported by the National Research Foundation of Ko-rea(NRF) grant funded by the Korea government(MEST) through the Center for Quan-tum Spacetime(CQUeST) of Sogang University with grant number 2005-0049409 and Ba-sic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education, Science and Technology(2010-0022369).
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