Gravity factorized

Andreas Blommaert1, Luca V. Iliesiu2 and Jorrit Kruthoff$\,{}^{2}$

1SISSA, Via Bonomea 265, 34127 Trieste, Italy

2Department of Physics, Stanford University, Stanford, CA 94305, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We find models of two dimensional gravity that resolve the factorization
puzzle and have a discrete spectrum, whilst retaining a semiclassical
description. A novelty of these models is that they contain non-trivially
correlated spacetime branes or, equivalently, nonlocal interactions in their
action. Such nonlocal correlations are motivated in the low-energy gravity
theory by integrating out UV degrees of freedom. Demanding factorization fixes
almost all brane correlators, and the exact geometric expansion of the
partition function collapses to only two terms: the black hole saddle and a
subleading “half-wormhole” geometry, whose sum yields the desired discrete
spectrum. By mapping the insertion of correlated branes to a certain double-
trace deformation in the dual matrix integral, we show that factorization and
discreteness also persist non-perturbatively. While in our model all wormholes
completely cancel, they are still computationally relevant: self-averaging
quantities, like the Page curve, computed in the original theory with
wormholes, accurately approximate observables in our theory, which accounts
for UV corrections. Our models emphasize the importance of correlations
between different disconnected components of spacetime, providing a possible
resolution to the factorization puzzle in any number of dimensions.

###### Contents

  1. 1 Introduction
  2. 2 JT gravity with correlated spacetime branes
    1. 2.1 JT gravity
    2. 2.2 Spacetime branes
    3. 2.3 Why correlated spacetime branes?
    4. 2.4 Geometric setup
  3. 3 Factorization and discreteness from correlated spacetime branes
    1. 3.1 All orders factorization
    2. 3.2 Uniqueness of the factorizing solution
    3. 3.3 Discreteness and exact half-wormholes
    4. 3.4 Magnitude of UV corrections
    5. 3.5 Additional comments
  4. 4 Factorization and discreteness from matrix integral localization
    1. 4.1 Localization concept
    2. 4.2 Argument
    3. 4.3 Additional comments
  5. 5 Including bulk matter
    1. 5.1 Factorization
    2. 5.2 Probe matter
    3. 5.3 Why wormholes are oftentimes good approximations
  6. 6 Discussion
  7. A Branes versus defects

##  1 Introduction

Understanding the duality between black holes and conventional quantum
mechanical systems remains an important problem in quantum gravity. The idea
behind this holographic program [1, 2, 3] is to consider known features of
quantum mechanical systems, and to determine whether black holes follow
similar rules. Much progress has been made in this direction in recent years
by studying non-perturbative wormhole contributions to the Euclidean
gravitational path integral. For example, trying to replicate the Page curve
[4] has thought us the importance of considering spacetime wormhole
contributions in the gravitational path integral in order to find a unitary
process of black hole evaporation [5, 6, 7].

Whilst the existence of spacetime wormholes has shed light on some important
problems in quantum gravity, it has also introduced puzzles. The goal of this
paper is to address two such puzzles:

  1. 1.

The factorization puzzle. The existence of wormholes in quantum gravity
immediately raises a tension with the dual quantum theory [8]. In the dual
theory, when we want to calculate moments of the partition function
$Z(\beta)$, we trivially find the product of partitions functions, thus
obtaining a factorizing answer. On the bulk side, there is an asymptotic
boundary for each $Z(\beta)$ and we are instructed to integrate over all
spacetimes consistent with these boundary conditions. This includes spacetimes
with wormholes that connect different asymptotic boundaries, which results in
a non-factorizing answer $Z(\beta_{1},\beta_{2})\neq
Z(\beta_{1})Z(\beta_{2})$.111In supergravity there are exceptions where the
wormhole contribution to various supersymmetry-protected observables happens
to vanish [9].  Thus the bulk and the boundary appear to be in disagreement.

  2. 2.

The discreteness puzzle. In known models in which the sum over bulk geometries
can be computed exactly, the gravitational partition function does not produce
a discrete energy spectrum.222Here, one should distinguish two toy models in
which the geometric expansion is (at least somewhat) under control. The first
is JT gravity where one can include the contribution of all wormholes (and
geometries) to still find a continuous density of states for the single-
boundary partition function. The second is pure 3d gravity, where all black
hole geometries can be included, but some more complicated topologies have yet
to be accounted for [10, 11, 12]. Both models appear to have continuous
spectra. A consequence of this lack of discreteness can be observed for
boundary correlation functions. Correlation functions in the dual quantum
mechanics with a discrete spectrum oscillate heavily around a non-zero
averaged value [13, 14]. Connected geometries and in particular spacetime
wormholes explain this non-zero averaged value [15, 16, 17, 18] but do not
explain the erratic oscillations in the dual quantum mechanics. So again, the
bulk and the boundary appear to disagree.

One way around both puzzles is to interpret gravity as dual to an ensemble of
quantum mechanical systems, instead of a single boundary theory [19, 16, 5, 6,
20, 21, 22, 18, 23, 24, 25, 26, 11, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40]. Simple models of gravity, like JT gravity and generalizations
thereof, and pure three-dimensional gravity or even supergravity, are all
seemingly dual to ensembles.

However these simple models are not realistic theories of the universe. There
is a widely held believe that all UV complete theories of quantum gravity in
AdS do factorize, and that they are each dual to one discrete quantum
mechanical systems. For example the Strominger-Vafa microstate counting [41]
is evidence of the discreteness of the spectrum, and recent work of Eberhardt
[42, 43, 44] addresses the factorization puzzle in a particular tensionless
AdS3 string theory. Furthermore, in AdS/CFT, starting from the boundary
perspective, in conventional CFT examples whose bulk duals are better
understood (such as $\mathcal{N}=4$ super Yang-Mills) it is unclear what
couplings one could even average over in order to get answers consistent with
multi-boundary wormholes.

We would like to understand how factorization and discreteness arise in the
UV, by modifying the tractable toy models discussed above. For this, we
consider JT gravity enriched with additional effective UV ingredients.

Suppose we start with some UV complete theory of quantum gravity that has a
low energy JT sector. For instance, we can consider one realization of the SYK
model [45] which, in its full glory, has string corrections in the bulk [46].
Alternatively, we can consider the AdS2 near-horizon region of a near-extremal
black hole in a higher dimensional UV complete theory, for which JT gravity is
a good effective theory [47, 48, 49, 50, 51, 52, 53, 54] but string
corrections need to be considered in order to determine the full spectrum of
black hole microstates. Now, instead of making any approximation that results
in JT, suppose one can integrate out exactly all the degrees of freedom,
except for the metric and dilaton. For low enough temperatures one should find
JT gravity but with complicated deformations in the action. Those deformations
would include deformations in the dilaton potential like those considered in
[55, 12, 56], but, in general, one also expects to get nonlocal deformations,
such as nonlocal dilaton potential terms. We can then apply a form of open-
closed string duality (but applied to spacetimes) where we interpret the
deformed dilaton gravity as JT gravity with spacetime branes [20]
inserted.333For example, JT gravity with FZZT branes (open string picture) is
equivalent to JT gravity with a deformed potential (closed string picture)
[57].  Similarly, nonlocal deformations turn out to correspond to inserting
correlated (or coupled) spacetime branes in JT theory.444As compared to [58],
who also introduced branes with the motivation to address the factorization
puzzle, the critical new ingredient that we introduced, and which actually
result in a factorizing theory of gravity, is precisely this nontrivial
coupling between the spacetime branes.

