# A definition of primary operators in $J\bar{T}$ \- deformed CFTs

Monica Guica

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

###### Abstract

$J\bar{T}$ \- deformed CFTs provide an interesting example of non-local, yet
UV-complete two-dimensional QFTs that are entirely solvable. They have been
recently shown to possess an infinite set of symmetries, which are a
continuous deformation of the Virasoro-Kac-Moody symmetries of the seed CFT.
In this article, we put forth a definition of primary operators in $J\bar{T}$
\- deformed CFTs on a cylinder, which are singled out by having CFT-like
momentum-space commutation relations with the symmetry generators in the
decompatification limit. We show - based on results we first derive for the
case of $J^{1}\wedge J^{2}$ \- deformed CFTs - that all correlation functions
of such operators in the $J\bar{T}$ \- deformed CFT can be computed exactly in
terms of the correlation functions of the undeformed CFT and are crossing
symmetric in the plane limit. In particular, two and three-point functions are
simply given by the corresponding momentum-space correlator in the undeformed
CFT, with all dimensions replaced by particular momentum-dependent conformal
dimensions. Interestingly, scattering amplitudes off the near-horizon of
extremal black holes are known to take a strikingly similar form.

###### Contents

  1. 1 Introduction
  2. 2 $J^{1}\wedge J^{2}$ warmup
    1. 2.1 Brief review of the $J^{1}\wedge J^{2}$ deformation
    2. 2.2 Flow of the states and of the symmetry generators
    3. 2.3 From states and generators of symmetries to operators
    4. 2.4 Correlation functions and a bootstrap check
  3. 3 Primary operators in $J\bar{T}$ \- deformed CFTs
    1. 3.1 Setup and general remarks
    2. 3.2 CFT Ward identities in momentum space
    3. 3.3 A proposal for primary operators in $J\bar{T}$ \- deformed CFTs
    4. 3.4 Correlation functions
  4. 4 Discussion
  5. A $SL(2,\mathbb{R})$ and Kac-Moody generator identities
  6. B $J^{1}\wedge J^{2}$ \- deformed free bosons
    1. B.1 Classical analysis
    2. B.2 Quantum analysis
  7. C The unflowed $J\bar{T}$ algebra

##  1 Introduction

A long-standing challenge in holography has been to find the microscopic
description of _generic_ extremal and non-extremal black holes, whose near-
horizon region does not contain an AdS3 factor. One can obtain important clues
into the nature of the holographic dual by studying the asymptotic symmetries
of the near-horizon backgrounds of interest [1], as well as scattering
amplitudes [2, 3], which in principle give access to the symmetries [4] and,
respectively, the correlation functions of the dual theory.

To date, the most progress has been made in the case of extremal black holes,
whose near-horizon region universally contains [5] a factor known as warped
AdS3: a deformation of AdS3 that preserves $SL(2,\mathbb{R})\times U(1)$
isometry. The asymptotic symmetries of this space-time enhance the $U(1)$
factor to one copy of the Virasoro algebra, leading to the so-called Kerr/CFT
proposal [6], which states that the near-horizon dynamics of extremal black
holes is described by a chiral half of a two-dimensional CFT, where the
chirality is due to taking the strict extremal limit [7].

The study of scattering off the near-extremal black hole geometry naively
appears to corroborate this claim, since it leads to a (scalar) momentum-space
two-point function of the form [8]

$\mathcal{G}_{\mbox{\tiny{$T_{L,R}$}}}(p,\bar{p})\sim
T_{L}^{2h(\bar{p})-1}e^{-\frac{p}{2T_{L}}}\left|\Gamma\left(h(\bar{p})+\frac{ip}{2\pi
T_{L}}\right)\right|^{2}\;\times\;T_{R}^{2h(\bar{p})-1}e^{-\frac{\bar{p}}{2T_{R}}}\left|\Gamma\left(h(\bar{p})+\frac{i\bar{p}}{2\pi
T_{R}}\right)\right|^{2}$ (1.1)

which precisely corresponds to the momentum-space two-point function in a two-
dimensional CFT at left and, respectively, right-moving temperatures
$T_{L,R}$. Similar results hold for higher-point functions [9].

This conclusion may nevertheless be a bit too fast, because the operator
dimensions that appear in the above formula, which are extracted from the
solution to the scalar wave equation in the near-horizon geometry, depend
explicitly on the momentum, $\bar{p}$, along the $U(1)$ direction. This fact
immediately implies that the “CFT” in the Kerr/CFT correspondence cannot be a
standard, local CFT111 Consequently, the asymptotic symmetry results of [6]
(or, more rigorously, [10]) should not be interpreted as suggesting that the
dual theory is a standard CFT, but rather that non-local theories can posses
Virasoro symmetry, a fact that was recently proven in [11, 12]. . Indeed, a
detailed study of the near-horizon geometry [13] reveals that the holographic
dual is instead a two-dimensional analogue of a dipole theory [14], obtained
by deforming the CFT by a finely-tuned set of irrelevant operators that
preserve the left $SL(2,\mathbb{R})$ symmetry and lead to a UV-complete
theory. This structure of the irrelevant deformation implies that the
resulting theory - sometimes called a “dipole CFT” - is local and conformal on
the left, but non-local on the right. Operators in these theories are best
described in a mixed position - momentum basis, and the left-moving piece of
their correlation functions has a form dictated by the $SL(2,\mathbb{R})_{L}$
conformal symmetry; however, the left conformal dimensions generically depend
on the right-moving momentum [15].

The fact that the dual theory is non-local raises some immediate questions
regarding (1.1). While the left-moving piece of this correlator is meaningful
- since it corresponds to the Fourier transform of a local, conformal two-
point function (albeit with a dimension that depends on the momentum along the
other direction) - its right-moving piece, which resides entirely on the non-
local direction, appears to have no intrinsic meaning, since it can always be
rescaled by an arbitrary function of $\bar{p}$ [16]. The question that we
would like to address in this article is: to what extent is an expression of
the form (1.1) in such a non-local theory meaningful?

That this correlator may actually be meaningful is suggested by the example of
dipole-deformed field-theories, whose non-locality is extremely constrained by
a star product structure. Operators in this theory and the allowed
counterterms must respect this structure, which effectively makes the theory
renormalizable [17], if the original QFT was so. Thus, in presence of
additional structures that constrain the non-locality, one may hope to be able
to assign an unambiguous meaning to an expression such as (1.1).

The main goal of this article is to identify such additional structures for
the case of two-dimensional non-local QFTs that may model the microscopic
physics of generic extremal black holes. Specifically, we will be working with
the example of $J\bar{T}$ \- deformed CFTs [18] \- a class of _universal_
irrelevant deformations of two-dimensional CFTs by an operator that is
bilinear in the stress tensor and a $U(1)$ current, which lead to a UV-
complete QFT. These theories belong to the more general class of Smirnov-
Zamolodchikov irrelevant current-current deformations [19], of which the
$T\bar{T}$ deformation [19, 20] is a particularly rich and interesting example
[21, 22, 23, 24]. While $J\bar{T}$ \- deformed CFTs are not exactly a model
for the Kerr/CFT correspondence because the deformation is double-trace - and
thus corresponds to AdS3 with mixed boundary conditions [25] \- a single-trace
variant [26, 27] of this deformation is. Notwithstanding, $J\bar{T}$ \-
deformed CFTs do appear to posses the correct non-local QFT structures that
are relevant for the Kerr/CFT correspondence, in addition to being highly
tractable. The concrete question that we would like to address in this article
is whether a formula such as (1.1) can be made sense of in the context of
$J\bar{T}$ \- deformed CFTs.

This question has two aspects. The first is that one should find a basis of
operators for which the correlation functions are expected to take a simple
form. In usual CFTs, these are local operators - in fact, primary operators,
if we want the higher-point functions to also be nicely constrained. In
$J\bar{T}$ \- deformed CFTs, the primary constraint is easily imposed on the
local left-moving side and yields correlation functions that are consistent
with $SL(2,\mathbb{R})$ invariance, the only change being that the left
conformal dimensions and $U(1)$ charges are shifted from their undeformed
values $\tilde{h},\tilde{q}$ by a momentum-dependent contribution [28]

$h(\bar{p})=\tilde{h}+\lambda\tilde{q}\bar{p}+\frac{\lambda^{2}k}{4}\bar{p}^{2}\;,\;\;\;\;\;q(\bar{p})=\tilde{q}+\frac{\lambda
k}{2}\bar{p}$ (1.2)

where $\lambda$ is the deformation parameter. Our (more non-trivial) goal is
to find a similar constraint that fixes the right-moving dependence of the
correlators.

The other aspect concerns the method used to compute the correlation
functions. Given the definition of the deformation in terms of an irrelevant
flow, conformal perturbation theory comes in naturally, and this was the
method used in [28] to analyse $J\bar{T}$ correlators, and in [29] (see also
[30, 31, 32]) for the more involved $T\bar{T}$ case. However, note that the
correlation functions computed with this method are UV divergent, and so need
to be regulated and renormalized; however, it is _a priori_ not clear whether
any choice of UV regulator and which choices of counterterms would be allowed.

In this article, we approach the computation of $J\bar{T}$ correlators in a
different way, which circumvents the issue of UV divergences and understanding
what are the allowed counterterms. Instead, we rely on the flow equation with
respect to $\lambda$ satisfied by the energy eigenstates, the symmetry
generators and an appropriately-defined set of operators to construct the
correlation functions of interest. Our approach is similar in spirit to the
one used in [33] to discuss $T\bar{T}$ correlators, though our basis of
operators is different and the correlation functions we compute are manifestly
finite throughout, as well as fully explicit.

More concretely, to fix the right-moving part of the correlator, we use the
fact that $J\bar{T}$ \- deformed CFTs were recently shown to possess an
infinite-dimensional “pseudo-conformal” symmetry [34], implemented - at the
classical level - by field-dependent generalizations of conformal and affine
$U(1)$ transformations

$v\;\rightarrow\;\bar{f}(v)\;,\;\;\;\;\;\;\;\;v\equiv V-\lambda\phi$ (1.3)

where $V$ is the right-moving coordinate and $\phi$ is the bosonisation of the
$U(1)$ current (with its zero mode removed, if working on the cylinder). The
corresponding symmetry generators have been constructed in [34, 12] at the
classical level, and in [11] at the full quantum level. The quantum
construction relied on the existence of an alternate basis for these
symmetries, denoted as the “flowed” representation, in which the generators -
denoted with a $\widetilde{}$ \- are simply defined to flow in the same way as
the energy eigenstates. By construction, they satisfy a Virasoro-Kac-Moody
algebra [35]; more non-trivially, they can also be shown to be conserved. A
similar construction holds on the left-moving side. The generators of the left
conformal and right pseudo-conformal (1.3) symmetries are given in terms of
the flowed symmetry generators by

$L_{n}=\tilde{L}_{n}+\lambda
H_{R}\tilde{J}_{n}+\frac{\lambda^{2}kH_{R}^{2}}{4}\,\delta_{n,0}\;,\;\;\;\;\;\;\bar{L}_{n}=\tilde{\bar{L}}_{n}+\lambda:H_{R}\tilde{\bar{J}}_{n}:+\frac{\lambda^{2}kH_{R}^{2}}{4}\,\delta_{n,0}$

$J_{n}=\tilde{J}_{n}+\frac{\lambda
k}{2}H_{R}\,\delta_{n,0}\;,\;\;\;\;\;\;\;\bar{J}_{n}=\tilde{\bar{J}}_{n}+\frac{\lambda
k}{2}H_{R}\,\delta_{n,0}$ (1.4)

where the $:\;:$ denote normal ordering, i.e. $\tilde{\bar{J}}_{n}$ is to the
right of $H_{R}$ for $n>0$, and to the left for $n<0$. Note the above relation
resembles a ‘spectral flow’ by the right-moving Hamiltonian, $H_{R}$. The
algebra of these generators is Virasoro-Kac-Moody on the left and a non-linear
deformation of it on the right; also, the left and right generators do not
commute.

It is interesting to ask whether this deformed (Virasoro-Kac-Moody)2 symmetry
can help us fix the form of correlation functions, as it does in usual CFTs.

In two-dimensional CFTs, one can define primary operators - of dimension
$(h,\bar{h})$ \- either through their transformation properties under finite
conformal transformations $z\rightarrow
z^{\prime}(z),\bar{z}\rightarrow\bar{z}^{\prime}(\bar{z})$

$\mathcal{O}^{\prime}(z^{\prime},\bar{z}^{\prime})=(\partial_{z}z^{\prime})^{-h}(\partial_{\bar{z}}\bar{z}^{\prime})^{-\bar{h}}\mathcal{O}(z,\bar{z})$
(1.5)

or through their commutation relations with the Virasoro generators

$[L_{n},\mathcal{O}(z)]=h(n+1)z^{n}\mathcal{O}+z^{n+1}\partial_{z}\mathcal{O}\;,\;\;\;\;\;\;n\geq-1$
(1.6)

and similarly on the right. For $n=\pm 1,0$ this relation, together with the
$SL(2,\mathbb{R})$ invariance of the vacuum, completely fixes the form of two
and three-point functions, and highly constrains higher-point ones.

In $J\bar{T}$ \- deformed CFTs, it is not clear how to define an analogue of
(1.5). One may naively attempt to simply replace $\bar{z}$ in (1.5) by its
field-dependent counterpart $\bar{z}-\lambda\phi$; however, this is in tension
with the known fact that the left-moving dimensions depend on the right-moving
momentum as in (1.2), which forces one to treat at least the right-movers in
momentum space. As we will show, it is nevertheless possible to come rather
close to a definition of ‘primary’ operators in $J\bar{T}$ \- deformed CFTs
that is analogous to the momentum-space counterpart of (1.6). The correlation
functions of the resulting operators are completely fixed in terms of the
corresponding undeformed CFT correlators and, interestingly, take a form that
highly resembles (1.1). To what extent our proposal is ‘the’ correct
definition of primary operators in these non-local CFTs is left for future
investigations.

This article is organised as follows. In section 2, we study a simple toy
model for the $J\bar{T}$ deformation, which also exhibits two sets of symmetry
generators related by an operator-dependent spectral flow. For this example we
show, using only flow equations and the interplay of the symmetry generators,
that general deformed primary correlators are completely determined by their
undeformed counterparts. In section 3, we apply the same technique to compute
general correlation functions in $J\bar{T}$ \- deformed CFTs, making
appropriate adjustments for the non-locality of the model. We conclude with a
discussion in section 4. Various technical details of the symmetry algebras
are detailed in the appendices, as well as a very explicit realisation of the
toy model of section 2 in terms of deformed free bosons.

##  2 $J^{1}\wedge J^{2}$ warmup

In this section, we would like to “warm up” for the construction of primary
operators in $J\bar{T}$ \- deformed CFTs by studying a simple toy model that
also exhibits two possible bases for the symmetry generators, related by an
operator-dependent spectral flow. This model is the so-called $J^{1}\wedge
J^{2}$ deformation of a two-dimensional CFT, which is the simplest possible
Smirnov-Zamolodchikov deformation - built from two $U(1)$ currents - and also
corresponds to a two-dimensional toy model for a four-dimensional gauge theory
in presence of a $\theta$ term [36].

Since the $J^{1}\wedge J^{2}$ deformation is exactly marginal, all the
powerful tools of conformal symmetry are still applicable to the deformed
theory; in particular, it is very clear what the primary operators are.
Throughout this section, we will make an effort to treat this deformation in a
language that is also applicable to the $J\bar{T}$ deformation, which will
provide a very useful guiding principle for how to proceed when the conformal
symmetry is deformed to the non-local, field-dependent symmetries (1.3).

###  2.1 Brief review of the $J^{1}\wedge J^{2}$ deformation

We consider a one-parameter $(\lambda)$ family of two-dimensional CFTs that
posses two global $U(1)$ conserved currents, $J^{1,2}$. The actions describing
the various members of this family are related via the flow equation

$\frac{\partial S}{\partial\lambda}=-\int
d^{2}x\,e^{\alpha\beta}\left(J^{1}_{\alpha}J^{2}_{\beta}\right)_{\lambda}$
(2.1)

where the bilinear operator appearing on the right-hand side is defined via
point-splitting [19] and the current components are computed in the deformed
CFT. The deformed spectrum has been understood in [36, 37], and certain
aspects of correlation functions have been analysed in [29]. In this
subsection, we will review some of these results, in a language that parallels
the $J\bar{T}$ analysis of [11].

