Physics Physical Sciences Doctor of Philosophy This dissertation is dedicated
to my mother T’iru.Epigraph Text

#  $T{\bar{T}}$TTbar and Holography

Asrat Demisē

(August 2021)

###### Abstract

###### Contents

  1. 0 Introduction
    1. 1 Wilsonian RG flow
    2. 2 Recent developments
    3. 3 Organization of the dissertation
  2. 1 $T{\bar{T}}$ deformation
    1. 1 $T{\bar{T}}$ deformation
      1. 1 Free massless scalar field
      2. 2 Deformed spectrum
      3. 3 Casimir energy
    2. 2 Single trace $T{\bar{T}}$ deformation
      1. 1 The Gauge/Gravity duality
      2. 2 NS5 - NS1 Branes System
      3. 3 Deformed spectrum
  3. 2 Correlation functions
    1. 1 Two point function in momentum space
    2. 2 Two point function in position space
  4. 3 Entanglement entropy and Entropic $c$-function
    1. 1 Single trace $T{\bar{T}}$, $J{\bar{T}}$ and $T{\bar{J}}$ deformations
    2. 2 Deformed string background
    3. 3 Holographic entanglement entropy
      1. 1 Case $\Psi=0:\lambda=0,\ \epsilon=0$
      2. 2 Case $\Psi=0:\lambda=\epsilon^{2}\neq 0$
      3. 3 Case $\Psi>0:\epsilon=0$
      4. 4 Case $\Psi>0:\epsilon\neq 0$
    4. 4 Entropic $c$–function
      1. 1 Case $\Psi=0:\lambda=0,\ \epsilon=0$
      2. 2 Case $\Psi=0:\lambda=\epsilon^{2}\neq 0$
      3. 3 Case $\Psi>0:\epsilon=0$
      4. 4 Case $\Psi>0:\epsilon\neq 0$
    5. 5 Discussion
  5. 4 Outlook
  6. 5 Fourier transform of primary operators in AdS
  7. 6 Intermediate relevant results

###### List of Figures

  1. 1 The plot shows the normalized effective central charge $c_{T{\bar{T}}}$ as a function of the size of the circle $R$ per the minimum size $L_{0}$.
  2. 1 Entropic c–function $C$ per $c_{0}$ as a function of interval length $L$ per $L_{0}$. Where $c_{0}={c\over 3}$ and $L_{0}={1\over 2}\sqrt{\pi t}$. The orange plot is for $\delta=0$, and the blue plot is for $\delta^{2}=1$. The (normalized) entropic c–function for the case $\delta=0$ diverges at $L_{0}$.

###### Acknowledgements.

It is a pleasure to acknowledge those who offered me their support,
encouragement and guidance during my PhD studies. First and foremost, I would
like to express my sincere gratitude to my advisor David Kutasov for his
continued feedback and guidance over the years. I also thank him for his
useful comments on the draft of the dissertation. I would like also to thank
Michael Levin, David Schmitz and Savdeep Sethi for serving on my dissertation
committee. I thank Savdeep Sethi for discussions on projects related to my
work and useful conversations. I also thank Jonah Kudler-Flam for inviting me
to collaborate on a project and continued discussions on related topics. In
the course of my PhD studies, I have met so many people who have supported,
encouraged and taught me. I am grateful to all the people in the Enrico Fermi
Institute and Physics Department. I am especially grateful to David Reid, Amy
Schulz, Robert Wald and Paul Wiegmann. I am indebted to the people at ICTP
from whom I learned so much about physics. I especially thank K. S. Narian who
was my thesis advisor at ICTP. Finally I would like to thank my friends in
Hyde Park and outside and my family for their time, constant support and
encouragement. I especially thank my mother.

##  Chapter 0 Introduction

In this chapter we present a brief summary of recent new developments in the
study of quantum field theories (QFTs) that are the foundations for the
dissertation. Along the way, we also discuss the motivations for the
dissertation and its main focuses. We end the chapter with a brief discussion
on the organization of the rest of the dissertation and on related published
works that are not included in the subsequent chapters. The study of QFTs is
generally the study of the different renormalization group (RG) flows. In what
follows, we begin with a brief discussion on Wilsonian RG flow which relates a
given quantum field theory (QFT) at different energy scales.

###  1 Wilsonian RG flow

A quantum field theory (QFT) is normally defined by specifying a set of
couplings and some cut-off scale. To study the theory at lower energy scales
we apply the renormalization group (RG) flow analysis. The RG is a group of
scale transformations, and it is equivalent to a redefinition of the cut-off
scale. Under the scale transformations, the couplings transform non-trivially.
This corresponds in the coupling space of the theory a trajectory, often
called flow, that connects the ultraviolet (UV) to the infrared (IR) regimes.
Thus, the RG flow compares the theory at different energy scales.

The contemporary understanding of mathematically well-defined QFTs is based on
fixed points of RG flow, and the Wilsonian idea is that a QFT is defined
through an RG flow that originates from a fixed point. Fixed points represent
scale invariant theories or conformal field theories (CFTs)111For a review on
the distinctions and/or relations between scale invariant and conformal field
theories see [1].. The (Wilsonian) RG flow is determined by specifying a set
of couplings at the fixed point. These couplings are associated with two
classes of operators.

At a fixed point operators are in general grouped based on their (scaling)
dimensions into three broad categories: irrelevant, relevant and marginal.

In general the number of relevant and marginal operators is finite, and these
are the operators for which we need to specify the couplings. These couplings
define the QFT and continue to describe all the possible RG flows to longer
distances. The general understanding after the work of Wilson and others (see
[2]) is that along an RG flow in the IR the theory is well-defined and/or
mathematically consistent and tractable. In general, the IR theory can be
either non-trivial, that is, interacting or free, or empty, that is, it can
have a mass gap. An example with a mass gap is non-abelian Yang-Mills (YM)
theory in four dimensions. This theory is believed to develop a mass gap in
the IR.

We must mention here, however, the phenomenon that along an RG flow it may
appear that an operator that was initially irrelevant (relevant) at the UV
fixed point builds up negative (positive) anomalous dimension, and eventually
becomes relevant (irrelevant) in the IR region. Such an operator that becomes
relevant (irrelevant) in the IR is sometimes referred to as a dangerously
irrelevant (harmlessly relevant) operator [3]. Thus, in general one needs also
to specify the couplings for the dangerously irrelevant operators since they
are relevant away from the UV fixed point.

On the other hand, the number of irrelevant operators at a given fixed point
is infinite. To study a QFT222The QFT is defined in the Wilsonian RG flow
sense discussed above by deforming the fixed point by a set of (marginally)
relevant operators. at high energies we specify in the theory the couplings
associated with the irrelevant operators. Thus, one has to consider a large
number of irrelevant operators. This makes, in general, difficult to apply the
RG analysis in a well organized and controlled way, and therefore, it limits
the predictive power or domain of validity of the theory. Note also that in
this case it is very likely that in the UV the theory may not be described by
the original fundamental field degrees of freedom. In general we may, but not
necessarily, attribute this to the existence of infinitely many irrelevant
operators at the fixed point. In general, whether one is able to consider one
or several of the infinitely many irrelevant operators, one will encounter
singularities or ambiguities and/or will be led to introduce (the other)
infinitely many couplings along the RG upflow towards the UV, and therefore,
in general observables are not UV finite. For the above reasons, the general
understanding has been, at least until recently, that RG upflows generated by
irrelevant operators do not lead to well-defined and mathematically tractable
quantum field theories.

###  2 Recent developments

In recent years, there have been two independent but related developments in
the study of irrelevant deformations in two dimensional QFTs [4, 5, 6, 7]. In
these developments deformations generated by special operators are considered.
Although the deformations are generated by irrelevant operators, several exact
results are obtained and observables are UV finite. Thus, the UV theories,
that is, the theories in the UV limit, are under better mathematical
tractability. However, they are distinct from conventional local QFTs. In the
dissertation, I will mainly study the second development which will be
discussed shortly below. In particular, I study the UV theory in the
corresponding irrelevant deformation using several observables as probes.

The first development is the deformation of a two dimensional QFT by the
determinant of the energy momentum stress tensor, commonly referred to as
$T{\bar{T}}$ deformation [4, 5, 6]. The generating $T{\bar{T}}$ operator is
irrelevant and it is well-defined (in the coincidence limit up to total
derivative terms). 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 coupling. This defines a trajectory in field theory space
parametrized by the deformation coupling. Therefore, it can be used to probe
and better understand the space of field theories.

In general, under an irrelevant deformation of a QFT, as we mentioned earlier,
the understanding is that at high energies or in the UV the theory is ill-
defined. The theory involves ambiguities and/or singularities and thus, it is
not solvable. The $T{\bar{T}}$ deformation, however, defines a theory that is
under better mathematical tractability and solvable. In particular, the
deformation preserves some of the symmetries of the original theory, and in
the resulting deformed theory, for several observables, exact and UV finite
results are obtained. The results include energy spectrum, S-matrix and
thermodynamic partition function [4, 5, 6, 8, 9, 11]. Other observables such
as correlation functions are also obtained in various limits [12, 13, 14]. At
high energies the deformed theory exhibits Hagdorn density of states and non-
locality. In particular, it is shown for a $T{\bar{T}}$ deformed CFT that
there exists an energy above which the specific heat is negative [15] and thus
the deformed theory has a large density of states [16]. These results indicate
that at high energies the deformed theory is not governed by a conventional
local fixed point.

The deformation can be thought of as coupling the theory to a two dimensional
gravity or a random geometry [17, 18, 9]. It can also be thought of as non-
trivial field dependent transformations or redefinitions of coordinates and
this, in particular, provides a simple explanation for the solvability of the
deformed theory [9, 10]. The deformation is also generalized in the presence
of supersymmetry [19] and to other irrelevant deformations for theories with
currents [20]. These latter deformations involving $U(1)$ currents are
commonly referred to as $J{\bar{T}}$ and $T{\bar{J}}$ deformations.

The second development is in two dimensional holographic CFTs which are dual
to string theory in asymptotically AdS spacetimes [7]. In this case the
deformation is commonly referred to as single-trace $T{\bar{T}}$ deformation.
The single-trace $T{\bar{T}}$ operator which generates the deformation is
irrelevant and present in any two dimensional holographic CFT. Thus, in two
dimensional holographic CFTs the deformation is universally applicable. This
deformation is closely related but distinct from the $T{\bar{T}}$
deformation333For a general discussion on single and multiple trace operators
in string theory see [21, 22, 23].. The single-trace $T{\bar{T}}$ operator, as
we will show in the next chapter, is given by a sum of $T{\bar{T}}$ operators
and therefore the name single-trace. The $T{\bar{T}}$ deformation in this case
is usually referred to as double-trace $T{\bar{T}}$ deformation for a reason
that will be clear in the next chapter. In general, however, it is not known
how to independently construct or define the single-trace $T{\bar{T}}$
operator in a generic two dimensional QFT that is not holographic and/or has
no tensor (or symmetric) product structure. The existence of such single-trace
operator in holography is related to a symmetric product structure of the
quantum field theories. I will comment further on this in Chapter 5.

The single-trace $T{\bar{T}}$ deformation corresponds in the bulk to a string
background that interpolates between $AdS_{3}$ in the IR and a linear dilaton
spacetime $\mathbb{R}^{(1,1)}\times\mathbb{R}$ in the UV. This background can
be constructed by taking the zero string coupling limit in some configuration
of fundamental strings and solitonic fivebranes. That is, the single-trace
$T{\bar{T}}$ deformation has interpretation in terms of branes. String theory
in the interpolating background is described by a deformed sigma model. The
deformation corresponds to adding an exact marginal current bilinear operator
to the IR Wess–Zumino–Witten (WZW) model. Thus, from the world-sheet
perspective it is more evident why the dual RG upflow is under better
mathematical tractability and that the dual deformed boundary theory is
solvable. In the dual string theory the deformation is also equivalent to
field dependent redefinitions or transformations of bulk coordinates or fields
and therefore this further provides a simple explanation as to why the
deformation in the boundary theory is solvable. The bulk description therefore
provides a way to study non-local theories non-perturbatively in a controlled
setting. Some quantities of interest such as energy spectrum, correlation
functions, thermodynamic partition function and entanglement entropy are
obtained [24, 25, 26, 27, 28].

At high energies the deformed boundary theory in this case also exhibits a
Hagedorn density of states and non-locality. This deformation is also
generalized to other irrelevant deformations for holographic theories with
currents [29]. These are commonly knows as single-trace $J{\bar{T}}$ and
$T{\bar{J}}$ deformations.

The single-trace $T{\bar{T}}$ deformation and its generalizations are useful
to better understand holography and gain insights into holography in flat
spacetimes. In the dissertation I present new results in the study of
holography in asymptotically non-AdS spacetimes.

###  3 Organization of the dissertation

The dissertation is mainly devoted to the study of single-trace
$T{\bar{T}},J{\bar{T}}$ and $T{\bar{J}}$ deformations in the context of
gauge/gravity duality. The organization of the dissertation is as follows.

In chapter two we introduce both $T{\bar{T}}$ and single-trace $T{\bar{T}}$
deformations in two dimensions. This chapter will be mainly a review. We begin
by defining the $T{\bar{T}}$ deformation. To demonstrate its relations to
string theory or quantum gravity we consider the deformation in the theory of
massless scalar fields444In the sense different from single-trace $T{\bar{T}}$
deformation. This will be clear in the next chapter.. In particular, we derive
the exact deformed Lagrangian density for a free massless scalar field at
finite values of the deformation coupling. We also obtain the spectrum for a
$T{\bar{T}}$ deformed QFT quantized on a circle. We discuss the main features
of this result in the case the initial theory is a CFT. In particular, we
discus the Casimir energy which defines in the deformed theory an effective
central charge as a function of the size of the circle. We next explain the
single-trace $T{\bar{T}}$ deformation in two dimensional holographic CFTs and
its relations to little string theory (LST). We review the gauge/gravity
duality and the NS5-NS1 branes configuration which in certain limits is
related to vacuum of LST. We also demonstrate that in the dual string theory
the single-trace $T{\bar{T}}$ deformation can also be interpreted as
equivalent to momentum dependent spectral flow or field dependent
redefinitions of bulk fields. We derive the single-trace $T{\bar{T}}$ deformed
spectrum from the dual string theory and discuss its relations to the
$T{\bar{T}}$ deformed spectrum.

In chapter three we discuss correlation functions in the single-trace
$T{\bar{T}}$ deformed boundary theory using the gauge/gravity duality. This
chapter is based on the paper [24]. The main question that we address is
concerning how the non-locality of the deformed theory is manifested in the
analytic structure of correlation functions of operators that are local in the
IR CFT. We in particular consider two point functions.

In chapter four we consider the general deformation involving the single-trace
operators $T{\bar{T}},J{\bar{T}}$ and $T{\bar{J}}$. This chapter is based on
my paper [27]. We study entanglement entropy in the boundary theory associated
to a spatial region of finite size from the dual string theory description. In
holographic field theories, entanglement entropy is encoded in certain
geometrical quantities in the bulk background, and the key observation of the
paper is to note that there is an isometry of the dual bulk string background
that preserves the boundary conditions imposed on the coordinates or fields.
This observation is very crucial to obtain exact results. We also compute the
Casin–Huerta entropic $c$–function. We study the monotonicity property of the
entropic $c$–function along the RG upflow and its independence of
regularization scheme that one introduces to regularize the UV divergence of
entanglement entropy. This provides further support that the RG upflow is
under better mathematical tractability. It also gives important insight into
the nature of the theory in the UV as it is not governed by a conventional
local UV fixed point. We also briefly comment on the holographic proposals for
the double-trace $T{\bar{T}},J{\bar{T}}$ and $T{\bar{J}}$ deformations.

In chapter five we comment on how one may define the single-trace $T{\bar{T}}$
deformation in a generic QFT. As we mentioned earlier, unlike the $T{\bar{T}}$
deformation, the single-trace $T{\bar{T}}$ deformation is defined only for
holographic theories or QFTs that have tensor product structures. In this
chapter we put forward an idea that can be useful in defining the single-trace
deformation in a generic QFT.

In the remainder of the current subsection I will briefly mention other
related research projects that are published but not included in the
dissertation.

In my paper [11] I studied the modular properties of Korteweg-De Vries (KdV)
charges correlation functions in a $T{\bar{T}}$ deformed two dimensional CFT.
A CFT has a symmetry algebra such as affine Lie algebra or Virasoro algebra.
The universal covering algebra contains an abelian subalgebra generated by KdV
charges. The $T{\bar{T}}$ deformation preserves the subalgebra. In the
undeformed CFT the KdV charges have well-defined modular properties. I showed
that this property also continues to exist in the resulting QFT after the
deformation. I found that correlation functions decompose into a direct sum of
two non-holomorphic but modular forms. I also obtained a general differential
equation that the KdV generalized torus partition function obeys. The
differential equation provides a non-perturbative description of $T{\bar{T}}$
deformed theories. I showed that the differential equation has a diffusion
like interpretation with reaction terms which depend on spins.

In the collaborative paper [30] we studied information theoretic quantities
both in single and double-trace $T{\bar{T}}$ deformations at zero and finite
temperatures. We computed mutual information and reflected entropy. For the
single-trace deformation we found that the mutual information and reflected
entropy diverge for disjoint intervals when the separation distance approaches
a minimum finite value. This implies that the mutual information fails to
serve as a geometric regulator which is related to the breakdown of the split
property at the inverse Hagedorn temperature. In contrast, for the double-
trace deformation we found all divergences to disappear including the standard
quantum field theory ultraviolet divergence that is generically seen as
disjoint intervals become adjacent. We furthermore computed reflected entropy
in conformal perturbation theory. While we found formally similar behavior
between bulk and boundary computations, we find quantitatively distinct
results. We commented on the interpretation of these disagreements and the
physics that must be altered to restore consistency. We also briefly discussed
the $T{\bar{J}}$ and $J{\bar{T}}$ deformations.

