# Miniature Wireless Power Transfer System
for Charging Vertically Oriented Receivers

Andrey Korzin1, Esmaeel Zanganeh1, Polina Kapitanova1
1 School of Physics and Engineering, ITMO University, Saint Petersburg,
197101, Russia

###### Abstract

Development of compact wireless power transfer (WPT) systems for charging
miniature randomly oriented electronic devices is quite a challenge.
Traditionally, WPT systems based on resonant magnetic coupling utilize face-
to-face aligned transmitter and receiver coils providing sufficient efficiency
at relatively large distances. However, with the presence of angular receiver
misalignment in a such system, the mutual coupling decreases resulting in a
low power transfer efficiency. Here we develop a compact WPT system for
wireless charging of miniature receivers vertically oriented with respect to
the transmitter. As a transmitter, we employ a butterfly coil that provides a
strong tangential component of the magnetic field. Thus, a vertically oriented
receiver located in the magnetic field can be charged wirelessly. We perform
numerical and experimental studies of the WPT system’s power transfer
efficiency as a function of the distance between the transmitter and the
receiver. The misalignment and rotation dependencies of power transfer
efficiency are also experimentally studied. We demonstrate the power transfer
efficiency of 60 $\%$ within transfer distance of 4 mm for a vertically
oriented receiver with an overall dimension of 20 mm $\times$14 mm at the
frequency of 6.78 MHz.

###### Index Terms:

wireless power transfer, butterfly coil, compact receiver, vertically oriented
receiver, smart glasses

##  I Introduction

Smart glasses are wearable technology devices that combine the graphic
abilities of mobile appliances, artificial intelligence, and augmented
reality. With smart glasses, one can make phone calls, get notifications, have
3D map navigation, graphic translation, virtual reality gaming, and more. It
is clear that all these utilities require some integrated electronics built
into these devices, which require a power source, which most often are
available in form of batteries. Traditionally, the smart glasses’ batteries
can be charged via cords or inside of a special case[1]. However, customers
always demand more freedom in the use of electronic devices, asking to
eliminate cords or cases. From our point of view, a comfortable way of
charging these devices would be the positioning of the smart glasses over a
charging pad (see Fig. 1) where a horizontally oriented transmitter (Tx) coil
is incorporated. The smart glasses have to be equipped with a receiver (Rx)
coil. Due to the compactness of the device [2] itself, there is limited space
to locate the Rx coil. Probably the most appropriate way is to incorporate it
inside the main part of the glasses’ temple, which is not bigger than 85 mm by
20 mm [2]. Thus, we come to the possible practical scenario when a user places
the smart glasses over the charging pad in a spatial position. With the higher
probability, we will get the situation when one needs to charge a miniature
vertically oriented Rx over the horizontally positioned Tx. However, the
angular misalignment is still possible (as shown in Fig. 1).

Wireless power transfer (WPT) has been rapidly becoming a significant
technology for many applications, such as electric vehicle charging, implanted
medical devices, robotic systems, and wearable electronics [3, 4, 5, 6, 7].
WPT technology based on magnetic resonant coupling has attracted great
attention due to its promise of safe and highly efficient mid-range charging
[8, 9, 10, 11, 12]. However, the power transfer efficiency (PTE) of magnetic
resonance WPT systems with ”face-to-face” locations of the Tx and Rx suffers
from the change of the Rx position, such as angle and axis misalignment [13,
14, 15]. It can be explained by the decrease in the coupling coefficient
between the Tx and Rx while the second one changes its spatial orientation.
Moreover, the degradation of the PTE can also be observed for the magnetic
resonance WPT systems with the Tx and Rx coils of different overall
dimensions. Therefore, a magnetic field shaping technology can be used to
control the magnetic flux at a particular location and enhance coupling
coefficient, thereby improving positional freedom and PTE simultaneously [16,
17]. For that purpose, one can utilize a butterfly Tx coil design [18] as well
as the changing of the current distribution in the planar Tx coils array [19,
20].

Figure 1:  Conceptual view of the miniature WPT system charging a vertically
oriented receiver of a smart glasses. Figure 2:  (a) Schematic view of the
miniature WPT system for charging vertically oriented receivers. (b) Simulated
magnetic field distribution of the WPT system at the frequency of 6.78 MHz
numerically obtained for 0.5 W of input power. Figure 3:  (a) Photograph of
the WPT system prototype connected to a VNA for measurements of the
S-parameters. (b) Simulated and measured PTE as a function of the distance
$d$. Measured s-parameters for: (c) d = 2 mm and (d) d = 4 mm. Figure 4:  PTE
extracted from the measured S-parameters for different rotation angles of the
Rx with respect to Tx: (a) $\alpha$ \- rotational angle along the z-axis, (b)
$\beta$ \- rotational angle along the y-axis with respect to the Rx’s lower
edge, and (c) $\gamma$ \- rotational angle along the y-axis when the Rx center
is aligned with the Tx center.

In this paper, we develop a miniature WPT system for charging vertically
oriented receivers. We introduce a butterfly-shaped Tx coil wich forms a
strong tangential component of the magnetic field, which is used to transfer
the power to a vertically oriented Rx coil. We perform numerical simulations
of the WPT system in CST Microwave Studio 2022 to study the S-parameters, PTE,
and near magnetic field distribution. To prove the numerical simulations,
experimental studies of the miniature WPT system are provided in the MHz
frequency range. The transfer distances, misalignment, and rotation
dependencies of the WPT system PTE are experimentally studied.

##  II WPT System Design

The miniature WPT system consists of a horizontally located Tx and a
vertically oriented Rx separated by a distance $d$, as shown in Fig. 2(a).
Both Tx and Rx coils are made of 5-turn rectangular copper wires with the
thickness of 18 $\mu$m implemented on both sides of FR-4 substrates
($\varepsilon_{r}$ = 4.5 and $tan(\delta)$ = 0.02) with the height of $h$=0.8
mm. The Tx coil is implemented as a _butterfly coil_ with the copper wires
winded in opposite directions. The wire width is $w$ = 0.3 mm, and the gap
between the wires is $g$ = 0.3 mm. The top and bottom layers of each coil are
connected with small copper vias with a radius of $r$= 1.5 mm. The overall
size of the Tx butterfly coil is 40 mm$\times$27.3 mm. In the Rx, the wire
width is $w$ = 0.2 mm, and the gap between the wires is $g$ = 0.2 mm. The
overall size of the Rx coil is 20 mm$\times$14 mm. The Tx and Rx coil
dimensions are selected to be compatible with smart glasses for possible
implementation of the WPT system in the future.

The numerical simulations of the WPT system characteristics are performed in
CST Microwave Studio 2022 with the help of the frequency domain solver. The Tx
coil is excited through the 50 ohm port, the Rx coil is loaded by 50 ohm port.
Due to the opposite wire wingdings of the Tx butterfly coil, the currents
flowing in the left and right sides generate a loop of magnetic field
penetrating the Tx surface, as shown in Fig. 2(b). Over the Tx surface, the
tangential component of the magnetic field is dominant. The location of the
vertically oriented Rx coil in the maximum of the magnetic field enables a
strong coupling and opens a way for power transmission. The numerically
obtained S-parameters for the Tx and Rx coils separation distance of $d$ = 3
mm revealed a poor matching of the WPT system. Thus, we employ matching
networks at the input and output of the system. They are based on series-
parallel capacitances and indicated in Fig. 2(a) as the capacitors $C_{1}$,
$C_{2}$, $C_{3}$, and $C_{4}$. To mount the capacitors, copper contact pads
are added at the input and output of the WPT system. During the numerical
simulations, we optimize the capacitance values to achieve zero reflection
coefficients on both the Tx ($|S_{11}|$) and Rx sides ($|S_{22}|$) at the
frequency of 6.78 MHz for the $d$= 3 mm. We reveal the minimums of the
reflection coefficients for the following values: $C_{1}$= 98 pF, $C_{2}$= 240
pF and $C_{3}$ = 98 pF, $C_{4}$ = 170 mm. The PTE of a WPT system can be
expressed in term of the S-parameters as follows:

$\eta=\frac{|S_{21}|^{2}}{1-|S_{11}|^{2}}\times 100\%$ (1)

We perform the numerical simulations of the reflection and transmission
coefficients for the different distances $d$ between the Tx and Rx with the
matching networks. Using the obtained S-parameters, we calculate the PTE as a
function of distance $d$ at the fixed frequency of 6.78 MHz (see Fig. 3(b)).
The maximum PTE of $77~{}\%$ is achieved for $d$= 0 mm. For the distance $d$=
3 mm, it reaches $65~{}\%$. When the distance between the Tx and Rx is
increased to $d$=10 mm, the PTE decays to 20$~{}\%$. It can be explained by
the decrease in the coupling coefficient between the Tx and Rx.

##  III Experimental Study of the PTE

To verify the results predicted numerically, the miniature WPT system
prototype is fabricated and demonstrated in Fig. 3(a). The Tx and Rx coils are
implemented utilizing printed circuit board technology using FR-4 substrates.
Surface mounted capacitors with the values obtained during the numerical
optimization are mounted on the contact pads. To connect the WPT system input
and output to the experimental equipment, SMA connectors are soldered. To
measure the S-parameters of the WPT system, a planar Vector Network Analyzer
(VNA) S5048 is used. To hold the Rx over the Tx, we use a specially designed
holder made of ABS plastic, but it is not shown in Fig. 3(a).

During the experimental investigation, we first measure the S-parameters for
separation distance $d$ from Tx to Rx ranging from 0 to 10 mm with a 1 mm
step. The measured S-parameters for distances of $d$ = 2 mm and $d$ = 4 mm are
shown in Fig. 3(c) and (d), respectively. For $d$ = 4 mm, the optimal coupling
condition is satisfied, and a single minimum of the reflection coefficient is
observed. For $d$ = 2 mm, we observe the frequency splitting phenomenon [21],
which can be explained by a stronger coupling than for the separation distance
of $d$ = 4 mm. The measured S-parameters are used to calculate the PTE of the
WPT system prototype by Eq. 1. The PTE extracted from the measured data is
compared with the simulated one in Fig. 3(b). A good agreement between the
measured data and simulated results is observed.

