Acta Polytechnica doi:10.14311/AP.2014.54.0142 Acta Polytechnica 54(2):142–148, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap A SIMPLE DERIVATION OF FINITE-TEMPERATURE CFT CORRELATORS FROM THE BTZ BLACK HOLE Satoshi Ohyaa, b a Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Pohraniční 1288/1, 40501 Děčín, Czech Republic b Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague, Břehová 7, 11519 Prague, Czech Republic correspondence: ohyasato@fjfi.cvut.cz Abstract. We present a simple Lie-algebraic approach to momentum-space two-point functions of two-dimensional conformal field theory at finite temperature dual to the BTZ black hole. Making use of the real-time prescription of AdS/CFT correspondence and ladder equations of the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R, we show that the finite-temperature two-point functions in momentum space satisfy linear recurrence relations with respect to the left and right momenta. These recurrence relations are exactly solvable and completely determine the momentum-dependence of retarded and advanced two-point functions of finite-temperature conformal field theory. Keywords: AdS/CFT correspondence, correlation functions, BTZ black hole. 1. Introduction and summary Conformal symmetry is powerful enough to constrain the possible forms of correlation functions in quantum field theory. It has been long appreciated that, for scalar (quasi-)primary operators, for example, SO(2,d) confor- mal symmetry completely fixes the possible forms of two- and three-point functions up to an overall normalization factor in any spacetime dimension d ≥ 1. This symmetry constraint works well in position space, however, its direct implication to momentum-space correlators are less obvious before performing the Fourier transform. Since momentum-space correlators are directly related to physical observables (e.g. the imaginary part of a retarded two-point function in momentum space gives the spectral density of many body systems), it is impor- tant to understand how directly conformal symmetry constrains the possible forms of momentum-space cor- relators. From the practical computational viewpoint, it is also important to develop efficient methods for computing momentum-space correlators directly through symmetry considerations, because Fourier transforms of position-space correlators are hard in general. In this short paper we continue our investigation [1] and present a novel Lie-algebraic approach to momentum- space two-point functions of conformal field theory at finite temperature by using the AdS/CFT correspon- dence. The AdS/CFT correspondence relates strongly- coupled conformal field theory to classical gravity in a one-higher spatial dimension. According to the cor- respondence, finite-temperature conformal field the- ory is dual to an asymptotically AdS spacetime that contains black holes. In this paper we focus on two- dimensional conformal field theory (CFT2) at finite temperature dual to the three-dimensional anti-de Sit- ter (AdS3) black hole (i.e. Bañados-Teitelboim-Zanelli (BTZ) black hole [2, 3]) and we give a simple derivation of retarded and advanced two-point functions of scalar operators of dual CFT2 by just using the real-time prescription of AdS/CFT correspondence à la Iqbal and Liu [4, 5] and the ladder equations of the Lie-algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R of the isometry group SO(2, 2) ∼= (SL(2,R)L ×SL(2,R)R)/Z2 of AdS3. In contrast to the