TsT, $T\bar{T}$ and black strings

We study the relationship between TsT transformations, marginal deformations of string theory on AdS$_3$ backgrounds, and irrelevant deformations of 2d CFTs. We show that TsT transformations of NS-NS backgrounds correspond to instantaneous deformations of the worldsheet action by the antisymmetric product of two Noether currents, holographically mirroring the definition of the $T\bar{T}$, $J\bar{T}$, $T\bar{J}$, and $J\bar{J}$ deformations of 2d CFTs. Applying a TsT transformation to string theory on BTZ $ \times S^3\times {\cal M}^4$ we obtain a general class of rotating black string solutions, including the Horne-Horowitz and the Giveon-Itzhaki-Kutasov ones as special cases, which we show are holographically dual to thermal states in single-trace $T\bar{T}$-deformed CFTs. We also find a smooth solution interpolating between global AdS$_3$ in the IR and a linear dilaton background in the UV that is interpreted as the NS-NS ground state in the dual $T\bar{T}$-deformed CFT. This background suggests the existence of an upper bound on the deformation parameter above which the solution becomes complex. We find that the worldsheet spectrum, the thermodynamics of the black strings (in particular their Bekenstein-Hawking entropy), and the critical value of the deformation parameter match the corresponding quantities obtained from single-trace $T\bar{T}$ deformations.


Introduction
The AdS d+1 /CFT d correspondence [1] has revealed a deep relationship between bulk theories of gravity in (d+1)-dimensional Anti-de Sitter spacetimes and conformal field theories in one dimension less. It has provided many insights into both strongly coupled CFTs and quantum gravity including black holes. One among many of its manifestations is the microscopic explanation of the entropy of certain higher-dimensional black holes with an AdS 3 factor in the near-horizon geometry [2,3]. Recently, progress has also been achieved for asymptotically AdS black holes in higher dimensions with the advent of localization techniques [4][5][6].
Various extensions of holographic dualities beyond the AdS/CFT proposal have appeared over the years. It is clear that most black holes including the real-world ones are not asymptotically AdS nor exhibit and AdS 3 factor in their near-horizon geometry, so AdS/CFT cannot be used as such. Various angles of attack to deal with more general black holes have been taken in recent years involving non-AdS geometries in the near-horizon region with both conformal [7][8][9] and non-conformal symmetries [10][11][12][13][14][15][16][17]. Although it has been suggested that conformal symmetry could play a role in understanding generic black holes, certain non-conformal symmetries have been able to account for the microscopic entropy of the corresponding solutions [13,14,18,19] in the spirit of [3]. To understand how holography works for non-AdS spacetimes or non-conformal quantum field theories, a useful strategy is to study deformations from known examples of AdS/CFT. Interesting examples involve the so-called β [20], noncommutative [21,22] and dipole deformations [23][24][25][26] of N = 4 SYM, in which the corresponding dual gravity backgrounds are identified with socalled TsT transformations (a succession of T-duality, shift and T-duality along U (1)×U (1) isometries) of the original geometry (for an overview see [27]). The implications of TsT transformations have also been investigated in the D1D5 system in [28,29].
Being able to depart from AdS spacetimes in the bulk is an important question as we stressed out above. We would like to find toy models of holographic dualities where we can compare quantities from both sides of the correspondence. In view of this, holographic dualities with d = 2 stand out as interesting starting points. In particular, string theory on AdS 3 backgrounds with pure NS-NS flux admits an exact worldsheet description in terms of an SL(2,R) WZW model that has been shown to be solvable [30][31][32]. The holographic dual to this theory has been discussed in [33][34][35][36][37][38][39][40][41] and most recently in [42][43][44][45]. An interesting feature of WZW models is that they allow for exact conformal deformations driven by integrable marginal current-current operators [46][47][48][49]. This allows us to reach a whole variety of new exact string theory backgrounds for which a worldsheet description is in principle available. These include in particular warped AdS 3 spaces, AdS 2 [50][51][52] and various black string solutions, in particular the Horne-Horowitz black string [53,54]. These geometries have the interesting feature that they appear in the near-horizon geometry of extreme four-dimensional black holes (Kerr or Reissner-Nordström), or represent close three-dimensional analogues of asymptotically flat black holes.
In parallel, it has been recently discovered that two-dimensional CFTs admit special classes of irrelevant deformations for which the theory remains solvable. These include the TT deformations [55][56][57], their higher spin generalizations and, for theories with a conserved U (1) current, the JT /TJ deformations [58]. These deformations enjoy remarkable properties and have been studied from various perspectives. For instance, the spectrum on the cylinder has been derived in [56][57][58][59] and the modular properties of the torus partition functions have been studied in [60][61][62][63]. The deformed theories have also been shown to be related to Nambu-Goto strings and QFTs coupled to gravity [64][65][66][67][68][69][70][71]. For additional references see the reviews [72,73]. The holographic interpretation of these irrelevant deformations include the double-trace version which changes the boundary conditions but does not change the local bulk geometry [74][75][76], as well as a single-trace version which also changes the local geometry [59,77,78]. In particular, the single-trace TT deformation is dual to a geometry that interpolates between AdS 3 and a linear dilaton background that is relevant in the study of Little String Theory [77,79,80]. While the single-trace JT /TJ deformation is holographically dual to the warped AdS 3 geometry that is relevant in the study of extremal black holes [59,77,78,81].
In this paper we investigate the relationship between marginal deformations of string theory on AdS 3 , deformations of the AdS 3 geometry, and the corresponding deformations of the dual CFT. TsT transformations play a central role in this story, which is analogous to the case of non-commutative, dipole, and β deformations, but with purely NS-NS backgrounds in the bulk. In particular, we find that TsT transformations of NS-NS backgrounds parametrized by the dimensionless constant λ can be described by the following deformation of the bosonic part of the worldsheet action S WS , (2) . (1.1) In eq. (1.1), k = ℓ 2 /ℓ 2 s where ℓ and ℓ s denote the AdS and string scales, respectively, the subscripts (1) and (2) correspond to the two target space coordinates X 1 and X2 along which the TsT transformation is performed, while j (1) and j (2) denote the worldsheet Noether one-forms generating shifts along these coordinates.
We conjecture that string theory on TsT-transformed AdS 3 × S 3 × M 4 backgrounds with NS-NS flux (M 4 is a compact four-dimensional manifold) whose worldsheet action is described by eq. (1.1), is holographically dual to single-trace deformation of the dual CFT. In particular, if the undeformed theory is described by a symmetric product, then the action S QFT of the deformed theory is defined by where µ is the deformation parameter and p is the total number of copies in the product CFT. In eq. (1.2), J i (1) and J i (2) denote the Noether one-forms generating symmetries dual to translations along the X 1 and X2 directions in the i th copy of the symmetric product. In order to obtain locally non-AdS 3 backgrounds at least one of the TsT coordinates must belong to the AdS 3 part of the AdS 3 × S 3 × M 4 background. Correspondingly, the holographic dual is expected to be described by an irrelevant deformation of the dual CFT. In particular, when both of the TsT coordinates are chosen from AdS 3 , we will show that the irrelevant deformation corresponds to the single-trace TT deformation. This corresponds to a finite temperature generalization of the proposal by [82], the latter of which is supported by matching the string theory spectrum on the Ramond vacuum. On the other hand, when only one of the U (1) directions belongs to the AdS 3 factor, the holographic dual is described by the single-trace JT /TJ deformation [81], a conjecture supported by matching the string theory spectrum at both zero [78,83] and finite temperature [81], and reproducing the thermodynamics of the bulk from the boundary side of the correspondence [81]. Finally, if both of the TsT directions are taken along U (1) isometries of the internal manifold then the deformation of the dual CFT is a marginal JJ deformation.
In this paper we provide further evidence for the correspondence between TsT transformations of AdS 3 backgrounds (1.1) and irrelevant deformations of the dual CFT (1.2). Applying a TsT transformation to string theory on BTZ ×S 3 × M 4 backgrounds we obtain finite temperature spacetimes that are dual to thermal states in single-trace TT -deformed CFTs. In particular, we show that (i ) the (long) string spectrum on the TsT-transformed background matches the spectrum of TT -deformed CFTs.
(ii ) The thermodynamics of the resulting black string solutions match the thermodynamics of TT deformations. In particular, we reproduce the Bekenstein-Hawking entropy of black strings from the density of states of the dual field theory.
(iii ) There is a critical value of the deformation parameter λ c such that, for λ ≤ λ c , the ground state of the TT -deformed CFT has a real energy and corresponds to a smooth solution in the bulk. On the other hand, for λ > λ c , the ground state energy becomes complex and, correspondingly, the bulk solution becomes complex as well.
(iv ) The TsT background describes other known black string solutions in special cases.
In particular, the Horne-Horowitz black string [53] corresponds to the non-rotating TsT background with λ = 1/2 after a gauge transformation. Similarly, the black string of [77] corresponds to the non-rotating TsT black string with λ = 1/2 after a shift of the dilaton and a gauge transformation.
The paper is organized as follows. In Section 2 we relate TsT transformations to marginal deformations of the worldsheet action, derive general expressions for the deformed spectrum, and discuss the relationship between TsT transformations and solvable irrelevant deformations. In Section 3 we perform a TsT transformation on the BTZ×S 3 × M 4 background, match the string theory spectrum to that of TT -deformed CFTs, and discuss the relationship between the TsT transformation and current-current deformations of the worldsheet. We compute the charges, entropy, and thermodynamic potentials of the TsT black string in Section 4. Therein, we also show that these quantities match the corresponding expressions derived in single-trace TT -deformed CFTs. In Section 5 we discuss the phase space of TsT-transformed solutions including the ground state, the TT flow of states, and the superluminal λ < 0 sign of the deformation. In Section 6 we describe the relationship between the TsT black string and the Horne-Horowitz and Giveon-Itzhaki-Kutasov backgrounds. In Appendix A we review the covariant formulation of gravitational charges while in Appendix B we consider alternative parametrizations of the black string.

