# Real-time methods in JT/SYK holography

Raúl<EMAIL_ADDRESS>Instituto de Física La Plata - CONICET
and Departamento de Física, Universidad Nacional de La Plata , C.C. 67, 1900,
La Plata, Argentina  Marcelo<EMAIL_ADDRESS>Instituto de Física La Plata - CONICET and Departamento de Física, Universidad
Nacional de La Plata , C.C. 67, 1900, La Plata, Argentina  Pedro J.
<EMAIL_ADDRESS>Centro Atómico Bariloche, R8402AGP,
San Carlos de Bariloche, Argentina

###### Abstract

We study the conventional holographic recipes and its real-time extensions in
the context of the correspondence between SYK quantum mechanics and JT
gravity. We first observe that only closed contours are allowed to have a 2d
space-time holographic dual. Thus, in any real-time formulation of the
duality, the boundaries of a classical connected geometry are a set of closed
curves, parameterized by a complex _closed_ time contour as in the Schwinger-
Keldysh framework. Thereby, a consistent extension of the standard holographic
formulas is proposed, in order to describe the correspondence between gravity
and boundary quantum models that include averaging on the coupling constants.
We investigate our prescription in different AdS1+1 solutions with Schwinger-
Keldysh boundary condition, dual to a boundary quantum theory at finite
temperature defined on a complex time contour, and consider also classical,
asymptotically AdS solutions (wormholes) with two disconnected boundaries.

In doing this, we revisit the so-called factorization problem, and its
resolution in conventional holography by virtue of some (non-local) coupling
between disconnected boundaries, and we show how in specific contexts, the
averaging proposal by-passes the paradox as well, since it induces a similar
effective coupling.

###### Contents

  1. 1 Introduction
  2. 2 SvR in general $d+1>2$ dimensions
    1. 2.1 Piece-wise holographic duality
    2. 2.2 Holographic excited states
  3. 3 Holographic recipes in 1+1 spacetime dimensions
    1. 3.1 The SvR formalism: No open paths in 2d-holography
    2. 3.2 Thermal (random) states as holographic dual of 2d spacetimes
    3. 3.3 Canonical quantization and BDHM correspondence in JT/SYK: excited states
    4. 3.4 The gluing conditions in JT geometry
    5. 3.5 Correlators in JT
  4. 4 Wormholes in holographic 2d gravity
    1. 4.1 On the factorization problem and its resolution
    2. 4.2 Random (rigid) deformations of SYK and wormholes
    3. 4.3 Wormhole correlators
  5. 5 Conclusions
  6. A Some realizations of wormholes in sourced JT
    1. A.1 Explicit coupling between operators on disconnected boundaries
    2. A.2 Averaging over random couplings with a rigid constraint
  7. B Scalar field in pure AdS2
    1. B.1 Real-time correlation functions
    2. B.2 Expectation values

##  1 Introduction

The AdS/CFT correspondence is a useful tool to study strongly coupled quantum
field theories through a gravitational model. In this sense, it allowed to
describe properties of many physical situations ranging from hydrodynamics to
quark gluon plasma and condensed matter theories [1]. In a nutshell, the
conjecture states the equivalence between the Hilbert space of the field
theory and the Hilbert space of its gravitational dual. Nevertheless a
complete map between these Hilbert spaces is not known. In recent years, a new
duality was found between a particular quantum mechanics model (called SYK)
and a particular two dimensional gravity theory (JT gravity). It is by
exploring this correspondence that an old puzzle of AdS/CFT [2, 3], dubbed the
factorization problem, could be revisited. The puzzle can be presented as
follows: the partition function of a set of $n$ non-interacting CFTs should
correspond to the product of each theory’s partition function, whereas the
bulk problem taking the $n$ CFTs as boundary conditions should a priori
consider all connected geometries in the expected sum over topologies required
for the gravitational path integral. A proposal to solve this problem,
inspired by this JT/SYK duality, is that the actual holographic dual to the
gravitational path integral is not a particular QFT but rather an ensemble of
QFTs, which effectively avoids the factorization in the CFT side of the
correspondence. See for example [4, 5] and references within for a review.

More precisely, SYK [6, 7] is a theory (or rather an ensemble of theories) of
$N$ Majorana fermions with polynomial all to all interactions of order $q$ via
a random coupling constant that has zero mean and Gaussian distribution. This
model is invariant under reparametrizations but in the IR this symmetry is
spontaneously broken down to $SL(2,\mathbb{R})$. This implies that at the IR
fixed point the Goldstone modes of SYK can be described by a one dimensional
conformal field theory [5]. Its holographic dual, called JT gravity [8, 9], is
a two dimensional model that couples a Dilaton scalar field with the geometry.
The action can be written in terms of a unique degree of freedom (a
reparametrization mode) and is $SL(2,\mathbb{R})$ invariant. Despite being a
two dimensional theory it can be obtained from higher dimensional near
extremal black hole solutions by dimensional reduction. By analyzing random
matrix models it can be seen that the partition function of both theories
coincide [10].

A particularly puzzling aspect of the JT/SYK duality is related to the
averaging process in the QM theory [4, 5] which is believed to be intimately
related to the existence of wormhole geometries. The standard picture of the
holography community is that the correspondence between CFTs and aAdS gravity
is a one to one map and many precision holography tests were carried
successfully in this front [1]. However, for JT computations to match most
predictions of SYK, one needed not to consider a single realization of coupled
fermions but actually an ensemble of these theories. If the averaging in
holography turns out to be mandatory for the higher dimensional scenarios as
well, this has some quite strong consequences. On one hand, this immediately
solves the factorization problem raised before, since the averaging
effectively make all theories interact with each other [11]. On the other
hand, this appears to be a quite unnatural limitation of AdS/CFT (see e.g.
[12] for discussion) and furthermore it conflicts with numerous non trivial
one to one realizations of the correspondence [1]. Whether the averaging is a
fundamental piece of AdS/CFT or a peculiarity of the JT/SYK example is still
an open debate [13, 14].

Even from a purely gravitational perspective, it is by now clear that JT
gravity provides a simpler but non trivial model to revisit old and pose new
gravitational path integral questions. In particular it has been capable of
reproducing the conjectured Page curve for the BH entropy [15, 16]. Many
studies and models on entanglement entropy as well as properties of the
partitions functions have been exhaustively studied in this context [10, 17,
18, 19].

It is important to stress that the study of the map between the QFT and
gravity Hilbert spaces requires a clear holographic prescription in real time,
such as to understand the possible physical states that can evolve in the
system. From the foundational works [20, 21] onward, the Euclidean AdS/CFT
prescription was ever-growing, and solid holographic dictionary entries were
built [1]. For simple enough systems, a Wick rotation of Euclidean results
suffices to extrapolate correct real-time predictions. However, the physical
interpretation of a Wick rotated Euclidean quantity often requires acute
physical intuition on the system’s phenomenology and Wick rotated Euclidean
correlators are often written as a sum over discrete Matsubara frequencies,
whose convergence and/or re-summation is hard to analyze [22].

Thus, a purely real-time prescription of holography is still mandatory to
fully understand the duality, even if it poses both conceptual and
computational challenges. Skenderis and van Rees (SvR) in [23, 22] developed a
holographic prescription in real-time holography based in finding gravity
duals to generic complex-time Schwinger-Keldysh (SK) contours. The use of
complex-time paths to represent real time physical systems is well documented
in the QFT literature [24, 25, 26] as well as for AdS/CFT applications, see
e.g. [27]. In short, the SvR prescription sets a gravitational path integral
with complex boundary conditions. If the CFT path integral over the SK path is
physically well understood and taking holography as a hypothesis, these
boundary conditions provide also a well posed and physical variational problem
for the gravity path integral. Its exact computation is in general out of
reach, but often for the cases of interest one can study approximations to
this problem.

The SvR prescription was already proven useful in expanding the map between
the dual gravity and QFT Hilbert spaces [28, 29, 30, 31, 32, 33]. In
particular, inserting boundary sources in the Euclidean segments can be seen
to build excited states over the fundamental configuration provided by the
saddle point of the sourceless Euclidean manifold. In the large-$N$ limit,
these excited states can be seen to become coherent excitations over the
vacuum with non-trivial expectation values for the sourced field [28, 29, 30],
but $1/N$ corrections progressively deform their coherence [34]. Furthermore,
the complexified nature of the SvR problem suggest that complex sources for
real fields yields relevant physical results. It is the case that coherent
states allow for complex parameters corresponding to non trivial expectation
values for both the field and its conjugated momentum [31, 32, 33]. Thereby, a
central piece of this paper is to focus on the study of these prescriptions
and methods in the specific arena of two dimensional gravity.

The main point of our work is to provide a systematic presentation of the SvR
prescription in the particular scenario of $1+1$ gravity. This well
established prescription provides a nice physical interpretation for all
observables and quantities computed. In doing so, we find some non-trivial
results that we summarize in the end of this section. We must mention that
approaches with similar motivations have risen recently in the literature, see
[35, 36, 37, 38], showing the community interest for this type of analysis and
developments. However, the focus of our work is different. Following a well
established real-time holographic prescription eliminates ambiguities that
rise upon direct analytic extensions of correlators.

We conclude this introductory section by listing some of the most noticeable
results and remarks achieved along the paper:

  * •

The holographic GKPW standard dictionary, and its real time extensions (the
SvR prescription), is extended to two-dimensional gravity. In the known model
of JT/SYK, it involves averaging on the quantum mechanics side.

  * •

Geometric arguments show that this extension can only be done by considering
_closed_ SK complex paths in the field theory, and the SvR extension of the
dictionary to real time is captured by the real time intervals of the path.

  * •

The arguments in [11] regarding traversable wormholes are revisited and
improved in the light of the JT/SYK duality. One can argue that by including a
(non local) interaction term between disconnected boundaries in the action
could solve the factorization issue. Moreover, it is consistent with the well-
established mechanism to construct traversable wormholes [11, 39].

  * •

A unified equation capturing the holographic real time prescriptions and
average ensemble for $b>1$ boundaries is presented in eq. (4.3).

  * •

The concern of factorization on eq. (4.3) implies certain restrictions on the
distribution of probability of the coupling constants on the boundaries. And
in addition, the existence of wormhole as classical gravity solution (dominant
saddle point) implies that the duality JT/SYK should be deformed a some non-
trivial way.

  * •

Novel real-time extensions of wormholes in JT with sources are studied, where
the time-ordered correlation functions are calculated.

The paper is organized as follows, in section 2 we will review the connection
between a SK path and holography using the SvR proposal in dimensions $d+1>2$.
In section 3 we will study the specific 2 dimensional case in the JT gravity
context dual to a single boundary SK path. We focus on the differences between
the $1+1$ and $d+1>2$ scenarios. We present a concrete example of a geometry
dual to a single SK path and explore its propagating modes and correlation
functions for probe fields. In section 4 we focus on the scenario of many SK
paths as boundaries and find constraints on the possible form of the ensemble
average by demanding consistency with factorization at large $N$. We also
present an example of a geometry dual to a pair of boundaries and explore its
real time dynamics and correlators. Finally, we summarize the results obtained
and mention some prospects for future work in section 5.

##  2 SvR in general $d+1>2$ dimensions

In this introductory section we review the Skenderis and van Rees prescription
as presented in $d+1>2$ standard AdS/CFT holography. In this context the
averaging is ignored. We will improve on this prescription in the upcoming
sections by a more suitable approach that captures this ingredient as well as
provides a tool to avoid the factorization problem. First, we review the
$d+1>2$ SvR [23, 22] construction, providing explicit gravity duals to generic
complex-time paths, which we collectively denote Schwinger-Keldysh (SK) paths.
These gravity duals can always be split in pure Euclidean or Lorentzian
segments. In this set-up we will see that it is natural to interpret
asymptotic sources (i.e. external sources insertions in the dual CFT) to the
Euclidean segments as excited states of the original theory. In general, these
excitations are only tractable as perturbations over a gravitational vacuum
and have been covered and studied in the bibliography in recent years [28, 29,
30, 31, 32, 33]. Their large $N$ phenomenology look like coherent excitations
over the reference vacuum. Real time correlations can be seen to be modified
by the insertions of these sources both to leading and subleading orders in
$1/N$. Interestingly, even complex sources for real fields have a natural
explanation in this set-up.

###  2.1 Piece-wise holographic duality

To present the SvR formalism, we find convenient to review a very simple
example of this prescription, being the case of a QFT scattering process. A
more detailed introduction can be found in [22]. The corresponding SK path for
our example is presented in Fig. 1(a), see [23, 28]. The dynamics take place
in the Lorentzian segment and (time-ordered) $n$-point functions are computed
via external sources inserted there. The initial and final states are defined
by the Euclidean regions and their asymptotic sources, the vacuum state being
defined as the Euclidean path integral with all sources in the Euclidean
segments turned off. The bulk dual is shown in Fig. 1(b): Euclidean half-
sphere sections and Lorentzian AdS cylinders are assigned to each Euclidean
and Lorentzian SK segment respectively and ${C}^{1}$-glued across
$\Sigma^{\pm}$, providing a candidate saddle for the full path integral.
Classical bulk fields configuration $\chi$ are fully determined in terms of
its asymptotic Lorentzian $\chi_{L}$ and Euclidean $\chi_{\pm}$ boundary
conditions. We write the holographic relation in this set-up as,

$\displaystyle
Z^{CFT_{d}}_{\chi_{-}\to\chi_{+}}[\chi_{L}]=\langle\chi_{+}|e^{-i\int{\cal
O}\chi_{L}}|\chi_{-}\rangle\equiv
Z^{AdS_{d+1}}_{\chi_{-}\to\chi_{+}}[\chi|_{\partial}=\chi_{L}],$ (2.1)

where $Z^{CFT_{d}}_{\chi_{-}\to\chi_{+}}[\chi_{L}]$ is the CFTd generating
function of correlators between the states defined by $|\chi_{-}\rangle\equiv
e^{-\int{\cal O}\chi_{-}}|0\rangle$ and the final state
$\langle\chi_{+}|\equiv(|\chi_{+}\rangle)^{\dagger}$, see [40, 28]. Real time
correlators are obtained by differentiation wrt $\chi_{L}$. On the gravity
side, we have

$\displaystyle
Z^{AdS_{d+1}}_{\chi_{-}\to\chi_{+}}[\chi|_{\partial}=\chi_{L}]=\left(\int_{\chi_{-}}{\cal
D}\chi\;e^{-I_{E}}\right)\left(\int_{\chi_{L}}{\cal
D}\chi\;e^{-iI_{L}}\right)\left(\int_{\chi_{+}}{\cal
D}\chi\;e^{-I_{E}}\right)$ (2.2)

with $I_{L/E}$ are the Lorentzian/Euclidean corresponding local gravity
actions on each segment. The smooth Israel gluing at $\Sigma^{\pm}$ that links
the factors in the rhs is left implicit. This expression suggests a _piece-
wise holographic recipe_ with intervals in which the dynamics take place
(Lorentzian) and intervals in which information on the system’s state is given
(Euclidean).

(a)

(b)

Figure 1: (a) In-Out SK path corresponding to a scattering amplitude
computation in QFT. Blue crosses represent operator insertions. (b) Bulk dual
of the In-Out SK path. The SvR prescription associates half Euclidean AdS
spheres and Lorentzian AdS cylinders respectively to the Euclidean and
Lorentzian segments of the path on the left. These are $C_{1}$ glued across
$\Sigma^{\pm}$. The resulting manifold provides a candidate saddle for the
gravitational path integral. Asymptotic sources are represented with blue
lumps.

###  2.2 Holographic excited states

After introducing the SvR formalism and its piece-wise holographic
prescription, we pay special attention to the Euclidean segment defining the
initial state, first factor on the rhs of eq. (2.2). Deformations on this
region (keeping the rest of the manifold including the $\Sigma^{-}$ surface
fixed) can be thought as providing a different initial state to the real time
evolution problem. When the deformation is given by turning on CFT external
sources, we collectively call these excitations holographic excited states
[28, 29, 34, 41, 42, 43, 30]. Its bulk wavefunction is obtained via a Hartle-
Hawking Euclidean path integral with non-trivial asymptotic boundary
conditions. On the CFT side, their interpretation is also natural as excited
states due to operator insertions in the Euclidean past. The sources turning
off softly near the $\Sigma$ gluing regions guarantees that the Hilbert space
at the meeting point is still given by the vacuum (reference) Hilbert space
and allows for the “excited state” nomenclature to make sense. To be explicit,
in (2.2) we have implicitly defined a state $|\chi_{-}\rangle$ on each side of
the dual pair as

$\text{\sf CFT:}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\langle A_{\partial\Sigma}|\chi_{-}\rangle\equiv\int_{A_{\partial\Sigma}}{\cal
D}A\;e^{-I_{CFT}-\int\chi_{-}\;{\cal
O}}\qquad\Longleftrightarrow\qquad\langle\chi_{\Sigma}|\chi_{-}\rangle=\int_{\chi_{\Sigma};\chi_{-}}{\cal
D}\chi\;e^{-I_{E}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ :\,\text{\sf AdS}$ (2.3)

Here $A$ denotes collectively the fundamental CFT fields and $\chi$ bulk
fields dual to CFT primaries ${\cal O}$. The integration on the Euclidean
gravitational path integral over the metric is left implicit. We have denoted
$A_{\partial\Sigma}$ and $\chi_{\Sigma}$ to the field-configuration basis at
$t=\tau=0$. Notice that this does not conflict with the standard Euclidean
intuition that external CFT sources $\chi$ translate into boundary conditions
for bulk fields $\chi$ under the holographic map.

