#  $J\bar{T}$ \- deformed CFTs as non-local CFTs

Monica Guica

_Université Paris-Saclay, CNRS, CEA,_ _Institut de Physique Théorique, 91191
Gif-sur-Yvette, France_

###### Abstract

Various holographic set-ups in string theory suggest the existence of non-
local, UV complete two- dimensional QFTs that possess Virasoro symmetry, in
spite of their non-locality. We argue that $J\bar{T}$\- deformed CFTs are the
first concrete realisation of such “non-local CFTs”, through a detailed
analysis of their classical and quantum symmetry algebra.

Classically, the symmetries consist of an infinite set of left-moving
conformal and affine $U(1)$ transformations that generate a Witt-Kac-Moody
algebra, as well as a set of non-local, field-dependent generalizations of
right-moving conformal and affine $U(1)$ transformations, whose algebra
depends on the chosen basis. Notably, there exists a basis, denoted as the
“flowed” representation, in which the right-moving charge algebra is simply
Witt-Kac-Moody. At the quantum level, we provide a concrete prescription for
constructing the symmetry generators via a combination of the flow equations
they satisfy and the Sugawara construction, and use this to explicitly resolve
the ordering ambiguities and the quantum corrections to the generators up to
second order in the $J\bar{T}$ coupling parameter. This construction naturally
produces the “flowed” generators, whose algebra is Virasoro-Kac-Moody to all
orders in the coupling, with the same central extension as that of the
undeformed CFT. We use this input to work out the quantum modifications to the
“unflowed” generator algebra.

A peculiarity of the Virasoro generators we study is that their zero mode does
not equal the Hamiltonian, but is a quadratic function of it; this helps
reconcile the Virasoro symmetry with the non-locality of the model. We argue
that also $T\bar{T}$ \- deformed CFTs posses Virasoro symmetry, and discuss
criteria for the existence of such a symmetry in more general non-local QFTs.

###### Contents

  1. 1 Introduction
  2. 2 Classical conserved charges and their algebra
    1. 2.1 Infinite classical symmetries of $J\bar{T}$ \- deformed CFTs on a cylinder
    2. 2.2 Poisson bracket algebra and flow of the symmetry generators
    3. 2.3 Spectral flow-invariant generators and a Sugawara-type construction
  3. 3 The quantum symmetry algebra
    1. 3.1 Definitions of the quantum symmetry generators
    2. 3.2 Perturbative construction of the quantum symmetry generators
    3. 3.3 Algebra of the unflowed generators
    4. 3.4 Comments on representation theory
  4. 4 More general non-local CFTs
  5. 5 Discussion
  6. A Properties of $\alpha_{m}^{l,r}$

##  1 Introduction

Decoupling limits play an important role in the derivation of holographic
dualities from string theory, most notably of the AdS/CFT correspondence [1].
However, there exist many decoupling limits in which the spacetime is not
asymptotically AdS [2, 3, 4]; rather, it corresponds to a non-normalizable
deformation of it. A naïve application of holographic reasoning then suggests
that such backgrounds are dual to irrelevant deformations of a CFT that
nevertheless produce UV complete QFTs. The resulting theory is non-local at
the scale set by the irrelevant deformation.

This idea has been realised concretely in a handful of examples, where the
irrelevant deformation could be rewritten as a star product deformation of the
original CFT Lagrangian [5, 6]. However, in general one does not have access
to such explicit additional structures and needs to deal directly with the
irrelevant deformation. This is extremely daunting, not least because
specifying the irrelevant flow is highly non-trivial. Not even maximal
supersymmetry can solve the problem [7].

A particularly interesting class of decoupled backgrounds correspond to
irrelevant deformations of a two-dimensional CFT by a $(1,2)$ operator [8, 9,
10]. These backgrounds are related to extremal black holes [10]. As mentioned,
it is notoriously hard to specify the flow driven by an irrelevant
deformation; however the existence of a decoupling limit suggests that the
resulting theory is UV complete. Interestingly, in all these set-ups,
asymptotic symmetry group analyses suggest the existence of two sets of the
infinite-dimensional Virasoro symmetry, even though the dual QFTs are
intrinsically non-local [11].

Symmetries, of course, provide a powerful tool of classification and control
over the behaviour of quantum field theories. Conformal symmetries, in
particular, allow one to in principle determine all correlation functions of
local operators in terms of a set of data encoded in two and three-point
functions [12]; these constraints are even more powerful in two dimensions,
because the conformal group is infinite-dimensional [13]. It would thus be
extremely interesting to establish whether the above conjecturally UV complete
QFTs do posses Virasoro symmetry, since it may provide the “additional
structure” needed to specify the irrelevant deformations. If it is present, we
will denote these theories as “non-local CFTs”111The example studied here, as
well as those of [8, 9, 10], posses an additional global $SL(2,\mathbb{R})$
symmetry. Relatedly, they are only ‘half non-local’. They have been previously
called “dipole CFTs”, in analogy with the four-dimensional example [6]. .

It is however not clear, _a priori_ , whether Virasoro symmetries are
compatible with non-locality. Since much of the evidence in favour of their
existence in non-local theories comes from asymptotic group analyses -
computations in the dual classical gravitational picture that suffer from
multiple limitations and ambiguities - it is certainly a logical possibility
that the Virasoro symmetry they predict is simply an artifact of the classical
gravity approximation or, worse, that the gravitational calculations miss
subtleties in the construction that ultimately make the charges ill-defined.
Thus, in order to be able to explore this interesting possibility, it is
crucial to have a well-defined, _QFT_ example of a non-local CFT.

The main claim of this article is that $J\bar{T}$ \- deformed CFTs [14], a
class of universal irrelevant deformations of two-dimensional CFTs that are in
principle well-defined from a quantum-field-theoretical point of view [15],
provide the _first concrete example_ of a non-local (dipole) CFT. We will
establish this fact through a detailed analysis of their classical and quantum
symmetries. Since certain properties of these symmetries may appear somewhat
peculiar, we start by reminding the reader a few basic facts about the
definition of a symmetry.

At the classical level, the simplest way to define a (continuous) symmetry is
as a transformation of the fields and, eventually, the coordinates, that
leaves the action invariant _off-shell_. One can then employ the algorithmic
Noether procedure to construct, for any given Lagrangian, a current and
associated charge corresponding to the given symmetry, which are conserved on-
shell.

Symmetries can also be defined in the Hamiltonian formalism, usually as
corresponding to functions on phase space whose Poisson bracket with the
Hamiltonian vanishes. This requirement must be appropriately modified in
presence of space-time dependent symmetries such as, for example, boosts,
whose Poisson bracket with the Hamiltonian does not vanish; instead

$\\{H,M^{0i}\\}=P^{i}$ (1.1)

i.e., the commutator is a linear combination of the generators of the Poincaré
group. In this case, one can build a conserved charge by including an explicit
time dependence in the generator222In the case of Galilean boosts, for
example, the conservation law of the explicitly time-dependent generator
equates the total mass $\times$ the position of the center of mass of the
system to the total momentum $\times$ time. . The general requirement for
function $Q$ on phase space to correspond to a symmetry is then

$\partial_{t}Q-\\{H,Q\\}=0$ (1.2)

_off-shell_. Note that the left-hand side of this equation simply equals
$\frac{dQ}{dt}$ upon using Hamilton’s equations of motion, thus expressing the
expected on-shell conservation of the charge. All this does, of course, is to
replicate the conserved charges obtained via the Noether procedure, which in
the case of space-time symmetries are often explicitly time-dependent, in the
Hamiltonian formalism.

The quantum-mechanical definition of symmetries parallels the discussion
above, in that the symmetry generators are usually expected to commute with
the Hamiltonian. However, in the case of space-time symmetries, this
requirement is relaxed to allow e.g. for commutators that are linear
combinations of the symmetry generators, by including some explicit time
dependence. The generators should then satisfy

$\left(\frac{\partial L_{S}}{\partial
t}\right)_{H}+\frac{i}{\hbar}[H,L_{H}]=0$ (1.3)

where the subscripts $S,H$ denote the Scrödinger and, respectively, the
Heisenberg picture operator. Upon using Heisenberg’s equation of motion, the
above condition encodes the time-independence of the Heisenberg picture
operator $L_{H}$, which implies that its expectation value is conserved in any
state.

As advertised, the focus of this article is on the symmetries of $J\bar{T}$ \-
deformed CFTs, a class of solvable [14, 16, 17, 18] irrelevant deformations of
two-dimensional CFTs by an operator bilinear in a $U(1)$ current and the
right-moving stress tensor. $J\bar{T}$ \- deformed CFTs correspond to a
particular case of the Smirnov-Zamolodchikov deformations [15], which also
include the well-studied $T\bar{T}$ deformation [15, 19, 20, 21, 22]. Our
reasons for concentrating on this particular deformation are two-fold: first,
$J\bar{T}$ \- deformed CFTs are interesting in their own right as solvable,
likely UV complete QFTs with non-standard UV behaviour, which are directly
relevant to understanding the holographic description of extremal black holes
[10, 23]; second, the structure of the deformed theory is particularly simple,
in that the deformation preserves an $SL(2,\mathbb{R})$ “left” subgroup of the
original conformal group. The resulting theory is thus local and conformal on
the left, while all the non-locality induced by the irrelevant deformation is
concentrated on the remaining null right-moving direction - a structure which
is significantly simpler and easier to control than, for example, that of
$T\bar{T}$.

The extended symmetries of (classical) $J\bar{T}$ \- deformed CFTs were first
investigated from a field-theoretical perspective in [24]. On the local left-
moving side, the global conformal and $U(1)$ symmetries were enhanced to form
a Witt-Kac-Moody algebra, as expected. Interestingly, also the non-local
right-moving side exhibited an infinite number of symmetries, implemented by a
field-dependent generalization of right-moving conformal and affine $U(1)$
transformations. These symmetries were denoted as “pseudo-conformal” in [24].
A subsequent analysis [25] noted that in finite volume, the generators of [24]
were inconsistent with charge quantization and proposed to slightly modify
them, which resolved the issue.

A peculiarity of the new finite-volume generators is that their Poisson
bracket with the Hamiltonian takes the form

$\\{H,Q\\}=f(H-P,J_{0})\,Q$ (1.4)

where $f$ is a known, non-trivial function of its arguments, which include the
momentum $P$ and a global $U(1)$ charge, $J_{0}$. Even if this bracket may
appear unusual, the charges are conserved off-shell via (1.2).

The modification proposed by [25] does affect the charge algebra, in a way
that will be spelled out explicitly in this article. Nevertheless, [25] showed
that a particular non-linear combination of the new generators, which highly
resembles a “spectral flow by the right-moving energy” and is singled out by
satisfying, in the classical limit, the same flow equation as the energy
eigenstates, still satisfies a Witt-Kac-Moody algebra. The zero mode of this
algebra is however not the Hamiltonian, but a non-linear function of it, which
provides perhaps the easiest way to understand the appearance of the non-
trivial commutator above. The analysis of [25] was classical, though fully
non-linear in $\lambda$.

In this article, we investigate the survival of these symmetries at the
quantum level. For this, we need to resolve any normal-ordering issues that
arise in defining the quantum generators, as well as to determine their
algebra, including central extensions and possible quantum corrections to the
structure constants, which are determined by their classical Poisson bracket
counterparts only up to leading order in $\hbar$. As in the classical case, a
peculiarity of the quantum generators we study is that their commutator with
the Hamiltonian takes the form (1.4). Even though this is not linear in the
generators, as we are accustomed to, it is trivial to give them an explicit
time dependence, so that they satisfy the conservation requirement (1.3). The
only difference with respect to usual spacetime-dependent symmetries, such as
boosts, is that the time-dependent multiplicative factors are now operator-
valued, whereas before they were c-numbers. In the classical limit, these
time-dependent generators reduce to the classical conserved charges, which are
known explicitly at full non-linear level in $\lambda$.

This paper is organised as follows. In section 2, we review the classical
symmetry generators in compact space, following [24, 25], and provide several
different useful ways to write their algebra. In section 3, we discuss
possible definitions of the quantum symmetry generators and use them to
explicitly construct the generators up to second order in the $J\bar{T}$
coupling parameter. We also work out the algebra of a set of generators
denoted as the “unflowed” ones to all orders in $\hbar$, finding non-trivial
modifications with respect to the semiclassical result, and discuss the
organisation of states into representations of the symmetry algebra. In
section 4, we extract a set of general criteria for the existence of Virasoro
symmetry in a non-local, UV complete QFT, and use them to argue that more
general theories, such as $T\bar{T}$ \- deformed CFTs, also posses Virasoro
symmetry. We conclude in 5 with a short discussion. In appendix A, we gather a
few useful formulae.

##  2 Classical conserved charges and their algebra

In this section, we review and clarify the results of [24] and [25] on the
infinite classical symmetries of $J\bar{T}$ \- deformed CFTs. In 2.1, we start
with a brief review of $J\bar{T}$ \- deformed CFTs, we then list the necessary
conditions for having a certain infinite set of symmetries and we display the
explicit expressions for the associated conserved charges. In section 2.2, we
present the Poisson bracket algebra of these charges and the flow equations
they satisfy; in 2.3, we display an equivalent way of encoding the same
information, which will be useful in the subsequent quantum analysis. We work
exclusively in the Hamiltonian formalism, which is best suited for studying
the algebra of the conserved charges.

###  2.1 Infinite classical symmetries of $J\bar{T}$ \- deformed CFTs on a
cylinder

In the Hamiltonian formalism, $J\bar{T}$ \- deformed CFTs are defined via the
flow equation

$\partial_{\lambda}\mathcal{H}=-\epsilon^{\alpha\beta}J_{\alpha}T_{\beta V}$
(2.1)

where $\mathcal{H}$ is the Hamiltonian density, $J$ is a $U(1)$ current,
$\lambda$ is the deformation parameter, and $T_{\beta V}$ is the generator of
translations in the null direction $V$. The theory is defined on the plane or
the cylinder (of circumference $R$), with coordinates $t,\sigma$. We will also
employ the null coordinates $U,V=\sigma\pm t$.

The components of the stress tensor are determined from the Hamiltonian in the
usual way. If we model the $U(1)$ current as a topologically conserved current
for a boson $\phi$, which should be generally possible, and assume that the
undeformed theory at $\lambda=0$ is a CFT, then the above equation can be
solved exactly, with solution333The derivation of [24] was for level $k=1$.
The result for general $k$ can be simply obtained via the rescaling
$\mathcal{J}_{+}\rightarrow\mathcal{J}_{+}/\sqrt{k}$ and
$\lambda\rightarrow\lambda\sqrt{k}$.  [24]

$\mathcal{H}_{R}=\frac{2}{\lambda^{2}k}\left(1-\lambda\mathcal{J}_{+}-\sqrt{(1-\lambda\mathcal{J}_{+})^{2}-\lambda^{2}k\mathcal{H}_{R}^{(0)}}\right)$
(2.2)

In the above, $\mathcal{H}_{L,R}=\frac{1}{2}(\mathcal{H}\pm\mathcal{P})$ are
the left- and, respectively, right-moving Hamiltonian densities,
$\mathcal{H}^{(0)}$ is the Hamiltonian of the undeformed CFT and
$\mathcal{J}_{\pm}=\sqrt{k}(\pi\pm\phi^{\prime})/2$, where
$\phi^{\prime}=\partial_{\sigma}\phi$ and $\pi$ is the conjugate momentum to
$\phi$. This is a universal formula for the classical $J\bar{T}$ \- deformed
Hamiltonian density to all orders in $\lambda$ that only depends on the
universal conserved currents of the seed theory.

The Poisson brackets of the Hamiltonian currents $\mathcal{H}_{L,R}$ follow
from the Poisson brackets of their undeformed counterparts, see [24] or [25]
for the explicit expressions. In particular, their commutation relations with
the total Hamiltonian $H=\int d\sigma(\mathcal{H}_{L}+\mathcal{H}_{R})$ are
given by

$\\{H,\mathcal{H}_{L}\\}=-\partial_{\sigma}\mathcal{H}_{L}\;,\;\;\;\;\;\;\;\\{H,\mathcal{H}_{R}\\}=\partial_{\sigma}\left(\mathcal{H}_{R}\,\frac{1+\lambda\mathcal{J}_{+}+\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}{1-\lambda\mathcal{J}_{+}-\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}\right)$
(2.3)

The first of these equations implies the holomorphic conservation of
$\mathcal{H}_{L}$ on-shell. There is also a holomorphic $U(1)$ current, with

$K_{t}=K_{\sigma}=\mathcal{J}_{+}+\frac{\lambda
k}{2}\mathcal{H}_{R}\;,\;\;\;\;\;\;\;\;\\{H,K_{t}\\}=-\partial_{\sigma}K_{t}$
(2.4)

We will also consider a right-moving $U(1)$ current $\bar{K}_{\alpha}$,
defined in [24], whose time component is given by

$\bar{K}_{t}=\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R}\;,\;\;\;\;\;\;\;\;\;\;\\{H,\bar{K}_{t}\\}=\partial_{\sigma}\left(\bar{K}_{t}\,\frac{1+\lambda\mathcal{J}_{+}+\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}{1-\lambda\mathcal{J}_{+}-\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}\right)$
(2.5)

As reviewed in the introduction, symmetries correspond to functions on phase
space that satisfy

$\partial_{t}Q-\\{H,Q\\}=0$ (2.6)

off-shell. It is easy to see that the quantities

$Q_{f}=\int d\sigma f(U)\mathcal{H}_{L}\;,\;\;\;\;\;\;\;P_{\eta}=\int
d\sigma\eta(U)\left(\mathcal{J}_{+}+\frac{\lambda k}{2}\mathcal{H}_{R}\right)$
(2.7)

labeled by the _arbitrary_ functions $f,\eta$ of the null coordinate
$U=\sigma+t$, do satisfy the requirement (2.6). These represent the expected
left conformal and affine $U(1)$ symmetries of $J\bar{T}$ \- deformed CFTs, as
can be checked by computing the change in $\phi$ induced by these
transformations. Their Poisson bracket algebra is the Witt-Kac-Moody algebra.
On a cylinder, which is the main setup of this article, we will work in the
Fourier basis $f_{n}=e^{inU/R}$, and denote the corresponding conserved
charges as $Q_{n},P_{n}$. The zero mode of $P_{\eta}$ will be denoted as
$Q_{K}=J_{0}+\lambda kE_{R}/2$.