TsT and irrelevant deformations
In this section we consider generic TsT transformations of IIB string theory with NS-NS background fields. We show that these TsT transformations are equivalent to instantaneous deformations of the worldsheet action by the antisymmetric product of two Noether currents. These deformations mirror the definition of the TT and JT /TJ deformations of 2d CFTs and we propose a precise correspondence for TsT transformations of AdS 3 ×S 3 ×M 4 backgrounds. Finally, provided that certain assumptions hold, we derive general expressions for the spectrum of strings winding on TsT-transformed backgrounds.

TsT as a worldsheet deformation
Let us begin by considering the bosonic sector of IIB string theory on an arbitrary background supported by NS-NS flux. The worldsheet action is given bỹ where ℓ s is the string scale, z = τ + σ andz = τ − σ are the worldsheet coordinates with ∂ ≡ ∂ z and∂ ≡ ∂z,M µν =G µν +B µν , and the worldsheet metric η ab and Levi-Civita symbol ǫ ab satisfy We assume that the background fieldsG µν ,B µν , and the dilatonΦ are independent of the target space coordinatesX n with n = 1,2 such that the action is invariant under two U (1) symmetries generated by 1 In particular, the Noether currents associated with the ξ (n) Killing vectors read These currents are generically not chiral and their conservation law corresponds to the equations of motion of theX n coordinates. Alternative expressions for the Noether currents that are useful in what follows includẽ where the bold notation serves to distinguish the vectorj (n) from the one-formj (n) representations of the current. 2 We now perform a TsT transformation of the background fields that consists of Tduality alongX 1 , followed by a shiftX2 = X2−2λX 1 withX 1 = X 1 , and another T-duality along the X 1 coordinate. The TsT transformation is characterized by a dimensionless parameterλ. After the TsT transformation, the transformed metric G µν , B µν field, and dilaton Φ can be obtained from the original fields via (see for example [20,84]) where Γ is defined by Note that when thej (1) andj (2) currents are both chiral or both antichiral,M has two rows or two columns of zeros. As a result, ΓM = 0 and the TsT transformation is trivial.
In this paper, we will only consider TsT transformations that lead to different worldsheet theories.
After the TsT transformation the worldsheet action is given by where X µ denote the target space coordinates after TsT. From eq. (2.7), it is not difficult to see that the background field M µν obeys the differential equation We note that the Noether currents j (n) of the deformed theory, which are given by eq. (2.4) withM µν → M µν andX µ → X µ , also satisfy Thus, eqs. (2.10) and (2.11) imply that the worldsheet action satisfies the following differential equation We conclude that TsT transformations correspond to instantaneous deformations of the worldsheet action by the antisymmetric product of two Noether currents. This is a universal result that applies to the action of strings on any TsT-transformed background supported by NS-NS flux, generalizing the observations made in [81]. 3 Note that the TsT-origin of eq. (2.12) guarantees that the deformation is exactly marginal to leading order in α ′ ∝ ℓ 2 s . This suggests that the analysis of ref. [46], which showed that chiral-antichiral currentcurrent deformations are exactly marginal, can be extended to the antisymmetric product of two Noether currents. Finally, we note that worldsheet deformation (2.12) mirrors the definition of the TT and JT /TJ deformations of 2d CFTs which are also instantaneous deformations by the antisymmetric product of two conserved currents [56][57][58]. The connection between TsT deformations and irrelevant deformations is discussed in detail in Section 2.3.

The worldsheet spectrum
We now derive general expressions for the spectrum of strings winding on TsT-transformed backgrounds that are valid provided that certain conditions, to be described momentarily, hold. An important feature of TsT transformations is that the equations of motion and Virasoro constraints of the deformed theory (2.9) can be obtained from those of the undeformed one by the following change of coordinates [86][87][88] where we sum over repeated indices and labels. Note that eq. (2.13) implies that theX n coordinates satisfy nonlocal boundary conditions, which distinguish the latter from the undeformedX n coordinates introduced earlier. In particular, since the background fields are assumed to be independent of theX n coordinates, we havê 14) The twisted boundary conditions satisfied by theX n coordinates can be deduced by inte- where the γ (n) parameters are given by In eq. (2.16) p (n) denotes the physical momenta of the string along the X n coordinate, i.e. the momenta of the string after the TsT transformation, which is defined by Note that the total derivative term in eq. (2.16) vanishes unless X n is compact, in which case the string can wind |w (n) | times along the X n coordinate For completeness, we assume arbitrary winding parameters w (n) along the TsT coordinates. The twisted boundary conditions (2.15) relate the spectrum of strings on the TsTtransformed background to the spectrum of strings before the TsT transformation. The spectrum is sensitive to the directions along which the strings are allowed to wind, and these may not coincide with the TsT coordinates. For example, this is the case in the holographic description of single-trace JT /TJ deformations where strings wind only along one of the TsT coordinates. For this reason, we will assume that the background features additional U (1) isometries. In order to derive general expressions for the worldsheet spectrum it is also useful to introduce two kinds of indices, X m and Xm, that are distinguished by the chiral properties of the Noether currents generating translations along these coordinates. Therefore, we will assume that the backgroundĜ µν ,B µν , andΦ fields are independent of the following coordinatesX The boundary conditions satisfied by these coordinates are parametrized bŷ 20) while the rest of the target space coordinatesX i , where i ∈ {N + 1, . . . , 10}, obey trivial boundary conditions. In particular, we will continue to use m = 1 andm =2 to denote the TsT directions such that γ (1) and γ (2) are momentum-dependent and given by eq. (2.16).
In contrast, the other γ (m) and γ (m) parameters satisfy where |w (m) | and |w (m) | are integers that count the number of times the string winds along the corresponding X m and Xm directions. We can implement the boundary conditions (2.20) by a shift of coordinates, which motivates the introduction of m andm indices. The constraint (2.23) is responsible for turning the global isometries of the background into infinite-dimensional sets of local symmetries on the worldsheet theory. Note that eq. (2.26) implies that the Noether currents associated with TsT transformations are chiral and antichiral up to topological terms. As a result, the instantaneous deformation (2.12) is closely related to the current-current deformations considered in ref. [46]. Finally, we note that theĥ (m) andĥ (m) currents shift by a constant under the change of coordinates (2.22), suggesting that the latter can be interpreted as a spectral flow transformation, an observation that can be verified explicitly for AdS 3 × S 3 × M 4 backgrounds. The shift of coordinates introduced in eq. (2.22) relates the components of the stress tensor T ab = 1 √ −γ δS δγ ab before and after the TsT transformation. In terms of the hatted variables, i.e. after TsT, the components of the stress tensor read Using the change of coordinates (2.22) we then find The zero modes of the chiral currentsĥ (m) andĥ (m) are related to the physical momenta of the string on the TsT-transformed background M µν . In order to see this we note that the nonlocal change of coordinates (2.13) implies that [88] where j (m) and j (m) are the Noether currents of the deformed theory. Thus, using the boundary conditions (2.20) and the definition of the momenta given in eq. (2.17) we find Consequently, the zero modes of the stress tensorL 0 ≡ − 1 2π then the worldsheet spectrum simplifies tô The constraint (2.35) may seem to be too restrictive but it turns out to be satisfied by all locally AdS 3 × S 3 × M 4 backgrounds provided that the TsT and winding directions are chosen appropriately. In particular, this background is relevant in the holographic description of the TT and JT /TJ deformations as we describe next.