As we mentioned above, most of the studies on these states were done in a
perturbative way, i.e.

$|\chi_{-}\rangle={\cal P}\\{e^{-\int{\cal
O}\chi_{-}}\\}|0\rangle\sim\left(1-\int{\cal O}\chi_{-}+\frac{1}{2}\int{\cal
P}\\{{\cal O}{\cal O}\\}\;\chi_{-}\;\chi_{-}+\dots\right)|0\rangle\;.$ (2.4)

Excitations of this sort can be built over any state of the theory but in
general one is often interested in reference states that have a known bulk
dual in the semi-classical limit. Over these, states (2.4) will also have a
semi-classical bulk dual as long as backreaction is under control. To clarify,
the reference state can be the vacuum as in the scattering example above, a
finite temperature state as it will be the case for most of this work, or any
other state accessible through an Euclidean path integral.

In the large $N$ limit, single trace operators $\cal O$ become generalized
free fields. This is explicitly realized by the BDHM [44] prescription, which
for a scalar operator ${\cal O}$ of conformal dimension $\Delta$ can be
written as

${\cal O}\equiv(2\Delta-d)\lim_{r\to\infty}r^{\Delta}\chi$ (2.5)

where $r\to\infty$ is taken to be the limit to the asymptotic boundary and
$r^{\Delta}$ allows to retain the leading $r^{-\Delta}$ contribution from the
canonically quantized field $\chi$ of mass $m^{2}=\Delta(\Delta-d)$. We should
stress the operator character of this equation: for weakly interacting bulk
matter fields it implies that ${\cal O}$ has a nice representation in terms of
the bulk ladder operators. Then, each term in the series in eq. (2.4) has a
natural $n$-particle state interpretation. In the strict $N\to\infty$ limit
the bulk matter fields become free and ${\cal O}$ becomes linear in the bulk
ladder operators. The state then becomes coherent [28] with $1/N$ corrections
gradually deforming their coherence property [34]. Once the connection between
(2.3) and coherent states is made one can import some intuition built from the
latter to the former. For our purposes, a particularly useful intuition is to
allow the sources $\chi_{-}$ to become complex valued and assign the real and
imaginary parts of $\chi_{-}$ its standard physical interpretation that
relates it to the $\chi$ and its conjugated momentum $\Pi_{\chi}$ expectation
values [31, 32]

$\langle\chi_{-}|\;\chi\;|\chi_{-}\rangle=\int_{\chi_{-}}{\cal
D}\chi\;\chi\;e^{-I_{E}}\propto\text{Re}[\chi_{-}],\qquad\qquad\langle\chi_{-}|\;\Pi_{\chi}\;|\chi_{-}\rangle=\int_{\chi_{-}}{\cal
D}\chi\;\Pi_{\chi}\;e^{-I_{E}}\propto\text{Im}[\chi_{-}]\;.$ (2.6)

where $\langle\chi_{-}|$ is built using Euclidean conjugation [40], i.e.
conjugation and time reflection on $\chi_{-}$. This guarantees that both
results are real even for $\chi_{-}\in\mathbb{C}$.

Note that we are taking a complex source $\chi_{-}\in\mathbb{C}$ for a real
bulk scalar field $\chi$ which seems to conflict with the counting of degrees
of freedom. What happens for a complex source is that this curve no longer
lies on the real axis, as did in the purely Lorentzian set-up.

The main take away from this section is that SvR provides a real time
holographic prescription in $d+1$ holography in which, for example, to study
real time correlations and build excited states over a reference state and
allows for both real and imaginary sources for real fields. Specifically,
imaginary sources for an operator $\cal O$ in the CFT translate in the bulk to
a non-trivial expectation value for the conjugate momentum $\Pi_{\chi}$.
However, notice that this prescription is unable clarify the factorization
problem as presented in the introduction. In the upcoming sections, we aim at
improving this real time holographic prescription in the 1+1 dimensional set-
up to account for this.

##  3 Holographic recipes in 1+1 spacetime dimensions

The main question addressed in this paper is if conventional holography holds
in the context of Jackiw-Teitelboim gravity in the same way as the standard
holographic prescriptions as GKPW, or its generalization to real-time: the SvR
formula eq. (2.1). The validity of the holographic map between operators eq.
(2.5) shall also be studied in this context [44]. In this section we will
study how to understand the SvR construction in two dimensions, showing that
open SK paths does not have a corresponding gravity dual state. Then we will
add a probe scalar field $\chi$ to the JT action to compute correlators of
quantum fields living on the boundary theory and will mention some features of
holographic excited states in this framework.

###  3.1 The SvR formalism: No open paths in 2d-holography

First, we are going to see that the SvR formula for one complex closed
boundary can be formulated in a conventional way simply by taking an evolution
operator $U:{\cal H}\to{\cal H}$, which is a path ordered (Hermitean)
operator, valued on a _closed_ oriented curve ${\cal C}$ embedded in the
complex plane $\mathbb{C}$ as

$Z_{QFT}=\text{Tr}\,\;U[\kappa,{\cal C}]\qquad\qquad U[\kappa,{\cal
C}]\equiv{\cal P}\,e^{-i\oint_{{\cal C}}d\theta\;(H+{\cal
O}\,\kappa(\theta))},\qquad\theta\in\mathbb{C},$ (3.1)

where $U[\kappa,{\cal C}]$ is the evolution operator for a boundary
Hamiltonian deformed by a source $\kappa(\theta)$. This operator acts on the
Hilbert space ${\cal H}$ of the suitable $0+1$-dimensional quantum field
theory.

Thereby, the conventional SvR formula takes the simple form

$\text{Tr}\,\;U[\kappa,{\cal C}]=Z_{grav}(\kappa)\qquad\qquad\partial
M\equiv{\cal C}\qquad\kappa=\chi|_{\cal C}$ (3.2)

where $\kappa$ are Dirichlet boundary conditions to probe matter fields $\chi$
defined on the (locally AdS) spacetime $M$, whose boundary is ${\cal C}$. This
is the formulation of [41, 42] in higher dimensions. In particular, this
formula also applies to the purely Euclidean set up [20, 21], where the SK
closed contour ${\cal C}$ corresponds to the Euclidean circle of length
$\beta$, and the dual geometry is the Euclidean AdS disc.

Notice that, eq. (3.2) can only be formulated for _closed_ complex boundaries.
In other words: _there is no states/evolution_ on open complex boundaries,
which are dual to some classical geometry. By strictly geometric arguments one
can show that the SvR(GKPW) recipe, in the context of two dimensional gravity,
can only be formulated for closed boundary curves; in fact, in order to have a
well defined rhs of (3.2), in the semiclassical limit, it has two dimensional
classical geometries $M$ as saddles of a well posed problem and it requires
that the boundary is given by closed curves444Actually, the argument does not
requires the semiclassical limit or saddle geometries; in fact, already the JT
partition function is well defined as a sum over Riemannian two dimensional
manifolds whose boundary is restricted to be $\sqcup^{b}{\cal C}$ . In other
words, in the simpler purely Euclidean version of eq. (3.2), $M$ is a general
Riemann surface with $b$ boundaries and $g$ handles where each boundary is
equivalent to a circle, i.e: $\partial M=S^{1}\sqcup\dots\sqcup S^{1}$, with
circumference lengths $\beta_{1}\dots\beta_{b}$ respectively.

####  3.1.1 A real-time prescription for JT/SYK: SvR with averaging

The Sachdev-Ye-Kitaev (SYK) model [6, 7] is a quantum mechanical theory for
$N\gg 1$ Majorana fermions with an all-to-all interaction. The action is
written in Euclidean time $\tau$ as

$I_{SYK}=\,\,\int
d\tau\left(\frac{1}{2}\sum_{i=1}^{N}\psi_{i}(\tau)\dot{\psi}_{i}(\tau)+\frac{1}{4!}\sum_{i,j,k,l=1}^{N}J_{ijkl}\psi_{i}(\tau)\psi_{j}(\tau)\psi_{k}(\tau)\psi_{l}(\tau)\right).$
(3.3)

The Majorana fermions are Hermitian and satisfy the canonical anticommutation
relation $\\{\psi_{i},\psi_{j}\\}=\delta_{ij},\,\,\,\,i,j=1,\ldots,N$. The
coupling $J_{ijkl}$ are taken randomly and in an independent way from a
Gaussian distribution function with the probability density

$G_{\sigma}(J_{ijkl})\equiv
e^{-\frac{J_{ijkl}^{2}}{2\sigma^{2}}}\qquad\text{where}\qquad\sigma^{2}\equiv\frac{3!\sigma^{2}_{0}}{N^{3}}$
(3.4)

In this distribution the mean coupling and the squared variance are defined

$\mu\equiv\langle J_{ijkl}\rangle\equiv\int
dJ\,G_{\sigma}(J)\,J_{ijkl}=0,\qquad\;\qquad\langle
J_{ijkl}^{2}\rangle\equiv\int
dJ\,G_{\sigma}(J)\,J_{ijkl}^{2}=\frac{3!\sigma^{2}_{0}}{N^{3}}\,.$

Note that $\sigma$ is a constant with mass dimensions.

Lastly, note the action (3.3) can be written with a more general
$q$-interaction term with coupling constant $J_{i_{1}\dots i_{q}}$ with $q$
indices, although for simplicity, we focus on the model $q=4$ in (3.3) and in
the rest of the paper.

Since SYK with random interactions is believed to be the dual of JT gravity
[5, 10], one should generalize the prescription of the previous Section to
describe averaged holographic theories. This might be done by correcting the
formula (3.2) on the left hand side as

$\text{Tr}\,\;U[J,\kappa,{\cal
C}]\qquad\to\qquad\langle\text{Tr}\,\;U[\kappa,{\cal C}]\rangle\equiv\int
dJ\,G(J)\,\text{Tr}\,\;U[J,\kappa,{\cal C}]$ (3.5)

where $U[J]$ denotes the evolution operator of the boundary field theory,
generated by the Hamiltonian $H[J;\psi,\dot{\psi}]$. In general, $J$ denotes
the set of parameters (coupling constants) labeling a family of boundary
quantum theories. The averaging is weighted by the function $G(J)$ which, in
particular in the SYK model, is taken to be the Gaussian distribution (3.4)
and $J\equiv J_{ijkl}$, with $i,j,k,l=1,\dots,N$

Thus, our proposal for the holographic map is similar to the SvR formula, with
the averaging on the coupling constants of the boundary quantum theory

$\langle\,\text{Tr}\,U\left[\kappa;\,{\cal C}\right]\rangle=Z_{JT}(\kappa),\;$
(3.6)

where ${\cal C}$ is closed, by virtue of the arguments of the previous
subsection, and the left hand side stands for an average on $\text{Tr}\,U$,
which consists of a path integral (3.1) on that curve. The generalization of
this formula to $b$ closed boundaries ${\cal C}_{1},\dots,{\cal C}_{b}$ is
straightforward, and shall be presented in Sec 4 for clearness. We shall see
below that this prescription has immediate consequences in the structure, and
interpretation of the states, as well as their dynamics.

The present prescription, and its generalization to many boundaries, are
consistent with the _annealed disorder_ framework555This approach is called
“annealed”, in contrast with the so called “quenched” disorder where the
averaging is taken on the free energy. In the annealed disorder the averaging
is taken directly on the partition function, see e.g. [4]. [4, 45]. They will
be tested in the forthcoming sections, in the specific arena of known JT (and
JT$+$scalar field) solutions.

###  3.2 Thermal (random) states as holographic dual of 2d spacetimes

The aim of this part is to describe the holographic states, namely, states of
the boundary quantum theory that are dual to some classical solution of the JT
model, with or without matter fields.

As shown above, the corresponding field theory path integrals, lhs of (3.2),
are defined on closed paths ${\cal C}$ which cannot be parameterized by only
one real-time interval $t\in[0,T]\subset\mathbb{R}\mapsto{\cal C}$, because it
would be a closed time-like curve where the evolution operator is defined on.
Instead, all SK path ${\cal C}$ necessarily has Euclidean intervals
$E_{\alpha}$ of length $l_{\alpha}$ respectively, and the total length is
$\beta\equiv\sum_{\alpha}l_{\alpha}$.

As explained before, each interval is endowed with an Euclidean evolution
operator

$U(J,E_{\alpha}):{\cal H}_{N}\to{\cal H}_{N}\;.$

The microscopic theory is SYK model, whose dynamics characterized by mean
$\mu=0$, and arbitrary parameters $q,\sigma_{0},N$. The random variables are
$2^{N/2}\times 2^{N/2}$ real coupling constants denoted by $J$. This operator
unambiguously characterizes a _microscopic_ pure state, that can be
represented as a ket $\Psi(J,E_{\alpha})$ in the tensor product of two-copies
of ${\cal H}_{N}$ 666 This can be easily seen by representing the matrix
elements of $U(J,l_{\alpha})$ as a path integral with fixed fields
($\psi_{i}$) at $\tau$ and $\tau+l_{\alpha}$. Then it is nothing but the wave
functional $\Psi[\psi_{i}(\tau),\psi_{i}(\tau+l_{\alpha})]$ .. This is similar
to the standard purification technique in the TFD formalism, see e.g. [41, 42]
for more details.

Therefore, we can naturally define the density operator by considering the
(path ordered) union of the Euclidean intervals

$\rho_{\beta}\left(J\,,\,\cup_{\alpha}E_{\alpha}\right)\equiv\,\;{\cal
T}\,\prod_{\alpha}\;U(J,E_{\alpha})=U(J,E_{1})\,U(J,E_{2})\,U(J,E_{3})\,\dots=U_{\beta}\left(J\,,\;\cup_{\alpha}E_{\alpha}\right)\;,$
(3.7)

where taking the trace $\rho_{\beta}\,\mapsto\,\text{Tr}\,\rho_{\beta}\,\equiv
Z(J,\beta)$ to compute the probability of the $J$-model (3.3), geometrically
represents to close the curve

$\bigcup_{\alpha}\,E_{\alpha}\,\mapsto\,S^{1}\;.$

In order to keep the probabilistic interpretation of this operator, we must
demand hermiticity

$\rho_{\beta}=\rho_{\beta}^{\dagger}\;.$ (3.8)

In addition, one can define the _effective_ , or macroscopic, density operator
by integrating out the random variable $J$ in this expression, namely

$\langle\rho_{\beta}\rangle\left(\cup_{\alpha}E_{\alpha}\right)\equiv\,\int\,dJ\;G_{\sigma}(J)\;\;\rho_{\beta}\left(J\,\,,\,\cup_{\alpha}E_{\alpha}\right)\;.$
(3.9)

This object is what, in averaged holography, should more naturally be
considered the corresponding dual to a classical Euclidean space-time.
Analogously to the standard SK closed-path contour in the complex plane it is
_periodic_ in the imaginary total time, by virtue of the identification of the
endpoints imposed by the trace. By composing this fact with the statement that
all two-dimensional geometry have closed boundaries ${\cal C}$ (which are
circles $S^{1}$ with possible insertions of segments parameterized by real
time intervals), we can remarkably conclude that _the states that are dual to
some (JT) geometry, are thermal_ , i.e: they can be described by a density
operator (3.7) with an associated inverse temperature $\beta$, in the same
sense that the conventional closed SK path.

####  3.2.1 TFD purification

Consider now just two equal Euclidean intervals
$l_{\alpha}\equiv\beta/2,\;\alpha=1,2$ on only one disconnected boundary
${\cal C}$, such that

$U_{2}(J\,;\,(0,\beta/2))=\left(U_{1}(J\,;\,(-\beta/2,0))\right)^{\dagger}\;,$

which guarantees (3.8). In other words, this means that the corresponding path
integral on the intervals $E_{1}\equiv(0,\beta/2)\;,\;E_{2}\equiv
E_{1}^{\dagger}=(-\beta/2,0)\;$, are reflected into each other [41, 42]. The
(microscopic) thermal density operator (3.7) is

$\rho_{\beta}(J)=U(J\,;\,(-\beta/2,\beta/2))=\,U_{1}(J\,;\,(-\beta/2,0))\,U_{2}(J\,;\,(0,\beta/2))\;=\,U_{1}(J\,;\,(-\beta/2,0))\,\left(U_{1}(J\,;\,(-\beta/2,0))\right)^{\dagger}.$

In this way, the operator $U_{1}(J\,;\,(0,\beta/2))$ itself can be identified
with the TFD state, and its thermal excitations can be systematically obtained
by taking a non trivial function $\kappa_{1}(\tau)\neq 0$ on $E_{1}$ [42].