##  Chapter 1 $T{\bar{T}}$ deformation

In this chapter we discuss both the $T{\bar{T}}$ deformation and the closely
related single-trace $T{\bar{T}}$ deformation. We obtain the corresponding
deformed spectrums. We begin with the $T{\bar{T}}$ deformation. We follow [4,
5, 6]. See also [31, 32, 33].

###  1 $T{\bar{T}}$ deformation

We work in two dimensional Cartesian coordinates $(x,y)$. We define the
complex variables

$({\textrm{holomorphic}})\ z=x+iy,\quad(\textrm{antiholomorphic})\
{\bar{z}}=z^{*}=x-iy.$ (1)

We take $y$ as the Euclidean time.

Consider a two dimensional quantum field theory (QFT)111The theory can have a
gravity dual description in the context of gauge/gravity duality. with energy
momentum stress tensor ${T}_{\mu\nu}$. Define the composite operator
${{\textrm{T}}{\bar{\textrm{T}}}}$ by the determinant of ${T}_{\mu\nu}$ as

${\textrm{det}}\left({T}_{\mu\nu}\right):=-{1\over\pi^{2}}{\textrm{T}}{\bar{\textrm{T}}}.$
(2)

The components of the energy momentum stress tensor are given in a particular
normalization by

$T:=-2\pi T_{zz},\quad{\bar{T}}:=-2\pi T_{\bar{z}\bar{z}},\quad\Theta:=2\pi
T_{z{\bar{z}}}.$ (3)

With this normalization the composite operator 2 that generates the
$T{\bar{T}}$ deformation takes the form222For a conformal field theory (CFT)
the trace $\Theta$ is zero. Thus,
${{\textrm{T}}{\bar{\textrm{T}}}}=T{\bar{T}}$.

${{\textrm{T}}{\bar{\textrm{T}}}}=T{\bar{T}}-\Theta^{2}.$ (4)

In the coincidence limit $z\to z^{\prime}$, as we will show shortly, the
operators $T(z,{\bar{z}}){\bar{T}}(z^{\prime},{\bar{z}^{\prime}})$ and
$\Theta(z,{\bar{z}})\Theta(z^{\prime},{\bar{z}^{\prime}})$ are both divergent
but their non-derivative divergent parts cancel each other and therefore
leaving the combination 4 finite up to total derivative terms. Thus, the
composite operator 4 is well-defined (up to total derivative terms), and as a
result 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 trajectory in field
theory space.

We parametrize the trajectory by $t$ and denote the Lagrangian density at each
point along the trajectory by ${\mathcal{L}}_{t}$. That is, each point $t$
along the trajectory represents a field theory described by a Lagrangian
density ${\mathcal{L}}_{t}$. Along the trajectory the Lagrangian density obeys
or flows according to the equation,

${d{\mathcal{L}}_{t}\over
dt}={{\textrm{det}}}{T}^{(t)}_{\mu\nu}:=-{1\over\pi^{2}}({\textrm{T}}{\bar{\textrm{T}}})^{(t)},$
(5)

where ${T}^{(t)}_{\mu\nu}$ is the energy momentum stress tensor of the theory
at the point labeled by $t$ along the trajectory. The parameter $t$ serves as
the deformation coupling. It has mass dimension $-2$. Thus, for instance, at
$t=0$ the composite operator is irrelevant in the Wilsonian RG flow sense. To
leading order in $t$ the $T{\bar{T}}$ deformation is

${\mathcal{L}}_{t}={\cal
L}_{0}-{t\over\pi^{2}}({\textrm{T}}{\bar{\textrm{T}}})^{(0)}+{\cal O}(t^{2}).$
(6)

We next show that the composite operator 5 is well-defined. The composite
operator 5 in terms of the components of ${T}^{(t)}_{\mu\nu}$ is given by

$({{\textrm{T}}{\bar{\textrm{T}}}})^{(t)}=T^{(t)}{\bar{T}}^{(t)}-\left(\Theta^{(t)}\right)^{2}.$
(7)

To show that it is well-defined we need to make the following two assumptions.
We need to assume conservation of the energy momentum stress tensor, that is,

${\bar{\partial}}T^{(t)}-\partial\Theta^{(t)}=0,\quad\partial{\bar{T}}^{(t)}-{\bar{\partial}}\Theta^{(t)}=0,$
(8)

and that for any two (local) operators ${\mathcal{O}}^{(t)}_{a}$ and
${\mathcal{O}}^{(t)}_{b}$ in the theory at the point $t$ along the trajectory
the Operator Product Expansion (OPE) takes the following form

${\mathcal{O}}^{(t)}_{a}(z_{1},{\bar{z}_{1}}){\mathcal{O}}^{(t)}_{b}(z_{2},{\bar{z}_{2}})=\sum_{c}C_{ab}^{c}(z_{1}-z_{2},\bar{z}_{1}-\bar{z}_{2}){\mathcal{O}}^{(t)}_{c}(z_{2},{\bar{z}_{2}}).$
(9)

That is, besides the conservation equations 8, we are also assuming that there
exists a complete set of (local) operators at each point $t$ along the
trajectory obeying the OPEs 9. In general the OPE coefficients depend on the
deformation coupling $t$ but for brevity we omit the explicit dependence on
it. See [12] for a discussion on how they evolve along the trajectory. Note
that the coefficients are invariant under global (rigid) translation.

We define the composite operator 7 by point splitting as

$({{\textrm{T}}{\bar{\textrm{T}}}})^{(t)}(z,{\bar{z}})=\lim_{z\to
z^{\prime}}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(z^{\prime},{\bar{z}^{\prime}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(z^{\prime},{\bar{z}^{\prime}})\right).$
(10)

Consider differentiating 10 with respect to $\bar{z}$. This gives

$\displaystyle\partial_{\bar{z}}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
(11)
$\displaystyle\partial_{\bar{z}}T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\partial_{\bar{z}}\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}}).$
(12)

Using the conservation equation 8 this becomes

$\displaystyle\partial_{\bar{z}}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
(13)
$\displaystyle\partial_{z}\Theta^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\partial_{\bar{z}}\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}}).$
(14)

Adding the second equation from 8 after multiplying it by $\Theta^{(t)}$ to 13
and rearranging terms we get

$\displaystyle\partial_{\bar{z}}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
$\displaystyle(\partial_{z}+\partial_{w})(\Theta^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}}))$
(15) $\displaystyle-$
$\displaystyle(\partial_{\bar{z}}+\partial_{\bar{w}})(\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})).$
(16)

Similarly, we have

$\displaystyle\partial_{z}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
$\displaystyle(\partial_{z}+\partial_{w})(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}}))$
(18) $\displaystyle-$
$\displaystyle(\partial_{\bar{z}}+\partial_{\bar{w}})(T^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})).$
(19)

Using the second assumption 9 in the OPEs on the right-hand sides of 15 and 18
we write

$\displaystyle\partial_{\bar{z}}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
$\displaystyle\sum_{i}A^{i}\partial_{w}{\mathcal{O}}^{(t)}_{i}(w,{\bar{w}})$
(21) $\displaystyle+$
$\displaystyle\sum_{i}B^{i}\partial_{\bar{w}}{\mathcal{O}}^{(t)}_{i}(w,{\bar{w}}),$
(22)
$\displaystyle\partial_{z}\left(T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\right)=$
$\displaystyle\sum_{i}C^{i}\partial_{w}{\mathcal{O}}^{(t)}_{i}(w,{\bar{w}})$
(23) $\displaystyle+$
$\displaystyle\sum_{i}D^{i}\partial_{\bar{w}}{\mathcal{O}}^{(t)}_{i}(w,{\bar{w}}),$
(24)

for some functions $A^{i},B^{i},C^{i},D^{i}$. We write also the OPE for the
composite operator 7 as

$T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})=\sum_{i}F^{i}(z-w,{\bar{z}}-{\bar{w}}){\mathcal{O}}^{(t)}_{i}(w,{\bar{w}}).$
(25)

for some functions $F^{i}$.

It follows from 21 and 23 that any operator ${\mathcal{O}}^{(t)}_{i}$
appearing in the expansion in 25 unless itself is a coordinate derivative of
another (local) operator, comes with a constant (i.e. coordinate independent)
coefficient $F^{i}$. In other words, the OPE 25 can be written as

$T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})-\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})=({{\textrm{T}}{\bar{\textrm{T}}}})^{(t)}(w,{\bar{w}})+{{\textrm{derivatives
\ terms}}}.$ (26)

Thus, in the coincidence limit it is well-defined up to total derivates terms
which will not be relevant unless the theory is defined on a spacetime with
non-trivial topology.

####  1 Free massless scalar field

We now consider as an example the theory of a free massless scalar field
$\phi$ in two dimensions and deform it with the $T{\bar{T}}$ operator. We show
that this leads to the Nambu-Goto Lagrangian for a string in three dimensions.
Therefore, in general the deformation is related to a quantum field theory
coupled with gravity or random metrics.

The dynamics of the free field theory is governed by the local action
functional

$S_{0}=\int d^{2}x\sqrt{g}{\mathcal{L}}_{0},\quad{\mathcal{L}}_{0}={1\over
2}X,\quad X=g^{\mu\nu}X_{\mu\nu},\quad
X_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi,$ (27)

where $g_{\mu\nu}$ is an auxiliary metric tensor that we will use to compute
the (Hilbert) energy momentum stress tensor. We replace the metric
$g_{\mu\nu}$ with the flat spacetime metric at the end. The deformed
Lagrangian density obeys the flow equation 5

${d{\mathcal{L}}_{t}\over
dt}={{\textrm{det}}}({T}^{(t)}_{\mu\nu}):=-{1\over\pi^{2}}({\textrm{T}}{\bar{\textrm{T}}})^{(t)}.$
(28)

Lorentz invariance implies the deformed Lagrangian density ${\mathcal{L}}_{t}$
depends only on the deformation parameter $t$ and the Lorentz invariant scalar
$X$. Therefore, we write ${\mathcal{L}}_{t}:={\mathcal{L}}(t,X)$. The energy
momentum stress tensor is given by

$T^{(t)}_{\mu\nu}=-{2\over\sqrt{g}}{\delta S_{t}\over\delta
g^{\mu\nu}}=g_{\mu\nu}{\mathcal{L}}_{t}-2{\partial{\mathcal{L}}_{t}\over\partial
g^{\mu\nu}},$ (29)

and using this the ${T{\bar{T}}}$ composite operator is given by 7

${\textrm{det}}({T}^{(t)}_{\mu\nu})={\mathcal{L}}^{2}_{t}-2{\mathcal{L}}_{t}g^{\mu\nu}{\partial{\mathcal{L}}_{t}\over\partial
g^{\mu\nu}}+2\epsilon^{\mu\nu}\epsilon^{\rho\sigma}{\partial{\mathcal{L}}_{t}\over\partial
g^{\mu\rho}}{\partial{\mathcal{L}}_{t}\over\partial g^{\nu\sigma}},$ (30)

where $\epsilon^{\mu\nu}$ is the two dimensional Levi-Civita symbol and
satisfies the identity

$\epsilon^{\mu\rho}\epsilon^{\nu\sigma}=g^{\mu\nu}g^{\rho\sigma}-g^{\rho\nu}g^{\mu\sigma}.$
(31)

After adding the $T{\bar{T}}$ operator the flow equation for the Lagrangian
density becomes

$\partial_{t}{\mathcal{L}}_{t}+(X\partial_{X}-1){\mathcal{L}}_{t}^{2}=0.$ (32)

The solution to this differential equation is given by

${\mathcal{L}}_{t}=-{1\over 2t}+{1\over
2t}\sqrt{1+4t{\mathcal{L}}_{0}}=-{1\over 2t}+{\mathcal{L}}_{NG},$ (33)

where ${\mathcal{L}}_{NG}$ is the Nambu-Goto string Lagrangian in the static
gauge in three dimensions, and the coupling $t$ plays the role of
$\alpha^{\prime}$ which is the square of the string length $l_{s}$. In the
Polyakov approach, the Nambu-Goto action can be simplified by introducing
fluctuating random metrics (see [34]). Thus, at the classical level we note
that the $T{\bar{T}}$ deformed theory is related to string theory or a quantum
theory coupled with gravity (or random metrics).

####  2 Deformed spectrum

We now obtain the energy spectrum for a $T{\bar{T}}$ deformed QFT quantized on
a circle with circumference $R$. We make the identification $x\sim x+R$.
Therefore, we are considering a $T{\bar{T}}$ deformed QFT on an infinite
cylinder of size $R$. To obtain the spectrum we first consider the expectation
value of the composite $({{\textrm{T}}{\bar{\textrm{T}}}})^{(t)}$ 7 operator
in the state $|n;t\rangle$

$C_{n}:=\langle
n;t|({{\textrm{T}}{\bar{\textrm{T}}}})^{(t)}|n;t\rangle=\langle
T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})\rangle_{n}-\langle\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\rangle_{n},$
(34)

where $|n;t\rangle$ is the eigenstate of the deformed Hamiltonian.

An important property of the object $C_{n}$ that is useful in obtaining the
spectrum is its independence of the coordinates. We now show this property. We
differentiate $C_{n}$ with respect to $\bar{z}$. This gives

$\partial_{\bar{z}}C_{n}=\langle\partial_{\bar{z}}T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})\rangle_{n}-\langle\partial_{\bar{z}}\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\rangle_{n},$
(35)

which using the conservation equation 8 becomes

$\partial_{\bar{z}}C_{n}=\langle\partial_{z}\Theta^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(w,{\bar{w}})\rangle_{n}-\langle\partial_{\bar{z}}\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\rangle_{n}.$
(36)

We assume that for any (local) field ${\mathcal{O}}_{i}$ the expectation value
$\langle{\mathcal{O}}_{i}\rangle_{n}$ is a constant independent of
$z,{\bar{z}}$. We also assume global translation invariance of OPE
coefficients or 9. These lead, making use of the second equation of 8, to

$\partial_{\bar{z}}C_{n}=-\langle\Theta^{(t)}(z,{\bar{z}})\partial_{w}{\bar{T}}^{(t)}(w,{\bar{w}})\rangle_{n}+\langle\Theta^{(t)}(z,{\bar{z}})\partial_{w}{\bar{T}}^{(t)}(w,{\bar{w}})\rangle_{n}=0.$
(37)

Similarly, we find

$\partial_{z}C_{n}=0,$ (38)

Therefore, the auxiliary object defined in 34 is independent of the
coordinates.

Another important property of $C_{n}$ is its factorization property. We next
show that $C_{n}$ or the expectation value of $T{\bar{T}}$ factorizes into
expectation values of the components of the energy momentum stress tensor. We
first note that the two point function decomposes by inserting the identity as

$\displaystyle\langle
T^{(t)}(z,{\bar{z}}){\bar{T}}^{(t)}(z^{\prime},{\bar{z}}^{\prime})\rangle_{n}=$
(39) $\displaystyle\sum_{n^{\prime}}\langle
n;t|T^{(t)}(z,{\bar{z}})|n^{\prime};t\rangle\langle
n^{\prime};t|{\bar{T}}^{(t)}(z,{\bar{z}})|n;t\rangle
e^{({\mathcal{E}}_{n}-{\mathcal{E}}_{n^{\prime}})|y-y^{\prime}|}e^{i(P_{n}-P_{n^{\prime}})(x-x^{\prime})}.$
(40)

We also find a similar decomposition for
$\langle\Theta^{(t)}(z,{\bar{z}})\Theta^{(t)}(w,{\bar{w}})\rangle_{n}$.
Therefore, for the combination in $C_{n}$ 34 to be independent of the
coordinates, all terms in these decompositions with $n^{\prime}\neq n$ must
cancel out. We assume the spectrum is non-degenerate333This assumption is not
necessary if we also consider the KdV charges.. Thus, we have the following
factorization formula

$C_{n}=\langle n;t|{\textrm{det}}(T^{(t)}_{\mu\nu})|n;t\rangle=\langle
n;t|T^{(t)}|n;t\rangle\langle n;t|{\bar{T}}^{(t)}|n;t\rangle-\langle
n;t|\Theta|^{(t)}n;t\rangle\langle n;t|\Theta^{(t)}|n;t\rangle.$ (41)

We mention here that for a QFT on a curved spacetime the factorization formula
does not hold since the metric is dynamical. In a $T{\bar{T}}$ deformed CFT on
a curved spacetime it holds only to leading order in the large central charge
limit [35].

From the definition of the energy-momentum tensor we have

$\langle n;t|T^{(t)}_{yy}|n;t\rangle=-{1\over
R}{\mathcal{E}}_{n}({\textrm{energy \ density}}),$ (42) $-\langle
n;t|T^{(t)}_{xx}|n;t\rangle={d\over
dR}{\mathcal{E}}_{n}({\textrm{pressure}}),$ (43) $-i\langle
n;t|T^{(t)}_{xy}|n;t\rangle={1\over R}P_{n}(\textrm{momentum \ density}).$
(44)

Since the theory is quantized on a circle, the momentum is quantized in units
of the size of the circle. That is,

$\partial_{t}P_{n}=0.$ (45)

Making use of the factorization formula 41 and the flow equation in the
Hamiltonian formulation444The Hamiltonian is similar to the Lagrangian in
Euclidean spacetime. which is

${1\over R}\partial_{t}{\mathcal{E}}_{n}=-\langle
n;t|{\textrm{det}}(T^{(t)}_{\mu\nu})|n;t\rangle,$ (46)

we find the equation

$\partial_{t}{\mathcal{E}}_{n}+{\mathcal{E}}_{n}\partial_{R}{\mathcal{E}}_{n}=-{P_{n}^{2}\over
R}.$ (47)

This partial differential equation (PDE) is related to the inviscid Burgers
equation that appear in the study of the theory of turbulence in fluid
mechanics [36]555 In the case in which $p_{n}=0$ the PDE is the inviscid
Burgers equation..

