Holographic correlators in AdS 3
aa r X i v : . [ h e p - t h ] M a r QMUL-PH-18-31
Holographic correlators in AdS Stefano Giusto O ++ ,O −− , Rodolfo Russo O + − and Congkao Wen O + − O ++ Dipartimento di Fisica ed Astronomia “Galileo Galilei”, Universit`a di Padova,Via Marzolo 8, 35131 Padova, Italy O −− I.N.F.N. Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy O + − Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom
Abstract
We derive the four-point correlators of scalar operators of dimension one in the supergravitylimit of the D1D5 CFT holographically dual to string theory on AdS × S × M , with M either T or K
3. We avoid the use of Witten diagrams but deduce our result from a limitof the heavy-heavy-light-light correlators computed in [1], together with several consistencyrequirements of the OPE in the various channels. This result represents the first holographiccorrelators of single-trace operators computed in AdS . e-mails: [email protected], [email protected], [email protected] Introduction
Recent years have seen a remarkable progress in the computation and the analysis ofconformal correlators at large ’t Hooft coupling and large N , where the problem be-comes tractable by using holographic techniques. For the paradigmatic example of four-dimensional N = 4 super Yang-Mills theory, dual to type IIB string theory on AdS × S ,we now know a general formula for the four-point correlators of half-BPS single-trace op-erators of arbitrary fixed dimension [2, 3], and similar results are available for theorieswith an AdS dual [4]. Surprisingly, not a single holographic correlator of single-trace op-erators has ever been computed in AdS . One of the reasons is that the standard Wittendiagram technique used to compute correlators in AdS d +1 cannot be naively extrapolatedto d = 2: simply setting d = 2 in the results for generic d leads to divergent expressionsfor the diagrams computing the exchange of a massless vector and graviton [5], and anappropriate procedure to define the d = 2 limit has not been formulated yet. Anotherissue is that the cubic couplings of type IIB supergravity on AdS × S are known [6],but not the quartic ones which are also needed for deriving a full 4-point correlator byusing Witten diagrams.Here we will focus on 4-point correlators in AdS /CFT of the type h O ( z , ¯ z ) O ( z , ¯ z ) O ( z , ¯ z ) O ( z , ¯ z ) i = 1 | z | | z | G ( z, ¯ z ) , (1.1)where we assume that the conformal dimensions satisfy ∆ = ∆ and ∆ = ∆ . Asusual, global conformal invariance implies that the 4-point correlators can be expressedin terms of a function of the cross-ratio z ≡ z z z z ⇒ − z = z z z z (1.2)as done in (1.1), so we will write our results in terms of the function G ( z, ¯ z ).Holographic correlators of the type (1.1) containing two light and two heavy operators,which we will dub HHLL correlators, have been studied from different points of view. Herethe heavy states are described by multi-trace operators that have a conformal dimensionproportional to the central charge c . The contribution to the HHLL correlators from the(large c ) Virasoro block of the identity was derived in [7] within a CFT approach andin [8] from a bulk perspective. In order to go beyond this approximation and obtain a fullHHLL CFT correlator it is necessary to specify the heavy operator more precisely. Whenthe dimension of the heavy multi-trace operators is of order of the central charge, theirgravitational backreaction effectively changes the background, replacing AdS × S witha non-trivial smooth geometry which approximates AdS × S only near the asymptoticboundary. The precise form of this regular solution encodes the choice of the heavy1tate. For half-BPS operators in the D1D5 CFT all such geometries are known [9, 10].The single-trace operators, on the other hand, are taken to have dimensions of order onein the large central charge limit: they thus represent perturbations of the backgroundand are described by linear wave equations. Then, for the HHLL case, it is possible tocalculate the four-point correlators of two single and two multi-trace operators bypassingthe difficulties affecting Witten diagrams for d = 2 [1, 11, 12]: they are extracted from anon-normalizable solution of the wave equation associated with the light operators in thegeometry sourced by the heavy ones. This method effectively reduces the computation ofthe HHLL four-point function to that of a two-point function in a non-trivial background,and thus it does not require evaluating formally divergent exchange diagrams.The question we address in this paper is how to reconstruct the correlators containingonly light single-trace operators, which will be referred to