Next, we mimic the spatial position of the Rx with respect to the Tx. We
experimentally study the PTE of the WPT system as a function of the Rx
rotational angle with respect to the Tx, as shown in the insets of Fig. 4. For
the fixed distance $d=3.5~{}mm$, we rotate the Rx along the z-axis with a
rotational angle $\alpha$ (see the inset of the Fig. 4(a)) and measure the
S-parameters for angle from $0$ to $90^{\circ}$ with a $10^{\circ}$ step. The
PTE extracted from the measured S-parameters as a function of rotational angle
$\alpha$ at the frequency of 6.78 MHz is depicted in Fig. 4(a). It is as high
as $65\%$ at $\alpha=0^{\circ}$ and slightly decreases with increasing of the
angle $\alpha$ up to 40∘. With further increase of the angle $\alpha$, it
decays faster. The PTE reaches zero for $\alpha=90^{\circ}$ since the magnetic
fields become parallel to the Rx surface and cannot induce current in the Rx
coil. For the fixed distance $d$ = 3.5 mm, we also rotate the Rx along the
y-axis with respect to the Rx’s lower edge (see inset of Fig. 4(b)) with a
rotational angle $\beta$ ranging from $0$ to $90^{\circ}$ with a $10^{\circ}$
step. The experimentally obtained PTE as a function of the $\beta$ is shown in
Fig. 4(b). As one can see, the PTE is greater than 65 $\%$ regardless of the
Rx rotational angle $\beta$. In this rotational scenario, the magnetic field
always penetrates the Rx whose lower side is positioned in the center of the
Tx and we do not observe the degradation of the PTE. We also study the
situation when the Rx center is positioned along the Tx center and the Rx is
rotated along y-axis (see the inset of Fig. 4(c)) with the angle $\gamma$. The
obtained PTE is illustrated in Fig. 4(c). One can see that the PTE decreases
from 64 $\%$ to 10 $\%$ by increasing the $\gamma$ from $0^{\circ}$ to
$90^{\circ}$.

Smart glasses can have many design variations. Some smart glasses allow you to
fully fold the temples and in this case, the Rx coil may be in a horizontal
position relative to the Tx butterfly coil. However, the misalignment of the
Rx coil regarding the Tx coil may remain. That is why we also experimentally
investigate the PTE dependency for the misalignment of the Rx with respect to
Tx, as shown in Fig. 5(a). We place the Rx at the separation distance of
$d$=3.5 mm parallel to the Tx and move it along the x-axis with a misalignment
parameter $S$. Then, we measure the PTE as a function of the misalignment
parameter $S$ that varies in the interval $-20~{}mm\leq S\leq 20~{}mm$. The
PTE extracted from the measured S-parameters at the frequency of 6.78 MHz is
higher than 67 $\%$ over the $-20~{}mm\leq S\leq-7~{}mm$ and $7~{}mm\leq S\leq
20~{}mm$ intervals of $S$ since the Rx strongly couples to the butterfly Tx
coil. For $S=0~{}mm$, the PTE drops to 20 $\%$ due to the compensation of the
currents in the Rx coil induced by opposite currents of the Tx coil. The PTE
dependence has a slight asymmetry due to the impact of the matching networks
of the Rx coil.

Figure 5:  (a) Schematic view of the WPT system with the Rx parallel to the Tx
and a misalignment parameter $S$. (b) Measured PTE as a function of the
misalignment parameter $S$.

##  IV Conclusion

We experimentally studied the PTE of the miniature WPT system composed of the
horizontally located Tx and vertically oriented Rx separated by a distance.
The transmitter was implemented as a butterfly coil that generates a strong
tangential component of the magnetic field, which was used to provide power
transfer to the Rx. The overall dimensions of the Tx $40~{}mm\times
27.3~{}mm\times 0.8~{}mm$ and the Rx $20~{}mm\times 14~{}mm\times 0.8~{}mm$
were selected to be compatible with the existing smart glasses. The PTE
extracted from the measured S-parameters of the WPT system is higher than 60
$\%$ over transfer distances up to 4 mm for vertically oriented Rx at the
frequency of 6.78 MHz. The PTE for rotation with respect to Rx’s lower edge is
higher than 60 $\%$ from $0^{\circ}$ to $90^{\circ}$. For rotation along the
y-axis with respect to the Rx centre, PTE holds higher than 60 $\%$ up to
$20^{\circ}$. Therefore, the transfer distances, misalignment, and rotation
dependencies of power transfer efficiency have been successfully demonstrated.
The proposed WPT system can be considered as a suitable platform for charging
compact devices like smart glasses with vertically oriented receivers.

This work was supported by the Russian Science Foundation (Project No.
21-79-30038)

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# Spacetime topology from holographic entanglement

Marcelo Botta Cantcheff<EMAIL_ADDRESS>Instituto de Física La
Plata - CONICET and Departamento de Física, Universidad Nacional de La Plata
C.C. 67, 1900, La Plata, Argentina

###### Abstract

An asymptotically AdS geometry connecting two or more boundaries is given by a
entangled state, that can be expanded in the product basis of the Hilbert
spaces of each CFT living on the boundaries. We derive a prescription to
compute this expansion for states describing spacetimes with general spatial
topology in arbitrary dimension. To large $N$, the expansion coincides with
the Schmidt decomposition and the coefficients are given by $n$-point
correlation functions on a particular Euclidean geometry.

We show that this applies to all spacetime that admits a Hartle-Hawking type
of wave functional, which via a standard hypothesis on the spatial topology,
can be (one to one) mapped to CFT states defined on the asymptotic boundary.
It is also observed that these states are endowed with quantum coherence
properties.

Applying this as holographic engineering, one can to construct an emergent
space geometry with certain predetermined topology by preparing an entangled
state of the dual quantum system. As an example, we apply the method to
calculate the expansion and characterize a spacetime whose initial spatial
topology is a (genus one) handlebody.

##  1 Introduction

The main goal of holographic gravity is to understand the rules to reconstruct
the spacetime geometry from states of some quantum theory defined on a fixed
timelike boundary. In the AdS/CFT realization of holography [1], the standard
interpretation is that the exact bulk geometry AdSd+1 corresponds to the
fundamental state $\left|0\right\rangle$ of the CFT Hilbert space ${\cal H}$
defined on its conformal boundary $S^{d-1}\times{\mathbb{R}}$, and all
classical asymptotically AdS (aAdS) spacetime with only one connected
boundary, should correspond to some exited state [3, 4].

By considering two non-interacting identical copies of this CFT (labeled by a
subindex $1,2$). The asymptotically $AdS_{d+1}$ spacetime with a eternal black
hole corresponds to the (entangled) state [2]:

$\left|\Psi(\beta)\right\rangle=\sum_{n}\,\frac{e^{-\frac{\beta}{2}\,E_{n}}}{Z^{1/2}}\left|E_{n}\right\rangle_{1}\otimes\left|E_{n}\right\rangle_{2}\leavevmode\nobreak\
\in{\cal H}_{1}\otimes{\cal H}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \beta\equiv(k_{B}T)^{-1},$ (1.1)

where the $\left|E_{n}\right\rangle$ are a complete basis of eigenstates of
the CFT Hamiltonian $H$, and $E_{n}$ are its eigenvalues. This is the TFD
state and describes a thermal state of the CFT quantum system at temperature
$T$ [5, 6, 7].

It has been shown that all classically connected spacetime should have a
similar (entangled) structure [8], thus it would be interesting to have some
precise recipe to describe this decomposition for more general states and dual
geometries. One of the important ingredients of this is that state above is a
quantum superposition of tensor products
$\left|E_{n}\right\rangle_{1}\otimes\left|E_{n}\right\rangle_{2}$, which are
assumed to correspond to a pair of disconnected classical aAdS spacetimes (see
fig. 1).

Figure 1:  (a) Penrose diagram of a maximally extended AdS-black hole. The
green line is a connected spacial slice (b) We schematically show the
interpretation of [8], where the resulting state (1.1) is a linear combination
of states $|E_{n}\rangle_{1}\otimes|E_{n}\rangle_{2}$ supposedly dual to aAdS
spacetimes. The blue lines represent the non-interacting CFT theories on the
two asymptotic boundaries.

For holographic uses of this description one must know the dictionary between
the product CFT basis and the gravitational dual, however we do not expect
that all the CFT energy eigenstates are dual themselves to some geometric
description, and if they are, it is unclear how to interpret their gravity
dual precisely. So from a holographic point of view, the expansion in the
energy basis might be senseless.

Since the TFD state is only a function of the (boundary) Hamiltonian, it is
diagonal in the energy basis, and so the rhs of (1.1) coincides with the so
called Schmidt form of the state. It has numerous applications in quantum
information theory, for example in the characterization of quantum
entanglement and purification of states, and in plasticity [9].

This is the Schmidt form of the state and its components are nothing but the
components of the propagator (the evolution operator) for an Euclidean time
$\beta/2$:

$e^{-\frac{E_{n}\beta}{2}}\,\delta_{nm}\,=\,\langle
E_{n}|U(\beta/2)|E_{m}\rangle\,,$

which can be represented as a path integral on the CFT fields on the Euclidean
cylinder $\Sigma\equiv S^{d-1}\times[0,\beta/2]$ (see fig 3).

The reason why this state corresponds to a classical geometry is that, because
of the standard duality recipes [10, 11], the same path integral can be
expressed as a partition function of gravity, and finally the classical
(Euclidean) geometry is recovered as saddle in the large $N$ approximation. In
this case, $\Sigma$ represents the (past) asymptotic boundary of the Euclidean
black hole geometry, and is homologous to the initial spacelike slice
$\Sigma_{0}$ of the maximally extended black hole: the Einstein-Rosen wormhole
(see fig 5 (left)).

The main objective of this paper is to construct a systematic recipe to
compute the multi-partite, manifestly entangled, descomposition of the CFT
state corresponding to some dual classical geometry, and as by-product: to
prepare entangled states dual to certain prefixed geometries.

The more fundamental question that still remains open, is which are the states
of a $b$-partite quantum system ${\cal H}_{1}\otimes\dots\otimes{\cal H}_{b}$
that have a dual geometric interpretation in terms of a connected/disconnected
classical geometry, and how general they are. In this sense, it has been
pointed out that entanglement is not sufficient and some other ingredient
participates of the emergence mechanism [12]. In [13] [14] was argued that
generic entangled states do not have a connected geometric dual, while in more
recent studies have shown that small deformations of the TFD double even
describe wormhole-like spacetimes [15][16][17]. The present approach aims to
address these issues in a more systematic way.

