Comments on single trace $T\bar{T}$ and other current-current deformations

Gaston Giribeta, Julio Olivab, Ricardo Stuardob

aDepartamento de Física, Universidad de Buenos Aires & IFIBA - CONICET

Ciudad Universitaria, pabellón 1 (1428) Buenos Aires, Argentina.

bDepartamento de Física, Universidad de Concepción

Casilla 160-C, Concepción, Chile.

String theory on AdS3 with NS-NS fluxes admits a solvable irrelevant
deformation which is close to the $T\bar{T}$ deformation of the dual CFT2.
This consists of deforming the worldsheet action, namely the action of the
$SL(2,\mathbb{R})$ WZW model, by adding to it the operator $J^{-}\bar{J}^{-}$,
constructed with two Kac-Moody currents. The geometrical interpretation of the
resulting theory is that of strings on a conformally flat background that
interpolates between AdS3 in the IR and a flat linear dilaton spacetime with
Hagedorn spectrum in the UV, having passed through a transition region of
positive curvature. Here, we study the properties of this string background
both from the point of view of the low-energy effective theory and of the
worldsheet CFT. We first study the geometrical properties of the semiclassical
geometry, then we revise the computation of correlation functions and of the
spectrum of the $J^{-}\bar{J}^{-}$-deformed worldsheet theory, and finally we
discuss how to extend this type of current-current deformation to other
conformal models.

##  1 Introduction

In the context of AdS3/CFT2 correspondence, it was shown in [1] that certain
type of $T\bar{T}$-deformation of the boundary CFT2, which can be regarded as
a single trace version of the one originally introduced in [2, 3, 4], gives
rise in the bulk to a string theory background that interpolates between AdS3
in the infrared limit and a flat linear dilaton background in the ultraviolet.
This construction was argued in [1] to provide a family of holographic pairs,
including a large class of string theory vacua with asymptotically linear
dilaton. The solvable irrelevant deformation of AdS3/CFT2 correspondence
studied in [1] was further studied in [5], where in particular its spectrum
was studied. It was observed that this type of deformation leads in the
ultraviolet to a theory with Hagedorn spectrum. This has been studied in [6,
7, 8, 9, 10, 11] and references therein and thereof; see also [12, 13, 14].

In [15, 16], the correlation functions in the deformed theory were studied,
and it provided an alternative way of studying the spectrum: The insertion of
the operator that realizes the deformation produces a logarithmic divergence
in the correlation functions, leading to the renormalization of the primary
operators. This yields an anomalous dimension that can be computed explicitly.
From this, one may determine the spectrum of the theory from the worldsheet
computation. The form of the correlation functions, on the other hand, permits
to investigate the properties of the dual theory [15].

The model studied in [1] was later investigated in many different context. The
entanglement entropy was first studied in [17]; in [18, 19] the theory was
studied in presence of boundaries; and the $J\bar{T}$ analog of it has also
been studied [20, 21, 22, 23]. Here, we study the properties of this string
background both from the point of view of the low-energy effective theory and
of the worldsheet CFT. In section 2, we study the geometrical properties of
the semiclassical geometry. We study the geometry as solution to the low-
energy effective field theory, its T-dual background, the main properties of
this specific deformation of AdS3, and the field probes in such an spacetime.
In section 3, we revise the computation of correlation functions of [15, 16]
and how it provides a direct way of studying the spectrum. Finally, in section
4 we discuss how to extend this type of deformation to other conformal models.

##  2 Low energy theory

### Interpolating background

Let us start by considering the effective theory describing the low-energy
limit of bosonic string theory. This is given by the field equations

$\displaystyle
R_{\alpha\beta}+2\nabla_{\alpha}\nabla_{\beta}\Phi-\frac{1}{4}H_{\alpha\mu\nu}H_{\beta}^{\mu\nu}=0,$
(1) $\displaystyle\nabla_{\alpha}\left(e^{-2\Phi}H^{\alpha\mu\nu}\right)=0,$
(2)
$\displaystyle\nabla^{\alpha}\nabla_{\alpha}\Phi-2\nabla_{\alpha}\Phi\nabla^{\alpha}\Phi+{2\alpha^{\prime}}+\frac{1}{12}H_{\alpha\mu\nu}H^{\alpha\mu\nu}=0.$
(3)

where $H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}$ is the field strengh
associated to the Kalb-Ramond $B$-field, and $\Phi$ is the dilaton. These
equations admit locally AdS3 solutions [24],

$ds^{2}=-\frac{r^{2}}{\ell^{2}}\,dt^{2}+\frac{\ell^{2}}{r^{2}}\,dr^{2}+r^{2}d\theta^{2},$
(4)

provided the other backgrounds fields take the form

$\Phi=\Phi_{0},\ \ \ \ H_{\mu\nu\rho}=\frac{2r}{\ell}\,\epsilon_{\mu\nu\rho}.$
(5)

The dilaton receives quantum (i.e. finite-$\alpha^{\prime}$) corrections.
Here, we will consider the convention $\alpha^{\prime}=1$, so that the
semiclassical limit corresponds to large
$k=\ell^{2}/\alpha^{\prime}=\ell^{2}$.

As (4) describes the universal covering of AdS3, we have $t\in\mathbb{R}$. The
radial coordinate is $r\in\mathbb{R}_{\geq 0}$, with the boundary of the space
being located at $r\rightarrow\infty$. If we take $\theta$ to be periodic with
a period $2\pi$, the metric above corresponds to that of the massless BTZ
geometry [25, 26]. It will be convenient to consider coordinates $r=\ell
e^{\phi}$ and $x=\ell\theta$. In these variables, the metric and the field
strength take the form

$\displaystyle ds^{2}=e^{2\phi}\left(-dt^{2}+dx^{2}\right)+\ell^{2}d\phi^{2},\
\ \ \
H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}=2e^{2\phi}\epsilon_{\mu\nu\rho},$
(6)

where we now consider the covering $x\in\mathbb{R}$. That is, the non-
vanishing component of the Kalb-Ramond field is $B_{xt}=e^{2\phi}$ and grows
when approaching the boundary at $\phi\rightarrow\infty$.

