it, one may use the $T\bar{T}$ trace relation in each copy

$T_{\sigma\sigma}^{I}(\sigma)=\mathcal{H}^{I}(\sigma)-2\mu\mathcal{O}_{T_{I}\bar{T}_{I}}(\sigma)$
(3.45)

which holds classically as is, and quantum-mechanically up to a total
derivative. Plugging this into (3.44) and acting with the whole equation on a
twisted sector, in which the expansion of the fields takes the form (3.22) (we
are ignoring for now the copies that do not participate in the twist), the
integral over $\mathcal{H}^{I}(\sigma)$ that multiplies
$\mathcal{O}_{T_{I}\bar{T}_{I}}$ in the second term yields a factor of
$H^{(w)}/w$, where $H^{(w)}$ is the Hamiltonian restricted to the $w$-twisted
sector and the index ‘$w$’ is now a shorthand for the individual $w$ copies
that enter the cycle. Ultimately we find, at the classical level

$\mathcal{X}_{T_{I}\bar{T}_{I}}^{(w),cls}=\frac{1}{1+2\mu H^{(w)}/(Rw)}\int
d\sigma
d\tilde{\sigma}\,G(\sigma-\tilde{\sigma})\mathcal{H}^{I}(\sigma)\mathcal{P}^{I}(\tilde{\sigma})$
(3.46)

This expression corresponds to nothing but the seed $\mathcal{X}_{T\bar{T}}$
on the covering space. One way of showing this is by expanding
$\mathcal{H}^{I}(\sigma)$ above as a function of the undeformed CFT generators
$\mathcal{H}^{(0)}_{I}(\sigma)$ and $\mathcal{P}^{I}(\sigma)$, using the
closed-form expression given in [39]. When acting upon a twisted-sector state,
which we choose again to be a single cycle of length $w$, each of the fields
can be expanded in a sum over fractional modes as in (3.22). We then note that
the integral over the sum of $\sigma$, $\tilde{\sigma}$ ensures that the $I$
dependence will drop out from the integral, leading to an overall $w$ factor.
The dependence on the Fourier modes of $\mathcal{H}^{(0)}$ and $\mathcal{P}$
will be the same as in the double-trace expression, but with $m\rightarrow
m/w$. Since each factor of $\mu$ in the expansion is accompanied by a factor
of $\mathcal{H}^{(0)}_{I}$ or $\mathcal{P}^{I}$, the overall factor of $w$
that multiplies the $\mu^{p}$ term in the expansion is $w^{-p}\times w\times
w$, where the last factor comes from the Fourier transform of the Green’s
function. The Fourier integrals also bring a factor of $R^{-(p-2)}$, by
dimensional analysis. We therefore notice that the $w$ and $R$ dependence is
such that it combines into precisely a dependence on $Rw$, which coincides,
upon lifting to the covering space, with the flow operator of the double-trace
$T\bar{T}$ \- deformed CFT. Including quantum corrections amounts to replacing
factors of $L_{m/w}$ in the expansion by factors of $n/wR$ that would result
from commuting two such fractional Virasoro modes, in perfect agreement with
the general counting. Thus, in the $w$-twisted sector,
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ acts just as the descent of
$\mathcal{X}_{T\bar{T}}$ on the covering space to the base cylinder.

More generally, if there are several twisted and untwisted sectors present -
as determined by the conjugacy class, $[g]$, of the permutation group - then
(3.44) reduces to a sum over sectors which, using the orthogonality of the
various subsectors of the Hilbert space, leads to the following expression for
the total flow operator $\mathcal{X}_{T_{I}\bar{T}_{I}}$ when acting on
$\mathcal{H}^{[g]}$

$\mathcal{X}^{[g]}_{T_{I}\bar{T}_{I}}=\sum_{\sigma\in
S_{N}}\mathcal{X}_{T_{I}\bar{T}_{I}}^{(\sigma(1)\ldots\sigma(w_{1}))}+\mathcal{X}_{T_{I}\bar{T}_{I}}^{(\sigma(w_{1}+1)\ldots\sigma(w_{1}+w_{2}))}+\ldots$
(3.47)

Here $w_{i}$ are the lengths of the cycles that appear in $[g]$, each
contributing with a flow operator constructed along the lines of the previous
paragraph, which lifts to the flow operator of the seed $T\bar{T}$ \- deformed
CFT on the cylinder of circumference $Rw_{i}$. The sum over permutations
considers all combinations in which the copies enter the cycles of $[g]$,
making the full operator well-defined on the Hilbert subspace
$\mathcal{H}^{[g]}$. Note that when all cycles have length one, this reduces
to the expression (3.40) for $\mathcal{X}$ in the untwisted sector.

Being obtained from the commutator (3.39), the above expression for
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ holds except possibly on the diagonal
degenerate subspaces. However, as we have just argued, we shouldn’t expect the
matrix elements on $\mathcal{H}_{D}$ to be fixed; our expression (3.47)
corresponds to a particular choice.

###  3.4 Symmetries of single-trace $T\bar{T}$ and $J\bar{T}$-deformed CFTs

Having discussed the flow operator, we would now like to study the extended
symmetries of single-trace $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs,
following the steps of the double-trace analysis. We therefore define a set of
Virasoro generators as171717Even though we use exactly the same notation as in
the double-trace case, it should be clear from the context that these are the
Virasoro generators in single-trace $T\bar{T}$/$J\bar{T}$ deformed CFTs.

$\partial_{\lambda}\widetilde{L}_{m}^{\lambda}=[\mathcal{X}_{T_{I}\bar{T}_{I}},\widetilde{L}_{m}^{\lambda}]\;,\;\;\;\;\;\widetilde{L}_{m}^{\lambda=0}=L_{m}^{\scriptscriptstyle{CFT}}$
(3.48)

#### Untwisted sector analysis

This equation is very easy to solve in the untwisted sector: since
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ is a single-trace operator and so is the
initial $L_{m}^{\scriptscriptstyle{CFT}}$, it follows that the flowed
$\widetilde{L}_{m}^{\lambda}$ will simply be

$\widetilde{L}_{m}^{\lambda}=\sum_{I}\widetilde{L}_{m}^{I,\lambda}$ (3.49)

where the $\widetilde{L}_{m}^{I,\lambda}$ represent the solution to the flow
equation in a single copy of a $T\bar{T}$ / $J\bar{T}$ \- deformed CFT. The
same definition can be extended to all the other Virasoro and Kac-Moody
generators of the seed symmetric product orbifold. Their algebra consists of
two-commuting copies of the Virasoro ($\ltimes$ Kac-Moody) algebra by
construction. As explained in the introductory subsection, the non-trivial
check that these generators represent symmetries of the theory is to prove
that they are conserved. For this, we compute their commutator with the
Hamiltonian

$[H,\widetilde{L}_{m}^{\lambda}]=\bigg{[}\sum_{I}H_{I},\sum_{J}\widetilde{L}_{m}^{J,\lambda}\bigg{]}=\sum_{I}\alpha_{m}^{I}\widetilde{L}_{m}^{I,\lambda}\;,\;\;\;\;\;\alpha_{m}^{I}\equiv\alpha_{m}(H_{I},P_{I})$
(3.50)

where the $\alpha_{m}$ for the cases of interest are given in (3.6). This
immediately implies that
$\sum_{I}e^{i\alpha_{m}^{I}t}\widetilde{L}_{m}^{I,\lambda}$ satisfies the
conservation equation (3.4). While one may worry that different operators
$\alpha_{m}^{I}$ appear in the time-dependent factors above, their expectation
value in any state of the symmetric orbifold is the same, as it cannot depend
on the particular copy.

#### Twisted sector analysis

In the twisted sectors on the cylinder, $\widetilde{L}_{m}^{\lambda}$ and
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ no longer take the form of a sum over copies.
As explained in the introductory subsection, in this sector we will
generically obtain fractionally-moded generators, of which the integer-moded
generators are a particular case. We will therefore discuss the preservation
of the fractional Virasoro and Kac-Moody symmetries generically. We again
define them via the flow equation

$\partial_{\lambda}\widetilde{L}^{\lambda}_{m/w}=[\mathcal{X}_{T_{I}\bar{T}_{I}},\widetilde{L}^{\lambda}_{m/w}]\;,\;\;\;\;\;\;\;\widetilde{L}_{m/w}^{\lambda=0}=\widetilde{L}_{m/w}^{\scriptscriptstyle{CFT}}$
(3.51)

This is a perfectly well-defined equation in the Hilbert space of the
$T\bar{T}$/$J\bar{T}$ symmetric orbifold. Since
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ belongs to the untwisted sector, it
infinitesimally takes a $w$-twisted sector operator to another one. The
standard Virasoro generators are obtained when $m$ is a multiple of $w$.

We would now like to show that the solution to this flow equation corresponds
precisely to the $T\bar{T}$ solution for the $\widetilde{L}_{m}^{\lambda}$ on
the covering space - which is a cylinder of size $Rw$ \- reinterpreted on the
base via (3.23) . As discussed at the beginning of this section, this map
makes no use of conformal invariance, but only sews together the different
copies. For simplicity, we assume again that the twist acts on the first $w$
copies of the seed, and as identity on the remaining $N-w$ ones. A summation
over $S_{N}$ will render the final result gauge-invariant.

As we have already discussed, in the $w$-twisted sector, the flow operator on
the base simply corresponds to the flow operator $\mathcal{X}_{T\bar{T}}$ of
the seed theory on the covering cylinder. Also, in the undeformed CFT, the
fractional Virasoro modes correspond to integer modes on the covering. It
follows that the solution to (3.51) will be the same as the solution to (3.3)
for the seed theory, but on a cylinder of radius $Rw$. Since we particularized
our discussion to a given number of copies on which the single-cycle twist
acts, only one of the $\mathcal{X}^{(w)}$ factors in (3.47) will act non-
trivially on the generator, but the final expressions will be symmetrized with
respect to $S_{N}$. For example, in the trivial case $w=1$, we obtain
$\widetilde{L}_{m}^{\lambda}$ in the seed for the first copy and the
symmetrization will yield the untwisted sector result (3.49). The same
procedure can be applied to the right-moving Virasoro generators, as well as
to the Kac-Moody currents.

In order for the solution to (3.51) to define a set of conserved operators, we
first need to compute its commutator with the Hamiltonian. This may be simply
evaluated on the cover, where it yields
$\alpha_{m}(H^{cov},...,wR)\widetilde{L}_{m}^{cov}$, where the dots stand for
the other conserved quantities that enter the definition (3.6) of $\alpha_{m}$
(i.e., momentum for $T\bar{T}$, and also $U(1)$ charge for $J\bar{T}$) and we
have dropped the index ‘$\lambda$’ from the generator, to lighten the
notation. Translating this to the base cylinder, we find
$\alpha_{m}(H^{(w)},...,wR)\widetilde{L}_{m/w}$. One may also derive these
expressions from the non-linear relation (2.56) between the undeformed and
deformed spectrum in the twisted sector, which implies a (non-linear and $w$
\- dependent) relation between $H^{(w)}$ and $\widetilde{L}_{0}^{(w)}$. That
this relation depends on the particular sector is not surprising, given that
the definition of $\widetilde{L}_{0}$ is sector-dependent.