These Witt-Kac-Moody symmetries were to be expected, based on the locality and
left-moving conformal invariance of the model. More non-trivially, [24] found
that the action was also invariant under a set of field-dependent coordinate
transformations, of the form

$V\rightarrow V-\epsilon\bar{f}(v)$ (2.8)

where $v$ is a possibly field-dependent coordinate. The generator of this
transformation is

$\bar{Q}_{\bar{f}}=\int d\sigma\bar{f}(v)\mathcal{H}_{R}$ (2.9)

as expected from the fact that $\mathcal{H}_{R}=-T_{tV}$. The condition (2.6)
that these transformations correspond to a symmetry for any function $\bar{f}$
can be written as a condition on $v$, and reads

$\frac{\partial_{t}v-\\{H,v\\}}{\partial_{\sigma}v}=-\frac{1+\lambda\mathcal{J}_{+}+\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}{1-\lambda\mathcal{J}_{+}-\frac{\lambda^{2}k}{2}\mathcal{H}_{R}}=-\frac{1-\lambda\\{H,\phi\\}}{1-\lambda\phi^{\prime}}$
(2.10)

where for the first equation we used (2.3), while the second is simply an
identity444This identity follows from the Poisson bracket
$\\{H,\phi\\}=-\frac{2\mathcal{J}_{-}+\lambda
k\mathcal{H}_{R}}{1-\lambda\mathcal{J}_{+}-\lambda^{2}k\mathcal{H}_{R}/2}-\mathcal{J}_{+}+\mathcal{J}_{-}$..
Thus, if $v$ satisfies the equation above, we obtain an infinite set of
classical symmetries, since the function $\bar{f}$ is arbitrary. We call these
symmetries pseudoconformal, since they are implemented by a field-dependent
modification of right-moving conformal transformations.

One can furthermore show that the quantities

$\bar{P}_{\bar{\eta}}=\int
d\sigma\bar{\eta}(v)\left(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R}\right)$ (2.11)

are also conserved, provided $v$ satisfies the constraint (2.10). These are
conserved charges associated to a set of field-dependent affine
transformations that generalize the right-moving $U(1)$. See [24] for more
details. The zero mode of $\bar{P}_{\bar{\eta}}$ will be denoted as
$\bar{Q}_{\bar{K}}=\bar{J}_{0}+\lambda kE_{R}/2$. Note that we have dropped
the superscript ‘KM’ with respect to the notation in [24].

The simplest solution to the equation (2.10) above is

$v=\sigma-t-\lambda\phi$ (2.12)

The Poisson brackets of the corresponding charges (2.9), (2.11) yield another
copy of the Witt-Kac-Moody algebra, which entirely commutes with the left-
moving one [24].

To study states and representations, it is natural to consider the charge
algebra of $J\bar{T}$ \- deformed CFTs on a cylinder, taken to have
circumference $R$. Charge conservation then requires that $\bar{f}$ be
periodic, which amounts to considering a basis of periodic functions
$\bar{f}_{n}=e^{-inv/R_{v}}$, where $R_{v}$ is the field-dependent radius of
$v$. From the definition (2.12), we have

$R_{v}=R-\lambda w$ (2.13)

where $w$ is the winding of the field $\phi$ around the circle. This
introduces explicit factors of $R_{v}$ in the algebra of Fourier modes, as
discussed in [24]. It is the rescaled generators $R_{v}\bar{Q}_{n}$ that
satisfy the usual Witt algebra at the level of Fourier modes.

However, in the case of $J\bar{T}$ \- deformed CFTs on a cylinder, it was
pointed out in [25] that the Poisson brackets of the right-moving generators
$\bar{Q}_{\bar{f}},\bar{P}_{\bar{\eta}}$ above with the global $U(1)$ charge
and the momentum are inconsistent with the quantization of these charges in
the deformed theory. The problem roughly stems from the fact that the field-
dependent coordinate $v$ above contains the zero mode
$\phi_{0}=\frac{1}{R}\int d\sigma\phi$ of the field $\phi$, whose non-trivial
commutation relations with the global $U(1)$ charges

$J_{0}\equiv\int d\sigma\,\mathcal{J}_{+}\;,\;\;\;\;\;\;\bar{J}_{0}\equiv\int
d\sigma\,\mathcal{J}_{-}$ (2.14)

lead to the above inconsistency.

This problem can be remedied by noting that the equation (2.10) only
determines $v$ up to a constant term. In [25], it was proposed to modify $v$
as

$v\rightarrow
v_{imp}=\sigma-t-\lambda\phi+\frac{\lambda\widetilde{\phi}_{0}R_{v}}{R-\lambda
Q_{K}}-\frac{\lambda(Q_{K}R_{v}+\bar{Q}_{\bar{K}}R)}{R(R-\lambda
Q_{K})}t=v+\frac{\lambda(\widetilde{\phi}_{0}-\alpha\,t)R_{v}}{R-\lambda
Q_{K}}$ (2.15)

where $\widetilde{\phi}_{0}$ is the spectral flow generator in the full
$J\bar{T}$ \- deformed CFT, which takes the form

$\widetilde{\phi}_{0}=\phi_{0}-\frac{\lambda}{R_{v}}\int
d\sigma\left(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R}\right)\,\hat{\phi}$ (2.16)

where $\hat{\phi}\equiv\phi-w\sigma/R$ is the scalar with its winding mode
removed, which is thus single-valued on the circle. As shown in [25] ,
$\widetilde{\phi}_{0}$ has remarkably simple Poisson brackets with the
symmetry generators (2.7), (2.9) and (2.11), namely

$\\{\widetilde{\phi}_{0},Q_{f}\\}=\frac{P_{f}}{R}\;,\;\;\;\;\;\\{\widetilde{\phi}_{0},P_{\eta}\\}=\frac{k}{2R}\int
d\sigma\eta\;,\;\;\;\;\;\\{\widetilde{\phi}_{0},\bar{Q}_{\bar{f}}\\}=\frac{\bar{P}_{\bar{f}}}{R_{v}}\;,\;\;\;\;\;\\{\widetilde{\phi}_{0},\bar{P}_{\bar{\eta}}\\}=\frac{k}{2R_{v}}\int
d\sigma\bar{\eta}(1-\lambda\phi^{\prime})$ (2.17)

Remarkably, the new field-dependent coordinate $v_{imp}$ still satisfies
(2.10), since the redefiniton (2.15) does not affect either the space, nor the
time derivative of $v$. The former is obvious; the latter follows from

$\partial_{t}\Delta v-\\{H,\Delta
v\\}=-\frac{\lambda(Q_{K}R_{v}+\bar{Q}_{\bar{K}}R)}{R(R-\lambda
Q_{K})}-\frac{\lambda R_{v}}{R-\lambda Q_{K}}\\{H,\widetilde{\phi}_{0}\\}=0$
(2.18)

derived using (2.17), where $\Delta v=v_{imp}-v$. Even if the change
$v\rightarrow v_{imp}$ looks rather innocuous in the Hamiltonian formalism,
its impact on the transformation of the fields such as $\phi$ is quite non-
trivial, due to the additional commutator $\\{v_{imp},\phi\\}$. This
introduces, in addition to the field-dependent coordinate shift by
$\bar{f}(v_{imp})$, a fully non-local term in the transformation law for
$\phi$, whose physical interpretation in the Lagrangian formalism is not very
clear. Leaving aside the highly non-local nature of the transformation, the
associated charge is conserved, as per the general arguments we gave.

Writing

$\hat{\phi}=\phi_{0}+\hat{\phi}_{nzm}$ (2.19)

one can show that $\phi_{0}$ entirely drops out form the expression for
$v_{imp}$; concretely

$v_{imp}=\frac{R_{v}}{R}\,\sigma-\lambda\hat{\phi}_{nzm}-\frac{\lambda^{2}}{R-\lambda
Q_{k}}\int d\sigma\left(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R}\right)\hat{\phi}_{nzm}-\frac{R_{v}(R+\lambda
Q_{K})}{R(R-\lambda Q_{K})}\,t$ (2.20)

and thus the commutators with the global $U(1)$ charges will now vanish.

To summarize, we find that the conserved charges whose action is well-defined
on the phase space of the $J\bar{T}$ \- deformed CFT on the cylinder are still
given by (2.9), but with $v$ replaced by $v_{imp}$. We will denote these
generators by barred capital calligraphic letters,
$\bar{\mathcal{Q}},\bar{\mathcal{P}}$, to differentiate them from
$\bar{Q},\bar{P}$, which are not well-defined on the cylinder. Explicitly,

$\bar{\mathcal{Q}}_{\bar{f}}=\int
d\sigma\bar{f}(v_{imp})\mathcal{H}_{R}\;,\;\;\;\;\;\;\;\bar{\mathcal{P}}_{\bar{\eta}}=\int
d\sigma\bar{\eta}(v_{imp})\left(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R}\right)$ (2.21)

These charges are: i) conserved and ii) consistent with semiclassical
quantization. The relationship between the two sets of charges is particularly
simple in the Fourier basis $\bar{f}_{n}=e^{-inv_{imp}/R_{v}}$, and reads

$\bar{\mathcal{Q}}_{n}=\bar{Q}_{n}\,e^{-\frac{in\lambda(\widetilde{\phi}_{0}-\alpha
t)}{R-\lambda
Q_{K}}}\;,\;\;\;\;\;\;\;\;\;\;\bar{\mathcal{P}}_{n}=\bar{P}_{n}\,e^{-\frac{in\lambda(\widetilde{\phi}_{0}-\alpha
t)}{R-\lambda Q_{K}}}$ (2.22)

Note that $\alpha$, defined in (2.15), only depends on $Q_{K}$ and
$\bar{Q}_{\bar{K}}$, and thus commutes with all the $\bar{Q}_{m},\bar{P}_{n}$.

###  2.2 Poisson bracket algebra and flow of the symmetry generators

The Poisson brackets of the left-moving generators were worked out in [24],
and read

$\\{Q_{m},Q_{n}\\}=-\frac{i(m-n)}{R}Q_{m+n}\;,\;\;\;\;\;\;\\{Q_{m},P_{n}\\}=\frac{in}{R}P_{m+n}\;,\;\;\;\;\;\;\;\\{P_{m},P_{n}\\}=-\frac{imk}{2}\delta_{m+n}$
(2.23)

The Poisson brackets of the right-moving generators can be derived either from
their explicit expressions (2.21), (2.2) in terms of the undeformed currents
or, equivalently, from the known algebra of the $\bar{Q}_{m},\bar{P}_{m}$ and
$\widetilde{\phi}_{0}$, using (2.22). We obtain the following non-linear
modification of the Witt-Kac-Moody algebra

$\displaystyle\\{\bar{\mathcal{P}}_{m},\bar{\mathcal{P}}_{n}\\}$
$\displaystyle=$ $\displaystyle-\frac{imk}{2}\delta_{m+n}-\frac{im\lambda
k}{2(R-\lambda Q_{K})}\delta_{n,0}\,\bar{\mathcal{P}}_{m}+\frac{in\lambda
k}{2(R-\lambda Q_{K})}\delta_{m,0}\,\bar{\mathcal{P}}_{n}$
$\displaystyle\\{\bar{\mathcal{Q}}_{m},\bar{\mathcal{P}}_{n}\\}$
$\displaystyle=$
$\displaystyle\frac{in}{R_{v}}\bar{\mathcal{P}}_{m+n}-\frac{im\lambda
k}{2(R-\lambda
Q_{K})}\bar{\mathcal{Q}}_{m}\delta_{n,0}+\frac{in\lambda}{R_{v}(R-\lambda
Q_{K})}\bar{\mathcal{P}}_{m}\bar{\mathcal{P}}_{n}$
$\displaystyle\\{\bar{\mathcal{Q}}_{m},\bar{\mathcal{Q}}_{n}\\}$
$\displaystyle=$
$\displaystyle-\frac{i(m-n)}{R_{v}}\bar{\mathcal{Q}}_{m+n}-\frac{im\lambda}{R_{v}(R-\lambda
Q_{K})}\bar{\mathcal{Q}}_{m}\bar{\mathcal{P}}_{n}+\frac{in\lambda}{R_{v}(R-\lambda
Q_{K})}\bar{\mathcal{P}}_{m}\bar{\mathcal{Q}}_{n}$ (2.24)

where we have reinstated the factors of the level in an appropriate way. The
commutation relations with the left-moving charges read

$\\{Q_{m},\bar{\mathcal{Q}}_{n}\\}=\frac{in\lambda}{R(R-\lambda
Q_{K})}\bar{\mathcal{Q}}_{n}P_{m}\;,\;\;\;\;\;\;\;\\{Q_{m},\bar{\mathcal{P}}_{n}\\}=\frac{in\lambda}{R(R-\lambda
Q_{K})}\bar{\mathcal{P}}_{n}P_{m}$

$\\{P_{m},\bar{\mathcal{Q}}_{n}\\}=\frac{in\lambda k}{2(R-\lambda
Q_{K})}\bar{\mathcal{Q}}_{n}\delta_{m,0}\;,\;\;\;\;\;\;\;\\{P_{m},\bar{\mathcal{P}}_{n}\\}=\frac{in\lambda
k}{2(R-\lambda Q_{K})}\bar{\mathcal{P}}_{n}\delta_{m,0}$ (2.25)

yielding again non-linear terms on the right-hand side.

We would now like to connect this to the results of [25], who showed that
particular non-linear combinations of the above generators satisfy a Witt-Kac-
Moody algebra. Concretely, these combinations are

$\widetilde{{P}}_{n}\equiv{P}_{n}-\frac{\lambda
kE_{R}}{2}\delta_{n,0}\;\;\;\;\mbox{and}\;\;\;\;\;\widetilde{{Q}}_{n}\equiv{Q}_{n}-\frac{\lambda
E_{R}}{R}{P}_{n}+\frac{\lambda^{2}kE_{R}^{2}}{4R}\,\delta_{n,0}$

$\widetilde{\bar{\mathcal{P}}}_{n}\equiv\bar{\mathcal{P}}_{n}-\frac{\lambda
kE_{R}}{2}\delta_{n,0}\;\;\;\;\mbox{and}\;\;\;\;\;\widetilde{\bar{\mathcal{Q}}}_{n}\equiv\frac{R_{v}}{R}\bar{\mathcal{Q}}_{n}-\frac{\lambda
E_{R}}{R}\bar{\mathcal{P}}_{n}+\frac{\lambda^{2}kE_{R}^{2}}{4R}\,\delta_{n,0}$
(2.26)

where $E_{R}=\int d\sigma\mathcal{H}_{R}$ is the right-moving energy,
alternativey denoted as $\bar{\mathcal{Q}}_{0}$. Its Poisson brackets are
given by

$\\{E_{R},\bar{\mathcal{P}}_{n}\\}=\frac{in}{R-\lambda
Q_{K}}\bar{\mathcal{P}}_{n}\;,\;\;\;\;\;\;\;\\{E_{R},\bar{\mathcal{Q}}_{n}\\}=\frac{in}{R-\lambda
Q_{K}}\bar{\mathcal{Q}}_{n}$ (2.27)

Using these, one can easily show that the commutation relations of (2.26),
which we will denote as the “flowed” generators, consist of two commuting
copies of the Witt-Kac-Moody algebra.

The relation (2.26) between the two sets of generators is reminiscent of
spectral flow (see the next subsection for a review), though the parameter of
the flow is not a number, but the right-moving energy, $E_{R}$. Due to this
fact, the transformation changes the algebra555It may be more appropriate to
rather call the relation between $\bar{Q},\bar{P}$ and
$\widetilde{\bar{\mathcal{Q}}},\widetilde{\bar{\mathcal{P}}}$ spectral flow,
i.e. conjugation by the spectral flow operator, as discussed in [25]. With
this definition, the algebra is naturally unchanged. In either case, the
terminology “flowed generators” will apply to the ones defined in (2.26)..
Note that at the level of the currents, the spectral flow operation simply
replaces the left-moving zero mode $Q_{K}=J_{0}+\lambda E_{R}k/2$ by $J_{0}$
and, similarly, $\bar{Q}_{\bar{K}}$ by $\bar{J}_{0}$ on the right-moving side.

As detailed in [25], the flowed and unflowed conserved charges satisfy
slightly different flow equations with respect to the parameter $\lambda$ of
the deformation. At $t=0$, we have

$\mathcal{D}_{\lambda}Q_{m}\equiv\partial_{\lambda}Q_{m}-i\\{\mathcal{O}_{cls},Q_{m}\\}=0$
(2.28)

where $\mathcal{O}_{cls}$ is the sum of two parts, $\hat{\mathcal{O}}_{cls}$
and $\Delta\mathcal{O}_{cls}$, given by666 Note we have changed the definition
of $\hat{\mathcal{O}},\Delta\mathcal{O}$ by a factor of $i$ with respect to
[25], in order to avoid dealing with explicit factors of $i$ at the quantum
level. We have also added the subscript ‘cls’ in order to differentiate these
classical functions on phase space from the corresponding operators we will
discuss in the next section.

$\hat{\mathcal{O}}_{cls}=i\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\phi^{\prime}(\sigma)\mathcal{H}_{R}(\tilde{\sigma})\;,\;\;\;\;\;\;\Delta\mathcal{O}_{cls}=i(w\chi_{0}-\phi_{0}E_{R})$
(2.29)

In the above, $G$ is the Green’s function on the cylinder and $\chi_{0}$ is
the zero mode of a field $\chi$ defined though
$\partial_{\sigma}\chi=\mathcal{H}_{R}$. The above flow equation is also
satisfied by $P_{m}$, $R_{v}\bar{Q}_{m}$ and $\bar{P}_{m}$. As explained, of
these, $\bar{Q}_{m},\bar{P}_{m}$ will not lift to well-defined quantum
symmetry generators on the cylinder, but only
$\bar{\mathcal{Q}}_{m},\bar{\mathcal{P}}_{m}$ will. The flow equations for
$\bar{\mathcal{Q}}_{m},\bar{\mathcal{P}}_{m}$, given in (2.22), follow from
the one above and the fact that $\mathcal{D}_{\lambda}\widetilde{\phi}_{0}=0$
at $t=0$, so only the $\lambda$ derivatives contribute. Concretely,

$\mathcal{D}_{\lambda}\bar{\mathcal{P}}_{m}=-\frac{im\widetilde{\phi}_{0}R}{(R-\lambda
Q_{K})^{2}}\bar{\mathcal{P}}_{m}\;,\;\;\;\;\;\;\;\;\mathcal{D}_{\lambda}(R_{v}\bar{\mathcal{Q}}_{m})=-\frac{im\widetilde{\phi}_{0}R}{(R-\lambda
Q_{K})^{2}}R_{v}\bar{\mathcal{Q}}_{m}$ (2.30)

One can check that these flow equations are consistent with the commutation
relations (2.24). Note that the explicit $\lambda$ dependence of the algebra
of these generators is directly related to the fact that the flow equations
they obey are not homogenous. On the other hand,
$\widetilde{Q}_{m},\widetilde{P}_{m},\widetilde{\bar{\mathcal{P}}}_{m}$ and
$R_{v}\widetilde{\bar{\mathcal{Q}}}_{m}$ satisfy

$\tilde{\mathcal{D}}_{\lambda}\widetilde{Q}_{m}\equiv\partial_{\lambda}\widetilde{Q}_{m}-i\left\\{\,\tilde{\mathcal{O}}_{cls},\widetilde{Q}_{m}\right\\}=0$
(2.31)

where
$\tilde{\mathcal{O}}_{cls}=\hat{\mathcal{O}}_{cls}+\Delta\tilde{\mathcal{O}}_{cls}$,
with $\hat{\mathcal{O}}_{cls}$ as given above and

$\Delta\tilde{\mathcal{O}}_{cls}\equiv
i\left[w(\chi_{0}-\widetilde{\chi}_{0})-\phi_{0}E_{R}+\frac{\widetilde{\phi}_{0}E_{R}R}{R-\lambda
Q_{K}}\right]$ (2.32)

which is the same flow equation (in the classical limit) as that satisfied by
the energy eigenstates777 In [25], $\Delta\tilde{\mathcal{O}}$ was denoted as
$\Delta\mathcal{O}-D$. The combination
$\widetilde{\chi}_{0}\equiv\chi_{0}+\frac{\lambda}{R_{v}}\int
d\sigma\mathcal{H}_{R}\hat{\phi}$, and we have rescaled $\phi_{0}$, $\chi_{0}$
by a factor of $R$.. Since the flow equation is homogenous and the algebra at
$\lambda=0$ is Witt-Kac-Moody, it follows that it will continue to be so at
nonzero $\lambda$. Our explicit check above confirms this more indirect
argument.