Marginal vs irrelevant deformations
Let us now consider string theory on where the AdS 3 part of the background may correspond to the NS vacuum, the massless BTZ black hole (Ramond vacuum), a general BTZ black hole or any other orbifold of AdS 3 that preserves two U (1) isometries. The worldsheet action on these backgrounds is described by a WZW model at level k = ℓ 2 /ℓ 2 s where ℓ and ℓ s denote the AdS and string scales, respectively. There is evidence that the long string sector of string theory on AdS 3 ×S 3 ×M 4 supported by NS-NS flux is dual to a symmetric product orbifold [36,43,89] where M 6k is a CFT with central charge c M = 6k that depends on the choice of the compact M 4 manifold, k is the number of NS5 branes, and p is the number of long strings generating the background. The structure of the dual CFT allows us to define both singletrace and double-trace bilinear deformations of the action. In particular, the single-trace TT and JT /TJ deformations of the dual CFT are instantaneous deformations defined by where µ denotes the deformation parameter in the field theory while J i (1) and J i (2) are Noether one-forms on the i th copy of M 6k that we define shortly. While the full holographic description of string theory on AdS 3 is not yet known, we will assume that single-trace deformations analogous to eq. (2.40) can still be defined (see [83] for a related discussion).
There is a striking similarity between the marginal deformation of the worldsheet action (2.12) and the irrelevant deformation of the dual CFT (2.40). Letting j (m) denote the Noether one-forms generating translations along the TsT directions on the worldsheet, and J (m) the corresponding one-forms in the dual field theory, we propose that string theories defined by the worldsheet action (2.12) which describes NS-NS background fields obtained by a TsT transformation of AdS 3 × S 3 × M 4 are holographically dual to single-trace deformations of a CFT of the form (2.40).
To obtain a non-AdS 3 geometry, at least one of the U (1) directions characterizing the TsT transformation must belong to the AdS 3 factor. Consequently, the holographic dual is expected to be described by an irrelevant deformation of a CFT. In particular, when both of the TsT directions are taken along translational U (1) isometries of AdS 3 , we will show that the irrelevant deformation corresponds to the single-trace TT deformation. This observation is supported by matching the string theory spectrum on the Ramond vacuum [82] and -as we will show in this paper -matching the string theory spectrum on finite-temperature backgrounds and reproducing the thermodynamics of TsT-transformed black strings from TT at finite temperature. On the other hand, when only one of the U (1) directions belongs to the AdS 3 factor, the holographic dual is described by the single-trace JT /TJ deformation [81], a conjecture supported by matching the string theory spectrum at both zero [78,83] and finite temperature [81], and reproducing the thermodynamics of the bulk from the boundary side of the correspondence [81]. 6 We now describe in more detail the proposed duality between TsT transformations of string theory on AdS 3 × S 3 × M 4 backgrounds and irrelevant deformations of 2d CFTs.

TT deformations
The single-trace TT deformation of the CFT dual to string theory on AdS 3 × S 3 × M 4 is defined by eq. (2.40) where µ → µ TT is a constant with dimensions of length squared and 6 If both of the TsT directions are taken along translational U (1) isometries of the internal manifold then the deformation of the dual CFT is a marginal JJ deformation. The spectrum of the deformed theory is partially captured by eqs. (2.36) and (2.37), with additional equations for the deformed U (1) charges that can be obtained from spectral flow of the symmetry algebra, as discussed in more detail in ref. [81].
the Noether one-forms J i (m) are given in terms of the stress tensor T i µν in the i th copy of the symmetric product by In eq. (2.41) x = ϕ + t andx = ϕ − t are the left and right-moving coordinates of the dual field theory, which also correspond to the lightcone coordinates of AdS 3 . In particular, the Noether currents associated with the J i (1) and J i (2) one-forms generate translations along the x andx coordinates, respectively. The single-trace TT deformation is dual to a TsT transformation of string theory on AdS 3 × S 3 × M 4 provided that the TsT coordinates X 1 and X2 correspond to translational U (1) isometries of AdS 3 , namely This TsT transformation is equivalent to the instantaneous deformation of the worldsheet action given in eq. (2.12) where the Noether currents corresponding to the j (1) and j (2) one-forms generate shifts along the target space coordinates X 1 and X2. As a result, there is a one-to-one correspondence between the j (n) and J i (n) one-forms featured in the worldsheet (2.12) and field theory deformations (2.40). Comparison of these equations suggests that the string and field theory deformation parametersλ and µ TT are related by where ℓ is the AdS scale of AdS and we have introduced another dimensionless parameter λ = kλ that is useful in what follows. Note that the AdS scale determines the size of the cylinder at the boundary and that ℓ and µ TT are the only dimensionful parameters in the dual field theory. Evidence for the correspondence between TsT transformations and TT deformations follows from matching their spectrum and thermodynamics. The spectrum on the TsTtransformed background is considered in detail in Section 3.3 while the thermodynamics of TT and TsT are derived in Sections 4.1 and 4.2. In this section we would like to provide a quick derivation of the spectrum using the general formulae given in eqs. (2.36) and (2.37). In order to do this it is necessary to choose, in addition to the TsT directions, the coordinates along which strings are allowed to wind. We will consider strings winding along the spatial circle of AdS 3 , the latter of which are associated to spectrally-flowed representations of SL(2, R). In this case the TsT and winding directions coincide such that the winding parameters featured in eqs. (2.36) and (2.37) satisfy Let us parametrize the dimensionless momenta along the TsT coordinates in terms of the dimensionful left and right-moving energies where E is the energy and J the angular momentum, Using eqs. (2.36) and (2.37), the zero modes of the worldsheet stress tensor are given bŷ Imposing the Virasoro constraints before and after the deformation allows us to express the deformed left and right-moving energies in terms of the undeformed ones such that Eq. (2.47) matches the spectrum of TT -deformed CFTs on a cylinder of size 2πℓ provided that w = −1 and λ satisfies eq. (2.43) (see Section 3.3 for more details). In particular, note that this derivation is valid for the TsT transformation of any orbifold of AdS 3 that preserves its translational U (1) isometries, including the massless BTZ black hole (Ramond vacuum) as well as generic BTZ black holes.

JT /TJ deformations
In analogy with the TT deformation, the single-trace JT deformation is defined by eq. (2.40) where µ → µ JT is a constant with dimensions of length and the Noether one-forms J i are given in terms of a U (1) current J i µ and the stress tensor T i µν by A similar definition exists for single-trace TJ deformations and, without loss of generality, we focus only on the JT case. In particular, the Noether currents corresponding to the J i (1) and J i (2) one-forms generate internal U (1) transformations and translations along thē x coordinate. As a result, the single-trace JT deformation is dual to a TsT transformation where the TsT coordinates X 1 and X2 are identified with translational U (1) isometries of the internal manifold and the AdS 3 factor, for example 7 For this choice of TsT coordinates, the Noether one-forms j (1) and j (2) in eq. (2.12) are also in one-to-one correspondence with J i (1) and J i (2) in eq. (2.40). This follows from the fact that the corresponding Noether currents generate U (1) transformations and translations along the X2 coordinate in the worldsheet and the field theory. The similarities between the worldsheet and field theory deformations suggest once again that the string and field theory parameters λ and µ JT are related by The correspondence between TsT transformations and single-trace JT deformations was proposed in [81] after matching the spectrum and thermodynamics in the bulk and boundary theories. In particular, one finstds that the string and field theory deformation parameters are indeed related by eq. (2.50). We conclude this section by showing that the general expressions for the spectrum of strings winding on the TsT transformed background given in eqs. (2.36) and (2.37) reproduce the spectrum of JT -deformed CFTs. As before, we would like to consider strings that wind along the spatial circle of AdS 3 , the latter of which is identified with the spatial circle in the dual field theory. This means that there is winding along one of the TsT directions (X2) and an additional U (1) direction denoted by X 3 ∈ AdS 3 . The winding parameters featured in eqs. (2.36) and (2.37) satisfy An important ingredient in the derivation of the spectrum is an appropriate identification of the deformed and undeformed U (1) charges. The latter were identified in [81] with the zero modes of the chiral currents generating U (1) transformations before and after the TsT transformation. This choice justifies the following parametrization of the momenta along the X 1 , X2 and X 3 coordinates where q denotes the U (1) charge before the deformation. Eqs. (2.36) and (2.37) then yield the following zero modes of the stress tensor in the deformed theorŷ Finally, imposing the Virasoro constraints before and after the TsT transformation we find which matches the spectrum of JT -deformed CFTs on a cylinder of size 2πℓ when w = −1 and λ satisfies eq. (2.50).

TsT black strings and TT deformations
We now consider the TsT transformation of the BTZ black hole that is relevant in the holographic description of TT -deformed CFTs at finite temperature. We derive the spectrum of strings winding on the TsT-transformed background and describe in more detail the holographic dictionary relating the worldsheet spectrum to that of TT -deformed CFTs. We also show that this background is closely related, by a change of coordinates and a gauge transformation, to another background that can be obtained from a chiral current-current deformation of the worldsheet action.