For instance in the SYK theory for the Grassmannian field $\psi(t)$, the
fundamental (TFD) state can be described by the wave functional

$\Psi(J,\psi(0),\psi(-\beta/2))\equiv\langle\psi(-\beta/2)|U_{1}(J)|\psi(0)\rangle\\\
=\int{\cal D}\psi\;e^{-\int_{\beta/2}^{0}d\tau\,\,{\cal L}_{SYK}[\psi,J]}$
(3.10)

which for an arbitrary constant $J$, is a well defined path integral with
fixed arbitrary data $\psi(0),\psi(-\beta/2)$ on the endpoints of $E_{1}$.
Therefore, by following the methods of [4] to integrate out $J$, one can
compute the components/matrix elements of the (macroscopic) fundamental state
(3.9)

$\langle\rho_{\beta}\rangle\equiv\int\,dJ\;G_{\sigma}(J)\;\;\langle\psi(-\beta/2)|U(J)|\psi(\beta/2)\rangle\\\
=\int{\cal
D}\psi\;e^{-\int_{-\beta/2}^{\beta/2}d\tau\,\frac{1}{2}\,\psi\;\dot{\psi}\;+\;\frac{3\sigma_{0}^{2}}{N^{3}}\int^{\beta/2}_{-\beta/2}d\tau\int^{\beta/2}_{-\beta/2}d\tau^{\prime}(\psi(\tau)\cdot\psi(\tau^{\prime}))^{4}}\;.$
(3.11)

This is a path integral that only depends on the initial/final
$\psi(\beta/2),\psi(-\beta/2)$ arbitrarily chosen on the endpoints of
$E_{1}\cup E_{2}$. It expresses what we call effective, or macroscopic thermal
state in SYK. This state should encode most of the features of the (Euclidean)
black hole solution of JT gravity.

###  3.3 Canonical quantization and BDHM correspondence in JT/SYK: excited
states

Let us consider $M$ with just one connected boundary curve $\partial
M\equiv{\cal C}$ with a set of real-time intervals in the same curve,
parameterized by $t_{\alpha}$ where $\alpha$ labels the real-time segments.

The simplest scenario to study probe excitations over a geometry is to solve a
Klein-Gordon equation for a (non back-reacting) scalar field $\chi$ of mass
$m^{2}=\Delta(\Delta-1)$, which is dual to a single trace (scalar) local
operator ${\cal O}(t)$ of conformal weight $\Delta$, which typically is only a
functional of the fundamental fields of the boundary theory, e.g. $\psi_{i}$
in SYK. Nevertheless, further generalizations are also possible where it also
depends on a random coupling constants (e.g [46]), that will be considered
below.

As indicated in the recipe (3.6), the boundary value of $\chi$ is what sources
the ${\cal{O}}$ insertions on ${\cal C}$, and allows to compute the generating
function of the boundary quantum theory (lhs of (3.6)). Furthermore, in[28,
42] it was shown that, by quantizing $\chi$ canonically, and applying the BDHM
rules [44] one can systematically construct the excited states, which
generally correspond to coherent states in the large-$N$ bulk Hilbert space.
The aim of this short section is precisely to reproduce this in the JT/SYK
context.

The vacuum solution of JT gravity with only one connected boundary ${\cal C}$
consists of exact AdS spacetime, with piece-wise signature as corresponds to
the different imaginary/real-time segments of ${\cal C}$, as discussed in Sec.
2.1. Consider an appropriate number of locally real time AdS pieces (charts)
$M_{\alpha}=\\{(t_{\alpha},r_{\alpha})\\}$ covering a time-independent
spacetime $M\equiv\\{(g_{\mu\nu}(r_{\alpha}),\Phi(r_{\alpha}))$. For a system
of coordinates covering AdS1+1 there are a complete set of normalizable
solutions of the KG equation on such geometry of frequencies $\omega_{k}$
labeled by $k$. Thereby, the general solution for $\chi$ can be expressed as

$\chi\;=\;\chi_{0}\;+\;\sum_{\alpha}\,\,\sum_{k}\,\Theta_{\alpha}\,\left(a_{\alpha,k}e^{i\omega_{k}t_{\alpha}}\,f_{k}(r_{\alpha})\,\pm\,a^{\dagger}_{\alpha,k}e^{-i\omega_{k}t_{\alpha}}\,f^{*}(r_{\alpha})\right),\qquad[a_{\alpha,k},a^{\dagger}_{\alpha,l}]=\delta_{kl}.$
(3.12)

where $\Theta_{\alpha}$ is the Heaviside distribution with support on
$M_{\alpha}$ and
$\chi_{0}\equiv\sum_{\alpha}\,\,\Theta_{\alpha}\,\chi_{0}(r_{\alpha},t_{\alpha})$
stands for the non-backreacting classical part of the solution that fulfills
the non vanishing (Dirichlet) boundary conditions

$\partial M\equiv{\cal C}\qquad\kappa=\chi_{0}|_{{\cal C}}\;.$

On the other hand, the (linear) fluctuations are canonically quantized by
promoting the coefficients of the general solution to operators as in the rhs
of (3.12). In order to have just one probe degree of freedom, the field $\chi$
can be taken taken hermitian (+) or anti-hermitian (-). The second choice
describes certain modified JT gravity [46], which admits interesting wormholes
solutions ,e.g. see Sec 3.3.1.

Summarizing, the BDHM recipe allows to connect this field to the operators
inserted on the boundary theory

${\cal
O}(t_{\alpha})\equiv(2\Delta-1)\lim_{r_{\alpha}\to\infty}\;r_{\alpha}^{\Delta}\,\left(\chi-\chi_{0}\right)\sim\sum_{k}\,\,\left(N_{\alpha,k}\,a_{\alpha,k}e^{-i\omega_{k}t_{\alpha}}\,\,\pm\,N_{\alpha,k}^{*}\,a^{\dagger}_{\alpha,k}e^{i\omega_{k}t_{\alpha}}\,\right)$
(3.13)

where the radial coordinate $r_{\alpha}\to\infty$ locally describes each
asymptotic region of $\partial M_{\alpha}$, so the relative coefficients in
the combination above are given by
$N_{\alpha,k}\equiv\lim_{r_{\alpha}\to\infty}r_{\alpha}^{\Delta}\;f_{\alpha,k}(r)$
and $N_{\alpha,k}^{*}\equiv\lim_{r_{\alpha}\to\infty}\;f_{\alpha,k}^{*}(r)$.
We show an explicit realization of this construction in Sec 3.5.

This relation is crucial to describe the excited states around the (thermal)
vacuum described in the previous section. Following the same procedure of ref
[42], plugging this into (3.1), and using the definition (3.7), one find that
these excitations consists of _thermal coherent states_ in the bulk Hilbert
space. Schematically

$\rho_{\beta}(\kappa)\sim\rho^{(JT)}_{\beta}\;\otimes\;\left(:\,e^{-\frac{\beta}{2}\,\hat{a}_{k}^{\dagger}\hat{a}_{k}+\kappa_{k}\,\hat{a}_{k}^{\dagger}+\kappa_{k}^{*}\,\hat{a}_{k}}\,:\right)\;$

for $\kappa$ small enough such that the back reaction is negligible and so,
the JT gravity sector factorizes from the $\chi$-excitations. The
$\alpha$-indices were omitted to simplify the notation,
$\hat{a}_{k},\hat{a}_{k}^{\dagger}$ denote the appropriate Bogoliubov’s
transformation (a linear combination of $a_{k},\hat{a}_{k}^{\dagger}$) that
diagonalizes the Hamiltonian, and $\kappa_{k},\kappa_{k}^{*}$ are related to
the (Wick-rotated) Fourier transform of the sources $\kappa(\tau)$ on the
imaginary-time segment $(0,\beta)$, see [42].

It is worth emphasizing that the standard holographic formula (2.5) [44]
implicitly assumes that the local operators in the lhs of (3.13) belong to the
SYK theory, i.e. they only depends on the fields of the theory and its time
derivatives, i.e.
${\cal{O}}={\cal{O}}(\psi_{i},\dot{\psi}_{i},\ddot{\psi}_{i},\dots)$. We will
show in a subsequent section that the inclusion of bulk fields with non-
trivial back reaction (sJT) might require a significant deformation of the
randomly coupled SYK model, and the standard BDHM formula should be properly
modified involving averaging. For instance, in presence of operators depending
on the (random) coupling constants (e. g. see the explicit example of Sec 4.2)
, the simplest modification of (3.13), which is consistent with (3.6), can be
expressed as

${\cal O}(t)\equiv\int dJ\,dM\,G(J)\,G(M)\,\;{\cal
O}(J,M,t)\,\,\,e^{-I[\psi,J,M,{\cal
C}]}=(2\Delta-1)\lim_{r\to\infty}\;r^{\Delta}\,(\chi(r,t)-\chi_{0})\,\,,$
(3.14)

The notation stands for an averaged _operator_ , where $M$ denotes all the
extra randomly distributed constants, introduced in the model through the
operator ${\cal O}$, see example below. This formula is also going to be
tested among the forthcoming examples.

####  3.3.1 SYK with random deformations

Let us illustrate a situation described above with the simplest toy example.
An interesting model inspired in [46] is described by the SYK Hamiltonian

$H[J]\equiv\frac{1}{2}(\psi_{i}\,\cdot\,\dot{\psi}^{i})+J_{ijkl}\psi^{i}\psi^{j}\psi^{k}\psi^{l},$
(3.15)

is the unperturbed microscopic Hamiltonian on a single $0+1$d boundary. The
type of perturbation proposed in ref [46] is

$V[M,t]\equiv\,i\;\kappa(t)\,{\cal O}[M]\,,$

where the operator generating the deformation is defined as

${\cal O}[M]\equiv M_{ijkl}\psi^{i}\psi^{j}\psi^{k}\psi^{l}\,,$ (3.16)

and the perturbation is sourced by an arbitrary function $\kappa$ along ${\cal
C}$. This scenario is slightly different from the general framework where the
formula (2.5) applies, since the operator that generates the perturbation is,
itself, associated to an (independent) random coupling $M$, independent from
$J$’s (see e. g. [46] and references therein). Essentially, this is nothing
but a standard SYK theory with $q=4$, by perturbing it with a purely imaginary
term

$J\;\mapsto\,\,j\equiv J+i\kappa M\;,$

where both $J,M$ denote real $2^{N/2}\times 2^{N/2}$ matrices. Thereby, the
model is SYK with _complex_ random coupling

$\langle U(j,{\cal C})\rangle\equiv\;\frac{\pi\sigma^{2}}{2}\int
dJ_{ijkl}\,dM_{ijkl}\,\,G_{\sigma}(J_{ijkl})\,\,G_{\sigma}(M_{ijkl})\;{\cal
T}\;e^{-\int_{{\cal
C}}d\theta\;\left(\frac{1}{2}(\psi\,\cdot\,\dot{\psi})+\frac{1}{4!}\sum_{i,j,k,l=1}^{N}\,j_{ijkl}\psi^{i}\psi^{j}\psi^{k}\psi^{l}\right)},$
(3.17)

where the double bracket represents the averaging on the two independent
random parameters $J,M$. By integrating out $M$, we recover the averaged
evolution operator

$\langle U(J)\rangle\equiv\int dJ_{ijkl}\,\,\,G_{\sigma}(J_{ijkl})\,\,\;{\cal
T}\,e^{-\int_{{\cal
C}}d\theta\;\left(\frac{1}{2}(\psi\,\cdot\,\dot{\psi})+\frac{1}{4!}\sum_{i,j,k,l=1}^{N}\,J_{ijkl}\psi^{i}\psi^{j}\psi^{k}\psi^{l}+\tilde{V}(\theta)\right)}$
(3.18)

which is the standard SYK model, but corrected with an _effective_ non-local
potential defined as

$\tilde{V}(\theta)\,\equiv\;-\;\frac{\sigma^{2}}{2}\;\;\int_{{\cal
C}}d\theta^{\prime}\,\kappa(\theta)\kappa(\theta^{\prime})(\psi(\theta)\cdot\psi(\theta^{\prime}))^{4}\;.$
(3.19)

Notice that, according to the considerations above, the corresponding bulk
dual could be modelled by a purely imaginary extra free scalar field $\chi$ on
a fixed JT background geometry [46]. The solutions are similar to the studied
in section 3.5 but with purely imaginary boundary values

$\lim_{r\to\infty}r^{-(\Delta-1)}\,\chi(r,t)=i\kappa(t)\,,$

and the canonical quantization of its fluctuations is to be implemented
according to (3.12), with the _minus_ choice. Thereby, the suitable
holographic extrapolation recipe may be realized by the formula (3.14). In
fact notice that by taking a derivative of the operator (3.17) with respect to
$\kappa(t)$, we obtain the lhs of that formula (3.14), while the counterpart
in the bulk, is given by the quantized field operator $\chi$ (or its spatial
derivatives), as standard in the holographic correspondence [47].

###  3.4 The gluing conditions in JT geometry

Before moving on to a specific example of (3.6), we consider the required
$C^{1}$ gluing conditions introduced in Sec. 2.1 for the particular scenario
of JT gravity. This is a fundamental tool to apply in the SvR method which
often requires to glue pieces of spacetime of different signature. The
analysis below shows that the quantum gravitational problem as defined on a SK
path leads to a well posed variational problem upon providing boundary
conditions only on all the asymptotic gravitational boundaries dual to the SK
path segments. The semiclassical analysis imposes a set of continuity
conditions: field continuity is required off shell to meet basic generating
functions properties and conjugated momenta continuity in the complex plane is
required on shell. It is necessary to demand $C^{1}$ continuity for the
metric, Dilaton and probe fields over the manifold, i.e. gluing of both the
field and its conjugated momenta in the complex plane.

####  3.4.1 Deriving the gluing conditions

Consider the Jackiw-Teitelboim gravity [8, 9] action defined on a generic
manifold $M$,

$16\pi
G_{N}I_{JT}=\Phi_{0}\left(\int_{M}\\!\\!\sqrt{g}R+2\int_{M}\\!\\!\sqrt{h}K\right)+\int_{M}\\!\\!\sqrt{g}\;\Phi\;(R-2\Lambda)+2\int_{\partial
M}\\!\\!\sqrt{h}\;\Phi\;(K-1)$ (3.20)

The terms accompanying $\Phi_{0}$ are topological and stand for the Euler
characteristic of the manifold under study, i.e. it is
$e^{-\Phi_{0}(2g+b-2)}$, with $g$ the genus and $b$ the number of boundaries.
For most purposes of this work, this term will play no major role. The
relevant terms for our analysis are the ones associated with the dynamic
Dilaton field $\Phi$. Notice from the second term that the equations of motion
for $\Phi$ fix the manifold to be pure AdS2, and the only remaining degree of
freedom is the Dilaton itself, which from the last term in (3.20) can be seen
to represent a reparametrization freedom between the actual physical time in
the dual $0+1$ quantum mechanics theory $u$ and the AdS2 time $t$. This
“boundary graviton” is the only degree of freedom in the theory. One can see
that the last term in (3.20) can be written explicitly in terms of the
reparametrization field $t(u)$ which have a Schwartzian action [4, 5]. We will
explore solutions of the Einstein-Dilaton equations of motion arising from the
action above and, in Sec. 4, deformations that allow for solutions describing
traversable wormholes.

For the purposes of this section, eq. (3.20) should be understood as the
manifold $M$ is a single segment of the SK path with defined signature, say
Lorentzian. This will contain at least one spacelike boundary $\Sigma$ and one
asymptotic boundary $\partial$. Its infinitesimal variation on a manifold with
a single timelike $\partial$ and a single spacelike $\Sigma$ boundary777The
existence of $\partial$ and $\Sigma$ boundaries implies that there is a
codim-2 boundary in which they meet. This type of terms have been taken into
account for SK paths in AdS/CFT in higher $d$ in [22] and in the specific JT
context in [48]. We disregard these type of contributions in what follows.
yields,

$\displaystyle 16\pi G_{N}\delta I_{JT}=$
$\displaystyle\int\sqrt{g}\left[\frac{1}{2}\;\Phi\;(R-2\Lambda)g^{\mu\nu}-R^{\mu\nu}\Phi+\nabla^{\mu}\nabla^{\nu}\Phi-g^{\mu\nu}\nabla^{2}\Phi\right]\delta
g_{\mu\nu}$ (3.21)
$\displaystyle\quad-\int\sqrt{g}\Phi_{0}\left[R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}\right]\delta
g_{\mu\nu}+\int\sqrt{g}(R-2\Lambda)\delta\Phi$ (3.22)
$\displaystyle\qquad+\int_{\partial}\sqrt{h}\left[2(K-1)\delta\Phi+\left(n^{\nu}\nabla_{\nu}\Phi-\Phi\right)h^{\alpha\beta}\delta
h_{\alpha\beta}\right]$ (3.23)
$\displaystyle\qquad+\int_{\Sigma}\sqrt{h}\left[2K\delta\Phi+n^{\nu}\nabla_{\nu}\Phi
h^{\alpha\beta}\delta h_{\alpha\beta}\right].$ (3.24)

An on shell analysis requires $\delta I_{JT}=0$ on the complete SK path. We do
this by imposing conditions that make the terms in (3.21), (3.22), (3.23)
trivial on their own for each segment individually. The terms in the first two
lines are bulk terms and are zero on shell by virtue of the equations of
motion. The third line correspond are boundary terms but do not lead to gluing
conditions between different pieces of the SK path. For example, fixing the
Dilaton and metric on the $\partial$ boundary sets these terms to zero on each
segment individually.