A nice feature of the flow equation (2.31) is that it does not contain any
$\phi_{0},\chi_{0}$. Explicitly,

$\Delta\tilde{\mathcal{O}}_{cls}=-\frac{iw\lambda}{R_{v}}\int
d\sigma\mathcal{H}_{R}\hat{\phi}_{nzm}-\frac{i\lambda E_{R}R}{R_{v}(R-\lambda
Q_{K})}\int
d\sigma(\mathcal{J}_{-}+\frac{\lambda}{2}\mathcal{H}_{R})\hat{\phi}_{nzm}$
(2.33)

where $\hat{\phi}_{nzm}$ is the scalar field with its winding and zero mode
removed (2.19). One can also work out the flow of the right-moving generators
with respect to $\tilde{\mathcal{D}}_{\lambda}$, finding

$\tilde{\mathcal{D}}_{\lambda}\bar{\mathcal{P}}_{m}=\frac{kRE_{R}}{2(R-\lambda
Q_{K})}\delta_{m,0}\;,\;\;\;\;\;\;\;\;\tilde{\mathcal{D}}_{\lambda}(R_{v}\bar{\mathcal{Q}}_{m})=\frac{RE_{R}}{R-\lambda
Q_{K}}\bar{\mathcal{P}}_{m}$ (2.34)

These equations are still non-homogenous, though perhaps simpler than (2.30).
Note that the second flow equation will suffer from ordering ambiguities at
the quantum level.

To summarize, we showed that classical $J\bar{T}$ \- deformed CFTs on a
cylinder posses two sets of symmetry generators of interest. The ones that are
the closest to the original conformal generators, which are given by (2.7),
(2.21), have the desireable property that their zero mode coincides with the
left- and, respectively, right-moving energy; however, their algebra is not
simple. One can also consider the spectrally flowed generators (2.26), whose
advantage is that they satisfy the well-studied Witt-Kac-Moody algebra. Their
zero modes, however, are non-linearly related to the left-/right-moving
energy. At the quantum level, this feature is responsible for resolving the
naïve tension one may perceive between having a Virasoro symmetry algebra and
the non-trivial $J\bar{T}$ \- deformed spectrum [25]. Both sets of generators
are rather different from their counterparts in the original CFT, as they both
have a non-trivial dependence on $\lambda$.

###  2.3 Spectral flow-invariant generators and a Sugawara-type construction

To better understand the relation between the two sets of generators and,
especially, to prepare the ground for their generalisation to the quantum
case, in this section we would like to remind a few facts about (quantum) CFTs
that posses a $U(1)$ current algebra in addition to Virasoro.

Denoting the Virasoro generators by $L_{m}$ and the Kac-Moody ones by $J_{m}$,
their algebra is

$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m+n}\;,\;\;\;\;\;\;\;[L_{m},J_{n}]=-nJ_{m+n}\;,\;\;\;\;\;\;[J_{m},J_{n}]=\frac{km}{2}\delta_{m+n}$
(2.35)

The modes $L_{m}$ of the stress tensor can be written as a Sugawara
contribution from the modes of the current, and a remainder $\hat{L}_{m}$

$L_{m}=\hat{L}_{m}+\frac{1}{k}\sum_{\ell}:J_{-\ell}J_{m+\ell}:$ (2.36)

where the ‘$:$’ denote creation-annihilation normal ordering, herein defined
as

$:J_{-\ell}J_{m+\ell}:\;\equiv\left\\{\begin{array}[]{ccc}J_{-\ell}J_{m+\ell}&if&\ell>0\\\
&&\\\ J_{m+\ell}J_{-\ell}&if&\ell\leq 0\end{array}\right.$ (2.37)

Of course, other choices of normal ordering are possible, but they are
irrelevant for this discussion. The algebra of $\hat{L}_{m}$ and $J_{m}$ is

$[\hat{L}_{m},\hat{L}_{n}]=(m-n)\hat{L}_{m+n}+\frac{c-1}{12}m(m^{2}-1)\delta_{m+n}\;,\;\;\;\;\;\;\;[\hat{L}_{m},J_{n}]=0\;,\;\;\;\;\;\;[J_{m},J_{n}]=\frac{km}{2}\delta_{m+n}$
(2.38)

The shift of the central charge by one is obtained through a careful analysis
of the normal ordering in (2.36). This term is not important in this section,
but will be in the quantum analysis of the later ones.

The algebra (2.35) admits a “spectral flow” outer automorphism labeled by a
parameter $\alpha$, under which [26]

$J_{m}\rightarrow J_{m}+\frac{\alpha
k}{2}\,\delta_{m,0}\;,\;\;\;\;\;\;\;L_{m}\rightarrow L_{m}+\alpha
J_{m}+\frac{k\alpha^{2}}{4}\delta_{m,0}$ (2.39)

The change in the modes of the stress tensor, $L_{m}$, under this
transformation comes entirely from the part of the stress tensor that is
associated to the current via the Sugawara construction, i.e. the
transformation above leaves $\hat{L}_{m}$ invariant. We will thus call the
$\hat{L}_{m}$ the spectral-flow-invariant generators.

The discussion above is fully quantum-mechanical. In this section, we will
only use the classical limit of this construction, which consists in dropping
the central extension of the Virasoro algebra (but not that of the current
algebra, which can be seen at the level of Poisson brackets), as well as the
normal ordering in the Sugawara stress tensor.

We would now like to understand whether the conserved (pseudo)conformal
charges in $J\bar{T}$ \- deformed CFTs can be similarly split into a spectral-
flow-invariant contribution and a Sugawara one, associated with the $U(1)$
current. To see this, it is best to start by looking at the single boson case
where, if this expectation is true, the entire stress tensor should be given
by the Sugawara contribution.

For a free boson ($k=1$), the Hamiltonian and momentum densities are

$\mathcal{H}^{(0)}=\frac{\pi^{2}+\phi^{\prime
2}}{2}\;,\;\;\;\mathcal{P}=\pi\phi^{\prime}\;\;\;\;\;\;\;\;\Rightarrow\;\;\;\;\;\;\;\;\mathcal{H}_{L,R}^{(0)}=\frac{(\pi\pm\phi^{\prime})^{2}}{4}=\mathcal{J}^{2}_{\pm}$
(2.40)

Plugging this into the general formula888A useful rewriting of this formula is
$\mathcal{H}_{R}\left(1-\lambda\mathcal{J}_{+}-\frac{\lambda^{2}k}{4}\mathcal{H}_{R}\right)=\mathcal{H}_{R}^{(0)}$
(2.41) (2.2) for the deformed Hamiltonian, we can show that

$\mathcal{H}_{L}=\left(\mathcal{J}_{+}+\frac{\lambda\mathcal{H}_{R}}{2}\right)^{2}=K_{t}^{2}\;,\;\;\;\;\;\;\mathcal{H}_{R}=\frac{\left(\mathcal{J}_{-}+\frac{\lambda\mathcal{H}_{R}}{2}\right)^{2}}{1-\lambda\phi^{\prime}}=\frac{\bar{K}_{t}^{2}}{v^{\prime}}$
(2.42)

where the left- and respectively, right-moving currents $K_{t}$, $\bar{K}_{t}$
were defined in (2.4), (2.5) and $v^{\prime}=v_{imp}^{\prime}$. Since
$\mathcal{H}_{L}$ is the square of the holomorphic current $K_{t}$, it
immediately follows that the left-moving conformal charges are precisely of
the Sugawara type, when one expresses this relation in terms of Fourier modes.
For the right-moving sector, a simple manipulation yields

$\bar{\mathcal{Q}}_{n}^{(fb)}=\int d\sigma
e^{-\frac{inv_{imp}}{R_{v}}}\frac{\bar{K}_{t}^{2}}{v^{\prime}}=\int d\sigma
d\tilde{\sigma}e^{-in\frac{v_{imp}}{R_{v}}}\bar{K}_{t}(\sigma)\frac{\delta(\sigma-\tilde{\sigma})}{v^{\prime}}\bar{K}_{t}(\tilde{\sigma})=\frac{1}{R_{v}}\sum_{m}\bar{\mathcal{P}}_{m}\bar{\mathcal{P}}_{n-m}$
(2.43)

where we rewrote the $\delta$ function as

$\frac{\delta(\sigma-\tilde{\sigma})}{v^{\prime}}=\delta(v_{imp}-\tilde{v}_{imp})=\frac{1}{R_{v}}\sum_{m}e^{\frac{im(v_{imp}-\tilde{v}_{imp})}{R_{v}}}$
(2.44)

and then performed the $\sigma$ and $\tilde{\sigma}$ integrals to obtain the
modes of the right-moving current. Thus, it is indeed true that the right-
moving pseudoconformal charges are given by a Sugawara-type expression in
terms of the right-moving $U(1)$ current, even though the stress tensor is not
itself the square of the current in position space.

It is easy to check that in the case of the free boson, the charges
$\bar{\mathcal{Q}}_{n}^{(fb)}$ given by the Sugawara expression above satisfy
the algebra (2.24). More generally, we would write

$Q_{n}=\hat{Q}_{n}+\frac{1}{kR}\sum_{m}P_{m}P_{n-m}\;,\;\;\;\;\;\;\;\bar{\mathcal{Q}}_{n}=\hat{\bar{\mathcal{Q}}}_{n}+\frac{1}{kR_{v}}\sum_{m}\bar{\mathcal{P}}_{m}\bar{\mathcal{P}}_{n-m}$
(2.45)

In terms of the integrated currents, the expression for $\hat{Q}_{m}$ is given
by

$\hat{Q}_{m}=\int d\sigma
e^{\frac{imU}{R}}\left(\mathcal{H}_{L}-\frac{1}{k}(\mathcal{J}_{+}+\frac{\lambda
k}{2}\mathcal{H}_{R})^{2}\right)=\int d\sigma
e^{\frac{imU}{R}}\left(\mathcal{H}_{L}^{(0)}-\frac{\mathcal{J}_{+}^{2}}{k}\right)$
(2.46)

where we used (2.41) and the fact that
$\mathcal{H}_{L}-\mathcal{H}_{R}=\mathcal{H}_{L}^{(0)}-\mathcal{H}_{R}^{(0)}=\mathcal{P}$.
We conclude that the spectral flow invariant left generators in the deformed
theory are identical to the undeformed ones, i.e.

$\hat{Q}_{m}(\lambda)=\hat{Q}_{m}^{(0)}$ (2.47)

For the right-movers, we find

$\hat{\bar{\mathcal{Q}}}_{m}=\int d\sigma
e^{-\frac{imv_{imp}}{R_{v}}}\left(\mathcal{H}_{R}-\frac{(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R})^{2}}{k(1-\lambda\phi^{\prime})}\right)=\int d\sigma
e^{-\frac{imv_{imp}}{R_{v}}}\frac{\mathcal{H}_{R}^{(0)}-\mathcal{J}_{-}^{2}/k}{1-\lambda\phi^{\prime}}$
(2.48)

What is nice about this expression is that in the last term, the numerator can
be expanded in Fourier modes using the original CFT expansion. We obtain

$\hat{\bar{\mathcal{Q}}}_{m}=\frac{1}{R_{v}}e^{\frac{imt(R+\lambda
Q_{K})}{R(R-\lambda Q_{K})}+\frac{im\lambda^{2}}{R_{v}(R-\lambda Q_{K})}\int
d\sigma(\mathcal{J}_{-}+\frac{\lambda}{2}\mathcal{H}_{R})\hat{\phi}_{nzm}}\sum_{n}\hat{\bar{\mathcal{Q}}}_{n}^{(0)}\,e^{-\frac{int}{R}}\int\frac{d\sigma}{1-\frac{\lambda\hat{\phi}^{\prime}R}{R_{v}}}\,e^{\frac{i(n-m)\sigma}{R}+im\lambda\frac{\hat{\phi}_{nzm}}{R_{v}}}$
(2.49)

which is a significantly simpler expression than that for
$\bar{\mathcal{Q}}_{m}$. The Poisson brackets of $\hat{\bar{\mathcal{Q}}}_{m}$
can be derived either from the explicit expression above, or by subtracting
the Sugawara contribution from the $\bar{\mathcal{Q}}_{m}$ commutators, and
are given by

$\\{\hat{\bar{\mathcal{Q}}}_{m},\hat{\bar{\mathcal{Q}}}_{n}\\}=-\frac{i}{R_{v}}(m-n)\hat{\bar{\mathcal{Q}}}_{m+n}\;,\;\;\;\;\;\;\\{\hat{\bar{\mathcal{Q}}}_{m},\bar{\mathcal{P}}_{n}\\}=-\frac{im\lambda}{2(R-\lambda
Q_{K})}\hat{\bar{\mathcal{Q}}}_{m}\,\delta_{n,0}$ (2.50)

Their Poisson bracket with $E_{R}$ is similar to that of
$\bar{\mathcal{P}},\bar{\mathcal{Q}}$, (2.27). Their commutators with the
modes of the currents are almost the usual ones, except the one with the zero
mode of $\mathcal{P}$. However, if we replace $\mathcal{P}$ by
$\tilde{\mathcal{P}}$, which amounts to modifying the zero mode in question,
then they are exactly the usual ones. The spectral-flow-invariant generators
satisfy certain homogenous flow equations, more precisely
$\mathcal{D}_{\lambda}\hat{Q}_{m}=\widetilde{\mathcal{D}}_{\lambda}\hat{Q}_{m}=0$
for the left-moving ones and
$\widetilde{\mathcal{D}}_{\lambda}\hat{\bar{\mathcal{Q}}}=0$ for the right-
moving ones.

As for the flowed generators (2.26), since they satisfy a Witt-Kac-Moody
algebra, they can automatically be written in the form (2.45) with the same
$\hat{Q}_{m}$, $\hat{\bar{Q}}_{m}$, but with $P_{m},\bar{\mathcal{P}}_{m}$
replaced by $\widetilde{P}_{m},\widetilde{\bar{\mathcal{P}}}_{m}$.

To summarize, all the left and right-moving generators, both flowed and
unflowed, can be written as the sum of a spectral-flow invariant contribution
and a Sugawara one. These expressions are classical. Of course, in the quantum
theory, one needs to be careful about operator ordering issues. This is the
subject of the next section.

##  3 The quantum symmetry algebra

###  3.1 Definitions of the quantum symmetry generators

A natural question is whether the classical symmetry generators described in
the previous section can be promoted to quantum ones and if so, what is the
symmetry algebra of the corresponding quantum generators. This can include,
for example, central extensions associated to anomalies, as well as quantum
corrections to the structure constants of the algebra.

The deformed generators will be denoted as
$L_{n}^{\lambda},K_{n}^{\lambda},\bar{L}_{n}^{\lambda}$ and
$\bar{K}_{n}^{\lambda}$. They are formally defined as999The correspondence
between the classical generators and the quantum ones is
$RQ_{m}\leftrightarrow L_{m}^{\lambda}\;,\;\;P_{m}\leftrightarrow
K_{m}^{\lambda}\;,\;\;R_{v}\bar{\mathcal{Q}}_{m}\leftrightarrow\bar{L}_{m}^{\lambda}\;,\;\;\bar{\mathcal{P}}_{m}\leftrightarrow\bar{K}_{m}^{\lambda}\;,\;\;E_{R}\rightarrow
H_{R}$ (3.1) as well as their tilded and hatted counterparts. The map between
Poisson brackets and commutators is $\\{\;,\,\\}_{P.B.}=-i[\;,\,]$.

$L_{n}^{\lambda}\equiv
R\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\int_{0}^{R}d\sigma\,e^{\frac{in(\sigma+t)}{R}}\mathcal{H}_{L}\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\;,\;\;\;\;\;\;\;K_{n}^{\lambda}\equiv\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\int_{0}^{R}d\sigma\,e^{\frac{in(\sigma+t)}{R}}(\mathcal{J}_{+}+\frac{\lambda
k}{2}\mathcal{H}_{R})\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}$

$\bar{L}_{n}^{\lambda}\equiv
R_{v}\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\int_{0}^{R}d\sigma\,e^{-\frac{inv_{imp}}{R_{v}}}\mathcal{H}_{R}\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\;,\;\;\;\;\;\;\;\bar{K}_{n}^{\lambda}\equiv\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}\int_{0}^{R}d\sigma\,e^{-\frac{inv_{imp}}{R_{v}}}(\mathcal{J}_{-}+\frac{\lambda
k}{2}\mathcal{H}_{R})\,\mbox{\tiny{$\stackrel{{\scriptstyle\circ}}{{\circ}}$}}$
(3.2)

i.e. they are given by the classical expressions (2.7), (2.21), subject to an
appropriate ordering prescription and to quantum corrections, altogether
denoted by ‘$\stackrel{{\scriptstyle\circ}}{{\circ}}$’. This prescription can
be established order by order in $\lambda$, after expanding the known fully
non-linear expression (2.2) for the deformed classical symmetry currents in
terms of the undeformed currents $\mathcal{H}_{L,R}^{(0)},\mathcal{J}_{\pm}$.
The first few terms in this expansion are

$\mathcal{H}_{L,R}^{\lambda}(\sigma)=\mathcal{H}_{L,R}^{(0)}(\sigma)+\lambda\mathcal{J}_{+}(\sigma)\mathcal{H}_{R}^{(0)}(\sigma)+\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)\left(\mathcal{J}_{+}^{2}(\sigma)+\frac{k\mathcal{H}_{R}^{(0)}(\sigma)}{4}\right)+\ldots$
(3.3)

The prescription $\stackrel{{\scriptstyle\circ}}{{\circ}}$ should be such that
the resulting quantum generators are well defined and obey a sensible algebra.
Note we have chosen to include an overall factor of $R_{v}$ in the definition
of the right-moving pseudoconformal charges, in order to simplify their
algebra. This implies, in particular, that the relation between the zero mode
of the $\bar{L}_{n}^{\lambda}$ algebra and the right-moving Hamiltonian,
$H_{R}$, is $\bar{L}_{0}^{\lambda}=R_{v}H_{R}$. The unrescaled generators are
simply obtained by dividing by $R_{v}$, since
$R_{v}=R-\lambda(J_{0}-\bar{J}_{0})$ commutes with all the charges.