Now, let us consider a deformation of (6), given by

$ds^{2}=\frac{e^{2\phi}}{\lambda
e^{2\phi}+1}(-dt^{2}+dx^{2})+\ell^{2}d\phi^{2},$ (7)

with $\lambda$ being a real parameter. This metric solves the field equations
(1) for arbitrary $\lambda$ provided the Kalb-Ramond field and the dilaton are
given by

$\displaystyle B_{xt}=\frac{2e^{2\phi}}{\lambda e^{2\phi}+1}\,,\ \ \ \
\Phi=\Phi_{0}-\phi-\frac{1}{2}\log(\lambda+e^{-2\phi}),$ (8)

respectively. Near the boundary, the dilaton becomes linear in $\phi$. We will
be mostly interested in the case $\lambda\geq 0$, as for $\lambda<0$ the
geometry exhibits a singularity at $\phi=-\frac{1}{2}\log|\lambda|$. In terms
of the double null coordinates ${{u}}=(x+t)/\ell$ and $\bar{{u}}=(x-t)/\ell$,
the fields take the following form

$ds^{2}=\ell^{2}d\phi^{2}+\frac{\ell^{2}d{{u}}\,d\bar{{{u}}}}{\lambda+e^{-2\phi}}\
,\ \ B=\frac{\ell^{2}d{{u}}\wedge d\bar{{{u}}}}{\lambda+e^{-2\phi}}.$ (9)

This solution has recently attracted much attention [1, 5, 8, 9, 10, 11, 17,
15, 16, 18, 19, 20, 21, 22, 23] as it appears as an exact string background
that corresponds to a marginal deformation of the worldsheet theory on
AdS${}_{3}\times\mathcal{N}$ which is closely related to the
$T\bar{T}$-deformation of the dual CFT.

Let us go back for a moment to the more familiar coordinates $r=\ell
e^{\phi}$, namely

$ds^{2}=-\frac{r^{2}}{r^{2}\lambda+\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}}dr^{2}+\frac{r^{2}}{r^{2}\lambda+\ell^{2}}dx^{2},$
(10)

in which it becomes evident that the geometry interpolates between AdS3 and
Minkowski space: While in the limit $r\ll\ell/\sqrt{\lambda}$ one recovers the
metric (4), in the limit $r\gg\ell/\sqrt{\lambda}$ one gets
$ds^{2}=-d\hat{t}^{2}+d\hat{x}^{2}+d\hat{y}^{2}$ where
$\hat{t}=t/\sqrt{\lambda}\ell$, $\hat{x}=x/\sqrt{\lambda}\ell$,
$\hat{y}=\ell\phi$. The local isometry group of spacetime (10) for arbitrary
value of $\lambda$ is $ISO(1,1)$ and is generated by the Killing vectors
$\partial_{t}$, $\partial_{x}$ and $x\partial_{t}-t\partial_{x}$. It gets
enhanced to the full $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ for $\lambda=0$
and to $ISO(2,1)$ in the limit $\lambda\to\infty$

The interpolating geometry (9) has very interesting properties. Apart from
being fascinating in that it describes the transition between AdS3 and the
type of linear dilaton background that appears in little string theory,
geometry (9) exhibits peculiar features: It admits a supersymmetric embedding
in type IIB SUGRA, since it appears in the S-dual frame of the D1/D5 system
[27]. Besides, it is solvable in different limits: On the one hand, despite
being a geometry of non-constant curvature, it turns out that the probe fields
are integrable on it, and thus enables to gain intuition from the
semiclassical analysis. On the other hand, the string worldsheet
$\sigma$-model is an exact string solution; being a marginal deformation of a
WZW model, can be solved explicitly in the sense that analytic expressions for
the correlation functions can be obtained and the spectrum can be written
down.

### T-duality

Let us study some of the properties of (9), starting by noticing that it is
dual to $pp$-waves on AdS3. Spacetime (6) happens to be invariant under $t$\-
and $x$-translations, and so we can apply T-duality transformations along the
direction generated by $\partial_{t}$ and $\partial_{x}$ in order to obtain
new solutions $\tilde{g}_{\mu\nu},\tilde{B}_{\mu\nu},\tilde{\Phi}$ to the
field equations (1)-(3). At the level of the low-energy effective action, this
amounts to applying the Buscher rules [28, 29]. For the $t$-direction, these
transformation rules read

$\tilde{g}_{tt}=\frac{1}{g_{tt}},\ \ \ \
\tilde{g}_{ti}=\frac{B_{ti}}{g_{tt}},\ \ \ \
\tilde{g}_{ij}=g_{ij}-\frac{g_{ti}g_{tj}}{g_{tt}}-\frac{B_{ti}B_{tj}}{g_{tt}},$
(11)

together with

$\tilde{B}_{ti}=\frac{g_{ti}}{g_{tt}},\ \ \ \
\tilde{B}_{ij}=B_{ij}-2\frac{g_{t[i}B_{j]t}}{g_{tt}},\ \ \ \
\tilde{\Phi}=\Phi-\frac{1}{2}\log\left(g_{tt}\right),$ (12)

where $i,j$ correspond to the coordinates other than $t$. After performing
these transformations and renaming variables as $u=x$, $v=t$, and
$y=\ell\phi$, one obtains

$d\tilde{s}^{2}=-F(y)\,dv^{2}+2\,dudv+dy^{2},\ \ \text{with}\ \
F(y)=\lambda+e^{-2{y}/{\ell}}$ (13)

together with $\tilde{B}_{\mu\nu}=0\,,\ \ \tilde{\Phi}=\tilde{\Phi}_{0}-\phi$.
Analogously, applying similar transformations in the $x$-direction, one gets

$d\tilde{s}^{2}=F(y)\,du^{2}+2\,dudv+dy^{2}\,.$ (14)

This geometry describes a 3-dimensional version of a $pp$-wave solution in
Brinkmann type coordinates with a wave profile $F$. The non-vanishing
component of the Riemann tensor for this geometry is

$R^{vy}_{\phantom{vy}uy}=-\frac{2}{l^{2}}e^{-2y/l}.$ (15)

This solution represents an exact string background.