The BTZ black hole
Let us begin by describing string theory on a BTZ black hole supported by NS-NS flux.
We are interested in the six-dimensional part of the AdS 3 × S 3 × M 4 background such that the metric, B-field, and dilaton are given by 8 where k = ℓ 2 /ℓ 2 s with ℓ s the string length and ℓ the radius of AdS, while dΩ 2 3 is the metric of the 3-sphere that can be written as The lightcone coordinates u = ϕ + t/ℓ and v = ϕ − t/ℓ of the AdS 3 factor satisfy which implies that the dual CFT lives in a cylinder of size 2πℓ. On the other hand, the coordinates of the 3-sphere satisfy The p and k variables parametrizing the dilaton in eq. (3.1) correspond to the electric and magnetic charges of the black string (see Appendix A for our conventions), the latter of which count the number of NS1 and NS5 branes generating the background On the other hand, the T u and T v parameters in eq. (3.1) are proportional to the dimensionless left and right-moving temperatures of the black string (after normalization by a factor of 1/π). In particular, the background left and right-moving energies can be computed using the covariant formalism reviewed in Appendix A and are given by where c is the central charge of the dual CFT which satisfies The metric in eq. (3.1) features a horizon at r h = 2T u T v with entropy and it is not difficult to verify that it satisfies the first law of thermodynamics.
The bosonic part of worldsheet action (2.1) reads where k will be assumed to be large (k ≫ 1) andS Ω denotes the contribution of the 3sphere. The action (3.9) features Noether currents generating translations along the u and v coordinates, the latter of which correspond to the two U (1) isometries of the AdS 3 part of the background. Using the general expression given in eq. (2.4) we find wherej ξ denotes the Noether current associated with the Killing vector ξ,j 1 andj 1 are chiral currents in the WZW formulation of the theory that are given by 9 and θ (f ) is a topological term that satisfies The topological θ (f ) terms contribute only total derivative terms to the corresponding Noether charges, i.e. to the momenta of the string along the u and v coordinates. Nevertheless, these terms are physically meaningful as they contribute in topologically nontrivial sectors of the theory such as the winding string sector. We have written eq. (3.10) in a way that makes manifest the fact that the Noether currentsj ∂u andj ∂v are chiral up to topological terms that are conserved off-shell. As discussed in Section 2, this is a consequence of the fact that the black string background satisfies the constraint (2.23) where m = 1 andm =2 correspond to theX 1 = u and X2 = v coordinates. In particular, when the temperatures of the BTZ black hole vanish, the Noether currents become chiral. This observation will help us relate TsT transformations to the chiral current-current deformations considered in [77].

The TsT black string
We now perform a TsT transformation along theX 1 = u andX2 = v coordinates of the BTZ black hole by T-dualizing along u, shifting v → v − 2λ k u, and T-dualizing along u once more. Note that we have parametrized the TsT transformation by the dimensionless parameter λ = kλ so that the deformed background fields do not depend explicitly on k. 9 In the WZW formulation of the theoryj 1 andj 1 correspond to the spacelike chiral currents where the SL(2, R) generators τ a can be chosen as τ 1 = 1 2 σ 1 , τ 2 = − i 2 σ 2 , τ 3 = 1 2 σ 3 and the SL(2, R) group element satisfies g = exp 2Tuuτ 1 exp log 1 2 The deformed string-frame metric, B-field, and dilaton are given by where φ 0 is a constant that may depend on the temperatures and Φ TsT is the value of the dilaton determined by Buscher's rule [90,91] where G µν andG µν denote the deformed and undeformed string-frame metrics. In analogy with the analysis of the warped BTZ black holes dual to JT /TJ deformations at finite temperature [81], we assume that the ϕ = 1 This identification of coordinates guarantees that the dual field theory is defined on the right cylinder for which the spectrum and thermodynamics of the TsT black string match the corresponding quantities in TT -deformed CFTs. It is interesting to note that it is possible to rescale λ away from eq. (3.13). This can be achieved by rescaling the coordinates and temperatures such that 16) while, at the same time, rescaling the size of the spatial circle ϕ ∼ ϕ+2π √ λ. Alternatively, these equations imply that backgrounds with different values of λ can be interpreted as describing the same solutions with rescaled spatial circles (3.15).
The supergravity equations of motion require the dilaton to satisfy Buscher's rule up to a constant, and we have used this freedom to parametrize Φ in a convenient way by including an additional factor of (1 − 4λ 2 T 2 u T 2 v )e −2φ 0 . In particular, the dilaton is real provided that which, as we will see in Section 4.2 reproduces the bound on the product of temperatures in TT -deformed CFTs. Interestingly, when φ 0 = 0 the dilaton in eq. (3.13) satisfies a similar equation to that dictated by Buscher's rule, except that the deformed metric G µν is replaced by another closely related background described in Section 3.4. We also note that when the temperatures and φ 0 vanish, the metric, B-field, and dilaton in eq. (3.13) reduce to the background fields described in ref. [77] which were obtained by a marginal current-current deformation of the worldsheet action (this was also observed in [92]).
The worldsheet action on the TsT-transformed background can be written as where S 0 denotes the original action given in eq. (3.9) while ∆S λ is the contribution from the TsT transformation which reads After the deformation, the Noether currents generating translations along u and v are respectively given by 20) wherej 1 andj 1 were defined in eq. (3.11). The fate of the chirally conserved currents of the undeformed theory is no longer manifest in these expressions. Nevertheless, the action (3.18) still features chiral currents but these are not associated with translations along the u and v coordinates. Instead, the chiral currents are associated with λ-dependent translations generated by such that the corresponding Noether currents are chiral up to topological terms Finally, we note that the deformed worldsheet action (3.18) satisfies where j ∂u and j ∂v are the Noether one-forms associated with the Noether currents j ∂u and j ∂v via eq. (2.6). This result is not surprising since we have shown that generic TsT transformations are equivalent to marginal deformations of the worldsheet action by the antisymmetric product of two Noether currents (2.12). In particular, since j ∂u and j ∂v generate shifts along the u and v coordinates -the lightcone coordinates of the dual field theory -we see that there is a one-to-one correspondence between the currents in the worldsheet deformation (3.25) and the currents in the TT deformation of the dual field theory (2.40).

The TT spectrum from the worldsheet
We now derive the spectrum of strings winding on the TsT-transformed background (3.13).
In particular, we describe in more detail the holographic dictionary relating the worldsheet spectrum to that of TT -deformed CFTs. Readers satisfied with the general derivation of the spectrum given in Section 2.2 may want to skip this section. A crucial feature of TsT transformations is that the equations of motion and Virasoro constraints of the deformed theory can be obtained from those of the undeformed theory by a nonlocal change of coordinates [86][87][88]. For the TsT transformation considered in this section this change of coordinates reads 10 whereû andv are the coordinates of the undeformed theory. The change of coordinates (3.26) leads to nonlocal boundary conditions for theû andv coordinateŝ where the γ (u) and γ (v) parameters satisfy and E L , E R denote the left and right-moving energies of the deformed theory, the latter of which are defined by In deriving eq. (3.28) we have assumed that the string winds along the compact ϕ = 1 2 (u+v) coordinate of the deformed background such that Classically, |w| counts the number of times the string winds along ϕ, while quantummechanically, w labels the spectrally-flowed representations of SL(2, R) in the undeformed theory. The boundary conditions (3.27) induce spectral flow of the underlying SL(2, R) L × SL(2, R) R symmetry algebra of the undeformed theory. In order to see this we note that the twisted boundary conditions (3.27) can be implemented viâ

31)
10 Recall that the Noether currents j ∂u and j ∂v of the deformed and undeformed theories are mapped into each other via eq. (3.26) (see eq. (2.30)). Interestingly, the same is true for the chirally conserved currents However, note that while the chiral currents on the left-hand side are related to translations along u and v, the chiral currents on the right-hand side are related to translations along u + 2λT 2 u v and v + 2λT 2 v u.
whereũ andṽ denote the coordinates of the undeformed theory satisfying trivial boundary conditions. Under spectral flow the chiral WZW currents (3.11) shift according tõ 32) such that, in terms of the energies and the winding, their zero modes satisfy Although spectral flow does not change the underlying symmetry algebra, it shifts the components of the stress-energy tensor bŷ Consequently, we find that the zero modesL T of the stress tensor are given bŷ whereL 0 andL 0 denote the zero modes before the deformation. The latter satisfỹ where j is the SL(2, R) weight of the state |ψ , (N,N ) denote the left and right levels, and (∆,∆) are the contributions from the S 3 × M 4 [30]. Not surprisingly, eq. (3.36) matches the general derivation of the zero modesL 0 andL 0 given in eq. (2.46). Imposing the Virasoro constraints before and after the deformation then yields the deformed spectrum reported in eq. (2.47). In our conventions w < 0 guarantees that the spectrum of the undeformed theory is bounded from below for both the discrete and principal continuous representations of SL(2, R). Furthermore, the twisted sector of the symmetric product orbifold conjectured to be dual to the long string sector of the undeformed theory is described by states with w < −1. This motivates us to consider the states with w = −1. Then, with the following choice of parameters we recover the spectrum of TT -deformed CFTs on a cylinder of size 2πR, where µ is the deformation parameter and E = E L + E R and J = E L − E R denote the energy and the angular momentum. Note that the identification of the string and field theory deformation parameters in eq. (3.38) matches the expression obtained by comparing the instantaneous deformations of the worldsheet and field theory actions (2.43).