Adding the appropriate time-dependence to ensure conservation, and taking into
account all the possible choices of copies that can enter the cycle, the full
answer for the fractional Virasoro generators then takes the form

$\widetilde{L}_{m/w}(t)=\sum_{\sigma\in
S_{N}}e^{i\alpha_{m}(H^{(\sigma(1)\ldots\sigma(w))},...,wR)t}\widetilde{L}_{m/w}^{(\sigma(1)\ldots\sigma(w))}$
(3.52)

where the superscripts indicate the particular copies entering the non-trivial
cycle. Note this reduces to the correct answer in the untwisted sector
($w=1$). The same holds for the right-moving Virasoro and Kac-Moody
generators. The algebra of the fractional Virasoro generators above is the
same as in the undeformed CFT, as follows from the definition (3.51). In
particular, for $m$ a multiple of $w$ we obtain the integer Virasoro modes,
which are present in any sector of the theory.

Note that in principle, these modes could also be obtained by flowing
$\mathcal{H}_{L,R}(\sigma)$ and then performing the Fourier transform, as in
[39]. Even though the operators we start from are single-trace operators,
$\sum_{I}\mathcal{H}_{L,R}^{I}(\sigma)$, the solution to the flow equation
looks differently in the different sectors because
$\mathcal{X}_{T_{I}\bar{T}_{I}}$ does. The result of the flow equation will
lead to a notion of emergent field-dependent coordinates, which will also be
defined in the twisted sectors; from the structure of the flow we see they
will correspond precisely to the field-dependent coordinates $u,v$ on the
covering space.

To conclude this section, we have explicitly shown that single-trace
$T\bar{T}$ and $J\bar{T}$ \- deformed CFTs contain operators that are
conserved and satisfy two commuting copies of the Virasoro ($\ltimes$ Kac-
Moody) algebra, showing they possess the corresponding symmetry. The central
charge and $U(1)$ level of the algebra are simply $N$ times those of the seed
theories. In twisted sectors, one finds also fractional Virasoro ($\ltimes$
Kac-Moody) conserved modes.

#### Other bases of generators

As discussed in section 3.1, the flowed Virasoro generators may not provide
the most natural basis of generators of the extended symmetries of these
theories. In single-trace $J\bar{T}$ \- deformed CFTs, the most natural basis
of left-moving generators is given by the standard generators of conformal and
affine $U(1)_{L}$ symmetries. In the untwisted sector, they take the form

$L_{m}=\sum_{I}L_{m}^{I}=\sum_{I}\widetilde{L}_{m}^{I}+\frac{\lambda}{R}H_{R}^{I}\tilde{J}_{m}^{I}+\frac{\lambda^{2}k}{8\pi
R}H_{R,I}^{2}\,\delta_{m,0}\;,\;\;\;\;\;\;J_{m}=\sum_{I}J_{m}^{I}=\sum_{I}\widetilde{J}_{m}^{I}+\frac{\lambda
k}{4\pi}H_{R}^{I}\,\delta_{m,0}$ (3.53)

and similarly for the right-moving generators, where we are now working in the
convention in which the $L_{m}$ are dimensionful. In the $w$-twisted sector,
one can write similar relations by considering the seed on the cylinder of
circumference $Rw$, as instructed by the solution of the flow equations

$L_{m/w}=\widetilde{L}_{m/w}+\frac{\lambda}{Rw}H_{R}^{(w)}\tilde{J}_{m/w}+\frac{\lambda^{2}k}{8\pi
Rw}H_{R}^{(w)2}\,\delta_{m,0}\;,\;\;\;\;\;\;J_{m/w}=\widetilde{J}_{m/w}+\frac{\lambda
k}{4\pi}H_{R}^{(w)}\,\delta_{m,0}$ (3.54)

where $H_{R}^{(w)}$ is the (globally-defined) right-moving Hamiltonian,
restricted to the $w$ \- twisted sector, with eigenvalues the $w$-twisted
sector energies (2.64). One may easily check that for $m=0$, these yield the
correct expression (2.64) for the deformed energies in the $w$ \- twisted
sector, taking into account the fact that the eigenvalue of
$\widetilde{L}_{0}$ is identical to that in the undeformed CFT. Note that,
throughout this section, the twisted generators will be built from the first
$w$ copies of the seed, and symmetrization is assumed only for the final
expressions.

Rewriting the relations above in terms of the effective Kac-Moody level in the
$w$-twisted sector $k^{(w)}=wk$, we obtain that the relation between the two
sets of generators is given by spectral flow with $\lambda H_{R}^{(w)}/w$.
Since the relation is non-linear, the Poisson algebra spanned by the untilded
generators is non-linear in these generators.

Note that for $m$ a multiple of $w$, we obtain the global generators of
extended symmetries, which correspond to the Fourier integrals of state (and
thus sector) - independent quantities over the base cylinder. This can be
established, at least classically, as follows: the solution for the generators
$L_{m},\bar{L}_{m}$ is the same as the double-trace solution, uplifted to the
covering space. E.g., for the left-movers, we have, using (3.23)

$L_{m}=L_{mw}^{cov}=\int_{0}^{wR}d\tilde{\sigma}\,e^{2\pi
im\tilde{\sigma}/R}\mathcal{H}_{L}(\tilde{\sigma})$ (3.55)

with $\mathcal{H}_{L}(\tilde{\sigma}+wR)=\mathcal{H}_{L}(\tilde{\sigma})$.
This integral can be reduced to the integral on an interval of size $R$ of
$\sum_{I=0}^{w-1}\mathcal{H}_{L}(\tilde{\sigma}+RI)$, which is periodic with
period $R$. Thus, the integral above becomes a local integral of the periodic
current on the base space. The same argument can also be applied to the right-
movers, the only difference being that the integrand is now a non-linear
function of the fields. We also expect it to extend to the quantum case; note
it implies that $L_{m},\bar{L}_{m}$ (whose most appropriate quantum definition
may or may not exactly coincide with (3.8), (3.9) with
$\lambda\rightarrow\lambda/w$) are the physical symmetry generators in the
full theory.

If $L_{m}$, $J_{m}$ and their right-moving counterparts are the global
symmetry generators in single-trace $J\bar{T}$ \- deformed CFTs then, given
that the relation between them and $\widetilde{L}_{m}$, $\widetilde{J}_{m}$,
etc. is that of spectral flow with parameter $\lambda H_{R}/w$, it follows
that the flowed generators are explicitly sector-dependent. This fact is not
suprising, given that the flow operator used to define these generators also
depends explicitly on the sector of the theory. It appears possible that
requiring at most implicit sector dependence of the symmetry generators in
single-trace $T\bar{T}$/$J\bar{T}$ \- deformed CFTs may provide a criterion
for selecting the physical basis of generators in these theories.

In single-trace $T\bar{T}$ \- deformed CFTs, one may argue, at least
classically, that the natural generators of “unrescaled” field-dependent
coordinate transformations in the untwisted sector of the symmetric product
orbifold should be

$Q_{m}=\sum_{I}Q_{m}^{I}=\sum_{I}R\,\widetilde{L}_{m}^{I}/R_{u}^{I}\;,\;\;\;\mbox{
with}\;\;\;\;\;\;R_{u}^{I}=R+2\mu H_{R}^{I}$ (3.56)

where the relevant classical expression for $\widetilde{L}_{m}$ for the
$T\bar{T}$ deformation is given in (3.12); dividing by the field-dependent
radius factors yields an expression for $Q_{m}$ that is the integral of a
quasi-local current. Once the appropriate quantum definition of the
“unrescaled” generators of the field-dependent symmetries is understood in the
seed theory, the result generalizes trivially in the untwisted sector as
above.

The algebra of these generators is a sum over copies of the non-linearly-
deformed Virasoro algebras obtained in the double-trace case, given up to
$\mathcal{O}(\hbar^{2})$ by

$[Q_{m},Q_{n}]=2\pi\hbar(m-n)\sum_{I}\frac{Q_{m+n}^{I}}{R+2\mu
H_{R}^{I}}+(m-n)\sum_{I}\frac{8\pi\hbar\,\mu^{2}H_{R}^{I}Q_{m}^{I}Q_{n}^{I}}{R\,R_{u}^{I}R_{H}^{I}}+\frac{\pi^{2}c\hbar\,m^{3}}{3}\sum_{I}\frac{1}{(R_{u}^{I})^{2}}\delta_{m+n}$
(3.57)

It is interesting to note that _new_ operators appear on the right-hand-side
in the single-trace case, rather than a product of operators that were already
among the generators, as in the double-trace. As we already discussed, it
would be good to understand more deeply whether this basis of operators may be
preferred for physical reasons and, if so, what is the significance of this
non-linear algebra.

In the $w$-twisted sector, one can introduce new fractional generators by
performing the division on the covering space of circumference $Rw$, with the
result

$\displaystyle Q_{m/w}=\frac{Rw\,\widetilde{L}_{m/w}}{R_{u}^{(w)}}$ (3.58)

with $R_{u}^{(w)}=Rw+2\mu H_{R}^{(w)}$. For $m$ multiple of $w$ we obtain the
integer versions on the base. These operators may be interpreted as
implementing field-dependent coordinate transformations on the covering space,
using a Fourier basis. The algebra of these operators is obtained by replacing
$Q^{I},H_{R}^{I}$ by $Q^{(w)},H_{R}^{(w)}$ in the above, $m$ by $m/w$, the
sums over copies with sums over cycles. Interestingly, if we take the
expectation value of this algebra (restricted to its integer modes) in a high-
energy state, the result precisely coincides with that of the asymptotic
symmetry group analysis of the asymptotically linear dilaton black hole
backgrounds performed in [40]. Note this effectively amounts to replacing
$\mu\rightarrow\mu/N$ in the double-trace algebra (3.15), which is precisely
what was found by the holographic analysis.

#### Higher spin currents

As discussed in the previous section, $T\bar{T}$ and $J\bar{T}$ \- deformed
CFTs also preserve the KdV charges associated with integrability. In their
single-trace version, it is natural to introduce the single-trace analogues of
the $\tilde{I}_{s}$, which in the untwisted sector are given as a sum over
copies of the corresponding expressions (3.16) - (3.17) with
$L_{n}^{I}\rightarrow\widetilde{L}_{n}^{I}$. Their conservation follows
trivially from the conservation of the flowed KdV charges in the seed
$T\bar{T}$/$J\bar{T}$ \- deformed CFT, given that the Hamiltonian is a sum
over the Hamiltonians in each copy. In the twisted sectors, their conservation
follows from that on the covering space.

The symmetric product orbifold of a two-dimensional CFT possesses, however,
many other higher spin conserved currents, associated to its much larger
symmetry algebra. As explained in e.g. [41], these higher-spin primary
currents only exist thanks to multi-trace contributions to an otherwise non-
primary field. One such higher-spin current that has been given as an example
therein is, for $R=2\pi$ and specialising to the untwisted sector

$(W_{4})_{n}=\sum_{I=1}^{N}\sum_{m}L_{m}^{I}L_{n-m}^{I}-\frac{3}{10}(n+2)(n+3)\sum_{I}L_{n}^{I}-\frac{\frac{22}{5c}+1}{N-1}\sum_{m,I\neq
J}L_{m}^{I}L_{n-m}^{J}$ (3.59)

One may wonder whether this current could be transported along the single-
trace $T\bar{T}$/$J\bar{T}$ flow in a similar manner to how the Virasoro and
KdV currents were transported, i.e. by requiring it to be covariantly constant
along the flow, which simply amonts to replacing the Virasoro modes by their
flowed counterparts. Taking $n=0$ for simplicity, we would like to show that
this simple procedure does not lead to a conserved charge.

