# Symmetries versus the spectrum of $J\bar{T}$ \- deformed CFTs

Monica Guica

_Institut de Physique Théorique, CEA Saclay, CNRS, 91191 Gif-sur-Yvette,
France_
_Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm,
Sweden_
_Nordita, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden_

###### Abstract

It has been recently shown that classical $J\bar{T}$ \- deformed CFTs possess
an infinite-dimensional Witt-Kač-Moody symmetry, generated by certain field-
dependent coordinate and gauge transformations. On a cylinder, however, the
equal spacing of the descendants’ energies predicted by such a symmetry
algebra is inconsistent with the known finite-size spectrum of $J\bar{T}$ \-
deformed CFTs. Also, the associated quantum symmetry generators do not have a
proper action on the Hilbert space. In this article, we resolve this tension
by finding a new set of (classical) conserved charges, whose action is
consistent with semiclassical quantization, and which are related to the
previous symmetry generators by a type of energy-dependent spectral flow. The
previous inconsistency between the algebra and the spectrum is resolved
because the energy operator does not belong to the spectrally flowed sector.

###### Contents

  1. 1 Introduction and statement of the problem
  2. 2 Flow of the eigenstates versus the symmetry generators
    1. 2.1 The flow of energy eigenstates
    2. 2.2 Flow of the symmetry generators
    3. 2.3 Relating the two
  3. 3 The spectrally flowed generators
    1. 3.1 Perturbative construction of the spectrally flowed generators
    2. 3.2 An all-orders proposal
  4. 4 Discussion
  5. A Summary of Poisson brackets in $J\bar{T}$ \- deformed CFTs
  6. B Poisson brackets of the non-local field $\chi$
    1. B.1 Poisson bracket of $\chi$ with $\mathcal{H}_{R}$
    2. B.2 Poisson bracket of $\chi$ with $\mathcal{J}_{-}$
    3. B.3 Other commutators
  7. C Flow equations for the currents and the charges

##  1 Introduction and statement of the problem

The discovery of the AdS/CFT correspondence [1] has marked a major step in our
current understanding of quantum gravity. While there are good reasons to
believe that gravity in general backgrounds is holographic, various clues
point towards the fact that for asymptotically flat spacetimes [2] or
spacetimes related to the near-horizon geometry of extremal black holes[3],
the dual QFT may be non-local. However, non-local quantum field theories are
still relatively poorly understood, in comparison with their local
counterparts.

In [4], Smirnov and Zamolodchikov (see also [5]) laid out the construction of
a set of tractable irrelevant deformations of two-dimensional local QFTs which
result into QFTs that are non-local, yet appear to be UV complete [6, 7].
Moreover, these theories are solvable, in the sense that one can compute their
spectrum, $S$-matrix and other observables [8, 9, 10, 11] in terms of the
corresponding quantities in the undeformed QFT. Even more interestingly,
certain single-trace analogues of the Smirnov-Zamolodchikov deformations have
been related to holography in non-asymptotically AdS spacetimes [12, 13, 14].

The Smirnov-Zamolodchikov deformations are constructed from bilinears of two
conserved currents. The best studied of these is the so-called $T\bar{T}$
deformation, constructed from the components of the stress tensor.
Deformations constructed from a $U(1)$ current and the stress tensor, such as
the $J\bar{T}$ [15] and the $JT_{a}$ [16] deformations are also relatively
well studied. Of these, the $J\bar{T}$ deformation of two-dimensional CFTs is
the simplest, as the non-locality of the deformed QFT is concentrated
exclusively to the right-moving side, and the theory stays local and conformal
on the left. The effect of performing several of these deformations
simultaneously has been studied in e.g. [17, 18].

It is interesting to better understand the structure of the Smirnov-
Zamolodchikov deformations from a quantum-field-theoretical point of view. It
has been recently shown [19] that at least at the classical level, $T\bar{T}$,
$J\bar{T}$ and $JT_{a}$ \- deformed CFTs all posses an infinite-dimensional
set of field-dependent symmetries, whose algebra consists of two commuting
copies of the Witt or the Witt-Kač-Moody algebra, if $U(1)$ currents are
present. This structure was suggested by the previous holographic analyses of
[20] for $T\bar{T}$ and [21] for $J\bar{T}$. These analyses also allowed for
the computation of the central extension of the symmetry algebra, which
becomes Virasoro-Kač-Moody. If these symmetries survive quantization, then we
would conclude that $T\bar{T}$, $J\bar{T}$ and $JT_{a}$ \- deformed CFTs
correspond to a non-local version of two-dimensional CFTs, with a similarly
rigid structure that would highly deserve further exploration.

There is, however, a problem, which can be seen already at the semiclassical
level. The symmetry analysis of [19] is valid on both the plane and the
cylinder. In the latter case, one immediately encounters a tension between the
equally-spaced energies of the Virasoro descendants predicted by the symmetry
analysis and the energies of the deformed eigenstates in $T\bar{T}$,
$J\bar{T}$ and $JT_{a}$ \- deformed CFTs, which take a square root form. In
this note, we address this issue for the simplest case of the $J\bar{T}$
deformation, where the locality of the left-moving side provides a useful
guiding principle for finding its resolution.

To state the problem explicitly, we start with a review of the relevant facts.
The finite-size energy spectrum of $J\bar{T}$ \- deformed CFTs placed on a
cylinder of circumference $R$ is given by [14, 21]

$E_{R}\equiv\frac{E-P}{2}=\frac{2}{\lambda^{2}}\left(R-\lambda
J_{0}-\sqrt{(R-\lambda
J_{0})^{2}-\lambda^{2}E_{R}^{(0)}}\right)\;,\;\;\;\;\;E_{L}=E_{R}+P$ (1.1)

where $\lambda$ is the deformation parameter (with dimensions of length),
$J_{0}$ is the left-moving charge in the undeformed CFT, $P$ is the quantized
momentum, $E^{(0)}_{R}$ is the undeformed right-moving energy and the chiral
anomaly coefficient has been set to $2\pi$. Note that while in the undeformed
CFT, the energies of the right-moving (Virasoro and Kač-Moody) descendats of a
primary state are equally-spaced, those of the corresponding
$J\bar{T}$-deformed descendants are not. The energies of the left-moving
descendants do have equal spacing, since they are obtained by raising
$E_{L}=E_{R}+P$ with $E_{R}^{(0)}$ and $J_{0}$ held fixed.

We note in passing that the relation between the undeformed and deformed
energies can be suggestively written as spectral flow [22], with a parameter
$\lambda E_{R}$ proportional to the right-moving energy

$E_{L}R=E_{L}^{(0)}R+\lambda
J_{0}E_{R}+\frac{\lambda^{2}}{4}E_{R}^{2}\;,\;\;\;\;\;\;\;E_{R}(R-\lambda
w)=E_{R}^{(0)}R+\lambda\bar{J}_{0}E_{R}+\frac{\lambda^{2}}{4}E_{R}^{2}$ (1.2)

Here, $\bar{J}_{0}$ is the right-moving $U(1)$ charge in the undeformed CFT,
and $w=J_{0}-\bar{J}_{0}$ is the winding charge. This observation will be
quite useful later. It has already been used in deriving the spectrum in
presence of a chiral anomaly [14] and for organising the conformal dimensions
of $J\bar{T}$-deformed CFTs on the plane [23].

The symmetries of $J\bar{T}$ \- deformed CFTs consist of, first, an infinite
set of left-moving conformal and $U(1)$ gauge symmetries that enhance the
$SL(2,\mathbb{R})_{L}\times U(1)_{L}$ global symmetries of the theory. These
symmetries are parametrized by two arbitrary functions of the left-moving
coordinate, $U=\sigma+t$. In the general Hamiltonian framework for $J\bar{T}$
\- deformed CFTs developed in [19], they are generated by

$Q_{f}=\int d\sigma\,f(U/R)\,\mathcal{H}_{L}\;,\;\;\;\;\;\;\;P_{\eta}=\int
d\sigma\,\eta(U/R)\,(\mathcal{J}_{+}+\frac{\lambda}{2}\mathcal{H}_{R})$ (1.3)

where, in order for the argument of the functions to have periodicity one, the
coordinate $U$ has been divided by the circumference of the circle.
$\mathcal{H}_{L}=\mathcal{H}_{R}+\mathcal{P}$ is the left-moving Hamiltonian
density, where the right-moving Hamiltonian density $\mathcal{H}_{R}$ is given
in terms of its undeformed counterpart $\mathcal{H}_{R}^{(0)}$ by a formula
entirely analogous to (1.1), with $J_{0}$ replaced by $\mathcal{J}_{+}$, the
left-moving current density. The commutation relations of the deformed
generators are then fixed by those of the undeformed currents, and one can
show that the left-moving charge algebra is precisely Witt-Kač-Moody

$\\{Q_{f},Q_{g}\\}=\frac{1}{R}\,Q_{fg^{\prime}-f^{\prime}g}\;,\;\;\;\;\;\;\\{Q_{f},P_{\eta}\\}=\frac{1}{R}\,P_{f\eta^{\prime}}\;,\;\;\;\;\;\;\\{P_{\eta},P_{\chi}\\}=\frac{1}{2}\int
d\sigma\chi\partial_{\sigma}\eta$ (1.4)

The factor of $R$ can be absorbed into a rescaling of the pseudoconformal
charges $\bar{Q}_{\bar{f}}$.

The second set of infinite-dimensional symmetries of $J\bar{T}$ \- deformed
CFTs are field-dependent, and are generated by functions of the field-
dependent coordinate

$v=\sigma-t-\lambda\phi$ (1.5)

where $\phi$ is related to the current $J$ via $J=\star d\phi$. The conserved
pseudo-conformal and U(1) charges are given by

$\bar{Q}_{\bar{f}}=\int
d\sigma\,\bar{f}\left(\frac{v}{R_{v}}\right)\,\mathcal{H}_{R}\;,\;\;\;\;\;\;\;\bar{P}^{KM}_{\bar{\eta}}=\int
d\sigma\,\bar{\eta}\left(\frac{v}{R_{v}}\right)\,(\mathcal{J}_{-}+\frac{\lambda}{2}\mathcal{H}_{R})$
(1.6)

where $R_{v}=R-\lambda w$ is the field-dependent radius of the field-depedent
coordinate $v$, and the particular combination $\bar{P}^{KM}$ of the right-
moving currents is singled out by its simple commutation relations.
Remarkably, these charges entirely commute with the left-moving ones, and the
charge algebra is still a functional Witt-Kač-Moody algebra

$\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}=\frac{1}{R_{v}}\bar{Q}_{\bar{f}^{\prime}\bar{g}-\bar{f}\bar{g}^{\prime}}\;,\;\;\;\;\;\\{\bar{Q}_{\bar{f}},\bar{P}_{\bar{\eta}}^{KM}\\}=-\frac{1}{R_{v}}\bar{P}^{KM}_{\bar{f}\bar{\eta}^{\prime}}\;,\;\;\;\;\;\;\\{\bar{P}^{KM}_{\bar{\chi}},\bar{P}_{\bar{\eta}}^{KM}\\}=-\frac{1}{2}\int
d\sigma\bar{\chi}\partial_{\sigma}\bar{\eta}$ (1.7)

The word ‘functional’ above refers to the fact that the factors of $R_{v}$ can
be absorbed into a redefinition of the ′ to mean ‘derivative with respect to
$v$’, rather than to the full rescaled argument of the functions
$\bar{f},\bar{\eta}$, as used above. In any case, when labeling the charges in
terms of the Fourier modes of $\bar{f},\bar{\eta}$, etc., the field-dependent
radius will explicitly appear in the algebra111In the $J\bar{T}$ case, we can
simply rescale the generators by $R_{v}$ to obtain a usual Witt-Kač-Moody
algebra.. While this algebra is not exactly Witt-Kač-Moody, it still predicts,
upon quantization, an equally-spaced spectrum of descendants, which is
incompatible with the energy formula (1.1).

It is in fact not hard to notice already from their classical Poisson
brackets, that the right-moving charges (1.6) will not have a proper action on
the semiclassical phase space of the theory, where the charges associated to
the global $U(1)$ symmetry and the momentum are quantized. Concretely, the
problem appears to lie in the commutators of the right-moving generators with
the $U(1)$ charges

$\\{\bar{Q}_{\bar{f}},J_{0}\\}=\frac{1}{R_{v}}\int_{0}^{R}d\sigma\bar{f}^{\prime}\left(\frac{v}{R_{v}}\right)\mathcal{H}_{R}(\sigma)\\{v(\sigma),J_{0}\\}=-\frac{\lambda}{2R_{v}}\,\bar{Q}_{\bar{f}^{\prime}}=\\{\bar{Q}_{\bar{f}},\bar{J}_{0}\\}$
(1.8)

and the analogous commutators of the right-moving $U(1)$ generators
$\bar{P}_{\bar{\eta}}$ with $J_{0}$ and $\bar{J}_{0}$. What this means is that
$J_{0}+\bar{J}_{0}$, which represents the global $U(1)$ charge of the
configuration and which should be quantized, is changed by a non-integer
amount (more precisely, $2\pi\lambda n/R_{v}$, with $n\in\mathbb{Z}$) by the
action of the semiclassically quantized right-moving generators on a state in
the deformed theory. A similar statement holds for the momentum, which from
(1.7) can be shown to satisfy

$\\{\bar{Q}_{\bar{f}},P\\}=-\frac{1}{R_{v}}\bar{Q}_{\bar{f}^{\prime}}\;,\;\;\;\;\;\;\;\\{\bar{P}_{\bar{\eta}},P\\}=-\frac{1}{R_{v}}\bar{P}_{\bar{\eta}^{\prime}}$
(1.9)

i.e. it is changed by units of $2\pi/R_{v}$, instead of $2\pi/R$. These
observations imply that the action of the right-moving generators (with the
exception of the global right-moving energy and charge) on a field
configuration is in tension with semiclassical quantization. Hence, the naive
quantum versions of the charges (1.6) do not act properly on the Hilbert space
of $J\bar{T}$ \- deformed CFTs on a cylinder.

While having an infinite set of symmetry generators that do not properly act
on the Hilbert space of the system is not very useful, an interesting question
is whether these generators can be modified in such a way that their algebra
is preserved, but their action on the Hilbert space is rectified. In this
note, we show that this is indeed possible, by explicitly constructing an
infinite set of charges that, upon quantization, would act on the deformed
finite-size Hilbert space in a way consistent with charge and momentum
quantization. These charges can therefore be used to organise the spectrum of
the deformed CFT.

To find them, we study the flow equation with respect to the deformation
parameter $\lambda$ of the various energy eigenstates and compare it to the
flow of the symmetry generators (1.3) and (1.6). Introducing a new set of
operators that relate deformed descendant states to the deformed primaries, we
find that they are related to the previously discussed symmetry generators by
a type of energy-dependent spectral flow transformation. The new symmetry
generators are conserved and satisfy a Witt-Kač-Moody algebra with a field-
independent radius. Their commutation relations with the energy and momentum
are non-trivial though, as the latter two operators belong to the unflowed
sector. This resolves the apparent tension between the symmetry algebra and
the spectrum of $J\bar{T}$ \- deformed CFTs. We should note that while our
analysis is mostly classical - i.e., at the level of Poisson brackets - the
quantum generalization of these generators now appears to be straightforward.