The key point is that in the above thought experiment we have made no
approximation whatsoever. Therefore, if it is true that UV theories do
factorize and are discrete, then there must be versions of JT gravity with
correlated spacetime branes or nonlocal corrections in the action which also
factorize, and have a discrete energy spectrum. Because UV complete theories
are quite scarce and special, we only expect this to be true for very specific
values for the correlation between spacetime branes and equivalently, for the
nonlocal deformations of the action.

### Summary and structure

We thus set out to find spacetime brane correlators which result in a
factorizing and discrete boundary dual. Our main results are the following:

  1. 1.

In section 2, we introduce JT gravity with correlated spacetime branes and
explain why these correlated branes are equivalent to nonlocal deformations in
the action which, in turn, can arise when integrating-out UV degrees of
freedom. We spell out the precise rules for including the contribution of
(correlated) branes to the gravitational partition function. The point is
there are additional boundaries for spacetimes to end on, and there can be
correlation between those additional boundaries. For example, the following
surfaces are a representative sample for the contribution to the partition
function

$Z(\beta)=\;\;\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.55}]{Zexample_intro.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.1)

where the blue wiggly lines represent the correlations (or couplings) between
the additional brane boundaries.

  2. 2.

Factorization happens because the brane interactions can cancel the
contribution of wormholes. For instance, in section 3.1 we find that the brane
two-point interaction must be tuned such that

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.55}]{Factorization_leading_cancellation.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,=\,\,0\,.$ (1.2)

This phenomenon was also observed by Saad, Shenker, Stanford and Yao [45, 40]
who investigated factorization to leading order in the genus
expansion.555Leading order factorization was also discussed in [59]. In our
setup, this brane two-point function implies factorization to all orders in
the genus expansion and for an arbitrary number of boundaries. Additionally,
we show that three or higher-point interactions between branes need to vanish.
In the UV setting, these brane interactions or more generally, the
interactions between disconnected components of the spacetime, can be thought
of as highly stringy, non-geometric connected configurations, non-geometric
wormholes. These cancel the geometric wormholes so that the full answer
factorizes.

  3. 3.

The brane two-point interaction (1.2) results in massive cancellations in the
$e^{\textsf{S}_{0}}$ expansion. The only contributions to the one-boundary
gravitational partition function which do not cancel, are the disk, and the
geometry with precisely one brane boundary inserted

$Z(\beta)=\Tr e^{-\beta\textsf{H}_{0}}=\leavevmode\hbox to152.4pt{\vbox
to47.81pt{\pgfpicture\makeatletter\hbox{\hskip
63.11937pt\lower-47.61113pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.45}]{Halfwormhole_contribution.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{-37.84187pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.78636pt}{-43.3059pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{black hole}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{45.80879pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.66982pt}{-42.33368pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{one-point function}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (1.3)

The one-point function of this brane encodes the microstructure of the dual
quantum mechanics: each Hamiltonian $\textsf{H}_{0}$ determines the one-point
function such that the above equation is exactly satisfied. As shown in
section 3.3, this answer is very similar to the half-wormhole picture [45],
with that crucial distinction that (1.3) is exact, to all perturbative and
non-perturbative orders in $e^{\textsf{S}_{0}}$. For typical draws of
$\textsf{H}_{0}$, the contribution of the half-wormhole in (1.3) is sub-
leading compared to the black hole. This is the reason why the classical black
hole is oftentimes a good approximation. The partition function in (1.3) can
be viewed as analogous to the whole expansion of strings on the disk
background. This includes configurations that look like classical geometries
(with wormholes) but also includes highly non-geometric configurations, some
of which again cancel the geometric wormholes.

  4. 4.

Wormholes re-emerge when we average over $\textsf{H}_{0}$, since they are
encoded in the statistical properties of the brane one-point function.
Denoting the ensemble average over $\textsf{H}_{0}$ by
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle}\dots\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rangle_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\text{conn.}}}$
we find that

$\leavevmode\hbox to78.68pt{\vbox
to48.37pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-48.16534pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.7}]{red_average_half_wormholes.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{64.87299pt}{-42.67957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.59518pt}{-44.83234pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\text{conn}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad=\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.7}]{n-wormhole.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\quad\text{higher
genus}\,.$ (1.4)

This equality shows that the expectation value of self-averaging
observables666Self-averaging quantities are defined by having variances that
are much smaller than their expectation values. in the theory (1.3) (where
wormholes cancel), are well approximated by computations in the original JT
gravity theory, where branes are absent but wormholes are present. In section
3.4 and 5, we exemplify how this occurs for some known self-averaging
quantities: the time-averaged spectral form factor and matter two-point
function, the volume of the black hole interior, and the entanglement entropy
of Hawking radiation (which follows the Page curve).

  5. 5.

In the dual matrix integral description of JT gravity [19], we use the map
between FZZT branes and determinants to investigate the effect of our
correlated branes. We find that the branes deform the ensemble partition
function to

$\displaystyle\mathcal{Z}(\textsf{H}_{0})$
$\displaystyle=\lim_{q\to\infty}\prod_{i=1}^{L}\int_{{\mathcal{C}}}\mathrm{d}\lambda_{i}\,\exp\bigg(q\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})\bigg{missing})=\prod_{i=1}^{L}\int_{{\mathcal{C}}}\mathrm{d}\lambda_{i}\,\bigg{(}\prod_{j=1}^{L}\delta(\lambda_{j}-E_{j})+\text{permutations}\bigg{)}\,,$

where the parameter $q$ is found to be large from a solution to a simple
Schwinger–Dyson equation. The large $q$ saddle-points of the emergent action
that we find are indeed

$\frac{\partial}{\partial\lambda_{i}}\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})=0\quad\Leftrightarrow\quad\lambda_{1}\dots\lambda_{L}=\text{permutations
of }E_{1}\dots E_{L}\,.$ (1.5)

This localizes the eigenvalues (up to permutations) to $\lambda_{i}=E_{i}$,
the eigenvalues of the Hamiltonian $\textsf{H}_{0}$ of the dual quantum
system. Thus, JT gravity with correlated branes is discrete and factorizing,
even non-perturbatively. We discuss this matrix integral perspective in
section 4.

  6. 6.

