Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite $\sqrt{T\bar{T}}$ deformations

The conformal symmetry algebra in 2D (Diff($S^{1}$)$\oplus$Diff($S^{1}$)) is shown to be related to its ultra/non-relativistic version (BMS$_{3}$$\approx$GCA$_{2}$) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT$_{2}$, the BMS$_{3}$ generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, $T$ and $\bar{T}$, closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS$_{3}$ becomes a bona fide symmetry once the CFT$_{2}$ is marginally deformed by the addition of a $\sqrt{T\bar{T}}$ term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT$_{2}$ because its energy and momentum densities fulfill the BMS$_{3}$ algebra. The deformation can also be described through the original CFT$_{2}$ on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to $T$ and $\bar{T}$. BMS$_{3}$ symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of $\mathrm{N}$ free bosons, which coincides with ultra-relativistic limits only for $\mathrm{N}=1$. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS$_{3}$ (or flat) versions.


Introduction
Conformal symmetries enhance those of special relativity, and become pivotal in the description of generic relativistic systems enjoying scale invariance; see e.g., [1,2]. Conformal field theories (CFT's), built in terms of these extended symmetries, are well-known to play a fundamental role in a broad variety of subjects [3][4][5][6][7][8]. Their power turns out to be particularly impressive in two spacetime dimensions, as a direct consequence of the fact that the conformal group exceptionally becomes infinite-dimensional. The conformal algebra in 2D is described by two copies of the Witt or centerless Virasoro algebra, being isomorphic to two copies of the algebra of diffeomorphisms on the circle (Diff(S 1 )⊕Diff(S 1 )), spanned by [L m , L n ] = (m − n) L m+n , L m ,L n = (m − n)L m+n , (1.1) with L m ,L n = 0 and m, n ∈ Z.

Map between relativistic and ultra/non-relativistic conformal algebras in 2D
Intriguingly, the conformal symmetry algebra in 2D (1.1) can be shown to be related to its ultra/non-relativistic version (1.2) by means of a precise nonlinear map of the generators, without the need of performing any sort of limiting process. In order to explicitly see the map it is useful to work in the continuum, so that the generators of the conformal algebra (1.1) can be trade by two arbitrary periodic functions defined on the circle, according to L m =´dφT (φ) e −imφ ,L m =´dφT (φ) e imφ . Thus, the conformal algebra is equivalently expressed as with T (ϕ) ,T (θ) = 0, and [·, ·] = i {·, ·}. Note that the continuous version of the conformal algebra (2.1) can be naturally interpreted as a Poisson structure. The searched for mapping is then defined as follows so that the corresponding brackets involving J and P can be readily found by virtue of the "fundamental" ones in (2.1), which exactly reproduce the continuous version of the BMS 3 algebra, given by with {P (ϕ) , P (θ)} = 0. In Fourier modes, J m =´dφJ (φ) e imφ , P m =´dφP (φ) e imφ , the algebra (2.3) then reduces to (1.2). Bearing in mind that supertranslation generators are defined up to a global scale factor, making P → αP with constant α in the map (2.2), yields the same result. Thus, for simplicity and later convenience, we keep assuming α = 1 afterwards.
In sum, the nonisomorphic conformal and BMS 3 algebras, in (2.1) and (2.3) respectively, are nonlinearly related by virtue of the map defined through (2.2), and it is worth highlighting that that no limiting process is involved in the mapping.

