$T\bar{T}/J\bar{T}$-deformed WZW models from Chern-Simons AdS$_3$ gravity with mixed boundary conditions

In this work we consider AdS$_3$ gravitational theory with certain mixed boundary conditions at spatial infinity. Using the Chern-Simons formalism of AdS$_3$ gravity, we find that these boundary conditions lead to non-trivial boundary terms, which, in turn, produce exactly the spectrum of the $T\bar{T}/J\bar{T}$-deformed CFTs. We then follow the procedure for constructing asymptotic boundary dynamics of AdS$_3$ to derive the constrained $T\bar{T}$-deformed WZW model from Chern-Simons gravity. The resulting theory turns out to be the $T\bar{T}$-deformed Alekseev-Shatashvili action after disentangling the constraints. Furthermore, by adding a $U(1)$ gauge field associated to the current $J$, we obtain one type of the $J\bar T$-deformed WZW model, and show that its action can be constructed from the gravity side. These results provide a check on the correspondence between the $T\bar{T}/J\bar{T}$-deformed CFTs and the deformations of boundary conditions of AdS$_3$, the latter of which may be regarded as coordinate transformations.


Introduction
Over the past few years, we have seen a surge of interest in deformed 2D conformal field theories [1][2][3][4][5][6][7][8][9][10]. Such theories are integrable, and in some cases allow a holographic description of 3D gravity. So far two kinds of deformations, namely the TT deformation and the JT deformation [2,[11][12][13], have been worked out in detail. It was proposed that the TTdeformed CFT corresponds to cutoff AdS 3 at a finite radius with the Dirichlet boundary condition [3,14,15]. There are some nontrivial checks on this proposal: the finite size spectrum turns out to be the same as quasilocal energy of the BTZ black hole at finite radius [3], and the TT flow equation coincides with the Hamilton-Jacobi equation governing the radial evolution of the classical gravity action in AdS 3 [16,17]. Based on this proposal, more holographic aspects of the TT -deformed CFT have been explored, such as entanglement entropy [18][19][20][21] and complexity [22]. Similarly, the JT deformation also have a holographic interpretation [8,23,24]. In addition to the above, the torus partition functions of the deformations were studied [25][26][27][28][29]. More recently, the correlation functions of TT and JT deformations have been computed [30][31][32][33][34][35]. As integrable quantum field theories, the deformed 2D CFTs still have infinitely many symmetries. These symmetries have also been studied from 3D gravity perception [36][37][38].
In the context of AdS 3 /CFT 2 , the boundary dynamics of AdS 3 gravity with the Brown-Henneaux boundary condition turns out to be a SL(2, R) WZW model. This result can be derived through the Chern-Simons form of AdS 3 gravity. In fact, the AdS 3 gravity can be reformulated as a SL(2, R) × SL(2, R) Chern-Simons theory, and the Brown-Henneaux boundary condition requires an extra boundary term. The Chern-Simons action with such a boundary term reduces to the sum of two chiral SL(2, R) WZW models. Furthermore, this boundary condition also gives certain constraints on the chiral WZW models, which lead to the reduction of the WZW model to the Liouville theory at the classical level [39] (for more details see the recent review [40]). More recently, it has been shown that the Chern-Simons AdS 3 gravity at quantum level is equivalent to the Alekseev-Shatashvili quantization of coadjoint orbit Diff(S 1 )/P SL(2, R) of the Virasoro group [41]. These considerations may be extended to the case of TT and JT deformation. There already has been some work on this topic, such as using Chern-Simons formalism [42,43] to study holographic aspects of TT /JT deformation, as well as the TT -deformed Liouville theory [44].
In this paper, we focus mainly on the boundary dynamics of AdS 3 associated with the TT /JT deformations. From the cutoff point of view, however, the boundary condition is defined at finite radius, which has no asymptotic degree of freedom. Nevertheless, it is shown that the Dirichlet boundary conditions at finite radius correspond to the mixed boundary conditions at infinity [45,46]. For the TT /JT deformation, these mixed boundary conditions were obtained in [23,36] through the variational principle approach. We shall take a close look at these boundary conditions in the Chern-Simons formalism, and derive the nontrivial boundary term. The energy of this system is obtained from the boundary term. As we shall see, these results agree precisely with the spectra of the TT /JT -deformed CFTs. Moreover, for the TT deformation, the total action allows the reduction to the constrained TT -deformed WZW model. After disentangling the constraints, we show the boundary dynamics are exactly the TT -deformed Alekseev-Shatashvili action. We will also derive one type of the constrained JT -deformed WZW model from the gravity side, in which the U(1) current is introduced by adding an extra Abelian gauge field to the Chern-Simons system. The resulting theory is also the JT -deformed conformal theory. We show that the asymptotic dynamics of AdS 3 gravity with the mixed boundary conditions are actually described by the deformed conformal theories. This paper is organized as follows: In section 2, we first review the mixed boundary condition of AdS 3 for the TT deformation. After rewriting this boundary condition in the Chern-Simons form, we obtain a nontrivial boundary term. The energy of the whole system can be read off from this boundary term, which matches the finite size spectrum of the TT deformation. In section 3, the boundary dynamics of AdS 3 with mixed boundary condition turns out to be the constrained TT -deformed WZW. We also show the equivalence between the sum of two opposite chiral WZW models and the standard non-chiral WZW model under the TT deformation. JT deformation is considered in section 4. Its spectrum is derived from Chern-Simons form by means of the surface integral. The boundary dynamics is also turned out to be a JT -deformed conformal theory. Finally, section 5 contains some conclusions and discussions.

