The dual geometries of $T\bar{T}$ deformed CFT$_2$ and highly excited states of CFT$_2$

In previous works, we have developed an approach to derive the pure AdS$_3$ and BTZ black hole from the entanglement entropies of the free CFT$_2$ and finite temperature CFT$_2$, respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the $T\bar{T}$ deformed CFT$_2$, which breaks the conformal symmetry. In terms of the physical arguments of the $T\bar{T}$ deformed CFT$_2$, the derived metric is a deformed BTZ black hole. The requirement that the CFT$_2$ lives on a conformally flat boundary leads to $r_{c}^{2}=\ 6R_{AdS}^{4}/(\pi c\mu)$ naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from $T\bar{T}$ deformed CFT$_2$. We furthermore calculate the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.


Introduction and summary
The TT deformed CFT 2 , discovered by the remarkable works [1][2][3], attracts increasing attention recently. It is a solvable example of quantum field theory. A conformal field theory (CFT), which is conformally invariant at different fixed points along the RG flow, can be deformed by relevant, irrelevant and marginal deformations. The irrelevant deformation only affects the physics in the UV region but not the IR region. TT deformation supports a class of solvable irrelevant deformations of CFT 2 by turning on the TT term. The integrability implies the theory contains an infinite element of conserved charges. The class of TT deformed CFT 2 is characterized by a length squared parameter µ ≥ 0 as dS µ /dµ = d 2 x(TT ) µ , where (TT ) µ = 1 8 T αβ T αβ − (T α α ) 2 µ is defined in terms of the stress tensor of the deformed theory. As µ → 0, the simplest example is It is clear that TT deformation is Lorentz invariant and breaks the conformal symmetry. McGough, Mezei, and Verlinde [4] proposed that the TT deformed CFT 2 is no longer located on the asymptotic boundary of AdS 3 but lives at the finite radial position r c 1 , 1 Note that to compare with [4], our µ should be replaced by µ/(4π 2 ) for notation difference. where R AdS is the radius of AdS 3 and c = 3R AdS

2G
is the central charge of the CFT 2 . Based on this picture, a new way to extend and study the AdS/CFT correspondence emerges. In a recent work [5], Chen and collaborators calculated the entanglement entropy of the TT deformed CFT 2 by the replica trick. They found that for a finite size system, there is no leading correction. But for a finite temperature system, they obtained It is curious why there is no correction to the finite size system. Without a correction, how do we tell the CFT lives at a finite radial distance for a finite size system by the entanglement entropy? We speculate the reason is that the entanglement entropy is defined for spacelike intervals only, which is not consistent with the Lorentz covariance. Since the finite size and finite temperature systems have the same cylindrical topology under exchanging t ↔ x, we are inspired to speculate that there exists a temporal entanglement entropy for a temporal interval (indicating entangling between the big bang and big crunch?), and the finite size system receives a correction in this temporal entanglement entropy. Unfortunately, up to now, there is no well-defined temporal entanglement entropy. We are thus led to believe, once a temporal entanglement entropy is correctly introduced, one would find the results as shown in Table (1): space-like time-like Finite size Finite temperature Table 1: The symbol marks a correction to the entanglement entropy caused by the TT deformation.
In the same paper, Chen etc. verified the holographic method (RT formula [6]) by calculating the geodesic length in the BTZ background. To match the entanglement entropy of the TT deformed CFT 2 , they found two endpoints of the geodesics must be fixed on the finite cut-off r c . The relevant paper [7] calculated entanglement entropy of the TT deformed CFT 2 between two antipodal points on a sphere. More recent works on the holographic studies of the TT deformed CFT 2 can be found in the refs. [8][9][10][11][12][13][14][15][16][17][18][19].
In previous works, we have developed a method to derive the dual geometries from the entanglement entropies of the CFT 2 without using AdS/CFT or imposing any constraints on the bulk geometry. It must be understood that neither kinematics nor dynamics (Einstein equation) can be used in the derivation, especially the latter one. Out of the general geometries, gravity is those respecting the Einstein equation. Therefore, using the Einstein equation in any place makes the derivations circular. On the other hand, it is easy to understand that the symmetry argument is a necessary but not a sufficient condition to determine the dual geometry of a CFT. Furthermore, obviously, the SO(2, 2) symmetry argument is no help for the TT deformed CFT 2 considered in this paper, since this particular CFT breaks the SO(2, 2) symmetry. Under these considerations, we successfully derived the asymptotic AdS 3 from the infinitely long system of CFT 2 [20], and 3D BTZ black hole from the finite temperature system of CFT 2 [21], both with time direction included as it should be.