as the LLLL correlators, fromthe HHLL correlators computed in [1, 12]. This is possible because the multi-trace oper-ators considered in [1, 12] depend on a free parameter, denoted as b in those references,which controls the number of their single-trace components and, hence, the dimensionof the heavy operators can be made small by taking the parameter b small. In this limitone thus naively expects that the HHLL correlators reduce to the LLLL ones. As wewill see, this naive expectation is not quite correct. To understand where the problemis, we remark that in the HHLL correlators no single-trace operator is exchanged in thechannels where a light and a heavy operator are fused together. Since this is true forany value of b , this feature survives the small b limit. On the contrary the OPE betweentwo single-trace operators does, in general, contain other single-trace operators. So, if weschematically denote by O L the single-trace operators appearing in the HHLL correlatorand by O ′ L the single-trace operators obtained by taking the small b limit of the originalheavy operators, then the OPE of O L and O ′ L should, in general, contain the contribu-tion of other single-trace operators, which is missed when we take the small b limit of theHHLL correlator. This implies that the result obtained from the HHLL correlator doesnot correctly describe the LLLL correlator in the limit in which O L and O ′ L are close.This problem, however, should not be relevant in the direct channel where the two O L operators are close: our fundamental assumption is that the small b limit of the HHLLcorrelator correctly captures the contributions to the LLLL correlator due to single-traceoperators exchanged in this channel. We will show how the contributions from the otherchannels can be unambiguously fixed by various consistency requirements.For example, when one of the O L operators is the same as one of the O ′ L operators, thesymmetry of the correlator under the exchange of O L and O ′ L can be used to recover thecorrect OPE expansion in one of the crossed channels. More generally, we will require thatthe holographic correlators have the expected singularities associated to the exchange ofprotected single-trace operators in all OPE channels. This requirement determines the2ull correlator up to contact terms, which holographically correspond to Witten diagramswithout internal propagators, and only affect the exchange of double-trace operators [13].In the examples we consider, the contact terms are fully determined by consistency withthe flat space limit and by matching the contribution of some protected double-traceoperator at weak and strong coupling. We apply this logic to the four-point functionof half-BPS single-trace operators of dimension (1 / , /
2) and find a single answer thatpasses all the consistency checks. We leave the generalization to correlators with higherdimensional operators for the future.The paper is organised as follows. In section 2 we characterise the half-BPS operatorsconsidered in this paper by using the free limit of the CFT description. Then we brieflyreview the results for the HHLL correlators in the supergravity limit that were derivedin [1, 12]. In section 3 we take the small b limit and obtain, for the case of a particularlysymmetric LLLL correlator, an expression that is valid in the direct channel limit z → z .Then we impose the consistency conditions mentioned above and obtain the result (3.10)for the full LLLL correlator considered. Finally in section 4 we generalise this result toother correlators involving operators of the same dimension (1 / , / N OPE data for the D1D5 CFT at strong coupling.
Exactly as it happens for N = 4 Super Yang-Mills, also the D1D5 Super CFT, which isdual to type IIB string theory on AdS × S × M , has a free locus in its superconformalmoduli space. On this locus the CFT is described by a collection free bosons and freefermions { ∂X A ˙ A ( r ) ( z ) , ψ α ˙ A ( r ) ( z ) , ¯ ∂X A ˙ A ( r ) (¯ z ) , ˜ ψ ˙ α ˙ A ( r ) (¯ z ) } , (2.1)where r = 1 , . . . , N = n n , c = 6 N , and A , ˙ A , α , ˙ α are SU (2) indices. To beprecise, the free description is in terms of an orbifold CFT, where the orbifold group isthe permutation group S N acting on the copy index ( r ). The SU (2) L × SU (2) R actingrespectively on α and ˙ α are part of the (affine) R-symmetry of the SCFT and will playan important role in our analysis. As usual we can characterise the (protected) chiralprimary operators in terms of this free field description and in this work we focus on the Here M can be either T or K