For this, we compute the commutator of $(\widetilde{W}_{4})_{0}$ with the
Hamiltonian. The terms that correspond to single-trace sums are conserved,
following the same steps as we used to show the conservation of the KdV
charges. However, the multitrace terms lead to a different result, due to the
fact that the commutator $[L_{m}^{I},\alpha_{n}^{J}]$ only takes the form
(3.11) if $I=J$, and is zero otherwise. Using this, we find, for $I\neq J$

$[\widetilde{L}_{m}^{I}\widetilde{L}_{-m}^{J},H]=\widetilde{L}_{m}^{I}\alpha_{-m}^{J}\widetilde{L}_{-m}^{J}+\alpha_{m}^{I}\widetilde{L}_{m}^{I}\widetilde{L}_{-m}^{J}=(\alpha_{m}^{I}+\alpha_{-m}^{J})\widetilde{L}_{m}^{I}\widetilde{L}_{-m}^{J}$
(3.60)

In a CFT, $\alpha_{m}^{I}=m\hbar=-\alpha_{-m}^{J}$ so this term vanishes;
however, in $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs, this is no longer the
case. To leading order in $\mu$, $\alpha_{m}^{I}=m\hbar-2\mu
m\hbar(H^{I}-P^{I})$, implying that the leading term breaking the conservation
is proportional to
$\mu\hbar\sum_{m}m(H_{R}^{I}-H_{R}^{J})\widetilde{L}_{m}^{I}\widetilde{L}_{-m}^{J}$,
a combination that does not appear to vanish. Similar results hold in single-
trace $J\bar{T}$ \- deformed CFTs, on the right-moving side. Thus, the naïve
flowed charge associated to this higher-spin symmetry is not conserved in the
deformed theory, thus confirming the idea that the conservation of the flowed
Virasoro generators and KdV charges is a special feature of these operators,
which does not extend to arbitrary currents in the theory. This analysis
points towards the conclusion that the associated higher spin symmetries are
broken by the single-trace $T\bar{T}$/ $J\bar{T}$ deformations; one should
ascertain though that it is not possible to construct corrections to the
charges that would restore their conservation.

##  4 Correlation functions

In this section, we would like to discuss correlation functions of operators
in symmetric product orbifolds of $T\bar{T}$ and, especially, $J\bar{T}$ \-
deformed CFTs. Since these theories are neither conformal, nor local, our
focus and methods will naturally be quite different from the standard
discussion of correlation functions in symmetric product orbifolds of CFTs
[72, 73, 75, 74], which is centered around computing correlators of twist
operators using non-trivial covering maps, and makes essential use of the
conformal transformation properties of the operators in question. We will
instead concentrate on momentum-space operators, which are natural to consider
in a non-local theory, and our main goal will be to understand how to choose a
special basis of these operators and compute their correlation functions in
terms of the correlators of the undeformed symmetric orbifold CFT.

Our main focus will be $J\bar{T}$ \- deformed CFTs, for whose double-trace
version [42] has proposed a concrete basis of “primary operator analogues” and
computed their correlation functions exactly. The main goal of this section
will be to adapt this prescription to the single-trace $J\bar{T}$ case and use
it to compute correlation functions of both untwisted and twisted-sector
operators. It is worth mentioning that, quite recently, [44] has also put
forth a special basis of operators in $T\bar{T}$ \- deformed CFTs and computed
their correlation functions, obtaining similar expressions. While a
generalisation of these results to the single-trace case would be both
interesting and likely possible using their formalism, we do not address this
problem here, mainly because it would require a very different method than the
one we use for the $J\bar{T}$ case.

The correlation functions in the symmetric product orbifold of $T\bar{T}$ and
$J\bar{T}$ \- deformed CFTs can then be compared to the correlation functions
evaluated using worldsheet techniques in the holographic setups that have been
related to these deformations. More precisely, they are expected to match the
correlation functions of vertex operators associated to long strings181818By
contrast, the short string sector is not described by a symmetric product
orbifold, and neither their spectrum, nor their correlation functions [21, 22,
23] match those in $T\bar{T}$/$J\bar{T}$ \- deformed CFTs. in these
backgrounds, which are well described by a symmetric product orbifold. Such
worldsheet correlation functions were recently computed in [25] for the case
of the asymptotically linear dilaton background, which is related to the
single-trace $T\bar{T}$ deformation; their large-momentum behaviour in the
untwisted sector agrees with that found by [44]. One may similarly compute
correlation functions of long string vertex operators in warped AdS3 by
adapting the results of [20] along the lines of [25], and then compare them
with our single-trace $J\bar{T}$ result; we find a slight disagreement on the
non-local side that we comment upon.

For completeness, we start this section with a review of the correlation
functions in double-trace $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs, focusing
on the explicit proposal of [42] for the latter case.

###  4.1 Review of correlation functions in $T\bar{T}$ and $J\bar{T}$ \-
deformed CFTs

In local QFTs, there is a special set of operators whose correlation functions
are interesting to study, namely local operators. This set is further
specialised in CFTs, where much of the focus is on operators that transform as
primaries under conformal transformations, as their correlation functions are
highly constrained by conformal invariance.

Since $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs are non-local, it is not a
priori clear which are the natural operators to consider, if a preferred basis
exists at all [76]. Indeed, due to the non-locality of these theories,
operators are best defined in momentum space. The computation of their
correlation functions using e.g. conformal perturbation theory yields
divergences, which need to be subtracted via counterterms. Since in a non-
local QFT the structure of the allowed counterterms is in general not known,
the finite part of the correlator may itself become ambiguous. The situation
is under better control in $J\bar{T}$-deformed CFTs, whose locality and
$SL(2,\mathbb{R})$ invariance on the left-moving side single out a set of
primary operators under these symmetries, for which at least the left-moving
part of the correlation function is fixed [43]; however, these questions
remain for the right-moving, non-local piece of the correlator.

Despite these concerns and complications, there has been much recent progress
in the computation of correlation functions of both $T\bar{T}$ and $J\bar{T}$
\- deformed CFTs. Upon making a judicious choice of the operators whose
correlation functions one would like to study, it can be shown that the end
result is a simple integral transform of the correlation functions of the
original CFT. For the case of two-and-three point functions, this integral
simply yields the CFT momentum-space correlator, with the conformal dimensions
replaced by certain momentum-dependent combinations. Thus, the deformed
correlators, though explicitly non-local, are directly and universally
determined by the correlation functions in the undeformed CFT, indicating that
both $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs have a structure that is as
rigid as that of two-dimensional CFTs.

####  $T\bar{T}$ \- deformed CFTs

The effect of the $T\bar{T}$ perturbation on correlation functions can be
studied at small coupling using conformal perturbation theory, where one can
easily note that it induces a momentum-dependent correction to the conformal
dimension [77]. The first all-orders analysis of the correlation functions of
$T\bar{T}$-deformed CFTs has been performed by [45], who also studied their
leading UV divergences and showed they can be absorbed into a non-local
renormalization of the operators. More recently, [44] approached this problem
using the path integral formulation of the $T\bar{T}$ deformation in terms of
coupling the undeformed CFT to JT gravity [78, 79]. As it is most clear from
its vielbein formulation, this description relates the $T\bar{T}$ \- deformed
dynamics to that of the original CFT, but seen through a set of “dynamical
coordinates” that are related to the $T\bar{T}$ ones in a universal, but
field-dependent fashion. These coordinates parametrize a flat target space,
identified with the space of the undeformed CFT. The basis of operators
considered in [44] corresponds to the original CFT operators on this space or,
alternatively, their Fourier transform with respect to the target space
coordinates. The correlation function of these momentum-space operators is
obtained by carefully performing the JT path integral and absorbing the UV
divergences into a renormalization of the operators. The procedure is rather
subtle and involved, and the final result takes the form

$\langle\mathcal{O}_{1}(p_{1})\ldots\mathcal{O}_{n}(p_{n})\rangle\propto\delta(\sum_{i=1}^{n}p_{i})\int\prod_{i}d^{2}\sigma_{i}\,\delta(\sigma_{1})e^{i\sum_{i}p_{i}\sigma_{i}}\mathcal{F}(p_{i})\langle\mathcal{O}_{1}(\sigma_{1})\ldots\mathcal{O}_{n}(\sigma_{n})\rangle_{CFT}\prod_{i<j}\left(\Lambda|\sigma_{ij}|\right)^{\frac{\mu
p_{i}\cdot p_{j}}{\pi}}$ (4.1)

where $\Lambda$ is a renormalization scale and $\mathcal{F}(p_{i})$ is a
smooth function of the momenta that grows at most polynomially at large
$p_{i}$. If one ignores this term and considers e.g. the two-point function,
one finds it corresponds precisely to the CFT momentum-space correlator

$\langle\mathcal{O}(p,\bar{p})\mathcal{O}(-p,-\bar{p})\rangle=\frac{(2\pi)^{2}}{2^{2(h+\bar{h})}\sin(\pi(h+\bar{h}))}\,\frac{p^{2h-1}\bar{p}^{2\bar{h}-1}}{\Gamma(2h)\Gamma(2\bar{h})}$
(4.2)

but with the dimensions replaced by the momentum-dependent combinations (in
our conventions)

$h=h(p,\bar{p})=h_{\scriptscriptstyle{CFT}}+\frac{\mu}{\pi}p\bar{p}\;,\;\;\;\;\;\;\;\bar{h}=\bar{h}(p,\bar{p})=\bar{h}_{\scriptscriptstyle{CFT}}+\frac{\mu}{\pi}p\bar{p}$
(4.3)

This is precisely the answer obtained in [25] by computing the correlation
functions of long string worldsheet vertex operators, as we review in section
4.3. It is interesting to ask whether a different renormalization of the
operators in [44] could reproduce this exact formula.

####  $J\bar{T}$ \- deformed CFTs

As already stated, the main goal of this section is to compute correlation
functions of both untwisted and twisted-sector operators in single-trace
$J\bar{T}$ \- deformed CFTs. For this, we will adapt the prescription of [42]
for defining appropriate analogues of primary operators and computing their
correlation functions. In order to facilitate the generalisation of these
results to the single-trace case, we present the construction of [42] in a
slightly different fashion, which we hope also makes its physical
interpretation more transparent.

This construction relies on the interplay of physical and flowed Virasoro
generators in $J\bar{T}$ \- deformed CFTs. Let us start by discussing the
left-moving sector, where the meaning of the various operators is very clear.
As we already explained, the standard $SL(2,\mathbb{R})_{L}$ conformal
symmetry of the theory unambigously identifies a set of left primary states
whose conformal dimensions become momentum-dependent due to the irrelevant
deformation [80]

$h(\bar{p})=h^{[0]}+\lambda
q^{[0]}\bar{p}+\frac{\lambda^{2}k\bar{p}^{2}}{4}\;,\;\;\;\;\;\;q(\bar{p})=q^{[0]}+\frac{\lambda
k\bar{p}}{2}$ (4.4)

where $h^{[0]},q^{[0]}$ are the undeformed left conformal dimension and
charge. Note this relation takes precisely the form of a spectral flow by
$\lambda\bar{p}$. To simplify the expressions in this section, we have
rescaled the deformation parameter $\lambda\to 2\pi\lambda$ with respect to
the previous ones. We also set $R=2\pi$.