In section 5 we include bulk matter and explain that factorization is again
resolved by fixing the brane two-point function. UV divergences associated to
wormholes with small necks are absent in our model. We then compute the probe
matter correlators, accounting for the full backreaction on the metric and,
for instance, find the expected two-point function for a theory with a
discrete spectrum. This addresses the puzzle posed in [13] concerning the
decay of the two-point function in the eternal black hole geometry.

  7. 7.

In section 6, we rewrite our model in several equivalent ways: as a theory
with correlated branes, a theory with some nonlocal dilaton potential, or a
theory in which the dilaton potential is picked from an ensemble average
distribution. Through these re-writings we emphasize that the critical
ingredient that is necessary to obtain a factorizing gravitational theory, is
the correlations between different disconnected components of spacetime. This
entices us to propose a similar mechanism for resolving the factorization
puzzle in higher dimensional theories, and forces us to question what the
origin of such correlations between disconnected components is in string
theory.

##  2 JT gravity with correlated spacetime branes

In this section we introduce JT gravity with correlated spacetime branes, this
sets the stage for section 3 where we will fine-tune these brane correlations
to ensure factorization and discreteness.

###  2.1 JT gravity

JT gravity is a two-dimensional gravity theory that involves a dilaton $\Phi$
and metric $g$, with action [60, 61]

$I[g,\Phi]=-\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\,\Phi\,(R+2)-\int_{\partial\Sigma}\mathrm{d}u\sqrt{h}\,\Phi\,(K-1)-\textsf{S}_{0}\chi(\Sigma)\,.$
(2.1)

With this boundary action one needs to impose Dirichlet boundary conditions on
the metric $h_{uu}=1/\epsilon^{2}$ and dilaton $\Phi=1/2\epsilon$, which fixes
the length of the Euclidean boundary circle to $\beta/\epsilon$ [62, 63, 64].
The extremal entropy $\textsf{S}_{0}$ multiplies the Euler character
$\chi(\Sigma)=2-2g-n$, and thus suppresses higher genus geometries. Path
integrating out the bulk dilaton $\Phi$ (after a contour rotation
$\Phi\to\mathrm{i}\Phi$) localizes the geometries to hyperbolic Riemann
surfaces (with boundary) with constant curvature $R+2=0$. We must include all
surfaces consistent with the boundary conditions [19].

A convenient way to organize the calculations, say for the case with one
asymptotic boundary, is by viewing the higher genus surface as consisting of
two parts: the trumpet, which has one asymptotic fixed length boundary, and
one geodesic boundary with fixed length $b$; and the remaining genus $g$
Riemann surface, which has one geodesic boundary of identical length $b$. We
must then integrate over the moduli space of genus $g$ Riemann surfaces with
geodesic boundary $b$, which produces the Weil-Petersson volume $V_{g,1}(b)$
[65, 66], and glue the trumpet to the genus $g$ Riemann surfaces, by
integrating over $b$. This gives the genus expansion of the JT gravity
partition function [19]

$\displaystyle
Z(\beta)=Z_{\text{disk}}(\beta)+\sum_{g=1}^{\infty}e^{\textsf{S}_{0}(1-2g)}\int_{0}^{\infty}\mathrm{d}b\,b\,V_{g,1}(b)\,Z_{\text{trumpet}}(\beta,b),$
(2.2)

where the disk and trumpet partition function can be computed by integrating
over boundary modes [67, 62, 63, 64]

$Z_{\rm
disk}(\beta)=\frac{e^{\textsf{S}_{0}}}{4\pi^{1/2}\beta^{3/2}}e^{\frac{\pi^{2}}{\beta}},\quad
Z_{\rm
trumpet}(\beta,b)=\frac{1}{2\pi^{1/2}\beta^{1/2}}e^{-\frac{b^{2}}{4\beta}}\,.$
(2.3)

This generalizes to the multi-boundary case (Weil-Petersson volumes count
connected Riemann surfaces)

$\displaystyle
Z(\beta_{1},\dots,\beta_{n})_{\text{conn.}}=\sum_{g=0}^{\infty}e^{\textsf{S}_{0}(2-2g-n)}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\dots\int_{0}^{\infty}$
$\displaystyle db_{n}b_{n}\,V_{g,n}(b_{1}\dots b_{n})$ $\displaystyle\quad
Z_{\rm trumpet}(\beta_{1},b_{1})\cdots Z_{\rm trumpet}(\beta_{n},b_{n})\,.$
(2.4)

This decomposition also makes it clear that JT gravity has a dual description
as a double scaled matrix integral [19], because the $V_{g,n}(b_{1}\dots
b_{n})$ satisfy the topological recursion relations (or loop equations) for a
double scaled Hermitian matrix integral with genus zero spectral density [68,
65]

$\rho_{0}(E)=\frac{e^{\textsf{S}_{0}}}{4\pi^{2}}\sinh 2\pi\sqrt{E}\,,$ (2.5)

which crucially matches also the spectrum associated with the JT gravity disk
partition function (2.3). Multi-boundary partition functions are computed in
the matrix integral as ensemble averages

$\left\langle Z(\beta_{1})\cdots
Z(\beta_{n})\right\rangle=\frac{1}{\mathcal{Z}}\int\mathrm{d}H\,e^{-L\Tr
V(H)}\,\Tr e^{\beta_{1}H}\cdots\Tr
e^{\beta_{n}H}=Z(\beta_{1},\dots,\beta_{n})\,,$ (2.6)

where the last equality states the JT-matrix integral duality. The potential
$V(H)$ will be specified when needed in section 4.

As stressed in [19], an important point is that we can really think of $H$ in
the matrix integral as the Hamiltonian of the boundary quantum mechanics. This
means that here we have a version of AdS/CFT that involves JT gravity theory
in the bulk and an ensemble of boundary Hamiltonians on the boundary.
Consequently, the JT gravity fixed thermal length gravitational partition
function (with an arbitrary number of boundaries) are dual to ensemble
averages of partition functions of quantum mechanical systems (with an
arbitrary number of $\Tr e^{-\beta H}$ insertions within the ensemble).

As motivated in the introduction, we now introduce additional ingredients in
gravity that eliminate this average, whilst retaining a geometric description.

###  2.2 Spacetime branes

Spacetime branes are natural candidates for these effective new UV
ingredients. They introduce extra boundaries for spacetimes to end on, and (as
mentioned in section 1) are known to have a dual description as deformations
in the JT gravity action [58, 57]. In this dual description, there are no
extra boundaries, this is essentially open-closed universe duality (similar to
open-closed string duality).

As in string theory, different boundary conditions define different spacetime
branes (branes simply from hereon), see [69] for a semiclassical
classification. We will encounter two types in particular.

First we have FZZT branes, the most commonly studied brane in minimal string
theory [70, 19, 71, 72, 69, 73, 74, 75, 76, 77, 78]. Here we should view the
worldsheets as spacetimes, so the FZZT branes are really spacetime branes.
Classically in JT gravity they correspond to fixed energy boundaries (and
fixed dilaton). In the dual matrix integral, one FZZT boundary corresponds
to777The significant potential term is related with the small $\beta$
divergence in the Laplace transform [19].

