Correlation functions in finite temperature CFT and black hole singularities
aa r X i v : . [ h e p - t h ] M a r ICCUB-21-002
Correlation functions in finite temperature CFTand black hole singularities
D. Rodriguez-Gomez a,b and J. G. Russo c,d a Department of Physics, Universidad de OviedoC/ Federico Garc´ıa Lorca 18, 33007 Oviedo, Spain b Instituto Universitario de Ciencias y Tecnolog´ıas Espaciales de Asturias (ICTEA)C/ de la Independencia 13, 33004 Oviedo, Spain. c Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)Pg. Lluis Companys, 23, 08010 Barcelona, Spain d Departament de F´ısica Cu´antica i Astrof´ısica and Institut de Ci`encies del CosmosUniversitat de Barcelona, Mart´ı Franqu`es, 1, 08028 Barcelona, Spain
ABSTRACT
We compute thermal 2-point correlation functions in the black brane
AdS backgrounddual to 4d CFT’s at finite temperature for operators of large scaling dimension. We finda formula that matches the expected structure of the OPE. It exhibits an exponentiationproperty, whose origin we explain. We also compute the first correction to the two-pointfunction due to graviton emission, which encodes the time travel to the black hole singu-larity. [email protected] [email protected] ontents x dependence . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The d = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 d = 2 The study of conformal field theories (CFT’s) at finite temperature is of general interest,not only for practical applications –as in real life critical points are at finite temperature–but also because finite temperature may reveal new aspects of the theories. On generalgrounds, we may explore the theory on different manifolds, and a common paradigm is S β × R d − . When the circle of length β is assumed to be the euclidean time direction, thisis equivalent, provided the appropriate boundary conditions are set, to the CFT at finitetemperature T = β − . Then, n -point functions probe new important properties not visibleon R d .In the following we will focus on the case of the 2-point function of an operator O ofdimension ∆, that is, on h O (0) O ( τ, ~x ) i . Introducing | x | = τ + ~x , at least in the regimewhere | x | ≪ β – i.e. when the distance between the two insertions is much smaller thanthe size of the thermal circle – it is justified to use the OPE. However, unlike the R d case,the size of the S provides a scale and allows operators to take a VEV. Thus, the 2-pointfunction unveils an interesting structure which permits to probe physical properties ofthe CFT. General features of the thermal 2-point function have been recently discussed in[1, 2], where it was argued that the 2-point function must admit an expansion in Gegenbauer2olynomials, which play the role of conformal blocks on S × R d − . The coefficients in thisexpansion can be related to microscopic details of the theory: the structure constants ofthe OPE, the coefficients of 2-point functions on R d and the VEVs which are non-vanishingon S β × R d − . This structure has been studied in weakly coupled theories including the O ( N ) model at large N [2] and free theories [3]. It was also studied using holography [4]on thermal AdS . However, because the stress tensor of the CFT acquires a VEV at finitetemperature, the proper gravity dual should be in terms of a black hole in
AdS . Motivatedby this, in section 3 we will compute, holographically, 2-point functions in 4d CFT’s dualto the black brane
AdS background. This computation is closely related to the calculationof 4-point correlation functions of 2 heavy and 2 light operators, discussed in [5]. In orderto simplify the problem, we will concentrate on operators of large conformal dimension forwhich we can use the geodesic approximation. The problem then becomes tractable andone can find a systematic expansion for the correlator in terms of the expected Gegenbauerpolynomials. The expansion exposes the corresponding coefficients, which are universal forany CFT at strong coupling with a holographic dual. In particular, that corresponding tothe energy-momentum tensor block can be compared with the expectations, reassuringlywith perfect agreement. One by-product of our analysis is that the leading terms in the T | x | ≪ ≫ d = 2 case (and in the free theoryin d = 4, albeit with a different exponentiated block). Similar exponentiation propertyfor 4-point functions was observed in CFT [6] (see also [7, 8]) and in CFT [9]. For thepresent case of 2-point functions, we shall see that one can get a detailed identification ofthe exponentiated block.Another important motivation for studying CFT’s at finite temperature is that theyprovide, through holography, a description of black holes. Indeed, a major challenge inAdS/CFT is understanding how detailed aspects of bulk physics, such as the physics in theblack hole interior, are ciphered in CFT correlation functions. Progress in this directionmay shed light on some of the most tantalizing problems in physics, regarding the nature ofspacetime singularities and other fundamental issues of quantum gravity. This problematicwas undertaken since the beginning of AdS/CFT and, since then, many important advanceshave been achieved (see e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18, 9, 19, 20, 21]).Recently, Grinberg and Maldacena [21] suggested that geometrical quantities such as theproper time from the event horizon to the singularity can be read off from the behavior ofthermal expectation values of CFT operators. It is natural to wonder whether higher-pointfunctions encode, in a similar manner, information about the singularity. This questionhas, of course, been asked in the past for 2-point functions [12, 13], where it was shownthat indeed 2-point functions contain, in a subtle way, information about the interior ofthe black hole.The observation in [21] that one-point functions are sensitive to the black hole interiorstems from the fact that the bulk theory contains a higher derivative coupling between thebulk field φ dual to O and W – the square of the Weyl tensor – which is non-zero on blackbrane backgrounds. This term is biggest in the vicinity of the singularity and the maincontribution to the one-point function comes from this region. As a result, the one-point3unction of O is proportional to e − i ∆ τ s , being τ s the proper time to the singularity. Thusone can read geometrical properties of the black hole interior just by looking at thermal1-point functions in the CFT. It is natural to expect that thermal 2-point functions, whichhave a more complicated structure, may contain additional information on the black holegeometry. Here we shall consider a model in the 2-point function mediated by a bulkcoupling | φ | W for four-dimensional CFT’s. We assume a U (1) symmetry, which impliesthat the first higher-derivative coupling must be of the suggested form. An example is abulk field φ representing a chiral primary operator which carries U (1) R charge. When φ isa bulk field of large mass, the corresponding operator has large conformal dimension andwe can use the geodesic approximation. Then, as we shall see, a similar mechanism as inGrinberg-Maldacena holds in the present case, by which the correlation function at largedistances seems to explore the vicinity of the singularity,The organization of this paper is as follows. In section 2 we review the computation of2-point functions in the geodesic approximation focusing on the T = 0 case, in particularre-formulating well-known results in a language which will be more suited for the T = 0case. In section 3 we then turn to correlators at non-zero temperature. After reviewingsome relevant facts from [2], we proceed to compute the 2-point correlation function atnon-zero T . We first do so for operators inserted at equal ~x , i.e. correlators with only(euclidean) time dependence, finding in particular the announced exponentiation. In orderto determine the full space dependence, we study the same correlator with general ~x - and τ -dependence in perturbation theory in a small temperature expansion. This allows usto explicitly identify the general expansion in terms of Gegenbauer polynomials. As aby-product, we cross-check our result with the coefficient of the energy-momentum tensorblock, finding perfect agreement. The result also shows that the exponentiation is of the fullenergy-momentum tensor block. We then show that this exponentiation takes place as wellin CFT (for completeness, we review the computation of the 2-point correlator in CFT through the geodesic approximation in appendix A). In section 4 we include the higher-derivative interaction ¯ φφW and argue that, in a suitable regime, the correction due to theinteraction permits to read the proper time to the singularity. It also exhibits resonanceseffects in graviton emission due to normal frequency modes. Finally, we conclude in section5 with a summary and comments. On general grounds, the correlation functions of a CFT with a holographic dual can besystematically constructed in an expansion in Witten diagrams, where the vertices canbe directly read off from the lagrangian of the bulk theory. In turn, the propagators –either bulk-to-boundary for the “external legs” or bulk-to-bulk for “internal lines”– areconstructed by solving the appropriate equations for the Green’s functions. As it is wellknown, for operators of large conformal dimension, such propagators can be approximatedby the exponential of minus the geodesic length between the corresponding points (seefor instance [10, 11]). Then, the Witten diagram looks like a collection of geodesic arcs4eeting at the vertices, which are integrated over all the bulk geometry. This picturecan be extended as well to 2-point functions of an operator O with conformal dimension∆. They are described by geodesics which penetrate the bulk departing from a boundarypoint x (1) b and ending at another boundary point x (2) b . However, it is possible to regardthis single geodesic as the junction of two geodesic arcs, one from x (1) b to some bulk point x and another from x to x (2) b upon integration over the junction point x , i.e. h O ( x (1) b ) O ( x (2) b ) i = Z bulk dx G ( x (1) b , x ) G ( x (2) b , x ) , (2.1)where G ( x ( i ) b , x ) = e − iS os ( x ( i ) b ,x ) , being S os ( x ( i ) b , x ) the product of the conformal dimension∆ times the (lorentzian) lenght of the geodesic from x ( i ) b to x ; or, equivalently, the on-shellaction for a particle of mass m ( ∼ ∆) which travels between x ( i ) b and x . Moreover, inthe limit of large conformal dimensions, the integral over the junction point x can be donethrough a saddle point approximation, which significantly simplifies the computation ofthe 2-point correlation function. As a warm-up, let us review the computation of 2-point correlation functions at zero tem-perature in
AdS d +1 [22, 23, 24] (see also [25, 26, 27] for applications to 3-point functions).The (euclidean) metric is ds = R z (cid:0) d~x d + dz (cid:1) . (2.2)Let us first construct each of the G ( x ( i ) b , x ) entering in (2.1). To that matter, using the SO ( d ) symmetry in the euclidean geometry, we consider a particle of mass m movingalong the x direction and penetrating in AdS d +1 . Note that the usual formula relatingthe conformal dimension to the mass, in the limit of large dimension, becomes ∆ = mR .Then, in the gauge where the trajectory is parametrized by z , the action is S = − i ∆ Z dz z − q ~x , (2.3)where dots stand for z -derivative. Note that ip x , the momentum canonically conjugatedto x , is conserved. The on-shell action is then obtained by integrating S up to a point z –which will eventually be the point where the two arcs meet. One finds S os = i ∆ log h ǫ z (cid:0) ∆ + p x s ∆ p x − z (cid:1)i . (2.4) The computation of the correlator requires to extracting a factor of ǫ , where ǫ is a regulator repre-senting the distance to the boundary, so that the ǫ →
