Milne Spacetime with conical defect: Some holographic Studies

Swayamsidha Mishra ***We are still grieving the loss of our friend and
collaborator Swayamsidha Mishra who passed away while this work was being
finished. Her invaluable contributions and dedication to this work will always
be remembered. 1,3, Sudipta Mukherji2,3 and Yogesh K. Srivastava1,3

1 National Institute of Science Education and Research (NISER),

Bhubaneswar, P.O. Jatni, Khurda, Odisha, India 752050

2Institute of Physics, Sachivalaya Marg,

Bhubaneswar, Odisha, India 751005

3Homi Bhabha National Institute, Training School Complex,

Anushakti Nagar, Mumbai India 400085

<EMAIL_ADDRESS><EMAIL_ADDRESS>
$\ ,$

###### Abstract

We initiate a holographic study of field theory in time dependent background
with a conical defect. We focus on the Milne spacetime to which, in the
absence of cosmological constant, at late time any hyperbolic Friedmann-
Robertson-Walker metric flows to. When the Milne vacuum is represented by the
adiabatic one, we are able to compute the two point correlators of operators
which are dual to the massive scalars in the bulk AdS-Milne background with a
defect. We find, for both twisted and untwisted operators, the correlators can
be represented as the sum over images. This sum can be carried out explicitly
to write the results in compact forms.

##  1 Introduction

AdS/CFT duality has been extremely useful in exploring various features of
strongly coupled quantum field theories as well as in providing insights into
the quantum theory of gravity, particularly in the context of black hole
physics. However, progress has been somewhat limited in exploiting the duality
in the cosmological context. This could be due to the fact that a cosmological
spacetime typically contains a space-like singularity and around that region,
classical gravity becomes unreliable. Nonetheless, since such a singularity
generally runs all the way to the boundary, one may wish to examine if the
boundary gauge theory could sense this singularity. Indeed in [1]-[7], in the
context of AdS cosmologies and in particular, for AdS-Kasner and AdS-FRW
cosmologies, such questions were addressed with various degrees of success.

Among the class of hyperbolic FRW geometries, the simplest is the Milne
spacetime where the scale factor has a linear dependence on time. On one hand,
since by a coordinate transformation the Milne spacetime can be expressed as
the future wedge of the Minkowski metric, the geometry is free of curvature.
On the other hand, as the time dependence of the metric is relatively simple,
one may hope to have some analytical handle [8], [9]. Further, in the absence
of a cosmological constant, we note that all the hyperbolic FRW geometries
approach the Milne spacetime at late time as the expansions in these models
drive the mass-energy density to zero. In this paper, our focus will be on the
Milne space-time. Within the holographic set up, we will represent the
boundary by the Milne geometry. The corresponding bulk dual will be the AdS-
Milne spacetime. The precise nature of the gauge theory on this boundary will
depend on the spacetime dimensions.

A generic difficulty that arises while working with quantum field theories on
a curved, especially time dependent, background is that the choice of the
vacuum becomes ambiguous. Consequently, the correlation functions constructed
out of the quantum fields start to depend on the choice of the vacuum. Even
though the Milne spacetime is a patch of Minkowski, there exists multiple
complete sets of modes which are related by Bogoliubov transformations. Fock
space built out of the corresponding creation and annihilation operators are
necessarily inequivalent and corresponding vacuum states carry different
physical properties. Two preferred choices for the vacuum states in the Milne
spacetime are the adiabatic and the conformal vacuum. Other less commonly used
vacua can also be defined, see for example [10]. Among these vacua, the
adiabatic vacuum is particularly appealing because of its similarity with the
familiar vauum in the Minkowski spacetime.

In the present work, to start with, we perform several computations in AdS-
Milne background and extract the corresponding gauge theory quantities. For a
massive minimally coupled scalar field we compute the bulk Wightman and the
other Green functions in the case when the Milne part of the vacuum is
represented by the adiabatic one. Subsequently, following the holographic
prescription, the corresponding boundary correlators are extracted. An
important issue here is whether the subregion duality [11] holds. We find
that, at least for the cases we study, the bulk and the boundary correlators
seem to respect the subregion duality. Next we turn our attention to the one
where the Milne part is represented by the conformal vacuum. Here, we have not
been able to find a closed form expression for the Wightman function. However,
we see that the retarded Green function has the same form as the one obtained
in the adiabatic vacuum. This is indeed expected as the retarded correlator is
supposed to be a state independent function. Equipped with these computations,
we turn our focus on the Milne spacetime in the presence of a conical defect.

Field theory on spacetime with a conical defect has been of interest for long
[12] \- [18]. Lately, with the advent of the holographic correspondence,
exploring the properties of a class of strongly coupled field theories on
conical spacetime has become possible [19] \- [22]. For example, this
correspondence can be exploited to compute the correlation functions of
operators in a strongly coupled field theory, admitting large $N$ expansion,
on a spacetime with a conical defect once the gravitational dual is known. As
a simple illustration, let us consider a field theory on a three dimensional
background given by

$ds^{2}=-dT^{2}+dR^{2}+R^{2}d\theta^{2},$ (1.1)

where the angular coordinate $\theta$ has a periodicity of $2\pi/q$. For any
value of $q$ other than $1$, there is a singularity at $R=0$. For $q>1$, to
which we would restrict to, the spacetime has an angle deficit. In the large
$N$ limit of the field theory, the leading contribution to the two point
correlation of a scalar operator would come from examining appropriate scalar
field on the AdS spacetime with the conical singularity. Written in the
Poincare coordinates, the bulk geometry in question is therefore

$ds^{2}=\frac{1}{z^{2}}\Big{[}dz^{2}-dT^{2}+dR^{2}+R^{2}d\theta^{2}\Big{]},$
(1.2)

where $z=\infty$ represents a horizon and $z=0$ is the boundary where the
gauge theory lives. The details of this theory follow from the $M2$ brane of
the M-theory compactified on $S_{7}$. Note that the singularity present on the
boundary now extends for all values of the radial coordinate $z$. The Wightman
function of a minimally coupled massive scalar $\Phi$ of mass $m$ on this
spacetime can be read out from [23]. Denoting the coordinates $(z,T,R,\theta)$
together as $x$ and restricting to the integer values of $q$, it is given by

$G_{+}(x,x^{\prime})=\langle
0|\Phi(x)\Phi(x^{\prime})|0\rangle=-\frac{1}{4\pi^{2}}\sum_{k=0}^{q-1}\frac{Q_{\nu-1/2}^{1}(w_{k})}{\sqrt{w_{k}^{2}-1}},$

where $Q_{\nu-1/2}^{1}(w_{k})$ is the associated Legendre function and

$w_{k}=1+\frac{R^{2}+R^{\prime 2}-2RR^{\prime}~{}{\rm
cos}(\theta-\theta^{\prime}-2\pi
k/q)+(z-z^{\prime})^{2}-(T-T^{\prime})^{2}}{2zz^{\prime}},$

and $\nu={\sqrt{9/4+m^{2}}}$. On the boundary, this field is dual to a scalar
primary operator $\cal{O}$ with scaling dimension $\Delta=\nu+3/2$. The
correlator can then be extracted following the BDHM prescription [24], namely,

$\displaystyle\langle\Psi|{\cal{O}}(T,R,\theta){\cal{O}}(T^{\prime},R^{\prime},\theta^{\prime})|\Psi\rangle$
$\displaystyle=$ $\displaystyle\lim\limits_{z,z^{\prime}\rightarrow
0}(zz^{\prime})^{-(\nu+3/2)}\Big{[}-\frac{1}{4\pi^{2}}\sum_{k=0}^{q-1}\frac{Q_{\nu-1/2}^{1}(w_{k})}{\sqrt{u_{k}^{2}-1}}\Big{]}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{q-1}\frac{C_{\Delta}}{[-(T-T^{\prime})^{2}+R^{2}+R^{\prime
2}-2RR^{\prime}~{}{\rm cos}(\theta-\theta^{\prime}-2\pi k/q)]^{\Delta}}$

where

$C_{\Delta}=\frac{2^{2\Delta-4}\Gamma(\Delta)\Gamma(\Delta-1)}{\pi^{2}\Gamma(2\Delta-2)}.$

Here $|\Psi\rangle$ is the appropriate boundary state of the field theory with
the conformal symmetry broken by the defect. Though the sum above can be
performed, written in this way, the correlator has an interpretation in terms
of the sum over images.

The metric (1.2) can arise due to the presence of a cosmic string. In the weak
field approximation and in the thin string limit, the parameter $q$ gets
related to the mass density of the string. If we wish to model the formation
of such a defect and examine the particle creation during the formation of the
defect, we need to go beyond this static spacetime and replace it by an
appropriate dynamical one [25, 26]. One possibility could be to consider a
geometry of the form

$ds^{2}=\frac{1}{z^{2}}\Big{[}dz^{2}-dT^{2}+dR^{2}+f(T)R^{2}d\theta^{2}\Big{]},$
(1.4)

where $f(T)=[1-~{}{\rm tanh}(T/T_{0})]/2$, and $\theta$ is a coordinate of
period $2\pi$. It represents a quench of the angular coordinate around $T=0$,
lasting for a time $T_{0}$. One may hope to gain some insights into the
behaviour of the strongly coupled field theory on the boundary by examining
the bulk representing this dynamic cone. Difficulty arises immediately however
due to the absence of an analytical handle. For example, for a generic
function $f(T)$, the Klein-Gordon equation would not admit a separation of
variables of the field. In order that to happen we need to assume that the
field is independent of the coordinate $\theta$. While, even with such a
cylindrical symmetry, it might be interesting to explore the dual field
theory, in this work we take a modest step. Here we intend to study aspects of
strongly coupled gauge theory on the Milne geometry in the presence of a
conical defect 222The possible occurrence of topological defects in the early
universe and various cosmological consequences due to their presence have been
an active area of research in the past, see for example [27]. . With the
choice of the adiabatic vacuum, the correlators of the operators dual to the
massive scalars in the bulk can be computed. The final form turns out to be
similar to the one given in (LABEL:adsc) written in the Milne coordinates but
we work this out explicitly starting with the quantization of minimal massive
scalar in AdS-Milne background. Because of the presence of the defect, on the
boundary we can have twisted scalar operators that are dual to the bulk scalar
fields with twisted boundary conditions (3.19). It follows from [28] and
subsequently from [29] that the twisted scalars can be defined on a non-simply
connected spacetime. These scalars satisfy the same equations of motion as
that of the untwisted scalars but differ in their boundary conditions. We end
our exploration with the computation of the boundary correlators involving the
twisted operators dual to these scalars.

##  2 AdS-Milne

We start out with the computation of the bulk and the boundary correlators in
the AdS-Milne spacetime. This, in turn, will set the stage for a similar, but
more involved, calculation of correlators in AdS-Milne spacetime with a
conical defect. This is analyzed in a subsequent section. Mode expasions of a
massive scalar field in this geometry turn out to be sensitive to the
spacetime dimensions. Therefore we carry out our study both in four and five
dimensions. The later has been provided in the appendix.