#### Classical analysis

It is useful to first understand the effect of the $J^{1}\wedge J^{2}$
deformation at the classical level. This is perhaps simplest to present in
Hamiltonian language. We thus consider a Hamiltonian density
$\mathcal{H}(\pi_{i},\phi_{i})$, which admits at least two $U(1)$ symmetries
that we associate to shifts in two scalars, $\phi_{1,2}$. As a result, the
Hamiltonian depends on these two canonical variables only through their
spatial derivatives, $\phi_{1,2}^{\prime}$. The two shift currents have
components

$J^{a}_{t}=\pi^{a}\,\sqrt{k}\;,\;\;\;\;\;\;J^{a}_{\sigma}=\partial_{\phi^{\prime}_{a}}\mathcal{H}\,\sqrt{k}\;,\;\;\;\;\;a=1,2$
(2.2)

where we have allowed for an arbitrary level222Even if $k$ is an anomaly
coeficient, for bosons it appears already at the level of the classical
Poisson brackets. This allows one to understand many properties of the
deformation with just a classical analysis. , $k$. We will also consider the
topologically conserved currents

$\tilde{J}^{a}_{t}=\phi^{\prime
a}\,\sqrt{k}\;,\;\;\;\;\;\;\;\tilde{J}^{a}_{\sigma}=\partial_{\pi_{a}}\mathcal{H}\,\sqrt{k}$
(2.3)

and will assume that in the undeformed CFT, with Hamiltonian density
$\mathcal{H}^{(0)}(\pi_{i},\phi_{i})$, the combinations
$J^{a}_{\alpha}\pm\tilde{J}^{a}_{\alpha}$ are (anti)chiral, which implies that

$\partial_{\pi^{a}}\mathcal{H}^{(0)}=\pi^{a}\;,\;\;\;\;\;\;\partial_{\phi^{\prime}_{a}}\mathcal{H}^{(0)}=\phi_{a}^{\prime}$
(2.4)

The flow equation obeyed by the deformed Hamiltonian reads (in the convention
$\epsilon_{t\sigma}=\epsilon^{\sigma t}=1$)

$\partial_{\lambda}\mathcal{H}=\epsilon^{\alpha\beta}J^{1}_{\alpha}J^{2}_{\beta}=k(\pi_{2}\partial_{\phi_{1}^{\prime}}\mathcal{H}-\pi_{1}\partial_{\phi_{2}^{\prime}}\mathcal{H})$
(2.5)

This equation can be solved by making the Ansatz

$\mathcal{H}(\lambda)=\widetilde{\mathcal{H}}(\lambda)+\frac{\lambda^{2}k^{2}}{2}(\pi_{1}^{2}+\pi_{2}^{2})+\lambda
k(\phi_{1}^{\prime}\pi_{2}-\phi_{2}^{\prime}\pi_{1})$ (2.6)

which implies that $\widetilde{\mathcal{H}}(\lambda)$ satisfies the equation

$\partial_{\lambda}\tilde{\mathcal{H}}=k\pi_{2}(\partial_{\phi_{1}^{\prime}}\widetilde{\mathcal{H}}-\phi_{1}^{\prime})-k\pi_{1}(\partial_{\phi_{2}^{\prime}}\widetilde{\mathcal{H}}-\phi_{2}^{\prime})$
(2.7)

This is solved by $\widetilde{\mathcal{H}}(\lambda)=\mathcal{H}^{(0)}$, using
the initial condition (2.4). One can easily check, following e.g. [38], that
the stress tensor computed from the resulting deformed Hamiltonian (2.6) is
both symmetric and traceless, and thus the deformed theory remains a CFT. In
this theory, it is useful we introduce the left/right Hamiltonian currents
($\mathcal{P}$ is the momentum density)

$\mathcal{H}_{L,R}\equiv\frac{\mathcal{H}\pm\mathcal{P}}{2}\;,\;\;\;\;\;\;\;\mathcal{P}=\sum_{i}\pi_{i}\phi^{\prime}_{i}$
(2.8)

We can now use the deformed Hamiltonian (2.6) and the definitions (2.2), (2.3)
to compute the components of the deformed conserved currents. A basis for the
currents that are now (anti)chiral is given by

$\mathcal{J}^{1}_{L,R}=\frac{\sqrt{k}}{2}(\pi_{1}\pm\phi^{\prime}_{1}\pm\lambda
k\pi_{2})\;,\;\;\;\;\;\;\;\mathcal{J}^{2}_{L,R}=\frac{\sqrt{k}}{2}(\pi_{2}\pm\phi^{\prime}_{2}\mp\lambda
k\pi_{1})$ (2.9)

where the above expressions represent their time components and, by
definition, $\mathcal{J}^{a}_{L,\sigma}=\mathcal{J}_{L,t}^{a}$,
$\mathcal{J}^{a}_{R,\sigma}=-\mathcal{J}^{a}_{R,t}$. The Poisson brackets of
the currents in this basis are diagonal and $\lambda$ \- independent

$\left\\{\mathcal{J}^{a}_{L}(\sigma),\mathcal{J}^{b}_{L}(\tilde{\sigma})\right\\}=-\left\\{\mathcal{J}^{a}_{R}(\sigma),\mathcal{J}^{b}_{R}(\tilde{\sigma})\right\\}=\frac{k}{2}\delta^{\prime}(\sigma-\tilde{\sigma})\;,\;\;\;\;\;\;\left\\{\mathcal{J}^{a}_{L}(\sigma),\mathcal{J}^{b}_{R}(\tilde{\sigma})\right\\}=0$
(2.10)

and their commutators with the deformed Hamiltonian currents
$\mathcal{H}_{L,R}(\lambda)$ yield the standard Witt-Kac-Moody algebra between
(anti)chiral currents in a CFT. It is also interesting to note that the
combinations

$\hat{\mathcal{H}}_{L}\equiv\mathcal{H}_{L}-\frac{1}{k}\sum_{a}(\mathcal{J}_{L}^{a})^{2}\;,\;\;\;\;\;\;\;\hat{\mathcal{H}}_{R}\equiv\mathcal{H}_{R}-\frac{1}{k}\sum_{a}(\mathcal{J}_{R}^{a})^{2}$
(2.11)

are independent of $\lambda$. Thus, they equal the corresponding quantities in
the undeformed CFT, which are nothing but the spectral-flow-invariant piece of
the Hamiltonian currents. This structure is reminiscent of that of the left-
moving Hamiltonian in $J\bar{T}$ \- deformed CFTs [11].

One final observation that will be useful shortly is that, if we define the
total currents
$\mathcal{J}^{a}_{\alpha}\equiv\mathcal{J}^{a}_{L,\alpha}+\mathcal{J}^{a}_{R,\alpha}$,
then the deforming operator can also be written in terms of them as

$\mathcal{O}_{J^{1}\wedge
J^{2}}=\epsilon^{\alpha\beta}J^{1}_{\alpha}J^{2}_{\beta}=2(\mathcal{J}_{L}^{1}\mathcal{J}^{2}_{R}-\mathcal{J}^{1}_{R}\mathcal{J}^{2}_{L})=\epsilon^{\alpha\beta}\mathcal{J}^{1}_{\alpha}\mathcal{J}^{2}_{\beta}$
(2.12)

#### Quantum analysis

We now move on to the quantum theory, and place the $J^{1}\wedge J^{2}$ \-
deformed CFT on a cylinder of radius $R$. Following [19], we consider
eigenstates $|n_{\lambda}\rangle$ of the energy and the charge, whose shift
and, respectively, winding charges are

$n^{a}=\int d\sigma\,\langle J^{a}_{t}\rangle\;,\;\;\;\;\;\;\;\;w_{a}=\int
d\sigma\,\langle\tilde{J}^{a}_{t}\rangle$ (2.13)

Since the deformation is integrable, it does not change the Hilbert space of
states on the cylinder, but it induces a flow of the energy eigenstates, of
the form

$\partial_{\lambda}|n_{\lambda}\rangle=\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}|n_{\lambda}\rangle$
(2.14)

where
$\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}=-\mathcal{X}^{\dagger}_{\mbox{\tiny{$J\bar{J}$}}}$
is a well-defined operator acting on the Hilbert space, which we will
determine shortly.

We would now like to understand how the energies and _chiral_ charges
$q^{a},\bar{q}^{a}$ \- i.e., the charges associated to the zero modes

$\mathcal{Q}^{a}=\int
d\sigma\,\mathcal{J}_{L}^{a}\;,\;\;\;\;\;\;\bar{\mathcal{Q}}^{a}=\int
d\sigma\,\mathcal{J}_{R}^{a}$ (2.15)

of the currents (2.9) - depend on $\lambda$. Using first order quantum-
mechanical perturbation theory and the factorization properties of the
Smirnov-Zamolodchikov operator in energy eigenstates, we find

$\displaystyle\partial_{\lambda}E_{n}^{\lambda}$ $\displaystyle=$
$\displaystyle\langle
n_{\lambda}|\partial_{\lambda}H|n_{\lambda}\rangle=R\left(\langle
n_{\lambda}|J^{1}_{\sigma}|n_{\lambda}\rangle\langle
n_{\lambda}|J^{2}_{t}|n_{\lambda}\rangle-\langle
n_{\lambda}|J^{1}_{t}|n_{\lambda}\rangle\langle
n_{\lambda}|J^{2}_{\sigma}|n_{\lambda}\rangle\right)$ (2.16)
$\displaystyle\hskip 14.22636pt=\;\frac{1}{R}\left[n_{2}(w_{1}+\lambda
kn_{2})-n_{1}(w_{2}-\lambda kn_{1})\right]$

where $\partial_{\lambda}H$ is the spatial integral of (2.5) over the circle
and we used (2.2) to compute the spatial components $J^{a}_{\sigma}$. Since
the shift and winding charges defined above are quantized, and thus cannot
flow with $\lambda$, this equation immediately integrates to the following
expression for the deformed energies

$E_{n}^{\lambda}=E_{n}^{(0)}+\frac{\lambda^{2}k}{2R}(n_{1}^{2}+n_{2}^{2})+\frac{\lambda}{R}(w_{1}n_{2}-w_{2}n_{1})$
(2.17)

While this expression is entirely analogous to (2.6), note that it does not
immediately follow from it.

The charges associated to the chiral conserved currents (2.9) are given by

$q^{a}=\tilde{q}^{a}+\frac{\lambda
k}{2}\epsilon^{ab}n_{b}\;,\;\;\;\;\;\;\bar{q}^{a}=\tilde{\bar{q}}^{a}-\frac{\lambda
k}{2}\epsilon^{ab}n_{b}$ (2.18)

where $\tilde{q}_{a},\tilde{\bar{q}}_{a}\equiv(n_{a}\pm w_{a})/2$ stand for
the undeformed (anti)chiral charges and $\epsilon_{12}=\epsilon^{12}=1$. Using
the state-operator correspondence to map the energies of eigenstates on the
cylinder to the conformal dimensions of local operators on the plane, we find
that the spectrum of left/right conformal dimensions in the $J^{1}\wedge
J^{2}$ \- deformed CFT depends on $\lambda$ as

$h=\tilde{h}+\lambda\epsilon_{ab}\tilde{q}^{a}n^{b}+\frac{\lambda^{2}k}{4}\,n_{a}n^{a}\;,\;\;\;\;\;\;\bar{h}=\tilde{\bar{h}}-\lambda\epsilon_{ab}\tilde{\bar{q}}^{a}n^{b}+\frac{\lambda^{2}k}{4}\,n_{a}n^{a}$
(2.19)

where $\tilde{h}$, $\tilde{\bar{h}}=(E_{n}^{(0)}\pm P_{n})R$ are the
undeformed conformal dimensions.

Thus, the effect of the $J^{1}\wedge J^{2}$ deformation on the spectrum of
energies or, equivalently, on the local operator dimensions is precisely that
of a simultaneous spectral flow in the two $U(1)$ directions, with charge-
dependent parameters $\eta^{a}=-\bar{\eta}^{a}=\lambda\epsilon^{ab}n_{b}$ that
are opposite on the left and the right. Note that the total momentum charge
$n^{a}=q^{a}+\bar{q}^{a}$ is unaffected. One can easily check that
$\hat{h}=h-(q_{a}q^{a})/k$ is left invariant, as expected.

###  2.2 Flow of the states and of the symmetry generators

We would now like to determine the form of the operator
$\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}$ entering the flow equation (2.14) for
the $J^{a}\wedge J^{b}$ \- deformed energy eigenstates at least in the
classical limit, in analogy with the results of [12] for $J\bar{T}$. Using
first order quantum-mechanical perturbation theory and assuming the CFT
degeneracies are dealt with, this operator can be read off from

$\partial_{\lambda}|n\rangle_{\lambda}=\sum_{m\neq n}\frac{\langle
m_{\lambda}|\partial_{\lambda}H|n_{\lambda}\rangle}{E_{n}^{\lambda}-E_{m}^{\lambda}}\,|m_{\lambda}\rangle$
(2.20)

where $\partial_{\lambda}H$ is the spatial integral of (2.5), performed on the
$t=0$ slice333As explained in [12], for $t\neq 0$ the flow operator will
receive additional contributions proportional to
$\partial_{\lambda}E_{n}^{\lambda}$.

$\partial_{\lambda}H=\int
d\sigma\,\epsilon^{\alpha\beta}J^{1}_{\alpha}J^{2}_{\beta}=\int
d\sigma\,\epsilon^{\alpha\beta}\mathcal{J}^{1}_{\alpha}\mathcal{J}^{2}_{\beta}$
(2.21)

where we used (2.12). To obtain a useful expression for
$\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}$, we follow the steps outlined in [33]
for the case of $T\bar{T}$. This involves splitting the two current insertions
in the deforming operator using a $\delta$ function, which is subsequently
rewritten in terms of the Green’s function on the cylinder, which satisfies

$\partial_{\sigma}G(\sigma-\tilde{\sigma})=\delta(\sigma-\tilde{\sigma})-\frac{1}{R}$
(2.22)

After these manipulations, we obtain

$\partial_{\lambda}H=\frac{2}{R}\,\epsilon_{ab}\mathcal{Q}^{a}\bar{\mathcal{Q}}^{b}-\partial_{t}\int
d\sigma
d\tilde{\sigma}\,G(\sigma-\tilde{\sigma})\mathcal{J}^{1}_{t}(\sigma)\mathcal{J}^{2}_{t}(\tilde{\sigma})$
(2.23)

where $\mathcal{Q}_{a},\bar{\mathcal{Q}}_{a}$ are the chiral charge operators
(2.15). The first term can also be written as a total time derivative by
bosonising the currents
$\mathcal{J}^{a}_{L,R}=\partial_{\pm}\varphi_{L,R}^{a}$ and noting that the
zero modes of the chiral scalars thus introduced satisfy

$[H,\varphi_{L,0}^{a}]=-i\mathcal{Q}^{a}\;,\;\;\;\;\;\;[H,\varphi_{R,0}^{a}]=-i\bar{\mathcal{Q}}^{a}$
(2.24)

Then,
$2\epsilon_{ab}\mathcal{Q}^{a}\bar{\mathcal{Q}}^{b}=[H,i\epsilon_{ab}(\varphi_{L,0}^{a}\bar{\mathcal{Q}}^{b}+\mathcal{Q}^{a}\varphi_{R,0}^{b})]=-i\frac{d}{dt}(\ldots)$.
The next step is to use an integral representation of the denominator in
(2.20) to rewrite it as

$\partial_{\lambda}|n_{\lambda}\rangle=-i\sum_{m\neq
n}\int_{-\infty}^{0}dt\,e^{t\epsilon}|m_{\lambda}\rangle\langle
m_{\lambda}|\partial_{\lambda}H(t)|n_{\lambda}\rangle$ (2.25)

where $\epsilon>0$ is an infinitesimal regulator. Performing the integral and
taking $\epsilon\rightarrow 0$, we find

$\partial_{\lambda}|n_{\lambda}\rangle=-i\sum_{m\neq
n}|m_{\lambda}\rangle\langle
m_{\lambda}\left|\frac{\epsilon_{ab}}{R}\left(\varphi_{L,0}^{a}\bar{\mathcal{Q}}^{b}+\mathcal{Q}^{a}\varphi_{R,0}^{b}\right)-\int
d\sigma
d\tilde{\sigma}\,G(\sigma-\tilde{\sigma})\mathcal{J}^{1}_{t}(\sigma)\mathcal{J}^{2}_{t}(\tilde{\sigma})\right|n_{\lambda}\rangle$
(2.26)

It is easy to argue [12] that the matrix elements of the first term will
vanish between different eigenstates, and thus will drop from the sum444Note
this derivation of the flow operator is significantly easier than its
$J\bar{T}$ [12] and $T\bar{T}$ (currently not understood) counterpart, where
the main difficulty lies is finding the projection of each of the two terms on
the energy eigenstates. . It is also easy to see, e.g. using a Fourier
decomposition, that the second term will only have non-zero matrix elements if
the eigenstates are different. Then, in the classical limit, the flow
‘operator’ for the energy eigenstates is simply given by

$\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}=i\int d\sigma
d\tilde{\sigma}G(\sigma-\tilde{\sigma})\mathcal{J}^{1}_{t}(\sigma)\mathcal{J}^{2}_{t}(\tilde{\sigma})$
(2.27)

We would now like to derive how the various currents flow with respect to
$\lambda$. For our purposes, it will be sufficient to understand this at the
classical level. If a classical current is left invariant by

$\tilde{\mathcal{D}}_{\lambda}\equiv\partial_{\lambda}-i\\{\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}},\;\cdot\;\\}$
(2.28)

then we will assume this implies that at the quantum level, it will flow in
the same way, (2.14), as the energy eigenstates. Introducing the total
momentum operator

$\mbox{\small{$\varPi$}}^{a}=\mathcal{Q}^{a}+\bar{\mathcal{Q}}^{a}=\sqrt{k}\int
d\sigma\,\pi^{a}$ (2.29)

we find that the various currents satisfy

$\tilde{\mathcal{D}}_{\lambda}\mathcal{J}_{L}^{a}=\frac{k}{2R}\epsilon^{ab}{\mbox{\small{$\varPi$}}}_{b}\;,\;\;\;\;\;\;\tilde{\mathcal{D}}_{\lambda}\mathcal{J}_{R}^{a}=-\frac{k}{2R}\epsilon^{ab}\mbox{\small{$\varPi$}}_{b}\;,\;\;\;\;\;\;\tilde{\mathcal{D}}_{\lambda}\mathcal{H}_{L}=\frac{1}{R}\epsilon_{ab}\mathcal{J}_{L}^{a}\mbox{\small{$\varPi$}}^{b}\;,\;\;\;\;\;\;\tilde{\mathcal{D}}_{\lambda}\mathcal{H}_{R}=-\frac{1}{R}\epsilon_{ab}\mathcal{J}_{R}^{a}\mbox{\small{$\varPi$}}^{b}$
(2.30)

Consequently, the following combinations

$\tilde{\mathcal{H}}_{L}\equiv\mathcal{H}_{L}-\frac{\lambda}{R}\epsilon_{ab}\mathcal{J}_{L}^{a}\mbox{\small{$\varPi$}}^{b}+\frac{\lambda^{2}k}{4R^{2}}(\mbox{\small{$\varPi$}}_{a})^{2}\;,\;\;\;\;\;\tilde{\mathcal{J}}_{L}^{a}\equiv\mathcal{J}_{L}^{a}-\frac{\lambda
k}{2R}\epsilon_{ab}\mbox{\small{$\varPi$}}^{b}$ (2.31)

$\tilde{\mathcal{H}}_{R}\equiv\mathcal{H}_{R}+\frac{\lambda}{R}\epsilon_{ab}\mathcal{J}_{R}^{a}\mbox{\small{$\varPi$}}^{b}+\frac{\lambda^{2}k}{4R^{2}}(\mbox{\small{$\varPi$}}_{a})^{2}\;,\;\;\;\;\;\tilde{\mathcal{J}}_{R}^{a}\equiv\mathcal{J}_{R}^{a}+\frac{\lambda
k}{2R}\epsilon_{ab}\mbox{\small{$\varPi$}}^{b}$ (2.32)

flow in the same way as the energy eigenstates. In terms of the
(dimensionless) Fourier modes of these generators, now seen as operators, we
have

$\tilde{J}_{m}^{a}=J_{m}^{a}-\frac{k\,\eta^{a}}{2}\delta_{m,0}\;,\;\;\;\;\;\tilde{L}_{m}=L_{m}-\eta_{a}J^{a}_{m}+\frac{k\eta_{a}\eta^{a}}{4}\delta_{m,0}$

$\tilde{\bar{J}}_{m}^{a}=\bar{J}_{m}^{a}-\frac{k\,\bar{\eta}^{a}}{2}\delta_{m,0}\;,\;\;\;\;\;\tilde{\bar{L}}_{m}=\bar{L}_{m}-\bar{\eta}_{a}\bar{J}^{a}_{m}+\frac{k\eta_{a}\eta^{a}}{4}\delta_{m,0}$
(2.33)

where we introduced the operator-dependent spectral flow parameter

$\eta_{a}=\lambda\epsilon_{ab}\mbox{\small{$\varPi$}}^{b}=-\bar{\eta}_{a}$
(2.34)

Thus, we find that in $J^{1}\wedge J^{2}$ \- deformed CFTs, there exist two
interesting bases for the symmetry generators, which are related by an
operator-dependent spectral flow. One basis consists of the generators
$L_{m},J_{m},\bar{L}_{m}$ and $\bar{J}_{m}$, which directly implement
conformal and affine $U(1)$ transformations. The other basis consists of the
generators $\tilde{L}_{m},\tilde{J}_{m}$ and their right-moving counterparts,
which have the property that they flow with $\lambda$ in the same way as the
energy eigenstates, namely

$|n_{\lambda}\rangle=U_{\lambda}|n_{0}\rangle\;,\;\;\;\;\;\;\tilde{L}_{m}(\lambda)=U_{\lambda}L_{m}(0)U^{-1}_{\lambda}$
(2.35)

where
$U_{\lambda}=\mathcal{P}e^{\int\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}}\,d\lambda}\,$
and $|n_{0}\rangle,L_{m}(0)$ are the energy eigenstates and, respectively, the
symmetry generators in the undeformed CFT. This structure exactly parallels
the one observed for $J\bar{T}$ -deformed CFTs [11].