In the paper [18] the prescription of [3][19] was applied to a class of
$2+1$-dimensional wormhole-like spacetimes [20] [21], stressing the
relationship with the Hartle-Hawking (HH) type of construction [22]. The
construction presented in this article is in line with this earlier work, and
makes use of more recently developed results and tools [4, 23, 24, 25]

Other previous works with different mehods, have also studied the
decomposition of entangled states dual to multi-boundary wormholes in $2+1$d
(and genus $g=0$) in a puncture limit [26] [27].

The manuscript is organized as follows. In Sec 2 we build and present the main
prescription, and show how to derive it from the standard holographic recipes.
Part of this consists of the homology and equivalence hypothesis, which is
argued from all known examples of holographic gravity at the end of the
Section. In Sec 3 we show how the same method can be generalized to capture
all the non trivial spacetime topologies in $2+1$d, and precisely _which_ CFT
states are actually involved in the emergence mechanism, observing (as a
result) that the ingredient of quantum coherence is actually a property of
these states [12]. In Sec 4 we discuss better which is the actual meaning of
dual geometric interpretation of a given quantum state, since the dominant
saddle geometry depends on the basis which the boundary state is project onto;
and furthermore, clarify the role of the Hartle-Hawking wave functional in the
present framework. In Sec 5 we follow and describe in detail the steps of the
prescription for the paradigmatic case of black holes. Finally in Sec 6 we
exemplify how to compute the Schmidt form for a non trivial spatial topology
($g=1$) using the present methods, and find how the coefficients encode the
genus. Concluding remarks are collected in Sec 7

##  2 The prescription

The idea that the connected spacetime geometry is encoded in the entanglement
pattern can be realized more concretely by projecting the states in the basis
of tensor products of (generalized) $n$-particle states:

$|n\rangle_{x_{1},x_{2},\dots,x_{n}}\equiv O(x_{1})O(x_{2})\dots
O(x_{n})|0\rangle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (2.1)

where $x_{1}....x_{n}$ are points of the asymptotic boundary where the CFT is
defined and $O$ are primary operators. For simplicity, we only go to consider
the subspace generated by states (2.1) built with a scalar primary operator
$O$, of conformal weight $\Delta$, dual to a massive scalar field $\Phi(x)$
with mass $\mu^{2}=\Delta(\Delta-d)$ defined on the bulk. In other words, we
will project states onto the CFT-sector, dual to the Fock space associated to
the scalar field $\Phi$.

In the large-$N$ approximation all the matter fields, so as metric
fluctuations (gravitons), behave as a free non back-reacting scalar field [28,
29]. On a fixed background spacetime $M$, this field can be canonically
quantized and following the holographic BDHM dictionary [28, 29, 4], we can
identify the CFT operators with
$\hat{\Phi}(x)=\sum_{\alpha}\,f_{\alpha}(x)\,a_{\alpha}^{\dagger}\,+\,h.c.$
near the AdS boundary, where $f_{\alpha}(x)$ are the normalizable modes
(labeled by $\alpha$) that solve the equation of motion, and
$a_{\alpha}\,(a^{\dagger}_{\alpha}\,)$ are the standard annihilation
(creation) operators.

Therefore, taking products of $O(x)$ we schematically have

$|n\rangle\sim(a^{\dagger})^{n}|0\rangle,$ (2.2)

where the subindices $\alpha_{1},...\alpha_{n}$ were dropped out for
simplicity. This is the reason why we refer to these states as (generalized)
$n$-particle states, and a (holographic) Fock space ${\cal F}$ can be defined
as the direct sum on the $n$-particle spaces. It is worth emphasizing that to
large $N$ this basis (approximately) diagonalizes the Hamiltonian, so that the
states $|E_{n}\rangle$ can be represented in this approximation as exact AdS
spacetimes with $n$ free particles.

So then, we are going to project the states in a (holographic) Fock space,
i.e. to decompose them in the basis of tensor products of (generalized)
$n$-particle states as follows:

$\left|\Psi\right\rangle=\sum_{nm}\,\Psi_{nm}\;\left|n\right\rangle_{1}\otimes\left|m\right\rangle_{2}\leavevmode\nobreak\
\in{\cal F}_{1}\otimes{\cal F}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (2.3)

considering first the tensor product of two CFTs, dual to spaces time with two
asymptotic boundaries, for simplicity. By standard holographic arguments [8],
all the relevant information on the emergent classical geometry (distance and
topology) is encoded in the components of this expansion, which manifestly
express the entanglement between two quantum systems living on the boundaries
(fig. 2) [31].

The TFD state is a particular example whose components in this basis are also
related to the components of the propagator [24, 25] :

$\Psi_{nm}(\beta)=\left\langle n\right|U(0,\beta/2)\left|m\right\rangle.$
(2.4)

In this way the evolution operator characterizes a state of the boundary gauge
theory, and one can generalize the TFD vacuum to (thermal) excited states by
substituting [25]

$U(0,\beta/2)\leavevmode\nobreak\ \to\leavevmode\nobreak\
U_{\lambda}(0,\beta/2)\equiv{\cal T}\,e^{\int^{\beta/2}_{0}d\tau\;{\cal
O}(\tau,x)\,\lambda(\tau,x)}\leavevmode\nobreak\ ,$ (2.5)

which expresses the sourced CFT propagator in the Interaction Picture
(I.P.)(${\cal T}$ is the euclidean time ordering operator). Below, we will
show the detailed mechanism because this family of states is dual to classical
aAdS geometries with two asymptotic boundaries, and derive a formula to
compute the coefficients of (2.3).

Figure 2: This figure schematically shows the terms of the expansion (2.3).
The red diagrams describe the result (2.9)

Consider the standard GKPW holographic formula [10, 11], conveniently
expressed in a piece-wise form (regarding Fig 3):

Figure 3:  The figure above represents the Schwinger-Keldysh contour in the
imaginary time associated to the path integral $Z(\lambda_{1},\lambda_{2})$,
and below, we depicted the boundary manifold for the prescription (2.6)

$\langle 0|\;e^{\int^{0}_{-\infty}d\tau\;{\cal
O}(\tau,x)\,\lambda_{1}(\tau,x)}\,U_{\lambda}(0,\beta/2)\,e^{-\int_{\beta/2}^{\infty}d\tau\;{\cal
O}(\tau,x)\,\lambda_{2}(\tau,x)}|0\rangle\,=\int_{\partial{\cal
M}=\Sigma^{c}}D{\cal M}\,D\Phi\,e^{-I[{\cal M},\Phi]}\,$ (2.6)

where the (Euclidean) time ordering and the integrals on $x\in S^{d-1}$ on the
left hand side are implicit. This formula relates the sourced CFT generating
functional $Z_{\lambda}(\lambda_{1},\lambda_{2})$ with a gravitational
partition function on the r.h.s., where $I$ consists of the Einstein-Hilbert
action suplemented with the Gibbons-Hawking boundary term plus terms governing
the matter fields. The boundary manifold $\Sigma^{c}$ is given by the surface
$\Sigma\equiv S^{d-1}\times[0,\beta/2]$ where _the state_ (2.5) is defined,
whose boundaries are glued to two auxiliary (semi-infinite) intervals. By
adding a point at the infinity $\tau=\pm\infty$ these auxiliary pieces are
compactified to the cups (half spheres) $C_{1,2}$ of figure 3, where one can
define non-vanishing sources $\lambda_{1},\lambda_{2}$. So on the left, the
operator (2.5) has been projected onto auxiliary states defined (in the I.P.)
as:

$|\lambda_{1,2}\rangle\equiv\,\,e^{\int_{C_{1,2}}\;{\cal
O}(\tau,x)\,\lambda_{1,2}(\tau,x)}|0\rangle.$ (2.7)

The right hand side of (2.6) stands for the standard gravitational path
integral on all the aAdS geometries ${\cal M}$ whose boundary is
$\Sigma^{c}\equiv C_{1}\cup\Sigma\cup C_{2}$ and Dirichlett b.cs. for all the
bulk fields $\Phi(x)$. One can evaluate it in the large $N$ approximation as
$Z_{\lambda}(\lambda_{1},\lambda_{2})\approx e^{-I(M,\phi(x))}$ by taking the
dominant (Euclidean) geometry saddle: $M$ 222Trivially, the exact Euclidean
AdSd+1 spacetime is a solution as long as the back-reaction due to the sources
$\lambda$ can be neglected., and the classical fields $\phi(x)$, that solve
the eqs.o.m. in terms of the boundary values $\lambda_{1,2}$ on $C_{1,2}$, and
$\lambda$ on $\Sigma$. This problem is well posed in general, and it is a
crucial part of the recipe to calculate the components of the expansion (2.3)
to large $N$.

It is worth remembering that to large $N$, the states (2.7) correspond to
coherent states in the bulk picture of the Hilbert space, i.e:
$|\lambda_{1,2}\rangle\propto
e^{\sum_{\alpha}\;\lambda_{1,2}(\alpha)a^{\dagger}_{\alpha}}|0\rangle\,$ (see
ref. [4] for details). This is useful for our purposes since the Fock
(orthonormal) basis can be systematically generated from these states by
taking $n$ derivatives with respect to $\lambda_{1,2}$ in $n$ arbitrary points
of the cups. Schematically:

$|n\rangle_{i}=\frac{(a_{i}^{\dagger})^{n}}{\sqrt{n!}}|0\rangle\sim\left.\frac{\partial^{n}}{\partial\lambda_{i}^{n}}|\lambda_{i}\rangle\right|_{\lambda_{i}=0}\qquad
i=1,2.$ (2.8)

In fact, differentiating $n_{1}+n_{2}$ times with respect to the sources
$\lambda_{1,2}$ to both sides of (2.6) in different points of $C_{1,2}$, and
using (2.4) and (2.5) to identify the l.h.s. in the limit
$\lambda_{1}=\lambda_{2}=0$, we derive our main prescription:

$\Psi_{n_{1}n_{2}}(\beta,\lambda)=\left.\frac{\partial^{n_{1}+n_{2}}}{\partial\lambda_{1}^{n_{1}}\partial\lambda_{2}^{n_{2}}}Z_{\lambda}(\lambda_{1},\lambda_{2})\right|_{\lambda_{1}=\lambda_{2}=0}$
(2.9)

where we have used (2.4) and (2.5) to identify the lhs in the limit
$\lambda_{1}=\lambda_{2}=0$. Recall that
$\partial\lambda_{i}^{n_{i}},\leavevmode\nobreak\ \leavevmode\nobreak\ i=1,2.$
denotes $\partial\lambda_{i}(x_{1})\dots\partial\lambda_{i}(x_{n_{i}})$ where
the points $x_{1}\dots x_{n_{i}}$ must finally be valued on the gluing spheres
$\partial C_{i}\sim S^{d-2}$. According to this result, the
entanglement/Schmidt components are given by functions of correlations between
insertions of the operators $O(x)$ on the regions $C_{1,2}$, through the
geometry $M$.