###  2.1 $3+1$ dimensional AdS-Milne

In the Poincare coordinates, the AdS-Milne metric in four dimensions takes the
form

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\left(-dt^{2}+t^{2}dr^{2}+t^{2}\sinh^{2}rd\theta^{2}+dz^{2}\right),$
(2.1)

$\theta$ being a periodic coordinate with period $2\pi$. We see that we have a
Milne spacetime for every value of the bulk radial coordinate $z$. The fact
that the AdS-Milne is a subregion or a patch in the AdS can be seen from the
coordinate transformations, $T=t\cosh r,~{}R=t\sinh r$. These transformations
cover only the part $T\geq 0,R\geq 0$ of the AdS. Even though the Poincare AdS
and AdS-Milne written in Poincare coordinates are related by coordinate
transformations, we must be careful while studying field theory in these
backgrounds as corresponding propagators need not be related by similar
coordinate transformations. Therefore, in the holographic set up, we should
independently define the Green functions in each coordinate system.

The equation of motion of a minimally coupled scalar of mass $m$ on AdS-Milne
is

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)-m^{2}\phi=0,$
(2.2)

Written explicitly, it takes the form

$\displaystyle-\frac{2z^{2}}{t}\partial_{t}\phi-z^{2}\partial_{t}^{2}\phi+\frac{z^{2}}{t^{2}}{\rm
coth}r\partial_{r}\phi+\frac{z^{2}}{t^{2}}\partial_{r}^{2}\phi+\frac{z^{2}}{t^{2}{\rm
sinh}^{2}r}\partial_{\theta}^{2}\phi-2z\partial_{z}\phi+z^{2}\partial_{z}^{2}\phi-m^{2}\phi=0.$
(2.3)

For generic values of $\nu=\sqrt{m^{2}+9/4}$, the solutions are either

$\displaystyle\phi_{\lambda\alpha n}(z,t,r,\theta)=C_{\lambda\alpha
n}\Big{[}z^{\frac{3}{2}}J_{\nu}(\lambda
z)\Big{]}\Big{[}\frac{H_{i\alpha}^{(2)}(\lambda
t)}{\sqrt{t}}\Big{]}P^{-n}_{i\alpha-\frac{1}{2}}({\rm cosh}r)e^{in\theta},$
(2.4)

or

$\displaystyle\phi_{\lambda\alpha n}(z,t,r,\theta)=\tilde{C}_{\lambda\alpha
n}\Big{[}z^{\frac{3}{2}}J_{\nu}(\lambda
z)\Big{]}\Big{[}\frac{J_{-i\alpha}(\lambda
t)}{\sqrt{t}}\Big{]}P^{-n}_{i\alpha-\frac{1}{2}}({\rm cosh}r)e^{in\theta}$
(2.5)

In these equations, $J_{\mu}(x),H_{i\alpha}^{(2)}(\lambda t)$ and
$P_{i\alpha-1/2}^{-n}(u)$ are the Bessel function, Hankel function of the
second kind and the associated Legendre function respectively. The constants
$\lambda\geq 0$, $\alpha\geq 0$ and $n$ takes integer values. In writing down
these solutions, we used the boundary conditions that the solutions are
regular at $z=0$ and $r=0$. Since these solutions are normalizable modes, they
are dual to operators of conformal dimension $\Delta=\Delta_{+}=3/2+\nu$.
Looking at the large $t$ behaviour of the Bessel and the Hankel functions, it
is easily seen that that the choice $H_{i\alpha}^{(2)}$ is similar to working
with the Minkowskian positive energy modes. The vacuum defined with respect to
these modes is called the adiabatic vacuum. On the other hand, if we take
$J_{-i\alpha}(\lambda t)$, we are in the conformal vacuum [8]. We first work
with the former. The normalization constants $C_{\lambda\alpha n}$ are
determined by the following condition,

$\displaystyle\int dzdrd\theta{\sqrt{-g}}g^{tt}[\phi_{\lambda\alpha
n}(z,t,r,\theta)\partial_{t}\phi^{*}_{\lambda^{\prime}\alpha^{\prime}n^{\prime}}(z,t,r,\theta)$
$\displaystyle-$
$\displaystyle\phi^{*}_{\lambda^{\prime}\alpha^{\prime}n^{\prime}}(z,t,r,\theta)\partial_{t}\phi_{\lambda\alpha
n}(z,t,r,\theta)]$ (2.6)
$\displaystyle=-i\delta(\lambda-\lambda^{\prime})\delta(\alpha-\alpha^{\prime})\delta_{nn^{\prime}}.$

Using (2.4), we get

$C_{\lambda\alpha n}=i\sqrt{\frac{{\alpha\lambda~{}{\rm
sinh}\pi\alpha}}{2\pi}}~{}\Gamma[i\alpha+1/2+n]\frac{e^{\frac{\pi\alpha}{2}}}{2}.$
(2.7)

To get to this, we have used the property (5.4). The canonical quantization
proceeds by defining the field operator

$\Phi=\sum_{i}[\phi_{i}~{}a_{i}+\phi_{i}^{*}~{}a^{\dagger}_{i}],$

where ${a_{l}}^{\dagger},a_{l}$ are the creation and the annihilation
operators respectively and $i$ includes the set of quantum numbers
$\lambda,\alpha,n$. The summation above represents integrations over
$\lambda,\alpha$ and a sum over $n$.

We start by computing the Wightman function. It is defined as

$G_{+}(z,t,r,\theta;z^{\prime},t^{\prime},r^{\prime},\theta^{\prime})=\langle
0|\Phi(z,t,r,\theta)\Phi(z^{\prime},t^{\prime},r^{\prime},\theta^{\prime})|0\rangle=\sum_{i}\phi_{i}(z,t,r,\theta)\phi_{i}^{*}(z^{\prime},t^{\prime},r^{\prime},\theta^{\prime}).$
(2.8)

Here, $|0\rangle$ includes the Minkowski vacuum for the Milne part. The other
Wightman function can be obtained from the relation
$G_{-}(x,x^{\prime})=G_{+}^{*}(x,x^{\prime})$. Denoting the set of coordinates
$(z,t,r,\theta)$ together as $x$, we get

$\displaystyle G_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}d\lambda
d\alpha\phi(z,r,t,\theta)\phi^{*}(z^{\prime},r^{\prime},t^{\prime},\theta^{\prime})=\frac{1}{4}\int
d\lambda
d\alpha\Big{(}\frac{(zz^{\prime})^{\frac{3}{2}}}{\sqrt{tt^{\prime}}}\alpha\lambda~{}{\rm
sinh}(\pi\alpha)e^{\pi\alpha}\Big{)}$ $\displaystyle\times$ $\displaystyle
J_{\nu}(\lambda z)J_{\nu}(\lambda z^{\prime})H_{i\alpha}^{(2)}(\lambda
t)H_{-i\alpha}^{(1)}(\lambda
t^{\prime})\sum_{n=-\infty}^{\infty}\Gamma[i\alpha+1/2+n]\Gamma[-i\alpha+1/2+n]$
$\displaystyle\times$ $\displaystyle
P_{i\alpha-1/2}^{-n}({\rm{cosh}}r)P_{i\alpha-1/2}^{-n}({\rm{cosh}}r^{\prime})e^{in(\theta-\theta^{\prime})}$
$\displaystyle=$
$\displaystyle\frac{(zz^{\prime})^{\frac{3}{2}}}{\pi\sqrt{tt^{\prime}}}\int_{0}^{\infty}d\lambda
d\alpha\alpha\lambda~{}{\rm tanh}(\pi\alpha)J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})K_{i\alpha}(i\lambda t)K_{i\alpha}(-i\lambda
t^{\prime})P_{i\alpha-1/2}({\rm cosh}\chi).$

The identities used to reach here have been presented in the appendix, see
(5.5). In the above, $K_{\nu}(z)$ is the modified Bessel function and $\chi$
is defined as

$\displaystyle{\rm cosh}\chi=~{}{\rm cosh}r~{}{\rm cosh}r^{\prime}-~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}~{}{\rm cos}(\theta-\theta^{\prime}).$ (2.10)

Equation (LABEL:midexp) can be further simplified using (5.6) to arraive at

$\displaystyle G_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\pi{\sqrt{2\pi}}}(zz^{\prime})^{\frac{3}{2}}\int_{0}^{\infty}d\lambda\lambda^{\frac{3}{2}}J_{\nu}(\lambda
z)J_{\nu}(\lambda
z^{\prime})\frac{K_{\frac{1}{2}}(\lambda{\sqrt{-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi}})}{{\sqrt{-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi}}}$ (2.11) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi^{2}(u^{2}-1)^{\frac{1}{2}}}Q_{\nu-1/2}^{1}(u),$

where

$u=\frac{1}{2zz^{\prime}}\Big{(}-t^{2}-t^{\prime 2}+2tt^{\prime}~{}{\rm
cosh}\chi+z^{2}+z^{\prime 2}+i\epsilon~{}{\rm sgn}(t-t^{\prime})\Big{)},$
(2.12)

the geodesic distance between two points in AdS-Milne spacetime and
$Q_{\nu-1/2}^{1}(u)$ is the associated Legendre function. To arrive at the
last line of (2.11), we have used a result from [34]. In the conformal limit,
when the mass $m^{2}=-2$ (and consequently $\nu=1/2$), the correlator
simplifies. Owing to the property that $Q_{0}^{1}=-1/\sqrt{u^{2}-1}$, we get
333We have calculated the Wightman function for normalizable modes that is
$G_{\Delta_{+}}$. There is a corresponding Wightman function $G_{\Delta_{-}}$
for the non-normalizable modes As shown in [36], it’s the sum
$G_{\Delta_{+}}+G_{\Delta_{-}}$ which corresponds to a simple power solution
in the conformal limit .

$G_{+}(x,x^{\prime})=\frac{1}{4\pi^{2}(u^{2}-1)}.$

Boundary correlator

Having obtained the bulk Wightman function, we can use the BDHM prescription
to construct the boundary correlator.