The state-operator correspondence, which is still valid [81], then predicts
the existence of a set of primary operators with these conformal dimensions,
which obey the standard Ward identities with respect to the true conformal
generators $L_{m},J_{m}$, and for which the left-moving part of the
correlation function is fixed by $SL(2,\mathbb{R})_{L}$ in the standard way.
To construct them, one starts by formally defining the ‘flowed’ operators [65]

$\partial_{\lambda}\widetilde{\mathcal{O}}^{\lambda}(\zeta,\bar{\zeta})=[\mathcal{X}_{J\bar{T}},\widetilde{\mathcal{O}}^{\lambda}(\zeta,\bar{\zeta})]$
(4.5)

where $\zeta,\bar{\zeta}$ are the coordinates on the cylinder and
$\mathcal{X}_{J\bar{T}}$ is the operator that drives the flow of the energy
eigenstates in the $J\bar{T}$ \- deformed CFT191919The flow is defined on a
fixed time slice, say $\tau=0$. The operator at $\tau\neq 0$ is simply defined
as
$\mathcal{O}_{\scriptscriptstyle{CFT}}(\tau,\sigma)=e^{iH_{\scriptscriptstyle{CFT}}\tau}\mathcal{O}_{\scriptscriptstyle{CFT}}(0,\sigma)e^{-iH_{\scriptscriptstyle{CFT}}\tau}$,
and the whole expression is then flowed. . By construction, their correlation
functions in the flowed vacuum state and their commutation relations with
$\widetilde{L}_{n},\widetilde{J}_{n}$ are identical to those in the undeformed
CFT

$[\widetilde{L}_{n},\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})]=e^{n\zeta}\bigg{(}n\,h^{[0]}\,\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})+\partial_{\zeta}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})\bigg{)}\;,\;\;\;\;\;\;[\widetilde{J}_{n},\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})]=e^{n\zeta}q^{[0]}\,\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})$
(4.6)

and similarly on the right-moving side. Despite this fact, the
$\tilde{\mathcal{O}}(\zeta,\bar{\zeta})$ do not correspond to physical
operators in the theory and are non-local even on the left-moving side; thus,
$\zeta,\bar{\zeta}$ should be simply viewed as labels inherited from the
undeformed CFT.

The operators we are interested in should instead be primary with respect to
the untilded Virasoro generators (3.8), which in our new conventions read

$L_{m}=\widetilde{L}_{m}+\lambda
H_{R}\widetilde{J}_{m}+\frac{\lambda^{2}k}{4}\,H_{R}^{2}\,\delta_{m,0}\;,\;\;\;\;\;\;J_{m}=\widetilde{J}_{m}+\frac{\lambda
k}{2}\,H_{R}\,\delta_{m,0}$ (4.7)

with left dimensions given by (4.4). In order for these dimensions to make
sense, they should also be eigenoperators of $H_{R}$, with eigenvalue
$\bar{p}$. We will find it convenient to work in a mixed basis,
$(\zeta,\bar{p})$. The operators should satisfy

$[H_{R},\mathcal{O}(\zeta,\bar{p})]=\bar{p}\,\mathcal{O}(\zeta,\bar{p})$ (4.8)

Since the relation between the undeformed and deformed dimension is given by
spectral flow, the relation between the physical and tilded operator should
involve dressing by an appropriate vertex operator. Note, however, that
$\widetilde{\mathcal{O}}$ already carries the correct charges with respect to
$J_{0}$ and $L_{0}$, implying that we should remove by hand the zero mode of
the vertex operator involved. Our Ansatz is, therefore

$\mathcal{O}(\zeta,\bar{p})=\int
d\bar{\zeta\,}e^{-\bar{\zeta}\bar{p}}:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):e^{\mathcal{Y}_{\mathcal{O}}\zeta+\bar{\mathcal{Y}}_{\mathcal{O}}\bar{\zeta}}$
(4.9)

where the normal-ordered dressed operator is given by

$:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})\\!:\;\equiv\;\widetilde{\mathcal{V}}_{\eta}^{+}(\zeta)\widetilde{\bar{\mathcal{V}}}_{\eta}^{+}(\bar{\zeta})\,\widetilde{\mathcal{O}}(\zeta,\bar{\zeta})\,\widetilde{\bar{\mathcal{V}}}^{-}_{\eta}(\bar{\zeta})\widetilde{\mathcal{V}}^{-}_{\eta}(\zeta)$
(4.10)

Here, $\widetilde{\mathcal{V}}^{\pm}$ represent the positive and,
respectively, negative-frequency parts of the dressing operator and the
$\,\widetilde{}\,$ stands for the fact that they are simply the $J\bar{T}$
flow of an identical expression in the undeformed CFT. With foresight, we have
included vertex operator dressings for both the left and right-movers, and
have left the spectral flow parameter arbitrary in order to be able to reuse
this computation in the single-trace case. The argument above yields the
following expression for the left dressing operators

$\widetilde{\mathcal{V}}_{\eta}^{+}(\zeta)=e^{\eta\sum_{n=1}^{\infty}\frac{1}{n}\widetilde{J}_{-n}e^{n\zeta}}\;,\;\;\;\;\;\;\;\widetilde{\mathcal{V}}_{\eta}^{-}(\zeta)=e^{\eta\left(\widetilde{J}_{0}\,\zeta-\sum_{n=1}^{\infty}\frac{1}{n}\widetilde{J}_{n}e^{-n\zeta}\right)}$
(4.11)

Note in (4.9) we have also allowed for a correction,
$\mathcal{Y}_{\mathcal{O}}$, to the zero mode of the current, which commutes
with all the left generators and will be fixed by the commutation relations of
$\mathcal{O}(\zeta,\bar{p})$ with the physical Virasoro generators.

In order to prepare the ground for generalizing these results to the single-
trace case, it will be useful to compute of the Ward identities obeyed by
$\mathcal{O}(\zeta,\bar{p})$ in two steps: first, we compute the commutation
relations of $:\\!\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}\\!\>$ with
the left-moving flowed generators using the explicit expressions (4.11) for
the dressings; then, we assemble them into commutation relations of the non-
linear combinations (4.7) with $\mathcal{O}(\zeta,\bar{p})$. The advantage of
performing the first step separately is that the commutation relations of
$:\\!\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}\\!:$ with the flowed
generators are _identical_ to the corresponding expressions in the undeformed
CFT, given that the flow acts by simple conjugation. The result is

$[\widetilde{J}_{m},\,:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):]=e^{m\zeta}\bigg{(}q^{[0]}+\frac{\eta
k}{2}(1-\delta_{m,0})\bigg{)}:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):$
(4.12)
$[\widetilde{L}_{m},\,:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):]=e^{m\zeta}\left(m(h^{[0]}+\eta
q^{[0]}+\frac{k\eta^{2}}{4})-\frac{\eta^{2}k}{4}(1-\delta_{m,0})+\partial_{\zeta}\right):\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):-\eta:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}(\zeta,\bar{\zeta}):\tilde{J}_{m}$

and corresponds, as it is clear from (4.11), to the CFT Ward identities for a
normal-ordered left vertex operator of charge $q^{[0]}+\eta k/2$, but with
part of its zero mode left out (so that the charge carried with respect to
$\widetilde{J}_{0}$ is just $q^{[0]}$). Note the
$\widetilde{\bar{\mathcal{V}}}_{\eta}$ (constructed from right-moving current
modes only) do not affect the left Ward identities.

It is then not hard to check that the commutation relations of
$\mathcal{O}(\zeta,\bar{p})$ with the generators (4.7) are precisely those of
a primary of the expected dimension and charge, provided we set
$\eta=\lambda\bar{p}$ and

$\mathcal{Y}_{\mathcal{O}}=\lambda
qH_{R}+\lambda\bar{p}q^{[0]}+\frac{k\lambda^{2}\bar{p}^{2}}{4}$ (4.13)

This can also be seen from the fact that the left-moving piece of
$\mathcal{O}(\zeta,\bar{p})$ organises into a left vertex operator of charge
$q$ in the deformed theory, i.e. constructed from the current modes $J_{m}$:
the dressings (4.11) shift the charge from $q^{[0]}$ to $q$ for the non-zero
modes, whereas the the first term in the expression for
$\mathcal{Y}_{\mathcal{O}}$ shifts the $\widetilde{J}_{0}$ term in (4.9)
(whose total coefficient is $2q/k$) to $J_{0}$. The last two terms correspond
to the difference $h(\bar{p})-h^{[0]}$ in the operator dimensions, which
becomes important when translating the operators defined on the cylinder to
those on the plane via the standard map
$\mathcal{O}_{(cyl)}(\zeta)=e^{h\zeta}\mathcal{O}_{(pl)}(z)$. Finally, the
total charge of the operator, as measured by $J_{0}$, is $q$. One may check
that, upon performing a conformal transformation to the plane, the left-moving
piece of (4.9) corresponds precisely to a local left vertex operator in the
deformed theory with charge $q$, with no mismatch between the coefficients of
the zero and non-zero modes.

Having understood in detail the construction of the left-moving piece of the
operator - which is entirely fixed by conformal symmetry - via the dressing
discussed above, the proposal of [42] consists in choosing the dressing
operators for the right-movers to have an identical form, but with all
quantities barred. Requiring that the resulting operator be an eigenoperator
of $H_{R}$202020In practice, this is obtained by writing the commutator of
$\mathcal{O}$ with $\widetilde{\bar{L}}_{0}$ in two different ways, see [42]
for details., we obtain the following expression for
$\bar{\mathcal{Y}}_{\mathcal{O}}$

$\bar{\mathcal{Y}}_{\mathcal{O}}=\lambda\bar{p}(\widetilde{J}_{0}-\widetilde{\\!\bar{J}}_{0})+\lambda
qH_{R}+\lambda\bar{p}q^{[0]}+\frac{k\lambda^{2}\bar{p}^{2}}{4}$ (4.14)

The right-moving piece of $:\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}:$
can again be interpreted as a vertex operator in the undeformed CFT, of charge
$\bar{q}^{[0]}+k\eta/2$, but with a discrepancy between the coefficient of the
zero and non-zero modes, which is then trivially flowed to the deformed
theory. A factor of $\lambda\bar{q}H_{R}$ in the expression for
$\bar{\mathcal{Y}}_{\mathcal{O}}$ could again be understood as a correction to
the zero mode of the right-moving current. However, since this term does not
commute with the right current generators, we no longer have a useful
interpretation for the full operator as a vertex operator in the deformed
theory. In addition, there are a number of discrepancies involving winding
between the operator we obtain and a naïve spectrally-flowed operator on the
right-moving side, at least at finite $R$. These discrepancies can presumably
be traced back to the fact that $\bar{L}_{0}$ and $H_{R}$ differ by winding
terms. In any case, given the explicit form (4.9) of the operator, the the
commutation relations with the generators (3.9) can be shown to take the form
of CFT Ward identities in the limit $R\rightarrow\infty$, with

$\bar{h}(\bar{p})=\bar{h}^{[0]}+\lambda\bar{q}^{[0]}\bar{p}+\frac{k\lambda^{2}\bar{p}^{2}}{4}\;,\;\;\;\;\;\;\bar{q}(\bar{p})=\bar{q}^{[0]}+\frac{k\lambda\bar{p}}{2}$
(4.15)

which resembles a spectral flow transformation by $\lambda\bar{p}$ of the
right-moving conformal dimensions and charges.

