General solutions in Chern-Simons gravity and 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}-deformations

We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a 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}-deformation.


Introduction
Three-dimensional gravity is a topological theory with no local degrees of freedom. This feature makes it a useful framework to study aspects of quantum and classical gravity, avoiding some of the complications we encounter in higher dimensions. The topological character of AdS 3 gravity is made manifest by rewriting it as a Chern-Simons theory with gauge group SO(2, 2) = SL(2, R) × SL(2, R) [1,2]. The Chern-Simons formulation has many advantages: the Einstein field equations map to a flatness condition on the connection, diffeomorphisms may be easily interpreted as gauge transformations [2], BTZ black holes emerge naturally as topological defects around which the gauge fields have nontrivial holonomies. Moreover, the Chern-Simons formulation simplifies in a great manner the study of theories of higher spin gravity [3][4][5][6][7]. In this paper, we study 3d gravity in Chern-Simons formalism. This theory should not be confused with Einstein gravity plus a Chern-Simons term since both theories are sometimes referred to as Chern-Simons gravity.

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Since there are no propagating degrees of freedom in the bulk, boundary conditions are especially relevant, because they encode the interesting physics. Boundary conditions have been extensively studied in the metric formulation [8][9][10][11]. The analysis is highly simplified by the fact that the most general asymptotically AdS 3 condition is known ij (x k ) z 2 + g (2) ij (x k ) + z 2 g (4) ij (x k ) , (1.1) where z is a radial coordinate, with boundary at z → 0, and i, j, k are 2-dimensional boundary indexes. Equation (1.1) is known as the Fefferman-Graham expansion of the metric. The parameters g (2) ij , and g (4) ij are related to g (0) ij through the asymptotic equations of motion [12]. Different boundary conditions correspond to fixing different components of the asymptotic metric. For example, g (0) ij can be fixed with Dirichlet boundary conditions, we could impose Neumann for g (2) ij , or consider a situation with mixed boundary conditions. These have a clear interpretation in the context of the AdS 3 /CFT 2 correspondence: when g (0) ij is fixed we can identify it with the CFT 2 metric [11], Neumann boundary conditions couple the boundary to 2d gravity, and mixed conditions are equivalent to double-trace deformations in CFT [13,14]. However, a systematic analysis of the Fefferman-Graham expansion (1.1) in Chern-Simons formalism is missing in the literature. This could be very helpful since the gauge theory description of 3d gravity is generally simpler than the metric case. In this paper, we solve the Chern-Simons equations of motion in a systematic way for a set of boundary conditions that reproduce the Fefferman-Graham gauge (1.1). Generalized boundary conditions for Chern-Simons AdS 3 gravity were already discussed in [15,16]. In section 4, we will comment on how our generalized connections relate to the results in [15,16].
Double-trace deformations in CFT have recently obtained much attention in the context of holography. This is mainly due to the proposal [17] which relates TT -deformed CFT 2 to AdS 3 gravity with a finite bulk cutoff. This proposal was generalized to include matter in [18], higher dimensions [19][20][21], and entanglement entropy calculations [22][23][24][25]. However, the formulation of double-trace deformations in AdS 3 in Chern-Simons formalism has remained unstudied until now. We will use our new generalized connections to define a variational principle with Dirichlet boundary conditions as in the metric formalism. This allows us to interpret the boundary values of the gauge fields as sources and expectation values of the dual operators in the CFT. This identification will prove very useful because it constitutes a departure point for the analysis of double-trace deformations in Chern-Simons formalism. We focus on one particular type of deformation that will be dual to a TT -deformation in the boundary CFT. We would like to emphasize that our analysis is purely classical. The identification between the boundary terms of the Chern-Simons theory and the deformed CFT action is done using known results from the metric formalism [37]. The precise understanding of the subtleties that might arise at the quantum level is left for future work.
This article is organized as follows. We start in section 2 by reviewing the Chern-Simons formalism of 3d gravity. In section 3, we solve the SL(2, R) × SL(2, R) Chern-Simons theory for a set of generalized boundary conditions, and construct a well-defined JHEP01(2021)039 variational principle. In section 4, we identify our connections with the most general solutions of asymptotically AdS 3 spaces in the Fefferman-Graham gauge and give a holographic interpretation to the boundary values of the connections. Finally, in section 5, we use the knowledge acquired in the previous sections to analyze double-trace deformations in the Chern-Simons formalism. We conclude in section 6, and in the appendices, we set our notation and review the vielbein formalism of general relativity, which is useful for our calculations.

