Thermal correlation functions in CFT and factorization

We study 2-point and 3-point functions in CFT at finite temperature for large dimension operators using holography. The 2-point function leads to a universal formula for the holographic free energy in $d$ dimensions in terms of the $c$-anomaly coefficient. By including $\alpha'$ corrections to the black brane background, one can reproduce the leading correction at strong coupling. In turn, 3-point functions have a very intricate structure, exhibiting a number of interesting properties. In simple cases, we find an analytic formula, which reduces to the expected expressions in different limits. When the dimensions satisfy $\Delta_i= \Delta_j+ \Delta_k$, the thermal 3-point function satisfies a factorization property. We argue that in $d>2$ factorization is a reflection of the semiclassical regime.


Introduction
Conformal Field Theories play a pivotal role in Quantum Field Theory as endpoints of trajectories of the Renormalization Group. They also govern the near-critical behavior of phase transitions occurring in relevant models of statistical mechanics and in physical theories. Introducing finite temperature is equivalent to considering a d-dimensional CFT placed in euclidean S 1 × R d−1 , which implies a periodic identification of the euclidean time, τ = τ + β. The response to this non-trivial geometry provides an extra probe of the structure of the theory beyond the usual quantization on R d (or its conformal cousins such as R × S d−1 ), which reveals important aspects of the dynamics of the theory.
If one considers two operators inserted within a ball of radius much smaller than β -so that effectively they see flat space around -it is possible to use the zero-temperature OPE. This was studied in general in [1] (see also [2]) for the case of the 2-point function, finding an elegant expansion in terms of Gegenbauer polynomials, whose coefficients incorporate both the OPE data as well as the non-zero vacuum expectation values in the thermal background. For strongly coupled theories with a gravity dual this structure naturally arises from the computation of the 2-point function in the black brane background, as recently shown for operators of large dimension in [3]. 3 It is interesting to note that, through the Eigenstate Thermalization Hypothesis, the thermal 2-point function is related to Heavy-Light-Light-Heavy 4-point correlators at T = 0 (e.g. [5,6,7,8]).
The main object of interest in this paper will be thermal (connected) 3-point functions in holographic theories. While at T = 0 the structure of three-point functions is completely fixed by conformal invariance (just like for 2-point functions), at finite temperature they have an extremely complicated structure, involving multiple regimes.
Holographic n-point functions can be computed by means of an expansion in Witten diagrams in the appropriate gravitational background. 3-point functions are distinguished from higher-point functions because there is only one possible Witten diagram, involving a cubic coupling. As a result, the value of the bulk cubic coupling enters only as an overall constant in the final result. Just as in [3], we will focus on correlators of operators with large dimension. In this case, the propagator can be computed in the WKB approximation, where it is represented as the exponential of the geodesic length of the trajectory for a particle moving between the boundary point where the operator is inserted up to the point of interaction. At T = 0, the 3-point function has been computed using this technique in e.g. [9,10,11]. A number of new interesting issues arise when extending the method to finite temperature, as we shall discuss.
Thermal correlation functions contain very interesting information about quantum systems including their chaotic nature (see e.g. [12] for a recent application). In holographic theories, it has been shown that they can probe the interior of the black hole in the gravitational dual (e.g. [13,14,15,16,17,18,19,20,21,3]). In particular, they are expected to shed new light on the nature of spacetime singularities, although this may require going far beyond the leading strong coupling order. We leave these studies for future work.
This paper is organized as follows. In section 2 we discuss the general set-up for the computation of 3-point functions for operators of large dimension using the geodesic approximation. We define ingoing and returning geodesics and describe the different contributions to the saddle-point equations. Details of the derivations are relegated to appendix A. In section 3 we adapt the discussion to the case of (connected) 2-point functions in d-dimensions. By matching with the expected form for the 2-point function in QFT we obtain a formula for the free energy of holographic theories as a function of their central charge c T . Using the known value of the free energy for holographic theories in [22], we test this formula by matching the central charges in d = 3, 4, 6 against the known values read off from e.g. [23,24,25]. For the d = 4 case, we discuss the leading α correction to the free energy, reproducing the results of [22]. In section 4 we initiate the study of 3-point functions, concentrating on a symmetric configuration, for which analytic results can be found. In section 5 we study the general three-point correlation function of scalar operators. We show that in an extremal limit they satisfy a remarkable factorization property.

