An integrable Lorentz-breaking deformation of two-dimensional CFTs

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times U(1)$ subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.


Introduction
Recently, Smirnov and Zamolodchikov [1] have studied an infinite class of irrelevant deformations of integrable two-dimensional QFTs (IQFTs), of the general double-trace form where T s , Θ s−2 and their barred counterparts correspond to the components of conserved spin s currents in the QFT 1 . These deformations are integrable, in the sense that there still exists an infinite set of local integrals of motion. They are also very interesting because they represent an unusual type of flow "up the RG" direction. The IQFT can also be taken to be a generic two-dimensional CFT, case in which the conserved currents and their associated commuting charges correspond to the KdV conserved integrals [2][3][4]. A particularly interesting deformation in this class is the TT deformation [1,5], built from the components of the stress tensor, which can be defined also for arbitrary twodimensional QFTs and for which the finite-size spectrum and thermodynamic equation of state have been obtained exactly at finite µ in terms of the spectrum and, respectively, the equation of state of the original QFT.
TT -deformed two-dimensional CFTs have a number of very interesting properties and applications. For positive µ, the high-energy spectrum exhibits Hagedorn behaviour, which can be understood from the close relation between the TT deformation of free bosons and the worldsheet theory of the bosonic string [6,7]. For negative µ, the theory appears to have a finite number of states and exhibits superluminal propagation [8,9]. The negative sign deformation has found a nice application in the AdS 3 /CFT 2 correspondence, as a proposed holographic dual to AdS 3 gravity with a finite bulk cutoff, r c = 1/ |µ| [9]. Another very interesting feature of TT -deformed QFTs is that, while their UV behaviour can be studied via their known S-matrix, it does not appear to correspond to a usual UV fixed point. Rather, it has been argued to correspond to a new type of UV behaviour, more characteristic of a theory of quantum gravity, termed "asymptotic fragility" [7,10]. A certain single-trace (in the AdS/CFT sense) variation on the TT deformation has also been argued to provide a holographic dual to string theory on a linear dilaton background [11]. For all of these reasons, the TT deformation is extremely interesting, in both field theory and holography.
It is natural to ask whether similarly integrable, universal double-trace deformations of two-dimensional CFTs exist. In this article, we focus on two-dimensional CFTs that posess an additional conserved U (1) current, J, and consider the Lorentz-breaking double-trace deformation (2.1) constructed from J andT , whereT is the current generatingz translations. As we will show, we can again obtain an exact formula for the finite-volume spectrum of the deformed theory and work out the thermodynamics.
Apart from its solvability, this deformation is interesting because it preserves an SL(2, R) L × U (1) R subgroup 2 of the original global conformal group, as can be seen from the fact that the deforming operator has dimension (1,2). Two-dimensional (local) QFTs with SL(2, R) L × U (1) R symmetry have been analysed in [12], where it was shown that, similarly to the case of two-dimensional CFTs [13], there is an infinite-dimensional enhancement of the global symmetry group. As one would expect, the left-moving conformal symmetry, SL(2, R) L , is enhanced to a full left-moving Virasoro algebra; the surprising finding of [12] was that the right-moving translations U (1) R are also enhanced to an infinite symmetry group, which can be either a left-moving U (1) Kač-Moody algebra or a right-moving Virasoro algebra, or both.
Two-dimensional QFTs with the former symmetry enhancement pattern are known as "warped CFTs" [14,15]. They are invariant under the change of coordinates where f (z), g(z) are two arbitrary holomorphic functions. Their properties have been studied in [14][15][16][17][18], and they were in particular shown to exhibit an interesting Cardy-like growth of their density of states. However, the theory defined via (2.1), to the extent that it can be understood as a quasilocal two-dimensional QFT (e.g., below the cutoff scale µ) such that the results of [12] apply, is rather expected to fall in the second category, where thez translations are enhanced to an infinite-dimensional right-moving Virasoro symmetry. The reason for this expectation is that the deformation parameter µ is continuous, and it would be quite surprising if the rightmoving Virasoro symmetry of the original CFT suddenly became a left-moving Kač-Moody symmetry as we infinitesimally turn on µ. This expectation can in principle be confirmed via holographic calculations [19] or directly, e.g. by using conformal perturbation theory. If the expectation is confirmed, then the theory defined via (2.1) would represent the first nontrivial example of a QFT with only SL(2, R) L × U (1) R global symmetry that is enhanced to Virasoro L × Virasoro R . However, in this article we will limit ourselves to defining the deformed theory, working out its spectrum and thermodynamics, and we leave the interesting question of symmetry enhancement to later work.
The plan of this paper is as follows. In section 2., we discuss the basics of the JT deformation and work out the spectrum and basic thermodynamic properties of the deformed theory in finite volume, after specializing to a purely (anti)chiral U (1) current. In section 3., we check our general prediction for the deformed spectrum in a concrete example involving deformed free fermions. Section 4. contains a discussion and future directions. In the appendix, we study an SL(2, R) L × U (1) R invariant deformation of the classical free boson and underline some of its interesting features.