This paper is organised as follows. In section 2, we derive the flow equation
satisfied by the energy eigenstates in a $J\bar{T}$ \- deformed CFT, by
adapting the method used in [11] to study the flow of states under the
$T\bar{T}$ deformation. We subsequently compare this to the flow equations
satisfied by the symmetry generators, and argue that the two sets of
generators must be related by a similarity transformation that we denote as
“spectral flow”. In section 3, we proceed to finding the flowed operators,
first perturbatively and then by making an all-orders proposal, whose
consistency we then check. The technical details of the very many Poisson
brackets we need are collected in the appendices.

##  2 Flow of the eigenstates versus the symmetry generators

###  2.1 The flow of energy eigenstates

Let $|n_{\lambda}\rangle$ be an energy (and momentum, and charge) eigenstate
in the theory deformed by an amount $\lambda$. As $\lambda$ is infinitesimally
changed, the change in the eigenstate is given by first-order quantum-
mechanical perturbation theory

$\partial_{\lambda}|n_{\lambda}\rangle=\sum_{m\neq n}\frac{\langle
m_{\lambda}|\partial_{\lambda}H|n_{\lambda}\rangle}{E_{n}^{\lambda}-E_{m}^{\lambda}}\,|m_{\lambda}\rangle$
(2.1)

where $\partial_{\lambda}H$ is the change in the Hamiltonian. For convenience,
we take the deforming operator to be $\tilde{J}\bar{T}$, rather than
$J\bar{T}$, where $\tilde{J}=\star d\phi$ is a topologically conserved
current. Its components are

$\tilde{J}_{t}=\phi^{\prime}\;,\;\;\;\;\;\;\;\tilde{J}_{\sigma}=\partial_{\pi}\mathcal{H}$
(2.2)

where $\pi$ is the canonical momentum conjugate to $\phi$. One can easily
check, using the method developed in [19], that the $\tilde{J}\bar{T}$
deformation leads to the same deformed Hamiltonian density as $J\bar{T}$.
Consequently, the change in the Hamiltonian is given by222We are using the
conventions of [19] throughout this article.

$\partial_{\lambda}H(\lambda)=-\int
d\sigma\mathcal{O}_{\tilde{J}\bar{T}}=-\int
d\sigma\,\epsilon^{\alpha\beta}\tilde{J}_{\alpha}(\sigma)T_{\beta V}(\sigma)$
(2.3)

To find the general solution for the deformed eigenstates, we will use the
technique proposed by [11]. On an equal-time slice, we write

$\int d\sigma\mathcal{O}_{\tilde{J}\bar{T}}=\int d\sigma
d\tilde{\sigma}\,\epsilon^{\alpha\beta}\tilde{J}_{\alpha}(\sigma)\delta(\sigma-\tilde{\sigma})T_{\beta
V}(\tilde{\sigma})$ (2.4)

It is useful to introduce the Green’s function on the cylinder of
circumference $R$

$G(\sigma)=\frac{1}{2\pi i}\sum_{m\neq 0}\frac{1}{m}e^{2\pi
im\sigma/R}=\frac{1}{2}\mbox{sgn}(\sigma)-\frac{\sigma}{R}$ (2.5)

which is single-valued and satisfies

$\partial_{\sigma}G(\sigma-\tilde{\sigma})=\delta(\sigma-\tilde{\sigma})-\frac{1}{R}$
(2.6)

Then, we can rewrite the deforming operator as

$\displaystyle\int d\sigma\mathcal{O}_{\tilde{J}\bar{T}}$ $\displaystyle=$
$\displaystyle\int d\sigma
d\tilde{\sigma}\left[\tilde{J}_{\sigma}(\sigma)\left(\frac{1}{R}+\partial_{\sigma}G(\sigma-\tilde{\sigma})\right)T_{tV}(\tilde{\sigma})-\tilde{J}_{t}(\sigma)\left(\frac{1}{R}-\partial_{\tilde{\sigma}}G(\sigma-\tilde{\sigma})\right)T_{\sigma
V}(\tilde{\sigma})\right]$ (2.7) $\displaystyle=$
$\displaystyle\frac{1}{R}\epsilon^{\alpha\beta}\tilde{J}^{0}_{\alpha}\,T^{0}_{\beta
V}-\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})(\partial_{\sigma}\tilde{J}_{\sigma}(\sigma)T_{tV}(\tilde{\sigma})+\tilde{J}_{t}(\sigma)\partial_{\tilde{\sigma}}T_{\sigma
V}(\tilde{\sigma}))$ $\displaystyle=$
$\displaystyle\frac{1}{R}\epsilon^{\alpha\beta}\tilde{J}^{0}_{\alpha}\,T^{0}_{\beta
V}-\partial_{t}\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\tilde{J}_{t}(\sigma)T_{tV}(\tilde{\sigma})$

where we introduced the notation

$\tilde{J}^{0}_{\alpha}\equiv\int
d\sigma\,\tilde{J}_{\alpha}\;,\;\;\;\;\;\;\;T^{0}_{\beta V}\equiv\int
d\sigma\,T_{\beta V}$ (2.8)

Naturally, the integral of the time components of the currents above will
yield the associated conserved charges, i.e. the winding, $w$, and
respectively (minus) the right-moving energy, $-E_{R}$. We further use the
manipulations of [11] to rewrite the regulated denominator of (2.1) as an
integral, in terms of which the flow equation for the states becomes

$\displaystyle\partial_{\lambda}|n_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle-i\sum_{m\neq
n}\int_{-\infty}^{0}dt\,e^{t\epsilon}\,|m_{\lambda}\rangle\langle
m_{\lambda}|\partial_{\lambda}H_{\lambda}(t)|n_{\lambda}\rangle$ (2.9)
$\displaystyle=$ $\displaystyle i\sum_{m\neq n}\,|m_{\lambda}\rangle\langle
m_{\lambda}\left|\,\frac{1}{R}\int_{-\infty}^{0}dt\,e^{t\epsilon}\epsilon^{\alpha\beta}\tilde{J}^{0}_{\alpha}\,T^{0}_{\beta
V}-\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\tilde{J}_{t}(\sigma)T_{tV}(\tilde{\sigma})\right|n_{\lambda}\rangle$

Here, $\epsilon>0$ is an infinitesimal regulator used to make the integral
converge, and the second term is evaluated on the $t=0$ slice. Since we are
working on the cylinder, the first term cannot be ignored. To evaluate it, we
need the explicit form of the spatial components of $\tilde{J}$ and the right-
moving translation generator, which can be worked out using the formulae given
in [19]

$\tilde{J}_{\sigma}=\partial_{\pi}\mathcal{H}=\phi^{\prime}+2\partial_{\pi}\mathcal{H}_{R}=\phi^{\prime}+2\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}$
(2.10)

and

$T_{\sigma
V}=2T_{VV}-\mathcal{H}_{R}=2\frac{\mathcal{H}_{R}}{\sqrt{}}-\mathcal{H}_{R}$
(2.11)

where the somewhat unusual notation $\sqrt{}\;$ is a shorthand for
$\sqrt{(1-\lambda\mathcal{J}_{+})^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}$.
Therefore,

$\epsilon^{\alpha\beta}\tilde{J}_{\alpha}^{0}T_{\beta V}^{0}=-2w\int
d\sigma\frac{\mathcal{H}_{R}}{\sqrt{}}-2E_{R}\int
d\sigma\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}$ (2.12)

In order to perform the time integral in (2.9), we would like to rewrite the
above operator as a time derivative, i.e. as a commutator with the
Hamiltonian. This can be achieved by introducing the zero modes

$\phi_{0}\equiv\int_{0}^{R}d\sigma\,\phi(\sigma)\;,\;\;\;\;\;\;\;\chi_{0}\equiv\int_{0}^{R}d\sigma\,\chi(\sigma)$
(2.13)

where the auxiliary non-local field $\chi$ is defined via

$\partial_{\sigma}\chi\equiv\mathcal{H}_{R}$ (2.14)

Such fields also made their appearance in the analysis of the charge algebra
for $T\bar{T}$ \- deformed CFTs in [19], though they were not given an
explicit name.

The Poisson brackets of the fields $\phi$ and $\chi$ (and, consequently, of
their zero modes $\phi_{0}$ and $\chi_{0}$) are fixed by the Poisson brackets
of the corresponding currents $\mathcal{J}_{\pm},\mathcal{H}_{R},\mathcal{P}$,
up to some possible integration functions. While the choice of these functions
is straightforward for the $\phi$ commutators, as $\phi$ is a local field, it
is however somewhat subtle for the case of the $\chi$ commutators, because
$\chi$ is non-local. In appendix B, we perform a rather detailed analysis of
the Jacobi identities that constrain these integration functions, with the
result

$\\{\chi_{0},H\\}=-2\int_{0}^{R}d\sigma\frac{\mathcal{H}_{R}}{\sqrt{}}+E_{R}+\frac{E_{R}R}{R_{v}}\;,\;\;\;\;\;\;\;\\{\phi_{0},H\\}=2\int_{0}^{R}d\sigma\frac{\mathcal{J}_{-}+\lambda/2\mathcal{H}_{R}}{\sqrt{}}+w$
(2.15)

where $E_{R}=\int d\sigma\mathcal{H}_{R}$ is the right-moving energy operator.
Consequently, we can write

$\epsilon^{\alpha\beta}\tilde{J}_{\alpha}^{0}T_{\beta
V}^{0}=-\\{H,w\chi_{0}-E_{R}\phi_{0}\\}-wE_{R}\frac{R}{R_{v}}=\frac{d}{dt}(w\chi_{0}-E_{R}\phi_{0})-w\frac{E_{R}R}{R_{v}}$
(2.16)

Plugging this into (2.9), the last term drops out, as it is evaluated between
two different energy-momentum eigenstates. The integral of the first term over
the half line yields, in the limit $\epsilon\rightarrow 0$

$\partial_{\lambda}|n_{\lambda}\rangle=i\sum_{m\neq
n}|m_{\lambda}\rangle\langle
m_{\lambda}\left|\,\frac{w\chi_{0}-E_{R}\phi_{0}}{R}+\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\phi^{\prime}(\sigma)\mathcal{H}_{R}(\tilde{\sigma})\,\right|n_{\lambda}\rangle$
(2.17)

We will be denoting these two contributions as $\Delta\mathcal{O}$ and
respectively $\hat{\mathcal{O}}$, defined as

$\Delta\mathcal{O}\equiv\frac{w\chi_{0}-E_{R}\phi_{0}}{R}\;,\;\;\;\;\;\;\;\;\hat{\mathcal{O}}\equiv\int
d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\phi^{\prime}(\sigma)\mathcal{H}_{R}(\tilde{\sigma})$
(2.18)

and their sum will be denoted as
$\mathcal{O}_{tot}=\Delta\mathcal{O}+\hat{\mathcal{O}}$. If we make use of the
identity

$\int
d\tilde{\sigma}\phi^{\prime}(\tilde{\sigma})G(\tilde{\sigma}-\sigma)=\left.\phi(\tilde{\sigma})G(\tilde{\sigma}-\sigma)\right|_{0}^{R}-\phi(\sigma)+\phi_{0}=-\hat{\phi}(\sigma)+\phi_{0}$
(2.19)

where $\hat{\phi}(\sigma)=\phi(\sigma)-w\sigma/R$ is the scalar field with its
winding mode removed (which is thus single-valued on the circle), then an
alternate expression for $\mathcal{O}_{tot}$ is

$\mathcal{O}_{tot}=\frac{w\chi_{0}}{R}-\int
d\sigma\hat{\phi}(\sigma)\mathcal{H}_{R}$ (2.20)

which is rather useful in computing its Poisson brackets.

As a final step of our manipulations, we use the assumed completeness of the
set of states to rewrite the flow equation for the energy eigenstates as

$\partial_{\lambda}|n_{\lambda}\rangle=i\mathcal{O}_{tot}|n_{\lambda}\rangle-i|n_{\lambda}\rangle\langle
n_{\lambda}|\mathcal{O}_{tot}|n_{\lambda}\rangle$ (2.21)

Introducing an operator, $D$, which is diagonal in the energy eigenbasis and
whose matrix elements are defined as $\langle
n_{\lambda}|D|n_{\lambda}\rangle=\langle
n_{\lambda}|\mathcal{O}_{tot}|n_{\lambda}\rangle$, we can rewrite the flow
equation for the eigenstates in its final form

$\boxed{\partial_{\lambda}|n_{\lambda}\rangle=i(\mathcal{O}_{tot}-D)|n_{\lambda}\rangle}$
(2.22)

Thus, to understand the flow of the states, we need to understand also which
parts of $\mathcal{O}_{tot}$ have non-zero expectation values in the energy
eigenstates. This is a quite non-trivial task for arbitrary values of the flow
parameter. We can nevertheless attempt to understand this problem
perturbatively. For example, at $\lambda=0$, we can use (2.5) to evaluate

$i\hat{\mathcal{O}}=\sum_{m\neq
0}\frac{1}{m}:\tilde{J}_{m}\bar{L}_{m}:+\ldots$ (2.23)

where $\tilde{J}_{m}=J_{m}-\bar{J}_{-m}$ and $\bar{L}_{m}$ are the Fourier
modes of $\phi^{\prime}$ and respectively $\mathcal{H}_{R}$ in the undeformed
CFT, and the colons denote normal ordering. Since the sum is strictly over
non- zero modes, it is clear that the expectation value of this operator in
any energy eigenstate of the undeformed CFT is zero. Thus, $\hat{\mathcal{O}}$
does not contribute to $D$, at least at $\lambda=0$. On the other hand, the
expectation value of $\Delta\mathcal{O}$ vanishes between any two different
energy eigenstates at $\lambda=0$, as one can see by evaluating

$\langle m|[H,\Delta\mathcal{O}]|n\rangle=(E_{m}-E_{n})\langle
m|\Delta\mathcal{O}|n\rangle=i\langle m|(J_{0}+\bar{J}_{0})E_{R}|n\rangle=0$
(2.24)

which implies that333For degenerate eigenstates, one can repeat the argument
for the commutator with other globally conserved charges.  $\langle
m|\Delta\mathcal{O}|n\rangle=0\;,\;$ $\forall m\neq n$. Thus, we find that at
$\lambda=0$, $D=\Delta\mathcal{O}$.

At higher orders in perturbation theory, $\Delta\mathcal{O}$ may start having
non-zero matrix elements between different eigenstates, which would therefore
not contribute to $D$. To understand what happens, we should study the change
with $\lambda$ of the matrix elements $\langle m|\Delta\mathcal{O}|n\rangle$

$\partial_{\lambda}\langle m|\Delta\mathcal{O}|n\rangle=\langle
m|\partial_{\lambda}\Delta\mathcal{O}-i[\mathcal{O}_{tot}-D,\Delta\mathcal{O}]|n\rangle=\langle
m|\hat{\mathcal{D}}_{\lambda}\Delta\mathcal{O}|n\rangle+i(D_{m}-D_{n})\langle
m|\Delta\mathcal{O}|n\rangle$ (2.25)

where the flow operator $\hat{\mathcal{D}}_{\lambda}$ is defined as

$\hat{\mathcal{D}}_{\lambda}\equiv\partial_{\lambda}-i[\mathcal{O}_{tot},\;\cdot\;]$
(2.26)

Using the explicit expression, (3.3), for
$\mathcal{D}_{\lambda}\Delta\mathcal{O}$ computed in the next section, we see
that at $\lambda=0$, the only contribution to $\langle
n|\mathcal{D}_{\lambda}\Delta\mathcal{O}|n\rangle$ comes from the terms
proportional to the zero modes of the fields $\phi$ and $\chi$. The $\lambda$
dependence of the diagonal matrix elements of $\hat{\mathcal{O}}$ can be
studied by plugging in the known expression for $\mathcal{H}_{R}(\lambda)$. At
first order in $\lambda$, it also does not look like this operator has non-
zero diagonal matrix elements in the energy eigenbasis444Using the formulae in
appendix C, it can be shown that $\hat{\mathcal{O}}$ satisfies the very simple
flow equation
$\mathcal{D}_{\lambda}\hat{\mathcal{O}}=w\,\hat{\mathcal{O}}/R_{v}$ (for
$a=-1$), which can be used to evaluate its contribution, if any, at higher
orders in $\lambda$., and thus it will not contribute to $D$.