5n turn, the conservation of p x gives a first order equation that can be easily solved: x = x + ∆ p x − s ∆ p x − z . (2.5)Here we have imposed the boundary condition that at the boundary z = 0, x (0) = x .Solving this equation for p x , one has p x = 2∆ ( x − x )( x − x ) + z . (2.6)This allows us to evaluate the action S from the boundary point (regulated at z = ǫ ) x = x , up to a generic z . We obtain (we now restore the full ~x d dependence by using SO ( d ) symmetry) S os = i ∆ log h z ǫ ( ~x − ~x ) + z i . (2.7)Then G ( ~x , x ) ≈ e − iS os . Finally, substituting this into (2.1), we get h O ( − ~x ) O ( ~x ) i = Z AdS d +1 dzz d d d x e ∆ log h z ǫ ( ~x − ~x z i +∆ log h z ǫ ( ~x + ~x z i . (2.8)where ( ~x, z ) is interpreted as the joining point over which we have to integrate. Since∆ ≫
1, we may use the saddle point approximation. It is easy to see that the saddlepoint lies at ~x = 0 and z = | ~x | . Note that ~x = 0 actually follows from symmetry: havingchosen the boundary points at ± ~x , the joining point will be at the symmetric point ~x = 0at some value of z . Using the above solution for the saddle-point equations, it follows that h O ( ~x ) O ( ~y ) i = 1 | ~x − ~y | , (2.9)just as expected. In the previous computation, the on-shell action (2.4) was obtained by using the conser-vation of p x , where p x is to be understood as the momentum corresponding to a particletravelling along the geodesic arc from a boundary point x to a generic bulk point ( x, z ).Both ( x ,
0) and ( x, z ) are to be understood as boundary conditions for our geodesic (whereit starts –which will be taken as fixed– and where it ends). By inverting the solution tothe equation of motion in (2.5) we can write p x in terms of x and ( x, z ) to obtain (2.7).In this way, S os is the action for the particle with fixed boundary conditions at one end Note that the argument of the logarithms is always smaller than 1. This ensures that, as ∆ ≫
1, theexponent goes to −∞ . = x at the boundary. Then, it is the evaluation of the integration by the saddle-pointmethod what fixes the joining point of the two geodesic arcs.As we will see, the explicit inversion of the solution to the equation of motion to findthe analog of (2.6) is not possible in the black brane background. Nevertheless, one canstill determine the joining point ( x, z ) by saddle-point equations. To see this, let us writethe total action as S T = i ∆ log h ǫ z (cid:0) ∆ + q ∆ − ( p (1) x ) z (cid:1)i + i ∆ log h ǫ z (cid:0) ∆ + q ∆ − ( p (2) x ) z (cid:1)i ;where p ( i ) x refers to the momentum associated with the geodesic from ( − x ,
0) to ( x, z ) andto the geodesic from ( x ,
0) to ( x, z ) respectively. These are, in principle, to be regardedas functions of the boundary conditions as p (1) x = p (1) x ( − x ; ( x, z )) , p (2) x = p (2) x ( x ; ( x, z )) . (2.10)Due to the symmetric choice of the boundary conditions, the joining point must lie at x = 0. Thus, we can use (2.5) to write equations for the coordinate z of the joining point,0 = − x + ∆ p (1) x − s ∆ ( p (1) x ) − z , x + ∆ p (2) x − s ∆ ( p (2) x ) − z . (2.11)This shows that p ( i ) x are only functions of z . Moreover, it is clear that p (2) x = − p (1) x (thiscould be anticipated from the fact that the total momentum after joining the two arcsmust be conserved). Thus, denoting p (1) x = p x , we have S T = 2 i ∆ log h ǫ z (cid:0) ∆ + p ∆ − p x z (cid:1)i ; 0 = − x + ∆ p x − s ∆ p x − z . (2.12)The saddle point equations are dS T dz = 0 and dS T dx = 0. As for the second equation,all the x -dependence of S could only arise through p x . However, since the latter are, asargued above, x -independent, dS T dx = 0 automatically follows (which is just a check of theconsistency of the choice x = 0). On the other hand, for the z -equation we have ∂S T ∂z + ∂S T ∂p x dp x dz = 0 . (2.13) dp x dz can be computed by differentiating the second equation in (2.12). Then (2.13) leads tothe following saddle-point equation: p ∆ − p x z = 0 −→ z = ∆ p x . (2.14)7herefore the on-shell version of (2.12) is S T (cid:12)(cid:12)(cid:12) os = 2 i ∆ log h p x ǫ i , − x + ∆ p x . (2.15)Thus, e − iS T = ǫ / | x | , which, upon restoring the full ~x dependence by using SO ( d )symmetry, reproduces the result in (2.9).Finally, it is worth noting that the location of the saddle point for z could have beenanticipated. Since we are computing a very simple “Witten diagram” where there is noinsertion, the resulting geodesic after joining the two arcs must be smooth. This meansthat dz/dx must vanish at the joining point, so that we have a U -shaped geodesic. Thus,immediate inspection of the equations of motion leads to the condition (2.14), which yieldsthe joining point at z = ∆ p x . At zero temperature the form of the 2-point correlation function of an operator O ofdimension ∆ in (2.9) simply follows from conformal invariance, up to an overall factor thatdepends on the normalization of the operators. The situation is much more interesting atnon-zero temperature. Considering the CFT at non-zero temperature amounts to puttingit on (euclidean) S β × R d − . For operators inserted at points separated by a distance muchsmaller than the length β of the thermal circle one can still use the OPE. A new feature ofnon-zero temperature is that operators can have a non-vanishing one-point function. As aresult, the 2-point function picks contributions from all operators with non-vanishing OPEcoefficients with O . This makes the structure of thermal 2-point functions very interesting.It turns out that the correlator h O (0) O ( τ, ~x ) i , within the regime of validity of the OPE √ ~x + τ ≪ β , admits the general expression [2] (see also [1]) h O (0) O ( τ, ~x ) i = X O∈ OPE[ O × O ] a O β ∆ O C ( d − J O ( η ) | x | ∆ O − , η = τ | x | , | x | = √ ~x + τ . (3.1)The sum in (3.1) runs over all operators O with dimension ∆ O and spin J O appearingin the O × O OPE. Here C ( d − J O ( η ) are Gegenbauer polynomials, and the combinations C ( d − J O ( η ) | x | ∆ O − can be regarded as conformal blocks on the cylinder. Finally, a O arenumerical coefficients, which can be related to microscopic quantities as a O = f OO O b O c O J !2 J ( d − J ; (3.2)8here f OO O is the corresponding OPE coefficient, c O is the normalization of the 2-pointfunction for O and b O is the coefficient of the one-point function of O (for instance, forscalars, hOi = b O T ∆ O ).Among the a O coefficients, that corresponding to the energy-momentum tensor is par-ticularly interesting. Note that the energy-momentum tensor T is a dimension d operatorof spin 2, and it thus corresponds to C ( d − ( η ) with ∆ O = 2. The OPE coefficient f OO O isdetermined by a Ward identity, f OO T = − dd − S d − ) , vol( S d − ) = 2 π d Γ (cid:0) d (cid:1) . (3.3)Moreover [2] b T = dd − h T i β T d . (3.4)Hence a T = − d ∆( d − ( d −