${{\mathcal{O}}}_{\rm
FZZT}(E)=-\int_{0}^{\infty}\frac{\mathrm{d}\beta}{\beta}\,e^{\beta
E}\,Z(\beta)=\Tr\log(E-H)-\frac{L}{2}V(E)\,,$ (2.7)

and FZZT branes are exponentials of boundaries, and so correspond in the
matrix ensemble to [70, 19, 22, 77]

$\psi_{\rm FZZT}(E)=\exp\left(-\int\frac{\mathrm{d}\beta}{\beta}e^{\beta
E}Z(\beta)\right)=\det(E-H)\,\exp\bigg(-\frac{L}{2}V(E)\bigg{missing})\,.$
(2.8)

Another semiclassical interpretation of FZZT boundaries, that we will use in
this work, was worked out in [77]. The basic idea is to decompose the FZZT
boundary in a complete set of geodesic boundaries. To do this, we need the
expansion coefficients, which can be found by Laplace transforming the trumpet
[34]

$\displaystyle-\int_{0}^{\infty}\frac{\mathrm{d}\beta}{\beta}\,e^{\beta
E}\,Z(\beta,\beta_{1},\dots,\beta_{n})_{\text{conn.}}$
$\displaystyle=\sum_{g=1}^{n}e^{\textsf{S}_{0}(1-2g-n)}\int_{0}^{\infty}\mathrm{d}b\,b\,M_{E}(b)\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\dots\int_{0}^{\infty}\mathrm{d}b_{n}b_{n}$
(2.9) $\displaystyle\qquad\qquad\qquad V_{g,n+1}(b,b_{1}\dots
b_{n})\,Z_{\text{trumpet}}(b_{1},\beta_{1})\dots
Z_{\text{trumpet}}(b_{n},\beta_{n})\,,$

where the expansion coefficient or wavefunction explicitly becomes

$M_{E}(b)=-\int_{0}^{\infty}\frac{\mathrm{d}\beta}{\beta}\,e^{\beta
E}\,Z_{\text{trumpet}}(b,\beta)=-\frac{1}{b}e^{-b(-E)^{1/2}}\,.$ (2.10)

This has a branchcut for positive real energies. The two values above and
below the cut define the two different FZZT boundary conditions for positive
energies, indeed this feature was already there in (2.7).

Formula (2.9) means that one can view the FZZT boundary as consisting of fixed
geodesic length $b$ boundaries with a wavefunction $M_{E}(b)$. In
semiclassical JT variables this wavefunction implements the Legendre transform
from geodesic boundary conditions to fixed energy boundary conditions [69].
The wavefunction can be thought of as coming from a cosmological constant on
the geodesic boundary. Semiclassically, with this extra boundary term in the
action, there is saddle for $b$ which depends on $E$.888The saddle-point can
generically be at arbitrary complex $b$. This is an important observation that
we discuss more in section 3.5.

The second type of branes in which we are interested are inserting geodesic
boundaries. In the dual matrix integral, it was found in [69] that inserting
one geodesic boundary corresponds with the operator999Here we are using a
hybrid notation, where the second term is a double scaled expression. However
it is just a function of $b$ and can easily be written in a finite $L$ form
just as the first term as we also do in section 4.

${{\mathcal{O}}}_{\rm
G}(b)=\frac{2}{b}\Tr\cos(bH^{1/2})-\int_{0}^{\infty}\mathrm{d}E\,\rho_{0}(E)\,\frac{2}{b}\cos(bE^{1/2})\,.$
(2.11)

The second term subtracts the naive disk contribution as there are indeed no
disk shaped geometries with these geodesic boundary conditions [69], in
particular

$\left\langle{{\mathcal{O}}}_{\rm
G}(b)\right\rangle=\sum_{g=1}^{\infty}e^{\textsf{S}_{0}(1-2g)}\,V_{g,1}(b)\,,\quad\left\langle{{\mathcal{O}}}_{\rm
G}(b_{1})\dots{\mathcal{O}}_{\rm
G}(b_{n})\right\rangle_{\text{conn}}=\sum_{g=0}^{\infty}e^{\textsf{S}_{0}(2-2g-n)}\,V_{g,n}(b_{1}\dots
b_{n})\,.$ (2.12)

Formula (2.11) can be checked from the matrix side directly, by computing the
inverse Laplace transform of the FZZT operator ${{\mathcal{O}}}(E)$, see
formula (2.7). Geodesic branes are exponentials of this operator

$\psi_{\rm
G}(b)=\exp\bigg(\frac{2}{b}\Tr\cos(bH^{1/2})-\int_{0}^{\infty}\mathrm{d}E\,\rho_{0}(E)\,\frac{2}{b}\cos(bE^{1/2})\bigg{missing})\,.$
(2.13)

The fact that ensemble averages of (2.11) return connected geometries (2.12)
will be important in section 3.4.

So far we mostly focused on the definition of branes in the open-string
picture and their natural appearance in the matrix integral. To get a better
sense of these branes in the closed-string picture, we need to understand what
their effect is on the dilaton potential. For the FZZT brane we already have
formula (2.9), which is a nice expansion in terms of additional geodesic
boundaries with a wavefunction on them, but we can also recast it in terms of
a change in the dilaton potential. The dilaton potential $U(\Phi)$ is defined
in the usual way (ignoring the boundary and topological term),

$I[g,\Phi]=-\frac{1}{2}\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g}\left(\Phi(R+2)+U(\Phi)\right)\,.$
(2.14)

The effect of adding an FZZT spacetime brane on the dilaton potential was
worked out in [57, 77]. They found

$U_{\rm
FZZT}(\Phi)=-e^{-\textsf{S}_{0}}e^{-2\pi\Phi}\frac{2z}{z^{2}+\Phi^{2}}\,,$
(2.15)

with $z^{2}=-E$ and where the non-zero shift in the threshold energy $E_{0}$
has been taken into account. For our purposes, we actually need to generalize
the notion of an FZZT brane slightly to a smeared version (see [12, 56] for
the defect case), which in the matrix integral corresponds to inserting

$\psi_{\rm
FZZT}(\lambda)=\exp\left(\int_{{\mathcal{C}}}\mathrm{d}z\,\lambda(z)\,\mathcal{O}_{\text{FZZT}}(z)\right)\,,$
(2.16)

with $\lambda$ a function (or distribution) integrated along some contour
$\mathcal{C}$ and again $z^{2}=-E$. All we do here is smear the brane in
target space, something we often do in higher-dimensional string theories as
well. The deformation in the dilaton potential changes then to

$U_{\rm
FZZT}(\Phi,\lambda(z))=-e^{-\textsf{S}_{0}}e^{-2\pi\Phi}\int_{{\mathcal{C}}}\mathrm{d}z\,\lambda(z)\frac{2z}{z^{2}+\Phi}\,,$
(2.17)