In either basis, the symmetry algebra consists of two commuting copies of the
Virasoro-Kac-Moody algebra, consistently with the fact that the spectral flow
parameter, even though operator-valued, commutes with all the modes of the
symmetry currents. The Hilbert space is organised into highest-weight
representations of this algebra, which can be built with respect to either
$L_{m}$ or $\tilde{L}_{m}$. Note that primary states $|h_{\lambda}\rangle$
with respect to one basis will also be primary with respect to the other;
however, the descendants in one basis will generally be a linear combination
of descendants of the same level in the other. Note also that, due to (2.35),
the eigenvalues of the zero modes of the flowed generators are independent of
$\lambda$, and thus will equal those of the undeformed CFT

$\tilde{L}_{0}|n_{\lambda}\rangle=\tilde{h}|n_{\lambda}\rangle\;,\;\;\;\;\tilde{J}_{0}|n_{\lambda}\rangle=\tilde{q}|n_{\lambda}\rangle\;,\;\;\;\;\;\;\;\tilde{\bar{L}}_{0}|n_{\lambda}\rangle=\tilde{\bar{h}}|n_{\lambda}\rangle\;,\;\;\;\;\tilde{\bar{J}}_{0}|n_{\lambda}\rangle=\tilde{\bar{q}}|n_{\lambda}\rangle$
(2.36)

The flow (2.19), (2.18) of the conformal dimensions and chiral charges is then
explained by the relation (2.33) between the flowed and the standard conformal
generators, where the operator-dependent spectral flow parameter takes on its
eigenvalue corresponding to the state under consideration.

###  2.3 From states and generators of symmetries to operators

We would now like to compute correlation functions of primary operators in the
$J^{1}\wedge J^{2}$ \- deformed CFT. Of course, since the deformed theory is
still a CFT and the conformal dimensions of primary operators are known
(2.19), one can immediately write down the primary two- and three-point
functions in this theory up to an overall normalization. In this section, we
will show that it is in fact possible to determine _all_ the correlation
functions in this model exactly in terms of the correlation functions of the
undeformed CFT. That this should have been possible is implied by the results
of [29] on the flow of correlation functions in $J^{1}\wedge J^{2}$ \-
deformed CFTs; our method allows, in addition, to write down an entirely
explicit expression for the relation between the deformed and undeformed
correlators of primary operators.

Since this exercise is supposed to serve as warm-up for the more difficult
$J\bar{T}$ case, we would like to phrase our computations entirely in terms of
states and symmetry generators on the cylinder, which are quantities that we
have access to also in $J\bar{T}$ \- deformed CFTs. In particular, we will do
our best to avoid resorting to radial quantization or the state-operator
correspondence, which have not (yet) been formulated for these theories. The
plan of this section is to slowly build some intuition for our construction;
for the actual proposal, the reader can skip to (2.48).

An observable that can be straightforwardly constructed from the above
building blocks is the cylinder two-point function, seen as the overlap of an
in- and an out-state created by acting with a (primary) operator on the vacuum

$\mathcal{O}(w)|0\rangle=e^{wh}e^{e^{w}L_{-1}}|h\rangle$ (2.37)

In the above, $|h\rangle$ is a primary state on the cylinder, $w=\tau+i\sigma$
is the complex coordinate on the cylinder ($\tau=it$) and $L_{\pm 1}=-e^{\pm
w}\partial_{w}$, $L_{0}=-\partial_{w}$ are the global conformal generators on
the cylinder, which satisfy the $SL(2,\mathbb{R})$ algebra with the usual
conventions. There is a completely analogous contribution from the right-
moving side, which we do not write to avoid cluttering.

The equation above is derived in several steps: first, one uses the state-
operator correspondence to map the primary state on the cylinder to a primary
operator inserted at the origin of the plane
$|h\rangle\rightarrow\mathcal{O}_{pl}(0)|0\rangle$. This may be understood as
the definition of the primary operator. Next, one can define an operator at an
arbitrary location $z$ on the plane by translating555Note that this is not a
unitarily-implemented translation, even though the prefactor does take the
familiar form
$zL_{-1}+\bar{z}\bar{L}_{-1}=-(z\partial_{z}+\bar{z}\partial_{\bar{z}})=-(t\partial_{t}+x\partial_{x})=iHt-
iPx\;\;\;\;\mbox{with}\;\;\;\;H=i\partial_{t}\;,P=-i\partial_{x}\;,\;z=x+it,$
because $L_{-1}$ is not Hermitean in radial quantization, and thus $H,P$ are
not, either. In fact, a translation is not a symmetry of the CFT in radial
quantization, because the latter singles out a special point - the origin of
the plane - where operators are inserted. The fact that $\mathcal{O}_{pl}(z)$
takes the form quoted in the text is implied by the Ward identities (1.6)
associated with translations, which are independent of the quantization we
choose [39].  it with $L_{-1}^{pl}=-\partial_{z}$, i.e.
$\mathcal{O}_{pl}(z)=e^{zL_{-1}^{pl}}\mathcal{O}_{pl}(0)e^{-zL_{-1}^{pl}}$.
When acting on the vacuum, which is annihilated by the right $L_{-1}$ factor,
we obtain the plane analogue of (2.37),
$\mathcal{O}_{pl}(z)|0\rangle=e^{zL_{-1}^{pl}}|h\rangle$. The final step is to
map the resulting expression back to the cylinder via $z=e^{w}$, using the
fact that in radial quantization, $L_{-1}^{pl}$ is identified with its
counterpart on the cylinder, as well as the relation
$\mathcal{O}_{(cyl)}(w)=e^{wh}\mathcal{O}_{pl}(z)$, which follows from the
transformation properties (1.5) of primary operators under conformal
transformations. Of course, almost none of these steps would hold666Some of
the complications that one encounters are: i) The map from the cylinder to the
plane, assuming it can be well-defined, will be field-dependent (1.3), and
thus $\tau\rightarrow-\infty$ and $\tau=0$ on the cylinder will not map to a
fixed location and, respectively, a fixed circle on the plane. Relatedly,
dilatations correspond to a field-dependent symmetry in $J\bar{T}$. This makes
it difficult to formulate a state-operator correspondence precisely, even if
intuitively such a map may exist; ii) The generator of right-moving
translations on the plane - which are standard symmetries - does not appear to
be identified with $\bar{L}_{-1}$ on the cylinder, which implements a field-
dependent transformation. This can be easily established by noting that
$L_{-1}$ and $\bar{L}_{-1}$ on the cylinder do not commute (C.6) (except when
$R\rightarrow\infty$), whereas they obviously do on the plane; iii) Instead of
mapping back to the cylinder, one could simply attempt to compute correlation
functions on the plane. However, in this case it is not clear how to define
Hermitean conjugation, given the general lack of understanding of radial
quantization in this theory. In particular, since $L_{1}^{pl}$ implements a
field-dependent symmetry, it is not clear whether the coordinate appearing in
the out bra should rather be a field-dependent coordinate. Using a different
quantization, such as [40], does not appear to help, either.  in $J\bar{T}$ \-
deformed CFTs, but the final result is a well-defined expression on the
cylinder, which we could simply use it to _define_ the operators that we would
like to consider.

It may in fact be possible to give an interpretation to (2.37) directly on the
cylinder, by thinking of the primary state as being created by an operator
insertion at $\tau=-\infty$, i.e.
$|h\rangle=\lim_{\tau\rightarrow-\infty}e^{-h\tau}\mathcal{O}(\tau)$, and of
the exponentiated $L_{-1}$ as implementing a conformal transformation that
brings the point at $-\infty$ to finite distance. Again, one needs to be
careful about the fact that $L_{-1}$ is not a Hermitean operator; however, as
we show in appendix A, its action on a primary state can be reproduced by the
action of a combination of the Hermitean operators $L_{1}+L_{-1}$ and
$i(L_{1}-L_{-1})$, with appropriately chosen coefficients. While this picture
does help avoid the map to radial quantization on the plane when constructing
the action of these operators, it does not necessarily help justify a
definition of the form (2.37) for $J\bar{T}$ \- deformed CFTs777The reason is
that in $J\bar{T}$ \- deformed CFTs, the transformation taking the point at
$-\infty$ to finite distance is field-dependent, and thus one may wonder
whether one should replace the field-independent label $e^{w}$ in the
definition (2.37) by a field-dependent one. In fact, one can easily check that
an operator defined via (2.37) in $J\bar{T}$ \- deformed CFTs lacks a number
of desirable properties - for example, the action of the field-independent
operator $e^{\alpha L_{0}}$ does not correspond to the translation
$w\rightarrow w+\alpha$ in the label of the operator, as expected, except in
the $R\rightarrow\infty$ limit. For more details, see appendix C. .

To compute correlators, we will also need the expression for the out state

$\langle 0|\mathcal{O}(w)=e^{-wh}\langle h|e^{e^{-w}L_{1}}$ (2.38)

which follows from the simple fact that on the cylinder, hermitean
conjugation888In radial quantization on the plane, one has instead $\langle
0|\mathcal{O}(z)\equiv(e^{z^{\prime}L_{-1}}|h\rangle)^{\dagger}=z^{-2h}\langle
h|e^{L_{1}/z}$, using $z^{\prime}=1/z$, which follows from the action of
hermitean conjugation on the cylinder and the map $z=e^{w}$.  sends
$w=i(t+\sigma)\rightarrow-w$. Taking the overlap, one obtains

$\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle=e^{-hw_{12}}\langle
h|e^{e^{-w_{1}}L_{1}}e^{e^{w_{2}}L_{-1}}|h\rangle=e^{-hw_{12}}e^{-2h\ln(1-e^{-w_{12}})}=\left(2\sinh\frac{w_{12}}{2}\right)^{-2h}$
(2.39)

where we used the relation (A.5) and the primary condition. This is of course
the correct result on the cylinder, where the dimension is given by (2.19).

As advertised, a nice feature of this method is that it recasts the
computation of the cylinder two-point function only in terms of states and
symmetry generators, which are in principle also accessible in $J\bar{T}$ \-
deformed CFTs999It is clear that the overlap of two states of the form (2.37)
and (2.38) can be evaluated also in in $J\bar{T}$ \- deformed CFTs, since
algebra of the unflowed generators is known; see e.g. appendix C. If the
operators in question only depend on the left-moving coordinate, then only the
commutation relations of the left-moving generators, $L_{\pm 1}$, are
relevant. These are simply $SL(2,\mathbb{R})$ commutation relations, and one
can proceed exactly as above to find the deformed left-moving piece of the
two-point function, which has the expected form. We are however interested in
its behaviour on the right-moving side. There, one encounters the complication
that the algebra of $\bar{L}_{\pm 1,0}$ does not close, but instead generates
the entire Kac-Moody tower. While the calculation is still in principle
doable, we will see in the sequel that this proposal is in fact _not_
equivalent to the one we finally settle for in the case of $J\bar{T}$ \-
deformed CFTs. . On the down side, this method is limited to two-point
functions only. Also, it is not clear whether (2.37) provides a satisfactory
definition for the primary operators in $J\bar{T}$ \- deformed CFTs, since we
were unable to motivate this particular choice for the action of the operator.

To proceed, it is useful to perform the calculation of the two-point function
via overlaps in a slightly different way, which explicitly involves the flowed
generators (2.33). In terms of them, the two-point function (2.39) reads

$\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle=e^{-hw_{12}}\langle
h_{\lambda}|e^{e^{-w_{1}}L_{1}}e^{e^{w_{2}}L_{-1}}|h_{\lambda}\rangle=e^{-hw_{12}}\langle
h_{\lambda}|e^{e^{-w_{1}}(\tilde{L}_{1}+\eta^{a}\tilde{J}^{a}_{1})}e^{e^{w_{2}}(\tilde{L}_{-1}+\eta^{a}\tilde{J}^{a}_{-1})}|h_{\lambda}\rangle$
(2.40)

where $\eta^{a}=\lambda\epsilon^{ab}\mbox{\small{$\varPi$}}_{b}$ is the
spectral flow operator (2.34) and we have reinstated the label $\lambda$ on
the state, to emphasize its flow properties. Since $\eta^{a}$ commutes with
all the modes of the currents and, inside this correlator, it is acting on the
state $|h_{\lambda}\rangle$, then we can simply replace it by its eigenvalue
$\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}=\lambda\epsilon^{ab}n_{b}$ in this
state. We then observe that the states and all the operators in the above
expression flow with $\lambda$ in exactly the same way (2.35), and thus this
correlator will be identical to the corresponding one in the undeformed CFT.
In particular, its $\lambda$ dependence is entirely due to the explicit
$\lambda$ \- dependence of $\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}$. Rather
than evaluating this correlator in the undeformed CFT, we will prefer to work
with the flowed states and generators in the deformed theory, keeping in mind
that the two computations are simply related by conjugation by the unitary
operator $U_{\lambda}$, defined in (2.35).

The correlator can now be evaluated using the following BCH-type formula

$e^{e^{w}(\tilde{L}_{-1}+\eta\tilde{J}_{-1})}e^{-e^{w}\tilde{L}_{-1}}=e^{\eta\sum_{n=1}^{\infty}\frac{1}{n}e^{nw}\tilde{J}_{-n}}$
(2.41)

derived in appendix A, which also holds if we sum over several currents. Using
this, the overlap is

$\displaystyle\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle$
$\displaystyle=$ $\displaystyle e^{-hw_{12}}\langle
h_{\lambda}|e^{e^{-w_{1}}\tilde{L}_{1}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{1}}\tilde{J}_{n}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}e^{nw_{2}}\tilde{J}_{-n}}e^{e^{w_{2}}\tilde{L}_{-1}}|h_{\lambda}\rangle$
$\displaystyle=$ $\displaystyle
e^{-(h-\tilde{h})w_{12}}e^{\frac{k\eta_{\mathcal{O}}^{2}}{2}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{12}}}\langle\tilde{\mathcal{O}}(w_{1})e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}e^{nw_{2}}\tilde{J}_{-n}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{1}}\tilde{J}_{n}}\tilde{\mathcal{O}}(w_{2})\rangle$

where in the first line we have used the fact that hermitean conjugation sends
$w\rightarrow-w$, in addition to its usual effect on the generators, and in
the second line we used the BCH identity $e^{A}e^{B}=e^{B}e^{A}e^{[A,B]}$,
valid if $[A,B]\propto I$, to commute the exponentials of the currents. We
additionally modeled the action of $e^{e^{w}\tilde{L}_{-1}}$ on the state
$|h_{\lambda}\rangle$ by the action of an auxiliary operator
$\tilde{\mathcal{O}}(w)$, acting on the vacuum

$\tilde{\mathcal{O}}(w)|0_{\lambda}\rangle=e^{w\tilde{h}}e^{e^{w}\tilde{L}_{-1}}|h_{\lambda}\rangle$
(2.43)

This relation follows from the corresponding relation in the undeformed CFT,
by conjugation with $U_{\lambda}$. The operator $\tilde{\mathcal{O}}$ is
simply _defined_ via the relation

$\tilde{\mathcal{O}}(w)\equiv U_{\lambda}\mathcal{O}_{CFT}(w)U^{-1}_{\lambda}$
(2.44)

and need not correspond to a physical operator in the deformed CFT. Given this
definition, it follows that $\tilde{\mathcal{O}}(w)$ satisfies the same Ward
identities with the flowed currents as the corresponding quantities in the
undeformed CFT, namely

$[\tilde{J}_{n}^{a},\tilde{\mathcal{O}}(w)]=\tilde{q}^{a}\,e^{nw}\tilde{\mathcal{O}}(w)\;,\;\;\;\;\;\;\;n\geq
0$ (2.45)

This in turn implies that, for any coefficients $\alpha_{n}$,

$e^{\alpha_{n}\tilde{J}^{a}_{n}}\tilde{\mathcal{O}}(w)e^{-\alpha_{n}\tilde{J}^{a}_{n}}=e^{\alpha_{n}\tilde{q}^{a}e^{nw}}\tilde{\mathcal{O}}(w)\;,\;\;\;\;\;\;\;e^{-\alpha_{n}\tilde{J}^{a}_{-n}}\tilde{\mathcal{O}}(w)e^{\alpha_{n}\tilde{J}_{-n}^{a}}=e^{-\alpha_{n}\tilde{q}^{a}e^{-nw}}\tilde{\mathcal{O}}(w)$
(2.46)

where the second relation is the hermitean conjugate of the first (using
$\mathcal{O}^{\dagger}_{q}(w)=\mathcal{O}_{-q}(-w)$). Using these relations to
commute the $\tilde{J}_{n}$ and $\tilde{J}_{-n}$ factors past the adjacent
operators (and noting the one on the left has charge $-\tilde{q}$), the final
answer that we obtain for the correlator is

$\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle=e^{w_{12}(\tilde{h}-h)}e^{\frac{k\eta_{\mathcal{O}}^{2}}{2}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{12}}+2\eta_{\mathcal{O}}\tilde{q}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{12}}}\langle\tilde{\mathcal{O}}(w_{1})\tilde{\mathcal{O}}(w_{2})\rangle=\frac{e^{-hw_{12}}}{(1-e^{-w_{12}})^{2(\tilde{h}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}+\frac{k}{4}\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2})}}$
(2.47)

where we used $\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw}=-\ln(1-e^{-w})$ and the
fact that the $\tilde{\mathcal{O}}$ two-point function is identical to the one
in the undeformed CFT, where the operator dimension was $\tilde{h}$. Thus,
this method precisely reproduces the shift of the operator dimensions due to
the $J^{1}\wedge J^{2}$ deformation inside the correlation function.

We are now ready to present our general construction. We define a set of
“operators” $\tilde{\mathcal{O}}(w)$ as solutions to the flow equation

$\partial_{\lambda}\tilde{\mathcal{O}}(w)=[\mathcal{X}_{\mbox{\tiny{$J\bar{J}$}}},\tilde{\mathcal{O}}(w)]$
(2.48)

with the initial condition that $\tilde{\mathcal{O}}(w)_{\lambda=0}$ equal the
CFT primary operators inserted at a point $w$ on the cylinder. One should
think of these operators as being defined on the $t=0$ slice101010For the
original CFT operators, we should therefore write
$\mathcal{O}(w)=e^{wL_{0}}\mathcal{O}(0)e^{-wL_{0}}$. , despite the $w$ label
(which has no physical meaning, except at $\lambda=0$). This flow equation can
certainly be integrated to an equation of the form (2.44), though it will not
in general produce a local operator in the deformed CFT (see appendix B for an
explicit example). Nevertheless, the correlation functions111111Even if our
notation will be mostly euclidean, we will tacitly consider the analytic
continuation to Wightman functions, in terms of which the flow picture makes
sense.  of the flowed operators will be identical to those in the undeformed
CFT, by virtue of the fact that they obey the same flow equation as the
deformed states. Such operators have been previously considered in [33]. One
of their nice features is that they can clearly also be defined in $J\bar{T}$
\- deformed CFTs.