2) vol( S d − ) h T i β c T T d ; (3.5)where c T stands for the coefficient of the energy-momentum tensor correlator normalizedas in [28]. For future reference, let us quote the d = 4 value a T = − π h T i β c T T . (3.6)On general grounds, thermal 2-point correlation functions must satisfy the KMS con-dition h O (0) O ( τ, ~x ) i ∼ h O (0) O ( τ + β, ~x ) i . Note that (3.1) does not manifestly exhibitsuch periodicity, which lies outside the regime of validity of the OPE (see [2] for furtherdevelopments). It is instructive to consider in detail the case of a massless free scalar field φ . In d = 4 thethermal 2-point function in position space is h φ (0) φ ( τ, ~x ) i = π β | ~x | h coth (cid:0) πβ ( | ~x | − iτ ) (cid:1) + coth (cid:0) πβ ( | ~x | + iτ ) (cid:1)i , (3.7)where we have chosen the normalization so that at zero temperature we recover (2.9)with ∆ = 1. Note that (3.7) manifestly satisfies the KMS condition of invariance under τ → τ + β .In the regime of √ τ + ~x ≪ β , (3.7) admits the expansion h φ (0) φ ( τ, ~x ) i = X n =0 ζ (2 n ) x n − β n C (1)2 n − ( η ) ; (3.8)9hich is precisely of the form given in (3.1) (and it reproduces the result in [2]). Inparticular, we recognize the contribution of the operators [ φφ ] n,ℓ = φ∂ µ · · · ∂ µ ℓ ( ∂ ) n φ foreven ℓ , which was anticipated in [2]. It should be noted that for a free field theory ∂ φ isa redundant operator and thus drops from (3.8).The contribution of the energy-momentum tensor can be read from the coefficientmultiplying the C (1)2 ( η ) Gegenbauer polynomial and it is 2 ζ (4) = π . This is the expectedvalue, since for free fields, h T i β = 2 dζ ( d )( d −
1) vol( S d − ) , c T = dd − (cid:16) vol( S d − ) (cid:17) . Let us consider a CFT d at finite temperature with an holographic description in terms ofthe black brane in AdS d +1 , ds = R z (cid:0) − f ( z ) dt + dz f ( z ) + d~x d − (cid:1) , f = 1 − z d z d , z = d π β . (3.9)Using the same recipe as in (2.1), we can compute holographically correlators for operatorsof large conformal dimension by approximating G ( x b , x ) by geodesic arcs in the black branebackground in (3.9). From now on, we specialize to the case d = 4 and we will choose unitswhere z = 1. For non-zero T , the background (3.9) exhibits SO ( d −
1) symmetry. Using this, it willbe sufficient to consider the motion of a particle in the geometry (3.9) with t = t ( z ), x ≡ x = x ( z ) and x i = 0, i = 1. The corresponding (lorentzian) action is S = − ∆ Z dz z − r f ˙ t − f − ˙ x . (3.10)It is clear that the canonical momenta conjugate to t and x are conserved. Let us denotethem by p t = ∆ µ and p x = ∆ ν , respectively. This allows to write the first order equationsas follows ˙ t = ∓ µ zf p − f + ( µ − f ν ) z , ˙ x = ± ν z p − f + ( µ − f ν ) z . (3.11)Then, the on-shell action takes the form S = − i ∆ Z dz z p f − µ z + f ν z . (3.12)10he equations of motion in (3.11) can be integrated explicitly in terms of elliptic inte-grals. We will restrict to the case of ν = 0, where the integrals simplify. This correspondsto insertions of operators at equal points on the R . The relevant equations then become S = − i ∆ Z dr r p − r − µ r , ˙ t = ∓ i µ (1 − r ) p − r − µ r , (3.13)where r = z . Note that close to the boundary ˙ t → ∓ i µ . Thus, for real µ , t is imaginary,recovering the Wick rotation ( c.f. the T = 0 case, where we worked directly in theeuclidean, and hence the momentum was purely imaginary).Integrating the action from the boundary at z = ǫ up to a generic bulk point z , we find S = 12 i ∆ log ǫ (cid:16) − µ z + 2 p − µ z − z (cid:17) z . (3.14)In turn, the solution to the equation of motion with boundary conditions t (0) = t is t − t = −
14 log µ + ( µ + 2) z + 2 iµ p − z − µ z + 1 − µ + 2 iµ −
2) (1 − z ) ! − i log µ − ( µ − z + 2 µ p − z − µ z + 1 + 2( µ + 2 µ + 2) ( z + 1) ! . (3.15)The second geodesic arc is obtained by changing t → − t and µ → − µ in (3.15). We can now compute the 2-point function h O ( t ) O ( − t ) i by joining two geodesic arcs: onefrom ( − t , ~ , z = 0) to the bulk point ( t, ~x, z ), the other from the bulk point ( t, ~x, z ) to theboundary point ( t , ~ , z = 0). Then we integrate over the bulk point where the two arcsare joined. Since (3.15) cannot be inverted to find the explicit form of G ( x b , x ), we willfollow the same steps as in the T = 0 case in section 2.2. Just like in the T = 0 case, thesaddle point will be at t = 0, ~x = 0 by symmetry. Then, (3.15) evaluated at the joiningpoint formally provides an equation for µ as a function of ± t and of the bulk joining point z . As in section (2.2), the bulk joining point is determined by the saddle point equationfor z , which is given by 0 = dSdz = ∂S∂z + ∂S∂µ dµdz , (3.16)where dµdz is read off from (3.15) evaluated at the joining point. One obtains the equation,0 = p − z − µ z . (3.17)11imilarly to the T = 0 case, this is the expected saddle-point equation, since it ensures asmooth ( U -shaped) geodesic, i.e. it corresponds to the turning point where dz/dt vanishes.The solution is z ± = − µ ± p µ . (3.18)The minus sign solution is complex, so we keep the plus sign, which gives us z ⋆ = z + ∈ (0 , h O ( t ) O ( − t ) i = e − iS (cid:12)(cid:12) os = 4 − ∆ (cid:0) µ + 4 (cid:1) ∆2 , (3.19)modulo a multiplicative numerical constant which we will neglect (we may fix it a posteriori by the normalization of the T → µ in (3.19) byits expression in terms of t . This can be in principle read off from (3.15) evaluated at thejoining point at t = 0 and z = z ⋆ . This gives the following expression: τ ≡ it = −