Throughout this work we will mostly focus on insertions of $\mathcal{O}_{\rm
G}(b)$. Geometrically, these correspond to an expansion like (2.9), but where
the wavefunction $M_{E}(b)$ is replaced by a more general wavefunction that we
denote by $Z_{\text{brane}}(b)$. This wavefunction is essentially the inverse
Laplace transform of the FZZT smearing function $\lambda(z)$

$Z_{\text{brane}}(b)=\int_{{\mathcal{C}}}\mathrm{d}z\,M_{z}(b)\,\lambda(z)=-\frac{1}{b}\int_{{\mathcal{C}}}\mathrm{d}z\,e^{-bz}\,\lambda(z)\,.$
(2.18)

The deformation in the dilaton potential corresponding to insertions of
(smeared) $\mathcal{O}_{\rm G}(b)$ operators is then

$U_{\rm
G}(\Phi,Z_{\text{brane}}(b))=e^{-\textsf{S}_{0}}\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{brane}}(b)\,e^{-2\pi\Phi}\,2\cos(b\Phi)\,.$
(2.19)

Indeed, inserting (2.18) and doing the integral over $b$ returns (2.17). In
summary, inserting smeared geodesic branes in the gravitational path integral
corresponds with the deformation

$\exp\bigg(\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{brane}}(b)\,{{\mathcal{O}}}_{\text{G}}(b)\bigg{missing})\Leftrightarrow\exp\bigg(e^{-\textsf{S}_{0}}\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{brane}}(b)\int_{\Sigma}\mathrm{d}^{2}x\sqrt{g(x)}\,e^{-2\pi\Phi(x)}\,\cos(b\Phi(x))\bigg{missing})\,.$
(2.20)

We expand on this relation in appendix A.

Notice that this looks similar to inserting smeared defects [12, 56], but with
the defect angle analytically continued, and including both analytic
continuations to $\mathrm{i}b$ and to $-\mathrm{i}b$. This was to be expected,
because defects and geodesic boundaries act as analytic continuations of one
another, this can be appreciated from several angles [12, 56, 55]. We
elaborate on this further in Appendix A. Notice also that we worked to order
$e^{-\textsf{S}_{0}}$, going to higher order would require working with the
string equation machinery. Since this is beyond the scope of this paper, we
will not pursue that here, but it would be interesting to consider smeared
branes in that setting.101010Furthermore, it would be worthwhile to understand
these potentials in more detail from the geometric point of view. Their
oscillatory behavior signals possible phase transitions [79].

###  2.3 Why correlated spacetime branes?

As explained in the introduction, we expect on general grounds that
integrating out UV degrees of freedom introduces nonlocal deformations in our
low-energy effective theory. In QFT, such nonlocalities can be typically
neglected in the low-energy effective theory since their effect is suppressed
by the energy scale of the original degrees of freedom; thus, at least in a
perturbative expansion around each saddle in the low-energy effective QFT such
nonlocal terms in the action play no role. In the low-energy effective theory
of gravity however, such terms need a more careful treatment; as we will
explain in the next section, the perturbative expansion of nonlocal terms in
the black hole saddle can have a competing effect to the sum over topologies
that occurs in the gravitational path integral. Thus, it will prove important
to keep track of such nonlocal deformations in the low-energy effective
theory. Below we will exemplify how such deformations can be re-expressed as
insertions of correlated or nonlocal spacetime branes. We now give an
intuitive explanation for why this is the case, and explain what we mean with
correlated spacetime branes.

Consider introducing the simplest nonlocal interaction in the dilaton gravity
path integral: a bilocal deformation of the dilaton potential. Making use of
the mapping of ${{\mathcal{O}}}_{\text{G}}(b)$ to metric variables one reads
off from (2.20) that we have

$\displaystyle\exp\bigg(\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\,{{\mathcal{O}}}_{\text{G}}(b_{1})\int_{0}^{\infty}\mathrm{d}b_{2}b_{2}\,{{\mathcal{O}}}_{\text{G}}(b_{2})\,Z_{\text{brane}}(b_{1},b_{2})\bigg{missing})$
$\displaystyle\Leftrightarrow\exp\bigg(-I_{\text{nonlocal}}[g,\Phi]\bigg{missing})=$
.

for some arbitrary function $Z_{\text{brane}}(b_{1},b_{2})$ whose meaning we
now discuss. When we expand out (2.3), what happens is that we are inserting
extra geodesic brane boundaries in the gravitational path integral, but now
there is a two-brane wavefunction component, which correlates two different
smeared spacetime branes through the function $Z_{\text{brane}}(b_{1},b_{2})$.
The function $Z_{\text{brane}}(b_{1},b_{2})$ can thus be viewed at the
connected component of the two-brane correlator, and the deformation to the JT
action (2.3) is manifestly bilocal whenever this brane correlation is non-
zero.111111The correlation we mention here is a correlation between boundaries
or in the universe field theory language correlators of (geodesic) boundary
creating operators, but we will often use the terminology _brane_ correlators,
even though these are exponentials of boundaries and live in a more abstract
target space.

The first effect of this deformation is that when computing the partition
function we have geometries with two extra correlated geodesic boundaries

$\displaystyle Z(\beta)\supset$
$\displaystyle\,\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\int_{0}^{\infty}\mathrm{d}b_{2}b_{2}\,Z_{\text{brane}}(b_{1},b_{2})\,\,\,\leavevmode\hbox
to46.91pt{\vbox to40.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-20.22653pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{ExampleSect2.3-2.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{39.83325pt}{12.51909pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.28743pt}{9.9491pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{39.83325pt}{-12.51909pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.28743pt}{-15.08908pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=\,\,\frac{1}{2}\;\;\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{ExampleSect2.3-1.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\int_{0}^{\infty}\mathrm{d}b_{2}b_{2}\,Z_{\text{brane}}(b_{1},b_{2})\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\,V_{0,3}(b_{1},b_{2},b)\,.$
(2.21)

In the second picture we introduced the notation that we will use for
correlated branes henceforth. Because this bilocal operator is in the
exponential in (2.3), there will also be contributions where the coupled
boundaries are inserted multiple times, for example

$Z(\beta)\supset\frac{1}{8}\;\;\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{ExampleSect2.3-3.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad\,.$ (2.22)

One might expect that the UV theory generates all kinds of multi-local
interactions, not just bilocal deformations (2.3) but also $n$-local
deformations of the type

$I_{\text{nonlocal}}[g,\Phi]\supset-\int_{\Sigma}\mathrm{d}^{2}x_{1}\sqrt{g(x_{1})}\dots\int_{\Sigma}\mathrm{d}^{2}x_{n}\sqrt{g(x_{n})}\,U_{\text{nonlocal}}(\Phi(x_{1})\dots\Phi(x_{n}))\,.$
(2.23)

These correspond to higher-point brane interactions, which we should thus
allow too. For example, the simplest effect of trilocal deformations are
geometries with three extra correlated geodesic boundaries