Mixed boundary condition for the TT deformation
In this section, we will study the mixed boundary condition of Chern-Simons AdS 3 gravity for the TT deformation. We first give a brief review of the mixed boundary condition. Then we put the mixed boundary condition in the Chern-Simons form. The nontrivial boundary term for mixed boundary condition is obtained. We will also show this boundary term gives exactly the energy of the system, which is in agreement with the spectrum of TT -deformed CFT.

Review of the mixed boundary condition
We start from the definition of TT -deformed CFT, whose action is given by the TT flow where the metric γ ij and stress tensor T ij are defined in the deformed theory. The deformed metric and stress tensor can be expressed in terms of the original ones through the variational principle approach. The basic procedure is to write the variation of the deformed action in terms of the deformed quantities. Then the TT flow (2.1) implies the flow equations Here we mainly draw attention to the flow equation of γ ij . The solution of γ ij flow equation can be expressed as where the superscript (0) denotes the quantities of the original theory. (2.3) indicates that the background metric of the deformed theory is corrected by the stress tensor of the original theory. If we consider a CFT in the flat spacetime, the deformed theory may not be in the flat spacetime because the background metric is also deformed. This approach was originally developed by Guica and Monten, see [23,36] for more details.
From the holographic point of view, γ ij is interpreted as the boundary metric of AdS 3 . Therefore, the deformed metric γ ij would imply the bulk boundary condition. In general, the solution of 3D gravity can be written in Fefferman-Graham gauge with the constraint jl .
(2.5) According to AdS 3 /CFT 2 dictionary, g (2) ij is proportional to the expectation value of the stress tensor of the boundary CFT [47] g (2) where the cosmological constant is set to Λ = −1/ℓ 2 = −1. We will use g ij to denote the leading order for the deformed bulk solution. Now, combining (2.5), (2.6) and (2.3), we arrive at the mixed boundary condition 1 ij . (2.7) Namely, the boundary metric of AdS 3 is given by (2.7) at infinity. This metric coincides with the boundary metric (expressed within the parentheses in (2.4)) at finite radius r = r c , provided the following relation [3] is invoked This asymptotic behavior allows us to write the bulk solution in the Fefferman-Graham gauge by replacing g (0) ij with g ij . Note that this mixed boundary condition differs in several respects from the Brown-Henneaux boundary condition [48]. Although this boundary condition is defined at infinity, the leading order of the boundary metric g ij is not a flat one. It also breaks the chiral boundary condition in Chern-Simons form. We therefore need a new boundary term to remove inconsistency in the variational principle approach. Besides, the leading order g ij fluctuates, which would inspire us to study the underlying asymptotic dynamics.
To keep our discussion explicit we consider the Bañados geometry, which constitutes the most general bulk solution of AdS 3 with g (0) ij = η ij . In holomorphic coordinates (z = θ+t,z = θ − t), the Bañados metric can be put in the form [49] ds 2 = dr 2 r 2 + r 2 dzdz + L(z)dz 2 +L(z)dz 2 + 1 r 2 L(z)L(z)dzdz, (2.9) where L(z) andL(z) are arbitrary functions depend on z andz, respectively. The mixed boundary condition would fix the boundary metric as Now, introduce the following new coordinates x ± such that the leading order of the boundary metric takes the manifestly flat form ds 2 c = dx + dx − , The deformed bulk solution is obtainable from (2.9) by performing the inverse of the coordinate transformation where we used the notations L µ ≡ L(z(µ, x + , x − )) andL µ ≡L(z(µ, x + , x − )). The concrete relation between L(x + ) and L µ (x + , x − ) may be found in several ways [36]. One of which is that the coordinate transformation (2.11) brings the deformed AdS 3 solution to the Bañados geometry. The horizon area or energy density should not change under such a coordinate transformation. So comparing these two metrics yields (2.13) As a result, we can write the deformed AdS 3 solution in terms of parameters L µ ,L µ through the coordinate transformation.
Moreover, it turns out that the TT -deformed theory can be mapped into the original theory via a field dependent coordinate transformation [50,51]. In terms of the differential form, the coordinate transformation reads According to the holographic dictionary, the parameters of Bañados geometry correspond to the stress tensor of the boundary Liouville theory through L(z) = 2T (z),L(z) = 2T (z) [40,49]. In this context, (2.12) is consistent with (2.14). Therefore, we can use the same coordinate transformation in the bulk to get the deformed AdS 3 solution.