$\lim\limits_{z,z^{\prime}\rightarrow
0}(zz^{\prime})^{-(\nu+3/2)}\langle\Phi(z,r,t,\theta)\Phi(z^{\prime},r^{\prime},t^{\prime},\theta^{\prime})\rangle_{\rm
AdS-
Milne}=\langle\Psi|{\cal{O}}(r,t,\theta){\cal{O}}(r^{\prime},t^{\prime},\theta^{\prime})|\Psi\rangle.$

Here ${\cal{O}}$ is the scalar primary operator of dimension $\Delta=3/2+\nu$,
dual to the bulk scalar $\Phi$. On the left the subscript indicates that the
bulk computation is done on AdS-Milne, and on the right, $|\Psi\rangle$ is the
corresponding state of the boundary conformal theory. Implementing the above,
we get

$\displaystyle\langle\Psi|{\cal{O}}(t,r,\theta){\cal{O}}(t^{\prime},r^{\prime},\theta^{\prime})|\Psi\rangle$
$\displaystyle=$ $\displaystyle\lim\limits_{z,z^{\prime}\rightarrow
0}(zz^{\prime})^{-(\nu+3/2)}G_{+}(x,x^{\prime})$ (2.13) $\displaystyle=$
$\displaystyle\frac{(\Delta-2)2^{\Delta}}{4\pi^{2}(-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi)^{\Delta}}\int_{0}^{\infty}\frac{{\rm
cosh}\beta}{(1+{\rm cosh}\beta)^{\Delta-1}}d\beta$ $\displaystyle=$
$\displaystyle\frac{C_{\Delta}}{(-t^{2}-t^{\prime 2}+2tt^{\prime}~{}{\rm
cosh}\chi)^{\Delta}},$

where

$C_{\Delta}=\frac{2^{2\Delta-4}\Gamma(\Delta)\Gamma(\Delta-1)}{\pi^{2}\Gamma(2\Delta-2)}.$
(2.14)

To arrive at the final line, we have used (5.7).

A similar computation can be carried out for the AdS-Milne spacetime in five
dimensions where the metric is

$ds^{2}=\frac{1}{z^{2}}\left[-dt^{2}+t^{2}dr^{2}+t^{2}\sinh^{2}rd\theta^{2}+t^{2}\sinh^{2}r\sin^{2}\theta
d\phi^{2}+dz^{2}\right].$ (2.15)

The details of the computation is provided in the appendix. The Wightman
function turns out to be

$\displaystyle
G_{+}(x,x^{\prime})=\frac{i}{(2\pi)^{5/2}(u^{2}-1)^{3/4}}Q_{\nu-1/2}^{3/2}(u),$
(2.16)

where

$\displaystyle u=\frac{1}{2zz^{\prime}}(-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\gamma+z^{2}+z^{\prime 2}),$

leading to the boundary correlator

$\displaystyle\langle\Psi|\mathcal{O}(t,r,\theta,\phi),\mathcal{O}(t^{\prime},r^{\prime},\theta^{\prime},\phi^{\prime})|\Psi\rangle$
$\displaystyle=$
$\displaystyle\frac{2^{2\nu-1}\Gamma(\frac{1}{2}+\nu)\Gamma(\nu+2)}{(\pi)^{\frac{5}{2}}\Gamma(2\nu+1)}\frac{1}{a^{\nu+2}}$
$\displaystyle=$
$\displaystyle\frac{2^{2\nu-1}\Gamma(\frac{1}{2}+\nu)\Gamma(\nu+2)}{(\pi)^{\frac{5}{2}}\Gamma(2\nu+1)}\frac{1}{(-t^{2}-t^{\prime
2}+2tt^{\prime}\cosh\gamma)^{\nu+2}}.$

Before proceeding to the next subsection, we end with the following note. In a
new coordinate system $\tau,\rho$, defined as $t=be^{\tau/b}$ and $r=\rho/b$
with $b$ being constant, the metric in (2.1) becomes

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\Big{[}dz^{2}+e^{2\tau/b}\Big{(}-d\tau^{2}+d\rho^{2}+b^{2}~{}{\rm
sinh}^{2}(\rho/b)d\theta^{2}\Big{)}\Big{]}.$

Now, in the limit $b\rightarrow\infty$, the boundary becomes flat. The
quantity $u$ defined in (2.12) reduces to
$u=\frac{1}{2zz^{\prime}}[-(\tau-\tau^{\prime})^{2}+\rho^{2}+\rho^{\prime
2}-2\rho\rho^{\prime}~{}{\rm cos}(\theta-\theta^{\prime})]$ and the boundary
correlator becomes the one in the Minkowski spacetime.

###  2.2 Comparison with Poincare-AdS

Since the metric as well as the vacuum state are invariant under full
Poincare-AdS symmetries, the final expression can be written in terms of the
geodesic distance, which facilitates a comparison with the Green functions in
Poincare-AdS.

For AdS-Milne in 3+1 dimensions we get the geodesic distance
$d(z,w)=\int_{z}^{w}ds$ as

$\displaystyle
d(z_{1},t_{1},r_{1},\theta_{1};z_{2},t_{2},r_{2},\theta_{2})=\ln\frac{1+\sqrt{1-\xi^{2}}}{\xi},$

where $\xi$ is given by

$\displaystyle\xi=\frac{2z_{1}z_{2}}{z_{1}^{2}+z_{2}^{2}-t_{1}^{2}-t_{2}^{2}+2t_{1}t_{2}\cosh\chi}=\frac{1}{u}.$

Using the relation between the hypergeometric function and associated Legendre
function of second kind, namely

$\displaystyle Q_{\nu}^{\mu}(z)$ $\displaystyle=$
$\displaystyle\frac{e^{\mu\pi
i}\Gamma(\nu+\mu+1)\Gamma(1/2)}{2^{\nu+1}\Gamma(\nu+3/2)}(z^{2}-1)^{\mu/2}z^{-\nu-\mu-1}~{}F\left(\frac{\nu+\mu+1}{2},\frac{\nu+\mu+2}{2};\nu+3/2;\frac{1}{z^{2}}\right),$

we find,

$\displaystyle Q_{\nu-1/2}^{1}\left(\frac{1}{\xi}\right)$ $\displaystyle=$
$\displaystyle-4\pi^{2}\left(\frac{1}{\xi^{2}}-1\right)^{1/2}G_{\nu+\frac{3}{2}}(\xi),$

where

$\displaystyle
G_{\nu+\frac{3}{2}}(\xi)=\frac{2^{-\nu-5/2}\Gamma\left(\nu+3/2\right)}{\pi^{3/2}\Gamma(\nu+1)}\xi^{\nu+3/2}F\left(\frac{3}{4}+\frac{\nu}{2},\frac{5}{4}+\frac{\nu}{2};\nu+1;\xi^{2}\right)$

is the scalar propagator for the normalizable modes in Poincare-AdS as given
in [35].

For AdS-Milne in 4+1 dimensions we can write the geodesic distance in a
similar way

$\displaystyle
d(z_{1},t_{1},r_{1},\theta_{1},\phi_{1};z_{2},t_{2},r_{2},\theta_{2},\phi_{2})=\ln\frac{1+\sqrt{1-\xi^{2}}}{\xi},$

where $\xi$ is now given by

$\displaystyle\xi=\frac{2z_{1}z_{2}}{z_{1}^{2}+z_{2}^{2}-t_{1}^{2}-t_{2}^{2}+2t_{1}t_{2}\cosh\chi}=\frac{1}{u}.$

Again, we can convert from associated Legendre functions to hypergeometric
function

$\displaystyle
Q_{\nu-1/2}^{3/2}\left(\frac{1}{\xi}\right)=-i(2\pi)^{5/2}\left(\frac{1}{\xi^{2}}-1\right)^{3/4}G_{\nu+2}(\xi),$

where

$\displaystyle G_{\nu+2}(\xi)$ $\displaystyle=$
$\displaystyle\frac{2^{-\nu-3}\Gamma\left(\nu+2\right)}{\pi^{2}\Gamma(\nu+1)}\xi^{\nu+2}F\left(1+\frac{\nu}{2},\frac{3}{2}+\frac{\nu}{2};\nu+1;\xi^{2}\right)$

is the scalar Green function given in [35].

###  2.3 AdS-Milne in conformal vacuum

As we discussed previously, working in conformal vacuum is equivalent to
choosing $J_{-i\alpha}(\lambda t)$ in the $t$ part instead of the Hankel
function. The basis of the mode expansion is therefore given by (2.5) rather
than (2.4)

$\displaystyle\phi(z,t,r,\theta)=\tilde{C}_{\lambda\alpha
n}\left[z^{3/2}J_{\nu}(\lambda z)\right]\left[\frac{J_{-i\alpha}(\lambda
t)}{\sqrt{t}}\right]P_{i\alpha-1/2}^{-n}(\cosh r)e^{in\theta},$ (2.18)

where the normalization constant, upto a constant phase factor, is given by

$\displaystyle\tilde{C}_{\lambda\alpha
n}=i\sqrt{\frac{\alpha\lambda}{4\pi}}~{}\Gamma\left[i\alpha+1/2+n\right].$

Now doing manipulations similar to what was done in the Minkowski vacuum, we
arrive at

$\displaystyle G_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int d\lambda
d\alpha\left[\frac{(zz^{\prime})^{3/2}}{\sqrt{tt^{\prime}}}\alpha\lambda\right]J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})$ (2.19)
$\displaystyle\sum_{n=-\infty}^{+\infty}\Gamma(i\alpha+1/2+n)\Gamma(-i\alpha+1/2+n)P_{i\alpha-1/2}^{-n}(\cosh
r)P_{i\alpha-1/2}^{-n}(\cosh r^{\prime})e^{in(\theta-\theta^{\prime})}$
$\displaystyle=$
$\displaystyle\frac{(zz^{\prime})^{3/2}}{4\sqrt{tt^{\prime}}}\int d\lambda
d\alpha\ \alpha\lambda J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})\frac{1}{\cosh\pi\alpha}P_{i\alpha-1/2}(\cosh\chi)$

where $\chi$ is defined by 2.10. The other Wightman function is given by
$G_{-}(x,x^{\prime})=G^{*}_{+}(x,x^{\prime})$.

In $4+1$ dimensions, the right solution of the scalar equation of motion turns
out to be

$\displaystyle\phi(z,t,r,\theta,\phi)=\tilde{C}_{\lambda\alpha}\left[z^{2}J_{\nu}(\lambda
z)\right]\left[\frac{J_{-i\alpha}(\lambda t)}{t}\right]Y_{\alpha
lm}(r,\theta,\phi)$

where the normalization constant is given by

$\displaystyle\tilde{C}_{\lambda\alpha}=\sqrt{\frac{\pi\lambda}{2\sinh\pi\alpha}}.$

The Wightman function is then

$\displaystyle G_{+}(x,x^{\prime})=\sum_{l,m}\int_{0}^{\infty}\lambda.d\lambda
d\alpha\frac{\pi}{2}\frac{(zz^{\prime})^{2}}{(tt^{\prime})^{2}\sinh\pi\alpha}J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})Y_{\alpha lm}(r,\theta,\phi)Y_{\alpha
lm}(r^{\prime},\theta^{\prime},\phi^{\prime}).$

As was done for the Minkowski vacuum, we can use the completeness relation of
the spherical harmonics (5.2) to cast the above function as

$\displaystyle G_{+}(x,x^{\prime})=\int_{0}^{\infty}\lambda d\lambda
d\alpha\frac{1}{4\pi}\frac{(zz^{\prime})^{2}}{(tt^{\prime})^{2}\sinh\pi\alpha}J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})\frac{\alpha\sin\alpha\gamma}{2\pi^{2}\sinh\gamma}.$

where $\gamma$ is given, as before, by (5.3). Unfortunately, we have not
succeeded in carrying out all the integrals in (2.19) and (2.3). However, we
could make some progress while considering the retarded propagator. This is
given by $G_{R}(x,x^{\prime})=\theta(t-t^{\prime})G(x,x^{\prime})$ where
$G(x,x^{\prime})=G_{+}(x,x^{\prime})-G_{-}(x,x^{\prime})$, difference between
the positive and the negative frequency Wightman functions.