Our task is now to relate the correlation functions of the physical primary
operators, $\mathcal{O}(w)$, in the deformed CFT to those of the unphysical
operators $\tilde{\mathcal{O}}(w)$, which simply equal the original CFT
correlators. For this, we need to relate $\mathcal{O}(w)$ and
$\tilde{\mathcal{O}}(w)$ in the deformed theory. This is straightforward in
$J^{1}\wedge J^{2}$ \- deformed CFTs, which are conformally invariant, and so
primary operators must obey the usual conformal Ward identity121212This
follows from the usual relation (1.6) on the plane, using $z=e^{w}$ and
$\mathcal{O}_{pl}(z)=e^{-wh}\mathcal{O}_{cyl}(w)$.

$[L_{n},\mathcal{O}(w)]=e^{nw}[\,nh\mathcal{O}(w)+\partial_{w}\mathcal{O}(w)]\;,\;\;\;\;\;\;n\geq-1$

$[J_{n}^{a},\mathcal{O}(w)]=q^{a}e^{nw}\mathcal{O}(w)\;,\;\;\;\;n\geq 0$
(2.49)

where, according to (2.19), (2.18)

$h=\tilde{h}+\eta^{a}_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}_{a}+\frac{k\,\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}\;,\;\;\;\;\;\;q^{a}=\tilde{q}^{a}+\frac{k\,\eta^{a}_{\mbox{\tiny{$\mathcal{O}$}}}}{2}\;,\;\;\;\;\;\;\eta^{a}_{\mbox{\tiny{$\mathcal{O}$}}}=\lambda\epsilon^{ab}n_{b}$
(2.50)

and similarly for the right-movers.

On the other hand, the flow equation applied to the original Ward identity
implies that

$[\tilde{L}_{n},\tilde{\mathcal{O}}(w)]=e^{nw}[\,n\tilde{h}\tilde{\mathcal{O}}(w)+\partial_{w}\tilde{\mathcal{O}}(w)]\;,\;\;\;\;\;\;n\geq-1$
(2.51)

together with (2.45). Using the relationship (2.33) between the flowed and the
unflowed generators, it is easy to show that the relation between
$\mathcal{O}(w,\bar{w})$ and $\tilde{\mathcal{O}}(w,\bar{w})$ is given by

$\boxed{\mathcal{O}(w,\bar{w})=\epsilon^{A_{\mbox{\tiny{$\mathcal{O}$}}}w+B_{\mbox{\tiny{$\mathcal{O}$}}}\bar{w}}e^{\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\sum_{n=1}^{\infty}\frac{1}{n}e^{nw}\tilde{J}^{a}_{-n}+\bar{\eta}^{a}_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}e^{n\bar{w}}\tilde{\bar{J}}^{a}_{-n}}\tilde{\mathcal{O}}(w,\bar{w})\,e^{-\eta_{\mathcal{O}}^{a}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw}\tilde{J}^{a}_{n}-\bar{\eta}_{\mathcal{O}}^{a}\sum_{n=1}^{\infty}\frac{1}{n}e^{-n\bar{w}}\tilde{\bar{J}}^{a}_{n}}}$
(2.52)

where

$A_{\mbox{\tiny{$\mathcal{O}$}}}=\eta_{a}q^{a}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\tilde{J}^{a}_{0}-\frac{k}{4}(\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a})^{2}-\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\tilde{q}^{a}\;,\;\;\;\;\;B_{\mbox{\tiny{$\mathcal{O}$}}}=\bar{\eta}^{a}\bar{q}_{a}+\bar{\eta}_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\tilde{\bar{J}}^{a}_{0}-\frac{k}{4}(\bar{\eta}_{\mbox{\tiny{$\mathcal{O}$}}}^{a})^{2}-\bar{\eta}_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\tilde{\bar{q}}^{a}$
(2.53)

and we have reinstated the right-movers. The subscript on the operators
$A_{\mbox{\tiny{$\mathcal{O}$}}},B_{\mbox{\tiny{$\mathcal{O}$}}}$ indicate
that they depend on the charges of the particular operator under
consideration. One should also be careful to distinguish the operators
$\eta^{a}$ from their eigenvalue $\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}$,
with
$[\eta^{a},\mathcal{O}(w)]=\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{a}\mathcal{O}(w)$.
Of course, in the case at hand we have $\bar{\eta}_{a}=-\eta_{a}$, since the
spectral flow (2.33) acts in opposite ways on the left- and the right-movers.

One can easily check that $\mathcal{O}(w)$ satisfies the usual Hermiticity
condition $\mathcal{O}_{q}^{\dagger}(-w)=\mathcal{O}_{-q}(w)$

$\mathcal{O}_{q}^{\dagger}(-w)=e^{-\eta^{a}_{\mathcal{O}}\sum\frac{1}{n}e^{nw}J_{-n}}\tilde{\mathcal{O}}_{q}^{\dagger}(-w)e^{\eta^{a}_{\mathcal{O}}\sum\frac{1}{n}e^{-nw}J^{a}_{n}-A_{\mathcal{O}}^{\dagger}w}=e^{-A_{\mathcal{O}^{\dagger}}w}\mathcal{O}_{-q}(w)e^{-A^{\dagger}_{\mathcal{O}}w}=\mathcal{O}_{-q}(w)$
(2.54)

where we used the fact that the charges of $\mathcal{O}^{{\dagger}}$ are
opposite from those of $\mathcal{O}$ and that $A_{\mathcal{O}}$ is hermitean.
We have again dropped the right-movers, for simplicity.

Thus, we find a rather simple, closed-form relation between the primary
operators of interest and the auxiliary operators $\tilde{\mathcal{O}}$ that
we defined through the flow equation. In appendix B, we present explicit
expressions for both sets of operators for the case of $J^{1}\wedge J^{2}$ \-
deformed free bosons, which make it clear that $\tilde{\mathcal{O}}(w)$ are
non-local, and thus do not correspond to physical operators that we would like
to consider otherwise.

In view of our previous discussion, note that the left prefactor in the
relation between $\mathcal{O}$ and $\tilde{\mathcal{O}}$ above can be
understood by acting with both sides of equation (2.52) on the vacuum, case in
which it can be mapped to the relation (2.41) between $L_{-1}$ and
$\tilde{L}_{-1}$. Heuristically, if one thinks of the primary state as being
created by the insertion of a primary operator at $\tau=-\infty$ on the
cylinder, then the action of $\mathcal{O}(w)$ can be obtained by conformally
mapping the point at infinity to finite distance using the standard conformal
generator $L_{-1}$, whereas the action of $\tilde{\mathcal{O}}(w)$ is obtained
by using instead the flowed generator $\tilde{L}_{-1}$. However, this
intuition does not help in understanding the right-hand factor in (2.52), nor
why is the spectral flow operator evaluated to
$\eta_{\mbox{\tiny{$\mathcal{O}$}}}$, even when not acting with (2.52) on the
vacuum, from either the left or the right. Therefore, while intuitively useful
and correct in the particular computation of the two-point function above, the
state overlap picture fails to identify the general map between the two
operators131313This observation will be particularly relevant in $J\bar{T}$ \-
deformed CFTs, where the spectral flow operator does not commute with the
modes of the current, and therefore it is important to establish whether it is
the operator or its eigenvalue that appears in the definition of
$\mathcal{O}$. . In the following subsection, we use the relation (2.52) to
compute arbitrary correlators in the $J^{1}\wedge J^{2}$ \- deformed CFT.

###  2.4 Correlation functions and a bootstrap check

Given the expression (2.52) for the primary operators in the $J^{1}\wedge
J^{2}$ \- deformed CFT in terms of the auxiliary operators
$\tilde{\mathcal{O}}(w)$, whose correlation functions are known, computing
correlation functions of $\mathcal{O}(w)$ becomes simply a matter of properly
commuting the current modes through.

Let us start with the two-point function. Evaluating two copies of (2.52) in
the vacuum, we find

$\displaystyle\langle\mathcal{O}(w_{1})O(w_{2})\rangle$ $\displaystyle=$
$\displaystyle
e^{-\sum_{i=1}^{2}(\frac{k}{4}\eta_{i}^{2}+\eta_{i}^{a}\tilde{q}^{a}_{i})w_{i}}\times$
$\displaystyle\times\;\langle
e^{(\eta_{a}q^{a}_{1}+\eta_{1}^{a}\tilde{J}^{a}_{0})w_{1}}\tilde{\mathcal{O}}(w_{1})e^{-\eta_{1}^{a}\sum_{n=1}^{\infty}\frac{1}{n}e^{-nw_{1}}\tilde{J}^{a}_{n}}e^{(\eta_{a}q^{a}_{2}+\eta_{2}^{a}\tilde{J}^{a}_{0})w_{2}}e^{\eta_{2}^{a}\sum_{n=1}^{\infty}\frac{1}{n}e^{nw_{2}}\tilde{J}^{a}_{-n}}\tilde{\mathcal{O}}(w_{2})\rangle$

where we have used the fact that the vacuum is annihilated from the left by
$\tilde{J}^{a}_{n}$ with $n>0$ and from the right by $\tilde{J}_{-n}^{a}$.
Next, we note that in the above, the leftmost $\eta_{a}$ and
$\tilde{J}_{0}^{a}$ will evaluate to zero, since they are acting on the
vacuum; as for the middle ones, they are evaluated on the eigenstate created
by $\tilde{\mathcal{O}}(w_{2})|0\rangle$, so they evaluate to $\eta_{2}^{a}$
and, respectively, $\tilde{q}^{a}_{2}$. The remaining manipulations are
identical to those performed in the previous section, and we obtain141414Note
that the action of a charge operator, e.g. $J_{0}$, on an out state yields
_minus_ its charge, since $J_{0}|h,q\rangle=q|h,q\rangle\;\;\Rightarrow\langle
h,-q|J_{0}=q\langle h,-q|=-(-q)\langle h,-q|$.

$\langle\mathcal{O}(w_{1})O(w_{2})\rangle=e^{-\sum_{i=1}^{2}(\frac{k}{4}\eta_{i}^{2}+\eta^{a}_{i}\tilde{q}^{a}_{i})w_{i}+\eta_{2}^{a}(q^{a}_{2}+\tilde{q}^{a}_{2})w_{2}}e^{-(\frac{k}{2}\eta_{1}^{a}\eta_{2}^{a}+\eta_{1}^{a}\tilde{q}^{a}_{2}+\eta^{a}_{2}\tilde{q}^{a}_{1})\sum_{n}\frac{1}{n}e^{-nw_{12}}}\langle\tilde{\mathcal{O}}(w_{1})\tilde{O}(w_{2})\rangle$
(2.56)

Performing the sum and using charge conservation, which sets
$-\eta_{1}=\eta_{2}\equiv\eta_{\mbox{\tiny{$\mathcal{O}$}}}$ and
$-\tilde{q}_{1}=\tilde{q}_{2}\equiv\tilde{q}$, we find

$\langle\mathcal{O}(w_{1})O(w_{2})\rangle=\frac{e^{-(\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}+\frac{k}{4}\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2})w_{12}}}{(1-e^{-w_{12}})^{\frac{k}{2}\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}+2\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}}}\cdot\left(e^{\frac{w_{12}}{2}}-e^{-\frac{w_{12}}{2}}\right)^{-2\tilde{h}}=\left(e^{\frac{w_{12}}{2}}-e^{-\frac{w_{12}}{2}}\right)^{-2(\tilde{h}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}+\frac{k}{4}\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2})}$
(2.57)

which is the correct result, including all the normalizations. A similar
computation hols on the right.

We can use the same kind of manipulations to compute the three-point function

$\displaystyle\langle\mathcal{O}_{1}(w_{1})\mathcal{O}_{2}(w_{2})\mathcal{O}_{3}(w_{3})\rangle$
$\displaystyle=$ $\displaystyle
e^{-\sum_{i=1}^{3}(\frac{k}{4}\eta_{i}^{2}+\eta_{i}^{a}\tilde{q}^{a}_{i})w_{i}}\langle\tilde{\mathcal{O}}_{1}(w_{1})e^{-\eta^{a}_{1}\sum\frac{1}{n}e^{-nw_{1}}\tilde{J}^{a}_{n}}e^{(\eta_{a}q^{a}_{2}+\eta^{a}_{2}\tilde{J}^{a}_{0})w_{2}+\eta^{a}_{2}\sum\frac{1}{n}e^{nw_{2}}\tilde{J}_{-n}^{a}}\times$
(2.58)
$\displaystyle\times\;\tilde{\mathcal{O}}_{2}(w_{2})\,e^{-\eta_{2}^{a}\sum\frac{1}{n}e^{-nw_{2}}\tilde{J}^{a}_{n}}e^{(\eta_{a}q^{a}_{3}+\eta_{3}^{a}\tilde{J}^{a}_{0})w_{3}+\eta^{a}_{3}\sum\frac{1}{n}e^{nw_{3}}\tilde{J}^{a}_{-n}}\tilde{\mathcal{O}}_{3}(w_{3})\rangle$

Inside this correlator, the operators $\eta_{a},\tilde{J}_{0}^{a}$ have the
following eigenvalues, from left to right:
$\eta^{a}=\eta^{a}_{2}+\eta^{a}_{3}=-\eta^{a}_{1}$,
$\tilde{J}^{a}_{0}=-\tilde{q}^{a}_{1}$ and $\eta^{a}=\eta^{a}_{3}$,
$\tilde{J}^{a}_{0}=\tilde{q}^{a}_{3}$. Upon commuting the current modes
through, we find

$\displaystyle\langle\mathcal{O}_{1}(w_{1})\mathcal{O}_{2}(w_{2})\mathcal{O}_{3}(w_{3})\rangle$
$\displaystyle=$ $\displaystyle
e^{-\frac{k\eta_{1}^{2}}{4}w_{1}-\frac{k\eta_{2}^{2}}{4}w_{2}+\frac{k\eta_{3}^{2}}{4}w_{3}-\eta_{1}^{a}\tilde{q}^{a}_{1}w_{1}-\eta_{1}^{a}q^{a}_{2}w_{2}+\eta^{a}_{2}\tilde{q}^{a}_{3}w_{2}+\eta^{a}_{3}\tilde{q}^{a}_{3}w_{3}}(1-e^{-w_{12}})^{(\eta_{2}^{a}\tilde{q}^{a}_{1}+\eta^{a}_{1}\tilde{q}^{a}_{2}+\frac{k}{2}\eta_{1}^{a}\eta^{a}_{2})}\times$
(2.59)
$\displaystyle\times\;(1-e^{-w_{23}})^{\eta^{a}_{3}\tilde{q}^{a}_{2}+\eta^{a}_{2}\tilde{q}^{a}_{3}+\frac{k}{2}\eta^{a}_{2}\eta^{a}_{3}}(1-e^{-w_{13}})^{\eta^{a}_{3}\tilde{q}^{a}_{1}+\eta^{a}_{1}\tilde{q}^{a}_{3}+\frac{k}{2}\eta^{a}_{1}\eta^{a}_{3}}\langle\tilde{\mathcal{O}}_{1}(w_{1})\tilde{\mathcal{O}}_{2}(w_{2})\tilde{\mathcal{O}}_{3}(w_{3})\rangle$

The expected form of this correlator is that of a primary three-point function
on the cylinder, namely

$\displaystyle\langle\mathcal{O}_{1}(w_{1})\mathcal{O}_{2}(w_{2})\mathcal{O}_{3}(w_{3})\rangle$
$\displaystyle\sim$
$\displaystyle\frac{e^{w_{1}h_{1}+w_{2}h_{2}+w_{3}h_{3}}}{(e^{w_{1}}-e^{w_{2}})^{h_{1}+h_{2}-h_{3}}(e^{w_{2}}-e^{w_{3}})^{h_{2}+h_{3}-h_{1}}(e^{w_{1}}-e^{w_{3}})^{h_{1}+h_{3}-h_{2}}}$
(2.60)
$\displaystyle=\;\frac{e^{-w_{1}h_{1}+w_{2}(h_{1}-h_{3})+w_{3}h_{3}}}{(1-e^{-w_{12}})^{h_{1}+h_{2}-h_{3}}(1-e^{-w_{23}})^{h_{2}+h_{3}-h_{1}}(1-e^{-w_{13}})^{h_{1}+h_{3}-h_{2}}}$

where the dimensions are given by (2.19). It is easy to check that the
exponents of the $(1-e^{-w_{ij}})$ factors in (2.59) precisely match these,
since

$h_{1}+h_{2}-h_{3}=\tilde{h}_{1}+\tilde{h}_{2}-\tilde{h}_{3}-\eta_{1}^{a}\tilde{q}^{a}_{2}-\eta_{2}^{a}\tilde{q}^{a}_{1}-\frac{k}{2}\eta^{a}_{1}\eta^{a}_{2}$
(2.61)

and cyclic permutations thereof, where we used charge conservation
$\tilde{q}^{a}_{3}=-\tilde{q}^{a}_{1}-\tilde{q}^{a}_{2}$,
$\eta^{a}_{3}=-\eta^{a}_{1}-\eta^{a}_{2}$. Moreover, it turns out that all the
prefactors in (2.59) combine precisely into the numerator of (2.60), with the
end result that the three-point function of primaries in the $J^{1}\wedge
J^{2}$ \- deformed CFT is given by (2.60) times the OPE coefficient,
$\tilde{C}_{123}$, in the _undeformed_ CFT.