The JT deformation
Consider a two-dimensional CFT with a U (1) symmetry generated by a conserved current J. We consider the following double-trace irrelevant deformation where J = J z andJ = Jz are the components of the U (1) current andT = Tzz and Θ = T zz are the components of the current generatingz translations. We assume, as in [1], that the resulting theory can be understood as a (quasi)local two-dimensional QFT below some scale, so the local currents (J,J ) and (T , Θ) continue to exist in the deformed theory. The currents satisfy the conservation equations The double-trace operator O JT is defined via the OPE This form of the OPE follows from the fact that both the z and z ′ derivatives of (2.3) can be shown, using the conservation equations (2.2), to only contain total derivative terms [20]. This allows for only one operator that is not a total derivative itself to appear in the OPE, and it must have a constant coefficient.
The theory also has a current T λ z associated with z translations, satisfying If the deformation preserves full SL(2, R) L invariance, then the current j λ L = T λ z · z is also conserved, implying that Tz z = 0, which is equivalent with holomorphy of T zz . The latter can in principle be checked using conformal perturbation theory, by evaluating the OPE The term linear in µ is obviously zero because the perturbing operator has h = 1. It seems reasonable that it will stay exactly marginal on the left at higher orders in µ, though it would be interesting to prove this statement, using techniques such as those of [21,22]. A holomorphic T = T zz implies the existence of an infinite-dimensional left-moving Virasoro symmetry enhancing the SL(2, R) L . Additionally, if the internal U (1) current is purely chiral (J = 0) or purely antichiral (J = 0), then we also expect an infinite enhancement of the U (1) symmetry to either a left or a right-moving U (1) Kač-Moody symmetry.

The finite-size spectrum
We now place the deformed theory on a cylinder of circumference R and study its spectrum, which in general will be discrete. We denote the cylinder coordinates as t, ϕ, with ϕ ∼ ϕ + R. We will mostly work in Euclidean signature (assuming we can Wick rotate), with Euclidean time τ = −it. The holomorphic coordinate z is given by 3 Following [1], we consider eigenstates of the Hamiltonian H, momentum operator P and charge Q which commute and thus can be simultaneously diagonalized. We denote these eigenstates collectively as |n H|n = E n |n , P |n = P n |n , Q|n = Q n |n (2.8) 3 The holomorphic coordinate on the cylinder is related to the non-compact coordinate on the plane via the usual map z pl = exp(−2πiz cyl /R), which is a symmetry of the theory. The map between the plane and the cylinder for SL(2, R) × U (1) invariant two-dimensional QFTs has been previously discussed in the context of warped CFTs in [16]. In that case, the second symmetry in (1.2) ensures that the theories on the plane and on the cylinder are equivalent; however, in our case this symmetry is absent, and it would be interesting to better understand the relationship between the two.
The expectation value of the deforming operator in the above eigenstate can be computed from the correlator C n (z, z ′ ) = n|J(z)T (z ′ ) −J(z)Θ(z ′ )|n (2.9) which can be shown to be independent of z, z ′ . In the limit z → z ′ , this correlator simply reduces to the expectation value of the deforming operator in the state |n , n|O JT |n . One can also evaluate C n (z, z ′ ) by inserting a complete set of energy-momentum-charge eigenstates |n ′ n ′ | in between the two operators Expanding around z, one can show that C n (z, z ′ ) can only be z ′ -independent if all contributions with n ′ = n cancel out [20]. Assuming the spectrum is non-degenerate 4 , we thus have It is useful to write this relation in terms of the components of the stress tensor along the Euclidean coordinates ϕ, τ . We have Note that since the theory is not Lorentz invariant, in general T ϕτ = T τ ϕ . However, due to the SL(2, R) L scaling symmetry we have argued for, we have instead Tz z = 0. This allows us to solve for T ϕτ in terms of the other components, so the above equations become 14) The current components read and the expectation value of O JT can be written as The addition of the operator O JT can be viewed as a deformation of the CFT Hamiltonian which implies that the energy levels E n will be changing as as we vary µ. As in [1,5], this will be the essential equation allowing us to compute the exact finite-size spectrum of the deformed theory. The expectation values of the current components in the translationally-invariant state |n are related to the corresponding conserved charges as Note that, in order to determine the spectrum, we still need an equation for n|J ϕ |n in terms of conserved quantities. However, unlike for the above, there does not appear to exist a general formula relating n|J ϕ |n to only the conserved charges. Therefore, we will determine the spectrum by making the additional simplifying assumption that the current is purely (anti)holomorphic. We treat each case separately below.