Review: AdS 3 as a Chern-Simons theory
There exists an alternative description of pure 3d gravity in terms of Chern-Simons (CS) gauge connections. In this section, we briefly review this formalism, and for further details, we refer the reader to the original articles [1,2] and more recently to, e.g., [26,27]. Let us start by considering the Chern-Simons action: where A is a gauge field that lives in a manifold M. The trace Tr(. . .) is a shortcut notation for the contraction using the Killing forms of the algebra. The gauge group is chosen to match the symmetry group of AdS 3 , which in Lorentzian signature is SO(2, 2) ∼ = SL(2, R) × SL(2, R). As a result, we consider the gauge connections A,Ā valued in two independent copies of sl(2, R), which are defined as The space-time metric can be recovered directly from the connections through where we are taking the trace in the fundamental representation of sl(2, R), with the conventions in (B.13). This provides a map between the solutions of the Chern-Simons theory with SL(2, R) × SL(2, R) and geometries in AdS 3 .

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The construction of locally AdS 3 solutions in the CS formalism s discussed in [26], whose results we summarize here. We start by gauge fixing the radial component of the connection to With this gauge choice, the connections can be parametrized without losing any generality [7] by Here ρ is the holographic radial direction, and x + , x − are boundary coordinates. The connections (2.8) are a solution of the Chern-Simons equations of motion (2.5) as long as The advantage of the parametrization in (2.8) is that we can isolate (a,ā), which act as the boundary values of the connections, and interpret the radial direction as emergent from a gauge transformation. The set of boundary conditions proposed in [26] leads to where L(x + ), andL(x − ) are generic functions will always fulfil the equations of motion (2.9). The connections described by (2.8) with (2.10) are known as Bañados solutions. These connections parametrize the space of all solutions that are asymptotically AdS 3 with a trivial flat boundary.

Generalized solutions in Chern-Simons theory
The Bañados solutions in (2.7)-(2.10) analysed in the previous section have boundary conditions A x − =Ā x + = 0. In this section, we depart from this choice and find solutions to SL(2, R) × SL(2, R) Chern-Simons gauge theories with generic boundary conditions. We also find a variational principle with Dirichlet boundary conditions for the gauge field.
Here, all the analysis is purely done in the Chern-Simons formalism of 3d gravity. However, in section 4, we will show that our solutions exactly reproduce the Fefferman-Graham expansion in metric formalism (4.1).

The connections
Here we assume the same gauge choice of the radial direction as in (2.7), but we will go away from the solution (2.10) with trivial boundary. We consider from now on = 1. We propose the following general boundary conditions for a andā is:

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where the index i = x + , x − . The only choice we made is to have equal L 0 components for both a andā The equations of motion (2.5), for each component in the Lie algebra indices, impose the following constraints for the parameters in (3.1) The left column corresponds to the constraints deduced from a, and the left column from a. We can write the previous equations more compactly, adding and subtracting different combinations of them Here, we have defined a new 2-dimensional flat index that has values a, b = {+, −}. The symbol ε ab is the Levi-Civita tensor defined in (B.1). 1 The appearance of this 2-dimensional indexing is a consequence of the gauge choice (2.7), where the radial dependence is taken care of L 0 . Its origin will be clear in section 4.2. It is interesting to use (2.8), together with (B.14), to analyse the full solution: where we have used the change z = e −ρ in the radial coordinate, with boundary located at z → 0. We would like to remark that the solution (3.7) is independent of the parametrization (2.8). Alternatively, one could solve the equations of motion (2.5) for A andĀ, with the gauge choice (2.7), and the most generic solution would also lead to (3.7), and truncate at some powers of z. The connections (3.7) have generalized boundary conditions respect to the Bañados solutions (2.10). In section 4, we will show that they reproduce the Fefferman-Graham expansion (4.1) in Chern-Simons formalism.