To explicitly calculate the retarded propagator, we first consider the $3+1$
dimensional AdS-Milne spacetime in conformal vacuum. In this case, we are able
to write

$\displaystyle G(x,x^{\prime})=\int d\lambda
d\alpha\frac{(zz^{\prime})^{3/2}}{4\sqrt{tt^{\prime}}}J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})\Big{(}J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})-J_{-i\alpha}(\lambda t^{\prime})J_{i\alpha}(\lambda
t)\Big{)}\frac{\alpha\lambda P_{i\alpha-1/2}(\cosh\chi)}{\cosh\pi\alpha}.$

We can straightforwardly show that the $G(x,x^{\prime})$ in this vacuum is
same as that in the Minkowski vacuum. To this end, we first write the Bessel
function in terms of the Hankel function using
$J_{\alpha}(x)=\frac{1}{2}\left(H^{(1)}_{\alpha}(x)+H^{(2)}_{\alpha}(x)\right)$.
Further, since $H^{(1,2)}_{-\alpha}=e^{\pm i\pi\alpha}H^{(1,2)}_{\alpha}$, we
can then write

$\displaystyle 4\left[J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})-J_{-i\alpha}(\lambda t^{\prime})J_{i\alpha}(\lambda
t)\right]=\sinh\pi\alpha\left(H^{(2)}_{i\alpha}(\lambda
t)H^{(1)}_{i\alpha}(\lambda t^{\prime})-H^{(2)}_{i\alpha}(\lambda
t^{\prime})H^{(1)}_{i\alpha}(\lambda t)\right).$ (2.20)

Therefore,

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$ $\displaystyle\int d\lambda
d\alpha\frac{(zz^{\prime})^{3/2}}{\sqrt{tt^{\prime}}}\alpha\lambda
J_{\nu}(\lambda z)J_{\nu}(\lambda z^{\prime})\times$
$\displaystyle\tanh\pi\alpha
P_{i\alpha-1/2}(\cosh\chi)\left(H^{(2)}_{i\alpha}(\lambda
t)H^{(1)}_{i\alpha}(\lambda t^{\prime})-H^{(2)}_{i\alpha}(\lambda
t^{\prime})H^{(1)}_{i\alpha}(\lambda t)\right).$

This is same as the one we get for the Minkowski vacuum. Likewise, in $4+1$
dimensions, in the Minkowski vacuum $G(x,x^{\prime})$ is

$\displaystyle
G(x,x^{\prime})=\frac{\pi(zz^{\prime})^{2}}{4tt^{\prime}}\int\lambda d\lambda
d\alpha J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})\left(H^{(2)}_{i\alpha}(\lambda t)H^{(1)}_{i\alpha}(\lambda
t^{\prime})-H^{(2)}_{i\alpha}(\lambda t^{\prime})H^{(1)}_{i\alpha}(\lambda
t)\right)\frac{\alpha\sin\alpha\xi}{2\pi^{2}\sinh\xi},$

and for conformal vacuum, the corresponding expression is

$\displaystyle
G(x,x^{\prime})=\frac{\pi(zz^{\prime})^{2}}{4tt^{\prime}}\int\lambda d\lambda
d\alpha J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})\left(J_{-i\alpha}(\lambda t)J_{i\alpha}(\lambda
t^{\prime})-J_{i\alpha}(\lambda t)J_{-i\alpha}(\lambda
t^{\prime})\right)\frac{\alpha\sin\alpha\gamma}{2\pi^{2}\sinh\pi\alpha\sinh\gamma}.$

Using (2.20), we immediately see that the last two expressions coincide. This
is not surprising. $G(x,x^{\prime})$, also known as the Pauli-Jordan function,
arises from a commutator of the fields and is supposed to be a state
independent function.

We can also give a general check that the retarded propagators or the Pauli-
Jordan functions are same in both the vacua. We start with

$G^{+}(x,y)=\sum_{i}\phi_{i}^{*}(x)\phi_{i}(y)\ \ ,\ \
G^{-}(x,y)=\sum_{i}\phi_{i}^{*}(y)\phi_{i}(x)$

Here $\phi_{i}$ are modes calculated in one vacuum (e.g. Minkowski vacuum) and
summation represents the integration over all continuous indices and the sum
over the discrete indices. Using the Bogoliubov coefficients, we can express
one set of modes $\phi_{i}$ in terms of other set $\tilde{\phi}$ as

$\phi_{i}(y)=\sum_{j}\left(A^{*}_{ji}\tilde{\phi}_{j}(y)-B_{ji}\tilde{\phi}_{j}^{*}(y)\right),\
\
\phi^{*}_{i}(x)=\sum_{k}\left(A_{ki}\tilde{\phi}^{*}_{k}(x)-B^{*}_{ki}\tilde{\phi}_{k}(x)\right),$
(2.21)

where Bogoliubov coefficients satisfy [8]

$\sum_{k}\left(A_{ik}A^{*}_{jk}-B_{ik}B^{*}_{jk}\right)=\delta_{ij},\ \
\sum_{k}\left(A_{ik}B_{jk}-B_{ik}A_{jk}\right)=0.$ (2.22)

It is easy to check that Bogoliubov coefficients in our case indeed satisfy
these identities. 444 In $3+1$ dimensions, we see that the transformation
equation is  $\tilde{\phi}_{\lambda\alpha
n}=\sum_{\lambda^{\prime}\alpha^{\prime}n^{\prime}}\left(A_{\lambda\alpha
n\lambda^{\prime}\alpha^{\prime}n^{\prime}}\phi_{\lambda^{\prime}\alpha^{\prime}n^{\prime}}+B_{\lambda\alpha
n\lambda^{\prime}\alpha^{\prime}n^{\prime}}\phi^{*}_{\lambda^{\prime}\alpha^{\prime}n^{\prime}}\right).$
Using 2.4 and 2.5 we see that the Bogoliubov coefficients are
$A_{\lambda\alpha
n\lambda^{\prime}\alpha^{\prime}n^{\prime}}=\delta_{\lambda\lambda^{\prime}}\delta_{nn^{\prime}}\delta_{\alpha\alpha^{\prime}}c(\alpha)\
\ ,\ \ B_{\lambda\alpha
n\lambda^{\prime}\alpha^{\prime}n^{\prime}}=\delta_{\lambda\lambda^{\prime}}\delta_{n,-n^{\prime}}\delta_{\alpha\alpha^{\prime}}d(\alpha).$
Using the relationship between Hankel and Bessel functions, along with
$H^{(2)}_{i\alpha^{\prime}}=e^{-\pi\alpha^{\prime}}H^{(2)}_{-i\alpha^{\prime}}$,
we get  $c(\alpha)=\frac{e^{\frac{\pi\alpha}{2}}}{\sqrt{2\sinh\pi\alpha}},\ \
d(\alpha)=-\frac{e^{-\frac{\pi\alpha}{2}}}{\sqrt{2\sinh\pi\alpha}}.$ Further,
putting the expansion (2.21) in the Wightman functions, we get

$G_{+}(x,y)=\sum_{i,j,k}\left(A_{ki}A^{*}_{ji}\tilde{\phi}_{k}^{*}(x)\tilde{\phi}_{j}(y)+B_{ji}B^{*}_{ki}\tilde{\phi}_{j}^{*}(y)\tilde{\phi}_{k}(x)-A_{ki}B_{ji}\tilde{\phi}_{k}^{*}(x)\tilde{\phi}^{*}_{j}(y)-B^{*}_{ki}A^{*}_{ji}\tilde{\phi}_{k}(x)\tilde{\phi}_{j}(y)\right).$

and

$G_{-}(x,y)=\sum_{i,j,k}\left(A_{ki}A^{*}_{ji}\tilde{\phi}_{k}^{*}(y)\tilde{\phi}_{j}(x)+B_{ji}B^{*}_{ki}\tilde{\phi}_{j}^{*}(x)\tilde{\phi}_{k}(y)-A_{ki}B_{ji}\tilde{\phi}_{k}^{*}(y)\tilde{\phi}^{*}_{j}(x)-B^{*}_{ki}A^{*}_{ji}\tilde{\phi}_{k}(y)\tilde{\phi}_{j}(x)\right).$

Subtracting the above two expressions and using (2.22), we get the Pauli-
Jordan function $G(x,y)$.

$G(x,y)=\sum_{i}\left(\phi_{i}^{*}(x)\phi_{i}(y)-\phi_{i}^{*}(y)\phi_{i}(x)\right)=\sum_{i}\left(\tilde{\phi}_{i}^{*}(x)\tilde{\phi}_{i}(y)-\tilde{\phi}_{i}^{*}(y)\tilde{\phi}_{i}(x)\right).$

Equipped now with these results, in the next section, we analyse scalar field
theory on the AdS-Milne spacetime containing a conical defect. Here this
defect runs all the way to the boundary. Our primary aim would be to find
boundary two point correlator of operators dual to the bulk field in a closed
form.

###  2.4 AdS-Milne in other vacua

So far we have focused on the adiabatic and the conformal vacua. Recently, in
[10], a more general class of vacua, similar to the alpha-vacua of the de
Sitter spacetime, was considered. We end this section with a computation of
the Wightman function in these vacua. Our result is similar to the one of
[10], namely, the Wightman function picks up a dependence on the coordinates
in a non-Poincare invariant manner.