This cannot be done for the terms in (3.24) that involve the fields on
$\Sigma$, which at best can be put so that they cancel upon a specific gluing
between adjacent segments. One could impose some conditions at $\Sigma$ even
without introducing any specific (local) action. Basic properties of the
generating function force the fields to be continuous on $\Sigma$, i.e. taking
two adjacent sample segments $a$ and $b$ from the SK path, one can split and
re-glue the partition function as

$Z_{SK}=\int
d\Phi_{\Sigma}\;Z_{a}\Big{[}\Phi_{a}|_{\Sigma}=\Phi_{\Sigma}\Big{]}\;Z_{b}\Big{[}\Phi_{b}|_{\Sigma}=\Phi_{\Sigma}\Big{]}\qquad\Rightarrow\qquad\Phi_{a}|_{\Sigma}=\Phi_{b}|_{\Sigma}$
(3.25)

This conditions is valid off-shell and allows in particular to take
$\delta\Phi_{a}|_{\Sigma}=\delta\Phi_{b}|_{\Sigma}$. As such, two adjacent
terms $\delta I_{a,b}$ of the actions above from terms in (3.24) combine as

$\delta I_{a}+\delta
I_{b}=\int_{\Sigma}\sqrt{h}\left[2(K_{a}-K_{b})\delta\Phi+(n^{\nu}\nabla_{\nu}\Phi_{a}-n^{\nu}\nabla_{\nu}\Phi_{b})h^{\alpha\beta}\delta
h_{\alpha\beta}\right]$ (3.26)

where the relative minus sign comes from the normal vectors being in opposite
directions [22]. These contributions can be arranged to cancel themselves by
demanding

$K_{a}=K_{b}\;,\qquad\qquad\Pi_{a}=\Pi_{b}\;,\qquad\qquad\text{ on
$\Sigma$}\;,$ (3.27)

where we have used that since $\Sigma$ is a spacelike codim-1 surface,
$n^{\nu}\nabla_{\nu}\Phi\equiv\Pi$. These conditions are enough to find a
candidate saddle to the path integral on the complete SK path upon gluing all
segments.

Now, the background solutions we will explore in this work will be time
independent on both signatures and of the form,

$ds^{2}=-h(r)dt^{2}+h(r)^{-1}dr^{2}\,,\qquad
ds^{2}=h(r)d\tau^{2}+h(r)^{-1}dr^{2}\,,\qquad\Phi=\Phi(r)\,,$ (3.28)

where $h(r)$ is the same function on both metrics and can be taken to be a
general function of the radius $r$ for our current purposes. Furthermore, we
will always be gluing at $t,\tau$ constant surfaces. Upon Wick rotation, the
induced metric and Dilaton in (3.28) remain unchanged at these surfaces so the
gluing conditions for the fields (3.25) are met. The conditions (3.27) are
also met by the solutions (3.28) at $t,\tau$ constant surfaces. The tensor
$K_{\Sigma}$ is

$(K_{\Sigma})_{\mu\nu}=\frac{1}{2}\mathcal{L}_{n}P_{\mu\nu}=\frac{1}{2}n^{\alpha}\partial_{\alpha}(g_{\mu\nu}-n^{2}n_{\mu}n_{\nu})+\partial_{\mu}(n^{\alpha})(g_{\alpha\nu}-n^{2}n_{\alpha}n_{\nu})=0\;,$
(3.29)

where $n_{\alpha}=\delta_{\alpha,\tau}\sqrt{h(r)}$. As for the Dilaton,

$\Pi_{\Sigma}\equiv n^{\nu}\nabla_{\nu}\Phi=\partial_{t}\Phi=0\;,$ (3.30)

such that we are gluing zero on both sides.

On top of these solutions we will study probe scalar fields $\chi$, which
satisfies the Klein-Gordon equations. The gluing conditions for $\chi$ is
exactly the same as in the general higher dimensional case so we refer the
reader to [23, 28]. We need to $C^{1}$ glue the field $\chi$ and conjugated
momenta $\Pi_{\chi}$. The continuous gluing will be explicit in our treatment.
For future reference we explicitly write the gluing conditions in SK path
ordered time,

$\chi_{a}=\chi_{b}\;,\qquad\qquad\Pi_{\chi,a}=\Pi_{\chi,b}\;,\qquad\qquad\text{
on $\Sigma$}\;.$ (3.31)

###  3.5 Correlators in JT

To conclude this section we present an example which cover interesting aspects
of the SvR construction reduced to the 1+1 JT scenario, eq. (3.6).
Interestingly, a naive dimensional reduction of Fig. 1 is not itself a
relevant example for holography: exact AdS2 (with global AdS timelike Killing
vector) does not correspond to a sensible Quantum Mechanics with finite energy
excitations [49, 50]. Accordingly, pure JT gravity does not allow a constant
Dilaton profile, which would be the equivalent of a time-like Killing vector
in the higher dimensional AdS/CFT examples [51]. In this context, we find
particularly relevant to provide a fully fledged real time example of JT dual
to a single SK path. Our example can be introduced as a dimensional reduction
of the solution presented in [41, 42] adapted to JT, i.e. including the
Dilaton field.

An important comment is due regarding our computation of correlators in JT/SYK
correspondence. From the foundational JT/SKY works [52, 53] it was recognized
that the physical correlators in SYK where not directly the ones written in
terms of AdS2 time foliation (say $t$) but rather the physical SYK time $u$.
Its relation $t(u)$ is defined implicitly by the boundary conditions imposed
on the JT problem. More specifically, by the distance at which the AdS2 space
is cutoff in each direction. However, it was also found out in these works
that since the degree of freedom in this case is a reparametrization mode, for
a semiclassical analysis, one can actually follow the general AdS/CFT
intuition of extrapolating CFT correlators directly from AdS computations, and
then performing a rescaling on the boundary time $t\to t(u)$ [53]. In what
follows, we will leave this final step implicit, since it is beyond the point
of the physics we want to emphasize, i.e. the real time geometries and
correlators allowed by the SvR prescription.

####  3.5.1 TFD evolution

Here we present a time TFD-like evolution of a thermal circle of length
$\beta$. The SK path associated to this problem can be seen in Fig.2(a) and
previous holographic work on this path for standard AdS/CFT can be found in
[54, 41, 42], see also [55] for a QFT introduction. The path extends forwards
in real time, evolves $-i\beta/2$ in imaginary time and then comes backwards
to the initial time before closing the thermal circle with a final $-i\beta/2$
evolution.

(a)

(b)

Figure 2: (a) SK path associated to a TFD construction in QFT. The total
imaginary time evolution is $-i\beta$. The red dots are identified. (b) Bulk
saddle point of the TFD SK path. The geometry can be understood as a full
Euclidean AdS2 manifold upon a time evolution using a Global Rindler AdS real
time evolution at a moment of time reflection symmetry.

The dual geometry is built from two Lorentzian AdS2 Schwarzschild BH exteriors
(or equivalently in AdS2, two Rindler patches) dubbed $R,L$ and an Euclidean
BH manifold halved in two pieces $I,F$ as shown in Fig.2(b). The metrics are

$ds^{2}=-(\rho^{2}-1)d\sigma^{2}+\frac{d\rho^{2}}{(\rho^{2}-1)}\;,\qquad\qquad
ds^{2}=(\rho^{2}-1)d\varsigma^{2}+\frac{d\rho^{2}}{(\rho^{2}-1)}\;,\qquad\qquad\rho\in[1,\infty)$
(3.32)

for the Lorentzian and Euclidean regions respectively. The Lorentzian time is
taken $T_{-}\leq\sigma\leq T_{+}$ for both pieces, a positive time evolution
in $R$ and $L$ running in opposite directions, see Fig. B.1, as mandated by
the standard TFD construction. This is equivalent to stating that
$H^{-}=H_{R}-H_{L}$ is taken as the global Hamiltonian for the system888For
the same SK path, the $H^{+}=H_{R}+H_{L}$ choice is difficult to approach in
the JT/SYK correspondence. Essentially, the JT EOMs do not support a solution
invariant wrt the AdS2 Global time Killing vector ( there is no constant
Dilaton solution) such that the Euclidean pieces that close the path at
$T_{I/F}$ would depend explicitly on $\Delta T$. Finding these explicit
solutions goes beyond the scope of this work. This is reminiscent of the
Global time evolution of the TFD state prepared by the gravity Euclidean path
integral for a Black Hole, where no Global $H^{+}$ time Killing vector is
present, see e.g. [56]. , see [55, 41, 42] for more details. The regions $I,F$
have the range $-\pi\leq\varsigma\leq 0$ and $0\leq\varsigma\leq\pi$
respectively. The two Euclidean halves are cut open at constant
$\varsigma$-curves and pieces are glued to the constant $\sigma$-curves, thus
meeting the continuity conditions (3.25) and (3.27). In this coordinates the
solution for the Dilaton is

$\Phi=\rho\;\phi_{b},$ (3.33)

where $\phi_{b}$ is a constant boundary value. The solution is time
independent, thus also meeting the continuity conditions. This completes the
description of the background gravity solution.

We now probe our construction with a real scalar field $\chi(\sigma,\rho)$
with action, equation of motion and boundary condition given by

$I_{KG}=-\int\sqrt{g}(\partial_{\mu}\chi\partial^{\mu}\chi+m^{2}\chi^{2})\>,\qquad\qquad\left(\square-m^{2}\right)\chi=0,\qquad\qquad\chi(\rho,\sigma)|_{\partial}\sim\rho^{\Delta-1}\chi_{R}(\sigma)$
(3.34)

respectively. The conformal dimension of the boundary operator $\Delta$ is
defined through the standard relation $m^{2}=\Delta(\Delta-1)$. By expanding
the solution in Fourier modes of frequency $\omega$ one gets the radial
equation,

$\left((\rho^{2}-1)\chi^{\prime}(\rho)\right)^{\prime}=\left(\Delta(\Delta-1)-\frac{\omega^{2}}{\rho^{2}-1}\right)\chi(\rho)\;.$
(3.35)

A solution meeting the boundary conditions (NN modes) can be written as,

$\chi_{NN}(\sigma,\rho)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\sigma^{\prime}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega(\sigma-\sigma^{\prime})}\left(R^{+}_{\omega}\;p^{+}_{\Delta\omega}(\rho)+R^{-}_{\omega}\;p^{-}_{\Delta\omega}(\rho)\right)\chi_{R}(\sigma^{\prime}),\qquad\qquad
R^{+}_{\omega}+R^{-}_{\omega}=1$ (3.36)

with the eigenfunctions $p^{\pm}_{\Delta\omega}$ defined as

$\displaystyle p^{\pm}_{\Delta\omega}(\rho)$ $\displaystyle\equiv
2^{\Delta-1}e^{\pm\frac{\pi\omega}{2}}\frac{\Gamma(\Delta)\Gamma(\Delta\pm
i\omega)}{\Gamma(2\Delta-1)}P_{\Delta-1}^{\mp i\omega}(\rho)\simeq
1\times\rho^{\Delta-1}+\dots+\frac{2^{1-2\Delta}\Gamma\left(\frac{1}{2}-\Delta\right)\Gamma(\Delta\pm
i\omega)}{\Gamma\left(\Delta-\frac{1}{2}\right)\Gamma(-\Delta\pm
i\omega+1)}\times\rho^{\Delta}+\dots$

where $P_{n}^{m}(\rho)$ are the associated Legendre polynomial and the custom
$p^{\pm}_{\Delta\omega}$ notation stands for the function containing its poles
only in the upper/lower half $\omega$ plane at $\omega_{n}=\pm i(\Delta+n)$.
The $R_{\omega}^{\pm}$ notation is convenient for our purposes and one can see
that $R_{\omega}^{+}/R_{\omega}^{-}$ can be interpreted as the relative weight
of outgoing and infalling modes through the horizon [41]. Notice that so far
there are no restrictions on the $R_{\omega}^{+}/R_{\omega}^{-}$ quotient,
which means that (3.36) is not yet uniquely defined as a solution. Fixing this
quotient amounts to choosing a particular time ordering for the real time
correlator [54, 22, 41].

Furthermore, a full set999The physical normalizable mode basis is actually not
continuous. A discrete set of N modes rises once the inner product between
functions is correctly orthonormalized, see e.g. [57]. of normalizable (N)
modes of the form

$\chi_{N}(\sigma,\rho)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega\sigma}N_{\omega}\left(p^{+}_{\Delta\omega}(\rho)-p^{-}_{\Delta\omega}(\rho)\right)$
(3.37)

with arbitrary $N_{\omega}$ can still be added to the solution.

Analogous general solutions with undetermined coefficients can be found for
each of the $I,R,F,L$ regions in terms of their own asymptotic sources. This
freedom in the solutions is completely determined once the solutions in the
different piece-wise bulk duals are glued together according to (3.31).

As an example, consider a problem in which only asymptotic boundary conditions
on $R$ are turned on and asymptotic boundary conditions are turned off in
$I,L,F$. For region $R$ the solution $\chi_{R}$ should be a sum of (3.36) and
(3.37). On the other hand, for the $I,L,F$ regions the solution should contain
only N modes similar to (3.37),

$\displaystyle\chi_{I}(\varsigma,\rho)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\;e^{-\omega\varsigma}I_{\omega}\left(p^{+}_{\Delta\omega}(\rho)-p^{-}_{\Delta\omega}(\rho)\right)$
$\displaystyle\chi_{L}(\sigma,\rho)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega\sigma}L_{\omega}\left(p^{+}_{\Delta\omega}(\rho)-p^{-}_{\Delta\omega}(\rho)\right)$
(3.38) $\displaystyle\chi_{F}(\varsigma,\rho)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\;e^{-\omega\varsigma}F_{\omega}\left(p^{+}_{\Delta\omega}(\rho)-p^{-}_{\Delta\omega}(\rho)\right)$

when no sources are present. Imposing $C^{1}$ field continuity (3.31) in this
foliation imposes [41],

$\displaystyle\chi_{R}=\chi_{I}\,,$
$\displaystyle\quad-i\partial_{t}\chi_{R}=\partial_{\varsigma}\chi_{I}\,,$ on
$\sigma=T_{I}$, $\varsigma=0$ $\displaystyle\chi_{R}=\chi_{F}\,,$
$\displaystyle\quad-i\partial_{t}\chi_{R}=\partial_{\varsigma}\chi_{F}\,,$ on
$\sigma=T_{F}$, $\varsigma=0$ $\displaystyle\chi_{L}=\chi_{I}\,,$
$\displaystyle\quad-i\partial_{t}\chi_{L}=\partial_{\varsigma}\chi_{I}\,,$ on
$\sigma=T_{I}$, $\varsigma=-\pi$ $\displaystyle\chi_{L}=\chi_{F}\,,$
$\displaystyle\quad-i\partial_{t}\chi_{L}=\partial_{\varsigma}\chi_{F}\,,$
$\displaystyle\qquad\text{on \quad$\sigma=T_{F}$,\quad$\varsigma=\pi$}\;.$
(3.39)

One finds for $R$

$R^{+}_{\omega}=\frac{-1}{e^{2\pi\omega}-1}\qquad\qquad
R^{-}_{\omega}=\frac{e^{2\pi\omega}}{e^{2\pi\omega}-1}\qquad\qquad
N_{\omega}=0$ (3.40)

and for the other regions

$I_{\omega}=-e^{-i\omega
T_{I}}\tilde{\phi}_{R;\omega}\frac{1}{e^{2\pi\omega}-1}\qquad
L_{\omega}=-\tilde{\phi}_{R;\omega}\frac{e^{2\pi\omega}}{e^{2\pi\omega}-1}\qquad
F_{\omega}=-e^{-i\omega
T_{F}}\tilde{\phi}_{R;\omega}\frac{e^{2\pi\omega}}{e^{2\pi\omega}-1}$ (3.41)

with $\tilde{\phi}_{R;\omega}$ the Fourier Transform of the
$\phi_{R}(\sigma)$. We stress that all coefficients are completely determined
after gluing. This fixes both the initial state of our theory (the TFD state
in this case) and the correlator time ordering (Feynman ordering) as we check
below. Notice that crucially $I_{\omega}$ and $F_{\omega}$ are exponentially
suppressed in $\omega\to\pm\infty$ such that the $\omega$ integral in the
anzats can be seen to converge.

Correlators can be computed using the prescribed eq. (3.6) with on shell
action eq. (3.34) after standard holographic renormalization, i.e.

$\displaystyle-iS^{0}$
$\displaystyle=\frac{i}{2}\int_{\partial}\sqrt{\gamma}\chi(n^{\mu}\partial_{\mu}\chi)$
$\displaystyle=\int d\sigma
d\sigma^{\prime}\chi_{R}(\sigma)\left[\frac{4^{-\Delta}\Gamma\left(\frac{3}{2}-\Delta\right)}{i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;\frac{e^{-i\omega(\sigma-\sigma^{\prime})}}{e^{2\pi\omega}-1}\times\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left.\left(-\;\frac{\Gamma(\Delta+i\omega)}{\Gamma(1-\Delta+i\omega)}+e^{2\pi\omega}\;\frac{\Gamma(\Delta-i\omega)}{\Gamma(1-\Delta-i\omega)}\right)\right]\chi_{R}(\sigma^{\prime})\;,$
(3.42)

where the (Feynman ordered) correlator is captured in the squared brackets
above. We will proceed with the computation in two steps.