In the case in which the original theory is a CFT the PDE 47 can be solved
exactly and the spectrum takes a simpler form. Solving the equation 47 with
the assumption that $t\geq 0$ and that at $t=0$ the theory is a CFT gives the
spectrum

${\mathcal{E}}_{n}(t)={R\over 2t}\left(\sqrt{1+{4tE_{n}\over
R}+{4t^{2}P_{n}^{2}\over R^{2}}}-1\right),\quad{\mathcal{E}}_{n}(t=0)=E_{n}.$
(48)

We note that the energy of a state in the deformed theory depends only on the
energy and momentum of the corresponding state in the undeformed CFT. We also
note that as a consequence of the factorization formula 41 there is no mixing
of states. Therefore, along the trajectory we are not creating or introducing
new states. The deformed energies ${\mathcal{E}}_{n}$ are in general positive
(and real) provided the coupling $t$ and the energies $E_{n}$ are positive
(and real). We note that states with energies

$E_{n}=P_{n},$ (49)

do not deform. That is, the energies do not flow ${\mathcal{E}}_{n}=E_{n}$. We
also note that in general in the case in which $t\gg R^{2}$ the spectrum is
independent of the size of the circle, and this suggests that in the UV the
theory is non-local (see [12]).

####  3 Casimir energy

In this section we discuss the deformed finite-size ground state energy or
Casimir energy of a CFT. For a CFT we have

$E_{n}(R)={2\pi\over R}\left(h_{n}+{\bar{h}}_{n}-{c\over 12}\right),\quad
P_{n}={2\pi\left(h_{n}-{\bar{h}}_{n}\right)\over R},$ (50)

where $h_{n}$ and $\bar{h}_{n}$ are the eigenvalues of the Virasoro generators
$L_{0}$ and $\bar{L}_{0}$, and $c$ is the central charge of the theory. The
ground state or Casimir energy of the deformed theory is given by

${\mathcal{E}}_{0}={1\over\pi\bar{R}\lambda}\left(\sqrt{1+2\lambda\pi\bar{R}E_{0}}-1\right)={1\over\pi\bar{R}\lambda}\left(\sqrt{1-{c\lambda\pi\over
6}}-1\right)=-{c_{T{\bar{T}}}(\lambda)\over 12\bar{R}},$ (51)

where the dimensionless deformation parameter $\lambda$ is defined as

$\quad\lambda={t\over\pi^{2}\bar{R}^{2}},\quad\bar{R}={R\over 2\pi},$ (52)

and the $c$-number $c_{T{\bar{T}}}(\lambda)$ defines an effective central
charge [37, 38, 39, 40]

$c_{T{\bar{T}}}(\lambda)={12\over\pi\lambda}\left(1-\sqrt{1-{c\lambda\pi\over
6}}\right),\quad c=c_{T{\bar{T}}}(\lambda=0).$ (53)

We assume $\lambda$ is positive. We note that the deformed Casimir energy
$\propto-c_{T{\bar{T}}}(\lambda)$ becomes complex unless the values of the
dimensionless coupling $\lambda$ are restricted to lie in the interval

$0\leq\lambda\leq{6\over\pi c}.$ (54)

Therefore, the Casimir energy becomes complex if the circumference or
perimeter of the circle is

$2\pi\bar{R}<L_{0}:=2\sqrt{\pi ct\over 6}.$ (55)

Thus, $T{\bar{T}}$ deformation introduces a minimum size or distance
$L_{0}\propto\sqrt{ct}$. In terms of the minimum size $L_{0}$ and the size of
the circle $R$ the effective central charge $c_{T{\bar{T}}}$ takes the form

${c_{T{\bar{T}}}(R)\over c}=2\cdot{R^{2}\over
L_{0}^{2}}\left(1-\sqrt{1-{L_{0}^{2}\over R^{2}}}\right).$ (56)

In the case the size of the circle $R<L_{0}$ the effective central charge
$c_{T{\bar{T}}}$ is complex and it has a positive real part and a negative
imaginary part. The interpretation of the imaginary part is not completely
clear666Purely imaginary central charges in general signal non-unitarity. In
$T{\bar{T}}$ however the effective central charge has a positive real part in
addition to an imaginary part. The real part is zero in the case $R<L_{0}$
only if the imaginary part is zero, and thus we do not expect a violation of
unitarity.. See Figure 1 for a plot of the effective central charge
$c_{T\bar{T}}$ as a function of the size of the circle $R$.

Figure 1: The plot shows the normalized effective central charge
$c_{T{\bar{T}}}$ as a function of the size of the circle $R$ per the minimum
size $L_{0}$.

Note that as we take the central charge $c$ large the range 54 of admissible
values of the coupling $\lambda$ shrinks more and more.

The existence of the minimum distance $L_{0}$ implies a maximum (also called
Hagedorn) temperature $T_{H}=1/\beta_{H}$777 The Hagedorn temperature was
first introduced in the thermodynamic studies of hadrons [41]., and
$\beta_{H}\propto L_{0}$. To show this relationship we consider the high
energy limit of 48. Since (we are assuming that) the energy eigenstates are
non-degenerate, we can use Cardy’s asymptotic entropy formula [38]. The
formula gives the high temperature behavior of the entropy. We find

$S=2\cdot 2\pi\sqrt{{c\over 6}\cdot{E\bar{R}\over 2}}\approx 4\pi\sqrt{{c\over
6}\cdot{t\over 4\pi}}{\mathcal{E}}=L_{0}{\mathcal{E}}=\beta_{H}{\mathcal{E}}.$
(57)

where $E$ is the total energy of the theory at $t=0$. We see a related result
in chapter four.

###  2 Single trace $T{\bar{T}}$ deformation

In this section we discuss the single-trace $T{\bar{T}}$ deformation and its
close relations with the double-trace $T{\bar{T}}$ deformation. We obtain the
single-trace $T{\bar{T}}$ deformed spectrum. We begin with a general and brief
review on gauge/gravity duality.

####  1 The Gauge/Gravity duality

The gauge/gravity duality or as often called the AdS/CFT correspondence is a
relationship between field theory and gravity [42, 43, 44, 45, 46]. The
general idea of the relationship is that string theory or M theory on some
background is related to and the same as some field theory. The relationship
is a duality because when the string theory is weakly coupled the field theory
is strongly coupled and vice versa. Historically, there are two motivations
and/or rationales for the relationship.

One motivation comes from ’t Hooft planar limit of certain (gauge) field
theories [47]. The limit involves taking the rank of the gauge group (or the
number of fundamental field degrees of freedom) to infinity while keeping the
(relevant) ’t Hooft coupling fixed. In this limit only planar Feynman diagrams
dominate scattering amplitudes and non-planar Feynman diagrams are suppressed
by the inverse of the rank of the gauge group with some exponents. This looks
similar to the loop (or genus) expansion of scattering amplitudes in string
theory after identifying the string coupling with the inverse of the rank of
the gauge group. These string amplitudes are also related to the amplitudes
that describe the dynamics of the (hadronic) strings that represent the flux
tubes of gluon fields connecting quarks.

The other motivation comes from two different views or descriptions of branes
in string theory [48, 49, 50, 51, 52]. One description treats them as non-
perturbative (solitonic) solutions of supergravity (to be more precise string
or M theory), in particular, as black holes with throat or near horizon
geometries. The supergravity backgrounds can be thought of as created by the
branes and matter. The other description treats the branes as
D(irichlet)-branes. D-branes are extended objects in spacetime where the
endpoints of open strings reside. They cary certain charges and have tensions.
Vector gauge and matter fields arise from strings that end on the branes.
Thus, these fields are confined on the branes world-volume. They, however,
interact with bulk fields such as graviton.

The two views are different descriptions of the same object and lead in the
low energy limit, where we keep the (relevant) coupling on the D-branes world-
volume fixed and send the Regge slope $\alpha^{\prime}$ and/or Planck length
$l_{p}$ to zero, to the correspondence or equivalence between the throat
region with a given boundary conditions and the non-gravitational field theory
on the D-branes world-volume. In this limit both the bulk and the Minkowski
spacetime far away from the throat are similar and decouple.

The simplest case of the correspondence is the equivalence between type IIB
string theory on $AdS_{5}\times S^{5}$ and ${\mathcal{N}}=4$ four dimensional
$U(N)$ SYM. The theory on the gravity side has two parameters. These are the
radius $R$ in string units $l_{s}$ or Planck units $l_{p}$ and the string
coupling $g_{s}$. These parameters are related on the field theory side to the
rank $N$ of the SYM gauge group and the SYM coupling $g$. The relations are

${\alpha^{\prime}\over
R^{2}}\sim{1\over\sqrt{\lambda}},\quad{\kappa^{2}_{10}\over R^{8}}\sim{1\over
N^{2}},\quad\alpha^{\prime}=l_{s}^{2},\quad\lambda=g^{2}N,\quad g^{2}\sim
g_{s},$ (58)

where $\kappa_{10}=\sqrt{8\pi G_{10}}\sim l_{p}^{4}$ is the ten dimensional
gravitational constant and $\lambda$ is the ’t Hooft coupling.

We note that $1/N$ corrections in gauge theory become quantum loop effects in
supergravity, while expanding in strong (’t Hooft) coupling amounts to
$\alpha^{\prime}$ corrections to supergravity. The gauge/supergravity theory
correspondence is reliable only when supergravity is valid. That is

$R\gg l_{s}\implies g\sqrt{N}\gg 1,\quad R\gg l_{p}\implies N\gg 1,$ (59)

so that the geometry is smooth. The correspondence is conjectured to be true
for all values of $\lambda$ and $N$.

The low energy supergravity solution corresponding to a type II Ramond-Ramond
(RR) charged black $p$-brane has in general a near horizon AdS geometry and
the correspondence is the statement that supergravity or to be more precise
string theory on a $d+1$ dimensional AdS spacetime times a compact manifold
with a given boundary conditions is dual to a $d$ dimensional conformal field
theory. AdS spaces are maximally symmetric solutions of the Einstein equations
with a negative cosmological constant. The isometry group $SO(2,d)$ of the
bulk AdS spacetime is the group of conformal symmetries of the one dimension
lower boundary. Therefore, it is useful to think of the conformal field theory
as living on the boundary of the AdS spacetime. The boundary is at a finite
conformal spatial distance and a light ray can reach the boundary in a finite
time. Therefore, AdS spacetime is a special box. The correspondence is also
holographic since the field theory is defined in at least one lower spacetime
dimension.

In the conformal field theory the natural/good observables are not S-matrices
as in QFT on flat spacetime but rather correlation functions of local
operators inserted on arbitrary boundary points emitting and absorbing
particles that propagate or are propagating in the interior (for related
discussions see [53, 54, 55]). In the correspondence, we identify generating
functional for correlation functions of local operators ${\mathcal{O}}^{I}$ in
the conformal field theory with the partition function of string or M theory
with some given boundary values for the fields $\Phi^{I}$,

$\langle
e^{\int{\mathcal{L}}+\Phi_{0}^{I}{\mathcal{O}}^{I}}\rangle=\langle\left.e^{W}\right|_{\Phi^{I}\to\Phi^{I}_{0}U^{\lambda_{I}}}\rangle_{ds^{2}},\quad
ds^{2}\propto{dU^{2}\over U^{2}}+U^{2}dx^{2},$ (60)

where $\lambda_{I}$ is determined by the mass of the field $\Phi^{I}$ which
act on the boundary as a source for the operator ${\mathcal{O}}^{I}$. The
coordinate $U$ labels the radial direction, and $\Phi^{I}_{0}$ is the boundary
value of the field $\Phi^{I}$ and it completely determines the solution to the
field equation of $\Phi^{I}$ in the AdS spacetime $ds^{2}$.

We next consider a brane configuration that is relevant for our discussion on
single-trace $T{\bar{T}}$ deformation.

####  2 NS5 - NS1 Branes System

S-duality888S-duality interchanges D1 with NS1, D5 with NS5, and it leaves D3
invariant [59]. or electric/magnetic duality requires the existence of NS5
brane [56]. NS5 is a 5-brane that is the magnetic dual of the elementary or
fundamental string that we denote here by NS1. NS1 couples with the Neveu-
Schwarz (NS) two-form. In type II string theories both the string and
fivebrane preserve half the supersymmetries.

The low energy type II supergravity solution corresponding to $k$ coincident
NS5 branes in the extremal limit is given by [57, 58, 59]

$ds^{2}=(-dt^{2}+dx_{1}^{2}\cdots+dx_{5}^{2})+H_{5}(dr^{2}+r^{2}d\Omega^{2}_{3}),$
(61) $e^{2\Phi}=g^{2}_{s}H_{5},\quad F_{3}=\star dH_{5},\quad
H_{5}=1+{R_{5}^{2}\over r^{2}},\quad{R_{5}^{2}\over\alpha^{\prime}}=k.$ (62)

The metric $ds^{2}$ is given in the string frame. The coordinates
$t,x_{1},x_{2},x_{3},x_{4},x_{5}$ label the directions along the fivebranes
and $d\Omega^{2}_{3}$ is the metric on the unit three dimensional sphere
$S^{3}$ in the space normal to the fivebranes. $\Phi$ is the dilaton field and
$g_{s}$ is the asymptotic value of the string coupling. The Hodge dual is
taken in the transverse space and $F_{3}$ is the Neveu-Schwarz–Neveu-Schwarz
(NSNS) 3-form flux.

We now consider taking the decoupling limit where we take the string coupling
$g_{s}$ and Planck length $l_{p}$ to zero. Note that we are not taking
$\alpha^{\prime}$ to zero so that the field theory coupling on the world-
volume of the branes is finite999On the NS5 branes the coupling
$g^{2}\sim\alpha^{\prime}$.. In this limit we obtain for the metric and
dilaton

$ds^{2}=(-dt^{2}+dx_{1}^{2}+\cdots+dx_{5}^{2}+d\phi^{2})+R_{5}^{2}d\Omega_{3}^{2},$
(63) $\Phi=-{1\over R_{5}}\phi.$ (64)

The boundary is at large positive $\phi$. We note that the string coupling
vanishes for large positive $\phi$ and it diverges as one approaches to large
negative $\phi$. Thus, studying string theory in the large negative $\phi$
region requires understanding the strong coupling behavior of the theory.

String theory in the linear dilaton background
$\mathbb{R}^{5,1}\times\mathbb{R}_{\phi}\times S^{3}$ 63 is conjectured to be
holographically dual to a six dimensional non-gravitionional theory on the
fivebranes world-volume. The six dimensional theory is both similar to and
different from the ten dimensional string theories. It is similar in that it
describes interacting strings. The strings have a string length proportional
to $l_{s}$ (see for example [60, 61]). It also exhibits Hagedorn spectrum and
duality symmetries. However, it is different because it is six-dimensional and
contains no graviton. For these reasons it is called little string theory
(LST) [62, 63, 64]. The LST can be defined with either type IIA or type IIB
fivebranes (see [65]).

We now consider adding $p$ NS1 strings along one of the coincident NS5 branes
spatial directions. We choose here the direction $x_{1}$. We compactify the
remaining spatial directions of the NS5 branes on a compact four dimensional
manifold ${\mathcal{N}}$. The simplest examples are the Calabi-Yau manifolds
$T^{4}$ (which breaks no supercharges) and $K_{3}$ (which breaks half of the
supercharges). The full background takes in this case the structure
${\mathcal{M}}_{3}\times{\mathcal{N}}\times S^{3}$. The relevant metric and
dilaton field then become [57, 58, 59]

$ds^{2}=H_{1}^{-1}(-dt^{2}+dx^{2}_{1})+R_{5}^{2}d\phi^{2},$ (65)
$e^{2\Phi}=R_{5}^{2}e^{-2\phi}H_{1}^{-1},\quad
H_{1}=1+R_{1}^{2}e^{-2\phi},\quad{R_{1}^{2}\over\alpha^{\prime}}={p\over\nu},$
(66)

where $\nu$ is proportional to the volume of the compact manifold
$\mathcal{N}$ in units of $\alpha^{\prime 2}$.

We note that for large negative $\phi$ the (six dimensional) string coupling
is $g_{6}=R_{5}/R_{1}$ (and the three dimensional string coupling is
$g_{3}^{2}=1/(p\sqrt{k})$) which can be made small by taking $p$ (and $k$)
large. Therefore we can apply perturbative string techniques (or free field
representations) to study physics in this region. We also note that in this
region the background ${\mathcal{M}}_{3}$ (65) is $AdS_{3}$ (with radius
$R_{5}$ after shifting the $\phi$) which is dual to a two dimensional
conformal field theory. For large positive $\phi$ the string coupling goes to
zero and the background ${\mathcal{M}}_{3}$ (65) is the three dimensional
linear dilaton background. The linear dialton background is holographically
dual to a two dimensional compactification of LST.

Thus, the string background ${\mathcal{M}}_{3}$ (65) interpolates between
$AdS_{3}$ in the IR or large negative $\phi$ and the three dimensional linear
dilaton background $\mathbb{R}^{1,1}\times\mathbb{R}_{\phi}$ in the UV or
large positive $\phi$. We next show that the background ${\mathcal{M}}_{3}$
corresponds to an exact marginal current bilinear deformation of the world-
sheet string theory on $AdS_{3}$. This defines in the boundary theory the
single-trace $T{\bar{T}}$ deformation [7].

The world-sheet string theory on the background $AdS_{3}$101010The string
background we obtained in the large negative $\phi$ limit of
${\mathcal{M}}_{3}$ (65). supported with the NSNS 3-form flux is described by
a sigma model on the $SL(2,R)$ group manifold [66, 67, 68, 69, 70, 71, 72, 73,
74]. The string theory on the three dimensional AdS spacetime is related to a
two dimensional conformal field theory residing on its one dimension lower
boundary via the AdS/CFT correspondence. The central charge of the conformal
field theory for large $k$ is given by the low energy gravity approximation
$c\sim pk$.

In the case in which the spatial direction of the boundary is compact, one can
impose either periodic or anti-periodic boundary condition for the fermions of
the boundary theory [75, 76, 77]. The periodic case gives the R sector. The R
sector ground state is described by massless BTZ which is a quotient of
$AdS_{3}$. The anti-periodic case gives to the NS sector. The NS sector ground
state is described by global $AdS_{3}$.

