Cutoff AdS3 versus TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T\overline{T} $$\end{document} CFT2 in the large central charge sector: correlators of energy-momentum tensor

In this article we probe the proposed holographic duality between TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T\overline{T} $$\end{document} deformed two dimensional conformal field theory and the gravity theory of AdS3 with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T\overline{T} $$\end{document} CFT in a Euclidean plane and a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in Euclidean AdS3 with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T\overline{T} $$\end{document} CFT parameters.


Introduction
The TT deformation of two dimensional quantum field theory has received intensive study in the past few years. As an irrelevant deformation, it leads to well-defined, albeit nonlocal, UV completion. In fact, it is a solvable deformation in many senses. It preserves integrability structures [1,2], deforms the scattering matrix by multiplying CDD factors [3,4], has solvable deformation of finite size spectrum [3,5] and preserves modular invariance of conformal field theory torus partition function [6,7]. The non-locality and solvability of the TT deformation can be understood from a different perspective by reformulation to random geometry [8], which also neatly derives the flow equation of the partition function. In addition, the TT deformation can be re-interpreted as coupling to Jackiw-Teitelboim gravity of the quantum field theory, which leads to the same flow equation of the partition function and CDD factors of the scattering matrix [4,9]. Correlators of TT deformed QFT or CFT were studied in [10][11][12]. While much of the work on the TT deformation has been done in the flat Euclidean plane or its quotient spaces such as cylinder and JHEP12(2020)168 torus, generalization to maximally symmetric spaces was considered in [13,14]. Further generalization to generic curved spaces was studied in [15,16], which has remarkably reproduced lots of result of previous study.
For a holographic CFT 2 , it's natural to ask what the holographic dual of its TT deformation is. It was proposed by McGough et al. [17] that for positive TT deformation parameter the holographic dual is a Dirichlet cutoff in the AdS 3 gravity, based on computation of signal propagation speed, quasi-local energy of BTZ blackhole and other physics quantities. It was followed by study on holographic entanglement entropy [18][19][20][21][22][23][24][25], generalization to higher or lower dimensions [26][27][28][29][30][31][32], and an interesting perspective from path integral optimization [33]. In addition, the proposal was examined by holographic computation of correlators of energy-momentum tensor in [34]. It was found that the large central charge perturbative correlators in TT CFT 2 agree with correlators of classical pure gravity in cutoff AdS 3 given a holographic dictionary that identifies gravity parameters with TT CFT parameters. But additional non-local double trace deformation must be supplemented to the TT deformation to reproduce correlators of scalar operators dual to matter fields added to gravity, in line with the general discussion of bulk cutoff in [35,36]. The possible limitation of the Dirichlet cutoff picture was echoed in [37], which showed that in the large central charge limit the holographic dual of TT CFT 2 in the Euclidean plane is in general AdS 3 gravity with mixed boundary condition, and only for positive deformation parameter and for pure gravity the mixed boundary condition can be reinterpreted as Dirichlet boundary condition at a finite cutoff, taking the original form proposed by McGough et al. . This article is to a large extent a follow-up of [34], and [38] which computed the correlators of energy-momentum tensor of TT CFT in a Euclidean plane beyond leading order in the large central charge limit. We start in section 2 by a brief review of TT deformation which highlights a trace relation formula. In section 3 we promote the trace relation to an operator identity and compute in the large central charge limit the correlators of energymomentum tensor for TT CFT in a Euclidean plane, a sphere and a hyperbolic space. In section 4 we compute correlators of energy-momentum tensor in classical pure gravity in Euclidean AdS 3 cut off by a Euclidean plane and a sphere. The gravity correlators are found to agree with TT CFT correlators given a dictionary between TT CFT parameters and gravity parameters. In section 5 we summarize our result and discuss related questions and possible directions of further research.