TsT and current-current deformations
We now show that the TsT black string considered in the previous section is closely related to another background by a change of coordinates and a gauge transformation. This background can be obtained from a gauged WZW model that we show is equivalent to an exactly marginal current-current deformation of the worldsheet action driven by the product of two chiral currentsj 1j1 . Let us start by deriving the background resulting in a current-current "symmetric" deformation driven byj 1j1 . Various approaches exist to determine the background fields of such a deformation. A first technique was developed in [47], where it was conjectured that O(d, d) transformations of the background fields of any WZW model correspond to marginal deformations of the WZW theory by an appropriate combination of left and right moving currents belonging to the Cartan subalgebra. Another way of implementing the symmetric deformation has been put forward in [48,93], suggesting that the whole deformation line can be realized as an axially and vectorially gauged (SL(2, R) × U (1)) /U (1) WZW model, in which the embedding of the dividing group has a component in both factors. We will take the latter approach.
We write a general element g ∈ SL(2, R) × U (1) with coordinates (u, v, r) for the SL(2, R) part (see footnote 9) and an additional coordinate X for the U (1) part. The gauged WZW action [94] in light-cone gauge can be written as where S 0 is the undeformed WZW action (3.9) while A ± andÃ ± are the gauge fields associated with a subgroup U (1) ⊂ SL(2, R) × U (1). The latter are expressed as where t L and t R are matrix representations of the Lie algebra of U (1) embedded in a representation of SL(2, R) × U (1) while a ± andã ± depend on the worldsheet coordinates. We perform a so-called axial gauging adapted to the parametrization in footnote 9, which corresponds to taking where κ is a real parameter. The action (3.40) is invariant under the action of left and right local symmetries g → e αt L ge −αt R , where α is arbitrary function on the worldsheet. Using this local symmetry, we can choose the gauge function α such that u T u + v T v = 0. After integrating out the gauge fields, and making the above gauge choice, we can read the background corresponding to the gauged model. For this, we perform a coordinate transformation by setting The resulting background is that of the symmetric deformation with κ 2 > 1. To reach the line κ 2 < 1, one would need to perform a vector gauging with t R = t L [54]. In both cases, we can further redefine such that the background fields read as Note that the dilaton is determined by conformal invariance from the one-loop β-function [48,91,95,96] and is given by [54] The background (3.44) is closely related to the TsT black string (3.13) considered in the previous section, and it can be obtained from the latter by the change of coordinates followed by the gauge transformation The value of the dilaton in (3.44) matches precisely the one given in eq. (3.13) with φ 0 = 0, which further justifies our parameterization of the dilaton in (3.13).
The worldsheet action simplifies significantly on the background (3.44) and is given by where S 0 denotes the action before the deformation (3.9) with the u ′ , v ′ . As described earlier, the deformed action S ′ λ can be interpreted as the result of an instantaneous deformation of the worldsheet by the product of two chiral currents where j 1 λ andj 1 λ have been appropriately normalized to satisfy In particular, note that the infinitesimal deformation of the action is driven byj 1j1 , wherẽ j 1 andj 1 are the chiral currents of the undeformed theory associated with translations along the u ′ and v ′ coordinates. Marginal deformations require the presence of (1, 1) operators and integrability ensures that the deformed theory, a finite deformation away from the original one, still retains such currents, namely j 1 λ andj 1 λ . The latter are no longer associated with translations along the u and v coordinates, however. Instead, up to topological terms, they correspond to Noether currents associated with the following Killing vectors When the temperatures vanish, the TsT black string (3.13) and the background (3.44) coincide. This is compatible with the fact that, in this case, the differential equations for the worldsheet action (3.25) and (3.49) agree with one another As a result, for the (zero temperature) Ramond vacuum the proposed holographic description of single-trace TT deformations is consistent with the analysis of [77], which considered current-current deformations of the worldsheet action driven byj 1j1 . A similar observation holds for the holographic interpretation of the JT /TJ deformations, the latter of which reduce to current-current deformations of the worldsheet when the temperatures vanish [81].
For generic values of the temperatures, the TsT black string and the background (3.44) differ on their global properties. This follows from the fact that the spatial circle of the original BTZ background (3.3) maps to different circles in the TsT-transformed solution (3.13) and in (3.44). In particular, the spatial circle determines along which direction the string is allowed to wind, which plays an important role in the derivation of the spectrum. Relatedly, while the gauge transformation (3.47) contributes only total derivative terms to the Noether charges, the non-trivial boundary conditions satisfied by winding strings render some of these contributions physical. As a result, the gauge transformation (3.47) induces shifts in the momenta of winding strings that affect the expressions for the spectrum. We also note that the choice of spatial circle affects the gravitational charges and the thermodynamics of the TsT-transformed background.
To summarize, the TsT black string (3.13) and the background (3.44) obtained from marginal deformations of the WZW model are locally equivalent. This connection is useful to understand the worldsheet CFT and the integrability of the deformation, perhaps even at the quantum level. At finite temperatures the two backgrounds feature different global properties that affect the winding string spectrum as well as their holographic interpretation. As we will show later, the global properties of (3.13) play an important role in matching the spectrum and thermodynamics of the black string to the corresponding quantities in TT -deformed CFTs. A similar phenomenon is also found in the single-trace JT /TJ deformations studied in [81].

Thermodynamics of TsT from TT
In this section we study the thermodynamics of TT -deformed CFTs and TsT-transformed backgrounds. We begin by deriving the entropy and thermal expectation values of the energies of TT -deformed CFTs in the canonical ensemble. We then turn to the TsT black string and compute its gravitational charges, entropy, and determine the thermodynamic potentials that are consistent with the first law of thermodynamics. Finally, we show that the thermodynamics of the TsT black string with fixed electric charge match the thermodynamics of single-trace TT -deformed CFTs.

Thermodynamics of TT -deformed CFTs
Let us begin by considering the thermodynamics of the TT -deformed symmetric product (M 6k ) p /S p . The action of the deformed theory is defined by are Noether one-forms generating translations along the chiral coordinates in the i th copy of the field theory. As a result, the left and right-moving energies of the deformed theory are given by where E i L (µ) and E i R (µ) denote the energies in the i th copy of the TT -deformed CFT, the latter of which can be conveniently written as [56,57] The deformed energies of the ground state are obtained by setting each of the undeformed energies in eq. (4.3) to their ground state values, RE i,vac In particular, the energies of the vacuum before the deformation are given by RE vac L (0) = RE vac R (0) = −pc M /24 = −c/24. When µ > 0, the spectrum of the deformed theory is well behaved provided that Otherwise, the energy of the ground state, and possibly a finite number of excited states, becomes complex. At the critical value µ c = 3R 2 /c M , the deformed ground state energies are real and given by RE vac L (µ c ) = RE vac R (µ c ) = −c/12. Using the holographic dictionary (3.38), we find that the critical value of the deformation parameter in the string theory formulation of the TT deformation is given by Note that since the string winding parameter described in Section 3.3 satisfies |w| > 0, the requirement λ ≤ λ c leads to a well-defined worldsheet spectrum in all of the winding sectors, cf. eq. (2.47). From the expressions for the spectrum in eq. (4.3) we see that there is no level crossing between states in the deformed theory, implying that the density of states does not change after the TT deformation. As a result, the Cardy entropy of the undeformed M 6k CFT can be written in terms of the deformed spectrum as The above formula is to be understood as the entropy of the TT -deformed M 6k CFT in the microcanonical ensemble. The total entropy of the TT -deformed CFT is obtained by summing over the contributions from each copy in the symmetric product such that It will prove convenient to derive the entropy in the canonical ensemble as well. Using the first law of thermodynamics, the temperatures T L ≡ (∂S i TT /∂E i L ) −1 and T R ≡ (∂S i TT /∂E i R ) −1 conjugate to the left and right-moving energies can be written as .

(4.11)
Note that the energies featured in the right-hand side of eqs. (4.10) and (4.11) correspond to thermal expectation values in TT -deformed CFTs at temperatures T L and T R . These equations also imply that, in the high E i L and E i R limits, the product of the left and right-moving temperatures is bounded from above by where the bound is saturated only when E i L , E i R → ∞. Using eqs. (4.10) and (4.11) we find that, in the canonical ensemble, the entropy of the i th copy of the TT -deformed product CFT satisfies where γ is defined by (4.14) The bound on the product of the temperatures (4.12) implies that 0 ≤ γ ≤ 1 such that the entropy is real when µ ≥ 0. Finally, the total entropy of single-trace TT -deformed CFTs in the canonical ensemble given by

Thermodynamics of TsT black strings
The TsT black string is a solution to the supergravity equations of motion for any constant values of T u , T v , φ 0 , and the deformation parameter λ. We are interested in computing the gravitational charges on the space of solutions parametrized by these variables. Using the covariant formulation of charges reviewed in Appendix A, we find that the left and right-moving energies associated with the Killing vectors ∂ u and −∂ v are integrable only for fixed values of λ. The finite expressions for these charges are given by where the factor of e 2φ 0 /(1−4λ 2 T 2 u T 2 v ) originates from the shift of the dilaton accompanying the e 2Φ TsT term in eq. (3.13). Note that the bound (3.17) guarantees that the energies of the black string do not become negative as we increase the temperatures. In Section 5.2 we will consider a different parametrization of the black string where the charges are manifestly well-behaved but the TsT origin of the spacetime is obscured.
The background (3.13) also features electric and magnetic charges that are given by We recall that before the TsT transformation, the BTZ black hole can be obtained from the near-horizon limit of k coincident NS5 branes and p coincident strings such that Q e = p and Q m = k. Thus, the values of the electric and magnetic charges are preserved after the TsT deformation provided that φ 0 = 0. This is one of the motivations behind our parametrization of the dilaton in eq. (3.13).
Let us now turn to the thermodynamics of the TsT black string. The background (3.13) features a horizon at r h = 2T u T v whose near-horizon (r − r h ≡ 4T u T v ǫ 2 → 0) geometry is described by Regularity of the Euclidean continuation of eq. (4.18) implies that the u and v coordinates satisfy the thermal identification (4.19) where the physical left and right-moving temperatures T L and T R satisfy From the near-horizon metric (4.18) we readily find that the entropy of the TsT black string in terms of the T u and T v parameters is given by Alternatively, we can write the entropy of the black string entirely in terms of the conserved charges such that We interpret eq. (4.22) as the entropy of the black string in the microcanonical ensemble. When neither the T u or T v parameters are zero, the energies (4.16) and entropy (4.21) of the black string are positive thanks to the bound T u T v ≤ 1/2λ given in eq. (3.17), where the equal sign corresponds to the case where both the energies and the entropy become infinite. This bounds leads to the following bound on the product of the physical temperatures which, using the holographic dictionary (3.38), is the same bound (4.12) obtained from the field theory analysis of TT -deformed CFTs at finite temperature. Hence, the parametrization of the black string in terms of the T u and T v variables is compatible with the fact that TT -deformed CFTs feature a maximum product of temperatures. If the electric charge is allowed to vary, i.e. if δφ 0 = 0, then the black string features a chemical potential conjugate to Q e that is given by [97] ν = πk(T u + T v ). (4.24) It is then not difficult to show that the thermodynamic potentials and their conjugate charges satisfy the first law of thermodynamics, Finally, we note that it is possible write the physical charges (4.16) in terms of the physical temperatures (4.20) without using the intermediate T u and T v variables. However, it turns out to be more convenient to express the temperatures in terms of the charges, the latter of which can be shown to satisfy

Matching the thermodynamics
We have seen that the TsT black string is characterized by three independent charges that correspond to the energy, angular momentum, and the electric charge. The black string also features conjugate thermodynamic potentials which suggest that this background is dual to a thermal state in the dual field theory. On the other hand, the thermodynamics of TTdeformed CFTs were derived in the absence of any additional thermodynamic potentials other than the temperature and the angular potential. For this reason, we restrict the analysis of this section to fixed values of the electric charge such that δφ 0 = 0. Without loss of generality, we assume that φ 0 = 0 which implies that the values of the electric and magnetic charges are unchanged by the deformation, i.e. they satisfy Q e = p and Q m = k. As a result, the central charge of the CFT dual to string theory on AdS 3 × S 3 × M 4 is preserved after the TsT transformation such that c = 6Q e Q m = 6pk. 11 Let us first consider the microcanonical ensemble. In this ensemble we identify the thermal expectation values of the dual TT -deformed CFT with the energies of the TsT black string. As a result, where E i L and E i R denote the expectation values in each copy of the TT -deformed product CFT. The canonical and microcanonical expressions for the entropy derived in Section 4.1 are valid in the Cardy regime where the energies and temperatures are assumed to be large. In this limit we expect the entropy to be homogeneously distributed among the p copies of the TT -deformed (M 6k ) p /S p CFT [77]. This implies, in particular, that the left and right-moving energies E i L and E i R are all equal and satisfy Consequently, we find that the Bekenstein-Hawking entropy of the black string given in eq. (4.22) matches the entropy of single-trace TT -deformed CFTs where we used the holographic dictionary (3.38) and each of the S i TT (E i L , E i R ) terms is given by the entropy formula (4.8) for the TT -deformed seed CFT.
We now turn to the canonical ensemble. We expect the TsT black string to correspond to a thermal state in the dual field theory whose left and right-moving temperatures T L and T R are determined from the horizon generator by eq. (4.20). Interestingly, we note that the structure of T L and T R is a special one as it guarantees that the γ parameter introduced in the field theory analysis in (4.14) is a rational function, namely where we used the dictionary relating the string and field theory variables given in (3.38). Let us assume once again that in the Cardy regime the entropy is distributed evenly among the p copies of the deformed symmetric product. We then find that the entropy of singletrace TT -deformed CFTs at temperatures (4.20) also matches the entropy of TsT black strings (4.21), where we used the holographic dictionary (3.38) and each of the entropies S i TT (T L , T R ) is given by eq. (4.13).
The fact that we are able to match the entropy in both the canonical and microcanonical ensembles suggests that the physical temperatures and the gravitational charges of the TsT black string are related via the thermodynamic relations (4.10) and (4.11) of TTdeformed CFTs. Indeed, it is not difficult to verify using the holographic dictionary (3.38) and eq. (4.29) that the physical temperatures and charges of the black string related via eqs. (4.26) and (4.27) reproduce the corresponding equations in TT -deformed CFTs. This further shows that the matching of the entropy in the canonical and microcanonical ensembles are indeed consistent with each other. Thus, we conclude that the TsT black string with fixed electric charge is dual to a thermal state in a single-trace TT -deformed CFT.

The phase space
In this section we take a closer look at the phase space of solutions obtained from a TsT transformation of the BTZ black hole. To begin, we find an everywhere smooth background that interpolates between the global AdS 3 vacuum in the IR and a linear dilaton background in the UV. This solution is obtained by analytic continuation of the T u and T v parameters and corresponds to the NS-NS vacuum of the dual TT -deformed CFT. We then show that, in order to recover the TT flow between states in the space of BTZ and TsT black strings solutions, it is necessary to consider λ-dependent T u and T v parameters. Finally, we comment on the solutions with λ < 0, which corresponds to the superluminal sign of the TT deformation in the field theory.

The ground state
In previous sections we had implicitly assumed that the T u and T v parameters are real so that the corresponding black string solutions describe thermal states in the dual TTdeformed CFT. In particular, the background with T u = T v = 0 features vanishing energies and describes the R-R vacuum in the dual field theory. A natural question to ask is what is the holographic dual to the NS-NS vacuum state characterized by left and right-moving energies given in eq. (4.4). In this section we find the NS-NS ground state and rederive the critical value λ c (4.6) that guarantees that the vacuum energy is real.
Let us begin by performing an analytic continuation of the temperatures after which the corresponding TsT-transformed background (3.13) no longer features a horizon or angular momentum. The corresponding background features a conical defect/surplus at r = 2ρ 2 0 unless the following condition is satisfied The solution to eq. (5.2) that guarantees smoothness of the TsT-transformed spacetime is given by where we have chosen the branch that goes to ρ 2 o → 1/4 as λ → 0, which corresponds to the value of global AdS 3 in this gauge.
Eq. (5.3) admits real solutions only when λ ≤ λ c = 1/2. This is the same critical value of the deformation parameter that guarantees a real ground state energy in the dual TTdeformed CFT, cf. eq. (4.6). This is not a coincidence, as the left and right-moving energies of the TsT-transformed background satisfying eqs. (5.1) and (5.3) match the energies of the vacuum state in TT -deformed CFTs. In order to see this, we note that the background left and right-moving energies satisfy where, following the discussion in Section 4.3, we have set φ 0 = 0. Using the holographic dictionary (3.38) we find that these expressions match the vacuum charges of TT -deformed CFTs given in eq.
In this gauge, it is clear that the string frame metric goes to R 1,1 × S 1 × S 3 as ρ → ∞, while the geometry caps off at ρ = 0 where it approaches global AdS 3 × S 3 in the region λρ 2 Λ1. Consequently, the background (5.6) is a smooth solution to the supergravity equations of motion interpolating between global AdS 3 in the IR and a linear dilaton in the UV. We also note that the solution (5.6) depends on ρ 0 only through the B field and a constant shift in the dilaton. In particular, when λ ≤ λ c all of the background fields are real. Interestingly, when λ > λ c eq. (5.6) remains a solution to the equations of motion but the B field and dilaton become complex. Finally, we note that at the critical value λ = λ c we have ρ 0 = 1 and the string coupling is zero everywhere. In this special case the supergravity approximation breaks down even in the IR (AdS 3 ) region and we need the little string theory description everywhere.