Let us now compute correlation functions of these operators. Since
$\mathcal{Y}_{\mathcal{O}},\bar{\mathcal{Y}}_{\mathcal{O}}$ only involve
$H_{R}$ and the winding and the $\mathcal{O}(\zeta,\bar{p})$ are
eigenoperators of both, the correlation function can be simplified to

$\displaystyle\langle\mathcal{O}_{1}(\zeta_{1},\bar{p}_{1})\ldots\mathcal{O}_{n}(\zeta_{n},\bar{p}_{n})\rangle$
$\displaystyle=\int\prod_{i=1}^{n}d\bar{\zeta}_{i}\,e^{-\sum_{i}\bar{p}_{i}\bar{\zeta}_{i}}\,e^{\sum_{i}(\lambda\bar{p}_{i}q_{i}^{[0]}+\frac{k\lambda^{2}\bar{p}_{i}^{2}}{4})(\zeta_{i}+\bar{\zeta}_{i})}\,e^{\lambda
H_{R}^{vac}q_{i}(\zeta_{i}+\bar{\zeta}_{i})}\times$
$\displaystyle\times\;e^{\sum_{i<j}(\lambda\bar{p}_{i}(q_{j}^{[0]}-\bar{q}_{j}^{[0]})\bar{\zeta}_{i}+\lambda
q_{i}\bar{p}_{j}(\zeta_{i}+\bar{\zeta}_{i}))}\,\langle:\\!\widetilde{\mathcal{V}}_{1}\widetilde{\mathcal{O}}_{1}(\zeta_{1},\bar{\zeta}_{1})\\!:\ldots:\\!\widetilde{\mathcal{V}}_{n}\widetilde{\mathcal{O}}_{n}(\zeta_{n},\bar{\zeta}_{n})\\!:\rangle$
(4.16)

where $H_{R}^{vac}$ represents the eigenvalue of $H_{R}$ in the flowed vacuum.
The last term in the integrand is almost a correlation function of vertex
operators of charges $q_{i}$ in the undeformed CFT, up to the missing zero
modes. Using the explicit expressions for the vertex operators to evaluate the
zero mode contribution, we obtain

$\displaystyle\langle:\\!\widetilde{\mathcal{V}}_{1}\widetilde{\mathcal{O}}_{1}(\zeta_{1},\bar{\zeta}_{1})\\!:\ldots:\\!\widetilde{\mathcal{V}}_{n}\widetilde{\mathcal{O}}_{n}(\zeta_{n},\bar{\zeta}_{n})\\!:\rangle\,=\,e^{-\sum_{i<j}\lambda\bar{p}_{j}(q_{i}\zeta_{i}+\bar{q}_{i}\bar{\zeta}_{i})}e^{-\sum_{i}(\lambda\bar{p}_{i}(q_{i}^{[0]}\zeta_{i}+\bar{q}_{i}^{[0]}\bar{\zeta}_{i})+\frac{k\lambda^{2}\bar{p}_{i}^{2}}{4}(\zeta_{i}+\bar{\zeta}_{i}))}\,\times$
$\displaystyle\hskip
1.42271pt\times\prod_{i<j}\left(e^{\frac{\zeta_{ij}}{2}}-e^{-\frac{\zeta_{ij}}{2}}\right)^{\frac{2}{k}(q_{i}q_{j}-q_{i}^{[0]}q^{[0]}_{j})}\left(e^{\frac{\bar{\zeta}_{ij}}{2}}-e^{-\frac{\bar{\zeta}_{ij}}{2}}\right)^{\frac{2}{k}(\bar{q}_{i}\bar{q}_{j}-\bar{q}^{[0]}_{i}\bar{q}^{[0]}_{j})}\langle\widetilde{\mathcal{O}}_{1}(\zeta_{1},\bar{\zeta}_{1})\ldots\widetilde{\mathcal{O}}_{n}(\zeta_{n},\bar{\zeta}_{n})\rangle$
(4.17)

where $\zeta_{ij}=\zeta_{i}-\zeta_{j}$,
$\bar{\zeta}_{ij}=\bar{\zeta}_{i}-\bar{\zeta}_{j}$. The second line simply
represents the spectrally-flowed correlation function on the cylinder. The
first exponential factor on the first line corresponds to the correction due
to the missing zero modes, while the second one accounts for the change in the
definition of the cylinder operators due to the shift in their dimensions. The
same result can be obtained by commuting the current modes until they
annihilate the (flowed) vacuum. Plugging in this expression into (4.16) and
reinstating the factors of the radius, the final result for the $n$-point
function simplifies to

$\displaystyle\langle\mathcal{O}_{1}(\zeta_{1},\bar{p}_{1})\ldots\mathcal{O}_{n}(\zeta_{n},\bar{p}_{n})\rangle$
$\displaystyle=\int\prod_{i=1}^{n}d\bar{\zeta}_{i}\,e^{-\sum_{i}\bar{p}_{i}\bar{\zeta}_{i}}\,e^{\frac{2\pi}{R}\mathcal{A}}\prod_{i<j}\left(e^{\frac{\pi\zeta_{ij}}{R}}-e^{-\frac{\pi\zeta_{ij}}{R}}\right)^{\frac{2}{k}(q_{i}q_{j}-q^{[0]}_{i}q_{j}^{[0]})}\,\times$
$\displaystyle\times\,\left(e^{\frac{\pi\bar{\zeta}_{ij}}{R}}-e^{-\frac{\pi\bar{\zeta}_{ij}}{R}}\right)^{\frac{2}{k}(\bar{q}_{i}\bar{q}_{j}-\bar{q}^{[0]}_{i}\bar{q}^{[0]}_{j})}\langle\mathcal{O}_{1}^{\scriptscriptstyle{CFT}}(\zeta_{1},\bar{\zeta}_{1})\ldots\mathcal{O}^{\scriptscriptstyle{CFT}}_{n}(\zeta_{n},\bar{\zeta}_{n})\rangle$
(4.18)

where we plugged in the fact that the correlation function of the
$\widetilde{\mathcal{O}}$ is identical to the corresponding undeformed CFT
correlator and the exponent $\mathcal{A}$ can be written, using charge
conservation, as

$\displaystyle\mathcal{A}$ $\displaystyle=\sum_{i=1}^{n}\lambda
H_{R}^{vac}q_{i}(\zeta_{i}+\bar{\zeta}_{i})+\sum_{i<j}\lambda\bar{p}_{j}(q_{i}-\bar{q}_{i})\bar{\zeta}_{ij}$
(4.19)

If we ignore this exponential factor, then the result of the integral is
simply the spectrally-flowed CFT correlation function in the mixed
$(\zeta_{i},\bar{p}_{i})$ basis. Note that the spectral flow only affects the
operator dimensions, but not the OPE coefficients of appropriate Virasoro-Kac-
Moody primaries. In particular, for the case of two-and three-point function,
the spectral flow will simply shift the operator dimensions to the momentum-
dependent combinations (4.4), (4.15) in the corresponding mixed
position/momentum-space correlator. Including the exponential factor will
simply shift the coefficients of the $\bar{\zeta}_{i}$ in the Fourier integral
by a factor that, importantly, is proportional to $1/R$. The result will be
the spectrally-flowed CFT momentum-space correlator evaluated at momenta that
differ from $\bar{p}_{i}$ by terms that vanish in the decompactification
limit.

Note that on the left-moving side, the only term that could potentially upset
the conformal invariance of the spectrally-flowed correlator is the shift
proportional to $H_{R}^{vac}\zeta_{i}$ in the expression for $\mathcal{A}$. As
discussed in [42], this discrepancy is related to the lack of standard
$SL(2,\mathbb{R})_{L}$ invariance of the flowed vacuum in finite size, which
also disappears in the $R\rightarrow\infty$ limit.

###  4.2 Correlation functions in single-trace $J\bar{T}$ \- deformed CFTs

Let us, for completeness, start with some generalities. There are several
classes of operators one may discuss in a symmetric product orbifold QFT:
untwisted or twisted, single-trace or multitrace. Operators that are untwisted
take the form

$\sum_{\sigma\in
S^{N}}\mathcal{O}_{\sigma(1)}^{1}\otimes\mathcal{O}^{2}_{\sigma(2)}\otimes\ldots\otimes\mathcal{O}^{N}_{\sigma(N)}$
(4.20)

where all insertions are at the same position $(\zeta,\bar{\zeta})$. In the
particular case where all but one operator are the identity, this reduces to
$\sum_{I}\mathcal{O}_{I}(\zeta,\bar{\zeta})$, which is a single-trace
untwisted operator. When more than one insertions are non-trivial, the
operator is called multitrace. The Fourier transform of these operators can be
written as a sum over copies, e.g. $\sum_{I}\mathcal{O}_{I}(p,\bar{p})$ in the
single-trace case, provided one acts on a state from the untwisted sector. The
multi-trace untwisted operators take a similar form, but with several sums
over copies and an integral over all the possible ways to distribute the
momenta among them: e.g., in the double-trace case, the relevant operators
take the form $\sum_{I,J}\int
d^{d}k\,\mathcal{O}^{1}_{I}(k)\mathcal{O}^{2}_{J}(p-k)$. The untwisted-sector
correlation functions of such operators can be straightforwardly computed
using those of the seed: in the single-trace case, they are simply
proportional to seed correlators; for multi-trace operators, one obtains a
momentum-space convolution of seed correlation functions, which may be
evaluated if the latter are known explicitly.

Thus, all the new correlators with respect to the seed lie in the twisted
sector. One may again classify the twisted-sector operators into single-trace
and multi-trace. For the single-trace operators, the twist involves only one
non-trivial cycle; for the multitrace ones, several cycles should be
considered. The end result should of course be symmetrized with respect to all
permutations of the various copies.

The construction of twisted-sector correlation functions in single-trace
$T\bar{T}$ and $J\bar{T}$ \- deformed CFTs should depend on the particular
method used to build the corresponding double-trace correlators. In the
$T\bar{T}$ case, the method of [44] uses the JT formulation of the
deformation, which involves maps from the space where the theory is defined to
a flat target space. Given the similarity between this formalism and one
involving the string worldsheet, for example in what regards the computation
of the finite-size spectrum [26, 27], it appears reasonable to assume that the
correlation functions in the twisted sectors will be captured by considering
“worldsheets” in the JT formulation that wind around the target space. The
string worldsheet computation of [25] suggests the effect of the winding on
the correlation functions is extremely straightforward, as it simply amounts
to replacing the $T\bar{T}$ coupling $\mu$ by $\mu/w$ in the twisted sectors.