String theory on $AdS_{3}$ contains sectors of short and long strings [71,
78]. The short strings are trapped in the interior of the AdS spacetime. They
correspond to physical states that belong to the positive energy discrete
representation of (the universal cover of) the $SL(2,R)$ current algebra and
its spectral flow. Long strings can extend to the boundary and/or are near the
boundary. They correspond to physical states that belong to the principal
continuous representation of the $SL(2,R)$ current algebra and its spectral
flow.

There are evidences that support the idea that the boundary theory has the
structure of a symmetric product [79, 80, 81]. In particular the long string
sector is believed to be described in the boundary theory by a conformal field
theory with a symmetric product structure. The single-trace deformation gives
further support to this idea in the long string sector. We will come back to
this point shortly. However, in general, the boundary theory has not been
fully understood, and in general it may not actually be described by a
symmetric product theory.

The bosonic part of the world-sheet Lagrangian is

$S={k\over 2\pi}\int
d^{2}z(\partial\phi\bar{\partial}\phi+e^{2\phi}\partial\bar{\gamma}\bar{\partial}\gamma),$
(67)

where $(\phi,\gamma,\bar{\gamma})$ are the coordinates on $AdS_{3}$. The
boundary of $AdS_{3}$ is located at $\phi=\infty$. The action possesses an
affine $SL(2,R)_{L}\times SL(2,R)_{R}$ symmetry, with independent generators
for the left and right movers. The left mover world-sheet currents are given
by

$J^{-}=e^{2\phi}\partial\bar{\gamma},\quad
J^{+}=-2\gamma\partial\phi-\partial\gamma+\gamma^{2}e^{2\phi}\partial\bar{\gamma},\quad
J^{3}=\gamma e^{2\phi}\partial\bar{\gamma}-\partial\phi,$ (68)

and similar expressions for the right movers. The left-moving currents
$J^{-},J^{3}$ and $J^{+}$ give in spacetime the left-moving global conformal
charges $L_{-1},L_{0}$ and $L_{+1}$, respectively, and similarly for the
right-movers. This can be generalized to the supersymmetric case. In our
discussions we only consider the bosonic part.

We introduce the auxiliary fields $\beta,\bar{\beta}$. The Lagrangian becomes,
taking into account effects on the measure of the functional integral and
rescaling the scalar fields $\phi,\gamma,\bar{\gamma}$,

$S={1\over 4\pi}\int
d^{2}z(\partial\phi\bar{\partial}\phi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-\beta\bar{\beta}e^{-2\phi/\alpha_{+}}-{2\over\alpha_{+}}\phi\sqrt{g}R),$
(69)

where $\alpha_{+}=\sqrt{2(k-2)}$ and $R$ is the curvature of the world-sheet.
Quantum corrections produce the linear dilaton term [82, 83, 85]. This
notation is useful to study the physics (in particular long strings) near the
boundary or large positive $\phi$ where the string coupling is small and the
theory becomes free. The left mover currents in this notation are given by
(see for example [67, 84, 85])

$J^{-}=\beta,\quad
J^{+}=\beta\gamma^{2}-2\alpha_{+}\gamma\partial\phi-\alpha^{2}_{+}\partial\gamma,\quad
J^{3}=\beta\gamma-\alpha_{+}\partial\phi.$ (70)

At large $k$ a single long string is described by a conformal field theory
${\mathcal{M}}$ with central charge $c=6k$ [86]. Since the number of
NS1-branes $p$ is fixed, we have at most $p$ long strings. The CFT of the long
string sector is believed to be given by the symmetric product [79, 80, 87]

$S^{p}({\mathcal{M}}):={\mathcal{M}}^{p}/S_{p},$ (71)

where ${\mathcal{M}}$ is the CFT of a single long string and $S_{p}$ denotes
the symmetric group, that is, the permutation group of $p$ elements. The
central charge of $S^{p}({\mathcal{M}})$ is given by $c=6pk$. A twisted sector
of the theory $S^{p}({\mathcal{M}})$ can describe fewer than $p$ long strings.
The reason is that the twisted sectors are organized by the conjugacy classes
of the symmetric or orbifold group $S_{p}$ [79, 88, 89].

The background ${\mathcal{M}}_{3}$ (65) corresponds to adding to the world-
sheet Lagrangian 69 a term proportional to the bilinear current
$J^{-}{\bar{J}}^{-}$ [7]. The deformed action is

$S_{\alpha_{1}}:={1\over 4\pi}\int
d^{2}z\left(\partial\phi\bar{\partial}\phi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-\beta\bar{\beta}e^{-2\phi/\alpha_{+}}-{2\over\alpha_{+}}\phi\sqrt{g}R+\alpha_{1}\beta\bar{\beta}\right),$
(72)

where $\alpha_{1}$ is the deformation coupling. The deformation breaks the
$SL(2,R)_{L}\times SL(2,R)_{R}$ current algebra. At $\alpha_{1}=0$ the zero
mode corresponding to $J^{-}$(${\bar{J}}^{-}$) gives rise to the spacetime
Virasoro generator $L_{-1}$(${\bar{L}}_{-1}$), and thus, it has the spacetime
scaling dimension $(1,0)$($(0,1)$). Therefore, the coupling $\alpha_{1}$ has
spacetime scaling dimension $(-1,-1)$ similar to the $T{\bar{T}}$ deformation
coupling.

String theory in $AdS_{3}\times{\mathcal{M}}$ contains an integrated vertex
operator $D(x,{\bar{x}})$ that depends on the coordinates $(x,\bar{x})$ of the
boundary spacetime, and thus, the operator $D(x,{\bar{x}})$ lives in the
boundary theory [90]. This operator is related to the deformation through [7]

$\int d^{2}zJ^{-}{\bar{J}}^{-}=t\int d^{2}xD(x,{\bar{x}}),$ (73)

here $t$ is some constant. Therefore, deformation of the world-sheet action by
adding the operator $\int d^{2}zJ^{-}{\bar{J}}^{-}$ is described in the dual
boundary theory by adding to the corresponding action the operator $\int
d^{2}xD(x,{\bar{x}})$. The operator $D(x,{\bar{x}})$ has spacetime scaling
dimension $(2,2)$. Thus, it is irrelevant.

In the symmetric product theory $S^{p}({\mathcal{M}})$ that describes the long
string sector, the operator $D(x,{\bar{x}})$ is identified with [87]111111To
be precise
$tD(x,{\bar{x}})=c\lim_{p\to\infty}\sum_{i=1}^{p}\int_{0}^{t}d\alpha({\rm
T\bar{T}})^{(\alpha)}_{i}(x,{\bar{x}}),$ (74) where $\alpha$ labels the
trajectory and $c$ is a constant. Thus, from the single-trace deformation
perspective, the $T{\bar{T}}$ deformation can be equivalently viewed as an RG
upflow.

$D(x,{\bar{x}})\equiv\sum_{i=1}^{p}(T\bar{T})_{i}(x,{\bar{x}}),$ (75)

where $(T{\bar{T}})_{i}$ is the $T{\bar{T}}$ operator in the $i$th copy of the
CFT ${\mathcal{M}}$. Therefore, it is given by taking the sum hence the name
single-trace $T{\bar{T}}$ operator. $D(x,{\bar{x}})$ transforms under $T(x)$
and ${\bar{T}}({\bar{x}})$ of $S^{p}({\mathcal{M}})$ as a quasi-primary
operator of dimension $(2,2)$. Its OPE’s with $T(x)$ and
${\bar{T}}({\bar{x}})$ of $S^{p}({\mathcal{M}})$ is similar to that of
$T{\bar{T}}(x,{\bar{x}})$ of $S^{p}({\mathcal{M}})$ however $D(x,{\bar{x}})$
is a single-trace operator and thus, it is distinct. Note that the
$T{\bar{T}}$ operator of the theory $S^{p}({\mathcal{M}})$ is

$T{\bar{T}}(x,{\bar{x}})=\left(\sum_{i=1}^{p}T_{i}\right)\left(\sum_{j=1}^{p}{\bar{T}}_{j}\right),$
(76)

where $T_{i}$ and ${\bar{T}}_{i}$ are the components of the energy momentum
tensor in the $i$th copy of the CFT ${\cal M}$. Therefore, it is obtained by
taking product of two sums or traces hence the name double-trace $T{\bar{T}}$
operator.

Therefore, it follows from the identification 75 that the spectrum for the
untwisted sector of the symmetric product theory $S^{p}({\mathcal{M}})$ 71
should be equal to that of the $T{\bar{T}}$ deformed spectrum of the copy
${\mathcal{M}}$. We next show this.

####  3 Deformed spectrum

We next show that string theory on the deformed background ${\mathcal{M}}_{3}$
65 is equivalent to string theory on $AdS_{3}$ with twisted boundary
conditions for the coordinates or fields. We use this interpretation to
compute the spectrum in the untwisted sector of the deformed symmetric product
theory.

We compactify the spatial coordinate of the boundary. Therefore, the topology
of the string background is $\mathbb{R}^{1,1}\times S^{1}$. In the boundary
theory the fermions can be either periodic or anti-periodic along the compact
direction. This leads to two distinct sectors. The R sector is described by
BTZ solutions and the NS sector is described by global AdS [75, 76, 77].

The bosonic part of the deformed world-sheet action is described by the action

$S=\frac{1}{2\pi}\int
d^{2}z\left(\partial\phi\bar{\partial}\phi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\left(\alpha_{1}-e^{-2\phi/\sqrt{k}}\right)\beta\bar{\beta}\right),$
(77)

where

$z:=\tau-\sigma,\quad{\bar{z}}:=\tau+\sigma,$ (78)

$\sigma$ is the spatial coordinate along the string and $\tau$ describes its
propagation in time. Note that we can set the deformation coupling
$\alpha_{1}=1$ by shifting the radial coordinate $\phi$. The boundary is at
$\phi=+\infty$. The null coordinates $\gamma$ and $\bar{\gamma}$ are given by

$\gamma=x-t,\quad\bar{\gamma}=x+t,\quad x\sim x+2\pi.$ (79)

The deformation 77 can also be understood as a T-duality-shift-T-duality (TsT)
(or more generally ${\cal O}(d,d)$) transformation [91, 92, 93, 94, 95]. TsT
transformation is a systematic procedure to generate a new supergravity
solution. Such TsT transformations can be equivalently viewed as twisted
boundary conditions [91, 95, 96, 97, 98, 99]. In this view the background
spacetime is $AdS_{3}$. In what follows we derive the twisted boundary
conditions corresponding to 77. We follow a similar approach used in [99]. We
then use the twisted boundary conditions to determine the spectrum in the
deformed boundary theory.

We vary the action to get the equations of motion for the bosonic string. We
find

$\partial\bar{\partial}\phi-e^{-2\phi/\sqrt{k}}\beta\bar{\beta}=0,$ (80)
$\bar{\partial}\gamma+(\alpha_{1}-e^{-2\phi/\sqrt{k}})\bar{\beta}=0,$ (81)
$\partial\bar{\gamma}+(\alpha_{1}-e^{-2\phi/\sqrt{k}})\beta=0,$ (82)
$\bar{\partial}\beta=0,$ (83) $\partial\bar{\beta}=0.$ (84)

The holomorphic (left-moving) and anti-holomorphic (right-moving) energy
momentum tensor components are given at classical level by

$-T=(\partial\phi)^{2}+\beta\partial\gamma,\qquad-{\bar{T}}=(\bar{\partial}\phi)^{2}+\bar{\beta}\bar{\partial}\bar{\gamma}.$
(85)

At quantum level normal ordering generates an improvement term for $\phi$.

We now make the following redefinition of the fields

$\bar{\partial}\hat{\gamma}=\bar{\partial}\gamma+\alpha_{1}\bar{\beta},\quad\partial\hat{\bar{\gamma}}=\partial\bar{\gamma}+\alpha_{1}\beta,\quad\partial\hat{\gamma}=\partial\gamma,\quad{\bar{\partial}}\hat{\bar{\gamma}}={\bar{\partial}}{\bar{\gamma}}.$
(86)

The solutions to these equations are given by

$\hat{\gamma}=\gamma+\alpha_{1}\mu_{+},\quad\hat{\bar{\gamma}}=\bar{\gamma}+\alpha_{1}\mu_{-},$
(87)

where

$\beta=\partial\mu_{-},\quad\bar{\beta}=\bar{\partial}\mu_{+},$ (88)

are the bosonizations of the currents $J^{-}:=i\beta$ and
${\bar{J}}^{-}:=-i\bar{\beta}$.

In terms of the new hatted variables we note that the local world-sheet
dynamics becomes

$\partial\bar{\partial}\phi-e^{-2\phi/\sqrt{k}}\beta\bar{\beta}=0,$ (89)
$\bar{\partial}\hat{\gamma}-e^{-2\phi/\sqrt{k}}\bar{\beta}=0,$ (90)
$\partial\hat{\bar{\gamma}}-e^{-2\phi/\sqrt{k}}\beta=0,$ (91)
$\bar{\partial}\beta=0,\qquad\partial\bar{\beta}=0.$ (92)

We also note that the components of the energy momentum tensor take the
following forms

$-T=(\partial\phi)^{2}+\beta\partial\hat{\gamma},\qquad-{\bar{T}}=(\bar{\partial}\phi)^{2}+\bar{\beta}\bar{\partial}\hat{\bar{\gamma}}.$
(93)

The improvement term is not affected since it only depends on $\phi$.
Therefore, the local dynamics of the string in the deformed background is
given by the $SL(2,R)$ WZW sigma model at level $k$. This is a particularly
important observation since we can now make use of the $SL(2,R)_{L}\times
SL(2,R)_{R}$ affine current algebra to construct vertex operators, obtain
spectrum and compute correlation functions.

We define the zero mode momenta $p_{L}$ and $p_{R}$ as

$p_{L}:=-\frac{1}{2\pi}\oint d\sigma\partial\mu_{-},\quad
p_{R}:=-\frac{1}{2\pi}\oint d\sigma\bar{\partial}\mu_{+},$ (94)

where the contour integral is at a fixed world-sheet time. The momenta $p_{R}$
and $p_{L}$ are related to the spacetime (boundary) energy $\bar{E}$ and
momentum $\bar{P}$ via the relations (see, for example, [100])

${\bar{P}}=p_{L}+p_{R},\quad{\bar{E}}=p_{R}-p_{L}.$ (95)

In the case in which $x$ is not compact the momentum $\bar{P}$ is not
quantized.

We note that, at fixed world-sheet time, using the identity

$\oint dz\partial\xi+\oint d\bar{z}\bar{\partial}\xi=\oint
d\sigma\partial_{\sigma}\xi=\xi(\sigma+2\pi)-\xi(\sigma):=\Delta\xi,$ (96)

we find the following twisted boundary conditions

$\Delta\hat{\gamma}=\Delta\gamma-2\pi\alpha_{1}p_{R}:=2\pi{\hat{w}}_{R}(p_{R}),\quad\Delta\gamma=2\pi
w_{R},$ (97)
$\Delta\hat{\bar{\gamma}}=\Delta\bar{\gamma}+2\pi\alpha_{1}p_{L}:=2\pi{\hat{w}}_{L}(p_{L}),\quad\Delta\bar{\gamma}=2\pi
w_{L},$ (98)

where $w_{L}$ ($=w_{R}:=w$) is the winding number of the string which counts
the number of times the string winds around the compact spatial direction. The
winding number $w$ can be positive or negative depending on the string
orientation.

Therefore, string theory on the deformed background ${\mathcal{M}}_{3}$ (65)
is the same as string theory on the undeformed background $AdS_{3}$ but with
twisted boundary conditions 97 and 98. As we will show shortly, this is
equivalent to a momentum dependent spectral flow. We also note that in general
${\hat{w}}_{L}\neq{\hat{w}}_{R}$. Thus, the fields ${\hat{\gamma}}$ and
$\hat{\bar{\gamma}}$ in general describe non-local dynamics. In chapter three
and four we examine the non-local behavior of the theory in more detail using
correlation functions and entanglement entropy.

We are in particular interested in the long string sector spectrum and
therefore we use the free field representation. In the free field
approximation we have

$\phi(z,\bar{z})\phi(w,\bar{w})\sim-\frac{1}{2}\ln|z-w|^{2},$ (99)
$\partial\mu_{-}(z)\hat{\gamma}(w)\sim-\frac{1}{z-w},\qquad\partial\hat{\gamma}(z)\mu_{-}(w)\sim-\frac{1}{z-w},$
(100)

and similar OPEs for the fields $\mu_{+},\hat{\bar{\gamma}}$.

We now construct world-sheet vertex operators $\hat{\cal
V}^{(w)}_{p_{L},p_{R}}$ that create states in a given winding sector $w$ with
definite momenta $p_{L}$ and $p_{R}$. The monodromy conditions 97 and 98 imply
the OPEs

$\hat{\gamma}(z_{1},\bar{z}_{1})\hat{\cal
V}^{(w)}_{p_{L},p_{R}}(z_{2},\bar{z}_{2})\sim
i\left(-w+\alpha_{1}p_{R}\right)\ln(z_{1}-z_{2})\hat{\cal
V}^{(w)}_{p_{L},p_{R}}(z_{2},\bar{z}_{2}),$ (101)
$\hat{\bar{\gamma}}(z_{1},\bar{z}_{1})\hat{\cal
V}^{(w)}_{p_{L},p_{R}}(z_{2},\bar{z}_{2})\sim-i\left(w+\alpha_{1}p_{L}\right)\ln(\bar{z}_{1}-\bar{z}_{2})\hat{\cal
V}^{(w)}_{p_{L},p_{R}}(z_{2},\bar{z}_{2}),$ (102)

This corresponds the dressing

$\hat{\cal
V}^{(w)}_{p_{L},p_{R}}=\hat{V}^{(w)}_{p_{L},p_{R}}e^{-i\alpha_{1}p_{R}\mu_{-}}e^{i\alpha_{1}p_{L}\mu_{+}},$
(103)

where $\hat{V}^{(w)}_{p_{L},p_{R}}$ is the world-sheet vertex operator on the
undeformed background $AdS_{3}$ [101, 100]. The vertex operator
$\hat{V}^{(0)}_{p_{L},p_{R}}:=\hat{V}_{p_{L},p_{R}}$ is obtained by taking the
Fourier transform of the primary operator $\hat{V}_{\hat{h},\hat{h}}$, see
appendix 5. It has, in the large $\phi$ and with our definitions in 94, the
form

$\hat{V}^{(w)}_{p_{L},p_{R}}=e^{ip_{L}\hat{\gamma}+iw\mu_{-}}e^{-ip_{R}\hat{\bar{\gamma}}+iw\mu_{+}}f_{\hat{h}}(\phi),$
(104)

where $f_{\hat{h}}(\phi)\sim e^{2{(\hat{h}}-1)\phi/\sqrt{k}}$ and the constant
$\hat{h}$121212We take the left $\hat{h}$ and right $\hat{\bar{h}}$ to be
equal. labels the (continuous) representations of the current algebra. Note
that the deformed vertex operators $\hat{\cal V}^{(w)}_{p_{L},p_{R}}$ are
mutually local since $\bar{P}$ is quantized and thus have no branch cut.