We thus find that, while the conformal dimensions and charges shift as in
(2.19), (2.18), the OPE coefficients, which contain the dynamical information
of the theory, are _unchanged_ by the deformation

$C_{ABC}=\tilde{C}_{ABC}$ (2.62)

Finally, we work out the four-point function, with the result

$\langle\mathcal{O}_{1}(w_{1})\mathcal{O}_{2}(w_{2})\mathcal{O}_{3}(w_{3})\mathcal{O}_{4}(w_{4})\rangle=e^{P}\prod_{i<j}(1-e^{-w_{ij}})^{\eta_{j}^{a}\tilde{q}^{a}_{i}+\eta^{a}_{i}\tilde{q}^{a}_{j}+\frac{k}{2}\eta^{a}_{i}\eta^{a}_{j}}\langle\tilde{\mathcal{O}}_{1}(w_{1})\tilde{\mathcal{O}}_{2}(w_{2})\tilde{\mathcal{O}}_{3}(w_{3})\tilde{\mathcal{O}}_{4}(w_{4})\rangle$
(2.63)

where the prefactor comes from evaluating the operators $A_{i}$ inside the
correlator, and reads

$P=-\sum_{i}(\frac{k}{4}\eta_{i}^{2}+\eta_{i}^{a}\tilde{q}^{a}_{i})w_{i}-(\eta^{a}_{1}q^{a}_{2}+\eta_{2}^{a}\tilde{q}^{a}_{1})w_{2}+[(\eta^{a}_{3}+\eta^{a}_{4})q^{a}_{3}+\eta^{a}_{3}(\tilde{q}^{a}_{3}+\tilde{q}^{a}_{4})]w_{3}+(\eta^{a}_{4}q^{a}_{4}+\eta^{a}_{4}\tilde{q}^{a}_{4})w_{4}$
(2.64)

Using this, one can check that the expression for the four-point function can
be simplified to

$\langle\mathcal{O}_{1}(w_{1})\mathcal{O}_{2}(w_{2})\mathcal{O}_{3}(w_{3})\mathcal{O}_{4}(w_{4})\rangle=\prod_{i<j}\left(e^{\frac{w_{ij}}{2}}-e^{-\frac{w_{ij}}{2}}\right)^{\frac{2}{k}(q_{i}q_{j}-\tilde{q}_{i}\tilde{q}_{j})}\langle\tilde{\mathcal{O}}_{1}(w_{1})\tilde{\mathcal{O}}_{2}(w_{2})\tilde{\mathcal{O}}_{3}(w_{3})\tilde{\mathcal{O}}_{4}(w_{4})\rangle$
(2.65)

It is also easy to see, using charge conservation, that any correlation
function of the primary operators $\mathcal{O}(w_{i})$ will be related through
a factor of exactly the same form to the corresponding correlator in the
undeformed CFT. Thus, the correlation functions of primary operators in the
deformed theory can be rather trivially expressed in terms of the undeformed
correlators.

Note that the expression (2.65) for the four-point function is crossing
symmetric, assuming crossing symmetry of the original CFT correlator. It is
interesting to rewrite this result in the language of conformal partial waves,
using the fact that in a two-dimensional CFT that possesses an affine $U(1)$
symmetry, the Virasoro-Kac-Moody conformal partial wave151515Here,
$z_{i},h_{i},q_{i}$ are the positions, dimensions and respectively $U(1)$
charges of the external operators, $h,q$ are the dimension and charge
(constrained by conservation) of the primary on whose family we project, and
$c$ is the central charge of the CFT. $\mathcal{W}_{h,q}(z_{i},h_{i},q_{i},c)$
can be written as a spectral-flow-invariant Virasoro block contribution
$\mathcal{V}_{\hat{h}}$ times an affine $U(1)$ block, $\mathcal{U}$

$\mathcal{W}_{h,q}(z_{i},h_{i},q_{i},c)=\mathcal{U}(z_{i},q_{i})\,\mathcal{V}_{\hat{h}}(z_{i},\hat{h}_{i},c-1)$
(2.66)

This was shown in [43] for the case of a neutral exchanged operator, and in
[41, 42] in the general case. In the above, $\hat{h}_{i}=h_{i}-q_{i}^{2}/k$
are the spectral-flow invariant pieces of the conformal dimensions and the
affine $U(1)$ block is given by

$\mathcal{U}(z_{i},q_{i})=\prod_{i<j}z_{ij}^{\frac{2q_{i}q_{j}}{k}}$ (2.67)

In a given channel, the four-point function can be written as an infinite sum
over conformal partial waves corresponding to the particular Virasoro-Kac-
Moody representations being exchanged

$\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\mathcal{O}_{3}(z_{3})\mathcal{O}_{4}(z_{4})\rangle=\sum_{h,q}C_{12h}C_{34h}\mathcal{W}_{h,q}(z_{i},h_{i},q_{i},c)$
(2.68)

Since the effect of the $J^{1}\wedge J^{2}$ deformation is to induce a charge-
dependent spectral flow transformation that leaves $\hat{h}$ invariant, the
only change in the Virasoro-Kac-Moody blocks will come from the change in
$\mathcal{U}(z_{i},q_{i})$, which only depends on the charges of the external
operators. Thus, the change in the conformal partial waves is

$\mathcal{W}_{h,q}(z_{i},h_{i},q_{i},c)\;\;\;\stackrel{{\scriptstyle\mbox{\tiny{$J^{1}\\!\\!\\!\wedge\\!\\!\\!J^{2}$}}}}{{\longrightarrow}}\;\;\;\prod_{i<j}z_{ij}^{\frac{2q_{i}q_{j}}{k}-\frac{2\tilde{q}_{i}\tilde{q}_{j}}{k}}\;\mathcal{W}_{h,q}(z_{i},h_{i},q_{i},c)$
(2.69)

irrespectively of which operator is being exchanged. Mapping this result from
the plane to the cylinder and using the fact that the OPE coefficients are
unchanged by the deformation, we can immediately reproduce the change (2.65)
in the four-point function. This is another way to check that crossing
symmetry is satisfied. The bootstrap equations of the deformed CFT are thus
trivially solved, given their solution in the undeformed CFT [41].

##  3 Primary operators in $J\bar{T}$ \- deformed CFTs

Armed with our understanding of the $J^{1}\wedge J^{2}$ \- deformed primaries
in a language that is in principle generalizable to $J\bar{T}$, we would now
like to present our proposal for defining primary operators in $J\bar{T}$ \-
deformed CFTs. To set up the stage, we start with a few general remarks about
the similarities and differences between the $J^{1}\wedge J^{2}$ and
$J\bar{T}$ cases.

###  3.1 Setup and general remarks

Let us summarize our current understanding of $J\bar{T}$ \- deformed CFTs that
is relevant for this question. On the cylinder, the $J\bar{T}$ \- deformed
energy eigenstates flow according to

$\partial_{\lambda}|n_{\lambda}\rangle=\mathcal{X}_{J\bar{T}}\,|n\rangle_{\lambda}$
(3.1)

where $\lambda$ now represents the $J\bar{T}$ flow parameter and
$\mathcal{X}_{J\bar{T}}$ is a presumably well-defined operator that was worked
out in [12] at the classical level, and discussed in [11] at the quantum
level. Given $\mathcal{X}_{J\bar{T}}$, one can define two commuting sets of
Virasoro-Kac-Moody generators $\tilde{L}_{n},\tilde{J}_{n}$ and
$\tilde{\bar{L}}_{n},\tilde{\bar{J}}_{n}$ via the flow equation

$\partial_{\lambda}\widetilde{L}_{n}=[\mathcal{X}_{J\bar{T}},\widetilde{L}_{n}]$
(3.2)

etc., which can be shown to be conserved [11]. This flow equation implies that
primary states in the undeformed CFT will flow to primaries with respect to
the $\tilde{L}_{n}$ and that the eigenvalues of $\tilde{L}_{0}$, etc., are
independent of $\lambda$ [35], and thus equal the corresponding eigenvalues in
the undeformed CFT

$\tilde{L}_{0}|h_{\lambda}\rangle=\tilde{h}|h_{\lambda}\rangle\;,\;\;\;\;\;\;\tilde{J}_{0}|h_{\lambda}\rangle=\tilde{q}|h_{\lambda}\rangle\;,\;\;\;\;\;\;\;\tilde{\bar{L}}_{0}|h_{\lambda}\rangle=\tilde{\bar{h}}|h_{\lambda}\rangle\;,\;\;\;\;\;\;\tilde{\bar{J}}_{0}|h_{\lambda}\rangle=\tilde{\bar{q}}|h_{\lambda}\rangle$
(3.3)

Thus, in terms of the flowed generators, the structure of the Hilbert space
looks the same as that of the undeformed CFT: primary states have the same
dimensions as in the seed CFT, and descendant states can be built by acting
with $\widetilde{L}_{-n}$, etc. on them.

As explained in [11], the generators that implement (pseudo)conformal and
affine $U(1)$ transformations in the deformed theory are given by

$L_{n}=\widetilde{L}_{n}+\lambda
H_{R}\widetilde{J}_{n}+\frac{\lambda^{2}kH_{R}^{2}}{4}\delta_{n,0}\;,\;\;\;\;\;\;J_{n}=\tilde{J}_{n}+\frac{\lambda
k}{2}H_{R}\,\delta_{n,0}$

$\bar{L}_{n}=\widetilde{\bar{L}}_{n}+\lambda\\!:\\!H_{R}\widetilde{\bar{J}}_{n}\\!:+\frac{\lambda^{2}kH_{R}^{2}}{4}\delta_{n,0}\;,\;\;\;\;\;\;\;\bar{J}_{n}=\tilde{\bar{J}}_{n}+\frac{\lambda
k}{2}H_{R}\,\delta_{n,0}$ (3.4)

whose relation to the $\tilde{L}_{n}$ resembles a spectral flow whose
parameter is proportional to the right-moving Hamiltonian. More specifically,
the left-moving generators $L_{n},J_{n}$ implement usual conformal and affine
$U(1)$ transformations, whereas the right-moving generators
$\bar{L}_{n},\bar{J}_{n}$ (with the exception of the global generators
$\bar{L}_{0},\bar{J}_{0}$) implement field-dependent conformal (1.3) and
affine $U(1)$ transformations. This structure is entirely analogous to the one
we have uncovered in $J^{1}\wedge J^{2}$ \- deformed CFTs; note, however, that
now the spectral flow “parameter” does not commute with the modes of the
right-moving currents, which adds a layer of complication to the problem.

Given our understanding of the states and symmetry generators on the cylinder,
the question is how to define a physical primary operator, $\mathcal{O}$, and
construct its correlation functions. Since the theory is local and conformal
on the left, $\mathcal{O}$ should obey the standard primary condition with
respect to the left-moving generators $L_{n},J_{n}$. The non-trivial part of
our task is to find an appropriate notion of a “primary condition” also on the
non-local right-moving side. This problem does not have a counterpart in
$J^{1}\wedge J^{2}$ \- deformed CFTs, where the Ward identities that primary
operators must satisfy are simply determined by the fact that the deformed
theory stays a CFT.

Since $H_{R}$ \- the generator of right-moving translations - enters the
relation (3.4) between the two sets of generators, it makes sense to choose a
basis of operators that diagonalizes its action, i.e. work in momentum space.
This is also instructed by the non-locality of the deformed CFT, and in
particular the fact that the left primary dimensions depend on the eigenvalue
of this operator as in (1.2). Thus, a first difference with the $J^{1}\wedge
J^{2}$ \- case is that we are forced to work in momentum space, at least as
far as the right-movers are concerned. For uniformity reasons, we will choose
this basis on both sides.

As in the previous section, we will construct candidate primary operators,
$\mathcal{O}$, based on an auxiliary operator $\tilde{\mathcal{O}}$ that is
defined to flow with the $J\bar{T}$ parameter in the same way as the energy
eigenstates. As before, we will attempt to relate the vacuum correlation
functions of the candidate $\mathcal{O}$ to those of $\tilde{\mathcal{O}}$,
which are identical to the correlation functions in the undeformed CFT, as
implied by the flow equation. Note, however, that now the ‘flowed vacuum’
state $|0_{\lambda}\rangle=U_{\lambda}|0\rangle_{\lambda=0}$, to which the
above argument applies, is not annihilated by one of the global
$SL(2,\mathbb{R})_{L}$ generators on the cylinder161616This problem does not
appear in $J^{1}\wedge J^{2}$ \- deformed CFTs, because the $\eta^{a}$ in
(2.34) does annihilate the flowed vacuum. . More specifically, the flow
equation implies that $\tilde{L}_{-1}|0_{\lambda}\rangle=0$ which, in terms of
the unflowed generators, translates into

$L_{-1}|0_{\lambda}\rangle=\lambda
J_{-1}H_{R}|0_{\lambda}\rangle=\frac{2}{\lambda
k}\left(R-\sqrt{R^{2}+\frac{\lambda^{2}kc}{24}}\right)J_{-1}|0_{\lambda}\rangle$
(3.5)

where we used the known result [27] for the flowed finite-size energy
eigenvalues (in this case, the ground state). Thus, on the cylinder, the
flowed vacuum is not $SL(2,\mathbb{R})$ invariant. While we see no obvious
reason that an $SL(2,\mathbb{R})$ \- invariant vacuum should not exist, this
state will clearly be different from $|0_{\lambda}\rangle$.

The reason we would like the vacuum to be annihilated by the
$SL(2,\mathbb{R})_{L}$ generators is that only then do we expect the primary
correlation functions to take the standard form dictated by conformal
symmetry. It is therefore important to construct this state of _a priori_
greater physical interest in $J\bar{T}$ \- deformed CFTs on the cylinder171717
By definition, this state would be annihilated by $L_{-1}$, which would
translate, as above, into an explicitly $\lambda$ \- dependent relation
between the action of $\tilde{L}_{-1}$ and $\tilde{J}_{-1}$ on it. This in
turn implies that this state will not satisfy a simple flow equation involving
$\mathcal{X}_{J\bar{T}}$, though one may be able to explicitly write it in the
basis (3.1), using $\lambda$ \- dependent coefficients. This state would also
need to satisfy an appropriate constraint with respect to the global right-
moving generators, whose algebra - which is not $SL(2,\mathbb{R})$ \- is
listed in appendix C.. Rather than addressing this interesting problem - which
appears somewhat complicated - we will simply avoid it by taking the
$R\rightarrow\infty$ limit, which forces the two candidate vacuum states to
coincide.

Our approach can thus be summarized as follows: given that out best
understanding of the states, symmetry generators and their flow is on the
cylinder, we will present our general construction of the candiate primary
operators and their correlation functions in this setting. However, this
construction is only expected to yield results consistent with
$SL(2,\mathbb{R})$ invariance (and its right-moving analogue) in the
$R\rightarrow\infty$ limit, where one is effectively working on the plane.
This limit will turn out to also resolve a consistency problem that we will
encounter along the way.

As we already mentioned, due to the non-locality of the model, we need to work
in momentum space. We will thus start by reviewing some basic results about
momentum-space Ward identities for primary fields in a CFT, before presenting
our proposal for the momentum-space primary operators.

###  3.2 CFT Ward identities in momentum space

Let us start by introducing the momentum-space operators

$\mathcal{O}(p)=\int dw\,e^{-pw}\mathcal{O}(w)$ (3.6)

where $w=i(t+\sigma)$ corresponds to a Lorentzian coordinate on the cylinder.
There is, as always, also a right-moving contribution, with mometum $\bar{p}$,
so that the spatial momentum $p-\bar{p}$ is quantized. Since all formulae in
this subsection are identical for the right-movers, we will omit writing them
explicitly.

Momentum-space CFT correlation functions can be computed by either taking the
Fourier transform of the corresponding position-space Wightman functions [44,
46], or by solving the conformal Ward identities directly in momentum space
[45]. While these studies were concerned with momentum-space correlators of
the CFT on the plane, for the problem at hand we are interested in the form of
the momentum-space conformal Ward identities on the cylinder. These can be
obtained by Fourier-transforming the position-space commutators (2.49) of the
$L_{n},J_{n}$, with the result

$[L_{n},\mathcal{O}(p)]=(n(h-1)+pR)\mathcal{O}\left(p-\frac{n}{R}\right)\;,\;\;\;\;\;\;[J_{n},\mathcal{O}(p)]=q\,\mathcal{O}\left(p-\frac{n}{R}\right)$
(3.7)

Despite their unusual form, these can in principle be used to fix the form of
the correlation functions181818For example, for a two-point function, they
give a constraint of the form  $\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle
0|L_{1}L_{-1}\mathcal{O}_{1}(p_{1})\mathcal{O}_{2}(p_{2})|0\rangle=[(1-h_{1}+p_{1}R)(h_{1}+p_{1}R)+(1-h_{2}+p_{2}R)(h_{2}+p_{2}R)]\mathcal{O}_{1}(p_{1})\mathcal{O}_{2}(p_{2})+$
$\displaystyle+\;(1-h_{1}+p_{1}R)(h_{2}-1+p_{2}R)\mathcal{O}(p_{1}+1/R)\mathcal{O}(p_{2}-1/R)+(1-h_{2}+p_{2}R)(h_{1}+p_{1}R-1)\mathcal{O}(p_{1}-1/R)\mathcal{O}(p_{2}+1/R)$
where two generator insertions are necessary in order to respect momentum
conservation. A solution to the above constraint is
$\langle\mathcal{O}_{1}(p_{1})\mathcal{O}_{2}(p_{2})\rangle\sim e^{\pm i\pi
R(p_{1}-p_{2})/2}\Gamma(h_{1}+p_{1}R)\Gamma(h_{2}+p_{2}R)\;,\;\;\;\mbox{where}\;\;\;p_{1}+p_{2}=0$
(3.9) which is proportional to the Fourier transform of the position space
two-point function..

While it would certainly be interesting to further explore the properties of
the solutions to these Ward identities on the cylinder, here we are mostly
interested in the limit $R\rightarrow\infty$, where they should reduce to the
well-known momentum-space Ward identities on the plane. To show how this comes
about, we need to relate the (dimensionless) $SL(2,\mathbb{R})$ generators on
the cylinder, $L_{n}=-R\,e^{\frac{nw}{R}}\partial_{w}$ for $n=\pm 1,0$ to the
planar ones, $L_{n}^{pl}=-w^{n+1}\partial_{w}$, as $R\rightarrow\infty$. This
can be simply achieved by expanding the cylinder generators at large $R$

$L_{0}=-R\,\partial_{w}=RL_{-1}^{pl}\;,\;\;\;\;\;L_{\pm 1}=-R\,\partial_{w}\mp
w\partial_{w}-\frac{w^{2}}{2R}\partial_{w}+\mathcal{O}(1/R^{2})$ (3.10)

The inverse relation reads

$L_{-1}^{pl}=\frac{1}{R}L_{0}\;,\;\;\;\;\;L_{0}^{pl}=\frac{1}{2}(L_{1}-L_{-1})+\mathcal{O}(1/R^{2})\;,\;\;\;\;\;\;L_{1}^{pl}=R(L_{1}+L_{-1}-2L_{0})+\mathcal{O}(1/R)$
(3.11)

Expanding the momentum-space commutators (3.7) of the operator
$\mathcal{O}(p)$ with the $L_{n}$, we find

$[L_{-1}^{pl},\mathcal{O}(p)]=p\,\mathcal{O}\;,\;\;\;\;\;\;[L_{0}^{pl},\mathcal{O}(p)]=(h-1)\mathcal{O}-p\,\partial_{p}\mathcal{O}+\mathcal{O}(1/R^{2})$

$[L_{1}^{pl},\mathcal{O}(p)]=p\,\partial_{p}^{2}\mathcal{O}+2(1-h)\partial_{p}\mathcal{O}+\mathcal{O}(1/R^{2})$
(3.12)

exactly as expected. One can check that the solution to these Ward identities,
e.g. for the two-point function, agrees with the $R\rightarrow\infty$ limit of
its cylinder counterpart (3.9).