Chern-Simons formalism and the boundary term
It is well-known that three dimensional Einstein gravity with a negative cosmological constant can be expressed as SL(2, R) × SL(2, R) Chern-Simons gauge theory [52], whose action is where The gauge fields A,Ā valued in two independent copies of SL(2, R), which are defined as the combination of vielbein and spin connection A a = ω a + e a ,Ā a = ω a − e a . It turns out that these equations are equivalent to first order gravitational field equations.
Let us first take a look at the Bañados geometry (2.9) in Chern-Simons form. The corresponding gauge fields can be calculated where L 0 , L ±1 are Lie-algebra generators of SL(2, R); see Appendix A for our convention. These gauge fields also can be obtained by solving (2.18) with the chiral boundary condition Az = 0,Ā z = 0 [49]. A useful trick to factor out the boundary degree of freedom is performing the following gauge transformation In this case, the reduced connections have the explicit form which depend on the boundary coordinates (z,z) only. For later discussion, we would like to use the coordinates θ = (z +z)/2, t = (z −z)/2 and impose the periodic condition θ ∼ θ + R.
Then the chiral boundary condition becomes A t = A θ andĀ t = −Ā θ . Now one can go through a consistent variational principle approach by adding some boundary terms to the action. The total action associated to the chiral boundary condition was found in [39], which takes the form In the Hamiltonian formalism, the supplementary boundary term plays the role of a surface integral, which implies the total energy of this system [53]. Inserting (2.19) and (2.20) into (2.23), the boundary term becomes For the BTZ black holes, L(z) = L 0 ,L(z) =L 0 , the boundary term (2.24) gives exactly the energy (or mass) of the black hole We now turn to the investigation of the mixed boundary condition for the TT deformation. As we shall see, this mixed boundary condition can be obtained from the Brown-Henneaux boundary condition through a field dependent coordinate transformation (2.12). Consequently, the gauge fields corresponding to the mixed boundary condition are given bỹ We use tilde symbols to denote the quantities in the deformed theory. One can clearly see that the deformed gauge fields obey instead of the chiral boundary condition. That is to say, the mixed boundary condition breaks the chiral boundary condition. However, the equation of motion still holds, because the deformed bulk solution also satisfies Einstein equation. In the coordinatesθ = (