The conditions 101 and 102 (or equivalently the dressing 103 with 104) are
equivalent to the spectral flow

${\cal L}_{m}=L_{m}-\left(\alpha_{1}p_{R}-w\right)J^{-}_{m},$ (105) $\bar{\cal
L}_{m}={\bar{L}}_{m}-\left(\alpha_{1}p_{L}+w\right)\bar{J}^{-}_{m},$ (106)

where $L_{m}$ (${\bar{L}}_{m}$) are the modes of the left (right) moving
component of the undeformed world-sheet energy momentum tensor and $J^{-}_{m}$
(${\bar{J}}^{-}_{m}$) are the modes of the left (right) moving (undeformed)
world-sheet current $J^{-}$(${\bar{J}}^{-}$).

The Virasoro constraint ${\cal L}_{0}=1$ ($\bar{\cal L}_{0}=1$) and the
relations in 95 lead to the equation

${\alpha_{1}\over 4}\left({\bar{P}}^{2}-{\bar{E}}^{2}\right)+{w\over
2}\left({\bar{E}}-{\bar{E}}_{0}\right)=0,\quad{\bar{E}}_{0}:={\bar{E}}(\alpha_{1}=0),\quad\partial_{\alpha_{1}}{\bar{P}}=0.$
(107)

This gives the spacetime spectrum

${\bar{E}}(\lambda)={\bar{P}}-{1\over 2A}\left(B+\sqrt{B^{2}-4AC}\right),$
(108)

where

$A=-{1\over 4}\lambda,\quad B=-w-{1\over 2}\lambda{\bar{P}},\quad
C=w\left({\bar{E}}_{0}-{\bar{P}}\right),$ (109)

and

$\alpha_{1}=-{1\over 2}\lambda,\quad\lambda={8\pi t\over
R^{2}},\quad{\bar{E}}={R{\mathcal{E}}\over 2\pi},\quad{\bar{P}}={RP\over
2\pi}.$ (110)

We note that when $w=1$ the single-trace $T{\bar{T}}$ deformed spectrum with
the relations 110 matches the spectrum of a $T{\bar{T}}$-deformed CFT obtained
earlier in 48. Recall that the CFT $\mathcal{M}$ in 71 is the dual theory that
describes a single long string. Therefore, the spectrum of (single particle)
states in the untwisted or $w=1$ sector of the dual single-trace deformed
symmetric product is given by the spectrum of the $T{\bar{T}}$ deformed theory
${\mathcal{M}}$ in 71. Thus, the single-trace deformation supports the
identification 75. This analysis can be extended for theories with currents
[114]. See also [25, 102].

In the case in which the spatial direction is not compact the dressed un-
flowed vertex operator $\hat{\cal V}_{p_{L},p_{R}}$ (103) has the world-sheet
scaling dimension given by

$-\frac{\hat{h}(\hat{h}-1)}{k}-\alpha_{1}p^{2},$ (111)

where $p^{2}:=p_{L}p_{R}$.

##  Chapter 2 Correlation functions

This chapter is based on the paper [24].

In this chapter the question that we address is concerning how the non-
locality and UV behavior of the deformed boundary theory is manifested on the
analytic structure of correlation functions of certain class of (gauge
invariant) operators [104]. We in particular compute two point function in the
case where the spatial direction is not compact. We use the momentum dependent
spectral flow.

The class of operators we consider has in the IR undeformed CFT the form

${\mathcal{O}}_{h,h}(x)=\int d^{2}zV_{h,h}(x;z){\mathcal{U}}(z),$ (1)

where $V_{h,h}$ (see 2) is a vertex operator in $AdS_{3}$ and
${\mathcal{U}}(z)$ is a vertex operator in the internal CFT on the compact
manifold ${\mathcal{N}}$. The coordinate $x$ labels the boundary and the
coordinate $z$ labels the world-sheet or the sphere. The operator $V_{h,h}$
has the spacetime scaling dimension $(h,h)$. The world-sheet scaling dimension
$(\Delta_{\mathcal{U}},\Delta_{\mathcal{U}})$ of the operator ${\mathcal{U}}$
is related to the world-sheet scaling dimension $(\Delta_{h},\Delta_{h})$ of
the operator $V_{h,h}$ through the mass-shell condition

$\Delta_{h}+\Delta_{\mathcal{U}}=1,\quad\Delta_{h}=-{h(h-1)\over k}.$ (2)

In $AdS_{3}$ (tree-level or sphere) two point function of vertex operators
$V_{h,h}$ in position space in a particular normalization is given by [70,
105],

$\langle
V_{h,h}(x,z)V_{h^{\prime},h^{\prime}}(y,w)\rangle=\delta(h-h^{\prime}){B(h)\over|x-y|^{4h}|z-w|^{4\Delta_{h}}},$
(3)

where the function $B$ is

$B(a)={k\over\pi}\nu^{1-2a}\gamma\left(1-\frac{2a-1}{k}\right),\quad\gamma(a)=\frac{\Gamma(a)}{\Gamma(1-a)},$
(4)

$\nu$ is some known function of the level $k$. In general, $B(h)$ can be made
smooth by choosing a particular normalization for the operator $V_{h,h}$ (of
course, unless it has a natural normalization), and we shall assume that this
is indeed the case. We normalize ${\mathcal{U}}$ such that
$\langle{\mathcal{U}}(z){\mathcal{U}}(w)\rangle=1/|z-w|^{\Delta_{\mathcal{U}}}$.
Therefore, the physics is captured by the simple power functions with the
exponents $4h$ and $4\Delta_{h}$. The two point function of the operators
${\mathcal{O}}_{h,h}:={\mathcal{O}}$ 1 is then given by

$\langle{\mathcal{O}}(x){\mathcal{O}}(y)\rangle={\delta(0)\over
V}{B(h)\over|x-y|^{4h}}={D(h)\over|x-y|^{4h}},$ (5)

where $V$ is the volume of the conformal group on the sphere and
$D(h)=(2h-1)B(h)$.

In the deformed theory the operator ${\mathcal{O}}$ deforms as

${\hat{\mathcal{O}}}_{\hat{h},\hat{h}}(x)=\int d^{2}z{\hat{\cal
V}}_{\hat{h},\hat{h}}(x;z){\mathcal{U}}(z),$ (6)

where ${\hat{\cal V}}_{\hat{h},\hat{h}}$ is the deformed vertex operator. The
deformation does not break the world-sheet conformal symmetry and in the
momentum dependent spectral flow interpretation we saw that a vertex operator
in the deformed theory has the world-sheet conformal dimension 111. The
conformal dimension does not depend on what interpretation one chooses. Thus,
in the deformed theory the mass-shell condition takes the form

$\hat{\Delta}_{{\hat{h}}}+\Delta_{\mathcal{U}}=1,\quad\hat{\Delta}_{{\hat{h}}}=-\frac{\hat{h}(\hat{h}-1)}{k}+{1\over
2}\lambda p^{2}.$ (7)

This equation can be solved for $\hat{h}:=\hat{h}_{p^{2}}(\lambda)$ as a
function of the deformation coupling $\lambda$. This gives using 2 the flow
equation

$2\hat{h}_{p^{2}}(\lambda)-1=\sqrt{(2h-1)^{2}+2k\lambda
p^{2}},\quad\hat{h}_{p^{2}}(\lambda=0):=h.$ (8)

As we also saw earlier, the local dynamics of strings in the deformed
background is described by the $SL(2,R)$ WZW sigma model at level $k$.
Therefore, (the un-flowed $w=0$) two point function of vertex operators in the
deformed theory is also given by 3 with $\Delta_{h}$ replaced by
$\hat{\Delta}_{{\hat{h}}}$ and $h$ replaced by ${\hat{h}}_{p^{2}}$. The
exponent $\hat{h}_{p^{2}}$ is related with the momentum $p^{2}$ via the
relation 8111Note that this implies a different choice of normalization
introduces a different momentum or equivalently coupling dependence.. Thus, in
the next section, we Fourier transform the two point function 5 with $h$
replaced by ${\hat{h}}_{p^{2}}$ and transform it back to position space so
that it only depends on position coordinates. We consider the two point
function separately in momentum and position spaces. See [24] for detailed
discussions where coset construction is used to study correlation functions.

###  1 Two point function in momentum space

We will do the calculation in Euclidean space. In momentum space we find using
5 for the two point function

$\langle{\hat{\mathcal{O}}}(p){\hat{\mathcal{O}}}(-p)\rangle=\pi
D({\hat{h}}_{p^{2}})\gamma(1-2{\hat{h}}_{p^{2}})\left(p^{2}\over
4\right)^{2{\hat{h}}_{p^{2}}-1},$ (9)

where $D(x)=(2x-1)B(x)$ and it is a smooth function. In particular for
$\lambda=0$ this gives the undeformed two point function in momentum space

$\langle{\mathcal{O}}(p){\mathcal{O}}(-p)\rangle=\pi
D(h)\gamma(1-2h)\left(p^{2}\over 4\right)^{2h-1}.$ (10)

The momentum space two point function 10 as a function of $h$ has poles for
$2h-1=n\in\mathbb{Z}_{+}$. They appear in the multiplicative factor that does
not depend on the momentum, and they can be removed or cancelled by
appropriate renormalization of the vertex operators. In this case where
$2h-1=n\in{\mathbb{Z}}_{+}$ the high momentum limit of the regularized two
point function behaves $p^{2n}\log\left(p^{2}/\mu^{2}\right)$ where $\mu$ is
some mass scale (see for example [106, 107, 108, 109]). In all other cases the
two point function behaves as power law $\left(p^{2}\right)^{2h-1}$. It
follows, therefore, that the non-trivial analytic properties of the two point
function 9 are fully captured by the function ${\hat{h}}_{p^{2}}$ 8.

In Euclidean momentum space the two point function 9 is real and has no branch
cut. In the (strictly) high Euclidean momentum limit it exhibits the hyper-
power law growth $p^{\alpha p}$ for some known positive constant $\alpha$.
Such growth is very different from the power law behavior we saw earlier.
However, in Minkowski momentum space the two point function is real and has no
branch cut only for spacelike momenta. For timelike momenta the flow equation
8 (or equivalently the momentum two point function 9) has a branch cut with
branch points given by

$p_{0}^{2}:=-{(2h-1)^{2}\over 2k\lambda}.$ (11)

.

In general branch cut in correlation functions or OPEs signals non-locality
and/or ambiguity. We next consider the two point function in position space.

###  2 Two point function in position space

In position or real space the two point function is given by

$\langle\hat{\mathcal{O}}(y)\hat{\mathcal{O}}(0)\rangle=2\pi\int_{0}^{\infty}d\mu^{2}\rho(\mu^{2};\lambda)K_{0}(\mu|y|),$
(12)

where $K_{0}$ is the Fourier transform of the propagator

$\int d^{2}p{e^{ip\cdot y}\over p^{2}+\mu^{2}}=2\pi K_{0}(\mu|y|),$ (13)

and $\rho(\mu^{2};\lambda)$ is the spectral density. Note that $K_{0}$ is
equal to the modified Bessel function. In the regime $\mu|y|\gg 1$ the Bessel
function decays exponentially $K_{0}(\mu|y|)\sim e^{-\mu|y|}/\sqrt{\mu|y|}$,
and in the regime $\mu|y|\ll 1$ it diverges logarithmically
$K_{0}\sim-\rm{log}(\mu|y|)$. The spectral density is defined in Kallen-
Lehmann representation by the equation

$\langle{\hat{\mathcal{O}}}(p){\hat{\mathcal{O}}}(-p)\rangle=\int_{0}^{\infty}d\mu^{2}{\hat{\rho}(\mu^{2};\lambda)\over\mu^{2}+p^{2}}.$
(14)

That is, the two point function is obtained by summing the free scalar
momentum space propagator or two point function over all masses $\mu$ with a
weight or probability density given by the spectral density
$\hat{\rho}(\mu^{2};\lambda)$222This identification of $\mu$ with mass should
not be taken literally; it is a mere observation. For example, in a CFT there
are no mass scales.. Using the momentum space two point function 9 in the
definition 14 we find that the spectral density $\hat{\rho}$ is given by

$\hat{\rho}(\mu^{2};\lambda)=\pi{D(\hat{h}_{-\mu^{2}})\over\Gamma^{2}(2{\hat{h}}_{-\mu^{2}})}\left({\mu^{2}\over
4}\right)^{2{\hat{h}}_{-\mu^{2}}-1}.$ (15)

For a CFT or at $\lambda=0$ this gives

$\rho(\mu^{2})=\pi{D(h)\over\Gamma^{2}(2h)}\left({\mu^{2}\over
4}\right)^{2h-1}.$ (16)

We note that spectral density is positive and continuous. It also does not
contain a delta function at $\mu=0$333See [110] for a discussion in relation
to Wightman and generalized free fields..

The spectral density 15 is complex for $\mu>\mu_{0}$ where
$\mu_{0}^{2}=-p_{0}^{2}$ is given by 11. It has a real and an imaginary part.
The interpretation of the imaginary part of the density is not completely
clear, however, we expect to be related to the non-local behavior of the
theory. It follows, therefore, that the two point function 12 in position
space is complex. In perturbation theory, keeping only the leading logarithmic
term at each order in the deformation coupling, the two point function 12 in
position space is given in compact form as

$\langle\hat{\mathcal{O}}(y)\hat{\mathcal{O}}(0)\rangle=e^{\frac{4\lambda
k}{2h-1}\ln\left(m^{2}|y|^{2}\right)\partial_{y}\partial_{\bar{y}}}\langle{\mathcal{O}}(y){\mathcal{O}}(0)\rangle,$
(17)

where $m$ is some arbitrary normalization scale and the two point function in
the undeformed theory is

$\langle{\mathcal{O}}(y){\mathcal{O}}(0)\rangle=\left(2\pi\right)^{2}|y|^{-4h},$
(18)

in some normalization. We in particular note that in perturbation theory the
two point function is real. In the regime where $\mu_{0}|y|\gg 1$ or
equivalently small $\lambda/|y|^{2}$ we find that the imaginary part goes like
$e^{-\sqrt{|y|^{2}/\lambda}}$. Thus, the imaginary part of the two point
function is non-perturbative in the deformation coupling $\lambda/|y|^{2}$. A
simple way to see this is by considering the large order behavior of the two
point function (see [111]). We write the two point function 12 as

$\langle\hat{\mathcal{O}}(y)\hat{\mathcal{O}}(0)\rangle={1\over|y|^{4h}}\sum_{n=0}^{\infty}c_{n}\left({\lambda\over|y|^{2}}\right)^{n}.$
(19)

The large order behavior of the coefficient $c_{n}$ is related to the
imaginary part. One obtains

$c_{n}\sim n^{\alpha}\cdot e^{\beta n}\cdot(2n)!,$ (20)

where $\alpha$ and $\beta$ are some constants. Such double factorial growth is
common in string theory and it corresponds to weak coupling non-perturbative
effects [112]. The imaginary part is directly related to the imaginary part of
the spectral density, and thus, it is related to the non-local behavior of the
theory.

##  Chapter 3 Entanglement entropy and Entropic $c$-function

In this chapter we study the holographic entanglement entropy of an interval
in a QFT obtained by deforming a holographic two–dimensional CFT via a general
linear combination of single-trace irrelevant operators $T{\bar{T}},\
J{\bar{T}}$ and $T{\bar{J}}$, and compute the Casin–Huerta entropic
$c$–function. In the UV, for a particular combination of the deformation
parameters, we find that the leading order dependence of the entanglement
entropy on the length of the interval is given by a square root but not
logarithmic term. Such power law dependence of the entanglement entropy on the
interval length is quite peculiar and interesting. We also find that the
entropic $c$–function is UV regulator independent, and along the RG upflow
towards the UV, it is non–decreasing. We show that in the UV the entropic
$c$–function exhibits a power law divergence as the interval length approaches
a minimum finite value determined in terms of the deformation parameters. This
value sets the non–locality scale of the theory. This chapter is based on the
paper [27].

###  1 Single trace $T{\bar{T}}$, $J{\bar{T}}$ and $T{\bar{J}}$ deformations

String theory on the background $AdS_{3}\times{\mathcal{X}}$ with
Neveu–Schwartz two–form $B$ field contains a sector of long strings that
extend to the boundary [71, 86, 80, 113]. The effective theory of $N$
coincident long strings on $AdS_{3}\times{\mathcal{X}}$ in the weak coupling
regime is believed to be well described by the symmetric product theory [80,
87]

${\mathcal{M}}^{N}/S_{N},$ (1)

where the conformal field theory ${\mathcal{M}}$ is the dual theory of a
single long string [86]. Note that the symmetric product theory 1 is not equal
to the full dual boundary (spacetime) theory since ${\mathcal{M}}$ only
describes the long string sector.

