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We study the TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\bar{T}$$\end{document} deformation of the chiral bosons and show the equivalence between the chiral bosons of opposite chiralities and the scalar fields at the Hamiltonian level under the deformation. We also derive the deformed Lagrangian of more generic theories which contain an arbitrary number of chiral bosons to all orders. By using these results, we derive the TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\bar{T}$$\end{document} deformed boundary action of the AdS3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {AdS}_3$$\end{document} gravity theory in the Chern–Simons formulation. We compute the deformed one-loop torus partition function, which satisfies the TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\bar{T}$$\end{document} flow equation up to the one-loop order. Finally, we calculate the deformed stress–energy tensor of a solution describing a BTZ black hole in the boundary theory, which coincides with the boundary stress–energy tensor derived from the BTZ black hole with a finite cutoff.


Introduction
The deformation by the TT operator [1] has drawn much attention, because of its solvability and the relation with gravity theory. Although the TT deformation is an irrelevant deformation, it is possible to derive the deformed Lagrangian, finite size spectrum and the S-matrix from the ones of the original theory [2][3][4], which does not require the integrability in many cases. Based on the finite size spectrum, one could compute the torus partition function of the TT deformation [5][6][7], which is still modular invariant but not conformal invariant. The TT deformation is related to the gravity theory in several aspects. On the one hand, the deformed theory can be interpreted as the original theory coupled to a topological gravity [5,8,9]. More concretely, one finds a one-to-one map between the equations of motion (EOM) in the deformed theories and those of the original theories [10,11], which enables one to derive the all-order deformed Lagrangians [12]. On the other hand, the two-dimensional TT deformed holographic CFT is proposed to correspond a e-mail: shuphy124@gmail.com (corresponding author) to the gravity theory with a finite cutoff under the Dirichlet boundary condition, where the cutoff is explicitly related to the deformation parameter [13]. More discussions on the holography under the TT deformation can be found in [14][15][16][17][18][19][20]. See also [21] for an interesting review and related topics.
The TT deformation of the two-dimensional scalar theory has been well studied. In particular, the all-order deformed action of the N massless free bosons has the form of the Nambu-Goto action in the static gauge of N + 2dimension [4]. The deformation of a scalar with an arbitrary potential was shown in [4,22] and more examples of Lagrangians of TT deformed theories was presented in [23]. In this paper, we study the chiral bosons, which are interesting in many aspects such as string theory and condensed matter. Even though the chiral bosons are not manifestly Lorentz invariant, the sum of a left and a right chiral bosons reproduces the scalar theory [24,25]. The TT deformed action of a general system of chiral bosons, scalars and fermions was studied in [26], where the first-order action for chiral bosons and canonical stress-energy tensor were used. We are interested here in the Floreanini-Jackiw action [27] and the covariant stress-energy tensor.
A remarkable connection between chiral Wess-Zumino-Witten (WZW) models and Chern-Simons theories was established in [28][29][30]. In particular, the AdS 3 Einstein gravity theory can be reformulated as a SL(2, R) × SL(2, R) Chern-Simons theory [31]. Much attention has been paid to the exact boundary action [32][33][34][35], due to its connection with two-dimensional conformal field theory [36]. The AdS 3 Chern-Simons action can be reduced to two chiral SL(2, R) Wess-Zumino-Witten (WZW) models on the boundary, and the AdS 3 boundary condition implements certain constraints on the chiral WZW model [32]. In [35], the exact boundary action was shown to be a quantum field theory of reparametrizations, which analogous to the Schwarzian action of the nearly AdS 2 gravity. Moreover, the torus parti-tion is one-loop exact and shows a shift in the central charge of 13. The present work aims to study the TT deformation of the boundary action. We derive the TT deformed Lagrangian of generic chiral boson theories by explicitly solving the flow equation. Then we focus on the TT deformed action of the constrained chiral WZW model associated with the AdS 3 Chern-Simons theory. We will see how the exact boundary action and the one-loop torus partition function change under the TT deformation. We also calculate the deformed stress-energy tensor of the boundary theory for the BTZ black hole, and compare it with the boundary stress-energy tensor derived from the BTZ black hole with a finite cutoff. Our results provide a concrete realization of the TT deformation on the boundary of the Chern-Simons AdS 3 gravity and may shed light on the holography dual of the deformation.
This paper is organized as follows. In Sect. 2, we present the TT deformed Lagrangian of chiral boson theories. We show the equivalence between the sum of two chiral bosons of opposite chiralities and a massless free non-chiral scalar under the TT deformation at the Hamiltonian level. In Sect. 3, we review the relation between the AdS 3 Chern-Simons theory and the sum of two constrained SL(2, R) chiral WZW model of opposite chiralities, and derive the corresponding deformed Lagrangian. We then compute the one-loop torus TT deformed partition function, which is found to satisfy the flow equations of TT deformation in all-order of deformation parameter up to one-loop level. We also compute the deformed stress-energy tensor for a solution describing a BTZ black hole in the deformed field theory and compare it with the boundary stress-energy tensor of the BTZ black hole at a finite cutoff. Section 4 is devoted to conclusions and discussions. In Appendix A, we consider the JJ and TJ deformation of the chiral bosons. In Appendix B, we study the solutions to the EOMs of TT deformed WZW models.