The main goal of this section is to put forth an appropriate basis of twisted-
sector operators in single-trace $J\bar{T}$ \- deformed CFTs and to compute
their correlation functions. We will be following step-by-step the procedure
used in the double-trace case. We will concentrate on single-trace operators,
which contain a twist corresponding to a single non-trivial cycle of length
$w$, inserted at their location. The operators of interest are primary on the
left-moving side, with the following momentum-dependent $SL(2,\mathbb{R})_{L}$
dimensions and $U(1)$ charges

$h^{(w)}(\bar{p})=h^{(w)}_{\scriptscriptstyle{CFT}}+\frac{\lambda\bar{p}}{w}\,q^{[0]}+\frac{k\lambda^{2}\bar{p}^{2}}{4w}\;,\;\;\;\;\;q=q^{[0]}+\frac{\lambda
k\bar{p}}{2}$ (4.21)

which were computed in section 2.3 via an infinite boost of the twisted-sector
deformed energies. $h^{(w)}_{\scriptscriptstyle{CFT}}$ is the dimension of the
twisted-sector operator in the undeformed symmetric product orbifold CFT,
which is related to a seed dimension by the holomorphic counterpart of (2.47).
Note $\lambda$ has been rescaled by a factor of $2\pi$ with respect to (2.65).
The extension to several non-trivial cycles is straightforward, as the
dimensions are additive.

The relation between the deformed and undeformed twisted-sector dimensions
corresponds to a transformation known as _fractional_ spectral flow [61, 62],
defined as the base space transformation that lifts to the standard spectral
flow on the covering space. It acts as

$h^{(w)}\rightarrow
h^{(w)}+\eta\,q^{(w)}+\frac{kw\,\eta^{2}}{4}\;,\;\;\;\;\;\;\;\;q^{(w)}\rightarrow
q^{(w)}+\frac{kw\,\eta}{2}$ (4.22)

on the base. In the standard discussions, $\eta$ is taken to be fractional,
hence the name. In our case, $\eta=\lambda\bar{p}/w$ is a continuous
parameter, so its fractionality is not very important; what is important is
that the level that enters the spectral flow transformation is the effective
level in the $w$-twisted sector, $k^{(w)}=kw$.

As argued in section 3.4, the physical generators of conformal symmetries in
the $w$ \- twisted sectors are the integer modes212121In principle, one could
also consider fractional Virasoro modes around the location of the twist
operator. The cylinder fractional modes appearing in (3.54) are associated to
a twist operator inserted at $-\infty$, whereas in this section we will be
considering twists inserted at finite distance on the cylinder. The two sets
of fractional generators can be related provided no other operator insertion
is present - an overly restrictive requirement. On the other hand, the
integer-moded Virasoro generators are everywhere well-defined. of (3.54),
which are related to the flowed generators in that sector by a transformation
resembling fractional spectral flow with parameter $\lambda H_{R}/w$

$L_{m}=\widetilde{L}_{m}+\frac{\lambda}{w}\,H_{R}\widetilde{J}_{m}+\frac{\lambda^{2}k}{4w}H_{R}^{2}\,\delta_{m,0}\;,\;\;\;\;\;\;\;J_{m}=\widetilde{J}_{m}+\frac{\lambda
k}{2}H_{R}\,\delta_{m,0}$ (4.23)

where we have again set $R=2\pi$ , rescaled $\lambda$ by a $2\pi$ factor and
dropped the index indicating the twist sector from $H_{R}$. Throughout this
section, we will work with operators corresponding to the first $w$ copies of
the seed QFT, and symmetrization is assumed only for the final expressions.
Our first task is to construct a basis of operators that are primary with
respect to the generators above, with conformal dimensions and charges given
in (4.21).

We may again proceed by defining a flowed operator
$\widetilde{\mathcal{O}}^{(w)}(\zeta,\bar{\zeta})$ via an equation of the form
(4.5), with the initial condition that at $\lambda=0$, the operator is a
$w$-twisted operator in the undeformed CFT222222Strictly speaking, since
$\mathcal{O}^{(w)}$ is only associated with the first $w$ copies of the QFT,
it should be flown with the flow operator associated with these copies - the
quantum analogue of (3.46). While this analogue of (4.5) is a well-defined
operator equation, it is difficult to lift it to the covering space due to the
presence of the twist operator at the location of
$\widetilde{\mathcal{O}}^{(w)}$. In particular, the mapping to the larger
cylinder used in section 3.4, which did not rely on conformal invariance,
cannot be used anymore. . This operator will have the same conformal dimension
and will satisfy the same Ward identities with respect to the flowed Virasoro
and Kac-Moody currents as in the undeformed theory. As in the previous
subsection, one should add an appropriate dressing to this operator, so it
becomes primary with respect to the standard conformal generators, with the
expected dimension (2.65). However, since the relation between the deformed
and undeformed dimension is now given by _fractional_ spectral flow, we can no
longer write a simple, explicit expression for the dressing vertex as in
(4.11); instead, any explicit expression would involve fractional current
modes. Since the fractional modes are defined only locally around operators of
generically different twists, it does not make sense to commute them as we did
in the double-trace case, and thus we need a different way to evaluate the
correlation functions of interest.

The approach that we will instead take will be to reformulate, to the largest
extent possible, the computations in terms of undeformed CFT ones - where the
lift to the appropriate covering space is standard - and a $J\bar{T}$ flow,
which does not affect the end result for correlators and commutators.
Concretely, we will write, in analogy with (4.9)

$\mathcal{O}^{(w)}(\zeta,\bar{p})=\int
d\bar{\zeta\,}e^{-\bar{\zeta}\bar{p}}:\\!\widetilde{\mathcal{V}}_{\eta}\widetilde{\mathcal{O}}^{(w)}(\zeta,\bar{\zeta})\\!:e^{\mathcal{Y}^{(w)}_{\mathcal{O}}\zeta+\bar{\mathcal{Y}}^{(w)}_{\mathcal{O}}\bar{\zeta}}$
(4.24)

where $:\\!\widetilde{\mathcal{V}}_{\eta}\widetilde{\mathcal{O}}^{(w)}\\!:$
now represents the single-trace $J\bar{T}$ flow of an operator in the
undeformed theory that is almost given by the fractional spectral flow, with
parameter $\eta=\lambda\bar{p}/w$, of the original $w$-twisted CFT operator,
up to some missing zero modes and some rescaling factors. The fractional
spectral flow in the undeformed CFT may be implemented by lifting to the
covering space, where it becomes usual spectral flow. Guided by the double-
trace case, one should subtract by hand232323Note that this procedure is not
as innocuous as it may sound. In the CFT, the transformation under the
covering map is understood for local operators, but once we remove some zero
modes e.g. from the operator on the covering space, it is no longer clear how
to map it back to the base.  from the result the zero mode of the dressing
vertex operator, so that its commutation relations with the undeformed CFT
generators - which, upon $J\bar{T}$ flow, will be the same as those of the
flowed generators with
$:\\!\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}\\!:$ \- take precisely the
form (4.12), but with $k\rightarrow kw$. Note that, in order to reach this
conclusion, we do not absolutely need the explicit form of
$\widetilde{\mathcal{V}}$.

Using the fact that, by assumption, $\mathcal{O}^{(w)}(\zeta,\bar{p})$ is an
eigenoperator of the right Hamiltonian $H_{R}$, with eigenvalue $\bar{p}$, and
requiring that the commutation relations with the generators (4.23) are CFT
Ward identities for operators with conformal dimension (2.65), we find
$\eta=\lambda\bar{p}/w$, and

$\mathcal{Y}^{(w)}_{\mathcal{O}}=\frac{1}{w}\bigg{(}\lambda
qH_{R}+\lambda\bar{p}q^{[0]}+\frac{k\lambda^{2}\bar{p}^{2}}{4}\bigg{)}$ (4.25)

Assuming that, as in the double-trace case, the commutation relation of
$\widetilde{\bar{L}}_{0}$ with
$:\\!\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}\\!(\zeta,\bar{\zeta}):$
still yields a simple $\bar{\zeta}$ derivative of the operator, we also deduce
that

$\displaystyle\bar{\mathcal{Y}}^{(w)}_{\mathcal{O}}=\frac{1}{w}\bigg{(}\lambda
qH_{R}+\lambda\bar{p}q^{[0]}+\frac{k\lambda^{2}\bar{p}^{2}}{4}+\lambda\bar{p}(\tilde{J}_{0}-\tilde{\bar{J}}_{0})\bigg{)}$
(4.26)

Thus, choosing $:\\!\widetilde{\mathcal{V}}\widetilde{\mathcal{O}}\\!:$ to be
a fractionally spectrally flowed operator on the left (with some zero modes
removed) reproduces the correct left Ward identities, even in absence of
explicit expressions for the dressing factors. Our proposal for the right-
moving piece of this operator - which will fix the basis (4.24) of operators
we consider in single-trace $J\bar{T}$ \- deformed CFTs - is that it behaves
exactly the same in the undeformed CFT, i.e. it is a fractional spectral flow
on the right with parameter $\lambda\bar{p}/w$. The commutation relations with
the flowed right-moving generators then follow from the CFT ones, and are
given by (4.12) with all quantities barred and $k\rightarrow kw$. The
commutation relations of $\mathcal{O}^{(w)}(\zeta,\bar{p})$ with
$\bar{L}_{m},\bar{J}_{m}$, whose relationship to the flowed right-moving
generators is given by (4.23) with all quantities barred, then follows
straightforwardly from their definitions. It is not hard to show that in the
$R\rightarrow\infty$ limit , they correspond to standard CFT right-moving Ward
identities with

$\bar{h}^{(w)}(\bar{p})=\bar{h}^{(w)}_{\scriptscriptstyle{CFT}}+\frac{\lambda\bar{q}^{[0]}\bar{p}}{w}+\frac{k\lambda^{2}\bar{p}^{2}}{4w}\;,\;\;\;\;\;\;\bar{q}(\bar{p})=\bar{q}^{[0]}+\frac{k\lambda\bar{p}}{2}$
(4.27)

which resembles a right-moving spectral flow with parameter
$\lambda\bar{p}/w$.

Let us now discuss the correlations functions of these operators, which take
the form

$\displaystyle\langle\mathcal{O}_{1}^{(w_{1})}(\zeta_{1},\bar{p}_{1})\ldots\mathcal{O}_{n}^{(w_{n})}(\zeta_{n},\bar{p}_{n})\rangle$
(4.28)

where each $\mathcal{O}_{i}^{(w_{i})}$ is now a gauge-invariant operator in
the $w_{i}$-twisted sector of the theory, and we assume that the twists are
such that the correlation function is non-vanishing. For simplicity, we will
consider only correlation functions of single-trace twist operators.