14 log ( µ − µ + 2 p µ + 4 ! + i p µ + 4 µ ( µ + 2 i ) − ! , (3.20)where we have introduced the euclidean time τ . The formulas (3.20) and (3.19) implicitlydefine the two-point function h O ( τ ) O ( − τ ) i . First discussed in [13] (note that our µ is i E in that reference), they encode a lot of information, including details of the black holeinterior, when explored in the complex µ plane.In the following we will be interested on real µ . Note that, for any real µ , the argumentof the log in the first term in (3.20) is always real and positive. In turn, the argument ofthe second log is a pure phase e iθ . Because of the different branches of the log, θ is definedup to multiples of 2 π . This implies that a given µ defines τ up to multiples of π ≡ β . In particular, it follows that h O ( τ ) O ( − τ ) i ∼ h O ( τ − β ) O ( − τ + β ) i , which is preciselythe KMS periodicity condition.Let us consider the small τ behavior, i.e. the limit of nearly coincident points. From(3.20) we find that, for large µ , τ = 1 µ − µ + O (cid:18) µ (cid:19) . (3.21)This regime thus represents the small τ limit. Moreover, since we are looking at the large µ range, this will correspond to the dominant solution ( c.f. (3.19)). Computing higherorders in (3.21), we find that, at nearly coincident points, the two-point function exhibits To restore the β -dependence, note that in d = 4, z = 1 = βπ . h O (0) O ( τ ) i = −
2∆ log τ + ∆ π T τ (cid:18) π τ T + 89 π τ T π τ T π τ T O (cid:0) τ (cid:1) (cid:19) , (3.22)where we have restored the factors of temperature. The terms in the first line agree withthe terms computed in [5]. From (3.22) we obtain h O (0) O ( τ ) i = 1 τ (cid:18) π T τ
40 + ∆ π T τ π T τ + 286 ∆ + 712) + O (cid:0) τ (cid:1) (cid:19) . (3.23)A particular large ∆ limit exists, with T τ → T τ ) . This limits keeps onlythe first subleading term in (3.22). Concretely, one obtains h O (0) O ( τ ) i = 1 τ (cid:18) π (∆ T τ )40 + π (∆ T τ ) π (∆ T τ ) · · · (cid:19) = 1 τ exp (cid:20) π ∆ T τ (cid:21) . (3.24)It should be noted that in this “double scaling limit” the correlator does not exhibit theKMS periodicity. This is expected, since the regime of the limit requires small T τ , whichclearly is not maintained under τ → τ + β .It is instructive to examine the analog limit for (completely connected) correlators of O = φ n in the free theory. The correlator is just the n -th power of that in (3.7). Expandedin powers it reads h O (0) i O ( τ, ~x ) i = 1 | x | n h ζ (2) n T | x | C (1)0 ( η )+ 2 n T | x | (cid:16) ζ (4) C (1)2 ( η ) + ( n − (cid:16) ζ (2) C (1)0 ( η ) (cid:17) (cid:17) + · · · i . Taking the large n ( ≡ ∆) limit of this expression and resumming the result we find h O (0) i O ( τ, ~x ) i = 1 | x | n e ζ (2) n T | x | C (1)0 ( η ) . (3.25)We note that this limit is dominated by the block associated with φ , which correspondsto the C (1)0 ( η ) Gegenbauer polynomial. 13 .2.3 Including x dependence Let us now consider the general two-point function h O (0) O ( τ, ~x ) i for operators of large di-mension. It can be computed in the geodesic approximation, just as we did for h O (0) O ( τ ) i ,by integrating (3.11). However, as the integrals lead to complicated expressions in termsof elliptic integrals, it is more illuminating to consider a small temperature expansion (byconformal invariance, this is equivalent to a short-distance expansion). We thus look forgeodesics that hit the boundary at ( t , x ) and end at the generic joining point which, bysymmetry, is located at ( t = 0 , ~x = 0 , z ). In the short-distance regime, the geodesic pene-trates very little into the bulk and remains close to the boundary. We have already seen in(3.21) that small τ corresponds to large µ , where z ⋆ ∼ µ . Consistently, the result in (3.23)is a series expansion around the T = 0 case in powers of T τ . Since the temperature entersthe equations of motion through z − ∼ T , the correlation function will be an expansionin powers of T . This is of course consistent with (3.23). In general, µ , ν have temperaturedependence and admit an expansion of the form µ = X i =0 µ i T i , ν = X i =0 ν i T i . (3.26)Next, we substitute the expansions (3.26) into the exact solution, expand the resultingexpression in powers of T , and solve, order by order, for the µ i and ν i as functions ofthe specified boundary conditions. In this way we construct G ( x b , x ) to any desired orderin T . We then substitute these expressions into the on-shell action. The resulting totalon-shell action still has to be extremized for z . Writing z = P i =0 z i T i , we can solve thecorresponding equation for the z i to the appropriate order in T . Finally, the on-shell actionevaluated on this z gives the two-point correlation function. We omit the intermediateexpressions which are very lengthy. The final result is given by h O (0) O ( τ ) i = (3.27)1 | x | h π T C (1)2 ( η ) | x | + ∆ π T (cid:16) C (1)4 ( η ) + C (1)2 ( η ) + C (1)0 ( η ) (cid:17) | x | + · · · i . where, as earlier, η ≡ τ / | x | , | x | = √ ~x + τ . As a cross-check, upon setting ~x = 0 in(3.27), we exactly recover (3.23).It is of interest to identify the operators contributing to (3.27). The first term clearlycorresponds to the energy-momentum tensor, which at non-zero T takes a VEV. Theremaining terms correspond to dimension 4 n and spin 2 n, n − , · · · , a T in (3.6). This leads to theformula: − h T i β T = 3 π c T . (3.28)14e claim that this is a universal relation which should be valid for any four-dimensionalCFT with a gravity dual. We can check it explicitly in the case of N = 4 super Yang-Millstheory. Using the formulas in [28] for the N = 4 field-theory content (in that language, N gauge fields, 6 N real scalars and “ N Dirac fermions”), one finds c T = N π . Thus(3.28) gives h T i β = − π N T , in nice agreement with the known result [29].Another evidence of the consistency of the formula (3.28) arises in the context of SUSYCFT’s corresponding to D CY cone over a base H , when the ten-dimensional near-brane geometry is AdS × H . In this example, using the fact that forCFT’s with a gravity dual the a = π H ) central charge equals c T , we can write h T i β h T i N =4 β = Vol( S )Vol( H ) . (3.29)Let us compare this prediction with the holographic computation of h T i β . The onlydependence on the actual geometry is through the value of the five-dimensional Newtonconstant G . This is proportional to the volume of the internal manifold H [30]. Thus h T i β ∼ Vol( H ) − , in agreement with (3.29).Remarkably, (3.27) can be resummed to give h O (0) O ( τ ) i = 1 | x | e ∆ π T C (1)2 ( η ) | x | , (3.30)with C (1)2 ( η ) = − η = 3 τ − ~x τ + ~x This completes the partial result (3.24), now including the full spacetime dependence. Itexhibits an extremely interesting exponentiation of the energy-momentum tensor block.This exponentiation is similar to that observed for 4-point functions in CFT [6, 7, 8] andin CFT [9]. However, in the present case of two-point functions, we have an explicitidentification of the exponentiated block. Note that expanding the exponential we couldread-off the coefficients for the corresponding blocks (each corresponding to suitably con-tracted powers of the energy-momentum tensor), thus providing an infinite sequence ofpredictions for such coefficients (see also [3, 31] for related discussions). d = 2 case In order to gain further insight on the exponentiation, it is instructive to consider the caseof 2d CFT’s. The thermal two-point correlation function can also be computed in termsof geodesics in two dimensions. Let us begin by considering the simplest case of includingonly time dependence, i.e. we consider h O (0) O ( τ ) i . In euclidean signature, one finds thatthe geodesic (to be precise, each of its half-arcs) which goes from − τ to τ in the boundaryis given by τ = i h cos τ − p z − sin τ cos τ + p z − sin τ i . (3.31)15o compute the correlator we consider these two arcs hitting the boundary at τ and − τ ,respectively, and meeting at a bulk point over which we integrate by the saddle-pointmethod. The turning point is at the symmetric point τ = 0 and at z = sin τ . Then asmall computation gives the following 2-point function: h O (0) O ( τ ) i = π T (cid:16) sinh( πT iτ ) (cid:17) . (3.32)One can similarly incorporate the x -dependence (see appendix A). In this way, one recoversthe general formula h O (0) O ( τ, x ) i = π T (cid:16) sinh( πT ( x − iτ )) sinh( πT ( x + iτ )) (cid:17) ∆ . (3.33)which agrees with the well-known expression for the thermal 2-point function in 2d CFT.Note that in the present case of two dimensions the geodesic approach gives the exactcorrelation function for any ∆.Let us now study the limit leading to exponentiation in the present d = 2 case. Nowthe appropriate limit requires T | x | →
0, ∆ → ∞ , with fixed ∆ T | x | . One finds h O (0) O ( τ, x ) i = e π T ∆ (2 η − | x | | x | . (3.34)From these expressions, one can derive a formula for the thermal expectation value ofthe energy-momentum tensor. Consider the leading correction. We have π T ∆ (2 η − | x | − . We may re-write this aslim d → β π ∆3 2!2 ( d − C ( d − ( η ) | x | − . (3.35)Comparing with the general formula in (3.1), one finds h T i β = − π β (2 πc T ) . (3.36)For a free boson one has c T = π , giving h T i β = − π T = − ζ (2) S , just as expected [2]. An important problem in holography is to describe physics of the black hole interior interms of boundary CFT correlation functions. In a recent paper [21], it was argued thatone can measure the proper time to the black hole singularity by examining the asymptoticbehavior of thermal expectation values of large charge operators. The expectation valueis induced by a gravitational coupling of the form φW , where W = W µνρσ W µνρσ is16he square of the Weyl tensor, which represents the first possible coupling in a derivativeexpansion. It was found that it has a factor of the form h O i ∝ e − mℓ , ℓ = Rd ( ± iπ + log 4) . (4.1)The parameter ℓ represents the renormalized proper lenght from the boundary to thesingularity, computed as ℓ = R lim ǫ → (cid:18)Z ∞ ǫ dzz √ − z d + log ǫ (cid:19) . (4.2)Thus the dependence of h O i on the conformal dimension ∆ ∼ m provides a measurementof ℓ .It is natural to expect that higher point thermal correlation functions may codify moreinformation on the black hole geometry, perhaps in an intricate way. In this section we shallconsider the thermal two-point function in the CFT and compute the leading correctioninvolving graviton emission for large dimension operators.We will consider scalar operators of large dimensions carrying a U (1) global charge. Inthis case the relevant part of the action for the dual bulk scalar field is I = 116 πG N Z d x √ g (cid:2) g µν ∂ µ ¯ φ∂ ν φ + m ¯ φφ + α ¯ φφW (cid:3) , ∆ ≈ mR ≫ . (4.3)The coupling ¯ φφW , where W is the square of the Weyl tensor, appears at leading orderin the derivative expansion. It leads to a new interaction vertex, which implies the exis-tence of a Witten diagram representing the following correction to the two-point function h O (0) ¯ O ( x ) i : I = α Z bulk G ( x (1) b , x ) G ( x (2) b , x ) W ( x ) . (4.4)In terms of r = z , in the black brane background, W = R r r .We shall consider the case corresponding to Green’s functions of operators inserted atcoincident spatial points but at different times t and t . Moreover, as we will be interestedin correlators of large scaling dimension, we can trade each of the Green’s functions G in(4.4) by the exponential of the geodesic length in (3.14). Finally, the integral in (4.4) canbe done through the saddle-point approximation. Choosing, with no loss of generality, t = − t , it is clear by symmetry that the saddle point of the t integration will again lieat the symmetric point t = 0 and ~x = 0. The general expression for t in terms of z and µ is given by (3.15). Therefore the integral to compute is I = c Z drr G ( x − x ′ ( z )) G ( x − x ′ ( z )) r , c = 72 R αr . (4.5)Due to the presence of the extra W term, the two geodesic arcs will now meet at a different r ⋆ , which can be obtained by studying the extrema of the integral (4.5) with respect to r .17o find it we follow the same strategy as in previous sections. To begin with, the integrandin (4.5) can be written as e − iS T , where S T = S ( z, µ ) + S ( z, µ ) + i log r . (4.6)It should be noticed that S T is a function which depends on the joining point r bothexplicitly and implicitly through µ , – the two rescaled momenta of each geodesic arc in(4.5) – since the variation is performed at fixed t . Then, similarly to the cases studiedpreviously, the saddle-point equation is0 = dS T dr = ∂S T ∂r + ∂S T ∂µ ∂µ ∂r + ∂S T ∂µ ∂µ ∂r . (4.7)The last two terms give an identical contribution, since µ = − µ ≡ µ . In order to find ∂µ∂r we differentiate (3.15) at t = 0 and constant t . Substituting the result into (4.7) weobtain the following equation: 0 = im p − r ( µ + r ) r ( r −
1) + ir . (4.8)This equation has three roots for r , with the following asymptotic behavior at large mass: r ± = ± i ∆ + µ O ( 1∆ ) ,r = 12 (cid:16)p µ + 4 − µ (cid:17) + O ( 1∆ ) . (4.9)We note that the leading term in r is the same as the saddle point (3.18) that appearedin the calculation of the two-point function by the geodesic approximation in section 3.2.2.As a result, to leading order, this gives rise to the same action: e − iS T ( r ) = 4 − ∆ (cid:0) µ + 4 (cid:1) ∆2 . (4.10)On the other hand, r ± give rise to complex geodesics with a complex action, e − iS T ( r ± ) = (cid:18) − µ ± i (cid:19) ∆ . (4.11)In the case of the geodesics that join at r , the parameter µ in (4.10) is again determinedby the condition (3.20) as a function of t .Now consider the contribution from the saddle points at r ± . Substituting r ± in (3.15)and expanding for large ∆, one finds The overall coefficient obtained by the saddle-point approximation is inexact because the logarithmicterm in (4.6) is not multiplied by a large coefficient. However, following [21], we will not be interestedin this coefficient but rather in the leading functional dependence, which is correctly reproduced by theaction. = − iτ = (cid:0) ±
14 + i (cid:1) log (cid:18) − µ + (1 ∓ i ) µ + (1 ∓ i ) (cid:19) + O (cid:18) (cid:19) . (4.12)This equation can be inverted to give µ = (1 ± i ) tan (cid:0) (1 ± i ) τ (cid:1) . (4.13)This determines µ in terms of τ for the saddle-point contribution at r ± , given by (4.11).Clearly, the contributions at r + and r − are complex conjugate and have equal weight | e − iS T | .Let us examine in detail the regime of validity of the approximation. Restoring R and z , r ± = ± iz ∆ + µ z R + O (cid:0) (cid:1) . (4.14)This shows that we must require ∆ ≫ µ z R . In turn, µ = (1 ± i ) Rz tan (cid:18) (1 ± i ) τz (cid:19) . (4.15)Then, the condition ∆ ≫ µ z R requires, (cid:12)(cid:12) tan (cid:18) (1 − i ) τz (cid:19) (cid:12)(cid:12) ≪ √ ∆ . (4.16)Since (cid:12)(cid:12) tan (cid:16) (1 − i ) τz (cid:17) (cid:12)(cid:12) is bounded (in fact < . ≫ e − iS T ( r ) and e − iS T ( r + ) + e − iS T ( r − ) , the dominant contributionfor a given τ is the one with largest modulus. At small τ , the dominant contribution isgiven by the saddle point sitting at r , since it will have large µ (see (3.21)) and thereforelarger weight factor | e − iS T | . This gives the expected 1 / | τ | m behavior for nearly coincidentpoints. However, there is a critical value of τ (namely τ c ∼ .
16) beyond which thedominant contribution is given by the saddle points at r ± , where the correction to the2-point function is given by e − iS T ( r + ) + e − iS T ( r − ) . Substituting (4.13) into (4.11), we obtain e − iS T ( r ± ) = const . − ∆ e ± iπ ∆2 (cid:0) cos (cid:0) ± i ( τ − τ ) (cid:1)(cid:1) . (4.17)Note the factor e − ℓ , with ℓ = ± iπ ∆2 + log 2, representing renormalized length (4.1) to theblack hole singularity. This is the squared of the similar factor appearing in the one-pointfunction in [21]. The reason of the squared is of course that this factor now accounts for An open problem is understanding the detailed physics, in particular the smoothness, of the transitionbetween the small/large τ (see [20] for a recent discussion). W correctionto the correlation function has a non-trivial time dependence, which is associated with theprobability of graviton emission.Indeed, one can get a deeper insight on the meaning of this contribution to the corre-lation function by expanding in exponential terms, e − iS T ( r ± ) = const . − ∆ e ± iπ ∆2 e − ∆ (1 ± i ) τ ∞ X n =0 ( − n n ! (∆ + n − − e − n (1 ± i ) τ . (4.18)The complex frequencies ω n = n ∆(1 ± i ) seem to be associated with quasinormal frequen-cies, which for a scalar field in the black brane background are ˆ ω = (∆ + k )(1 ± i ) for∆ ≫ n is not surprising sincethe coupling ¯ φφW involves multi-graviton emission. In this paper we have studied thermal 2-point functions for operators O of large scalingdimension in four-dimensional CFT’s with a holographic dual. The leading contribution tothe 2-point function comes from a “Witten diagram” where one has two bulk-to-boundarypropagators, one from a boundary point x (1) b to a bulk point x and another one from x toa boundary point x (2) b , and the point x where they meet is integrated over the whole bulk.In the limit of large scaling dimension one can use the geodesic approximation and tradeeach propagator by a geodesic arc with the given endpoints and perform the integrationover x through a saddle-point approximation [22, 23, 24]. Even though this representsa big simplification, one is still left with the problem of writing the on-shell action foreach geodesic arc as a function of the endpoints. The relevant equations are typicallytranscendental equations which cannot be inverted explicitly; consequently, the correlationfunction is determined parametrically by a set of equations. Explicit expressions can beobtained by a systematic expansion in powers of T | x | (in particular, within the regimeof validity of