$\displaystyle Z(\beta)\supset$
$\displaystyle\,\frac{1}{6}\int_{0}^{\infty}db_{1}b_{1}\dots\int_{0}^{\infty}db_{3}b_{3}\,Z_{\text{brane}}(b_{1},b_{2},b_{3})\,\,\leavevmode\hbox
to43.19pt{\vbox to59.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-29.9002pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{ExampleSect2.3-5.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{35.28094pt}{22.19276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.90181pt}{19.62277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{1}\,$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{35.28094pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.90181pt}{-2.56999pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{2}\,$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{35.28094pt}{-22.19276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.90181pt}{-24.76276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{3}\,$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\,\,\frac{1}{6}\,\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{ExampleSect2.3-4.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\quad$
$\displaystyle=\frac{1}{6}\int_{0}^{\infty}db_{1}b_{1}\dots\int_{0}^{\infty}db_{3}b_{3}\,Z_{\text{brane}}(b_{1},b_{2},b_{3})\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\,V_{0,4}(b_{1},b_{2},b_{3},b)\,.$
(2.24)

We will show in section 3 that actually, factorization (to all orders in
$e^{-\textsf{S}_{0}}$) requires $Z_{\text{brane}}(b_{1}\dots b_{n})=0$
whenever $n>2$, but the two-point function $Z_{\text{brane}}(b_{1},b_{2})$
crucially must be nonzero.

One important consequence of having nonlocal interactions in the gravitational
path integral is that closed universes121212By closed universes we mean
geometries without an asymptotic boundary. They could still have a geodesic
boundary, like the torus with one geodesic boundary. no longer trivially
cancel. For example, in (2.3) we could have $x_{1}$ being a point on the disk
and $x_{2}$ being a point on a separate torus. Without this bilocal
interaction the torus would be an irrelevant vacuum diagram, but here it can
become part of the connected Feynman graph. The first correction of this type
is

$Z(\beta)\supset\quad\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.85}]{ExampleSect2.3-6.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(2.25)

Throughout section 3 we do not include degenerate cylindrical surfaces
stretching between two geodesic boundaries. This makes sense, just because re-
introducing those cylinders just renormalizes the brane correlators
$Z_{\text{brane}}(b_{1}\dots b_{n})$, as we discuss in section 4.2, where we
will need the renormalized correlators.

More generally, taking into account all possible brane correlators, the full
expansion for $Z(\beta)$ becomes

$Z(\beta)=Z_{\rm
disk}(\beta)+\sum_{g=1}^{\infty}e^{\textsf{S}_{0}(1-2g)}\int_{0}^{\infty}\mathrm{d}b\,b\,V_{g,1}(b)\,Z_{\rm
trumpet}(\beta,b)+\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\rm
trumpet}(\beta,b)\,X(b)\,,$ (2.26)

where $X(b)$ encodes all corrections from the spacetime brane
correlators131313We implicitly exclude the degenerate cylinders $g^{c}=0$,
$n^{c}=2$ from the sums on the second line, as mentioned above.

$\displaystyle X(b)$
$\displaystyle=\sum_{g=0}^{\infty}\sum_{n=1}^{\infty}\frac{1}{n!}\,e^{\textsf{S}_{0}(1-2g-n)}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\dots\int_{0}^{\infty}\mathrm{d}b_{n}b_{n}\,V_{g,n+1}(b,b_{1},\dots,b_{n})\,Z_{\text{brane}}(b_{1},\dots,b_{n})$
(2.27)
$\displaystyle+\sum_{g=0}^{\infty}\sum_{n=1}^{\infty}\sum_{g^{c}=0}^{\infty}\sum_{n^{c}=1}^{\infty}\frac{1}{n!}\frac{1}{n^{c}!}\,e^{\textsf{S}_{0}(1-2g-2g^{c}-n-n^{c})}\,\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\dots\int_{0}^{\infty}\mathrm{d}b_{n}b_{n}\,V_{g,n+1}(b,b_{1},\dots,b_{n})$
$\displaystyle\hskip
113.81102pt\int_{0}^{\infty}\mathrm{d}b_{1}^{c}b_{1}^{c}\dots\int_{0}^{\infty}\mathrm{d}b_{n}^{c}b_{n}^{c}\,V_{g^{c},n^{c}}(b_{1}^{c},\dots,b_{n}^{c})\,Z_{\text{brane}}(b_{1},\dots,b_{n},b_{1}^{c},\dots,b_{n^{c}}^{c})$
$\displaystyle+\text{branes connecting to multiple closed universes}$ (2.28)

The first line are all cases where the brane correlators do not connect to
closed universes, on the second line we have all contributions connected to
one closed universe (hence the superscript $c$), etcetera. Therefore, the
brane correlator
$Z_{\text{brane}}(b_{1},\dots,b_{n},b_{1}^{c},\dots,b_{n^{c}}^{c})$ should be
read as having at least one connected contribution between a brane on the
surface with the asymptotic boundary and a brane on the extra closed universe.

###  2.4 Geometric setup

To summarize, the rules for constructing all geometries such that we account
for the contribution of the correlated branes are as follows:

  1. 1.

Besides the asymptotic boundary, the partition function includes a sum over an
arbitrary number of geodesic boundaries, for example one more contributing
geometry is

$\text{Example of new geometry with multiple one-point
functions:}\quad\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{Description2.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,.$ (2.29)

  2. 2.

The extra brane boundaries are smeared with wavefunctions, and the joint
connected wavefunction for $n$ boundaries is
$Z_{\text{brane}}(b_{1},\dots,b_{n})$. We denote this connected component of
the brane-correlator, as in for example (2.24) and (2.25).

$\text{Notation for connected brane correlators:}\quad
Z_{\text{brane}}(b_{1},\dots,b_{n})=\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.30}]{nPtCorrelator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.30)

These wavefunctions are integrated over $b_{1},\dots,b_{n}$ using the Weil-
Peterson measure.

  3. 3.

Brane correlators can connect spacetime components with asymptotic boundaries
(open universes) to closed Euclidean spacetime components (closed universes),
which would otherwise factor out of boundary observables. In addition to the
closed universe connected through the brane correlator to an open universe as
in (2.25), this also generates extra connected contributions, where closed
universes connect to multiple open universes (and potentially also to each
other).

$\text{Examples how closed universes contribute: }\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.75}]{Description1.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,.$ (2.31)

The closed universes can alternatively be absorbed in a renormalization of the
brane correlations, see section 3.5.