3, as we will focus on a sector that is common to the two cases. O α ˙ α = N X r =1 − iǫ ˙ A ˙ B √ N ψ α ˙ A ( r ) ˜ ψ ˙ α ˙ B ( r ) . (2.2)This operator is in the untwisted sector of the orbifold CFT and is manifestly symmetricunder S N ; it is straightforward to see that it is chiral primary operator, since its conformalweights are ( h, ¯ h ) = (1 / , /
2) and its charges under the generators ( J , ˜ J ) in SU (2) L × SU (2) R are ( j, ¯ j ) = ( ± / , ± /
2) depending on the values of α and ˙ α . On the bulk sideof the duality (2.2) corresponds to a supergravity fluctuation of AdS × S , so we willrefer to this type of operators as single particle or equivalently single trace in analogywith the nomenclature used in the case of N = 4 Super Yang-Mills.An important property of 2d CFTs with extended supersymmetry is the possibilityto perform a spectral flow transformation. Here we follow the conventions of [15], thenthe minimal spectral flow, connecting the NSNS and the RR sector of theory, maps theNSNS SL (2 , C ) invariant vacuum into the RR ground state with ( j, ¯ j ) = ( N/ , N/
2) andof course ( h, ¯ h ) = ( c/ , c/ j, ¯ j ) = ( − h, − ¯ h )and are obtained just by inserting many copies of the same single-particle antichiral pri-mary operator at the same point. In particular we are interested in the multiparticlestate obtained by multiplying the operator (2.2) N b times O NSNS b = (cid:0) O ++ (cid:1) N b , ¯ O NSNS b = (cid:0) O −− (cid:1) N b . (2.3)After spectral flowing this state to the RR sector one obtains a heavy state ( O NSNS b → ¯ O H and ¯ O NSNS b → O H , according to the conventions of [1]) that is described by anothersupersymmetric type IIB solution , see for instance Eqs. (3.1) and (B.1) of [1]. Theonly feature of this solution we need to recall here is that it depends on a continuousparameter b which is related to the number of single particle constituents in the heavystate (when considered before the spectral flow).In the supergravity description the operator (2.2) corresponds to a fluctuation of thebackground encoded in a scalar and a 3-form of the 6d supergravity obtained from thestandard KK reduction of type IIB on M [1]. By studying the equations of motionof this perturbation in the background, it is possible to extract the HHLL correlators To be precise the supergravity solution is dual to a coherent state with a weighted sum over allpossible numbers of single particle constituents which, in the large N limit, is sharply peaked [19]. O H ¯ O H O α ˙ α O β ˙ β i , see Eqs. (3.58) and (F.3) [1]. For convenience we report those resultshere after spectral flowing back to the NSNS sector where the heavy operators becomethe corresponding multiparticle states: the correlator h ¯ O NSNS b O NSNS b O ++ O −− i reads G HHLL ( z, ¯ z ) = (cid:20) b a (cid:18) | z | π ˆ D −
12 + N | − z | (cid:19)(cid:21) + O ( b ) , (2.4)while for the correlator h ¯ O NSNS b O NSNS b O + − O − + i we have G HHLL ( z, ¯ z ) = (cid:20) b a (cid:18) zπ ˆ D − (cid:19)(cid:21) + O ( b ) , (2.5)where b / (2 a ) = N b /N and the definition of the D -functions is summarised in ap-pendix A. In momentum space it is also possible to write an explicit expression for thesecorrelators that is exact in b [12], but we will not need it here since we will focus on thesmall b limit. The general picture that has emerged from the recent developments in holographic cor-relators [2, 3, 14, 20–25] is that the contributions of the single-trace operators exchangedin the various channels, together with the constraints coming from supersymmetry, de-termine the correlators uniquely. We will adopt a similar approach here and reconstructthe correlators with four single-trace operators from the information extracted from theHHLL correlators reviewed above, supplemented by some basic consistency requirementsof the OPE in the various channels.We start from the analysis of the small b limit of the HHLL correlator (2.4) whichshould capture the contributions of the single-trace operators exchanged in the s -channel z → z → z ) for the correlator h O −− ( z , ¯ z ) O ++ ( z , ¯ z ) O ++ ( z , ¯ z ) O −− ( z , ¯ z ) i = 1 | z z | G ( z, ¯ z ) . (3.1)Naively one would expect to recover this LLLL correlator by setting N b = N b / (2 a ) = 1in the HHLL result (2.4). It is however clear that this cannot lead to the correct answer:for once, the N b = 1 limit of (2.4) is not symmetric under the exchange of z and z , as Of course these results are reliable in the supergravity regime where one takes N to be large and anappropriate strong coupling limit. More precisely the contributions of the single-trace operators that are degenerate with double-traceoperators are not needed, and hence all the information on holographic correlators is obtained, in thesupergravity limit, from