the gauge fieldsÃ andĀ have the following relations
The r dependence of the deformed gauge fields can also be eliminated through the gauge transformation (2.21). Thus, we get the reduced connections for deformed theorỹ This is the mixed boundary condition in Chern-Simons form. In order to have a well-defined variational principle, we have to add a supplementary boundary term. It turns out that the corrected boundary term is where we have invoked (2.30) and (2.31) in the last step. The detailed derivation of this nontrivial boundary term is given in Appendix B.
Here we give some comments about this boundary term. This term reduces to the limiting case (2.24) when µ → 0. Unlike the limiting case where the chiral boundary condition holds, the boundary term (2.32) in general cannot be separated into a chiral part depending only onã and an antichiral part depending only onā. One may see this more clearly by writing L µ ,L µ in terms of the reduced connections. As a consequence, the chiral action I(A) and the antichiral action I(Ā) in Chern-Simons theory are coupled to each other through the boundary interaction term (2.32), as long as µ = 0. This is the effect of TT deformation in Chern-Simons gravity.
The boundary term also gives rise to the total energy of this system. Working in the Hamiltonian formalism, the surface integral reads which is consistent with the result derived from the bulk stress tensor [36]. For the BTZ black holes, we can work out the total energy with the help of (2.13) are the mass and the angular momentum of the black hole, respectively. The total energy of this system is in agreement with the spectrum of the TT -deformed CFT. E precisely matches the quasi-local energy of the BTZ black hole due to µ = 1/r 2 c . This result is consistent with the cutoff point of view [3]. However, the mixed boundary condition considered in this paper is actually an asymptotic boundary condition, which is defined at infinity rather than at the finite radius r = r c . The advantage of this mixed boundary condition is that we can study the boundary dynamics directly in Chern-Simons theory, as we shall discuss in the next section.

From Chern-Simons theory to TT -deformed WZW model
In this section, we would like to study the boundary dynamics of AdS 3 with the certain mixed boundary condition. We first take a short look at the chiral boundary condition . It is shown that the Chern-Simons action can be reduced to the WZW model [54] where g and G take values in SL(2, R). It turns out that (3.2) produces a non-chiral SL(2, R) WZW model, and the latter allows a further reduction to the Liouville theory classically [55]. At the quantum level, the Chern-Simons gravity is equivalent to the Alekseev-Shatashvili quantization of Virasoro group [41]. In other words, the asymptotic dynamics of AdS 3 with the Brown-Henneaux boundary condition can be described by the conformally invariant theory.
The above consideration can be extended to the case where the mixed boundary condition is imposed. As we shall see, the corresponding boundary term (2.32) leads to a coupling between two opposite chiral WZW models, and the resulting theory is equivalent to the TT -deformed non-chiral WZW model. Moreover, the mixed boundary condition also gives constraints on the TT -deformed WZW models, which would give a further reduction to the TT -deformed Alekseev-Shatashvili action.

Reduction to a sum of two coupled chiral WZW actions
Given that an action with the well-defined variational principle, we are ready to reduce the Chern-Simons action to the TT -deformed WZW model. The main difference with the CFT case is the boundary term. Firstly, we would like to express the boundary term in terms of the gauge fields. In the following we find it is convenient to define According to (2.30) and (2.31), one can write L µ in terms of Xθθ andXθθ as well as a similar expression forL µ . It is straightforward to derive the following identity Comparing this with the first line of (2.32), the boundary term B can be expressed as It follows that the total Chern-Simons action consistent with the mixed boundary condition may reduce to This is exactly the TT -deformed chiral WZW action, which was derived from the TT flow equations [43]. Here we derive the TT -deformed WZW model based on the Chern-Simons AdS 3 gravity with the mixed boundary condition.
In order to see the effect of TT deformation, one may expand (3.7) as a Taylor series with respect to µ. The first few terms of this expansion read The leading order reproduces the sum of two decoupled chiral WZW actions, as presented in (3.2). The deformation contributes to higher order terms of µ. Clearly, such higher order terms can no longer be written as the sum of a left-moving part and a right-moving part. In other words, the TT deformation provides a coupling between two opposite chiral degrees of freedom.