TT deformed Lagrangian of chiral bosons
In this section, we will study the TT deformation of chiral bosons. The Floreanini-Jackiw action [27] of a left-moving chiral boson is As a warm-up, we will first consider the simple case of two chiral bosons of opposite chiralities and solve the flow equation induced by the TT deformation. More complicated theories of chiral bosons will also be considered, which will be useful in the study of the AdS 3 Chern-Simons theory.

Two chiral bosons of opposite chiralities
Let us begin with the undeformed Lagrangian of a left and a right chiral boson The stress-energy tensor of the deformed theory becomes from which we obtain the corresponding energy and momentum The Hamiltonian density of the system can be written as where π = 1 2 ∂ θ φ andπ = − 1 2 ∂ θφ are the canonical momenta of the fields.
Let us turn to the TT deformed free massless non-chiral scalar. The Lagrangian is given by [4] L scalar The associated Hamiltonian density is where the canonical moment of ϕ is defined as One can check that the Hamiltonian densities (16) and (18) are equivalent via the relation The TT deformed Lorentz invariant free massless scalar is related to the undeformed model via a field dependent coordinate transformation [10,11]. To obtain a solution to the deformed theory, one can start with a solution wherex ± =θ ±t, to the equation of motion of the undeformed model Then one need to solve the equations to expressx ± in terms of t and θ , where the derivatives of F and G are the components of stress-energy tensor in the specific classical solution A solution to the equation of motion of the deformed model is then given by Let us return to the chiral boson model. Though we have not found a coordinate transformation which maps the equations of motion of the model (11) directly to those of the undeformed model, one can check that is a solution to the equations of motion. Here t, θ are still related tox ± by (23). h(t) andh(t) are arbitrary functions of t. We show this in a more general case in Appendix B. The energy and momentum corresponding to the solution (26) are We now put the deformed model on a circle of length L. Then the fields should be periodic in coordinate θ . We take periodicities of f and g to be L and consider the solutions with the following form where we introduce n(λ) and m(λ) such that the periodicity of ϕ is L in coordinate θ . It is not difficult to show that Then we have Using Eqs. (23) and Finally we get which is a classical version of the general quantum spectrum in [3,4]. The significance of the sign of the deformation parameter λ is well-known in the literature. When λ > 0, the deformed energy can become complex if H 0 is large. This regime of λ is related to holography. For λ < 0 the deformed energy spectrum is real and there are Hagedorn growth of density of states [38].

General theory of chiral bosons
To solve the flow equation, the field contents and details of the potentials are not important. We can study the TT deformed Lagrangian of more general model of chiral bosons with the initial translational invariant Lagrangian where K ± , W ± and V ± are the functions of the fields. We require that the equations of motion are consistent with the conservation of the stress-energy tensor defined by (5). We can again solve the flow equation (6) using a perturbative approach. The all-order solution can be written as with As a particular example, we consider a generalized chiral bosons theory where G andḠ are non-degenerate matrices. The TT deformation of the chiral boson theory (38) is therefore with One can also add an arbitrary number of Weyl-Majorana fermions to the theory (38). The undeformed Lagrangian is where ψ andψ are Weyl-Majorana left and right fermions respectively. The deformed theory is then given by (41) with This action differs from the one obtained in [26] using the canonical stress-energy tensor. It was argued in [26] that there should be a field redefinition which would make the TT deformed action driven by the canonical stress-energy tensor coincide with the one driven by the covariant stressenergy tensor. It would be interesting to find such a field redefinition explicitly.