Purely holomorphic current
Assuming that the current is purely holomorphic, which implies J ϕ = −iJ τ , we find Thus, when changing µ infinitesimally, the energy levels change as which is the equation that we need to solve. Note that since P n is quantized, P n R ∈ Z, it cannot vary with µ. It is useful to introduce the left/right-moving energies E L,R In terms of E R n , the level equation is The left-moving energy is simply given by In the original undeformed CFT, the left/right energies are given by where h,h (and Q) label the undeformed CFT spectrum with h −h ∈ Z/2 and we dropped the index 'n'. Consequently, in the deformed theory the left/right energies will be Thus, the energies of all states that carry a non-trivial left-moving charge Q are modified: they grow for µQ > 0 and decrease for µQ < 0. States with Q = 0 are undeformed; in particular, all the left-moving ground states, which have Q = h = 0, are unchanged. Since in a generic CFT containing a U (1) current the lowest-lying charged state must have h < c/α + O(1) with α > 8 [23], the spectrum will necessarily be modified above this conformal dimension. The effective local description breaks down for R < µQ/2, or equivalently Q > 2R/µ. Under the natural assumption that the CFT spectrum is symmetric under Q → −Q, we find that the deformed finite-size theory breaks down at this radius for either sign of µ. This is quite different from the TT case, where one obtains radically different behaviour for each of the signs of µ. Also note that unlike in TT , we do not have a branch point singularity in the energy spectrum, but rather a pole at a finite energy. It would be interesting to understand the physical interpretation of this behaviour.

Purely antiholomorphic current
If we now assume that the current is purely antiholomorphic (J ϕ = +iJ τ ), we find which yields the following equation for the energy levels However, since in a CFT ε n is independent of R, we find that in this case the spectrum is entirely undeformed. This can be understood from the fact that in the original CFT, the operator T zz that enters the deformation is zero inside correlation functions, so all corrections to the partition function in conformal perturbation theory vanish.