To summarize, up to first order in $\lambda$, we expect that

$D=\Delta\mathcal{O}-\lambda(\mathcal{D}_{\lambda}\Delta\mathcal{O})_{no\,z.m.}+\mathcal{O}(\lambda^{2})$
(2.27)

i.e., we are subtracting all the off-diagonal contributions to
$\Delta\mathcal{O}$ up to this order. Performing this analysis to higher order
looks increasingly cumbersome, and we may need a better method.

The discussion so far holds for states defined on the $t=0$ slice. It is
interesting to also consider the flow equation for states defined at a time
$t$ instead of $t=0$. Our derivation of the flow operator (2.17) still holds,
except that it should now be evaluated at time $t$, rather than $t=0$. Since
the states at $t$ are related to the states at $t=0$ by a $\lambda$ \-
dependent energy factor, the flow equation is best written as

$\partial_{\lambda}|n_{\lambda}(t)\rangle=i(\mathcal{O}_{tot}(t)-\partial_{\lambda}E_{n}\,t-D(t))|n_{\lambda}(t)\rangle\;,\;\;\;\;\;\;\partial_{\lambda}E=2\frac{E_{R}Q_{K}}{R-\lambda
Q_{K}}$ (2.28)

where $Q_{K}\equiv J_{0}+\lambda E_{R}/2$, and the expression for
$\partial_{\lambda}E$ is obtained from (1.1). The matrix elements of the
operator $D(t)$ are defined as the expectation values of
$\mathcal{O}_{tot}(t)$ in energy eigenstates.

###  2.2 Flow of the symmetry generators

Having understood the flow of the energy eigenstates with respect to
$\lambda$, we would now like to discuss the corresponding flow of the symmetry
generators $Q_{f},P_{\eta},\bar{Q}_{\bar{f}}$ and $\bar{P}_{\bar{\eta}}$. It
is useful to compute the action of the operator $\mathcal{D}_{\lambda}$
defined above on these generators. Our analysis will be classical, and thus we
will be using the Poisson bracket counterpart of this flow operator, i.e

$\mathcal{D}_{\lambda}\equiv\partial_{\lambda}+\\{\mathcal{O}_{tot},\;\cdot\;\\}$
(2.29)

obtained by the usual replacement $[\;,\;]\rightarrow i\\{\;,\;\\}$. To
compute the action of $\mathcal{D}_{\lambda}$, we will need the Poisson
brackets of the various currents in the $J\bar{T}$-deformed CFT, which were
derived in [19] and are collected for convenience in appendix A. We will also
need the Poisson brackets of the various symmetry currents with the zero modes
of $\chi$ and $\phi$.

The commutators of the zero mode of $\phi$ are obtained by simply integrating
the corresponding commutators of the field $\phi(\sigma)$, and we obtain

$\\{\phi_{0},\mathcal{H}_{R}(\sigma)\\}=\frac{\mathcal{J}_{-}+\frac{\lambda}{2}\mathcal{H}_{R}}{\sqrt{(1-\lambda\mathcal{J}_{+})^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}\;,\;\;\;\;\;\;\\{\phi_{0},\mathcal{P}(\sigma)\\}=\phi^{\prime}(\sigma)\;,\;\;\;\;\;\;\\{\phi_{0},\mathcal{J}_{\pm}\\}=\frac{1}{2}$
(2.30)

Note that in the CFT limit, $\\{\phi_{0},\mathcal{H}_{L}\\}=\mathcal{J}_{+}$
and $\\{\phi_{0},\mathcal{H}_{R}\\}=\mathcal{J}_{-}$, so the exponential of
this operator is precisely what generates spectral flow for the left- and the
right-movers.

The Poisson brackets of the zero mode $\chi_{0}$ are significantly more
involved, due to the fact that the ancillary field $\chi$ is non-local, being
defined as the integral of the local current $\mathcal{H}_{R}$. Consequently,
its Poisson brackets are defined only up to certain integration functions,
whose form is non-trivially constrained by various Jacobi identities. These
constraints are analysed in detail in appendix B, and the end result for the
various commutators of $\chi_{0}$ is

$\displaystyle\\{\chi_{0},\mathcal{J}_{+}\\}$ $\displaystyle=$
$\displaystyle-\frac{\lambda
R}{2}\partial_{\sigma}\left[\frac{\mathcal{H}_{R}}{\sqrt{}}\left(1+a-\frac{\lambda\hat{\phi}}{R_{v}}\right)\right]-\frac{\lambda
R}{2R_{v}}\mathcal{H}_{R}$ (2.31)
$\displaystyle\\{\chi_{0},\mathcal{J}_{-}\\}$ $\displaystyle=$ $\displaystyle
R\,\partial_{\sigma}\left[\frac{\mathcal{J}_{-}}{\sqrt{}}\left(1+a-\frac{\lambda\hat{\phi}}{R_{v}}\right)\right]-\frac{\lambda
R}{2R_{v}}\mathcal{H}_{R}$ (2.32)
$\displaystyle\\{\chi_{0},\mathcal{H}_{R}\\}$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{H}_{R}}{\sqrt{}}+\frac{R}{R_{v}}\mathcal{H}_{R}+R\partial_{\sigma}\left(\frac{\mathcal{H}_{R}}{\sqrt{}}\left(1+a-\frac{\lambda\hat{\phi}}{R_{v}}\right)\right)$
(2.33) $\displaystyle\\{\chi_{0},\mathcal{P}\\}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{R}-\frac{R}{R_{v}}\mathcal{H}_{R}-R\partial_{\sigma}\left(\frac{\mathcal{H}_{R}}{\sqrt{}}\left(1+a-\frac{\lambda\hat{\phi}}{R_{v}}\right)\right)$
(2.34) $\displaystyle\\{\chi_{0},\phi\\}$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}R\left(1+a-\frac{\lambda\hat{\phi}}{R_{v}}\right)$
(2.35)

where, as before, $\hat{\phi}=\phi-w\sigma/R$ equals $\phi$ with its winding
mode removed. The terms proportional to the constant $a$ are allowed by all
the Jacobi identities we have studied555This does not mean that there cannot
exist other Jacobi identities that constrain the value of $a$, or that require
the introduction of new terms in the commutators above. Our analysis is thus
valid up to this caveat.. Since its value does not seem to be fixed and,
moreover, it drops out from most of our subsequent computations, we will
henceforth fix it to the convenient value $a=-1$.

Using these, one can compute the flow equations for the various currents,
which are spelled out for convenience in appendix C, and from them derive the
flow of the conserved charges. One finds that the left-moving charges are
simply constant with respect to $\mathcal{D}_{\lambda}$

$\mathcal{D}_{\lambda}Q_{f}=\mathcal{D}_{\lambda}P_{\eta}=0$ (2.36)

while the right-moving ones satisfy

$\mathcal{D}_{\lambda}\bar{Q}_{\bar{f}}=\frac{w}{R_{v}}\bar{Q}_{\bar{f}}-\frac{wt}{R_{v}^{2}}\bar{Q}_{\bar{f}^{\prime}}\;,\;\;\;\;\;\;\;\mathcal{D}_{\lambda}\bar{P}^{KM}_{\bar{\eta}}=-\frac{wt}{R_{v}^{2}}\bar{P}^{KM}_{\bar{\eta}^{\prime}}$
(2.37)

Note that the first term on the right-hand side of the $\bar{Q}_{\bar{f}}$
flow is necessary in order for the flow equation to be compatible with the
charge algebra (1.7), as the latter contains an explicit factor of $1/R_{v}$,
whose $\lambda$ derivative does not vanish. If we consider instead the
rescaled dimensionless charges $R_{v}\bar{Q}_{\bar{f}}$, they satisfy a flow
equation analogous to that of $\bar{P}^{KM}$. Their algebra is also the
standard Witt-Kač-Moody algebra.

The explicit time dependence appearing on the right-hand side of (2.37) can be
understood by computing the time derivative of e.g. $\bar{P}^{KM}$, where as
usual $\frac{d}{dt}=\partial_{t}-\\{H,\,\cdot\,\\}$. One finds that
$\frac{d}{dt}\bar{P}^{KM}=\\{\mathcal{D}_{\lambda}H,\bar{P}_{\bar{\eta}}\\}\neq
0$ because, as a result of the first equation, $\mathcal{D}_{\lambda}H=\omega
E_{R}/R_{v}$ .

Given the above form of the flow equations, it is convenient to define

$\mathcal{D}_{\lambda}^{\prime}=\mathcal{D}_{\lambda}-\frac{wE_{R}t}{R_{v}}$
(2.38)

which annihilates all of the (rescaled) conserved charges.

###  2.3 Relating the two

To summarize, we found that the quantum version of the rescaled conserved
charges $Q_{f},P_{\eta},R_{v}\bar{Q}_{\bar{f}}$ and
$\bar{P}_{\bar{\eta}}^{KM}$, which we will collectively denote as
$\mathcal{L}$, are annihilated by the operator (2.29)

$\hat{\mathcal{D}}_{\lambda}\mathcal{L}=\partial_{\lambda}\mathcal{L}-i[\mathcal{O}_{tot},\mathcal{L}]=0$
(2.39)

(or, $\hat{\mathcal{D}}_{\lambda}^{\prime}\mathcal{L}=0$, if we work at $t\neq
0$). On the other hand, the states satisfy the flow equation (2.22), which
involves an additional diagonal operator $D$, which can be rather complicated.

We now consider two energy eigenstates, $|n_{0}\rangle$ and
$|n_{0}^{\prime}\rangle$, that in the undeformed CFT are related by the action
of a symmetry generator,
$|n_{0}^{\prime}\rangle=\mathcal{L}_{(\lambda=0)}|n_{0}\rangle$, which can be
any of the Virasoro or Kač-Moody generators. Our goal is to find a new
operator, $\widetilde{\mathcal{L}}$, that relates the corresponding flowed
states in the deformed CFT, i.e.
$|n^{\prime}_{\lambda}\rangle=\widetilde{\mathcal{L}}\,|n_{\lambda}\rangle$.
The flow equation (2.22) for the states then implies that the flow equation
for the corresponding operators is

$\partial_{\lambda}\widetilde{\mathcal{L}}-i[\mathcal{O}_{tot}-D,\widetilde{\mathcal{L}}]=0$
(2.40)

The solutions to the two flow equations are related by
$\widetilde{\mathcal{L}}=e^{X}\mathcal{L}e^{-X}$ where $X$ must satisfy

$[\widetilde{\mathcal{L}},(\partial_{\lambda}e^{X}-i[\mathcal{O}_{tot},e^{X}])e^{-X}-D]=0$
(2.41)

for _any_ $\widetilde{\mathcal{L}}$. This implies that the second argument
either vanishes, or it is proportional to the identity or some other operator
that commutes with all the $\widetilde{\mathcal{L}}$. Assuming for simplicity
that it vanishes, we can write

$D=(\partial_{\lambda}e^{X}-i[\mathcal{O}_{tot},e^{X}])\,e^{-X}$ (2.42)

Noting that

$(\partial_{\lambda}e^{X})\,e^{-X}=\partial_{\lambda}X+\frac{1}{2}[X,\partial_{\lambda}X]+\frac{1}{3!}[X,[X,\partial_{\lambda}X]]+\ldots$
(2.43)

$e^{X}\mathcal{O}_{tot}e^{-X}=\mathcal{O}_{tot}+[X,\mathcal{O}_{tot}]+\frac{1}{2}[X,[X,\mathcal{O}_{tot}]]+\ldots$
(2.44)

the above equation can be written as

$D=\hat{\mathcal{D}}_{\lambda}X+\frac{1}{2}[X,\hat{\mathcal{D}}_{\lambda}X]+\frac{1}{3!}[X,[X,\hat{\mathcal{D}}_{\lambda}X]]+\ldots\;,\;\;\;\;\;\;\hat{\mathcal{D}}_{\lambda}X\equiv\partial_{\lambda}X-i[\mathcal{O}_{tot},X]$
(2.45)

This result gives us a way to construct $X$, and therefore
$\widetilde{\mathcal{L}}$, if we know $\mathcal{L}$ and $D$. If we work at
$t\neq 0$, then $\mathcal{D}_{\lambda}$ should be replaced by
$\mathcal{D}_{\lambda}^{\prime}$, $\mathcal{O}_{tot}$ by
$\mathcal{O}_{tot}(t)-wE_{R}t/R_{v}$ and $D$ by $D_{tot}(t)\equiv
D(t)+\partial_{\lambda}E\,t$, as follows from (2.28).

As we already explained, finding $D$ to all orders is a rather difficult task,
but we can certainly attempt this exercise perturbatively. Since at
$\lambda=0$, $D=\Delta\mathcal{O}=(w\chi_{0}-E_{R}\phi_{0})/R$, where
$\phi_{0}$ is known to implement spectral flow in a CFT, we will henceforth
denote the $\widetilde{\mathcal{L}}$ as the “spectrally flowed” operators, in
this case by an energy-dependent amount. This connection will be made
significantly more precise in the next section.