conventional approaches to momentum- space CFT correlators (such as Fourier-transform of position-space correlators or original real-time AdS/CFT prescription [4–6], which requires bulk field equations to be solved explicitly), our Lie-algebraic method is quite simple and clarifies the role of conformal symmetry in momentum-space correlators in a direct way: For finite-temperature two-point functions in momentum space, conformal symmetry manifests itself in the form of recurrence relations which are exactly solvable and, up to an overall normalization factor, completely determine the momentum dependence of two-point functions. The rest of the paper is organized as follows. In section 2 we briefly review the AdS3 black hole based on the quotient construction [3, 7]: The AdS3 black hole is a locally AdS3 spacetime and is given by a quotient space of AdS3 with an identification of points under the action of a discrete subgroup Z of the isometry group SO(2, 2) of AdS3. Though not so widely ap- preciated, the AdS3 black hole is a quotient space of AdS3 with a particular coordinate patch in which both the time- and angle-translation generators generate the one-parameter subgroup SO(1, 1) ⊂ SO(2, 2).1 In 1This is true for a non-extremal black hole with positive mass. Time- and angle-translation generators generate other one-parameter subgroups for the zero-mass limit of a black hole (or a black hole vacuum), the extremal black hole and the negative mass black hole (or black hole with naked sin- gularity). For example, in the case of a black hole vacuum, 142 http://dx.doi.org/10.14311/AP.2014.54.0142 http://ojs.cvut.cz/ojs/index.php/ap vol. 54 no. 2/2014 A Simple Derivation of Finite-Temperature CFT Correlators section 3 we introduce a coordinate realization of the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R realized in scalar field theory on the AdS3 black hole background. We then demonstrate in section 4 a simple Lie-algebraic method for computing the retarded and advanced CFT2 two-point functions by just using the ladder equations of the Lie algebra so(2, 2) ∼= sl(2,R)L⊕sl(2,R)R in the basis in which SO(1, 1) ×SO(1, 1) ⊂ SO(2, 2) gener- ators become diagonal. We will see that our method correctly reproduces the known results [5, 8, 9]. 2. AdS3 black hole: Locally AdS3 spacetime in the SO(1, 1) ×SO(1, 1) diagonal basis Let us start with the following non-rotating BTZ black hole described by the metric ds2AdS3 BH = − ( ρ2 R2 − 1 ) dτ2 + dρ2 ρ2/R2 − 1 + ρ2dθ2, (1) where τ ∈ (−∞, +∞), ρ ∈ (0,∞), θ ∈ [0, 2π), and R > 0 is the AdS3 radius. In this paper we simply call (1) the AdS3 black hole and focus on the region outside the horizon ρ > R. For the following discussions it is convenient to introduce a new spatial coordinate x as follows: ρ = R coth(x/R), (2) where x ranges from 0 to ∞. Note that the black hole horizon ρ = R corresponds to x = ∞, while the AdS3 boundary ρ = ∞ corresponds to x = 0. A straightforward calculation shows that in the coordinate system (τ,x,θ) the black hole metric (1) takes the following form: ds2AdS3 BH = −dτ2 + dx2 + R2 cosh2(x/R)dθ2 sinh2(x/R) . (3) For the sake of notational brevity, we will hereafter work in the units in which R = 1. Several comments are in order: (1.) BTZ black hole. The above AdS3 black hole (1) is locally isometric to the rotating BTZ black hole [2, 3] and obtained by suitable change of spacetime coordinates. Indeed, it is easy to show that the BTZ black hole metric ds2BTZ = − (r2 −r2+)(r2 −r2−) r2 dt2 + r2dr2 (r2 −r2+)(r2 −r2−) + r2 ( dφ− r+r− r2 dt )2 , (4) the time- and angle-translation generators generate the sub- group E(1) ×E(1) ⊂ SO(2, 2) ∼= (SL(2,R)L ×SL(2,R)R)/Z2 prior to making the Z-identification. (Note that SL(2,R) contains three distinct one-parameter subgroups: the rota- tion group SO(2), the Lorentz group SO(1, 1) and the Eu- clidean group E(1).) For detailed discussions of the quo- tient construction, we refer to the original paper [3] (see also [7]). where r+ and r− are the outer and inner horizons, respectively, is reduced to the AdS3 black hole (1) by the following coordinate change [10]: ρ = √ r2 −r2− r2+ −r2− , τ = r+t−r−φ, θ = r+φ−r−t. (5) Note that the light-cone coordinates satisfy the rela- tions τ ±θ = (r+ ∓r−)(t±φ). (2.) Local coordinate patch of AdS3. The AdS3 black hole is a locally AdS3 spacetime and is obtained from the AdS3 spacetime with a suitable periodic identification [3, 7]. To see this, let us first note that the AdS3 spacetime can be embedded into the four-dimensional ambient space R2,2 and defined as the following hypersurface with constant negative curvature −1 (= −1/R2): AdS3 = { (X−1,X0,X1,X2) ∈ R2,2 ∣∣ − (X−1)2 − (X0)2 + (X1)2 + (X2)2 = −1 } . (6) The AdS3 black hole (3) is given by the following local coordinate patch of the hypersurface: (X−1,X0,X1,X2) = ( coth x cosh θ, sinh τ sinh x , coth x sinh θ, cosh τ sinh x ) , (7) with the periodic identification θ ∼ θ + 2nπ (n ∈ Z). In fact, it is straightforward to show that the induced metric ds2AdS3 = −(dX −1)2 − (dX0)2 + (dX1)2 + (dX2)2 ∣∣ (X−1,X0,X1,X2)∈AdS3 on the hyper- surface takes the following form: ds2AdS3 = −dτ2 + dx2 + cosh2 xdθ2 sinh2 x . (8) It should be emphasized that the periodic identi- fication θ ∼ θ + 2nπ makes the metric (8) black hole. As mentioned in [3], without such identification the metric (8) just describes a portion of AdS3 and the horizon is just that of an accelerated observer. (Roughly speaking, (7) is the AdS3 counterpart of the Rindler coordinate patch of Minkowski spacetime.) (3.) SO(1, 1) × SO(1, 1) global symmetry. As is well-known, the AdS3 spacetime (6) has an alternative equivalent description as the SL(2,R) group manifold defined as follows: AdS3 = { X = ( X−1+X2 X1−X0 X1+X0 X−1−X2 ) ∣∣∣ det X = 1}. (9) With this definition it is obvious that the AdS3 spacetime (9) is invariant under left- and right- multiplications of SL(2,R) matrices, X 7→ X′ = gLXgR, where gL ∈ SL(2,R)L and gR ∈ SL(2,R)R with the Z2-identification (gL,gR) ∼ (−gL,−gR). 143 Satoshi Ohya Acta Polytechnica (Note that (gL,gR) and (−gL,−gR) give the same X′.) In the local coordinate patch (7) the 2×2 matrix X = ( X−1+X2 X1−X0 X1+X0 X−1−X2 ) takes the following form: X = ( coth x cosh θ+ cosh τsinh x coth x sinh θ− sinh τ sinh x coth x sinh θ+ sinh τsinh x coth x cosh θ− cosh τ sinh x ) . (10) Now it is easy to see that the time-translation τ 7→ τ′ = τ + � is induced by the noncompact SO(1, 1) ⊂ SO(2, 2) group action X 7→ X′ = gLXgR given by the matrices gL = g−1R = ( cosh �2 sinh � 2 sinh �2 cosh � 2 ) ∈ SO(1, 1). (11) Likewise, the spatial-translation θ 7→ θ′ = θ + � is induced by another SO(1, 1) ⊂ SO(2, 2) group action X 7→ X′ = gLXgR given by the matrices gL = gR = ( cosh �2 sinh � 2 sinh �2 cosh � 2 ) ∈ SO(1, 1). (12) Hence, prior to making the periodic identification θ ∼ θ + 2nπ, the time- and spatial-translation gen- erators i∂τ and −i∂θ must be given by two distinct SO(1, 1) generators of the