As in the previous section, we can evaluate the correlation function in two
steps: first, we use the fact that (4.24) are eigenoperators of
$\mathcal{Y}^{(w_{i})}_{\mathcal{O}},\bar{\mathcal{Y}}^{(w_{i})}_{\mathcal{O}}$
to reduce the computation to the evaluation of a correlation function of
“almost CFT vertex operators”, as in (4.16)

$\displaystyle\langle\mathcal{O}_{1}^{(w_{1})}(\zeta_{1},\bar{p}_{1})\ldots\mathcal{O}_{n}^{(w_{n})}(\zeta_{n},\bar{p}_{n})\rangle$
$\displaystyle=$
$\displaystyle\int\prod_{i=1}^{n}d\bar{\zeta}_{i}\,e^{-\sum_{i}\bar{p}_{i}\bar{\zeta}_{i}}\,e^{\sum_{i}(\lambda\bar{p}_{i}q_{i}^{[0]}+\frac{k\lambda^{2}\bar{p}_{i}^{2}}{4})\frac{(\zeta_{i}+\bar{\zeta}_{i})}{w_{i}}}\,e^{\lambda
H_{R}^{vac}q_{i}\frac{(\zeta_{i}+\bar{\zeta}_{i})}{w_{i}}}\times$ (4.29)
$\displaystyle\times\;e^{\sum_{i<j}\frac{1}{w_{i}}(\lambda\bar{p}_{i}(q_{j}^{[0]}-\bar{q}_{j}^{[0]})\bar{\zeta}_{i}+\lambda
q_{i}\bar{p}_{j}(\zeta_{i}+\bar{\zeta}_{i}))}\,\langle:\\!\widetilde{\mathcal{V}}_{1}\widetilde{\mathcal{O}}_{1}^{(w_{1})}(\zeta_{1},\bar{\zeta}_{1})\\!:\ldots:\\!\widetilde{\mathcal{V}}_{n}\widetilde{\mathcal{O}}_{n}^{(w_{n})}(\zeta_{n},\bar{\zeta}_{n})\\!:\rangle$

The computation of the correlator of
$:\widetilde{\mathcal{V}}_{i}\widetilde{\mathcal{O}}_{i}:$ can then be traced
back (by inverting the $J\bar{T}$ flow) to the cylinder correlation function
of the original CFT operators, fractionally spectrally flowed by an amount
$\lambda\bar{p}_{i}/w_{i}$, up to some missing zero modes. Note that, unlike
for the standard spectral flow, whose action on a general correlation function
is known in closed form (4.1), for fractional spectral flow involving
operators from different twisted sectors no such formula appears to exist. On
the other hand, there does exist a well-defined prescription for computing
such a correlation function by mapping the CFT correlators to the covering
space, so we may consider this part of the problem as being in principle
solved.

The additional issue of subtracting the zero modes is complicated by the fact
that: i) unlike in the double-trace case, we no longer have a globally defined
explicit expression for the dressing vertex operators and ii) the maps to the
covering space are for local operators, but locality is spoiled by the removal
of the zero modes. Note one constraint on the corrections is that they should
be consistent with translation invariance on the cylinder, which the prefactor
in (4.29) currently violates, as did the analogous prefactor in (4.16). We
will not attempt to estimate the zero-mode contribution herein. Instead, we
argue, guided by the explicit expressions in the double-trace case, that the
corrections from such zero modes will take the form of exponential factors,
which are all suppressed in the $R\rightarrow\infty$ limit, and will thus not
contribute to the final correlation function on the plane. Thus, in this limit
we are left with usual CFT $n$-point function in mixed position/ momentum
space, for twisted-sector operators with fractionally spectrally flowed
conformal dimensions given by (4.21), (4.27)

$\displaystyle\langle\mathcal{O}_{1}^{(w_{1})}(\zeta_{1},\bar{p}_{1})\ldots\mathcal{O}_{n}^{(w_{n})}(\zeta_{n},\bar{p}_{n})\rangle_{plane}$
$\displaystyle=$
$\displaystyle=\;\int\prod_{i=1}^{n}d\bar{\zeta}_{i}\,e^{-\sum_{i}\bar{p}_{i}\bar{\zeta}_{i}}\,\langle:\\!(\mathcal{V}_{1}\mathcal{O}_{1}^{(w_{1})})_{\scriptscriptstyle{CFT}}(\zeta_{1},\bar{\zeta}_{1})\\!:\ldots:\\!(\mathcal{V}_{n}\mathcal{O}_{n}^{(w_{n})})_{\scriptscriptstyle{CFT}}(\zeta_{n},\bar{\zeta}_{n})\\!:\rangle_{plane}$

Thus, we have succeeded in evaluating correlation functions of a particular
basis of of twisted-sector operators in single-trace $J\bar{T}$ \- deformed
CFTs in terms of correlation functions of the undeformed theory. In order to
compare with the holographic computation of the next section, it is useful to
fully transform the result to momentum space and directly consider the Fourier
transform of the euclidean correlator. For the specific case of the two-point
function, the result is

$\langle\mathcal{O}^{(w){\dagger}}(p,\bar{p})\mathcal{O}^{(w)}(-p,-\bar{p})\rangle=\frac{(2\pi)^{2}}{2^{2(h^{(w)}+\bar{h}^{(w)})}\sin(\pi(h^{(w)}+\bar{h}^{(w)}))}\,\frac{p^{2h^{(w)}-1}\bar{p}^{2\bar{h}^{(w)}-1}}{\Gamma(2h^{(w)})\Gamma(2\bar{h}^{(w)})}$
(4.31)

where $h^{(w)},\bar{h}^{(w)}$ stand for the momentum-dependent combinations
(4.21), (4.27).

###  4.3 Comparison with holographic results

The single-trace $T\bar{T}$ and $J\bar{T}$ deformations have been linked to
holography for certain non-AdS backgrounds, namely an asymptotically linear
dilaton spacetime for the case of $T\bar{T}$ [8], and warped AdS3 for
$J\bar{T}$ [9, 10]. Both of these backgrounds are supported by pure NS-NS flux
and flow to AdS3 in the interior; the full spacetimes can be viewed as non-
normalizable deformations thereof. Perturbative worldsheet string theory in
these backgrounds is given by the $SL(2,\mathbb{R})$ WZW model (for level
$N_{5}>1$), deformed by a certain class of exactly marginal current-current
operators. Such deformations are exactly solvable. In certain examples, the
deformed string background can be thought of as the near-horizon geometry of a
stack of $N_{5}$ NS5 branes and $N_{1}\gg 1$ F1 strings [8, 82].

For $N_{5}>1$, the full CFT dual to the infrared AdS3 is known to _not_ be
described by a symmetric product orbifold. It is a somewhat singular theory
[5], due to the presence of states with a continuous spectrum known as long
strings, which wind around the asymptotic AdS3 boundary. Its long string
subsector has, on the other hand, been argued to be described by a symmetric
orbifold [5, 83]; note, however, that it captures only a small fraction of the
system, at least in the regime of interest [83]. Given this structure, it
should be clear that the symmetric product orbifold of
$T\bar{T}$/$J\bar{T}$-deformed CFTs cannot be exactly dual to these non-AdS
backgrounds for $N_{5}>1$, because the remaining sectors do not possess a
symmetric orbifold structure242424Nevertheless, there exists an apparently
independent link between asymptotically linear dilaton backgrounds and single-
trace $T\bar{T}$, suggested by the fact that black hole entropy [8, 82, 84]
and the infinite asymptotic symmetries of the linear dilaton spacetime [40]
precisely agree with those of single-trace $T\bar{T}$. Since only universal
quantities match, this suggests the dual could be part of a “universality
class” of $T\bar{T}$-deformed CFTs. ; at the same time, the long string
subsector survives in the deformed backgrounds and, moreover, its dynamics are
well-described by single-trace $T\bar{T}$ and, respectively, $J\bar{T}$ \-
deformed CFTs. Concretely, the spectrum of long strings has been shown [8, 18,
9, 10] to perfectly match that of single-trace $T\bar{T}$/ $J\bar{T}$ (as
derived in this article, or also in [30]); more recently, correlation
functions of long string vertex operators in the asymptotically linear dilaton
background have been computed [30] and appear to match well with the recent
$T\bar{T}$ results of [44].

The aim of this section is to compare the correlation functions of our
proposed set of “primary operator analogues” in single-trace $J\bar{T}$ \-
deformed CFTs with the those of long string vertex operators in warped AdS3.
The latter will be estimated by adapting the short-string computation of [20]
for the same background to long strings, along the lines of [25]. The two
results turn out to not exactly match. To explain this, we first review some
relevant prior work.

Let us start by considering worldsheet vertex operators of type II superstring
theory in $AdS_{3}\times\mathcal{N}$ in the presence of pure NS-NS flux. Here
$\mathcal{N}$ is a 7-dimensional compact manifold. Long strings correspond to
worldsheet vertex operators that belong to the continuous series
representation of $SL(2,\mathbb{R})$. Their worldsheet dimension is given by

$\displaystyle\Delta=-\frac{j(j+1)}{N_{5}}-w\left(h+\frac{N_{5}w}{4}\right)+\Delta_{\mathcal{N}}+N$
(4.32)
$\displaystyle\bar{\Delta}=-\frac{j(j+1)}{N_{5}}-w\left(\bar{h}+\frac{N_{5}w}{4}\right)+\bar{\Delta}_{\mathcal{N}}+\bar{N}$

where $j\in-1/2+i\mathbb{R}$ labels the Casimir of the global
$SL(2,\mathbb{R})$ algebra, $w\geq 1$ denotes the integer spectral flow in
$AdS_{3}$ \- identified with the winding of the long string around the AdS3
boundary, $(\Delta_{\mathcal{N}},\bar{\Delta}_{\mathcal{N}})$ are left/right
vertex operator dimensions of the worldsheet CFT in $\mathcal{N}$,
$(N,\bar{N})$ are the left/right oscillator numbers in $AdS_{3}$ and, finally,
$(h,\bar{h})$ represent the eigenvalues of the $J^{3},\bar{J}^{3}$ zero modes
of the worldsheet $SL(2,\mathbb{R})$. Note that for continuous series
representations, these are unrelated to the eigenvalue of the Casimir. In
global AdS3, $(h,\bar{h})$ are identified with the left/right energies of the
state on the cylinder, and thus to the dual operator dimensions via the
standard state-operator map. The physical on-shell condition for these
superstring vertex operators is $\Delta=\bar{\Delta}=1/2$. They can be
constructed explicitly [85], and their correlation function takes the
form252525In this article, we normalize the worldsheet operators such that the
two-point function of the dual CFT operators takes the form
$x_{12}^{-2h}\bar{x}_{12}^{-2\bar{h}}$. Note this normalization is different
from the standard convention in string theory in $AdS_{3}$.

$\langle
V(z_{1},x_{1})V(z_{2},x_{2})\rangle=\frac{1}{z_{12}^{2\Delta}\bar{z}_{12}^{2\Delta}x_{12}^{2h}\bar{x}_{12}^{2\bar{h}}}$
(4.33)

where $x,\bar{x}$ are auxiliary coordinates that become identified with the
space where the dual CFT lives. Integrating over the worldsheet coordinates,
one obtains the standard correlation function of CFT operators on the
boundary. One may also perform a Fourier transform of the latter, to obtain
the momentum-space boundary correlator. For example, the momentum space two-
point function takes the form

$\langle
V(z_{1},p)V(z_{2},-p)\rangle=\frac{(2\pi)^{2}p^{2h-1}\bar{p}^{2\bar{h}-1}}{2^{2(h+\bar{h})}\sin(\pi(h+\bar{h}))\Gamma(2h)\Gamma(2\bar{h})}\frac{1}{z_{12}^{2\Delta}\bar{z}_{12}^{2\bar{\Delta}}}$
(4.34)

which will be useful in this section.