a finite number of terms. Only the leading term in N , given by 1 + | − z | , has the expected symmetry. D -functions, while the LLLL correlator in(3.1) obviously is. In hindsight, the failure of this naive expectation is not surprising: theHHLL result (2.4) has been derived by first taking N b to scale like N in the large N limit(so that O NSNSb can be treated as a heavy operator and be represented by a geometry),and then by sending the ratio N b /N to zero; the LLLL correlator, on the other hand,should be computed by setting N b = 1 from the start, and then sending N to infinity.There is a priori no reason that the these two different limits agree, and indeed they donot. The question is if we can still learn something on the LLLL correlator from (2.4). Ourfundamental assumption will be that the small b limit of the HHLL correlator correctlycaptures only the contributions to the LLLL correlator associated with the exchange ofsingle-trace operators in the s -channel. If we denote by G s ( z, ¯ z ) this contribution, wethen deduce from (2.4) that G s ( z, ¯ z ) = 1 + 1 N (cid:20) π | z | | − z | ˆ D − (cid:21) . (3.2)Note that we have not included in the expression above the term | − z | , which comesfrom the exchange of the identity in the u -channel, and, for later convenience, we haveexpressed ˆ D in terms of ˆ D , using the identity ˆ D = | − z | ˆ D which followsfrom (A.11).A consistency check on the above form for G s is obtained by looking at the light-conelimit ¯ z →
1, where one expects that the only exchanged states are the affine descendantsof the identity (i.e. the descendants generated by the Virasoro and the R-symmetrygenerators), since these are the only protected states with ¯ h = 0. One can indeed checkthat in this limit G s ¯ z → −→ − N (cid:18) z − z log z (cid:19) , (3.3)which is the identity affine block expanded up to order 1 /N . This can be seen as follows: The identity Virasoro block at order 1 /N is equivalent to the globalblock of the stress tensor; for external states of dimension h L = 1 / V V = 1 + 112 N (1 − z ) F (2 , ,
4; 1 − z ) = 1 − N (cid:20) z − z ) log z (cid:21) ;the U (1)-affine block for external states of charge q L = 1 / V A = z N = 1 + 12 N log z + O ( N − ) . The identity affine block is then V = V V V A = 1 − N (cid:18) z − z log z (cid:19) + O ( N − ) .
6n the full correlator (3.1), single-trace operators are also exchanged in the u -channel z → ∞ (or z → z ). This u -channel contribution is easily determined by imposing thesymmetry under the exchange of z and z :1 | z z | G ( z, ¯ z ) = 1 | z z | G ( z ′ , ¯ z ′ ) with z ′ ≡ − z z z z = zz − , (3.4)which implies G ( z, ¯ z ) = | − z | G (cid:18) zz − , ¯ z ¯ z − (cid:19) . (3.5)An obvious way to obtain a correlator with the correct symmetry is to add to G s theterm G u ( z, ¯ z ) = | − z | G s (cid:18) zz − , ¯ z ¯ z − (cid:19) = | − z | + | − z | N (cid:20) π | z | ˆ D − (cid:21) . (3.6)In deriving the above equation we have used the transformation property of the ˆ D -functions explained in (A.12).Since in the t -channel z → z → z ) there is no exchange of single-trace operatorswith dimensions less than ( h, ¯ h ) = (1 , G s and G u should correctly include all the single-trace exchanges. This,however, does not completely identify the correlator, as one still has the freedom to addterms that only affect the contributions of the double-trace operators. From the bulkpoint of view, such terms originate from Witten diagrams associated to quartic contactinteractions, and have first been studied in [13]. They are in principle determined bysupersymmetry, which completely fixes the supergravity action at the two-derivativelevel. We follow here a simpler route, and resolve this ambiguity by requiring consistencywith the flat space limit and with the t -channel OPE. The constraint coming from the flatspace limit is easier to formulate in Mellin space [27, 28]: the allowed contact interactionsshould be functions of the Mellin variables scaling at most linearly when all the variablesare taken to infinity. Moreover, the requirement that the contact interactions do notintroduce single-trace exchanges implies that their Mellin transform have no poles. Onecan show that these constraints, together with the symmetry under the exchange ofthe s and u -channels, leave only two possible contact terms. In Mellin space, they areproportional to t and a constant. As we show in appendix B, they correspond to thefollowing two contributions , when re-expressed in the space-time coordinates: G cont . ( z, ¯ z ) = 2 π N | − z | (cid:16) c ˆ D + c | z | ˆ D (cid:17) . (3.7) The fact that G cont . in (3.7) satisfies the symmetry (3.5) follows from (A.12).