This is a manifest relation between the components of the (TFD) state and
correlation functions computed in the classical geometry dominating
$Z_{\lambda}$, and it has been derived by _only_ assuming the GKPW
prescription. In other words, the formulas (2.6) and (2.9) explicitly show how
certain entangled bi-partite form of the state is dual to a classical
spacetime from the basic dictionary.

In this sense, we achieve as a result that deformations from the TFD state by
$\lambda\neq 0$ also have geometric dual interpretation. Then according to
(2.8), by considering a perturbative expansion in $\lambda$ of the state
(2.5), one systematically captures the deviations from the TFD vacuum
described by local insertions $O(x)$ at points of $\Sigma$ [15, 16, 17].

Homology and Equivalence Hypothesis

The formulae derived above require a complementary constraint to link the
space(time) topology with the data $\Sigma,\lambda$ characterizing the state.

Let consider a classical Lorentzian aAdS spacetime $M_{L}[\Sigma_{0}]$, which
evolves from an initial totally geodesic spacelike hypersurface $\Sigma_{0}$
(i.e, its extrinsic curvature $K_{ij}=0$), one would like to describe as an
entangled state of the boundary field theory by computing the Schmidt
coefficients.

Then, in the Hartle-Hawking (HH) approach to AdS quantum gravity [22, 2, 3, 4,
25], the initial wave functional is a path integral on the Euclidean compact
manifolds ${\cal M}[\Sigma_{0}]$ bounded by the (past) asymptotic boundary
$\Sigma$ and an initial surface $\Sigma_{0}$, which is also totally geodesic
with respect to ${\cal M}[\Sigma_{0}]$, and
$\partial\Sigma=\partial\Sigma_{0}$ . These geometries ${\cal M}[\Sigma_{0}]$
can be called Euclidean cobordisms between the conformal boundary $\Sigma$ and
$\Sigma_{0}$ [32], which are _homologous_ surfaces.

In addition to this, we will assume that $\Sigma$ is topologically equivalent
to $\Sigma_{0}$, as can be observed in many examples. In the AdS black hole,
the initial _space_ $\Sigma_{0}$ is the connected (AdS) Einstein-Rosen
wormhole, equivalent to $\equiv\Sigma\sim S^{d-1}\times[0,\beta/2]$, and
anchored by two disconnected spheres $\partial\Sigma\sim S_{(1)}^{d-1}\sqcup
S_{(1)}^{d-1}$ [2] (see fig 5 (left)). The other known examples are general
aAdS3 spacetimes that can be described as a foliation
$\Sigma_{(b,g)}\times\mathbb{R}$, with metrics
$ds^{2}=d\tau^{2}+\cosh^{2}\tau\,d\Sigma_{(b,g)}^{2}$, where $\Sigma_{(b,g)}$
is an arbitrary 2d Riemann surface such that both $\Sigma_{0}$ (totally
geodesic) and $\Sigma$ are copies of $\Sigma_{(b,g)}$ at $\tau=0$ and
$\tau=-\infty$ respectively.

This is what we will refer to as the hypothesis of _homology and equivalence_
, which almost all examples of the AdS/CFT literature seem fulfill, and it
will be complementary to the formulas (2.6) and (2.9) in order to connect an
arbitrary topology of a spatial slice $\Sigma_{0}$ with the Schmidt
coefficients 333The HH wave functional for a global $AdS_{d+1}$ spacetime, for
instance, is given by a path integral on the Euclidean cobordisms between a
cup $C\equiv\Sigma$, which is conformal to the (euclidean) boundary, and an
equivalent disk $C\sim\Sigma_{0}$ anchored by the (blue) sphere $S^{d-1}$ (see
one of the two cups drawn in the fig. 2 )[3]..

In the current prescription, this constraint works as an important part of the
recipe to build up some specific space(time) geometry from quantum
entanglement: by defining $\lambda(x)$ as the pullback of an Euclidean
classical solution of the bulk fields (a saddle of the HH wave functional)
$M_{0},\phi_{0}$ onto $\Sigma$, which unambiguously defines the state (2.5).
The final step is to use the above formulas ((2.6) and (2.9)) to compute the
Schmidt coefficients explicitly. In the next Section, we will show how this
recipe generalizes to capture arbitrary number of boundaries and non-trivial
topologies. In Sec. 4 this will be elaborated more in depth.

Finally, it is worth noticing that the method formulated here connects the
Schwarszchild-AdS spacetime with the TFD state in all details. It is similar
to the original argument of [2] about homology, but differs by the fact that
the formula (2.9) explicitly provides the state from direct calculations in
the bulk (see Sec. 5), which is key to further generalizations.

Observe also that the geometries $M_{0}$ and $M$, the saddles of the HH wave
functional and of $Z_{\lambda}(\lambda_{1},\lambda_{2})$ (r.h.s. of (2.6))
respectively, can be different although the state is the same. This motivates
the discussion of Sec. 4. on the connection between these geometries other
dual descriptions of a same CFT state. In Sec 4 it also will be argued that
some of this restriction must be relaxed in suitable contexts.

##  3 More general states with holographic dual

The formula (2.6) can be generalized to arbitrary number of boundaries and
topologies of $\Sigma$ such that the operator $U_{\lambda}$ must be
substituted by a more general object (See fig 4). For instance, in AdS2+1/CFT2
the state is characterized by the operator $U_{b,g,\lambda}$ on behalf of
$U_{\lambda}(\beta/2)$, where $g$ is the genus and $b$ the number of
boundaries of a general Riemman surface $\Sigma_{b,g}$. Thus, we can see that
all the states described as CFT path integrals on these surfaces (with local
sources $\lambda$), expressed by

$U_{b,g,\lambda}(\eta_{1}\dots\eta_{b})=\int_{\eta_{1}\dots\eta_{b}}[D\eta]\;e^{-S_{CFT}[\eta]\,+\,\int_{\Sigma_{(b,g)}}\lambda\,O}\;,$
(3.1)

where $\eta_{1}\dots\eta_{b}$ denote the Dirichlet boundary conditions for the
CFT fields on the boundaries of $\Sigma_{(b,g)}$, _are dual_ to a bulk
geometry. Since a prescription similar to (2.6) can be formulated, one can
find the dominant saddle $M$ of the path integral to its right.

Figure 4:  The prescription (2.6) for more general CFT states (eq. (3.1)) is
depicted: gluing the pieces on the blue spheres yields a surface homologous
and equivalent to $\Sigma^{c}_{g}$.

In fact, according to the state-operator correspondence in CFT’s in the path
integral representation, eq (3.1) defines a very general class of CFT states,
and consequently, a holographic formula like (2.6) can always be written to
make contact with a dual geometry (see fig 4). The precise recipe is to glue
$b$ cups $C_{1}...C_{b}$ (with sources $\lambda_{1}\dots\lambda_{b}$) on each
boundary of $\Sigma_{(b,g)}$, which yields the compactified boundary
$\Sigma^{c}_{g}$. Therefore, the generalization of the prescription (2.6) can
be expressed as

$\int[D\eta]\;e^{-S_{CFT}[\eta]\,+\,\int_{\Sigma_{(b,g)}}\lambda
O+\,\int_{C_{1}}\lambda_{1}O+\dots+\,\int_{C_{b}}\lambda_{b}O}\,=\int_{\partial{\cal
M}=\Sigma_{g}^{c}}D{\cal M}\,D\Phi\,e^{-I[{\cal M},\Phi]}\,,$ (3.2)

where the left path integral sums over all the CFT field configurations on the
closed Riemann surface $\Sigma^{c}_{g}$. This is nothing but the _wave
functional_ $U_{b,g,\lambda}(\lambda_{1}\dots\lambda_{b})$ of the state (3.1)
in the particular basis of states (2.7), which is generally overcomplete [4,
25].

Then on the right hand side, one can find the dual geometry by finding a
classical aAdS solution $M,\phi$ for the specific boundary conditions:
$\Sigma^{c}_{g}\,,\,\lambda,\lambda_{1}\dots\lambda_{b}$. If there is more
than one saddle, the geometric dual is understood to be the dominant one.

Finally by taking $n_{1}+...+n_{b}$ functional derivatives with respect to
$\lambda_{1}\dots\lambda_{b}$ on different points of the respective $C_{i}$s
to both sides of (3.2), we obtain a expression similar to (2.9) for the
coefficients $\Psi_{n_{1}\dots n_{b}}$, which are correlation functions
computed on the spacetime defined as the solution $M$ in the limit
$\lambda_{1}\dots\lambda_{b}\to 0$. So the gravity dual of the state in this
basis might be viewed as the Witten diagram itself (with $n_{1}+...+n_{b}$
external lines) within such background geometry. In the forthcoming section,
we will discuss more these statements in the context of the state
representations and wave functionals.

As stated in the previous section, our prescription consists of the formulas
(3.2) and the corresponding generalization of (2.9) to $b$ boundaries,
complemented with the constraint of topological equivalence argued before. It
provides the components of the $b$-partite entangled state, whose (Lorentz)
gravity dual evolves from an initial _spacelike hypersurface $\Sigma_{0}$_,
which is totally geodesic and _equivalent to_ $\Sigma_{(b,g)}$, and such that
$\partial\Sigma_{(b,g)}=\partial\Sigma_{0}$. Although (3.2) is expressed in
$2+1$d for concreteness, the prescription holds for arbitrary spacetime
dimension and topology of the compact surface $\Sigma$.