In [87] the authors considered deforming the Lagrangian of the world-sheet
string theory on the background $AdS_{3}\times{\mathcal{X}}$ by a general
linear combination of truly marginal current–current operators

$\lambda
J^{-}{\bar{J}}^{-}+\epsilon_{+}K{\bar{J}}^{-}+\epsilon_{-}{\bar{K}}J^{-}.$ (2)

They showed that, in the long string sector, this deformation is equivalent to
the deformation of the theory ${\cal M}$ (with spectral flow or winding number
$w=1$) by a general linear combination of the recently much studied irrelevant
operators [4, 5, 6, 20]

$-tT{\bar{T}}-\mu_{+}J{\bar{T}}-\mu_{-}{\bar{J}}T,$ (3)

where $J$ and ${\bar{J}}$ are the left and right moving $U(1)$ currents,
respectively, and $T$ and ${\bar{T}}$ are the left and right moving stress
tensor components, respectively, of the conformal field theory ${\cal M}$.

String theory on the background $AdS_{3}\times{\cal X}$ contains a class of
(integrated) vertex operators ${\bar{A}},\ A$ and $D$ constructed in [90],
which depends on the coordinates $(x,{\bar{x}})$ of the dual conformal field
theory. It is shown in [7, 87, 114] that the deformation 2 in the action
amounts to adding a linear combination of the operators $\int
d^{2}x{\bar{A}},\ \int d^{2}xA$ and $\int d^{2}xD$. In the boundary
(spacetime) theory the operators ${\bar{A}},\ A$ and $D$ have the same scaling
dimensions as $T{\bar{J}},\ J{\bar{T}}$ and $T{\bar{T}}$, respectively,
however, they are single-trace operators, in the sense that, in the long
string sector, they are interpreted as sum over the $N$ copies of the
operators $T{\bar{J}},\ J{\bar{T}}$ and $T{\bar{T}}$, respectively, of the
field theory ${\cal M}$.

The operators $J{\bar{T}}$ and $T{\bar{J}}$ have left and right scaling
dimensions $(1,2)$ and $(2,1)$, respectively, and therefore, the deformation 3
results in a theory that breaks Lorentz invariance [20, 29, 114].

The spacetime couplings $t,\ \mu_{+}$ and $\mu_{-}$ are related to the world-
sheet dimensionless couplings $\lambda,\ \epsilon_{+}$ and $\epsilon_{-}$ via
the relations [87],

$t=\pi\alpha^{\prime}\lambda,\quad\mu_{\pm}=2\sqrt{2\alpha^{\prime}}\epsilon_{\pm},\quad\alpha^{\prime}=l_{s}^{2},$
(4)

where $\alpha^{\prime}$ is the Regge slope, and $l_{s}$ is the intrinsic
string length.

The deformation 3 is irrelevant and therefore the couplings grow as we ascend
the renormalization group. In general, under an irrelevant deformation of a
quantum field theory, in the ultraviolet, the associated coupling is large and
the description of the theory in terms of the original infrared degrees of
freedom often breaks down. The theory also suffers from ambiguities and/or
arbitrariness. It is also often the case that quantum corrections generate an
infinite number of irrelevant operators. Under the deformation 3 it is shown,
however, that the theory is solvable, in the sense that the spectrum on an
infinite cylinder [4, 5, 6, 20, 25, 102] and the partition function on a torus
[9, 18, 28, 115, 116, 117] can be computed exactly. The theory also does not
acquire new couplings. On a torus modular invariance can be employed to
constrain the theory [11, 115, 116, 117]. In the case in which the coupling
$t$ is positive and $\mu_{\pm}=0$ the theory involves no ambiguities [116,
117]. We obtain in this chapter the exact entanglement entropy and entropic
c–function in the deformation 3. This increases the number of non–trivial
quantities that one can exactly solve and study in this class of theories.

It is shown in [87] that in the space of couplings in which the combination

$F={t\over\pi}-{(\mu_{+}+\mu_{-})^{2}\over 8},$ (5)

is positive, $F>0$, the energies of states are real, and the density of states
asymptotically exhibits Hagedorn growth. In the limit $F\to 0^{+}$ however the
theory appears to be distinct. The density of states (in a fixed charge
sector) asymptotically exhibits an intermediate growth between Cardy and
Hagedorn growths. We also show later in the chapter that in this limit the Von
Neumann entanglement entropy at short distances exhibits a square root area
law correction but not logarithmic correction. Such square root correction
term is quite peculiar and interesting. In the case in which $F<0$ the
energies become complex above a scale fixed by the couplings and the
corresponding bulk geometry is singular and/or it has either closed timelike
curves or no timlike direction. The signature of the bulk metric switches
signs beyond a finite radial distance where the singularity occurs. We will
not consider this case in this chapter as it is not clear how to consistently
apply the Ryu and Takayanagi holographic prescription and its covariant
generalization. We comment on this later in the discussion section.

We now briefly mention the holographic proposals for the closely related
double-trace deformations. In this class of deformations $T$ and $J$ are the
left moving energy momentum tensor and current, respectively, of the full
boundary theory. The right moving energy momentum tensor and current are
denoted by $\bar{T}$ and $\bar{J}$ respectively. In these holographic
proposals the coupling $t$ is negative and therefore $F<0$. For negative $t$
with $\mu_{\pm}=0$ the dual bulk spacetime of a $T{\bar{T}}$ deformed two
dimensional holographic conformal field theory is proposed to be $AdS_{3}$
with a Dirichlet boundary at finite radial distance fixed by the coupling $t$
[118]. For either sign of $\mu_{+}$ (and $\mu_{-}=0$ or vice versa) and $t=0$
it is shown in [119] that the dual bulk spacetime of a $J{\bar{T}}$ (or
$T{\bar{J}}$) deformed two dimensional holographic conformal field theory is
$AdS_{3}$ with boundary conditions that mix the metric and a gauge field dual
to the current $J$ (or $\bar{J}$). In either of these cases in the field
theory side we have states with complex energies.

In this chapter we study the (Von Neumann) entanglement entropy for a spatial
interval of length $L$ in the deformed (full) spacetime theory with $F\geq 0$
from its bulk string theory description and compute the Casin–Huerta entropic
$c$–function. We study the monotonicity property of the entropic c–function
along the renormalization group upflow and its independence of regularization
scheme that one introduces to regularize the ultraviolet divergence of
entanglement entropy. This provides further support that the renormalization
group upflow is under better control. It also gives important insight into the
nature of the theory in the ultraviolet as it is not governed by an
ultraviolet fixed point [17].

The rest of the chapter is organized as follows. In section 3.2 we review the
corresponding bulk string theory background obtained under the deformation 2.
In section 3.3 we compute the entanglement entropy in the deformed dual
spacetime theory using the holographic prescription. Following this, we
discuss its large and small $L$ limits. In section 3.4 we compute the entropic
$c$–function and study its ultraviolet and infrared limits. In section 3.5 we
discuss the main results and future research directions. We comment on the
entropic $c$–function of a double-trace deformed field theory. We also comment
on the case where $F<0$. In appendix 6 we collect some of the intermediate
results that are required in section 3.3.

###  2 Deformed string background

We begin with string theory on

$AdS_{3}\times S^{1}\times{\cal N},$ (6)

with Neveu–Schwartz two–form $B$ field. Where the component ${\cal N}$ is an
internal six dimensional compact manifold. Its presence is irrelevant in our
discussion. 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
${\bar{J}}$ [66, 90].

The bosonic part of the world-sheet theory on $AdS_{3}\times S^{1}$ is
described by the action

$S={k\over 2\pi}\int
d^{2}z(\partial\phi{\bar{\partial}}\phi+e^{2\phi}\partial{\bar{\gamma}}{\bar{\partial}}\gamma+{1\over
k}\partial\psi{\bar{\partial}\psi}),$ (7)

where $(\phi,\gamma,{\bar{\gamma}})$ are the coordinates on $AdS_{3}$, and
$\psi\sim\psi+2\pi$ is the coordinate on $S^{1}$. The boundary of $AdS_{3}$ is
located at $\phi=+\infty$. The coordinates $\gamma$ and $\bar{\gamma}$ are

$l_{s}\gamma=t+x,\quad l_{s}\bar{\gamma}=-t+x.$ (8)

The level $k$ is given by

$l^{2}=l_{s}^{2}k,$ (9)

where $l$ is the radius of curvature of $AdS_{3}$.

The action has an affine $SL(2,R)_{L}\times SL(2,R)_{R}\times U(1)_{L}\times
U(1)_{R}$ symmetry with left mover world-sheet currents

$J^{-}=e^{2\phi}\partial\bar{\gamma},\quad
J^{+}=-2\gamma\partial\phi-\partial\gamma+\gamma^{2}e^{2\phi}\partial\bar{\gamma},\quad
J^{3}=\gamma e^{2\phi}\partial\bar{\gamma}-\partial\phi,\quad K=\partial\psi,$
(10)

and similar expressions for the right movers ${\bar{J}}^{-},{\bar{J}}^{+}$,
${\bar{J}}^{3}$ and ${\bar{K}}$.

Consider deforming the world-sheet theory 7 by adding to its Lagrangian the
deformation 2. The deformation 2 is truly marginal, and therefore, it
preserves the conformal symmetry. It breaks the affine $SL(2,R)_{L}\times
SL(2,R)_{R}\times U(1)_{L}\times U(1)_{R}$ world-sheet symmetry down to
$U(1)_{L}\times U(1)_{R}\times U(1)_{L}\times U(1)_{R}$ affine symmetry.

The deformation corresponds to a deformation of the metric $g$, dilaton
$\Phi$, and Neveu–Schwartz two–form $B$ [87, 94]. We shall refer to the
deformed background as ${\cal M}_{4}$,

$ds^{2}=d\phi^{2}+hd\gamma d{\bar{\gamma}}+{2h\epsilon_{+}\over\sqrt{k}}d\psi
d{\bar{\gamma}}+{2h\epsilon_{-}\over\sqrt{k}}d\psi d\gamma+{1\over
k}hf^{-1}d\psi^{2},$ (11) $e^{2\Phi}=g_{s}^{2}e^{-2\phi}h,$ (12)
$B_{\gamma{\bar{\gamma}}}=g_{\gamma{\bar{\gamma}}},\quad
B_{\gamma\psi}=g_{\gamma\psi},\quad
B_{\psi{\bar{\gamma}}}=g_{{\bar{\gamma}}\psi},$ (13)

where

$h^{-1}=e^{-2\phi}+\lambda-4\epsilon_{+}\epsilon_{-},\quad
f^{-1}=h^{-1}+4\epsilon_{+}\epsilon_{-}.$ (14)

For $\lambda=0,\ \epsilon_{\pm}=0$ ${\cal M}_{4}$ reduces to our starting
background $AdS_{3}\times S^{1}$. For $\lambda=0$ ${\cal M}_{4}$ reduces to a
warped $AdS_{3}\times S^{1}$ background [29, 114]. For $\epsilon_{\pm}=0$
${\cal M}_{4}$ reduces to a background that is asymptotically $AdS_{3}\times
S^{1}$ for large negative $\phi$ and
$\mathbb{R}_{\phi}\times\mathbb{R}^{1,1}\times S^{1}$ for large positive
$\phi$ [7].

It is shown in [87] that for a combination of the couplings

$\Psi=\lambda-(\epsilon_{+}+\epsilon_{-})^{2},$ (15)

with $\Psi\geq 0$ the geometry is smooth and it has no closed timelike curves.
This positivity condition on 15 is the dual analogue of the positivity
condition on 5.

In this chapter we consider the case in which

$\epsilon_{+}={\epsilon\over 2},\quad\epsilon_{-}={\epsilon\over
2},\quad\Psi\geq 0.$ (16)

In this case the background ${\cal M}_{4}$ is

$ds^{2}=\alpha^{\prime}d\phi^{2}-hdt^{2}+h\left(dx+{\epsilon\sqrt{\alpha^{\prime}\over
k}}d\psi\right)^{2}+{\alpha^{\prime}\over k}d\psi^{2},\quad
e^{2\Phi}=g_{s}^{2}e^{-2\phi}h,\quad h^{-1}=e^{-2\phi}+\Psi.$ (17)

In what follows we work on this background to compute the entanglement entropy
and the entropic $c$–function for an interval of length $L$ in the deformed
dual conformal field theory using the holographic prescription. We note that
for $\lambda<0$ or equivalently $\Psi<0$ the function $h$ changes sign beyond
a finite radial distance and therefore the metric signature also changes
signs. It is not clear in this case how to consistently apply the holographic
prescription. We will not consider this case in this chapter. We comment on
this later in the discussion section.

###  3 Holographic entanglement entropy

In this section we compute the entanglement entropy in the deformed spacetime
conformal field theory for a spatial interval of length $L$ with endpoints at
$x=-L/2$ and $x=+L/2$. It is defined as the Von Neumann entropy,

$S=-{\rm Tr}\rho\log\rho,$ (18)

corresponding to the reduced density matrix $\rho$ of the subregion $L$. The
reduced density matrix $\rho$ is obtained by tracing the global density matrix
over the states of the complement of the subregion $L$. The entanglement
entropy measures the amount of entanglement between the subregion $L$ and its
complement. Due to short range correlations near the boundary of the subregion
$L$ the entropy is, however, divergent. To regulate these ultraviolet
divergence we introduce a finite cutoff. The entropy is an intrinsic property
of the subregion $L$. Thus, it is a useful tool to characterize a field theory
along a renormalization group flow. An equally useful and well defined
quantity that we will discuss in detail in the next section is the entropic
$c$–function which, in any local and Lorentz invariant quantum field theory,
is ultraviolet regulator independent and finite.

In holographic field theories, entanglement entropy is encoded in certain
geometrical quantities in the bulk geometry [120, 121, 122, 123, 124, 125,
126, 127]. In the context of the AdS/CFT correspondence [42, 43, 44],
entanglement entropy is given by the area of a co–dimension two minimal
surface in the bulk geometry [120, 121, 125]. See [122] for a covariant
generalization. In this chapter we work under the assumption that the
holographic entanglement prescription is also valid for quantum field theories
that have dual gravity or string theory descriptions. In what follows we begin
by briefly stating this prescription.

Suppose we have a $d$–dimensional holographic quantum field theory. Suppose
also the dual string theory is on a background ${\cal M}_{d+1}$. We assume the
background ${\cal M}_{d+1}$ is static. This is the case in the theory we are
considering. To compute the entanglement entropy $S_{\cal R}$ of a given
spatial region ${\cal R}$ in the boundary quantum field theory we first find a
co–dimension two static surface ${\cal K}$ in the bulk geometry ${\cal
M}_{d+1}$ that ends on the boundary of ${\cal R}$. The surface ${\cal K}$ is
homologous to ${\cal R}$ and minimizes the area functional. The entanglement
entropy $S_{\cal R}$ in the $d$–dimensional boundary quantum field theory is
then given by

$S_{\cal R}={{\rm{Area}}({\cal K})\over 4G_{\rm N}^{(d+1)}},$ (19)

where $G^{(d+1)}_{\rm N}$ is the $d+1$–dimensional Newton’s constant of the
${\cal M}_{d+1}$ geometry.

Following the holographic prescription we now compute the entanglement entropy
for the spatial interval of length $L$ in the deformation dual to 2. The bulk
string geometry 17 at a moment of time is

$ds^{2}=\alpha^{\prime}d\phi^{2}+hdy^{2}+{\alpha^{\prime}\over
k}d\psi^{2},\quad h^{-1}=e^{-2\phi}+\Psi,$ (20)

where $y=x+{\epsilon\sqrt{\alpha^{\prime}/k}}\ \psi$.

We now look for a two–dimensional surface $\phi(y,\psi)$ in the geometry 20
(and wrapping the internal ${\cal N}$ space) that minimizes globally (in the
space of functions) the area functional 19 which taking into account the
dilaton [120, 124] is111Here we rescaled the metric 20 by the level $k$.

$S={\sqrt{k\alpha^{\prime}}\over
4G_{N}^{(4)}}\int_{0}^{2\pi}d\psi\int_{-{L\over
2}+{\epsilon\sqrt{\alpha^{\prime}/k}}\ \psi}^{+{L\over
2}+{\epsilon\sqrt{\alpha^{\prime}/k}}\ \psi}dy\ e^{2\phi}\sqrt{{1\over
h}\left(1+{\alpha^{\prime}\over
h}\left(\partial_{y}\phi\right)^{2}+k\left(\partial_{\psi}\phi\right)^{2}\right)},$
(21)

with the boundary conditions

$\phi(\pm L/2+{\epsilon\sqrt{\alpha^{\prime}/k}}\
\psi,\psi)=\infty,\quad\phi(y,0)=\phi(y+{\epsilon\sqrt{\alpha^{\prime}/k}}\
2\pi,2\pi),$ (22)

where $\psi$ is on $S^{1}$ that is $\psi\sim\psi+2\pi$.

We note that under the following continuous and discrete spacetime
transformations

$\psi\to\psi+\delta,\quad y\to y+{\epsilon\sqrt{\alpha^{\prime}\over
k}}\delta,\quad{\rm{and}}\quad\psi\to-\psi+2\pi,\quad
y\to-y+{\epsilon\sqrt{\alpha^{\prime}\over k}}2\pi,$ (23)

where $\delta$ is an arbitrary constant, the bulk background 20 and the
boundary conditions 22 are invariant. The surface

$\phi(y,\psi)=\phi(y-{\epsilon\sqrt{\alpha^{\prime}/k}}\
\psi)=\phi(-y+{\epsilon\sqrt{\alpha^{\prime}/k}}\ \psi),$ (24)

is invariant under the above symmetry transformations 23 and thus we expect
that it minimizes the area functional. The minimal surface 24 is generated by
translating the curve, for example at $\psi=0$, $\phi(y)$, along the line
$y={\epsilon\sqrt{\alpha^{\prime}/k}}\ \psi$. This curve has the parity
symmetry $y\to-y$. The entanglement entropy is then obtained using 21. We find

$S={\sqrt{k}\over 4G_{N}^{(3)}}\int_{-{L\over 2}}^{+{L\over
2}}dx\sqrt{H(U)}\sqrt{1+\beta(U)(\partial_{x}U)^{2}},$ (25)

where $G^{(3)}_{\rm N}=G^{(4)}_{\rm N}/2\pi l_{s}$, and

$U=e^{\phi},\quad U^{2}h^{-1}=1+U^{2}\left(\lambda-\epsilon^{2}\right),\quad
U^{-2}H(U)=U^{2}h^{-1},\quad U^{4}\beta(U)=(1+U^{2}\lambda)\alpha^{\prime}.$
(26)

The boundary conditions 22 now take the form

$U(\pm L/2)=U_{\infty},$ (27)

where $U_{\infty}$ is an ultraviolet cutoff.