Our main task is thus to determine the ordering prescription
‘$\stackrel{{\scriptstyle\circ}}{{\circ}}$’. A first and rather non-trivial
requirement is that the algebra of the resulting generators close. This
requirement is insufficient, however, to fully fix the quantum generators and
their commutation relations, as finite correction terms may be independently
added to them to achieve e.g., a more familiar-looking algebra. The resulting
symmetry algebra may thus appear arbitrary, and potentially correlated
corrections in the different generators would be missed. Another drawback of
this approach is that it is not systematic, in the sense that it is not clear
_a priori_ which operator ordering will generate a sensible, closed algebra.

#### Flow equations

A potentially better way to proceed is to use the quantum counterpart of the
classical flow equations (2.31) that the symmetry generators satisfy to fix
the quantum corrections. That the quantum counterpart exists follows from the
very definition [25] of the operator $\tilde{\mathcal{O}}$ as the operator
appearing in the flow that relates energy eigenstates in the various deformed
theories101010Note that with this definition, $\tilde{\mathcal{O}}$ is anti-
Hermitean.

$\partial_{\lambda}|n_{\lambda}\rangle=\tilde{\mathcal{O}}\,|n_{\lambda}\rangle$
(3.4)

The explicit expression for $\tilde{\mathcal{O}}$ follows from first order
quantum-mechanical perturbation theory [27, 25] and is directly derived from
the $J\bar{T}$ deforming operator $\mathcal{O}_{J\bar{T}}$, via [28]

$[H,\tilde{\mathcal{O}}]=\mathcal{O}_{J\bar{T}}^{off-diag.}=\int
d\sigma\,\epsilon^{\alpha\beta}J_{\alpha}T_{\beta
V}+\sum_{n}\partial_{\lambda}E^{(n)}|n_{\lambda}\rangle\langle n_{\lambda}|$
(3.5)

and the requirement that its diagonal elements in the energy eigenbasis
vanish. The $J\bar{T}$ operator itself is defined via point splitting [15]. In
the classical limit, $\tilde{\mathcal{O}}_{cls}$ is given by the expressions
(2.29), (2.33), which hold on the $t=0$ slice. If, moreover, one believes that
$J\bar{T}$ \- deformed CFTs on a cylinder exist111111One potential worry is
the fact that the energies of $J\bar{T}$ \- deformed CFTs on a cylinder become
imaginary at large enough initial dimension, for any fixed value of
$\lambda/R$, suggesting that the finite-size theory is non-perturbatively ill-
defined. This is due to the presence of superluminal modes, which in compact
space give rise to closed timelike curves [29]. On the other hand, finite size
provides a nice and simple set of observables to study that certainly make
sense perturbatively in $\lambda$, to any desired order. It is in this
restricted sense that we mean the well-definiteness of $J\bar{T}$ \- deformed
CFTs on a cylinder.  and the deformation does not affect the Hilbert space of
states on the cylinder, but only the Hamiltonian and its associated
eigenstates - which are smoothly modified as a function of $\lambda$ \- one
concludes that $\tilde{\mathcal{O}}$, given above, should be well defined at
the full quantum level.

Given the full quantum operator $\tilde{\mathcal{O}}$, we can simply _define_

$\partial_{\lambda}\widetilde{L}_{m}^{\lambda}\equiv[\tilde{\mathcal{O}},\widetilde{L}_{m}^{\lambda}]\;,\;\;\;\;\;\;\;\partial_{\lambda}\widetilde{K}_{m}^{\lambda}\equiv[\tilde{\mathcal{O}},\widetilde{K}_{m}^{\lambda}]$

$\partial_{\lambda}\widetilde{\bar{L}}{}_{m}^{\lambda}\equiv[\tilde{\mathcal{O}},\widetilde{\bar{L}}{}_{m}^{\lambda}]\;,\;\;\;\;\;\;\;\partial_{\lambda}\widetilde{\bar{K}}{}_{m}^{\lambda}\equiv[\tilde{\mathcal{O}},\widetilde{\bar{K}}{}_{m}^{\lambda}]$
(3.6)

Since the flow equations are homogenous, it immediately follows (assuming that
$\tilde{\mathcal{O}}$ satisfies all necessary Jacobi identities) that the
$\widetilde{L}^{\lambda}_{m}$ etc. generators so defined satisfy the same
algebra as at $\lambda=0$, i.e. two commuting copies of the Virasoro-Kac-Moody
algebra, with the same central extension, $c$, as that of the undeformed CFT.
It is similarly easy to show [30] that energy eigenstates are also eigenstates
of $\widetilde{L}_{0}^{\lambda},\widetilde{\bar{L}}{}^{\lambda}_{0}$, with the
same eigenvalue as in the undeformed CFT121212Note this does not imply that
$\widetilde{L}_{0}^{\lambda},\widetilde{\bar{L}}{}^{\lambda}_{0}$ coincide
with the undeformed left-/right-moving Hamiltonians $L_{0},\bar{L}_{0}$ as
operators: even if their eigenvalues are the same, their eigenvectors are
clearly different. . This construction is essentially the same as that of the
spectrum-generating operators of [30], though it is not clear whether the
authors considered their $\widetilde{L}_{m}^{\lambda}$ analogues to be
_symmetry_ generators, as we will argue at the end of this section.

Given that we were originally interested in the unflowed generators (3.2), we
note that it may also be possible to use a flow equation to define the left-
moving generators as

$\partial_{\lambda}L_{m}^{\lambda}=[\mathcal{O},L_{m}^{\lambda}]\;,\;\;\;\;\;\;\;\partial_{\lambda}K_{m}^{\lambda}=[\mathcal{O},K_{m}^{\lambda}]$
(3.7)

where $\mathcal{O}$ equals (2.29) in the classical limit. At the quantum
level, it may be possible to define $\mathcal{O}$ as the sum of an appropriate
generalization of the two terms in (2.29), once all quantum corrections to the
currents from which they are built are understood. Note that $\mathcal{O}$
does not, strictly speaking, correspond to a well-defined operator, due to the
terms proportional to the zero modes $\chi_{0},\phi_{0}$ it contains, which
make its action on the Hilbert space ill-defined sometimes. However, its
action on the left-moving generators does appear to make sense, which
motivates the definition above. This definition immediately implies that
$L_{m}^{\lambda},K_{m}^{\lambda}$ obey a Virasoro-Kac-Moody algebra with
central charge $c$, thus fulfilling our expectation.

Other generators that obey simple flow equations are the spectral-flow-
invariant operators $\hat{L}^{\lambda}_{m}$ and $\hat{\bar{L}}^{\lambda}_{m}$,
defined via

$\hat{L}_{m}^{\lambda}=\widetilde{L}^{\lambda}_{m}-\frac{1}{k}\sum_{\ell}:\widetilde{K}^{\lambda}_{-\ell}\widetilde{K}^{\lambda}_{m+\ell}:\;,\;\;\;\;\;\;\;\;\hat{\bar{L}}{}_{m}^{\lambda}=\widetilde{\bar{L}}{}^{\lambda}_{m}-\frac{1}{k}\sum_{\ell}:\widetilde{\bar{K}}{}^{\lambda}_{-\ell}\widetilde{\bar{K}}{}^{\lambda}_{m+\ell}:$
(3.8)

with $\widetilde{L}^{\lambda}_{m}$ etc., obtained by solving (3.6). It
immediately follows that they satisfy

$\tilde{\mathcal{D}}_{\lambda}\hat{L}_{m}^{\lambda}=\tilde{\mathcal{D}}_{\lambda}\hat{\bar{L}}_{m}^{\lambda}=0$
(3.9)

where
$\tilde{\mathcal{D}}_{\lambda}\equiv\partial_{\lambda}-[\tilde{\mathcal{O}},\,\cdot\,]$.
Conversely, one could start with $\hat{L}_{m}^{\lambda}$,
$\widetilde{\mathcal{P}}_{m}^{\lambda}$ that satisfy the
$\tilde{\mathcal{D}}_{\lambda}$ flow equation and show that the
$\widetilde{L}_{m}^{\lambda}$ defined through the Sugawara procedure also
satisfies the flow equation. As discussed in the sequel, the left-moving
$\hat{L}_{m}^{\lambda}$ can be alternatively defined as solutions to
$\mathcal{D}_{\lambda}\hat{L}_{m}^{\lambda}=0$. Note that the “unflowed”
right-moving generators $\bar{L}^{\lambda}_{m}$ are not as naturally defined
via flow equations, because the latter have an inhomogenous piece that suffers
from ordering ambiguities.

Of course, from a practical point of view, the exact quantum-mechanical
expressions for $\tilde{\mathcal{O}}$ and $\mathcal{O}$ need to be calculated,
since the classical expressions (2.29), (2.33) only determine them up to
ordering ambiguities, as well as possible quantum corrections. The ordering of
these operators is in principle fixed by their definition in terms of the
$J\bar{T}$ deforming operator, which is well defined [15]; however, it can
itself receive quantum corrections131313These quantum corrections can in
principle be determined systematically, order by order in $\lambda$, by using
the flow equations. That is, knowledge of the quantum-corrected $J\bar{T}$
operator up to a certain order would determine $\tilde{\mathcal{O}}$ up to the
same order, using (3.5), which in turn can be used to determine the right-
moving generators at the _next_ order, using (3.6). These should in turn
determine the next-order correction to the right-moving current
$\mathcal{H}_{R}$, which enters the definition of the $J\bar{T}$ operator. The
last step does not appear straightforward, however, because the right-moving
generators are not simply the Fourier modes of $\mathcal{H}_{R}$, as in a
usual CFT (3.32), and thus in $J\bar{T}$ \- deformed CFTs, extracting
$\mathcal{H}_{R}$ from the right-moving conserved charges is complicated, even
though in principle, they encode the same information. , which need to be
incorporated. In addition, there could still be ambiguities in solving (3.5),
especially when working in perturbation theory. Note, however, that these
ambiguities are significantly fewer than individual ambiguities in each of the
generators, and they lead to correlated corrections, which simultaneously
affect all symmetry generators, without affecting the symmetry algebra - at
least as long as $\tilde{\mathcal{O}}$ is a well-defined operator. These
features will be exemplified in the following section, where we use the flow
equations to explicitly compute the symmetry generators up to second order in
the $J\bar{T}$ deformation parameter.

#### Unflowed generators and the Sugawara construction

Given the flowed operators $\widetilde{L}_{m}^{\lambda}$, etc., one can
construct the quantum counterparts of the standard “unflowed” ones by
appropriately generalising the classical relations (2.26) to the quantum case.
For the affine $U(1)$ generators and the left-moving Virasoro ones, the
natural, unambiguous generalisation is

$K_{m}^{\lambda}\equiv\widetilde{K}_{m}^{\lambda}+\frac{\lambda
k}{2}H_{R}\,\delta_{m,0}\;,\;\;\;\;\;\;\;\bar{K}_{m}^{\lambda}\equiv\widetilde{\bar{K}}_{m}^{\lambda}+\frac{\lambda
k}{2}H_{R}\,\delta_{m,0}$ (3.10)

$L_{m}^{\lambda}\equiv\widetilde{L}_{m}^{\lambda}+\lambda\widetilde{K}_{m}H_{R}+\frac{k\lambda^{2}}{4}H_{R}^{2}\,\delta_{m,0}$
(3.11)

In the case of the left-movers, one should be able to check, in principle,
whether the generators so defined coincide at the full quantum level with
those obtained via the flow equation (3.7).

The situation is a bit more complicated in the case of the right-moving
pseudoconformal generators, for which an ordering prescription is necessary.
One may attempt to motivate one by using the Sugawara construction. For this,
one can first construct the spectral-flow-invariant operators
$\hat{L}_{m}^{\lambda},\hat{\bar{L}}^{\lambda}_{m}$ at the full quantum level
using the flow equation (3.9); note that for completeness, we have also
included the left-moving generators in the discussion. The resulting
generators would thus obey a Virasoro algebra with central charge $c-1$, by
construction. The unflowed (pseudo)conformal generators are then defined as

$L_{m}^{\lambda,S}\equiv\hat{L}_{m}^{\lambda}+\frac{1}{k}\sum_{\ell}:K_{-\ell}^{\lambda}K^{\lambda}_{m+\ell}:\;,\;\;\;\;\;\;\bar{L}_{m}^{\lambda,S}\equiv\hat{\bar{L}}{}_{m}^{\lambda}+\frac{1}{k}\sum_{\ell}:\bar{K}_{-\ell}^{\lambda}\bar{K}^{\lambda}_{m+\ell}:$
(3.12)

with ‘$:$’ some normal ordering prescription and
$K^{\lambda}_{m},\bar{K}^{\lambda}_{m}$ given by (3.10), by assumption.
Rewriting the above in terms of $\widetilde{\bar{L}}{}_{m}^{\lambda}$ and
$\widetilde{\bar{K}}{}_{m}^{\lambda}$, we find that $L_{m}^{\lambda,S}$
coincides with $L_{m}^{\lambda}$ given in (3.11). As for the right-movers, if
we use the normal ordering spelled out in (2.37), we find

$\bar{L}_{m}^{\lambda,S}=\widetilde{\bar{L}}{}_{m}^{\lambda}+\left\\{\begin{array}[]{ccc}\frac{\lambda}{2}(H_{R}\widetilde{\bar{K}}{}^{\lambda}_{m}+\widetilde{\bar{K}}{}^{\lambda}_{m}H_{R})&if&m\geq
0\\\ &&\\\
\lambda\widetilde{\bar{K}}{}^{\lambda}_{m}H_{R}&if&m<0\end{array}\right.\;\;\;\;\;\;+\;\frac{k\lambda^{2}}{4}H_{R}^{2}\,\delta_{m,0}$
(3.13)

More generally, we will have

$\bar{L}_{m}^{\lambda}=\widetilde{\bar{L}}{}_{m}^{\lambda}+\lambda\widetilde{\bar{K}}{}^{\lambda}_{m}H_{R}+\lambda\xi_{m}[H_{R},\widetilde{\bar{K}}{}^{\lambda}_{m}]+\frac{k\lambda^{2}}{4}H_{R}^{2}\delta_{m,0}$
(3.14)

where $\xi_{m}$ depends rather sensitively on the ordering prescription used
in (3.12). For example, another possible choice of normal ordering is

$\sum_{\ell}:A_{-\ell}A_{m+\ell}:^{\prime}\equiv\sum_{m+\ell>-\ell}A_{-\ell}A_{m+\ell}+\sum_{m+\ell\leq-\ell}A_{m+\ell}A_{-\ell}$
(3.15)

It is easy to check that, using $:^{\prime}$ instead of $:$ in e.g. (3.8) has
no bearing on the tilded generator algebra, but only yields a constant shift
in $\widetilde{\bar{L}}{}_{0}^{\lambda}$. On the other hand, if
$A_{m}=\bar{K}^{\lambda}_{m}$, we obtain

$\bar{L}_{m}^{\lambda,S^{\prime}}=\widetilde{\bar{L}}{}_{m}^{\lambda}+\left\\{\begin{array}[]{ccc}\lambda
H_{R}\widetilde{\bar{K}}{}^{\lambda}_{m}&if&m\geq 0\\\ &&\\\
\lambda\widetilde{\bar{K}}{}^{\lambda}_{m}H_{R}&if&m<0\end{array}\right.\;\;\;\;\;\;+\;\frac{k\lambda^{2}}{4}H_{R}^{2}\delta_{m,0}$
(3.16)

i.e. $\xi_{m}=\Theta(m)$, which is different from (3.13). Thus, because
$\widetilde{\bar{K}}{}^{\lambda}_{m}$ and $H_{R}$ do not commute, different
normal ordering prescriptions will yield different expressions for the
$\bar{L}_{m}^{\lambda}$, which will in turn result in different algebras for
the unflowed generators, as we will show in more detail in section 3.3.

The flow equation obeyed by the generators (3.14) is

$\tilde{\mathcal{D}}_{\lambda}\bar{L}^{\lambda}_{m}=\bar{K}_{m}^{\lambda}\tilde{\mathcal{D}}_{\lambda}(\lambda
H_{R})+\xi_{m}[\tilde{\mathcal{D}}_{\lambda}(\lambda
H_{R}),\bar{K}_{m}^{\lambda}]$ (3.17)

where $\tilde{\mathcal{D}}_{\lambda}H_{R}$ is computed below in (3.20). Thus,
the ambiguity in the ordering of (3.12) translates into an ambiguity in the
ordering of the inhomogenous term in the flow equation.

To fix this ambiguity, one may recall that the original CFT generators satisfy
a Hermiticity condition, $\bar{L}_{m}^{\dagger}=\bar{L}_{-m}$. Using the fact
that $\tilde{\mathcal{O}}$ is anti-Hermitean, one can show that this condition
is preserved along the flow, namely that

$(\bar{L}_{m}^{\lambda})^{\dagger}=\bar{L}_{-m}^{\lambda}$ (3.18)

Imposing this condition on (3.14) and using the fact that the
$\widetilde{\bar{L}}{}_{m}^{\lambda}$ already satisfy this relation, as it
follows from their definition and the anti-hermiticity of
$\tilde{\mathcal{O}}$, one finds that the value of $\xi_{m}$ for which this
condition is satisfied is the one given by the normal ordering (3.15), i.e.
$\xi_{m}=\Theta(m)$.

Please note that even though (3.11), (3.14) are well-defined quantum-
mechanical definitions of the unflowed symmetry generators, we do not
currently have a way of checking whether this is what we mean by (3.2). It is
envisageable that the full quantum definition of these operators could contain
corrections to (3.11), (3.14) of higher order in $\hbar$, which would be
invisible in the classical expressions. For the left-movers, we can check
whether such corrections are present by using their alternate definition in
terms of the flow equation (3.7), which contains in principle all quantum
corrections to $\mathcal{O}$, and thus to the $L_{m}^{\lambda}$. This,
however, does not so straightforwardly apply to the right-movers, because the
inhomogenous terms in their flow equation could themselves receive quantum
corrections at higher order in $\hbar$. Thus, unlike the
$\widetilde{L}_{m}^{\lambda}$ and their barred counterparts, which are in
principle well-defined at the full quantum level, the unflowed generators
appear, at least from the perspective of this particular construction, as a
less natural basis of symmetry generators.