Equivalence to TT -deformed non-chiral WZW action
As is well known, the sum of left and right chiral WZW actions is equivalent to the standard non-chiral WZW action [39]. It is natural to expect that (3.7) is equivalent to a TT -deformed version of the non-chiral WZW model. By using the usual technique in [39,40], we will verify this in this subsection. First, we combine the gauge fields g,ḡ and introduce the new variables 10) The sum of Wess-Zumino terms becomes We then write the TT -deformed chiral WZW action (3.7) in terms of the new variables Π and k −1 dk where we used the notation k ′ = k −1 ∂θk andk = k −1 ∂tk.
The auxiliary variable Π can be eliminated by the equation of motion. Varying the action (3.14) with respect to Π, we obtain the equation of motioṅ where Ω is introduced for convenience. According to the above equation, we get the relations One can express the Π-dependent quantities in terms of k-dependent quantities by solving these equations above. The solutions show Tr (k ′ Π) = Tr(kk ′ ) Tr (3.21) Substituting these relations back into the action (3.14), we arrive at an action depending on k only In the light cone coordinates, this action finally becomes where X ij is defined by This is exactly the action for the TT -deformed non-chiral WZW model, which is first derived from TT flow equation in [58]. Therefore, we have verified that the equivalence between the sum of two chiral WZW models and the standard non-chiral WZW model still holds under the TT deformation.

Constraints on the TT -deformed WZW model
This mixed boundary condition also gives constraints on the TT -deformed WZW model. In order to study the constraints, we consider the Gauss decomposition of SL(2, R) Then the gauge fieldsÃ,Ā can be expressed as Comparing with (2.26) and (2.27), we see that the fields are fixed at r → ∞ as follows: e 2φ ∂θF =ηr, ∂θφ = e 2φ Ψ∂θF, (3.29) e 2φ ∂θF =ηr, ∂θφ = e 2φΨ ∂θF , (3.30) where the parameters η,η take the form It is useful to write Xθθ,Xθθ in terms of the parameters According to the constraints (3.29) and (3.30), we express φ ′ ,φ and Ψ ′ ,Ψ as where the overdot and prime denote the derivative with respect tot andθ. Similar relations for theφ ′ ,φ andΨ ′ ,Ψ can also be obtained. For the Brown-Henneaux boundary condition, the parameters η,η are both equal to 1. Then, the constraints can reduce the WZW model to Alekseev-Shatashvili action. However, when the deformation is turned on, the parameters η,η appear in the constraints. In order to make a further reduction, we have to find the relations between η,η and F,F .
In fact, one can rewrite Xθθ andXθθ in Gauss parametrization. As a consequence, (3.33) implies the differential equations for η andη where {f ;θ} represents Schwarzian derivative defined by Although it is difficult to get the exact solutions, we can find the perturbation solutions in the first few orders of small µ η = 1 + µ{F ;θ} + O(µ 2 ),η = 1 + µ{F ;θ} + O(µ 2 ). (3.40) In the Gauss parametrization, we can reduce the TT -deformed WZW model into where η,η are determined by the equations (3.37) and (3.38). Moreover, it is useful to parametrize the boundary value of F andF as such that ξ,ξ are valued in Diff(S 1 )/P SL(2, R) [41]. Then we find the relationṡ as well as the similar relations for the barred quantities. In order to see whether the resulting theory is a TT -deformed conformal theory, we can consider the perturbation form of this action. Plugging (3.40) into the action (3.41) and dropping some total derivative terms, we finally arrive at The leading order is exactly the sum of left-moving and right-moving Alekseev-Shatashvili quantization of coadjoint orbit Diff(S 1 )/P SL(2, R) of the Virasoro group [41,56,57]. The first order correction is nothing but coupling these two copies through the TT deformation, since the stress tensors of chiral Alekseev-Shatashvili actions are exactly given by Therefore, the boundary dynamics of AdS 3 with mixed boundary condition is described by the action (3.45), which is a TT -deformed conformal theory in first order as expected. In [43], very similar results were obtained from a boundary WZW model through the TT flow. These results may give a precise check on the correspondence between the TT -deformed CFT and AdS 3 gravity with the mixed boundary condition.

JT deformation
Another interesting integrable deformation is the JT deformation [12]. In this section, we would like to study the JT deformation. We firstly give a brief review for the boundary condition for JT -deformed CFT. In Chern-Simons form, this boundary condition implies a certain nontrivial boundary term. The spectrum of JT -deformed CFT is obtained from this boundary term in the Hamiltonian form. We will also show that the asymptotic boundary dynamics is described by one type of the JT -deformed chiral WZW model.