Thermodynamics
Let us start with the case of a purely holomorphic current, where the energy levels are nontrivially displaced, as in (2.29). Since the spectrum is continuously deformed, we expect that the degeneracy is still given by the CFT formula 5 S = 2π which holds for h,h >> c. Here k is the level of the left Kač-Moody chiral algebra. Setting for simplicity the total momentum to zero, we have Therefore, in terms of E, Q, the expression for the entropy is The first law of thermodynamics reads Thus, the temperature is given by Notice this blows up as R approaches µQ/2. The "pressure" is as expected. Finally, the chemical potential Thus, we see that all the thermodynamic quantities diverge at R = µQ/2, as the gap between energy levels becomes infinite at small enough but finite radius. Effectively, it looks like the theory lives on a circle of radius R − µQ/2, rather than R. This observation can be formalized by noticing that, at least at a perturbative level, the JT deformation can be induced via a field-dependent diffeomorphism performed on the original two-dimensional CFT 6,7 Note this transformation would be a symmetry of warped CFTs, though it is not a symmetry here. 7 The variation of the action under the coordinate transformation x a → x a + ξ a (x) is given by which agrees with the expression used in the following sections, but differs from the usual definition via the coupling to a background vielbein, δS = d d x e Tµ a δe µ a , by a factor of e = √ g. (Since the stress tensor is not symmetric, it most naturally couples to a background vielbein).
Since the coordinates of the deformed theory are identified as z ∼ z + R,z ∼z + R, the identifications in the undeformed picture are which tells us that the right-movers experience a different size of the circle. We can also understand this as a change of the metric on which the CFT is placed [24] Assuming that the current is constant, J = Q/R, we find that the boundary metric develops closed timelike curves for R < µQ/2, which is another way to see the breakdown of the theory at this radius. The deformation also leads to a modification of the propagation speed of excitations, which can now become superluminal around certain backgrounds. The speed of the left/right-moving excitations in the metric (2.42) is Notice that the propagation speed for the right-movers is superluminal for µQ > 0. This can be understood from the fact that the interaction is repulsive in this range 8 [8,24]. Note that superluminal propagation does not in itself pose a problem, because the deformed theory is not Lorentz invariant. However, it does lead to problems in the finite volume theory, as (2.42) shows that points in the same constant time slice cannot be identified if Q > 2R/µ.
For the case of a purely right-moving current, we have at zero momentum which is identical to the entropy in a CFT with a right-moving Kač-Moody current of level k.
Notice that this deformation can be induced by the field-dependent coordinate transformation which is entirely antiholomorphic. Under it, the right-moving stress tensor picks up a factor proportional to µcJ ′′ (z). Since the latter integrates to zero, the energy levels are unchanged.
Thus, we find that the spectrum and thermodynamics of the deformed theory depend on the properties of the current by which we deform, e.g. chiral vs. anti-chiral. Below, we study some simple examples.

A simple example: deformed free fermions
In this section, we would like to exemplify and check our general findings from the previous section.
The simplest examples of theories where the U (1) current can be made purely chiral/ antichiral are fermionic ones. We treat the case of a purely chiral (left-moving) and purely antichiral (right-moving) U (1) current separately.

Purely left-moving current
We consider the following action describing two complex fermions At µ = 0, this simply describes a free left-moving complex femion ψ L and a free right-moving complex fermion ψ R . The purely holomorphic conserved current that we will be considering for the deformation is associated to the symmetry that rotates the left-moving fermions ψ L → e iα ψ L , ψ ⋆ L → e −iα ψ ⋆ L with the right-moving ones ψ R , ψ ⋆ R inert The components of the stress tensor are 9 Note that since the current is exactly holomorphic, the perturbation of the free fermion action takes the form of precisely JT . The equations of motion arē and their starred counterparts. Note that on-shell we have Tz z = 0, as expected. Also, we find that T zz = µ 2 J L Tzz. We start by solving for the currents J L , T R , which satisfȳ with the general solution The solution for the fermions themselves is To find the spectrum, we expand the fermions in modes, upon imposing appropriate boundary conditions. These will be either Ramond or Neveu-Schwarz The mode expansion for ψ R takes the form where b n represent fermionic creation/annihilation operators, satisfying and the constants γ n are normalization factors that will be determined shortly. The sum runs over n integer in the Ramond sector, and over n integer plus a half in the NS one. The shift in the radius in the denominator comes from the fact that under ϕ → ϕ + R, the argument of ψ R shifts by R − µQ/2, where we are considering states with fixed ψ where the momentum canonically conjugate to ψ R is and similarly for its complex conjugate. Since the operator J L commutes with ψ ⋆ R , there is no ordering ambiguity. We find 10 Finally, we can now check whether the energy spectrum agrees with what we have derived on general grounds. The right-moving energy is given by 10 We used the identity Note that as far as the non-zero modes of T R are concerned, the integrand equals ∂ ϕ S, where S has been defined in (3.12). Consequently, the energy only gets contributions from the zero modes, and reads Plugging in the mode expansion (3.14), the final expression for the right-moving energy takes the form where the expectation values is computed in an energy eigenstate, obtained by acting with a number of fermionic creation operators on the vacuum. We thus find a nice match with the general prediction (2.26). We can also match the prediction (2.29) for the left-moving energy Plugging in the solution (3.12) ψ L into T zz , we find It is easy to check it satisfies∂T zz = 0. Requiring (anti)-periodicity of the left-moving fermion solution (3.12), we find that ψ which only receives contributions from the zero modes of S. Requiring the absence of winding modes fixes S z.m. = T R z.m. (z − z). Since it involves (free) fermions of two different types, the correlator factorizes and we find which agrees with (2.27), obtained via the general analysis.