Boundary terms
In pure 3-dimensional gravity, the boundary conditions are especially important because we do not have bulk degrees of freedom and, therefore, all the interesting information is located at the boundary. In section 3.1, we have fixed a radial gauge for our solution, and now we need to make sure we can tie it to a well defined variational principle at the boundary. This is crucial in the context of the AdS/CFT correspondence, since AdS action on-shell is related to the CFT partition function. We start by varying the CS action in (2.3), which on-shell is

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where ∂M is the boundary of the manifold M. We consider light-cone coordinates (x + , x − ) for the boundary as in the previous sections, and we can expand the wedge products as: (3.9) For the Bañados connections in (2.10), we have A x + = 0, andĀ x − = 0. Therefore, the variation of the action (3.9) is zero if we also consider δA + x and δĀ x − fixed at the boundary, and hence we would have a well-defined variational principle. However, this argument does not hold for (3.7), because we have all non-zero components of the connection at the boundary. We need to define a new variational principle for these background solutions. We begin by expressing the variation of the action (3.8) in terms of the parameters in (3.7) where the parameter ω i cancels out due to the negative sign between the barred and non-barred sector of the actions. It also remarkable that there is no explicit dependence on the radial coordinate z, unlike in the metric formalism, where the action needs to be renormalized, due to divergences at the boundary (see section 4.1). This is due to the gauge choice (2.7), and the subsequent gauge reparametrization (2.8). Note, however, that the connections (3.7) still diverge at the boundary with where the dots stand for less dominant terms as z → 0. We define a variational principle with Dirichlet boundary conditions for the connections, which will remain fixed at the boundary. This means that the variation of parameter e a is null, and f a is free to vary. With these considerations, the variation (3.10) is not zero, but we have the freedom to add a new boundary term to the action that accomplish it. We propose where S bd is defined as: The term S bd is designed such that the total action has zero variation when e a is held fixed, and, in fact, we see that (3.14) We have found then a variational principle for the connections (3.7) with Dirichlet boundary conditions. This is necessary to interpret the connections holographically. The AdS/CFT dictionary tells us that we can identify (3.14) with the variation of the CFT the action, which schematically is d 2 x O(x) δJ(x) [28]. Therefore, we see that the vielbein e a acts as JHEP01(2021)039 a source, whose variation vanishes at the boundary, and f a as the expectation value the dual operators.
We would like to remark that our variational principle is different from the one used for the Bañados solutions (2.10). As explained in (3.9), this is due to the form of the connections (3.7) which have A x − ,Ā x + = 0 at the boundary. In [29], they also analyse the variation of the Chern-Simons action (3.8) for connections with all components not null at the boundary. However, their definition of variational principal is different from ours, and they do not need to add any counterterm such our (3.13). In higher spin gravity, it is as well usual to consider connections with no null boundary conditions for a x − ,ā x + . Our identification of sources and vevs also differs from the one that is normally used in the higher spin literature. See, for example, [30].

Solutions with non-trivial curved boundary
In this section, we show that the Chern-Simons connections (3.7) represent geometries in the Fefferman-Graham expansion, which are the most generic solutions with boundaries that are asymptotically AdS 3 . We first review these solutions in metric formalism in section 4.1. In section 4.2, we identify the Chern-Simons parameters in (3.7) as CFT quantities: e i , ω i , and f i are related to a 2-dimensional vielbein, spin connection, and stress tensor, respectively.
We would like to comment that general boundary conditions for AdS 3 Chern-Simons theories have been also studied in [15]. However, their solutions do not have a regular Fefferman-Graham expansion (1.1) when translated to the metric formalism, and the metric contains off-diagonal terms such as g x ± z . They find that the asymptotic symmetry algebra associated to their gauge condition consists of two copies of affine sl(2) k -algebras. Moreover, similar boundary conditions to (3.1) have been discussed in [16]. They also relate the dominant term in their connections to a 2-dimensional vielbein and find a well-defined variational principle at the boundary. However, our identification of the parameters in the connections (3.7) as CFT quantities is more systematic, which will be useful for future applications. For example, the relation we find between f i and the boundary stress tensor will be crucial in section 5 for the analysis of double-trace deformations.