One of the distinct ingredients of our approach is to use Synge's world function [22] to extract the spacetime metric from given geodesic lengths: where x i = (x, y), and L (x, x ) is the length of the geodesic connecting two points x i and x i in the unknown bulk geometry. It turns out that, in order to determine the function form of the bulk geodesic length L (x, x ), we need to use the entanglement entropies in UV and IR (mass gap) regions. These entanglement entropies can fix the behaviors along the boundary directions x, t where the CFT lives, and the holographic direction y which is created by the energy cut-off of the CFT. In order to verify the universality and correctness of our method, we apply it to the TT deformed CFT 2 at finite temperature to derive its dual geometry in this paper.
The derived metric turns out to be a deformed BTZ metric in the planar coordinate, where "deformed" refers to a prefactor of dx 2 in contrast to the standard BTZ metric: where β H ≡ T −1 = β 2π . It should be emphasized that x and t in eqn. (5) are the physical (proper) arguments of the TT deformed CFT 2 . Though after a redefinitionx = (1 − πµc 12β 2 H + . . .)x, the metric (6) is the standard BTZ,x is the physical argument of the original undeformed CFT. The advantage of the metric (5) is that, once requiring the CFT lives on the flat boundary of the dual geometry, we immediately deduce that the geometry covers y c ≤ y ≤ β H , with y 2 c = πµc 6 . After transforming y c to the global coordinate, we get r 2 c = 6R 4 AdS πcµ , which is in perfect agreement with eqn. (2). In [4], in order to check the duality, quantities such as energy spectrum, thermodynamic properties and propagation speeds were calculated based on the standard BTZ metric (6), and they were compared with the results from the TT deformed CFT 2 to determine the value of r c . Nevertheless, using the deformed metric (5), in addition to determining r 2 c = 6R 4 AdS πcµ immediately, we straightforwardly calculate the energy spectrum and propagation speeds, which are the same as those from the TT deformed CFT 2 . It is important to note that all these conclusions cannot be obtained without the metric component g tt .
As a further check, we next apply our approach to the highly excited states of CFT 2 with large c and sparse spectrum of low dimensional operators. Using the entanglement entropies of such states of a finite size system, given in [23][24][25], we derive the dual geometry where β Ψ = ( 12 h c − 1) −1/2 . When h < c 12 , the metric represents a conical defect in the center of the global AdS 3 . On the other hand, when h > c 12 , the inserted operator is heavy enough to produce a BTZ black hole at temperature T = β −1 Ψ . The reminder of this paper is outlined as follows. In section 2, we show how to derive the deformed BTZ spacetime from the entanglement entropies of the TT deformed CFT 2 , and calculate the energy spectrum and propagation speeds with the deformed metric. In section 3, we address the highly excited states of CFT 2 and derive its dual geometry from the entanglement entropy. Section 4 is for conclusion.
2 The dual geometry of the TT deformed CFT 2 To the leading order correction, the UV entanglement entropy of TT deformed CFT 2 at finite temperature was calculated in [5], where β H ≡ T −1 = β 2π and a is a UV cut-off. As µ/β 2 H → 0, it reduces to the entanglement entropy of the undeformed CFT 2 at finite temperature: Since the correction is a function of x/β H and µ/β 2 H , it is tempting to check if it can be captured by an effective correction to x. This step is consistent with the fact that irrelevant deformations only affect the physics in the UV region. Therefore, we set where F (µ, x) is a correction to x. It is remarkable to notice for small F (µ, ∆x), the Taylor expansion is Therefore, comparing with eqn. (8), we can get So, the UV entanglement entropy of TT deformed CFT 2 can be written as: The time-dependent entanglement entropy of the undeformed CFT 2 at finite temperature is: As t = 0, it reduces to eqn. (9). For the TT deformation, we only need to replace x by x 1 − πµc and identifying this entanglement entropy with the length of the geodesic anchored on the boundary in the dual geometry, one has From the holographic principle, the energy cut-off a generates a holographic dimension y, say a 2 → yy (1 + . . .).