As a further result, it is worth noticing that a general property of the
states (3.1) (which have geometric dual), is the quantum coherence from the
bulk point of view, in the large-$N$ approximation [12]. This can be verified
using the [28] recipe, and it has been explicitly checked in the cases of pure
AdS and BTZ geometries [4, 25].

##  4 Dual geometric interpretation and the Hartle-Hawking wave functional

In general it can be established that a state has dual geometric
interpretation if, when projected onto some suitable basis, its wave
functional is dominated by some classical geometry.

Going further, although the duality to a classical geometry is a feature of
the state, the specific dual geometry shall depend on the basis which the
state is projected on. For instance as explained above, eq. (3.2) shows this
in the particular basis of all the field configurations $\lambda_{i}(x)$ on
the cups $C_{i}$; and the generalization of the formula (2.9) to $b$
boundaries expresses the projection of the state onto the complete Fock basis
(2.7). So consistently, the states (3.1) satisfies the previous definition,
and has (dual) geometric interpretation.

In such a context, the Hartle-Hawking (HH) w.fs. are nothing but the
projection of the states (3.1) onto the set of configurations of all the bulk
fields $\\{h_{\mu\nu}(x),f(x),x\in\Sigma_{0}\\}$ on certain surfaces
$\Sigma_{0}$ anchored by $\partial\Sigma$. It is defined as the path integral
involving the local fields of the gravitational theory and compact topologies
[22]:

$\Psi_{(\lambda,\Sigma)}[\Sigma_{0},f]=\int_{\partial{\cal
M}=\Sigma_{0}\cup\Sigma}D{\cal
M}\,\int_{\Phi(\Sigma)=\lambda,\Phi(\Sigma_{0})=f}D\Phi\,\leavevmode\nobreak\
\leavevmode\nobreak\ \,e^{-I[{\cal M},\Phi]}\qquad.$ (4.1)

Let $M[\Sigma_{0}]$ denote the dominant saddle geometry, which is a function
of all the boundary conditions: $\phi(x)\equiv\lambda(x)$ on $\Sigma$ 444Near
the asymptotic surface $\Sigma$, the metric is locally AdS, and the induced
metric and matter fields are $h_{\mu\nu}(x),f(x)$ on $\Sigma_{0}$, but here we
would like to emphasize the importance of the dependence in the choice of
$\Sigma_{0}$.

Notice that an important general property that characterizes the configuration
basis is that the surfaces $\Sigma_{0}$ must complement $\Sigma$ to form a
_closed_ (compact) manifold $\Sigma_{0}\cup\Sigma$, in order to have a well
posed Dirichlet problem to determine $M[\Sigma_{0}]$. This explains the
homology statement. Moreover, $\Sigma_{0}$ is to be identified with the
initial spatial surface of the proper Lorentzian spacetime
$M^{L}[\Sigma_{0}]$, which is generally obtained by analytical continuation
from the Euclidean saddle geometry $M[\Sigma_{0}]$.

On the other hand, different elements of the basis give place to different
dominant geometries since the Dirichlett b.cs. are in fact different; however
in order to define a classically more significant HH geometric dual, a sort of
double saddle point analysis can be done since the wave functional has a peak
on

$\frac{\delta\Psi_{(\lambda,\Sigma)}}{\delta h_{\mu\nu}}=0\qquad,$

which -to large $N$\- implies that the amplitude maximizes in a saddle such
that $(K_{\mu\nu}-Kh_{\mu\nu})|_{\Sigma_{0}}=0$, which becomes a Newman b.c.
on the surface $\Sigma_{0}$, trivially fulfilled by choosing a totally
geodesic surface $K_{\mu\nu}|_{\Sigma_{0}}=0$. In this case $\Sigma_{0}$ is a
moment of time reflection symmetry, and let denote the corresponding saddle as
$M_{0}$.

This geometry is particularly useful for many purposes; e.g, in real-time
applications [2, 3, 22], this is the solution to be glued to the Lorenzian
piece $M^{L}[\Sigma_{0}]$, and the (squared) norm of the state
$\langle\Psi|\Psi\rangle$ is computed by a path integral whose saddle point is
the geometry obtained by joining two copies of $M_{0}$ across their common
boundary $\Sigma_{0}$ [4]. This has an obvious time-reflection symmetry with
respect to $\Sigma_{0}$ [32].

Observations regarding the equivalence constraint and the gravitational space
of states

In the most familiar example of the state (2.5), there are two very different
HH wave functionals as we have two possibilities for totally geodesic
$\Sigma_{0}$ such that $\Sigma_{0}\cup\Sigma$ is closed, namely:

1. $\Sigma_{0}\sim S^{1}\times I\,$ where $I$ is a real interval, then $\Sigma_{0}$ is connected, homologous, and _equivalent_ to $\Sigma$, and the saddle geometry $M[\Sigma_{0}]$ corresponds to the BTZ solution (see fig 5 (left)). Notice that the manifold $M[S^{1}\times I]$ is a solid thorus, and there are curves which cannot be continuously contracted in it.

2. $\Sigma_{0}\sim D^{2}\sqcup D^{2}$, it consists of a disconnected pair of discs, which are homologous _but not_ equivalent to $\Sigma$ as claimed in Sec 2, and the solution in the interior $M[D^{2}\sqcup D^{2}]$ corresponds to a solid cylinder filled with the global AdS solution (referred to as thermal AdS ). In contrast with the previous case, every closed curve in $M[D^{2}\sqcup D^{2}]$ is contractible.

This is the simplest context where we observe that the equivalence constraint
does not hold; therefore, it depends on the regime/moduli parameters that
characterize the state, i.e, the asymptotic surface $\Sigma$, and $\lambda$.

As is well known, this feature is described by the Hawking-Page (HP)
transition, governed by whether some of these geometric saddles dominates the
partition function

$Z(\beta)\equiv\text{Tr}\;U(\beta/2)U^{\dagger}(\beta/2)=\langle\Psi|\Psi\rangle.$
(4.2)

Notice that this is a sum over the probabilities of the different classical
geometries (1\. 2., in this case).

Nevertheless, as argued above, even at the level of amplitudes one can compute
all the wave functionals for each possible choice of $\Sigma_{0}$, and
determine which one dominates in each regime. The result should then be
consistent with the standard description of the HP-transition.

Thereby, the bulk representations of the state should be formally captured as
a ”sum” over the choices of (topologically different) $\Sigma_{0}$ such that
$\Sigma_{0}\cup\Sigma$ is closed. Thus, in order to describe this consistently
with (4.2), two obvious requirements must be met _to large $N$_:

I - There are no cross terms in (4.2), that is, the (gravity) Hilbert space is
such that the configuration basis $\\{(\Sigma_{0},f(x))\,\forall
f(x)/x\in\Sigma_{0}\\}$ on topologically different $\Sigma_{0}$’s are
orthogonal, at least at the semiclassical level, schematically:

$\Sigma_{0}\neq\Sigma^{\prime}_{0}\Longrightarrow\langle\Sigma_{0},f|\Sigma^{\prime}_{0},f^{\prime}\rangle\approx
0,$

while for two equivalent slices $\Sigma_{0}=\Sigma^{\prime}_{0}$, the inner
product reduces to the usual definition for fields $\langle
f|f^{\prime}\rangle_{\Sigma_{0}}$.

II - Only two topologies (1\. 2.) contribute to (4.2).

A natural and consistent proposal that collects these facts is that the states
in the bulk Hilbert space, are represented as a direct sum over all the
topologically inequivalent initial slices $\Sigma_{0}$. For a generic state
$\Psi_{(\lambda,\Sigma)}\in{\cal H}^{\otimes b}$, where $b$ is the number of
connected components of $\partial\Sigma$:

$\Psi_{(\lambda,\Sigma)}=\bigoplus_{\Sigma_{0}}\;\Psi[\Sigma_{0}]\qquad/\;\Sigma_{0}\cup\Sigma=\text{closed}\;\;,\;\partial\Sigma_{0}=\partial\Sigma,$
(4.3)

which, can be represented as (3.1) in the boundary theory. This can be
alternatively formulated by claiming that the identity operator
$\mathbb{1}_{b}$ on ${\cal H}^{\otimes b}$, is represented in the bulk-gravity
Hilbert space as:

$\mathbb{1}_{b}^{(grav)}\equiv\sum_{\Sigma_{0}}\;\left(\sum_{f}\,|\Sigma_{0},f\rangle\langle\Sigma_{0},f|\right)\,.$
(4.4)

This formally _projects_ the state $|\Psi_{(\lambda,\Sigma)}\rangle$ onto the
topologically different spatial slices, and the standard description of the HP
transition turns out be manifest as this expression is inserted into the rhs
of (4.2)

In the specific case discussed above ($d=1,b=2$) eq. (4.3) becomes:

$\Psi_{(\lambda,\Sigma)}=\;\Psi[S^{1}\times I]\oplus\Psi[D^{2}\sqcup
D^{2}]\oplus\;\dots,$ (4.5)

where $\dots$ denote direct sum on surfaces $\Sigma_{0}$ with higher genus
$g(\Sigma_{0})\geq 1$, that because of (II) should be negligible; for
instance, their wave functionals might depend on the genus, _e.g._ as $\sim
e^{-cg^{q}}\;,q\in\mathbb{N}$, where $c$ is a high positive number

As conclusion, note that one could relax the equivalence hypothesis from our
prescription of Sec. 2, and evaluate all the wave functionals according to eq.
(4.3) to compare them. Moreover, the equivalence property can be _derived_ in
this framework if the component $\Sigma_{0}\equiv\Sigma$, i.e.
$\Psi_{(\lambda,\Sigma)}[\Sigma,f]$, finally dominates the expansion (4.3) at
least in a suitable regime of $\lambda$, and moduli of $\Sigma$. This specific
subject shall be further investigated in a future work.