##  3 The spectrally flowed generators

###  3.1 Perturbative construction of the spectrally flowed generators

In this section, we attempt to solve the equation (2.45) for $X$
perturbatively, for $D$ given in (2.27), and use the solution to find the
first few terms in the $\lambda$ expansion of the flowed generators. This can
be done by assuming $X$ has an expansion of the form

$X=\lambda\mathcal{O}_{1}+\lambda^{2}\mathcal{O}_{2}+\ldots\;\;\;\;\Rightarrow\;\;\;\;\;\mathcal{D}_{\lambda}X=\mathcal{O}_{1}+\lambda\mathcal{D}_{\lambda}\mathcal{O}_{1}+2\lambda\mathcal{O}_{2}+\lambda^{2}\mathcal{D}_{\lambda}\mathcal{O}_{2}+\ldots$
(3.1)

where the $\mathcal{O}_{n}$ are in general non-linear functions of $\lambda$.
We would moreover like to work at $t\neq 0$ so, according to our previous
discussion, $\mathcal{D}_{\lambda}$ should be replaced by
$\mathcal{D}^{\prime}_{\lambda}$ and

$D_{tot}(t)\equiv
D(t)+\partial_{\lambda}E\,t=D_{0}-\lambda(\mathcal{D}^{\prime}_{\lambda}D_{0})_{no\;z.m.}+\mathcal{O}(\lambda^{2})+\frac{2Q_{K}E_{R}}{R-\lambda
Q_{K}}\,t$ (3.2)

where $D_{0}=\Delta\mathcal{O}-wE_{R}t/R_{v}$. Since we know $D_{tot}$ to
first order in $\lambda$, we can thus find $X$, and consequently
$\widetilde{\mathcal{L}}$, to second order. Evaluating

$\mathcal{D}^{\prime}_{\lambda}D_{0}=\frac{w}{R_{v}}(\Delta\mathcal{O}-wE_{R}\frac{t}{R_{v}})-\frac{w}{R}\int
d\sigma\frac{\mathcal{H}_{R}}{\sqrt{}}\frac{wt+R\hat{\phi}(\sigma)}{R_{v}}-\frac{E_{R}}{R}\int
d\sigma\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}\frac{wt+R\hat{\phi}(\sigma)}{R_{v}}$
(3.3)

gives us

$D(t)=\frac{w\chi_{0}-E_{R}\phi_{0}}{R}-wE_{R}\frac{t}{R_{v}}+\frac{\lambda
w}{R_{v}}\int
d\sigma\frac{\mathcal{H}_{R}}{\sqrt{}}\left(\hat{\phi}-\frac{\phi_{0}}{R}\right)+\frac{\lambda
E_{R}}{R_{v}}\int
d\sigma\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}\left(\hat{\phi}-\frac{\phi_{0}}{R}\right)+\ldots$
(3.4)

It is extremely useful to note that to this order, $D(t)$ can be written as

$D(t)\approx\frac{w\widetilde{\chi}_{0}-E_{R}\widetilde{\phi}_{0}}{R}-wE_{R}\frac{t}{R_{v}}-\frac{\lambda\phi_{0}E_{R}Q_{K}}{R^{2}}+\mathcal{O}(\lambda^{2})$
(3.5)

where the “improved” zero modes $\widetilde{\phi}_{0}$ and
$\widetilde{\chi}_{0}$ are defined as

$\widetilde{\phi}_{0}\equiv\phi_{0}-\frac{\lambda R}{R_{v}}\int
d\sigma(\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2)\hat{\phi}\;,\;\;\;\;\;\;\;\;\widetilde{\chi}_{0}\equiv\chi_{0}+\frac{\lambda
R}{R_{v}}\int d\sigma\mathcal{H}_{R}\hat{\phi}$ (3.6)

The usefulness of introducing these quantities stems from the extremely simple
Poisson brackets they satisfy, to _all orders_ in $\lambda$. The Poisson
brackets of $\widetilde{\phi}_{0}$ with the left-movers are

$\left\\{\widetilde{\phi}_{0},K_{U}\right\\}=\frac{1}{2}\;,\;\;\;\;\;\;\;\;\left\\{\widetilde{\phi}_{0},\mathcal{H}_{L}\right\\}=K_{U}$
(3.7)

and with the right-movers

$\\{\widetilde{\phi}_{0},\,\bar{Q}_{\bar{f}}\\}=\frac{R}{R_{v}}\bar{P}^{KM}_{\bar{f}}\;,\;\;\;\;\;\;\\{\widetilde{\phi}_{0},\,\bar{P}^{KM}_{\bar{\eta}}\\}=\frac{R}{2R_{v}}\int
d\sigma\bar{\eta}(1-\lambda\phi^{\prime})$ (3.8)

which implies that $\widetilde{\phi}_{0}$ is the corrected operator
implementing spectral flow in the $J\bar{T}$ \- deformed CFT. The field
$\widetilde{\chi}_{0}$ commutes with $K_{U},\mathcal{H}_{L},E_{R},\bar{J}_{0}$
and, for our particular choice of the constant $a$ in the Poisson brackets,
with everything else666For $a\neq-1$, its non-zero commutators are
$\\{\widetilde{\chi}_{0},\mathcal{H}_{R}\\}=(1+a)R\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\;,\;\;\;\;\;\;\;\\{\widetilde{\chi}_{0},\mathcal{J}_{-}\\}=(1+a)R\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}\;,\;\;\;\;\;\;\;\\{\widetilde{\chi}_{0},\phi\\}=-(1+a)R\frac{\mathcal{J}_{-}+\lambda/2\mathcal{H}_{R}}{\sqrt{}}$
(3.9) which implies
$\\{\widetilde{\chi}_{0},\bar{Q}_{\bar{f}}\\}=-\frac{R}{R_{v}}\bar{Q}_{\bar{f}^{\prime}}\left(1+a\right)\;,\;\;\;\;\;\;\;\;\\{\widetilde{\chi}_{0},\bar{P}^{KM}_{\bar{\eta}}\\}=-\frac{\bar{P}_{\bar{\eta}^{\prime}}^{KM}R}{R_{v}}\left(1+a\right)\;,\;\;\;\;\;\;\;\\{\widetilde{\phi}_{0},\widetilde{\chi}_{0}\\}=\frac{R^{2}\bar{Q}_{\bar{K}}}{R_{v}}(1+a)$
(3.10) .

It is also interesting to check the flow equations that $\widetilde{\phi}_{0}$
and $\widetilde{\chi}_{0}$ satisfy777More generally,
$\mathcal{D}_{\lambda}\widetilde{\chi}_{0}=\mathcal{D}^{\prime}_{\lambda}\widetilde{\chi}_{0}=\frac{w}{R_{v}}\widetilde{\chi}_{0}+\frac{wR}{R_{v}}E_{R}(1+a)\;,\;\;\;\;\;\;\;\;\;\mathcal{D}_{\lambda}\widetilde{\phi}_{0}=-\frac{R\bar{Q}_{\bar{K}}}{R_{v}}w(1+a)\;,\;\;\;\;\;\;\;\;\;\mathcal{D}^{\prime}_{\lambda}\widetilde{\phi}_{0}=\frac{wtR}{R_{v}^{2}}\bar{Q}_{\bar{K}}$
(3.11)

$\mathcal{D}^{\prime}_{\lambda}\widetilde{\chi}_{0}=\frac{w}{R_{v}}\widetilde{\chi}_{0}\;,\;\;\;\;\;\;\;\;\;\mathcal{D}^{\prime}_{\lambda}\widetilde{\phi}_{0}=\frac{wtR}{R_{v}^{2}}\bar{Q}_{\bar{K}}$
(3.12)

Given the rewriting (3.5) of $D(t)$, it is best to consider a different
identification of what is considered ‘zeroth’ versus ‘first’ order in its
expansion with respect to $\lambda$, namely

$\widetilde{D}_{0}=\frac{w\widetilde{\chi}_{0}-E_{R}\widetilde{\phi}_{0}}{R}-wE_{R}\frac{t}{R_{v}}+\mathcal{O}(\lambda^{2})\;,\;\;\;\;\;\;\;\widetilde{D}_{1}=-\frac{\phi_{0}E_{R}Q_{K}}{R^{2}}+\mathcal{O}(\lambda)$
(3.13)

Then, the coefficients of the perturbative expansion (3.1) of $X$, obtained
using (2.45), are given by

$\mathcal{O}_{1}=\frac{w\widetilde{\chi}_{0}-E_{R}\widetilde{\phi}_{0}}{R}-wE_{R}\frac{t}{R_{v}}+\frac{2Q_{K}E_{R}t}{R}\;,\;\;\;\;\;\;\mathcal{D}^{\prime}_{\lambda}\mathcal{O}_{1}=\frac{w}{R_{v}}\frac{w\widetilde{\chi}_{0}-E_{R}\widetilde{\phi}_{0}}{R}+\frac{2Q_{K}wE_{R}\,t}{R^{2}}-\frac{wtE_{R}(Q_{K}+\omega)}{R^{2}}$
(3.14)

which implies that, to zeroth order in $\lambda$

$\displaystyle R^{2}\mathcal{O}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\phi_{0}E_{R}Q_{K}-\frac{w(w\widetilde{\chi}_{0}-E_{R}\widetilde{\phi}_{0})}{2}-Q_{K}wE_{R}t+Q_{K}^{2}E_{R}t+\frac{1}{2}wtE_{R}(Q_{K}+\omega)$
(3.15) $\displaystyle\approx$
$\displaystyle-\frac{1}{2}\phi_{0}E_{R}\bar{Q}_{\bar{K}}-\frac{\omega^{2}}{2}\chi_{0}+E_{R}Q_{K}\bar{Q}_{\bar{K}}t+\frac{1}{2}wtE_{R}(Q_{K}+\omega)$

The above expressions for $\mathcal{O}_{1,2}$ give us the classical limit of
the operator $X$ entering the similarity transformation, up to
$\mathcal{O}(\lambda^{3})$.

We would now like to check the effect of the similarity transformation on the
various conserved charges. To pass from the quantum commutators to Poisson
brackets, we note that the operator $X$ should have a factor of $\hbar^{-1}$
in front, which cancels against the $\hbar$ factors in the commutators to
yield the classical result

$\widetilde{\mathcal{L}}=e^{X}\mathcal{L}e^{-X}\;\;\leftrightarrow\;\;\widetilde{\mathcal{L}}_{cls}=\mathcal{L}+\lambda\\{\mathcal{O}_{1},\mathcal{L}\\}+\lambda^{2}\\{\mathcal{O}_{1},\mathcal{L}\\}+\frac{\lambda^{2}}{2}\\{\mathcal{O}_{1},\\{\mathcal{O}_{1},\mathcal{L}\\}\\}+\mathcal{O}(\lambda^{3})$
(3.16)

Let us first work out the effect of this transformation on $K_{U}$. Using

$\\{\mathcal{O}_{1},K_{U}\\}=-\frac{E_{R}}{2R}\;,\;\;\;\;\;\;\;\\{\mathcal{O}_{2},K_{U}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{2R^{2}}\\{\phi_{0},K_{U}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{4R^{2}}\;,\;\;\;\;\;\;\;\\{\mathcal{O}_{1},E_{R}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{R}$
(3.17)

we can readily show that

$\boxed{\widetilde{K}_{U}=K_{U}-\frac{\lambda
E_{R}}{2R}+\mathcal{O}(\lambda^{3})}$ (3.18)

Note in particular that the zero mode of $\widetilde{K}_{U}$ is just $J_{0}$.
This implies that the spectrally flowed generators will commute with $J_{0}$,
since the commutator
$[J_{0},e^{X}\bar{Q}_{\bar{f}}e^{-X}]=e^{X}[e^{-X}J_{0}e^{X},\bar{Q}_{\bar{f}}]e^{-X}=e^{X}[Q_{K},\bar{Q}_{\bar{f}}]e^{-X}=0$.

Next, we would like to check what happens to the left-moving energy current
$\mathcal{H}_{L}$. We evaluate

$\\{\mathcal{O}_{1},\mathcal{H}_{L}\\}=-\frac{E_{R}K_{U}}{R}\;,\;\;\;\;\;\;\;\\{\mathcal{O}_{2},\mathcal{H}_{L}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{2R^{2}}\\{\phi_{0},\mathcal{H}_{L}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{2R^{2}}K_{U}$
(3.19)

$\\{\mathcal{O}_{1},-E_{R}K_{U}\\}=\frac{E_{R}^{2}}{2R}+\frac{E_{R}K_{U}\bar{Q}_{\bar{K}}}{R}$
(3.20)

which in turn implies that

$\boxed{\widetilde{\mathcal{H}}_{L}=\mathcal{H}_{L}-\frac{\lambda
E_{R}K_{U}}{R}+\frac{\lambda^{2}E_{R}^{2}}{4R^{2}}+\mathcal{O}(\lambda^{3})}$
(3.21)

Note that the transformation of $K_{U}$ and that of $\mathcal{H}_{L}$
correspond precisely to a spectral flow transformation with parameter $\lambda
E_{R}$. The zero mode of $\widetilde{\mathcal{H}}_{L}$ equals the left-moving
energy $E_{L}^{(0)}$ in the undeformed CFT.

Let us now turn to the right-movers, starting with the Kač-Moody generators
$\bar{P}^{KM}_{\bar{\eta}}$ in (1.6). Using (3.8), we compute

$\\{\mathcal{O}_{1},\bar{P}^{KM}_{\bar{\eta}}\\}=-\frac{E_{R}}{2R_{v}}\int
d\sigma\bar{\eta}(1-\lambda\phi^{\prime})+\frac{1}{R_{v}}\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)\bar{P}_{\bar{\eta}^{\prime}}^{KM}$
(3.22)
$\\{\mathcal{O}_{2},\bar{P}^{KM}_{\bar{\eta}}\\}\approx-\frac{E_{R}\bar{Q}_{\bar{K}}}{4R_{v}R}\int
d\sigma\bar{\eta}(1-\lambda\phi^{\prime})+\frac{\bar{Q}_{\bar{K}}}{2R^{2}}(\widetilde{\phi}_{0}-2Q_{K}t)\bar{P}_{\bar{\eta}^{\prime}}^{KM}-\frac{wt}{2R^{3}}(Q_{K}+\omega)\bar{P}_{\bar{\eta}^{\prime}}^{KM}$
(3.23)

It is useful to treat separately the case in which $\eta=I$ (the identity),
for which

$\tilde{\bar{P}}_{I}^{KM}=\left(\bar{J}_{0}+\frac{\lambda
E_{R}}{2}\right)-\frac{\lambda
E_{R}}{2}-\frac{\lambda^{2}}{4R}E_{R}\bar{Q}_{\bar{K}}+\frac{\lambda^{2}}{2}\\{\mathcal{O}_{1},-E_{R}/2\\}=\bar{J}_{0}+\mathcal{O}(\lambda^{3})$
(3.24)

as expected from spectral flow. For $\bar{\eta}\neq I$, we compute

$R^{2}\\{\mathcal{O}_{1},\\{\mathcal{O}_{1},\bar{P}^{KM}_{\bar{\eta}}\\}\\}=\left(\frac{\phi_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)\bar{Q}_{\bar{K}}\bar{P}_{\bar{\eta}^{\prime}}^{KM}+\left(\frac{\phi_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)^{2}\bar{P}_{\bar{\eta}^{\prime\prime}}^{KM}$
(3.25)

The transformed $\bar{P}^{KM}_{\bar{\eta}}$ is then, for $\bar{\eta}\neq I$

$\displaystyle\tilde{\bar{P}}_{\bar{\eta}}^{KM}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\bar{P}_{\bar{\eta}}^{KM}\\!+\frac{\lambda}{R_{v}}\\!\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)\\!\bar{P}_{\bar{\eta}^{\prime}}^{KM}\\!+\frac{\lambda^{2}}{2R^{2}}\\!\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)^{2}\\!\\!\\!\\!\bar{P}_{\bar{\eta}^{\prime\prime}}^{KM}\\!+\frac{\lambda^{2}}{R^{3}}\\!\left[\bar{Q}_{\bar{K}}(\phi_{0}-2Q_{K}t)-\omega^{2}t\right]\\!\\!\bar{P}_{\bar{\eta}^{\prime}}^{KM}\\!+\mathcal{O}(\lambda^{3})$
(3.26) $\displaystyle\approx$
$\displaystyle\\!\\!\\!\bar{P}_{\bar{\eta}}^{KM}+\hat{\lambda}(1+\hat{\lambda}Q_{K})\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t+\omega
t-\hat{\lambda}\omega\bar{Q}_{\bar{K}}t}{R}\right)\bar{P}_{\bar{\eta}^{\prime}}^{KM}+\frac{\hat{\lambda}^{2}}{2}\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t+\omega
t}{R}\right)^{2}\\!\\!\bar{P}_{\bar{\eta}^{\prime\prime}}^{KM}+\mathcal{O}(\lambda^{3})$

where we have introduced the dimensionless $J\bar{T}$ coupling
$\hat{\lambda}=\lambda/R$. It is not hard to check that this expression is
conserved to the given order in $\lambda$, using

$\\{H,\widetilde{\phi}_{0}\\}=-Q_{K}-\frac{\bar{Q}_{\bar{K}}R}{R_{v}}$ (3.27)

Using
$\\{P,\widetilde{\phi}_{0}\\}=-Q_{K}+\frac{\bar{Q}_{\bar{K}}R}{R_{v}}\approx-w(1-\hat{\lambda}\bar{Q}_{\bar{K}})$
and the approximate expansion above, we also find

$\\{P,\tilde{\bar{P}}_{\bar{\eta}}^{KM}\\}=\frac{1}{R}\,\tilde{\bar{P}}_{\bar{\eta}^{\prime}}^{KM}+\mathcal{O}(\lambda^{3})$
(3.28)

i.e., acting with the spectrally flowed right-moving generator preserves the
quantization of the momentum.