the OPE in field theory). In the case of correlation functions with only τ -dependence, we have computed the correlator (3.23) up to the order ( T τ ) , and it iseasy to extend this expansion to any desired order. The full spacetime dependence ofthe thermal correlation function can be studied by a perturbative expansion in powers of T . Following this approach, we explicitly found the expected expansion of the correlatorin terms of Gegenbauer polynomials anticipated in [2]. In the limit that we considered,all contributions have been identified to correspond to appropriately contracted powers ofthe energy-momentum tensor. In this expansion the term corresponding to the energy-momentum tensor itself is particularly interesting, as the coefficient can be compared Stricto sensu it is not a Witten diagram, as the propagators are not meeting at a vertex read off fromthe bulk theory lagrangian. ≫ T | x | ≪ admits the same exponentiation of the energy momentum block. In fact, wecan see a similar phenomenon even in the free theory, although in that case the exponen-tiated block is different. The exponentiation of the block of the energy-momentum tensorappears to be reminiscent of the exponentiation of Virasoro blocks in 4-point functions inCFT as in [6] (see also [7, 8]). The exponentiation can be traced to the existence of asemiclassical limit (with ∆ − playing the role of ~ ). It should be noted that the exponenthas no coupling dependence in the limit. While from the holographic point of view theexponential form admits a simple explanation, from a purely field-theoretic point of view(for instance, by explicit diagrammatic computation of the 2-point function) it is a highlynon-trivial prediction of strong coupling. This is very similar to the recent large chargestudies in the O ( N ) model [34, 35, 36, 37]: for correlation functions of large charge ( ∼ dimension) operators a semiclassical expansion in ~ ∼ ∆ − emerges, so that there is anautomatic exponentiation (see also [38]).One of the most fascinating aspects of thermal correlation functions in a CFT is thatthey can encode information of the black hole interior through the holographic correspon-dence. In section 4 we have considered a massive bulk scalar field φ , dual to a CFT scalarfield operator O of large dimension, which couples to the Weyl tensor through an interac-tion | φ | W . This is the first possible interaction in a derivative expansion assuming U (1)symmetry. This could be, for instance, the case of chiral primaries in N = 4 SYM, whichare charged under a U (1) R symmetry. This vertex gives rise to a correction to the 2-pointthermal correlation function which is contributed by complex geodesics entering deeplyinto the black hole interior. The correction carries a phase factor of the form e − ℓ , where ℓ is the proper lenght to the singularity, which is twice the similar factor appearing in theone-point function computed in the model of [21]. In addition, it has a time-dependencewhich carries information about quasi-normal frequencies, which seem to be associatedwith multi-graviton emission. Clearly, it would be extremely interesting to decipher de-tailed aspects of quantum black holes hidden in more general CFT thermal correlationfunctions. Acknowledgements
D.R-G is partially supported by the Spanish government grant MINECO-16-FPA2015-63667-P. He also acknowledges support from the Principado de Asturias through the grant21C-GRUPIN-IDI/2018/000174 J.G.R. acknowledges financial support from projects 2017-SGR-929, MINECO grant PID2019-105614GB-C21, and from the State Agency for Re-search of the Spanish Ministry of Science and Innovation through the “Unit of ExcellenceMar´ıa de Maeztu 2020-2023” (CEX2019-000918-M).
A Thermal two-point correlation function in d = 2 In this appendix we compute the thermal two-point correlation function in two-dimensionalCFT with general spacetime dependence. The solution for the geodesic is obtained byintegrating (3.11). We find t − t = −
12 log (cid:18) µ − ν − z ( µ + ν + 1) + 2 iµ p (1 − z )(1 + ν z ) − z µ (1 − z ) ( − ν + ( µ + i ) ) (cid:19) ,x − x = 12 log (cid:18) µ + ν (2 z −
1) + 1 + 2 iν p (1 − z )(1 + ν z ) − z µ µ − ( ν − i ) (cid:19) ,S = i z ( − µ + ν −
1) + 2 p (1 − z )(1 + ν z ) − z µ z ! . (A.1)At z = 0, the expressions for x and t satisfy the boundary condition x = x , t = t .The second geodesic arc has a similar solution but with the boundary condition at z = 0, x = − x , t = − t , and µ → − µ , ν → − ν . By symmetry, they meet at some point z where x = 0, t = 0. The action is the same for both geodesics. The point where they meet isdetermined by demanding that it is an extremum of the action, i.e. it satisfies ∂S∂z + ∂S∂µ ∂µ∂z + ∂S∂ν ∂ν∂z = 0 . (A.2)The partial derivatives ∂ z µ and ∂ z ν are obtained by differentiating the above expressionsfor t and x at the joining point t = x = 0 and fixed t , x . This leads to a system of twolinear equations with two unknowns. Substituting the (lengthy) solution into (A.2), we getthe condition p − z − ν z + z ( ν − µ ) z (1 − z ) = 0 . (A.3)As expected, the point z that extremizes the action corresponds to the turning point ofthe geodesic obtained by the union of the two geodesic arcs, where ˙ t and ˙ x diverge. Wehave followed this longer derivation through extremization of S because this derivationalso applies in the presence of a vertex operator in the Witten diagram where the turningpoint is no longer smooth. The solution is z ∗ = q ( ν − µ − + 4 ν − µ + ν − ν . (A.4)22his gives the following formula at the extreme point: e − iS ( z ∗ ) = q ( µ − ν + 1) + 4 ν . (A.5)Substituting the value of z ∗ into t and x , one can solve the resulting equations for µ and ν in terms of the initial values ( ± t , ± x ) at z = 0. We find µ = − sin(2 τ )cos(2 τ ) − cosh(2 x ) , ν = i sinh(2 x )cos(2 τ ) − cosh(2 x ) , τ = it . (A.6)These expressions can be substituted into z ∗ to express the turning point in terms of τ and x , z ∗ = 12 cosh ( x ) (cosh(2 x ) − cos(2 τ )) . 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