7e can fix the coefficients c and c by looking at the z → O ++ O ++ :,which results from the fusion of two chiral primaries of the same chirality and, hence, itmust have a vanishing anomalous dimension and its three-point coupling with the externaloperators should be the same as in the free theory. The vanishing of the anomalousdimension requires that the correlator does not contain terms of the type log | z | for z →
0: it is immediate to see that this condition imposes c = 0. One should also requirethat the coefficient of the term of order ( z ¯ z ) , which encodes the square of the three-point function h O −− O −− : O ++ O ++ : i , equals the free theory result. The free correlatoris given by G free ( z, ¯ z ) = 1 + | − z | + 1 N | z | − − | − z | − N + O ( z ) + O (¯ z ) , (3.8)while expanding the correlator at strong coupling (with c = 0) one finds G s ( z, ¯ z ) + G u ( z, ¯ z ) + G cont . ( z, ¯ z ) = 2 − − c N + O ( z ) + O (¯ z ) , (3.9)and thus this determines c = 1.Our final result for the correlator (3.1) is then G ( z, ¯ z ) = (cid:18) − N (cid:19) (1 + | − z | ) + 2 π N | z | | − z | ( ˆ D + ˆ D + ˆ D ) . (3.10) One can generalize the result of the previous section by considering correlators of generaloperators O α ˙ α in the same R-symmetry multiplet. We can introduce the general notation h O α ˙ α ( z , ¯ z ) O α ˙ α ( z , ¯ z ) O α ˙ α ( z , ¯ z ) O α ˙ α ( z , ¯ z ) i = 1 | z z | G ( α ˙ α )( α ˙ α )( α ˙ α )( α ˙ α ) ( z, ¯ z ) , (4.1)so that the correlator in (3.1) (or (3.10)) will be renamed G ( −− )(++)(++)( −− ) . Essentially the onlyother correlator to compute is G ( −− )(++)(+ − )( − +) : we could deduce this correlator from its HHLL“parent” (2.5), using arguments similar to the ones outlined in the previous section, or byusing the Ward identity that relates correlators within the same R-symmetry multiplet.We will follow this second route in this section.A straightforward way to derive the Ward identity is to write one of the O −− operatorsin the correlator (3.1) as O −− = [ J − , O + − ] and to move the current J − on the otheroperators by the usual argument of deforming the integration contour. Using [ J − , O −− ] = We thank Agnese Bissi for suggesting this possibility to us. J − , O ++ ] = − O − + (where the minus sign is needed to have O − + = ( O + − ) † ), oneimmediately finds G ( −− )(++)(++)( −− ) ( z, ¯ z ) = G ( −− )(++)( − +)(+ − ) ( z, ¯ z ) + G ( −− )( − +)(++)(+ − ) ( z, ¯ z )= G ( −− )(++)(+ − )( − +) (cid:18) z , z (cid:19) + | − z | G ( −− )(++)(+ − )( − +) (cid:18) z − z , ¯ z − z (cid:19) , (4.2)where in the second step we have expressed the r.h.s. in terms of the single correlator G ( −− )(++)(+ − )( − +) by exchanging the positions of the operators. One can check, using the trans-formation properties of the ˆ D -functions in (A.11) and (A.12), that a solution of the Wardidentity (4.2) is G ( −− )(++)(+ − )( − +) ( z, ¯ z ) = 1 − N + 2 π N z | − z | ( ˆ D + ˆ D + ˆ D ) . (4.3)The above solution passes several consistency checks: it reduces to the appropriate iden-tity affine blocks in the light-cone limits ¯ z → z → z N for ¯ z → z − N for z → G ( −− )(++)(+ − )( − +) free ( z, ¯ z ) = 1 + 12 N (cid:18) | − z | ¯ z + 1 − ¯ z ¯ z − (1 − z ) (cid:19) . (4.4)Note that for this correlator the first non-protected operators appear at dimension( h, ¯ h ) = (1 ,