Review of the boundary condition for the JT deformation
By the definition of JT deformation, its action could be written as For convenience, we have written it in vielbein form. In this model, we have to consider the CFT involving stress tensor T a i and the conserved current J i , which are canonically conjugate to the boundary vielbein e i a and the gauge field Φ i . Then the variation of the original CFT action would be When the deformation is turned on, we may suppose the variation takes the following form The deformed quantities are marked with a tilde. In [23], by using the JT flow equation (4.1), the JT -deformed variables were constructed from the original theorỹ We focus mainly on the deformed vielbeinẽ i a and the gauge fieldΦ i , which could help us to fix the boundary condition of AdS 3 .
On the gravity side, we have to introduce a U(1) Chern-Simons gauge field coupling with AdS 3 gravity. Therefore, the total action associated with the JT deformation should be where k ′ is the U(1) Chern-Simons level. Generally, the U(1) charge is introduced by adding a Maxwell term, such as the charged black hole. Since we are working in an odddimensional spacetime, this gauge field have the U(1) Chern-Simons form. In order to ensure the variational process, we add the Gibbons-Hawking boundary term for the gravitational part. As for the gauge field part, the boundary term turns out to be where γ ij is the induced metric on the boundary ∂M. Then the variation of total action in the bulk becomes where T grav ij is the Brown-York stress tensor [59,60], T U(1) ij comes from the U(1) Chern-Simons boundary term and J i is the U(1) conserved current. This is the basic structure in AdS 3 /CFT 2 correspondence with additional U(1) charge [61].
In Fefferman-Graham gauge, the deformed vielbein (4.4) corresponds to fixing the g which can be obtained from the Bañados geometry through a coordinate transformation Therefore, the deformed solution is parametrized by L µ ,L µ , J We use similar notations for the JT deformation, these notations should not be confused with the TT deformation. A very similar boundary condition for AdS 3 has been considered in [62], when they studied SL(2, R) × U(1) symmetries in AdS 3 .
In addition, we also need to fix the gauge fieldΦ. From (4.10), the gauge fieldΦ can be written asΦ − =F (x − , x + ), (4.14) Comparing the deformed gauge fieldΦ with (4.4), we can identify where T ij is the total stress tensor of the system This means that the additional boundary term of the U(1) Chern-Simons action have a backreaction for the formalism of deformed gauge field. Finally, one arrives at the equation We summarize the mixed boundary conditions to complete this subsection. The mixed boundary condition for JT deformation includes fixing AdS 3 metric as well as U(1) gauge field. The AdS 3 metric is determined by a coordinate transformation (4.12). The gauge field refers to the stress tensor of the whole system through (4.14) and (4.18). As a result, we can express the metric and gauge field in terms of L, F , J. Moreover, this mixed boundary condition would imply the asymptotic dynamics because it is defined at infinity.

Chern-Simons formalism and the boundary term
Now we put the mixed boundary condition in the Chern-Simons formalism to find out the associated boundary term. As mentioned above, the total action in the bulk consists of the gravitational part and the U(1) Chern-Simons gauge field part. For the gravitational part, the action can be formulated in SL(2, R) × SL(2, R) Chern-Simons theory. Therefore, the total action would be By using the coordinate transformation (4.12), we obtain the SL(2, R) gauge fields (4.22) which still satisfy the equations of motion. After eliminating the radial coordinates, we write down the induced connections Clearly, the left chiral boundary condition is maintained, but the right chiral boundary condition is broken. Besides, the U(1) gauge fieldΦ is fixed in (4.14) and (4.15). In the coordinatesθ = (x + + x − )/2,t = (x + − x − )/2, the mixed boundary condition becomes This boundary condition requires a boundary term be added to the action (4.20), which turns out to be The detailed derivation of this boundary term is given in Appendix C. This boundary term also reduces to the CFT case when µ → 0. In addition, it provides a coupling between the right chiral Chern-Simons theory and a U(1) gauge field, but keeps the left chiral Chern-Simons action unchanged.
In the Hamiltonian form, this boundary term gives the surface integral We consider the BTZ black holes, in which L andL are constants. After rescaling the coordinates [23], we can identify (4.31) Up to a coefficient, the surface integral ends up with which is the spectrum of the JT deformed CFT in [8,23], as expected. Here we reproduce the spectrum from gravity side using the surface integral method. Just as in the case of TT deformation, the boundary term is defined at infinity. From the holographic point of view, the JT deformation corresponds actually to a deformation of the boundary condition of AdS 3 , which can be treated as a coordinate transformation. This asymptotic boundary condition may imply the boundary dynamics, and we would like to discuss this in later subsections.