Finally, moving on to the right-moving pseudoconformal generators, we find

$\\{\mathcal{O}_{1},\bar{Q}_{\bar{f}}\\}=-\frac{E_{R}\bar{P}^{KM}_{\bar{f}}}{R_{v}}+\frac{1}{R_{v}}\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)\bar{Q}_{\bar{f}^{\prime}}$
(3.29)

$\\{\mathcal{O}_{2},\bar{Q}_{\bar{f}}\\}=-\frac{E_{R}\bar{Q}_{\bar{K}}}{2R_{v}R}\bar{P}_{\bar{f}}^{KM}+\frac{\bar{Q}_{\bar{K}}}{2R_{v}R^{2}}(\widetilde{\phi}_{0}-2Q_{K}t)\bar{Q}_{\bar{f}^{\prime}}-\frac{wt(Q_{K}+\omega)}{2R^{3}}\bar{Q}_{\bar{f}^{\prime}}$
(3.30)

$\\{\mathcal{O}_{1},\\{\mathcal{O}_{1},\bar{Q}_{\bar{f}}\\}\\}=-\frac{E_{R}}{R_{v}}\\{\mathcal{O}_{1},\bar{P}_{\bar{f}}^{KM}\\}+\frac{E_{R}\bar{Q}_{\bar{K}}}{R_{v}R}\bar{P}_{\bar{f}}^{KM}+\frac{\frac{\phi_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}}{R_{v}}(\bar{Q}_{\bar{K}}\bar{Q}_{\bar{f}^{\prime}}-E_{R}\bar{P}_{\bar{f}^{\prime}}^{KM})+\frac{(\frac{\phi_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}})^{2}}{R_{v}^{2}}\bar{Q}_{\bar{f}^{\prime\prime}}$
(3.31)

Let us first check the case $f=const$. Remembering that the orginal right-
moving pseudoconformal generator that satisfies
$\mathcal{D}_{\lambda}^{\prime}\mathcal{L}=0$ is $\bar{Q}_{\bar{f}}R_{v}$,
applying the similarity transformation to it yields

$\widetilde{E}_{R}\,R=E_{R}R_{v}-\lambda
E_{R}\bar{J}_{0}-\frac{\lambda^{2}}{4}E_{R}^{2}+\mathcal{O}(\lambda^{3})$
(3.32)

in perfect agreement with our expectation, $RE_{R}^{(0)}$. The factor of $R$
on the left-hand side has been included for dimensional reasons. For general
$f$, we find

$\displaystyle\widetilde{\bar{Q}}_{\bar{f}}\,R$ $\displaystyle=$
$\displaystyle\bar{Q}_{\bar{f}}R_{v}-\lambda
E_{R}\bar{P}^{KM}_{\bar{f}}+\frac{\lambda^{2}E_{R}^{2}}{4}\delta_{f=I}+\lambda\left(\frac{\widetilde{\phi}_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)\bar{Q}_{\bar{f}^{\prime}}+\frac{\lambda^{2}}{2R^{2}}(\phi_{0}-2Q_{K}t)(2\bar{Q}_{\bar{K}}\bar{Q}_{\bar{f}^{\prime}}-2E_{R}\bar{P}^{KM}_{\bar{f}^{\prime}})$
(3.33) $\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{2R}\left(\frac{\phi_{0}-2Q_{K}t}{R}+\frac{wt}{R_{v}}\right)^{2}\bar{Q}_{\bar{f}^{\prime\prime}}-\frac{\lambda^{2}\omega
t}{2R^{2}}(Q_{K}+\omega)\bar{Q}_{\bar{f}^{\prime}}+\frac{\lambda^{2}\omega
t}{2R^{2}}(\bar{Q}_{\bar{K}}\bar{Q}_{\bar{f}^{\prime}}-2E_{R}\bar{P}_{\bar{f}^{\prime}}^{KM})$

As before, this can be organised as the following perturbative expansion

$\displaystyle\widetilde{\bar{Q}}_{\bar{f}}\,R$ $\displaystyle=$
$\displaystyle
R_{v}\left(\bar{Q}_{\bar{f}}+\hat{\lambda}(1+\hat{\lambda}Q_{K})\,\frac{\widetilde{\phi}_{0}-2Q_{K}t+\omega
t(1-\hat{\lambda}\bar{Q}_{\bar{K}})}{R}\bar{Q}_{\bar{f}^{\prime}}+\frac{\hat{\lambda}^{2}}{2R^{2}}(\phi_{0}-2Q_{K}t+\omega
t)^{2}\bar{Q}_{\bar{f}^{\prime\prime}}\right)$ (3.34) $\displaystyle-\lambda
E_{R}\left(\bar{P}_{\bar{f}}^{KM}+\frac{\lambda(\phi_{0}-2Q_{K}t+\omega
t)}{R^{2}}\bar{P}_{\bar{f}^{\prime}}^{KM}\right)+\frac{\lambda^{2}E_{R}^{2}}{4}\delta_{f=I}+\mathcal{O}(\lambda^{3})$

This has precisely the correct form to yield a conserved charge and an
integer-quantized momentum, as one can check by computing its Poisson brackets
with $H$ and $P$.

###  3.2 An all-orders proposal

The result of the perturbative analysis we have just performed is that the
symmetry generators that act properly on the eigenstates of the system are
given by a kind of energy-dependent spectral flow. While the form of the
resulting left-moving generators, $\widetilde{K}_{U}$ and
$\widetilde{\mathcal{H}}_{L}$ in (3.18) and (3.21), matches precisely to what
we expect from spectral flow with parameter $\lambda E_{R}$, the form of the
right-moving generators (3.26) and (3.34) is significantly more involved. In
particular, while it is nothing but natural that (3.18) and (3.21) should
represent the full expressions for the left-moving flowed generators to all
orders in $\lambda$, it is also clear that the $\lambda$ expansion of the
right-moving generators will contain an infinite number of terms.

In this section, we will make a proposal for an all-orders (formal) expression
for the right-moving generators, starting from the assumption that (3.18) and
(3.21) are the correct expression for the flowed left-moving currents to all
orders in $\lambda$. Our main tool will be the fact that the charge algebra is
preserved by the flow (2.40), and therefore the spectrally flowed left and
right generators should commute to all orders in $\lambda$.

Our analysis will proceed in two steps. First, we will construct combinations
of the right-moving conserved charges that commute with the left-moving
spectrally flowed currents, and show that these building blocks satisfy the
correct Poisson brackets with the energy, momentum and the global $U(1)$
charges to have, upon quantization, a consistent action on the Hilbert space.
Then, we find linear combinations of these blocks that satisfy the expected
flow equation, with an operator $D$ we will similarly derive.

Let us start by analysing the building block,
$\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}$, for the right-moving $U(1)$
generator. The requirement that it commute with all the left-moving charges
(or, alternatively, the currents), reads

$\\{K_{U}-\frac{\lambda}{2R}E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=0\;,\;\;\;\;\;\;\;\\{\mathcal{H}_{L}-\frac{\lambda
E_{R}K_{U}}{R}+\frac{\lambda^{2}}{4R^{2}}E_{R}^{2},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=0$
(3.35)

where $\bar{\mathcal{P}}_{\bar{\eta}}$ can in fact be _any_ right-moving
current. Note the second equation follows from the first if

$\\{\mathcal{H}_{L},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=2K_{U}\\{K_{U},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}$
(3.36)

which can _a posteriori_ be checked to be the case. Remembering that

$\\{\widetilde{\phi}_{0},K_{U}\\}=\frac{1}{2}\;,\;\;\;\;\;\\{\widetilde{\phi}_{0},E_{R}\\}=\frac{\bar{Q}_{\bar{K}}R}{R_{v}}\;,\;\;\;\;\;\\{E_{R},\bar{P}_{\bar{\eta}}\\}=-\frac{1}{R_{v}}\bar{P}_{\bar{\eta}^{\prime}}$
(3.37)

a natural Ansatz for $\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}$ (for
$\eta\neq I$) is

$\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}={\bar{P}}_{\bar{\eta}}+\hat{\lambda}a_{1}\widetilde{\phi}_{0}\,\bar{P}_{\bar{\eta}^{\prime}}+\frac{\hat{\lambda}^{2}a_{2}}{2}\,\widetilde{\phi}_{0}^{2}\,\bar{P}_{\bar{\eta}^{\prime\prime}}+\ldots$
(3.38)

Plugging into (3.35), we find the recursion relation

$a_{n+1}=\frac{a_{n}}{R-\lambda Q_{K}}\;,\;\;\;\;\;a_{0}=1$ (3.39)

which implies that the solution is simply

$\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}={\bar{P}}_{\bar{\eta}}+\frac{\hat{\lambda}\widetilde{\phi}_{0}}{R-\lambda
Q_{K}}{\bar{P}}_{\bar{\eta}^{\prime}}+\frac{\hat{\lambda}^{2}\widetilde{\phi}_{0}^{2}}{2(R-\lambda
Q_{K})^{2}}{\bar{P}}_{\bar{\eta}^{\prime}}+\ldots$ (3.40)

Note that in the case of the Kač-Moody current $P^{KM}_{\bar{\eta}}$, the
first three terms agree precisely with the result (3.26) of the perturbative
analysis of the previous section, at $t=0$. One can also check that (3.36)
holds, using (3.7), (A.13) and the commutators of $\phi$ with
$\mathcal{H}_{L}$ and $K_{U}$.

Note the above is a formal expression in that $\phi_{0}$, and thus
$\widetilde{\phi}_{0}$, is not a well-defined operator. However, its
exponential is expected to be. Using the $\widetilde{\phi}_{0}$ Poisson
brackets

$\\{\widetilde{\phi}_{0},J_{0}\\}=\\{\widetilde{\phi}_{0},\bar{J}_{0}\\}=R\,\frac{R-\lambda
Q_{K}}{2R_{v}}$ (3.41)

with $Q_{K}\equiv J_{0}+\lambda E_{R}/2$ and
$\bar{Q}_{\bar{K}}\equiv\bar{J}_{0}+\lambda E_{R}/2$ that we presented
earlier, we can easily check that the charges of this combination are as
expected, namely

$\\{J_{0},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=\\{\bar{J}_{0},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=\frac{\lambda}{2R_{v}}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}-\frac{(R-\lambda
Q_{K})}{2R_{v}}\frac{\lambda}{R-\lambda
Q_{K}}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}=0$ (3.42)

The Poisson bracket with the right-moving energy is

$\\{E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=-\frac{\lambda}{R-\lambda
Q_{K}}\frac{\bar{Q}_{\bar{K}}}{R_{v}}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}-\frac{1}{R_{v}}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}=-\frac{1}{R-\lambda
Q_{K}}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}$ (3.43)

while the commutator with $\mathcal{H}_{L}$ is given by

$\\{\mathcal{H}_{L},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=\frac{\lambda
K_{U}}{R}\\{E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=-\frac{\lambda
K_{U}}{R(R-\lambda Q_{K})}\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}$
(3.44)

These together imply that the Poisson bracket with the total momentum is

$\\{P,\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=\\{E_{L}-E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=-\frac{1}{R}\,\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}$
(3.45)

and thus the action of $\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}$ in the
Fourier basis increases the momentum by an integer amount in units of the
radius, which is now consistent with semiclassical quantization. The total
energy is given by

$\\{E_{L}+E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\\}=-\frac{R+\lambda
Q_{K}}{R-\lambda
Q_{K}}\frac{\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}}{R}$ (3.46)

To ensure conservation of the charges, one should, for $t\neq 0$, replace
$\widetilde{\phi}_{0}$ by the block

$\widetilde{\phi}_{0}-\left(Q_{K}+\frac{\bar{Q}_{\bar{K}}R}{R-\lambda
w}\right)t$ (3.47)

which is conserved by itself. This agrees precisely with what happened in our
previous perturbative analysis, and makes it manifest that each term in the
sum (3.40) is separately conserved.

It is also interesting to note that (3.43) implies that the spectrally flowed
right-moving energy $E_{R}^{(0)}$ is also changed by an integer amount

$\left\\{E_{R}^{(0)},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\right\\}=\left\\{E_{R}-\frac{\lambda
J_{0}E_{R}}{R}-\frac{\lambda^{2}E_{R}^{2}}{4R^{2}},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\right\\}=\left\\{E_{R},\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}\right\\}\left(1-\frac{\lambda
Q_{K}}{R}\right)=-\frac{1}{R}\,\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}^{\prime}}$
(3.48)

as expected from the fact that it is the global mode
$(\widetilde{\bar{L}}_{0})$ of the spectrally flowed algebra.

An identical analysis for the case of the pseudoconformal generators shows
that they must appear in the combination

$\widetilde{\bar{\mathcal{Q}}}_{\bar{f}}=\bar{Q}_{\bar{f}}+\frac{\hat{\lambda}\widetilde{\phi}_{0}}{R-\lambda
Q_{K}}\bar{Q}_{\bar{f}^{\prime}}+\frac{\hat{\lambda}^{2}\widetilde{\phi}_{0}^{2}}{2(R-\lambda
Q_{K})^{2}}{\bar{Q}}_{\bar{f}^{\prime\prime}}+\ldots$ (3.49)

and their commutation relations are exactly analogous with those of
$\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}$.

As already explained, the formal expressions
$\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}$ and
$\widetilde{\bar{\mathcal{Q}}}_{\bar{f}}$ are not exactly the spectrally
flowed $\widetilde{\mathcal{L}}$ generators, as the commutation requirement
(3.35) is a weaker condition than the flow equation (2.40). Instead, they
represent the building blocks of the spectrally flowed generators. To find
which linear combination of them represents the $\widetilde{\mathcal{L}}$, we
now turn to the flow equation they satisfy.

It turns out that the flow operator $D$ can also be fixed by its commutation
relations with $K_{U}$ and $\mathcal{H}_{L}$, upon choosing a judicious
Ansatz. Assuming that the spectrally flowed left-moving generators are given
precisely by (3.18) and (3.21), the flow equation (2.40) they are expected to
satisfy and the fact (C.2) that they are annihilated by
$\mathcal{D}_{\lambda}$ fix the commutation relations of $D$ to to all orders
to

$\\{D,K_{U}-\frac{\lambda}{2R}E_{R}\\}=\mathcal{D}_{\lambda}(K_{U}-\frac{\lambda}{2R}E_{R})=-\frac{1}{2R_{v}}E_{R}\;,\;\;\;\;\;\;\;\\{D,\mathcal{H}_{L}-\frac{\lambda
E_{R}K_{U}}{R}+\frac{\lambda^{2}E_{R}^{2}}{4R^{2}}\\}=-\frac{E_{R}}{R_{v}}(K_{U}-\frac{\lambda
E_{R}}{2R})$ (3.50)

We make the following Ansatz for $D$

$D=a(\lambda)\widetilde{\phi}_{0}+b(\lambda)\widetilde{\chi}_{0}+c(\lambda)$
(3.51)

where the operators $a,b,c$ commute with both $K_{U}$ and $E_{R}$. The first
equation implies that

$a-\frac{\lambda}{R}\\{D,E_{R}\\}=-\frac{E_{R}}{R_{v}}\;,\;\;\;\;\mbox{where}\;\;\\{D,E_{R}\\}=\frac{a\bar{Q}_{\bar{K}}R}{R-\lambda
w}$ (3.52)

thus yielding

$a=-\frac{E_{R}}{R-\lambda Q_{K}}$ (3.53)

in perfect agreemnet with our perturbative solution (3.5). While the
coefficient $b$ is not fixed, this is not very important, since with our
choice of $\chi_{0}$ Poisson brackets, $\widetilde{\chi}_{0}$ commutes with
all operators. We will set $b=w/R$ to match with the perturbative answer, and
$c=0$, at least on the $t=0$ slice.