What makes the unflowed generators special is, of course, the fact that their
zero mode is simply related to the left- and, respectively, right-moving
Hamiltonian. Concretely141414The relationship between $H_{R}$ and
$\widetilde{\bar{L}}{}^{\lambda}_{0}$ appears very similar to that between
$\mathcal{H}_{R}$ and $\mathcal{H}_{R}^{(0)}$, given in (2.2). One should note
though that (2.2) is a relation between local currents, whereas (3.19) is a
relation between their zero modes, which does not follow in any simple way
from the local relation. See also comments in the previous footnote. , we have

$L_{0}^{\lambda}=H_{L}=\widetilde{L}{}_{0}^{\lambda}+\lambda
J_{0}H_{R}+\frac{k\lambda^{2}}{4}H_{R}^{2}\;,\;\;\;\;\;\;\;\bar{L}_{0}^{\lambda}=R_{v}H_{R}=\widetilde{\bar{L}}{}_{0}^{\lambda}+\lambda\bar{J}_{0}H_{R}+\frac{k\lambda^{2}}{4}H_{R}^{2}$
(3.19)

According to an argument given in [30], this relation is expected to be exact,
because it follows from the universal expression for the finite-size spectrum
of of $J\bar{T}$ \- deformed CFTs on a cylinder, which is believed to be an
exact formula. Note this implies in particular that

$\widetilde{\mathcal{D}}_{\lambda}H_{R}=\frac{H_{R}Q_{K}}{R-\lambda Q_{K}}$
(3.20)

at the full quantum level, where we used (3.6). Whether the remaining
$\bar{L}_{m}^{\lambda}$ receive further quantum corrections beyond (3.14) is
still an open question, as we already discussed.

#### Conservation

The final question that we would like to address in this section is why the
flowed generators $\widetilde{L}^{\lambda}_{m}$ etc., which are defined to
satisfy the same flow equation as the energy eigenstates, should correspond to
_symmetries_ of the system. The reader might be rightfully skeptical of this
definition, which appears devoid of any physical content: indeed, any
(integrable) deformation of the theory, including massive ones, that smoothly
deforms the states of the system will admit a Virasoro algebra, according to
this formal construction.

In order for the Virasoro generators to actually correspond to symmetries of
the system, one should additionally be able to construct an infinite number of
_conserved quantities_ from them. In the classical theory, this is indeed the
case, and we provided the explicit form (2.7), (2.21) of the conserved
charges. We would now like to argue that also in the quantum theory, it is
possible to add an explicit time dependence to the generators (which matches
the time dependence of the classical generators in the $\hbar\rightarrow 0$
limit), such that they satisfy the conservation requirement (1.3).

For this, we need to know the commutation relations of the flowed generators
with the Hamiltonian, $H=H_{L}+H_{R}=2H_{R}+P$. The commutators of the left-
moving generators with the Hamiltonian are

$[H,\widetilde{L}_{m}^{\lambda}]=[H_{L}+H_{R},\widetilde{L}_{m}^{\lambda}]=-\frac{m\hbar}{R}\widetilde{L}_{m}^{\lambda}$
(3.21)

where we used the expression (3.19) for $H_{L}=L^{\lambda}_{0}$. Even though
this does not vanish, it implies that conjugating $L_{m}^{\lambda}$ by the
time evolution operator will simply yield $L_{m}^{\lambda}$ multiplied by a
c-number function of time. By adding an explicit time dependence to the
Schrödinger-picture generator, of the form

$\widetilde{L}_{m,S}^{\lambda}(t)=e^{\frac{imt}{R}}\widetilde{L}_{m,S}^{\lambda}(0)$
(3.22)

one obtains an operator whose expectation value is conserved, i.e., the
Heisenberg picture operator satisfies (1.3). Same holds for
$\widetilde{K}_{m}^{\lambda}$.

The commutation relations of the
$\widetilde{\bar{L}}{}_{m}^{\lambda},\widetilde{\bar{K}}{}_{m}^{\lambda}$ with
the Hamiltonian can be computed by assuming they take the following form

$[\widetilde{\bar{L}}{}_{m}^{\lambda},H_{R}]=\widetilde{\bar{L}}{}_{m}^{\lambda}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{L}}{}_{m}^{\lambda}$
(3.23)

inspired by the classical Poisson brackets (2.27), where $\alpha_{m}^{l,r}$
are functions of the right-moving Hamiltonian and the global $U(1)$ charges.
Note that $\alpha_{m}^{l,r}$ need not commute with
$\widetilde{\bar{L}}{}_{m}^{\lambda}$, but do commute with $H_{R}$, and can in
principle be different. Their explicit expressions, worked out in appendix A,
follow from the commutation relations of the
$\widetilde{\bar{L}}{}_{m}^{\lambda}$ and the expression (3.19) for its zero
mode in terms of $H_{R}$. They read

$\alpha_{m}^{l,r}(H_{R})=\pm 2\frac{R-\lambda Q_{K}-\sqrt{(R-\lambda
Q_{K})^{2}\mp\hbar km\lambda^{2}}}{k\lambda^{2}}\approx\frac{m\hbar}{R-\lambda
Q_{K}}\pm\frac{km^{2}\lambda^{2}\hbar^{2}}{4(R-\lambda
Q_{K})^{3}}+\mathcal{O}(\hbar^{3})$ (3.24)

where $Q_{K}=J_{0}+\frac{\lambda k}{2}H_{R}$. This reproduces the classical
result (2.27) to leading order in $\hbar$. One can alternatively write
$\alpha_{m}^{l,r}$ in terms of $\widetilde{\bar{L}}_{0}{}^{\lambda}$, whose
eigenvalues are simply the right-moving dimensions in the undeformed CFT, by
using the relation

$R-\lambda Q_{K}=\sqrt{(R-\lambda
J_{0})^{2}-\lambda^{2}k\widetilde{\bar{L}}{}^{\lambda}_{0}}$ (3.25)

If the eigenvalue of $\widetilde{\bar{L}}{}^{\lambda}_{0}$ is such that the
square root above is real, then note that $\alpha_{m}^{l}$ is well-defined for
large negative $m$, whereas $\alpha_{m}^{r}$ is well-defined for large $m>0$.
Otherwise, one can simply see (3.24) as perturbative expressions, which hold
up to any desired order in $\lambda$.

The commutator of $\widetilde{\bar{K}}{}^{\lambda}_{m}$ with $H_{R}$ is
similarly given by

$[\widetilde{\bar{K}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{K}}{}^{\lambda}_{m}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{K}}{}^{\lambda}_{m}$
(3.26)

Using certain identities derived in the appendix, one can show that the
relation between $\alpha^{l}_{m}$ and $\alpha_{m}^{r}$ is consistent with the
relation (3.23) between them, and that all the Jacobi identities involving
$\widetilde{\bar{L}}{}^{\lambda}_{m},\widetilde{\bar{K}}{}^{\lambda}_{m}$ and
$H_{R}$ are satisfied.

The commutator of the right-moving generators with the full Hamiltonian
$H=2H_{R}+P$, where
$P=H_{L}-H_{R}=(\widetilde{L}_{0}^{\lambda}-\widetilde{\bar{L}}{}_{0}^{\lambda})/R$
is

$[H,\widetilde{\bar{L}}{}_{m}^{\lambda}]=\left(\frac{m\hbar}{R}-2\alpha_{m}^{l}\right)\widetilde{\bar{L}}{}_{m}^{\lambda}$
(3.27)

In analogy to the procedure employed in the case of the usual conformal
transformations, one can simply include an explicit time dependence in the
Schrödinger-picture generator

$\widetilde{\bar{L}}{}_{m,S}^{\lambda}(t)=\widetilde{\bar{L}}{}_{m,S}^{\lambda}(0)\,e^{i(\frac{2\alpha_{m}^{r}}{\hbar}-\frac{m}{R})t}=e^{i(\frac{2\alpha_{m}^{l}}{\hbar}-\frac{m}{R})t}\widetilde{\bar{L}}{}_{m,S}^{\lambda}(0)$
(3.28)

to make its expectation value conserved at the full quantum level. We thus
conclue that the infinite number of generators
$\widetilde{\bar{L}}{}^{\lambda}_{m}$ defined via the flow equations (3.6)
correspond to _symmetries_ of the system, since we have just explicitly
constructed an infinite set of conserved quantities from them.

One can check, using the special properties of the functions
$\alpha^{l,r}_{m}$, that the algebra of the
$\widetilde{\bar{L}}{}^{\lambda}_{m}$ is time-independent, as expected. The
time dependence that one needs to attach to the right-moving flowed current
generators $\widetilde{\bar{K}}{}^{\lambda}_{m}$, as well as to the unflowed
generators $\bar{L}_{m}^{\lambda},\bar{K}_{m}^{\lambda}$ in order for them to
satisfy (1.3) is identical to the one above. It is interesting to compare this
time dependence to that of the classical expressions (3.2), as one can easily
note the presence of an infinite series of corrections in $\hbar$. It is
interesting to ask whether (3.2) receives any additional quantum corrections.

#### Summary

The upshot of this discussion is that the definition of the
$\widetilde{L}{}^{\lambda}_{m}$, etc. via the flow equation, together with the
fact that their zero modes (3.19) are simple, known functions of the
Hamiltonian of the system, allows one to show that
$\widetilde{L}_{m}^{\lambda}$ etc., correspond to an infinite set of conserved
charges, which implies they are _symmetries_ of $J\bar{T}$ \- deformed CFTs.
These symmetries organise into two commuting copies of a Virasoro-Kac-Moody
algebra. This argument is extremely general, and it immediately applies to
other integrable deformations, such as the $T\bar{T}$ one; see section (4) for
details.

If the discussion in this section may have appeared somewhat abstract, in the
following one we embark on a very explicit calculation of the corrections to
the quantum generators up to $\mathcal{O}(\lambda^{2})$, using the various
definitions we provided. Readers not interested in the technical details can
skip directly to section 3.3.

###  3.2 Perturbative construction of the quantum symmetry generators

To proceed, let us first set up the notation. The undeformed symmetry
generators will be denoted as $L_{m},J_{m},\bar{L}_{m},\bar{J}_{m}$, defined
via

$L_{n}=R\int_{0}^{R}\,e^{\frac{in\sigma}{R}}\mathcal{H}_{L}^{(0)}(\sigma)\;,\;\;\;\;\;\;\;J_{n}=\int_{0}^{R}\,e^{\frac{in\sigma}{R}}\mathcal{J}_{+}(\sigma)$
(3.29)

$\bar{L}_{n}=R\int_{0}^{R}d\sigma
e^{-\frac{in\sigma}{R}}\mathcal{H}_{R}^{(0)}(\sigma)\;,\;\;\;\;\;\;\;\;\bar{J}_{n}=\int_{0}^{R}d\sigma
e^{-\frac{in\sigma}{R}}\mathcal{J}_{-}(\sigma)$ (3.30)

where $\mathcal{H}_{L,R}^{(0)},\mathcal{J}_{\pm}$ now represent operators in
the undeformed CFT. We have included an overall factor of $R$, so that the
generators be dimensionless. The (quantum) commutation relations of these
modes are the standard ones, given in (2.35). The inverse relations read

$\mathcal{H}_{L}^{(0)}=\sum_{n}\frac{L_{n}}{R^{2}}e^{-in\sigma/R}\;,\;\;\;\;\;\;\;\mathcal{J}_{+}=\sum_{n}\frac{J_{n}}{R}e^{-in\sigma/R}$
(3.31)
$\mathcal{H}_{R}^{(0)}=\sum_{n}\frac{\bar{L}_{n}}{R^{2}}e^{in\sigma/R}\;,\;\;\;\;\;\mathcal{J}_{-}=\sum_{n}\frac{\bar{J}_{n}}{R}e^{in\sigma/R}$
(3.32)

The generators above are defined on the $t=0$ slice. If we decided to work at
$t\neq 0$, we would simply replace $\sigma\rightarrow\sigma+t$ in the left-
moving generators and $\sigma\rightarrow\sigma-t$ in the right-moving ones. In
any case, these generators are time-independent.

We will also need the expansion of $\phi$ on the $t=0$ slice, which is given
by

$\phi(\sigma)=\phi_{0}+\frac{w\sigma}{R}+\hat{\phi}_{nzm}=\phi_{0}+\frac{\sigma}{R}(J_{0}-\bar{J}_{0})+i\sum_{m\neq
0}\frac{1}{m}(J_{m}-\bar{J}_{-m})\,e^{-\frac{im\sigma}{R}}$ (3.33)

where we have separated out the winding from the zero mode and the non-zero
ones. The commutation relations of the zero mode are

$[L_{n},\phi_{0}]=-iJ_{n}\;,\;\;\;\;\;[\bar{L}_{n},\phi_{0}]=-i\bar{J}_{n}\;,\;\;\;\;\;[J_{n},\phi_{0}]=-\frac{ik}{2}\delta_{n,0}\;,\;\;\;\;\;[\bar{J}_{n},\phi_{0}]=-\frac{ik}{2}\delta_{n,0}$
(3.34)

Note that the expansion of $\mathcal{J}_{\pm}(\sigma)$ and $\phi(\sigma)$ on
the $t=0$ slice will be the same in the deformed theory as in the undeformed
one, by definition. However, in the deformed theory, time evolution to $t\neq
0$ will be implemented by a much more complicated operator, so the expression
for these fields on a later slice will not be as easily obtained from the
$t=0$ ones as above. In this section we will therefore construct the expansion
of the quantum generators on the $t=0$ slice only, and simply rely on the
general result (3.22), (3.28) for their expression at a different time. We
also set $R=1$ thoughout.

#### The left-moving generators

Our discussion of the left-moving generators will be rather detailed, in order
to exemplify how the calculations are performed and highlight the technical
issues that arise. The expression for the deformed left-moving generators is
given in (3.2). The expansion of $\mathcal{H}_{L}$ in powers of $\lambda$ is
given up to $\mathcal{O}(\lambda^{2})$ in (3.3), while that of the holomorphic
current is

$\mathcal{J}_{+}+\frac{\lambda
k}{2}\mathcal{H}_{R}=\mathcal{J}_{+}+\frac{\lambda
k}{2}\mathcal{H}_{R}^{(0)}+\frac{\lambda^{2}k}{2}\mathcal{J}_{+}\mathcal{H}_{R}^{(0)}+\ldots$
(3.35)

Plugging in the peturbative expansion of the currents into the generators,
performing the Fourier transform and promoting the classical fields to quantum
operators, we obtain the following “uncorrected” expressions

$L_{n}^{\lambda,uncorr}=L_{n}+\lambda\sum_{m}\bar{L}_{m}J_{n+m}+\lambda^{2}\sum_{m,\ell}\bar{L}_{m}J_{\ell}J_{n+m-\ell}+\frac{\lambda^{2}k}{4}\sum_{m}\bar{L}_{m}\bar{L}_{-n-m}+\ldots$
(3.36)

$K_{m}^{\lambda,uncorr}=J_{m}+\frac{\lambda
k}{2}\left(\bar{L}_{-m}+\lambda\sum_{\ell}J_{m+\ell}\bar{L}_{\ell}+\ldots\right)$
(3.37)

for which we now need to provide a $\stackrel{{\scriptstyle\circ}}{{\circ}}$
prescription. Note that there are no ordering ambiguities up to
$\mathcal{O}(\lambda)$ in $L_{n}^{\lambda}$, and up to
$\mathcal{O}(\lambda^{2})$ in $K_{n}^{\lambda}$. Our current task is thus to
determine an appropriate ordering of the $\mathcal{O}(\lambda^{2})$ terms in
(3.36), as well as potential quantum corrections to both formulae.

Let us start by analysing the $K_{m}^{\lambda}$. The algebra of the
uncorrected operators is

$[K_{m}^{\lambda,uncorr},K_{n}^{\lambda,uncorr}]=\frac{mk}{2}\delta_{m+n}-\frac{\lambda^{2}c\,k^{2}}{48}m(m^{2}-1)\delta_{m+n}$
(3.38)

where $c$ is the central charge of the undeformed CFT. While the generators
(3.37) are certainly well-defined up to this point, we do however expect the
algebra of the $K_{m}^{\lambda}$ to be simply Kac-Moody, as part of the usual
enhancement of the global left-moving $SL(2,\mathbb{R})\times U(1)$ symmetry
of $J\bar{T}$ \- deformed CFTs. It is thus highly desirable to absorb the
unexpected second term into a shift

$K_{m}^{\lambda,uncorr}\;\rightarrow\;K_{m}^{\lambda,corr}=K_{m}^{\lambda,uncorr}+\frac{\lambda^{2}c\,k}{48}(m^{2}-1)J_{m}$
(3.39)

One could in principle also add $\mathcal{O}(\lambda^{2})$ terms that commute
with $J_{m}$, which cannot be fixed by just this commutator. Since this
correction is proportional to the central charge, it could not have been
predicted by the classical analysis.

Our choice to shift the current in order to obtain the desired algebra may
however seem arbitrary, so below we would like to show that it is in fact
implemented naturally by the solution to the flow equation (3.6) or (3.7) for
the Kac-Moody generators. Since the flow operator $\tilde{\mathcal{O}}$
entering (3.6) is better defined, we will start by analysing this equation
which, according to (3.10), will construct for us all components of
$K_{m}^{\lambda}$, except $m=0$.

The operator $\tilde{\mathcal{O}}$ is defined directly in terms of the
$J\bar{T}$ deforming operator via (3.5). An important point is that the
$J\bar{T}$ operator defined in [15] is _not_ normal ordered, i.e. its
perturbative expansion in terms of modes is

$\mathcal{O}_{J\bar{T}}=\epsilon^{\alpha\beta}J_{\alpha}T_{\beta
V}=-2\left(\sum_{m}J_{m}\bar{L}_{m}+2\lambda\sum_{m,p}J_{m}J_{p}\bar{L}_{m+p}+\frac{\lambda
k}{2}\sum_{m}\bar{L}_{-m}\bar{L}_{m}+\ldots\right)$ (3.40)

where the $\ldots$ contain higher corrections in $\lambda$, as well as
potential quantum corrections at $\mathcal{O}(\lambda)$. In deriving this
expansion, we have simply placed the mode expansion of $J_{\alpha}$ to the
left of the mode expansion of $T_{\beta V}$. The classical expressions for all
needed components can be found in [25]. The result is that the last two terms,
which could in principle have had an ordering that depended on the sign of the
mode number, do not, which appears to be related to the special factorization
properties of the $J\bar{T}$ operator. The fact that the perturbative
expansion of the $J\bar{T}$ operator is _not_ creation-annihilation ordered is
important in showing, for example, that the expectation value of the
$J\bar{T}$ operator in energy eigenstates is finite151515To see this, one can
take for example a deformed primary state at $\mathcal{O}(\lambda)$, which is
given by
$|h\rangle_{\lambda}=|h\rangle-\lambda\sum_{\ell=1}^{\infty}\frac{1}{\ell}J_{-\ell}\bar{L}_{-\ell}|h\rangle+\ldots$
where we assumed $\tilde{\mathcal{O}}$, too, is normal ordered. If the terms
in (3.40) were creation-annhilation ordered, then the expectation value
${}_{\lambda}\langle h|\mathcal{O}_{J\bar{T}}|h\rangle_{\lambda}$ would be
infinite at $\mathcal{O}(\lambda)$. On the other hand, one can easily check
that the non-normal ordered expression (3.40) does produce a finite answer, up
to a term proportional to $c$. More precisely  ${}_{\lambda}\langle
h|\mathcal{O}_{J\bar{T}}|h\rangle_{\lambda}=\frac{\lambda
kc}{12}\sum_{\ell=1}^{\infty}\ell(\ell^{2}-1)$ (3.41) This term can be
absorbed into an $\mathcal{O}(\lambda c)$ correction to the $J\bar{T}$
operator (3.40). .