TT flow from TsT
We have seen that the entropy and thermodynamics of the TsT black string match those of TT -deformed CFTs at finite temperature. As discussed in Section 4.1, the density of states is not expected to change after the deformation provided that the energies are related by eq. (4.3). An interesting question to ask is how the deformed TT spectrum (4.3) arises in the bulk. A naive guess is to compare the energies before and after the TsT transformation with fixed values of T u and T v . Given the parametrization of the deformed energies in eq. (4. 16), the values of the undeformed charges required to satisfy the TT spectrum are λ-dependent and given by where, following the discussion in Section 4.3, we have set φ 0 = 0. The equations above differ from the charges of the original BTZ black hole before the TsT transformation, see eq. (3.6). In other words, while the TsT transformation maps the phase space of the undeformed theory to the phase space of a TT -deformed CFT, the flow induced by λ with fixed (λ-independent) values of T u and T v does not agree with the TT flow of states. In order to see the TT flow, the T u and T v variables parametrizing the black string must depend on λ such that , , (5.8) where the E L (λ) and E R (λ) variables implement the TT flow via The TsT black string (3.13) is written in terms of E L (λ) and E R (λ) in eqs. (B.4) and (B.6) of Appendix B. The change of variables (5.8) allows us to express the background fields directly in terms of the deformed energies since the background charges are given by Q L (λ) = c 6 E L (λ) and Q R (λ) = c 6 E R (λ). We then note that eqs. (5.8) and (5.9) allow us to describe the TT flow of an undeformed BTZ black hole with charges (3.6), to a deformed black string solution whose energies are given by where Q ∂t (0) ≡ Q L (0)+Q R (0) and Q ∂ϕ (0) ≡ Q L (0)−Q R (0) denote the undeformed energy and angular momentum. Using the holographic dictionary (3.38) and assuming that each copy in the TT deformation of the symmetric product contributes the same amount to the total energy (see eq. (4.29)), it is not difficult to show that eqs. (5.11) and (5.12) reproduce the spectrum of single-trace TT -deformed CFTs.
Although the TsT origin of the background is no longer manifest in terms of the new variables E L and E R , this parameterization of the black string has other advantages beyond making the TT flow manifest. In particular, note that eq. (5.8) satisfies the bound T u T v < 1/2λ described in Section 4.2 for all positive values of E L and E R , and hence the apparent pathologies in the charges (4.16) and the entropy (4.21) of the black string are no longer present in terms of these variables. In addition, eq. (5.8) is valid for any choice of φ 0 , including values of φ 0 that may depend on T u and T v . Finally, it is important to note that the change of variables (5.8) does not affect the results of previous sections since the matching of the worldsheet spectrum and the thermodynamics of the black string to TT -deformed CFTs is independent of how we parametrize the T u and T v variables.
We conclude by noting that the NS-NS vacuum discussed in Section 5.1 can be phrased in a very natural and simple way using the parameterization (5.8). Indeed, the smoothness condition (5.2) for the NS-NS ground state translates into E L (0) = E R (0) = −1/4 which implies undeformed energies Q L (0) = Q R (0) = −c/24 that correspond to the energies of the global AdS 3 × S 3 vacuum. Using eq. (5.9), we then find that the deformed energies parameterized by E L (λ) = E R (λ) = − 1 4λ 1 − √ 1 − 2λ match the energies of the TTdeformed ground state. Thus, the background (5.6) corresponds to the parametrization of the black string that is aligned with the TT flow. Note that the derivation of the NS-NS vacuum was done independently of the change of variables (5.8) by imposing smoothness on the TsT-transformed background. Hence, the results of Section 5.1 provide additional justification for the parametrization of the black string described in this section.

TsT black strings with λ < 0
Let us briefly comment on the negative sign of the TsT parameter λ which corresponds to the superluminal sign (µ < 0) of the TT deformation in the dual field theory. We first note that when λ < 0, the TsT black strings (3.13) are still identified with thermal states in single-trace TT -deformed CFTs. This follows from the fact that the gravitational charges (4.16), the physical temperatures (4.20), and the entropy (4.21) are valid for both positive and negative values of λ. Similarly, the holographic dictionary relating the bulk and boundary charges, temperatures, and entropy works in the same way for either sign of the deformation parameter.
We have seen in Section 4.1 that for positive signs of the TT deformation, potential complex energy states appear at low energies unless µ ≤ ℓ 2 /2k, and that there is an upper bound on the temperatures given by T L T R ≤ 1/8π 2 kµ. Using the holographic dictionary λ = µk/ℓ 2 , we have shown that these features of TT -deformed CFTs can be reproduced from TsT black strings in the bulk side of the correspondence. When the deformation parameter is negative, we can also reproduce the critical energy of TT -deformed CFTs which corresponds to the maximum values of the energy before the spectrum becomes complex. Assuming that the angular momentum vanishes and requiring that the energy in each copy of the TT -deformed product CFT is real, we find that the critical energy is given by On the gravity side, if we require the TsT background with λ < 0 to have positive temperatures and a real value of the dilaton, we obtain the following bounds on the T u and T v parameters . (5.14) When the angular momentum vanishes, the bound (5.14) implies an upper bound on the gravitational energy , (5.15) where, following the discussion of Section 4.3, we have set φ 0 = 0. Using the holographic dictionary (3.38) and c = 6pk, we find that the critical energy of the black string (5.15) matches the critical energy in TT -deformed CFTs (5.13). The critical energy (5.13) can also be understood in a geometrical way. First, note that the scalar curvature of the TsT black string is given, in the Einstein frame, by Thus, when λ < 0, the background features a curvature singularity at From the metric in eq. (3.13) it is clear that closed timelike curves also develop at radius r = r c , in agreement with the observations made in zero temperature backgrounds in [72,83]. This suggests that, in analogy with the double trace deformation [74], a UV cutoff at r = r c can be used to avoid these pathologies in the space of black string solutions with fixed λ. 12 We now note that the bound (5.14) implies an upper bound for the horizon that reads The critical energy of the TsT black string (5.15) was obtained when the bound on the T u and T v parameters (5.14) is saturated, which in turn implies that Hence, the critical energy (5.15) is reached by black strings whose event horizon coincides with the cutoff radius. Note that black strings with r h > 1/|λ| still lie within the cutoff since r h ≤ r c for all values of T u and T v , but these solutions feature complex values of the dilaton.

Relationship to other backgrounds
In this section we consider the Horne-Horowitz black string constructed in [53] and generalize it to include angular momentum. We show that after a change of coordinates this background corresponds to the TsT black string with λ = 1/2, the critical value that guarantees a well-behaved spectrum for all states. As a result, the Horne-Horowitz black string with fixed electric charge is dual to a thermal state in a TT -deformed CFT. Finally, we show that the background considered by Giveon, Itzhaki and Kutasov in [77] also corresponds to the TsT black string with λ = 1/2 after a change of coordinates, a gauge transformation, and a choice of φ 0 . This background reproduces the thermodynamics of TT but with thermodynamic potentials that differ from those derived in Section 3.