1) in the s -channel, but only at dimension h + ¯ h ≥ t and u -channels,because multi-trace operators like : O ++ O + − : or : O ++ O − + : preserve supersymmetryin the left or in the right sector and are thus protected. We believe that there are nocontact terms with correct symmetries that could be added to G ( −− )(++)(+ − )( − +) without spoilingthe Ward identity (4.2) or modifying the contributions from the exchange of protectedoperators.Analogue to the case of N = 4 SYM, to keep track of the R-symmetry it is convenientto introduce two-dimensional vectors and to define O i = A iα ¯ A i ˙ α O α ˙ α ( z i , ¯ z i ) , (4.5)where, for each value of i , the vectors A iα are given by either A + = (cid:18) (cid:19) or A − = (cid:18) (cid:19) , (4.6)and the same for ¯ A i ˙ α . With this set up, the general four-point correlator of the R-symmetry multiplet is now defined as h O O O O i = | A · A A · A | | z z | G ( α c , ¯ α c , z, ¯ z ) , (4.7)9here α c is the cross ratio α c = A · A A · A A · A A · A , (4.8)and similarly for ¯ α c . The dot in the above formulas denotes the contraction with ǫ αβ or ǫ ˙ α ˙ β . With the help of these R-symmetry variables, we find that the free correlators cannow be expressed as G free ( α c , ¯ α c , z, ¯ z ) = 1 + | − z | | − α c | (cid:18) | α c | + 1 | z | (cid:19) ++ 12 N (cid:18) − zz (1 − α c ) + α c (1 − z )1 − α c − α c | − z | ¯ z | − α c | + c.c. (cid:19) (4.9)where c.c. represents the conjugate terms with α c ↔ ¯ α c , z ↔ ¯ z . Whereas for the AdS correlator G ( α c , ¯ α c , z, ¯ z ), the main interest of the paper, we find that the result is givenby G ( α c , ¯ α c , z, ¯ z ) = G + 1 N | − α c z | | − α c | (cid:20) π | − z | (cid:16) ˆ D + ˆ D + ˆ D (cid:17)(cid:21) , (4.10)where G contains only rational functions of z, ¯ z and is given by G = (cid:18) − N (cid:19) (cid:20) | − z | | − α c | (cid:18) | α c | + 1 | z | (cid:19)(cid:21) . (4.11)Note the leading- N term comes from G free in (4.9). It is straightforward to check that G ( α c , ¯ α c , z, ¯ z ) and G free ( α c , ¯ α c , z, ¯ z ) have all the correct symmetries and that, in the limitsof { α c → −∞ , ¯ α c → −∞} and { α c → −∞ , ¯ α c → } , (4.10) reduces to (3.10) and (4.3)while (4.9) reduces to (3.8) and (4.4). Finally, we comment that both G free ( α c , ¯ α c , z, ¯ z )and G ( α c , ¯ α c , z, ¯ z ) satisfy ∂ ¯ z (cid:16) G free ( α c , ¯ α c , z, ¯ z ) (cid:12)(cid:12) ¯ α → / ¯ z (cid:17) = 0 , ∂ ¯ z (cid:16) G ( α c , ¯ α c , z, ¯ z ) (cid:12)(cid:12) ¯ α → / ¯ z (cid:17) = 0 (4.12)which takes exactly the same form as the superconformal Ward identity in the case of N = 4 SYM [29, 30]. The main result of this note is summarised in eqs. (4.10), (4.11), which give the holo-graphic correlators of the scalar operators of dimension (1 / , /
2) defined, at the freepoint of the CFT, in (2.2). We hope that this result may pave the way for a systematicconstruction of holographic correlators in AdS .10n the much better studied AdS case, the knowledge of holographic correlators haslead to uncover a very rich structure, in particular for what concerns the spectrum ofdouble-trace operators of N = 4 SYM at strong coupling (see for example [21,25,31,32]).An analogous information should be encoded also in AdS correlators. Considering forexample the correlator in (3.1), the s -channel OPE contains a series of double-traceoperators of the form O m, ¯ m = : O −− ∂ m ¯ ∂ ¯ m O ++ : with m, ¯ m = 0 , , . . . , (5.1)whose conformal dimensions ( h m, ¯ m , ¯ h m, ¯ m ) receive quantum corrections at order 1 /N : h m, ¯ m = 1 + m + γ m, ¯ m N , ¯ h m, ¯ m = 1 + ¯ m + γ m, ¯ m N . (5.2)As usual the anomalous dimensions γ m, ¯ m can be extracted from the terms containinglog | − z | in the z → O m, ¯ m , and all theseoperators could mix away from the free orbifold point. The knowledge of the correlatorsderived in this note is not