To proceed, we start with the general solution of (2.3)

$\displaystyle\phi_{\lambda\alpha
n}(z,t,r,\theta)=\Big{[}z^{\frac{3}{2}}J_{\nu}(\lambda
z)\Big{]}\Big{[}\frac{C_{\lambda\alpha n}H_{i\alpha}^{(2)}(\lambda
t)+\tilde{C}_{\lambda\alpha n}H_{i\alpha}^{(1)}(\lambda
t^{\prime})}{\sqrt{t}}\Big{]}P^{-n}_{i\alpha-\frac{1}{2}}({\rm
cosh}r)e^{in\theta}.$ (2.23)

The normalization condition (2.1) for the above solution gives

$|C_{\lambda\alpha n}|^{2}e^{-\pi\alpha}-|\tilde{C}_{\lambda\alpha
n}|^{2}e^{\pi\alpha}=-\frac{\alpha\lambda}{8\pi}|\Gamma(n+\frac{1}{2}+i\alpha)|^{2}\sinh\pi\alpha.$
(2.24)

It can be checked that the choice

$C_{\lambda\alpha
n}=\frac{i}{2}\sqrt{\frac{\alpha\lambda}{4\pi}}e^{\pi\alpha}\Gamma(i\alpha+\frac{1}{2}+n)\
\ ,\ \tilde{C}_{\lambda\alpha
n}=\frac{i}{2}\sqrt{\frac{\alpha\lambda}{4\pi}}e^{-\pi\alpha}\Gamma(i\alpha+\frac{1}{2}+n)$
(2.25)

gives the solution (2.18) in the conformal vacuum with the correct
normalization factor. Related to the adiabatic modes is a two parameter family
of modes, called alpha-modes. We can write the normalization constants of
these in terms of a two-parameter family of constants labelled by
$\rho,\sigma$ as

$\displaystyle C_{\lambda\alpha
n}=i\cosh\rho\sqrt{\frac{\alpha\lambda\sinh\pi\alpha}{8\pi}}e^{\frac{\pi\alpha}{2}}\Gamma(i\alpha+\frac{1}{2}+n)$
$\displaystyle\tilde{C}_{\lambda\alpha n}=i\sinh\rho\
e^{i\sigma}\sqrt{\frac{\alpha\lambda\sinh\pi\alpha}{8\pi}}e^{-\frac{\pi\alpha}{2}}\Gamma(i\alpha+\frac{1}{2}+n)\sqrt{\sinh\pi\alpha}$
(2.26)

For $\rho,\sigma$ set to zero, we get back the adiabatic modes (2.4) with the
correct normalization constant. The above normalization constants satisfy the
relation (2.24) . Since $\rho,\sigma$ are constants, independent of $\alpha$,
we can see that this family does not contain the conformal modes given by
(2.25) . We can calculate the Wightman function for these alpha-modes by
putting (2.23) in the relation (2.8) along with the normalization constants in
(2.26) and using the appropriate alpha vacua states. The Wightman function for
the field in the alpha vacua then comes out as

$\displaystyle G_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{\cosh^{2}\rho\
Q_{\nu-1/2}^{1}(u_{1})}{4\pi^{2}(u_{1}^{2}-1)^{\frac{1}{2}}}-\frac{\sinh^{2}\rho\
Q_{\nu-1/2}^{1}(u_{2})}{4\pi^{2}(u_{2}^{2}-1)^{\frac{1}{2}}}+i\sinh\rho\cosh\rho\frac{e^{-i\sigma}\
Q_{\nu-1/2}^{1}(u_{3})}{4\pi^{2}(u_{3}^{2}-1)^{\frac{1}{2}}}$ $\displaystyle+$
$\displaystyle\frac{i\sinh\rho\cosh\rho~{}e^{i\sigma}\
Q_{\nu-1/2}^{1}(u_{4})}{4\pi^{2}(u_{4}^{2}-1)^{\frac{1}{2}}},$

where

$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2zz^{\prime}}\Big{(}-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi+z^{2}+z^{\prime 2}+i\epsilon~{}{\rm
sgn}(t-t^{\prime})\Big{)},$ $\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2zz^{\prime}}\Big{(}-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi+z^{2}+z^{\prime 2}-i\epsilon~{}{\rm
sgn}(t-t^{\prime})\Big{)},$ $\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2zz^{\prime}}\Big{(}-t^{2}-t^{\prime
2}-2tt^{\prime}~{}{\rm cosh}\chi+z^{2}+z^{\prime 2}+i\epsilon\Big{)},$
$\displaystyle u_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2zz^{\prime}}\Big{(}-t^{2}-t^{\prime
2}-2tt^{\prime}~{}{\rm cosh}\chi+z^{2}+z^{\prime 2}-i\epsilon)\Big{)}.$

First term is just $\cosh^{2}\rho$ times the Wightman function in the
adiabatic vacuum (2.11). Second term is $\sinh^{2}\rho$ times the Wightman
function in the adiabatic vacuum, but with $t\leftrightarrow t^{\prime}$.
Other two terms are not Poincare invariant and are analogous to the dependence
on antipodal distance as discussed in [10].

##  3 AdS-Milne with a conical defect

AdS-Milne background in the presence of a conical defect has the same form as
before except that the the angular coordinate has a different periodicity. In
$3+1$ dimensions, the metric is given by

$ds^{2}=\frac{1}{z^{2}}\Big{[}dz^{2}-dt^{2}+t^{2}dr^{2}+t^{2}~{}{\rm
sinh^{2}r}~{}d\theta^{2}\Big{]},$ (3.1)

where $0\leq\theta<2\pi/q$. We will take $q>1$.

The above geometry may arise in the presence of an infinite string in a four
dimensional AdS-Milne spacetime. This can be seen by closely following the
treatment of [22]. Consider the Nambu-Goto action

$S_{NG}=-\mu\int d\sigma_{0}d\sigma_{1}\sqrt{-detP[g_{ab}]},\ \
P[g_{ab}]=g_{\mu\nu}(X)\frac{\partial
X^{\mu}(\sigma)}{\partial\sigma^{a}}\frac{\partial
X^{\nu}(\sigma)}{\partial\sigma^{b}}.$ (3.2)

Here $\mu$ is the tension associated with the string. We choose a gauge such
that $\sigma^{0},\sigma^{1}=t,z$ and consider the string extended along the
$z$ direction. Embedding coordinates will then be
$X^{\mu}(t,z)=\left(t,z,r(t,z),\theta(t,z)\right)$. In general, we can vary
Nambu-Goto action with respect to embedding coordinates to get the equations
of motion. But we are interested in a particular solution corresponding to
string at the origin of $(r,\theta)$ plane, represented by a delta-function
source $\delta(r)$.

Corresponding Nambu-Goto action and its variation with respect to the metric
is given by

$S=-\frac{\mu}{\pi}\int dtdzdrd\theta\sqrt{-g_{tt}g_{zz}}\delta(r),\ \ \delta
S=\frac{\mu}{2\pi}\int dtdzdrd\theta\left(\frac{g_{zz}\delta
g_{tt}+g_{tt}\delta g_{zz}}{\sqrt{-g_{tt}g_{zz}}}\right)\delta(r).$ (3.3)

Comparing this with the general relation between metric variation and stress
tensor

$\delta S=\frac{1}{2}\int dtdzdrd\theta\sqrt{-g}T^{\mu\nu}\delta g_{\mu\nu},$
(3.4)

we can get read off the stress tensor corresponding to the string
configuration. Equivalently, we can do a coordinate transformation of the
result given in [22]. We get

$T^{\mu\nu}=-\frac{\mu\delta(r)t}{\pi\sqrt{g_{rr}g_{\theta\theta}}}\begin{pmatrix}g_{tt}^{-1}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&g_{zz}^{-1}\end{pmatrix}.$ (3.5)

Taking this as the stress tensor for the string, we solve the coupled
Einstein-Hilbert and the Nambu-Goto action

$S=\frac{1}{16\pi G_{4}}\int
d^{4}x\sqrt{-g}\left(R+\frac{6}{L^{2}}\right)-\mu\int
d\sigma_{0}d\sigma_{1}\sqrt{-detP[g_{ab}]}.$ (3.6)

Variation of the above leads to

$R^{\mu\nu}-\frac{R}{2}g^{\mu\nu}-\frac{3}{L^{2}}g^{\mu\nu}=8\pi
G_{4}T^{\mu\nu}.$ (3.7)

Solving this system of equations we get the metric

$ds^{2}=\frac{1}{z^{2}}\Big{[}dz^{2}-dt^{2}+t^{2}dr^{2}+t^{2}~{}{\rm
sinh^{2}r}~{}d\theta^{2}\Big{]},$ (3.8)

with $0\leq t,r\leq\infty$ and $0\leq\theta\leq 2\pi(1-4\mu G_{4})$. This is
same as (3.1) once we identify $q=(1-4\mu G_{4})$.

We now turn to the scalar propagator on this geometry in the Minkowski vacuum.
In the following we will primarily restrict $q$ to be an integer. The scalar
field now needs to satisfy $\phi(z,t,r,\theta)=\phi(z,t,r,\theta+2\pi/q)$.
Solving the equation of motion, we find

$\displaystyle\phi(z,t,r,\theta)=\frac{i}{2}\sqrt{\frac{q\alpha\lambda~{}{\rm
sinh}\pi\alpha}{2\pi}}\Gamma(i\alpha+\frac{1}{2}+qn)e^{\pi\alpha/2}\Big{[}z^{\frac{3}{2}}J_{\nu}(\lambda
z)\Big{]}\Big{[}\frac{H_{i\alpha}^{(2)}(\lambda
t)}{\sqrt{t}}\Big{]}P^{-qn}_{i\alpha-\frac{1}{2}}({\rm cosh}r)e^{iqn\theta}.$

Therefore, now the Wightman function takes the form

$\displaystyle G^{q}_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(zz^{\prime})^{\frac{3}{2}}}{8\pi\sqrt{tt^{\prime}}}\int\sum_{n=-\infty}^{\infty}\lambda~{}d\lambda~{}\alpha~{}d\alpha~{}e^{\pi\alpha}~{}{\rm
sinh}(\pi\alpha)J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})H_{i\alpha}^{(2)}(\lambda t)H_{-i\alpha}^{(1)}(\lambda t^{\prime})$
$\displaystyle\times$
$\displaystyle|\Gamma(i\alpha+1/2+qn)|^{2}P_{i\alpha-1/2}^{-qn}({\rm
cosh}r)P_{i\alpha-1/2}^{-qn}({\rm
cosh}r^{\prime})e^{iqn(\theta-\theta^{\prime})}.$

Further, using

$\displaystyle H_{i\alpha}^{(2)}(\lambda t)H_{-i\alpha}^{(1)}(\lambda
t^{\prime})=\frac{4}{\pi^{2}}e^{-\pi\alpha}K_{i\alpha}(i\lambda
t)K_{i\alpha}(-i\lambda t^{\prime}),$

we can rewrite

$\displaystyle G^{q}_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(zz^{\prime})^{\frac{3}{2}}}{2\pi^{3}\sqrt{tt^{\prime}}}\int\sum_{n=-\infty}^{\infty}\lambda~{}d\lambda~{}\alpha~{}d\alpha~{}e^{\pi\alpha}~{}{\rm
sinh}(\pi\alpha)J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})K_{i\alpha}(i\lambda t)K_{i\alpha}(-i\lambda t^{\prime})$ (3.9)
$\displaystyle\times$
$\displaystyle|\Gamma(i\alpha+1/2+qn)|^{2}P_{i\alpha-1/2}^{-qn}({\rm
cosh}r)P_{i\alpha-1/2}^{-qn}({\rm
cosh}r^{\prime})e^{iqn(\theta-\theta^{\prime})}.$

Using (5.8), we can simplify (3.9) as,

$\displaystyle G^{q}_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(zz^{\prime})^{\frac{3}{2}}}{4\pi\sqrt{2\pi
tt^{\prime}}}\int_{0}^{\infty}\lambda~{}d\lambda\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}dx~{}x^{-1/2}J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})e^{\Big{(}\frac{t^{2}+t^{\prime
2}}{2tt^{\prime}}x-\frac{\lambda^{2}tt^{\prime}}{2x}\Big{)}}$ (3.10)
$\displaystyle\times$ $\displaystyle I_{|n|q}(x~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime})e^{-x~{}{\rm cosh}r~{}{\rm
cosh}r^{\prime}}e^{iqn(\theta-\theta^{\prime})}.$