We would now like to show that

$\boxed{\widetilde{\bar{P}}^{KM}_{\bar{\eta}}\equiv\widetilde{\bar{\mathcal{P}}}_{\bar{\eta}}+\frac{\lambda}{2}\widetilde{\bar{\mathcal{Q}}}_{\bar{\eta}}-\frac{\lambda
E_{R}}{2}\,\delta_{\bar{\eta}=I}}$ (3.54)

and

$\boxed{R\,\widetilde{\bar{Q}}_{\bar{f}}\equiv
R_{v}\widetilde{\bar{\mathcal{Q}}}_{\bar{f}}-\lambda
E_{R}\widetilde{\bar{\mathcal{P}}}^{KM}_{\bar{f}}\\!\\!-\frac{\lambda^{2}E_{R}^{2}}{4}\,\delta_{\bar{f}=I}}$
(3.55)

precisely satisfy the flow equation (2.40) for the above choice of $D$. We
compute, for $\eta\neq I$ and at $t=0$

$\mathcal{D}_{\lambda}\widetilde{\bar{P}}^{KM}_{\bar{\eta}}=\mathcal{D}_{\lambda}\left(\frac{\hat{\lambda}\widetilde{\phi}_{0}}{R-\lambda
Q_{K}}\right)\widetilde{\bar{P}}^{KM}_{\bar{\eta}^{\prime}}=\frac{\widetilde{\phi}_{0}}{(R-\lambda
Q_{K})^{2}}\widetilde{\bar{P}}^{KM}_{\bar{\eta}^{\prime}}=\left\\{\frac{w\widetilde{\chi}_{0}}{R}-\frac{E_{R}\widetilde{\phi}_{0}}{R-\lambda
Q_{K}},\widetilde{\bar{P}}^{KM}_{\bar{\eta}}\right\\}$ (3.56)

The case $\eta=I$, for which
$\widetilde{\bar{P}}^{KM}_{\bar{I}}\\!\\!\\!=\bar{J}_{0}$ needs to be worked
out separately, and it is easy to check that

$\mathcal{D}_{\lambda}\bar{J}_{0}=\\{D,\bar{J}_{0}\\}$ (3.57)

which justifies the constant shift by $-\lambda E_{R}/2$. As far as the
pseudoconformal charges are concerned, we find that, for $\bar{\eta}\neq I$

$\mathcal{D}^{\prime}_{\lambda}\widetilde{\bar{\mathcal{Q}}}_{\bar{\eta}}-\left\\{\frac{w\widetilde{\chi}_{0}}{R}-\frac{E_{R}\widetilde{\phi}_{0}}{R-\lambda
Q_{K}},\widetilde{\bar{\mathcal{Q}}}_{\bar{\eta}}\right\\}=\frac{\omega}{R_{v}}\widetilde{\bar{\mathcal{Q}}}_{\bar{\eta}}+\frac{R\,E_{R}}{R_{v}(R-\lambda
Q_{K})}\widetilde{\bar{\mathcal{P}}}^{KM}_{\bar{\eta}}$ (3.58)

We can now easily show that the linear combination (3.55) is exactly
annihilated by $\mathcal{D}_{\lambda}-D$. The constant shift follows from the
fact that $\\{\mathcal{D}_{\lambda}-D,E_{R}\\}=E_{R}Q_{K}/(R-\lambda Q_{K})$,
which implies in particular that

$\left\\{\mathcal{D}_{\lambda}-D,E_{R}R_{v}-\lambda
E_{R}\bar{J}_{0}-\frac{\lambda^{2}}{4}E_{R}^{2}\right\\}=0$ (3.59)

The equations (3.54) and (3.55) encode the effect of the spectral flow by an
amount proportional to the right-moving energy on the right-moving charges.
Due to the flow equation they obey, the spectrally flowed generators satisfy
the same algebra as at $\lambda=0$, i.e. two commuting copies of the Witt-Kač-
Moody algebra. The charges, defined at $t\neq 0$ via the replacement (3.47),
are conserved. The generators of the global part of the algebra are given by

$\widetilde{Q}_{f=I}=E_{L}^{(0)}\;,\;\;\;\;\;\;\widetilde{\bar{Q}}_{\bar{f}=I}=E_{R}^{(0)}$
(3.60)

whose sum _does not_ correspond to the energy operator; rather, it is non-
linearly related to it via (1.2). This implies that while the Fourier modes of
$\widetilde{\bar{Q}}$ and $\widetilde{\bar{P}}$ are integer-spaced with
respect to $E_{L}^{(0)}$ and $E_{R}^{(0)}$ (as follows from the algebra), they
have nontrivial commutation relations with $E_{L}$ and $E_{R}$, of the form
(3.43), (3.44). This resolves the tension mentioned at the beginning of this
article, between the integer spacing of the Virasoro descendants and the non-
linear dependence (1.1) of the $J\bar{T}$ \- deformed energy spectrum on the
initial energy.

##  4 Discussion

In this article, we have constructed a new, infinite set of symmetry
generators in $J\bar{T}$ \- deformed CFTs. Compared to the conserved charges
found in [19], these generators still form two commuting copies of the Witt-
Kač-Moody algebra, but now their Poisson brackets with the global $U(1)$
charge and the momentum are consistent with semiclassical quantization. It is
therefore expected that in the quantum $J\bar{T}$ \- deformed CFT on a
cylinder, these generators will be acting properly on the Hilbert space of the
theory, and thus the spectrum will organise into Virasoro - Kač-Moody
representations.

The new symmetry generators are related to those of [19] by a type of energy-
dependent spectral flow transformation, whose action on the various generators
is given in (3.18), (3.21), (3.54) and (3.55). The expressions (3.40) and
(3.49) for the building blocks are somewhat formal and should be resummed in
order to make sense. While the descendants obtained via the action of these
symmetry generators will have integer-spaced spectrally flowed energies
$E^{(0)}_{L,R}$, there is no tension with the known formula (1.1) for the
deformed energies, as the latter are measured with respect to the non-flowed
generators.

As is clear from (1.1), at large initial right-moving energy, the finite-size
spectrum of $J\bar{T}$ \- deformed CFTs will become complex. While this is
only a problem for the theory on the cylinder, and not on the plane [23], it
is interesting to remark that even on the cylinder, this problem is invisible
at the level of the algebra and the spectrum of the flowed generators, which
simply coincide with those of the undeformed CFT. Rather, the complex energies
arise solely as a result of the expression (A.1) for the Hamiltonian, and
signal the breakdown of this square root formula. This suggests that the
states of the deformed CFT on a cylinder may be well-behaved, and the only
problem is with the operator we use to measure the energy (which, however, is
also the one used to evolve the system).

There are several interesting future directions. First, one should be able to
pass from the classical construction of the symmetry generators to the quantum
one by including the appropriate normal ordering prescriptions and check, at
least perturbatively, whether their quantum commutators work out as expected.
A particularly interesting issue is that of the central extension, given that
the classical Witt algebra is naturally expected to become Virasoro at the
quantum level. The holographic analysis of [21] indicates a field-dependent
central extension of the form $c(R/R_{v})^{2}$ for the generators defined in
[19], where $c$ is the central charge of the original CFT. Given that the
spectrally flowed generators are rescaled by a factor of $R_{v}$ with respect
to the original ones suggests that the central extension in the flowed sector
has a chance of being identical to that of the original CFT, in agreement with
the fact that the $J\bar{T}$ deformation does not change the number of states.
It would be very interesting verify this, at least perturbatively.

Other relevant questions are to understand the physical meaning of the
spectrally flowed generators, and whether there is a geometric symmetry that
they implement. It would also be interesting to see whether the new symmetry
generators may have applications also in the deformed CFT on the plane, e.g.
in the construction of more complicated observables such as correlation
functions. Finally, it would be interesting to understand the role and
physical interpretation of the improved zero mode $\widetilde{\chi}_{0}$,
whose action on the Hilbert space is surprisingly minimal.

#### Acknowledgements

MG would like to thank Ruben Monten for collaboration on related subjects.
This research was supported in part by the ERC Starting Grant 679278 Emergent-
BH and the Swedish Research Council grant number 2015-05333.

##  Appendix A Summary of Poisson brackets in $J\bar{T}$ \- deformed CFTs

In this appendix, we collect the results of [19] on the Poisson brackets of
the various currents in the $J\bar{T}$ \- deformed CFTs. These were computed
using the expression they derived by solving the flow equation for the
deformed right-moving Hamiltonian density $\mathcal{H}_{R}$

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

in terms of the undeformed one, $\mathcal{H}_{R}^{(0)}$, together with the
commutation relations of the undeformed currents,
$\mathcal{H}_{R}^{(0)},\mathcal{J}_{\pm}$ and $\mathcal{P}$. The currents
$\mathcal{J}_{\pm}$ represent the time components of the linear combinations
$(J_{\alpha}\pm\tilde{J}_{\alpha})/2$, where $J$ is represented as a $U(1)$
shift current for a scalar field $\phi$, and $\tilde{J}=\star d\phi$ is the
corresponding topologically conserved current. In Hamiltonian language

$\mathcal{J}_{\pm}=\frac{\pi\pm\phi^{\prime}}{2}$ (A.2)

where $\pi$ is the momentum conjugate to $\phi$. Note the expression for
$\mathcal{H}_{R}$ is symmetric under $\pi\leftrightarrow\phi^{\prime}$, since
it only depends on $\mathcal{J}_{+}$. From this, we conclude that the
$J\bar{T}$ and $\tilde{J}\bar{T}$ deformations, which differ by precisely this
exchange, lead to the same deformed theory, at least at the classical level.

The commutation relations derived in [19] are

$\\{\mathcal{H}_{R}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=-\left(\frac{\mathcal{H}_{R}(\sigma)}{\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}}+\frac{\mathcal{H}_{R}(\tilde{\sigma})}{\sqrt{\left(1-\lambda\mathcal{J}_{+}(\tilde{\sigma})\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\tilde{\sigma})}}\right)\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.3)
$\\{\mathcal{P}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=\left(\mathcal{H}_{R}(\sigma)+\frac{\mathcal{H}_{R}(\tilde{\sigma})}{\sqrt{(1-\lambda\mathcal{J}_{+}(\tilde{\sigma}))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\tilde{\sigma})}}\right)\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.4)

$\\{\mathcal{H}_{R}(\sigma),\mathcal{J}_{+}(\tilde{\sigma})\\}=\frac{\lambda\mathcal{H}_{R}(\sigma)}{2\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.5)

$\\{\mathcal{H}_{R}(\sigma),\mathcal{J}_{-}(\tilde{\sigma})\\}=-\frac{\mathcal{J}_{-}(\sigma)}{\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.6)

Using the fact that $\mathcal{J}_{+}-\mathcal{J}_{-}=\phi^{\prime}$ and that
the right-hand side of the last two commutators are total $\tilde{\sigma}$
derivatives, we can deduce the commutator of $\mathcal{H}_{R}$ with $\phi$

$\\{\mathcal{H}_{R}(\sigma),\phi(\tilde{\sigma})\\}=\frac{-\mathcal{J}_{-}(\sigma)-\lambda\mathcal{H}_{R}(\sigma)/2}{\sqrt{(1-\lambda\mathcal{J}_{+}(\sigma))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}}\delta(\sigma-\tilde{\sigma})$
(A.7)

Since this commutator is obtained by integration, in principle we could add an
integration function of $\sigma$ to the right-hand side; however, such an
addition would be quite unnatural, given that the commutator is local (i.e.,
proportional to a $\delta$ function).

The commutators of the momentum and the currents are the same as in the
undeformed CFT

$\\{\mathcal{P}(\sigma),\mathcal{P}(\tilde{\sigma})\\}=\left(\mathcal{P}(\sigma)+\mathcal{P}(\tilde{\sigma})\right)\partial_{\sigma}\delta(\sigma-\tilde{\sigma})\;,\;\;\;\;\;\;\;\;\;\\{\mathcal{J}_{\pm}(\sigma),\mathcal{J}_{\pm}(\tilde{\sigma})\\}=\pm\frac{1}{2}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.8)

$\\{\mathcal{P}(\sigma),\mathcal{J}_{+}(\tilde{\sigma})\\}=\mathcal{J}_{+}(\sigma)\partial_{\sigma}\delta(\sigma-\tilde{\sigma})\;,\;\;\;\;$
(A.9)

From here, one can deduce that

$\\{\mathcal{P}(\sigma),\phi(\tilde{\sigma})\\}=-\phi^{\prime}(\sigma)\delta(\sigma-\tilde{\sigma})\;,\;\;\;\;\;\;\;\;\\{\mathcal{J}_{\pm}(\sigma),\phi(\tilde{\sigma})\\}=-\frac{1}{2}\delta(\sigma-\tilde{\sigma})$
(A.10)

We note that the zero modes $J_{0}=\int d\sigma\mathcal{J}_{+}$,
$\bar{J}_{0}=\int d\sigma\mathcal{J}_{-}$ commute with all the other currents
in the theory, and their only non-zero commutator is with $\phi$. The winding
charge $w=J_{0}-\bar{J}_{0}$ also commutes with $\phi$. This implies in
particular that the field-dependent radius $R_{v}$ commutes with all the
operators we consider.

Finally, one can work out the commutators of the chiral current
$K_{U}=\mathcal{J}_{+}+\frac{\lambda}{2}\mathcal{H}_{R}$ and of the left-
moving Hamiltonian $\mathcal{H}_{L}=\mathcal{H}_{R}+\mathcal{P}$, which take
the very simple form

$\\{K_{U}(\sigma),K_{U}(\tilde{\sigma})\\}=\frac{1}{2}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})\;,\;\;\;\;\;\;\\{\mathcal{H}_{L}(\sigma),K_{U}(\tilde{\sigma})\\}=K_{U}(\sigma)\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.11)

$\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{L}(\tilde{\sigma})\\}=(\mathcal{H}_{L}(\sigma)+\mathcal{H}_{L}(\tilde{\sigma}))\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(A.12)

and their commutators with $\mathcal{H}_{R}$ are

$\\{\mathcal{H}_{R}(\sigma),K_{U}(\tilde{\sigma})\\}=-\frac{\lambda\tilde{\mathcal{H}}_{R}}{2\sqrt{\tilde{}}}\delta^{\prime}\;,\;\;\;\;\;\;\\{\mathcal{H}_{R}(\sigma),\mathcal{H}_{L}(\tilde{\sigma})\\}=\left(\tilde{\mathcal{H}}_{R}-\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\right)\delta^{\prime}$
(A.13)

which are total $\sigma$ derivatives. In particular, this implies that
$\\{E_{R},K_{U}\\}=\\{E_{R},\mathcal{H}_{L}\\}=0$.