Holographic 3-point functions: setting up the stage
We are interested on thermal correlation functions of scalar operators in holographic ddimensional CFT's. In particular, we will examine two-point functions and the three-point function, To simplify the the problem, we will mostly focus on the case where operators are inserted at the same spatial point, which, with no loss of generality can be taken to be 0, that is, we will consider x i = (t i , 0). On general grounds, the finite temperature state is dual to a black brane in AdS d+1 We reserve t for the time coordinate with Lorentzian signature and τ for the Wick-rotated euclidean coordinate. Each operator is dual to a fluctuating scalar field φ i in the black brane in AdS d+1 with its mass related by the standard holographic formula to the dimension of the dual operator, We shall consider operators of large scaling dimensions. For such operators, ∆ ≈ mR when mR 1. The bulk lagrangian is then obtained by expanding the corresponding gravitational action in the fluctuations, where terms higher-than-quadratic in the fluctuations -suppressed by 1 N in terms of CFT variables-give rise to bulk Witten diagrams contributing to higherpoint correlators in the boundary theory. Note that 3-point functions are somewhat special in that only cubic vertices contribute to them. Hence the relevant part of the bulk lagrangian looks like The λ ijk couplings depend on the particular model. However, since such couplings will only appear as overall constants in front of the 3-point function, their precise value will not be relevant in this paper. In order to compute the 3-point function holographically starting with (2.3) we should evaluate the corresponding 3-leg Witten diagram. Essentially where G ∆ i (x i , u) is the boundary-to-bulk propagator for a field dual to an operator of dimension ∆ from x i in the boundary to the bulk point u = ( x, z). This propagator comes from solving, with the appropriate boundary conditions, the Klein-Gordon equation in the black brane background. Such computation dramatically simplifies in the case of ∆ i 1. In that case one can approximate G ∆ (x, u) by the WKB approximation, which leads to being S ∆ (x, u) the action for a particle of mass m ∼ R −1 ∆ which departs from the boundary at x and arrives to the bulk point u. Alternatively, this can be thought of as the length (x, u) of a geodesic from x to u. Thus, for ∆ i 1 (but still, ∆ i c T ), the three-point correlation function of interest can be computed as where u now represents the intersecting point of the three geodesic arcs. Under the assumption ∆ i 1, the integral over the bulk point u is determined by a saddle-point calculation. Thus, one finds where S os stands for the "action" S = ∆ 1 (x 1 , u I ) + ∆ 2 (x 2 , u I ) + ∆ 3 (x 3 , u I ) evaluated on-shell -i.e. at the saddle-point u I , solving the equation dS/du = 0.
Our first task is to compute the boundary-to-bulk propagator in the black brane background, a.k.a. the geodesic arcs for the case of ∆ 1. Since we consider all operators inserted at 0, it is clear that the spatial part of u I at the saddle point will be at the origin. We can thus focus on geodesics which move in the plane (z, t). The action for a massive particle moving along t whose worldline is parametrized by z is given by whereṫ = dt/dz. Since the action does not depend on t, the corresponding conjugate momentum -which will be denoted as p t = ∆ µ -is conserved. This gives a first-order equationτ where we have introduced the euclidean time τ = it. The signs ± in (2.9) respectively describe either a geodesic going towards the black hole or a geodesic going towards the boundary.
The parameter µ can be viewed as an integration constant, which is determined by demanding that the geodesic departs from (τ = τ 1 , z = 0) and reaches the intersection point (τ I , z I ). At this point, This can be viewed as a condition that implicitly defines µ = µ(τ I , z I ). The equation (2.10) cannot be inverted explicitly to find an explicit formula for µ(τ I , z I ), except in the case of d = 2. Nevertheless, the implicit definition implied by the condition (2.10) is enough in order to determine the saddle point and thereby the three-point correlation function. From (2.9), it is clear that the maximum reach inside the bulk of the geodesic, i.e. the turning point where dz/dτ = 0, is given by the equation In (2.10) we assumed that the sign describing the geodesic going towards the black hole. These will be called ingoing geodesics, that is, geodesics departing from the boundary at x and arriving to u I before getting to z max . For the calculation of the three-point function, configurations where geodesics reach z max and then meet u I in their return way will also be relevant. These will be called returning geodesics. These two types of geodesics are shown in fig.(1). The analysis of each case is carried out in appendix A. The contribution to the action for each type of geodesic is, respectively, given by (2.13) As discussed above, here µ is to be understood as an integration constant, which, through the equation of motion (see appendix A), has to be tuned so that the geodesic arc meets the desired boundary conditions: departing the boundary from the corresponding point x and arriving at the intersection point u I .
The total action in (2.7) will be a sum of geodesic arcs, each of them either ingoingcontributing as (A.5)-or returning -contributing as (2.13). The choice of ingoing/returning is determined by the dynamics.
The formula (2.7) for the three-point function requires to evaluate the u integral through the saddle-point method, which implies an extremization of the total action in the τ and z direction. Using the formulas of appendix A, we obtain (2.14) where = 1 for ingoing geodesics and = −1 for returning geodesics.