TT deformation and trace relation
The TT deformation with the continuous deformation parameter µ is defined by a flow of action in the direction of TT operator dS dµ = dV TT (2.1) The TT operator is a covariant quadratic combination of energy-momentum tensor 1

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where the energy-momentum tensor is defined in the convention δS = 1 2 dV T ij δg ij (2.3) It was shown in [5] that the composite TT operator has an unambiguous and UV finite definition modulo derivative of local operators by limit of point splitting for quantum field theory in the Euclidean plane with a conserved and symmetric energymomentum tensor. This point splitting definition can be generalized to maximally symmetric spaces by carrying over Zamolodchikov's argument, but it was found that the factorization property of the expectation value is lost in general [5,13]. We refer interested readers to Jiang's note [39] and other references for many interesting properties of TT CFT. Here we focus on the trace relation crucial for computation in the following sections When regarded as a classical field equation it was discovered in free scalar theory [3], and was later proved for TT CFT 2 in generic curved spaces in [15]. Actually, we have a very basic argument for theories with Lagrangian density L as an algebraic function of the metric. 2 For these theories, the energy-momentum tensor takes the form and we have the TT flow equation for the Lagrangian density And the trace relation takes the form

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To prove the trace relation, we take derivative of µ∂ µ L + L − g ij ∂L ∂g ij with respect to µ. Using (2.8) we get At each point, the expression µ∂ µ L+L−g ij ∂L ∂g ij is a function of µ, the metric and other fields at that point. The formula above can be viewed as a first order partial differential equation of this function. We have µ∂ µ L + L − g ij ∂L ∂g ij = 0 at µ = 0 as the initial condition because the energy-momentum tensor in CFT is traceless. Then by the first order differential equation we have µ∂ µ L + L − g ij ∂L ∂g ij = 0 for all µ, that is, the trace relation holds for all µ. For quantum theory we expect quantum corrections to the trace relation, it depends on how TT deformation is defined for quantum field theory in curve spaces. 3 In our work we assume it holds as an operator identity within connected correlators, at least in the large central charge limit, and the TT operator is given by the point splitting definition since we work in maximally symmetric spaces.

Correlators of energy-momentum tensor of TT deformed CFT 2 in the large central charge limit
In this section we use the trace relation (2.6) to compute the correlators of energymomentum tensor in the large central charge limit, a limit of large degrees of freedom similar to the large N limit in gauge theory. More precisely it's a limit with a large central charge c of the undeformed CFT, but finite µc where µ is the TT deformation parameter. A detailed discussion of the large c limit can be found in [38]. Inspired by the work in [34] and [38], we first compute up to four point correlators of energy-momentum tensor for TT CFT in the two dimensional Euclidean plane E 2 . Then we consider TT CFT in the two dimensional sphere S 2 and the two dimensional hyperbolic space H 2 to compute up to three point correlators.

Large c correlators of TT CFT in E 2
In principle, our tools to compute correlators of energy-momentum tensor in this section are the trace relation, the conservation equation, dimensional analysis, Bose symmetry, CFT limit and other physical considerations. The conservation equation of energy-momentum tensor is

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It holds in a correlator except for contact terms. In the Euclidean plane the metric takes the form in the complex coordinates z,z and the conservation equation is We have vanishing one point correlator and it's shown in [38] that two point correlators remain the same as in the undeformed CFT in the large c limit 4,5,6,7 It's sometimes convenient to use the normalization of energy-momentum tensor in CFT T = 2πT zz ,T = 2πTzz, Θ = 2πT zz (3.6) and the two point correlators now take the form To compute the three point correlators, we start with T ( w)Θ( v)T ( u) c where the superscript c means connected correlators. 8 Using the trace relation (2.6) in the Euclidean plane We use a vector like w to denote a point in the two dimensional Euclidean plane, with holomorphic and anti-holomorphic coordinates being w andw respectively. 5 Here the superscript (0) on T indicates it's the energy-momentum tensor in the undeformed CFT, and the superscript (0) on the expectation value means it's evaluated in the undeformed CFT, for example, by path integral with the undeformed CFT action. By this convention we should add superscript like (µ) for the energy-momentum tensor and the expectation value in the TT deformed CFT with deformation parameter µ, but we choose to omit it for simplicity of the text. 6 For simplicity we omit correlators that can be simply inferred by symmetry, e.g. Tzz( w)Tzz( v) = c 8π 2 1 (w−v) 4 . 7 A bit abuse of notation, we use the equality sign even if it's only equal in the large c limit, because we exclusively work in this limit. 8 In the Euclidean plane, two and three point correlators are equal to the connected counterparts because one point correlator vanishes.