TT deformation and Chern-Simons gravity
The three-dimensional Einstein gravity theory with a negative cosmological constant can be reformulated as a Chern-Simons action with a gauge group SL(2, R)× SL(2, R) [31]. It was shown in [32] that the Chern-Simons action is equivalent to two copies of constrained SL(2, R) chiral WZW models of opposite chiralities on the boundary, which can be combined into a non-chiral Liouville field theory. The chiral description is more convenient to deal with the zero modes and leads to geometric actions associated with coadjoint orbits of the Virasoro group [34,35]. In this section, we will focus on the TT deformed action of the constrained chiral WZW model. Since the original Lagrangian is a special case of (38), we can get the all-order TT deformation of the boundary action using the results in the previous section. 1

AdS 3 Chern-Simons theory
Let us recall the connection between AdS 3 gravity and the chiral WZW model derived in [32]. The AdS 3 Einstein grav-ity with metric can be reformulated as the Chern-Simons action [35] where k = 1 4G and we couple the boundary terms to the boundary zweibein E a . We will take The gauge fields A andĀ are expressed by using the SL(2) generators and related with the bulk dreibein e a and the bulk spin connection ω In this action, A = A 0 dt +Ã i dx i andĀ =Ā 0 dt +Ã i dx i are separated into the temporal and spatial parts. The boundary conditions of the gauge fields are fixed to be which are chosen to match the asymptotics of the AdS 3 geometry where is the boundary spin connection. For simplicity, we consider = 0 in this paper. The boundary term S bdy is necessary for a consistency variation principle.
Since the spatial field strengthF is flat, one can parametrize theÃ andÃ as whered is the spatial exterior derivative. g andḡ are elements of SL (2) and can be written in the Gauss parameterization: The gauge fields can be written as The action (46) thus can be evaluated as where L WZW 0 has the form of Eq. (38) with The fields in the expression of g andḡ are not independent. The boundary condition (3.1) imposes the constrains on the AdS boundary. The constrains can be expressed as By using the conditions (55), we could express the action S in terms of F andF, which we parameterize as where φ andφ are elements of Diff(S 1 )/P SL(2, R) and we get two copies of the Alekseev-Shatashvili quantization of coadjoint orbit Diff(S 1 )/P SL(2, R) of the Virasoro group [40]. If we parameterize F andF as with α = n, n ∈ Z, we get the orbit Diff(S 1 )/U (1). See [41] for further discussion.

TT deformation of the boundary action
With the solution (41) to the flow equation induced by the TT deformation at hand, we are now ready to get the all-order TT deformation of the boundary action. Simply plugging (53) into (41), we obtain a TT deformed WZW model denoted by L WZW λ . 2 However, the action (52) is constrained. It is a non-trivial question whether the constrains are deformed by the TT . To treat the constraints carefully, we introduce Lagrange multipliers in the undeformed action: Here we keep E ± θ unfixed which is necessary when we apply the solution (41). The terms with coefficient E ± θ can be view as potentials. By using (41) again with V = − √ 2ra + and V = √ 2rā − , we find the deformed constrained Lagrangian where F μν are given in (53) and we have set Plugging A 3 θ =Ā 3 θ = 0 into the rest two constraints, we get which are similar to (5.16) in [42]. The explicit expressions are where s ands are halves of the Schwarzian derivatives defined by Suppose the solution p andp satisfying (62) is known, we then substitute this solution to the L cWZW λ and obtain the all-order TT deformed Lagrangian where the overdot and prime denote the derivative with respect to t and θ respectively. The deformed stress-energy tensor is given by Using the parameterization We have where f (n) denotes the nth derivative of f with respect to θ . We also define Dropping total derivatives, the Lagrangian can be written as where p andp are determined by s ands through the constraints (62).
Though the constraints (62) are difficult to solve to all orders in λ, we can solve p andp in the first few orders of small λ which leads to When λ = 0, this reproduces the original Lagrangian. The first order term ss is nothing but the TT operator of the undeformed action.
At the end of this subsection, let us comment on the deformed constrains (61) and their relation with finite cutoff AdS. Since (55) is derived from the boundary condition (3.1) of gauge fields (or metric), it is natural to guess that the boundary condition will also be transformed non-trivially under the TT deformation. Let us suppose the new boundary is at r = r c with a large enough r c . If we identify r 2 c λ = 1, (61) leads to 2e ± θ = r c , which is consistency with the metric (45) at finite cutoff r = r c . In Sect. 3.4, we will check the identification r 2 c λ = 1 in more details by calculating the boundary stress-energy tensor.