The first step is by noticing that the $\omega$ integral can be explicitly
carried out via Residue Theorem, closing the complex path from above or below
depending on the sign of $(\sigma-\sigma^{\prime})$. The singularity structure
of the integrand contains two families of poles: (i) at $\omega=\pm
i(\Delta+n)$ coming from the $\Gamma(\Delta\pm i\omega)$ functions and (ii) at
$\omega=\pm in$ coming from the $e^{2\pi\omega}-1$ denominator. We now show
that the second set of poles never contribute to the correlator regardless of
the sign of $(\sigma-\sigma^{\prime})$. To see this, consider closing the
integral from below ($(\sigma-\sigma^{\prime})>0$) and take the $\omega=-in$,
$n\in\mathbb{N}$. The residue of $(e^{2\pi\omega}-1)^{-1}$ is 1 for all $n$,
so their contribution can be isolated as

$\sum_{n\geq
0}e^{n(\sigma-\sigma^{\prime})}\left(-\;\frac{\Gamma(\Delta+n)}{\Gamma(1-\Delta+n)}+\frac{\Gamma(\Delta-n)}{\Gamma(1-\Delta-n)}\right)=\sum_{n\geq
0}\frac{\pi\;e^{n(\sigma-\sigma^{\prime})}(-1)^{n}}{\Gamma(1-\Delta-n)\Gamma(1-\Delta+n)\sin(\pi\Delta)}\left(-1+1\right)=0$
(3.43)

which is identically zero term by term. A similar cancellation on the set of
“thermal” poles was seen in [22, 41] and more recently in [36]. As for the (i)
$\omega=-i(\Delta+n)$ poles, for $(\sigma-\sigma^{\prime})>0$ we only pick the
second term in brackets in (3.42) to obtain,

$\displaystyle\langle{\cal O}_{R}(\sigma){\cal O}_{R}(\sigma^{\prime})\rangle$
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;\frac{e^{-i\omega(\sigma-\sigma^{\prime})}}{e^{2\pi\omega}-1}\left(-\;\frac{\Gamma(\Delta+i\omega)}{\Gamma(1-\Delta+i\omega)}+e^{2\pi\omega}\;\frac{\Gamma(\Delta-i\omega)}{\Gamma(1-\Delta-i\omega)}\right)$
(3.44)
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega(\sigma-\sigma^{\prime})}\;\frac{e^{2\pi\omega}}{e^{2\pi\omega}-1}\;\frac{\Gamma(\Delta-i\omega)}{\Gamma(1-\Delta-i\omega)}\;,\qquad\qquad(\sigma-\sigma^{\prime})>0$
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\frac{2\pi
i}{1-e^{2i\pi\Delta}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{e^{-(n+\Delta)\left(\sigma-\sigma^{\prime}\right)}}{\Gamma(1-n-2\Delta)}$
$\displaystyle=\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\frac{1}{\left[\cosh\left(\sigma-\sigma^{\prime}\right)-1\right]^{\Delta}}$

The $(\sigma-\sigma^{\prime})<0$ scenario follows similarly (one rather picks
the first term in brackets in (3.42) in closing the integral from above) and
yields the same result.

The second step is to determine the causal propagation the SK path is
prescribing. Notice that we have already shown that the integral is non
trivial whenever $|\sigma-\sigma^{\prime}|\neq 0$, so our correlator is
neither retarded nor advanced. One can see that the results we have obtained
so far are equivalent to consider an exponential in time regulated as
$(\sigma-\sigma^{\prime})(1-i\epsilon)$, $\epsilon>0$, which corresponds to
standard Feynman ordering. One can confirm this claim by going back to (3.42)
and check that for $(\sigma-\sigma^{\prime})>0$ one can still drop the first
term going to the second line. Then, the factors
$\exp\\{-i\omega(\sigma-\sigma^{\prime})(1-i\epsilon)\\}\sim\exp\\{-\omega\epsilon\\}$
correctly regulate the $\omega^{-(2\Delta-1)}$ divergence as
$\omega\to+\infty$ in the integrand. For $(\sigma-\sigma^{\prime})<0$ the
Feynman regulator behaves as $e^{+\omega\epsilon}$, correctly regulating the
integrand oscillations as $\omega\to-\infty$. Finally we get

$\displaystyle\langle{\cal O}_{R}(\sigma){\cal O}_{R}(\sigma^{\prime})\rangle$
$\displaystyle=\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\frac{1}{\left[\cosh\left((\sigma-\sigma^{\prime})(1-i\epsilon)\right)-1\right]^{\Delta}}$
(3.45)

Similar computations can be done by turning on asymptotic sources for the
other regions $I,L,R$. The linearity of the problem allows to solve for only
one source turned on at a time, and then find the general solution by
addition. As reviewed in Sec. 2, boundary sources on the Lorentzian regions
are usually put to zero after derivation and are thought as a tool to compute
correlators. On the other hand, Euclidean sources can be left turned on to
consider holographic excited states in the geometry and probe their real time
evolution. We will not carry all the computations since most of the important
pieces stem from the example above. We thus only present the results for
correlators and expectation values of the available observables, i.e. ${\cal
O}_{R},{\cal O}_{L}$, in this simple model.

The correlators are

$\displaystyle\langle{\cal O}_{L}(\sigma){\cal
O}_{L}(\sigma^{\prime})\rangle=\langle{\cal O}_{R}(\sigma){\cal
O}_{R}(\sigma^{\prime})\rangle^{*}\;,\qquad\langle{\cal O}_{L}(\sigma){\cal
O}_{R}(\sigma^{\prime})\rangle=\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\frac{1}{\left[\cosh\left(\sigma-\sigma^{\prime}\right)+1\right]^{\Delta}}$

where the resulting time ordering in the $\langle{\cal O}_{L}(\sigma){\cal
O}_{L}(\sigma^{\prime})\rangle$ correlator results reverse-time ordered.
Notice that correlators with operators inserted on opposite sides does not
have any singularities, signaling the lack of causal communication between
boundaries.

Expectation values of the operators yield via BDHM prescription (3.14),

$\displaystyle\langle\chi_{F}|{\cal O}_{R}(\sigma)|\chi_{I}\rangle$
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;\frac{e^{-i\omega\sigma}}{e^{2\pi\omega}-1}\frac{\Gamma(\Delta+i\omega)}{\Gamma(1-\Delta+i\omega)}\times$
(3.46)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left(-\int_{-\pi}^{0}d\varsigma\chi_{I}(\varsigma)e^{-\omega\varsigma}+\int_{0}^{\pi}d\varsigma\;e^{2\pi\omega}\chi_{F}(\varsigma)e^{-\omega\varsigma}\right)\;$
$\displaystyle=\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\left(\int_{-\pi}^{0}d\varsigma\frac{\chi_{I}(\varsigma)}{\left[\cosh\left(\sigma-i\varsigma\right)-1\right]^{\Delta}}+\int_{0}^{\pi}d\varsigma\frac{\chi_{F}(\varsigma)}{\left[\cosh\left(\sigma-i\varsigma\right)-1\right]^{\Delta}}\right)$
(3.47) $\langle\chi_{F}|{\cal
O}_{L}(\sigma)|\chi_{I}\rangle=\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\left(\int_{-\pi}^{0}d\varsigma\frac{\chi_{I}(\varsigma)}{\left[\cosh\left(\sigma+i\varsigma\right)-1\right]^{\Delta}}+\int_{0}^{\pi}d\varsigma\frac{\chi_{F}(\varsigma)}{\left[\cosh\left(\sigma+i\varsigma\right)-1\right]^{\Delta}}\right)$
(3.48)

which can be computed for any profile of Euclidean sources. The first
expression in (3.46) should be contrasted with (3.14) shows that one can
decompose the excitation in the N modes of the geometry and follow them
individually.

We now make two comments on the results of the example above:

#####  $\bullet$ Only the trivial solution in absence of sources as a check of
the gluing conditions:

A generic SK path split in several real and imaginary time pieces leads to
many bulk region pieces which should be $C^{1}$ glued to each other. We
propose a consistency check for these gluing conditions based on the fact that
(at least near the boundary, since the bulk interior might develop a non-
trivial topology) one is implementing a $C^{1}$ gluing on a closed complex-
time path of known period $-i\beta$.

The trick is to study the problem of finding a bulk solution when all
asymptotic boundary sources are turned off. This amounts to considering a
solution in $R$ as (3.37) and solutions eq. (3.38) for the other regions. The
only freedom in these solutions are its coefficients
$N_{\omega},I_{\omega},L_{\omega},F_{\omega}$ that are fixed by eqs. (3.5.1).

Since the SK problem is well posed, its solution being unique upon giving
boundary conditions on the asymptotic boundary, and we have turned off all
sources, it stems that the anzats $\chi=0$ must be the only consistent
solution. We can use this condition to check our derived gluing conditions. If
correct, the gluing of all N modes should impose

$N_{\omega}=e^{-\beta\omega}N_{\omega}$ (3.49)

and the same condition repeated for all other coefficients. Its solutions are
either $-i\beta\,\omega\sim 2\pi n$ or $N_{\omega}=0$. The former solution,
i.e. an expansion on the Matsubara frequencies of the pure Euclidean scenario,
is crucially inconsistent in real time. The reason for it is that pure
imaginary $\omega$ in the real time scenario forces the asymptotic behaviour
of the solution to change such that it no longer decays at the conformal
boundary, i.e. the N modes become divergent at the boundary. More precisely,
$p^{\pm}_{\Delta\omega}(\rho)\sim e^{\pm\omega\ln\rho}$ at $\rho\to\infty$
such that for imaginary frequencies a cancellation between these, as we have
built for the real time scenario is no longer possible. This inconsistency
also appears for any dimension $d>1$, see e.g. [41, 42]. We thus conclude that
the only consistent solution is
$N_{\omega}=I_{\omega}=L_{\omega}=F_{\omega}=0$, i.e. there are no pure N
modes in the geometry. We conclude that the real time pieces of the problem
impose non-trivial conditions that can help to check if we have written our
gluing conditions correctly in our preferred time foliation.

#####  $\bullet$ Other correlator orderings:

The in-depth analysis on the location of the complex poles in the field
solution we carried in this example allows to go beyond the Feynman
propagator. Specifically, by coming back at eq. (3.44) one can extract both
the retarded and advanced correlation functions

$\displaystyle Ret\\{\langle{\cal O}_{R}(\sigma){\cal
O}_{R}(\sigma^{\prime})\rangle\\}$
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega(\sigma-\sigma^{\prime})}\frac{e^{2\pi\omega}}{e^{2\pi\omega}-1}\;\frac{\Gamma(\Delta-i\omega)}{\Gamma(1-\Delta-i\omega)}$
$\displaystyle=\Theta(\sigma-\sigma^{\prime})\times\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\frac{1}{\left[\cosh\left(\sigma-\sigma^{\prime}\right)-1\right]^{\Delta}}$
$\displaystyle Adv\\{\langle{\cal O}_{R}(\sigma){\cal
O}_{R}(\sigma^{\prime})\rangle\\}$
$\displaystyle=\frac{2\Gamma\left(\frac{3}{2}-\Delta\right)}{4^{\Delta}i\pi\Gamma\left(\Delta-\frac{1}{2}\right)}\int_{-\infty}^{\infty}d\omega\;e^{-i\omega(\sigma-\sigma^{\prime})}\frac{(-1)}{e^{2\pi\omega}-1}\;\frac{\Gamma(\Delta+i\omega)}{\Gamma(1-\Delta+i\omega)}$
$\displaystyle=\Theta(\sigma^{\prime}-\sigma)\times\frac{\left(2\Delta-1\right)\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\;\frac{1}{\left[\cosh\left(\sigma-\sigma^{\prime}\right)-1\right]^{\Delta}}$

which can be checked to be correct by noticing that both provide the correct
expression for the correlator poles only on the upper/lower half
$\sigma$-plane, i.e. providing the correct Heaviside $\Theta$ functions.

##  4 Wormholes in holographic 2d gravity

In this Section we are going to study many aspects of two dimensional gravity,
as being dual to suitable quantum systems defined on disconnected boundaries.
Starting from the fact already observed in the previous Section, that any
classical 2d spacetime $M$ (with arbitrary piecewise signature) have $b$
closed curves as boundaries $\partial M={\cal C}^{1}\sqcup\dots\cal C^{1}$, we
construct and study the real-time holographic prescriptions for $b>1$,
involving (or not) averaging on coupling constants. In doing this, one must
deal with the so-called factorization problem and emphasize its implications
on that holographic prescriptions and dual quantum theories.

The natural generalization of the proposal JT/SYK is to consider the tensor
product of $b$ copies of the SYK models, but in order to get interesting
wormhole spacetimes consistent with holography, we must consider suitable
generalizations of JT model as dual gravity.

###  4.1 On the factorization problem and its resolution

The so-called factorization problem can be summarized as follows. For
simplicity, consider a manifold with two boundaries $b=2$, the prescription
(3.2) reads,

$\text{Tr}\,\;U[{\cal C}_{1}]\;U[{\cal
C}_{2}]=Z_{grav}(\kappa_{1},\kappa_{2})\qquad\qquad\partial M\equiv{\cal
C}_{1}\sqcup{\cal C}_{2}\qquad\kappa_{1,2}=\chi|_{{\cal C}_{1,2}}$ (4.1)

Since the operators $U[{\cal C}_{1}]$, $U[{\cal C}_{2}]$ act on the Hilbert
spaces ${\cal H}_{1}$, ${\cal H}_{2}$ respectively, the left hand side of this
equation _factorizes_ as

$\text{Tr}\,\;U[{\cal C}_{1}]\;U[{\cal C}_{2}]=(\text{Tr}_{1}\,U[{\cal
C}_{1}])\;(\text{Tr}_{2}\,\;U[{\cal C}_{2}]\,)\;,$

while inconsistently, the rhs involves _a sum_ over different spacetime
topologies $M_{1}\sqcup M_{2}$, plus all the connected manifolds with two
boundaries and genus $g\geq 1$, which _does not factorize_. It is important
emphasize here that eq. (4.1) stands for the exact expression, valid to all
quantum level/order, and the gravitational path integral must not be
understood in terms of any semiclassical or saddle point approximation.

This is nothing but a more refined form of the argument proposed in
[11]101010In that old version of the argument it was shown that, in certain
wormhole AdS5-spacetimes, the CFT state ($\sim U^{\otimes 2}$) should be
described by a thermal density matrix $e^{-\beta H}$, where $H$ should contain
a term _coupling_ both boundaries.  to conclude that the assumption $U[{\cal
C}_{1}]$, $U[{\cal C}_{2}]$ acting on the Hilbert spaces ${\cal H}_{1}$,
${\cal H}_{2}$ separately should be incorrect, and therefore, at least a part
of the total Hamiltonian ${\cal H}_{1}$ should involve operators acting on
$\;{\cal H}_{2}$ and vice-versa. It is equivalent to the presence of
(coupling) terms involving operators of the two field theories in the
Hamiltonian, e.g. $\propto{\cal{O}}_{1}{\cal{O}}_{2}$. Later, this argument
was drastically enforced in [39], where it was shown that these type of
(double trace) terms in the boundary Hamiltonian slightly deform the dual
geometry to allow traversable wormholes.

In more recent years, it has been argued that the factorization problem would
be absent in the context of averaging theories as SYK in $0+1d$, where the
boundary field theory, supposed to be dual to JT gravity (or at least certain
effective dof´s), is obtained as a suitable average on certain family of
randomly coupled theories [58]. In what follows we are going to formulate more
precisely the prescriptions for this type of holographic duality, and will try
to explain how the averaging mechanism can avoid the factorization problem, in
fact, we will show that theories ${\cal H}_{1}$, ${\cal H}_{2}$ with random
parameters on each boundary give rise to an effective coupling between them.

In summary, there are at least two ways to solve this apparent conflict,
namely: a) in a purely holographic theories (without any averaging) one must
accept the presence of terms in the theory that couple the fields on
disconnected components of the asymptotic boundary [11, 39]; otherwise, b) one
should relax the standard holographic dictionary by assuming some proper type
of averaging on the lhs of the formula (3.2). However, we are going to finally
show that in the SYK case the second option yields to the first one in an
effective sense.

####  4.1.1 SYKb/JT: the real-time prescription for $b$ disconnected
boundaries

We have argued in Sec. 3.1 that any 2d-spacetime $M$ with $b$ boundaries, must
be holographically described by $b$ closed curves where the (averaged)
evolution operators are defined on. Thus, the prescription (3.6) can be
generalized by doing the following natural replacement

$\text{Tr}\Bigg{\\{}\bigotimes_{A=1}^{b}\;U[J_{A},\kappa_{A},{\cal
C}_{A}]\Bigg{\\}}\leavevmode\nobreak\ \to\leavevmode\nobreak\
\langle\text{Tr}\,U[\kappa,{\cal C}]\rangle\equiv\int dJ_{1}\dots
dJ_{b}\,G(J_{1},\dots,J_{b})\,\text{Tr}\Bigg{\\{}\bigotimes_{A=1}^{b}\;U[J_{A},\kappa_{A},{\cal
C}_{A}]\Bigg{\\}}$ (4.2)

On the the lhs the tensor product is because we are initially assuming that
the operators $U[{\cal C}_{A}]$ act on their own Hilbert spaces ${\cal H}_{A}$
respectively, $A=1,\dots,b$. So the novel modified holographic formula, that
substitutes the SvR recipe capturing the field theory averaging, $\langle$ SvR
$\rangle$, expresses as follows

$\langle\,\text{Tr}\,U\left[\kappa_{1},\dots,\kappa_{b}\,;\,{\cal
C}_{1}\sqcup\dots\sqcup{\cal
C}_{b}\right]\rangle=Z_{JT}(\kappa_{1},\dots,\kappa_{b})$ (4.3)

The left hand side stands for an average on $\text{Tr}\,U$’s which is
_equivalent_ to a path integral on a closed curve. In forthcoming Sections,
this prescription is going to be implemented using some specific 2d-wormhole
solutions already studied in the literature.