BMS generators within CFT 2
The mapping in (2.2) naturally makes one wondering about how precisely the BMS 3 algebra manifests itself for a generic (nonanomalous) classical CFT 2 . Indeed, the mapping directly prescribes a way in which BMS 3 generators emerge as composites of those of the conformal symmetries. Nonetheless, it can be shown that the composite generators do not span a Noetherian symmetry of the CFT 2 .
In order to see that, let us consider a generic CFT 2 on a cyllinder. In the conformal gauge, using null coordinates x = t + φ andx = t − φ, the canonical generators of the conformal symmetries are given by (see e.g., [1,2]) being conserved (Q CF T = 0) by virtue of the (anti-)chirality of the components of the stress-energy tensor and the parameters (∂T =∂T = ∂¯ =∂ = 0). The transformation laws of T andT then read from the conformal algebra (2.1), since The nonlinear map (2.2) implies that the generators (3.1) and transformation laws (3.2), can be expressed as where prime stands for ∂ φ , while the parameters J , P , relate to and¯ through Thus, the generators and transformation laws in (3.3), (3.4), acquire the expected form of those for the BMS 3 algebra (see e.g., [22,25]) 2 . Legitimate BMS 3 generators are obtained when the parameters ,¯ are no longer chiral, but instead, being manifestly state-dependent according to (3.5). Hence, at fixed time slices, the parameters J , P can be consistently assumed to be state-independent arbitrary functions, so that the Poisson brackets of the generators clearly close according to the BMS 3 algebra by virtue of (2.3). It is worth emphasizing that since J stands for the momentum density, superrotation generators yield the corresponding conserved charges. Nevertheless, supertranslation generators are not conserved, as it can bee seen from the time evolution of P , that can be obtained from that of the (anti-)chiral T andT by virtue of the map (2.2), given bẏ Therefore, supertranslations do not correspond to Noetherian symmetries of the CFT 2 .
2 A warning note is in order: if the parameters and¯ were still assumed to be chiral, this would be just a mirage; because in that case, the new ones, J , P , would become state-dependent, and hence, this would only amount to an alternative way of expressing the original conformal algebra generators and transformation laws in (3.1), (3.2), in terms of different variables.

BMS 3 symmetries from √ TT deformations
According to the map (2.2), the supertranslation density P can be seen as a finite nontrivial marginal deformation of the CFT 2 energy density H = T +T . Hence, a simple way to achieve conservation of supertranslations consists in deforming the original Hamiltonian of the CFT 2 to coincide with the supertranslation generator. Thus, starting from the CFT 2 in the conformal gauge, the simplest deformation is implemented through the Hamiltonian densityH = H + 2 √ TT = P , so that the deformed action reads Note that since only the Hamiltonian was deformed, the Poisson brackets remain the same as those of the original CFT 2 in (2.1). Hence, the time evolution of supertranslation and superrotation densities can be readily obtained from (2.3) withH =´dφP ; so that the canonical BMS 3 generators (3.6) are now manifestly conserved (Q = 0) provided that the parameters fulfill˙ P = J and˙ J = 0, being apparently state independent. Therefore, the BMS 3 generators (3.6) span a bona fide Noetherian symmetry of the deformed action (4.1).
It is also worth pointing out that the deformed theory (4.1) retains the integrability properties of the original CFT 2 , since the universal enveloping algebra of BMS 3 also contains an infinite number of independent commuting (KdV-like) charges [25] 3 .
For a generic gauge choice, the deformation (4.1) can be written as where it is implicitly assumed that I CF T is written in Hamiltonian form, and T µν stands for the stress-energy tensor of the undeformed CFT 2 . Remarkably, the action (4.2) keeps being invariant under diffeomorphisms and local scalings, but it is no longer a CFT 2 because the energy and momentum densities of the deformed theory yield to generators that fulfill the BMS 3 algebra (2.3) instead of the conformal one in (2.1).
In order to see that, let us consider the original CFT 2 in a generic (non-conformal) gauge, so that in a local patch, the two-dimensional metric can be brought to the same conformal class as the following one c.f., [108] where N and N φ stand for the lapse and shift functions, respectively 4 . The total Hamiltonian of the CFT 2 then reads The deformation in (4.2) has the net effect of deforming the energy density of the CFT 2 to be that of a supertranslation, i.e., H → P , so that the total Hamiltonian deforms as Supertranslation and superrotation densities evolution is then spanned by the deformed HamiltonianH , which by virtue of (2.3) readṡ In absence of global obstructions, the canonical generators become expressed as an integral over the spatial circle precisely as in (3.6), but now being conserved provided that the state-independent parameters fulfill Thus, the transformation law of supertranslation and superrotation densities is given by (3.4), corresponding to Noetherian BMS 3 symmetries.