### Geometric properties of the interpolating spacetime

As said, spacetime (9) interpolates between AdS3 in the limit
$\phi\rightarrow-\infty$ and a flat linear dilaton background in the opposite
limit. The geometry thus has non-constant curvature. In fact, it can be shown
to have an infinite region of positive curvature. To see this, we can compute
the scalar curvature

$R=\frac{2(4\lambda r^{2}-3\ell^{2})}{(\lambda r^{2}+\ell^{2})^{2}},$ (16)

which, indeed, happens to be positive for $r>\ell\sqrt{3/(4\lambda)}$, all the
way to infinity. $R$ has a global maximum at
$r_{\text{max}}=\ell\sqrt{5/(2\lambda)}$ with a maximum value
$R_{\text{max}}=8/(7\ell^{2})$ that does not depend on the deformation
parameter $\lambda$. Figure 1 depicts the function $R$ as a function of the
radial coordinate $r$ for different values of $\lambda$.

Figure 1: Ricci scalar for different values of $\lambda$ (here, $\ell=1$).

Other curvature invariants of (9) are

$\displaystyle R^{\mu\nu}R_{\mu\nu}=\frac{4(6\lambda^{2}r^{4}-8l^{2}\lambda
r^{2}+3l^{4})}{(\lambda r^{2}+l^{2})^{4}},$ (17) $\displaystyle
R^{\mu}_{\phantom{\mu}\nu}$ $\displaystyle
R^{\nu}_{\phantom{\nu}\rho}R^{\rho}_{\phantom{\rho}\mu}=\frac{-8(-10\lambda^{3}r^{6}+18l^{2}\lambda^{2}r^{4}-12l^{4}\lambda
r^{2}+3l^{6})}{(\lambda r^{2}+l^{2})^{6}},$ (18)

and we see from these, and from (16), that the geometry is actually singular
at $r=\ell/\sqrt{-\lambda}$. It is important noticing that it is sufficient to
give the three curvature invariant $R$, $\text{Tr}(R_{\mu\nu}^{2})$ and
$\text{Tr}(R_{\mu\nu}^{3})$ to characterize them all, since any higher
curvature scalar can be obtained as a combination of powers of the latter
three quantities by virtue of the three-dimensional identities

$\delta^{\mu_{1}\cdots\mu_{n}}_{\nu_{1}\cdots\nu_{n}}\tilde{R}^{\nu_{1}}_{\
\mu_{1}}\cdots\tilde{R}^{\nu_{n}}_{\ \mu_{n}}\equiv 0\ ,n>3\ ,$ (19)

where $\tilde{R}^{\nu_{1}}_{\ \mu_{1}}$ is the traceless part of the Ricci
tensor. Despite being of non-constant curvature, geometry (9) yields vanishing
Cotton tensor

$C_{\mu\nu}=\epsilon_{\mu}^{\
\alpha\beta}\nabla_{\alpha}(R_{\nu\beta}-\frac{1}{4}Rg_{\nu\beta})=0$ (20)

which implies that it is locally conformally flat. This property makes Weyl
invariant probes being integrable on this background, as we will show below.
This permits to gain a semiclassical intuition. The conformal factor that
allows to write the interpolating background (10) in a manifestly conformally
flat form, defines an improper Weyl transformations and, therefore, imposing
boundary conditions and asymptotic behaviors on probe fields is non-trivially
related to their flat counterpart.

### Probes on the deformed geometry

Consider a conformally coupled scalar field on the geometry (9). The
corresponding equation is

$\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\Phi-\frac{1}{8}{\sqrt{-g}}R\Phi=0.$
(21)

We consider the separable ansatz

$\Phi(t,r,x)=e^{-i\omega t}e^{i\kappa x}\varphi(r).$ (22)

This problem is exactly solvable. However, we can first gain intuition from
the well-known small and large $r$ regimes, where it reduces to the AdS3 and
to the flat space computation, respectively. This does not mean that the
solution to the complete problem will be a simple junction of the two constant
curvature problems. The transmission coefficients may actually change due to
the different boundary conditions that have to be satisfied in the
$\lambda$-deformed background.

Let us consider first the case $\kappa^{2}-\omega^{2}<0$. In this case, the
solution for $\varphi(r)$ takes the form

$\begin{split}\varphi(r)&=\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}\left(A_{1}\,e^{i\sqrt{\omega^{2}-\kappa^{2}}\,\chi(r)}+B_{1}\,e^{-i\sqrt{\omega^{2}-\kappa^{2}}\,\chi(r)}\right),\end{split}$
(23)

where

$\chi(r)=\sqrt{\lambda}\log\left(\lambda
r+\sqrt{(r^{2}\lambda+1)\lambda}\right)-\frac{\sqrt{r^{2}\lambda+1}}{r}$ (24)

and where $A_{1}$ and $B_{1}$ are two constant to be determined by requiring
appropriate boundary conditions. In order to impose conditions at infinity, it
is convenient to solve (21) on the nearly flat metric

$ds^{2}\simeq-\frac{1}{\lambda}dt^{2}+\frac{dr^{2}}{r^{2}}+\frac{1}{\lambda}dx^{2}$
(25)

which is the large $r$ limit of (10) (here, we set $\ell=1$ for short). The
radial dependence of the conformal scalar on this metric reads
$\varphi(r)\propto
A_{1}\,e^{i\sqrt{\lambda}\sqrt{\omega^{2}-\kappa^{2}}\log(r)}+B_{2}\,e^{-i\sqrt{\lambda}\sqrt{\omega^{2}-\kappa^{2}}\log(r)}$.
We want to impose outgoing boundary conditions at infinity. Since for
$\lambda\neq 0$ we have

$\displaystyle\Phi(t,r,x)$ $\displaystyle\sim
A_{1}\,e^{-i\omega(t-\sqrt{\lambda}\log(r))}+B_{1}\,e^{-i\omega(t+\sqrt{\lambda}\log(r))},$
(26)

by expanding at infinity we find that imposing outgoing boundary conditions
corresponds to $B_{1}=0$. This is confirmed by considering the flux of
particles defined by the $U(1)$-current
$j^{\mu}=-i\left(\varphi^{*}\partial^{\mu}\varphi-\varphi\partial^{\mu}\varphi^{*}\right)$.
Thus,

$\varphi(r)=A_{1}\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}e^{i\sqrt{\omega^{2}-\kappa^{2}}\left(\sqrt{\lambda}\log\left(\lambda
r+\sqrt{(r^{2}\lambda+1)\lambda}\right)-\frac{\sqrt{r^{2}\lambda+1}}{r}\right)}.$
(27)

Now, let us study the behavior near $r=0$. To do so, we expand the expression
around the origin, where we find that the dominant part goes like
$\varphi(r)\sim{r^{-1/2}}{e^{-i{\sqrt{\omega^{2}-\kappa^{2}}}/{r}}}$. At this
point, we are interested in making connection between this result and the
well-known result for AdS3 (i.e. $\lambda=0$) when $\omega^{2}>\kappa^{2}$;
namely

$\varphi_{\lambda=0}(z)=A_{2}\,z\,J_{\frac{1}{2}}\left(\sqrt{\omega^{2}-\kappa^{2}}z\right)+B_{2}\,z\,J_{-\frac{1}{2}}\left(\sqrt{\omega^{2}-\kappa^{2}}z\right)$
(28)

where $z=1/r$ and where $J_{\pm 1/2}$ are Bessel functions111These Bessel
functions actually reduce to elementary functions, which is due to the fact
that on AdS our problem reduces to that of a scalar with the conformal mass
$m^{2}\ell^{2}=m_{\text{conf}}^{2}\ell^{2}=-3/4$. For arbitrary values of the
mass the Bessel functions are replaced by $J_{\pm\nu}$ with
$\nu=\sqrt{1+m^{2}}$.. In order to relate (28) with the complex exponentials
in (23) one can use

$J_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\sin(x)}{\sqrt{x}}\ ,\ \ \
J_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\cos(x)}{\sqrt{x}}$ (29)

and, therefore, in terms of $r$ this reads

$\varphi_{\lambda=0}(r)=(A_{2}+B_{2})\,\frac{e^{i\frac{\sqrt{\omega^{2}-\kappa^{2}}}{r}}}{r^{1/2}}-(A_{2}-B_{2})\,\frac{e^{-i\frac{\sqrt{\omega^{2}-\kappa^{2}}}{r}}}{r^{1/2}}\,.$
(30)

We see that asymptotic behavior of (27) is a particular linear combination of
the solutions in (30), namely with $B_{2}=-A_{2}$. This is equivalent to
setting very special mixing boundary conditions in the AdS3 region $r\ll
1/\sqrt{\lambda}$. This makes it a difference with respect to the AdS3 case
$\lambda=0$.

Now, let us see what happens in the case $\kappa^{2}-\omega^{2}>0$, where the
solution reads

$\varphi(r)=\left(\frac{r^{2}\lambda+1}{r^{2}}\right)^{1/4}\left(C_{1}e^{-\sqrt{\kappa^{2}-\omega^{2}}\chi(r)}+D_{1}e^{\sqrt{\kappa^{2}-\omega^{2}}\chi(r)}\right)$
(31)

As the wave function is now confined, we want the solution to vanish at
infinity; therefore, we set $D_{1}=0$. As before, one can first take a look at
the solution in the case $\lambda=0$; namely

$\varphi_{\lambda=0}(r)=\frac{C_{2}}{r}K_{\frac{1}{2}}\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right)+\frac{D_{2}}{r}I_{\frac{1}{2}}\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right);$
(32)

and then rewrite the modified Bessel functions $K_{1/2}$, $I_{1/2}$ as

$K_{\frac{1}{2}}(x)=\sqrt{\frac{\pi}{2}}\frac{e^{-x}}{\sqrt{x}}\ \ \ \ \
I_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi}}\frac{\sinh(x)}{\sqrt{x}}$ (33)

to make contact with the solution for $\lambda=0$,

$\displaystyle\varphi_{\lambda=0}(r)$
$\displaystyle=\frac{C_{2}}{r^{1/2}}e^{-\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}}+\frac{D_{2}}{r^{1/2}}\sinh\left(\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}\right).$
(34)

Expanding the $\lambda\neq 0$ solution near $r=0$, one obtains

$\varphi(r)=A_{1}e^{\frac{\sqrt{\kappa^{2}-\omega^{2}}}{r}}(r^{-1/2}\,{e^{-\frac{1}{2}\sqrt{k^{2}-\omega^{2}}\sqrt{\lambda}\log(\lambda)}}+\mathcal{O}(r^{1/2})\,)\,;$
(35)

so, we see that (35) corresponds to the linear combination $C_{2}=D_{2}$
above. Notice that, while (27) diverges as $\sim r^{-1/2}$ when $r$ tends to
zero, (35) is exponentially divergent in that limit. Condition $C_{2}=D_{2}$
also looks from the AdS3 viewpoint ($r\ll 1/\sqrt{\lambda}$) as mixed boundary
conditions which are induced by the presence of the unusual asymptotic of the
deformed theory.

As the conformally coupled scalar, the Dirac action is also Weyl invariant;
therefore, it is natural to study a spin-$1/2$ probe on the deformed,
conformally flat background (10). Explicitly, the Dirac equation reads

$\left(\gamma^{a}e_{a}^{\mu}\partial_{\mu}+\frac{1}{2}\omega_{\mu}^{ab}\gamma^{c}e_{c}^{\mu}J_{ab}\right)\Psi=0\,.$
(36)

The spinor $\Psi$ will split in two components
$\Psi^{T}=\left(\Psi_{1},\Psi_{2}\right)$. The defomed metric (10) is of the
form

$ds^{2}=-f^{2}(r)dt^{2}+g^{2}(r)dr^{2}+f^{2}(r)dx^{2}\,,$ (37)

for which we can choose dreibein and compute the spin connections, leading
respectively to

$e^{0}=f\,dt\text{, }e^{1}=g\,dr\text{ and }e^{2}=f\,dx\,,$ (38)
$\omega^{01}=\frac{f^{\prime}}{g}dt\text{ and
}\omega^{12}=-\frac{f^{\prime}}{g}dx\ .$ (39)