Thermal 2-point functions in d dimensions
The 2-point functions of operators with ∆ 1 can be computed in terms of a single geodesic departing from the boundary from a point x 1 , making an excursion into the bulk and coming back to another point −x 1 . One writes (3.1) The geodesic is uniquely determined in terms of x 1 . Alternatively, the two-point function can also be computed using the formalism developed above. We view the geodesic as two geodesic arcs joining at some bulk point u (see e.g. [26,27,28]). Then the two-point function can be computed as the integral The integral can be evaluated by a saddle-point approximation, with the action being S = ∆ (x 1 , u) + ∆ (x 2 , u). By symmetry, the saddle-point must be located at ( 0, z I ).
Since both geodesics are ingoing, 1 = 2 = 1. Thus we get the condition This is precisely the maximum reach for the geodesic, see (2.11), which implies that the arcs are just the arms of a single U-shaped geodesic, as expected, in this way reproducing the result obtained by a direct calculation of the action (3.1).
Let us now consider the explicit calculation of the correlator . Then, using the formulas from appendix A, we have The correlation function is found by substituting into exp(−iS) the expression of z I in terms of τ I , implicitly defined by (3.5). Since (3.5) cannot be inverted in general, we can work in perturbation theory in the temperature. This is equivalent to a short-distance expansion of the correlator and it amounts to expanding (3.5) in powers of z 2 . We find This gives where τ = τ 1 2 .

Universal formula for the free energy
The thermal 2-point function, in the regime T |x| 1, contains interesting information The key point is that the S 1 × R d−1 geometry provides, through the circle, both a special direction and a scale allowing for non-zero VEVs of the form, where η ≡ τ /|x| and C (ν) ). As noted in [3] for the d = 2, 4 cases, the thermal 2-point function (3.7) becomes exact in the double-scaling limit, In this limit, the O (T τ ) d terms in (3.7) vanish. The full spacetime dependence can now be restored using the consistency with the OPE (3.11). This implies that the exponent must be the x = 0 limit of the second Gegenbauer polynomial (this was shown explicitly to be the case in d = 2, 4 in [3]). Therefore we can write Let us consider explicitly the case of d = 4. Expanding the exponential, one finds where the numerical coefficients C J,n turn out to be given as well in terms of Gegenbauer polynomials as if n, J = 0 (3.15) Comparing with the general form of the correlator, each term in the series in (3.13) represents the contribution of a spin J and dimension 4n operator constructed as the appropriate contraction of n factors of energy-momentum tensors, which we will denote schematically as T n . Clearly, the spin of such contraction is at most 2n. Consistently, the coefficients in (3.15) vanish if J > 2n. Moreover, the contribution of each term scales with ∆ n = ∆ ∆ T n 4 , which is precisely the large ∆ scaling expected [5] (see also [29]). Note that this extends to arbitrary dimensions, as the expansion of (3.13) involves ∆ n = ∆ The leading term in the expansion of (3.13) corresponds to the J = 2 term in (3.11). This yields the identification a T = A d . The OPE coefficient f OOT with the energymomentum tensor is fixed by a Ward identity. One has [1] .
Thus, combining with (3.11), we find a universal formula for the vacuum expectation value for the energy density This formula can be alternatively derived from the thermodynamics of black holes supplemented with the AdS/CFT dictionary relating the Newton constant and c T as in e.g. [30].
Using the standard thermodynamic relation, F = E + T dF dT , we find We can now combine this formula with the general expression for the free energy given in [22] (see also [31]) to provide an independent derivation of the central charge for ABJM, N = 4 SYM and the 6d (2, 0) theory for d = 3, 4, 6 respectively. We find Note that these central charges are in the convention where a free real scalar contributes as (3.20) For N = 4 SYM, using [23] with n φ = 6 real scalars, n ψ = 4 2 Dirac fermions and one gauge field in d = 4, one precisely recovers the expression of c N =4 T given above. Consider now ABJM theory. It is convenient to divide by c free scalar T , so that we find the central charge in the conventions where a free real scalars contributes 1 -we will denote it with a hat. We findĉ in agreement with [25] for k = 1 (see eq. (5.11)) (see also [24]). Finally, for the 6d (2, 0) theory, note that in our conventions, a free tensor multiplet contributes as [24]  Thus, in units where the free tensor multiplet contributes 1, we havê reproducing the known result.