enough to resolve this mixing problem. The problem simplifiesconsiderably if we limit ourselves to compute the anomalous dimensions averaged overall the operators with the same bare dimension, which in the following we will denote as h γ m, ¯ m i . In general, this computation could be performed using two equivalent approaches[24]: the large spin perturbation theory of [20,21,23] or the Lorentzian inversion formula of[14,25]. The power of these methods is that they allow to deduce the averaged OPE datain one channel solely from the singularities of the correlator in the crossed channels whichat large N are determined by the exchange of the protected single trace operators. Thus,to compute the anomalous dimensions of the double-trace operators exchanged in the s -channel of the correlator in (3.1), one has to consider only the single-trace contributionsin the u -channel (since in this correlator no single-particle operator is exchanged in the t -channel). As explained in section 3, this protected part is just the affine block of theidentity. The derivation of the averaged anomalous dimensions in a two-dimensionalcorrelator where the identity and the graviton are the only exchanged single-particlestates has already been performed in [33] using the Lorentzian inversion method. Theonly difference with our correlator is that we also have the contribution of the R-symmetrycurrents: it is easy to see how this new contribution modifies the relevant integrand in [33]and so we can obtain the result relevant for our correlator (3.1) from that of [33]. Wefind h γ m, ¯ m i = − ( n + n + 1) with n = min( m, ¯ m ) . (5.3)We verified that these agree with the anomalous dimensions derived by directly expandingthe correlator (3.10) for z → ℓ = | m − ¯ m | >
2. It is known [14,34] that in the large N -expansion the Lorentzianinversion formula can fail to work for ℓ = 0 , ,
2, and indeed we find that the simpleformula (5.3) fails to reproduce the averaged anomalous dimensions for these values of ℓ . For example from the z → h γ , i = h γ , i = h γ , i = − / h γ , i = h γ , i = − / Acknowledgements
We would like to thank Agnese Bissi and Leonardo Rastelli for discussions and corre-spondence, and Luis F. Alday for several helpful comments on a preliminary verion ofthis paper. This work was partially supported in part by the Science and TechnologyFacilities Council (STFC) Consolidated Grant ST/L000415/1
String theory, gauge the-ory & duality . C.W. is supported by a Royal Society University Research Fellowship No.UF160350
A The ˆ D -functions The contact Witten diagram in AdS d +1 with external operators of dimension ∆ i , usuallydenoted as the D -function, is given by D ∆ ∆ ∆ ∆ ( ~z , ~z , ~z , ~z ) = Z d d =1 w √ g Y i =1 K ∆ i ( w ; ~z i )= Γ ˆ∆ − d ! π d/ Z ∞ Y i (cid:20) dt i t ∆ i − i Γ(∆ i ) (cid:21) e P i,j =1 z ij titj , (A.1)where ˆ∆ = P i ∆ i , z ij = ( ~z i − ~z j ) , g is the determinant of the AdS d +1 metric in EuclideanPoincar´e coordinates w ≡ ( w , ~w ) ds = dw + P di =1 dw i w , (A.2)and K ∆ ( w ; ~z ) is the bulk-to-boundary propagator for a scalar field of conformal dimension∆: K ∆ ( w, ~z ) = (cid:20) w w + ( ~w − ~z ) (cid:21) ∆ , (A.3)12ith ~z , ~w points on the d -dimensional boundary. One can define ¯ D -functions, which areindependent of the dimension d and depend on the cross-ratios z and ¯ z (1 − z )(1 − ¯ z ) = z z z z , z ¯ z = z z z z , (A.4)as¯ D ∆ ∆ ∆ ∆ ( z, ¯ z ) = 2 Q i =1 Γ(∆ i ) π d/ Γ (cid:16) ˆ∆ − d (cid:17) | z | ˆ∆ − | z | | z | ˆ∆ − − | z | ˆ∆ − − D ∆ ∆ ∆ ∆ ( ~z , ~z , ~z , ~z ) . (A.5)The ˆ D -functions which we use in the bulk of the