It is useful to use the explicit, real representation of the Dirac matrices
$\gamma^{0}=i\sigma^{2},\gamma^{1}=\sigma^{1},\gamma^{2}=\sigma^{3}$. With
these expressions at hand, Dirac equation leads to the coupled system

$\displaystyle\frac{1}{f}\partial_{t}\Psi_{2}+\frac{1}{g}\partial_{r}\Psi_{2}+\frac{1}{f}\partial_{x}\Psi_{1}+\frac{f^{\prime}}{fg}\Psi_{2}$
$\displaystyle=0,$ (40)
$\displaystyle\frac{1}{f}\partial_{t}\Psi_{1}-\frac{1}{g}\partial_{r}\Psi_{1}+\frac{1}{f}\partial_{x}\Psi_{2}-\frac{f^{\prime}}{fg}\Psi_{1}$
$\displaystyle=0.$ (41)

Defining $\Psi_{i}\left(t,r,x\right)=e^{-i\omega t+i\kappa
x}\psi_{i}\left(r\right)$ with $i=1,2$, we can integrate the radial profiles
for the spinor as

$\displaystyle\psi_{1}\left(r\right)=\left(\frac{r^{2}\lambda+l^{2}}{r^{2}}\right)^{1/4}\varphi(r)\text{
and
}\psi_{2}(r)=\frac{\omega}{\kappa}\psi_{1}(r)-\frac{i}{\kappa}\frac{1}{g(r)}\frac{d}{dr}\left(f(r)\psi_{1}(r)\right)\
,$ (42)

where $\varphi(r)$ is given by (23) or (31) depending on the sign of
$\kappa^{2}-\omega^{2}$, as before. Consequently, the asymptotic behavior for
the fermion is inherited by that of the scalar. When the momentum along the
direction $x$ vanishes, namely when $\kappa=0$ the integration for the Dirac
field is simpler and leads to

$\displaystyle\psi_{1}(r)$
$\displaystyle=C_{1}\left(\frac{1+r^{2}\lambda}{r^{2}}\right)^{1/2}\left(\sqrt{\lambda}r+\sqrt{1+r^{2}\lambda}\right)^{-i\omega\sqrt{\lambda}}e^{\frac{i\omega\sqrt{1+r^{2}\lambda}}{r}}$
(43) $\displaystyle\psi_{2}(r)$
$\displaystyle=C_{2}\left(\frac{1+r^{2}\lambda}{r^{2}}\right)^{1/2}\left(\sqrt{\lambda}r+\sqrt{1+r^{2}\lambda}\right)^{i\omega\sqrt{\lambda}}e^{-\frac{i\omega\sqrt{1+r^{2}\lambda}}{r}}$
(44)

The two independent solution $C_{1}=0$ or $C_{2}=0$, respectively describe an
ingoing or outgoing flux of particles. Again, from the perspective of AdS3,
these would correspond to mixed boundary conditions.

##  3 String theory

Now, let us study the string worldsheet theory. The worldsheet action on the
background (9) takes the form

$S=\frac{1}{2\pi}\int
d^{2}z\left(\frac{1}{2}\partial\phi\bar{\partial}\phi+\frac{1}{2}\partial\bar{u}\bar{\partial}u(\lambda+e^{-\sqrt{2/(k-2)}\phi})^{-1}\right),$
(45)

together with an extra linear dilaton term $-({1}/{2\pi})\int
d^{2}z\sqrt{{2}/({k-2})}{R}\phi$ with ${R}$ being here the worldsheet
curvature. We see here that, provided $\lambda\neq 0$, in the limit
$\phi\to\infty$ one recovers the free theory with a background charge term. In
the case $\lambda=0$, in contrast, the theory at $\phi\to\infty$ exhibits a
non-trivial coupling between $\phi$ and the $u,\bar{u}$ dependence. This can
be regarded as an effective potential in the $\phi$-direction. This potential
vanishes for holomorphic configurations such that
$\partial\bar{u}\bar{\partial}u=0$. This configurations are closely related to
the so-called long strings, which form a continuum in the spectrum in AdS3.

### Strings on AdS${}_{3}\times\mathcal{N}$

Let us begin by reviewing the undeformed theory ($\lambda=0$), namely bosonic
string theory on AdS${}_{3}\times\mathcal{N}$. This theory corresponds to the
level-$k$ WZW model on $SL(2,\mathbb{R})$, and so it has $\hat{sl}(2)_{k}$
affine Kac-Moody symmetry, which is generated by local currents whose modes
are usually denoted $J_{n}^{\pm}$, $J_{n}^{3}$, along with their anti-
holomorphic counterparts. Virasoro symmetry follows from the Sugawara
construction. We consider primary operators of the form

$V_{h}(p|z)=Z_{0}|p|^{2-2h}\,e^{ip{{u}}(z)+i\bar{p}\bar{{{u}}}(\bar{z})}e^{\sqrt{2/(k-2)}(h-1)\phi(z,\bar{z})}\,\times\,...$
(46)

where the ellipsis stand for the contributions of internal part $\mathcal{N}$.
These are the vertex operators of the theory. $p$ and $\bar{p}$ are the
momenta conjugate to directions $u$ and $\bar{u}$, while $h$ is related to the
radial momentum. The factor $Z_{0}|p|^{2-2h}$ stands for a normalization. The
worldsheet conformal dimension of these operators are

$\displaystyle\Delta_{\lambda=0}=\frac{h(1-h)}{k-2}+\Delta_{\mathcal{N}}+N\,$
(47)

An analogous expression holds for $\bar{\Delta}_{\lambda=0}$ with
$\bar{\Delta}_{\mathcal{N}}$ and $\bar{N}$. $\Delta_{\mathcal{N}}$ stand for
the conformal dimension of the operators of the CFT on the internal space
$\mathcal{N}$, and $N$ is the string excitation number. As just said, $p$ and
$\bar{p}$ represent the momentum in the boundary, and they relate to the
momentum in (22) as follows

$\kappa=\frac{p+\bar{p}}{\ell}\ ,\ \ \ \omega=\frac{\bar{p}-p}{\ell}\,.$ (48)

In the Euclidean theory, $t\to it$ and $\bar{p}$ is the complex conjugate of
$p$. The index $h$ labels the representations of $SL(2,\mathbb{R})$. We focus
on the long string states, which belong to the continuous series
representations, having

$h=\frac{1}{2}+is\,,\ \ \text{with}\ \ \ \ s\in\mathbb{R}.$ (49)

These long strings can reach the boundary due to the coupling to the
$B$-field. They have a continuous energy spectrum, which depends on the
spectral flow variable $w\in\mathbb{Z}_{\geq 0}$ that accounts for the winding
number of the string around the boundary. To analyze the spectrum of the
theory on AdS${}_{3}\times\mathcal{N}$ in the momentum space, it is convenient
to consider the operator basis

$V_{h,m,\bar{m}}(z)=\frac{\Gamma(h+m)}{\Gamma(1-h-\bar{m})}\int\frac{d^{2}p}{|p|^{2}}\,p^{-m}\bar{p}^{-\bar{m}}\,V_{h}(p|z)\,.$
(50)

Performing spectral flow transformation on the states created by these
operators, one obtains the states of the sector $w$, whose conformal
dimensions are

$\displaystyle\Delta_{\lambda=0}=\frac{h(1-h)}{k-2}-mw-\frac{k}{2}w^{2}+\Delta_{\mathcal{N}}+N\,.$
(51)

where the energy is given by $m+\bar{m}+kw$ and the angular momentum by
$m-\bar{m}$.

The 2-point function in the theory on AdS${}_{3}\times\mathcal{N}$ is well-
known. For long strings in the basis $V_{h}(p|z)$, this takes the form

$\langle
V_{\frac{1}{2}+is_{1}}(p_{1}|z_{1}=0)V_{\frac{1}{2}+is_{2}}(p_{2}|z_{2}=1)\rangle_{\lambda=0}=\frac{2s_{1}}{\pi
k}\,Z_{0}^{2}\nu(k)^{2is_{1}}|p_{1}|^{4is_{1}}\,\delta^{(2)}(p_{1}+p_{2})\,\delta(s_{1}-s_{2})\,e^{2i\varphi}$
(52)

where

$e^{2i\varphi}={\Gamma}/{\Gamma^{*}}\,,\ \ \text{with}\ \ \ \
\Gamma=\Gamma(-2is)\Gamma({-2is}/({k-2}))$ (53)

and

$\nu(k)=\frac{\Gamma(\frac{1}{k-2})}{\Gamma(1-\frac{1}{k-2})}.$ (54)

The subscript ${\lambda=0}$ in (52) refers to the fact that the quantity
corresponds to the undeformed AdS${}_{3}\times\mathcal{N}$ background. In the
deformed theory, the 2-point function has been computed in [15, 16], yielding

$\langle
V_{h_{1}}(p_{1}|z_{1}=0)V_{h_{2}}(p_{2}|z_{2}=1)\rangle_{\lambda}=\delta^{(2)}(p_{1}+p_{2})\delta_{h_{1}-h_{2}}\,|p_{1}|^{4h_{1}-2}\,B(h_{1})$
(55)

with

$B(h_{1})=\frac{\nu(k)^{2h_{1}-1}}{\pi}\,\frac{\Gamma(1-2h_{1})\Gamma(1-\frac{2h_{1}-1}{k-2})}{\Gamma(2h_{1}-1)\Gamma(\frac{2h_{1}-1}{k-2})},$
(56)

and where the spectrum of the theory is given by

$h=\frac{1}{2}\pm\frac{1}{2}\sqrt{8(k-2)\lambda|p|^{2}-s^{2}}\,.$ (57)

This reduces to the 2-point function of the $SL(2,\mathbb{R})_{k}$ WZW model
in the limit $\lambda=0$. Eq. (57) follows from imposing the Virasoro
constraint $\Delta_{\lambda=0}=1$ on (47).

### Turning on the deformation

To see in detail how to obtain the correlator (55)-(56) and (57), we may first
rewrite action (45) by adding auxiliary fields $v,\,\bar{v}$, yielding the
equivalent action

$S_{\lambda}=\frac{1}{2\pi}\int
d^{2}z\left(\frac{1}{2}\partial\phi\bar{\partial}\phi-{{v}}\bar{\partial}{{u}}-\bar{{{v}}}\partial\bar{{{u}}}-2{{v}}\bar{{{v}}}e^{-\sqrt{2/(k-2)}\phi}-2\lambda{{v}}\bar{{{v}}}\right),$
(58)

where now the pair $(v,u)$ forms a commuting, dimension-(1,0) $(\beta,\gamma)$
ghost system. As we work in the conformal gauge, we are omitting here a
background charge that represents the dilaton term. For $\lambda=0$, equation
(58) is, indeed, the WZW model written in Wakimoto variables [30].
Nevertheless, we prefer to keep the notation $v,\,u$ to make contact with the
spacetime interpretation (9). The action of the $\lambda$-deformed theory is
thus given by $S_{SL(2,\mathbb{R})\text{ WZW}}-2\lambda\int
dz^{2}\,{{v}}\bar{{{v}}}$. This corresponds to a current-current deformation
of the WZW model, with the deformation being realized by the operator
$\lambda{{v}}\bar{{{v}}}$. This is consistent with the fact that the specific
Kac-Moody current in these variables reads $J^{-}(z)=v(z)$.

When trying to compute a correlation function such as $\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}$ in the path
integral approach, namely

$\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}=\int\mathcal{D}\phi\mathcal{D}^{2}u\mathcal{D}^{2}v\,e^{-S_{\lambda}}\,V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\,,$
(59)

the presence of the operator $\lambda\int d^{2}z\,v\bar{v}$ induces an
ultraviolet divergent term in the effective action after integrating the
fields $u,\bar{u}$ (see [16] for more details). This makes the contribution of
the correlators coming from the undeformed theory to factorize, and the
deformation ends up contributing with an exponential that contains the
conformal integral

$I_{0}=\int d^{2}z\,|z-z_{1}|^{-2}|z-z_{2}|^{-2}.$ (60)

This divergent integral appears frequently in quantum field theory
calculations. For instance, it appears in the one-loop computation of the
anomalous dimension of the composite operator $\bar{\psi}\psi$ in the Thirring
model. The result of it is logarithmically divergent and it can be regularized
using different methods. By introducing a regulator $\epsilon$, this can be
resolved as

$I_{\epsilon}=\left(1+2\epsilon\log|z_{1}-z_{2}|+\mathcal{O}(\epsilon^{2})\right)\left(\frac{2\pi}{\epsilon}+\mathcal{O}(\epsilon^{0})\right),$
(61)

and, after renormalizing the vertex operators by choosing
$Z_{\epsilon}=e^{-2\lambda|p|^{2}/\epsilon}$, one obtains

$\langle
V_{h_{1}}(p_{1}|z_{1})V_{h_{2}}(p_{2}|z_{2})\rangle_{\lambda}\sim\,|z_{1}-z_{2}|^{-4\Delta_{0}-4\lambda|p_{1}|^{2}}$
(62)

From this, it is possible to read the anomalous dimension induced by the
deformation; namely

$\Delta_{\lambda=0}\rightarrow\Delta_{{\lambda}}=\Delta_{\lambda=0}+\lambda|p|^{2}.$
(63)

Finally, imposing the Virasoro constraint $\Delta_{{\lambda=1}}$ and writing
it in terms of the quanties of the undeformed theory that satisfied
$\Delta_{\lambda=0}=1$, one gets (57), which reduces to (49) in the case
$\lambda=0$. We observe that $h\in\mathbb{R}$ provided
$-4|p|\sqrt{k\lambda}\geq s\geq+4|p|\sqrt{k\lambda}$; and
$h\in\frac{1}{2}+i\mathbb{R}$ provided $|s|>4|p|\sqrt{k\lambda_{0}}$. The
overall factor $|p_{1}|^{4h_{1}-2}$ in the 2-point function and the dependence
of $h_{1}$ on $\lambda$ has been studied in detail in [15] to investigate the
properties of the dual theory, especially its non-locality encoded in a branch
cut that the 2-point function of the $\lambda$-deformed theory exhibits.

##  4 Generalizations

The advantage of the computation of the anomalous dimension described above is
that it admits a straightforward generalization to other models, such as the
$SL(N,\mathbb{R})$ WZW models or their supersymmetric extensions. Despite not
in all such cases one has a string $\sigma$-model interpretation of the CFT,
this is still interesting from the CFT point of view as it provides a set of
solvable non-rational models. The simplest extension of this sort is the
$SL(N,\mathbb{R})$ WZW model. In that case, the action can in principle be
written as a sum of a Gaussian piece and an interaction piece $S_{I}$; namely

$S_{SL(N,\mathbb{R})\text{ WZW}}=\frac{1}{2\pi}\int
d^{2}z\left(\left(\partial\phi,\bar{\partial}\phi\right)-\sum_{a=1}^{N(N-1)/2}({{{v}}}_{a}\bar{\partial}{{{u}}}_{a}+{\bar{{{v}}}}_{a}\partial{\bar{{{u}}}}_{a})\right)+S_{I}.$
(64)

This involves a set of $N-1$ scalars and $N(N-1)$ copies of $\beta,\gamma$
systems, which here we keep denoting by $v_{a},u_{a}$ with
$a=1,2,...N(N-1)/2$. The scalars, $\phi_{i}$, with $i=1,2,...N-1$ form a
vector in the space of roots of $sl(N)$. We denote $(\,.\,,\,.\,)$ the product
in this space of roots, which is defined in terms of the Cartan matrix
$K_{ij}=(e_{i},e_{j})$ with $e_{1},\,e_{2},\,...,\,e_{N-1}$ being the simple
roots, with the $N-1$ fundamental weights $w_{i}$ satisfying
$(w_{i},e_{j})=\delta_{ij}$. $\rho$ is the Weyl vector, i.e. the half-sum of
all positive roots. The Lagrangian also includes a background charge term
$\int d^{2}z\,(\rho,\phi){R}/\sqrt{k-N}$.

As before, in the appropriate basis a solvable family of current-current
deformation of the theory is given by the addition of the marginal operator

$\sum^{N-1}_{i=1}\frac{\lambda_{i}}{\pi}\int
d^{2}z\,{{v}}_{i}\bar{{{v}}}_{i},$ (65)

where the field $J^{-}_{i}(z)=v_{i}(z)$ with $i=1,2,...N-1$ correspond to the
Abelian subalgebra formed by $N-1$ lowering operators. If we denote
$H=(J^{3}_{1},J^{3}_{2},...J^{3}_{N-1})$ the generators of the Cartan
subalgebra, and $E=(J^{+}_{1},J^{+}_{2},...J^{+}_{N(N-1)/2})$ and
$F=(J^{-}_{1},J^{-}_{2},...J^{-}_{N(N-1)/2})$ the raising and lowering
operators, respectively, then there exists an ordering such that the first
$N-1$ elements $F\supset(J^{-}_{1},J^{-}_{2},...J^{-}_{N-1})$ form an Abelian
subalgebra. More importantly, there exists a free field representation such
that these $N-1$ fields are given by $J^{-}_{i}=v_{i}$ with $i=1,2,...N-1$.
For the case $A_{N-1}$ with $N=2,3,4,5$ these representations have been
explicitly constructed in the literature [31, 32, 33], and the generic case
has been extensively discussed [34, 35, 36]. Let us first show how the
argument goes for generic $N$ and then consider an illustrating particular
case.