Since we are interested in corrections up to $\mathcal{O}(\lambda^{2})$ in
$K_{m}^{\lambda}$, we need the expansion of $\tilde{\mathcal{O}}$ up to
$\mathcal{O}(\lambda)$. Remember that $\tilde{\mathcal{O}}$ consists of two
parts, $\hat{\mathcal{O}}$ and $\Delta\tilde{\mathcal{O}}$, whose classical
limits are (2.29) and, respectively, (2.33). Given these classical limits and
the definition (3.5), a natural proposal for the quantum operators
$\hat{\mathcal{O}}$ and $\Delta\tilde{\mathcal{O}}$ up to
$\mathcal{O}(\lambda)$ is given by

$\;\;\;\hat{\mathcal{O}}=\frac{1}{2}\left(\sum_{\ell\neq
0}\frac{1}{\ell}(J_{\ell}-\bar{J}_{-\ell})(\bar{L}_{\ell}+\lambda\sum_{p}J_{p}\bar{L}_{\ell+p}+\ldots)-\mbox{
h.c.}\right)$

and

$\Delta\tilde{\mathcal{O}}=\frac{1}{2}\left(w\lambda\sum_{\ell\neq
0}\frac{1}{\ell}(J_{\ell}-\bar{J}_{-\ell})\bar{L}_{\ell}+\lambda\sum_{\ell\neq
0}\frac{1}{\ell}(J_{\ell}-\bar{J}_{-\ell})\bar{J}_{\ell}\,\bar{L}_{0}-\mbox{h.c.}\right)$
(3.42)

where ‘h.c’ denotes hermitean conjugation. This last term is needed in order
to ensure that the full $\tilde{\mathcal{O}}$ is anti-hermitean. The operators
above reproduce the necessary quantum correction (3.41) to the $J\bar{T}$
deforming operator161616It is a valid question whether $\tilde{\mathcal{O}}$
itself receives quantum corrections proportional to $c$ at order $\lambda$, as
we have found was the case for the $J\bar{T}$ operator. In principle, this
question can be answered systematically by computing the
$\mathcal{O}(\lambda)$ correction to all the generators and plugging it back
into the components of the stress tensor that enter the $J\bar{T}$
deformation. In practice, this is somewhat complicated to implement because,
unlike in usual CFTs (3.32), $\mathcal{H}_{R}$ is not simply related to the
conserved right-moving charges via a Fourier transform. Given that (3.42)
already produces the needed correction (3.41) to the $J\bar{T}$ operator, we
will assume that no further corrections are necessary at this order. . Despite
not being normal ordered, the expectation value of $\tilde{\mathcal{O}}$ in
energy eigenstates is finite - more precisely, zero - by construction.

It is interesting to note that the expression for $\tilde{\mathcal{O}}$ given
above is not the only possible solution to (3.5) up to this order. In fact, if
one uses a creation-annihilation normal-ordered $\tilde{\mathcal{O}}$, which
equals (twice) the first term in (3.42) for $\ell>0$, and (twice) the
hermitean conjugate term for $\ell<0$, one still satisfies (3.5) for the
_same_ $J\bar{T}$ operator (3.40). It is interesting to investigate what is
the impact of this alternate choice of flow operator, which we will denote as
$:\tilde{\mathcal{O}}:$, on the symmetry generators and their algebra171717The
question motivating this exercise is: if we made a mistake in computing
$\tilde{\mathcal{O}}$, (how) would we discover it at the level of the
generators? This question is non-trivial because _a priori_ any choice of
$\tilde{\mathcal{O}}$ will lead to a Virasoro-Kac-Moody algebra by
construction, and the only way this will fail is if $\tilde{\mathcal{O}}$ does
not, for example, satisfy the Jacobi identity. . We will thus keep both
options for now, and derive their respective consequences. Of course, other
orderings may also be compatible with (3.5) at this order in perturbation
theory, but we will not attempt to be exhaustive. One should keep in mind,
however, that the non-normal-ordered option is a priori preferred.

The current $K_{m}^{\lambda}$ can be built iteratively, stating from $J_{m}$
at $\lambda=0$. At zeroth order, $\tilde{\mathcal{O}}=\hat{\mathcal{O}}$, and
the solution to the flow equation (3.6) is

$\widetilde{K}_{m}^{\lambda}=K_{m}^{\lambda}=J_{m}+\frac{\lambda
k}{2}\bar{L}_{-m}\;,\;\;\;\;m\neq 0$ (3.43)

and $\widetilde{K}_{0}^{\lambda}=J_{0}$ for $m=0$. It is easy to see that this
solution for the zero mode will hold to all orders in $\lambda$, since neither
$\hat{\mathcal{O}}$, nor $\Delta\tilde{\mathcal{O}}$ contain any terms that do
not commute with $J_{0}$. The first order solution for $K_{m}^{\lambda}$ is
also given by the above, but now for all $m$. Plugging it into the flow
equation at the next order, we obtain to $\mathcal{O}(\lambda)$

$\displaystyle[\hat{\mathcal{O}},K_{m}^{\lambda}]$ $\displaystyle=$
$\displaystyle\left(\frac{k}{2}\bar{L}_{-m}+\frac{\lambda
k}{2}\sum_{\ell}J_{\ell}\bar{L}_{\ell-m}+\frac{\lambda
kc}{24}(m^{2}-1)(J_{m}-\bar{J}_{-m})\right)_{m\neq 0}+\frac{\lambda
k}{2}\sum_{\ell\neq 0}J_{\ell}\bar{L}_{\ell-m}+$ (3.44) $\displaystyle\hskip
28.45274pt+\;\frac{\lambda
k}{4}(2\bar{J}_{0}\bar{L}_{-m}-\bar{J}_{-m}\bar{L}_{0}-\bar{L}_{0}\bar{J}_{-m})$

and

$[\Delta\tilde{\mathcal{O}},K_{m}^{\lambda}]=\frac{\lambda
k}{4}(2w\bar{L}_{-m}+\bar{J}_{-m}\bar{L}_{0}+\bar{L}_{0}\bar{J}_{-m})\;,\;\;\;\;\;\;m\neq
0$ (3.45)

Adding up the two contributions and integrating with respect to $\lambda$, we
find

$K_{m}^{\lambda}=J_{m}+\frac{\lambda
k}{2}\bar{L}_{-m}+\frac{\lambda^{2}k}{2}\sum_{\ell}J_{\ell}\bar{L}_{\ell-m}+\left.\frac{\lambda^{2}kc}{48}(m^{2}-1)(J_{m}-\bar{J}_{-m})\right|_{m\neq
0}$ (3.46)

Strictly speaking, we have only shown this for $m\neq 0$, since only these
components satisfy $\tilde{\mathcal{D}}_{\lambda}K_{m}^{\lambda}=0$. However,
comparison with the uncorrected expression (3.37) indicates that this
expression will hold for all $m$. We will also rederive it in the sequel,
using the flow equation (3.7).

Had we used $:\tilde{\mathcal{O}}:$ instead to flow, we would have obtained an
additional shift

$K_{m}^{\lambda}(:)=K_{m}^{\lambda}+\frac{\lambda^{2}km(m-1)}{8}\bar{J}_{-m}$
(3.47)

due to the normal ordering prescription. This term does not affect the
$K_{m}^{\lambda}$ commutator, but does contribute to the one with
$\bar{K}_{n}^{\lambda}$.

We can also compare the solution (3.46) to that of the flow equation generated
by $\mathcal{O}$, which has the advantage that it involves $K_{m}^{\lambda}$
for all $m$. For this, we need the quantum expression for the operator
$\Delta\mathcal{O}$

$\Delta\mathcal{O}=\frac{i}{2}(w\chi_{0}^{\lambda}-\phi_{0}H_{R})-\mbox{h.c.}$
(3.48)

This Ansatz is based on the classical expression (2.29) and the ordering of
the $J\bar{T}$ operator. The superscript of $\chi_{0}^{\lambda}$ indicates
this quantity receives corrections at higher orders in $\lambda$, as compared
with $\phi_{0}$, which does not. The right-moving Hamiltonian entering (3.48)
is, to this order

$H_{R}=\bar{L}_{0}+\lambda\sum_{p}J_{p}\bar{L}_{p}+\ldots$ (3.49)

The commutator $[\chi_{0}^{\lambda},K_{m}^{\lambda}]=-i\lambda
k/2\bar{L}_{-m}+\ldots$ follows from the Poisson bracket computed in [25].
Plugging all this into (3.48), we find

$[\Delta\mathcal{O},K_{m}^{\lambda}]=\frac{\lambda
k}{2}(J_{0}-\bar{J}_{0})\bar{L}_{-m}+\frac{k}{2}\left(\bar{L}_{0}+\lambda\sum_{\ell}J_{\ell}\bar{L}_{\ell}\right)\delta_{m,0}+\frac{\lambda
k}{4}(\bar{J}_{-m}\bar{L}_{0}+\bar{L}_{0}\bar{J}_{-m})$ (3.50)

Adding this up to (3.44), we obtain exactly the same expression (3.46) up to
$\mathcal{O}(\lambda^{2})$, now for all $m$. This exercise also serves as a
cross check that the only difference between $K_{m}^{\lambda}$ and
$\widetilde{K}_{m}^{\lambda}$, as defined via their respective flow equations,
is indeed consistent with (3.10).

The commutator of the corrected $K_{m}^{\lambda}$ is simply

$[K_{m}^{\lambda},K_{n}^{\lambda}]=\frac{mk}{2}\delta_{m+n}+\mathcal{O}(\lambda^{3})$
(3.51)

as expected from our general arguments in the previous section.

Let us now turn to the $L_{m}^{\lambda}$, given in (3.36). To first order in
$\lambda$, there is no ordering ambiguity and it can be easily checked that
the generators satisfy the Virasoro algebra with central extension $c$.
Understanding the appropriate ordering is however necessary at
$\mathcal{O}(\lambda^{2})$. As discussed in section 3.1, there exist at least
two ways of defining the left-moving conformal generators. The simplest way to
construct a well-defined $L_{m}^{\lambda}$ while avoiding to explicitly deal
with the ordering is to use the Sugawara construction (3.12). If the
$\hat{L}_{m}^{\lambda}$ and the $K_{m}^{\lambda}$ are well-defined and satisfy
the expected algebra, then the well-definiteness and the algebra of the
$L_{m}^{\lambda}$ follows.

As we saw in the previous section, at the classical level the spectral-flow-
invariant left generators in the deformed theory are identical to the
undeformed ones (2.47). The most natural generalization of this observation to
the quantum level is to take

$\hat{L}_{m}^{\lambda}=\hat{L}_{m}\;,\;\;\;\;\;\;\forall\lambda$ (3.52)

This in particular implies that the $\hat{L}_{m}^{\lambda}$ satisfy a Virasoro
algebra with central charge $c-1$. The above equality also follows from
solving the flow equation
$\tilde{\mathcal{D}}_{\lambda}\hat{L}^{\lambda}_{m}=0$, since $\hat{L}_{m}$
commutes with all the operators that appear in the definition of
$\tilde{\mathcal{O}}$. For the same reason, $\hat{L}_{m}^{\lambda}$ entirely
commute with $K_{n}^{\lambda}$,

$[\hat{L}_{m}^{\lambda},K_{n}^{\lambda}]=0\;,\;\;\;\;\;\;\forall\lambda$
(3.53)

The deformed left-moving generators obtained via the Sugawara construction
then satisfy a Virasoro algebra with central charge $c$, since the building
blocks satisfy (2.38). They also satisfy the flow equation
$\mathcal{D}_{\lambda}L^{\lambda}_{m}=0$, since both $K_{m}^{\lambda}$ and
$\hat{L}^{\lambda}_{m}$ do. The ordering we chose for the Sugawara
construction is automatically giving us a well-defined ordering prescription
for the generators, which take the explicit form

$\displaystyle L_{n}^{\lambda,S}$ $\displaystyle=$ $\displaystyle
L_{n}+\lambda\sum_{\ell}J_{n+\ell}\bar{L}_{\ell}+\frac{\lambda^{2}k}{4}\left(\sum_{\ell>0}\bar{L}_{\ell}\bar{L}_{-n-\ell}+\sum_{\ell\geq
0}\bar{L}_{\ell-n}\bar{L}_{-\ell}\right)+\frac{\lambda^{2}}{2}\sum_{\ell>0,p}(J_{-\ell}J_{n+\ell+p}+J_{-\ell+p}J_{n+\ell})\bar{L}_{p}+$
(3.54) $\displaystyle+$ $\displaystyle\frac{\lambda^{2}}{2}\sum_{\ell\geq
0,p}(J_{n-\ell}J_{\ell+p}+J_{n-\ell+p}J_{\ell})\bar{L}_{p}+\frac{\lambda^{2}c}{48}\left(\sum_{\ell>0}(\ell^{2}-1)(J_{-\ell}-\bar{J}_{\ell})J_{n+\ell}+\sum_{\ell\geq
0}(\ell^{2}-1)J_{n-\ell}(J_{\ell}-\bar{J}_{-\ell})\right)$ $\displaystyle+$
$\displaystyle\frac{\lambda^{2}c}{48}\left(\sum_{\ell>0}((n+\ell)^{2}-1)J_{-\ell}(J_{n+\ell}-\bar{J}_{-n-\ell})_{\ell\neq-n}+\sum_{\ell\geq
0}((n-\ell)^{2}-1)(J_{n-\ell}-\bar{J}_{\ell-n})_{\ell\neq n}J_{\ell}\right)$

Note that this does _not_ correspond to the usual creation-annihilation normal
ordering. In particular, the terms in the first paranthesis are anti-ordered.

One can alternatively construct the $L_{m}^{\lambda}$ by integrating the flow
equation (3.7). If one uses the non-normal-ordered $\mathcal{O}$ to flow the
seed $L_{m}$, one finds

$\partial_{\lambda}L_{m}^{\lambda}=\sum_{\ell}J_{\ell+m}\bar{L}_{\ell}+\frac{\lambda
k}{2}\sum_{\ell}\bar{L}_{-\ell-m}\bar{L}_{\ell}+2\lambda\sum_{\ell,q}J_{\ell}J_{q+m}\bar{L}_{\ell+q}+\frac{\lambda
c}{24}\sum_{\ell\neq
0}(\ell^{2}-1)[(J_{\ell}-\bar{J}_{-\ell})J_{m-\ell}+J_{m-\ell}(J_{\ell}-\bar{J}_{-\ell})]$
(3.55)

Since $L_{m}^{\lambda,S}$ also obeys the flow equation, the two end results
must be the same, by consistency. Indeed, (3.55) can be shown to be equivalent
to the Sugawara precription (3.54), through subtle cancellations of infinite
terms. One can similarly show that the $L_{m}^{\lambda,S}(:)$ constructed via
the Sugawara procedure, using $K_{m}^{\lambda}(:)$ instead of
$K_{m}^{\lambda}$, is equivalent to the $L_{m}^{\lambda}(:)$ generated via the
flow equation driven by $:\mathcal{O}:$. Thus, despite the slight ordering
ambiguity we have decided to keep at this stage, the algebra of the left-
moving generators is always Virasoro-Kac-Moody, with the same central charge
as that of the original CFT. This fact follows, of course, from the definition
of the generators via a homogenous flow equation, which will produce a
Virasoro-Kac-Moody algebra for any choice of operator $\mathcal{O}$ that
satisfies the Jacobi identity.

Given that the unflowed left-moving generators $L_{m}^{\lambda}$ can be
defined by flowing the seed $L_{m}$ using (3.7), while the flowed generators
$\widetilde{L}_{m}^{\lambda}$ are defined by flowing the same seed with
$\tilde{\mathcal{O}}$, we can directly check the proposed relation (3.11)
between the two up to $\mathcal{O}(\lambda^{2})$, but up to any order in
$\hbar$. At the first order, we obtain

$\widetilde{L}_{m}^{\lambda}=L_{m}^{\lambda}-\lambda J_{m}\bar{L}_{0}$ (3.56)

while at the next order, we find

$\partial_{\lambda}\widetilde{L}_{m}^{\lambda}=\partial_{\lambda}L_{m}^{\lambda}-J_{m}(\bar{L}_{0}+2\lambda\sum_{p}J_{p}\bar{L}_{p})-\lambda
k\bar{L}_{-m}\bar{L}_{0}+\frac{\lambda k}{2}\bar{L}_{0}^{2}\delta_{m,0}$
(3.57)

which is precisely consistent with (3.11).

Finally, it is instructive to contrast the algebra of the generators obtained
via the Sugawara construction or by integrating the flow equation with the
algebra of a set of generators defined using a simple-minded creation-
annihilation normal ordering for the operators $L^{\lambda,uncorr}_{m}$. The
commutator of two $:\;:$ ordered left generators is181818Note these are
different from the $L_{m}^{\lambda}(:)$ operators generated by flowing with
$:\mathcal{O}:$, which are given by (3.55), to which the ordering prescription
(2.37) is applied.

$\displaystyle[:L_{m}^{\lambda,uncorr}:,:L_{n}^{\lambda,uncorr}:]$
$\displaystyle=$
$\displaystyle(m-n)\left[:L_{m+n}^{\lambda}:-\frac{k\lambda^{2}}{4}(m+n)\bar{L}_{-m-n}\sum_{1}^{\infty}1+\frac{\lambda^{2}k}{2}\bar{L}_{-m-n}(m^{2}+n^{2}-mn-1)\right]+$
(3.58)
$\displaystyle+\;\frac{\lambda^{2}c}{12}\sum_{\ell}\ell(\ell^{2}-1)J_{m+\ell}J_{n-\ell}+\frac{c}{12}\left(m(m^{2}-1)-\sum_{1}^{\infty}\ell^{2}(\ell^{2}-1)+m\sum_{1}^{\infty}\ell(\ell^{2}-1)\right)\,\delta_{m+n}$

The terms proportional to the central charge, though infinite, are not of
concern so far, since corrections proportional to $c$ to the generators are to
be expected. It is, however, concerning that the algebra above does not appear
to close among well-defined, finite generators; in particular, the infinite
sums are indicative of an inappropriate ordering. Indeed, it can be easily
seen that $L_{n}^{\lambda}$ and $:L_{n}^{\lambda,uncorr}:$ differ by infinite
terms proportional to $\lambda^{2}\bar{L}_{-n}$. It is interesting to ask
whether a redefinition of $:L_{n}^{\lambda}:$ can bring us to a well-defined
ordering, e.g. (3.54) or (3.55), by absorbing the divergent terms proportional
to $\lambda^{2}\bar{L}_{-m-n}$ into a redefinition of $L_{m+n}^{\lambda}$. It
still seems difficult though to obtain generators that satisfy a Virasoro
algebra, due to the last term on the first line, which is not of the correct
form. In any case, while it may be possible to recover the ordering (3.54) or
(3.55) by appropriately cancelling the infinite terms above, there does not
appear to exist any systematic way to do so, which could be extrapolated to
higher orders in perturbation theory.