###  3.3 A proposal for primary operators in $J\bar{T}$ \- deformed CFTs

We are now ready to construct a set of momentum-space operators in $J\bar{T}$
\- deformed CFTs that come as close as possible to being primary, in the sense
of (3.7). As we explained, we will start by working on the cylinder and then
take the $R\rightarrow\infty$ limit in which, at least for a CFT, the
momentum-space Ward identities and their solutions seamlessly translate from
the cylinder to the plane.

We start by introducing again a set of operators
$\tilde{\mathcal{O}}(w,\bar{w})$ that formally satisfy the flow equation

$\partial_{\lambda}\tilde{\mathcal{O}}(w,\bar{w})=[\mathcal{X}_{J\bar{T}},\tilde{\mathcal{O}}(w,\bar{w})]$
(3.13)

with the initial condition that they equal the local operator
$\mathcal{O}(w,\bar{w})$ in the undeformed CFT. This definition automatically
implies that the correlation functions of $\tilde{\mathcal{O}}(w,\bar{w})$
will be identical to those in the undeformed CFT, irrespectively of the non-
locality of the deformed theory, provided we evaluate them in the flowed
vacuum state, $|0_{\lambda}\rangle$. We re-emphasize that these operators -
previously discussed in [33] \- are just auxiliary, formal constructs with no
particular physical significance, as they do not correspond to local operators
even on the local, left-moving side. In particular, $w,\bar{w}$ are simply
labels, corresponding to the position of the initial local CFT operator that
we flow, which have no particular meaning in the deformed theory.

Since both $\tilde{\mathcal{O}}(w,\bar{w})$ and $\tilde{L}_{n},$ etc. flow in
the same way with $\mathcal{X}_{J\bar{T}}$, it follows that
$\tilde{\mathcal{O}}(w,\bar{w})$ will satisfy the usual Ward identities

$[\tilde{L}_{n},\tilde{\mathcal{O}}(w,\bar{w})]=e^{nw}[n\tilde{h}\tilde{\mathcal{O}}(w,\bar{w})+\partial_{w}\tilde{\mathcal{O}}(w,\bar{w})]\;,\;\;\;\;\;\;[\tilde{J}_{n},\tilde{\mathcal{O}}(w,\bar{w})]=e^{nw}\tilde{q}\,\tilde{\mathcal{O}}(w,\bar{w})$
(3.14)

$[\tilde{\bar{L}}_{n},\tilde{\mathcal{O}}(w,\bar{w})]=e^{n\bar{w}}[n\tilde{\bar{h}}\tilde{\mathcal{O}}(w,\bar{w})+\partial_{\bar{w}}\tilde{\mathcal{O}}(w,\bar{w})]\;,\;\;\;\;\;\;[\tilde{J}_{n},\tilde{\mathcal{O}}(w,\bar{w})]=e^{n\bar{w}}\tilde{\bar{q}}\,\tilde{\mathcal{O}}(w,\bar{w})$
(3.15)

where, again, $w,\bar{w}$ are just labels inherited from the undeformed
theory.

As explained at the beginning of this section, our candidate primary operators
should be constructed in momentum space, and so they satisfy

$[H_{R},\mathcal{O}(p,\bar{p})]=\bar{p}\,\mathcal{O}(p,\bar{p})$ (3.16)

In addition, they should be primary with respect to the unflowed left-moving
generators $L_{n},J_{n}$

$L_{n}=\tilde{L}_{n}+\eta\tilde{J}_{n}+\frac{k\eta^{2}}{4}\delta_{n,0}\;,\;\;\;\;\;J_{n}=\tilde{J}_{n}+\frac{k\eta}{2}\delta_{n,0}\;,\;\;\;\;\;\;\eta\equiv\lambda
H_{R}$ (3.17)

with the expected eigenvalues

$h=\tilde{h}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}\;,\;\;\;\;\;\;\;q=\tilde{q}+\frac{k}{2}\eta_{\mbox{\tiny{$\mathcal{O}$}}}\;,\;\;\;\;\;\;\eta_{\mbox{\tiny{$\mathcal{O}$}}}=\lambda\bar{p}$
(3.18)

This simply translates into the constraint (3.7), where $h,q$ are given above.
The reason we introduced the above notation is to highlight the similitude
with the $J^{1}\wedge J^{2}$ example.

Given our experience with the $J^{1}\wedge J^{2}$ deformation, we can easily
construct a solution for the left-moving piece of $\mathcal{O}(p,\bar{p})$
that satisfies (3.7). As for its right-moving piece, we expect a similar
constraint to hold in terms of the right-moving generators
$\bar{L}_{n},\bar{J}_{n}$; however, given that these are not standard
conformal generators, we do not know exactly what relation to impose. We thus
resort to simply guessing an appropriate right-moving factor, work out its
properties, and then justify our choice _a posteriori_ via its rather
reasonable predictions in the $R\rightarrow\infty$ limit.

Our proposed definition of an operator that has the required commutation
relations with the left-moving generators and possibly reasonable commutators
with the right-moving ones is

$\boxed{\mathcal{O}(p,\bar{p})=\int
dwd\bar{w}\,e^{-pw-\bar{p}\bar{w}}\epsilon^{A_{\mathcal{O}}w+B_{\mathcal{O}}\bar{w}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}(e^{nw}\tilde{J}_{-n}+e^{n\bar{w}}\tilde{\bar{J}}_{-n})}\tilde{\mathcal{O}}(w,\bar{w})e^{-\eta_{\mathcal{O}}\sum_{n=1}^{\infty}\frac{1}{n}(e^{-nw}\tilde{J}_{n}+e^{-n\bar{w}}\tilde{\bar{J}}_{n})}}$
(3.19)

where $A_{\mbox{\tiny{$\mathcal{O}$}}}$ and $B_{\mbox{\tiny{$\mathcal{O}$}}}$
are operators - which we will determine shortly - that depend on the conserved
quantum numbers of $\mathcal{O}$ and - as in the $J^{1}\wedge J^{2}$ case -
are assumed to be linear combinations of $\tilde{J}_{0},\tilde{\bar{J}}_{0}$
and $H_{R}$. Note that, unlike in the $J^{1}\wedge J^{2}$ example, these
operators do not commute with exponential factor that follows. The split
between $A_{\mbox{\tiny{$\mathcal{O}$}}},B_{\mbox{\tiny{$\mathcal{O}$}}}$ and
the $p,\bar{p}$ factors we pulled out is of course completely arbitrary, but
will be soon fixed in a convenient fashion. For the time being, we are still
working on the cylinder and have set $R=1$; the factor of the radius can be
easily reinstated by dimensional analysis: $w\rightarrow w/R$ and
$p\rightarrow pR$, with $\eta$ fixed.

Given this explicit expression, we can simply compute the commutators of
$\mathcal{O}(p,\bar{p})$ with the various generators. The simplest such
commutator is with the left-moving current, which reads

$[J_{n},\mathcal{O}(p,\bar{p})]=\left(\tilde{q}+\frac{k}{2}\eta_{\mbox{\tiny{$\mathcal{O}$}}}\right)\mathcal{O}(p-n,\bar{p})$
(3.20)

exactly as expected. Note that for $n\neq 0$, the shift in the charge comes
from the coefficient of $\tilde{J}_{n}$ in the exponent of (3.19), which
therefore needs to be a number. For $n=0$, it comes from the commutator with
$H_{R}$, which thus sets $\eta_{\mbox{\tiny{$\mathcal{O}$}}}=\lambda\bar{p}$,
with $\bar{p}$ defined in (3.16).

The commutator with $L_{n}$ reads

$\displaystyle[L_{n},\mathcal{O}(p,\bar{p})]$ $\displaystyle=$
$\displaystyle\left[n\left(\tilde{h}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}-1\right)+p\right]\mathcal{O}(p-n,\bar{p})+\left(\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{J}_{0}+\eta
q-A_{\mbox{\tiny{$\mathcal{O}$}}}-\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}-\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}\right)\mathcal{O}(p-n,\bar{p})+$
(3.21) $\displaystyle+$
$\displaystyle(\lambda\bar{p}-\eta_{\mbox{\tiny{$\mathcal{O}$}}})\mathcal{O}(p,\bar{p})\tilde{J}_{n}$

where the last term vanishes, for the reason we just stated. The primary
condition with respect to $L_{n}$ then fixes

$A_{\mbox{\tiny{$\mathcal{O}$}}}=\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{J}_{0}+\eta
q-\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{q}-\frac{k}{4}\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}$
(3.22)

which is entirely analogous to the expression (2.53) we obtained in
$J^{1}\wedge J^{2}$ \- deformed CFTs.

We now turn to the commutation relations with the right-movers, still for $R$
finite. The commutation relations with the right-moving $U(1)$ current are

$[\bar{J}_{n},\mathcal{O}(p,\bar{p})]=\left(\tilde{\bar{q}}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}}{2}\right)\mathcal{O}(p,\bar{p}-n)+\tilde{\bar{J}}_{n}[\mathcal{O}(p,\bar{p})-\mathcal{O}(p+a\alpha_{n}^{r},\bar{p}+b\alpha_{n}^{r})]$
(3.23)

where $a,b$ are the coefficients of $H_{R}$ inside
$A_{\mbox{\tiny{$\mathcal{O}$}}},B_{\mbox{\tiny{$\mathcal{O}$}}}$ and
$\alpha_{n}$, given in (C.2), is defined through the commutator

$[\tilde{\bar{J}}_{n},H_{R}]=\tilde{\bar{J}}_{n}\alpha_{n}$ (3.24)

The somewhat suspicious-looking operator-valued shift of the arguments in the
last term is simply a shorthand for the corresponding Fourier-space
expression, and follows from the contribution of terms of the form

$\tilde{\bar{J}}_{n}-e^{cH_{R}}\tilde{\bar{J}}_{n}e^{-cH_{R}}=\tilde{\bar{J}}_{n}(1-e^{-c\alpha_{n}})$
(3.25)

with $c=aw$ or $b\bar{w}$ to the commutator. Interestingly, this shift affects
both the left and the right-moving side. Note also that (3.22) implies that
$a=\lambda q$.

Notwithstanding the second term - which will turn out to be negligible at
large $R$ \- (3.23) takes precisely the form of a CFT Ward identity between an
affine Kac-Moody current and an operator of right-moving charge

$\bar{q}=\tilde{\bar{q}}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}}{2}$ (3.26)

which exactly mirrors the behaviour on the left-moving side. The fact that all
the $\bar{J}_{n}$ modes have this behaviour, and not just the $n=0$ one (which
corresponds to the global right-moving charge), is an interesting output of
our construction.

Let us now also fix the operator $B_{\mbox{\tiny{$\mathcal{O}$}}}$, e.g. by
computing the commutator of $\mathcal{O}(p,\bar{p})$ with
$\tilde{\bar{L}}_{0}=H_{R}(R-\lambda\tilde{J}_{0}-\frac{\lambda^{2}k}{4}H_{R})$
which, using (3.16), yields191919Note we could have chosen the eigenvalue of
$H_{R}$ in (3.16) to be different from the $\bar{p}$ factor appearing in the
definition (3.19). This would have simply resulted in an expression for
$B_{\mbox{\tiny{$\mathcal{O}$}}}$ that depended on both constants, as only
$\bar{p}-B_{\mbox{\tiny{$\mathcal{O}$}}}$ is fixed.

$[\tilde{\bar{L}}_{0},\mathcal{O}(p,\bar{p})]=\left(\bar{p}+\lambda\tilde{q}\bar{p}+\frac{\lambda^{2}k}{4}\bar{p}^{2}-\lambda\tilde{J}_{0}\bar{p}-\lambda
qH_{R}\right)\mathcal{O}(p,\bar{p})$ (3.27)

This commutator can also be evaluated by using (3.19) and the Ward identity
(3.15) for $\tilde{\mathcal{O}}$, with the result

$[\tilde{\bar{L}}_{0},\mathcal{O}(p,\bar{p})]=(\bar{p}-B_{\mbox{\tiny{$\mathcal{O}$}}})\mathcal{O}(p,\bar{p})$
(3.28)

Equating the two expressions, we find that the solution for
$B_{\mbox{\tiny{$\mathcal{O}$}}}$ is identical202020One can also check that
with this choice, $\mathcal{O}(p,\bar{p})$ satisfies the expected Hermiticity
conditions.  to the one for $A_{\mbox{\tiny{$\mathcal{O}$}}}$, (3.22). This
may seem a bit surprising, as we would have expected the right-moving
$B_{\mbox{\tiny{$\mathcal{O}$}}}$ to depend on the right-moving charges and
current operators, as it does in $J^{1}\wedge J^{2}$ \- deformed CFTs (2.53),
and not on the left-moving ones. The reason for this dependence can be traced
back to the fact that $\bar{L}_{0}$ \- the dimensionless right-moving
generator that is related to $\tilde{\bar{L}}_{0}$ by spectral flow - does not
equal $RH_{R}$, but rather $\bar{L}_{0}=R_{v}H_{R}$, where
$R_{v}=R-\lambda(J_{0}-\bar{J}_{0})$ is the field-dependent radius of the
field-depedent right-moving coordinate. This field-dependent rescaling has no
counterpart in $J^{1}\wedge J^{2}$ \- deformed CFTs, and we could not see any
simple modification of (3.19) that would yield a
$B_{\mbox{\tiny{$\mathcal{O}$}}}$ of the expected form, without spoiling the
rather pleasing commutation relations we have obtained so far. One could, of
course, consider working in terms of the field-dependent coordinate on the
right-moving side - which natually involves factors of $R_{v}$ \- but the
resulting expressions are significantly more complicated than (3.19).

The reason that this particular expression for
$B_{\mbox{\tiny{$\mathcal{O}$}}}$ is problematic is due to its effect on the
correlation functions of $\mathcal{O}(p,\bar{p})$ on the cylinder, which will
be discussed in the next subsection. However, as we will show, this effect is
subleading at large $R$. Since, as we explained earlier, the
$R\rightarrow\infty$ limit is also needed to resolve the problem with the
choice of vacuum in $J\bar{T}$ \- deformed CFTs, we will simply continue to
use the proposed expression (3.19), despite its drawbacks at finite $R$, and
show that it does indeed yield very reasonable predictions as
$R\rightarrow\infty$.

Let us now finally compute the commutation relations of
$\mathcal{O}(p,\bar{p})$ with the unflowed right-moving generators,
$\bar{L}_{n}$, given in (3.4). We find

$\displaystyle[\bar{L}_{n},\mathcal{O}(p,\bar{p})]$ $\displaystyle=$
$\displaystyle\left[n\left(\tilde{\bar{h}}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{\bar{q}}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}-1\right)+\bar{p}\right]\mathcal{O}(p,\bar{p}-n)+(\tilde{\bar{L}}_{n}+:\eta\,\tilde{\bar{J}}_{n}:)[\mathcal{O}(p,\bar{p})-\mathcal{O}(p+a\alpha_{n},\bar{p}+b\alpha_{n})]$
(3.29) $\displaystyle+$
$\displaystyle[(\eta_{\mbox{\tiny{$\mathcal{O}$}}}-\eta)(\tilde{q}-\tilde{\bar{q}})-\eta_{\mbox{\tiny{$\mathcal{O}$}}}(\tilde{J}_{0}-\tilde{\bar{J}}_{0})]\mathcal{O}(p,\bar{p}-n)$

The first term on the right-hand side looks exactly like a momentum-space
conformal Ward identity in a CFT, where the right-moving conformal dimension
of the operator is given by

$\bar{h}=\tilde{\bar{h}}+\eta_{\mbox{\tiny{$\mathcal{O}$}}}\tilde{\bar{q}}+\frac{k\eta_{\mbox{\tiny{$\mathcal{O}$}}}^{2}}{4}$
(3.30)

Using $\eta_{\mbox{\tiny{$\mathcal{O}$}}}=\lambda\bar{p}$, this exactly
corresponds to a momentum-dependent spectral flow of the right-moving
dimensions. Note that, unlike for the left-movers, this expression does _not_
follow from the flow of the right-moving energy eigenvalues on the cylinder,
as the latter involve a factor of $\tilde{q}$, rather than $\tilde{\bar{q}}$.

The second term can be written as $\bar{L}_{n}$ multiplying the term in
parantheses (with $a=b=\lambda q$), since the latter vanishes for $n=0$. As we
will argue, this term will drop out in the $R\rightarrow\infty$ limit. The
last term is required by the definition of the right-moving generators, which
include a factor of the field-dependent radius - e.g. for $n=0$,
$\bar{L}_{0}=H_{R}R_{v}$. Since we have chosen to diagonalize $H_{R}$, and not
$\bar{L}_{0}$, this term is a simple consequence of the non-trivial commutator
of $\mathcal{O}$ with the winding operator appearing in $R_{v}$.

Let us now discuss the $R\rightarrow\infty$ limit of these commutators. We
note that the “operator-valued shift” of $p,\bar{p}$ in (3.23) and (3.29)
scales with $R$ as

$p+\frac{\lambda q\alpha_{n}}{R}\approx p+\frac{\lambda
qn\hbar}{R^{2}}\;\;\;\;\;\mbox{as}\;R\rightarrow\infty$ (3.31)

Thus, the contribution of this term to the commutators of $\mathcal{O}$ with
$\bar{L}_{n},\bar{J}_{n}$ scale as $1/R^{2}$ in the large $R$ limit. It is
easy to see from (3.12) that these terms will consequently not contribute to
the Ward identities on the plane, at least as far as $\bar{L}_{\pm 1,0}$ are
concerned. As for the last term in (3.29), we note that it is suppressed by
$1/R$ in the commutator with $\bar{L}_{0}/R\approx L_{-1}^{pl}$, and it drops
out from the combinations $\bar{L}_{1}-\bar{L}_{-1}$ and
$R(\bar{L}_{1}+\bar{L}_{-1}-2\bar{L}_{0})$, which are in principle identified
with the plane generators.

To summarize, while at finite $R$ our candidate primary operators obey
precisely CFT Ward identities with respect to the left-moving generators and
certain ‘CFT-like’ Ward identities with respect to the right-moving ones, in
the $R\rightarrow\infty$ limit all Ward identities appear to reduce to exactly
CFT ones, at least as far as the global conformal and Kac-Moody generators are
concerned. This is consistent with the fact that the right-moving algebra
becomes Virasoro-Kac-Moody in the strict $R\rightarrow\infty$ limit. One
should be cautious, however, about the presence of subtle contributions to the
Ward identities in this limit - related to the momentum dependence of the
conformal dimensions - and thus a more careful study is called for.

While the CFT-like form of the Ward identities (3.23) and (3.29) looks rather
appealing, especially as $R\rightarrow\infty$, note that we have by no means
derived it from first principles. For that, one would need to better
understand how operators transform under the pseudo-conformal symmetries
generated by $\bar{L}_{n},\bar{J}_{n}$, taking into account the fact that
their algebra is not Virasoro-Kac-Moody at finite $R$. One can alternatively
use the Ward identities we worked out as a _definition_ of what is to be meant
by a primary operator in the non-local $J\bar{T}$ \- deformed CFT, whose
solution is, of course, (3.19). However, it seems hard to justify the choice
(3.23), (3.29), especially at finite $R$. It is, nevertheless, intriguing that
our candidate primary operators (3.19) satisfy such simple-looking Ward
identities with respect to the pseudo-conformal generators, which likely hint
towards a much richer, pseudo-local structure of $J\bar{T}$ \- deformed CFTs.
This intuition is further supported by our results for the correlation
functions, which are presented in the next subsection.

