Holographic Euclidean thermal correlator

In this paper, we compute holographic Euclidean thermal correlators of the stress tensor and $U(1)$ current from the AdS planar black hole. To this end, we set up perturbative boundary value problems for Einstein's gravity and Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with appropriate gauge fixing and regularity boundary conditions at the horizon of the black hole. The linearized Einstein equation and Maxwell equation in the black hole background are related to the Heun equation of degenerate local monodromy. Leveraging the connection relation of local solutions of the Heun equation, we partly solve the boundary value problem and obtain exact two-point thermal correlators for $U(1)$ current and stress tensor in the scalar and shear channels.


Introduction
As an embodiment of the holographic principle [1,2], the Anti-de Sitter gravity/conformal field theory (AdS/CFT) correspondence [3][4][5] establishes a connection between a quantum gravity theory in AdS space and a conformal field theory on the boundary.This equivalence is encapsulated in the Gubser-Klebanov-Polyakov-Witten (GKPW) relation, where the partition function of the conformal field theory with operator sources equals the gravity partition function with prescribed boundary conditions In the most helpful limit to exploit this correspondence, the classical gravity on-shell action becomes the generating functional of connected correlators of the strongly-coupled CFT Correlators are computed by functional differentiation of the generating functional, which amounts to solving the perturbative boundary value problem for the bulk fields' equation of motion.This involves varying the boundary value of the bulk fields and solving for the corresponding variation of the on-shell configuration in the bulk.The near-boundary behavior is well-established, allowing the extraction of holographic correlators [6][7][8][9].However, solving the global boundary value problem is generally intricate, exemplified in cases like pure gravity [10].
Although the prescription is clear, explicit computation of holographic Euclidean correlators in the GKPW approach has been limited to pure AdS space and its quotient spaces, such as thermal AdS where the method of images can be applied (e.g., see [11] for thermal bootstrap emphasis).In our prior work [12], we computed holographic torus correlators of the stress tensor.This study focuses on Euclidean thermal two-point correlators of the stress tensor and U (1) current in four-dimensional CFTs.Beyond the Hawking-Page transition [13], the thermal state holographically corresponds to a five-dimensional Euclidean AdS planar black hole [14].Correlators are derived by solving perturbative boundary value problems in Einstein's gravity and Maxwell theory for the U (1) gauge field in the black hole background.Two important steps are involved.The first is to appropriately fix the gauge and impose regularity boundary conditions at the horizon, ensuring a unique solution.The second step identifies the equations of motion as the Heun equation [15], and solves the boundary value problems with the connection relation of local solutions.The general connection relation was established in [16], and it has been applied to exact thermal correlators in Minkowski signature [17,18], and employed in various black hole perturbation problems [19][20][21][22][23][24].In our case, the Heun equations feature degenerate local monodromy, with characteristic exponents differing by an integer.We compute the connection relation by taking a limit of the generic case.Ultimately, we obtain exact two-point correlators for the U (1) current and stress tensor in the scalar and shear channels (as defined in [25]).
Thermal two-point correlators, also known as thermal spectral functions, have many important applications and have been studied in [25] using gauge invariants in each channel.In the final discussion section, we comment on our approach to holographic computation and relevant applications of thermal two-point correlators.

Holographic setup
We start by reviewing the basics of holographic computation independent of the bulk background geometry.For Einstein's gravity, it's customary to work in the Fefferman-Graham gauge [6,26] near the conformal boundary and in dimension four, we have the series expansion The background metric of the holographic field theory γ ij corresponds to g ij , and the one-point correlator of the stress tensor 1 , with appropriate renormalization, is given by [7,27,28] where ij .In our case, the holographic field theory lives on a flat background, so the terms of Schouten tensor do not contribute.We have 1 Our convention for the stress tensor is, for classical field theories, δγ S = 1 2 dV Tijδγ ij .For quantum theories Tij = 2 δI δγ ij .The integration measure is assumed to be included in the definition of the functional derivative.
The Einstein equation near the conformal boundary determines the series (2.2) in terms of g (0) ij and g (4) ij or equivalently the one point correlator T ij , and imposes holographic Ward identities of conservation and the Weyl anomaly on T ij .Near boundary solutions of the Einstein equation are in one-to-one correspondence to the pair (γ ij , T ij ).The global geometry of the bulk spacetime fully determines the one-point correlator as a functional of the boundary metric, from which we can compute multi-point correlators by functional differentiation.In particular for two-point correlator we have Similarly, for the U (1) gauge field, we can put it in the radial gauge near the conformal boundary using the Fefferman-Graham coordinates of the bulk metric with the series expansion The one-point correlator with appropriate renormalization is given by The Maxwell equation near the conformal boundary determines the series (2.7) in terms of A i and A i or equivalently the one point correlator J i , and imposes the holographic Ward identity of conservation on J i .The global geometry of the bulk spacetime fully determines J i .Multi-point correlators are computed by taking functional derivative with respect to the source A, for two point correlator we have Now we specialize in the holographic correlators from the five-dimensional AdS planar black hole.The black hole is a solid cylinder B 2 × R 3 with the metric The period of Euclidean time t, namely the inverse temperature, is β = πρ 0 .The conformal boundary is at ρ = 0, and the horizon is at ρ = ρ 0 , being the R 3 axis of the cylinder.The standard Fefferman-Graham radial coordinate r is related to ρ by For simplicity, we set ρ 0 = 1 in the metric, effectively working in the unit of ρ 0 , and we will recover ρ 0 dependence when final results are obtained.As a convention, we label bulk spacetime coordinate indices by Greek alphabets µ, ν, ρ, . .., the boundary spacetime coordinate indices by Roman alphabets i, j, k, . . .and the boundary space indices by a, b, c, . ...