##  Appendix B Poisson brackets of the non-local field $\chi$

In this appendix, we derive the commutators of the non-local field $\chi$,
defined through $\partial_{\sigma}\chi=\mathcal{H}_{R}$, with the various
other fields in the theory.

The Poisson brackets of $\chi$ are obtained by integrating the corresponding
commutators of $\mathcal{H}_{R}$. Two of these Poisson brackets, namely

$\\{\chi(\sigma),K_{U}(\tilde{\sigma})\\}=-\frac{\lambda\tilde{\mathcal{H}}_{R}}{2\sqrt{\tilde{}}}\delta\;,\;\;\;\;\;\;\\{\chi(\sigma),\mathcal{H}_{L}(\tilde{\sigma})\\}=\left(\tilde{\mathcal{H}}_{R}-\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\right)\delta$
(B.1)

are local, being proportional to $\delta$ functions, and thus we do not
include an integration function. Other commutators, however, are significantly
more involved, and require working out the consistency conditions imposed by
the various Jacobi identities that they satisfy.

In the following, we will frequently use the notation
$\sqrt{}=\sqrt{(1-\lambda\mathcal{J}_{+}(\sigma))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}$
and
$\sqrt{\tilde{}}=\sqrt{(1-\lambda\mathcal{J}_{+}(\tilde{\sigma}))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\tilde{\sigma})}$,
as well as the abbreviations $\tilde{A}=A(\tilde{\sigma})$ and
$\delta^{\prime}=\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$.

###  B.1 Poisson bracket of $\chi$ with $\mathcal{H}_{R}$

The $\\{\chi,\tilde{\mathcal{H}}_{R}\\}$ Poisson bracket is given by
integrating the $\\{\mathcal{H}_{R},\tilde{\mathcal{H}}_{R}\\}$ commutator

$\\{\chi(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=-\frac{2\mathcal{H}_{R}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})+\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\,\Theta(\sigma-\tilde{\sigma})+A(\tilde{\sigma})$
(B.2)

where $A(\tilde{\sigma})$ is an integration function. This function needs to
have winding, in order to cancel the dependence on the starting point in

$\\{\chi(\sigma),H_{R}\\}=-\frac{\mathcal{H}_{R}}{\sqrt{}}-\frac{\mathcal{H}_{R}(0)}{\sqrt{(0)}}+\int_{0}^{R}d\tilde{\sigma}A(\tilde{\sigma})\equiv-\frac{\mathcal{H}_{R}}{\sqrt{}}+\int_{0}^{R}d\tilde{\sigma}A_{p}(\tilde{\sigma})$
(B.3)

where $E_{R}=\int\mathcal{H}_{R}d\sigma$ is the total right-moving energy.
This equation defines $A_{p}$, the periodic part of $A$, with the winding
subtracted. To fix this function, we need to analyse the constraints coming
from the various Jacobi identities that the Poisson bracket (B.2) satisfies.

#### Constraints from time evolution

A first constraint on $A$ comes from analysing the time dependence of the
above commutator

$\frac{d}{dt}\\{\chi(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=\partial_{t}A(\tilde{\sigma})-\\{H,\\{\chi(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}\\}=-\\{\\{H,\chi\\},\tilde{\mathcal{H}}_{R}\\}-\\{\chi,\\{H,\tilde{\mathcal{H}}_{R}\\}\\}$
(B.4)

where $H=E_{L}+E_{R}$ is the total Hamiltonian. Making use of

$\\{H,\mathcal{H}_{R}\\}=\partial_{\sigma}(2\mathcal{H}_{R}/\sqrt{}-\mathcal{H}_{R})\;,\;\;\;\;\;\;\\{H,\mathcal{J}_{+}\\}=-\partial_{\sigma}(\mathcal{J}_{+}+\lambda\mathcal{H}_{R}/\sqrt{})\;,\;\;\;\;\;\\{H,\mathcal{H}_{R}/\sqrt{}\\}=\frac{1+\lambda
K_{U}}{1-\lambda K_{U}}\partial_{\sigma}(\mathcal{H}_{R}/\sqrt{})$ (B.5)

It can be easily shown that the terms proportional to $\Theta,\delta^{\prime}$
and $\delta$ functions cancel, and the constraint that we obtain on the
function $A$ is

$\partial_{t}A-\\{H,A\\}=\int_{0}^{R}d\tilde{\sigma}\\{A_{p}(\tilde{\sigma}),\mathcal{H}_{R}\\}-\partial_{\sigma}\left(\frac{1+\lambda
K_{U}}{1-\lambda K_{U}}A\right)$ (B.6)

where $A_{p}$ represents the part of $A$ without the winding contribution. If
we choose $A$ such that this term is absent, we find several qualitatively
different solutions to the remaining equation. For example

$RA\equiv\partial_{\sigma}\left(\sigma\frac{\mathcal{H}_{R}}{\sqrt{}}\right)+R\partial_{\sigma}\hat{A}=\sigma\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}+\frac{\mathcal{H}_{R}}{\sqrt{}}+Ra\,\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}$
(B.7)

solves the equations for an arbitrary constant $a$. Another solution can be
obtained by noting that the field-dependent coordinate
$v=\sigma-t-\lambda\phi$ satisfies

$\frac{dv}{dt}=-1-\\{H,v\\}=-\frac{1+\lambda K_{U}}{1-\lambda
K_{U}}v^{\prime}$ (B.8)

and thus a general solution to the equation with $A_{p}=0$ and the correct
winding is

$A=\partial_{\sigma}\left(\frac{v}{R_{v}}\frac{\mathcal{H}_{R}}{\sqrt{}}\right)+a\,\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}$
(B.9)

where the total derivative is necessary in order to ensure that $A_{p}=0$
above.

#### Constraints from the Jacobi identity with $K_{U}$

To distinguish between the two solutions, we can check the Jacobi identity for
$\\{K_{U},\\{\chi_{0},\tilde{\mathcal{H}}_{R}\\}\\}$. Using the expression for
the $\\{\chi,\tilde{K}_{U}\\}$ commutator, we obtain

$\\{K_{U},(R-\tilde{\sigma})\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}+R\tilde{A}\\}+\frac{\lambda\delta^{\prime}}{2\sqrt{}}\left[(R-\sigma)\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}+RA\right]=\frac{\lambda}{2\sqrt{}}\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})$
(B.10)

Taking into account the fact that888This is most simply computed by using
$\sqrt{}=1-\lambda K_{U}$ and the equivalence of distributions spelled out in
footnote 10.

$\\{K_{U},\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}=\frac{\lambda}{2\sqrt{}}\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})$
(B.11)

we can reduce this to the following simple constraint on $A$

$\\{K_{U},\tilde{A}\\}+\frac{\lambda}{2\sqrt{}}A\delta^{\prime}=0$ (B.12)

It is easy to check, using (B.11), that the first Ansatz, (B.7), does _not_
satisfy the consistency requirement (B.12). On the other hand, using the
commutator

$\\{K_{U},\tilde{v}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}=-\lambda\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\{K_{U},\tilde{\phi}\\}+\tilde{v}\\{K_{U},\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}=\frac{\lambda}{2\sqrt{}}\partial_{\sigma}(v\mathcal{H}_{R}/\sqrt{})\delta$
(B.13)

we can show that the second Ansatz does identically satisfy the consistency
condition999 Making use (or not) of the relation
$\\{K_{U},\tilde{\hat{\phi}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}=\frac{\lambda\delta(\sigma-\tilde{\sigma})}{2\sqrt{}}\partial_{\sigma}\left(\hat{\phi}\frac{\mathcal{H}_{R}}{\sqrt{}}\right)-\frac{R_{v}\mathcal{H}_{R}\delta}{2R(\sqrt{})^{2}}$
(B.14) . One can also check that the term
$a\partial_{\sigma}\mathcal{H}_{R}/\sqrt{}$ also satisfies it, so this does
not fix $a$.

#### Constraints from the Jacobi identity with $\mathcal{H}_{R}$

We now look at the $\\{\chi_{0},\mathcal{H}_{R}\\}$ Jacobi identity

$\\{\mathcal{H}_{R},\\{\chi_{0},\tilde{\mathcal{H}}_{R}\\}\\}-\\{\tilde{\mathcal{H}}_{R},\\{\chi_{0},\mathcal{H}_{R}\\}\\}+\\{\chi_{0},\\{\tilde{\mathcal{H}}_{R},\mathcal{H}_{R}\\}\\}=0$
(B.15)

We will also need

$\\{\mathcal{H}_{R},\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}=-\frac{2\mathcal{H}_{R}^{(0)}}{\sqrt{}^{3}}\delta^{\prime}(\sigma-\tilde{\sigma})+\left(\frac{1}{\sqrt{}}\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}-2\,\partial_{\sigma}\frac{\mathcal{H}_{R}^{(0)}}{\sqrt{}^{3}}\right)\delta(\sigma-\tilde{\sigma})$
(B.16)

where, from (A.1),
$\mathcal{H}_{R}^{(0)}=\mathcal{H}_{R}(1-\lambda\mathcal{J}_{+}-\frac{\lambda^{2}}{4}\mathcal{H}_{R})$.
Plugging in the expression for

$\\{\chi_{0},\mathcal{H}_{R}\\}=-2\mathcal{H}_{R}/\sqrt{}+(R-\sigma)\partial_{\sigma}(\mathcal{H}_{R}/\sqrt{})+RA$
(B.17)

we find the intermediate equation

$\displaystyle
R\left(\\{\mathcal{H}_{R},\tilde{A}\\}-\\{\tilde{\mathcal{H}}_{R},A\\}+\frac{A\delta^{\prime}}{\sqrt{}}+\frac{\tilde{A}\delta^{\prime}}{\sqrt{\widetilde{}}}\right)+(R-\tilde{\sigma})\\{\mathcal{H}_{R},\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}-(R-\sigma)\\{\tilde{\mathcal{H}}_{R},\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\\}+$
$\displaystyle\hskip
56.9055pt+\left(\frac{R-\sigma}{\sqrt{}}\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}+\frac{2\mathcal{H}_{R}^{(0)}}{\sqrt{}^{3}}+(\sigma\rightarrow\tilde{\sigma})\right)\delta^{\prime}=0$
(B.18)

The difference

$(R-\tilde{\sigma})\\{\mathcal{H}_{R},\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\\}-(R-\sigma)\\{\tilde{\mathcal{H}}_{R},\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\\}$
(B.19)

can be manipulated using the criteria for when two distributions of the form
$E(\sigma,\tilde{\sigma})\delta^{\prime\prime}+F(\sigma,\tilde{\sigma})\delta^{\prime}+G(\sigma)\delta$
are equivalent101010These crieria are obtained by integrating against a test
function $g(\tilde{\sigma})$, with the result
$g^{\prime\prime}E+g^{\prime}(2\partial_{\tilde{\sigma}}E+F)+g(\partial_{\tilde{\sigma}}^{2}E+\partial_{\tilde{\sigma}}F+G)$
(B.20) and respectively $f(\sigma)$, for which
$f^{\prime\prime}E+f^{\prime}(2\partial_{\sigma}E-F)+f(\partial_{\sigma}^{2}E-\partial_{\sigma}F+G)$
(B.21) Two distributions are equivalent if the terms multiplying $f$, $g$ and
their various derivatives are the same . . At the end of the day, we find the
very simple constraint

$\\{\mathcal{H}_{R},\tilde{A}\\}-\\{\tilde{\mathcal{H}}_{R},A\\}+\frac{A\delta^{\prime}}{\sqrt{}}+\frac{\tilde{A}\delta^{\prime}}{\sqrt{\widetilde{}}}=0$
(B.22)

Let us now write $A=A_{0}+\hat{A}$, where
$A_{0}\equiv\partial_{\sigma}\left(\frac{v}{R_{v}}\frac{\mathcal{H}_{R}}{\sqrt{}}\right)$.
Evaluating

$R_{v}\left(\\{\mathcal{H}_{R},A_{0}\\}+\frac{A_{0}\delta^{\prime}}{\sqrt{}}\right)=\frac{2v\mathcal{H}_{R}^{(0)}}{\sqrt{}^{3}}\delta^{\prime\prime}+\frac{\mathcal{H}_{R}}{\sqrt{}}\delta^{\prime}+2\partial_{\sigma}\left(v\frac{\mathcal{H}_{R}^{(0)}}{\sqrt{}^{3}}\right)\delta^{\prime}$
(B.23)

the constraint we obtain on $\hat{A}$ is then, simply

$\\{\mathcal{H}_{R},\tilde{\hat{A}}\\}-\\{\tilde{\mathcal{H}}_{R},\hat{A}\\}+\frac{\hat{A}\delta^{\prime}}{\sqrt{}}+\frac{\tilde{\hat{A}}\delta^{\prime}}{\sqrt{\widetilde{}}}=-\frac{1}{R_{v}}\left(\frac{\mathcal{H}_{R}}{\sqrt{}}+\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\right)\delta^{\prime}$
(B.24)

This is solved by $\hat{A}=\mathcal{H}_{R}/R_{v}+\ldots$, where the $\ldots$
are periodic solutions to the homogenous equation, such as the
$a\partial_{\sigma}\mathcal{H}_{R}/\sqrt{}$ term.

Note however that for this value of $\hat{A}$, we need to revisit the
conservation equation (B.6), which receives a new contribution from

$\int
d\tilde{\sigma}\\{A_{p}(\tilde{\sigma},\mathcal{H}_{R}(\sigma)\\}=\frac{1}{R_{v}}\\{E_{R},\mathcal{H}_{R}\\}=\frac{1}{R_{v}}\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}$
(B.25)

This contribution is very easy to cancel by including an explicit time-
dependent term, $t/R_{v}\partial_{\sigma}\mathcal{H}_{R}/\sqrt{}$. This term
is also consistent with the $K_{U}$ and $\mathcal{H}_{R}$ Jacobi identities.
Therefore, the final solution we find for the integration function $A$ is

$\boxed{A=\partial_{\sigma}\left[\left(\frac{v}{R_{v}}+a\right)\frac{\mathcal{H}_{R}}{\sqrt{}}\right]+\frac{1}{R_{v}}\left(\mathcal{H}_{R}+t\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\right)=\partial_{\sigma}\left[\left(\frac{v+t}{R_{v}}+a\right)\frac{\mathcal{H}_{R}}{\sqrt{}}\right]+\frac{\mathcal{H}_{R}}{R_{v}}}$
(B.26)

which has the nice feature of not being explicitly time-dependent, since
$v+t=\sigma-\lambda\phi$.

###  B.2 Poisson bracket of $\chi$ with $\mathcal{J}_{-}$

Another commutator that requires special attention is that of $\chi$ with
$\mathcal{J}_{-}$. Integrating the
$\\{\mathcal{H}_{R},\tilde{\mathcal{J}}_{-}\\}$ commutator we obtain

$\\{\chi,\tilde{\mathcal{J}}_{-}\\}=-\frac{\mathcal{J}_{-}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})+\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{J}}_{-}}{\sqrt{\tilde{}}}\,\Theta(\sigma-\tilde{\sigma})+B(\tilde{\sigma})$
(B.27)

In order for the commutator with $\bar{J}_{0}$ to be independent of the
starting point of the interval, we need the winding of $B$ to equal
$\mathcal{J}_{-}(0)/\sqrt{}(0)$. $B$ should also satisfy all the relevant
Jacobi identities.