##  3 Factorization and discreteness from correlated spacetime branes

The purpose of this section is the present the bulk gravitational description
of one quantum mechanical system with Hamiltonian
$\textsf{H}_{0}$.141414Earlier work in this direction includes [22, 18, 57,
80, 45, 40, 59]. The dual gravitational theory has a discrete spectrum (the
eigenvalues $E_{1},\dots,E_{L}$ of $\textsf{H}_{0}$) and factorizes

$Z(\beta_{1},\dots,\beta_{n})=Z(\beta_{1})\dots Z(\beta_{n})\,.$ (3.1)

We claim that the sought-after gravitational theory is nothing but JT gravity
with correlated spacetime branes, where the connected spacetime brane
correlators take the specific values

Factorization $\displaystyle\Leftrightarrow\quad
Z_{\text{brane}}(b_{1},b_{2})=-\frac{1}{b_{1}}\delta(b_{1}-b_{2})\,,\,\,\text{
and }\,\,\,Z_{\text{brane}}(b_{1}\dots b_{n})=0\text{ for $n>2$}$ Discreteness
$\displaystyle\Leftrightarrow\quad
Z_{\text{brane}}(b)=\sum_{i=1}^{L}\frac{2}{b}\cos(bE_{i}^{1/2})-\int_{0}^{\infty}dE\,\rho_{0}(E)\,\frac{2}{b}\cos(bE^{1/2})\,.$
(3.2)

The first line follows from demanding factorization, and the second line from
the requirement that the theory has the discrete spectrum $E_{i}$ of
$\textsf{H}_{0}$.

We will first prove that (3.2) is a solution to the factorization puzzle,
after which, in section 3.2, we prove it is the unique solution. In section
3.3, we derive the formula for the one-point function that is needed to
observe a discrete spectrum, and we quantify the magnitude of this one-point
function in section 3.4. In section 4 we further stack this up by proving that
these brane correlators also have the desired effect in the dual matrix
integral description non-perturbatively in $e^{-\textsf{S}_{0}}$.

###  3.1 All orders factorization

To understand why (3.2) resolves factorization, notice first that in
$Z(\beta_{1},\beta_{2})$ the genus zero wormhole is precisely canceled by two
trumpets connected by a brane two-point function

$\leavevmode\hbox to125.12pt{\vbox
to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-6.55522pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.85}]{Factorization_leading_cancellation.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{113.8093pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{106.03154pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (3.3)

or explicitly in formulas (the trumpet partition function is in (2.3) and the
wormhole partition function is equation (135) in [19])

$Z_{\text{wormhole}}(\beta_{1},\beta_{2})+\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\int_{0}^{\infty}\mathrm{d}b_{2}b_{2}\,Z_{\rm
trumpet}(\beta_{1},b_{1})Z_{\rm
trumpet}(\beta_{2},b_{2})\,Z_{\text{brane}}(b_{1},b_{2})=0\,.$ (3.4)

This indeed vanishes when we insert (3.2) and compute the integrals.

We can prove that (3.2) is a solution to the factorization puzzle to all
orders in the genus expansion, for two boundaries, by noticing that can group
together geometries in such a way that partition functions that include branes
on cycles which are homotopic to the asymptotic boundary, cancel the
contributions of geometries that have no brane on these same cycles.

Explicitly, using (3.2) (and gluing branes to the asymptotic boundary using
the Weil-Peterson measure) one can remark that, for any surface $\Sigma$,
which may or may not have extra branes, and which may or may not connect the
two asymptotic boundaries (either via geometries or via extra brane
correlators on geometries), we have the cancellation

(3.5)

All surfaces having two asymptotic boundaries are among those listed in (3.5),
and therefore $Z(\beta_{1},\beta_{2})=Z(\beta_{1})Z(\beta_{2})$ indeed.

To show that the gravitational path integral factorizes when considering
$n$-asymptotic boundaries, we can generalize (3.5) by grouping geometries in
essentially the same way. When the boundaries have temperatures $\beta_{1}$,
$\dots$, $\beta_{n}$, then, for any geometry $\Sigma$ which might include
handles or an arbitrary number of brane correlators, we have the cancellation

$\displaystyle\sum_{k=0}^{n}\sum_{\sigma_{k}^{n}}\;\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{nBdyConnectedRelation.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad\;\;=\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{nBdyConnectedRelation_Copy.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\sum_{k=0}^{n}(-1)^{k}{n\choose
k}=0\,.$ (3.6)

Above, the sum over $\sigma_{k}^{n}$ is over all combinations diving the set
of $n$ boundaries of $\Sigma$ into $k$ boundaries that end on an asymptotic
boundary and $n-k$ boundaries ending on a brane. Generically all boundaries
can have different inverse temperatures $\beta_{i}$. The first equation
follows by inserting $Z_{\text{brane}}(b_{1},b_{2})$ as given in (3.2) for all
the blue brane correlators, and gluing the branes using the Weil-Petersson
measure. One finds that each term in the sum is proportional to the same
gravitational partition function, and the prefactors add up to zero. Notice
that the case $n=1$ implies that all higher genus corrections to $Z(\beta)$
are canceled.

The only exceptions to these enormous cancellations are the genus zero disk,
and the disk with a brane one point function (3.15), which do not have
geodesic (that are not a boundary) homotopic to the asymptotic boundary,
therefore the above argument does not go through. In summary, when we include
an exponential of brane two-point function (3.2) in JT gravity, all wormhole
geometries are cancelled to all orders in $e^{-\textsf{S}_{0}}$, and only
factorizing contributions remain.

###  3.2 Uniqueness of the factorizing solution

Let us now prove that (3.2) is the unique solution for the brane correlators,
if we _demand_ factorization. In this section we put $Z_{\text{brane}}(b)=0$,
as the one-point function does not impact the factorization discussion.

To have a genus expansion that still makes sense, it will be important to
think of the brane correlators as having an $e^{-\textsf{S}_{0}}$ expansion
too

$Z_{\text{brane}}(b_{1}\dots b_{n})=\quad\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.30}]{nPtCorrelator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad=\sum_{k=-\infty}^{2-n}\bigg{(}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.30}]{nPtCorrelator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{k}\,,$ (3.7)

where the terms on the right side are assumed to scale as
$e^{k\textsf{S}_{0}}$. The upper bound $k_{\text{max}}=2-n$ is fixed by
demanding that the connected $n$-point partition function vanishes at order
$O(e^{k\textsf{S}_{0}})$ when $k>2-n$.151515Since the leading brane-free
connected geometry is $O(e^{\textsf{S}_{0}(2-n)})$, we indeed can not have
$n$-brane vertices for $k>2-n$.