The conclusions of this analysis are that the flow equation, even keeping a
slight ambiguity in the definition of the flow operator, nicely fixes the
ordering and the correction terms to the left-moving generators in a
correlated manner. Not only does it make sure that the resulting algebra is
closed, but it also guarantees it to be Virasoro-Kac-Moody. Once the Kac-Moody
generators are built, the conformal ones can be alternatively constructed
using the Sugawara procedure, which yields equivalent results and shortcuts
some of the technical steps.

#### The right-moving generators

The right-moving generators are given (3.2) by the corresponding classical
expressions upon application of an appropriate ordering procedure. As before,
it is convenient to first construct the algebra of the rescaled spectral-flow-
invariant generators and of the right-moving currents,
$\bar{K}_{m}^{\lambda}$, and then use the Sugawara construction to produce the
full right-moving generators.

The classical expression for the spectral-flow-invariant operators
$\hat{\bar{\mathcal{Q}}}_{m}^{\lambda}$ is given in (2.49). Setting $t=0$ and
remembering that $\hat{\bar{L}}_{m}^{\lambda}$ correspond to the rescaled
charges $R_{v}\hat{\bar{\mathcal{Q}}}_{m}$, the expansion of the general
expression (2.49), up to second order in $\lambda$, in terms of the undeformed
Fourier modes is

$\displaystyle\hat{\bar{L}}_{m}^{\lambda,uncorr}$ $\displaystyle=$
$\displaystyle\hat{\bar{L}}_{m}+\frac{\lambda}{R_{v}}\sum_{\ell\neq
0}\frac{\ell-m}{\ell}\hat{\bar{L}}_{m+\ell}(J_{\ell}-\bar{J}_{-\ell})+\lambda^{2}\sum_{\ell}\hat{\bar{L}}_{m+\ell}\sum_{p\neq
0,\ell}\left(1+\frac{m^{2}}{2p(\ell-p)}-\frac{m}{p}\right)(J_{p}-\bar{J}_{-p})\times$
(3.59) $\displaystyle\times$
$\displaystyle\\!\\!\\!\\!(J_{\ell-p}-\bar{J}_{p-\ell})-\frac{m\lambda^{2}}{2}\sum_{\ell\neq
0}\frac{1}{\ell}[(J_{\ell}-\bar{J}_{-\ell})\bar{J}_{\ell}+\bar{J}_{\ell}(J_{\ell}-\bar{J}_{-\ell})]\,\hat{\bar{L}}_{m}+\ldots$

where $1/R_{v}\approx 1+\lambda w$ in this expansion, and we have written the
last term in a symmetrized way that will be convenient later. Interestingly,
note that except for the last term, there are no ordering issues up to this
order, since $\hat{\phi}$ commutes with $\phi^{\prime}$ and
$\hat{\bar{L}}{}^{\lambda}_{m}$ to all orders in $\lambda$. The only source of
ordering ambiguities stems from the terms descending from
$\widetilde{\phi}_{0}$, which may perhaps be fixed by requiring that it remain
the spectral flow generator in the full quantum theory. In any case, this is a
separate issue.

The most efficient way to fix the quantum corrections to
$\hat{\bar{L}}^{\lambda}_{m}$ \- which are all proportional to the central
charge - is by integrating the flow equation (3.9) up to
$\mathcal{O}(\lambda^{2})$. If we use the non-normal-ordered
$\tilde{\mathcal{O}}$ to drive the flow up to $\mathcal{O}(\lambda^{2})$ , we
obtain precisely the above expression (with the particular ordering we
displayed), plus a correction proportional to

$\displaystyle\Delta\hat{\bar{L}}_{m}^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{\lambda(c-1)}{12R_{v}}(m^{2}-1)(J_{-m}-\bar{J}_{m})_{m\neq
0}+\frac{\lambda^{2}(c-1)}{24}\\!\sum_{\ell\neq
0,-m}\\!\left[\frac{\ell-m}{\ell}((\ell+m)^{2}-1)-\frac{m}{\ell}(m^{2}-1)\right]\times$
(3.60) $\displaystyle\times$
$\displaystyle(J_{\ell}-\bar{J}_{-\ell})(J_{-\ell-m}-\bar{J}_{\ell+m})$

Note that there is a correction already at $\mathcal{O}(\lambda)$, which
contributes to the flow equation at the next order. If we flow instead using
the normal-ordered $:\tilde{\mathcal{O}}:$, the only change is to replace the
last term in (3.59) by its normal-ordered version.

One can check that the algebra of the corrected
$\hat{\bar{L}}^{\lambda}_{m}=\hat{\bar{L}}^{\lambda,uncorr}_{m}+\Delta\hat{\bar{L}}^{\lambda}_{m}$
is precisely Virasoro with central extension $c-1$. We also compute the
commutation relations with the left-moving generators. First, it is obvious
that

$[\hat{\bar{L}}_{m}^{\lambda},\hat{L}_{n}^{\lambda}]=0\;,\;\;\;\;\forall\lambda$
(3.61)

to all orders in $\lambda$, due to the fact that the expansion of the right-
moving generators only contains modes that commute with
$\hat{L}_{n}=\hat{L}_{n}^{\lambda}$. The commutators with the current are
however non-trivial

$[\hat{\bar{L}}_{m}^{\lambda},K_{n}^{\lambda}]=\frac{\lambda km}{2}(1+\lambda
J_{0})\hat{\bar{L}}_{m}^{\lambda}\delta_{n,0}$ (3.62)

and agree with the expansion of the corresponding classical expression to
second order in $\lambda$. The commutator with $L_{n}^{\lambda}$ follows from
the two above, via the Sugawara construction.

The expression for the right-moving $U(1)$ charges, obtained by expanding
(3.2) to second order, is

$\displaystyle\bar{K}_{m}^{\lambda,uncorr}$ $\displaystyle=$
$\displaystyle\bar{J}_{m}+\frac{\lambda k}{2}\bar{L}_{m}-\frac{\lambda
m}{R_{v}}\sum_{\ell\neq
0}\frac{1}{\ell}(J_{\ell}-\bar{J}_{-\ell})\left(\bar{J}_{\ell+m}+\frac{\lambda
k}{2}\bar{L}_{\ell+m}\right)+\frac{\lambda^{2}k}{2}\sum_{\ell}J_{\ell}\bar{L}_{m+\ell}+$
(3.63) $\displaystyle+$
$\displaystyle\frac{m^{2}\lambda^{2}}{2}\sum_{\ell,p\neq 0}\frac{1}{\ell
p}(J_{\ell}-\bar{J}_{-\ell})(J_{p}-\bar{J}_{-p})\bar{J}_{\ell+p+m}-\lambda^{2}m\sum_{\ell\neq
0}\frac{1}{\ell}(J_{\ell}-\bar{J}_{-\ell})\bar{J}_{\ell}\bar{J}_{m}$

To determine the ordering and other corrections, we use the flow equation
iteratively twice, taking into account the fact that it is no longer
homogenous. If we use the non-normal-ordered flow operator
$\tilde{\mathcal{O}}$ to fix all $\bar{K}_{m}^{\lambda}$ with $m\neq 0$, we
obtain the expression above (with the terms entering the $\ell$ sums
appropriately symmetrized, as e.g. in (3.59)), plus a correction

$\Delta\bar{K}_{m}^{\lambda}=\frac{\lambda^{2}kc}{48}(m^{2}-1)(J_{-m}-\bar{J}_{m})_{m\neq
0}$ (3.64)

The flow equation obeyed by the corrected
$\bar{K}_{m}^{\lambda}\equiv\bar{K}_{m}^{\lambda,uncorr}+\Delta\bar{K}_{m}^{\lambda}$
is

$\tilde{\mathcal{D}}_{\lambda}\bar{K}_{m}^{\lambda}=\frac{k}{2}(\bar{L}_{0}+\lambda\sum_{p}J_{p}\bar{L}_{p}+\lambda
J_{0}\bar{L}_{0})\delta_{m,0}$ (3.65)

in perfect agreement with the classical result (2.34). The commutator of two
such currents is

$[\bar{K}_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=\frac{km}{2}\delta_{m+n}+(1+\lambda
J_{0})\left(\frac{\lambda
km}{2}\bar{K}_{m}^{\lambda}\delta_{n,0}-\frac{nk\lambda}{2}\bar{K}_{n}^{\lambda}\delta_{m,0}\right)+\mathcal{O}(\lambda^{3})$
(3.66)

again in perfect agreement with the classical prediction (2.24), up to this
order in perturbation theory. Next, we compute

$[\hat{\bar{L}}_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=\frac{\lambda
km}{2}(1+\lambda J_{0})\hat{\bar{L}}_{m}^{\lambda}\delta_{n,0}$ (3.67)

in agreement with (2.50), with no additional corrections from the central
terms. It is also trivial to show that
$[\bar{K}_{m}^{\lambda},\hat{L}_{n}^{\lambda}]=0$ to all orders in $\lambda$,
with the expected consequences for the
$[\bar{K}^{\lambda}_{m},L^{\lambda}_{n}]$ commutator. Finally, we also find

$[\bar{K}_{m}^{\lambda},K_{n}^{\lambda}]=\frac{\lambda km}{2}(1+\lambda
J_{0})\bar{K}_{m}^{\lambda}\delta_{n,0}$ (3.68)

which agrees with (2.25).

It is interesting to ask what happens if we use $:\tilde{\mathcal{O}}:$
instead to drive the flow. The correction term becomes

$\Delta\bar{K}_{m}^{\lambda}=\frac{\lambda^{2}kc}{48}(m^{2}-1)(J_{-m}-\bar{J}_{m})_{m\neq
0}-\frac{\lambda^{2}k}{8}m(m-1)\bar{J}_{m}-\frac{\lambda^{2}mk}{4}(J_{-m}-\bar{J}_{m})\sum_{1}^{\infty}1$
(3.69)

in addition to normal-ordering all the terms in (3.63) that require
it191919The last term in (3.63) is ordered as
$:(J_{\ell}-\bar{J}_{-\ell})\bar{J}_{\ell}:\bar{J}_{m}$. . The last term above
is problematic, because it contributes an infinite central term to the
$[\bar{K}_{m}^{\lambda},K_{n}^{\lambda}]$ commutator (the finite second term
induced by the normal ordering cancels out.) We thus conclude that the normal-
ordered flow operator $:\tilde{\mathcal{O}}:$, which was in principle
consistent up to at least $\mathcal{O}(\lambda)$ with the defining equation
(3.5), does not produce symmetry generators that have a well-defined algebra.
On the other hand, the non-normal-ordered $\tilde{\mathcal{O}}$ does appear to
yield well-defined results, at least up to this order.

The full unflowed generators can be built using the Sugawara construction
(3.12), using the normal ordering (3.15) that preserves the usual hermiticity
condition (3.18). Their commutation relations follow from those of the
building blocks. It is interesting to compare $\bar{L}_{m}^{\lambda,S}$ with
the expression that would be obtained via the flow equation. Since the
analysis is rather cumbersome, we only perform the comparison up to
$\mathcal{O}(\lambda)$. We find

$\bar{L}_{m}^{\lambda}=R_{v}\bar{L}_{m}+\lambda\sum_{p}J_{p}\bar{L}_{p+m}-\frac{m\lambda}{2}\sum_{\ell\neq
0}\frac{1}{\ell}[(J_{\ell}-\bar{J}_{-\ell})\bar{L}_{\ell+m}+\bar{L}_{\ell+m}(J_{\ell}-\bar{J}_{-\ell})]+\frac{\lambda
c}{12}(m^{2}-1)(J_{-m}-\bar{J}_{m})_{m\neq 0}$ (3.70)

This ordering is generated by flowing with the $\hat{\mathcal{O}}$ (since
$\Delta\tilde{\mathcal{O}}$ vanishes at leading order), and by fixing the
inhomogenous terms in the flow equation to be

$\tilde{\mathcal{D}}_{\lambda}\bar{L}_{m}^{\lambda}=\frac{1}{2}(\bar{J}_{m}\bar{L}_{0}+\bar{L}_{0}\bar{J}_{m})$
(3.71)

One can show that the above expression formally coincides with the Sugawara
one, if one plugs in the explicit expressions for $\hat{L}_{m}^{\lambda}$ and
$\bar{K}_{m}^{\lambda}$ to this order, through subtle cancellations of
infinite terms. The match is however not perfect: in the term proportional to
the central charge, we could not get the shift from $c-1$ in (3.60) to $c$ in
(3.70).

It is an interesting question which method: the flow equation or the Sugawara
construction, is best for defining $\bar{L}_{m}^{\lambda}$, and whether they
produce equivalent results. In our opinion, the flow equation appears more
suitable, despite the fact that the exact non-homogenous terms are not known a
priori, because it uniformly applies the same ordering. The Sugawara
construction, on the other hand, provides a simpler way to achieve finite
results that only differ from the expected expression by a finite number of
terms.

We have so far mostly concentrated our attention on the unflowed generators.
The flowed right-moving generators are found by solving the homogenous flow
equation (3.6). In the case of $\widetilde{\bar{K}}{}_{m}^{\lambda}$, the
result simply amounts to replacing the zero mode of $\bar{K}_{m}^{\lambda}$ by
$\bar{J}_{0}$. Therefore, their algebra is given by (3.66), except for
$m,n=0$, when one obtains instead a trivial commutator. Consequently,

$[\widetilde{\bar{K}}{}_{m}^{\lambda},\widetilde{\bar{K}}{}_{n}^{\lambda}]=\frac{km}{2}\delta_{m+n}+\mathcal{O}(\lambda^{3})\;,\;\;\;\;\;\;\;\;\;\;[\hat{\bar{L}}{}_{m}^{\lambda},\widetilde{\bar{K}}{}_{n}^{\lambda}]=0+\mathcal{O}(\lambda^{3})$
(3.72)

One can in principle arrive at the same result starting from the definition
(3.10), by subtracting from (3.66) the commutation relations for $H_{R}$. This
method is significantly more cumbersome, but can be used as a cross-check. One
should in particular be careful to include quantum correction terms to $H_{R}$
(e.g. of the form (3.60)), without which one does not obtain the correct
result.

As for the flowed right-moving pseudoconformal generators, they can be
constructed from $\hat{\bar{L}}{}^{\lambda}_{m}$ and
$\widetilde{\bar{K}}{}^{\lambda}_{m}$ via the Sugawara procedure (3.8). Since
$\hat{\bar{L}}{}^{\lambda}_{m}$ commutes with
$\widetilde{\bar{K}}{}^{\lambda}_{m}$ and the former satisfy a Virasoro
algebra with central charge $c-1$, it follows that
$\widetilde{\bar{L}}{}^{\lambda}_{m}$ , $\widetilde{\bar{K}}{}^{\lambda}_{m}$
satisfy a Virasoro-Kac-Moody algebra with central charge $c$. Of course, we
could have arrived at the same expression by solving the flow equation (3.6)
for these generators, but the computation would have been significantly more
cumbersome than the one presented herein.

The computations worked out in this section exemplify how our somewhat
abstract definitions for the symmetry generators make very concrete
predictions for their ordering and quantum corrections. Up to
$\mathcal{O}(\lambda)$, our formulae for the flowed generators are similar to
those worked out in [30]; however, their expressions correspond to a slightly
different choice of $J\bar{T}$ flow operator, i.e. one constructed from the
chiral $U(1)$ current, as opposed to the topological current we chose herein.

###  3.3 Algebra of the unflowed generators

In section 3.1, we have given a general argument that the algebra of the
flowed generators consists of two commuting copies of the Virasoro-Kac-Moody
algebra, with the same central extension as that of the original CFT, to all
orders in $\lambda$ and $\hbar$. We have moreover argued that the following
relations between the flowed and unflowed operators hold:

$L_{m}^{\lambda}\equiv\widetilde{L}_{m}^{\lambda}+\lambda\widetilde{K}{}^{\lambda}_{m}H_{R}+\frac{k\lambda^{2}}{4}H_{R}^{2}\delta_{m,0}\;,\;\;\;\;\;\;K_{m}^{\lambda}=\widetilde{K}^{\lambda}_{m}+\frac{\lambda
k}{2}H_{R}\delta_{m,0}$

$\bar{L}_{m}^{\lambda}\equiv\widetilde{\bar{L}}{}_{m}^{\lambda}+\lambda\widetilde{\bar{K}}{}^{\lambda}_{m}H_{R}-\lambda\xi_{m}\widetilde{\bar{K}}{}_{m}^{\lambda}\alpha_{m}^{r}+\frac{k\lambda^{2}}{4}H_{R}^{2}\delta_{m,0}\;,\;\;\;\;\;\;\bar{K}_{m}^{\lambda}=\widetilde{\bar{K}}{}^{\lambda}_{m}+\frac{\lambda
k}{2}H_{R}\delta_{m,0}$ (3.73)

The term proportional to $\xi_{m}$ in the right-moving pseudoconformal
generator represents the different ordering prescription for this generator
for different values of $m$, discussed in section 3.1. Using the explicit form
of the commutators

$[\widetilde{\bar{L}}{}_{m}^{\lambda},H_{R}]=\widetilde{\bar{L}}{}_{m}^{\lambda}\alpha_{m}^{r}\;,\;\;\;\;\;\;\;[\widetilde{\bar{K}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{K}}{}_{m}^{\lambda}\alpha_{m}^{r}$
(3.74)

this ordering difference can be written in the form above, where
$\alpha_{m}^{r}$ is defined in (3.24). As explained, the value
$\xi_{m}=\Theta(m)$ is the one consistent with the usual Hermiticity condition
(3.18) of the generators; however, in this section we leave $\xi_{m}$
arbitrary, so as to capture as many ordering choices as possible.