Lie group SO(2, 2) ∼= (SL(2,R)L × SL(2,R)R)/Z2. After the periodic identification θ ∼ θ + 2nπ, on the other hand, gL and gR in Eq. (12) should be regarded as an element of the coset SO(1, 1)/Z,2 where the Z- identification is defined by � ∼ � + 2nπ (n ∈ Z). Hence, the AdS3 black hole is given by the quotient space AdS3/Z, where the identification subgroup Z = {gn | n = 0,±1,±2, · · ·} ⊂ SO(2, 2) is gener- ated by the matrix g = gL = gR = ( cosh π sinh π sinh π cosh π ) . It should be emphasized here that the fact that the time-translation generator generates the non- compact Lorentz group SO(1, 1) is a manifestation of thermodynamic aspects of a black hole: If we work in Euclidean signature, the noncompact Lorentz group SO(1, 1) becomes the compact rotation group SO(2) ∼= S1 such that the frequencies conjugate to the imaginary time are quantized and hence give rise to the Matsubara frequencies. (4.) Two-point function. As we have seen, the AdS3 black hole (1) is a locally AdS3 spacetime but its global structure is quite different from AdS3. This global difference of course leads to a big difference between the structure of two-point functions of CFT2 living on the boundary of the AdS3 black hole and those living on the boundary of AdS3 [10]. To see this, let GAdS3 BH(τ,θ) be a scalar two-point function of CFT2 dual to the AdS3 black hole and let GAdS3 (τ,θ) 2Note that the parameter space of SO(1, 1) (more precisely, SO+(1, 1), i.e. the connected component to the identity element) is the whole line R. Hence the parameter space of SO(1, 1)/Z is R/Z, which is isomorphic to a circle S1. be a scalar two-point function of CFT2 living on the AdS3 boundary without periodic identification. Then, once we get GAdS3 (τ,θ), the scalar two-point function of CFT2 dual to the AdS3 black hole is given by the coset construction (or the method of images [10]): GAdS3 BH(τ,θ) = ∑ n∈Z ρ(n)GAdS3 (τ,θ + 2nπ), (13) where ρ : Z → U(1) is a scalar (i.e. one-dimensional) unitary representation of the identification subgroup Z and given by ρ(n) = einα. Here α is a real parameter and its value depends on the model. For example, for scalar operator O(τ,θ) that satisfies the periodic boundary condition O(τ,θ + 2π) = O(τ,θ), α is zero (i.e. ρ is the trivial representation). (Basically, α is a boundary condition parameter for O(τ,θ) with respect to angle θ.) We emphasize that, regardless of the value of α, thus constructed two-point function (13) indeed satisfies the periodic boundary condition GAdS3 BH(τ,θ + 2π) = GAdS3 BH(τ,θ). For simplicity throughout this paper we will focus on GAdS3 (i.e. the zero-winding sector of GAdS3 BH), because GAdS3 BH can be constructed from the knowl- edge of GAdS3 . Hence in what follows we do not need to worry about the subtleties of periodic identification and global difference between the AdS3 black hole and the AdS3 spacetime.3 Let us next consider a massive scalar field φ of mass m on the background spacetime (8) (without periodic identification) that satisfies the Klein-Gordon equation (�AdS3 −m2)φ = 0, where the d’Alembertian is given by �AdS3 = sinh 2 x [ −∂2τ + ∂2x − 1 sinh x cosh x∂x − −∂2θ cosh2 x ] . In order to get CFT two-point functions via real-time AdS/CFT prescription, we need to find a solution to the Klein-Gordon equation whose τ- and θ-dependences are given by the plane waves, φ(τ,x,θ) = φω,k(x)e−iωτ+ikθ; that is, we need to know a simultaneous eigenfunction of the d’Alembertian �AdS3, the time-translation