Geometric aspects
Since the deformed action is manifestly invariant under diffeomorphisms ξ = ξ µ ∂ µ , it is reassuring to verify that the Noether current j µ =T µ ν ξ ν , with is conserved (∂ µ j µ = 0) provided that the evolution equations of the energy and momentum densities (4.6), as well as those of the parameters in (4.7) hold. The precise form of the diffeomorphisms is then identified as which close in the Lie brackets, [ξ 1 , ξ 2 ] = ξ 3 , with Note that one might be tempted to extract an stress-energy tensor Θ µ ν from the corresponding density in (5.1) by making use of the metric of the undeformed theory g µν in (4.3), according toT µ ν = √ −gΘ µ ν . However, this tensor is not conserved (∇ µ Θ µ ν = 0), reflecting the fact that the metric the of CFT 2 is not preserved under BMS 3 diffeomorphisms ξ µ up to a local scaling, i.e., Hence, the metric of the undeformed CFT 2 is not a suitable object to describe the geometric properties of the deformed theory. An appropriate Riemannian metric for the geometric description of the deformation is obtained as follows. Note that the total deformed Hamiltonian (4.5) is a homogeneous functional of T andT of degree one, so that it fulfills the following identitỹ Therefore, the deformed theory can be equivalently described by placing the original CFT 2 on a state-dependent curved metric, whose lapse and shift functions,Ñ andÑ φ , are respectively given by the variation of the deformed Hamiltonian with respect to the energy and momentum densities of the undeformed theory, i.e., 5 The mapping (2.2) allows to express the deformed metric (5.6) in terms of the supertranslation and superrotation densities, so that it reads where N and N φ correspond to the (state-independent) lapse and shift functions of the original undeformed metric in (4.3), respectively 6 . It must be emphasized that the manifest state dependence of the lapse and shift functions (or Beltrami differentials) of the deformed metric (5.7) provides a local obstruction to gauge them away, preventing the possibility of choosing the standard conformal gauge once the theory is deformed.
being automatically symmetric and traceless, while its conservation implies the evolution equations of supertranslation and superrotation densities (4.6). Therefore, the canonical BMS 3 generators (3.6) can be written in manifestly covariant way as with ξ µ given by (5.2), and according to the deformed metricg µν in (5.7), the unit timelike normal is given byñ µ = (Ñ , 0), andγ = 1. The geometric description of the deformed theory is then suitably carried out in terms of the two relevant structures,g µν andΘ µ ν , being inextricably intertwined. In fact, since both objects are state dependent, they acquire nontrivial functional variations when acting on them under diffeomorphisms, given by Therefore, since the functional variations (5.10) must be taken into account, BMS 3 symmetries geometrically arise from diffeomorphisms ξ that preserve the form of both relevant structures up to a local scaling, i.e., from the solutions of the following deformed conformal Killing equations∇ where L ξ stands for the Lie derivative. It is amusing to verify that starting from scratch with the deformed metric and stressenergy tensor,g µν andΘ µ ν , the deformed conformal Killing equations (5.11) can be exactly solved. Indeed, the solution is precisely given by the BMS 3 diffeomorphisms ξ µ in (5.2) with parameters P , J fulfilling (4.7), where the transformation law of supertranslation and superrotation densities is also found to be given by (3.4).
Note that the geometric interpretation also allows to find the transformation law of the fields in the deformed theory from those of the original undeformed (primary) fields, collectively denoted by χ, by writing them in a manifestly covariant way, and then acting with the Lie derivative along BMS 3 symmetries spanned by ξ, i.e., δ ξ χ = L ξ χ.