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or we get Working in the large c limit in which connected correlators of energy-momentum tensor scale as c, the correlator on the right hand side only contribute in the large c limit by factorization into two correlators We now compute another three point correlator. By the conservation equation ) modulo a function holomorphic in v. By Bose symmetry it must be holomorphic in w as well, then it should be holomorphic in u by translational symmetry. However it has to be anti-holomorphic in u by another conservation equation and the fact T ( w)T ( v)Θ( u) c = 0. So it cannot depend on u at all. Using the cluster decomposition principle when u is far from w and v we see it must be zero. Other correlators can also be computed in this way except for is holomorphic by the conservation equation and it has the CFT limit c However, it was proved in [38] that n point correlators are polynomial in µ of degree n − 2, so we can rule out possible additional terms dependent on µ like µ 3 c 4 4 . To summarize we list non-zero three point correlators Compared to previous work some clarification is needed. These expressions for three point correlators have been obtained in [34] as the leading order in µ result, by using the trace relation to the leading order in µ. We get the same expression as large c result, that is, c times arbitrary functions of µc. On the other hand, large c correlators were first systematically studied in [38]. Using Cardy's trick of "random geometry" path integration and Ward identities, [38] fixed the universal part of large c correlators of energy-momentum tensor. But non-universal terms in higher point correlators, which involve derivative of the energy momentum tensor with respect to the deformation parameter dT ij dµ , remained undetermined. Those non-universal terms depend on the model of undeformed CFT because we have to specify the model to compute dT ij dµ . A sample computation for free scalars was given in [38] and exactly the same expression as ours for three point correlators were JHEP12(2020)168 obtained. The key difference is we do not assume the specific model of the undeformed CFT, but we do assume the operator identity promoted from the trace relation. In other words, dT ij dµ can still play a role in our computation, but they are constrained to satisfy the trace relation, instead of being computed from a specific model.
In a similar way, we computed two four point correlators One can continue in this procedure to obtain higher point correlators.

Large c correlators of TT CFT in S 2 and H 2
Now we study correlators of TT CFT in a two dimensional sphere of radius r or a hyperbolic space of radius r. In a maximally symmetric space, one point correlator of energymomentum tensor is proportional to the metric The coefficient can be determined by the trace relation in vacuum expectation value supplemented by a trace anomaly term [17,18], and by using large c factorization, we get For sphere with radius r the scalar curvature is R = 2 r 2 , we find For hyperbolic space with radius r the scalar curvature is R = − 2 r 2 , we find We note a square root singularity occurs at µ = 24πr 2 c .

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Higher point correlators are a bit more complicated in a curved space. They are multipoint tensors based on the (co)tangent spaces at those points. Because the sphere and the hyperbolic space are maximally symmetric, two point correlators must be maximally symmetric bi-tensors, that is, bi-tensors covariant with the isometry group. Maximally symmetric bi-tensor has been studied in [40] exactly in the context of tensorial two point correlators, and it has already been used in [41] to study correlators of energy-momentum tensor in maximally symmetric spaces. Recently it was reviewed in [13] to study expectation value of TT operator in maximally symmetric spaces in general dimensions. Following their analysis and assuming the energy-momentum tensor is traceless in connected correlators in the undeformed CFT, we get two point correlators of undeformed CFT in S 2 and H 2 . Details of computation are left to the appendix A. Two point correlators of energymomentum tensor of CFT in S 2 take the form in the complex stereographic projection coordinates of the sphere, 9 in which the metric is It's related to the spherical coordinates by z = 2 cot θ 2 e iφz = 2 cot θ 2 e −iφ . 10 And two point correlators of energy-momentum tensor of CFT in H 2 take the form in the complex Poincare disk coordinates of the hyperbolic space, in which the metric is In an alternative coordinate system z = 2 tanh σ 2 e iφz = 2 tanh σ 2 e −iφ , the metric takes the form For TT CFT in S 2 and H 2 , we can use trace relation to show the energy-momentum tensor is traceless in connected two point correlators in the large c limit, so the analysis in appendix A can be carried over to show two point correlators are determined up to a factor as a function of µ