generator i∂τ and the spatial-translation generator −i∂θ. For such a simultaneous eigenfunction the Klein-Gordon equation reduces to the following differential equation:4 ( −∂2x + 1 sinh x cosh x ∂x + ∆(∆ − 2) sinh2 x + k2 cosh2 x ) φω,k = ω2φω,k, (14) where ∆ = 1 + √ m2 + 1 is one of the solutions to the quadratic equation ∆(∆ − 2) = m2. Note that near the 3Actually, the momentum-space two-point functions com- puted in Refs. [5, 8, 9] are nothing but the momentum-space representation of GAdS3 rather than GAdS3 BH (or GBTZ). 4Redefining the field as φ 7→ φ̃ = (coth x)−1/2φ, one sees that the differential equation (14) reduces to the Schrödinger equation with hyperbolic Pöschl-Teller potential( −∂2x + (∆ − 1)2 − 1/4 sinh2 x + k2 + 1/4 cosh2 x ) φ̃ω,k = ω2φ̃ω,k. 144 vol. 54 no. 2/2014 A Simple Derivation of Finite-Temperature CFT Correlators AdS3 boundary x = 0 the differential operator in the left- hand side of (13) behaves as −∂2x+ 1 x ∂x+ ∆(∆−2) x2 +O(1). Hence the general solution has the following asymptotic near-boundary behavior: φ(τ,x,θ) ∼ A∆(ω,k)x∆e−iωτ+ikθ + B∆(ω,k)x2−∆e−iωτ+ikθ as x → 0, (15) where A∆(ω,k) and B∆(ω,k) are integration constants which may depend on ∆, ω and k. The real-time prescription of AdS/CFT correspondence tells us that the retarded and advanced two-point functions are given by the ratio [4, 5] G R/A ∆ (ω,k) = (2∆ − 2) A∆(ω,k) B∆(ω,k) , (16) where the retarded two-point function GR∆ is obtained by the solution that satisfies the in-falling boundary conditions at the horizon, whereas the advanced two- point function GA∆ is obtained by the solution that satisfies the out-going boundary conditions at the horizon [6]. The goal of this paper is to compute the ratio (15) in a Lie-algebraic fashion without solving the Klein-Gordon equation explicitly. 3. Lie algebra sl(2, R)L ⊕ sl(2, R)R in the SO(1, 1) × SO(1, 1) diagonal basis In order to get the momentum-space two-point func- tions, we need to find a simultaneous eigenfunction of the d’Alembertian �AdS3, the time-translation gen- erator i∂τ and the spatial-translation generator −i∂θ. As we will see below, the d’Alembertian is given by the quadratic Casimir of the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R. On the other hand, as we have seen in the previous section, the time- and spatial- translations are induced by two distinct noncompact SO(1, 1) group actions such that i∂τ and −i∂θ must be given by SO(1, 1) generators of the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R. Hence we need to work in the basis in which the noncompact SO(1, 1) generators become diagonal. We note that unitary repre- sentations of the Lie algebra sl(2,R) in the noncompact SO(1, 1) basis have been studied in the mathematical literature [11, 12], and are known to be a bit com- plicated. In this paper we will not touch upon these mathematical subtleties and will not discuss which of the unitary representations are realized in the scalar field theory on the background (8) (without periodic identification).5 Instead, we will present a rather heuris- tic argument that reproduces the known results by just 5One way to avoid these subtleties is to Wick-rotate both the time τ and angle θ. In such Euclidean-like signature, the noncompact SO(1, 1) ×SO(1, 1) symmetry becomes the compact SO(2) ×SO(2) symmetry such that we can use standard unitary representations of the Lie-algebra so(2, 2) ∼= sl(2,R)L⊕sl(2,R)R in the SO(2)×SO(2) diagonal basis. In this approach, computations