In order to apply eqn. (4) to get the metric, this ending on boundary geodesic length eqn. (16) is obviously not sufficient and we need to push the endpoints into the bulk, with dependence on the holographic dimension y, to describe bulk geodesic connecting arbitrary endpoints (t, x, y) and (t , x , y ). On the other hand, as shown by eqn. (8), when β H → ∞, the entanglement entropy becomes the free CFT 2 one, whose dual geometry is the pure AdS 3 in Poincare patch, as we derived in [20] by our approach. The geodesic length of pure AdS 3 in Poincare patch connecting (t, x, y) and (t , x , y ) is To meet these two requirements (16) and (17) under different limits, we only have one possibility to generalize the boundary anchored expression (16) to the general bulk geodesic length: where f x, x , µ ; y, y ; t, t , and then we can apply eqn. (4) to get the metric.
Step 1: When β H y = y = a (geodesics anchored on the boundary), L bulk in eqn. (18) must reduce to L boundary , given by equation (16), where i , ρ i ,¯ i andρ i are regular and bounded functions.
Step 2: As β H → ∞, or µ/β 2 H 1 and β H x, t, y and y , the general expression (18) must match the pure AdS 3 background (17). From step 1, we know the leading term of f and g is the unit. So, we have where we only keep the leading order. When keeping the higher orders, we suppose to get the metric of asymptotic AdS by using equation (4). Or equivalently speaking, since f x, x , µ and given by the last two terms of (22).
Since the correction x β H 1 − µc and g y β H , y β H , we are free to set µ = 0 in the following steps and the derivations are same as what were done in our previous work [21] for the case of finite temperature CFT. After fixing f y β H , y β H and g y β H , y β H , we restore µ = 0.
Step 3: When two endpoints of a geodesic coincide, the geodesic length vanishes exactly. Plugging x = x , y = y and t = t into eqn. (24), we get Step 4: We now consider the length of the segment in Fig. (1) as ∆x → ∞. There are three ways to calculate it. The left picture is given in [26,27] from boundary conformal field theory (BCFT), Figure 1: The left picture is given by BCFT with ∆x → ∞. The middle picture is obtained from L bulk by setting y = a, y = β H and x/2 → ∞. The right picture is also given by L bulk from a different point of view, by setting y = y = a and ∆x → ∞. The solid lines in all three pictures describe the same object.
On the other hand, by using eqn. (24), we have two other ways to calculate it. The first way is to straightforwardly substitute y = a, y = β H , ∆x/2 → ∞ into (24) to get It is easy to understand that this length is one half of that connecting y = y = a, ∆x → ∞. So the second way is These three lengths (28), (29) and (30) Step 5: An important lesson we learned from the free CFT 2 case in [20] is that, in order to completely determine the dual geometry, we need to know the geodesic length between a and β H with x = 0, i.e. the vertical geodesic. To be consistent, this particular geodesic length must be provided by the CFT 2 entanglement entropy. In the free CFT 2 , the IR entanglement entropy precisely fits the requirement. In the finite temperature CFT 2 , there is no available such IR entanglement entropy. To solve this problem, we map the finite temperature system to a finite size (L S ) system by replacing β → L S and impose the periodic boundary condition 2 Therefore, the geodesic length between a and β H = β 2π in the finite temperature system can be obtained from the geodesic length connects a and L 2π in the finite size system, finite temperature finite size L geodesic a, β 2π Noting that L S 2π is the radius of the finite size system with the circumference L S , therefore, this geodesic goes from boundary to the center of the circle, as illustrated in Fig. (2). Figure 2: The geodesic between y = a and y = β H at a finite temperature system is mapped to a finite size system, corresponding to the radius of a circle from y = a to y = L S 2π .
We know that the entanglement entropy of a finite size system is 3 The maximal entanglement entropy is achieved by splitting the circle into two equal regions, ∆x = L S /2. The corresponding geodesic is nothing but a diameter It is then easy to get what we need 2 To make the discussion simpler, we do not choose the equivalent replacement L S = iβ H . 3 According to [5], the entanglement entropy of a finite size system receives no correction.