Review: metric formulation
Let us start by considering a set of coordinates where the metric is in Fefferman-Graham gauge: Solutions to Einstein's field equations with negative cosmological constant truncate in three dimensions [12], which gives ij (x k ) + z 2 g ij (x k ) , ij is interpreted as the metric of the boundary theory. The elements of g (4) ij are fully determined in terms of g (2) ij , and g (0) ik g (0) kl g (2) lj .
ij fulfils the following conditions: The indices have been raised and lowered with the metric g ij , and R (0) and ∇ ij . In [11], they interpret these geometries holographically. For that, it is necessary to define a variational principle with Dirichlet boundary condition for the metric: when g (0) ij is held fixed, and the subleading terms can vary at the boundary. Initially, the metric (4.2) is not an extremum of the Einstein-Hilbert action, because its variation contains linear terms in δg (2) ij , and divergences at the boundary. To circumvent this problem, it is necessary to add an appropriate counterterm: For a review on this analysis, see e.g. [31]. After adding (4.5) to the Einstein-Hilbert action, with a Gibbons-Hawking term, the on-shell variation of the total action is: where the boundary stress tensor T ij is defined as: It is interesting to see that the counterterm (4.5) is divergent at the boundary when we consider the expansion (4.2), as opposed to what happened in the boundary terms in the Chern-Simons formalism studied in section 3.2. The quantity T ij is a symmetric tensor that obeys the properties of stress-energy tensor on the CFT [11]. For example, using (4.4) we find The first equality shows that T ij is conserved, and the second recovers the CFT Weyl anomaly, T i i = − c 24π R, from which we can deduce the central charge c = 3 2G 3 . This illustrates that g (0) ij is the metric of the CFT 2 living on the boundary of the AdS 3 .

Connections with asymptotically curved boundary
In this section, we show that the asymptotically AdS 3 metric solutions presented in section 4.1 are equivalent to the Chern-Simons connections studied in section 3.1. The first

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indication is that we have chosen the same gauge conditions: the choice (2.7) is equivalent to the Fefferman-Graham gauge in (4.1). (ω). One can easily see this using (2.6) to relate the metric and the connections. Moreover, we will now show that the parameters e i and ω i in (3.7) represent the vielbein and the spin connection of the theory living on the curved boundary. We start seeing this, identifying g (0) in (2.6) from the expansion (4.2): where in the second equality we have defined the scalar product as an abbreviation for the contraction with the metric η ab , defined in (B.1). Equation ij is the boundary metric, we identify e a i as a boundary vielbein. The boundary vielbein e a i should not be confused with the 3-dimensional bulk vielbein E m µ used in section 2, see appendix B. We can see that e i controls the dominant term of E i , as defined from (2.2): We have seen that the connections (3.7) and the metric (4.2) are both in Fefferman-Graham gauge, and allow for boundary theories with non-trivial metric. However, to ultimately check that the two solutions are the same but in different formalisms, we now prove that the equations of motion (3.5)-(3.6) are equivalent to the constraints (4.3)-(4.4). Using (2.6), we can find g (2) , and g (4) in terms of the parameters of the connections: where we have used the right equation in (3.5) to write the expression for g (2) ij . We start reproducing the constraint (4.3) with where ε ij is the levi-civita tensor with curved indices defined in (B.4). In the first equality, we used the expression in (B.6) for the inverse of g ij . For the second equality, we used the relation (B.7). The left constraint in (4.4) is recovered by (4.14) The previous equation, together with (A.3), are enough to reproduce the differential condition over g (2) in (4.4). In section 3.2, we found a variational principle for the solutions (3.7) that keeps the boundary value of the CS connections fixed. We will now show that this variational principle is equivalent to the one in the metric formulation reviewed in section 4.1. To start the comparison, we translate the term (4.6) to the vielbein formalism using (4.9): where we used the relation between determinants below (B.4). Comparing (3.14) with (4.15), we can identify the components of the stress-energy tensor. Expanding the wegde product, its compact expression turns out to be It is interesting to compute the trace of the stress tensor: where we have used the left constraint in (3.6), and identification of Chern-Simons level (2.4), and with = 1. This is the CFT Weyl anomaly we have recovered in the metric formulation in (4.8). Moreover, we can show that (4.7) is equal to (4.16) by changing to curved indices appropriately contracting with the vielbein. Therefore, the definitions of the variational principle coincide in the metric and Chern-Simons solutions. To finish, we would like to comment that, as expected, the Bañados connections (2.10) represent solutions with flat metric at the boundary, and have null Weyl anomaly.

TT -deformation on the boundary of Chern-Simons AdS 3 gravity
In this section, we use the general AdS 3 solutions (3.7) in Fefferman-Graham gauge, and the variational principle defined in (3.14) to study double-trace operators in the dual theories of SL(2, R) × SL(2, R) Chern-Simons gauge theory. We will analyze a specific deformation in the boundary of the Chern-Simons theory that will turn out to be TT -deformations in dual CFT.