article are instead defined in terms ofthe D -functions with d = 2 asˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) = lim z →∞ | z | D ∆ ∆ ∆ ∆ ( z = 0 , z , z = 1 , z = z ) , (A.6)where it is understood that we parametrize a 2-dimensional point ~z i by the complexnumber z i . The relation between ˆ D and ¯ D -functions is thusˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) = π Γ (cid:16) ˆ∆ − (cid:17) Q i =1 Γ(∆ i ) | z | ˆ∆ − − | − z | ˆ∆ − − ¯ D ∆ ∆ ∆ ∆ ( z, ¯ z ) . (A.7)The ˆ D -functions relevant for this article can be reconstructed fromˆ D ( z, ¯ z ) = 2 πiz − ¯ z D ( z, ¯ z ) , (A.8)andˆ D ( z, ¯ z ) = − πi ( z − ¯ z ) (cid:20) z + ¯ zz − ¯ z D ( z, ¯ z ) + log | − z | i + z + ¯ z − z ¯ z i | − z | log | z | (cid:21) , (A.9)where D ( z, ¯ z ) is the Bloch-Wigner dilogarithm D ( z, ¯ z ) = 12 i (cid:20) Li ( z ) − Li (¯ z ) + 12 log | z | log (cid:18) − z − ¯ z (cid:19)(cid:21) . (A.10)The ˆ D -functions have simple transformation properties under exchange of the variouspoints z i . The identities used in the article areˆ D ∆ ∆ ∆ ∆ (cid:18) z , z (cid:19) = | z | ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) , ˆ D ∆ ∆ ∆ ∆ (1 − z, − ¯ z ) = ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) , ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) = | z | ∆ − ∆ − ∆ +∆ ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) , (A.11)13hich also imply, for instance,ˆ D ∆ ∆ ∆ ∆ (cid:18) zz − , ¯ z ¯ z − (cid:19) = | − z | ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) , ˆ D ∆ ∆ ∆ ∆ (cid:18) z − z , ¯ z − z (cid:19) = | z | ˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) . (A.12) B The Mellin amplitudes
Following the convention of ref. [3], the Mellin amplitude of the connected part of acorrelation function is defined through G ( U, V ) = π Z ds πi dt πi U s V t − ∆232 M ( s, t ) Γ (cid:18) ∆ − s (cid:19) Γ (cid:18) ∆ − s (cid:19) × Γ (cid:18) ∆ − t (cid:19) Γ (cid:18) ∆ − t (cid:19) Γ (cid:18) ∆ − u (cid:19) Γ (cid:18) ∆ − u (cid:19) , (B.1)where the cross ratios U, V are related to z, ¯ z via U = | − z | , V = | z | , and M ( s, t ) isthe Mellin amplitude. We have defined ∆ ij = ∆ i + ∆ j . Here s, t, u are variables in Mellinspace, which satisfy the constraints s + t + u = P i =1 ∆ i ≡ ˆ∆. In the flat space limit,they play the role of Mandelstam variables of scattering amplitudes.To compute the Mellin amplitudes of the two terms in (3.7) in the main text, we willuse the Mellin transformation of the ˆ D -function, which is given byˆ D ∆ ∆ ∆ ∆ ( z, ¯ z ) = Γ ˆ∆ − d ! π d/ Q j =1 Γ(∆ j ) Z ds πi dt πi U s V t Γ (cid:16) − s (cid:17) Γ (cid:18) − t (cid:19) (B.2) × Γ (cid:18) ∆ + s + t (cid:19) Γ (cid:18) ∆ − ∆ − s (cid:19) Γ (cid:18) ∆ − ∆ − t (cid:19) Γ (cid:18) ∆ − ∆ + s + t (cid:19) , where ∆ ijk = ∆ i + ∆ j + ∆ k . Starting with the term proportional to ˆ D , we have | − z | ˆ D ( z, ¯ z ) = π Z ds πi dt πi V t U s +1 Γ (cid:16) − s (cid:17) Γ (cid:18) − t (cid:19) Γ (cid:18) s + t (cid:19) . (B.3)Shifting the integration variables by t → t − s → s −
2, we obtain | − z | ˆ D ( z, ¯ z ) = π Z ds πi dt πi V t − U s Γ (cid:16) − s (cid:17) Γ (cid:18) − t (cid:19) Γ (cid:16) − u (cid:17) . (B.4)Comparing with (B.1), we see that the Mellin amplitude of | − z | ˆ D ( z, ¯ z ) is simply 1.14et us now consider the other contribution in (3.7): | − z | | z | ˆ D ( z, ¯ z ) = π Z ds πi dt πi V t +1 U s +1 Γ (cid:16) − s (cid:17) Γ (cid:18) − t (cid:19) × Γ (cid:18) s + t (cid:19) Γ (cid:18) − t − (cid:19) . (B.5)A similar change of integration variables ( t → t − s → s −
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