###  3.4 Correlation functions

The computation of correlation functions of the candidate primary operators
(3.19) in $J\bar{T}$ \- deformed CFTs proceeds in direct analogy to its
$J^{1}\wedge J^{2}$ counterpart, which we detailed in section 2.4. We will
first evaluate the correlation functions at finite $R$, using the flowed
vacuum $|0_{\lambda}\rangle$ \- in which the correlators of the auxiliary
$\tilde{\mathcal{O}}$ operators reduce to the ones in the undeformed CFT - and
only take the $R\rightarrow\infty$ limit at the end. This way of proceeding
will make it clear that the contribution associated with the non-CFT-like term
in (3.29) drops out from the correlation function in the decompactification
limit.

The momentum-space two-point function of the candidate primary operators reads

$\displaystyle\langle\mathcal{O}_{-q}(p_{1},\bar{p}_{1})\mathcal{O}_{q}(p_{2},\bar{p}_{2})\rangle$
$\displaystyle=$ $\displaystyle\int
d^{2}w_{1}d^{2}w_{2}e^{-\sum_{i}(p_{i}w_{i}+\bar{p}_{i}\bar{w}_{i})}e^{-\eta_{vac}q(w_{12}+\bar{w}_{12})-\eta_{\mbox{\tiny{$\mathcal{O}$}}}(\tilde{q}-\tilde{\bar{q}})\bar{w}_{12}}\times$
(3.32) $\displaystyle\hskip
17.07164pt\times\;\langle\mathcal{O}_{h_{1},\bar{h}_{1}}(w_{1},\bar{w}_{1})\mathcal{O}_{h_{2},\bar{h}_{2}}(w_{2},\bar{w}_{2})\rangle$

where $\bar{p}_{1}=-\bar{p}_{2}=-\bar{p}$ by momentum conservation, imposed as
usual by the integral over the center of mass position. The correlation
function on the last line is a usual position-space CFT two-point function,
with the same normalization as in the undeformed CFT, but with conformal
dimensions given by

$h_{i}(\bar{p}_{i})=\tilde{h}_{i}+\lambda\tilde{q}_{i}\bar{p}_{i}+\frac{\lambda^{2}k}{4}\bar{p}_{i}^{2}\;,\;\;\;\;\;\bar{h}_{i}(\bar{p}_{i})=\tilde{\bar{h}}_{i}+\lambda\tilde{\bar{q}}_{i}\bar{p}_{i}+\frac{k\lambda^{2}}{4}\bar{p}_{i}^{2}$
(3.33)

where $\tilde{h}_{i},\tilde{\bar{h}}_{i}$ are the conformal dimensions of the
respective operator in the undeformed CFT. The momentum-dependent shift in the
dimensions, which corresponds to a spectral flow with parameters
$\eta_{i}=\bar{\eta}_{i}=\lambda\bar{p}_{i}$ for each of the operators, occurs
through exactly the same mechanism as for the $J^{1}\wedge J^{2}$ two-point
function (2.57). The only differences with this previous case are that: the
spectral flow operator no longer annihilates the flowed vacuum, so we defined

$\eta|0_{\lambda}\rangle=\lambda
H_{R}|0_{\lambda}\rangle=\eta_{vac}|0_{\lambda}\rangle\neq 0$ (3.34)

where the actual eigenvalue can be read off from (3.5). This term contributes
to the correlator as indicated above. The second difference is due to the
expression (3.22) for $B_{\mbox{\tiny{$\mathcal{O}$}}}$ which, as discussed,
depends on the left-moving $U(1)$ charges, instead of the right-moving ones.
This leads to an explicit additional dependence on the winding charge, also
indicated in (3.32).

We now write the
$\langle\mathcal{O}_{1}(w_{1},\bar{w}_{1})\mathcal{O}_{2}(w_{2},\bar{w}_{2})\rangle$
correlator in terms of its Fourier transform,
$\mathcal{G}_{h_{i},\bar{h}_{i}}(p,\bar{p})\,\times$
$\delta^{2}(p_{1}+p_{2})$, and perform the trivial integral over
$w_{i},\bar{w}_{i}$. We obtain

$\displaystyle\langle\mathcal{O}_{-q}(-p,-\bar{p})\mathcal{O}_{q}(p,\bar{p})\rangle$
$\displaystyle=$ $\displaystyle\int
dp^{\prime}d\bar{p}^{\prime}\delta(p^{\prime}-p+\eta_{vac}q)\delta(\bar{p}^{\prime}-\bar{p}+\eta_{vac}q+\eta_{\mbox{\tiny{$\mathcal{O}$}}}(\tilde{q}-\tilde{\bar{q}}))\mathcal{G}_{h_{i}(\bar{p}),\bar{h}_{i}(\bar{p})}(p^{\prime},\bar{p}^{\prime})$
(3.35) $\displaystyle=$
$\displaystyle\mathcal{G}_{h_{i}(\bar{p}),\bar{h}_{i}(\bar{p})}\big{(}p-\eta_{vac}q,\bar{p}-\eta_{vac}q-\eta_{\mbox{\tiny{$\mathcal{O}$}}}(\tilde{q}-\tilde{\bar{q}})\big{)}$

where $q$ itself depends on $\bar{p}$ as in (3.18). Thus, the momentum-space
two-point function of the operators (3.19) is precisely given (up to some
trivial shifts in the arguments) by a momentum-space CFT two-point function,
but with the conformal dimensions replaced by their momentum-dependent
counterparts212121 Note that when writing this expression, one should replace
the dimensions by their momentum-dependent counterparts not only in the
functional part of the correlator, but also in the prefactors, which contain
factors of e.g. $\Gamma(h)$. This resonates with the behaviour of the
correlation functions discussed in the single-trace version of
$T\bar{T}$-deformed CFTs, which are computed using worldsheet string theory
[47]. (3.33). Remarkably, this is exactly the same behaviour that we observed
in (1.1) for scattering amplitudes off near-extremal black holes!

Let us now study the three-point function of $\mathcal{O}(p,\bar{p})$. We
similarly obtain

$\displaystyle\langle\mathcal{O}_{1}(p_{1},\bar{p}_{1})\mathcal{O}_{2}(p_{2},\bar{p}_{2})\mathcal{O}_{3}(p_{3},\bar{p}_{3})\rangle$
$\displaystyle=$ $\displaystyle\int
d^{2}w_{i}\,e^{-\sum_{i}(w_{i}p_{i}+\bar{w}_{i}\bar{p}_{i})}e^{\eta_{vac}\sum_{i}q_{i}(w_{i}+\bar{w}_{i})}\times$
$\displaystyle\times\;e^{-\eta_{1}(q_{1}-\bar{q}_{1})\bar{w}_{1}+[\eta_{2}(q_{3}-\bar{q}_{3})-\eta_{1}(q_{2}-\bar{q}_{2})]\bar{w}_{2}+\eta_{3}(q_{3}-\bar{q}_{3})\bar{w}_{3}}\langle\mathcal{O}_{h_{1},\bar{h}_{1}}(w_{1},\bar{w}_{1})\mathcal{O}_{h_{2},\bar{h}_{2}}(w_{2},\bar{w}_{2})\mathcal{O}_{h_{3},\bar{h}_{3}}(w_{3},\bar{w}_{3})\rangle$

where the three-point function appearing on the last line corresponds
precisely to a CFT three-point function in position space, with conformal
dimensions given by the momentum-dependent expressions (3.33) and is derived
through exactly the same steps as for the $J^{1}\wedge J^{2}$ \- deformed
three-point function (2.59). Fourier-transforming this (CFT) expression and
performing the $w_{i},\bar{w}_{i}$ integrals, one again obtains a result that
corresponds to the original momentum-space CFT three-point function with the
operator dimensions replaced by (3.33), and with slightly shifted arguments,
as in (3.35). Note this implies that the OPE coefficients, appropriately
defined, do not change with the deformation, as was the case in $J^{1}\wedge
J^{2}$.

The computation of higher-point functions on the cylinder proceeds in an
identical manner. In the case of the four-point function, we note that while
the position-space four-point function that appears in the integrand is
entirely crossing symmetric, as we showed in section 2.4 for the analogous
$J^{1}\wedge J^{2}$ deformation, the winding-dependent prefactors due to the
unexpected form of $B_{\mathcal{O}}$ are not, and thus spoil the crossing
symmetry of the result. Consequently, our proposal is not quite correct in
finite size.

This is however easy to fix by taking the $R\rightarrow\infty$ limit. Noting
that the only way in which these winding terms enter the correlator is through
the shift of the argument of the momentum-space correlator, as in (3.35), it
is clear that they can be dropped in the $R\rightarrow\infty$ limit, since
they scale as
$\eta_{\mbox{\tiny{$\mathcal{O}$}}}(\tilde{q}-\tilde{\bar{q}})/R$ with respect
to $\bar{p}$. A similar comment applies to the $\eta_{vac}q$ term, which can
also be dropped. The resulting four-point functions are simply the Fourier
transform of position-space correlators of the form (2.65), which are
manifestly crossing symmetric and are entirely determined by the corresponding
four-point function in the undeformed CFT. These correlation functions should
be considered in the $R\rightarrow\infty$ limit, in which they simply become
correlators on the plane. Identical comments apply to higher-point functions.

The fact that all the correlation functions of our candidate $J\bar{T}$
primary operators are entirely determined by the original CFT correlators in
such a strikingly simple manner strongly suggests that $J\bar{T}$ \- deformed
CFTs possess a very similar structure to that of standard two-dimensional
CFTs, which simply awaits for the right language to be uncovered. We hope that
some of the tools proposed in this article will be helpful in making progress
on this interesting issue.

##  4 Discussion

In this article, we have argued that, despite their non-locality, $J\bar{T}$
\- deformed CFTs do allow for a notion of primary operators with respect to
the generators of the field-dependent symmetries that act on the non-local
side. We moreover showed how to compute arbitrary correlation functions of
these operators _exactly_ in terms of the correlators of the undeformed CFT.
These correlation functions appear consistent (i.e., they obey crossing
symmetry) in the decompactification limit.

As discussed in the introduction, that special non-local theories may posses a
structure that is sufficiently rigid to completely fix the form of low-point
correlation functions is extremely interesting, as such theories could provide
a microscopic dual to generic near-extremal (and, possibly, also non-extremal
[48]) black holes. It thus seems worthwhile to better understand this
structure, as well as its possible generalizations, and compare it to the
results of scattering in black hole backgrounds.

A basic question is to understand from “first principles” the Ward identities
that primary operators should satisfy, by relating them to the expected
(position-space) transformation properties of the operator under field-
dependent coordinate transformations. This “first principles” derivation
should also be able to determine whether there are corrections to the primary
operator that involve the field-dependent coordinate, an effect that we could
perhaps not see due to the large $R$ limit.

A related task is to directly work out the general constraints that these Ward
identities impose on correlation functions, i.e. without appealing to the
auxiliary construction involving the $\tilde{\mathcal{O}}$ operators, but
rather paralleling the usual argument used for standard CFTs. This question
can be asked either on the cylinder or on the plane, and each case presents
its own challenges: on the cylinder, one first needs to undestand the
properties of the $SL(2,\mathbb{R})$ \- invariant vacuum which, as explained,
is different from the flowed one, and thus the result for the correlation
functions could be rather different from those discussed in the previous
section; on the plane, the momentum-space Ward identities should receive
contributions from the explicit momentum dependence of the conformal
dimensions, which is not clear how to recover from the $R\rightarrow\infty$
limit. Nonetheless, the final result that we have obtained for the correlation
functions suggest that one should obtain as many constraints as there are in
usual CFTs, though the language in which they are expressed may be different.

On the more technical side, an interesting issue that we have encountered
concerns the possible definitions of a vacuum state for $J\bar{T}$ \- deformed
CFTs on a cylinder, and its $SL(2,\mathbb{R})$ invariance properties. For the
two possible choices of vacuum we discussed in section 3.1, it would be
interesting to work out their definition and exact relation, both at finite
radius and in the $R\rightarrow\infty$ limit. Another interesting technical
question is to understand how the construction presented in this article works
in the specific case of conserved currents, for example the stress tensor, and
how to express these symmetry currents in terms of the associated conserved
charges. This may also clarify how the field-dependent coordinate, which is
essential for the definition of the charges, may fit in with the flow
equations and momentum-space picture for the operators used herein.

Finally, it would be very interesting to extend these results to other special
non-local theories, such as $T\bar{T}$ \- deformed CFTs. An essential input
for our present construction was the existence, in $J\bar{T}$ \- deformed
CFTs, of two different bases for the right-moving symmetries: as a set of
generators that flow in the same way as the energy eigenstates or, as the
generators that directly implement pseudoconformal transformations. In
$T\bar{T}$ \- deformed CFTs, only the first set have been shown to exist at
the full quantum level [11]; as for the generators of field-dependent
symmetries, they are currently understood only at a classical level and on the
plane [34]. One may nevertheless hope that the needed relation between the two
will eventually be found (though, most likely, it will not correspond to a
spectral flow, since the existence of a $U(1)$ current is not required in this
case) and can be used to define an analogous set of primary operators. It is
interesting to note that the most naïve guess - based on the analogy with
$J\bar{T}$ \- for how an appropriately defined ‘primary’ two-point function
will be changed - namely, via a momentum-dependent shift in the conformal
dimensions - matches the behaviour found in [29].

Other interesting extensions of this work would be to the single-trace
versions of the $T\bar{T}$ and $J\bar{T}$ deformation, where one should, in
addition, be able to compare the proposed definition of the primary operators
with the expectation from worldsheet string theory [47, 49]. One may hope
that, by extending these type of symmetries and their consequences to an ever
larger class of theories, one would ultimately be able to conjecture a set of
axioms (e.g., for the correlation functions) that all “dipole CFTs” - or, more
generally, all “non-local CFTs” - should obey, and that this definition would
be general enough to capture, in a holographic sense, the near-horizon
dynamics of all extremal and non-extremal black holes.

### Acknowledgements

The author would like to thank Miguel Paulos, Sylvain Ribault and especially
Balt van Rees for useful conversations. She is especially grateful to
Alessandro Bombini and Andrea Galliani for collaboration on a related project.
This research was supported in part by the ERC starting grant 679278 Emergent-
BH.

##  Appendix A $SL(2,\mathbb{R})$ and Kac-Moody generator identities

In this appendix, we derive a few identities that are useful in the
manipulations of section 2.3.

####  $\bullet\;\;SL(2,\mathbb{R})$

Given the $SL(2,\mathbb{R})$ generators $L_{\pm 1,0}$, which satisfy the usual
commutation relations, we would like to find the relations between the
coefficients $(a,b,c)$ and $(\tilde{a},\tilde{b},\tilde{c})$ in

$e^{aL_{-1}}e^{2bL_{0}}e^{cL_{1}}=e^{\tilde{a}L_{1}}e^{2\tilde{b}L_{0}}e^{\tilde{c}L_{-1}}$
(A.1)

which correspond to two different ways of parametrizing the same group
element. This identity can be derived by using the following representation of
$L_{\pm 1,0}$

$L_{-1}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\;,\;\;\;\;\;\;\;L_{0}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\;,\;\;\;\;\;\;L_{1}=\left(\begin{array}[]{cc}0&0\\\
-1&0\end{array}\right)$ (A.2)

Using this, we find

$e^{aL_{-1}}e^{2bL_{0}}e^{cL_{1}}=\left(\begin{array}[]{cc}1&a\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}e^{b}&0\\\
0&e^{-b}\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-c&1\end{array}\right)=\left(\begin{array}[]{cc}e^{b}-ace^{-b}&ae^{-b}\\\
-ce^{-b}&e^{-b}\end{array}\right)$ (A.3)

and

$e^{\tilde{a}L_{1}}e^{2\tilde{b}L_{0}}e^{\tilde{c}L_{-1}}=\left(\begin{array}[]{cc}e^{\tilde{b}}&\tilde{c}e^{\tilde{b}}\\\
-\tilde{a}e^{\tilde{b}}&e^{-\tilde{b}}-\tilde{a}\tilde{c}e^{\tilde{b}}\end{array}\right)$
(A.4)

This leads to the relation

$b=-\ln(e^{-\tilde{b}}-\tilde{a}\tilde{c}e^{\tilde{b}})\;,\;\;\;\;\;\;a=\frac{\tilde{c}}{e^{-2\tilde{b}}-\tilde{a}\tilde{c}}\;,\;\;\;\;\;\;c=\frac{\tilde{a}}{e^{-2\tilde{b}}-\tilde{a}\tilde{c}}$
(A.5)

We also note the identity

$e^{\alpha L_{1}+2\beta L_{0}+\gamma L_{-1}}\equiv e^{M}=\cosh(\sqrt{\det
M})I+\sinh(\sqrt{\det M})\frac{M}{\sqrt{\det M}}$ (A.6)

which we can use to show that ($z=Re\,z+iIm\,z=|z|e^{i\phi}$)

$e^{zL_{-1}-\bar{z}L_{1}}=e^{e^{i\phi}\tanh|z|L_{-1}}e^{-2L_{0}\ln\cosh|z|}e^{-e^{-i\phi}\tanh|z|L_{1}}$
(A.7)

Thus, if we let $e^{w}=e^{i\phi}\tanh|z|$ and act on $|h\rangle$, we obtain
precisely (2.37), up to certain overall factors coming from the action of the
middle term. Note that $|e^{w}|<1$ or $Re\,w<0$, which implies that this
interpretation only applies to operators that are to the past of the $\tau=0$
slice, where the state is defined.

Another possibly useful identity is

$e^{\alpha L_{0}}e^{e^{w}L_{-1}}=e^{e^{w+\alpha}L_{-1}}e^{\alpha L_{0}}$ (A.8)

which can be used to check that $L_{0}$ implements a translation of the state
created by acting with $\mathcal{O}(w)$ on the vacuum
$\mathcal{O}(w)|0\rangle=e^{wh}e^{e^{w}L_{-1}}|h\rangle$.

####  $\bullet\;\;SL(2,\mathbb{R})$ \- Kac-Moody

A more interesting identity to derive is the following

$e^{e^{w}(\tilde{L}_{-1}+\eta\tilde{J}_{-1})}e^{-e^{w}\tilde{L}_{-1}}=e^{\eta\sum_{n=1}^{\infty}\frac{1}{n}e^{nw}\tilde{J}_{-n}}$
(A.9)

claimed in the same section, where $\eta$ is a constant or an operator that
commutes with all other operators that appear in this expression.