Reverse space(time) engineering from CFT states

The homology and equivalence constraint defines a one to one holographic
dictionary between the spatial topology and the asymptotic boundary $\Sigma$,
in the proper range of parameters characterizing the state (read the
discussion in the next section). According to this, one can reciprocally
reconstruct the initial data on a spatial slice of the spacetime from a
general state defined in the CFT theory as (3.1), then, and one
straightforwardly obtains the HH (main) geometric dual by solving a
classically well posed problem.

In fact one must solve the Euclidean Einstein equations for a compact $M_{0}$,
the cobordism between $\Sigma_{(b,g)}$ and the equivalent (and totally
geodesic) Riemann surface $\Sigma_{0}$. They join on the $b$ holes to form a
closed surface, where one gives Dirichlett b.cs. $\phi\equiv\lambda$ (and aAdS
metric) on $\Sigma_{(b,g)}$, and the canonical momenta
$\Pi_{\Phi}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\ K_{ij}=0$ on
$\Sigma_{0}$.

Otherwise, regarding the framework discussed above, one could drop the
equivalence assumption and compute the wave functions
$\Psi_{(b,g)}[\Sigma_{0}]$ in the semi-classical approximation, _for all_
$\Sigma_{0}$, by evaluating the action on the corresponding classical
solutions $M_{0}\leavevmode\nobreak\ /\,\partial
M_{0}=\Sigma_{(b,g)}\cup\Sigma_{0}$. The expectation is that
$\Sigma_{(b,g)}\sim\Sigma_{0}$ dominates in the proper region of parameters.

Figure 5: (left) The eternal black hole is represented: we see that
$\Sigma_{0}$ is a totally geodesic surface, homologous to $\Sigma$ and they
both share the boundaries (red points). The shaded (blue) region shows the
euclidean space time dominating the Hartle-Hawking wave functional. (right)
The same region is depicted regarding Secs 2 and 4: it shows that the HH-
saddle $M_{0}$ described in Sec. 4, the shaded blue region, is topologically a
solid thorus $D^{2}\times S^{1}$ containing non contractible curves.

##  5 The preliminary example: the TFD state from bulk computations

The goal of this section is to show how the prescription works in the simplest
case: the AdS black hole spacetime. Since the formulas (2) (2.9) were derived
from the fundamental holographic prescriptions [10, 11], we do not need to
check them explicitly. Nevertheless, the holographic computation indicated on
the r.h.s. of (2.9) can be illuminating about many conceptual and technical
details; for instance, the definition and treatment of the Fock basis, its
analytical extensions, the central role of the two-point functions in
characterizing the Schmith coefficients, as well as some general remarks on
the method.

To high black hole mass/temperature ($\beta<1/T_{Hawking-Page}$), the HH wave
functional must be dominated by the euclidean geometry shown in the lower part
of fig. 5 (left), and the initial surface $\Sigma_{0}$ is totally geodesic
($K_{ij}=0$) with respect to both: $M_{0}$ and $M_{L}$. This surface is the
(aAdS) Einstein-Rosen wormhole $\Sigma_{0}\sim S^{d-1}\times I$, whose
boundary are two disconnected spheres $\partial\Sigma\sim S_{(1)}^{d-1}\sqcup
S_{(2)}^{d-1}$ [2] (fig. 5 (right)).

Precisely, the first step of the recipe of Sec 2 is to use the equivalence
constraint to put $\Sigma\equiv\Sigma_{0}$ in the formula (2). Remarkably, to
make the present calculation we will not use any other detail on the Euclidean
AdS-Schwarschild solution $M_{0}$.

Then, following the prescription (2.6), one must find the (dominant) saddle
geometry $M$ filling $\Sigma^{c}\equiv C_{1}\cup\Sigma\cup C_{2}$ (see fig.
3). This is obtained by smoothly gluing the three pieces $M_{1},N,M_{2}$ of
global AdS together, on totally geodesic discs denoted by ${\cal S}_{1,2}(\sim
D^{d})$, such that $\partial M_{1,2}=C_{1,2}\cup{\cal S}_{1,2}\;,\partial
N={\cal S}_{1}\cup\Sigma\cup{\cal S}_{2}$ are closed. Taking the fundamental
state $\lambda\equiv\Phi|_{\Sigma}=0$, the resulting geometry $M$ is the
Euclidean $AdS_{d+1}$ 555Actually, it is slightly deformed by the auxiliary
sources $\lambda_{1,2}$.. Therefore, from the general solution for a massive
scalar field in $AdS_{d+1}$ in global coordinates, one can compute the
boundary-to-boundary correlators following the standard methods.

For our present purposes it is convenient to obtain the Euclidean two-points
function on the cylinder of fig. 4 by analytical continuation, from the
propagator in the Lorentzian $AdS_{d+1}$ spacetime. By using a suitable system
of global coordinates (see ref. [33]), the Feynman propagator is obtained by
integrating on the standard contour in the complex $\omega$-plane [3, 33]:

$\langle 0|{\cal T}_{L}\,{\cal O}(t,\Omega){\cal
O}(t^{\prime},\Omega^{\prime})|0\rangle=\sum_{lm}\sum_{k=0}^{\infty}{\cal
N}_{lmk}\,Y_{lm}(\Omega)Y_{lm}^{\ast}(\Omega^{\prime})\,e^{-i|t-t^{\prime}|\omega_{kl}}\;,$
(5.1) ${\cal
N}_{lmk}(\Omega,\Omega^{\prime})=2\Delta\;\frac{2^{(\Delta-d)}\,\Gamma(1-\nu)}{\Gamma(1+\nu)}\,\frac{(-1)^{k}}{k!}\frac{\Gamma(k+l+\Delta)}{\Gamma(k+l+\frac{d}{2})\Gamma(-(k+\nu))}\;,$

where $\Omega$ denotes the coordinates on the $d-1$ unit sphere 666The
spherical harmonics $Y_{lm}$ on $S^{d-1}$ satisfy
$\nabla^{2}Y_{lm}=-q^{2}Y_{lm}$ with $q^{2}=l(l+d-2),\,l=0,1,\ldots$, as well
as standard relations of orthogonality and normalization.,
$\nu\equiv\sqrt{\mu^{2}+d^{2}/4}$, and $\omega_{kl}\equiv 2k+l+\Delta\,$,
$k,l=0,1,2,\ldots$ are the normal frequencies.

By performing the analytical extension

$t^{\prime}\equiv t_{1}-i\tau_{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ t\equiv t_{2}-i(\tau_{2}+\beta/2)$

such that $\tau_{1}\leq 0$ and $\tau_{2}\geq 0$, describe the positions of the
insertions on the respective cups $C_{1,2}$. Using that the CFT vacuum is
invariant under arbitrary (complex) time translations
$U(z)|0\rangle=0\,,\forall z\in\mathbb{C}$, we obtain:

$\displaystyle\langle 0|{\cal
T}\,O(t_{1}-i\tau_{1},\Omega_{1})U(\beta/2)O(t_{2}-i\tau_{2},\Omega_{2})|0\rangle=$
$\displaystyle=\sum_{lm}\sum_{k=0}^{\infty}{\cal
N}_{lmk}\,Y_{lm}(\Omega_{1})Y_{lm}^{\ast}(\Omega_{2})\,e^{-\omega_{kl}\beta/2}e^{-i(t_{1}-t_{2})\omega_{kl}}e^{(\tau_{1}-\tau_{2})\omega_{kl}}\,.$
(5.2)

Since the regions $C_{1,2}$ are auxiliary and would be causally independent,
the time ordering of the real time variables $t_{1,2}$ does not matter for the
purpose here, hence the modulus has been conveniently dropped out from this
expression.

Regarding the formula (2.9), the l.h.s. of this equation exactly expresses the
matrix elements of the operator $U(\beta/2)$ in the (analytically extended)
Fock basis $O(t_{1,2}-i\tau_{1,2},\Omega_{1,2})|0\rangle$, but in fact, the
most familiar TFD form of these coefficients is to be recovered when one
(Fourier) transforms them into the space of real frequencies/energies
$\omega_{1},\omega_{2}$; thereby, by multiplying this by
$e^{i(t_{1}-t_{2})\omega}$ and integrating out the auxiliary variables
$t_{1},t_{2}$, we obtain:

$\displaystyle\Psi_{11}(\omega_{1},\Omega_{1};\omega_{2},\Omega_{2})=\langle
0|\,O_{\omega_{1}}(-i\tau_{1},\Omega_{1})U(\beta/2)O_{\omega_{2}}(-i\tau_{2},\Omega_{2})|0\rangle=$
$\displaystyle\sum_{lm}\sum_{k=0}^{\infty}{\cal
N}_{lmk}\,Y_{lm}(\Omega_{1})Y_{lm}^{\ast}(\Omega_{2})\,e^{-\omega_{kl}\beta/2}\,e^{(\tau_{1}-\tau_{2})\omega_{kl}}\,\delta(\omega_{kl}-\omega_{1})\delta(\omega_{kl}-\omega_{2})\,.$
(5.3)

We could achieve the same expression by using the BDHM dictionary [28, 29]

$O(t-i\tau,\Omega)=\sum_{lm}\sum^{\infty}_{k=0}\,\,\left(O_{\omega_{kl}}(-i\tau,\Omega)e^{-i\omega_{kl}t}+h.c.\right)=\qquad\qquad$
$=\sum_{lm}\sum_{k=0}^{\infty}\int^{\infty}_{0}d\omega\,\left(\,O_{\omega}(-i\tau,lm)\,Y^{\ast}_{lm}(\Omega)\,e^{-i\omega
t}\delta(\omega_{kl}-\omega)+h.c.\right)\;\;,\,\omega_{kl}>0\,.$

Regarding the initial discussion of Sec. 2 on the Fock spaces, and the
relation (2.4), eq. (5) expresses the components of the state in the basis:

$O_{\omega_{kl}}(-i\tau,\Omega)|0\rangle=\sum_{lm}\,e^{-\tau\omega_{kl}}\;Y_{lm}^{\ast}(\Omega)\,{\cal
N}^{1/2}_{lmk}\,\,|\omega_{kl},lm\rangle\;\left(=\sum_{lm}\;Y_{lm}^{\ast}(\Omega)\,\;\,O_{\omega_{kl}}(-i\tau,lm)|0\rangle\right)\,,$
(5.4)

where $|lm\rangle$ is nothing but the standard (orthonormal) basis of the
Hilbert space of one-particle on the sphere $S^{d-1}$. It is worth remembering
that in addition to the complex conjugate, the corresponding _bra_ involves
the Euclidean time reflection $\tau\to-\tau\,$ [4]. Notice from the definition
of the basis (5.4), that the pre-factor $e^{-\tau\omega}$ can be removed by
choosing $\tau=0$. This observation plays a role in our prescription and
applies to more general cases (e.g. Sec 6), where one evaluates the Euclidean
propagator on the boundaries of $C_{1,2}$.