As before, we first look at the time derivative of this commutator, and try to
fix $B$ by requiring that the Jacobi identity hold. We make use of

$\\{H,\mathcal{J}_{-}/\sqrt{}\\}=\frac{1+\lambda K_{U}}{1-\lambda
K_{U}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}$ (B.28)

and find that $B$ satisfies

$\partial_{t}B-\\{H,B\\}=-\partial_{\sigma}\left(\frac{1+\lambda
K_{U}}{1-\lambda
K_{U}}B\right)+\frac{1}{R_{v}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}$
(B.29)

There are several solutions to this equation that have the correct winding,
such as

$B(\sigma)=\partial_{\sigma}\left(\frac{\sigma}{R}\,\frac{\mathcal{J}_{-}}{\sqrt{}}\right)+b\,\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}+\frac{t}{R_{v}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}$
(B.30)

or

$B(\sigma)=\partial_{\sigma}\left(\frac{v}{R_{v}}\frac{\mathcal{J}_{-}}{\sqrt{}}\right)+b\,\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}+\frac{t}{R_{v}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}$
(B.31)

for some constant $b$. To find which solution is correct, we need to analyse
some further Jacobi identities.

#### Constraint from the commutator with $K_{U}$

We first check the Jacobi identity for
$\\{K_{U},\\{\chi_{0},\tilde{\mathcal{J}}_{-}\\}\\}$. The consistency
condition we obtain is

$\\{K_{U},(R-\tilde{\sigma})\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{J}}_{-}}{\sqrt{\tilde{}}}+R\tilde{B}\\}+\frac{\lambda\delta^{\prime}}{2\sqrt{}}\left[(R-\sigma)\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}+RB\right]=\frac{\lambda}{2\sqrt{}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})$
(B.32)

Using the fact that

$\\{K_{U},\frac{\tilde{\mathcal{J}}_{-}}{\sqrt{\tilde{}}}\\}=\frac{\lambda}{2\sqrt{}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}\delta(\sigma-\tilde{\sigma})$
(B.33)

and the criterion for the equivalence of two distributions, we can reduce the
above equation to

$\\{K_{U},\tilde{B}\\}+\frac{\lambda\delta^{\prime}}{2\sqrt{}}B=0$ (B.34)

It is easy to see that the terms proportional to
$\partial_{\sigma}\mathcal{J}_{-}/\sqrt{}$ simply drop out of this equation.
We then check that the first Ansatz does not solve this equation, whereas the
second one does.

#### Constraint from the $\mathcal{H}_{R}$ commutator

The Jacobi identity reads

$\\{\mathcal{H}_{R},\\{\chi_{0},\tilde{\mathcal{J}}_{-}\\}\\}+\\{\\{\chi_{0},\mathcal{H}_{R}\\},\tilde{\mathcal{J}}_{-}\\}-\\{\chi_{0},\\{\mathcal{H}_{R},\tilde{\mathcal{J}}_{-}\\}\\}=0$
(B.35)

and from its form, it is easy to see it will relate the two integration
functions $A$ and $B$. To simplify the constraint, we use

$\\{\mathcal{H}_{R},\frac{\tilde{\mathcal{J}}_{-}}{\sqrt{\tilde{}}}\\}=-\frac{\mathcal{J}_{-}(1-\lambda\mathcal{J}_{+})}{\sqrt{}^{3}}\delta^{\prime}+\left(\frac{1}{\sqrt{}}\partial_{\sigma}\frac{\mathcal{J}_{-}}{\sqrt{}}-\partial_{\sigma}\frac{\mathcal{J}_{-}(1-\lambda\mathcal{J}_{+})}{\sqrt{}^{3}}\right)\delta(\sigma-\tilde{\sigma})$
(B.36)

$\\{\frac{\mathcal{H}_{R}}{\sqrt{}},\tilde{\mathcal{J}}_{-}\\}=-\frac{\mathcal{J}_{-}(1-\lambda\mathcal{J}_{+})}{\sqrt{}^{3}}\delta^{\prime}$
(B.37)

and find that $A$ and $B$ must satisfy the simple relation

$\\{\mathcal{H}_{R},\tilde{B}\\}+\\{A,\tilde{\mathcal{J}}_{-}\\}+\frac{B}{\sqrt{}}\delta^{\prime}=0$
(B.38)

Letting $A=A_{0}+\hat{A}$, $B=B_{0}+\hat{B}$ with
$A_{0}=\partial_{\sigma}(v\mathcal{H}_{R}/R_{v}\sqrt{})$ and
$B_{0}=\partial_{\sigma}(v\mathcal{J}_{-}/R_{v}\sqrt{})$, we find the
following constraint

$\\{\mathcal{H}_{R},\tilde{\hat{B}}\\}+\\{\hat{A},\tilde{\mathcal{J}}_{-}\\}+\frac{\hat{B}}{\sqrt{}}\delta^{\prime}=\frac{\lambda}{2R_{v}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\delta^{\prime}-\frac{1}{R_{v}}\frac{\mathcal{J}_{-}}{\sqrt{}}\delta^{\prime}$
(B.39)

The already known $\mathcal{H}_{R}/R_{v}$ contribution to $\hat{A}$ accounts
for the last term on the right-hand side. However, we need a new term in
$\hat{B}$ to account for the first term. It is clear that

$\hat{B}=-\frac{\lambda\mathcal{H}_{R}}{2R_{v}}$ (B.40)

does the job, and one can check that it is also consistent with the previous
consistency conditions.

Finally, it is not hard to check that any term in $B$ proportional to
$\partial_{\sigma}\mathcal{J}_{-}/\sqrt{}$ and in $A$ with
$\partial_{\sigma}\mathcal{H}_{R}/\sqrt{}$ (with the same proportionality
coefficient) automatically satisfies this equation. This sets $a=b$. We can
also check the time dependence matches exactly.

To summarize, the solution that we have found for B that is consistent with
all the Jacobi identities we have checked is

$\boxed{B(\sigma)=\partial_{\sigma}\left[\left(\frac{v+t}{R_{v}}+a\right)\frac{\mathcal{J}_{-}}{\sqrt{}}\right]-\frac{\lambda\mathcal{H}_{R}}{2R_{v}}}$
(B.41)

for the same arbitrary constant $a$ as in (B.26).

###  B.3 Other commutators

The Poisson brackets of $\chi$ with all the remaining fields, such as
$\mathcal{J}_{+}$ or $\mathcal{P}$, are determined by its commutators with
$\mathcal{H}_{R}$ and $\mathcal{J}_{-}$ (i.e., the functions $A,B$) and its
commutators with $\mathcal{H}_{L}=\mathcal{H}_{R}+\mathcal{P}$ and
$K_{U}=\mathcal{J}_{+}+\lambda\mathcal{H}_{R}/2$, which are local. We thus
find

$\\{\mathcal{P},\tilde{\chi}\\}=-\left(\mathcal{H}_{R}+\frac{\mathcal{H}_{R}}{\sqrt{}}\right)\delta(\sigma-\tilde{\sigma})+\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{}}\,\Theta(\tilde{\sigma}-\sigma)+A(\sigma)$
(B.42)
$\\{\chi,\tilde{\mathcal{J}}_{+}\\}=\frac{\lambda\mathcal{H}_{R}}{2\sqrt{}}\delta(\sigma-\tilde{\sigma})-\frac{\lambda}{2}\partial_{\tilde{\sigma}}\frac{\tilde{\mathcal{H}}_{R}}{\sqrt{\tilde{}}}\,\Theta(\sigma-\tilde{\sigma})-\frac{\lambda}{2}A(\tilde{\sigma})$
(B.43)

Combining the latter with the $\\{\chi,\tilde{\mathcal{J}}_{-}\\}$ commutator
and integrating, we find the $\\{\chi,\tilde{\phi}\\}$ commutator

$\\{\chi,\tilde{\phi}\\}=-\frac{\tilde{\mathcal{J}}_{-}+\lambda\tilde{\mathcal{H}}_{R}/2}{\sqrt{\tilde{}}}\left(\Theta(\sigma-\tilde{\sigma})+\frac{\tilde{v}+\tilde{t}}{R_{v}}+a\right)+C(\sigma)$
(B.44)

The $\\{\mathcal{H}_{R},\tilde{\phi}\\}$ commutator requires that
$C^{\prime}(\sigma)=0$, so $C=c$, a constant, which we will set to zero.

##  Appendix C Flow equations for the currents and the charges

Let $\mathcal{D}_{\lambda}=\partial_{\lambda}+\\{\mathcal{O}_{tot},\cdot\\}$,
where for the purposes of computing commutators, the expression

$\mathcal{O}_{tot}=\frac{w\chi_{0}}{R}-\int d\sigma\mathcal{H}_{R}\hat{\phi}$
(C.1)

is significantly easier to use. Using the commutators with the zero modes, we
find that at _classical_ level, the currents satisfy the following flow
equations

$\mathcal{D}_{\lambda}K_{U}=\mathcal{D}_{\lambda}\mathcal{H}_{L}=0$ (C.2)

The equation for $\mathcal{H}_{R}$ is

$\mathcal{D}_{\lambda}\mathcal{H}_{R}=\frac{w}{R_{v}}\mathcal{H}_{R}+\partial_{\sigma}\left[\frac{\mathcal{H}_{R}}{\sqrt{}}\left(w(1+a)-\frac{R\hat{\phi}}{R_{v}}\right)\right]$
(C.3)

which implies the following flow equation for $\chi$

$\mathcal{D}_{\lambda}\chi=\frac{\mathcal{H}_{R}}{\sqrt{}}\left(w(1+a)-\frac{R\hat{\phi}}{R_{v}}\right)+\frac{w}{R_{v}}\chi$
(C.4)

The flow equation for $\mathcal{J}_{-}$ is given by

$\mathcal{D}_{\lambda}\mathcal{J}_{-}=-\frac{1}{2}\mathcal{H}_{R}-\frac{w\lambda}{2R_{v}}\mathcal{H}_{R}+\partial_{\sigma}\left[\frac{\mathcal{J}_{-}}{\sqrt{}}\left(w(1+a)-\frac{R\hat{\phi}}{R_{v}}\right)\right]$
(C.5)

which, together with (C.3), implies that

$\mathcal{D}_{\lambda}(\mathcal{J}_{-}+\frac{\lambda}{2}\mathcal{H}_{R})=\partial_{\sigma}\left[\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}\left(w(1+a)-\frac{R\hat{\phi}}{R_{v}}\right)\right]$
(C.6)

Finally, the flow equation for $\phi$ is

$\mathcal{D}_{\lambda}\phi(\sigma)=-\frac{\mathcal{J}_{-}+\lambda\mathcal{H}_{R}/2}{\sqrt{}}\left(w(1+a)-\frac{R\hat{\phi}}{R_{v}}\right)$
(C.7)

We subsequently use these flow equations to compute the flow of the conserved
charges. We trivially have
$\mathcal{D}_{\lambda}Q_{f}=\mathcal{D}_{\lambda}P_{\eta}=0$. As for the
right-moving charges, we obtain

$\mathcal{D}_{\lambda}\bar{Q}_{\bar{f}}=\frac{w}{R_{v}}\bar{Q}_{\bar{f}}-\frac{w}{R_{v}}\left(a+1+\frac{t}{R_{v}}\right)\bar{Q}_{\bar{f}^{\prime}}\;,\;\;\;\;\;\;\;\mathcal{D}_{\lambda}\bar{P}^{KM}_{\bar{\eta}}=-\frac{w}{R_{v}}\left(a+1+\frac{t}{R_{v}}\right)\bar{P}^{KM}_{\bar{\eta}^{\prime}}$
(C.8)

## References

  * [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999), 1113-1133 arXiv:hep-th/9711200 [hep-th].
  * [2] W. Li and T. Takayanagi, “Holography and Entanglement in Flat Spacetime,” Phys. Rev. Lett. 106 (2011), 141301 arXiv:1010.3700 [hep-th].
  * [3] S. El-Showk and M. Guica, “Kerr/CFT, dipole theories and nonrelativistic CFTs,” JHEP 12 (2012), 009 arXiv:1108.6091 [hep-th].
  * [4] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B 915 (2017), 363-383 arXiv:1608.05499 [hep-th].
  * [5] A. Cavaglià, S. Negro, I. M. Szécsényi and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016), 112 arXiv:1608.05534 [hep-th].
  * [6] S. Dubovsky, R. Flauger and V. Gorbenko, “Solving the Simplest Theory of Quantum Gravity,” JHEP 09 (2012), 133 arXiv:1205.6805 [hep-th].
  * [7] S. Dubovsky, V. Gorbenko and M. Mirbabayi, “Natural Tuning: Towards A Proof of Concept,” JHEP 09 (2013), 045 arXiv:1305.6939 [hep-th].
  * [8] J. Cardy, “The $T\overline{T}$ deformation of quantum field theory as random geometry,” JHEP 10 (2018), 186 arXiv:1801.06895 [hep-th].
  * [9] S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, “$T\overline{T}$ partition function from topological gravity,” JHEP 09 (2018), 158 arXiv:1805.07386 [hep-th].
  * [10] J. Cardy, “$T\bar{T}$ deformation of correlation functions,” JHEP 19 (2020), 160 arXiv:1907.03394 [hep-th].
  * [11] J. Kruthoff and O. Parrikar, “On the flow of states under $T\overline{T}$,” arXiv:2006.03054 [hep-th].
  * [12] A. Giveon, N. Itzhaki and D. Kutasov, “$\mathrm{T}\overline{\mathrm{T}}$ and LST,” JHEP 07 (2017), 122 arXiv:1701.05576[hep-th].
  * [13] L. Apolo and W. Song, “Strings on warped AdS3 via $\mathrm{T}\bar{\mathrm{J}}$ deformations,” JHEP 10 (2018), 165 arXiv:1806.10127 [hep-th].
  * [14] S. Chakraborty, A. Giveon and D. Kutasov, “$J\overline{T}$ deformed CFT2 and string theory,” JHEP 10 (2018), 057 arXiv:1806.09667 [hep-th].
  * [15] M. Guica, “An integrable Lorentz-breaking deformation of two-dimensional CFTs,” SciPost Phys. 5 (2018) no.5, 048 arXiv:1710.08415 [hep-th].
  * [16] T. Anous and M. Guica, “A general definition of $JT_{a}$ – deformed QFTs,” arXiv:1911.02031 [hep-th].
  * [17] S. Frolov, “$T{\overline{T}}$, $\widetilde{J}J$, $JT$ and $\widetilde{J}T$ deformations,” J. Phys. A 53 (2020) no.2, 025401 arXiv:1907.12117 [hep-th].
  * [18] B. Le Floch and M. Mezei, “Solving a family of $T\bar{T}$-like theories,” arXiv:1903.07606 [hep-th].
  * [19] M. Guica and R. Monten, “Infinite pseudo-conformal symmetries of classical $T\bar{T}$, $J\bar{T}$ and $JT_{a}$ \- deformed CFTs,” arXiv:2011.05445 [hep-th].
  * [20] M. Guica and R. Monten, “$T\bar{T}$ and the mirage of a bulk cutoff,” arXiv:1906.11251 [hep-th].
  * [21] A. Bzowski and M. Guica, “The holographic interpretation of $J\bar{T}$-deformed CFTs,” JHEP 01 (2019), 198 arXiv:1803.09753 [hep-th].
  * [22] A. Schwimmer and N. Seiberg, “Comments on the N=2, N=3, N=4 Superconformal Algebras in Two-Dimensions,” Phys. Lett. B 184 (1987), 191-196
  * [23] M. Guica, “On correlation functions in $J\bar{T}$-deformed CFTs,” J. Phys. A 52 (2019) no.18, 184003 arXiv:1902.01434 [hep-th].