Including α corrections in the d = 4 case
One of the famous problems in AdS/CFT concerns the match between the holographic free energy and the free energy derived from N = 4 super Yang-Mills theory. As known, the weak-coupling free energy differs from the strong-coupling (holographic) result by a factor of 3/4. It is believed that both limits arises from a monotonic function f (λ), which interpolates between 1 and 3/4. The leading strong-coupling correction to f (λ) was computed in [22] by computing the explicit modifications to the black brane geometry induced by the α 3 R 4 corrections to the type IIB effective action. In the present formalism, we may similarly include corrections by considering the geodesic moving in the α corrected background. We will ignore other possible sources of α corrections, such as possible higherderivative interactions of the bulk scalar field, assuming that the leading-order correction comes from the α corrected background. The first observation is that α corrections arise in powers of z 4 . Thus the net effect of the α corrections will be to change the a OOO coefficients, as opposed to turning on new terms (for instance, a term proportional to C (1) 0 corresponding to a dimension 2 scalar is not induced, as it would require a z 2 z 2 0 correction). As a result of corrections to the background, the relation between z 0 and the temperature also gets α corrections. It becomes term in the expansion. Therefore, the leading α correction to the energy density can be obtained from the O(z 4 ) correction in the metric. Formally, one gets the same metric as in the unperturbed case, by trading z 4 0 by z 4 0,corrected (1 + 75γ) −1 , where we have included the subscript to emphasize that this is the α corrected z 0 given in (3.24). Using (3.24), we obtain the re-scaling, πT (1 + 15γ) 1 4 .

(3.25)
Thus, This leads to the free energy This is precisely the leading correction at strong coupling found in [22].

Thermal 3-point correlation functions with reflection symmetry
Let us now consider thermal 3-point functions at coincident spatial points. According to the previous discussion, this is computed holographically as where S = i=1,2,3 S ∆ i (τ i , u I ), being S ∆ i (τ i , u I ) the contribution of an ingoing/returning geodesics from τ i at the boundary to the intersection point u I -to be fixed through extremization of S. With no loss of generality, one can set τ 3 = 0. Then the relevant geodesic configuration is dynamically fixed and it depends on the values of τ 1 , τ 2 and ∆ 1,2,3 (more specifically, on two ratios, e.g. ∆ 1 /∆ 3 , ∆ 2 /∆ 3 ). The key equation is with ∆ 1 µ 1 + ∆ µ 2 + ∆ 3 µ 3 = 0. For given values of the parameters, this equation will have a unique real solution z I ∈ (0, 1), for a specific choice of signs 1,2,3 (and other choices of sign will give no real solution z I ∈ (0, 1)). This determines which geodesics are ingoing and which ones are returning. For example, for τ 1 sufficiently close to τ 2 , the geodesics for operators 1 and 2 must be ingoing and the geodesic 3 must be returning. On the other hand, if we choose τ 3 = 0, τ 1 and τ 2 sufficiently separated and ∆ 3 < ∆ 1 + ∆ 2 , geodesics 1 and 2 are returning and 3 is outgoing. This will be illustrated by several examples in the following sections.
We begin with a case where calculations dramatically simplify. This is the case of a  By symmetry, τ I = 0, therefore the ingoing geodesic corresponding to O ∆ (0) is actually a straight line at constant τ = 0 that extends from the boundary up to z I . This fixes the corresponding µ = 0, and, through the second equation in (2.14), sets µ 1 = −µ 2 ≡ µ. Therefore, we are left with the first equation, which becomes

Symmetric three-point function in d = 2
As a warm-up, we start considering the symmetric thermal three-point function in the simpler d = 2 case. We consider the configuration of fig. (2). In two dimensions, the integral defining the geodesic τ (z, µ) gives simple logarithmic functions, which permits to find an explicit closed formula for µ in terms of the boundary coordinate τ . The saddlepoint equation for z is solved by Substituting into the equation for the geodesic, at the intersection point we have 0 = τ (z I , µ). This can be explicitly solved for µ. We find (z 0 = 1) Substituting into the action, we obtain where the temperature has been restored, recalling that z 0 = 1/(2πT ). This agrees with the result of [32]. Note that, in the "extremal" limit ∆ 0 → 2∆, one obtains 4 which is, up to a ∆-dependent constant, the product of two 2-point functions, We will return to this factorization property in sec. 5. Beyond the extremal limit, when ∆ > 2∆, the saddle point no longer lies within the region 0 < z < z 0 . The correlation function can be defined by analytic continuation and it is of the same form (4.6) (for 2∆ < ∆ < 2∆/z max , the saddle point is imaginary; for ∆ > 2∆/z max , it lies inside the horizon).