Mixed boundary conditions in Chern-Simons formalism
In section 3.2, we found a variational principle with Dirichlet boundary conditions for the CS connections (3.7). This allows us to interpret the Chern-Simons boundary action (3.14) in the CFT, with the quantities e a and f a acting as sources and expectation values of the JHEP01(2021)039 dual operator, respectively. In this section, we will use this knowledge to study the addition of double-trace combinations of f a to the boundary action. We focus on double-trace deformations of the type: So far, we have not motivated the choice of this double-trace operator, but it will become clear in section 5.2. It is well known [13,14] that performing double-trace deformations in a CFT with large N is equivalent to a change of the boundary conditions of the dual bulk quantities. From the variational principle, one can find modified relations between the asymptotic value of the bulk fields, and the sources and the expectation values [32,33]. For a more recent review of the procedure, see [34]. We start by writing the variation of the action in the deformed theory as the variation of a new on-shell action, where the sources and operators depend on the deformation. In our case, this reads where δS is the action of the original theory (3.14), and λ is the deformation parameter. The label inside the parenthesis indicates if the quantities belong to the original or the deformed theory. We need to solve (5. 2) for f a (λ) and e b (λ) to find how the vevs and sources change in term of the deformation parameter. This is equivalent to solving the following equation: where e a , and f a depend on the deformation parameter λ, but we leave the dependence implicit in this equation for simplicity in the notation. The second line in (5.4) allows us to see that (5.3) is solved by: These are the equations for the flow of the CS parameters under a TT -deformation. The flow of the spin-connection ω can be found from the constraints in (3.6) using the flow of the other CS parameters. As we found, the left equation in (3.6) is the torsionless condition (A.4) in 2 dimensions in vielbein formalism, whose solution we know for the spin connection [35]: where the symbol * de a represents the Hodge dual of the 2-form de a .

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Now, we consider the following initial conditions at the beginning of the flow: which means we have choose a flat theory at the boundary, because the Ricci scalar (4.17) is zero. It is easy to solve the equations (5.5), and with this boundary conditions: We see that the parameter f a (λ) is constant under the flow, and e a (λ) has a linear behaviour in λ. As expected, after the addition of the deformation, the boundary conditions mix the sources and expectations of the expectation values. We can deduce the flow of spinconnection using (5.6) with (5.8): where in the last equality we have used that de a = df a = 0, since the initial parameters e a and f a follow the constraints (3.6) for ω i = 0. We see that if we start with a theory with a flat boundary theory, it will remain flat after evolution under a f + f − -deformation.

f + f − as a TT -deformation
In section 5.1, we analysed the flow of under the double trace deformation (5.1). Now, we will see that f + f − is equivalent to the usual TT operator in CFT. Remember the definition [36]: where in the second equality we have conveniently written it in terms of the flat indices using the vielbein. We consider the action of this double-trace operator and using the holographic expression for the stress tensor in (4.16) to write the this action in terms of the Chern-Simons parameters we obtain Therefore the TT -deformation is equivalent to the is equivalent to the f + f − -deformations (5.1). It is instructive to compare with the metric formalism. In [37], they use the variational principle method to analyse the flow of operators and sources after a TT -deformation in a CFT. They then interpret this results holographically, and find that the metric field has mixed boundary conditions after the deformation. Here, we will see that the mixed boundary conditions found in [37], are equivalent to (5.8). For that we use (4.9), which we write as ij (λ) = 2 e i (λ) · e j (λ) = 2 e i · e j + 2

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Another candidate to reproduce the TT deformation in terms of the connections A, andĀ: This term is designed to exactly reproduce (5.1), up to an overall normalization constant, for (3.7) with ω i = 0. The advantage of this term is that we do not have explicit zdependence, so it is natural to consider it as a pure boundary deformation. However, this advantage brings a more complicated combination of A, andĀ, as a downside. In particular, we need an explicit dependence in the determinant of the vielbein. This is similar to the what happens the Chern-Simon U(1) gauge field in [39], where we need to couple to the boundary metric to find a well variation principle. However, it is strange from the point of view of Chern-Simons as a gravitational theory, since all the information about the metric should be encoded in the connections.
We have proposed two terms that could perform a TT -deformation in terms of the connections A, andĀ. However, it is not clear how natural they are from the point of view of the Chern-Simons variables.