Purely right-moving current
We now consider the model where ψ 1 is a real two-dimensional fermion, ψ 2 is a complex fermion and J is the current associated to the symmetry which is purely antiholomorphic. The components of the stress tensor read The equations of motion imply that ψ 1,2 are anti-holomorphic, so T zz = 0 on-shell. It is not hard to check that the deformation takes the form of a JT -type deformation, since the Lagrangian satisfies [5] upon taking into account the Grassman nature of ψ 2 . The action (3.27) differs from the free fermionic action for ψ 1,2 by the purely antiholomorphic coordinate transformation 11z Note thatz ′ ∼z ′ + R ′ , where R ′ = R − µQ/2 in the superselection sector of charge Q = − R 0 dϕJ . It is clear that the contribution of ψ 2 to the energy levels is the same as in the free theory. To find the contribution of ψ 1 , we expand it in modes with m an integer for Ramond boundary conditions and an integer plus a half for NS ones. The contribution of ψ 1 to the right-moving energy is then The two factors of 1+µJ/2 are cancelled by two corresponding factors of R ′−1 = (R−µQ/2) −1 , one from the normalization of ψ 1 and one from thez ′ derivative. Therefore, we find that the energy spectrum is identical to that in the undeformed model, as expected.

Discussion and future directions
We have studied an irrelevant, yet integrable deformation of a general two-dimensional CFT possessing a chiral U (1) current and worked out the finite-size spectrum and the thermodynamics of the resulting quasilocal QFT, in the special cases of a purely chiral/anti-chiral U (1) current. In the chiral case, we found that the deformation acts non-trivially on the spectrum and may be induced via a field-dependent coordinate transformation on the original CFT that mixes left and right-movers. In the anti-chiral case, the spectrum is unchanged, and this can be understood from the fact that the deformation corresponds to a field-dependent, but purely antiholomorphic coordinate transformation of the original CFT, which is a symmetry. The discussion below will thus refer only to the non-trivial chiral case. As in the TT case, the field-dependent coordinate transformation induces a change in the speed of sound, which can now be made superluminal for both signs of the deformation parameter µ. While superluminality is not in itself a problem due to the lack of Lorentz invariance, note that this can lead to closed timelike curves in the finite-volume theory; in addition, we find that various thermodynamic quantities diverge if the circle on which the theory is placed has radius R < µQ/2. Since both problems disappear as R → ∞, we can still hope that the theory on the plane makes sense.
Given that the deformation involves following the RG flow upwards, understanding the UV behaviour of the deformed theory is non-trivial. In the TT case, much progress has been made using the S-matrix approach; more precisely, it was shown that S-matrix of the deformed theory only differs by an energy-dependent phase factor from the original one [?]. This phase factor modifies the high-enegy asymptotic behaviour of the S-matrix, preventing the UV completion from being a usual local QFT; however, it has been argued that such an asymptotic behaviourtermed asymptotic fragility -does not obviously lead to an inconsistency and should sometimes be allowed, e.g. in a theory of quantum gravity. It would be very interesting to investigate whether the effect of the JT deformation on the S-matrix can be similarly encompassed by a phase, and understand the type of asymptotic UV behaviour to which it leads.
Better understanding the UV behaviour of the deformed theory is also important from a technical point of view. In particular, our derivation of the deformed spectrum assumed that the spacetime and internal symmetries are associated to local conserved currents, i.e. that the deformed CFT can be approximated as a quasilocal QFT. This approximation is expected to break down at high enough energies, and we should gain a better understanding of its regime of validity. This limitation of our derivation exactly parallels that of [1] for the TT case, which is also expected to break down when the non-local nature of the deformed theory takes over. One important difference with respect to TT is that in our case the deformation parameter is a null vector, so one may expect that the non-localities are only restricted to the x − direction, while locality in x + is preserved. Note also that in the TT case, even though the derivation of [1] of the deformed spectrum may break down at scales of order µ, the definition of the deformed theory via the S-matrix does appear to hold up to arbitrarily high energies, which is another reason that it would be worth finding such an alternative definition in the JT case.