We denote the value at which the curve $U$ takes its minimum value by $U_{0}$.
This value is related to the length of the interval $L$. This follows from the
Euler’s variational equation of the action 25 with the boundary conditions 27.
One finds

$L(U_{0})=2\sqrt{H(U_{0})}\int_{U_{0}}^{U_{\infty}}dU{\sqrt{\beta(U)}\over\sqrt{H(U)-H(U_{0})}}.$
(28)

The entropy using the Euler’s equation of motion becomes

$S={\sqrt{k}\over 2G_{\rm
N}^{(3)}}\int_{U_{0}}^{U_{\infty}}dU\sqrt{\beta(U)\over H(U)-H(U_{0})}H(U).$
(29)

We rewrite the expression 28 of the interval length $L$ in terms of the
minimum value $U_{0}$ as

$L(U_{0})={\sqrt{\alpha^{\prime}}\over U_{0}}\int_{1}^{x_{\infty}}{dx\over
x}\sqrt{{(1+\alpha x)(1+\alpha_{-})\over x(x-1)(\alpha_{-}x+\alpha_{-}+1)}},$
(30)

where

$\alpha=\lambda U^{2}_{0},\quad\alpha_{-}=\Psi U^{2}_{0},\quad
x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}}.$ (31)

The integral 30 is ultraviolet convergent and it solves to

${L\over
2\sqrt{\alpha^{\prime}\lambda}}=\sqrt{1+\alpha\over\alpha}E\left(\arcsin\sqrt{{1+\alpha_{-}\over
1+2\alpha_{-}}},\sqrt{1+2\alpha_{-}\over(1+\alpha)(1+\alpha_{-})}\right),$
(32)

where $E(\varphi,k)$ is the incomplete elliptic integral of the second
kind222Here and in what follows please note our notation of elliptic
integrals. There are different notations of elliptic integrals in the
literature.,

$E(\varphi,k)=\int_{0}^{\varphi}d\theta\sqrt{1-k^{2}\sin^{2}\theta}.$ (33)

In the limit in which $\Psi\to 0^{+}$ or, equivalently $\alpha_{-}\to 0^{+}$,
we note that the interval $L$ 32 takes the following simpler form

${L\over
2\sqrt{\alpha^{\prime}\lambda}}=\sqrt{1+\alpha\over\alpha}E\left(\sqrt{1\over
1+\alpha}\right),$ (34)

where $E(k)=E(\pi/2,k)$ is the complete elliptic integral of the second kind.
Sending $\alpha\to 0^{+}$ in 34 yields

${L\over\sqrt{\alpha^{\prime}}}={2\over U_{0}}.$ (35)

We rewrite the entanglement entropy 29 as

$S={\sqrt{k\alpha^{\prime}}\over 4G_{\rm
N}^{(3)}}\int_{1}^{x_{\infty}}dx\sqrt{{\alpha x+1\over
x(x-1)(\alpha_{-}x+\alpha_{-}+1)}}\cdot(\alpha_{-}x+1).$ (36)

We note that for $\alpha_{-}\neq 0$ the entropy diverges as $S\sim
x_{\infty}$, and in the limit in which we take $\alpha_{-}\to 0^{+}$ with
$\alpha\neq 0$ it diverges as $S\sim\sqrt{x_{\infty}}$. We also note that in
the case in which we take both $\alpha\to 0^{+},\ \alpha_{-}\to 0^{+}$ it
diverges as $S\sim{\rm{Log}}(x_{\infty})$. The integral 36 solves with the
ultraviolet cutoff $x_{\infty}$ to

$\displaystyle S={\sqrt{k\alpha^{\prime}}\over 2G_{\rm
N}^{(3)}}{1\over\sqrt{(\alpha+1)(\alpha_{-}+1)}}\left\\{\left({\alpha_{-}}+{\alpha}-\alpha\alpha_{-}{d\over
d\xi}\right)\right.$ (37)
$\displaystyle\left.\left[{1\over\xi+1}\cdot\Pi\left(\arcsin\sqrt{{\alpha_{-}+1\over
2\alpha_{-}+1}\cdot\left({1-{1\over
x_{\infty}}}\right)},{2\alpha_{-}+1\over(\xi+1)(\alpha_{-}+1)},\sqrt{{2\alpha_{-}+1\over(\alpha+1)(\alpha_{-}+1)}}\right)\right]_{\xi=0}\right.$
(38) $\displaystyle\left.+\ F\left(\arcsin\sqrt{{\alpha_{-}+1\over
2\alpha_{-}+1}\cdot\left(1-{1\over
x_{\infty}}\right)},\sqrt{{2\alpha_{-}+1\over(\alpha+1)(\alpha_{-}+1)}}\right)\right\\},$
(39)

where $\Pi(\varphi,n,k)$ is the incomplete elliptic integral of the third
kind, and $F(\varphi,k)=\Pi(\varphi,0,k)$ is the incomplete elliptic integral
of the first kind,

$\Pi(\varphi,n,k)=\int_{0}^{\varphi}{d\theta\over(1-n\sin^{2}\theta)\sqrt{1-k^{2}\sin^{2}\theta}}.$
(40)

In the limit $\Psi\to 0^{+}$ or, equivalently $\alpha_{-}\to 0^{+}$, the
entropy 37 gives

$\displaystyle S=$ $\displaystyle{\sqrt{k\alpha^{\prime}}\sqrt{1+\alpha}\over
2G_{\rm N}^{(3)}}\left[F\left(\arcsin\sqrt{1-{1\over x_{\infty}}},\sqrt{1\over
1+\alpha}\right)-E\left(\arcsin\sqrt{1-{1\over x_{\infty}}},\sqrt{1\over
1+\alpha}\right)\right]$ (41) $\displaystyle+{\sqrt{k\alpha^{\prime}}\over
2G_{\rm N}^{(3)}}\sqrt{(\alpha x_{\infty}+1)\cdot\left(1-{1\over
x_{\infty}}\right)}.$ (42)

Taking $\alpha\to 0^{+}$ in 41 gives

$S={\sqrt{k\alpha^{\prime}}\over 2G_{\rm
N}^{(3)}}{\rm{Log}}(2\sqrt{x_{\infty}}).$ (43)

In the rest of the current section we study the above results in turns.

####  1 Case $\Psi=0:\lambda=0,\ \epsilon=0$

In this case we have for the interval length $L$ and entanglement entropy $S$
from 35 and 43

${L\over\sqrt{\alpha^{\prime}}}={2\over U_{0}},\quad
S={\sqrt{k\alpha^{\prime}}\over 2G_{\rm
N}^{(3)}}{\rm{Log}}\left(2{U_{\infty}\over U_{0}}\right).$ (44)

We write the entropy as

$S={c\over 3}{\rm{Log}}\left(2{L\over
L_{\Lambda}}\right),\quad{L_{\Lambda}\over\sqrt{\alpha^{\prime}}}:={2\over
U_{\infty}},\quad c={3\sqrt{k\alpha^{\prime}}\over 2G_{\rm N}^{(3)}},$ (45)

where $L_{\Lambda}$ is an ultraviolet cutoff. This result is the well–known
entanglement entropy for a two–dimensional holographic conformal field theory
with (Brown–Henneaux) central charge $c$ that is dual to pure $AdS_{3}$ [121,
128, 129].

####  2 Case $\Psi=0:\lambda=\epsilon^{2}\neq 0$

In this case we have from 34 and 41 that the interval length $L$ and
entanglement entropy $S$ are given by

${L\over
2\sqrt{\alpha^{\prime}\lambda}}=\sqrt{1+\alpha\over\alpha}E\left(\sqrt{1\over
1+\alpha}\right),$ (46) $\displaystyle S=$
$\displaystyle{\sqrt{k\alpha^{\prime}}\sqrt{1+\alpha}\over 2G_{\rm
N}^{(3)}}\left[F\left(\arcsin\sqrt{1-{1\over x_{\infty}}},\sqrt{1\over
1+\alpha}\right)-E\left(\arcsin\sqrt{1-{1\over x_{\infty}}},\sqrt{1\over
1+\alpha}\right)\right]$ (47) $\displaystyle+{\sqrt{k\alpha^{\prime}}\over
2G_{\rm N}^{(3)}}\sqrt{(\alpha x_{\infty}+1)\cdot\left(1-{1\over
x_{\infty}}\right)},\quad\alpha=\lambda U_{0}^{2},\quad
x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}}.$ (48)

Note that this case corresponds taking $F=0$ in 5.

We find from 46 that in the large $U_{0}$ limit the interval length $L$
asymptotes to a minimum value which we denote by $L_{0}$. It takes the value

$L_{0}=\pi\sqrt{\alpha^{\prime}\lambda}=\sqrt{\pi t}.$ (49)

We find using 46 the following large $U_{0}$ expansion of the interval length
$L$,

${L\over L_{0}}=1+{1\over 4}\cdot{1\over\alpha}-{3\over
64}\cdot{1\over\alpha^{2}}+{\cal
O}\left({1\over\alpha^{3}}\right),\quad\alpha=\lambda U_{0}^{2}.$ (50)

Inverting the above equation one finds

$\alpha={1\over 4\xi}-{3\over 16}+{\cal O}(\xi),\quad\xi={L\over L_{0}}-1.$
(51)

We will use this result momentarily to write the entropy in the ultraviolet in
terms of the length $L$.

The small $U_{0}$ expansion of the interval length $L$ is

${L\over\sqrt{\alpha^{\prime}}}={2\over U_{0}}\left(1-{1\over
4}\alpha\ln\alpha+{1\over 32}\alpha^{2}\ln\alpha+{\cal
O}(\alpha^{3})\right),\quad\alpha=\lambda U_{0}^{2}.$ (52)

The leading term corresponds to the deep $AdS_{3}$ geometry (see 44).
Therefore, in the large $L$ limit the surface is deep inside the bulk. We note
also that the correction starts at order $\lambda=\epsilon^{2}$.

Inverting equation 52 we find

$\alpha=\left({2\over\pi}\cdot{L_{0}\over
L}\right)^{2}\left[1-\left({2\over\pi}\cdot{L_{0}\over
L}\right)^{2}{\rm{Log}}\left({2\over\pi}\cdot{L_{0}\over L}\right)+{\cal
O}\left({2\over\pi}\cdot{L_{0}\over L}\right)^{4}\right].$ (53)

We can similarly study the large and small $U_{0}$ or, equivalently, the large
and small $L$ limits of the entanglement entropy 47. One finds in the large
interval length $L$ limit

$S={c\over 3}\left[\sqrt{\alpha x_{\infty}}-{1\over
2}{\rm{Log}}(\alpha)-{\alpha\over 8}{\rm{Log}}(\alpha)+{\cal
O}\left(\alpha^{2}\right)\right],\quad\alpha=\lambda U_{0}^{2},\quad
x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}},$ (54)

which upon using 53 gives

$S={c\over 3}\left[{L_{0}\over
L_{\Lambda}}-{\rm{Log}}\left({2\over\pi}\cdot{L_{0}\over L}\right)+{1\over
4}\left({2\over\pi}\cdot{L_{0}\over
L}\right)^{2}{\rm{Log}}\left({2\over\pi}\cdot{L_{0}\over L}\right)+{\cal
O}\left({2\over\pi}\cdot{L_{0}\over L}\right)^{4}\right],$ (55)

where

$L_{\Lambda}:={\pi\sqrt{\alpha^{\prime}}\over U_{\infty}},$ (56)

and $L_{\Lambda}$ is an ultraviolet cutoff. The leading logarithmic term is
the contribution from the $AdS_{3}$ region found deep inside the bulk. The
coefficient of this term is $-c/3$, as expected. We note that the logarithmic
terms depend on the ratio $L/L_{0}$, and thence $L_{0}$ sets the non–locality
scale of the theory. This will become even more evident in the next section.

As we approach $L_{0}$ we find

$S={c\over 3}\left[\sqrt{\alpha x_{\infty}}+{\pi\over
4}{1\over\alpha^{1/2}}-{\pi\over 32}{1\over\alpha^{3/2}}+{\cal
O}\left({1\over\alpha^{5/2}}\right)\right],\quad\alpha=\lambda U_{0}^{2},\quad
x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}},$ (57)

which simplifies using 51 to

$S={c\over 3}\left[{L_{0}\over L_{\Lambda}}+{\pi\over 2}\left({L\over
L_{0}}-1\right)^{{1\over 2}}-{\pi\over 16}\left({L\over
L_{0}}-1\right)^{{3\over 2}}+{\cal O}\left(\left({L\over
L_{0}}-1\right)^{5\over 2}\right)\right].$ (58)

We note that there is no a logarithmic correction term. The entanglement
entropy instead shows a square root area law correction at next–to–leading
order. The entanglement entropy scales as the square root of the length of the
interval. Such power law scaling is quite peculiar and interesting. In local
and Lorentz–invariant even–dimensional quantum field theories, however, in
general the presence of a logarithmic correction term is generic, and its
coefficient is expected to be universal [130]. It would be interesting to
understand this theory better. We discuss its entropic $c$–function in the
next section.

####  3 Case $\Psi>0:\epsilon=0$

This case is studied in [26]. Setting $\alpha=\alpha_{-}$ in 32 and 37 we find
for the length $L$ and entanglement entropy $S$

${L\over
2\sqrt{\alpha^{\prime}\lambda}}=\sqrt{1+\alpha\over\alpha}E\left(\arcsin\sqrt{{1+\alpha\over
1+2\alpha}},\sqrt{1+2\alpha\over 1+2\alpha+\alpha^{2}}\right),$ (59)
$\displaystyle S={\sqrt{k\alpha^{\prime}}\over 2G_{\rm
N}^{(3)}}{1\over\alpha+1}\left\\{\left(2{\alpha}-\alpha^{2}{d\over
d\xi}\right)\right.$ (60)
$\displaystyle\left.\left[{1\over\xi+1}\Pi\left(\arcsin\sqrt{{\alpha+1\over
2\alpha+1}\left({1-{1\over
x_{\infty}}}\right)},{2\alpha+1\over(\xi+1)(\alpha+1)},\sqrt{{2\alpha+1\over\alpha^{2}+2\alpha+1}}\right)\right]_{\xi=0}\right.$
(61) $\displaystyle\left.+\ F\left(\arcsin\sqrt{{\alpha+1\over
2\alpha+1}\left(1-{1\over
x_{\infty}}\right)},\sqrt{{2\alpha+1\over\alpha^{2}+2\alpha+1}}\right)\right\\},\quad\alpha=\lambda
U_{0}^{2},\quad x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}}.$ (62)

We find that in the large $U_{0}$ limit the interval length $L$ asymptotes to
a minimum value which we denote by $L_{0}$ (this should cause no ambiguity)

$L_{0}={\pi\sqrt{\alpha^{\prime}\lambda}\over 2}={1\over 2}\sqrt{\pi t}.$ (63)

We note that there is a factor of 2 difference between 49 and 63. We find from
59 that the interval length $L$ has the following large $U_{0}$ expansion

${L\over L_{0}}=1+{2\over\pi\alpha}+{3\pi-16\over 16\pi\alpha^{2}}+{\cal
O}\left({1\over\alpha^{3}}\right),\quad\alpha=\lambda U_{0}^{2}.$ (64)

Inverting the above equation one finds

$\alpha={2\over\pi\xi}+{3\pi-16\over 32}+{\cal O}(\xi),\quad\xi={L\over
L_{0}}-1.$ (65)

The small $U_{0}$ expansion takes the form

${L\over\sqrt{\alpha^{\prime}}}={2\over U_{0}}\left[1-{\alpha^{2}\over
4}{\rm{Log}}(\alpha)+{3\over 8}\alpha^{3}{\rm{Log}}(\alpha)+{\cal
O}\left(\alpha^{4}\right)\right],\quad\alpha=\lambda U_{0}^{2}.$ (66)

We note that for a long interval the surface is deep inside the bulk in the
$AdS_{3}$ region. We also note that the term linear in $\alpha$ is zero.
Therefore, the correction starts, in this case, at order $\lambda^{2}$.

Inverting the above equation we find

$\alpha=\left({4\over\pi}{L_{0}\over
L}\right)^{2}\left[1-\left({4\over\pi}{L_{0}\over
L}\right)^{4}{\rm{Log}}\left({4\over\pi}{L_{0}\over L}\right)+{\cal
O}\left(\left({4\over\pi}{L_{0}\over L}\right)^{6}\right)\right].$ (67)

In the large $L$ limit we find that the entanglement entropy $S$ has the
following series expansion

$S={c\over 3}\left[{1\over 2}\alpha x_{\infty}-{1\over
2}{\rm{Log}}(\alpha)-{1\over 4}\alpha+{\cal
O}\left(\alpha^{2}{\rm{Log}}(\alpha)\right)\right],\quad\alpha=\lambda
U_{0}^{2},\quad x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}},$ (68)

which using 67 gives

$S={c\over 3}\left[{1\over 2}\cdot{L_{0}^{2}\over
L^{2}_{\Lambda}}\cdot{16\over\pi^{2}}-{\rm{Log}}\left({L_{0}\over
L}\cdot{4\over\pi}\right)-{1\over 4}\left({L_{0}^{2}\over
L^{2}}\cdot{16\over\pi^{2}}\right)+{\cal O}\left({L_{0}^{2}\over
L^{2}}\cdot{16\over\pi^{2}}\right)^{2}\right],\quad
U_{\infty}:={2\sqrt{\alpha^{\prime}}\over L_{\Lambda}}.$ (69)

The (leading) logarithmic term is due to the deep $AdS_{3}$ region in the
bulk. The coefficient of this term is $-c/3$, as expected. We also note that
the arguments of the leading logarithmic terms in 55 and 69 are also equal
despite the different $L_{0}$ values.