Let us remark here a couple of important constraints on the distribution
$G_{b}\equiv G(J_{1},\dots,J_{b})\,$ that appears in this formula: notice
first that in the large $N$ limit, the right hand side is given by the saddle
point approximation, thus, if the bulk theory is strictly JT gravity, whose
fields are Dilaton and metrics with no additional back-reacting fields, the
right hand side factorizes as

$Z_{JT}(\kappa_{1},\dots,\kappa_{b})\approx e^{iI_{JT}(\kappa_{1},{\cal
C}_{1})}\dots e^{iI_{JT}(\kappa_{b},{\cal C}_{b})},$ (4.4)

and each of these theories is dual to a single SYK model (3.5), which implies
that to large $N$ the leading contribution to $G_{b}$ must be a product of $b$
normal distributions as (3.4). On the other hand, one can be tempted to take
this as the more natural distribution even at quantum level; nevertheless, if
the field theories on each disconnected boundary are decoupled among them, the
lhs of (4.2) would factorize, and again, it would contradict the quantum (JT)
gravity path integral on the right hand side. Therefore, in order to avoid
this paradox we conclude that in SYKb/JT duality, the distribution _only_ can
factorize in the large $N$ limit.

These two facts can be summarized in the following expression,

$G(J_{1},\dots J_{b})\,\to\,G(J_{1})\dots G(J_{b})\qquad\text{as}\qquad N\gg
1\;,$ (4.5)

where each factor on the right is given by (3.4), while other (sub-leading in
$1/N$) contributions cannot factorize111111However, we will see (4.5) that
although $G_{b}$ can be written exactly as a product of distributions, the
factorization issue can be avoided by introducing more random couplings in the
SYK model. Notice, however, that there are more general frameworks where the
two dimensional gravity admit connected wormholes as the dominant classical
solution. In these scenarios the first part of this argument fails because the
saddle point approximation (4.4) does not factorize. In this case the leading
contribution (to large N) must be very different from (4.5). We discuss some
realizations of ensembles that violate (4.5) below.

####  4.1.2 Random SYKb models and the rigidity constraint

If we consider that the quantum (random) systems living on different
(disconnected) boundaries are similar, it is quite natural that the general
properties and constants that characterize each one are the same. Thus, the
function $G(J_{1},\dots,J_{b})$, as well as the actions, should factorize in
$b$ equal random models characterized by
$\sigma_{1},\dots,\sigma_{b}\equiv\sigma$ and vanishing mean values. However
as we just argued, this runs into trouble with the averaging-resolution of the
factorization puzzle.

In that sense, one can assume some specific form of the function $G_{b}$,
implementing some constraint between the random couplings $J$s defined on the
different boundaries. In fact, they can be related in some simple way, e.g,
through a set of linear relations

$\sum_{B=1}^{b}\;c_{AB}\,J_{B}+c_{A}=0\;\;\qquad A,B=1,\dots b\;.$

which we can define as _rigidity relations_. This is a further requirement,
whose origin from fundamental features of the SYKb quantum system is not
within the scope of this article. The simplest form of this constraint is

$J_{1}=\dots=J_{b}\,.$ (4.6)

These properties can be summarized and described in the generalized
$\langle$SvR$\rangle$ recipe (4.3), by defining

$G(J_{1},\dots,J_{b})\equiv\;\;\prod^{b-1}_{A=1}\,\int_{-\infty}^{\infty}\,dJ_{A}\;G_{\sigma}(J_{A})\;\delta(J_{A}-J_{A+1})\;$
(4.7)

which from now on, will be simply referred to as _random rigidity_. We are
going to see below and in the forthcoming toy models that this type of
distributions induce effective couplings between the boundaries, which would
be consistent with wormhole-like gravitational saddle points. Note that this
distribution does not satisfy (4.5).

#####  $\bullet$ Rigid random models and dual wormhole geometry:

Consider the simplest case $b=2$. The specific SYK model on each boundary is
described by the action

$I[J_{A},\Psi_{A}]\equiv\int_{{\cal
C}_{A}}d\theta\;\left(\frac{1}{2}\psi_{A}\;\cdot\;\dot{\psi}_{A}+\frac{1}{4!}\sum_{i,j,k,l=1}^{N}\,J_{Aijkl}\psi_{A}^{i}\psi_{A}^{j}\psi_{A}^{k}\psi_{A}^{l}\right)\;;\qquad\qquad
A=1,2\;.$ (4.8)

on a closed complex-path ${\cal C}_{A}$. The dot of the kinetic term denotes
sum over repeated $i=1,\dots,N$, (i.e $(x_{i}\cdot
y_{i})=\sum_{i=1}^{N}x_{i}y_{i}$). The lhs of the recipe (4.3) in this case
reads

$\displaystyle\langle\,\text{Tr}\,U\left[\kappa_{1},\kappa_{2}\,;\,{\cal
C}_{1}\sqcup{\cal C}_{2}\right]\rangle$ $\displaystyle=\int\prod_{A}{\cal
D}\psi_{A}\int
dJ_{1}\,dJ_{2}\,\delta(J_{1}-J_{2})\,G_{\sigma}(J_{1})\,G_{\sigma}(J_{2})\;\,e^{-(I[J_{1},\psi_{1}]+I[J_{2},\psi_{2}])}$
(4.9) $\displaystyle=\int\prod_{A}{\cal D}\psi_{A}\;\int
dJ\,e^{-\frac{J^{2}}{2\sigma^{2}}}\,\,\,e^{-(I[J,\psi_{1}]+I[J,\psi_{2}])}$
(4.10)

where we have used the rigidity relations (4.7). The final result is the
effective path integral for the quantum boundary theory

$\langle\,\text{Tr}\,U\left[\kappa_{1},\kappa_{2}\,;\,{\cal C}_{1}\sqcup{\cal
C}_{2}\right]\rangle=\;\sigma{\sqrt{\frac{\pi}{2}}}\;\int\prod_{A}{\cal
D}\psi_{A}\;\,\,\;e^{-(I_{eff}[\psi_{1}]+I_{eff}[\psi_{2}])+I_{int}(\psi_{1},\psi_{2})}$
(4.11)

where

$I_{eff}[\psi]\equiv\,\,\int_{{\cal
C}}d\theta\;\;\frac{1}{2}\psi\;\cdot\;\dot{\psi}\;\;+\;\;\frac{\sigma^{2}}{2}\int_{{\cal
C}}d\theta\int_{{\cal
C}}d\theta^{\prime}\,(\psi(\theta)\cdot\psi(\theta^{\prime}))^{4}\;,$ (4.12)
$I_{int}[\psi_{1},\psi_{2}]\equiv\;\;\sigma^{2}\;\int_{{\cal
C}_{1}}d\theta_{1}\int_{{\cal
C}_{2}}d\theta_{2}\,(\psi_{1}(\theta_{1})\cdot\psi_{2}(\theta_{2}))^{4}$
(4.13)

The case with local insertions $\kappa_{1},\kappa_{2}\neq 0$ corresponds to
add the term $\int_{{\cal C}}d\theta\kappa(\theta){\cal O}(\theta)\,$ to (4.8)
and (4.12). The action (4.12) stands for an effective SYK theory on each
boundary.

Thereby, the main conclusion of this calculation is that two (or $b$) SYK
quantum models, with random coupling constants related by a rigidity
constraint (4.7), behave as two effective SYK models on each boundary, with an
effective coupling term between them. Thus, a dominant wormhole solution is
able to exist in the dual gravitational theory, and in addition, this
effective boundary theory cannot factorize because of the term (4.13). The
(effective) coupling constant is $\sigma^{2}\equiv 3!\sigma_{0}^{2}/N^{3}>0$,
where the dependence with $N$ turns this coupling negligible in the semi-
classical limit. As shown in ref. [11], in conventional (pure) holography, the
presence of a coupling term as (4.13) is a necessary condition to have a
wormhole dual geometry, while the positiveness of this term is closely related
to the possibility of having traversable (or not) dual wormholes [39].

#####  $\bullet$ The fundamental state of SYKb with holographic JT dual:

Going deeper into the arguments of Sec 4.1.1, the fundamental state of $b$
independent copies of SYK, namely SYKb, must be a tensor product of states
(3.7)

$\rho^{\otimes
b}=\rho_{\beta_{1}}(J_{1})\otimes\dots\otimes\rho_{\beta_{b}}(J_{b})$ (4.14)

associated to each disconnected (closed) piece of the boundary. This is
because, the holographic dual of the fundamental state shall be a classical
solution of _Euclidean_ JT gravity121212For further details on this claim,
read [59] and the discussion related in [60]., but _there are no_ such
Euclidean wormhole solutions connecting two or more boundaries in pure JT
gravity. Thus, the correlation functions between different boundaries must
vanish, as precisely described by the state (4.14). In averaged field
theories, eq. (4.3), the same geometric argument requires that (4.5) is met.

As we will see later, there can be deformations of _pure_ JT gravity by
including other local fields in the gravity theory, or simply matter fields
which can back-react or modify the global structure (topology) of the space-
time non trivially. In what follows we will generically refer to these
theories as _sourced_ JT gravity (denoted as sJT), which might admit classical
wormholes connecting two (or more) asymptotic boundaries. Let us denote the
additional bulk local fields by $\chi$, and the back reaction is controlled by
the $||T_{\mu\nu}(\chi)||$ scale: e.g, if this is much less than $1/G_{N}$ the
back reaction is negligible, and eq. (4.5) approaches the distribution. A non-
trivial correction of (4.5) in this scale/parameter, should contain non
factorizable terms.

Although (4.5) is a first non-trivial condition on the distribution, there is
no general prescription to determine how SYK should be deformed, or its impact
on the gravity side. For example, in Section 4.2 we will consider a particular
realization with those features, where the pure JT is deformed with a an
imaginary free scalar field, which admits classical wormhole solutions and its
dual is suggested to be a particular (rigid) deformation of SYK [46].

###  4.2 Random (rigid) deformations of SYK and wormholes

Now consider two disconnected boundaries labeled by $A=1,2$ and the specific
model proposed in [46] whose phenomenology is suggestively similar to wormhole
solutions of certain sJT. The boundary theory consists of two independent SYK
actions (4.8), defined on each of them. Then it will be deformed by adding the
purely imaginary potential

$V_{A}[M]\equiv\,(-1)^{A}\;i\kappa\;{\cal O}[M]\,$

on the respective boundary $A=1,2$. It is similar to the model of Sec 3.3.1:
the operator ${\cal O}[M]$ was already defined in (3.16) and the source
$\kappa$ is taken to be independent on time for simplicity, although in
general it can be time dependent. This situation is slightly different from
the framework defined in Sec 3.3.1, since the operator that generates the
perturbation is, itself, associated to an independent random coupling $M$.

Let us consider previously the following toy model. The action on each
boundary is described by the action

$I[J_{A},\Psi_{A}]\equiv\int_{{\cal
C}_{A}}d\theta\;\left(\frac{1}{2}\psi_{A}\;\cdot\;\dot{\psi}_{A}+j_{A;ijkl}\psi_{A}^{i}\psi_{A}^{j}\psi_{A}^{k}\psi_{A}^{l}\right)\;$
(4.15)

on a closed complex-path ${\cal C}_{A}$, where $\,j_{A}\equiv
J_{A}+(-1)^{A}i\kappa M_{A}$. The lhs of the recipe (4.3) in this case reads

$\int\prod_{A}{\cal D}\psi_{A}\int
dJ_{1}\,dJ_{2}\,\,G_{\sigma}(J_{1})\,G_{\sigma}(J_{2})\int
dM_{1}\,dM_{2}\,\delta(M_{1}-M_{2})\,G_{\sigma}(M_{1})\,G_{\sigma}(M_{2})\;\,e^{-(I[j_{1},\psi_{1}]+I[j_{2},\psi_{2}])}$
(4.16) $\\\ =\int\prod_{A}{\cal D}\psi_{A}\;\int
dJ_{1}\,dJ_{2}\,\,G_{\sigma}(J_{1})\,G_{\sigma}(J_{2})\int
dM\,\,\,e^{-\frac{M^{2}}{2\sigma^{2}}}\,\,e^{-(I[J_{1}+i\kappa_{1}M,\psi_{1}]+I[J_{2}+i\kappa_{2}M,\psi_{2}])}$

where the rigidity relations (4.7) were assumed only for the (real) coupling
constants $M$, and $\kappa_{1}=\kappa_{2}\equiv\kappa$ being a _non-vanishing_
constant.

Notice that here we did not impose rigidity on the respective SYK coullings
$J_{1},J_{2}$ such that they are considered independent; therefore as
$\kappa\to 0$, the theory becomes a couple of independent SYK models on each
boundary, and then the dual gravity would become strictly JT. However, as
argued around eq. (4.5), the remaining averaging in $K$ is necessary to avoid
the factorization paradox, thus in this model, one must demand that
$\kappa\neq 0$.

The final result is the effective path integral for the quantum boundary
theory

$\langle\,\text{Tr}\,U\left[\kappa\,;\,{\cal C}_{1}\sqcup{\cal
C}_{2}\right]\rangle=\sigma{\sqrt{\frac{\pi}{2}}}\;\int
dJ_{1}\,dJ_{2}\,\,G_{\sigma}(J_{1})\,G_{\sigma}(J_{2})\int\prod_{A}{\cal
D}\psi_{A}\;e^{-(I[J_{1},\psi_{1}]+I[J_{2},\psi_{2}])+I_{int}(\psi_{1},\psi_{2})}$
(4.17)

which consists of two decoupled SYK models, plus an effective potential term

$I_{int}[\psi_{1},\psi_{2}]\equiv\;\;\frac{(i\kappa)^{2}\sigma^{2}}{2}\;\left(\sum_{A}\int_{{\cal
C}_{A}}d\theta\int_{{\cal
C}_{A}}d\theta^{\prime}\,(\psi_{A}(\theta)\cdot\psi_{A}(\theta^{\prime}))^{4}-2\int_{{\cal
C}_{1}}d\theta_{1}\int_{{\cal
C}_{2}}d\theta_{2}\,(\psi_{1}(\theta_{1})\cdot\psi_{2}(\theta_{2}))^{4}\right)$

where the last term is a genuine coupling between the boundaries as expected.
It is worth noticing that as $N\to\infty$ this term is negligible, and one
recovers two copies of the (exact) SYK/JT correspondence as expected. Another
remarkable fact with this is that now the (effective) coupling constant is
$\kappa^{2}\;3!\sigma_{0}^{2}/N^{3}\,>0$ which suggests the traversability of
the dual wormhole. As shown in ref. [11], in conventional (pure) holography,
the presence of a coupling term as (4.13) is a necessary condition to have a
wormhole dual geometry, while the positiveness of this term is closely related
to the possibility of having traversable (or not) dual wormholes [39]. The
model proposed in [46], however, consists of imposing the additional rigidity
constraint to $J_{1},J_{2}$. This is described by the action

$S[J_{A},\Psi_{A}]\equiv\int_{{\cal
C}_{A}}d\theta\;\left(\frac{1}{2}\psi_{A}\;\cdot\;\dot{\psi}_{A}+j_{Aijkl}\psi_{A}^{i}\psi_{A}^{j}\psi_{A}^{k}\psi_{A}^{l}\right)\;$
(4.18)

on a closed complex-path ${\cal C}_{A}$, where $\,j_{A}\equiv
J_{A}+(-1)^{A}i\kappa M_{A}$. The lhs of the recipe (4.3) in this case reads

$\int\prod_{A}{\cal D}\psi_{A}\int
dJ_{1}\,dJ_{2}\,\delta(J_{1}-J_{2})\,G_{\sigma}(J_{1})\,G_{\sigma}(J_{2})\int
dM_{1}\,dM_{2}\,\delta(M_{1}-M_{2})\,G_{\sigma}(M_{1})\,G_{\sigma}(M_{2})\;\,e^{-(I[j_{1},\psi_{1}]+I[j_{2},\psi_{2}])}$
(4.19) $\\\ =\int\prod_{A}{\cal D}\psi_{A}\;\int dJ\,\int
dM\,e^{-\frac{J^{2}}{2\sigma^{2}}}\,\,e^{-\frac{M^{2}}{2\sigma^{2}}}\,\,e^{-(I[J+i\kappa_{1}M,\psi_{1}]+I[J+i\kappa_{2}M,\psi_{2}])}$

where we have used the rigidity relations (4.7) for both real coupling
constants $J,M$, and $\kappa_{1}=\kappa_{2}\equiv\kappa$ is constant. The
final result is the effective path integral for the quantum boundary theory

$\langle\,\text{Tr}\,U\left[\kappa\,;\,{\cal C}_{1}\sqcup{\cal
C}_{2}\right]\rangle=\;\;\frac{\pi\sigma^{2}}{2}\;\int\prod_{A}{\cal
D}\psi_{A}\;\,\,\;e^{-(I_{eff}[\psi_{1}]+I_{eff}[\psi_{2}])+I_{int}(\psi_{1},\psi_{2})}$
(4.20)

where

$I_{eff}[\psi]\equiv\,\,\int_{{\cal
C}}d\theta\;\;\frac{1}{2}\psi\;\cdot\;\dot{\psi}\;\;+\;\;\frac{(1+(i\kappa)^{2})\sigma^{2}}{2}\int_{{\cal
C}}d\theta\int_{{\cal
C}}d\theta^{\prime}\,(\psi(\theta)\cdot\psi(\theta^{\prime}))^{4}\;,$ (4.21)
$I_{int}[\psi_{1},\psi_{2}]\equiv\;\;(1-(i\kappa)^{2})\sigma^{2}\;\int_{{\cal
C}_{1}}d\theta_{1}\int_{{\cal
C}_{2}}d\theta_{2}\,(\psi_{1}(\theta_{1})\cdot\psi_{2}(\theta_{2}))^{4}$
(4.22)

Now the (effective) coupling constant is
$3!\sigma_{0}^{2}/N^{3}(1+\kappa^{2})>0$ [61]. Notice that as $\kappa\to 0$,
one recovers the model (4.11). The dependence of this coupling with $N$ is
crucial to get consistency with the requirement (4.5), since this coupling
vanishes in the large $N$ limit, however regarding the duality SYK/JT, the
leading boundary theory effectively behaves as a system of free fermions. In
this sense, an interesting modification similar to the previous toy model
might be considered by simply relaxing the rigidity in $J$s (eq. (4.17)).