of momentum-space two-point functions are essentially reduced to those presented in Ref. [1]. using the ladder equations of the Lie algebra so(2, 2) in the SO(1, 1) ×SO(1, 1) diagonal basis. To begin with, let us first recall the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R, which is spanned by six self-adjoint generators {A0,A1,A2,B0,B1,B2} that satisfy the following commutation relations: [A0,A1] = iA2, [A1,A2] = −iA0, [A2,A0] = iA1, (17a) [B0,B1] = iB2, [B1,B2] = −iB0, [B2,B0] = iB1, (17b) with other commutators vanishing, [Aa,Bb] = 0 (a,b = 0, 1, 2). We note that A0 and B0 are com- pact SO(2) generators, whereas A1, A2, B1, B2 are noncompact SO(1, 1) generators. Note also that the standard classification of unitary representations of the Lie algebra so(2, 2) is based on the Cartan-Weyl ba- sis {A0,A1 ± iA2,B0,B1 ± iB2}, where A1 ± iA2 and B1±iB2 play the role of ladder operators that raise and lower the eigenvalues of A0 and B0 by ±1. For the fol- lowing discussions, however, it is convenient to introduce the hermitian linear combinations A± = A2 ∓A0 and B± = B2 ∓B0, which also play the role of “ladder” op- erators; see next section. In the basis {A1,A±,B1,B±} the commutation relations (17a) and (17b) are cast into the following forms: [A1,A±] = ±iA±, [A+,A−] = 2iA1, (18a) [B1,B±] = ±iB±, [B+,B−] = 2iB1. (18b) In the problem of scalar field theory on the background (8) (without periodic identification), these symmetry generators turn out to be given by the following first- order differential operators: A1 = i 2 (∂τ + ∂θ), (19a) A± = − i 2 e±(τ+θ) [ sinh x∂x ± ( cosh x∂τ + 1 cosh x ∂θ )] , (19b) B1 = i 2 (∂τ −∂θ), (19c) B± = + i 2 e±(τ−θ) [ sinh x∂x ± ( cosh x∂τ − 1 cosh x ∂θ )] , (19d) which indeed satisfy the commutation relations (18a) and (18b). The quadratic Casimir of the Lie algebra so(2, 2) ∼= sl(2,R)L⊕sl(2,R)R yields the d’Alembertian on the AdS3 black hole C2(so(2, 2)) = 2C2(sl(2,R)L) + 2C2(sl(2,R)R) = sinh2 x ( −∂2τ + ∂ 2 x − 1 sinh x cosh x ∂x − −∂2θ cosh2 x ) , (20) where the quadratic Casimir of each sl(2,R) is given by C2(sl(2,R)L) = (A0)2−(A1)2−(A2)2 = −A1(A1±i)− 145 Satoshi Ohya Acta Polytechnica A∓A± and C2(sl(2,R)R) = (B0)2 − (B1)2 − (B2)2 = −B1(B1 ± i) −B∓B±. A straightforward calculation shows that C2(sl(2,R)L) and C2(sl(2,R)R) coincide and are given by C2(sl(2,R)L) = C2(sl(2,R)R) = 1 4 C2(so(2, 2)). (21) Asymptotic near-boundary algebra. We are in- terested in the asymptotic near-boundary behavior of the solution to the Klein-Gordon equation (14). To analyze this, let us introduce the boundary symmetry generators defined as the limit x → 0 of (19a)–(19d): A01 := lim x→0 A1 = i 2 (∂τ + ∂θ), (22a) A0± := lim x→0 A± = − i 2 e±(τ+θ) [ x∂x ± (∂τ + ∂θ) ] , (22b) B01 := lim x→0 B1 = i 2 (∂τ −∂θ), (22c) B0± := lim x→0 B± = + i 2 e±(τ−θ) [ x∂x ± (∂τ −∂θ) ] , (22d) which still satisfy the commutation relations of the Lie algebra so(2, 2) ∼= sl(2,R)L ⊕ sl(2,R)R [A01,A 0 ±] = ±iA 0 ±, [A 0 +,A 0 −] = 2iA 0 1, (23a) [B01,B 0 ±] = ±iB 0 ±, [B 0 +,B 0 −] = 2iB 0 1. (23b) The quadratic Casimir of this asymptotic near-boundary algebra, which we denote by so(2, 2)0 ∼= sl(2,R)0L ⊕ sl(2,R)0R, takes the following simple form: C2(so(2, 2)0) = 4C2(sl(2,R)0L) = 4C2(sl(2,R)0R) = x 2∂2x −x∂x =: C 0 2. (24) As we have repeatedly emphasized, we are interested in si- multaneous eigenstates of the d’Alembertian �AdS3 x→0→ C02 , the time-translation generator i∂τ = B01 + A01 and the spatial-translation generator −i∂θ = B01 −A01. Let |∆,kL,kR〉0 be a simultaneous eigenstate of C02 , A01 and B01 that satisfies the following eigenvalue equations: C02|∆,kL,kR〉 0 = ∆(∆ − 2)|∆,kL,kR〉0, (25a) A01|∆,kL,kR〉 0 = kL|∆,kL,kR〉0, (25b) B01|∆,kL,kR〉 0 = kR|∆,kL,kR〉0. (25c) In the coordinate realization these eigenvalue equations become the following differential equations:( −∂2x + 1 x ∂x + ∆(∆ − 2) x2 ) φ0∆,kL,kR = 0, (26a) (i∂xL −kL)φ 0 ∆,kL,kR = 0, (26b) (i∂xR −kR)φ 0 ∆,kL,kR = 0, (26c) where xL and xR are light-cone coordinates given by xL = τ + θ and xR = τ −θ, and (kL,kR) and (ω,k) are related by kL = (ω −k)/2 and kR = (ω + k)/2. These differential equations are easily solved with the result φ0∆,kL,kR(τ,x,θ) = A∆(kL,kR)x ∆e−ikLxLe−ikRxR + B∆(kL,kR)x2−∆e−ikLxLe−ikRxR, (27) which precisely coincides with the asymptotic near- boundary behavior of the solution (15). 4. Recurrence relations for finite-temperature CFT2 two-point functions As mentioned in the previous section, A± and B± (and also A0± and B0±) play the role of “ladder” operators. To see this, let us consider states A0±|∆,kL,kR〉0 and B0±|∆,kL,kR〉0. The commutation relations [A01,A0±] = ±iA0± and [B01,B0±] = ±iB0± give A01A0±|∆,kL,kR〉0 = (kL±i)A0±|∆,kL,kR〉0 and B01B0±|∆,kL,kR〉0 = (kR± i)B0±|∆,kR,kL〉0, which imply that A0± and B0± raise and lower the eigenvalues kL and kR by ±i:6 A0±|∆,kL,kR〉 0 ∝ |∆,kL ± i,kR〉0, (28a) B0±|∆,kL,kR〉 0 ∝ |∆,kL,kR ± i〉0. (28b) In the coordinate realization (22a) and (22c) with the solution (27), the left-hand sides become A0±φ 0 ∆,kL,kR = i ( − ∆ 2 ± ikL ) A∆(kL,kR) ×x∆e−i(kL±i)xLe−ikRxR + i (∆ 2 − 1 ± ikL ) B∆(kL,kR) ×x2−∆e−i(kL±i)xLe−ikRxR, (29a) B0±φ 0 ∆,kL,kR = −i ( − ∆ 2 ± ikR ) A∆(kL,kR) ×x∆e−ikLxLe−i(kR±i)xR − i (∆ 2 − 1 ± ikR ) B∆(kL,kR) ×x2−∆e−ikLxLe−i(kR±i)xR, (29b) which should be proportional to φ0∆,kL±i,kR and φ0∆,kL,kR±i, respectively. In other words, the inte- gration constants should satisfy the recurrence rela- tions (−∆2 ± ikL)A∆(kL,kR) ∝ A∆(kL ± i,kR) and ( ∆2 − 1 ± ikL)B∆(kL,kR) ∝ B∆(kL ± i,kR), and simi- lar expressions for kR. Hence the two-point function G∆(kL,kR), which is given by the ratio G∆(kL,kR) = 6One may wonder why the eigenvalues of the self-adjoint operators A01 and B 0 1 take the complex values kL ± i and kR ± i. The reason is that, even if the state |∆,kL,kR〉0 lies inside the domain in which the operators A01 and B 0 1 become self-adjoint, the states A0±|∆,kL,kR〉 0 and B0±|∆,kL,kR〉 0 turn out to lie outside the self-adjoint domain of A01 and B 0 1 . (For rigorous mathematical discussions we refer to the literature [11, 12].) As we will see below, however, a naive use of the “ladder” equations (28a) and (28b) correctly yields the retarded and advanced two-point functions. 146 vol. 54 no. 2/2014 A Simple Derivation of Finite-Temperature CFT Correlators (2∆ − 2)A∆(kL,kR)/B∆(kL,kR), should satisfy the following recurrence relations: G∆(kL,kR) = ∆ 2 − 1 ± ikL −∆2 ± ikL G∆(kL ± i,kR), (30a) G∆(kL,kR) = ∆ 2 − 1 ± ikR −∆2 ± ikR G∆(kL,kR ± i). (30b) These recurrence relations are linear such that they are easily solved by iteration. But how should we iden- tify the solutions to these recurrence relations with the retarded and advanced two-point functions? A standard prescription to get the retarded (advanced) two-point functions via AdS/CFT is to use the so- lution to the Klein-Gordon equation that satisfies the in-falling (out-going) boundary conditions at the horizon x = ∞ [6]. Here we present an alterna- tive approach to get the retarded