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for S 2 and for H 2 . For S 2 , the factor can be determined by using the one point correlator of energymomentum tensor in the replica sphere obtained in [18] to compute Renyi entropy of antipodal points Taking a variation in n, the replica number, which can be interpreted as a variation of the metric, we have Setting n = 1 we return to the regular sphere, and by plugging in Since it's an integrated formula we need to work out two point correlators with contact terms included. Starting with the known correlator 4 , and by repeated use of Ward identity of conservation of energy momentum tensor we obtain is the delta function with respect to the measure i 2 dv ∧ dv. Plugging in these correlators and completing the integration, we finally get By the same token, we need to work out one point correlator of energy-momentum tensor in the replica hyperbolic space

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to find the factor g(µ) for H 2 . We play the same trick as in [18], that is, we solve (3.15) together with the conservation equation in the replica hyperbolic space with the conical singularity smoothed. 11 We find we get Now we compute three point correlators in S 2 . Using the trace relation Similarly for H 2 we get

Correlators of energy-momentum tensor of Einstein gravity in cutoff AdS 3
In this section we compute correlators of energy-momentum tensor of Einstein gravity in cutoff AdS 3 . In the holographic setup, the large c partition function of the TT CFT living on the cutoff surface as the boundary of the bulk gravity, as a functional of the boundary metric h, is related to the on-shell action of the gravity by The action for the Euclidean Einstein gravity is The first term is the Einstein-Hilbert action, the second term is the Gibbons-Hawking term where K = h ij K ij is the trace of the extrinsic curvature K ij on the boundary surface, and the third term is the counter term with other possible addition of local functions of the boundary metric omitted. Taking a functional derivative of (4.1) with respect to the boundary metric, we get one point correlator of energy-momentum tensor in TT CFT on the left hand side, and the Brown-York tensor on the right hand side . . . (4.4) Therefore in order to compute gravity correlators of energy-momentum tensor, we have to compute functional derivatives of the extrinsic curvature with respect to the boundary metric. To this end, we solve the variation of the bulk metric in response to variation of the boundary metric, then compute the extrinsic curvature from the bulk metric.
To begin with, we gauge-fix the metric to be in Gaussian normal coordinates by diffeomorphism, that is, the radial coordinate is the arclength parameter along the geodesic normal to the cutoff surface. For a variation of the boundary metric δh ij = f ij where is the infinitesimal parameter, the bulk metric takes the form Here ρ is the radial coordinate and x i 's are transverse coordinates. In this gauge there are only three independent components of the metric. At the cutoff surface ρ = ρ 0 , the extrinsic curvature is given by The Einstein's equation for the AdS 3 gravity is 12 It's shown in the appendix B that the Einstein's equation for AdS 3 can be decomposed into three equations, the Gauss equation JHEP12(2020)168 and the radial equation Solving these three equations order by order, we obtain the Brown York tensor order by order to compute the correlators of energy-momentum tensor. In fact, the Einstein's equation for AdS 3 can be further simplified to partial differential equations in the transverse two dimensional space, because the form of the radial dependence of the metric can be solved independently from the boundary metric, following the spirit of [42]. Here we show the results of the gravity correlators and compare them to the correlators in TT CFT, leaving details of the computation to appendix C.

E 2 as the cutoff surface
Pure gravity in Euclidean AdS 3 with a cutoff y = y 0 in the Poincare patch was proposed to be the holographic dual to TT CFT in the cutoff Euclidean plane. In the appendix C, we computed one point correlators two point correlators 4 (4.14) three point correlators and four point correlators

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After a rescaling of the coordinates z → y 0 l zz → y 0 lz to bring the metric in the plane ds 2 = l 2 dzdz y 2 0 back to form ds 2 = dzdz, we find the gravity correlators agree with the TT CFT correlators given the holographic dictionary (4.17)

S 2 as the cutoff surface
Pure gravity in Euclidean AdS 3 with a cutoff ρ = ρ 0 in the patch is proposed to be the holographic dual to TT CFT in the cutoff sphere. We computed one point correlator, which is just the Brown-York tensor two point correlators and three point correlators We find the gravity correlators agree with the TT CFT correlators given the dictionary 13

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which takes the same form as TT CFT in a Euclidean plane. The sphere has its intrinsic scale r, so the second line can also be replaced by which relates TT deformation parameter to the location of the bulk cutoff.