Given the input that $\widetilde{\bar{L}}{}^{\lambda}_{m}$ and
$\widetilde{\bar{K}}{}^{\lambda}_{m}$ satisfy a Virasoro-Kac-Moody algebra
with central charge c to all orders, we would like to derive the algebra
satisfied by the generators (3.73) to all orders in $\lambda,\hbar$. Using
(3.74), the algebra of the deformed operators is

$[\bar{K}_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=\frac{km\hbar}{2}\delta_{m+n}-\frac{\lambda
k}{2}\bar{K}_{n}\alpha_{n}^{r}\delta_{m,0}+\frac{\lambda
k}{2}\bar{K}_{m}\alpha_{m}^{r}\delta_{n,0}$ (3.75)
$\displaystyle[\bar{L}_{m}^{\lambda},\bar{K}_{n}^{\lambda}]$ $\displaystyle=$
$\displaystyle-n\hbar\bar{K}_{m+n}^{\lambda}-\lambda\bar{K}_{m}^{\lambda}\bar{K}_{n}^{\lambda}\alpha_{n}^{r}+\frac{\lambda
k}{2}\bar{L}_{m}^{\lambda}\alpha_{m}^{r}\delta_{n,0}-\frac{\lambda^{2}k}{4}\bar{K}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}+$
(3.76) $\displaystyle+$
$\displaystyle\lambda\xi_{m}\left[\bar{K}_{m}^{\lambda}\bar{K}_{n}^{\lambda}(\alpha^{r}_{m}+\alpha^{r}_{n}-\alpha^{r}_{m+n})-\frac{km\hbar}{2}\alpha_{m}^{r}\delta_{m+n}\right]$
$\displaystyle[\bar{L}_{m}^{\lambda},\bar{L}_{n}^{\lambda}]$ $\displaystyle=$
$\displaystyle(m-n)\hbar\,\bar{L}_{m+n}^{\lambda}+\frac{c\hbar}{12}m(m^{2}-1)\delta_{m+n}-\lambda\bar{K}_{m}^{\lambda}\bar{L}_{n}^{\lambda}\alpha_{n}^{r}+\lambda\bar{K}_{n}^{\lambda}\bar{L}_{m}^{\lambda}\alpha_{m}^{r}-\frac{k\lambda^{2}}{4}\bar{L}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}+$
(3.77) $\displaystyle+$
$\displaystyle\frac{k\lambda^{2}}{4}\bar{L}_{m}^{\lambda}(\alpha_{m}^{r})^{2}\delta_{n,0}+\lambda(\xi_{m}\bar{K}_{m}^{\lambda}\bar{L}_{n}^{\lambda}-\xi_{n}\bar{K}_{n}^{\lambda}\bar{L}_{m}^{\lambda})(\alpha^{r}_{m}+\alpha^{r}_{n}-\alpha^{r}_{m+n})+\lambda\hbar(m-n)\xi_{m+n}\bar{K}_{m+n}^{\lambda}\alpha^{r}_{m+n}+$
$\displaystyle+$
$\displaystyle\lambda\hbar\bar{K}_{m+n}^{\lambda}(n\xi_{n}\alpha^{r}_{n}-m\xi_{m}\alpha^{r}_{m})+\frac{\lambda^{2}m\hbar
k}{2}\,\xi_{m}\xi_{n}\alpha_{m}^{r}\alpha_{n}^{r}\delta_{m+n}$

It is trivial to show that the left-moving generators
$L_{m}^{\lambda},K_{m}^{\lambda}$ satisfy a Virasoro-Kac-Moody algebra with
central charge $c$. The commutation relations between the left- and the right-
moving generators are

$[K_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=-\frac{\lambda
k}{2}\bar{K}_{n}^{\lambda}\alpha_{n}^{r}\delta_{m,0}\;,\;\;\;\;\;\;[L_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=-\lambda
K_{m}^{\lambda}\bar{K}_{n}^{\lambda}\alpha_{n}^{r}-\frac{\lambda^{2}k}{4}\bar{K}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}$

$[K_{m}^{\lambda},\bar{L}_{n}^{\lambda}]=-\frac{\lambda
k}{2}\bar{L}_{n}^{\lambda}\alpha_{n}^{r}\delta_{m,0}\;,\;\;\;\;\;\;[L_{m}^{\lambda},\bar{L}_{n}^{\lambda}]=-\lambda
K_{m}^{\lambda}\bar{L}_{n}^{\lambda}\alpha_{n}^{r}-\frac{\lambda^{2}k}{4}\bar{L}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}$
(3.78)

which agree with the classical commutators (2.25) at leading order in $\hbar$.

Let us now discuss a few properties of this algebra. In the appendix, it is
shown that the commutators of $H_{R}$ with the
$\widetilde{\bar{L}}{}^{\lambda}_{m}$, $\widetilde{\bar{K}}{}^{\lambda}_{m}$
operators satisfy the Jacobi identities. Since sums and products of operators
satisfying the Jacobi identities also satisfy them, it follows from the
definitions (3.73) that the above algebra is consistent, for any choice of the
parameters $\xi_{m}$. Note that the potential ordering ambiguity parameterized
by the latter only enters the algebra at $\mathcal{O}(\hbar^{2})$, and is thus
consistent with the classical results.

One may try to simplify the above expressions in the special case
$\xi_{m}=\Theta(m)$, obtaining e.g.

$[\bar{L}_{m}^{\lambda},\bar{K}_{n}^{\lambda}]=-n\hbar\bar{K}_{m+n}^{\lambda}+\frac{\lambda
k}{2}\bar{L}_{m}^{\lambda}\alpha_{m}^{r}\delta_{n,0}+\left\\{\begin{array}[]{lll}-\lambda\bar{K}_{n}^{\lambda}\alpha_{n}^{r}\bar{K}_{m}^{\lambda}+\frac{\lambda^{2}k}{4}\bar{K}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}&\mbox{for}&m\geq
0\\\ &&\\\
-\lambda\bar{K}_{m}^{\lambda}\bar{K}_{n}^{\lambda}\alpha_{n}^{r}-\frac{\lambda^{2}k}{4}\bar{K}_{n}^{\lambda}(\alpha_{n}^{r})^{2}\delta_{m,0}&\mbox{for}&m\leq
0\end{array}\right.$ (3.79)

which effectively amounts to an ordering prescription of the non-linear terms
in the algebra that depends on the sign of $m$. The
$[\bar{L}^{\lambda}_{m},\bar{L}^{\lambda}_{n}]$ commutator does not appear to
significantly simply, so we will omit rewriting it. It is worth noting though
that the last term in (3.77) vanishes for this choice of $\xi_{m}$, since
$m,n$ cannot be simultaneousy positive.

The algebra (3.77) can also be rewritten in terms of $\alpha_{m}^{l}$, using
the identity (3.26). Remembering that $\alpha_{m}^{r}$ is better defined for
large positive $m$ and $\alpha_{m}^{l}$ for large negative $m$, one can choose
the best rewriting, depending on the particular signs of sign of $m,n$.

Finally, note that, unlike the Virasoro algebra, which has a finite-
dimensional subalgebra generated by $L_{0,\pm 1}$, the algebra of the unflowed
generators $\bar{L}^{\lambda}_{\pm 1,0}$ does not close; instead, they
generate the entire tower of $U(1)$ generators, $\bar{K}_{m}^{\lambda}$. In
particular, the only notion of $SL(2,\mathbb{R})$ Casimir is the one
associated to the precise combinations of $\bar{L}^{\lambda}_{\pm 1,0}$ and
$\bar{K}_{\pm 1,0}^{\lambda}$ that correspond to the flowed generators of the
$SL(2,\mathbb{R})$ subalgebra.

###  3.4 Comments on representation theory

The definition of the flowed generators $\widetilde{L}^{\lambda}_{m}$, etc.,
via the same flow equation as that obeyed by the energy eigenstates implies
that the states in $J\bar{T}$ \- deformed CFTs are organised in the same way
as in usual CFTs, namely in highest-weight representations of the Virasoro
algebra202020We will assume that these are the only relevant representations
in the seed CFT.. One thus has primary states, which satisfy

$\widetilde{L}_{m}^{\lambda}|h\rangle_{\lambda}=\widetilde{K}_{m}^{\lambda}|h\rangle_{\lambda}=\widetilde{\bar{L}}{}_{m}^{\lambda}|h\rangle_{\lambda}=\widetilde{\bar{K}}{}_{m}^{\lambda}|h\rangle_{\lambda}=0\;,\;\;\;\;\;\forall
m>0$ (3.80)

and

$\widetilde{L}_{0}^{\lambda}|h\rangle_{\lambda}=h|h\rangle_{\lambda}\;,\;\;\;\;\;\;\widetilde{K}_{0}^{\lambda}|h\rangle_{\lambda}=q|h\rangle_{\lambda}\;,\;\;\;\;\;\widetilde{\bar{L}}{}_{0}^{\lambda}|h\rangle_{\lambda}=\bar{h}|h\rangle_{\lambda}\;,\;\;\;\;\;\;\widetilde{\bar{K}}{}_{0}^{\lambda}|h\rangle_{\lambda}=\bar{q}|h\rangle_{\lambda}$
(3.81)

i.e., their eigenvalues are identical to those of the corresponding states in
the undeformed CFT. Descendant states are then obtained by acting with
$\widetilde{L}_{-m}^{\lambda}$, etc, and they generate the entire Verma
module. Note that the norm of these states is identical to the norm of the
corresponding undeformed ones, which are real and positive, and none of the
problems of $J\bar{T}$ \- deformed CFTs on a cylinder, such as the existence
of imaginary energies, are easily visible from this picture. It certainly
appears to be a possibility that, if $|h\rangle_{\lambda}$ are well defined
vectors in the Hilbert space, these problems could be dissociated from the
theory, to only belong to the Hamiltonian operator212121To illustrate this
point, consider at fixed $\lambda$ and $R$ a primary state whose energy is
real. There is no reason to believe that the corresponding state vector
$|h\rangle_{\lambda}$ is not well-defined. Consider next the descendant state
obtained by acting with $\widetilde{\bar{L}}{}^{\lambda}_{-n}$ on this state.
There is no apparent reason that this new state vector should be ill-defined,
either; yet, its energy, as measured by
$H=L_{0}^{\lambda}/R+\bar{L}_{0}^{\lambda}/R_{v}$, is imaginary for large
enough $n$. This appears to be more of a problem with the operator measuring
the energy, than with the state itself. .

The primary states $|h\rangle_{\lambda}$ are clearly primary also with respect
to the unflowed generators $\bar{L}_{m}^{\lambda},\bar{K}_{m}^{\lambda}$.

$L_{m}^{\lambda}|h\rangle_{\lambda}=K_{m}^{\lambda}|h\rangle_{\lambda}={\bar{L}}_{m}^{\lambda}|h\rangle_{\lambda}={\bar{K}}_{m}^{\lambda}|h\rangle_{\lambda}=0\;,\;\;\;\;\;\forall
m>0$ (3.82)

which follows from their definition (3.73). One can thus equally well generate
the spectrum of the $J\bar{T}$ \- deformed CFT using
$L^{\lambda}_{-m},K^{\lambda}_{-m}$, etc. However, the states so generated do
not have well-defined norms; for example, we have

${}_{\lambda}\langle
h|L_{m}^{\lambda}L_{-m}^{\lambda}|h\rangle_{\lambda}=2m\hbar\;{}_{\lambda}\langle
h|L_{0}^{\lambda}|h\rangle_{\lambda}+\frac{c\hbar}{12}m(m^{2}-1)$ (3.83)

${}_{\lambda}\langle
h|\bar{L}_{m}^{\lambda}\bar{L}_{-m}^{\lambda}|h\rangle_{\lambda}=2m\hbar\;{}_{\lambda}\langle
h|\bar{L}_{0}^{\lambda}|h\rangle_{\lambda}+\frac{c\hbar}{12}m(m^{2}-1)$ (3.84)

so their norm becomes imaginary at large enough initial dimension. The
resulting states thus need to be treated perturbatively in $\lambda$. Again,
one cannot exclude the possibility that the state vectors are well-defined,
but the $L_{m}^{\lambda}$ etc., are simply not a very natural set of symmetry
generators to define, in addition to being significantly more cumbersome to
use than the $\widetilde{L}^{\lambda}_{m}$.

##  4 More general non-local CFTs

The argument that we have ultimately given in section 3.1 in favour of the
existence of Virasoro symmetry in $J\bar{T}$ \- deformed CFTs is strikingly
simple and general: consider a family of non-local two-dimensional QFTs,
assumed to be UV complete, which are obtained via an integrable irrelevant
deformation of a CFT2 (of central charge $c$), labeled by a continuous
parameter $\lambda$. In order to have Virasoro symmetry, all that is required
is:

  * i)

that the deformation does not change the Hilbert space, but only the
Hamiltonian and its associated eigenstates. The latter should satisfy a smooth
flow equation

$\partial_{\lambda}|n\rangle_{\lambda}=\tilde{\mathcal{O}}|n\rangle_{\lambda}$
(4.1)

Then, one can always formally define a set of generators222222In this section,
we drop the bars and the tildes from the operators, for simplicity.
$L_{m}^{\lambda}$ via the flow equation (3.6). By construction, the
$L_{m}^{\lambda}$ will obey a Virasoro _algebra_ , with central extension $c$.
Note this does not by itself imply that the theory has Virasoro symmetry.

  * ii)

if $L_{0}^{\lambda}$ is a known function of just the Hamiltonian (and,
possibly, of the momentum and other conserved charges that commute with all
the generators), it immediately follows that

$[L_{m}^{\lambda},H]=\alpha_{m}(H,P,Q)L_{m}^{\lambda}$ (4.2)

The generators $L_{m,S}^{\lambda}(t)\equiv
e^{i\alpha_{m}(H,P,Q)t}L_{m,S}^{\lambda}(0)$ then satisfy the conservation
requirement (1.3). Therefore, the Virasoro generators defined above actually
correspond to _symmetries_.

Of course, explicitly defining the non-local QFT in terms of the irrelevant
deformation is a daunting task, as is proving the the resulting theory is UV
complete. Since the generators $L_{m}^{\lambda}$ are defined using the flow
operator $\tilde{\mathcal{O}}$, which in turn is directly related via (3.5) to
the deforming operator that defines the QFT, it is equally challenging to
construct the Virasoro generators explicitly. However, for many purposes, such
as studying the constraints imposed by the symmetry on various observables,
the explicit form of the generators is not needed, but only the fact that they
_exist_. This follows directly from the well-definiteness of the deformation,
which in the cases of interest is related to the existence of the decoupling
limit.

One piece of information that one does need to know explicitly in order to
establish the Virasoro generators as symmetries is the relation between
$L_{0}^{\lambda}$ and the Hamiltonian of the theory, which determines the
commutator of the $L_{m}^{\lambda}$ with $H$. In $J\bar{T}$ and $T\bar{T}$ \-
deformed CFTs, this relation can be inferred from the complete knowledge of
the deformed energy spectrum, as nicely explained in [30]. Of course, the
simplicity of these relations is a direct consequence of the fact that the
deformed energies are given by a universal formula, which only depends on the
initial energies. It is an interesting question whether similar, universal
relations can be obtained in more general theories, and what is the most
general relation between $L_{0}^{\lambda}$ and $H$ (and possibly other
operators) that allows for the construction of an infinite set of conserved
charges.

In the following, we will discuss several examples of non-local, UV complete
QFTs that satisfy the conditions i) and ii) above, denoted as _non-local
CFTs_. We will allow ourselves to relax condition ii) as needed, only
requiring that it should be possible to add an appropriate time dependence to
the Virasoro generators, so that they satisfy the conservation equation (1.3).

_$T\bar{T}$ \- deformed CFTs_

The conditions i) and ii) above are straightforwardly satisfied by the
$T\bar{T}$ deformation. First, $T\bar{T}$ \- deformed CFTs are believed to be
well-defined two-dimensional QFTs obtained by specifying the irrelevant flow
[15]; they were in addition given a fully non-perturbative definition in [22].
Second, their finite-size spectrum is given by

$E_{\mu}(R)=\frac{\sqrt{1+4\mu(h+\bar{h})+4\mu^{2}(h-\bar{h})^{2}}-1}{2\mu}$
(4.3)

where $h,\bar{h}$ are the original conformal dimensions, $\mu$ is the
$T\bar{T}$ deformation parameter, and we have set $R=1$. The Virasoro
generators, $L^{\mu}_{m}$, can then be constructed via the $T\bar{T}$ analogue
of (3.6). Using the argument of [30] to extract the relation between $H,P$ and
$L_{0}^{\mu},\bar{L}_{0}^{\mu}$, the above equation implies that the zero
modes of this Virasoro algebra are related to the Hamiltonian and momentum
operators as

$L_{0}^{\mu}=\frac{H+P}{2}(1+\mu(H-P))\;,\;\;\;\;\;\;\bar{L}_{0}^{\mu}=\frac{H-P}{2}(1+\mu(H+P))$
(4.4)

These relations immediately determine the commutators of the Virasoro
generators with the Hamiltonian, which take the form

$[L_{m}^{\mu},H]=\alpha_{m}(H,P)L_{m}^{\mu}\;,\;\;\;\;\;\;\;[\bar{L}_{m}^{\mu},H]=\bar{\alpha}_{m}(H,P)\bar{L}_{m}^{\mu}$
(4.5)

where

$\alpha_{m}(H,P)=\frac{1}{2\mu}\left(\sqrt{(1+2\mu H)^{2}+4\mu m\hbar(1+2\mu
P)+4\mu^{2}m^{2}\hbar^{2}}-(1+2\mu H)\right)$ (4.6)

and $\bar{\alpha}_{m}$ is given by the same expression, but with
$P\rightarrow-P$. The operators

$L_{m,S}^{\mu}(t)\equiv
e^{i\alpha_{m}(H,P)t}L_{m,S}^{\mu}(0)\;,\;\;\;\;\;\;\bar{L}_{m,S}^{\mu}(t)\equiv
e^{i\bar{\alpha}_{m}(H,P)t}\bar{L}_{m,S}^{\mu}(0)$ (4.7)

then have conserved expectation value in any state.

From this, we immediately conclude that $T\bar{T}$ \- deformed CFTs have a
Virasoro symmetry of a similar type to the one studied in detail in this
article. The only qualitative difference between the two theories is that
while in $J\bar{T}$ \- deformed CFTs, the left-moving Virasoro symmetry is of
a standard type, i.e. implemented by local conformal transformations, in the
$T\bar{T}$ case both Virasoro symmetries will be implemented by fully non-
local transformations. This is directly related to the fact that the
$J\bar{T}$ deformation produces a ‘half - nonlocal’ theory (more precisely, a
dipole CFT), whereas the $T\bar{T}$ deformation produces a fully non-local
one.

While the quantum-mechanical arguments we have just provided apply equally
well to $T\bar{T}$ \- deformed CFTs as they do to $J\bar{T}$ \- deformed ones,
in the latter case we have additionally been able to provide a completely
explicit, fully non-linear classical realisation of the conserved charges (as
well as a perturbative quantum construction). By contrast, in classical
$T\bar{T}$ \- deformed CFTs, Virasoro symmetries have currently been shown to
exist in the non-compact case only, when the subtleties associated with
boundary conditions can be ignored232323The conserved charges studied in [24]
are only well defined in infinite volume. In finite volume, the field-
dependent coordinates proposed therein do not satisfy all possible Jacobi
identities [31]. An encouraging sign coming from the study of $J\bar{T}$ \-
deformed CFTs suggests that the zero mode of the field-dependent coordinate,
which is the one that prevents the Jacobi identities from being satisfied in
$T\bar{T}$, may in fact drop out from the expression for the improved finite-
size charges. . It would be certainly interesting to find the classical
conserved charges for $T\bar{T}$ \- deformed CFTs in compact space, e.g. by
following the steps outlined in [25] for the case of $J\bar{T}$.

_Single-trace $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs_

Another set of theories that are likely to satisfy the two conditions we