and advanced two- point functions without knowing the boundary condi- tions at the horizon x = ∞. A key is the generic causal properties of two-point functions: The re- tarded two-point function has support only on the future light-cone, whereas the advanced two-point func- tion has support only on the past light-cone. Let us first focus on the case where the point (τ,θ) on the AdS3 boundary (∂AdS3) lies inside the future light-cone xL = τ + θ > 0 and xR = τ − θ > 0. In this case the state (A0−)n(B0−)mφ0∆,kL,kR ∝ e−i(kL−in)xLe−i(kR−im)xR converges as n,m → ∞ such that (A0−)n(B0−)mφ0∆,kL,kR would be well-defined. Hence it would be natural to expect that the ladder equations A0−φ0∆,kL,kR ∝ φ 0 ∆,kL−i,kR and B0−φ 0 ∆,kL,kR ∝ φ 0 ∆,kL,kR−i would lead to the retarded two-point function. Indeed, iterative use of the re- lations G∆(kL,kR) = ∆ 2 −1−ikL −∆2 −ikL G∆(kL − i,kR) and G∆(kL,kR) = ∆ 2 −1−ikR −∆2 −ikR G∆(kL,kR − i) gives GR∆(kL,kR) = Γ( ∆2 − ikL) Γ(1 − ∆2 − ikL) × Γ( ∆2 − ikR) Γ(1 − ∆2 − ikR) gR(∆), (31) where gR(∆) is a normalization factor given by gR(∆) = limn,m→∞GR∆(kL − in,kR − im). This is the retarded two-point function with the desired analytic structure: GR∆(kL,kR) is analytic in the upper-half complex kL- and kR-planes and has simple poles at kL = −i2πT ( ∆2 + n) and kR = −i2πT ( ∆2 +m) (n,m ∈ Z≥0) on the lower- half complex kL- and kR-planes, where T = 12π (= 1 2πR ) is the Hawking temperature with respect to time τ. Let us next derive the retarded two-point function of CFT2 dual to the rotating BTZ black hole (4). To this end, let pL and pR be momenta conjugate to the BTZ light-cone coordinates t±φ. Since τ ±θ and t±φ are related as τ ±θ = (r+ ∓r−)(t±φ), we have kL = 1r+−r− pL and kR = 1r++r− pR, from which we get GR∆(pL,pR) = Γ(hL − ipL2πTL ) Γ(h̄L − ipL2πTL ) Γ(hR − ipR2πTR ) Γ(h̄R − ipR2πTR ) gR(∆), (32) where TL and TR are the Hawking temperature for left- and right-moving sectors with respect to the BTZ time t and given by TL = r+ −r− 2π and TR = r+ + r− 2π . (33) hL and hR are conformal weights for a scalar operator of dual CFT2 given by hL = hR = ∆ 2 with h̄L = h̄R = 1 − ∆ 2 . (34) Note that Eq. (32) precisely coincides with the known results [8] (see also [5, 9] for the case of fermionic operators.) Let us next move on to the case where the point (τ,θ) ∈ ∂AdS3 lies inside the past light- cone xL = τ + θ < 0 and xR = τ − θ < 0. In this case the state (A0+)n(B0+)mφ0∆,kL,kR ∝ e−i(kL+in)xLe−i(kR+im)xR converges as n,m →∞ such that (A0+)n(B0+)mφ0∆,kL,kR would be well-defined. Iter- ative use of the relations G∆(kL,kR) = ∆ 2 − 1 + ikL −∆2 + ikL G∆(kL + i,kR), (35a) G∆(kL,kR) = ∆ 2 − 1 + ikR −∆2 + ikR G∆(kL,kR + i), (35b) then gives the advanced two-point function GA∆(pL,pR) = Γ(hL + ipL2πTL ) Γ(h̄L + ipL2πTL ) Γ(hR + ipR2πTR ) Γ(h̄R + ipR2πTR ) gA(∆), (36) where gA(∆) = limn,m→∞GA∆(pL + in,pR + im). We note that, since in general the retarded and advanced two-point functions are related by complex conjugate GA∆(pL,pR) = [G R ∆(pL,pR)] ∗, the normalization con- stants must be related by gA(∆) = [gR(∆)]∗. Acknowledgements The author is supported in part by ESF grant CZ.1.07/2.3.00/30.0034 References [1] S. Ohya, “Recurrence relations for finite-temperature correlators via AdS2/CFT1,” JHEP 12 (2013) 011, arXiv:1309.2939 [hep-th]. doi: 10.1007/JHEP12(2013)011 [2] M. Bañados, C. Teitelboim, and J. Zanelli, “Black hole in three-dimensional spacetime,” Phys. Rev. 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