Summary and discussion
In this article we have computed large c correlators of energy-momentum tensor for TT CFT in a Euclidean plane, a sphere and a hyperbolic space using an operator identity version of the trace relation. To examine the cutoff AdS holographic proposal by McGough et al. [17], we have computed correlators in pure Einstein gravity in Euclidean AdS 3 cut off by the Euclidean plane and the sphere, and found agreement with the TT CFT correlators given the same dictionary for both cases relating gravity parameters G, l to TT CFT parameters c, µ. The cutoff AdS picture was derived from first principle by Guica et al. [37] as a pure gravity special case of more general holographic description as AdS 3 gravity with mixed boundary condition, for TT CFT in a Euclidean plane in the large c limit. Our computation suggests a generalization of Guica's derivation to the case of a sphere. For further research it's also natural to consider correlators of other operators dual to matter fields added to the bulk, and examine the more general holographic description.
Apart from holography, TT CFT in a sphere and hyperbolic space deserves further study in its own right. TT deformation in a Euclidean plane was shown to be an integrable deformation, but the holographic proposal by McGough et al. [17], the work on partition function and entanglement entropy in [18] and our computation of two point correlators of energy-momentum tensor seems to indicate that large c TT flows to trivial in a sphere. On the other hand, correlators of energy-momentum tensor in TT CFT in the hyperbolic space blow up and run into a square root singularity when µ = 24πr 2 c , that may be an indication of failure of the notion of a local energy-momentum tensor. In general, we expect TT in curved spaces to be qualitatively different from TT in a Euclidean plane in many ways, even though for maximally symmetric spaces the definition of TT is somewhat similar. Further study on correlators and entanglement entropy will shed more light on this issue.
We have restricted our work to maximally symmetric spaces. The symmetry does not only greatly reduce the complexity of the computation, but also provides an unambiguous definition of the TT operator, assuming the existence of a conserved symmetric energymomentum tensor. Perhaps the most important open question is to generalize TT to generic curved spaces, which has been studied in [15,16] and some good results have been obtained, including a derivation of Guica's mixed boundary condition and the large c sphere partition function. It would be interesting to see how the new formalism works at the level of correlators, of energy-momentum tensor and other operators, in and beyond large c limit.

Acknowledgments
We would like to thank Ofer Aharony, Per Kraus, Yunfeng Jiang, Ken Kikuchi and Talya Vaknin for discussion. We also want to express our gratitude to health workers on front line fighting covid-19. YZ is supported by NSFC grant 11905033.

A Maximally symmetric bi-tensor and CFT correlators of energy-momentum tensor in S 2 and H 2
In this appendix we briefly discuss maximally symmetric bi-tensor and derive two point correlators of energy-momentum tensor of CFT in S 2 and H 2 , loosely following the notation in [13]. Roughly speaking, the direction along the geodesic connecting the two points is the only special direction in the (co)tangent spaces of the two points. As a result, it was shown in [40] that the natural basis for maximally symmetric bi-tensors based on two points w and v are the operators of parallel transport along the geodesic I ij ( w, v), the metric at each point g ij ( w), g k l ( v) and the unit tangent vectors to the geodesic at each point where L( w, v) denotes the geodesic length and the differentiations are with respect to the point w and v, respectively. 14 As is shown in [41], two point correlator of energy-momentum tensor in a d-dimensional maximally symmetric space is a linear combination of five independent bi-tensor structures with coefficients being functions of the geodesic length L This bi-tensor structure is further constrained by conservation of energy-momentum tensor, which by identities reduces to three equations

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for sphere, and for hyperbolic space. In addition, the second, the third, the fourth and the fifth bi-tensor structures are linearly dependent in two dimensional space, so we can set A 4 = 0 in our cases of S 2 and H 2 . For undeformed CFT we assume the energy-momentum tensor is traceless within connected correlators, as a result we get two additional constraints for the correlator