Symmetric three-point function in d = 4
In d = 4 the solution to (4.4) is Here we have chosen units where 1 = z 0 = β/π. We assume δ < 1, that is, the 'nonextremal case' ∆ < 2∆. As in the d = 2 case, for ∆ > 2∆ the correlator is defined by analytic continuation. Substituting (4.9) into the geodesic equation τ = τ (z, µ), we find the condition at the intersection point, 0 = τ (z I , µ). This can be solved for µ in series of small τ 1 . We get (4.10) Substituting this into S one finally finds where the temperature has been restored. As a sanity check, in the T = 0 limit we find , which is precisely the expected result (cf. (5.1)).
Let us now consider different limits. For the hierarchy ∆ ∆ , to leading order we have where τ = 2τ 1 . This approaches the 2-point function On the other hand, in the 'extremal' limit ∆ = 2∆, we find 5 General three-point correlation function and factorization limit

Factorization
In the previous section, we considered the symmetric 3-point function . The symmetric configuration makes it possible to find an analytic formula, which is exact in the d = 2 case, and in the d = 4 case it describes a T |τ | 1 regime. Both in d = 2, 4 the three-point function satisfies a surprising factorization property in the "extremal" limit ∆ → 2∆. In this subsection we will clarify the origin of this factorization and study it for general three-point functions.
At T = 0, the factorization property is manifest from the generic form of the 3-point function, which is entirely fixed by conformal symmetry, (5.1) Setting, for example, ∆ 3 = ∆ 1 + ∆ 2 , one gets However, at T = 0 the form of the 3-point function is not fixed by symmetries and it has an extremely complicated general structure.
To start with, let us first consider a generic three-point correlation function with only time-dependence. With no loss of generality, we can place one of the operators at τ = 0. Thus we consider correlators of the form O ∆ 1 (−τ 1 ) O ∆ 3 (0) O ∆ 2 (τ 2 ) and, for concreteness, we will initially assume that with τ 1 > 0 and τ 2 > 0. For large dimensions, ∆ i 1, the correlation function is represented by three geodesic arcs meeting at some point (τ I , z I ) in the bulk. An examination of the saddle-point equation shows that, for ∆ 3 < ∆ 1 + ∆ 2 , and ∆ 3 sufficiently close to ∆ 1 + ∆ 2 , the operator O ∆ 3 is described by an ingoing geodesic while O ∆ 1,2 are described by returning geodesics, generalizing the picture of fig. 2. The saddle point equations then read Other choices of relative signs do not have real solution 0 < z I < 1 in the region ∆ 3 = ∆ 1 + ∆ 2 − . The saddle-point equation (5.6) can be solved explicitly in terms of one of the roots of the quartic polynomial for X ≡ z 2 , obtained by squaring the saddle-point equation. For sufficiently small τ 1 and τ 2 , as compared with the scale of the thermal circle β, there is a unique root that represents the dominant saddle point and describes the expected short-distance behavior of the correlation function.
In the region ∆ 3 ∼ ∆ 1 + ∆ 2 , it has the behavior Note that z I is real for ∆ 3 ≤ ∆ 1 + ∆ 2 , and imaginary for ∆ 3 > ∆ 1 + ∆ 2 . At the critical point ∆ 3 = ∆ 1 + ∆ 2 , we have z I = 0, and the three point correlation function exhibits the remarkable factorization property: where µ k (τ k ), k = 1, 2, are implicitly defined by The factorization limit is illustrated in figs. 3 and 4. One can see that the geodesic describing the operator O ∆ 3 shrinks to zero as ∆ 3 ∼ ∆ 1 +∆ 2 . In this limit, the configuration reduces to two individual geodesics ending at the same point τ 3 = 0, which represent the