Another interesting possibility to be studied is to re-scale the relative
coupling, e.g. $\kappa\to 1/N$, in order to turn the deformation manifestly
sub-leading with respect to the coupled SYK model, eq. (4.11).

###  4.3 Wormhole correlators

The models discussed so far are suitable deformations of SYK consistent with
the averaged holographic formulas (eq. (4.3)), the large $N$ limit, and are
able to describe dual gravity models with wormholes as dominant saddles. In
real time, it is particularly interesting to get traversable wormholes.
Thereby, the next step is to analyze the existence and properties of these
solutions in 2d gravity; namely, sourced JT models that we have defined above.

For multiple boundaries we consider two disconnected SK paths in complex time
and assuming the presence of bulk interactions or extra bulk fields that
support the wormhole we explore real time gravitational geometries. We take
these manifolds as background and compute real time correlation functions for
probe scalar fields. The problem we are set to solve is the two boundary
version of (4.3),

$Z_{SYK}\equiv\int_{{\cal C}_{1};{\cal C}_{2}}e^{-I_{g}-I_{KG}}$ (4.23)

where $\int_{{\cal C}_{1};{\cal C}_{2}}$ represents an integration over all
fields with boundary conditions fixed on the asymptotic boundaries defined by
the curves ${\cal C}_{1},{\cal C}_{2}$ to be specified below. $I_{g}$
represents a particular gravitational theory that supports the wormhole as a
saddle point. The precise way in which the wormholes are made stable is not of
particular importance. For example, one can consider any of the set-ups
described in App. A, for which our geometries would be saddle point solutions.
The $I_{KG}$ is taken to be an action for a massive probe scalar $\chi$ of
which we compute boundary correlators in real time and expectation values on
holographic excited states.

An important comment is due regarding our examples and Hamiltonian choices
$H^{\pm}\equiv H_{1}\pm H_{2}$ in the boundary theories. Recalling our
discussion below (3.32), we have that the prescribed time evolution for a TFD
system is generated by $H^{-}\equiv H_{1}-H_{2}$ [55] over the SK path in Fig.
2. The corresponding bulk evolution in this scenario for a single SK boundary
involved only the exterior Rindler patches, covered by the time-like Killing
vector that does not penetrate into the horizon. As mentioned in footnote 8,
the complementary $H^{+}$ scenario for the same SK path was beyond the scope
of our work. Essentially it lacks a Global time Killing vector which would
make the gravitational dual a complicated complex signature manifold. As a
consequence, this scenario was disregarded in Sec. 3.5.

However, we show that upon considering two SK paths as boundary conditions, a
real time saddle associated with the $H^{+}$ boundary Hamiltonian choice opens
up. The $H^{+}$ scenario will involve an AdS2 version of the real time Thermal
AdS scenario [22, 41] or Thermal Wormhole and, as such, segments of Lorentzian
Global AdS2 geometry. This time, the $I_{g}$ wormhole saddle do provide a
constant Dilaton profile and the Global time Killing vector is recovered.
Conversely, the $H^{-}$ scenario is no longer able to provide a consistent
time independent Dilaton profile upon inclusion of the sources and real time
segments, thus reversing the situation that we had for the single SK path
scenario.

####  4.3.1 Thermal wormhole

We now present the example of a wormhole geometry dual to a couple of Thermal
SK path as in Fig. 3 with $H^{+}$ as the boundary Hamiltonian. The resulting
bulk saddle is also shown in Fig. 3. The geometry is composed of two
Lorentzian and two Euclidean sections of AdS2 of length $\Delta T$ and
$-i\beta/2$ respectively. It can be understood as a dimensional reduction of a
real time Thermal AdS solution [22, 41].

Thermal AdS is usually interpreted in higher dimensions as a state of thermal
equilibrium between the spacetime and matter, in which the matter is not hot
enough to collapse to a Black Hole [62]. In our scenario, an analogous phase
transition occurs at high temperatures to a disconnected geometry [51, 46].
Furthermore, there is no exact zero temperature Global AdS2 scenario relevant
for holography [52, 53].

In any case, as in the higher dimensional thermal AdS scenario, the physical N
modes of the geometry retain the pure AdS normal frequencies and its
dependence in the temperature appear as a rescaling in the mode normalization.
Mathematically, the equations of motion for the probe scalars are identical to
the ones in Global AdS2 so the propagating modes must be the same and the
topological periodicity can be forced via images method. We refer to App. B
for a study of the correlators and N modes of a massive scalar $\chi$ over
pure AdS2.

The metrics that cover the geometry are

$ds^{2}=-(r^{2}+1)dt^{2}+\frac{dr^{2}}{r^{2}+1}\;,\quad
ds^{2}=(r^{2}+1)d\tau^{2}+\frac{dr^{2}}{r^{2}+1}\;,\quad
r,t\in(-\infty,\infty)\quad\tau\in(-\beta/2,\beta/2]$ (4.24)

and the Dilaton profile is

$\Phi=\frac{2\phi_{b}}{\pi}(1+r\arctan(r))$ (4.25)

where $\phi_{b}$’s exact expression depends on the model that make the
wormhole stable, see e.g. (A.2) and (A.6). We denote the real time boundaries
in front of Fig. 3 $R_{1},R_{2}$ and the ones in the back as $L_{1},L_{2}$.

Figure 3: On the sides of the figure we show the two SK paths dubbed ${\cal
C}_{1}$ and ${\cal C}_{2}$. These paths have an effective interaction such
that their dynamics are coupled. In the middle of the figure the geometry dual
to two SK interacting SK paths is shown. It contains segments of pure
Euclidean and Global Lorentzian AdS2.

We now present scalar real time correlators in the geometry shown in Fig. 3
for a Klein Gordon scalar field, see (B.3) over metric (4.24). As mandated by
the SvR prescription, correlators are computed by: putting boundary sources,
solving the equations of motion that meet the gluing conditions, computing the
on-shell action and deriving wrt to the external sources to find the
correlators using (4.23).

The math involved in the computations for this section are highly derivative
of those in App. B and as such we mostly present results. To put an example,
for a single source $\chi_{R_{1}}(t)$ source on $R_{1}$ after solving the
equations of motion and meeting the gluing conditions with all other regions
one gets, cf. with (B.8),

$\displaystyle\chi_{R}(t,r)$ $\displaystyle=\sum_{n=-\infty}^{\infty}\int
dt^{\prime}\int_{{\cal{F}}}\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime}+in\beta)}q_{1;\omega}(r)\chi_{R_{1}}(t^{\prime})$
(4.26)
$\displaystyle=\frac{2^{-\Delta}\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\sum_{n=-\infty}^{\infty}\int
dt^{\prime}\frac{\chi_{R_{1}}(t^{\prime})}{\left(\sqrt{r^{2}+1}\cos((t-t^{\prime})(1-i\epsilon)+in\beta)-r\right)^{\Delta}}\;.$
(4.27)

From the many possible correlators available to compute, we are mainly
interested in the ones that cross the real time wormhole. To fix notation, we
state that the same side $R_{1},R_{1}$ correlator is

$\langle{\cal O}_{R_{1}}(t){\cal
O}_{R_{1}}(t^{\prime})\rangle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\sum_{n=-\infty}^{\infty}\frac{1}{\left(\cos((t-t^{\prime})(1-i\epsilon)+in\beta)-1\right)^{\Delta}}\;,$
(4.28)

The correlators that cross from $R_{1},R_{2}$ is

$\langle{\cal O}_{R_{1}}(t){\cal
O}_{R_{2}}(t^{\prime})\rangle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\sum_{n=-\infty}^{\infty}\frac{1}{\left(\cos((t-t^{\prime})(1-i\epsilon)+in\beta)+1\right)^{\Delta}}\;.$
(4.29)

whilst the one that crosses both the wormhole and into the second copy
$R_{1},L_{2}$,

$\langle{\cal O}_{R_{1}}(t){\cal
O}_{L_{2}}(t^{\prime})\rangle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-1/2\right)}\sum_{n=-\infty}^{\infty}\frac{1}{\left(\cos(t-t^{\prime}+i(n+\frac{1}{2})\beta)+1\right)^{\Delta}}\;.$
(4.30)

The gravitational saddle shows that (4.29) must still contain lightcone
singularities, but our precise SvR real time prescription mandates that the
Lorentzian Thermal correlator is Feynman ordered. On the other hand, (4.30) is
no longer required to have a regulator. Notice interestingly that boundary
correlators on the same path such as

$\langle{\cal O}_{R_{1}}(t){\cal
O}_{L_{1}}(t^{\prime})\rangle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\sum_{n=-\infty}^{\infty}\frac{1}{\left(\cos(t-t^{\prime}+i(n+\frac{1}{2})\beta)-1\right)^{\Delta}}\;.$
(4.31)

have also lost its lightcone divergences due to the interaction between paths.
One should confront correlators (4.28) and (4.31) with the standard matrix
correlator for isolated SK paths, see [54, 41].

As for the expectations values of boundary operators in holographic excited
states, one gets that the solution on $R$ for sources in the Euclidean past
and future respectively are

$\displaystyle\chi_{I}(t,r)$
$\displaystyle=\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}e^{-i(\Delta+n)(t+im\beta)}\left(\bar{\chi}_{I_{1}}(\Delta+n)+\bar{\chi}_{I_{2}}(\Delta+n)\right)\mathop{\mathrm{Res}}_{\omega=\Delta+n}f_{L}(\omega,r)$
(4.32) $\displaystyle\chi_{F}(t,r)$
$\displaystyle=\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}e^{+i(\Delta+n)(t+im\beta)}\left(\bar{\chi}_{F_{1}}(\Delta+n)+\bar{\chi}_{F_{2}}(\Delta+n)\right)\mathop{\mathrm{Res}}_{\omega=\Delta+n}f_{L}(\omega,r)$
(4.33)

where we have denoted,

$\bar{\chi}_{I_{1/2}}(\omega)=\int_{-\frac{\beta}{2}}^{0}d\tau
e^{\omega\tau}\phi_{I_{1/2}}(\tau)\qquad\qquad\bar{\chi}_{F_{1/2}}(\omega)=\int^{\frac{\beta}{2}}_{0}d\tau
e^{-\omega\tau}\phi_{F_{1/2}}(\tau)$ (4.34)

and $\phi_{I_{1/2}}(\tau)$ and $\phi_{F_{1/2}}(\tau)$ are the Euclidean
sources on the I and F regions, defined to be non-trival only on
$\tau\in[-\beta/2,0]$ and $\tau\in[0,\beta/2]$ respectively. The expectation
value of boundary operators can now be obtained via the BDHM dictionary, i.e.

$\displaystyle\langle\chi_{F}|{\cal O}_{R}(t)|\chi_{I}\rangle$
$\displaystyle\equiv(2\Delta-1)\lim_{r\to+\infty}r^{\Delta}\left(\chi_{I}(t,r)+\chi_{F}(t,r)\right)$
(4.35)
$\displaystyle=\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}\left[e^{-i(\Delta+n)t}\left(\bar{\chi}_{I_{1}}(\Delta+n)+\bar{\chi}_{I_{2}}(\Delta+n)\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.+e^{+i(\Delta+n)t}\left(\bar{\chi}_{F_{1}}(\Delta+n)+\bar{\chi}_{F_{2}}(\Delta+n)\right)\right]F^{\beta}_{\Delta,n}$
(4.36)
$\displaystyle=\sum_{m=-\infty}^{\infty}\left(\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}}d\tau\frac{\chi_{I_{1}}(\tau)+\chi_{F_{1}}(\tau)}{\left[\cos(t+i\tau-
im\beta)-1\right]^{2\Delta}}+\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}}d\tau\frac{\chi_{I_{2}}(\tau)+\chi_{F_{2}}(\tau)}{\left[\cos(t+i\tau-
im\beta)+1\right]^{2\Delta}}\right)$ (4.37)

where we have written the ressumed expression in (4.37) to make explicit that
the sum of modes in $n$ makes the expression convergent. Our main result is
(4.36), where the propagating modes of the solution are made explicit and the
inherited mode normalization from holography in this finite temperature
scenario is,

$\displaystyle F^{\beta}_{\Delta,n}$
$\displaystyle=\frac{\Gamma\left(\frac{3}{2}-\Delta\right)\Gamma(n+2\Delta)}{4^{\Delta-1}\pi\Gamma\left(\Delta-\frac{1}{2}\right)\Gamma(n+1)}e^{-(\Delta+n)m\beta}\;.$
(4.38)

An analogous result for $\langle\chi_{F}|{\cal O}_{L_{1}}(t)|\chi_{I}\rangle$
can be found upon exchanging 1$\leftrightarrow$2 in (4.36). We must emphasize
that this set of correlators, expectation values and their attached geometry
Fig. 3 dual to a couple of Schwinger Keldysh paths, alongside their precise
time-orderings can be taken as a nice corollary of our work.

##  5 Conclusions

In this paper we have extended the Skenderis and van Rees and the BDHM
prescription for real time holography in the context of the JT/SYK
correspondence, and explored its consequences on both the factorization
problem and the role of the ensemble averaging in SYK.

In doing so, we were able to show a number of novel properties of real time
holography in the JT/SYK correspondence. For example, topological arguments
show that for the holographic gravitational problem to be consistent in 1+1
dimensions, the SK path defining the QM theory in the boundary must be closed.
Put in other words, the dynamics of holography for a single path in JT/SYK is
always exploring physics of a thermal initial state or perturbations of such a
state. In particular, we should highlight eq. (4.3) as an improved
$\langle$SvR$\rangle$ prescription which corresponds to a unified equation
capturing the holographic real time prescriptions and average ensemble for an
arbitrary number of boundaries. By construction, the prescription in eq. (4.3)
is free of the factorization problem and the role of ensemble averaging is
explored. Among other things, we also revisited well-established mechanisms to
construct traversable wormholes [11, 39, 51] in this context and updated the
criteria defined in [11].

To illustrate our construction, we presented a couple of relevant examples
both for single and multiple boundary SK paths. The first example provides a
real time saddle point solution dual to a single SK path. The geometry and its
phenomenology can be understood as a dimensional reduction of the solution
found in [41, 42] and shows that there are real time scenarios that hold from
traditional to JT/SYK holography. The second example presented is a novel real
time holographic dual to a couple of either entangled or interacting theories
defined over two SK paths. The complete manifold is seen to contain segments
of real time Global AdS2 (it can be seen as a dimensional reduction of Thermal
AdS2) and is checked to be relevant in the context of (modified) JT/SYK
holography. Thus, the canonically quantized modes travelling the system are
the same as in Global AdS2, adequately rescaled by the temperature. Explicit
results for correlators and expectation values of boundary operators over
holographic excited states are also studied and a number of interesting
properties are presented. Most notably, it can be seen that the effective
coupling between the disconnected boundaries modifies the correlation between
real time segments on the same path. Interestingly, the examples also serve to
illustrate an important choice on boundary time $H^{\pm}=H_{R}\pm H_{L}$ which
we have not seen emphasized elsewhere in the literature.

As for future directions, it would be interesting to find further novel
connected solutions for a $b>1$ set of boundaries and study the correlations
that such a geometry implies between the theories. If done systematically, one
could in principle try to derive constraints on the $G(J_{1},\dots,J_{b})$
that define the ensemble average on the QFT side of the duality, see e.g.
(4.5). We leave this possibility for future work.

## Acknowledgments

We thank J.Russo for relevant feedback on this work. RA is supported by UNLP
and CONICET grant PIBAA (2022 - 2023), MBC is supported by UNLP and CONICET
grants X791, PIP 2017-1109 and PUE Búsqueda de nueva Física. PJM is supported
by CONICET, CNEA, and Universidad Nacional de Cuyo, Argentina.

##  Appendix A Some realizations of wormholes in sourced JT

The models discussed in 4.2 show that there are suitable deformations of SYK,
that because of fundamental requirements of consistency of the holographic
formulas (e.g. factorization), are able to describe dual gravity models with
wormholes as dominant saddles. In real time $1+1$ gravity, it is particularly
interesting to study traversable wormholes. This is mostly due to its
mathematical simplicity that allows for explicit solutions. We now present two
realizations of these wormholes studied in detail in [51] and [46] that
provide physical context to the correlator computation that we will be
computing in this section below. In particular for [46] we will be able to
provide some physical insight to some expectation values computations that
were unclear before.