Combining (A.3) and (A.6), we get
The solution for S 2 is where a 1 and b 5 are two constants. Because the energy-momentum tensor is symmetric and traceless within connected correlators, it's natural to use the complex stereographic projection coordinates for the sphere, in which the metric takes the form

B Geometry of hypersurfaces and Einstein's equation in cutoff AdS 3
For self-containedness we offer a basic introduction to the geometry of hypersurface to derive the equations used to compute correlators of energy-momentum tensor in Einstein gravity in cutoff AdS 3 . A hypersurface Σ in a (Pseudo)Riemannian manifold M can be defined as the zero set of a smooth function Σ = {p ∈ M, f (p) = 0}. The canonical normal vector is defined by If ζ is a null vector, then it's also a tangent vector of the hypersurface. If ζ is either spacelike or timelike, the tangent space can be decomposed as the direct sum of the tangent space of the hypersurface and the one-dimensional space N spanned by ζ, T p M = T p Σ N p . In this case we can also define the unit normal n = ζ √ |g(ζ,ζ)| which is normalized to g(n, n) = with = 1 for spacelike normal and = −1 for timelike normal. Now we consider the extrinsic geometry of the hypersurface. The operator of projection to T p Σ, denoted simply by P , takes the form in the coordinate basis The first fundamental form is given by the induced metric The Weingarten map is defined as X → ∇ X n and the second fundamental form, also known as the extrinsic curvature, is given by for X, Y ∈ T Σ, with the assumption that the connection is Levi-Civita, that is metric compatible and torsion free An alternative definition of the extrinsic curvature is given by Lie derivative of the metric in the normal direction

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To work out the extrinsic curvature in coordinate basis, we have to do projection onto T Σ first since the coordinate basis doesn't all lie in T Σ. We find Now we study the relation between the intrinsic and extrinsic geometry of hypersurfaces. A covariant derivative of a vector can be decomposed into a sum of the part in T p Σ and the part in N p For X, Y ∈ T Σ, we define the covariant derivative in the hypersurface aŝ Because the projection operator P commutes with linear combination over C ∞ (M ) and tensor product,∇ is also a connection. Furthermore, for X, Y, Z ∈ T Σ so∇ is also Levi-Civita. Needless to say, it coincides with the unique Levi-Civita connection we would have derived from the intrinsic geometry, namely the induced metric. It's natural to define the Riemann curvature tensor in the hypersurfacê The decomposition of the equation above into T p Σ and N p gives us Gauss and Codazzi equation, respectively. For W ∈ T p Σ

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and Or in coordinate basisR The Einstein's equation for the AdS d+1 gravity takes the form where l is the AdS radius. We choose a Gaussian normal coordinate patch in which the metric takes the form By definition Using n = ∂ ρ and setting X = ∂ i , Y = ∂ j , we find a simple formula for the extrinsic curvature in this coordinate system By a double contraction the Gauss equation is reduced to 15 By a single contraction the Codazzi equation is reduced tô To derive the radial equation, we proceed as 15 In all of our cases the normal of the cutoff surface is spacelike, so =1.

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where R ρjρi is computed to be

By a single contraction over roman indices of the Gauss equation we have
Using the fact thatR ij =R 2 g ij in two dimensional space, we eliminateR ij to get a radial equation more practical for computation We use these three equations (B.24)(B.25)(B.28), the same set of equations used in [34], to compute gravity correlators. However further simplifications are possible. Following the spirit of [42], we can fix the radial dependence of the bulk metric and reduce the Einstein's equation to partial differential equations in the two dimensional transverse space. For three dimensional space, the Einstein's equation (B.20) fixes the metric to be locally AdS We set l = 1 for simplicity here and from now on in the appendix. Using (B.23), the radial equation now reads where " " denotes derivative with respect to ρ. It's straightforward to verify, by changing to Fefferman-Graham coordinatesρ = e −2ρ that the radial equation and the uncontracted Gauss and Codazzi equation are equivalent to equation (7), (8) and (9) in [42]. The radial equation can be integrated to give so these three equations are further reduced to equation (15) in [42] as partial differential equations in the two dimensional transverse space. In the standard context of AdS/CFT, g (0) as the metric on the conformal boundary is given, we solve for g (2) to compute various holographic physics quantities as we study holographic Weyl anomaly, holographic renormalization etc. [43,44]. In our context of cutoff AdS/TT CFT, we fix the metric at a finite cutoff surface as a function of g (0) and g (2) , but still three equations for three independent variables.