Adding x dependence
Let us now incorporate the x dependence. The details are given in appendix A. In the general case, we find the following saddle-point equations where, again i = ±1 according to whether the corresponding geodesic is ingoing or returning at the point z I .
Let us assume that O 1 (τ 1 , x 1 ) and O 2 (τ 2 , x 2 ) are represented by the returning geodesics, and O 3 (τ 3 , x 3 ) by an ingoing geodesic. The factorization limit then occurs when z I → 0. The saddle-point equation for z I then requires In such a case the ingoing geodesic shrinks to zero and one has Thus the factorization of the three-point function seems to be a general property, which occurs in the limit 3 i=1 i ∆ i → 0. As the location points of the operators move around in the boundary, there can be transitions where a returning geodesic becomes ingoing, and vice-versa (see fig. 5). A critical point occurs whenever the intersection point coincides with one of the z max of the three geodesics. A simple examination shows that this never happens in the symmetric case of section 4. For example, the condition, z Assuming µ 2 < 0, the geodesic 1 is returning for µ 1 < µ c 1 and ingoing for µ 1 > µ c 1 . For the symmetric case µ 1 = −µ 2 , ∆ 1 = ∆ 2 , and it is not possible to have a transition (geodesic 1 is always returning) It would be very interesting to study this phenomenon in more detail, in particular, the analytic properties of such transitions in the infinite ∆ i limit. On general grounds, one expects that the three-point function should be analytic in x 1 , x 2 , x 3 away from the coinciding points. This is also supported by the fact that ingoing-returning transitions also take place at T = 0, with the same critical value µ c 1 . In the T = 0 case, the well-known 3-point function has a simple form, as being completely fixed by conformal invariance. Clearly, it is an analytic function of x 1,2,3 away from the coinciding points.
In addition to ingoing-returning transitions, as the separation of the operators along the thermal circle increases and the geodesics go further into the bulk, other saddle points have to be considered. This is analogous to what happens for 2-point functions in [14,3]. Although in this paper we restricted the considerations to distances less than the critical distance corresponding to possible saddle-point jumps, it would clearly be very interesting to study these aspects in detail.

Factorization property as a sign of a semiclassical limit
At T = 0, extremal correlators have been extensively discussed, albeit mostly in connection with subtleties related to their prefactors: in top-down constructions from type IIB string theory [33], the overall coefficient typically vanishes while at the same time the coupling constant diverges in such a way that these effects cancel leaving a finite result [34,35].
As far as their spacetime dependence is concerned, because their form is completely fixed by conformal symmetry, factorization arises as a trivial consequence. It is striking that the factorization property of extremal correlators still seems to hold in the general T = 0 theory, where three-point functions, as well as two-point functions, have a much more complicated structure.
To investigate the implications of this property, let us suppose that we start with a general CFT and take, as a fact, that extremal three-point (connected) correlation functions factorize as where all operators are assumed to be primaries. Consider the x 2 → x 1 limit. In this limit we can replace O ∆ 1 (x 1 ) O ∆ 2 (x 2 ) by the OPE, which would express the RHS of (5.9) as a sum over two-point functions.
Let us make the extra assumption that two-point functions of non-identical operators vanish. 5 The relevant term of the OPE is Then, substituting into (5.9), one obtains

(5.11)
This is precisely the form of the 2-point function that arises from the geodesic approximation. Thus we see that the factorization property implicitly requires the semiclassical approximation of large ∆. At finite ∆, correlation functions do not have, in general, the form (5.11). An exception is d = 2, where the exact thermal two-point function is of the form (5.11), in which case the factorization (5.9) also holds for operators of small dimension. For d > 2, the factorization property for extremal correlators does not generally hold beyond the large ∆ approximation.
On general grounds, (x) must be of the form 1], the f δ can naturally be expressed in terms of Gegenbauer polynomials, which are orthogonal functions in [−1, 1]. In this way one recovers the general form discussed in [1]. A consequence of the general form (5.11) is that the double scaling limit of [3], T |x| → 0, ∆ → ∞, with fixed ∆(T |x|) δ 0 , exists and is well defined (in the strong coupling expansion, one has δ 0 = d as in (3.12)).
The action for returning geodesics can be computed by considering the action of a complete geodesic departing the boundary, entering the bulk and coming back to the boundary, subtracting the last piece from z I to the boundary. Thus, we can write In computing these derivatives, we must take into account the fact that µ and ν are functions of z I , τ I , x I (and of τ 1 , x 1 ) defined by the conditions (A.6). We will need to use the following identities: In addition, by taking the differential of (A.6) (or of the analog equations for the returning case) one obtains the following "constitutive" relations, Using now the explicit expressions of S, I τ and I x , we finally get