U(1) current
Now, we work on the boundary value problem of the U (1) gauge field, beginning with gauge-fixing.We can put it in the radial gauge A ρ = 0 in the region 0 ≤ ρ < 1 (excluding the horizon) by a U (1) gauge transformation.For a global solution, its restriction to the region 0 ≤ ρ < 1 must have a regular limit going to the horizon ρ = 1.Therefore, we formulate the boundary value problem in the radial gauge, with the boundary condition that the solution has a regular limit as ρ → 1 after a gauge transformation.To work out the explicit form of the boundary condition, we introduce the "cylindrical radial coordinate" s Near the horizon ρ → 1 or s → 0, the metric takes the form of Euclidean metric in "cylindrical coordinates" and the horizon is properly covered by the "Cartesian coordinates" The gauge field is regular at the horizon if and only if its components in a coordinate chart that properly covers the horizon, for example the "Cartesian coordinates", are regular.
That is, we have and there exists a U (1) gauge transformation Λ, such that The components on the right-hand side can only depend on x because the t-circle shrinks to a point as s → 0. We find as a turned-on source on the CFT side, determine a unique solution to the Maxwell equation as we will see.
We utilize the translational symmetry in t, x direction and work with Fourier modes Ãi with Matsubara frequency ω = 2m, m ∈ Z and spatial momentum p.For simplicity, we also rotate the spatial momentum to the x 1 direction.The Maxwell equation then decouples to the transverse channel for Ã2 , Ã3 and the longitudinal channel for Ãt , Ã1 .
For the transverse component Ã2 (and the same for Ã3 ) we have where we used the convenient coordinate z = ρ 2 .This is an ordinary differential equation with four regular singularities z = 0, 1, −1, ∞.By the substitution Ã2 (z) = (1−z 2 ) − 1 2 w(z), we get a Heun equation in the normal form for w(z) with the Heun equation parameters We refer the readers to Appendix A for a brief review of Fuchsian differential equations, the Heun equation, its connection problem, and notational conventions.By the boundary condition (3.9), Ã2 is regular at z = 1, so it must be proportional to the solution of exponent |m| 2 at z = 1.The constant of proportionality is determined by the boundary condition Ã2 | z=0 = Ã2 and the connection relation (A.12).We find For the longitudinal components Ãt , Ã1 , we have When m = 0, the solution to this third-order differential equation is determined by the three boundary conditions ) can be transformed to the normal Heun equation By (3.19) the solution must be proportional to w + for Ã1 to be regular at z = 1.The constant of proportionality can be further determined by using the connection relation (A.12) and evaluating (3.17) at z = 0. We find Then we integrate to obtain Ãt with the constant of integration given by the boundary value Ãt We get Ã1 by plugging Ãt back to (3.19) When m = 0, we get Ã1 = Ã1 from (3.19).We still solve for z Then we plug it into (3.17) and evaluate at z = 1.We have Ãt (m = 0)| z=1 = 0 from the boundary condition (3.10), and we find z + , the same as the previous case when m = 0. So, we can carry over the results for m = 0 and set m = 0 in the expression.
To obtain the holographic correlators, we recover the dependence on ρ 0 or the inverse temperature β = πρ 0 , and read off A (2) i from the bulk gauge field A i (in our case the coefficient of z 1 ) Ã( 2) where We compute two-point correlators by the formula for renormalized one-point correlators (2.8).Rotating the spatial momentum to a general direction, we find Gauge fixing and regularity boundary conditions at the horizon for Einstein's gravity follow the same line as the Maxwell theory.We can make the solid cylinder coordinates ρ, t, x the Fefferman-Graham coordinates of the perturbed bulk metric in the region 0 ≤ ρ < 1 by a diffeomorphism.Then, the boundary value problem is formulated in this gauge with the boundary condition that the metric has a regular limit as ρ → 1 after a diffeomorphism.For a first-order perturbation of the bulk metric, we have And to the first order, the diffeomorphism is characterized by a vector V , then the regularity boundary condition at the horizon is the variation of the bulk metric has a regular limit as ρ → 1 (or s → 0), that is, its components in the "Cartesian coordinates" (3.3) are regular.We find Similar to the case of U (1) gauge field, we work in Fourier modes and rotate the spatial momentum to the x 1 direction.And for simplicity, we use the variable h ij = ρ 2 δg ij which on the conformal boundary equals the variation of the CFT background metric 12) The linearized Einstein equation decouples to the scalar channel of h23 and h22 − h33 , the shear channel of ht2 , h12 and ht3 , h13 , and the sound channel of htt , h11 , h22 + h33 , ht1 .In the scalar channel, we have and in the shear channel, we have The computation in these two channels is very similar to that of the transverse channel and longitudinal channel of the U (1) gauge field in the previous section, so we will be brief.
For the scalar channel, by the substitution h23 (z) = z 1 2 (1 − z 2 ) − 1 2 w(z) we obtain Heun equation in the normal form With the boundary conditions and the connection relation (A.13) we find For the shear channel, plugging (4.17) into ∂ z (4.15), we obtain By the substitution ∂ z ht2 = (1 − z 2 ) − 1 2 w(z), the equation is transformed to Heun equation of normal form With boundary conditions and the connection relation (A.13) we find and furthermore ht2 = δγ t2 (1 + . ..) + p 2 + 4m 2 32 (2mp The coefficient of z 2 in the solution of perturbed bulk metric corresponds to the variation of g ij and hence the variation of the one-point correlator by (2.4).Then we can read off the two-point correlators with The unsolved part for the stress tensor is the sound channel.We have We don't know how to analytically solve the boundary value problem here.For future reference, we can reduce the sound channel to a five-dimensional first-order equation of variables htt , h11 , h22 + h33 2 , ht1 , ∂ z ht1 (a similar equation can be found in [25]), and by the substitution we can transform the equation into a Fuchsian system of normal form2 with