From Chern-Simons theory to JT -deformed WZW model
We then follow the method used in TT deformation to study the asymptotic dynamics for this mixed boundary condition. By using (4.25), (4.26) and (4.27), one gets Plugging into (4.28), the boundary term becomes Finally, the total Chern-Simons action with this certain boundary term can be reduced to This is actually one type of the JT -deformed WZW action, which can also be got from JT flow equation by adding an extra U(1) gauge field, see Appendix D for details. The effect of JT deformation is coupling the right-moving SL(2, R) WZW model with left-moving U(1) gauge field. From the perspective of holography, the boundary dynamics of AdS 3 with the mixed boundary condition can be described by (4.36), namely a JT -deformed conformal theory.
We give some comments about the JT -deformed WZW model. The difference between the JT -deformed scalar field and the JT -deformed WZW model is the definition of U(1) current J. In the latter one, the current J is introduced through adding an extra U(1) gauge field. Of course, one can do the deformation by using one component of SL(2, R) current J a , such as J 0 . However, there will be another boundary condition for AdS 3 instead of the mixed one. We will not discuss this case in this paper.

Constraints on the JT -deformed WZW model
We now consider constraints on the JT -deformed WZW model. We will use the same notation as in the TT deformation. By using the Gauss decomposition (3.27) and (3.28), the boundary condition (4.21) and (4.22) imply the constraints e 2φ ∂θF = r, ∂θφ = e 2φ Ψ∂θF, (4.37) e 2φ ∂θF =ζr, ∂θφ = e 2φΨ ∂θF , (4.39) The left-moving part remains unchanged, but the right-moving part is deformed because of ζ = 1. From these constraints, one can express φ ′ ,φ and Ψ ′ ,Ψ in terms of F According to these relations, we get the differential equation forζ (4.45) The solutions of this equation allow us to express the parameterζ in terms ofF andΦ. The perturbation solution in the first few orders of small µ is which can be used to give a further reduction of the deformed WZW action.
Finally, the total action (4.36) can be expressed in Gauss parametrization Again, one can parametrize the F andF to the angular variables ξ andξ. Substituting the perturbation solution (4.46) into the action, we arrive at The leading order of this action is the sum of two opposite chiral Alekseev-Shatashvili actions with an additional U(1) gauge field. The first order correction is just the coupling of the right-moving Alekseev-Shatashvili action and the left-moving U(1) gauge field through the JT operator. Consequently, the asymptotic boundary dynamics of AdS 3 with this mixed boundary condition is described by one type of JT -deformed Alekseev-Shatashvili action. However, since our construction depends on introduction of a gauge fieldsΦ, the resultant theory should differ from the standard JT deformation. The latter is the coupling of two opposite chiral Alekseev-Shatashvili actions without additional gauge fields.