Now the $\lambda$ integral can be performed using 5.9 to get, after a little
algebra555Some of our manipulations here are similar to that of [23].,

$\displaystyle G^{q}_{+}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(zz^{\prime}/tt^{\prime})^{\frac{3}{2}}}{4\pi\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{iqn(\theta-\theta^{\prime})}\int_{0}^{\infty}dx\sqrt{x}I_{|n|q}(x~{}{\rm
sinh}r~{}{\rm
sinh}r^{\prime})I_{\nu}\Big{(}\frac{zz^{\prime}x}{tt^{\prime}}\Big{)}$ (3.11)
$\displaystyle\times$ $\displaystyle~{}{\rm
exp}\Big{[}-\frac{x}{2tt^{\prime}}(z^{2}+z^{\prime 2}-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}r~{}{\rm cosh}r^{\prime})\Big{]}$ $\displaystyle=$
$\displaystyle q\Big{(}\frac{zz^{\prime}}{4\pi}\Big{)}^{\frac{3}{2}}\
\int_{0}^{\infty}\frac{ds}{s^{4}}e^{-\frac{{\cal{V}}^{2}}{4s^{2}}}S_{q}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm
sinh}r^{\prime}}{2s^{2}}\Big{)}I_{\nu}\Big{(}\frac{zz^{\prime}}{2s^{2}}\Big{)}.$

In arriving at the last equation, we have used $s^{2}=\frac{tt^{\prime}}{2x}$
and

$\displaystyle S_{q}\Big{(}\frac{tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}}{2s^{2}}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}e^{inq(\theta-\theta^{\prime})}I_{|n|q}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}}{2s^{2}}\Big{)}$ (3.12) $\displaystyle=$
$\displaystyle 2\sum_{n=0}^{\infty~{}^{\prime}}~{}{\rm
cos}[nq(\theta-\theta^{\prime})]I_{nq}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}}{2s^{2}}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{q}\sum_{k=0}^{\infty}~{}{\rm
exp}\Big{[}\Big{(}\frac{tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}}{2s^{2}}\Big{)}~{}{\rm
cos}\Big{(}\theta-\theta^{\prime}-\frac{2k\pi}{q}\Big{)}\Big{]}.$ (3.13)

In (3.11),

${\cal{V}}^{2}=z^{2}+z^{\prime 2}-t^{2}-t^{\prime 2}+2tt^{\prime}~{}{\rm
cosh}r~{}{\rm cosh}r^{\prime}.$

The prime on the sum in equation (3.12) means that the contribution from $n=0$
term should be halved. Since we are working with integer values of $q$, the
$k$ sum will take integer values from $0$ to $q-1$. We can now carry out the
integration over $s$ using 5.10 to get the final expression of the Wightman
function

$G^{q}_{+}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\sum_{k=0}^{q-1}\frac{Q_{\nu-1/2}^{1}(u_{k})}{\sqrt{u_{k}^{2}-1}}.$
(3.14)

Here $u_{k}$ is defined as

$\displaystyle u_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2zz^{\prime}}(z^{2}+z^{\prime 2}-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi_{kq}),$ (3.15)

and

$\displaystyle{\rm cosh}\chi_{k,q}$ $\displaystyle=$ $\displaystyle~{}{\rm
cosh}r~{}{\rm cosh}r^{\prime}-~{}{\rm sinh}r~{}{\rm sinh}r^{\prime}~{}{\rm
cos}(\theta-\theta^{\prime}-2\pi k/q).$

The expression for the Wightman function is reminiscent of the method of
images. In the coincident limit, $G^{q}_{+}(x,x^{\prime})$ diverges. This
divergence comes from the $k=0$ term of the sum. We can define a renormalized
function subtracting this contribution and write

$G^{q}_{+R}(x,x^{\prime})=-\frac{1}{4\pi^{2}}\sum_{k=1}^{q-1}\frac{Q_{\nu-1/2}^{1}(u_{k})}{\sqrt{u_{k}^{2}-1}}.$
(3.16)

Boundary correlator

The boundary correlator is constructed as before

$\displaystyle\langle\Psi_{q}|{\cal O}(t,r,\theta){\cal
O}(t^{\prime},r^{\prime},\theta^{\prime})|\Psi_{q}\rangle$ $\displaystyle=$
$\displaystyle~{}\lim\limits_{z,z^{\prime}\rightarrow
0}(zz^{\prime})^{-(\nu+3/2)}G^{q}_{+}(x,x^{\prime})$ (3.17) $\displaystyle=$
$\displaystyle\sum_{k=0}^{q-1}\frac{C_{\Delta}}{(-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi_{k,q})^{\Delta}},$

where $C_{\Delta}$ is a constant defined in (2.14).

The summation over $k$ can be performed and the final result comes out as

$\displaystyle\langle\Psi_{q}|{\cal O}(t,r,\theta){\cal
O}(t^{\prime},r^{\prime},\theta^{\prime})|\Psi_{q}\rangle=\frac{(-1)^{\Delta-1}C_{\Delta}}{\Gamma(\Delta)(2tt^{\prime}~{}{\rm
sinh}r~{}{\rm
sinh}r^{\prime})^{\Delta}}\Big{(}\frac{\partial^{\Delta-1}}{\partial\gamma^{\Delta-1}}\Big{)}I_{1}(q,\gamma,\theta-\theta^{\prime}),$
(3.18)

where,

$\displaystyle
I_{1}(q,\gamma,\theta-\theta^{\prime})=\Big{[}\frac{q[(\gamma+\sqrt{\gamma^{2}-1})^{2q}-1]}{\sqrt{\gamma^{2}-1}[1+(\gamma+\sqrt{\gamma^{2}-1})^{2q}-~{}2{\rm
cos}q(\theta-\theta^{\prime})(\gamma+\sqrt{\gamma^{2}-1})^{q}]}\Big{]},$

and

$\displaystyle\gamma=\frac{-t^{2}-t^{\prime 2}+2tt^{\prime}~{}{\rm
cosh}r~{}{\rm cosh}r^{\prime}}{2tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}}.$

The details are provided in the appendix.

Twisted scalar

Having come thus far, we end this section with a study of the correlators
involving twisted fields. These fields satisfy the same equations as the
untwisted scalars but differ in their boundary conditions. The quasi-periodic
boundary condition that the twisted scalars obey is given by

$\phi(z,t,r,\theta+\frac{2\pi}{q})=e^{-2\pi i\beta}\phi(z,t,r,\theta),$ (3.19)

where $0\leq\beta\leq 1$. Such twisted scalar fields arise, for example, when
we consider a charged scalar field in AdS in the presence of a cosmic string
carrying internal magnetic flux. As is well known (see, for example [43] and
references therein), the corresponding gauge field component can be eliminated
by a gauge transformation and then one is left with a scalar with twisted
boundary conditions. For earlier studies along this direction, see for example
[17], [32].

Normalized solution of the equation of motion is now

$\displaystyle\phi(z,t,r,\theta)=C_{\lambda\alpha
n}z^{\frac{3}{2}}J_{\nu}(\lambda z)\frac{H^{2}_{i\alpha}(\lambda
t)}{\sqrt{t}}P^{q(n-\beta)}_{i\alpha-1/2}({\rm cosh}r)e^{iq(n-\beta)\theta},$

where

$\displaystyle C_{\lambda\alpha
n}=\frac{ie^{\pi\alpha/2}}{2}\sqrt{\frac{q\alpha~{}{\rm
sinh}(\pi\alpha)}{2\pi}}\Gamma[q(n-\beta)+i\alpha+1/2].$

Calculation is quite similar to the previous case of $\beta=0$ and as before
the Wightman function takes the form

$W_{\beta}^{q}(x,x^{\prime})=q\Big{(}\frac{zz^{\prime}}{4\pi}\Big{)}^{\frac{3}{2}}\int_{0}^{\infty}\frac{ds}{s^{4}}e^{-\frac{{\cal{V}}^{2}}{4s^{2}}}S_{q\beta}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm
sinh}r^{\prime}}{2s^{2}}\Big{)}I_{\nu}\Big{(}\frac{zz^{\prime}}{2s^{2}}\Big{)},$
(3.20)

where now

$\displaystyle S_{q\beta}\Big{(}\frac{tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}}{2s^{2}}\Big{)}=\sum_{n=-\infty}^{\infty}e^{i(n-\beta)q(\theta-\theta^{\prime})}I_{|n-\beta|q}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}}{2s^{2}}\Big{)},$

and

$\displaystyle{\cal{V}}^{2}=z^{2}+z^{\prime 2}-t^{2}-t^{\prime
2}+2tt^{\prime}{\rm cosh}r~{}{\rm cosh}r^{\prime}.$

We can now use the following relation to carry out the $s$ integration [33]:

$\displaystyle\sum_{n=-\infty}^{\infty}e^{iq(n-\beta)(\theta-\theta^{\prime})}I_{|n-\beta|q}\Big{(}\frac{tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}}{2s^{2}}\Big{)}=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\sum_{n}\Big{[}\frac{1}{q}e^{[{tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}~{}{\rm cos}(2\pi
n/q-\theta+\theta^{\prime})}]/{2s^{2}}}e^{i\beta(2\pi
n-q\theta+q\theta^{\prime})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\frac{1}{2\pi i}\sum_{j=+,-}je^{ji\pi
q\beta}\int_{0}^{\infty}dye^{-{tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}~{}{\rm
cosh}y}/{2s^{2}}}f(y)\Big{]}e^{-iq\beta(\theta-\theta^{\prime})},$ (3.21)

where

$\displaystyle f(y)=\frac{{\rm cosh}[qy(1-\beta)]~{}-~{}{\rm cosh}(q\beta
y)e^{-iq(\theta-\theta^{\prime}+j\pi)}}{{\rm cosh}(qy)~{}-~{}{\rm
cos}[q(\theta-\theta^{\prime}+j\pi)]}.$

Here the sum over $n$ runs as

$-\frac{q}{2}+\frac{(\theta-\theta^{\prime})q}{2\pi}\leq
n\leq+\frac{q}{2}+\frac{(\theta-\theta^{\prime})q}{2\pi}.$ (3.22)

Once substituted (3.21) into (3.20), we can integrate over $s$ using

$\displaystyle\int_{0}^{\infty}\frac{ds}{s^{4}}e^{-[\frac{{\cal{V}}^{2}}{2}-tt^{\prime}~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime}~{}{\rm cos}(2\pi
n/q-\theta+\theta^{\prime})]/(2s^{2})}I_{\nu}\Big{(}\frac{zz^{\prime}}{2s^{2}}\Big{)}=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\frac{2}{(zz^{\prime})^{3/2}\sqrt{\pi}\sqrt{u_{n}^{2}-1}}Q^{1}_{\nu-1/2}(u_{n}),$

where $u_{n}$ is defined as in (3.15), to finally arrive at

$\displaystyle W_{\beta}^{q}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{2}}\sum_{n}e^{-2i\pi\beta
n}\frac{Q_{\nu-1/2}^{1}(u_{n})}{\sqrt{u_{n}^{2}-1}}$ (3.23) $\displaystyle-$
$\displaystyle\frac{iq}{(2\pi)^{3}}e^{-iq\beta(\theta-\theta^{\prime})}\sum_{j=+,-}je^{ji\pi
q\beta}\int_{0}^{\infty}dyf(y)\frac{Q_{\nu-1/2}^{1}(u_{y})}{\sqrt{u_{y}^{2}-1}},$

where $u_{y}$ has the same structure of (3.15) with ${\rm cosh}\chi_{kq}$
replaced by ${\rm cosh}y$. This is the general form of the Wightman function.