Another technical point that would deserve a proper proof is to show that the deformed theory has exact SL(2, R)× U (1) invariance, which is equivalent to showing that the deforming operator is exactly marginal with respect to the left conformal symmetries to all orders in conformal perturbation theory. We leave such a proof to later work.
As we already mentioned, one reason that JT -deformed CFTs are interesting is that they may provide a first example of an SL(2, R)×U (1)-invariant QFT where the U (1) is enhanced to a right-moving Virasoro symmetry. This expectation relies on our ability to treat the deformed CFTs as local two-dimensional QFTs (see discussion above), such that the results of [12] apply. That the right-moving translation symmetry should be enhanced to a full Virasoro (which in particular includes rescalings) sounds extremely counter-intuitive. In order to bring some support for this claim, in the appendix we work out a simple example involving a deformed classical free boson. We show that, indeed, the stress tensor can be made purely antiholomorphic on-shell, which is consistent with full Virasoro enhancement in the quantum theory. It would be extremely interesting to find this additional Virasoro in conformal perturbation theory.
It is worth emphasizing that, even if we find the above Virasoro symmetry, the underlying field theory is not a usual two-dimensional CFT, as can be concluded from its behaviour in finite volume. However, due to the potential non-localities discussed above, it remains to be seen to what extent this Virasoro symmetry exists and how it acts on the space of states.
Some other physical and technical points that would be worth understanding further are how to derive the spectrum in the case of more general U (1) current, which is neither chiral nor anti-chiral, and also consider more general, e.g. non-abelian currents. Also, it would be interesting to better understand the relationship between the theory on the cylinder and that on the plane and how JT -deformed CFTs relate to warped CFTs.
Finally, let us end with some speculations on the possible implications of the JT deformation for holography. Two-dimensional holographic QFTs with SL(2, R) × U (1) invariance and an infinitely-enhanced symmetry group have been extensively discussed in connection with the Kerr/CFT correspondence [25], a proposed microscopic description of maximally spinning black holes in our galaxy. The near-horizon region of the Kerr black hole is captured by a particular deformation of AdS 3 known as "warped AdS 3 ". A holographic study of this spacetime suggests that its holographically dual QFT is a deformation of a two-dimensional CFT by an irrelevant (1, 2) operator [26,27] and, just like in the case of the TT -type deformations discussed by [1], one is supposed to "go up" the RG flow. It is currently not understood what singles out the particular trajectory "up the RG flow" relevant for the Kerr/CFT case beyond the leading order in the deformation parameter µ, though one piece of information is that it preserves the Cardy form of the thermal entropy. Also, the analysis of the asymptotic symmetries of warped AdS 3 spacetimes show an enhancement of the U (1) right-moving translation symmetry to a full Virasoro symmetry.
The irrelevant operator used in defining the Kerr/CFT-type deformation is not JT , since it needs to be a single-trace operator 12 . However, the two deformations do appear to have many features in common, such as the flow up the RG direction, the Cardy form of the entropy, the likely Virasoro enhancement of right-moving translations and the appearance of closed timelike curves in the finite volume theory [28]. Given that the JT deformation is defined also for finite µ, we may hope to learn important lessons about Kerr/CFT by studying this much simpler theory. For example, as in JT , the Cardy formula may simply follow from the fact that the spectrum is continuously deformed as a function of µ; as for the Virasoro symmetry, we hope to obtain a concrete handle on establishing its existence, the degree to which it is a true symmetry, and compare with its holographic realisation.