Deformed free bosons
Let us see how the deformation works in a simple and concrete example, given by the action of N free bosons with flat target metric, Before implementing the generic deformation (4.2), it is useful to express the background metric g µν in the gauge choice (4.3), so that the Hamiltonian action reads where Π I = δL δΦ I , and H = 1 2 Π I Π I + Φ I Φ I , J = Π I Φ I . The deformed Hamiltonian action is then given byĨ The transformation law of the fields and their momenta under BMS 3 symmetries spanned by ξ in (5.2) are then found to be . Moreover, the stress-energy tensor of the deformed theory is obtained fromT µ ν = √ −gΘ µ ν withT µ ν andg µν respectively given by (5.1) and (5.7). It is worth highlighting that the deformed action (6.3) clearly cannot be obtained from any standard limiting process of the undeformed one for N > 1. The peculiarity of the deformed single free boson (N = 1) stems from the fact that the supertranslation density simplifies as P = Π 2 , so that the momentum can be eliminated from its own field equation, and the deformed action (6.3) can be written in Lagrangian form as where V µ = ( √ −g) 1/2 n µ stands for a vector density of weight 1/2, constructed out from the metric g µν in (4.3) of the undeformed theory. Noteworthy, this vector density is invariant under the BMS 3 symmetries spanned by ξ in (5.2), since L ξ V µ = 0. Therefore, the deformed action of a single free boson (6.5) coincides with the ultra-relativistic limit of the undeformed theory (6.1) for N = 1, when the Carrollian limit is taken in a similar way as for the tensionless string [16,[113][114][115][116].
Additionally, the vector density can be reexpressed as V µ = 1 √ 2 e 1/2 τ µ , where e and τ µ correspond to the einbein and the dual of the "clock one-form" of a Carrollian geometry [117], respectively; so that action of the deformed free boson agrees with the Carrollian one found in [118].
Remarkably, the action (6.5) can be understood in terms of two inequivalent geometric structures. One of them is Riemannian and described through the state-dependent metric g µν in (5.7), while the remaining structure stands for a Carrollian manifold.

Ending Remarks
Since the map (2.2) possesses a square root, our results also carry out for its negative branch, i.e., when the supertranslation density is given by In particular, the deformed action of a single free boson for the negative branch reads with P (−) = Φ 2 . Curiously, the deformed actionĨ (−) agrees with an inequivalent ultrarelativistic limit of the single free boson defined by Φ → Φ/c, Π → cΠ, when c → 0. This limit coincides with that needed to pass from the standard Liouville theory to its "flat" version [82]. Indeed, starting from a single free boson in the conformal gauge (N = 1, N φ = 0) the deformed free boson in the negative branch corresponds to the kinetic term of the flat Liouville theory. It is also worth to pointing out that the centrally extended conformal algebra (given by two copies of the Virasoro algebra) can be shown to be related to BMS 3 with central extensions, in terms of a map that is necessarily nonlocal. Nevertheless, if only zero modes are involved, the local nonlinear map in (2.2) still holds. Thus, blindly applying the map (2.2) for the zero modes, the Cardy formula once expressed in terms of left and right groundstate energies (L 0 ,L 0 ), given by reduces to its BMS 3 (or flat) versioñ when the deformed energy and momentum of the groundstate P 0 and J 0 are expressed in terms of the BMS 3 central charges [119][120][121][122]. Noteworthy, the hypotheses that ensure positivity of the Cardy formula (7.3) (L 0 < 0, L 0 < 0, L > 0,L > 0), by virtue of both branches of the map, imply that the deformed entropy (2.2) is also positive (S > 0). Furthermore, the map between the chemical potentials follows the same rule as that of the parameters in (3.5), with ( ,¯ )→(β,β) and ( P , J )→(β,θ), where left and right temperatures relate to the modular parameter of the torus as τ = β/2π, andβ,θ stand for the temperature and chemical potential of the deformed theory. Therefore, around equilibrium, the S-modular transformation τ → −1/τ precisely maps into its BMS 3 (flat) version [119,120].β → 4π 2β θ 2 ,θ → − 4π 2 θ . (7.5) As a closing remark, since the uplift of the deformed action (4.2) to higher dimensions is clearly invariant under diffeomorphisms and local scalings, it would be worth exploring whether the D-dimensional deformed theories might be invariant under the conformal Carrollian algebra, which is known to be isomorphic to BMS D+1 [12].