Thereby, (5) can be expressed as a state in the sector $n_{1}=n_{2}=1$ of the
Fock space in this basis, resulting its Schmidt form:

$|\Psi_{11}\rangle=\sum_{lm}\sum_{k=0}^{\infty}\,{\cal
N}_{lmk}\,\,e^{-\omega_{kl}\beta/2}\,\,|\omega_{kl},lm\rangle_{1}|\omega_{kl},lm\rangle_{2}\,.$
(5.5)

Comparing this with (5.1) we see that all the information of this manifestly
entangled expansion is captured in the (Euclidean) two-points function. For
the simplest case of the BTZ black hole ($d=2+1$), this adopts a simple form:

$\Psi_{11}(\varphi_{1},\varphi_{2})=\langle 0|{\cal
T}\,O(0,\varphi_{1})U(i\beta/2)O(0,\varphi_{2})|0\rangle=\frac{(\Delta-1)^{2}}{2^{\Delta-1}\pi}\Big{(}\cosh(\beta/2)-\cos(\varphi_{1}-\varphi_{2})\Big{)}^{-\Delta}\,,$
(5.6)

where $\tau_{1,2}=0$, and $\varphi_{1,2}$ are the coordinates on the circles
$b_{1,2}\sim S^{1}$. This interprets as the wave function of an entangled pair
of particles created at the cups $C_{1,2}$, and in the large-$N$
approximation, the coefficients $\Psi_{nn}$ of the expansion (2.3) is a
product of $n$ factors like this. As a check for very large $\beta$, the
Schmith coefficients reduce to $n$ powers of it

$\Psi_{nn}\propto e^{-n\Delta\beta/2}$ (5.7)

which agrees with the TFD component of the $n\Delta$-energy level, since to
low temperature, only contribute pairs of particles in the fundamental state
$\omega\approx\Delta$.

This shows how the state dual to a AdS-Schwarschild spacetime, the TFD double,
can be straightforwardly found by holographic computations (eq. (2.9)).

##  6 Characterizing entanglement for non-trivial spatial topology

In 2+1 spacetime dimensions, the spatial slices of a BTZ black hole have a
cylindrical topology, while in the wormholes the spatial slices are general
two-dimensional Riemann surfaces with $b$ boundaries (holes). From each
asymptotic region, these geometries look like the BTZ solution, and the other
boundaries as well as the non-trivial topology are always hidden behind a
horizon [34].

The goal of this final section is to estimate the SD coefficients of
spacetimes with non-trivial spacelike topology, in order to characterize them
through the entanglement pattern between the quantum systems defined on the
asymptotic boundaries. As an application of the presented method, let us study
the simplest example: the state of a bipartite quantum system, whose $2+1$d
dual spacetime starts from a spatial hypersurface $\Sigma_{0}$ with a handle
($g=1$). This solution can be properly interpreted as a _two-way wormhole_.

According to the homology and equivalence hypothesis, in the appropriate
regime of parameters (moduli), $\Sigma_{0}$ shall be conformal to a surface
topologically equivalent to $\Sigma_{(2,1)}$ ($b=2$, $g=1$), and anchored by
two disconnected circles: $\partial\Sigma_{(2,1)}=S^{1}\sqcup S^{1}$. In this
example, we take $\lambda|_{\Sigma_{(2,1)}}=0$ for simplicity, which can be
interpreted as the fundamental state related to that topology.

By gluing $\Sigma_{2,1}$ with two cups on its boundaries, the resulting closed
surface is equivalent to a thorus:

$\Sigma^{c}_{2,1}\equiv T^{2}=S^{1}\times S^{1}$

As highlighted in the TFD-state example, a computational advantage of the
proposed recipe is that the Hartle-Hawking geometry $M[\Sigma_{0}]$, filling
$\Sigma_{0}\cup\Sigma_{(2,1)}$, might be unknown. What we really need to
compute the SD components is the saddle geometry $M$ that fits into
$C_{1}\cup\Sigma_{(2,1)}\cup C_{2}$ and dominates the rhs of eq. (3.2).

The Euclidean bulk geometry $M$ can be built up by joining two pieces
$M_{1,2}$ such that $\partial M_{1,2}\equiv C_{1,2}\cup D^{2}$, to $N$ whose
boundary is $D^{2}\cup\Sigma_{(2,1)}\cup D^{2}$, glued across the common
totally geodesic surfaces ${\cal S}_{1,2}\sim D^{2}$ ($K_{\mu\nu}|_{{\cal
S}_{1,2}}=0$). So we have:

$M\equiv M_{1}\cup N\cup M_{2}$

where all the pieces are locally AdS3, then $\partial M\equiv M_{1}\cup N\cup
M_{2}=C_{1}\cup\Sigma_{(2,1)}\cup C_{2}$, which is _conformal_ to
$\partial\tilde{M}=T^{2}$. Therefore, the metrics on both closed surfaces
shall be related by some conformal factor
$h_{\mu\nu}=\Omega^{2}(x)\tilde{h}_{\mu\nu}$, and since $\Sigma_{(2,1)}$ is _a
piece of_ $T^{2}$, $\Omega^{2}(x)$ might be different from one only at points
of the cups $C_{1,2}$, then, by attaching them to the holes of
$\Sigma_{(2,1)}$ demanding continuity, one gets

$\Omega^{2}(x)\approx 1\,,\;\forall x\in\partial C_{1,2}.$ (6.1)

The classical spacetime $\tilde{M}$ that fits into $T^{2}$ is isometric (and
homeomorphic) to $M$, so the correlators can be computed there. This geometry
can be described by

$d\tilde{s}^{2}=\tilde{r}^{2}d\tilde{\varphi}^{2}+\frac{d\tilde{r}^{2}}{\tilde{r}^{2}+1}+(\tilde{r}^{2}+1)d\tilde{\theta}^{2}\;,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{\varphi}\in[0,2\pi]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{\theta}\in[0,2\pi
R],\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ r\geq 0$ (6.2)

The periodic identifications of this manifold are

$\tilde{\theta}\sim\tilde{\theta}+2\pi R\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\varphi}\sim\tilde{\varphi}+2\pi\leavevmode\nobreak\ .$

A circle along the angle $\tilde{\theta}$ is non-contractible, and it has a
minimal proper length:

$l(\tilde{r})=\int_{0}^{2\pi
R}\;\sqrt{\tilde{r}^{2}+1}\;d\tilde{\theta}\;=\;2\pi
R\sqrt{\tilde{r}^{2}+1}\;\geq\;2\pi R\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (6.3)

as expected for a closed curve around the handle.

Since $\lambda=0$ on $\Sigma_{(2,1)}$, the prescription (3.2) for this problem
may be expressed as

$\int[D\tilde{\eta}]\;e^{-S_{CFT}[\tilde{\eta}]\,+\,\int_{\tilde{C}_{1}}\Omega^{-\Delta}\lambda_{1}\,O+\,\int_{\tilde{C}_{2}}\Omega^{-\Delta}\lambda_{2}\,O}\,=\int_{\partial{\cal
M}=T^{2}}D{\cal M}\,D\Phi\,e^{-I[{\cal M},\Phi]}\,,$ (6.4)

where $\tilde{C}_{1,2}=T^{2}-\Sigma_{(2,1)}$ denote the regions of the thorus
that conformally map onto $C_{1,2}$. To large $N$, the r.h.s. of this
expression can be approximated by the saddle $\tilde{M},\tilde{\phi}$. The
boundary conditions
$\tilde{\lambda}_{1,2}\equiv\tilde{\phi}|_{\tilde{C}_{1,2}}$ must transform
accordingly, then we have that
$\tilde{\lambda}_{1,2}=\Omega^{-\Delta}_{1,2}\lambda_{1,2}$ as appears on the
l.h.s. (see for instance [4] and appendix B of [30]).

Therefore, according to (2.9) the state components are obtained by taking
$n_{1}+n_{2}$ derivatives with respect to the sources
$\lambda_{1},\lambda_{2}$, that to large $N$, are given by products of two-
point functions

$\langle{\cal O}(\tilde{\varphi}_{1},\tilde{\theta}_{1}){\cal
O}(\tilde{\varphi}_{2},\tilde{\theta}_{2})\rangle=-\frac{\delta^{2}I[\tilde{M}]}{\delta\lambda_{1}\delta\lambda_{2}}=\Omega^{\Delta}(\varphi_{1},\theta_{1})\Omega^{\Delta}(\varphi_{2},\theta_{2})\langle{\cal
O}(\varphi_{1},\theta_{1}){\cal O}(\varphi_{2},\theta_{2})\rangle$ (6.5)

but this shall be valued on $\partial\Sigma_{(2,1)}$ that consists of two
separated circles on the thorus parameterized by $s_{1,2}\in S^{1}$. Using
(6.1), the final result is777More details on this calculus can be found in
refs [24, 25]. For simplicity of the notation, we have expressed it in the
original coordinates $\theta,\phi$.

$\Psi_{1,1}\approx\langle{\cal O}(\varphi(s_{1}),\theta(s_{1}))\,{\cal
O}(\varphi(s_{2}),\theta(s_{2}))\rangle=\sum_{j\in\mathbb{Z}}\frac{(\Delta-1)^{2}}{2^{\Delta-1}\pi}\left[\cosh(\Delta\theta+2\pi
Rj)-\cos(\Delta\varphi)\right]^{-\Delta}\,.$ (6.6)

where $\Delta\varphi\equiv\varphi(s_{2})-\varphi(s_{1})$ and
$\Delta\theta\equiv\theta(s_{2})-\theta(s_{1})$.

As prominent feature of the state components characterizing the
$(b=2,g=1)$-spatial topology (to large $N$), we can observe the appearing of
the integer number $j$. In terms of geodesic distance between two points of
different boundaries, this is related to the contribution from a minimal
length around the handle ($\sim 2\pi R$), such that one shall sum over the
geodesics encircling it $j$ times.