Conclusions
In this article, we studied SL(2, R) × SL(2, R) Chern-Simons gauge theories in AdS 3 , and we presented two main results. The first one is the solution of the Chern-Simons equations of motion with generalized boundary conditions (3.7). We have shown that these connections are equivalent to the Fefferman-Graham expansion (4.2) in metric formalism. Moreover, we proved that we can interpret the different components of the gauge field as CFT quantities: e i , ω i , and f i are related to a 2-dimensional vielbein, spin connection, and stress tensor, respectively. We have done this by proposing a variational principle with Dirichlet boundary conditions for the gauge connections. This allowed us to study how the boundary of the Chern-Simons theory behaves under a specific type double-trace deformation introduced in (5.1). Using the variational principle method, we have found the mixed boundary after the deformation (5.8) in the Chern-Simons formalism. This is our second result. We showed that this deformation corresponds to a TT -deformation from the point of view of the CFT.
Higher spin theories of gravity are very straight-forwardly generalized from the description of AdS 3 gravity as Chern-Simons gauge theory. The generalization ammounts to extend the gauge group from SL(2, R) to SL(N, R). In [7], they find solutions to these theories that are considered asymptotically AdS 3 with flat trivial boundary. Assuming (2.8) these are . It would be interesting to generalise these solutions to have more generic boundary conditions of the type (3.7), as we proposed in this paper for the sl(2, R) case. This, together with a well-defined variational principle, could allow us to formulate double-trace deformations involving spinning fields in Chern-Simons theory. We leave this idea as material for future work.

A Vielbein formalism
This appendix is a very short review, and collection of useful identities, of the vielbein formulation of general relativity. For some more context, see, for example [40].
In the vielbein formalism, the choice of coordinates x i on a manifold is replaced by the election of a local basis. This basis is formed by the d-dimensional vectors, e a = e a i dx i , known as vielbeins. We can relate the metric of a curved manifold g ij to a flat (noncoordinate) metric η ab using the vielbein: The vielbein is used to change the indices of the tensors from curved to flat, or viceversa. For covariant derivatives of tensors with flat indices, the role of the Christofel symbol is replaced by the spin connection, characterized by ω a i b . For example, the necessary contractions in the case of a tensor with mixed indices X a j are: The following identity is required by construction of the vielbein formalism: which is known as the tetrad postulade. Notice that (A.3) defines the spin connection in terms of the Christoffel symbols, and the vielbein. Moreover, the zero torsion condition, i.e., Γ λ ij = Γ λ ji can be written in vielbein formalism as: The condition over the flat metric ∇ i η ab = 0 imposes the so-called metricity condition: where ω iab = η ac ω c i b . It is also useful to write the curvature tensor with flat indixes as:

B Conventions and useful identities
The review of vielbein formalism made in appendix A is generic to any dimension. However, in this paper we have considered two different specific types of vielbein theories: a 3-dimensional one representing the bulk, and 2-dimensional one in the boundary. This appendix summarizes our conventions for both cases, and collects identities that are relevant for our manipulations.

B.1 Vielbein formalism in 2 dimensions
In the main text we described a 2-dimensional theory in vielbein formarlism, whose and zero otherwise. In two dimensions, the metricity condition (A.5) imposes that where ω i is the only independent component of the 2-dimensional spin connection. The symbol ε ab is the Levi-Civita defined as One can raise and lower indices of the Levi-Civita tensor with the flat metric, as, for example: ε a b = η ab ε b b . It is useful also to define ε ij as Levi-Civita tensor with curved indices: where g = det(g) and e = det(e) are the determinants of metric and the vielbein, which are related via det(g) = −4 det(e) 2 . The definition of determinant we used, for a generic tensor X a i : We use the Levi-Civita symbols to write the inverse of the metric as g ij = ε ii ε jj g i j . (B.6) Other useful relations using the Levi-Civita symbols are the following contractions of vielbeins: We can use them, for example, to write find a compact expression for the Ricci scalar from the Riemann curvature tensor (A.6). In 2d, using (B.2), we have We find the trace of R a b ij by contracting its indices with the vielbein: where we have used the contraction of vielbein in the right of (B.7).

B.2 Vielbein formalism in 3 dimensions
In 3-dimensions, we used the following definitions for the vielbein and spin connection: where the capital the letters E and, Ω to make a distinction with the 2-dimensional case described in section B.1. To avoid confusion, we used also different nomenclature for the 3-dimensional coordinate indices {µ, ν}, and the flat indices {m, n, l} = 0, ±1, which in this case are directly related to the sl(2, R) algebra: