Moving holographic boundaries

Meseret Asrat

International Center for Theoretical Sciences

Tata Institute of Fundamental Research

Bengaluru, KA 560089, India

In this paper, we show that for one sign of the deformation coupling single-
trace $T{\overline{T}}$ deformation moves the holographic screen in Gödel
universe radially inward. For the other sign of the coupling it moves the
holographic screen radially outward. We (thus) argue, on general grounds, that
in holography (single-trace) $T{\overline{T}}$ deformation can be generally
thought of as either moving the holographic boundary into the bulk or washing
it away to infinity. In Anti-de Sitter this breaks the spacetime conformal
symmetry. We further note that moving timelike holographic boundary into bulk
creates a curvature singularity. In the boundary the singularity is understood
by states with imaginary energies. To make the theory sensible we introduce an
ultraviolet cutoff and thereby move the boundary into the bulk. In this paper
we first obtain the Penrose limit of the single-trace $T{\overline{T}}$
deformed string background and then perform $T$-duality along a space-like
isometry to obtain a class of deformed Gödel universes. The string background
we consider is $AdS_{3}\times S^{3}\times{\cal M}_{4}$. The single-trace
$T{\overline{T}}$ deformation is a particular example of the more general
$O(d,d)$ transformations.

04/23

1\. Introduction

In recent years, there have been two independent but related developments in
the study of irrelevant deformations in two dimensional quantum field theories
(QFTs). In these developments the deformations are generated by special
operators [[1]1[2][3]--[4]4]. The deformations do not mix states, i.e., a
deformed state depends only on the corresponding undeformed state. The
deformations also do not create or introduce new states. Thus, different
sectors of a Hilbert space deform independently. Therefore, they do not mix
among themselves [[5]5[6]--[7]7]. In general, depending on the sign of the
deformations couplings, the energy spectrum of the deformed theory (quantized
on a circle) is either real or complex. We will comment on this later in the
paper.

The first development is the deformation of a two dimensional QFT by the
determinant of the energy momentum stress tensor, commonly referred to as
$T{\overline{T}}$ deformation [[1]1[2]--[3]3]. The generating
$T{\overline{T}}$ operator is irrelevant (in the Wilsonian renormalization
group flow sense††1 Note irrelevance is defined with respect to a fixed point.
An irrelevant operator is less important in the infrared but become more
important in the ultraviolet.) and it is well-defined (in the coincidence
limit up to total derivative terms††2 Operator product expansion coefficients
are naturally interpreted as renormalization counter terms [[8]8,, [9]9].
Thus, $T{\overline{T}}$ (is expected to) results in a theory which is well-
behaved at high energies.) [1]. Since it is built out of (the components of)
the stress tensor which is present and conserved in any QFT with spacetime
translation invariance, the operator exists universally, and therefore also
the deformation is universally applicable. Given that the operator is well-
defined in any QFT, it can be used to define a one parameter family of
theories by iteratively adding it to and updating the Lagrangian density in a
small increment of the deformation coupling. This defines a curve or
trajectory in field theory space (parametrized by the coupling). Therefore, it
can be used to probe and better understand the space of field theories.

In general, under an irrelevant deformation of a conformal field theory (CFT),
the understanding is that at high energies or in the ultraviolet (UV) the
(quantum) theory is ill-defined. The theory involves ambiguities and/or
singularities and thus, the deformation is not solvable. In general we may,
but not necessarily, attribute this to the existence of infinitely many
irrelevant operators at the fixed point. See [7] for further discussion.

The $T{\overline{T}}$ deformation, however, defines or results in a theory
that is under better mathematical tractability. In particular, the deformation
preserves some of the symmetries of the original theory, and in the resulting
deformed theory, for several quantities including quantum observables (e.g.,
Lagrangian density, currents, etc.), exact and finite or non-singular results
are obtained. Among these, the principal quantity of interest is the energy
(or the partition function). Thus, in this sense the deformation is solvable.

The energy spectrum (and hence the usual partition function) of the deformed
theory (quantized on a circle††3 There is a minimum size of the circle below
which the Casimir energy or the effective central charge becomes complex, see
[7] and references therein.) is obtained using the operator product expansion
(OPE) technique [8] (in particular in the case the original theory is a CFT).
See also equation (C.7) in Appendix C. At short distances or high energies the
deformed theory exhibits Hagedorn density of states and non-locality. Thus, at
high energies the deformed theory is not governed by a conventional local
fixed point, see [7] and references therein.

In the case the undeformed theory is a holographic CFT i.e., dual to string
theory in $AdS_{3}$ (times some compact internal space), the dual spacetime
after $T{\overline{T}}$ deformation††4 For one sign of the coupling. In our
notation the sign is positive. See also [[10]10,, [11]11] for discussions in
the case the sign of the coupling is negative. is proposed to be $AdS_{3}$
with a Dirichlet boundary condition at a finite radial distance which is fixed
by the deformation coupling. Thus, in this sense, $T{\overline{T}}$
deformation moves the boundary where the CFT is thought to live into the bulk
[12].

The second development is in two dimensional holographic field theories which
are dual to string theory in asymptotically Anti-de Sitter (AdS) spacetimes
[4]. In this latter development, the deformation is commonly referred to as
single-trace $T{\overline{T}}$ deformation for a reason that we will explain
momentarily. It is also irrelevant at the infrared (IR) fixed point. Unlike
the usual $T{\overline{T}}$ deformation (discussed above), however, as we will
explain shortly, it is defined from the bulk (worldsheet) string theory side.
Thus, it is only (or at least it is) applicable or defined in two dimensional
holographic CFTs. This deformation is closely related but distinct from the
usual $T{\overline{T}}$ deformation (discussed above).

In $AdS_{3}$ the long string sector is believed to be described at weak
coupling or near the boundary and large $N$ by a symmetric product CFT.
Therefore, in this sector the single-trace $T{\overline{T}}$ operator is
(believed to be) given by sum of $T{\overline{T}}$ operators and therefore the
adjective single-trace.††5 Its existence is related to a (symmetric) product
structure of the (dual) boundary theory. In general in the case the
holographic theory is described by a symmetric product, the single-trace
$T{\overline{T}}$ operator is (believed to be) given by a sum of
$T{\overline{T}}$ operators. The sum is over the copies of the defining theory
of the symmetric product. In this case the usual $T{\overline{T}}$ operator is
given by product of two sums. The two factors in the product are the two
components of the total energy momentum tensor of the symmetric product CFT.
Thus, the usual $T{\overline{T}}$ deformation is also sometimes referred to as
double-trace $T{\overline{T}}$ deformation.††6 See [[13]13,, [14]14] for a
discussion on multi-trace operators and holography.

In general, irrespective of whether the boundary theory (which is dual to the
(combined) short and long string sectors) is described by a symmetric product
or not, in the bulk, the deformation is understood as a
$T$-duality–shift–$T$-duality (TsT) transformation††7 Such TsT transformations
can be equivalently viewed as twisted boundary conditions, see [7] and
references therein. This will become more evident in section two. In this view
the background spacetime remains locally $AdS_{3}$. Thus, the local dynamics
of strings in the deformed background is described by the $SL(2,R)$ Wess-
Zumino-Witten (WZW) sigma model at level $k$. This observation was noted and
used to compute spectrum and correlation functions in [7]. or as a particular
example of the more general $O(d,d)$ transformations [[15]15,, [16]16,,
[17]17[18][19][20]--[21]21]. $O(d,d)$ transformations are used to generate new
string theory backgrounds from known ones. They correspond to deforming the
worldsheet theory by a linear combination of exactly marginal current bilinear
operators [22]. We will review $O(d,d)$ transformation in Appendix D.

Note that the identification of the bulk marginal current bilinear deformation
with the irrelevant single-trace boundary deformation is at the level of the
respective theories actions [4]. This is crucial for the identification to
make sense. Thus, the equivalence is perfectly sensible. In general, in
classical (super)gravity (or $1/N$ expansion) a bulk operator $\phi(x,\rho)$
whose asymptotic behavior is $\rho^{-\Delta}$ is identified via the
gauge/(super)gravity duality [[23]23[24]--[25]25] with a boundary operator
${\cal O}(x)\equiv\lim_{\rho\to\infty}\rho^{\Delta}\phi(x,\rho)$ whose
(scaling) dimension is $\Delta$, where $x$ denotes collectively the boundary
spacetime coordinates.††8 A scalar $\phi$ of mass $m$ in $AdS_{d+1}$ has in
general the asymptotic expansion of the form $\phi\approx{\tenrm source\
term}\cdot\rho^{-(d-\Delta)}+{\tenrm expectation\ value}\cdot\rho^{-\Delta}$
where $\Delta=d/2+\sqrt{d^{2}/4+m^{2}l^{2}}$, and $l$ is the radius of
$AdS_{d+1}$. See [[26]26[27]--[28]28] for special cases and more discussion.

Of the (several) classes of operators ${\cal O}(x)$, the class (consisting) of
single trace operators is of particular interest. The (gauge invariant) single
trace boundary operators ${\cal O}(x)$ can be equivalently given as integrated
(physical) vertex operators ${\cal V}(z,x)$ on the worldsheet, i.e., ${\cal
O}(x)\equiv\int d^{2}z{\cal V}(z,x)$. Therefore, one can write $\int
d^{2}x{\cal O}(x)\equiv\int d^{2}z{\cal K}(z)$, where we defined ${\cal
K}(z):=\int d^{2}x{\cal V}(x,z)$.††9 In general, however we cannot exchange
order of integrations. Adding $\int d^{2}z{\cal K}(z)$ to the worldsheet
action or equivalently ${\cal K}(z)$ to the worldsheet Lagrangian is then
(expected to be) equivalent via the duality to adding $\int d^{2}x{\cal O}(x)$
to the boundary theory action or equivalently ${\cal O}(x)$ to the boundary
theory Lagrangian. It is in this sense the equivalence or identification is
made in [4]. Note that ${\cal K}(z)$ is obtained by integrating ${\cal
V}(x,z)$ over all the boundary spacetime.

The (gauge invariant) single trace operator ${\cal O}(x)$ that is considered
in [4] is studied earlier in [29]. We follow the notations of [[4]4,, [29]29]
and denote the operator by $D(x)$, see Appendix C for its expression. The
corresponding operator ${\cal K}(z)$ is a current bilinear and exactly
marginal [4]. It is (related to) the screening charge at strictly large level
(of the $sl(2)$ affine (current) Lie algebra), i.e., $\int d^{2}z{\cal K}=\int
d^{2}z\beta{\overline{\beta}}$, see e.g., [30].††10 The (relevant) screening
charge is given by $Q_{k}=\int
d^{2}z\beta\overline{\beta}e^{-{2\over\alpha_{+}}\phi}$, where
$\alpha_{+}=\sqrt{2(k-2)}$ and $k$ is the level. The operator
$e^{-{2\over\alpha_{+}}\phi}$ has scaling dimensions zero. $D(x)$ is the
dilaton vertex operator and has the same quantum numbers as the composite
operator $T{\overline{T}}(x)$ [29]. This is one of the primary reasons for
identifying $D(x)$ with a/the single trace $T{\overline{T}}(x)$. However,
$D(x)$ is different from (the canonical) $T{\overline{T}}(x)$ since the latter
is given (via the duality) by a product of two integrated (physical) vertex
operators corresponding to $T$ and ${\overline{T}}$.††11 If we write
$T(x)=\int d^{2}z{\cal V}_{T}(z,x)$ and ${\overline{T}}(y)=\int d^{2}z{\cal
V}_{\overline{T}}(z,y)$, then the usual operator $T{\overline{T}}(x)$ is given
by $T{\overline{T}}(x)=\lim_{y\to x}\int d^{2}z\int d^{2}w{\cal
V}_{T}(z,x){\cal V}_{\overline{T}}(w,y)$. This implies $\int
d^{2}xT{\overline{T}}(x)=\int d^{2}z{\cal M}_{T\overline{T}}(z)$ provided
${\cal M}_{T\overline{T}}(z)$ exists. The operator ${\cal
M}_{T\overline{T}}(z):=\lim_{\epsilon\to 0}\int d^{2}w\int d^{2}x{\cal
V}_{T}(z,x){\cal V}_{\overline{T}}(w,x+\epsilon)$. In view of the
$T{\overline{T}}$ operator, we expect ${\cal M}_{T\overline{T}}(z)$ to exist
and be well-defined although it is obtained by integrating over all worldsheet
and spacetime coordinates, see [[14]14,, [31]31] for related discussions.
However, in the (long string) untwisted sector $D(x)$ should agree with the
usual $T{\overline{T}}(x)$ (of the defining theory of the orbifold theory),
assuming the identification is correct. As we mentioned earlier, the existence
of $D(x)$ is related to the orbifold structure of the boundary theory.

The single-trace $T{\overline{T}}$ deformation, for one sign of the coupling
††12 In our notation the sign is negative. and in Poincaré coordinates,
corresponds in the bulk to a string background that interpolates between
$AdS_{3}\times{\cal M}_{7}$ in the IR and a linear dilaton spacetime (or
vacuum of compactified little string theory (LST)) ${\tenrm
I\kern-1.79993ptR}^{(1,1)}\times{\tenrm I\kern-1.79993ptR}\times{\cal M}_{7}$
in the UV, where ${\cal M}_{7}$ is a compact seven dimensional (internal)
manifold. In this paper, we determine the deformed string background in global
coordinates and discuss both signs.

The single-trace deformation is solvable in the sense that the exact energy
spectrum can be obtained. See equation (C.7) in Appendix C. At high energies
the deformed theory also exhibits Hagedorn density of states and non-locality,
see [7] and references therein.

In this paper we study Gödel (type) universes [[32]32[33]--[34]34] in the
single-trace $T{\overline{T}}$ deformed string theory, or more generally, in
string theory obtained by $O(d,d)$ transformations [19]. We use Gödel (type)
universe (as a toy model) to shed light on (single-trace) $T{\overline{T}}$
deformation in holography. As a consequence of (the) factorization property
(of the $T{\overline{T}}$ operator), the $T{\overline{T}}$ deformation does
not mix different sectors of a Hilbert space and also does not create or
introduce new states. This is also the case in single-trace $T{\overline{T}}$
deformation (in particular in the long string or untwisted sector), see e.g.,
[7] and references therein. In string theory Gödel (type) universes are
(usually) obtained following these two steps: (1) we take the Penrose limit,
i.e. restrict the Hilbert space to a specific sector and (2) we perform
$T$-duality along a spacelike isometry [[35]35--[36]36]. See also [37].

The Gödel (type) universe is characterized by a preferred (cylindrical)
surface, which is referred to as a holographic screen [[35]35,, [38]38]††13 In
the rest of the paper we will use this terminology to refer to the preferred
surface., and closed timelike curves. The closed timelike curves do not
constitute geodesics and (must) intersect the region beyond the holographic
screen [39]. Therefore, the holographic screen, as defined by an inertial
observer, either hides i.e., screens, the closed timelike curves or cuts them
into parts. Thus, in classical (super)gravity it prevents the development of
closed timelike curves.

It is conjectured in [[35]35] that in general holography protects causality
[[40]40]. A different view is proposed in the paper [41] by studying
(cylindrical) supertube probes [42]. It shares the same underlying concept
with the enhançon approach [43], but also has some (contextual) differences.
It replaces the spacetime beyond the holographic screen by a spacetime which
is free of closed timelike curves (and singularities). However, its status is
not clear, see e.g., [[44]44,, [45]45].

In this paper we will not be concerned on understanding the closed timelike
curves. They do not directly play any role and also they do not pose any
problem in our discussion. We will leave that to a future work. For
discussions on quantization of strings in Gödel (type) universe see also
[[44]44,, [46]46].

In this paper we show that, depending on the sign of the deformation coupling,
single-trace $T{\overline{T}}$ deformation corresponds to moving the
holographic screen either into the bulk or out to infinity. Therefore, in
holography in general it is natural to view the single-trace $T{\overline{T}}$
deformation as moving the boundary into the bulk or washing it away to
infinity. We explain in what sense. This observation is one of the main
results of the paper.

The paper is organized as follows. In section two we obtain the single-trace
$T{\overline{T}}$ deformed string theory background in global coordinates by
performing TsT transformation on $AdS_{3}\times S^{3}\times{\cal M}_{4}$. We
also show that in the Poincaré (patch) limit it reduces to the result given in
[[4]4,, [17]17,, [47]47]. In section three we briefly review the Penrose
limit. Specifically, we obtain a Penrose limit of the deformed string theory.
In section four we perform $T$-duality along a spacelike isometry and get the
single-trace $T{\overline{T}}$ deformed Gödel (type) universe. We discuss how
the location of the holographic screen depends on the coupling of the
deformation. We also discuss the implications for (single-trace
$T{\overline{T}}$ deformation and) holography in section five. In section six
we discuss future research directions. In Appendix A we review the T-duality
transformation in Type II bosonic string theory. In Appendix B we review the
(single trace) $D$ operator which is identified in the long string sector to
the single trace $T{\overline{T}}$ operator. In Appendix C we discuss the
deformation in the worldsheet and give the deformed boundary spacetime
spectrum. In Appendix D we briefly review $O(d,d)$ transformation and obtain
the supergravity solutions in the (straightforward) generalizations of
$T{\overline{T}}$ deformation to deformations involving $U(1)$ currents. In
Appendix E we provide the Maurer-Cartan one form on the three sphere. We use
this in section four.

2\. The $T{\overline{T}}$ deformed $AdS_{3}$ string background

We consider string theory on $AdS_{3}\times S^{3}\times{\cal M}_{4}$ where
${\cal M}_{4}$ is a compact four dimensional manifold. The simplest examples
are the Calabi-Yau manifolds $T^{4}$ (which breaks no supercharges) and
$K_{3}$ (which breaks half of the supercharges). The background in general
supports admixture of Neveu-Shwartz–Neveu-Schwartz (NSNS) and Ramond-Ramond
(RR) fluxes. In this paper we only consider bosonic string theory.

The $AdS_{3}$ part of the worldsheet string action in the presence of NSNS
flux is given by the Wess-Zumino-Witten (WZW) sigma model on the group
manifold $SL(2,R)$††14 In general suprstring on $AdS_{3}\times S^{3}$
supported with NSNS and RR fluxes is described by the sigma model on the
supergroup $PSU(1,1|2)\cong SU(1,1|2)/C^{*}$ where $C^{*}$ is the
multiplicative group of complex numbers [[48]48,, [49]49]. $C^{*}$ is
isomorphic to the product $R^{*}_{+}\times U(1)$ where $R^{*}_{+}$ is the
multiplicative group of positive real numbers. In general the bosons (even
subspace) and fermions (odd subspace) are coupled (except, e.g. with no RR
flux, at strictly large level $k$ and/or $N$). However, the single-trace
$T{\overline{T}}$ deformation acts only on the $SU(1,1)\cong SL(2,R)$ part of
the even subspace. It doesn’t alter the interaction term, e.g., in the case
the number of RR fluxes is zero and $k$ is finite.

$S^{SL(2,R)}_{k}={k\alpha^{\prime}\over 2\pi}\int
d^{2}z\left[\partial\widetilde{\theta}\overline{\partial}\widetilde{\theta}-{\tenrm
cosh}^{2}\widetilde{\theta}\partial\widetilde{\varphi}\overline{\partial}\widetilde{\varphi}+{\tenrm
sinh}^{2}\widetilde{\theta}\partial\widetilde{\psi}\overline{\partial}\widetilde{\psi}-{1\over
2}{\tenrm
cosh}(2\widetilde{\theta})\left(\overline{\partial}\widetilde{\varphi}\partial\widetilde{\psi}-\overline{\partial}\widetilde{\psi}\partial\widetilde{\varphi}\right)\right],$

where $k$ is the level and $l_{s}=\sqrt{\alpha^{\prime}}$ is the string
length. The angle $\widetilde{\psi}$ has $2\pi$ periodicity.

The $S^{3}$ part of the worldsheet string action is given by the WZW sigma
model on the group manifold $SU(2)$,

$S_{k}^{SU(2)}={k\alpha^{\prime}\over 2\pi}\int
d^{2}z\left[\partial\theta\overline{\partial}\theta+\cos^{2}\theta\partial\varphi\overline{\partial}\varphi+\sin^{2}\theta\partial\psi\overline{\partial}\psi+{1\over
2}\cos(2\theta)\left(\overline{\partial}\varphi\partial\psi-\overline{\partial}\psi\partial\varphi\right)\right].$

See appendix E for details of the parametrization of the group $SU(2)$ we are
using.

The single-trace $T{\overline{T}}$ deformed string action is obtained as
follows. We will ignore the $S^{3}\times{\cal M}_{4}$ part of the action for a
moment. We first perform $T$-duality along $\widetilde{\psi}$
[[50]50[51]--[52]52,, [53]53]. Along the string
$\widetilde{\psi}(\tau,\sigma+2\pi)\sim\widetilde{\psi}(\tau,\sigma)+2\pi w$,
where $w$ is some integer. See Appendix A for details on how the background
fields and action transform under the T-duality. This gives

$\eqalign{S_{k}={k\alpha^{\prime}\over 2\pi}\int
d^{2}z\left[\partial\widetilde{\theta}\overline{\partial}\widetilde{\theta}+\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\partial\widetilde{\varphi}\overline{\partial}\widetilde{\varphi}+{1\over{\tenrm
sinh}^{2}\widetilde{\theta}}\partial\widetilde{\chi}\overline{\partial}\widetilde{\chi}\right.\cr-\left.{1\over
2}{{\tenrm cosh}(2\widetilde{\theta})\over{\tenrm
sinh}^{2}\widetilde{\theta}}(\overline{\partial}\widetilde{\varphi}\partial\widetilde{\chi}+\overline{\partial}\widetilde{\chi}\partial\widetilde{\varphi})\right].}$

We next shift the coordinate $\widetilde{\varphi}$ as

$\widetilde{\varphi}\to\widetilde{\varphi}+{1\over 2}\lambda\widetilde{\chi}.$

The parameter $\lambda$ is dimensionless. It is related to the (single-trace)
$T{\overline{T}}$ deformation coupling $\widehat{\lambda}$ by
$\widehat{\lambda}=\lambda l_{s}^{2}$ (up to a constant factor). The action
now becomes

$\eqalign{S_{k}={k\alpha^{\prime}\over 2\pi}\int
d^{2}z\left\\{\partial\widetilde{\theta}\overline{\partial}\widetilde{\theta}+\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\partial\widetilde{\varphi}\overline{\partial}\widetilde{\varphi}\right.\cr+\left[-{1\over
2}{{\tenrm cosh}(2\widetilde{\theta})\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\right]\left(\overline{\partial}\widetilde{\varphi}\partial\widetilde{\chi}+\overline{\partial}\widetilde{\chi}\partial\widetilde{\varphi}\right)\cr+\left.\left[{1\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{{\tenrm
cosh}(2\widetilde{\theta})\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\right)\right]\overline{\partial}\widetilde{\chi}\partial\widetilde{\chi}\right\\}.}$

We finally do another $T$-duality along $\widetilde{\chi}$. This leads to the
single-trace $T{\overline{T}}$ deformed string action in global coordinates

$S_{k}={k\alpha^{\prime}\over 2\pi}\int d^{2}z\partial
Y^{a}\Sigma_{ab}\overline{\partial}Y^{b},$

where
$Y^{a}=(Y^{0},Y^{1},Y^{2})=(\widetilde{\varphi},\widetilde{\psi},\widetilde{\theta})$
and the background fields $\Sigma_{ab}$ are

$\matrix{\Sigma_{22}=1,\cr\Sigma_{11}=\left[{1\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{{\tenrm
cosh}(2\widetilde{\theta})\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\right)\right]^{-1}={{\tenrm
sinh}^{2}(\widetilde{\theta})\over 1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})},&\cr\Sigma_{01}=-\Sigma_{10}=-\Sigma_{11}\left[-{1\over
2}{{\tenrm cosh}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\right]=-{1\over 2}{{\lambda\over
4}-{\tenrm cosh}(2\widetilde{\theta})\over 1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})},&\cr\Sigma_{00}=-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}-\Sigma_{11}\left[-{1\over 2}{{\tenrm
cosh}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}+{\lambda\over 2}\left(-{\tenrm
cosh}^{2}\widetilde{\theta}+{1\over 4}{{\tenrm
cosh}^{2}2\widetilde{\theta}\over{\tenrm
sinh}^{2}\widetilde{\theta}}\right)\right]^{2}=-{{\tenrm
cosh}^{2}(\widetilde{\theta})\over 1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})}.&}$

The deformed two form $B$ field is now given by

$B={1\over 2}\sum_{a,b=0}^{2}B_{ab}dY^{a}\wedge
dY^{b}=B_{01}d\widetilde{\varphi}\wedge d\widetilde{\psi},\quad
B_{01}=k\alpha^{\prime}\Sigma_{01}=-{1\over 2}k\alpha^{\prime}{{\lambda\over
4}-{\tenrm cosh}(2\widetilde{\theta})\over 1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})}.$

The three form flux $H$ is then obtained by taking exterior derivative. We get
$H=dB=H_{012}d\widetilde{\varphi}\wedge d\widetilde{\psi}\wedge
d\widetilde{\theta}$, where $H_{012}=\partial
B_{01}/\partial\widetilde{\theta}$. It is given by

$H_{012}=k\alpha^{\prime}{\left[1-\left({\lambda\over
4}\right)^{2}\right]{\tenrm
sinh}(2\widetilde{\theta})\over\left[1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})\right]^{2}}.$

The dilaton $\phi$ is now given by

$\phi={1\over 2}\ln\left|{g^{2}_{s}\over 1+\left({\lambda\over
4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})}\right|,$

where $g_{s}$ is the string coupling and $\lambda<4$ since
$\widetilde{\theta}$ is positive. We comment on the values the coupling
$\lambda$ take in section four. The $S^{3}\times{\cal M}_{4}$ part of the
action is not affected by the deformation since we are only considering
deforming the AdS part.

The series of transformations we performed to get the $T{\overline{T}}$
deformed AdS background (2.1) are summarized pictorially in Fig. 1. The first
and the last transformations, i.e., the T-duality transformations that take us
from plot $a)$ to $b)$ and from plot $c)$ to $d)$, are symmetries and thus do
not change the respective theories physics. The second transformation, i.e.,
the shift that takes us from plot $b)$ to $c)$, however, changes the boundary
conditions and/or periodicities of $\widetilde{\varphi}$ and/or
$\widetilde{\psi}$, see [7]. The plots are at constant $\widetilde{\varphi}$.

Fig. 1: $a)$ is a plot of the AdS space at constant $\widetilde{\varphi}$. The
compact direction is $\widetilde{\psi}$. The distance from the symmetry axis
to the surface is ${\ninerm sinh}(\widetilde{\theta})$, see (2.1). Along the
symmetry axis, the values of $\widetilde{\theta}$ increases from $0$ to
$+\infty$. Plot $b)$ denotes the space after making a T-duality
transformation. The compact direction is $\widetilde{\chi}$, see (2.1). The
distance from the symmetry axis is $1/{\ninerm sinh}(\widetilde{\theta})$.
Plot $c)$ denotes the space with the metric (2.1) obtained after making the
shift (2.1). For positive coupling $\lambda$, this brings the tip of the cone
shaped surface from infinity to a finite value of $\widetilde{\theta}$. For
negative coupling $\lambda$, the tip remains at $\widetilde{\theta}=+\infty$.
However, the rate at which the surface contracts to the tip is slower compared
to $b)$. Plot $d)$ denotes the space (2.1) after making the final T-duality
transformation. The compact direction is $\widetilde{\psi}$. The distance from
the axis of the symmetry is $(e^{\phi}/g_{s}){\ninerm
sinh}(\widetilde{\theta})$, where $\phi$ is the dilaton (2.1). In the plots
$c)$ and $d)$ we used the value $\lambda=2$.

The action in Poincaré coordinates is obtained (as usual) by taking the large
$\widetilde{\theta}$ limit of (2.1). To be precise, we take
$\rho:=e^{\widetilde{\theta}}$ to infinity and $\lambda$ to zero but we keep
$\lambda\rho^{2}$ finite.††15 Upon taking, in the Ricci scalar in global
coordinates, this limit, i.e., we take $\rho\to\infty,\ \lambda\to 0,\
\lambda\rho^{2}\to{\tenrm const.}$, we obtain the correct Ricci scalar in
Poincaré coordinates. Thus, $\lambda\rho^{2}$ provides a natural coordinate to
work with. As we will see later in the paper, it is of crucial importance to
work in a (natural) coordinate system in which $\lambda\rho^{2}$ is the radial
axis in interpreting the deformation as moving holographic boundaries. Upon
taking this limit, the single-trace $T{\overline{T}}$ deformed string action
becomes

$S_{k}={k\alpha^{\prime}\over 2\pi}\int
d^{2}z\left[\partial\widetilde{\theta}\overline{\partial}\widetilde{\theta}+{1\over-\lambda+4e^{-2\widetilde{\theta}}}\partial(\widetilde{\psi}+\widetilde{\varphi})\overline{\partial}(\widetilde{\psi}-\widetilde{\varphi})\right],$

which is in agreement with [[4]4,, [17]17,, [47]47].††16 An action identical
to (2.1) can be also obtained by performing the TsT transformation on
$AdS_{3}$ in Poincaré coordinates. For this reason we sometimes refer the
coordinates in (2.1) as Poincaré coordinates. The dilaton field depends only
on the radial coordinate $\widetilde{\theta}$. The exact expression is
obtained making use of (2.1) or (A.9). After taking the limit we get

$e^{2\phi}={g^{2}_{s}\over 1-\left({\lambda\over
4}\right)e^{2\widetilde{\theta}}},$

Here in general $\widetilde{\theta}$ runs along the real line since (2.1),
i.e., the metric (with the dilaton and two-form fields), is an exact (in
$\alpha^{\prime}$) solution to string theory (equations of motion). We will be
precise in section five. The Ricci scalar of the metric (see (2.1)) is given
by††17 The Ricci scalar has mass dimension two. Here we set $k\alpha^{\prime}$
to one.

$R=-32{3+\lambda\rho^{2}\over(-4+\lambda\rho^{2})^{2}},$

where $\rho:=e^{\widetilde{\theta}}$ is positive. We observe that the Ricci
scalar $R$ depends on $\rho$ only through the combination $\lambda\rho^{2}$.
The components of the metric can be also made to depend on $\rho$ only through
the combination $\lambda\rho^{2}$ by suitable redefinition of coordinates. We
will use this crucial observation later in the paper. Note that the null
energy condition (NEC) requires $\lambda\leq 0$ [[54]54[55]--[56]56]. In a
large class of theories, in general, violation of the NEC implies instability
[57]. Note also that the norm of the coupling $\lambda$ can be set to one by
shifting the radial coordinate.††18 And by rescaling $\widetilde{\varphi},\
\widetilde{\psi}$. In a $T{\overline{T}}$ deformed CFT quantized on a circle
this corresponds to changing the size of the circle. We denote the deformed
part of the spacetime as ${\cal A}_{3}$. We next take, in the following
section, the Penrose limit of the deformed string background ${\cal
A}_{3}\times S^{3}$. The manifold ${\cal M}_{4}$ will not play any role in our
discussion and thus we simply ignore it.

3\. Penrose limit of $T{\overline{T}}$ deformed $AdS_{3}\times S^{3}$

In this section we study the Penrose limit of the six dimensional spacetime
${\cal A}_{3}\times S^{3}$ which results in a plane wave background. Every
non-pathological or physically reasonable spacetime has a Penrose limit
[[58]58,, [59]59]. The limit restricts the Hilbert space of the theory to a
specific sector. We in particular consider the Berenstein, Maldacena and
Nastase (BMN) plane wave limit [59].

The ${\cal A}_{3}$ part of the metric is given by

$\eqalign{ds_{{\cal
A}_{3}}^{2}&=k\alpha^{\prime}g_{\mu\nu}dY^{\mu}dY^{\nu},\quad
g_{\mu\nu}={\Sigma_{\mu\nu}+\Sigma_{\nu\mu}\over
2},\cr&=k\alpha^{\prime}\left(d\widetilde{\theta}^{2}-e^{2\phi}{\tenrm
cosh}^{2}\widetilde{\theta}d\widetilde{\varphi}^{2}+e^{2\phi}{\tenrm
sinh}^{2}\widetilde{\theta}d\widetilde{\psi}^{2}\right),}$

where the background field $\Sigma_{\mu\nu}$ is given by (2.1), and
$Y^{\mu}=(Y^{0},Y^{1},Y^{2})=(\widetilde{\varphi},\widetilde{\psi},\widetilde{\theta})$.
The dilaton $\phi$ is given by (2.1).††19 Here we set $g_{s}=1$ for simplicity
and convenience. The $S^{3}$ part of the metric is given by

$ds_{s^{3}}^{2}=k\alpha^{\prime}\left(d\theta^{2}+\cos^{2}\theta
d\varphi^{2}+\sin^{2}\theta d\psi^{2}\right).$

The plane wave background is obtained by expanding the metric around, or by
zooming in on, the geodesic of a light, or nearly so, particle that is moving
along the equator of $S^{3}$ given by $\theta=0$, which is the $\varphi$
direction, and positioned in ${\cal A}_{3}$ at $\widetilde{\theta}=0$. At
$\widetilde{\theta}=0$ the metric (3.1) reduces to $ds_{{\cal
A}_{3}}^{2}=k\alpha^{\prime}\Sigma_{00}d\widetilde{\varphi}^{2}$. The null
geodesic equation can be solved by
$\varphi=\sqrt{-\Sigma_{00}}\cdot\tau+{\tenrm const.}$ and
$\widetilde{\varphi}=\tau$, where $\tau$ is an affine parameter.

We now focus on or blow up the geometry near the geodesic. To do this we
introduce the coordinates,

$\widetilde{\theta}={r\over\sqrt{k\alpha^{\prime}}},\quad\theta={y\over\sqrt{k\alpha^{\prime}}},$

and take the large volume limit, i.e., $k\to\infty$.

We first redefine $\varphi$ by rescaling it as
$\varphi\to\varphi/(1-{\lambda\over 4})$. Thus, in the BMN limit we find

$\eqalign{ds^{2}=-{2k\alpha^{\prime}\over(1-{\lambda\over
4})^{2}}d\widetilde{x}^{-}dx^{+}-{1\over(1-{\lambda\over
4})^{2}}\left[{(1+{\lambda\over 4})^{2}\over(1-{\lambda\over
4})^{2}}r^{2}+y^{2}\right](d{x^{+}})^{2}+dr^{2}+{1\over(1-{\lambda\over
4})^{2}}\cdot
r^{2}d\widetilde{\psi}^{2}\cr+dy^{2}+y^{2}d\psi^{2}-k\alpha^{\prime}\left[{1\over{(1-{\lambda\over
4})^{2}}}+{(1+{\lambda\over 4})^{2}\over(1-{\lambda\over
4})^{4}}\cdot{r^{2}\over
k\alpha^{\prime}}\right]\left(d{\widetilde{x}}^{-}\right)^{2}-2{(1+{\lambda\over
4})^{2}\over(1-{\lambda\over 4})^{4}}r^{2}d\widetilde{x}^{-}dx^{+}+\cdots,}$

where

${\widetilde{x}}^{-}=\widetilde{\varphi}-\varphi,\quad x^{+}=\varphi,$

and the ellipsis consists of terms sub-leading in $k$. After rescaling
${\widetilde{x}}^{-}$ and $x^{+}$ ††20 That is, we redefine
$\widetilde{x}^{-}$ by making the replacement
${\widetilde{x}}^{-}\to(1-{\lambda\over 4}){\widetilde{x}^{-}}$. Similarly,
$x^{+}\to(1-{\lambda\over 4})x^{+}$., the six dimensional plane wave geometry
takes the form

$ds^{2}=-2dx^{-}dx^{+}-\left(\mu^{2}r^{2}+y^{2}\right)d{x^{+}}^{2}+dr^{2}+{1\over(1-{\lambda\over
4})^{2}}\cdot r^{2}d\widetilde{\psi}^{2}+dy^{2}+y^{2}d\psi^{2},$

where $x^{-}=k\alpha^{\prime}\widetilde{x}^{-}$ and the parameter $\mu$
depends on the deformation coupling $\lambda$. $\mu$ is given by

$\mu^{2}:={(1+\gamma)^{2}\over(1-\gamma)^{2}},\quad\gamma={\lambda\over 4}.$

The metric (3.1) has conical singularity at $r=0$ unless
$\widetilde{\psi}/(1-{\lambda\over 4})$ is periodic with a period $2\pi$, see
[60]. We redefine $\widetilde{\psi}$ by rescaling it as
$\widetilde{\psi}\to(1-{\lambda\over 4})\widetilde{\psi}$. This restore the
periodicity of $\widetilde{\psi}$ to $2\pi$, see appendix A and the discussion
in [22]. The metric now becomes

$ds^{2}=-2dx^{-}dx^{+}-\left(\mu^{2}r^{2}+y^{2}\right)d{x^{+}}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2}+dy^{2}+y^{2}d\psi^{2}.$

We note that in the case the deformation coupling $\lambda$ is zero, the plane
wave background is

$ds^{2}=-2dx^{-}dx^{+}-\left(r^{2}+y^{2}\right)d{x^{+}}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2}+dy^{2}+y^{2}d\psi^{2}.$

The two form $B$ field has in general the form

$B=B_{-a}d{\widetilde{x}}^{-}\wedge dZ^{a}+B_{+a}dx^{+}\wedge dZ^{a}+{1\over
2}B_{ab}dZ^{a}\wedge dZ^{b},$

where
$Z^{a}=(\widetilde{\varphi},\widetilde{\psi},\widetilde{\theta},\varphi,\psi,\theta)$.
In general, we can use the gauge freedom to set $B_{+a}=0$, see e.g. [61]. We
first use this gauge condition and then take the BMN limit. In this limit the
$B$ field (modulo a closed two-form) takes the form

$\eqalign{B&=B_{r\widetilde{\psi}}dr\wedge d\widetilde{\psi}+B_{y\psi}dy\wedge
d\psi,\cr&=-2\mu x^{+}rdr\wedge d\widetilde{\psi}-2x^{+}ydy\wedge d\psi.}$

The three-form flux $H$ is given by $H=dB$. In general, we can consider
admixture of NSNS and RR fluxes [[61]61].

In the BMN limit the dilaton (2.1) is given by

$e^{2\phi}={g_{s}^{2}\over(1-\gamma)^{2}}.$

It depends only on the coupling.

In the next section by applying $T$-duality on the plane wave geometry (3.1)
we obtain Gödel (type) universes. In superstring theory, $T$-duality changes
the chirality of fermions. It changes, for example, type IIB string theory to
type IIA string theory. In this paper, we will mainly be concerned with
bosonic string theory [[62]62].

4\. Gödel universes in the $T{\overline{T}}$ deformed string theory

In string theory Gödel (type) universes are $T$-dual of plane wave solutions
[[35]35--[36]36]. See also [[44]44,, [46]46]. In this section we obtain the
Gödel (type) universe from the plane wave spacetime (3.1). The spacetime (3.1)
with $\mu^{2}\neq 1$ is viewed as the single-trace $T{\overline{T}}$
deformation of the spacetime (3.1).

In this section we consider the deformed spacetime with the metric (3.1). We
will assume that no NSNS two form $B$ field is turned on [36].††21 In the
presence of the $B$ field (3.1) the spacetime (3.1) is invariant under $T$
duality (after making the trivial change $\chi\to-\chi$, where $\chi$ is the
T-dual coordinate), i.e., it is self T-dual. The relevant metric (3.1), which
we write again below for convenience, is

$ds^{2}=-2dx^{-}dx^{+}-\left(\mu^{2}r^{2}+y^{2}\right)d{x^{+}}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2}+dy^{2}+y^{2}d\psi^{2}.$

The spacetime (4.1) is homogeneous [62]. The Ricci scalar $R$ is independent
of $\mu$ and equals zero. We first define the coordinates

$\widetilde{x}^{1}=r\cos\widetilde{\psi},\quad\widetilde{x}^{2}=r\sin\widetilde{\psi},\quad\widetilde{x}^{3}=y\cos\psi,\quad\widetilde{x}^{4}=y\sin\psi.$

In these coordinates the metric becomes††22 In the variables
$\widetilde{x}^{i}$, the $B$ field (3.1) is given by  $B=-2x^{+}\left(\mu
d{\widetilde{x}}^{1}\wedge d{\widetilde{x}}^{2}+d{\widetilde{x}}^{3}\wedge
d{\widetilde{x}}^{4}\right).$ The three form flux is
$H=dB=-2dx^{+}\wedge\left(\mu d{\widetilde{x}}^{1}\wedge
d{\widetilde{x}}^{2}+d{\widetilde{x}}^{3}\wedge d{\widetilde{x}}^{4}\right)$.

$ds^{2}=-2dx^{-}dx^{+}-\left[\mu^{2}\left((\widetilde{x}^{1})^{2}+(\widetilde{x}^{2})^{2}\right)+((\widetilde{x}^{3})^{2}+(\widetilde{x}^{4})^{2})\right]d{x^{+}}^{2}+(d\widetilde{x}^{1})^{2}+(d\widetilde{x}^{2})^{2}+(d\widetilde{x}^{3})^{2}+(d\widetilde{x}^{4})^{2}.$

We further make the following change of variables††23 In general we can make
$\mu$ positive by sending $x^{\pm}\to-x^{\pm}$. Thus, we will assume $\mu$ to
be positive. We will come again to this point momentarily.

$\widetilde{x}^{1}+i\widetilde{x}^{2}=(x^{1}+ix^{2})e^{-i\mu
x^{+}},\quad\widetilde{x}^{3}+i\widetilde{x}^{4}=(x^{3}+ix^{4})e^{-ix^{+}}.$

In this new set of coordinates the metric takes the form [63]

$\eqalign{ds^{2}&=-2dx^{-}dx^{+}+\sum_{i=1}^{4}(dx^{i})^{2}-2\sum_{i,j=1}^{4}J_{ij}x^{i}dx^{j}dx^{+},\cr&=-2dx^{+}\left(dx^{-}+\sum_{i,j=1}^{4}J_{ij}x^{i}dx^{j}\right)+\sum_{i=1}^{4}\left(dx^{i}\right)^{2},\cr&=-2dx^{+}\left(dx^{-}-\sum_{i=1}^{4}b_{i+}dx^{i}\right)+\sum_{i=1}^{4}\left(dx^{i}\right)^{2},\quad
b_{i+}:=\sum_{j=1}^{4}J_{ij}x^{j},}$

where $J_{ij}=-J_{ji}$ and the nonzero entries of $J_{ij}$ are††24 In the new
variables, the $B$ field is now given by (modulo a closed two-form)
$B=-J_{ij}x^{i}dx^{j}\wedge dx^{+},$ where summation over repeated indices is
assumed.

$J_{12}=-J_{21}=\mu,\quad J_{34}=-J_{43}=1.$

For convenience, in what follows, we work with dimensionless coordinates. We
redefine $x^{-},x^{i}$ as $x^{-}/l_{s}^{2},x^{i}/l_{s}$. We define

$x^{+}=t+w,\quad 2x^{-}=t-w,$

and rewrite the metric as

$ds^{2}=-dt^{2}+dw^{2}+\sum_{i=1}^{4}(dx^{i})^{2}-2\sum_{i,j=1}^{4}J_{ij}x^{i}dx^{j}(dt+dw).$

We note that the coordinate $w$ is spacelike and an isometry of the spacetime.
Thus, we can perform $T$-duality along $w$ [64]. This gives the Gödel (type)
universe

$ds^{2}=-\left(dt+\sum_{i,j=1}^{4}J_{ij}x^{i}dx^{j}\right)^{2}+\sum_{i=1}^{4}(dx^{i})^{2}+dq^{2},$

and the two-form $B$ field (modulo a closed two-form) given by

$B=B_{iq}dx^{i}\wedge dq,\quad B_{iq}=-J_{ij}x^{j},$

where summation over repeated indices is assumed. The three-form flux is given
by $H=dB=\sum_{i,j=1}^{4}H_{ijq}dx^{i}\wedge dx^{j}\wedge dq$ and
$H_{ijq}=J_{ij}$. The curvature scalar is not invariant under T-duality. The
curvature scalar of the metric (4.1) is $R=2(1+\mu^{2})$. The dilaton is given
by $e^{2\phi}=g_{s}^{2}/(1-\gamma)^{2}$, see (3.1).

The Gödel type universe (4.1) exhibits the essential properties or attributes
of the four dimensional Gödel universe obtained by Kurt Gödel. In particular,
we expect the universe (4.1) to contain light-like surface and closed timelike
curves. The space (4.1) is homogeneous, i.e., no point is preferred and any
point can be reached from any other point by a symmetry transformation [65].
It has the following Killing vector fields which act transitively.

$V_{q}=\partial_{q},\quad V_{t}=\partial_{t},\quad
B_{i}=\partial_{i}-\sum_{j=1}^{4}J_{ij}x^{j}\partial_{t},$

where $\partial_{q}=\partial/\partial q,\ \partial_{t}=\partial/\partial t$,
and $\partial_{i}=\partial/\partial x^{i}$. It also has the Killing vector
fields

$L_{12}=x^{1}\partial_{2}-x^{2}\partial_{1},\quad
L_{34}=x^{3}\partial_{4}-x^{4}\partial_{3},$

which generate a $U(1)\times U(1)$ symmetry, see footnote 26. The non-trivial
commutation relations are

$[B_{i},B_{j}]=2J_{ij}V_{t},\quad[L_{ij},B_{k}]=-\sum_{l=1}^{4}(\delta_{ik}\delta_{jl}-\delta_{jk}\delta_{il})B_{l}.$

We hope to study the causal structure of the spacetime (4.1) in detail in a
future work. We will take, without loss of generality, $J_{12}$ to be
positive. That is, we have

$-1\leq\gamma\leq 1,$

see (3.1) and (4.1).

The interval (4.1) covers the whole positive $\mu$ line. All other values of
the coupling can be mapped onto the domain (4.1). In this domain $\gamma$ is
connected to zero continuously. As we will discuss momentarily, the
restriction (4.1) is related to the curvature singularity of the underlying
string theory (2.1), e.g. see the Ricci scalar (2.1). It can also be
understood by examining the sign of the
$\overline{\partial}\widetilde{\chi}\partial\widetilde{\chi}$ coefficient in
(2.1).

We note that the section of the universe in which $x^{1}={\tenrm const.}$ and
$x^{2}={\tenrm const.}$ is independent of the coupling, and thus, there are
string states that do not deform. These are string states that live on the
sphere. For simplicity and convenience we consider the section of the universe
where $x^{3}={\tenrm const.}$ and $x^{4}={\tenrm const.}$. It has the topology
of a cylinder times a circle which we uncompactify. The line subspace is
parametrized by $q$. In what follows we will consider an observer positioned
at $x^{3}={\tenrm const.}$ and $x^{4}={\tenrm const.}$.

Therefore, for constant $x^{3}$ and $x^{4}$, the Gödel (type) universe metric
(4.1) is four dimensional and it is the direct sum of the metric on the
manifold with coordinates $(t,x^{1},x^{2})$ which is given by

$ds^{2}=-\left[dt+\mu\left(x^{1}dx^{2}-x^{2}dx^{1}\right)\right]^{2}+\sum_{i=1}^{2}(dx^{i})^{2},$

and the metric on the line which is given by

$ds^{2}=dq^{2}.$

Therefore, we set, without loss of generality, $q=0$, i.e., we will simply
ignore the line subspace in our following discussion as it is also done in
[39] for obvious reason. In order to describe the essential characteristic
properties of the universe under single-trace $T{\overline{T}}$ deformation
considering this section is sufficient since the spacetime (4.1) is
homogeneous [[65]65,, [66]66], and other choices do not alter the
conclusions.††25 This section corresponds equivalently to setting $y$ to zero,
see equations (3.1), (4.1). This can alternatively be viewed as setting in
(4.1) $J_{34}=-J_{43}=0$. In cylindrical coordinates

$x^{1}=r\sin\xi,\quad x^{2}=r\cos\xi,$

the metric (4.1) becomes††26 The three dimensional subspace of Gödel universe
metric obtained by Kurt Gödel is given by, see [39]
$ds^{2}=2\omega^{-2}\left(-dt^{2}+dr^{2}-\left({\tenrm sinh}^{4}r-{\tenrm
sinh}^{2}r\right)d\phi^{2}+2\sqrt{2}{\tenrm sinh}^{2}rd\phi dt\right),$ where
$-\infty<t<\infty,0\leq r<\infty$, and $0\leq\phi\leq 2\pi$. $\phi=0$ is
identified with $\phi=2\pi$. The Ricci scalar $R=2\Lambda$, where $\Lambda$ is
the cosmological constant. It has the value $\Lambda=-\omega^{2}$. It
describes a rotating spacetime ($g_{\phi t}\neq 0$) filled with noninteracting
dust particles of positive energy density $\rho=-R/\kappa$, where $\kappa=8\pi
G_{N}$ is the gravitational constant. In general we can put the relevant part
of the metric (4.1) into the form
$\matrix{ds^{2}=-\left(dt-\mu\left(r^{2}_{1}d\varphi_{1}+r_{2}^{2}d\varphi_{2}\right)\right)^{2}+dr_{1}^{2}+r_{1}^{2}d\varphi_{1}^{2}+\mu\left(dr_{2}^{2}+r_{2}^{2}d\varphi_{2}^{2}\right),\cr=-\left(dt-r^{2}\left(\mu\cos^{2}\theta
d\varphi_{1}+\sin^{2}\theta
d\varphi_{2}\right)\right)^{2}+dr^{2}+r^{2}\left(d\theta^{2}+\cos^{2}\theta
d\varphi_{1}^{2}+\sin^{2}\theta d\varphi_{2}^{2}\right),}$ where $0\leq
r_{1},r_{2}<\infty$, and $0\leq\varphi_{1},\varphi_{2}\leq 2\pi$. $0\leq
r<\infty$ and $0\leq\theta\leq\pi/2$. $\theta=0$ is a great circle on the
three unit sphere, and it is the case $x^{3}={\tenrm const.}$ and
$x^{4}={\tenrm const.}$. In terms of the left and right invariant Maurer-
Cartan one-forms of the three sphere we can write the metric as
$ds^{2}=-\left[dt-r^{2}\left(\mu_{-}\omega_{3}+\mu_{+}\overline{\omega}_{3}\right)\right]^{2}+dr^{2}+r^{2}d\Omega_{3}^{2},$
where $d\Omega_{3}^{2}={1\over 4}\sum_{i=1}^{3}\omega_{i}^{2}={1\over
4}\sum_{i=1}^{3}\overline{\omega}_{i}^{2}$ is the metric on the unit three
sphere. $\mu_{\pm}=(\mu\pm 1)/4$. $\omega_{i}$ are the left invariant one-
forms and $\overline{\omega}_{i}$ are the right invariant one-forms, see
Appendix E. The form (4.1) clearly manifests the $U(1)\times U(1)$ symmetry.
We note that in the case $\mu_{-}=0$ the symmetry enhances to $SU(2)\times
U(1)$.

$ds^{2}=-\left(dt-\mu r^{2}d\xi\right)^{2}+dr^{2}+r^{2}d\xi^{2}.$

It is known that every homogeneous spacetime admits a geodesic which is an
orbit of a one-parameter group of isometries [66]. Therefore, consider the
curve $\gamma(s)=\\{(t(s),r(s),\xi(s))\\}$ with $t={\tenrm const.}$ and
$r={\tenrm const.}$. Here $s$ is a point in ${\tenrm I\kern-1.79993ptR}$ (or
an interval in ${\tenrm I\kern-1.79993ptR}$). The norm of the Killing vector
$L_{12}:=v=((\partial/\partial\xi)^{\mu})=(0,0,1)$ which is tangent to this
curve is

$v^{2}:=v_{\mu}v^{\mu}=g_{\xi\xi}=r^{2}(1-\mu^{2}r^{2}).$

For a spacelike tangent vector $v^{2}>0$. Since $v^{2}\leq 0$ for $r\geq
r_{c}$, where $r_{c}$ is the critical radius given by

$r_{c}:=1/\mu,$

there exists closed timelike and null curves. However, we stress that there
are no closed timelike and null geodesics. The critical radius is the
outermost radial point a geodesic which started at the origin can reach. The
Gödel spacetime is homogeneous.††27 Note that, on the contrary to the usual
assumption in cosmological models, the spacetime is not isotropic. Thus, for
every point of the spacetime a closed timelike curve can be found that passes
through it. However, every closed timelike curve must intersect the outer
region $r>r_{c}$.

Light rays emitted from the origin follow a spiral trajectory [39]. In a
finite affine parameter they reach the surface at $r=r_{c}$. The surface at
$r_{c}$ is a trivial bundle over a null circle and it is referred to as
velocity-of-light surface. Then, the light rays begin to reconverge and, in
the same affine parameter, they reach the origin. The behavior of timelike
geodesics is similar. They reach a maximum value of $r$ less than $r_{c}$ and
then reconverge at the origin with same affine parameter. Light emitted from
beyond the critical radius $r>r_{c}$ can never reach the origin.

The holographic screen is obtained by maximizing the area of constant $t$ and
$r$ [38]. The area is one dimensional and it is given by

$A(r)=2\pi\cdot r(1-\mu^{2}r^{2})^{1\over 2}.$

This gives the location of the holographic screen

$r_{\cal H}={1\over 2^{1\over 2}\mu}={1\over 2^{1\over 2}}r_{c},$

which is less than the critical radius. Therefore, the closed timelike curves
are either hidden from the observer by the screen or they are cut into causal
parts. The area (4.1) at the holographic screen radius $r_{\cal H}$ is given
by

$A(r_{\cal H})={\pi\over\mu}.$

This gives a bound on the entropy $S$ [[38]38,, [67]67,, [68]68]. It is given
by

$S={A(r_{\cal H})\over 4}={\pi\over 4\mu}.$

We note that $S$ is positive and at $\gamma=1$ it is zero. The angular speed
$\Omega_{{\cal G}}$ with which Gödel (type) universe rotates is given by
[[32]32,, [69]69],

$\Omega_{{\cal G}}:={1\over r_{\cal{H}}}.$

Thus, $r_{\cal H}$ determines the intrinsic rotation (or vorticity) of the
universe and also defines the holographic screen. In string units, we have
$l_{s}=1/2^{1\over 2}$, thus, we can write $r_{\cal H}=(1/\mu)l_{s}.$††28 For
convenience one may find it is useful to think of $r_{\cal H}$ as the radial
location of the AdS boundary. This will become more evident momentarily.

Consider the case in which the coupling $\gamma$ is positive (4.1). That is,
$0\leq\gamma\leq 1$. Thus, in this case as $\gamma$ becomes larger and larger,
$\mu$ goes to infinity, and hence $r_{\cal H}$ goes to zero. The holographic
screen moves into the bulk. The other case is when the coupling $\gamma$ is
negative. In this case $-1\leq\gamma\leq 0$. We note that as the norm of
$\gamma$ gets larger and larger, $\mu$ goes to zero, and hence $r_{\cal H}$
goes to infinity. Thus, the holographic screen moves away from the bulk. That
is, it pushes the closed timelike curves further away to infinity. Therefore,
in view of these facts and the uniqueness/factorization property (see e.g.,
[[5]5,, [6]6]), in holography, it is natural to view the single-trace
$T{\overline{T}}$ deformation as moving the boundary either into the bulk or
out to infinity. In what follows, we explain these observations and draw some
conclusions on single-trace $T{\overline{T}}$ deformation. For simplicity we
will mainly use the Poincaré coordinates (2.1).

5\. Moving holographic boundaries with $T{\overline{T}}$

We noted earlier in section two that, depending on the sign of the coupling,
we have two inequivalent classes of string theory backgrounds, see (2.1).
However, once we decide which sign to consider, the value of the coupling can
be fixed to a preferred value, i.e., $\gamma=\pm 1$, by shifting the radial
coordinate $\widetilde{\theta}$. In this section we use the observations made
in the previous section to better understand the (single trace)
$T{\overline{T}}$ deformation.

First, let us consider the case in which $\gamma$ equals zero. The boundary
(or spacetime) theory is conformal. The holographic boundary is located at
large $\rho$. We recall that in general $\rho$ enters in the Ricci scalar (see
(2.1)) through the combination $\gamma\rho^{2}$. Thus, in general the natural
variable or coordinate to work with or better suited is not $\rho$, but
$\gamma\rho^{2}$. Recall that we obtained the metric by taking $\gamma\to 0$,
$\rho\to\infty$ and keeping $\gamma\rho^{2}$ finite constant. Since the Ricci
scalar of AdS is constant (and negative), at the boundary we in particular
also have $|\gamma|\rho^{2}:=b_{0}={\tenrm finite\ const.}$, see (2.1).††29
Putting back the dimensions, this is
$|{\widehat{\gamma}}|{\widehat{\rho}}^{2}=b_{0}$ where
$\widehat{\rho}=\rho/l_{s}$ and $\widehat{\gamma}=\gamma l_{s}^{2}$.

In general the constant $b_{0}$ can take any positive value. We note that for
small $b_{0}$ the metric (2.1) can be put (up to an overall positive constant)
into the form

$ds^{2}=k\alpha^{\prime}\left[{dr^{2}\over
r^{2}}+r(-dt^{2}+dx^{2})\right],\quad 0\leq r\leq b_{0}\ll 1,$

where $r=|\gamma|\rho^{2},\ t={\widetilde{\varphi}}/\sqrt{|\gamma|},\
x={\widetilde{\psi}}/\sqrt{|\gamma|}$. The Ricci scalar $R=-3/2$. Note $R$
does not depend on $r$ and thus $b_{0}$. However, since (5.1) is an exact
solution to string theory we can extend the range or the value of $b_{0}$ to
one. Therefore, we set $b_{0}=1$.††30 In general $b_{0}$ can take any positive
finite value, specifically it is given by the pole of the Ricci scalar $R$
(see (2.1)). In general $\gamma=\alpha\cdot\lambda$ for some positive real
number $\alpha$, see the discussion on the positive coupling case below. Thus,
at the boundary $\rho=\lim_{|\gamma|\to 0}1/\sqrt{|\gamma|}$. It is (useful
and convenient to think of $b_{0}$ as) the position of the (conformal)
boundary in string units. The coordinate $r$ has a natural interpretation as
“conformal” coordinate. This is complementary to our comment in footnote $7$.

In the case in which the coupling $\gamma$ is negative, we note that the
deformed string metric (2.1) is well-defined and has a non-singular Ricci
scalar (2.1). We choose, without loss of generality, $\gamma=-1$; this is our
preferred value. We can always achieve this by redefining the coordinates.

We note that this choice of value for $\gamma$ corresponds, in our discussion
in the earlier section, to placing Gödel’s holographic screen at infinity, see
(4.1). Thus, at $\gamma=-1$ there are no closed timelike curves in the
subspacetime (4.1). In general it corresponds to a slowly rotating Gödel
(type) universe (4.1). Thus, in view of this, the timelike boundary of AdS is
washed away to infinity. The boundary is at $-\gamma\rho^{2}:=b_{-1}=\infty$
in string units.

For positive coupling, on the other hand, the Ricci scalar (see (2.1)) is
always singular. Thus, for positive values of the coupling $\gamma$, the
boundary††31 At $\gamma=0$ the boundary is at $\gamma\rho^{2}=b_{0}=1$ in
string units. turns into a singularity. Therefore, at the boundary or
singulaity $\gamma\rho^{2}=1$, and thus, as we increases $\gamma$, $\rho$
decreases, see plot $c)$ in Fig. 1. Note we are not assuming a specific value
for the coupling. It can take any positive value.††32 In global coordinates
$\gamma<1$ since the radial coordinate is positive. We expect this also to be
the case in the usual $T{\overline{T}}(x)$ deformation as well, since, as we
explained in the introduction, in the untwisted sector the single trace and
ordinary $T{\overline{T}}(x)$ deformations are the same.

In general a positive coupling $\gamma$, as we noted in the discussion in the
earlier section, corresponds to a relatively fast rotating Gödel (type)
universe (4.1). As we increase $\gamma$ to one, the holographic screen moves
towards the origin. Thus, at $\gamma=1$ it terminates or ceases to exist and
therefore, the Gödel (type) universe becomes pathological [39].

Had we taken the Penrose limit, around a different $\widetilde{\theta}$ value
or null geodesic, the interval (4.1) which we write here again for convenience

$-4\leq\lambda\leq 4,$

and the value of $\lambda$ at which the holographic screen ceases to exist,
i.e., $\lambda=4$, would have been different. That is to say, the coupling
$\lambda$ can take any value.††33 In global coordinates $\gamma$ is less than
1 since $\widetilde{\theta}$ is positive. In Poincaré coordinates, however,
$\widetilde{\theta}$ can take negative values, see e.g. (2.1), thus $\gamma$
is not restricted to a certain region on the real line. Therefore, in general,
positive values of the coupling $\lambda$ do not lead to a sensible theory.
However, we still would have the same interval (4.1) for $\gamma$, except that
we would now have to normalize $\lambda$ differently. The value of $\gamma$ at
which the holographic screen terminates or dissolves would still also be the
same, i.e., $\gamma=1$. Thus, we still would have the same conclusions.

It is interesting to note that the presence of the curvature singularity (see
(2.1)) is captured in the T-dual of the BMN limit by the dissolution of the
holographic screen at $\gamma=1$. It is also interesting to note that since
$\gamma$ can take any value between zero and one (see (4.1)), and also since
they give equivalent string backgrounds (see (2.1)), the discussion above
implies, moving the AdS boundary into the bulk creates a curvature
singularity. In this case the timelike boundary ceases to exist, i.e., it
dissolves. Therefore, the metric (2.1) loses its physical meaning in the same
way the Gödel (type) universe does, i.e., the holographic screen closes in on
the observer and the universe becomes empty (and pathological).

In general, to make the theory sensible and avoid the curvature singularity we
need to introduce a UV cutoff. We require
$\rho\leq\rho_{0}<e^{\widetilde{\theta}_{0}}$ where
$\widetilde{\theta}_{0}:=-{1\over 2}\ln\gamma$. We also need to provide
suitable boundary conditions for the bulk fields at $\rho_{0}$. The radial
position of the boundary is $\gamma\rho_{0}^{2}:=b_{1}$ in string units. Thus,
we have $b_{1}<b_{0}$. Therefore, this way single-trace $T{\overline{T}}$
deformation is said to move the boundary into the bulk.

These observations are in agreement with one’s expectations. In the case
$\gamma\leq 0$ we expect the deformed spacetime theory to be free of
singularities and/or ambiguities. In particular, since the NEC is satisfied,
we expect the energies to be real.††34 One exception to this, e.g., is the
Casimir vacuum [70]. Indeed, for negative coupling, the spectrum of the
deformed spacetime theory dual to the long string sector (quantized on a
circle) is real, see [7] and references therein. However, the boundary
(radial) position $b_{0}$ is now washed away to infinity $b_{-1}$, i.e.,
$b_{0}<b_{-1}=\infty$. In the case $\gamma>0$ we expect a pathology/feature
analogous or dual to the curvature singularity. In general, for positive
coupling, the spectrum is complex, see [7] and references therein. Thus, the
creation of the curvature singularity is accompanied in the deformed theory by
the energies of some states becoming imaginary. In this case the timelike
boundary ceases to exist, i.e., it dissolves. To make the theory well-defined,
in general, we may need to impose a UV cutoff, i.e., $b_{1}<b_{0}$, and a set
of suitable boundary conditions at $b_{1}$. However, we stress that it is not
clear whether such a set exists.

6\. Discussion

In this paper we studied Gödel (type) universes (4.1) in single-trace
$T{\overline{T}}$ deformation to gain insights into (single-trace)
$T{\overline{T}}$ deformation and holography. The single-trace
$T{\overline{T}}$ deformed string background (2.1) is obtained by performing
TsT transformation on $AdS_{3}$. In global coordinates, to the best of my
knowledge, the single-trace $T{\overline{T}}$ deformed string background
(2.1), which we again write here for convenience, has not been reported
elsewhere in the literature before. The deformed string action in global
coordinates is

$S_{k}={k\alpha^{\prime}\over 2\pi}\int d^{2}z\partial
Y^{a}\Sigma_{ab}\overline{\partial}Y^{b},\quad
k\alpha^{\prime}\Sigma_{ab}=g_{ab}+B_{ab},$

where
$Y^{a}=(Y^{0},Y^{1},Y^{2})=(\widetilde{\varphi},\widetilde{\psi},\widetilde{\theta})$,
$g_{ab}$ is the metric and $B_{ab}$ is the NSNS two form field. The background
fields $\Sigma_{ab}$, the dilaton $\phi$ and the NSNS three form flux $H$ are
given by

$\eqalign{\Sigma_{22}&=1,\cr\Sigma_{11}&=(e^{2\phi}/g_{s}^{2}){\tenrm
sinh}^{2}\widetilde{\theta},\cr\Sigma_{00}&=-(e^{2\phi}/g_{s}^{2}){\tenrm
cosh}^{2}\widetilde{\theta},\cr\Sigma_{01}&=-\Sigma_{10}=-{1\over
2}(e^{2\phi}/g_{s}^{2})\left({\lambda\over 4}-{\tenrm
cosh}(2\widetilde{\theta})\right),\cr B&=B_{01}d\widetilde{\varphi}\wedge
d\widetilde{\psi},\quad B_{01}=k\alpha^{\prime}\Sigma_{01},\cr
H&=dB=H_{012}d\widetilde{\varphi}\wedge d\widetilde{\psi}\wedge
d\widetilde{\theta},\quad H_{012}=\partial
B_{01}/\partial\widetilde{\theta},\cr e^{2\phi}&={g_{s}^{2}\over
1+\left({\lambda\over 4}\right)^{2}-2\left({\lambda\over 4}\right){\tenrm
cosh}(2\widetilde{\theta})}.}$

The coupling $\lambda<4$ since $\widetilde{\theta}$ is positive. The radial
location of the curvature singularity is given by the zero of $e^{-2\phi}$. At
$\lambda=4$, the singularity is at $\widetilde{\theta}=0$.

To examine the situation in which $\lambda$ is very close to $4$ or
equivalently $\widetilde{\theta}$ is very small, we first write
$\gamma=\lambda/4=1-\epsilon^{2}$ and
$\widetilde{\theta}^{2}=\epsilon^{4}(1-\alpha^{4})/4$, where $\epsilon\ll 1$.
We assume, without loss of generality, $\alpha$ is, at least, of the order of
$\epsilon^{1/4}$. This gives the metric $ds^{2}={1\over
16\epsilon^{4}}\left(d\epsilon^{4}\alpha^{4}\right)^{2}-{1\over\epsilon^{4}\alpha^{4}}d\widetilde{\varphi}^{2}+{\epsilon^{4}\over
4}\cdot{1\over\epsilon^{4}\alpha^{4}}d\widetilde{\psi}^{2}$. The $B$ field is
$B_{01}=k\alpha^{\prime}/(2\epsilon^{2}\alpha^{4})$. The dilaton is given by
$e^{2\phi}=g_{s}^{2}/(\epsilon^{4}\alpha^{4})$. The background fields can be
put into the form

$\eqalign{ds^{2}&=k\alpha^{\prime}\left(dr^{2}-{1\over r}dt^{2}+{1\over
r}d\eta^{2}\right),\cr&=k\alpha^{\prime}\left(dr^{2}-{dx^{+}dx^{-}\over
r}\right),\cr B&={k\alpha^{\prime}\over r}\cdot dt\wedge
d\eta,\cr&={k\alpha^{\prime}\over r}\cdot{1\over 2}\cdot dx^{-}\wedge
dx^{+},\cr\phi&=-{1\over 2}\ln
r+\kappa_{\epsilon},\quad\kappa_{\epsilon}={1\over 2}\ln\left(g_{s}^{2}\over
4\epsilon^{2}\right),}$

where $r=\epsilon^{2}\alpha^{4}/4\ll 1$,
$t=\left(1/2\epsilon\right)\cdot\widetilde{\varphi}$,
$\eta=\left(\epsilon/4\right)\cdot\widetilde{\psi}$ and
$x^{\pm}=t\pm\eta$.††35 In (super)gravity the metric, flux and dilaton (6.1)
are governed by the action [71]  $S={1\over 2\kappa^{2}}\int
d^{3}x\sqrt{-g}e^{-2\Phi}\left[R+4\left(\nabla\Phi\right)^{2}-{1\over
12}H^{2}\right],$ where $x^{a}=(x^{0},x^{1},x^{2})=(t,\eta,r)$ and
$e^{\Phi}={e^{\phi}\over g_{s}},\quad{1\over 2\kappa^{2}}={1\over
2\kappa_{0}^{2}g_{s}^{2}}={1\over 16\pi G_{N}},$ where $G_{N}$ is the
gravitational coupling constant in three spacetime dimensions. The background
fields (6.1) solve the equations of motion that follow from the action (6.1)
$\eqalign{R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-{1\over
4}H_{\mu\rho\sigma}H_{\nu}^{\rho\sigma}&=0,\cr\nabla^{\rho}\left(e^{-2\Phi}H_{\mu\nu\rho}\right)&=0+\varepsilon^{\alpha\beta}g_{\mu\alpha}g_{\nu\beta}{\cal
O}(\epsilon^{2}),\cr R+4\nabla^{2}\Phi-4\left(\nabla\Phi\right)^{2}-{1\over
12}H^{2}&=0,}$ where $\varepsilon^{01}=-\varepsilon^{10}=-1$ and
$\varepsilon^{21}=-\varepsilon^{12}=0$ is the totally antisymmetric tensor.
The Ricci scalar is $R=-7/(2r^{2})=-56/(\epsilon^{4}\alpha^{8})$. Note that
$\eta$ parameterizes a non-contractable loop, and thus the metric has no
conical singularity at $r=0$. Note for a given $\epsilon$,
$\widetilde{\theta}$ increases to $\epsilon^{2}/2\ll 1$ as we decrease $r$ to
zero. The geometry the metric describes looks like a funnel, see Fig. 2. Thus,
the strings get stretched more and more as $r$ gets closer and closer to zero.

Fig. 2: The plot depicts the space the metric $ds^{2}/{k\alpha^{\prime}}$
(6.1) describes at constant $t$. The compact direction is $\eta$. The distance
from the symmetry axis to the surface is $1/r$. Along the symmetry axis $r$
increases.

Note that if we naively take $\lambda=4$ in (2.1), (2.1) and (2.1) the
background fields reduces to

$\eqalign{ds^{2}=k\alpha^{\prime}\left(d\widetilde{\theta}^{2}+{1\over
4}\coth^{2}\widetilde{\theta}d\widetilde{\varphi}^{2}-{1\over
4}d\widetilde{\psi}^{2}\right),\cr g_{s}^{2}e^{-2\phi}=4\cdot{\tenrm
sinh}^{2}\widetilde{\theta}.}$

The three form NSNS flux $H$ is zero. The metric describes the would be
exterior spacetime had it exist or been physical. Thus, we need to apply Wick
rotations. We now apply a Wick rotation on $\widetilde{\psi}$ to analytical
continue to a Euclidean signature. There is no $B$ field. Thus, we simply set
$\widetilde{\psi}=-i\xi$. The result of the rotation and the trivial
redefinition $\widetilde{\theta}\to\widetilde{\theta}/2$ is

$\eqalign{ds^{2}={k\alpha^{\prime}\over
4}\left[d\widetilde{\theta}^{2}+\coth^{2}(\widetilde{\theta}/2)d\widetilde{\varphi}^{2}+d\xi^{2}\right],\cr
g_{s}^{2}e^{-2\phi}=4\cdot{\tenrm sinh}^{2}(\widetilde{\theta}/2).}$

The Ricci invariant is $R=-4\cdot{\tenrm csch}^{2}(\widetilde{\theta}/2)$. We
could compactify $\widetilde{\varphi}$ on a unit circle such that
$0\leq\widetilde{\theta}<\infty$, and $0\leq\widetilde{\varphi},\ \xi\leq
2\pi$. The metric has the topology of $D^{2}\times S^{1}$, where $D^{2}$ is a
two-disc. At $\widetilde{\theta}=0$ it has a curvature singularity. Note that
$\widetilde{\varphi}$ parameterizes a non-contractable loop, and thus it need
not be periodic to avoid a conical singularity at $\widetilde{\theta}=0$. The
geometry which the metric describes looks like a funnel, see Fig. 2. It is
wider at $\widetilde{\theta}=0$. It asymptotes for large $\widetilde{\theta}$
to a cylinder. The T-duality transformation on $\widetilde{\varphi}$ gives

$\eqalign{ds^{2}={k\alpha^{\prime}\over
4}\left[d\widetilde{\theta}^{2}+\tanh^{2}(\widetilde{\theta}/2)d\iota^{2}+d\xi^{2}\right],\cr
g_{s}^{2}e^{-2\phi}=4\cdot{\tenrm cosh}^{2}(\widetilde{\theta}/2),}$

where $0\leq\widetilde{\theta}<\infty$, and $0\leq\iota,\ \xi\leq 2\pi$. The
Ricci invariant is $R=4\cdot{\tenrm sech}^{2}(\widetilde{\theta}/2)$. We note
in particular that the T-dual solution has no singularity, i.e., the curvature
and dilaton are regular, however it has a horizon [72]. The manifold that the
metric describes at constant $\xi$ looks like a semi-cigar [[73]73,, [74]74],
see Fig. 3. The tip of the cigar is at $\widetilde{\theta}=0$. Far away from
the tip the manifold asymptotes to a cylinder.

Fig. 3: The plot depicts the space the metric $ds^{2}/{k\alpha^{\prime}}$
(6.1) describes at constant $\xi$. The compact direction is $\iota$. The
distance from the symmetry axis to the surface is
$\tanh(\widetilde{\theta}/2)$. Along the symmetry axis $\widetilde{\theta}$
increases.

We note that a further Wick rotation on $\widetilde{\varphi}$ in (6.1) gives

$\eqalign{ds^{2}={k\alpha^{\prime}\over
4}\left[d\widetilde{\theta}^{2}-\coth^{2}(\widetilde{\theta}/2)d\tau^{2}+d\xi^{2}\right],\cr
g_{s}^{2}e^{-2\phi}=4\cdot{\tenrm sinh}^{2}(\widetilde{\theta}/2).}$

This is the result obtained in [[75]75[76]--[77]77] in the semi-classical
large $k$ limit. This describes the singularity. Another way to get the result
(6.1) from (6.1) is by performing the following three steps.

1) We first make a T-duality transformation along the $\widetilde{\varphi}$
coordinate. This introduces the T-dual coordinate $\iota$. The solution we get
has no curvature and/or dilaton singularity. However, it has a regular
horizon. The result is

$\eqalign{ds^{2}=k\alpha^{\prime}\left(d\widetilde{\theta}^{2}+4\cdot\tanh^{2}\widetilde{\theta}d\iota^{2}-{1\over
4}\cdot d\widetilde{\psi}^{2}\right),\cr g_{s}^{2}e^{-2\phi}={\tenrm
cosh}^{2}\widetilde{\theta},}$

where $\iota$ is periodic with period $\pi$.

2) We next apply Wick rotations on the T-dual coordinate $\iota$ and
$\widetilde{\psi}$. This is allowed since the metric and dilaton are not
singular (and also there is no $B$ field). We simply set $\iota=-i\upsilon$
and $\widetilde{\psi}=-i\xi$. This changes the metric signature. We find

$\eqalign{ds^{2}=k\alpha^{\prime}\left(d\widetilde{\theta}^{2}-4\cdot\tanh^{2}\widetilde{\theta}d\upsilon^{2}+{1\over
4}\cdot d\xi^{2}\right),\cr g_{s}^{2}e^{-2\phi}={\tenrm
cosh}^{2}\widetilde{\theta}.}$

3) We finally apply inverse T-duality transformation along the Wick rotated
T-dual coordinate $\upsilon$, see [78]. This introduces the dual coordinate
$\tau$. This gives

$\eqalign{ds^{2}=k\alpha^{\prime}\left(d\widetilde{\theta}^{2}-{1\over
4}\cdot\coth^{2}\widetilde{\theta}d\tau^{2}+{1\over 4}\cdot
d\xi^{2}\right),\cr g_{s}^{2}e^{-2\phi}=4\cdot{\tenrm
sinh}^{2}\widetilde{\theta}.}$

The curvature is singular at $\widetilde{\theta}=0$ and thus $\tau$ need not
be compact and periodic. This is the result (6.1), whereupon $\tau$ is
uncompactified.

To describe, in the sense of [79], the interior of the singularity, we
analytically continue $\widetilde{\theta}$. This gives

$\eqalign{ds^{2}={k\alpha^{\prime}\over
4}\left[-d\chi^{2}+\cot^{2}(\chi/2)d\tau^{2}+d\xi^{2}\right],\cr
g_{s}^{2}e^{-2\phi}=4\cdot\sin^{2}(\chi/2),}$

where $-\pi\leq\chi\leq 0$. The Ricci invariant is $R=4\csc^{2}(\chi/2)$. At
$\chi=0$, the dilaton and curvature diverge. Thus, the solutions (6.1) and
(6.1) matches at $\chi=0=\widetilde{\theta}$. $\chi=-\pi$ corresponds to the
horizon. Therefore, the black hole solution is naked. The observation is in
agreement with one’s expectation and in general positive $\lambda$ leads to
unphysical phenomena. Note $\chi$ and $\widetilde{\theta}$ are different
coordinates and together they do not correspond a single solution [79]. We
will discuss in detail (6.1) and its interior solution (6.1) in a separate
paper.

In string theory, Gödel (type) universes are (usually) obtained first by
taking the Penrose limit and then by performing $T$-duality. The Penrose limit
of the spacetime ${\cal A}_{3}\times S^{3}$ with the metric given by the
direct sum of (6.1) and the metric on the three sphere (2.1) is given by
(3.1), which we again write below for convenience

$ds^{2}=-2dx^{-}dx^{+}-\left(\mu^{2}r^{2}+y^{2}\right)d{x^{+}}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2}+dy^{2}+y^{2}d\psi^{2},$

where $\mu$ is given by (3.1). It is interesting to note that in the case the
coupling $\gamma=-1$, i.e. $\mu=0$, the spacetime (6.1) reduces to

$\eqalign{ds^{2}&=-2dx^{-}dx^{+}-y^{2}d{x^{+}}^{2}+dy^{2}+y^{2}d\psi^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2},\cr&=-2dx^{+}\left(dx^{-}+y^{2}d\widehat{\psi}\right)+dy^{2}+y^{2}d\widehat{\psi}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2},\cr&=-2dx^{+}\left(dx^{-}-b_{\widehat{\psi}+}d\widehat{\psi}\right)+dy^{2}+y^{2}d\widehat{\psi}^{2}+dr^{2}+r^{2}d\widetilde{\psi}^{2},\quad
b_{\widehat{\psi}+}:=-y^{2}.}$

Here we have made the replacement $\psi=\widehat{\psi}+x^{+}$, and also we
have defined $b:=y^{2}dx^{+}\wedge d\psi$ (see (3.1)), see [80] for a
discussion on field theory and the dynamics of a particle with fixed $p_{-}$
on the metric (6.1), where $p_{-}$ is the momentum that is canonically
conjugate to $x^{-}$. The T-dual of the spacetime (6.1) is the Gödel (type)
universe given in (4.1), or equivalently, in which we write again below

$ds^{2}=-\left[dt-r^{2}\left(\mu_{-}\omega_{3}+\mu_{+}\overline{\omega}_{3}\right)\right]^{2}+dr^{2}+r^{2}d\Omega_{3}^{2}+dq^{2},$

where $\mu_{\pm}=(\mu\pm 1)/4$, and $\omega_{i}$ are the left invariant one-
forms and $\overline{\omega}_{i}$ are the right invariant one-forms of the
unit three sphere with metric $d\Omega_{3}^{2}$.

We showed that depending on the sign of the deformation coupling single-trace
$T{\overline{T}}$ deformation either speeds up or slows down the intrinsic
rotation (or vorticity) of the universe. That is, the holographic screen (4.1)
is either moved radially into the bulk or out to infinity relative to its
original position (which is $r_{\cal H}=l_{s}$ in string units). This
naturally suggests or shows that in holography (single-trace) $T\overline{T}$
deformation can be viewed, in general, as either moving the holographic
boundary into the bulk or washing it away to infinity. In AdS this breaks the
spacetime conformal symmetry. We noted that moving timelike boundary into bulk
creates a curvature singularity. Thus, the timelike boundary ceases to exist.
The creation of the singularity is understood or noted in the deformed
spacetime theory by the energies of some states becoming imaginary.

It is interesting to note that, in general, cosmological models or spacetimes,
which are flat or nearly so, do not have (observer independent) timelike
boundaries, and hence, as in the Gödel (type) universe, single-trace
$T{\overline{T}}$ deformation is not expected to generate a curvature
singularity (in particular upon moving the boundaries into the bulk).
Therefore, it is not expected to involve inconsistencies and pathological
results. It may be useful in constructing (physical) cosmological universes
and to gain insights into black hole singularity.

In a future work, we hope to study the string theory spectrum in the plane
wave background (3.1). In particular, we would like to study the properties of
the theory as a function of the parameter $\mu$ and better understand the
constraint (4.1). We also would like to study (single trace) $T{\overline{T}}$
deformation in the presence of fermions [[81]81,, [82]82]. In particular we
would like to know whether it can be understood in terms of fermionic
$T$-dualities and shift [[53]53,, [83]83]. We also hope to study, in a future
work, the causal structure of the single-trace $T{\overline{T}}$ deformed
Gödel (type) universe with black hole [84]. This can teach us also the causal
structure of the related Kerr spacetime under (single-trace) $T{\overline{T}}$
deformation. We plan also to study the asymptotic symmetry algebra of the
background (6.1) with a black hole in a future work.

Acknowledgements: I thank Nava Gaddam for reading the draft and useful
comments. I also thank Nadav Drukker and David Kutasov for useful comments and
email correspondence. I thank also the String and Quantum Gravity group at
ICTS for their valuable comments and suggestions. This work is supported by
the Department of Atomic Energy under project no. RTI4001.

Appendix A. Bosonic T-duality

We briefly review T-duality in Type II bosonic string theory. See
[[50]50[51]--[52]52,, [53]53] for detailed discussions.

Consider the string sigma model action

$S:=\int d^{2}z{\cal L},\quad{\cal L}=\partial
X^{m}\Sigma_{mn}(X)\overline{\partial}X^{n}.$

The background fields are organized as

$\quad\Sigma_{mn}:=G_{mn}+B_{mn},\quad G_{mn}=G_{nm},\quad B_{mn}=-B_{nm},$

where $G_{mn}$ are the components of the metric $G$ and $B_{mn}$ are the
components of the Kalb-Ramond NSNS two-form $B$ field. We omitted a term
linear in the dilaton $\Phi$ field. The dilaton enters at the quantum level,
see, e.g., [85].

We assume the background fields $\Sigma_{mn}$ and $\Phi$ are independent of
$X^{1}$. Thus, the transformation

$X^{1}\to X^{1}+c,$

is a symmetry of the action, where $c$ is a constant. Therefore, we can couple
the theory to a gauge field $A_{a}$ on the worldsheet. We replace $\partial
X^{1}\to\partial X^{1}+A$ and
$\overline{\partial}X^{1}\to\overline{\partial}X^{1}+{\overline{A}}$. The
gauged Lagrangian density is

${\cal L}=\partial X^{m}\Sigma_{mn}\overline{\partial}X^{n}+\partial
X^{m}\Sigma_{m1}{\overline{A}}+A\Sigma_{1n}\overline{\partial}X^{n}+A\Sigma_{11}{\overline{A}}+\chi(\partial{\overline{A}}-\overline{\partial}A),$

where the field $\chi$ is a Lagrange multiplier. The equation of motion of
$\chi$ forces $A_{a}$ to be the derivative of a scalar, i.e.,
$F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}=0$. Therefore, we get back the
original same action after fixing the gauge, e.g. by assuming $X^{1}=0$, and
integrating out $\chi$. In general, however, $F_{ab}=0$ does not necessarily
imply the gauge field potential $A_{c}$ can be written globally as the
derivative of a scalar field. That is, it does not necessarily imply the gauge
potential is exact globally, i.e. there could be obstructions. $F_{ab}=0$ only
implies the gauge field is, at minimum, a pure gauge which is, by definition,
a local statement.

We assume $X^{1}$ is periodic and compact, i.e., at fixed worldsheet time it
parametrizes a circle. Thus, the symmetry (A.1) is a global $U(1)$ (zero-form)
symmetry. See [[50]50[51]--[52]52,, [53]53] for discussions in the case in
which $X^{1}$ is not compact, and in the case in which T-duality does not
apply or is not an exact symmetry.

We fix the gauge by assuming $X^{1}={\tenrm const.}$. This gives

$S=\int d^{2}z{\cal L},\quad{\cal L}=\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}1}{\overline{A}}+A\Sigma_{1{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+A\Sigma_{11}{\overline{A}}+\chi(\partial{\overline{A}}-\overline{\partial}A),$

where $\widehat{m}$ takes all values of $m$ except $m=1$. Integrating out the
field $\chi$, i.e., eliminating it using its equation of motion, gives back
(A.1). The equation of motion of $\chi$ is $F_{ab}=0$. This is solved by
$A=\partial\chi^{1}$ and ${\overline{A}}={\overline{\partial}}\chi^{1}$, where
$\chi^{1}$ is identified with $X^{1}$. The first de Rham cohomology group of
$S^{1}$ (or punctured plane) is $H^{1}(S^{1})={\tenrm I\kern-1.79993ptR}$.††36
The $r$th de Rham cohomology group $H^{r}(M)$ of a differentiable manifold $M$
is defined by the quotient $Z^{r}(M)/B^{r}(M)$, where $Z^{r}(M)$ is the set of
closed $r$ forms and $B^{r}(M)$ is the set of exact $r$ forms. Each set is a
group. Thus, in general a one form on $S^{1}$ is not exact. However, since
$X^{1}$ is compact and thus it winds around in either directions integer
times, we can still write the gauge potential on $S^{1}$ as the gradient of a
compact scalar and make the above identifications. This will get us back to
the original action. Note the gauge potential is not exact but it is closed
since the scalar can wind.

If we instead perform integration by parts, it leads to

$S^{\prime}=\int d^{2}z{\cal L}^{\prime},\quad{\cal L}^{\prime}=\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+A\Sigma_{11}{\overline{A}}-\left(\partial\chi-\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}1}\right){\overline{A}}+A\left(\Sigma_{1{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+\overline{\partial}\chi\right).$

Integrating out the gauge fields $A$ and $\overline{A}$ using their equations
of motion gives the T-dual Lagrangian density

${\cal L}^{\prime}=\partial
X^{\widehat{m}}\sigma_{{\widehat{m}}{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+\partial\chi\sigma_{\chi{\widehat{m}}}{\overline{\partial}X}^{\widehat{m}}+\partial
X^{\widehat{m}}\sigma_{{\widehat{m}}\chi}\overline{\partial}\chi+\partial\chi\sigma_{\chi\chi}\overline{\partial}\chi,$

where

$\sigma_{\chi\chi}={1\over\Sigma_{11}},\quad\sigma_{\chi\widehat{m}}={\Sigma_{1\widehat{m}}\over\Sigma_{11}},\quad\sigma_{{\widehat{n}}\chi}=-{\Sigma_{{\widehat{n}}1}\over\Sigma_{11}},\quad\sigma_{{\widehat{m}}{\widehat{n}}}=\Sigma_{{\widehat{m}}{\widehat{n}}}-{\Sigma_{{\widehat{m}}1}\Sigma_{1{\widehat{n}}}\over\Sigma_{11}}.$

The Jacobian factor introduced in the path integral measure by the change of
variables leads to a shift of the dilaton. The new dilaton $\phi$ is

$\phi=\Phi-{1\over 2}\log\left(\Sigma_{11}\right),\quad\Sigma_{11}=G_{11}.$

The new background fields are arranged as

$\sigma_{{\widehat{m}}{\widehat{n}}}=g_{{\widehat{m}}{\widehat{n}}}+b_{{\widehat{m}}{\widehat{n}}},\quad
g_{{\widehat{m}}{\widehat{n}}}=g_{{\widehat{n}}{\widehat{m}}},\quad
b_{{\widehat{m}}{\widehat{n}}}=-b_{{\widehat{n}}{\widehat{m}}},$

and similar expressions for $\sigma_{\chi{\widehat{n}}},\
\sigma_{{\widehat{n}}\chi},\ \sigma_{\chi\chi}$. Thus, the components of the
new metric $g$ and two-form $b$ field are given in terms of those in (A.1) by
the relations

$g_{\chi\chi}={1\over G_{11}},\quad
g_{\chi{\widehat{m}}}={B_{1{\widehat{m}}}\over G_{11}},\quad
g_{{\widehat{n}}\chi}=-{B_{{\widehat{n}}1}\over G_{11}},\quad
g_{{\widehat{m}}{\widehat{n}}}=G_{{\widehat{m}}{\widehat{n}}}-{1\over
G_{11}}\left(G_{{\widehat{m}}1}G_{1{\widehat{n}}}+B_{{\widehat{m}}1}B_{1{\widehat{n}}}\right),$

and

$b_{\chi{\widehat{m}}}={G_{1{\widehat{m}}}\over G_{11}},\quad
b_{{\widehat{n}}\chi}=-{G_{{\widehat{n}}1}\over G_{11}},\quad
b_{{\widehat{m}}{\widehat{n}}}=B_{{\widehat{m}}{\widehat{n}}}-{1\over
G_{11}}\left(G_{{\widehat{m}}1}B_{1{\widehat{n}}}+B_{{\widehat{m}}1}G_{1{\widehat{n}}}\right).$

The relations (A.1), (A.1) and (A.1) are also known as the Buscher rules.

Note that the field $\chi$ in (A.1) started its life as a Lagrange multiplier
and in general we are free to multiply it by a field independent factor, e.g.,
to remove a conical singularity see e.g. (3.1) and (6.1).

We note that in the case in which $\Sigma_{11}$ has zero scaling dimensions
the operator $A{\overline{A}}$ is exactly marginal. It does not acquire
anomalous dimensions since the fields $A$ and $\overline{A}$ are not
dynamical. Thus, we can deform (A.1) by adding $A{\overline{A}}$. This gives

$S^{\prime\prime}=\int d^{2}z{\cal L}^{\prime\prime},\quad{\cal
L}^{\prime\prime}=\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+A\Sigma^{\prime}_{11}{\overline{A}}-\left(\partial\chi-\partial
X^{\widehat{m}}\Sigma_{{\widehat{m}}1}\right){\overline{A}}+A\left(\Sigma_{1{\widehat{n}}}\overline{\partial}X^{\widehat{n}}+\overline{\partial}\chi\right),$

where

$\Sigma^{\prime}_{11}=\Sigma_{11}+\lambda,$

and $\lambda$ is the deformation coupling.

Appendix B. The single trace $T{\overline{T}}$ operator

We briefly review string theory on $AdS_{3}$. We also give the vertex operator
that defines the single trace $T{\overline{T}}$ deformation. We primarily
follow the discussion presented in [29].

We consider bosonic string theory on $AdS_{3}$. It is described by the Wess-
Zumino-Witten (WZW) sigma model on the group manifold $SL(2,R)$. The
worldsheet action possesses the affine current (Kac-Moody) Lie alegrba
$sl(2)\times sl(2)$. The chiral currents obey the OPE

$J^{a}(z_{1})J^{b}(z_{2})\sim{k\over
2}{\eta^{ab}\over(z_{1}-z_{2})^{2}}+{f^{ab}_{c}\over
z_{1}-z_{2}}J^{c}(z_{2}),$

where $\eta^{ab}$ is the Killing metric, $f^{ab}_{c}$ are the structure
constants and $k$ is the level of $sl(2)$. We will assume the level $k$ is
sufficiently large. Similar expressions for the ${\overline{J}}^{a}$. The
worldsheet stress-energy-momentum tensor is given by the Sugawara construction

$T^{\tenrm ws}(z)={1\over k-2}\eta_{ab}J^{a}J^{b}(z).$

Similar expression for the ${\overline{T}}(\overline{z})$.

Physical operators are represented by vertex operators. The vertex operators
and in general primary operators are eigenfunctions of the Laplacian on AdS.
The eigenvalues give the worldsheet scaling dimensions. For a vertex operator
to create a physical state it must have weight $(1,1)$††37 In general after
multiplying it by an appropriate vertex operator from the internal CFT., and
thus, it must be marginal. Most important vertex operators are in general
exactly marginal.

The function

$\Phi_{h}(z;x)={1\over\pi}\left(|\gamma-x|^{2}e^{\phi}+e^{-\phi}\right)^{-2h},$

is an eigenfunction of the Laplacian operator in the coordinates
$(\phi,\gamma,{\overline{\gamma}})$. The variables $(x,{\overline{x}})$ are
auxiliary and labels points on a two dimensional manifold. Thus, we regard
them as position coordinates on the auxiliary space.

The primary operator $\Phi_{h}(z;x)$ has the worldsheet scaling dimensions
$(\Delta_{h},{\overline{\Delta}}_{h})$

$\Delta_{h}=\overline{\Delta}_{h}={-h(h-1)\over k-2}.$

In momentum space, see [7], we have

$\Phi_{h}(z;p)={2^{-2h+2}\over\Gamma(2h)}\cdot
e^{i{\vec{p}}\cdot{\vec{\gamma}}}\cdot e^{2(h-1)\phi}\cdot\alpha^{2h-1}\cdot
K_{-2h+1}(\alpha),\quad\alpha:=|{\vec{p}}|e^{-\phi},$

where $K_{\nu}$ is the modified Bessel function of the second kind and
${\vec{p}}\cdot{\vec{\gamma}}={\overline{p}}\cdot{\gamma}+{p}\cdot{\overline{\gamma}}$.
$K_{\nu}$ has the property that $K_{-\nu}=K_{\nu}$ for all $\nu$. Near
$\alpha=0$, we have

$\eqalign{K_{i\nu}(\alpha)&\approx-\log(\alpha/2)\left(1-{1\over
6}\nu^{2}\log^{2}\left(\alpha/2\right)\right)+{\cal O}(\nu^{4}),\quad\nu\geq
0\cr K_{\nu}(\alpha)&\approx{1\over 2}\Gamma(\nu)\left({\alpha\over
2}\right)^{-\nu},\quad{\tenrm Re(\nu)}>0.}$

We note using the property of $K_{\nu}$ that

$G_{-h+1}(z;p)={\Gamma(2h)\over\Gamma(-2h+2)}G_{h}(z;p),$

where $G_{\nu}$ is defined as

$G_{\nu}(z;p):=\left({|{\vec{p}}|^{2}\over 4}\right)^{-\nu}\Phi_{\nu}(z;p).$

Thus, for real $h$ we only need to know $\Phi_{h}(z;x)$ for $h\geq 1/2$. For
complex $h={1/2}+is$, it relates $s>0$ with $s<0$. Thus, the operators
$\Phi_{1/2+is}$ and $\Phi_{1/2-is}$ are not independent. They are related as

$G_{{1\over 2}-is}(z;p)={\Gamma(1+2is)\over\Gamma(1-2is)}G_{{1\over
2}+is}(z;p).$

In the large $\phi$ limit we find

$\Phi_{h}(z;p)={1\over 2h-1}\cdot e^{i{\vec{p}}\cdot{\vec{\gamma}}}\cdot
e^{2(h-1)\phi}={1\over
2j+1}e^{2j\phi+i{\vec{p}}\cdot{\vec{\gamma}}},\quad{\tenrm Re}(h)>{1\over 2}.$

where $j=h-1$. Using the series representation of $K_{\nu}(\alpha)$, in the
small $\alpha$ limit, the first few relevant or dominant terms are

$\Phi_{h}(z;p)={1\over 2h-1}\cdot e^{i{\vec{p}}\cdot{\vec{\gamma}}}\cdot
e^{2(h-1)\phi}+{\Gamma(-2h+1)\over\Gamma(2h)}\cdot
4^{-2h+1}\cdot|{\vec{p}}|^{4h-2}\cdot e^{i{\vec{p}}\cdot{\vec{\gamma}}}\cdot
e^{-2h\phi}+\cdots.$

We rewrite this as

$\Phi_{h}(z;p)={1\over 2h-1}\left(e^{2(h-1)\phi}+R_{h}\cdot
e^{-2h\phi}\right)e^{i{\vec{p}}\cdot{\vec{\gamma}}}+\cdots,\quad
R_{h}={\Gamma(-2h+1)\over\Gamma(2h-1)}\left(|{\vec{p}}|^{2}\over
4\right)^{2h-1}.$

Note that in momentum space two point function of operators ${\cal
O}_{h,h}:={\cal O}$ of conformal dimensions $(h,h)$ is $\pi\cdot
R_{h}/(2h-1)$, see [7]. In position space the operators are normalized such
that $\langle{\cal O}(x){\cal O}(y)\rangle=1/|x-y|^{4h}$. Thus, in the large
$\phi$ limit

$\Phi_{h}(z;p)={1\over\pi}\left({2\pi\over
2\Delta-d}\cdot\rho^{-(d-\Delta)}+\langle{\cal O}(p){\cal
O}(-p)\rangle\cdot\rho^{-\Delta}\right)e^{i{\vec{p}}\cdot{\vec{\gamma}}},$

here $d=2$, $\Delta=2h$ and $\rho=e^{\phi}$. We hope to investigate the analog
of (B.1) in the theory (6.1) in a future work.

We note that in the case $h=1/2$ the two terms are equivalent and $R_{h}=-1$.
For exact result one can use (B.1) and the first equation in (B.1). We get
(see also [27] for a related discussion)

$\eqalign{\Phi_{1/2}(z;p)&=-\log\left(|{\vec{p}}|e^{-\phi}/2\right)\cdot
e^{-\phi}\cdot
e^{i{\vec{p}}\cdot{\vec{\gamma}}},\cr&={1\over\pi}\left[\pi\log\left(2\cdot
e^{\phi}\right)\cdot e^{-\phi}-{\pi\over
2}\log\left(|{\vec{p}}|^{2}/4\right)\cdot
e^{-\phi}\right]e^{i{\vec{p}}\cdot{\vec{\gamma}}}.}$

In the large $\phi$ limit $\Phi_{1/2}$ goes to zero. In general in the case
$2h-1=n\in{\tenbf Z}_{+}$, $R_{h}$ is divergent since the gamma function has
simple poles at negative integers and it needs to be regularized, see [7] for
further discussion.

We also note that for ${\tenrm Re}(h)>1$ $\Phi_{h}$ is divergent. The natural
norm $\int|\Phi_{h}|^{2}\Omega_{3}={1\over 2}\int d\phi d\gamma
d\overline{\gamma}e^{2\phi}|\Phi_{h}|^{2}$ implies $\Phi_{h}$ is non-
normalizable for ${\tenrm Re}(h)\geq 1/2$, where $\Omega_{3}$ is the invariant
volume element. Therefore, for ${\tenrm Re}(h)\geq 1/2$ it corresponds on the
boundary to a local operator. For $h=1/2+is$ with $s>0$, the norm ${1\over
2}\int d\phi d\gamma
d\overline{\gamma}e^{2\phi}\Phi^{*}_{h}\cdot\Phi_{h^{\prime}}\propto\delta^{(2)}(x-x^{\prime})\cdot\delta(s-s^{\prime})/s^{2}$,
see (B.1).††38 Note that one can normalize the operator by some function of
only $h$ such that the norm is
$\delta^{(2)}(x-x^{\prime})\cdot\delta(s-s^{\prime})$. Thus, in this case
$\Phi_{h}$ is delta-function normalizable. The operators $\Phi_{h}$ with
$h={1\over 2}+is$ belong to the principal continuous representations of
$SL(2,R)$. They correspond to (normalizable) states in the boundary CFT.

The global chiral symmetry is generated by the zero modes. On functions they
are represented by the differential operators

$J^{-}_{0}=\partial_{\gamma},\quad J^{3}_{0}=\gamma\partial_{\gamma}-{1\over
2}\partial_{\phi},\quad
J^{+}_{0}=\gamma^{2}\partial_{\gamma}-\gamma\partial_{\phi}-e^{-2\phi}\partial_{\overline{\gamma}}.$

Similar expressions for the ${\overline{J}}^{a}_{0}$. Near the (conformal)
boundary $\phi=+\infty$, we note, e.g. see (B.1), that

$J^{-}_{0}\equiv{\cal D}^{-}_{0}:=-\partial_{x},\quad J^{3}_{0}\equiv{\cal
D}^{3}_{0}:=-x\partial_{x}-j,\quad J^{+}_{0}\equiv{\cal
D}^{+}_{0}:=-x^{2}\partial_{x}-2xj,$

where $j=h-1$. They generate global conformal transformations on the auxiliary
space $(x,{\overline{x}})$. Thus, we interpret the auxiliary space as the
boundary spacetime and $(x,{\overline{x}})$ as the boundary spacetime
coordinates.

The primary operator $\Phi_{h}$ has spacetime scaling dimensions $(h,h)$. In
particular the primary operator $\Phi_{1}(z;x)$ has spacetime scaling
dimensions $(1,1)$. Near the boundary of AdS

$\Phi_{1}(z;x)=\delta^{(2)}(\gamma-x),$

see e.g. (B.1) or (B.1). Thus, we write near the boundary of AdS

$\Phi_{j+1}(z;x)={1\over 2j+1}\delta^{(2)}(\gamma-x)e^{2j\phi},\quad{\tenrm
Re}(j)>-{1\over 2}.$

For convenience we now organize the chiral currents. We first introduce a
near-boundary worldsheet current by taking some linear combination. Consider
the chiral current

$J(z;y):=-J^{+}(z;-y)=-e^{y{\cal D}_{0}^{-}}J^{+}(z;0)e^{-y{\cal
D}_{0}^{-}}=-J^{+}(z)+2yJ^{3}(z)-y^{2}J^{-}(z),$

where $J^{+}(z):=J^{+}(z;0)$. We note that $J$ has spacetime scaling dimension
$(-1,0)$, see (B.1). Similarly we then note that

$J^{3}(z;-x)={1\over 2}\partial_{x}J(z;x),\quad J^{-}(z;-x)=-{1\over
2}\partial^{2}_{x}J(z;x)=J^{-}(z).$

Similar expressions for the
${\overline{J}}^{a}({\overline{z}};-{\overline{x}})$.

The worldesheet stress-tensor (B.1) is given in terms of the worldsheet
current $J$ by

$T^{\tenrm ws}(z)={1\over 2(k-2)}\left[J\partial^{2}_{x}J-{1\over
2}(\partial_{x}J)^{2}\right].$

Here we set $x=0$. Similar expression for the ${\overline{T}}^{\tenrm
ws}({\overline{z}})$.

Primary fields near the boundary are defined with respect to $J$ via the OPE

$J(z_{1};x_{1})\Phi_{h}(z_{2};x_{2})\sim{2j(x_{2}-x_{1})+(x_{2}-x_{1})^{2}\partial_{x_{2}}\over
z_{1}-z_{2}}\Phi_{h}(z_{2};x_{2}),$

where $j=h-1$. Similar OPE expression with
${\overline{J}}({\overline{z}}_{1},{\overline{x}}_{1})$.

The near boundary gravition or spin 2 vertex operator polarized along the
boundary of AdS and corresponding to diffeomorphisms that preserve the
asymptotic form of AdS is given by

$T(z;x)={1\over
2(k-2)}\left(\partial_{x}J\partial_{x}\Phi_{1}+2\partial^{2}_{x}J\Phi_{1}\right){\overline{J}}.$

In three spacetime dimensions the graviton has no propagating degrees of
freedom. Thus, the graviton resides solely on the boundary. The spacetime
energy-momentum tensor is obtained by integrating the (near-boundary) graviton
vertex operator

$T(x)=\int d^{2}zT(z;x).$

Similar expression for ${\overline{T}}({\overline{x}})$.

The dilaton or spin zero vertex operator near the boundary is given by

$D(z;x)=\left(\partial_{x}J\partial_{x}+2\partial^{2}_{x}J\right)\left(\partial_{\overline{x}}{\overline{J}}\partial_{\overline{x}}+2\partial^{2}_{\overline{x}}{\overline{J}}\right)\Phi_{1}.$

The boundary dilaton or spin zero operator is then

$D(x)=\int d^{2}zD(z;x).$

It obeys the relation

$\int d^{2}xD(x,{\overline{x}})=4\int
d^{2}zJ^{-}{\overline{J}}^{-}(z,{\overline{z}}).$

This follows by changing order of integrations and performing integration by
parts. In the Wakimoto realization of the $sl(2)$ Lie algebra, the conserved
current $J^{-}:=\beta$, see Appendix C. $D(x,{\overline{x}})$ has the
spacetime scaling dimensions $(2,2)$. The equality (B.1) defines the single
trace $T{\overline{T}}$ deformation. See Appendix C for more discussion on
this.

Appendix C. Single trace $T{\overline{T}}$ deformation

We briefly review the single trace deformation and give the deformed spacetime
spectrum in a given winding or twisted sector. We mainly follow the discussion
presented in [7].

Bosonic string theory on $AdS_{3}$ can be equivalently described in the
Wakimoto realization of $SL(2,R)$ by the worldsheet action

$S={1\over 2\pi}\int
d^{2}z\left(\partial\phi{\overline{\partial}}\phi-{2\over\alpha_{+}}\phi
R^{(2)}+\beta{\overline{\partial}}\gamma+{\overline{\beta}}\partial{\overline{\gamma}}-\beta\overline{\beta}e^{-{2\over\alpha_{+}}\phi}\right),$

where $\alpha_{+}=\sqrt{2(k-2)}$, $k$ is the level and $R^{(2)}$ is the
curvature of the wordsheet. At $\phi=+\infty$, the null coordinates
$\gamma=x-t$ and $\overline{\gamma}=x+t$. Therefore, the zero mode of the
current $J^{-}$ associated with the Killing vector field $\partial_{\gamma}$
is related to the spacetime energy $E$ and momentum $P$. The fields $\beta$
and $\overline{\beta}$ can be identified with the background fields $A$ and
$\overline{A}$ in (A.1). Since the operator $e^{-{2\over\alpha_{+}}\phi}$ has
zero worldsheet scaling dimensions, as we noted in (A.1), we can deform the
action (C.1) by adding $\beta\overline{\beta}$. This gives

$S_{\lambda}={1\over 2\pi}\int
d^{2}z\left[\partial\phi{\overline{\partial}}\phi+\beta{\overline{\partial}}\gamma+{\overline{\beta}}\partial{\overline{\gamma}}-\beta\overline{\beta}\left(e^{-{2\over\alpha_{+}}\phi}-\lambda\right)\right].$

We omitted the dilaton term. The coupling $\lambda$ is dimensionless.
Integrating out the fields $\beta$ and $\overline{\beta}$ using their
equations of motion we get

$S_{\lambda}={1\over 2\pi}\int
d^{2}z\left(\partial\phi{\overline{\partial}}\phi+{e^{{2\over\alpha_{+}}\phi}\over
1-\lambda
e^{{2\over\alpha_{+}}\phi}}\partial{\overline{\gamma}}{\overline{\partial}}\gamma\right),$

which is in agreement with (2.1) and equations (D.35) and (D.37) in the next
Appendix. The dilaton is determined by requiring worldsheet conformal
invariance (to all orders in $\alpha^{\prime}$).

The Hilbert space of string theory on $AdS_{3}$ is given by a tensor product
of a short and long string sectors. Near the asymptotic boundary, i.e., in
perturbation theory in the string coupling $g_{s}^{2}\propto 1/p\propto 1/N$,
the long string sector is believed to be described in the dual boundary theory
by a symmetric product $S^{N}({\cal M}):={\cal M}^{N}/S_{N}$, where $p$ is the
(maximal) number of long strings, $N$ is the degree of the symmetric group
$S_{N}$ and ${\cal M}$ is the theory dual to a single long string.

In the long string sector in the case the theory is quantized on a circle the
deformed spacetime spectrum is given by [7],

${\overline{E}}(\lambda)={\overline{P}}-{1\over
2A}\left(B+\sqrt{B^{2}-4AC}\right),$

where

$A={1\over 2}\lambda,\quad B=-w+\lambda{\overline{P}},\quad
C=w({\overline{E}}_{0}-{\overline{P}}),$

and

${\overline{E}}={R{\cal E}\over 2\pi},\quad{\overline{P}}={RP\over
2\pi},\quad{\overline{E}}_{0}={\overline{E}}(\lambda=0),$

where $w$ is the winding number of the long string which counts the number of
times the string winds around the compact spatial direction. We rewrite the
spectrum (C.1) as

${\overline{E}}(\alpha)={1\over\alpha}\left(\sqrt{1+2\alpha{\overline{E}}_{0}+\alpha^{2}{\overline{P}}^{2}}-1\right),\quad\alpha=-{\lambda\over
w}.$

Note ${\overline{E}_{0}}$ is the energy of the undeformed state in the winding
or twisted sector $w$ and ${\overline{P}}$ is its momentum. In the case $w=1$
or untwisted sector the spectrum (C.1) is that of a $T{\overline{T}}$ deformed
CFT. This is in agreement with the identification that in the long string
sector the single trace deformation $D(x,{\overline{x}})$ is equivalent to a
symmetric product of the $T{\overline{T}}$ deformed CFT ${\cal M}$.

We note that in the $w$ twisted sector the states with
${\overline{E}}_{0}={\overline{P}}$ do not flow. That is,
${\overline{E}}(\alpha)={\overline{E}}_{0}$.

Appendix D. Single trace deformations as $O(d,d)$ transformations

We give the (super)gravity solutions of single trace deformation in the most
general case. We also give the deformed currents along the flow. We consider
single trace $T{\overline{T}},\ J{\overline{T}},\ T{\overline{J}}$ and
$J{\overline{J}}$ deformations. We follow [19].

We begin with string theory on $AdS_{3}\times S^{1}\times{\cal X}$. The
$S^{1}$ can be part of, e.g., $T^{4}$ or $S^{3}$. The $S^{1}$ component gives
rise in the boundary conformal field theory to a $U(1)$ current algebra
generated by the spacetime currents $J$ and $\overline{J}$. We will simply
ignore ${\cal X}$ since it is not relevant for our discussion.

The metric $g$ and two-form $B$ field are

$ds^{2}=g_{mn}dX^{m}dX^{n}=d\rho^{2}+e^{2\rho}d\gamma
d\overline{\gamma}+{1\over k}d\psi^{2},\quad B={1\over 2}B_{mn}dX^{m}\wedge
dX^{n}={1\over 2}e^{2\rho}d\gamma\wedge d\overline{\gamma},$

where $(\phi,\gamma,\overline{\gamma})$ are the coordinates on $AdS_{3}$ and
$\psi\sim\psi+2\pi$ is the coordinate on $S^{1}$. The dilaton field $\Phi$ is
constant (since the isometry group of the bulk AdS spacetime is the group of
conformal symmetries of the boundary spacetime).

The classical string sigma model action is

$S=\int d^{2}z\overline{\partial}X^{m}\left(g+B\right)_{mn}\partial X^{n}.$

The dilaton enters at one loop in $\alpha^{\prime}$.

Suppose $V_{a}=v_{a}^{m}\partial_{m},\ a=i,{\overline{j}}$ are the generators
of the target space isometries, i.e., Killing vector fields

${\cal L}_{V_{a}}g=0,\quad{\cal L}_{V_{a}}\Phi=0,\quad{\cal
L}_{V_{a}}B=-d{\widetilde{V}}_{a},\quad{\widetilde{V}}_{a}={\widetilde{v}}_{am}dX^{m},$

where ${\widetilde{V}}_{a}$ are one-forms.††39 Thus, $\delta
X=\epsilon_{a}\left(v_{a}^{m}\partial_{m}X-X^{m}\partial_{m}V_{a}\right)$ and
$\delta
B_{mn}=-\epsilon_{a}\left(\partial_{m}{\widetilde{v}}_{an}-\partial_{n}{\widetilde{v}}_{am}\right)$.
The action variation is $\delta S=\int
d^{2}z\left(\overline{\partial}\epsilon_{a}\cdot
J_{a}+\partial\epsilon_{a}\cdot{\overline{J}}_{a}\right)+{\tenrm equations\
of\ motion}$. This defines the currents. Note, since $d^{2}=0$, we have
$d{\cal L}_{V_{a}}B={\cal L}_{V_{a}}dB={\cal L}_{V_{a}}H=0$, where $H=dB$ is
the three-form NSNS flux. The corresponding associated currents are given by

$J_{i}/\sqrt{{\cal N}_{i}}=v_{i}^{m}(g+B)_{mn}\partial
X^{n}+{\widetilde{v}}_{im}\partial
X^{m},\quad{\overline{J}}_{\overline{i}}/\sqrt{{\cal
N}_{\overline{i}}}=v_{\overline{i}}^{m}(g-B)_{mn}{\overline{\partial}}X^{n}-{\widetilde{v}}_{{\overline{i}}m}{\overline{\partial}}X^{m}:=J_{\overline{i}},$

where ${\cal N}_{i}$ and ${\cal N}_{\overline{i}}$ are normalization
constants. In general in a CFT one can choose different normalization
constants for $J_{i}$ and ${\overline{J}}_{\overline{i}}$. We will below
choose ${\cal N}_{i}=1$ and ${\cal N}_{\overline{i}}=1$.

We introduce a generalized Killing vector field
$V_{a}^{M}=(v^{m}_{a},{\widetilde{v}}_{am})$. We will assume the generalized
Killing vectors $V_{a}^{M}$ form a closed algebra with structure constants
$f_{ab}^{c}$.

Consider $2q$ generalized Killing vector fields ${\cal
V}_{i}=(V_{i}^{M},V_{\overline{i}}^{M}),\ i=1,2,\cdots,q$. Define the set of
matrices

$\left(T_{i{\overline{j}}}\right)_{M}^{\quad
N}=\left(t_{i{\overline{j}}}\right)_{M}^{\quad
N}-\left(t_{{\overline{j}}i}\right)_{M}^{\quad
N},\quad\left(t_{i{\overline{j}}}\right)_{M}^{\quad
N}=V_{iM}V_{\overline{j}}^{N},$

where

$V_{aM}=\eta_{MN}V_{a}^{N},\quad\eta_{MN}=\pmatrix{0&\delta_{m}^{n}\cr\delta_{n}^{m}&0}.$

Thus,

$\left(t_{i{\overline{j}}}\right)_{M}^{\quad
N}=\pmatrix{{\widetilde{v}}_{im}v_{\overline{j}}^{n}&{\widetilde{v}}_{im}{\widetilde{v}}_{{\overline{j}}n}\cr
v_{i}^{m}v_{\overline{j}}^{n}&v_{i}^{m}{\widetilde{v}}_{{\overline{j}}n}}.$

The matrix $T_{i{\overline{j}}}$ satisfy the algebra of the group $O(d,d)$,
i.e., $\eta\cdot T_{i{\overline{j}}}+\eta\cdot T_{i{\overline{j}}}^{t}=0$,
where $T^{t}_{i{\overline{j}}}$ is the transpose of $T_{i{\overline{j}}}$ and
$\eta$ is the $O(d,d)$ invariant metric given in (D.1).

Define the $O(d,d)$ matrix

$h_{M}^{\quad
N}=e^{-r^{i{\overline{j}}}\left(T_{i{\overline{j}}}\right)_{M}^{\quad
N}}=e^{-2r^{i{\overline{j}}}\left(t_{i{\overline{j}}}\right)_{M}^{\quad
N}},\quad r^{i{\overline{j}}}=-r^{{\overline{j}}i},$

where $r^{i{\overline{j}}}$ is an antisymmetric matrix of couplings. For
constant $T_{i{\overline{j}}}$, the $O(d,d)$ transformation

${\cal H}^{\prime}_{MN}=h_{M}^{\quad P}{\cal
H}_{PQ}\left(h^{t}\right)^{Q}_{\quad N},\quad
e^{-2\Phi}\sqrt{-g}=e^{-2\Phi^{\prime}}\sqrt{-g^{\prime}},$

is a symmetry of string theory [19]. The generalized metric ${\cal H}$ is
defined††40 To see how the generalized metric ${\cal H}$ naturally arises,
consider the generalized vector $V^{M}=(W_{\sigma}^{m},\Pi^{\tau}_{m})$
$V=W_{\sigma}^{m}\partial_{m}+\Pi^{\tau}_{m}dX^{m},\quad
W_{\sigma}^{m}={1\over 2\pi}\partial_{\sigma}X^{m},\quad\Pi^{\tau}_{m}={1\over
2\pi\alpha^{\prime}}{\partial{\cal
L}\over\partial{\dot{X}}^{m}},\quad{\dot{X}}^{m}=\partial_{\tau}X^{m},$ where
$(\tau,\sigma)$ are worldsheet coordinates. $\sigma$ is the spatial coordinate
along the string and $\tau$ describes its propagation in time.
$\Pi^{\tau}_{m}=G_{mn}\Pi^{\tau n}$ is the canonical momentum density
conjugate to $X^{m}$. In a situation in which the spatial dimension $X^{m}$ is
compactified on a circle $S^{1}$ of radius $R$, i.e.
$X^{m}(\tau,\sigma+2\pi)=X^{m}(\tau,\sigma)+2\pi Rw$, the zero mode
$\int_{0}^{2\pi}d\sigma W_{\sigma}^{m}=wR$ gives (e.g., with $R=1$ in units of
length) the winding number $w$ of the string and
$\int_{0}^{2\pi}d\sigma\Pi^{\tau}_{m}=n/R$ gives (e.g., with $R=1$ in units of
length) the quantized integer $n$ which is also called the Kaluza-Klein
excitation number. At the classical level, the Hamiltonian density is then
given by $H=\Pi^{\tau}_{m}{\dot{X}}^{m}-{\cal L}=V^{M}{\cal
H}_{MN}\left(V^{t}\right)_{N}$. The Virasoro constraints are $V^{M}{\cal
H}_{MN}\left(V^{t}\right)_{N}=0$ and $V^{M}\eta_{MN}\left(V^{t}\right)^{N}=0$.
${\cal H}$ is an element of the group $O(d,d)$, i.e. ${\cal H}\eta{\cal
H}^{t}=\eta$.

${\cal
H}_{MN}=\pmatrix{g_{mn}-(BgB)_{mn}&\left(Bg\right)_{m}^{n}\cr-\left(Bg\right)_{n}^{m}&g^{mn}}.$

Thus, for constant $T_{i{\overline{j}}}$, $O(d,d)$ gives (super)gravity
solutions. In our discussion below $T_{i{\overline{j}}}$ are constant
matrices. The transformation (D.1) corresponds on the worldsheet to current
bilinear deformations.

In the cases the $r$ matrix with entries $r^{i{\overline{j}}}$ is a solution
of the Classical Yang-Baxter Equation (CYBE),

$f_{ab}^{c_{1}}r^{c_{2}a}r^{c_{3}b}+f_{ab}^{c_{2}}r^{c_{3}a}r^{c_{1}b}+f_{ab}^{c_{3}}r^{c_{1}a}r^{c_{2}b}=0,$

the deformation, at least to first order in perturbation theory in the
couplings, is marginal. The CYBE (D.1) is trivially satisfied in the case the
structure constants $f_{{\overline{i}}{\overline{j}}}^{k}=0$.

In the cases we shortly examine, not only $T_{i{\overline{j}}}$ are constant
matrices but the $r$ matrix satisfies the stronger equation,

$\sum_{i,j}r^{i{\overline{l}}}r^{j{\overline{m}}}f_{ij}^{k}=\sum_{{\overline{i}}{\overline{j}}}r^{l{\overline{i}}}r^{m{\overline{j}}}f_{{\overline{i}}{\overline{j}}}^{\overline{k}}=0,$

and thus, the deformation is expected to be marginal at finite values of the
couplings [86]. The condition (D.1) is sufficient for exact marginality of the
deformation only for constant $T_{i{\overline{j}}}$. The deformed worldsheet
Lagrangian density ${{\cal L}^{\prime}}$ obeys the flow equation

${\partial{\cal
L}^{\prime}\over\partial\lambda^{i{\overline{j}}}}=J^{\prime}_{i}J^{\prime}_{\overline{j}}=J^{\prime}_{i}{\overline{J}^{\prime}}_{\overline{j}},\quad\lambda^{i{\overline{j}}}={1\over
2}\left(r^{i{\overline{j}}}-r^{{\overline{j}}i}\right)=r^{i{\overline{j}}}.$

The deformed currents $J^{\prime}_{a}$ along the flow are given by

$J^{\prime}_{i}/\sqrt{{\cal
N}_{i}^{\prime}}=v_{i}^{m}(g^{\prime}+B^{\prime})_{mn}\partial
X^{n}+{\widetilde{v}}_{im}\partial
X^{m},\quad{\overline{J}^{\prime}}_{\overline{i}}/\sqrt{{\cal
N}_{\overline{i}}^{\prime}}=v_{\overline{i}}^{m}(g^{\prime}-B^{\prime})_{mn}{\overline{\partial}}X^{n}-{\widetilde{v}}_{{\overline{i}}m}{\overline{\partial}}X^{m}:=J_{\overline{i}},$

where ${\cal N}_{i}^{\prime}={\cal N}_{\overline{i}}^{\prime}$ is in general a
smooth function of the couplings and
$V_{a}^{M}=(v_{a}^{m},{\widetilde{v}}_{am}),\ a=i,{\overline{j}}$ are the
Killing vector fields (D.1). In general the generalized Killing fields
$V_{a}^{M}$ do not remain Killing fields under an $O(d,d)$ transformation. We
will assume the normalization ${\cal N}_{i}^{\prime}={\cal
N}_{\overline{i}}^{\prime}=1$ unless there is a different natural choice, see
e.g. subsection D.4. below.

The generalized Killing vector fields $V^{M}_{a}$ of the (super)gravity string
background (D.1) are

$\matrix{V_{1}^{M}=\left(\partial_{\gamma},0\right),&V^{M}_{\overline{1}}=(\partial_{\overline{\gamma}},0)\cr
V_{2}^{M}=\left(-\gamma\partial_{\rho}+\gamma^{2}\partial_{\gamma}-e^{-2\rho}\partial_{\overline{\gamma}},-\gamma
d\rho-d\gamma\right),&V^{M}_{\overline{2}}=\left(-\overline{\gamma}\partial_{\rho}+\overline{\gamma}^{2}\partial_{\overline{\gamma}}-e^{-2\rho}\partial_{\gamma},-\overline{\gamma}d\rho-d\overline{\gamma}\right)\cr
V_{3}^{M}=\left(-{1\over 2}\partial_{\rho}+\gamma\partial_{\gamma},-{1\over
2}d\rho\right),&V^{M}_{\overline{3}}=\left(-{1\over
2}\partial_{\rho}+\overline{\gamma}\partial_{\overline{\gamma}},{1\over
2}d\rho\right)\cr V_{4}^{M}={k\over 2}\left(\partial_{\psi},{1\over
k}d\psi\right),&V_{\overline{4}}^{M}={k\over 2}\left(\partial_{\psi},-{1\over
k}d\psi\right)}.$

They satisfy (in the sense of the Courant bracket [87], a generalization of
the Lie bracket) the algebra

$\matrix{[V^{M}_{1},V^{M}_{2}]=2V^{M}_{3},&[V^{M}_{3},V^{M}_{1}]=-V^{M}_{1},&[V^{M}_{3},V^{M}_{2}]=V^{M}_{2}\cr[V^{M}_{\overline{1}},V^{M}_{\overline{2}}]=2V^{M}_{\overline{3}},&[V^{M}_{\overline{3}},V^{M}_{\overline{1}}]=-V^{M}_{\overline{1}},&[V^{M}_{\overline{3}},V^{M}_{\overline{2}}]=V^{M}_{\overline{2}}\cr[V_{i}^{M},V_{i}^{M}]=0,&[V_{\overline{i}}^{M},V_{\overline{i}}^{M}]=0,&[V_{i}^{M},V_{\overline{j}}^{M}]=0}.$

Consider the Killing vector fields

$V_{1}^{M}=\left(\partial_{\gamma},0\right),\quad
V^{M}_{\overline{1}}=(\partial_{\overline{\gamma}},0),\quad V_{4}^{M}={k\over
2}\left(\partial_{\psi},{1\over k}d\psi\right),\quad
V_{\overline{4}}^{M}={k\over 2}\left(\partial_{\psi},-{1\over k}d\psi\right).$

The associated currents are (D.1)

$J_{1}:=J^{-}=e^{2\rho}\partial\gamma,\quad
J_{\overline{1}}:={\overline{J}}^{-}=e^{2\rho}{\overline{\partial}}\gamma,\quad
J_{4}:=J=\partial\psi,\quad
J_{\overline{4}}:={\overline{J}}={\overline{\partial}}\psi.$

We consider the deformation

$\alpha_{1}J^{-}{\overline{J}}^{-}+\alpha_{2}J^{-}{\overline{J}}+\alpha_{3}{\overline{J}}^{-}J+\alpha_{4}J{\overline{J}}.$

Note that the currents commute among themselves. The matrix
$r^{i{\overline{j}}}$ is

$r^{i{\overline{j}}}={1\over 2}\pmatrix{0\ &-\alpha_{1}\ &0\
&-\alpha_{2}\cr\alpha_{1}\ &0\ &\alpha_{3}\ &0\cr 0\ &-\alpha_{3}\ &0\
&-\alpha_{4}\cr\alpha_{2}\ &0\ &\alpha_{4}\ &0\cr}.$

Note that $v_{a}^{m}$ and ${\widetilde{v}}_{a}$ are constants (D.1). Thus,
$T_{i{\overline{j}}}$ are constant matrices. Therefore, the transformation
(D.1) is guaranteed to give a (super)gravity solution. The $r$ matrix (D.1)
also satisfies the condition (D.1), and thus the deformations are exactly
marginal.

The $O(4,4)$ transformation (D.1) gives the new (super)gravity solutions
$g^{\prime},\ B^{\prime}$ and $\Phi^{\prime}$. The new metric $g^{\prime}$ is

$\eqalign{g^{\prime}_{\rho\rho}&=1,\cr
g^{\prime}_{\gamma\overline{\gamma}}&=-{e^{2\rho}k\alpha_{4}^{2}\over
2e^{2\rho}\left(k\alpha_{1}\alpha_{4}^{2}+\alpha_{2}\alpha_{3}\left(-k\alpha_{4}+e^{k\alpha_{4}}-1\right)\right)-2k\alpha_{4}^{2}}=g^{\prime}_{\overline{\gamma}\gamma},\cr
g^{\prime}_{\gamma\psi}&={1\over-{2\alpha_{2}\over\alpha_{4}}-{2k\left(\alpha_{2}\alpha_{3}+\left(e^{-2\rho}-\alpha_{1}\right)\alpha_{4}\right)\over\alpha_{3}-e^{k\alpha_{4}}\alpha_{3}}}=g^{\prime}_{\psi\gamma},\cr
g^{\prime}_{\overline{\gamma}\psi}&={1\over-{2\alpha_{3}\over\alpha_{4}}-{2k\left(\alpha_{2}\alpha_{3}+\left(e^{-2\rho}-\alpha_{1}\right)\alpha_{4}\right)\over\alpha_{2}-e^{k\alpha_{4}}\alpha_{2}}}=g^{\prime}_{\psi\overline{\gamma}},\cr
g^{\prime}_{\psi\psi}&={e^{2\rho}\alpha_{2}\alpha_{3}+e^{k\alpha_{4}}\left(k\alpha_{4}^{2}+e^{2\rho}\left(\alpha_{2}\alpha_{3}(k\alpha_{4}-1)-k\alpha_{1}\alpha_{4}^{2}\right)\right)\over
k\left(k\alpha_{4}^{2}+e^{2\rho}\left(\alpha_{2}\alpha_{3}\left(k\alpha_{4}-e^{k\alpha_{4}}+1\right)-k\alpha_{1}\alpha_{4}^{2}\right)\right)}.}$

The new two-form $B^{\prime}$ field has the components

$B^{\prime}_{\gamma\overline{\gamma}}=g^{\prime}_{\gamma\overline{\gamma}},\qquad
B^{\prime}_{\gamma\psi}=g^{\prime}_{\gamma\psi},\qquad
B^{\prime}_{\psi\overline{\gamma}}=g^{\prime}_{\overline{\gamma}\psi}.$

The new dilaton $\Phi^{\prime}$ is given by

$e^{2\Phi^{\prime}}=g_{s}^{2}\sqrt{{\alpha_{4}^{4}k^{2}e^{\alpha_{4}k}\over\left(\alpha_{4}^{2}k-e^{2\rho}\left(\alpha_{1}\alpha_{4}^{2}k+\alpha_{2}\alpha_{3}\left(\alpha_{4}(-k)+e^{\alpha_{4}k}-1\right)\right)\right)^{2}}},$

where $g_{s}=e^{\Phi}={\tenrm const.}$ is the string coupling.

The deformed currents are given by

$J^{\prime}_{i}=v_{i}^{m}(g^{\prime}+B^{\prime})_{mn}\partial
X^{n}+{\widetilde{v}}_{im}\partial
X^{m},\quad{\overline{J}^{\prime}}_{\overline{i}}=v_{\overline{i}}^{m}(g^{\prime}-B^{\prime})_{mn}{\overline{\partial}}X^{n}-{\widetilde{v}}_{{\overline{i}}m}{\overline{\partial}}X^{m}.$

where $V_{a}^{M}=(v_{a}^{m},{\widetilde{v}}_{am}),\ a=i,{\overline{j}}$ are
the Killing vector fields (D.1).

In the remainder of this appendix we consider different cases.

D.1. $T{\overline{T}},\ J{\overline{T}},\ T{\overline{J}}$ deformations

Consider the case in which we turn on $\alpha_{1}$, $\alpha_{2}$ and
$\alpha_{3}$. The above metric (D.1) takes in this case the following form

$\eqalign{g^{\prime}_{\rho\rho}&=1,\cr
g^{\prime}_{\gamma\overline{\gamma}}&={e^{2\rho}\over 2}\cdot
m_{+}=g^{\prime}_{\overline{\gamma}\gamma},\cr
g^{\prime}_{\gamma\psi}&={e^{2\rho}\over
2}\cdot\alpha_{3}m_{+}=g^{\prime}_{\psi\gamma},\cr
g^{\prime}_{\overline{\gamma}\psi}&={e^{2\rho}\over
2}\cdot\alpha_{2}m_{+}=g^{\prime}_{\psi\overline{\gamma}},\cr
g^{\prime}_{\psi\psi}&={1\over k}\cdot{m_{+}\over m_{-}},}$

where

$m_{\pm}={1\over 1-\left(\alpha_{1}\pm{1\over
2}k\alpha_{2}\alpha_{3}\right)e^{2\rho}},$

The dilaton is

$e^{2\Phi^{\prime}}=g_{s}^{2}\sqrt{m_{+}^{2}}.$

The new two form field $B^{\prime}$ is given by (D.1). The new currents are
given by (D.1). They are given by

$\eqalign{J^{\prime-}&=m_{+}e^{2\rho}\partial\left(\overline{\gamma}+\alpha_{3}\psi\right)\cr&=m_{+}J^{-}+\alpha_{3}m_{+}e^{2\rho}J,\cr{\overline{J}^{\prime}}^{-}&=m_{+}e^{2\rho}\overline{\partial}\left(\gamma+\alpha_{2}\psi\right)\cr&=m_{+}{\overline{J}}^{-}+\alpha_{2}m_{+}e^{2\rho}{\overline{J}},\cr
J^{\prime}&={1\over 2}\left(1+{m_{+}\over m_{-}}\right)\partial\psi+{1\over
2}k\alpha_{2}m_{+}e^{2\rho}\partial\overline{\gamma}\cr&={1\over
2}\left(1+{m_{+}\over m_{-}}\right)J+{1\over
2}k\alpha_{2}m_{+}J^{-},\cr{\overline{J}^{\prime}}&={1\over
2}\left(1+{m_{+}\over m_{-}}\right)\overline{\partial}\psi+{1\over
2}k\alpha_{3}m_{+}e^{2\rho}\overline{\partial}\gamma\cr&={1\over
2}\left(1+{m_{+}\over m_{-}}\right){\overline{J}}+{1\over
2}k\alpha_{3}m_{+}{\overline{J}}^{-}.}$

The deformed worldsheet Lagrangian density ${\cal L}^{\prime}$ flows

${\partial{\cal
L}^{\prime}\over\partial\alpha_{1}}=J^{\prime-}{\overline{J}}^{\prime-},\quad{\partial{\cal
L}^{\prime}\over\partial\alpha_{2}}=J^{\prime-}{\overline{J}}^{\prime},\quad{\partial{\cal
L}^{\prime}\over\partial\alpha_{3}}=J^{\prime}{\overline{J}}^{\prime-}.$

D.2. $T{\overline{T}},\ J{\overline{T}}$ deformations

Consider the case in which we turn on $\alpha_{1}$ and $\alpha_{3}$. The
metric (D.1) takes in this case the following form

$\eqalign{g^{\prime}_{\rho\rho}&=1,\cr
g^{\prime}_{\gamma\overline{\gamma}}&={e^{2\rho}\over 2}\cdot
m=g^{\prime}_{\overline{\gamma}\gamma},\cr
g^{\prime}_{\gamma\psi}&={e^{2\rho}\over
2}\cdot\alpha_{3}m=g^{\prime}_{\psi\gamma},\cr
g^{\prime}_{\overline{\gamma}\psi}&=0=g^{\prime}_{\psi\overline{\gamma}},\cr
g^{\prime}_{\psi\psi}&={1\over k},}$

where

$m={1\over 1-\alpha_{1}e^{2\rho}},$

The dilaton is

$e^{2\Phi^{\prime}}=g_{s}^{2}\sqrt{m^{2}}.$

The new two form field $B^{\prime}$ is given by (D.1). The new currents are
given by (D.1). They are given by

$\eqalign{J^{\prime-}&=me^{2\rho}\partial\left(\overline{\gamma}+\alpha_{3}\psi\right)\cr&=mJ^{-}+\alpha_{3}me^{2\rho}J,\cr{\overline{J}^{\prime}}^{-}&=me^{2\rho}\overline{\partial}\gamma=m{\overline{J}}^{-},\cr
J^{\prime}&=\partial\psi=J,\cr{\overline{J}^{\prime}}&=\overline{\partial}\psi+{1\over
2}k\alpha_{3}me^{2\rho}\overline{\partial}\gamma\cr&={\overline{J}}+{1\over
2}k\alpha_{3}m{\overline{J}}^{-}.}$

D.3. $T{\overline{T}}$ deformation

Consider the case in which we turn on only $\alpha_{1}$. The metric (D.1)
takes in this case the following form

$\eqalign{g^{\prime}_{\rho\rho}&=1,\cr
g^{\prime}_{\gamma\overline{\gamma}}&={e^{2\rho}\over 2}\cdot
m=g^{\prime}_{\overline{\gamma}\gamma},\cr
g^{\prime}_{\gamma\psi}&=0=g^{\prime}_{\psi\gamma},\cr
g^{\prime}_{\overline{\gamma}\psi}&=0=g^{\prime}_{\psi\overline{\gamma}},\cr
g^{\prime}_{\psi\psi}&={1\over k},}$

where

$m={1\over 1-\alpha_{1}e^{2\rho}},$

The dilaton is

$e^{2\Phi^{\prime}}=g_{s}^{2}\sqrt{m^{2}}.$

The new two form field $B^{\prime}$ is given by (D.1). The new currents are
given by (D.1). They are given by

$\eqalign{J^{\prime-}&=me^{2\rho}\partial\overline{\gamma}=mJ^{-},\cr{\overline{J}^{\prime}}^{-}&=me^{2\rho}\overline{\partial}\gamma=m{\overline{J}}^{-},\cr
J^{\prime}&=\partial\psi=J,\cr{\overline{J}^{\prime}}&=\overline{\partial}\psi={\overline{J}}.}$

D.4. $J{\overline{J}}$ deformation

Consider the case in which we turn on only $\alpha_{4}$. The metric (D.1)
takes in this case the following form

$\eqalign{g^{\prime}_{\rho\rho}&=1,\cr
g^{\prime}_{\gamma\overline{\gamma}}&={e^{2\rho}\over
2}=g^{\prime}_{\overline{\gamma}\gamma},\cr
g^{\prime}_{\gamma\psi}&=0=g^{\prime}_{\psi\gamma},\cr
g^{\prime}_{\overline{\gamma}\psi}&=0=g^{\prime}_{\psi\overline{\gamma}},\cr
g^{\prime}_{\psi\psi}&={e^{k\alpha_{4}}\over k},}$

The dilaton is

$e^{2\Phi^{\prime}}=g_{s}^{2}e^{k\alpha_{4}/2}.$

Note that it is independent of the coordinates, i.e., it is a constant. The
new two form field $B^{\prime}$ is given by (D.1). The new currents are given
by (D.1). They are given by

$\eqalign{J^{\prime-}&=e^{2\rho}\partial\overline{\gamma}=J^{-},\cr{\overline{J}^{\prime}}^{-}&=e^{2\rho}\overline{\partial}\gamma={\overline{J}}^{-},\cr
J^{\prime}&={1\over 2}\left(1+e^{k\alpha_{4}}\right)\partial\psi={1\over
2}\left(1+e^{k\alpha_{4}}\right)J,\cr{\overline{J}^{\prime}}&={1\over
2}\left(1+e^{k\alpha_{4}}\right)\overline{\partial}\psi={1\over
2}\left(1+e^{k\alpha_{4}}\right){\overline{J}}.}$

The worldsheet deformed Lagrangian density ${\cal L}^{\prime}$ obeys the
equation

${\partial{\cal
L}^{\prime}\over\partial\alpha_{4}}=e^{k\alpha_{4}}J{\overline{J}}={\tenrm
sech}^{2}\left({k\alpha_{4}\over 2}\right)J^{\prime}{\overline{J}^{\prime}}.$

This flow equation is not of the form (D.1). However, the currents
$J^{\prime}$ and ${\overline{J}}^{\prime}$ are conserved independently and
thus, they can be redefined independently by multiplying each in general by a
different smooth function of the coupling $\alpha_{4}$. This equivalently
corresponds to choosing different values for the normalizations ${\cal
N}^{\prime}_{i}$ and ${\cal N}^{\prime}_{\overline{i}}$ in (D.1) for the
currents $J^{\prime}$ and ${\overline{J}}^{\prime}$. In a particular choice
for ${\cal N}^{\prime}_{4}$ and ${\cal N}^{\prime}_{\overline{4}}$ we get

${\partial{\cal
L}^{\prime}\over\partial\alpha_{4}}=J^{\prime}{\overline{J}^{\prime}}.$

This has the form (D.1).

Appendix E. The Maurer-Cartan form on the three sphere

The three sphere $S^{3}$ is the group manifold of $SU(2)$. The generators of
the Lie algebra $su(2)$ are

$a_{1}={1\over 2}i\sigma_{1},\quad a_{2}={1\over 2}i\sigma_{2},\quad
a_{3}={1\over
2}i\sigma_{3},\quad[a_{i},a_{j}]=-\epsilon_{ijk}a_{k},\quad\epsilon_{123}=+1.$

where $\epsilon_{ijk}$ is the totally antisymmetric tensor and $\sigma_{1},\
\sigma_{2}$ and $\sigma_{3}$ are the Pauli spin matrices. They are given by

$\sigma_{1}=\pmatrix{0&1\cr 1&0},\quad\sigma_{2}=\pmatrix{0&-i\cr
i&0},\quad\sigma_{3}=\pmatrix{1&0\cr
0&-1},\quad\sigma_{i}\sigma_{j}=\delta_{ij}I_{2\times
2}+i\epsilon_{ijk}\sigma_{k},\quad\epsilon_{123}=+1.$

Here and below a sum over repeated indices is assumed.

We parametrize the group element $g_{S^{3}}\in SU(2)$ as

$g_{S^{3}}=e^{i\varphi\sigma_{3}/2}e^{i\theta\sigma_{1}/2}e^{i\psi\sigma_{3}/2}=\pmatrix{e^{i(\varphi+\psi)/2}\cos(\theta/2)&ie^{i(\varphi-\psi)/2}\sin(\theta/2)\cr
ie^{-i(\varphi-\psi)/2}\sin(\theta/2)&e^{-i(\varphi+\psi)/2}\cos(\theta/2)},$

where the parameters $\varphi,\ \theta,\ \psi$ are the Euler angles. They
satisfy the conditions

$0\leq\varphi<2\pi,\quad 0\leq\theta<\pi,\quad-2\pi\leq\psi<2\pi.$

The inverse is given by

$g_{S^{3}}^{-1}=\pmatrix{e^{-i(\varphi+\psi)/2}\cos(\theta/2)&-ie^{i(\varphi-\psi)/2}\sin(\theta/2)\cr-
ie^{-i(\varphi-\psi)/2}\sin(\theta/2)&e^{i(\varphi+\psi)/2}\cos(\theta/2)}.$

The left invariant Maurer-Cartan one forms $\omega_{i}$ are given by

$\omega=g_{S^{3}}^{-1}dg_{S_{3}}=a_{i}\omega_{i},$

where

$\omega_{1}=\cos\psi d\theta-\sin\psi\sin\theta
d\varphi,\quad\omega_{2}=-\sin\theta\cos\psi d\varphi-\sin\psi
d\theta,\quad\omega_{3}=d\psi+\cos\theta d\varphi.$

The right invariant Maurer-Cartan one forms $\overline{\omega}_{i}$ are given
by

$\overline{\omega}=dg_{S^{3}}g_{S_{3}}^{-1}=a_{i}\overline{\omega}_{i},$

where

$\overline{\omega}_{1}=\cos\varphi d\theta+\sin\varphi\sin\theta
d\psi,\quad\overline{\omega}_{2}=\sin\theta\cos\varphi d\psi-\sin\varphi
d\theta,\quad\overline{\omega}_{3}=d\varphi+\cos\theta d\psi.$

The metric $d\Omega_{3}^{2}$ on the unit three sphere $S^{3}$ is given by

$\eqalign{d\Omega_{3}^{2}&=-{\tenrm
tr}\left(g_{S^{3}}^{-1}dg_{S_{3}}g_{S^{3}}^{-1}dg_{S_{3}}\right),\cr&={1\over
4}\sum_{i=1}^{3}\omega_{i}^{2}={1\over
4}\sum_{i=1}^{3}{\overline{\omega}}_{i}^{2},\cr&={1\over
4}\left(d\theta^{2}+d\varphi^{2}+d\psi^{2}+2d\varphi d\psi\cos\theta\right).}$

We rewrite this as

$d\Omega_{3}^{2}={1\over 4}\left(d\theta^{2}+d\varphi^{2}+d\psi^{2}+2d\varphi
d\psi\cos\theta\right)=d\widetilde{\theta}^{2}+\cos^{2}\widetilde{\theta}d\widetilde{\theta}_{+}^{2}+\sin^{2}\widetilde{\theta}d\widetilde{\theta}^{2}_{-},$

where

$\widetilde{\theta}={1\over 2}\theta,\quad\widetilde{\theta}_{+}={1\over
2}\left(\varphi+\psi\right),\quad\widetilde{\theta}_{-}={1\over
2}\left(\varphi-\psi\right).$

We also note that

${1\over 4}\mu(\overline{\omega}_{3}+\omega_{3})+{1\over
4}(\overline{\omega}_{3}-\omega_{3})=\mu\cos^{2}\widetilde{\theta}d\widetilde{\theta}_{+}+\sin^{2}\widetilde{\theta}d\widetilde{\theta}_{-}.$

References

relax[relaxrelaxreference11] A. B. Zamolodchikov, “Expectation value of
composite field T anti-T in two-dimensional quantum field theory,” [arXiv:hep-
th/0401146 [hep-th]]. relax[relaxrelaxreference22] F. A. Smirnov and A. B.
Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B
915, 363-383 (2017) doi:10.1016/j.nuclphysb.2016.12.014 [arXiv:1608.05499
[hep-th]]. relax[relaxrelaxreference33] A. Cavaglià, S. Negro, I. M. Szécsényi
and R. Tateo, “$T\overline{T}$-deformed 2D Quantum Field Theories,” JHEP 10,
112 (2016) doi:10.1007/JHEP10(2016)112 [arXiv:1608.05534 [hep-th]].
relax[relaxrelaxreference44] A. Giveon, N. Itzhaki and D. Kutasov,
“$T{\overline{T}}$ and LST,” JHEP 07, 122 (2017) doi:10.1007/JHEP07(2017)122
[arXiv:1701.05576 [hep-th]]. relax[relaxrelaxreference55] O. Aharony, S.
Datta, A. Giveon, Y. Jiang and D. Kutasov, “Modular invariance and uniqueness
of $T\overline{T}$ deformed CFT,” JHEP 01, 086 (2019)
doi:10.1007/JHEP01(2019)086 [arXiv:1808.02492 [hep-th]].
relax[relaxrelaxreference66] M. Asrat, “KdV charges and the generalized torus
partition sum in $T\overline{T}$ deformation,” Nucl. Phys. B 958, 115119
(2020) doi:10.1016/j.nuclphysb.2020.115119 [arXiv:2002.04824 [hep-th]].
relax[relaxrelaxreference77] M. Asrat, “$T{\overline{T}}$ and Holography,”
doi:10.6082/uchicago.3365 [arXiv:2112.02596 [hep-th]].
relax[relaxrelaxreference88] K. G. Wilson and W. Zimmermann, “Operator product
expansions and composite field operators in the general framework of quantum
field theory,” Commun. Math. Phys. 24 (1972), 87-106 doi:10.1007/BF01878448
relax[relaxrelaxreference99] J. H. Lowenstein, “Normal products in the
thirring model,” Commun. Math. Phys. 16, 265-289 (1970) doi:10.1007/BF01646535
relax[relaxrelaxreference1010] L. Apolo, P. X. Hao, W. X. Lai and W. Song,
“Glue-on AdS holography for $T\overline{T}$-deformed CFTs,” JHEP 06, 117
(2023) doi:10.1007/JHEP06(2023)117 [arXiv:2303.04836 [hep-th]].
relax[relaxrelaxreference1111] J. Manschot and S. Mondal, “Supersymmetric
black holes and $T{\overline{T}}$ deformation,” Phys. Rev. D 107, no.12,
L121903 (2023) doi:10.1103/PhysRevD.107.L121903 [arXiv:2207.01462 [hep-th]].
relax[relaxrelaxreference1212] L. McGough, M. Mezei and H. Verlinde, “Moving
the CFT into the bulk with $T\overline{T}$,” JHEP 04, 010 (2018)
doi:10.1007/JHEP04(2018)010 [arXiv:1611.03470 [hep-th]].
relax[relaxrelaxreference1313] E. Witten, “Multitrace operators, boundary
conditions, and AdS / CFT correspondence,” [arXiv:hep-th/0112258 [hep-th]].
relax[relaxrelaxreference1414] O. Aharony, M. Berkooz and E. Silverstein,
“Multiple trace operators and nonlocal string theories,” JHEP 08, 006 (2001)
doi:10.1088/1126-6708/2001/08/006 [arXiv:hep-th/0105309 [hep-th]].
relax[relaxrelaxreference1515] O. Lunin and J. M. Maldacena, “Deforming field
theories with U(1) x U(1) global symmetry and their gravity duals,” JHEP 05,
033 (2005) doi:10.1088/1126-6708/2005/05/033 [arXiv:hep-th/0502086 [hep-th]].
relax[relaxrelaxreference1616] S. Frolov, “Lax pair for strings in Lunin-
Maldacena background,” JHEP 05, 069 (2005) doi:10.1088/1126-6708/2005/05/069
[arXiv:hep-th/0503201 [hep-th]]. relax[relaxrelaxreference1717] D. Israel, C.
Kounnas and M. P. Petropoulos, “Superstrings on NS5 backgrounds, deformed
AdS(3) and holography,” JHEP 10, 028 (2003) doi:10.1088/1126-6708/2003/10/028
[arXiv:hep-th/0306053 [hep-th]]. relax[relaxrelaxreference1818] L. F. Alday,
G. Arutyunov and S. Frolov, “Green-Schwarz strings in TsT-transformed
backgrounds,” JHEP 06, 018 (2006) doi:10.1088/1126-6708/2006/06/018
[arXiv:hep-th/0512253 [hep-th]]. relax[relaxrelaxreference1919] T. Araujo, E.
Ó. Colgáin, Y. Sakatani, M. M. Sheikh-Jabbari and H. Yavartanoo, “Holographic
integration of $T\overline{T}$ & $J\overline{T}$ via $O(d,d)$,” JHEP 03, 168
(2019) doi:10.1007/JHEP03(2019)168 [arXiv:1811.03050 [hep-th]].
relax[relaxrelaxreference2020] L. Apolo, S. Detournay and W. Song, “TsT,
$T\overline{T}$ and black strings,” JHEP 06, 109 (2020)
doi:10.1007/JHEP06(2020)109 [arXiv:1911.12359 [hep-th]].
relax[relaxrelaxreference2121] A. Sfondrini and S. J. van Tongeren,
“$T\overline{T}$ deformations as $TsT$ transformations,” Phys. Rev. D 101,
no.6, 066022 (2020) doi:10.1103/PhysRevD.101.066022 [arXiv:1908.09299 [hep-
th]]. relax[relaxrelaxreference2222] S. F. Hassan and A. Sen, “Marginal
deformations of WZNW and coset models from O(d,d) transformation,” Nucl. Phys.
B 405, 143-165 (1993) doi:10.1016/0550-3213(93)90429-S [arXiv:hep-th/9210121
[hep-th]]. relax[relaxrelaxreference2323] J. M. Maldacena, “The Large N limit
of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2,
231-252 (1998) doi:10.4310/ATMP.1998.v2.n2.a1 [arXiv:hep-th/9711200 [hep-th]].
relax[relaxrelaxreference2424] S. S. Gubser, I. R. Klebanov and A. M.
Polyakov, “Gauge theory correlators from noncritical string theory,” Phys.
Lett. B 428, 105-114 (1998) doi:10.1016/S0370-2693(98)00377-3 [arXiv:hep-
th/9802109 [hep-th]]. relax[relaxrelaxreference2525] E. Witten, “Anti-de
Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253-291 (1998)
doi:10.4310/ATMP.1998.v2.n2.a2 [arXiv:hep-th/9802150 [hep-th]].
relax[relaxrelaxreference2626] T. Regge and C. Teitelboim, “Role of Surface
Integrals in the Hamiltonian Formulation of General Relativity,” Annals Phys.
88, 286 (1974) doi:10.1016/0003-4916(74)90404-7 relax[relaxrelaxreference2727]
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, “Asymptotically anti-de
Sitter spacetimes and scalar fields with a logarithmic branch,” Phys. Rev. D
70, 044034 (2004) doi:10.1103/PhysRevD.70.044034 [arXiv:hep-th/0404236 [hep-
th]]. relax[relaxrelaxreference2828] M. Henneaux, C. Martinez, R. Troncoso and
J. Zanelli, “Asymptotic behavior and Hamiltonian analysis of anti-de Sitter
gravity coupled to scalar fields,” Annals Phys. 322, 824-848 (2007)
doi:10.1016/j.aop.2006.05.002 [arXiv:hep-th/0603185 [hep-th]].
relax[relaxrelaxreference2929] D. Kutasov and N. Seiberg, “More comments on
string theory on AdS(3),” JHEP 04, 008 (1999)
doi:10.1088/1126-6708/1999/04/008 [arXiv:hep-th/9903219 [hep-th]].
relax[relaxrelaxreference3030] K. Becker and M. Becker, “Interactions in the
SL(2,IR) / U(1) black hole background,” Nucl. Phys. B 418, 206-230 (1994)
doi:10.1016/0550-3213(94)90245-3 [arXiv:hep-th/9310046 [hep-th]].
relax[relaxrelaxreference3131] O. Aharony, Z. Komargodski and S. S. Razamat,
“On the worldsheet theories of strings dual to free large N gauge theories,”
JHEP 05, 016 (2006) doi:10.1088/1126-6708/2006/05/016 [arXiv:hep-th/0602226
[hep-th]]. relax[relaxrelaxreference3232] K. Godel, “An Example of a new type
of cosmological solutions of Einstein’s field equations of graviation,” Rev.
Mod. Phys. 21, 447-450 (1949) doi:10.1103/RevModPhys.21.447
relax[relaxrelaxreference3333] M. M. Som and A. K. Raychaudhuri,
“Cylindrically symmetric charged dust distributions in rigid rotation in
general relativity,” Proc. R. Soc. Lond. A 304 81-86 (1968)
doi:10.1098/rspa.1968.0073 relax[relaxrelaxreference3434] J. P. Gauntlett, J.
B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, “All supersymmetric
solutions of minimal supergravity in five- dimensions,” Class. Quant. Grav.
20, 4587-4634 (2003) doi:10.1088/0264-9381/20/21/005 [arXiv:hep-th/0209114
[hep-th]]. relax[relaxrelaxreference3535] E. K. Boyda, S. Ganguli, P. Horava
and U. Varadarajan, “Holographic protection of chronology in universes of the
Godel type,” Phys. Rev. D 67, 106003 (2003) doi:10.1103/PhysRevD.67.106003
[arXiv:hep-th/0212087 [hep-th]]. relax[relaxrelaxreference3636] T. Harmark and
T. Takayanagi, “Supersymmetric Godel universes in string theory,” Nucl. Phys.
B 662, 3-39 (2003) doi:10.1016/S0550-3213(03)00349-3 [arXiv:hep-th/0301206
[hep-th]]. relax[relaxrelaxreference3737] D. Brecher, P. A. DeBoer, D. C. Page
and M. Rozali, “Closed time - like curves and holography in compact plane
waves,” JHEP 10, 031 (2003) doi:10.1088/1126-6708/2003/10/031 [arXiv:hep-
th/0306190 [hep-th]]. relax[relaxrelaxreference3838] R. Bousso, “The
Holographic principle for general backgrounds,” Class. Quant. Grav. 17,
997-1005 (2000) doi:10.1088/0264-9381/17/5/309 [arXiv:hep-th/9911002 [hep-
th]]. relax[relaxrelaxreference3939] S. W. Hawking and G. F. R. Ellis, “The
Large Scale Structure of Space-Time,” Cambridge University Press, 2023, ISBN
978-1-00-925316-1, 978-1-00-925315-4, 978-0-521-20016-5, 978-0-521-09906-6,