The asymptotically linear dilaton and warped AdS3 backgrounds can both be
obtained from AdS3 $\times\;\mathcal{\mathcal{N}}$ via a transformation known
as TsT: T-duality, shift, T-duality. In both cases, the effect of the TsT
transformation can be encoded in a non-local coordinate transformation, which
is equivalent to twisting the boundary conditions of the fields on the AdS3
worldsheet theory in a charge-dependent way [86]. This mildly affects the
relationship between $\Delta$ and $h$, by adding

$\delta_{T\bar{T}}\Delta=\delta_{T\bar{T}}\bar{\Delta}=\frac{\mu}{\pi}\,p\bar{p}\;,\;\;\;\;\;\;\delta_{J\bar{T}}\Delta=\delta_{J\bar{T}}\bar{\Delta}=\lambda\bar{p}\left(q^{[0]}+\frac{\lambda
k\bar{p}}{4}\right)$ (4.35)

on the left-hand side of (LABEL:vir1). If $(h,\bar{h})$ are interpreted as the
left/right global energy, then these shifts yield the correct deformed energy
formulae to match to single-trace $T\bar{T}$/$J\bar{T}$.

The vertex operators in the deformed theory may be expressed in terms of the
vertex operators in the undeformed theory, with some appropriate dressing.
This has been worked out in [20] for the warped $AdS_{3}$ background and in
[25] for the asymptotically linear dilaton one; see also [24] for relevant
prior results. It turns out that the correlation function of the appropriate
vertex operators still takes the form (4.34), but the relationship between $h$
and $\Delta$ is modified by the shifts (4.35). Since $\Delta$ must equal
$1/2$, this translates into a shift of $h,\bar{h}$ that takes the form:

$\displaystyle T\bar{T}$ $\displaystyle:h\rightarrow
h+\frac{\mu}{w\pi}p\bar{p}\;,\;\;\;\;\hskip
51.21504pt\bar{h}\rightarrow\bar{h}+\frac{\mu}{w\pi}p\bar{p}$ (4.36)
$\displaystyle J\bar{T}$ $\displaystyle:h\rightarrow h+\frac{\lambda
q^{[0]}\bar{p}}{w}+\frac{\lambda^{2}k\bar{p}^{2}}{4w}\;,\;\;\;\;\;\;\;\;\bar{h}\rightarrow\bar{h}+\frac{\lambda
q^{[0]}\bar{p}}{w}+\frac{\lambda^{2}k\bar{p}^{2}}{4w}$

and can be obtained by combining (LABEL:vir1) with the appropriate shift in
(4.35). Thus, we obtain a two-point function that is identical to the
momentum-space two-point function in a CFT, but with shifted
dimensions262626Note that this argument appears to imply that, if the
normalization of the undeformed AdS3 vertex operators is rescaled by some
function of the dimensions, then all these factors would become momentum-
dependent. This is _not_ what happens if we perform a similar rescaling in the
field-theory analysis of section 4.1, where the only functions of $h$ that
acquire a momentum dependence are those that result from the Fourier
transform. This suggests that the argument of [25] may be more subtle than it
naïvely appears. . The $T\bar{T}$ shift of $h,\bar{h}$ nicely agrees with [44]
for the case $w=1$. More generally, one obtaines the shifts in the twisted
sectors to be the same, but controlled by $\mu/w$.

Note that the shift in the left-moving single-trace $J\bar{T}$ dimension
precisely agrees with our previous analysis (2.65), including the $w$
dependence. However, the right-moving piece of the correlator (4.34) does not
agree with (4.31), which involves $\bar{q}^{[0]}$ instead of $q^{[0]}$ in the
right-moving dimension $\bar{h}^{(w)}(\bar{p})$. It would be interesting to
understand the origin of this mismatch. Remember, in particular, that in our
field-theory analysis we did encounter shifts of the right-movers that
involved $J_{0}$ instead of $\bar{J}_{0}$; however, the discrepancies produced
by these terms disappeared in the $R\rightarrow\infty$ limit272727The reason
was that these discrepancies only appeared in a single exponential factor. Had
they appeared in a $\sinh$ factor, they would have contributed to the shift in
dimension, which would have then agreed with the long string result.. It would
be interesting to perform a more careful worldsheet analysis, possibly on the
cylinder, in order to track this discrepancy. Of course, it could also be that
the string theory vertex operators simply are a different set of operators
from those for which we computed correlation functions in field theory. Our
criterion for fixing the right-moving piece of the operators in field theory
was based on symmetries, i.e. by requiring that they satisfy CFT-like Ward
identities with respect to the right-moving generators (3.9), which were
related to the flowed right-moving Virasoro generators by an operator-
dependent spectral flow. Since these Virasoro symmetries are not yet
understood from the worldsheet perspective, the analogous way to fix the
operator basis is not yet available on the string theory side. Conversely, it
could be that consistency of the worldsheet vertex operators (e.g., mutual
locality) would single out a different set of constraints on the operators
that would be natural to impose. In any case, it would be interesting to
further explore the properties of the two sets of operators and check whether
one may be preferred to the other.

Finally, let us mention that exactly the same method may be used to compute
correlation functions of the short string vertex operators [21, 22, 23]; one
simply needs to reinterpret the relation between $\Delta$ and $h$ for the
discrete $SL(2,\mathbb{R})$ representations on the worldsheet. In this case,
one most commonly consisders $w=0$ short strings. Since now the representation
is lowest-weight, the spacetime dimension is related to the worldsheet Casimir
as $h=j+1$. Taking into account the shift in the relation between $\Delta$ and
the worldsheet dimension, one finds

$h_{ALD}=\frac{1}{2}+\sqrt{\left(h-\frac{1}{2}\right)^{2}+\frac{\mu
N_{5}}{2\pi}\,p\bar{p}}\;,\;\;\;\;\;h_{wAdS}=\frac{1}{2}+\sqrt{\left(h-\frac{1}{2}\right)^{2}+\lambda
N_{5}q^{[0]}\,\bar{p}+\frac{\lambda^{2}N_{5}k}{4}\,\bar{p}^{2}}$ (4.37)

Since the information about the $T\bar{T}$ deformed symmetric product is only
captured by the long string sector , it is not surprising that the momentum
dependence of the conformal dimensions is different from [44]. The same
comment applies to $J\bar{T}$, where the short string sector conformal
dimensions do not match with our symmetric product orbifold results.

##  5 Conclusions

In this article, we have studied various properties of symmetric product
orbifolds of $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs - namely the spectrum,
the symmetries and the correlation functions - from a purely field-theoretical
perspective. Our derivations relied mostly on Hilbert space techniques and
made no use of conformal invariance, which is not present in these models.

The first observable we discussed was the torus partition function. We showed
that the group-theoretical techniques [33, 34, 35] that were previously
developed to determine the partition function of a symmetric product orbifold
of _C_ FTs in terms of that of the seed can be easily generalised to two-
dimensional _Q_ FTs (not necessarily Lorentz-invariant) by appropriately
taking into account the dependence of the partition function on the size of
the circle on which the theory is defined. We also showed that the modular
properties of the symmetric product orbifold followed from those of the seed
theory. We then applied these results to the symmetric product orbifold of
$T\bar{T}$ and $J\bar{T}$ \- deformed CFTs, reproducing the finite-size
spectra that were previously computed using worldsheet techniques. It would be
interesting to find other classes of UV-complete QFTs with dimensionful
couplings whose symmetric product orbifold can be studied with this method.

Our second result was a proof that the full Virasoro $\times$ Virasoro
($\ltimes$ Kac-Moody2) symmetries of the symmetric product orbifold CFT,
including their fractional counterparts, survive the single-trace $T\bar{T}$/
$J\bar{T}$ deformation. As in the double-trace case [38], the argument was
based on transporting the extended symmetry generators of the undeformed CFT
along the irrelevant flow, and then showing that they remain conserved. The
“physical” symmetry generators may now be singled out by the fact that they
correspond to integrals of quasi-local current densities descended from the
covering space, whereas the flowed ones explicitly depend on the twist sector.
We further exploited these symmetries - following the steps of the double-
trace analysis [42] \- to single out a special basis of operators in single-
trace $J\bar{T}$ \- deformed CFTs, both from the untwisted and the twisted
sector, and compute arbitrary correlation functions thereof. It seems
reasonable to hope that similar techniques may be used in the future to
construct the correlation functions of single-trace $T\bar{T}$ \- deformed
CFTs, provided the construction of correlation functions in the double-trace
$T\bar{T}$-deformed CFTs can be recast in the same language as in the
$J\bar{T}$ case, namely using an interplay of the symmetries and the flow
equation.

Our results show that the ‘QFT data’ of single-trace $T\bar{T}$ and $J\bar{T}$
\- deformed CFTs are rigidly determined by the corresponding observables in
the seed double-trace-deformed theories, which in turn are universally
determined by those of the undeformed CFT. Thus, from the point of view of the
program set forth by [11] \- of understanding the space of UV-complete282828We
replaced the integrability requirement of [11] by UV-completeness, because for
just the standard $T\bar{T}$/ $J\bar{T}$ deformations, the universal
modification of the S-matrix does not require integrability of the underlying
theory. two-dimensional quantum field theories - these theories are not
significantly more general than their double-trace counterparts, which
themselves can be understood as mostly kinematical deformations of the
underlying CFT [87]. On the other hand, symmetric product orbifold CFTs do
sometimes allow for non-universal exactly marginal deformations that break the
symmetric orbifold structure; if such deformations could also be applied to a
$T\bar{T}$/$J\bar{T}$ symmetric product orbifold in such a way that its UV
completeness is preserved, then this would have the potential of significantly
enlarging the space of known UV-complete, yet non-local QFTs.

Another important motivation for understanding the detailed properties of
single-trace $T\bar{T}$/$J\bar{T}$ \- deformed CFTs is their application to
non-AdS holography. The exact symmetric orbifolds we studied should be
holographically dual to a highly stringy spacetime, corresponding to $N_{5}=1$
in our analysis of section 4.3. The worldsheet theory for a string propagating
in such a background is no longer described by the RNS formalism, but could in
principle be studied with the methods of [88, 89], which would be very
interesting to adapt to this non-AdS setting. Alternatively, one could
concentrate on the dual description of the weakly-curved spacetimes that are
usually of most interest in holography, which involve deformations of these
theories that break the symmetric orbifold structure. It would be very
interesting to understand which features of the single-trace
$T\bar{T}$/$J\bar{T}$ \- deformed CFTs we studied - the entropy, the
symmetries, the structure of the correlators - remain universal once one moves
off the symmetric orbifold point. The results of [8, 40] suggest that at least
the entropy and the extended symmetries should be part of the list.

#### Acknowledgements

We are grateful to Luis Apolo, Brando Bellazzini, Alex Maloney, Sylvain
Ribault and especially Alex Belin for insightful conversations. The work of SC
received funding under the “Horizon 2020” Framework Program for Research and
innovation under the Marie Sklodowska-Curie grant agreement number 945298. The
work of SG is supported by the PhD track fellowship of the École
Polytechnique. The work of SG and MG was supported in part by the ERC starting
grant 679278 Emergent-BH.

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