To prove this, we first use the Baker-Campbell-Hausdorff formula, which states
that

$e^{X}e^{Y}=e^{\sum_{n=1}^{\infty}\frac{1}{n}(-1)^{n-1}\sum_{r_{i},s_{i}}\\!\\!c(r_{i},s_{i})[X^{r_{1}}Y^{s_{1}}\ldots
X^{r_{n}}Y^{s_{n}}]}$ (A.10)

where $i\in\\{1,\ldots n\\}$, $r_{i}+s_{i}>0$, $c(r_{i},s_{i})$ are some
numerical coefficients determined from these numbers, and the term in brackets
is a nested commutator of $X$’s and $Y$’s. In our case

$X=e^{w}(\tilde{L}_{-1}+\eta\tilde{J}_{-1})\;,\;\;\;\;Y=-e^{w}\tilde{L}_{-1}$
(A.11)

so $[X,Y]=\eta e^{2w}\tilde{J}_{-2}$, and all further commutators with either
$X$ or $Y$ will decrease the level of $\tilde{J}_{-n}$ by one and add a
multiplicative factor of $e^{w}$, times some numerical coefficient.
Consequently, the term on the right-hand side must take the form

$e^{e^{w}(\tilde{L}_{-1}+\eta\tilde{J}_{-1})}e^{-e^{w}\tilde{L}_{-1}}=e^{\eta\sum_{n=1}^{\infty}c_{n}e^{nw}\tilde{J}_{-n}}$
(A.12)

for some numerical coefficients $c_{n}$ that we will now determine. This can
be done by comparing the two ways of computing the
$\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle$ overlap presented in
section 2.3.

One way consists of simply repeating the steps in (2.3), using (A.12) instead

$\displaystyle\langle\mathcal{O}(w_{1})\mathcal{O}(w_{2})\rangle$
$\displaystyle=$ $\displaystyle e^{-hw_{12}}\langle
h_{\lambda}|e^{e^{-w_{1}}(\tilde{L}_{1}+\eta_{\mathcal{O}}\tilde{J}_{1})}e^{e^{w_{2}}(\tilde{L}_{-1}+\eta_{\mathcal{O}}\tilde{J}_{-1})}|h_{\lambda}\rangle$
(A.13) $\displaystyle=$ $\displaystyle
e^{w_{12}(\tilde{h}-h)}\langle\tilde{\mathcal{O}}(w_{1})e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}c_{n}e^{-nw_{1}}\tilde{J}_{n}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}c_{n}e^{nw_{2}}\tilde{J}_{-n}}\tilde{\mathcal{O}}(w_{2})\rangle$
$\displaystyle=$ $\displaystyle
e^{w_{12}(\tilde{h}-h)}e^{\frac{k\eta_{\mathcal{O}}^{2}}{2}\sum_{n}nc_{n}^{2}e^{-nw_{12}}}\langle\tilde{\mathcal{O}}(w_{1})e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}c_{n}e^{nw_{2}}\tilde{J}_{-n}}e^{\eta_{\mathcal{O}}\sum_{n=1}^{\infty}c_{n}e^{-nw_{1}}\tilde{J}_{n}}\tilde{\mathcal{O}}(w_{2})\rangle$
$\displaystyle=$ $\displaystyle
e^{w_{12}(\tilde{h}-h)}e^{\frac{k\eta_{\mathcal{O}}^{2}}{2}\sum_{n}nc_{n}^{2}e^{-nw_{12}}+2\eta_{\mathcal{O}}\tilde{q}\sum_{n}c_{n}e^{-nw_{12}}}\langle\tilde{\mathcal{O}}(w_{1})\tilde{\mathcal{O}}(w_{2})\rangle$

However, the alternate computation we performed, using just the commutation
relations of the unflowed generators, yields directly (2.39). Comparing the
two expressions, we conclude that $c_{n}=\frac{1}{n}$.

##  Appendix B $J^{1}\wedge J^{2}$ \- deformed free bosons

In this appendix, we work out in detail the case of $J^{1}\wedge J^{2}$ \-
deformed free bosons, which should help concretize the general analysis of
section 2 . Parts of this analysis have previously appeared in [37, 25].

###  B.1 Classical analysis

We start with a variation on the calculation in appendix A of [25]. Consider
the action

$S=-\kappa\int
dx^{+}dx^{-}\,[\partial\phi_{1}\bar{\partial}\phi_{1}+\partial\phi_{2}\bar{\partial}\phi_{2}-\lambda(\partial\phi_{1}\bar{\partial}\phi_{2}-\bar{\partial}\phi_{1}\partial\phi_{2})]=\int
d\sigma dt\,\mathcal{L}$ (B.1)

where $x^{\pm}=\sigma\pm t$ and
$\partial,\bar{\partial}=\frac{1}{2}(\partial_{\sigma}\pm\partial_{t})$. The
components of the two conserved shift currents
$J^{\alpha,a}=-\frac{\partial\mathcal{L}}{\partial(\partial_{\alpha}\phi_{a})}$
are

$J^{1}_{+}=\kappa\,(\partial\phi_{1}+\lambda\partial\phi_{2})\;,\;\;\;\;\;J^{1}_{-}=\kappa\,(\bar{\partial}\phi_{1}-\lambda\bar{\partial}\phi_{2})$
(B.2)

$J^{2}_{+}=\kappa\,(\partial\phi_{2}-\lambda\partial\phi_{1})\;,\;\;\;\;\;J^{2}_{-}=\kappa\,(\bar{\partial}\phi_{2}+\lambda\bar{\partial}\phi_{1})$
(B.3)

Note that the action satisfies a flow equation of the Smirnov-Zamolodchikov
type222222Our conventions are $\epsilon^{\sigma t}=\epsilon_{t\sigma}=1$.,
$\partial_{\lambda}\mathcal{L}=-\epsilon^{\alpha\beta}J_{\alpha}^{1}J^{2}_{\beta}$,
provided we choose

$\kappa=\frac{1}{1+\lambda^{2}}$ (B.4)

Using this, we can construct chiral currents by taking linear combinations of
$J^{1,2}_{\alpha}$ and the components of the topologically conserved current
$\tilde{J}^{a}$, with

$\tilde{J}^{a}_{+}=\partial\phi_{a}\;,\;\;\;\;\;\;\;\tilde{J}^{a}_{-}=-\bar{\partial}\phi_{a}$
(B.5)

A basis for these currents is

$J^{1}_{L}=\frac{J^{1}+\kappa\tilde{J}^{1}-\lambda\kappa\tilde{J}_{2}}{2}=\kappa\partial\phi_{1}\;,\;\;\;\;\;\;J^{1}_{R}=\frac{J^{1}-\kappa\tilde{J}^{1}-\lambda\kappa\tilde{J}_{2}}{2}=\kappa\bar{\partial}\phi_{1}$
(B.6)

$J^{2}_{L}=\frac{J^{2}+\kappa\tilde{J}^{2}+\lambda\kappa\tilde{J}_{1}}{2}=\kappa\partial\phi_{2}\;,\;\;\;\;\;\;J^{2}_{R}=\frac{J^{2}-\kappa\tilde{J}^{2}+\lambda\kappa\tilde{J}_{1}}{2}=\kappa\bar{\partial}\phi_{2}$
(B.7)

We would now like to compute the Poisson brackets of the chiral currents. For
this, we work out the canonical momenta

$\pi_{1}=\kappa\,(\dot{\phi}_{1}+\lambda\partial_{\sigma}\phi_{2})\;,\;\;\;\;\;\;\pi_{2}=\kappa\,(\dot{\phi}_{2}-\lambda\partial_{\sigma}\phi_{1})$
(B.8)

which satisfy the canonical equal-time commutation relations
$\\{\phi_{a}(\sigma),\pi_{b}(\sigma^{\prime})\\}=\delta_{ab}\,\delta(\sigma-\sigma^{\prime})$,
and express the currents in terms of them. We find that the Poisson brackets
of the above chiral currents are diagonal in this basis, but are proportional
to a factor of $\kappa$, which represents the level of the chiral algebra. It
is desirable to work instead with the combinations

$\mathcal{J}^{1}_{L,R}=J^{1}_{L,R}\pm\lambda
J^{2}_{L,R}\;,\;\;\;\;\;\;\;\mathcal{J}^{2}_{L,R}=J^{2}_{L,R}\mp\lambda
J^{1}_{L,R}$ (B.9)

which have level one. Their expression in terms of the canonical variables is

$\mathcal{J}^{1}_{L,R}=\frac{1}{2}(\pi_{1}\pm\phi^{\prime}_{1}\pm\lambda\pi_{2})\;,\;\;\;\;\;\;\;\mathcal{J}^{2}_{L,R}=\frac{1}{2}(\pi_{2}\pm\phi^{\prime}_{2}\mp\lambda\pi_{1})$
(B.10)

The Hamiltonian density is given by

$\mathcal{H}=\pi_{a}\dot{\phi}_{a}-\mathcal{L}=\frac{1+\lambda^{2}}{2}(\pi_{1}^{2}+\pi_{2}^{2})+\frac{1}{2}(\phi_{1}^{\prime
2}+\phi_{2}^{\prime
2})+\lambda(\phi_{1}^{\prime}\pi_{2}-\phi_{2}^{\prime}\pi_{1})$ (B.11)

in agreement with our general result (2.6).

###  B.2 Quantum analysis

The shift in the chiral charges and the energies of (primary) states on the
cylinder with momenta $n^{a}$ and windings $w_{a}$ were worked out in full
generality in section 2.1. Using the state-operator correspondence, these
states correspond to (primary) vertex operators that carry these charges, of
the form

$\mathcal{V}(z,\bar{z})=\;:e^{ic^{a}_{L}\phi^{a}_{L}(z)+ic^{a}_{R}\phi^{a}_{R}(\bar{z})}:$
(B.12)

where $\phi^{a}_{L,R}$ are the left- and right-moving pieces of the above
scalar fields and $c^{a}_{L,R}$ are coefficients that we would like to
determine. For simplicity, we will concentrate on the left-moving piece of the
vertex operator and drop the ‘$L$’ index. The action (B.1) implies that the
OPEs of the scalars is

$\phi_{a}(z)\phi_{b}(z^{\prime})=-\frac{\delta_{ab}}{2\kappa}\ln(z-z^{\prime})$
(B.13)

The currents $\partial\phi_{a}$ are primary with dimension $1$, as can be
checked by computing their OPE with the stress tensor,
$T_{zz}=\kappa\sum_{a}(\partial\phi_{a})^{2}$. The OPE of the above vertex
operator with the chiral left currents
$\mathcal{J}^{a}=\kappa(\partial\phi^{a}+\lambda\epsilon^{ab}\partial\phi_{b})$
is thus

$\mathcal{J}^{a}(z)\mathcal{V}(0)\sim\frac{c^{a}+\lambda\epsilon^{ab}c_{b}}{2z}\mathcal{V}(0)$
(B.14)

Equating the coefficients of $z^{-1}$ with the flowed charge $q^{a}$ in
(2.18), we find

$c^{a}=2\kappa(q^{a}-\lambda\epsilon^{ab}q_{b})$ (B.15)

For the right-moving coefficients, the sign of $\lambda$ is switched and
$q^{a}\rightarrow\bar{q}^{a}$. The associated dimensions are given by the
(primary) OPEs

$T(z)\mathcal{V}(0)\sim\frac{1}{4z^{2}}\sum_{a}\frac{c_{a}^{2}}{\kappa}\mathcal{V}(0)+\frac{1}{z}\partial\mathcal{V}(0)=\frac{\sum_{a}q_{a}^{2}}{z^{2}}\mathcal{V}(0)+\frac{1}{z}\partial\mathcal{V}(0)$
(B.16)

The left conformal dimension of this operator is $\sum_{a}(q^{a})^{2}$, in
perfect agreement with (2.19). It is useful to rewrite the exponent as

$c^{a}\phi_{a}=2q^{a}\kappa(\phi_{a}+\lambda\epsilon_{ab}\phi^{b})\equiv
2q^{a}\varphi_{a}$ (B.17)

and similarly on the right, where $\varphi^{a}_{L,R}$ simply correspond to the
bosonisation of $\mathcal{J}^{a}_{L,R}$. The vertex operator thus takes the
form

$\mathcal{V}(z,\bar{z})=\;:e^{2iq^{a}\varphi^{a}_{L}(z)+2i\bar{q}^{a}\varphi^{a}_{R}(\bar{z})}:$
(B.18)

The mode expansion of $\varphi^{a}_{L}$ is given in terms of the modes
$J^{a}_{n}$ of $\mathcal{J}^{a}$, i.e.

$\varphi^{a}_{L}(z)=\varphi^{a,0}_{L}-iJ_{0}^{a}\,\ln z+i\sum_{n\neq
0}\frac{\tilde{J}^{a}_{n}}{n}z^{-n}$ (B.19)

and similarly on the right, where we used the fact that
$J_{n}^{a}=\tilde{J}_{n}^{a}$ for $n\neq 0$ to render the expression more
familiar. Note that the zero modes of the left and right chiral bosons are
independent. It is easy to see that these operators, even inserted at zero,
will flow with $\lambda$. To completely specify them, we need to spell out the
normal ordering - that is, we put all the annihilation operators to the right
of the creation ones

$\mathcal{V}(z)=:e^{2iq_{a}\varphi^{a}_{L}}:=e^{2iq_{a}\varphi^{0,a}_{L}+2q_{a}\sum_{n=1}^{\infty}\frac{\tilde{J}_{-n}^{a}}{n}z^{n}}e^{2q_{a}J_{0}^{a}\ln
z-2q_{a}\sum_{n=1}^{\infty}\frac{\tilde{J}^{a}_{n}}{n}z^{-n}}$ (B.20)

The OPE of two such operators can be computed using the BCH formula and the
current commutation relations. We find

$\displaystyle\mathcal{V}_{q_{1}}(z_{1})\mathcal{V}_{q_{2}}(z_{2})$
$\displaystyle=$ $\displaystyle
e^{4iq_{1}^{a}q_{2}^{b}[J_{0}^{a},\varphi_{0}^{b}]\ln
z_{1}-2q_{1}^{a}q_{2}^{a}\sum_{n=1}^{\infty}\frac{1}{n}(z_{2}/z_{1})^{n}}\\!\\!:\mathcal{V}_{q_{1}}(z_{1})\mathcal{V}_{q_{2}}(z_{2}):=e^{2q_{1}^{a}q_{2}^{a}(\ln
z_{1}+\ln(1-z_{2}/z_{1}))}\cdot$ (B.21) $\displaystyle\cdot$
$\displaystyle:\mathcal{V}_{q_{1}}(z_{1})\mathcal{V}_{q_{2}}(z_{2}):=(z_{1}-z_{2})^{2q_{1}\cdot
q_{2}}:\mathcal{V}_{q_{1}}(z_{1})\mathcal{V}_{q_{2}}(z_{2}):$

where we used $[\varphi^{0,a},J^{b}_{0}]=\frac{i}{2}\delta^{ab}$. This yields
the correct OPE of vertex operators. Note that the zero mode contribution
plays an essential role in rendering the correlator translationally-invariant.
Including the right-moving piece, we also see that these operators are
mutually local, since $q_{1}^{a}q_{2}^{a}-\bar{q}_{1}^{a}\bar{q}_{2}^{a}$ is
$\lambda$-independent.

We would now like to construct the free boson realisation of the flowed
operator $\tilde{\mathcal{V}}(z)$ and compare it to the above operator. At
$\lambda=0$, this operator is simply
$\mathcal{V}_{CFT}(z)=:e^{2i\tilde{q}_{a}\phi_{a}(z)}\\!:$, where
$\phi_{a}(z)$ has a decomposition of the form (B.19) in terms of the
_undeformed_ current modes. At finite $\lambda$, $\tilde{\mathcal{V}}$ is
given by integrating the flow equation (2.48). The flow operator in this
theory is (2.27), which in terms of Fourier modes reads

$\mathcal{X}_{J\bar{J}}=\sum_{n\neq 0}\frac{1}{2\pi
n}(J^{1}_{n}J^{2}_{-n}-\bar{J}^{1}_{n}\bar{J}^{2}_{-n}+J^{1}_{n}\bar{J}^{2}_{n}-\bar{J}^{1}_{n}J^{2}_{n})$
(B.22)

As one can see from (2.30), the action of this operator on the non-zero modes
$\phi_{a}^{nzm}(z)$ inside $\mathcal{V}_{CFT}$ is to simply turn them into
modes of the deformed current, $\tilde{J}_{n}$. On the other hand, since the
zero mode of the current - which equals the $\lambda$ \- independent
$\tilde{J}_{0}^{a}$ \- commutes with $\mathcal{X}_{J\bar{J}}$, it will not
flow. The scalar zero mode part cannot be inferred from the simple classical
flow equation; however, we do know it should be simply
$2iq_{a}\varphi^{a}_{0}$, because the flowed state carries charge $q_{a}$, and
the charge is carried entirely by the zero mode. Given all this, a candidate
$\tilde{\mathcal{V}}(z)$ operator is

$\tilde{\mathcal{V}}(z)=:\\!e^{2iq_{a}\varphi_{0}^{a}+2i\tilde{q}_{a}(\varphi_{nzm}^{a}(z)-i\tilde{J}_{0}^{a}\ln
z)}\\!:$ (B.23)

This is in perfect agreement with the relation (2.52) between the primary and
the flowed operator, upon conformally transforming from $w$ to $z$. The fact
that the coefficients of the zero and non-zero modes are different clearly
indicates that $\tilde{\mathcal{V}}(z)$ is not a local operator. On the other
hand, if we compute the OPE of two such operators, we find

$\tilde{\mathcal{V}}(z_{1})\tilde{\mathcal{V}}(z_{2})=\left(1-\frac{z_{2}}{z_{1}}\right)^{2\tilde{q}_{1}\cdot\tilde{q}_{2}}\\!\\!\\!e^{4i\tilde{q}_{1}^{a}\ln
z_{1}q_{2}^{b}[\tilde{J}_{0}^{a},q_{2}^{b}\varphi_{L,0}^{b}+\bar{q}_{2}^{b}\varphi_{R,0}^{b}]}:\\!\tilde{\mathcal{V}}(z_{1})\tilde{\mathcal{V}}(z_{2})\\!:=(z_{1}-z_{2})^{\frac{1}{2}\tilde{q}_{1}\cdot\tilde{q}_{2}}\\!:\\!\tilde{\mathcal{V}}(z_{1})\tilde{\mathcal{V}}(z_{2})\\!:$
(B.24)

where we used the fact that the commutator
$[\tilde{J}_{0}^{\alpha},\varphi_{L/R,0}^{b}]=-\frac{i}{2}(\delta^{ab}\delta_{L}-\frac{\lambda}{2}\epsilon^{ab})$
and that $q^{a}+\bar{q}^{a}=n^{a}$. Therefore, these operators have the same
OPE as the original CFT local operators, even though they are not themselves
local.

##  Appendix C The unflowed $J\bar{T}$ algebra

The algebra of the unflowed generators in $J\bar{T}$ \- deformed CFTs was
spelled out in [11], and is rather involved. For the purposes of this article,
we would like to only collect its subalgebra that contains the global
$SL(2,\mathbb{R})$ generators $L_{\pm 1,0}$ and their right-moving
counterparts $\bar{L}_{\pm 1,0}$. Unlike the case of standard CFTs, here the