One-loop torus partition function
The partition function in the undeformed theory was obtained and shown to be one-loop exact in [35]. We now compute the one-loop torus partition function in the deformed theory. Let us Wick-rotate to the Euclidean time t = −iy and put the boundary theory on a torus of complex structure τ . The Euclidean action is S E = −i S. On the torus, the coordinate z = θ +iy has the identifications z ∼ z+2π and z ∼ z+2πτ . We first focus on the Diff(S 1 )/P SL(2, R) case. The fields φ andφ satisfy the boundary condition We consider the saddle point of the Euclidean Lagrangian where τ 1 and τ 2 are real and imaginary part of τ respectively and we will use instead of λ to avoid square root. Expanding φ andφ in fluctuations around the saddle the fluctuations of p andp depend on δφ and δφ via the constraints. We have p = p 0 + p 1 + p 2 + · · · ,p =p 0 +p 1 +p 2 + · · · (78) where p 1 (p 1 ) and p 2 (p 2 ) are linear and quadratic terms in the fluctuation fields δφ and δφ respectively. We then expand the Lagrangian and the constraints around the saddle, and express every term by using δφ and δφ. On the torus, the fluctuation fields δφ and δφ can be expand as where we have set the zero modes to zero and the functions where Finally, the quadratic action is given by where M m,n is a 2 × 2 matrix M m,n = n(n 2 − 1) Then following the procedure in [35], we obtain the classical partition function and the one-loop torus partition function (90) Note that the partition function is not modular invariant even in the undeformed theory. The spectrum should become complex when λ > 0. In this case the one-loop torus partition function should be understood as an analytic continuation from the regime of λ < 0. The singularity at λ = −1 is related to the Hagedorn divergence. It is easy to check that the classical partition satisfies the flow equation on the torus while the one-loop partition function satisfies the flow equation up to the one-loop where the first term on the right hand side is order O(C). The one-loop torus function satisfy the flow equation up to the one loop order O(C 0 ). This suggests that the TT deformed partition function should not be one-loop exact as in the undeformed theory. It is worth to note that the flow equations (91) and (92) are satisfied for all order in λ, which provide evidence for our all-order TT deformed Lagrangian. One can also compute the partition function of the Diff(S 1 )/U (1) case, where with α = n, n ∈ Z. One can repeat the same steps and finally get

TT deformation and BTZ black hole
Following the same procedure in Sect. 3.1, one can describe the BTZ black hole in the formalism of the Chern-Simons theory. In this subsection, we compute the stress-energy tensor of the TT deformed boundary theory of the BTZ background. For simplicity, we will focus on the classical solution of the BTZ Chern-Simons theory. We also compare the associated stress-energy tensor with the "boundary stress-energy tensor" of the BTZ gravity with a finite cutoff. The BTZ black hole is described by the metric To describe the BTZ black hole in the Chern-Simons formulation, it is convenient to define Then the metric can be written as The associated classical gauge fields are , and the group elements are where . The BTZ metric leads to same boundary condition (3.1) of gauge fields at boundary. In the same way as in Sect. 3.1, we could derive the boundary action and the constrains of the BTZ black hole, which have the same form as (52) and (55) respectively. However, instead of (56), the fields in the BTZ black hole are which provides the orbit Diff(S 1 )/U (1). To describe an BTZ black hole, we require b 2 < 0 andb 2 < 0. When b =b ∈ (0, 1/2) we have a conical defect rather than a BTZ black hole. See [35] for more discussions.
In Appendix B, we show that the solutions to the EOM of the deformed theory can be obtained from the ones of original theory. The deformed solution associated with g (0) andḡ (0) is wherẽ Here we introduce b λ andb λ such that the boundary condition are undeformed. The deformed stress-energy tensor in terms (115) which matches (104) up to a factor under the identification r 2 c λ = 1. As in [13,14], to compare with the energy obtained on the QFT side one should multiply the energy by the circumference of the circle L = 2πr c to get a dimensionless "proper energy" When r c is large, we have E = 2π M + O(r −1 c ). The same result can be derived in the Chern-Simons formulation. We assume that the boundary term on a finite cutoff surface has the same form as that at infinity The boundary conditions consistent with the variational principle are To obtain the boundary stress-energy tensor we need to insert back the zweibein. The zweibein on the cutoff surface are Therefore the on-shell boundary term should be interpreted as (120) Then we get the boundary stress-energy tensor, which equals (110) up to a factor.

Conclusions and discussions
In this paper, we have studied the TT deformation of chiral bosons. In particular, the TT deformation of two chiral bosons of opposite chiralities is equivalent to that of a non-chiral free scalar theory at the Hamiltonian level. Furthermore, we have obtained the all-order TT deformed Lagrangian of more general theories which contain an arbitrary number of chiral bosons with potentials. Based on these results, we study the TT deformation of the boundary theory in Chern-Simons AdS 3 gravity which is a constrained chiral WZW model. We have derived the all-order TT deformed Lagrangian and computed the one-loop torus partition function of the deformed theory, which satisfies the flow equation of general TT torus partition function up to one-loop order. Our result suggests that the one-loop torus partition function is not one-loop exact under the TT deformation, which is unlike the situation in the undeformed theory [35]. Moreover, we have computed the stress-energy tensor of the solution associated with a BTZ black hole in the deformed theory, which matches the boundary stress-energy tensor of the BTZ black hole at a finite radial location on the bulk side. Let us comment on future research directions. It would be interesting to start with the Chern-Simons theory describing the AdS 3 gravity with a finite cutoff to derive the TT deformed boundary action. This will help us to realize the holography under TT deformation more explicitly. Moreover, the original exact boundary action of Chern-Simons AdS 3 gravity can be applied to compute the four-point functions in the light-light and heavy-light limit [35]. Recently, many studies have been devoted to the correlators in general TT deformed CFTs [44][45][46][47][48]. It would be interesting to compute correlators in our deformed model and compare them with these results. It would also be interesting to generalize our analysis to higher spin theories of gravity formulated in terms of SL(N , R) Chern-Simons theory [49].

Data Availability Statement
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Appendix A: JJ and TJ deformation of two chiral bosons
In this Appendix, we consider the JJ and TJ deformation of the chiral bosons.

A.1 JJ deformation
Consider the Lagrangian of two chiral bosons of opposite chiralities To define currents J andJ , we couple the chiral bosons to gauge fields We define the currents as In the undeformed theory When A andĀ are closed, J andJ are conserved. The JJ operator in the deformed theory is defined as Solving the flow equation We get Finally setting A =Ā = 0, we get A.2 TJ deformation We couple the left chiral boson to the zweibein and the left chiral boson to a gauge field We define the currents as The TJ operator in the deformed theory is defined as Solving the flow equation We get Finally we set E + θ − 1 = E + t − 1 =Ā = 0 and obtain

Appendix B: TT deformed chiral WZW model
We consider the sum of a left and a right chiral WZW model with where g andḡ are group elements of group G andḠ respectively. We define The equations of motions are Consider the TT deformed chiral WZW model with S = 1 − 2(tr(A θ A θ ) + tr(Ā θĀθ ))λ + (tr(A θ A θ ) − tr(Ā θĀθ )) 2 λ 2 . (141) The equations of motions are When λ = 0, a solution to the equation of motion is g = h(t)g 0 (x + ),ḡ =h(t)ḡ 0 (x − ).
satisfies the equation of motion for the deformed theory. Using one can check that (142) and (143) are satisfied. One should also consider boundary condition so in general g 0 andḡ 0 should depend on λ.