As we approach $L_{0}$ the entropy $S$ takes the form

$S={c\over 3}\left[{1\over 2}\alpha x_{\infty}-{1\over
2}{\rm{Log}}(\alpha)+{\cal
O}\left({1\over\alpha}\right)\right],\quad\alpha=\lambda U_{0}^{2},\quad
x_{\infty}={U_{\infty}^{2}\over U_{0}^{2}}.$ (70)

Using 65 this gives

$S={c\over 3}\left[{1\over 2}\cdot{16\over\pi^{2}}\cdot{L^{2}_{0}\over
L^{2}_{\Lambda}}+{1\over 2}{\rm{Log}}\left({\pi\over 2}\left({L\over
L_{0}}-1\right)\right)+{\cal O}\left({L\over L_{0}}-1\right)\right].$ (71)

In this limit the geometry is a linearly varying dilaton background. We note
that in this case we have a logarithmically divergent term as opposed to the
former $\Psi=0$ case. The coefficient of this term is $c/6$.

####  4 Case $\Psi>0:\epsilon\neq 0$

In the large $U_{0}$ limit we find from 32 that the interval length approach a
minimum value $L_{0}$

$L_{0}={\pi\sqrt{\alpha^{\prime}\lambda}\over 2}={1\over 2}\sqrt{\pi t},$ (72)

that is determined solely by $\lambda$.

The length $L$ has the following large $U_{0}$ expansion

${L\over
L_{0}}=1+{2-\delta^{2}\over\pi(1-\delta^{2})\alpha}+{3\pi-16+2(4-\pi)\delta^{2}-\pi\delta^{4}\over
16\pi(1-\delta^{2})^{2}\alpha^{2}}+{\cal
O}\left({1\over\alpha^{3}}\right),\quad\delta^{2}={\epsilon^{2}\over\lambda},\quad\alpha=\lambda
U_{0}^{2}.$ (73)

Inverting this we find

$\alpha={2-\delta^{2}\over\pi(1-\delta^{2})\xi}+{3\pi-16+2(4-\pi)\delta^{2}-\pi\delta^{4}\over
32-48\delta^{2}+16\delta^{4}}+{\cal O}(\xi),\quad\xi={L\over L_{0}}-1.$ (74)

The small $U_{0}$ expansion takes the form

${L\over\sqrt{\alpha^{\prime}}}={2\over U_{0}}\left[1-{\delta^{2}\over
4}\alpha{\rm{Log}}(\alpha)-{\alpha^{2}\over
4}(1+\delta^{2}p){\rm{Log}}(\alpha)+{\cal
O}(\alpha^{3})\right],\quad\delta^{2}={\epsilon^{2}\over\lambda},\quad\alpha=\lambda
U_{0}^{2},$ (75)

here $p$ is a polynomial in $\delta^{2}$. The leading term corresponds to the
deep $AdS_{3}$ region. We note that in this case the correction starts at
order $\epsilon^{2}$.

Inverting the above equation one finds

$\displaystyle\alpha=\left({4\over\pi}{L_{0}\over
L}\right)^{2}\left[1-\delta^{2}\left({4\over\pi}{L_{0}\over
L}\right)^{2}{\rm{Log}}\left({4\over\pi}{L_{0}\over
L}\right)-\left({4\over\pi}{L_{0}\over
L}\right)^{4}(1+\delta^{2}p){\rm{Log}}\left({4\over\pi}{L_{0}\over
L}\right)\right.$ (76) $\displaystyle\left.+{\cal
O}\left(\left({4\over\pi}{L_{0}\over L}\right)^{6}\right)\right].$ (77)

In the large $L$ limit we find the following expansion for the entropy $S$

$S={c\over 3}\left[{1\over 2}\alpha x_{\infty}\sqrt{1-\delta^{2}}-{1\over
2}{\rm{Log}}(\alpha)-{1\over 4}\alpha\left({1\over(1-\delta^{2})^{2}}+{1\over
2}\delta^{2}{\rm{Log}}(\alpha)\right)+{\cal O}\left(\alpha^{2}\right)\right],$
(78)

which upon using 76 gives

$\displaystyle S={c\over 3}\left\\{\left({4L_{0}\over\pi
L_{\Lambda}}\right)^{2}\sqrt{1-\delta^{2}\over
4}-{\rm{Log}}\left({4L_{0}\over\pi L}\right)-{1\over 4}\left({4L_{0}\over\pi
L}\right)^{2}\left[{1\over(1-\delta^{2})^{2}}-\delta^{2}{\rm{Log}}\left({4L_{0}\over\pi
L}\right)\right]\right.$ (79) $\displaystyle+\left.{\cal
O}\left(\left({L_{0}\over L}\right)^{4}\right)\right\\},\ $ (80)

where $L_{\Lambda}$ is defined in 69. The leading logarithmic term as in the
previous cases corresponds to the $AdS_{3}$ region found deep inside the bulk.
The coefficient of this term is $-c/3$, as expected.

As we approach $L_{0}$ we find

$S={c\over 3}\left[{1\over 2}\alpha x_{\infty}-{2-\delta^{2}\over
4\sqrt{1-\delta^{2}}}{\rm{Log}}(\alpha)+{\cal
O}\left({1\over\alpha}\right)\right],$ (81)

which using 74 gives

$S={c\over 3}\left[{1\over 2}\cdot{16\over\pi^{2}}\cdot{L^{2}_{0}\over
L^{2}_{\Lambda}}\cdot\sqrt{1-\delta^{2}}+{2-\delta^{2}\over
4\sqrt{1-\delta^{2}}}{\rm{Log}}\left(\left({L\over
L_{0}}-1\right)\cdot{\pi(1-\delta^{2})\over 2-\delta^{2}}\right)+{\cal
O}\left({L\over L_{0}}-1\right)\right].$ (82)

We note that the leading logarithmic term has a coefficient that depends on
$\delta^{2}$.

In the next section we study in the above cases the Casin–Huerta entropic
$c$–function.

###  4 Entropic $c$–function

In quantum field theory entanglement entropy is ultraviolet divergent. It
requires an ultraviolet cutoff to regularize the divergence. However, in
two–dimensional local and Lorentz–invariant quantum field theories the
Casin–Huerta entropic $c$–function [131] which is derived from the
entanglement entropy as

$C:=L{\partial S\over\partial L},$ (83)

is independent of the ultraviolet cutoff and finite. The interval length $L$
is interpreted as a renormalization group scale.

The entropic $c$–function is also a monotonic function of $L$, and at fixed
points of renormalization group flow it is proportional to the corresponding
central charges. For scale invariant theories the entropic $c$–function is
constant; it is independent of $L$.

The entropic $c$–function is a useful tool to probe phase transitions. In this
section we study the entropic $c$–function for the different cases we studied
in the former section. We study in each case its monotonicity as a function of
$L$ and its independence of the ultraviolet regulator. We also examine its
behavior in the ultraviolet regime.

####  1 Case $\Psi=0:\lambda=0,\ \epsilon=0$

In this case we have

$C={c\over 3}.$ (84)

This is the result for a two–dimensional holographic conformal field theory
with central charge $c$ [131]. It is independent of the interval length $L$.
The entropic $c$–function is non–negative and constant.

####  2 Case $\Psi=0:\lambda=\epsilon^{2}\neq 0$

In this case we find that the entropic $c$–function $C(\alpha)$ is given by

$C={c\over 3}\sqrt{1+\alpha}E\left(\sqrt{1\over
1+\alpha}\right),\quad\alpha=\lambda U_{0}^{2}.$ (85)

We study the small and large $U_{0}$ limit, or equivalently the large and
small $L$ limits of the entropic $c$–function 85.

In the large $L$ limit we find

$C={c\over
3}\left(1+{2\over\pi^{2}\xi^{2}}{\rm{Log}}\left({2\over\pi}\cdot\xi\right)+{\cal
O}\left({1\over\xi^{4}}\right)\right),\quad L:=\xi L_{0},\quad
L_{0}=\pi\sqrt{\alpha^{\prime}\lambda}=\sqrt{\pi t}.$ (86)

In the small $L$ limit we find

$C={c\over 3}\left({\pi\over 4}\cdot{1\over\xi^{1\over 2}}+{5\pi\over
32}\cdot\xi^{1\over 2}+{\cal O}(\xi^{3/2})\right),\quad\xi={L\over L_{0}}-1.$
(87)

One can think of $L$ as a renormalization group scale. We note that the
entropic $c$–function increases as we ascend the renormalization group, and it
diverges in the ultraviolet at $L_{0}$. At short distances it diverges as

$C\sim\xi^{-{1\over 2}},\quad\xi={L\over L_{0}}-1.$ (88)

This is the case since in the ultraviolet the theory is not governed by a
fixed point. We also note that the entropic c–function is independent of the
ultraviolet cutoff that we introduced to regularize the entanglement entropy.
The variable $\alpha$ can be expressed using the result 46 in terms of the
interval length $L$ and the coupling $t$ to write 85 as a function of only $L$
and $t$.

####  3 Case $\Psi>0:\epsilon=0$

This case is studied in [26]333It is also studied in closely related works
[132, 133].. In this case we find that the entropic $c$–function $C(\alpha)$
in closed–form is given by

$C={c\over 3}(1+\alpha)E\left(\arcsin\sqrt{1+\alpha\over
1+2\alpha},\sqrt{1+2\alpha\over(1+\alpha)^{2}}\right),\quad\alpha=\lambda
U_{0}^{2}.$ (89)

Using the results 65 and 67 for $U_{0}$ we find in the large $L$ limit

$C={c\over 3}\left(1+{8\over\pi^{2}\xi^{2}}+{\cal
O}\left({1\over\pi\xi}\right)^{4}\right),\quad\xi:={L\over L_{0}},\quad
L_{0}={\pi\sqrt{\alpha^{\prime}\lambda}\over 2}={1\over 2}\sqrt{\pi t},$ (90)

and in the small $L$ limit

$C={c\over 3}\left({1\over 2}\left(1-{1\over\xi}\right)^{-1}+{\cal
O}\left(1-{1\over\xi}\right)\right),\quad\xi:={L\over L_{0}}.$ (91)

We also note in this case that the entropic $c$–function is non–negative and
increasing. At short distances it diverges as

$C\sim\xi^{-1},\quad\xi={L\over L_{0}}-1.$ (92)

Our results 90 and 91 are in agreement with the corresponding analyses in
[26]. Our result 89 gives a non–perturbative answer and it can be written as a
function of the interval length $L$ and the coupling $t$. We also note from 89
that the entropic c–function is independent of the ultraviolet cutoff.

####  4 Case $\Psi>0:\epsilon\neq 0$

In this case we find that the $c$–function $C(\alpha,\chi)$ is given by

$C={c\over
3}\sqrt{(1+\alpha)(1+\alpha\chi)}E\left(\arcsin\sqrt{1+\alpha\chi\over
1+2\alpha\chi},\sqrt{1+2\alpha\chi\over(1+\alpha)(1+\alpha\chi)}\right),$ (93)

where

$\alpha=\lambda U_{0}^{2},\quad\Psi=\lambda\chi.$ (94)

In this case the large $L$ limit of the entropic $c$–function takes the form

$C={c\over
3}\left(1+{8\over\pi^{2}\xi^{2}}\left({1\over{\chi^{2}}}+(1-\chi){\rm{Log}}\left({\pi\xi\over
4}\right)\right)+{\cal O}\left({1\over\pi\xi}\right)^{4}\right),$ (95)

where

$\xi:={L\over L_{0}},\quad L_{0}={\pi\sqrt{\alpha^{\prime}\lambda}\over
2}={1\over 2}\sqrt{\pi t}.$ (96)

In the small $L$ limit we find

$C={c\over 3}\left({1+\chi\over
4\sqrt{\chi}}\left(1-{1\over\xi}\right)^{-1}+{\cal
O}\left(1-{1\over\xi}\right)\right),\quad\xi:={L\over L_{0}}.$ (97)

In this case also the entropic $c$–function is non–negative, ultraviolet
cutoff independent and increasing. In the ultraviolet it diverges as

$C\sim\chi^{-{1\over 2}}\cdot\xi^{-1},\quad\xi={L\over L_{0}}-1.$ (98)

We note that setting $\chi=1$ in 93 gives 89, and setting $\chi=0$ gives 85.
At $\alpha=0$ it gives 84. The entropic $c$–function $C$ 93 and the interval
length $L$ 32 satisfy the curious relation

${C\over c_{0}}={L\over l_{0}}\cdot\sqrt{1+\alpha\chi},\quad
l_{0}={2\sqrt{\alpha^{\prime}}\over U_{0}},\quad c_{0}={c\over
3},\quad\alpha=\lambda U_{0}^{2},$ (99)

where $l_{0}$ and $c_{0}$ can be thought of as the interval length $L$ and the
entropic $c$–function $C$ at $\lambda=0$ or in the infrared, respectively.

###  5 Discussion

In this chapter we computed the (Von Neumann) entanglement entropy and the
entropic $c$–function for an interval of length $L$. We found that the
entropic $c$–function is given by

$C={c\over
3}\sqrt{(1+\alpha)(1+\alpha\chi)}E\left(\arcsin\sqrt{1+\alpha\chi\over
1+2\alpha\chi},\sqrt{1+2\alpha\chi\over(1+\alpha)(1+\alpha\chi)}\right),$
(100)

where444In terms of the field theory side deformation parameters $\delta^{2}$
is given by ${\pi\mu^{2}\over 8t}$, and $\mu=2\mu_{+}=2\mu_{-}$.

$\alpha=\lambda
U_{0}^{2},\quad\chi=1-\delta^{2},\quad\delta^{2}={\epsilon^{2}\over\lambda}={\pi\mu^{2}\over
8t}.$ (101)

The variable $\alpha$ is related to the interval length $L$ via

${L\over
2\sqrt{\alpha^{\prime}\lambda}}=\sqrt{1+\alpha\over\alpha}E\left(\arcsin\sqrt{{1+\alpha\chi\over
1+2\alpha\chi}},\sqrt{1+2\alpha\chi\over(1+\alpha)(1+\alpha\chi)}\right),\quad\alpha^{\prime}\lambda={t\over\pi}.$
(102)

We found that the entropic c–function is non–negative and ultraviolet cutoff
independent. This is required for a theory that is internally consistent and
it is a non–trivial consistency check. Therefore, this provides further
evidence that the deformed theory is very sound and under control. We also
found that along the renormalization group upflow towards the ultraviolet it
is non–decreasing. At long distances it is proportional to the central charge
of the original conformal field theory. At short distances it diverges. This
is the case since in the ultraviolet the deformed theory is not governed by an
ultraviolet fixed point. The minimum distance at which the entropic c–function
diverges sets the non-locality scale of the theory.

Figure 1: Entropic c–function $C$ per $c_{0}$ as a function of interval length
$L$ per $L_{0}$. Where $c_{0}={c\over 3}$ and $L_{0}={1\over 2}\sqrt{\pi t}$.
The orange plot is for $\delta=0$, and the blue plot is for $\delta^{2}=1$.
The (normalized) entropic c–function for the case $\delta=0$ diverges at
$L_{0}$.

In 1 we show plots of the entropic c–function $C$ 100 as a function of the
interval length $L$ 102 for $\delta=0$ and $\delta^{2}=1$. In the plots we
normalized $C$ with respect to $c_{0}={c\over 3}$ and $L$ with respect to
$L_{0}={1\over 2}\sqrt{\pi t}$.

In the case in which $F>0$, the entropic $c$–function diverges in the
ultraviolet as

$C\sim\chi^{-{1\over 2}}\cdot\xi^{-1},\quad\xi={L\over L_{0}}-1,\quad
L_{0}:={\pi\sqrt{\alpha^{\prime}\lambda}\over 2}={1\over 2}\sqrt{\pi t}.$
(103)

Note that $L_{0}$ is determined solely by $\lambda$; it is independent of
$\epsilon$.

In the case in which $F=0$, the entropic $c$–function diverges at short
distances as

$C\sim\xi^{-{1\over 2}},\quad\xi={L\over L_{0}}-1,\quad
L_{0}:=\pi\sqrt{\alpha^{\prime}\lambda}=\sqrt{\pi t}.$ (104)

Note that the minimum interval length $L_{0}$ in this case is twice larger
than the corresponding length in the former case. This is depicted in 1 with
$\delta=0$ and $\delta^{2}=1$.

We note that the exponent of $\xi$ in the case in which $F>0$ is $-1$, and in
the case in which $F=0$ it is $-{1/2}$. This is due to the fact that at short
distances the entanglement entropy for the case in which $F>0$ 82 contains a
logarithmically divergent term (with a coefficient that depends on $\chi$)
whereas in the case in which $F=0$ 58 it does not contain a logarithmically
divergent term. In the latter case the entanglement entropy area law exhibits
a square root correction 58. The entanglement entropy scales as the square
root of the length of the interval,

$S={c\over 3}\left[{L_{0}\over L_{\Lambda}}+{\pi\over 2}\left({L\over
L_{0}}-1\right)^{{1\over 2}}+{\cal O}\left(\left({L\over
L_{0}}-1\right)^{3\over 2}\right)\right],$ (105)

where $L_{\Lambda}$ is the ultraviolet cutoff. Such power law scaling is quite
distinct and interesting.

It would be interesting to compute the entropic $c$–function directly in
$J{\bar{T}},\ T{\bar{J}},\ T{\bar{T}}$ deformed field theory, for example in
perturbation theory, and compare it with the bulk calculation result 100
(where $\alpha$ is given by 102). It would be also nice to understand better
the theory corresponding to the case in which $F=0$. We leave these for future
work.

In this chapter we mainly focused on the case in which $F\geq 0$. As we
mentioned in the introduction, in the case $F<0$ or equivalently $\Psi<0$,
however, the bulk geometry is singular and/or it has either closed timelike