Summary and discussion
In our study, we calculated holographic Euclidean thermal correlators of the U (1) current and stress tensor for four-dimensional CFTs using the AdS 5 planar black hole, following the approach of GKPW.By utilizing the connection relation of local solutions of the Heun equation, we obtained exact correlators for the U (1) current and stress tensor in the scalar and shear channels.
Extensive research has focused on thermal two-point correlators (thermal spectral functions).Notably, [25] demonstrated the presence of gauge invariants in each channel that diagonalize coupled differential equations.These invariants and their derivatives render the on-shell action quadratic.Thermal two-point correlators have been computed using this formalism numerically or analytically by approximations [31,32].For example in the longitudinal channel the gauge invariant is E L = p Ãt − ω Ã1 and we have This is a Fuchsian differential equation with six singularities.The two singularities z = ± 1 + ω 2 p 2 are apparent singularities since they don't appear in the equation of the fields.One can verify that these apparent singularities cannot be transformed away by a substitution E L (z) = P (z)f (z) where P (z) is a meromorphic function that does not introduce new singularities.In essence, these apparent singularities remain inherent to the equation.We don't know how to relate this equation to the Heun equation and obtain the exact holographic correlators.From the technical standpoint, we want to work with equations of fields, and in the Euclidean signature, the boundary conditions of fields with gauge/diffeomorphism symmetry are clearly specified.This is the technical reason for our approach of holographic computation, in addition to giving an illustrative example of Euclidean boundary value problems.
Thermal two-point correlators find diverse applications.They encode the information in operator product expansion (OPE) of holographic CFTs.For instance, [33][34][35] computed holographic correlators in the OPE limit via near-boundary analysis, extracting OPE coefficients for multi-stress tensors.For integer operator dimension with operator mixing, exact two-point correlators are necessary for complete OPE coefficient extraction.Moreover, if we can analytically continue to the Lorentzian signature, we will get better understanding of the linear response to perturbations in thermal equilibrium, and compute transport coefficients such as shear viscosity, thermal conductivity, and electric conductivity [36,37], and higher order transport coefficients (see [38,39] for formula of second order coefficients in terms of two-point correlators and holographic computation).In addition, we can probe the chaotic dynamics by studying the pole-skipping of the correlators [40][41][42].In addition, we consider spherical thermal correlators (scalar case solved in [17]) and stress tensor correlators from a higher derivative gravity theory as interesting generalizations of our work.