We now map L S → β to get the geodesic length between a and β H = β 2π in the finite temperature system: which agrees with that obtained through holographic duality in [28]. Therefore, from the general expression (24), as x = x , t = t , y = a and y = β H , we have We thus obtain For convenience, we summarize all the constraints we have obtained for the general expression (24) of bulk geodesic length: From eqn. (41) and (43), we get Since a is a varying quantity, y or y = β H must be a zero of g(y/β H , y /β H ) . Moreover, g must be symmetric for y and y . So, the function form must be On the other hand, from eqn. (41) and (42), one gets It then easy to fix n = 2 and κ = 1/2 and Similarly, from eqn. (41), we get where m, δ > 0 are some numbers. The story is not over yet. In order to match eqn. (40), there must be σ 1 = θ 1 . When applying eqn. (4) to calculate the metric, noting a limit ∆x, ∆y, ∆t → 0 is going to be imposed after making the derivatives, it is easy to see that terms proportional to ∆y β H 2 only contribute to g yy , but not to g xx and g tt . So, looking at eqn. (24), (47) and (48), without altering the derived metric, equivalently, we are free to pack all the corrections into g(y, y ) and simply set f (y, y ) = 1. Finally, we substitute f (y, y ) and g(y, y ) into eqn. (24) to get the bulk geodesic length, Using (4), we obtain the deformed BTZ metric, Note the coordinate x and t here are the physical arguments of the TT deformed CFT 2 . In contrast, in the refs [4,5], the authors started with the standard BTZ metric, and then the boundary coordinates in their works are the physical arguments of the undeformed CFT. To have the deformed CFT living on a conformally flat boundary, from this deformed BTZ metric (50), there ought to be So the boundary of the dual geometry is located at y 2 c = πcµ 6 + . . ., which is finite. Transforming the planar coordinate to the global one by y = R 2 AdS /r, we get which completely agrees with (2) as conjectured in ref. [4]. One of the advantages of the deformed metric (50) is that it naturally indicates that the dual geometry is a finite region y c ≤ y ≤ β H . For simplicity, we rewrite the metric by using eqn. (52) as To study the propagation speed, let us consider the induced metric at some point y = y * : where we define The right and left propagation speeds of the wave are given by Considering an observer sitting at y 2 c = µπc 6 (or r 2 c = 6R 4 AdS πcµ ), when 0 < y 2 * < y 2 c = µπc 6 + . . . ., the speed is less than the speed of light (subluminal). When y 2 * > y 2 c = µπc 6 + . . . ., the speed is superluminal, which again agrees with the results of TT deformed CFT [4]. More discussions can be found in [29,30].
As a further check, we compute the proper energy by using the deformed BTZ metric eqn. (50). Transforming the planar coordinate to the global one, setting R AdS = 1 for simplicity, we then have with Since we already determined the boundary of the geometry at r c , when calculating the quasi-local gravitational energy, we should integrate along r = r c . Using 3 2Gc = R AdS = 1, the proper energy [31][32][33] is where L = dφ √ g φφ | r=rc is the proper size on the boundary, and the non-vanishing components are This energy completely agrees with the energy spectrum of TT deformed CFT 2 , given in [4,33].
3 The dual geometry of the highly excited states in CFT 2 For a highly excited state of CFT 2 , . This entanglement entropy is identical to that of a finite temperature system with T = β −1 Ψ . As three dimensional gravity has no propagation, the excited states of the dual CFT 2 only leads to local defects and global topologies in the bulk. This is why the entanglement entropies of finite size system, finite temperature system and their excited states take the same form, and their dual metrics are connected by simple transformations. It is then easy to understand that we really only need to consider a single representative of CFTs with distinct topologies. So, we can safely use the same procedure which we have used in [21] to derive the dual geometries of CFT 2 at finite temperature to get the geodesic length: and the metric Setting R AdS = 1 for simplicity, the metric in globe coordinate system is When h < c 12 , β Ψ is imaginary, it is a finite size system with rescaled length. The dual geometry describes a conical defect placed in the center of the global AdS. The defect is caused by the back-reaction of a massive particle. When h > c 12 , the primary state |ψ h approximates to a thermal state and the dual excited state in AdS is heavy enough to form a black hole. These conclusions are in consistent with the holographic calculations [24,25].

Conclusion
In this paper, we derived the deformed BTZ black hole metric from the entanglement entropy of TT deformed CFT 2 . The metric shows explicitly that the dual region 6R 4 AdS cµ ≥ r 2 ≥ r 2 H is finite in the bulk. We used the deformed BTZ metric to calculate the propagation speed and energy spectrum. Both results match perfectly with those calculated in the TT deformed CFT 2 , and no identification is necessary. We then showed how to get the dual geometry of highly excited states of CFT 2 . The metric describes a conical defect located in the center of the global AdS for h < c 12 , or covers a BTZ black hole at temperature T = β −1 Ψ = 12 h c − 1 as h > c 12 .
discussions and suggestions. This work is supported in part by the NSFC (Grant No. 11875196, 11375121 and 11005016).