Conclusion and discussion
In this paper, we study the holographic aspects of TT /JT -deformed CFTs in Chern-Simons formalism. It is shown that the deformed CFTs correspond to AdS 3 with mixed boundary conditions. Based on the mixed boundary condition, the certain boundary terms are obtained. We also show that the boundary dynamics of Chern-Simons AdS 3 gravity turns out to be the TT /JT -deformed WZW model.
Unlike the cutoff point of view, the mixed boundary condition for the TT deformation is defined at infinity. We find that this boundary condition implies a nontrivial boundary term in Chern-Simons formalism. The boundary term gives rise to total energy of this system, which matches with the spectrum of TT -deformed CFT. This spectrum is exactly the quasi-local energy of BTZ black hole, if we identify µ = 1/r 2 c . After writing the boundary term in terms of gauge fields, the total action can reduce to TT -deformed two chiral WZW models. The effect of TT deformation is coupling the two chiral WZW models. Moreover, the mixed boundary condition also gives the constraints on TT -deformed WZW model. By disentangling the constraints, the boundary theory turns out to be the TT -deformed Alekseev-Shatashvili quantization of coadjoint orbit of the Virasoro group. Finally, we show that the TT -deformed standard non-chiral WZW model is equivalent to the TT -deformed two chiral WZW models.
As for the JT deformation, the holographic interpretation is also AdS 3 gravity but with an extra U(1) Chern-Simons gauge field coupling to the gravity. After rewriting the gravitational action in Chern-Simons formalism, we also obtain the associated boundary term. As expected, this boundary term precisely gives the spectrum of JT -deformed CFT. In addition, based on this nontrivial boundary term, the boundary dynamics is also studied. It turns out that the boundary dynamics of AdS 3 can be described by one type of constrained JT -deformed WZW model. This type of JT -deformed WZW model can also be obtained from the JT flow equation through adding a supplementary U(1) gauge field. However, this type of JT -deformed WZW model turns out to be a coupling of the right-moving Alekseev-Shatashvili action to a U(1) gauge field. The standard JT deformation should be the coupling of two opposite chiral Alekseev-Shatashvili actions via the JT operator. Regarding to this, it would be interesting to find another boundary condition in the bulk and perform a holographic check.
Furthermore, we show that the effect of TT deformation is the coupling of two opposite chiral SL(2, R) WZW models, and the effect of JT deformation is coupling a right-moving SL(2, R) WZW model with a U(1) WZW model. It would be interesting to consider SL(N, R) WZW models and couple two WZW models through higher spin currents deformation, since SL(N, R) WZW models correspond to higher spin gravity [63][64][65]. This will be helpful to understand the holographic aspects of higher spin gravity under the integrable deformation.

A Conventions
In this paper, we use the generators of SL(2, R) The commutation relations are Its Cartan-Killing metric is

B Boundary term for TT deformation
In this appendix, we will derive the boundary term (2.32) for TT deformation. Firstly, we expect the variation of the total action behaves like the form which vanishes due to the mixed boundary condition. Therefore, the variation of boundary term can be identified as According to (2.30) and (2.31), we can get the variation ofã,ã with respect to L µ ,L µ Besides, it is straightforward to obtain Substituting these relations into (B.2), it yields The right hand side of this equation is a total derivative. The expected primitive function of this boundary term variation could be In addition, the boundary term could be written into another form As a consequence, the boundary term for TT deformation is just (2.32).

C Boundary term for JT deformation
In this appendix, we will derive the boundary term for JT deformation. According to the boundary condition (4.25), (4.26) and (4.27), we can write down the expected variation of total action. We would like to consider the gravitational part and U(1) gauge field part separately. For the gravitational action, its variation should take the following The variation of U(1) gauge field action should be Both of them vanish because of the boundary condition. Then, we can read off the variation of the boundary terms δB grav = − κ 4π ∂M dtdθ Tr(ãθδãθ) + 1 + µJ 1 − µJ Tr (āθδāθ) , (C.3) Plugging these relations into the boundary term and noting (4.18), we can write these boundary terms in terms of L, J and F One can verify the variation of each boundary term is not a total derivative. However, combining the gravitational part and U(1) gauge field part, we can get a total derivative. This might imply the boundary term coupling the gravity with U(1) gauge field. The variation of total boundary term is (C.10) Integrate the above formula, we arrive at the expected boundary term In order to define the stress tensor, we put the right-moving SL(2, R) WZW model in a curved background whose metric is g tt = 0, g tθ = g θt = 1 2 , g θθ = h. This Lagrangian becomes chiral WZW action of left-moving copy if setting h = −1, and h = 1 for the right-moving copy. We then couple U(1) WZW model with gauge field B, such that the Lagrangian becomes Following the technique used for chiral Bosons [43,66,67], we finally obtain the improved action Then the conserved stress tensorT i a and conserved current J i can be defined bȳ We identity this action as the original theory.
Therefore, the JT -deformed Lagrangian L µ satisfy the flow equation with the initial condition Solving the JT flow equation (D.11), and setting e − t = e − θ = 1, B t = B θ = 0, one can get the deformed Lagrangian Finally, the total action for JT -deformed WZW model is (D.14)