Remarkably, in view of (5.6), the result (6.6) matches with the following
entangled state:

$\left|\Psi(b=2,g=1)\right\rangle=\sum_{j\in\mathbb{Z}}\,\left|TFD(\beta_{j})\right\rangle\leavevmode\nobreak\
=N^{1/2}\,\sum_{\alpha}\,\sum_{j\in\mathbb{Z}}\,e^{-\frac{\beta_{j}}{2}\,E_{\alpha}}\left|E_{\alpha}\right\rangle_{1}\otimes\left|E_{\alpha}\right\rangle_{2}\leavevmode\nobreak\
,$ (6.7)

where

$\frac{\beta_{j}}{2}\equiv\Delta\theta+2\pi Rj\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j\in\mathbb{Z}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$

$N^{1/2}$ is a normalizing constant, and $\alpha$ labels the eigenstates of
(one copy of) the CFT Hamiltonian. In the forthcoming subsection, we will
argue this expression from a geometric perspective using the torus symmetries.

This state depends on the parameters $R$ and $\Delta\theta$, that is bounded
because of the thorus size: $0\leq\Delta\theta\leq\pi R$. Note that if $R$ is
large compared with the AdS-radious scale ($R\gg 1$), only $j=0$ contributes,
and the state is in agreement with the standard TFD double at inverse
temperature $\beta_{0}\sim 2\Delta\theta$. The interpretation in this case may
be that the correlators are dominated by curves that do not probe the handle.
The simplicity of this result immediately suggests the general ansatz for
arbitrary $b,g$, but it should be checked properly.

The CFT operator/state from identification symmetries

In this brief subsection we add a more detailed discussion and a derivation of
the state dual to the handled spatial geometry $\Sigma_{(2,1)}$ based on
geometric aspects and symmetries. As explained before, the completed
$\Sigma^{c}_{(2,1)}$ is a 2d torus, and the path integral (6.4) corresponds to
the projection of the state (3.1), $U_{2,1,(\lambda=0)}$, onto coherent states

$U_{\cal
T}[\lambda_{1},\lambda_{2}]\equiv\langle\lambda_{1}|U_{2,1,(\lambda=0)}|\lambda_{2}\rangle,$
(6.8)

where ${\cal T}\in\mathbb{C}$ denotes the modulus of the torus (6.4), that
represents the wave functional. Then, the formula (2.9) provides the
projection on a Fock basis. For the holographic calculations, we have fixed
the boundary torus such that there are two independent symmetries of
identification

$(\theta,\varphi)\cong(\theta+2\pi R,\varphi+2\pi)\qquad,$ (6.9)

which must be manifest at the level of _the state_ itself. This is a very
important property to be used in the following analysis.

It is worth clarifying that this is rather different than the partition
function $Z_{2,0}(\beta)=\text{Tr}\;U_{2,0}$ defined also as a path integral
on the torus, but describing the _thermal partition function_ for (CFT)
degrees of freedom on $S^{1}$, where the (thermal) state
$\rho_{\beta}(S^{1})=U_{2,0}$ enjoys the explicit periodicity of the circle
$\varphi(\cong\varphi+2\pi)$ that manifestly appears in the correlations
functions. While the other periodicity, in $\beta\equiv 2\pi R$, only arises
upon _taking trace_ by virtue of the gluing of both cylinder extremes. In such
familiar case, by identifying the generator of translation $P_{\theta}$ with
the Hamiltonian, the operator describing the state can be expressed by

$\;U_{2,0}(\beta)=\;e^{-\beta\,H}\,\,\,,$ (6.10)

which commutes with $P_{\varphi}$, the generator of the spatial translations
on $S^{1}$. Although this shares geometric properties, the state we are
interested in, $U_{2,1}$, _is not_ thermal, and it is described by the
operator (6.8) endowed with the symmetries (6.9).

In fact, by virtue of the formula (2.9), we can impose $\,U_{\cal
T}(\theta+2\pi R)\equiv U_{\cal T}(\theta)$ directly on the components of the
operator in the Fock basis (2.1), and obtain the conditions:

$\langle 0|{\cal O}(0,0)\,U_{\cal T}(2\pi R){\cal
O}(\theta,0)|0\rangle=\langle 0|{\cal O}(0,0)\,{\cal
O}(\theta,0)|0\rangle\qquad,\qquad\dots$ (6.11)

where we have used that $\,U_{\cal T}(\pm 2\pi R)|0\rangle=|0\rangle$. This
looks like KMS relations for the state $U_{\cal T}$, and $(\dots)$ express
that similar relations hold for all the $n_{1}+n_{2}$\- point functions, which
completely expand the operator in the (holographic) Fock space of Sec 2. The
correlation functions in (6.11) appear valued at fixed (the same) $\varphi$,
but they trivially generalize to arbitrarily different points by composing
$U_{\cal T}$ with $e^{\,i\varphi\,P_{\varphi}\,}$.

From the direct holographic calculus in a AdS3 spacetime whose boundary is the
torus, one obtains that the right hand side of these equations is (6.6), for
the sector $n_{1}=n_{2}=1$ of the space of states, as well as the proper ones
for arbitrary $n_{1},n_{2}$. On the other hand, noticing that that each ($j$)
term of the sum (6.6) corresponds to the matrix elements of (thermal-like)
operators as (6.10) (see Sec. 5):

$\frac{(\Delta-1)^{2}}{2^{\Delta-1}\pi}\left[\cosh(\theta+2\pi
Rj)-\cos(\varphi)\right]^{-\Delta}=\langle{\cal O}(0,0)\,U_{2,0}(2\pi Rj){\cal
O}(\theta,\varphi)\rangle\,,$ (6.12)

the solution of the equations (6.11), that also fulfills (6.6), can be
expressed as

$\;U_{\cal T}(2\pi R)=\sum_{j\in\mathbb{Z}}\;\,U_{2,0}(2\pi
Rj)\,=\,\sum_{j\in\mathbb{Z}}\;e^{-(j2\pi R)\,P_{\theta}}\,.$ (6.13)

In addition, this can be composed with the relative displacements
$\Delta\theta,\Delta\varphi$ generated by $P_{\theta}$ and $P_{\varphi}$
respectively, and finally obtain

$\;U_{\cal T}=\,\sum_{j\in\mathbb{Z}}\;e^{-(\Delta\theta+j2\pi
R)\,P_{\theta}+\,i\Delta\varphi\,P_{\varphi}\,}\,.$ (6.14)

Because of the infinite sum, this operator is manifestly invariant under the
replacement $\theta\to\theta+2\pi R$, and thereby, it captures both
identification symmetries (6.9) as expected.

The operator (6.14) represents exactly the CFT state whose holographic dual
has a initial space $\Sigma_{(2,1)}$, and then, identifying $P_{\theta}$ once
more with the Hamiltonian of the boundary quantum system, one can express this
state in a purified form as (6.7).

##  7 Concluding remarks

We presented a formalism that systematically connects explicitly entangled CFT
states with their emergent aAdS spacetimes in both senses: given a classical
AdS spacetime one can obtain the (multi-partite) entanglement pattern in terms
of $n$-point correlation functions computed on that geometry; and
reciprocally, given a very general CFT state, one can find a dual classical
Euclidean spacetime, and the initial spatial geometry where the Hartle-Hawking
wave functional is defined [22], by solving a well posed classical problem.
The multi-partite expansion is realized here in a basis with a dual bulk
interpretation (in terms of particles on a fixed geometry), which is
convenient for perturbative treatments.

As had been argued and conjectured in [12], it is worth emphasizing the
coherent nature of the states (3.1) (in the sense of [4]) that have a
classical dual geometry, as an essential ingredient for the holographic
emergence in addition to the quantum entanglement.

Many crucial aspects of the present construction are based on some advances
and tools developed in recent years [4, 25]. As a first check we show how the
method encounters the TFD state from the Schwarschild-AdS geometry [2], and
generalize this for deviations from this state/geometry [15, 16, 17], captured
by inserting general sources at the Euclidean asymptotic boundary.

One of the motivating objectives of this research was to clarify how and what
information about the geometry (particularly its topology) is encoded in the
multi-partite, manifestly entangled, decomposition of the state. We found that
the presence of a handle in the emergent spatial geometry is related to
entanglement coefficients involving a sum on an integer number $j$,
interpreted as a winding number. In terms of geodesic distance, this can be
viewed as the minimal curve encircling the handle $j$ times; hence, the genus
$g$ might be related to the number of parameters associated to different
winding numbers. Suggestively, the resulting dual state of this geometry
agrees with a superposition of TFD states.

In such a sense, these results do generalize the van Raamsdonk’s observation,
since more specific aspects of topology than space (time) connectivity can be
encoded in Schmidt coefficients.

The explicit entangled state is useful for studying certain salient aspects of
holographic (quantum) gravity and CFT for future work. In emergent gravity,
the main application of this is space-time engineering, it is useful for
calculating and studying the characteristics of the coefficients based on the
most general spatial topologies, and then classifying them; thereby, one would
get a reverse engineering program to build up different (non-trivial)
classical spacetimes by suitably preparing the quantum state. Regarding CFT,
since the obtained entangled state is in its diagonal (Schmidt) form for large
N, quantities such as Rényi and von Neumann entropy can be calculated
directly, as well as studying other features of quantum information. We hope
that non-trivial topological aspects (e.g. $g\neq 0$) can provide non-trivial
contributions to them.

The contexts where the hypothesis about topological equivalence is relaxed and
how the recipe may even work should be further explored in future research. In
fact, more studies can be done on the conditions and the range of parameters
so that different initial spatial topologies dominate the geometric dual. The
present prescription can be also applied to study replica wormholes
geometries, and some aspects of traversable wormholes. A potentially
interesting framework for it is the Jackiw-Teitelboim gravity in $1+1$d
spacetimes [35]. Although for clearness we have formulated the recipe (3.2) in
the general $2+1$d case, the prescription holds for any spacetime dimension.

Acknowledgements The author is grateful to CONICET for financial support.
Thanks are due to Pedro J. Martínez for the help with the figures and the
fruitful comments and discussions.

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