Acknowledgements
The author is grateful to C. Aron, J.

A. Notes on the deformed free boson
In this appendix, we study the JT deformation of a two-dimensional free scalar, X, where J is the current associated to shifts in X. This model is more complicated than the fermionic ones we studied in the main text -in particular, J does not have definite chirality -and for now we do not have a solution for its spectrum. However, we find it worth pointing out a few facts, such as: i) the model is again related via a field-dependent coordinate transformation to the original free scalar; ii) at least for a simple class of solutions, parametrized by momentum and winding, we can explicitly check that the deformed energy spectrum obeys (2.18); iii) the right-moving stress tensor can always be improved (via a local on-shell redefinition) such that it becomes purely antiholomorphic.
The last property applies to all the models described by the action (A.1) and brings preliminary evidence, so far at a purely classical level, that the right-moving translations are enhanced to a full right-moving Virasoro symmetry, in agreement with the results of [12].
We start from the action where F is some arbitrary function with F(0) = 1 and which admits a Taylor expansion around zero. By construction, this theory has SL(2, R) L × U (1) R invariance. The stress tensor is given by As expected from the SL(2, R) L invariance, the component Tz z = 0. The equation of motion is and is not hard to check that T zz is holomorphic on-shell.

The JT deformation
Let us now consider the U (1) current associated to the shift symmetry X → X + const If we would like the theory (A.1) to correspond to the JT deformation, then we need which should be true for any λ. Plugging in the expressions for J a and T ab , we find that F ′ = 1 4 F 2 , which allows us to solve for F Note that this form of the Lagrangian can be obtained from the free boson CFT by performing the field-dependent coordinate transformation For this particular choice of F, the equation of motion simplifies to To find the classical solutions, we note the above can also be written as∂(∂XF) = 0, from which we find that A general solution is where f, g are arbitrary functions.

Basic spectrum check
In principle, we could expand the above solution in modes (imposing the appropriate periodicity conditions), compute the conserved charges Q, P and E and check whether the µ dependence of the energy, at fixed P, Q, obeys (2.18). However, this appears tedious in practice, given the non-linearity of (A.11). We will therefore concentrate on a very simple solution λ 2 Q 2 − 2λQR (A.14) 13 In order to gain an extra parameter (winding), we are considering compact X.

Improvement of the stress tensor
As is well known, the Noether procedure for constructing the stress tensor as the conserved current associated to translations may not yield a tensor with all the desired symmetry properties. It is possible in certain cases to 'improve' the stress tensor, enhancing its symmetries while leaving the conservation equations untouched. We will be interested in whether the right-moving part of the stress tensor can be improved T zz →T zz = T zz − ∂A , Tzz →Tzz = Tzz +∂A (A. 17) such thatT zz = 0. In that case,Tzz will be entirely antiholomorphic on-shell. The fact that the right-moving stress tensor can be improved such that it is antiholomorphic on-shell is true in all the theories of the form (A.1), not just for the JT deformation. To show this, we assume F(λ∂X) has a power series expansion of the form F(x) = 1 + ax + bx 2 + cx 3 + . . .

(A.19)
The zeroth order solution is given by .20) and the higher order ones etc, where we have set to zero the purely (anti)holomorphic solutions at higher order. The left-moving stress tensor, evaluated on this solution, is simply and is purely holomorphic, as expected. Next, we plug these formulae into the series expansion of T zz and find that the deformation is easily integrable with respect to z, if we choose The new antiholomorphic stress tensor is 25) This confirms, at the classical level, the prediction of [12] that it is possible to redefine the stress tensor current so as to have a full RM Virasoro symmetry. Thus, the Virasoro symmetry does appear to exist explicitly. However, a curious feature of this redefinition is that while the energy is unchanged, the fact thatT zz = 0 implies thatT xx = −T τ τ , or ∂ R E = −E/R, which the deformed spectrum is unlikely to satisfy.