When $q$ and $q\beta$ are both integers, the second term vanishes and we get

$\displaystyle W_{\beta}^{q}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{2}}\sum_{n}e^{-2i\beta\pi
n}\frac{Q_{\nu-1/2}^{1}(u_{n})}{\sqrt{u_{n}^{2}-1}}.$ (3.24)

The twisted bulk scalar is dual to a twisted scalar scalar primary
${\cal{O}}_{\beta}$ of dimension $\Delta$ with the same periodicity (3.19)
along $\theta$ inherited from $\phi$. The boundary correlator is therefore,

$\langle\Psi_{\beta q}|{\cal O}_{\beta}(t,r,\theta){\cal
O}_{\beta}(t^{\prime},r^{\prime},\theta^{\prime})|\Psi_{\beta
q}\rangle=\sum_{n}\frac{C_{\Delta}e^{-2i\beta\pi n}}{(-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\chi_{n,q})^{\Delta}},$ (3.25)

where the constant $C_{\Delta}$ has been defined earlier. This series can be
summed to get

$\displaystyle\langle\Psi_{\beta q}|{\cal O}_{\beta}(t,r,\theta){\cal
O}_{\beta}(t^{\prime},r^{\prime},\theta^{\prime})|\Psi_{\beta
q}\rangle=\frac{(-1)^{\Delta-1}C_{\Delta}}{\Gamma(\Delta)(2tt^{\prime}~{}{\rm
sinh}r~{}{\rm
sinh}r^{\prime})^{\Delta}}\Big{(}\frac{\partial^{\Delta-1}}{\partial\gamma^{\Delta-1}}\Big{)}J^{q}_{\beta}(\gamma,\theta-\theta^{\prime}),$
(3.26)

where, $J^{q}_{\beta}(\gamma,\theta-\theta^{\prime})=J$ is given by

$\displaystyle
J=\frac{qe^{-iq\beta(\theta-\theta^{\prime})}[(\gamma+\sqrt{\gamma^{2}-1})^{2q}-(\gamma+\sqrt{\gamma^{2}-1})^{q\beta}+2e^{iq(\theta-\theta^{\prime})}(\gamma+\sqrt{\gamma^{2}-1})^{q}\sinh(q\beta\ln(\gamma+\sqrt{\gamma^{2}-1})]}{\sqrt{\gamma^{2}-1}[1+(\gamma+\sqrt{\gamma^{2}-1})^{2q}-~{}2{\rm
cos}q(\theta-\theta^{\prime})(\gamma+\sqrt{\gamma^{2}-1})^{q}]]},$

and

$\displaystyle\gamma=\frac{-t^{2}-t^{\prime 2}+2tt^{\prime}~{}{\rm
cosh}r~{}{\rm cosh}r^{\prime}}{2tt^{\prime}~{}{\rm sinh}r~{}{\rm
sinh}r^{\prime}}.$

The details are provided in the appendix.

##  4 Conclusions

We have initiated a holographic study of field theory on the time dependent
background with a conical defect. In this work, our focus has been on the
Milne spacetime to which, in the absence of cosmological constant, any
hyperbolic FRW metric flows to at late times. When the Milne vacuum is chosen
to be the adiabatic one, we are able to compute the two point correlators of
operators which are dual to the massive scalars in the bulk AdS-Milne
background with defect. We find, for both twisted and untwisted operators, the
correlators can be represented as the sum over images in the covering space.
This sum can be carried out explicitly to write the results in compact forms.
If we restrict ourselves to the adiabatic vacuum, our computations suggest
that the field theory defined on a part of the boundary of AdS is dual to a
subregion in the AdS bulk. Though it may not be entirely obvious, there are
indication that the subregion duality may hold in general [44]-[46].

Evidently, our exploration is incomplete. First of all, Milne spacetime offers
another natural vacuum known as the conformal vacuum. We have not been able
get a closed form expression of correlators in this vacuum. Further, the
renormalized stress tensors, which one calculates from the two point
correlators, for the conformal vacuum are known to be non-trivial in Milne
spacetime. This remains to be computed in the presence of the defect in this
holographic framework.

In case of the AdS-Rindler foliation of AdS, interesting progress has been
made in understanding the relationship between entanglement and spacetime
[41]. Since AdS-Milne is another foliation of AdS, albeit, time-dependent, it
will be instructive to study the relationship between spacetime and
entanglement structure for the time-dependent situation. Apart from these
considerations, our results also connect with recent studies of conical
defects in $AdS_{3}$ , for example, [42]. We believe our results will also be
useful in the study of entanglement entropy in time-dependent situations in a
holographic context.

We are currently exploring some of these issues and hope to report on them in
the near future.

Acknowledgements: We would like to especially thank Souvik Banerjee for the
initial collaboration and many insightful discussions throughout the project.

##  5 Appendix

$4+1$ dimensional AdS-Milne

Since mode expansions of a massive scalar is sensitive to the spacetime
dimensions, it is instructive to carry out a computation in $4+1$ dimensions.
The metric is given in (2.15). The relevant solution of the equation of motion
for the Minkowski vacuum turns out to be

$\displaystyle\phi(z,t,r,\theta,\phi)=C_{\alpha lm}z^{2}J_{\nu}(\lambda
z)\frac{H_{i\alpha}^{(2)}(\lambda t)}{t}Y_{\alpha lm}(r,\theta,\phi),$

where $\nu=\sqrt{4+m^{2}}$ and we have defined [37]\- [38]

$\displaystyle Y_{\alpha
lm}(r,\theta,\phi)=\frac{\Gamma(i\alpha+l+1)}{\Gamma(i\alpha+1)}\frac{\alpha}{\sqrt{\sinh
r}}P_{i\alpha-1/2}^{-l-1/2}(\cosh r)Y_{lm}(\theta,\phi)$

The spherical harmonics $Y_{lmp}(\chi,\theta,\phi)$, for $0<p<\infty$, form a
complete orthonormal set of square integrable functions on the unit
hyperboloid [37]-[38]

$\displaystyle\int_{0}^{\infty}d\chi\sinh^{2}\chi\int d\Omega
Y_{lmp}(\chi,\Omega)Y_{l^{\prime}m^{\prime}p^{\prime}}^{*}(\chi,\Omega)=\delta(p-p^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}.$

where the harmonic functions satisfy

$\displaystyle\int_{0}^{\infty}d\chi\sinh\chi
P_{ip-1/2}^{-l-1/2}(\cosh\chi)P_{ip\prime-1/2}^{-l^{\prime}-1/2}(\cosh\chi)=\frac{|\Gamma(ip)|^{2}}{|\Gamma(ip+l+1)|^{2}}\delta(p-p^{\prime}).$

The normalization constant $C_{\alpha lm}$ turns out to be

$\displaystyle C_{\alpha lm}=i\frac{\sqrt{\pi\lambda}}{2}e^{\pi\alpha/2}.$

Putting everything together, we therefore have

$\displaystyle\phi(z,t,r,\theta,\phi)=i\int_{0}^{\infty}d\lambda
d\alpha\sum_{l,m}\frac{\sqrt{\pi\lambda}}{2}e^{\pi\alpha/2}z^{2}J_{\nu}(\lambda
z)\frac{H_{i\alpha}^{(2)}(\lambda t)}{t}Y_{\alpha lm}(r,\theta,\phi).$ (5.1)

Consequently the Wightman function is

$\displaystyle G_{+}(x,x^{\prime})=-\sum_{l,m}\int_{0}^{\infty}d\lambda
d\alpha\frac{\pi\lambda}{4}\frac{(zz^{\prime})^{2}}{tt^{\prime}}J_{\nu}(\lambda
z)J_{\nu}(\lambda z^{\prime})H_{i\alpha}^{(2)}(\lambda
t)H_{i\alpha}^{(1)}(\lambda t^{\prime})Y_{\alpha lm}(r,\theta,\phi)Y_{\alpha
lm}(r^{\prime},\theta^{\prime},\phi^{\prime}).$

Using the completeness of the spherical harmonics

$\displaystyle\sum_{lm}Y_{\alpha lm}(\chi,\Omega)Y_{\alpha
lm}(\chi^{\prime},\Omega^{\prime})=\frac{\alpha~{}{\rm
sin}\alpha\gamma}{2\pi^{2}{\rm sinh}\gamma},$ (5.2)

we can write the Wightman function as

$\displaystyle
G_{+}(x,x^{\prime})=\frac{\pi(zz^{\prime})^{2}}{4tt^{\prime}}\int\lambda
d\lambda d\alpha J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})H_{i\alpha}^{(2)}(\lambda t)H_{i\alpha}^{(1)}(\lambda
t^{\prime})\frac{\alpha\sin\alpha\gamma}{2\pi^{2}\sinh\gamma},$

where

$\displaystyle{\rm cosh}\gamma={\rm cosh}\chi~{}{\rm cosh}\chi^{\prime}-{\rm
sinh}\chi~{}{\rm sinh}\chi^{\prime}{\rm cos}\omega,$ $\displaystyle{\rm
cos}\omega={\rm cos}\theta~{}{\rm cos}\theta^{\prime}+{\rm sin}\theta~{}{\rm
sin}\theta^{\prime}~{}{\rm cos}(\phi-\phi^{\prime}).$ (5.3)

Further, doing the $\alpha$ integral, we get

$\displaystyle
G_{+}(x,x^{\prime})=-\frac{i(zz^{\prime})^{2}}{4\pi^{2}}\int_{0}^{\infty}\lambda^{2}d\lambda
J_{\nu}(\lambda z)J_{\nu}(\lambda
z^{\prime})\frac{K_{1}\Big{(}\lambda\sqrt{-t^{2}-t^{\prime
2}+2tt^{\prime}~{}{\rm cosh}\gamma}\Big{)}}{\sqrt{t^{2}+t^{\prime
2}-2tt^{\prime}~{}{\rm cosh}\gamma}}.$

Finally, after completing the $\lambda$ integral we reach at (2.16).