Consider the operators

$V_{h}(p,z)=Z_{0}\,e^{\sqrt{2/(k-N)}(h,\phi(z))}e^{i\sum_{a=1}^{{N(N-1)}/{2}}(p^{a}u_{a}(z)+\bar{p}^{a}\bar{u}_{a}(z))}$
(66)

where $h=(h_{1},h_{2},...h_{N-1})$ is the vector of the space of roots, and
$p=(p^{1},p^{2},...p^{N(N-1)/2})$ are the momentum associated to the
directions $u_{a}$; and consider the correlation functions

$\left\langle
V_{h}(p_{1}|z_{1})V_{h}(p_{2}|z_{2})\right\rangle_{\lambda_{1},...\lambda_{N-1}}=\int\prod_{i=1}^{N-1}\mathcal{D}\phi_{i}\prod_{a=1}^{N(N-1)/2}\mathcal{D}^{2}u_{a}\mathcal{D}^{2}v_{a}\,e^{-S_{\lambda_{1},...,\lambda_{N-1}}}\,V_{h}(p_{1}|z_{1})V_{h}(p_{2}|z_{2})\,.$
(67)

After integrating in $u_{i}$ for some $i$ (those that correspond to the fields
$u_{i}$ that do not appear other than in the kinetic term) the action
(64)-(65) being linear in these fields, one obtains

$\bar{\partial}{{v}}_{i}=2\pi
i(p^{i}_{1}\delta^{2}(z-z_{1})+p^{i}_{2}\delta^{2}(z-z_{2})),$ (68)

The solution is

${{v}}_{i}(z)=\frac{ip^{i}_{1}}{z-z_{1}}-\frac{ip^{i}_{1}}{z-z_{2}},$ (69)

where we have used that, on the sphere, $p_{1}+p_{2}=0$ in virtue of the
Riemann-Roch theorem. This can now be inserted back in (65). When doing so,
one observes that a logarithmically divergent integral similar to (60)
appears, yielding an anomalous correction to the conformal dimension of
operators (66). To see this in detail, let us consider the case $N=3$, in
which the undeformed theory is given by the WZW model on $SL(3,\mathbb{R})$,
whose action reads

$\displaystyle S_{SL(3,\mathbb{R})\text{ WZW}}=\frac{1}{2\pi}\int d^{2}z$
$\displaystyle\left(\left(\partial\phi,\bar{\partial}\phi\right)-\sum_{a=1}^{3}({{v}}_{a}\bar{\partial}{{u}}_{a}+{\bar{v}}_{a}{\partial}{\bar{u}}_{a})+\right.$
(70)
$\displaystyle\left|{{v}}_{2}+{{v}}_{1}{{u}}_{3}\right|^{2}e^{\sqrt{2/(k-3)}\left(e_{2},\phi\right)}-{{v}}_{3}\bar{{{v}}}_{3}e^{\sqrt{2/(k-3)}\left(e_{3},\phi\right)}-$
$\displaystyle\left.{{v}}_{1}\bar{{{v}}}_{1}e^{\sqrt{2/(k-3)}\left(\rho,\phi\right)}\right).$

together with a background charge term $\int d^{2}z\ (\rho,\phi)R/\sqrt{k-3}$.
As before, this action can be written in terms of the Wakimoto variables in
such a way that two commuting currents take a simple form
$J^{-}_{1}(z)={{v}}_{1}(z)$ and $J^{-}_{2}(z)={{v}}_{2}(z)$. Therefore, in the
spirit of the deformation for $SL(2,\mathbb{R})$, we deform the
$SL(3,\mathbb{R})$ WZW model by adding to it two quadratic operators, for
${{v}}_{1}$ and ${{v}}_{2}$; namely

$S_{\lambda_{1},\lambda_{2}}=S_{SL(3,\mathbb{R})\text{
WZW}}-\frac{\lambda_{1}}{\pi}\int
d^{2}z\,{{v}}_{1}\bar{{{v}}}_{1}-\frac{\lambda_{2}}{\pi}\int
d^{2}z\,{{v}}_{2}\bar{{{v}}}_{2}.$ (71)

We consider operators (66) with $N=3$, $h=(h_{1},h_{2})$ and
$p=(p_{1},p_{2},p_{3})$, and the correlation function $\left\langle
V_{h}(p,z_{1})V_{h}(-p,z_{2})\right\rangle_{\lambda_{1},\lambda_{2}}$. After
integrating on $u_{i}$, one finds the solutions for ${{v}}_{1}$ and
${{v}}_{2}$ to be

${{v}}_{1}(z)=\frac{ip^{1}(z_{1}-z_{2})}{(z-z_{1})(z-z_{2})}\ ,\ \ \
{{v}}_{2}(z)=\frac{ip^{2}(z_{1}-z_{2})}{(z-z_{1})(z-z_{2})}.$ (72)

which, when replaced in the action, yields

$\displaystyle S_{\lambda_{1},\lambda_{2}}=\frac{1}{2\pi}\int d^{2}z$
$\displaystyle\left(\left(\partial\phi,\bar{\partial}\phi\right)-{{v}}_{3}\bar{\partial}{{u}}_{3}-\bar{{{v}}_{3}}\partial\bar{{{u}}}_{3}-{{v}}_{3}\bar{{{v}}}_{3}e^{(\phi_{2}-\sqrt{3}\phi_{3})/\sqrt{k-3}}\right.$
(73)
$\displaystyle\left.+\left|p^{2}+p^{1}{{u}}_{3}\right|^{2}e^{(\phi_{2}+\sqrt{3}\phi_{3})/\sqrt{k-3}}-|p^{1}|^{2}e^{2\phi_{2}/\sqrt{k-3}}\right)$
$\displaystyle+\frac{1}{\pi}(\lambda_{1}|p^{1}|^{2}+\lambda_{2}|p^{2}|^{2})\,|z_{1}-z_{2}|^{2}\,I_{0},$

with $I_{0}$ given by (60). Regularizing as in (61) and renormalizing the
vertices accordingly, one obtains the corrected conformal dimension

$\Delta_{\lambda_{1},\lambda_{2}}=\Delta_{\lambda_{1}=\lambda_{2}=0}+\lambda_{1}\left|p^{1}\right|^{2}+\lambda_{2}\left|p^{2}\right|^{2}.$
(74)

This follows from the last line in (73), which contains the logarithmic
dependence in (61). This manifestly shows that the method of [16] can be
straightforwardly adapted to higher-rank.

Acknowledgments

G.G. thanks Edmundo Lavia and Matías Leoni for discussions and collaborations.
This work has been partially supported by CONICET through the grant PIP
1109-2017 and by Proyecto de Cooperación Internacional 2019/13231-7
FAPESP/ANID. The work of R.S. is funded by ANID Fellowships 22191591. The work
of J.O. partially funded by FONDECYT grant 1181047.

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