The Horne-Horowitz black string
The Horne-Horowitz black string is a three-dimensional, asymptotically zero-curvature background of string theory supported by NS-NS flux that is obtained by gauging an SL(2, R) × U (1) WZW model. This construction is similar to the one discussed in Section 3.4 except that the gauged U (1) symmetry corresponds to a different subgroup of SL(2, R). The Horne-Horowitz background is characterized by a mass M and a charge Q such that, when M > |Q| > 0, the black string features a regular inner horizon and a timelike singularity. These are also features of four-dimensional Reissner-Nordström black holes, which makes the Horne-Horowitz background an interesting toy model for asymptotically-flat charged black holes. As we will see, another motivation to study this class of solutions is that they correspond to thermal states in TT -deformed CFTs.
Let us comment on the space of black string solutions parametrized by (6.1). As discussed in Section 4.2, λ = 1/2 is the critical value of the deformation parameter for which the spectrum of winding strings, as well as the spectrum of of the dual TT -deformed CFT, are well-defined for all values of the energies including the ground state. We also note that in the Horne-Horowitz parametrization of the black string the expressions for the conserved charges and the entropy are both regular, and Q is allowed to be positive or negative. The existence of the horizon requires that (M − |Q| + J)(M − |Q| − J) ≤ 0, which is equivalent to non-negativity of both the left and the right moving energies Q L = 1 2 (Q ∂t + Q ∂ϕ ) and Q R = 1 2 (Q ∂t − Q ∂ϕ ). Relatedly, we see from eq. (6.5) that the Horne-Horowitz black string features an event horizon provided that the background before the TsT transformation corresponds to a BTZ black hole at finite temperature. In analogy with [53], solutions with M − |Q| < |J| do not have a horizon and in general feature conical singularities. One possible exception is the analog of the ground state we discussed in Section 5.1. However, with the identification ϕ ∼ ϕ+2π, the would-be smooth solution corresponds to M = J = 0, which makes the metric (6.1) ill defined. 14 In order to describe the NS-NS ground state, we need to go back to the r coordinate using eq. (6.4) followed by another radial redefinition to write the metric in the form (5.6) with e 2Φ HH = 0.
We also note that the parameters appearing in the solution (6.1) have slightly different origins in the Horne-Horowitz construction and the one presented here. In the former case, the solution is built as a deformation of pure AdS 3 featuring no parameters, except the cosmological constant that is being kept fixed, and the constant expectation value Φ 0 of the dilaton. Turning on the deformation introduces another parameter κ, which can be understood as a combination of λ and the temperature. The parameters (Φ 0 , κ) are then related to the mass M and charge Q of the black string. In the present paper, the black string is obtained as a deformation of a BTZ black hole, and is characterized by four parameters: the mass and angular momentum of the original solution, the expectation value Φ 0 of the dilaton, and the deformation parameter λ. The solution (6.1) is then obtained at fixed λ = 1 2 , which leaves us with three parameters for three conserved charges that depend on M , J and Q. Further restricting to the non-rotating black strings, we obtain the same two parameter phase space parameterized by M and Q. The two approaches are consistent as the deformation parameter in the original Horne-Horowitz background [53] can be interpreted as an arbitrary mass parameter at fixed λ.
Not surprisingly, the thermodynamics of the Horne-Horowitz black string with fixed electric charge Q match the thermodynamics of TT -deformed CFTs with central charge c Q ≡ 6pkQ. Indeed, the entropy of the Horne-Horowitz background can be written in terms of the left and right-moving energies as which makes the relationship to TT -deformed CFTs manifest. Thus, we conclude that the Horne-Horowitz black string with fixed electric charge is dual to a thermal state in a 14 As noted in [53], we can choose an identification that depends on M and Q to make the solution smooth.

The Giveon-Itzhaki-Kutasov black string
Let us now consider the black string studied in refs. [77] (see also [98]) which was argued to reproduce the thermodynamics of TT -deformed CFTs. We will show that, after a change of coordinates and a gauge transformation, this background corresponds to the non-rotating TsT black string with λ = 1/2 and a temperature-dependent value of φ 0 . Although δQ e = 0 for this background, the expression for the entropy can be written in terms of the energy in a way that resembles the asymptotic density of states of TT -deformed CFTs. The finite-temperature background considered in [77] is given in the string frame by where the u and v coordinates satisfy In our conventions the constants ρ 2 0 and ρ 2 1 are dimensionless with the latter given by ρ 2 1 = 8pkG 3 / kℓ 2 s = 2. Using the covariant formulation of charges, we find that up to a constant the energy is given by while the angular momentum vanishes. Crucially, the background features an energydependent electric charge, which plays an important role in reproducing the thermodynamics of TT -deformed CFTs. The GIK black string is closely related to the TsT black string studied in Section 3. Indeed, after the change of coordinates we find that the background (6.7) is given in terms of the TsT metric, B-field, and dilaton in eq. (3.13) by where, in addition to λ = 1/2, we set T u = T v = T and identify 15 In terms of T , the energy of the GIK black string (6.9) can be written as This expression does not match the charge Q L + Q R of the TsT background computed in Section 4.2, the latter of which corresponds to the first term in eq. (6.14). This follows from the fact that (i) there is a shift of the B-field between the GIK and TsT backgrounds that is responsible for the second term in eq. (6.14); and (ii) there is a constant shift of the energy (the third term) that is not determined when integrating the infinitesimal charges. The temperature-dependent value of φ 0 is responsible for making the shift of the gauge field contribute to the energy. Otherwise, when φ 0 is constant, as in the Horne-Horowitz black string with fixed electric charge, the shift of the gauge field translates into a constant shift of the energy. The difference between the charges of the GIK and TsT black strings shows that, although related via a gauge transformation and a shift of the dilaton, these backgrounds describe different physical solutions. The entropy of the GIK background is given in terms of the energy by which matches the entropy of single-trace TT -deformed CFTs with RE = Q ∂t at λ = 1/2 as originally observed in [77]. Note that the background (6.7) features an electric charge that depends on the energy. Hence, we expect the holographic dictionary relating the bulk and boundary quantities to differ from our discussion in Section 4.3, where we matched the sector of solutions with fixed electric charge (δQ e ∝ δφ 0 = 0) to TT deformed CFTs. 16 In order to illustrate the differences in the holographic dictionary, we first note that the Hawking temperature T H and the chemical potential ν are given by 12 . 16 Note that the reason why it is possible to reproduce the thermodynamics of TT with a dilaton that differs from the one considered in Section 4.2 is that expression for the energy has also changed. and lead to a consistent first law of thermodynamics, δS GIK = 1 ℓT H δQ ∂t + νδQ e . (6.17) As a consequence of the additional chemical potential, the Hawking temperature T H cannot be matched to the temperature of TT -deformed CFTs given in eqs. (4.10) and (4.11).
The reason for this is that the GIK background (6.7) describes a one-parameter space of solutions where the energy Q ∂t and the electric charge Q e are not independent variables. Within this one-one-parameter phase space, the correct potential conjugate to the energy can be determined by expressing Q e in terms of Q ∂t such that the first law becomes δS GIK = 1/ℓT eff H δQ ∂t where T eff H is the effective temperature. The latter is given by and matches the temperature of TT -deformed CFTs.
To summarize, the GIK black string [77] can be embedded into our general TsT black string solution (3.13) up to an appropriate choice of φ 0 and a constant gauge transformation. In particular, the results of Section 4.3 and ref. [77] map different sectors of the phase space (3.13) to single-trace TT -deformed CFTs.
background. In the covariant formulation of conserved charges [99,100], the infinitesimal variation of the gravitational charge associated with ξ is given by where ∂Σ denotes the boundary of a codimension-1 spacelike surface. The antisymmetric tensor K µν ξ featured in eq. (A.2) is defined by where k µν ξ,f denotes the contribution of the field f = {g, B, Φ} to the variation of the charge. For the action (A.1), the antisymmetric tensors k µν ξ,f are given by [97,99] k µν ξ,g = 1 2 ξ ν ∇ µ δg α α − ξ ν ∇ α δg αµ + ξ α ∇ ν δg αµ + where H = dB, δf α 1 ...αn = g α 1 λ 1 . . . g αnλn δf λ 1 ...λn for any tensor field f α 1 ...αn , and we will only consider zero mode variations of the background such that δ ≡ δT u ∂ Tu +δT v ∂ Tv +δλ∂ λ . The conserved charges are not sensitive to the choice of spacelike surface but they do depend on the global properties of the spacetime. We will assume that the coordinates are identified as in (3.15), such that the variation of the charge may be written as where we recall that t = 1 2 (u − v) and ϕ = 1 2 (u + v). The BTZ and TsT black strings also feature electric and magnetic charges which count the number of coincident fundamental strings and NS5 branes generating the original background. The electric and magnetic charges are respectively denoted by Q e and Q m and are given by

B Alternative parametrizations of the black string
In this appendix we consider alternative parametrizations of the TsT black string (3.13). These backgrounds are obtained by expressing the T u and T v parameters in terms of the E L (λ) and E R (λ) variables , Using the following rescaling of the radial coordinate we find that the expressions for the metric, B-field, and dilaton are given by where we used the fact that the undeformed variables E L (0) and E R (0) satisfy eq. (5.9). Note that the introduction of λ dependence in the T u and T v parameters means that the TsT origin of the black string (3.13) is no longer manifest in eq. (B.4). As discussed in detail in Section 5.2, this is necessary to describe the TT flow from the BTZ to the TsT black string. An alternative parametrization of the black string consists of using the change of variables (B.1) together with the following shift and rescaling of the radial coordinate The change of coordinates (B.5) is valid only when λ = 0. As a result, in this gauge it is not obvious how to recover the background before the TsT transformation. The TsT black string obtained after the change of variables and coordinates given in eqs. (B.1) and (B.5) is given by where E ≡ E L (λ) + E R (λ) and J ≡ E L (λ) − E R (λ) are proportional to the deformed energy and angular momentum. In particular, note that when λ = 1/2 this background reduces to the generalization of the Horne-Horowitz black string described in Section 6.1.