Deriving (3.18) and (3.26)

We start with the formula [31]

$\frac{1-p^{2}}{1-2p~{}{\rm cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=1+2\sum_{m=1}^{\infty}p^{m}~{}{\rm
cos}[m(\Delta\theta-\frac{2\pi k}{q})].$

From the above, it follows that

$\sum_{k=0}^{q-1}\frac{1-p^{2}}{1-2p~{}{\rm cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=q+2\sum_{k=0}^{q-1}\sum_{m=1}^{\infty}p^{m}~{}{\rm
cos}[m(\Delta\theta-\frac{2\pi k}{q})].$

Now since

$\sum_{m=-\infty,\neq 0}^{\infty}p^{|m|}e^{im(\Delta\theta-2\pi
k/q)}=2\sum_{m=1}^{\infty}p^{m}~{}{\rm cos}[m(\Delta\theta-\frac{2\pi
k}{q})],$

we get

$\sum_{k=0}^{q-1}\frac{1-p^{2}}{1-2p~{}{\rm cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=q+\sum_{k=0}^{q-1}\sum_{m=-\infty,\neq
0}^{\infty}p^{|m|}e^{im\Delta\theta}e^{-\frac{2i\pi mk}{q}}.$

Further, using

$\sum_{k=0}^{q-1}e^{-\frac{2i\pi
mk}{q}}=q\sum_{n=-\infty}^{\infty}\delta_{-m,nq},$

we arrive at

$\displaystyle\sum_{k=0}^{q-1}\frac{1-p^{2}}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi k}{q})+p^{2}}$ $\displaystyle=$ $\displaystyle
q+q\sum_{n=-\infty}^{\infty}~{}~{}\sum_{m=-\infty,\neq
0}^{\infty}p^{|m|}e^{im\Delta\theta}\delta_{-m,nq}$ $\displaystyle=$
$\displaystyle q+q\sum_{n=-\infty,\neq
0}^{\infty}p^{q|n|}e^{-inq\Delta\theta}$ $\displaystyle=$ $\displaystyle
q+2q\sum_{n=1}^{\infty}p^{qn}~{}{\rm cos}(qn\Delta\theta)$ $\displaystyle=$
$\displaystyle q+q\Big{[}\frac{1-p^{2q}}{1-2p^{q}~{}{\rm
cos}(q\Delta\theta)+p^{2q}}-1\Big{]}$ $\displaystyle=$
$\displaystyle\frac{q(1-p^{2q})}{1-2p^{q}~{}{\rm cos}(q\Delta\theta)+p^{2q}}.$

Therefore,

$\sum_{k=0}^{q-1}\frac{1}{\frac{1+p^{2}}{2p}-~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})}=\frac{2qp(1-p^{2q})}{(1-p^{2})(1-2p^{q}~{}{\rm
cos}(q\Delta\theta)+p^{2q})}.$

Taking $\frac{1+p^{2}}{2p}=\gamma$, we get (3.18).

Similarly, we can sum the series for the twisted scalar to get (3.26). Since
$\beta=0,1$ cases reduce to untwisted scalar case, we will restrict to
$0<\beta<1$. Again, we start with

$\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}(1-p^{2})}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi k}{q})+p^{2}}=\sum_{k=0}^{q-1}e^{-2\pi i\beta
k}+2\sum_{k=0}^{q-1}\sum_{m=1}^{\infty}e^{-2\pi i\beta k}p^{m}~{}{\rm
cos}[m(\Delta\theta-\frac{2\pi k}{q})].$

First sum on the RHS vanishes for the range of $\beta$ that we are considering
while the second term can be evaluated using the same methods as before to get

$\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}(1-p^{2})}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=q\sum_{n=-\infty,\neq-\beta}^{\infty}p^{q|-n-\beta|}e^{-i(nq+q\beta)\Delta\theta}\\\
.$

Since $n$ takes integer values while $\beta$ is not an integer, there is no
restriction on the sum. Expanding the sum, we get

$\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}(1-p^{2})}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=qp^{q\beta}e^{-i(q\beta)\Delta\theta}+q\sum_{n=-\infty}^{1}p^{q|-n-\beta|}e^{-i(nq+q\beta)\Delta\theta}+q\sum^{\infty}_{n=1}p^{q|-n-\beta|}e^{-i(nq+q\beta)\Delta\theta}\\\
$

Relabelling $n\rightarrow-n$ in the second term, we get

$\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}(1-p^{2})}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=qp^{q\beta}e^{-iq\beta\Delta\theta}+qe^{-iq\beta\Delta\theta}\left(p^{-q\beta}\sum^{\infty}_{n=1}p^{qn}e^{inq\Delta\theta}+p^{q\beta}\sum^{\infty}_{n=1}p^{qn}e^{-inq\Delta\theta}\right)\\\
$

Now we can do the final sums using the formula

$\sum_{n=0}^{\infty}s^{n}e^{inx}=\frac{1-se^{-ix}}{1-2s\cos x+s^{2}}$

$\displaystyle\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}(1-p^{2})}{1-2p~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})+p^{2}}=qp^{q\beta}e^{-iq\beta\Delta\theta}+qp^{-q\beta}e^{-iq\beta\Delta\theta}\left(\frac{1-p^{q}e^{-iq\Delta\theta}}{1-2p^{q}\cos(q\Delta\theta)+p^{2q}}-1\right)$
$\displaystyle+qp^{q\beta}e^{-iq\beta\Delta\theta}\left(\frac{1-p^{q}e^{iq\Delta\theta}}{1-2p^{q}\cos(q\Delta\theta)+p^{2q}}-1\right)$

Simplifying we finally get

$\sum_{k=0}^{q-1}\frac{e^{-2\pi i\beta k}}{\frac{1+p^{2}}{2p}-~{}{\rm
cos}(\Delta\theta-\frac{2\pi
k}{q})}=\frac{2qpe^{-iq\beta\Delta\theta}(p^{q\beta}-p^{2q}-2p^{q}e^{iq\Delta\theta}\sinh(q\beta\ln
p))}{(1-p^{2})(1-2p^{q}~{}{\rm cos}(q\Delta\theta)+p^{2q})}.$

Taking $\frac{1+p^{2}}{2p}=\gamma$, we get (3.26).

Identities used in the main text

Here we collect a number of identities used in the main text.

First, to get the normailzation of $C_{\lambda\alpha n}$, we have used the
following properties of $P_{i\alpha-1/2}^{-n}(x)$ [30]:

$\displaystyle
P_{iz-1/2}^{m}(u)=\frac{\Gamma[-iz+m+1/2]}{\Gamma[-iz-m+1/2]}P_{iz-1/2}^{-m}(u)=\frac{\Gamma[iz+m+1/2]}{\Gamma[iz-m+1/2]}P_{iz-1/2}^{-m}(u)$

and

$\displaystyle\int_{1}^{\infty}duP_{iz^{\prime}-1/2}^{-m}(u)P_{iz-1/2}^{-m}(u)=\frac{\pi}{z~{}{\rm
sinh}(\pi z)}\frac{\delta(z-z^{\prime})}{|\Gamma(m+1/2+iz)|^{2}}.$ (5.4)

To get to (LABEL:midexp), we have used

$\displaystyle
H_{i\alpha}^{(2)*}(x)=H_{-i\alpha}^{(1)},~{}H_{i\alpha}^{(2)}(\lambda
t)H_{-i\alpha}^{(1)}(\lambda
t^{\prime})=\frac{4}{\pi^{2}}e^{-\pi\alpha}K_{i\alpha}(i\lambda
t)K_{-i\alpha}(-i\lambda t^{\prime}),$

and

$\displaystyle 2\sum_{n=1}^{\infty}(-)^{n}~{}{\rm
cos}n(\theta-\theta^{\prime})P_{i\alpha-1/2}^{-n}({\rm{cosh}}r)P_{i\alpha-1/2}^{n}({\rm{cosh}}r^{\prime})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+P_{i\alpha-1/2}({\rm{cosh}}r)P_{i\alpha-1/2}({\rm{cosh}}r^{\prime})=P_{i\alpha-1/2}({\rm{cosh}}\chi).$
(5.5)

To reach to equation (2.11), we used [31]

$\displaystyle\int_{0}^{\infty}d\alpha~{}\alpha~{}{\rm
sinh}(\pi\alpha)\frac{\pi}{{\rm
cosh}\pi\alpha}K_{i\alpha}(a)K_{i\alpha}(b)P_{i\alpha-\frac{1}{2}}({\rm
cosh}\chi)$
$\displaystyle~{}~{}~{}~{}~{}=\sqrt{\frac{\pi}{2}}\Big{(}\frac{ab}{\sqrt{a^{2}+b^{2}+2ab~{}{\rm
cosh}\chi}}\Big{)}^{\frac{1}{2}}K_{\frac{1}{2}}(\sqrt{a^{2}+b^{2}+2ab~{}{\rm
cosh}\chi}).$ (5.6)

This identity is valid when $|arg(a)|<\pi/2$ and so we replace $t$ with
$te^{-i\epsilon}$, with limit $\epsilon\rightarrow 0$.

Equation (2.14) uses the identity

$\displaystyle\int_{0}^{\infty}\frac{{\rm cosh}2yt}{{\rm
cosh}^{2x}bt}dt=\frac{4^{x-1}}{b}B(x+y/b,x-y/b),$ (5.7)

Here $B(x,y)$ represents the beta function.

We used relation [39]

$\displaystyle|\Gamma(i\alpha+nq+1/2)|^{2}P_{i\alpha-1/2}^{-qn}({\rm
cosh}r)P_{i\alpha-1/2}^{-qn}({\rm cosh}r^{\prime})$
$\displaystyle~{}~{}~{}~{}~{}=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}dxx^{-1/2}I_{|n|q}(x~{}{\rm
sinh}r~{}{\rm sinh}r^{\prime})K_{i\alpha}(x)e^{-x~{}{\rm cosh}r~{}{\rm
cosh}r^{\prime}},$

where $I_{\nu}(x)$ is the modified Bessel function and [40]

$\displaystyle\int_{0}^{\infty}x~{}{\rm sinh}(\pi
x)K_{ix}(a)K_{ix}(b)K_{ix}(y)dx=\frac{\pi^{2}}{4}e^{-\frac{y}{2}\Big{(}\frac{a}{b}+\frac{b}{a}+\frac{ab}{y^{2}}\Big{)}},$
(5.8)

to get (3.10).

Using [31]

$\displaystyle\int_{0}^{\infty}e^{-\alpha
x}J_{\nu}(2\beta\sqrt{x})J_{\nu}(2\gamma\sqrt{x})dx=\frac{1}{\alpha}I_{\nu}\Big{(}\frac{2\beta\gamma}{\alpha}\Big{)}e^{-\Big{(}\frac{\beta^{2}+\gamma^{2}}{\alpha}\Big{)}},$
(5.9)

we can perform the $\lambda$ integral in (3.10)to get (3.11).

Using the known result [31]

$\displaystyle\int_{0}^{\infty}e^{-xu}I_{\nu}(x)x^{\mu-1}dx=\sqrt{\frac{2}{\pi}}\frac{e^{-i(\mu-1/2)\pi}}{\sqrt{u^{2}-1}}Q_{\nu-1/2}^{\mu-1/2}(u),$
(5.10)

we perform the integral over $s$ to go from (3.11 )to (3.14).

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