In the context of our present work and previous criteria on existence of
wormhole solutions [11], we can classify these solutions as being dual to: a)
an explicit coupling between operators on disconnected boundaries in pure
holography, or b) averaging over random couplings with a rigid constraint.

###  A.1 Explicit coupling between operators on disconnected boundaries

A natural set-up to realize wormhole solutions was the approach studied in
[51] on which the Gao-Yafferis-Wall mechanism [39] in higher dimensions was
adapted to a JT scenario. The method consists on explicitly coupling the QM
dual theories on each side in a particular way such that an effective negative
energy density arises in the bulk interior. This opens up a window of time in
which the wormhole between the causally disconnected theories become
connected. To be concrete, one adds the interaction $g\sum_{i}\int du\;{\cal
O}^{i}_{1}(u){\cal O}^{i}_{2}(u)$, $g>0$ to (3.20), where ${\cal
O}^{i}_{1/2}(u)$, $i=1,\dots,N$, are $N$ boundary operators dual to free
fields in AdS corresponding to the 1 and 2 boundaries, all with conformal
dimension $\Delta$. One then takes a saddle point approximation on the effect
of the interaction, i.e.

$\langle e^{ig\sum_{i}^{N}\int du\;{\cal O}^{i}_{1}(u){\cal
O}^{i}_{2}(u)}\rangle\sim e^{ig\sum_{i}\int du\;\langle{\cal O}_{1}{\cal
O}_{2}\rangle}\;,$ (A.1)

with $t_{1/2}(u)$ on their own have Schwarzian actions. Since all operators in
the boundary interaction share conformal dimension $\Delta$, the sum over $i$
ends up putting an extra $N$ in front of the interaction, enhancing its effect
and making $gN$ the effective coupling constant, which is kept fixed as
$N\to\infty$ and $g\to 0$. As anticipated, this interacting action leads to
equations of motion that allow for a traversable wormholes as a saddle point
with Dilaton profile,

$\Phi=gN^{2\Delta}\Big{(}1+r\arctan(r)\Big{)}$ (A.2)

The throat size is linear in the interaction’s coupling constant $g>0$. In
(A.2) we have disregarded a Casimir energy contribution, which will always be
subleading in the parameter regimes we are interested in. This wormhole is
presented directly as a non-local relevant deformation of the quantum
mechanics action and as such can be immediately analytically extended to a
generic SK closed contour $\cal C$.

###  A.2 Averaging over random couplings with a rigid constraint

A less standard approach within sJT to produce Euclidean wormhole as saddle
point solutions to the Euclidean version of (3.20) was presented in [46]. The
authors study a system of two non-interacting SYK theories but impose a rigid
constraint between the (complex) couplings of both theories, which leads to an
effective coupling between them after averaging. They find phases of this SYK
system that are reminiscent to wormholes on the gravity side.

An effective gravitational dual model in JT is proposed by the authors. A
marginal ($\Delta=1$) scalar is added to the JT action with pure imaginary
boundary sources. We choose $\xi$ rather than $\chi$ for this particular
scalar field since it would be taken as part of the gravity theory rather than
a probe field over a solution. This coupled gravity equations of motion are
solved exactly and the imaginary sources provide the effective negative energy
density in the bulk to support the Euclidean wormhole. We will focus on this
effective gravity model regardless of the precise SYK dual.

The extra term added to the action in (3.20) is

$I_{\xi}=\frac{1}{2}\int\\!\\!\sqrt{g}\;\partial_{\mu}\xi\partial^{\mu}\xi,$
(A.3)

with $\xi\to(-1)^{A}i\kappa$ at the asymptotic boundaries $A=1,2$. The
boundary conditions are pure imaginary sources $\pm i\kappa$,
$\kappa\in\mathbb{R}$ at the asymptotic boundaries given by thermal cycles
${\cal C}_{1,2}$, each of physical length $\beta$, c.f. with Sec. 4.2. The
coupled equations of motions derived from these actions can be consistently
solved by

$ds^{2}=(r^{2}+1)d\tau^{2}+\frac{dr^{2}}{r^{2}+1},\qquad\tau\in[0,\beta)\qquad
r\in(-\infty,\infty)$ (A.4)

where $r\to\mp\infty$ are identified with $1,2$ boundaries respectively. The
EOMs for $\xi$ are standard KG equations on fixed AdS2, with a unique solution
that meets the boundary conditions (A.3)

$\square\xi=0\qquad\Rightarrow\qquad\xi=\frac{2i\kappa}{\pi}\arctan(r)$ (A.5)

The metric equations of motion, now coupled to the massless scalar, provide a
differential equation for $\Phi$ whose solution can be written as

$\Phi=\frac{2\kappa^{2}}{\pi^{2}}\Big{(}1+r\arctan(r)\Big{)}\;,$ (A.6)

We now describe some of its properties. The most important role is played by
$\kappa\in\mathbb{R}$ providing a net positive contribution for $\Phi$ at
$r=0$ opening up the wormhole. We stress that this is an exact solution to the
full nonlinearly coupled JT+$\xi$. However, the wormhole geometry as a saddle
depends strongly on two conditions, i.e. $\xi$ being massless and the sources
being imaginary. As in (A.2), in eq. (A.6) we have dropped a contribution
which is subleading in our regime of interest [46]. Classical solution (A.6)
should be identified with the $\chi_{0}$ classical solution in the context of
eq. (3.12).

####  A.2.1 A comment on imaginary sources

To conclude this Appendix, we briefly comment on imaginary sources for real
scalar fields in the light of the discussion in Sec. 2.2. A property of this
wormhole solution due to (A.5) is that the imaginary sources induce an
imaginary result for the path integral

$W_{\pm;\kappa}=\lim_{r\to\pm\infty}\;\;\int_{\pm i\kappa}{\cal
D}\xi\;\xi\;e^{-I_{JT}-I_{\xi}}=\pm i\kappa$ (A.7)

in which integration over the gravitational field is implicit and which in
[46] is interpreted as an imaginary expectation value for the boundary
operators. In the language of Sec. 3.2.1 this can be rewritten as

$W_{\pm;\kappa}=\lim_{r\to\pm\infty}\;\;\text{Tr}\\{U_{+i\kappa;-i\kappa}\,\xi\,U_{+i\kappa;-i\kappa}\\}=\lim_{r\to\pm\infty}\;\;\text{Tr}\\{U_{+i\kappa;-i\kappa}\,U_{+i\kappa;-i\kappa}\,\xi\\}=\pm
i\kappa$ (A.8)

where $U_{+i\kappa;-i\kappa}$ is defined as the Hartle-Hawking gravity wave-
functional preparing an initial state with boundary conditions $\pm i\kappa$
on the boundaries, see Fig. 4(a). Now, considering the pure imaginary result,
it cannot be the case that $U_{+i\kappa;-i\kappa}\,U_{+i\kappa;-i\kappa}$ and
$\xi$ are both Hermitian operators. We have two choices. Either a) $\xi$ is
canonically quantized as an anti-Hermitian operator, see eqs (3.12) (3.14)).
This amounts to rescale $\xi\to i\xi$ off shell and reinterpret this
construction as a saddle with real fields but an overall minus sign in the
action of the scalar eq. (A.3). From this perspective, the appearance of a
wormhole saddle is not entirely surprising as one is coupling exotic negative
energy matter fields to gravity, see e.g. [63]. Option b) is that the field
$\xi$ is kept Hermitian, and the $\pm i\kappa$ sources are retained as
imaginary sources for a real field. Then, a careful Euclidean conjugation of
$U_{+i\kappa;-i\kappa}$ [40], implies a reinterpretation of (A.7) as a matrix
element of $\xi$ rather than an expectation value, which would also explain a
complex result in the computation. This would be more in line with the general
$d>1$ discussion in Sec. 2.2. In this language, the expectation value of the
operator requires to reflect the sources on the same boundary, and a Hermitian
state is defined as, see Fig. 4(b),

$H_{\pm;\kappa}\equiv\lim_{r\to\pm\infty}\;\;\text{Tr}\\{U_{+i\kappa;-i\kappa}\,U_{-i\kappa;+i\kappa}\,\xi\\}\qquad\qquad(U_{+i\kappa;-i\kappa}\,U_{-i\kappa;+i\kappa})^{\dagger}=(U_{+i\kappa;-i\kappa}\,U_{-i\kappa;+i\kappa})$
(A.9)

Notice that this is a different computation than $P_{\pm;\kappa}$. Based on
our previous developments reviewed in Sec. 2.2 one would expect that

(a)

(b)

Figure 4: (a) A graphical representation of $P_{\pm;\kappa}$ is presented (b)
Taking $\xi$ as a Hermitian operator upon quantization, an Euclidean Time
Reflection operation leads to a second object $H_{\pm;\kappa}$ to provide the
$\xi$ field expectation value. A saddle point for $H_{\pm;\kappa}$ is unknown.

$\lim_{r\to\pm\infty}\;\;\text{Tr}\\{U_{+i\kappa;-i\kappa}\,U_{-i\kappa;+i\kappa}\,\xi\\}=0\qquad\qquad\lim_{r\to\pm\infty}\;\;\text{Tr}\\{U_{+i\kappa;-i\kappa}\,U_{-i\kappa;+i\kappa}\,\Pi_{\xi}\\}\propto\pm\kappa$
(A.10)

resembling (2.6), where $\Pi_{\xi}$ is taken to be the conjugated momentum to
the bulk field $\xi$. Notice, however, that we are not claiming that a
wormhole saddle (or any other geometric saddle) exists for the boundary
conditions in (A.10). This is emphasized in Fig. 4(b). Furthermore, the abrupt
sign flip $+i\kappa\to-i\kappa$ on each boundary should require high frequency
modes for $\xi$, and thus a novel (and time dependent) solution for the non-
linearly coupled JT+$\xi$ equations of motion, which is beyond the scope of
this work. We stress that (A.10) must be taken as a conjecture based on higher
dimensional results and not as an explicit computation. We also point out that
this analysis was entirely made from a gravitational JT+$\xi$ point of view.
We leave a study of the consequences of this conjecture from the QFT side for
future work.

We conclude by a making a final comment on this construction. It is
interesting to note that $U_{+i\kappa;-i\kappa}$ interpreted as an holographic
excited state with imaginary sources would be the first case in which the
coupled gravity + matter equations of motion were exactly solved and the
$\kappa\neq 0$ state is able to provide a different topology than the
$\kappa=0$ case. This would be the first non-perturbative study in this
regard. It would be interesting to consider higher dimensional analogues of
this mechanism.

##  Appendix B Scalar field in pure AdS2

In this App. present the relevant AdS2 coordinates for the main body of the
text and some relevant classical solutions for a massive scalar field $\chi$
over pure Global AdS2. We review why this set-up on its own is not adequate
for holography in 1+1 dimensions, but it is still useful for our purposes on
the light of Sec. (4.3.1).

We define AdS2 as a hypersurface defined over $1+2$ flat spacetime

$-U^{2}-V^{2}+X^{2}=-1\qquad ds^{2}=-dU^{2}-dV^{2}+dX^{2}\;,$

and define the two foliations,

$U=\sqrt{r^{2}+1}\cos(t)=\rho\qquad
V=\sqrt{r^{2}+1}\sin(t)=\sqrt{\rho^{2}-1}\sinh(\sigma)\qquad
X=r=\sqrt{\rho^{2}-1}\cosh(\nu)$ (B.1)

We will also use the Euclidean analytic extensions of these metrics
$t\to-i\tau$ and $\sigma\to-i\varsigma$. We show how these coordinates cover
the AdS2 and Euclidean AdS (H2) manifolds in Fig. 5.

For this Appendix we need the Global Lorentzian AdS2 foliation in terms of
$\\{r,t\\}$ in (B.1),

$ds^{2}=-(r^{2}+1)dt^{2}+\frac{dr^{2}}{r^{2}+1},\qquad r,t\in(-\infty,\infty)$
(B.2)

Figure 5: The relevant foliations used in this work are shown both for
Lorentzian AdS2 and Euclidean AdS2, i.e. H2. The coordinates are defined in
(B.1).

A solution of the JT EOMs, see eq. (3.20), requires also a Dilaton $\Phi$
profile on top of the metric (B.2). Profiting from examples that reduce to JT
upon dimensional reduction, it is customary to associate the magnitude of the
Dilaton $\Phi$ as the size of the wormhole at any given Cauchy slice, see e.g.
[53]. Thus, Global AdS2 (an eternally traversable wormhole) should also
support a non-trivial time independent Dilaton solution to be considered
relevant for the JT/SYK correspondence. It can be shown that there exists no
such Dilaton solution on pure Global AdS2 [51].

However, we found in Sec. 4.3.1 that the saddle point geometry dual to two
disconnected SK paths yields a manifold saddle that can be understood as a
Thermal AdS2 solution, over which we aim to compute probe scalar correlators.
We thus find useful to study a massive scalar KG field in Global AdS2, to
introduce the relevant notation and present useful analysis. We will denote
for future convenience the two AdS1+1 boundaries as $R_{1}$ and $R_{2}$ and
also retain standard holography notation such as
$\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{1}}(t^{\prime})\rangle$ and
$\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{2}}(t^{\prime})\rangle$ for AdS
boundary correlators even if there is no known realizations of a QM ${\cal
O}_{R_{1/2}}(t)$ operators dual to only pure AdS2 for they will become
physical in the Thermal AdS2 scenario.

###  B.1 Real-time correlation functions

We take a probe scalar, $\chi(t,r)$ in the Lorentzian metric (B.2) with action

$I_{KG}=-\int\sqrt{g}(\partial_{\mu}\chi\partial^{\mu}\chi+m^{2}\chi^{2})\>,\qquad\qquad\left(\square-m^{2}\right)\chi=0\,,$
(B.3)

One can expand the solution in a Fourier basis $\chi(t,r)=e^{i\omega t}q(r)$
to obtain the radial equation,

$\left((r^{2}+1)q^{\prime}(r)\right)^{\prime}=\left(\Delta(\Delta-1)-\frac{\omega^{2}}{r^{2}+1}\right)q(r)$
(B.4)

with $m^{2}=\Delta(\Delta-1)$. A solution to this equation is

$q_{1;\omega}(r)=C_{1}P_{\Delta-1}^{\omega}(ir)+C_{2}Q_{\Delta-1}^{\omega}(ir)\,,\qquad
C_{2}=\frac{2C_{1}}{i\pi}\,,\qquad
C_{1}=\frac{i\sqrt{\pi}e^{-\frac{1}{2}i\pi(\Delta+\omega)}\Gamma(\Delta-\omega)}{2^{\Delta}\Gamma\left(\Delta-\frac{1}{2}\right)}\,,$
(B.5)

where $P_{a}^{b}(x)$ and $Q_{a}^{b}(x)$ are the Legendre functions of the
first and second type respectively. The coefficients are chosen so that the
solution diverges at $R_{1}$, $r\to+\infty$ and is regular at $R_{1}$,
$r\to-\infty$, i.e.

$\displaystyle q_{1;\omega}(r)$ $\displaystyle\sim 1\times r^{\Delta-1}+{\cal
O}(r^{-\Delta})\,,\qquad\qquad r\rightarrow\infty,$ (B.6) $\displaystyle
q_{1;\omega}(r)$ $\displaystyle\sim{\cal
O}(r^{-\Delta})\,,\qquad\qquad\qquad\qquad r\rightarrow-\infty.$ (B.7)

As defined, $q_{1;\omega}$ has poles in the complex $\omega$ plane at
$\omega=\pm(\Delta+n)$, where $n\in\mathbb{N}$. With $q_{1;\omega}(r)$ at
hand, one can define a solution with boundary conditions $\chi(t,r)\sim
r^{\Delta-1}\chi_{R_{1}}(t)$ at $r\to+\infty$ as

$\chi_{R}(t,r)=\int
dt^{\prime}\int_{{\cal{F}}}\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}q_{1;\omega}(r)\chi_{R_{1}}(t^{\prime})=\frac{2^{-\Delta}\Gamma(\Delta)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\frac{1}{\left(\sqrt{r^{2}+1}\cos((t-t^{\prime})(1-i\epsilon))-r\right)^{\Delta}}\;,$
(B.8)

where the sub-index ${\cal{F}}$ means that we have chosen a Feynman path for
the $\omega$ integral. An analogous solution with sources at $r\to-\infty$ can
be defined as $\chi_{R}(t,r)\to\chi_{R}(t,-r)$.

By looking at the speciffic frequencies $\omega=\pm(\Delta+n)$ one finds that
there also exist an infinite set of solutions that decay at both
$r\to\pm\infty$, i.e. the AdS2 N modes. These can be added to any real time
solution without altering the asymptotic boundary conditions. The most general
set of these modes can be written,

$q_{N}(t,r)=\sum_{n=0}^{\infty}\left(C_{n}e^{-i(\Delta+t)n}\mathop{\mathrm{Res}}_{\omega=\Delta+n}q_{1,\omega}(r)+D_{n}e^{i(\Delta+t)n}\mathop{\mathrm{Res}}_{\omega=\Delta+n}q_{1,\omega}^{*}(r)\right).$
(B.9)

We will find these modes of use for computing expectation values on excited
states below.

The Feynman correlator for the theory on the $R_{1}$ boundary can be obtained
by considering the on-shell action in (B.3) over a solution consisting of
(B.8) and with all N modes in (B.9) turned off. This yields

$I^{0}_{11}\equiv-\frac{i}{2}\int