A Fuchsian ODE, the Heun equation and connection problem
In this appendix, we briefly review Fuchsian differential equations, the Heun equation, and its connection relation we used in the computation in the main text.
An ordinary differential equation (ODE) is called Fuchsian if the coefficients are rational functions and all singularities are regular.Eigenvectors of local monodromy constitute a natural basis of local solutions around singularities.When eigenvalues of the local monodromy are all distinct, eigenvectors span the space of local solutions, and they take the form of a series where z 0 is the singularity, k labels the local solution and the prefactor (z − z 0 ) ρ k captures the local monodromy.The characteristic exponents ρ k are computed as the roots of the indicial equation.We usually adopt the normalization that c 0 = 1.When we have repeated eigenvalues of the local monodromy, that is some characteristic exponents differ by integers, we may need generalized eigenvectors to span the space of local solutions, and they are expressed as series with logarithms.For a second order ODE, we label the two characteristic exponents as ρ + , ρ − , with Reρ + ≥ Reρ − .There is always a series solution without logarithm w − .For computational convenience, we choose the convention that the coefficient of the power (z − z 0 ) ρ + is zero in w (z 0 ) − .The Heun equation is the second-order Fuchsian ODE with four regular singularities.By Möbius transformation and substitutions, we can bring it to the normal form The four singularities with exponents at these points are We adopt the convention that Rea 0 ≥ 0 etc., so the exponents with plus sign will be the exponent with greater real part ρ + .The connection relation of the local solutions in the generic case (that is, characteristic exponents do not differ by an integer) was studied in [16] by relating the Heun equation to the Belavin-Polyakov-Zamolodchikov (BPZ) equation [43] satisfied by conformal blocks with degenerate insertion 4 in the Liouville field theory in the semiclassical limit.By the Alday-Gaiotto-Tachikawa (AGT) correspondence, the Liouville correlators can be exactly computed by localization in supersymmetric gauge theories [46][47][48][49][50]. Without losing generality, let z = 0 and z = 1 be two adjacent singularities, the connection relation between local solutions around these two points is where and F is the Nekrasov-Shatashvili function, defined as power series in 1 t with combinatorially defined rational functions of other parameters as the coefficients, see Appendix B for details.The exchange momentum a is to be implicitly determined from the relation In our computation, the masslessness of the bulk fields leads to a degenerate local monodromy of the Heun equation at z = 0 (the conformal boundary), that is, two exponents differ by an integer (a 0 becomes a half-integer).This degenerate scenario can be derived as a limit of the generic case, as a specific solution to the Heun equation continuously depends on the parameters.The emergence of logarithm and the discontinuity of the local monodromy basis reflect a qualitative change of the local monodromy, rather than a specific solution.The solution w + remains well-defined and continuously depends on parameters including a 0 , even when a 1 approaches half-integers 5 .We proceed to take the limit a 0 → N 2 , N ∈ N while keeping other parameters, such as t, a 1 , a t , a ∞ , a, fixed 6 .For a 0 = 0 we have 4 One can also refer to the relevant studies offered by [44,45]. 5Meanwhile w − is not continuous when a1 approaches half-integers.When both a0 and a1 are halfintegers, the complete connection relation is computed by solving two linear equations obtained from the limits of w (0) + and w (1) + . 6Another curve in the parameter space can also be chosen to approach the limit, such as fixing u, an explicit parameter in the Heun equation, instead of a.However, as the connection coefficients explicitly depend on a, fixing a yields a relatively simple expression for the limit.
The quantity in the square bracket must vanish when a 0 = 0 for the limit to exist.It indeed vanish because ∂ a 0 F | a 0 =0 = 0 with F being an even function of a 0 .Then the limit becomes the derivative with respect to a 0 , and we get where ψ denotes the digamma function.For a 0 = 1 2 we find w + = lim The quantity in the square bracket must vanish when a 0 = 1 2 for the limit to exist, that is, we must have one can verify (A.10) holds to the order of expansion.Again, the limit becomes the derivative with respect to a 0 and we find w  In general, the coefficient c N in the series solution z + on the right hand side of (A.4) simultaneously take a 0 = N 2 as a pole, so the limit a 0 → N 2 always becomes a differentiation with respect to a 0 .For example, for a 0 = 1 we have w We use Mathematica to compute the connection relation in the degenerate case for higher values of N .

B The Nekrasov-Shatashvili function
The Nekrasov-Shatashvili function is defined as where L stands for the leg length and A stands for the arm length of the site in the tableau.
Our definition of the Nekrasov-Shatashvili function is essentially the definition in [17], with a t and a 1 are swapped and t replaced by 1  t in the argument.That's because we consider the connection relation between local solutions around z = 0 and z = 1, while in [17] the pair z = 0 and z = t was considered.

(z 0 ) + with the exponent ρ + 3 .
If two exponents differ by an integer, the other solution w(z 0 ) − to form a basis may contain logarithm.There is also no canonical choice of w (z 0 ) − since we can add any constant multiple of w (z 0 ) + to w (z 0 )

1 2 −a 0 ∞
k=0 c k z k and the connection coefficient for w (0)