C Perturbative solutions to Einstein gravity in cutoff AdS 3 and correlators of energy-momentum tensor
When the cutoff surface is the two dimensional Euclidean plane E 2 , it's natural to use Poincare patch for AdS 3

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with the cutoff surface at y = y 0 . Consider a variation of the boundary metric where η is the flat metric, which takes the form η ij = δ ij in the Cartesian coordinates and η zz = ηz z = 1 2 η zz = ηzz = 0 in the complex coordinates. In response to the variation of boundary metric, the bulk metric now takes the form subject to the boundary condition ij (y 0 , x) = 0 . . . (C.5) Now we work out g ij (y, x) order by order by solving the Einstein's equation. We will give explicit formula for computation to the second order, while computation to the third order and higher, heavily aided by Mathematica, is too complicated to show explicit and complete expression. The inverse of the metric is computed to be g ij = y 2 η ij − y 4 η ik η jl g (1) kl + 2 y 6 η ik η mn η lj g (1) km g (1) nl − 2 y 4 η ik η jl g (2) kl + . . . (C. 6) and the extrinsic curvature is computed to be ij − 1 2 2 y∂ y g (2) ij + . . .
Plugging these quantities into the radial equation (B.28), to order O( ) we have the equation for g (1) y∂ y + 1 2 y∂ y y∂ y g (1) ij −

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where B ij is subject to the constraint η ij B ij = 0, or B zz = Bz z = 0 in complex coordinates, as well as A ij . Plugging this expression into the Codazzi equation (C.11) we get Therefore we have with the integrability condition So the trace part takes the form where ∂zA zz (z,z)dz+∂ z Azz(z,z)dz represents the primitive function whose partial derivatives with respect to z andz are ∂zA zz (z,z) and ∂ z Azz(z,z), respectively. Next by plugging (C.15) into the Gauss equation (C.13), we get where x = η ij ∂ 2 ij = 4∂ z ∂z is the Laplacian in the two dimensional Euclidean space. With (C. 19) plugged in, the equation reduces to The connected two point correlator of energy-momentum tensor is given by is the boundary metric. We have the boundary condition

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Using (C.28) we find with other two point correlators being zero. Because one point correlators all vanish, connected two point correlators are equal to two point correlators. Comparing with the standard CFT result we get the Brown-Henneaux central charge relation where we restored the AdS radius l that was previously set to one in our computation.
To compute three point correlators we need to obtain the bulk metric to the second order. Plugging the expression of g (1) into the radial equation (C.10) we obtain So g (2) takes the form g (2) ij (y, with η ij G ij = η ij H ij = 0. Plugging this expression into the Codazzi equation (C.12) we get Therefore we have 1 + 1 2 y∂ y F = (4∂zG zz + P z )dz + (4∂ z Gzz + Pz)dz + y 2 Q z dz + Qzdz (C.38)

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Therefore the solution for g (2) can be written as We now use the solution of g (2) and the formula to compute three point correlators, where the variation of the extrinsic curvature to the second order is given by δK ij = δg ij − 1 + 1 2 y∂ y g (1) ij − 2 1 + 1 2 y∂ y g (2) ij | y=y 0 (C.47) To compute T (z)T (w)T (v) , we only turn on f zz while keeping other components of the variation of the boundary metric zero for computational simplicity, and we find from (C.45) where we only show terms of the form of double integral of two f 's which contribute to the three point correlator. Substituting it into the equation (C.46), we get or in the more usual normalization

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where X, Y, Z, Ω are functions of g (1) and g (2) . By counting the powers in y, this equation reduces to four equations, three being consistency equations satisfied by the solution of g (1) and g (2) , and one being the propagation equation Using the equation In principle one can continue in this way to compute higher point correlators. Computation for the case of a spherical cutoff surface is similar.