The idea is now to impose factorization, order per order in the
$e^{-\textsf{S}_{0}}$ expansion, and for any number of boundaries, and use
this to find the expansion coefficients in (3.7). Demanding
$Z(\beta_{1},\beta_{2})=0$ at order $O(1)$ gives the equation (3.3), which in
our current mindset fixes the leading order propagator to

$\bigg{(}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8}{0.0}{0.0}{0.8}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{propagator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{0}=-\frac{1}{b_{1}}\delta(b_{1}-b_{2})\,.$
(3.8)

Continuing to focus on the two-boundary gravitational partition function, at
next order $O(e^{-\textsf{S}_{0}})$ we have no contributions other than that
coming from subleading contributions to the two-brane correlator, and so
factorization of $Z(\beta_{1},\beta_{2})$ at order $O(e^{-\textsf{S}_{0}})$
demands that

$\bigg{(}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8}{0.0}{0.0}{0.8}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{propagator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{-1}=0\,.$
(3.9)

At order $O(e^{-2\textsf{S}_{0}})$ there are several diagrams that can
contribute, so factorization becomes less trivial161616Here we will write
symmetry factors as if all asymptotic boundaries have the same temperature, to
avoid cluttering. Furthermore, the brane correlators in drawings always denote
the leading order ones, unless explicitly specified otherwise.

(3.10)

The other six diagrams involve only leading order brane two-point correlators.
Using (3.5), we learn that most diagrams cancel, except the first on the first
line and the second on the second line. To determine the
$O(e^{-2\textsf{S}_{0}})$ contribution to the two-brane correlator, we must
hence first determine the leading order $3$-brane correlator. Setting
$Z(\beta_{1},\beta_{2},\beta_{3})=0$ at (first non-trivial) order
$O(e^{-\textsf{S}_{0}})$ gives Using (3.6) for $n=3$, this equation imposes
that the leading order brane $3$-point vertex vanishes, which, in turn, can be
inserted in (3.10) to prove that the second-subleading order propagator also
vanishes

$\bigg{(}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.16}{0.0}{0.0}{0.16}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.45}]{3ptFactorization5_2.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{-1}=0\,,\quad\bigg{(}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8}{0.0}{0.0}{0.8}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{propagator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{-2}=0\,.$
(3.12)

Similarly, demanding that $Z(\beta_{1}\dots\beta_{n})=0$ at leading non-
trivial order, and using (3.6), we prove (by induction) that the leading non-
trivial order $n$-point brane vertices must all vanish in a factorizing theory

$\bigg{(}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.55}{0.0}{0.0}{0.55}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.30}]{nPtCorrelator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}_{2-n}=0\quad\text{for
}n>2\,.$ (3.13)

This can in turn be used (again in combination with (3.6)) to prove
inductively that subleading lower-point brane vertices must all vanish. In
summary, we have proven that demanding factorization uniquely fixes the brane
correlators to be Gaussian (3.2), all higher point brane correlators must
vanish

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8}{0.0}{0.0}{0.8}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.38}]{propagator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\frac{1}{b_{1}}\delta(b_{1}-b_{2})\,,\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.55}{0.0}{0.0}{0.55}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.30}]{nPtCorrelator.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0\quad\text{for
}n>2\,.$ (3.14)

###  3.3 Discreteness and exact half-wormholes

We now turn on some non-zero brane one-point function $Z_{\text{brane}}(b)$.
For most geometries with one-point functions we can still use equation (3.6),
the only new exception is the geometry with only one $Z_{\text{brane}}(b)$
inserted.171717Remember from section 2.3 that we do not allow degenerate
cylinders. Including them does not affect this argument, it only renormalizes
the brane correlators, see section 4.2. Therefore our theory remains extremely
simple, the partition function becomes _exactly_

$\displaystyle
Z(\beta)=\,Z_{\text{disk}}(\beta)+\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\,Z_{\text{brane}}(b)=\leavevmode\hbox
to152.4pt{\vbox to47.81pt{\pgfpicture\makeatletter\hbox{\hskip
63.11937pt\lower-47.61113pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.45}]{Halfwormhole_contribution.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{-37.84187pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.78636pt}{-43.3059pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{black hole}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{45.80879pt}{-39.83368pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.66982pt}{-42.33368pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{one-point function}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (3.15)

The disk captures the contribution of the black hole saddle. The only other
remaining contribution can be identified as the half-wormhole discussed in
Saad, Shenker, Stanford and Yao [45, 40] with the important distinction that
here we proved that the partition function receives no other perturbative
contributions in $e^{-\textsf{S}_{0}}$. We will prove in section 4 that this
remains exactly true non-perturbatively in $e^{-\textsf{S}_{0}}$.

In this sense what we have done is prove that in this setup the half-wormhole
approximation is exact.

We now want to check that the expression for $Z_{\text{brane}}$ in (3.2)
results indeed in a theory with discrete spectrum $E_{1},\dots,E_{L}$. Using
the Gaussian integral

$\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\,\frac{2}{b}\,\cos(bE^{1/2})=\frac{1}{2\pi^{1/2}\beta^{1/2}}\int_{-\infty}^{+\infty}\mathrm{d}b\,\exp(-\frac{b^{2}}{4\beta}+ibE^{1/2})=e^{-\beta
E}\,,$ (3.16)

we find that the half-wormhole geometry in (3.15) contributes

$\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\,Z_{\text{disk}}(b)=\sum_{i=1}^{L}e^{\beta
E_{i}}-\int_{0}^{\infty}dE\,\rho_{0}(E)\,e^{-\beta E}=\Tr
e^{-\beta\textsf{H}_{0}}-Z_{\text{disk}}(\beta)\,,$ (3.17)

and hence this gravitational theory indeed has a discrete spectrum determined
by the eigenvalues of $\textsf{H}_{0}$

$Z(\beta)=\Tr e^{-\beta\textsf{H}_{0}}\,.$ (3.18)

Multi-boundary partition functions become just products of this, because all
the wormholes still cancel.

Note that using (3.16), we can alternatively write the full partition function
of the gravity theory as

$\displaystyle Z(\beta)$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\sum_{i=1}^{L}\frac{2}{b}\cos(bE_{i}^{1/2})$
$\displaystyle=-\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet}}(\beta,b)\sum_{i=1}^{L}\bigg{(}M_{\mathrm{i}E_{i}^{1/2}}(b)+M_{-\mathrm{i}E_{i}^{1/2}}(b)\bigg{)}\,.$
(3.19)

In particular we recognize (both sheets of) the wavefunction of FZZT
boundaries (2.10) in the allowed region. Hence, the full partition function
can be viewed as a sum of cylinder geometries ending on FZZT boundaries, and
the wavefunction $M_{\mathrm{i}E_{i}^{1/2}}(b)+M_{-\mathrm{i}E_{i}^{1/2}}(b)$
can be interpreted as preparing the microstate with energy $E_{i}$.
Pictorially

$\Tr
e^{-\beta\textsf{H}_{0}}=-\sum_{i=1}^{L}\quad\int_{0}^{\infty}\mathrm{d}b\,b\,\bigg{(}\,\,\,\leavevmode\hbox
to47.47pt{\vbox to27.85pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-27.64616pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.55}{0.0}{0.0}{0.55}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.25}]{Half_wormhole_fixed_energy.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{28.95093pt}{-18.77887pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.96304pt}{-20.74458pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$M_{\mathrm{i}E_{i}^{1/2}}(b)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\quad\leavevmode\hbox
to50.75pt{\vbox to27.85pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-27.64616pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.55}{0.0}{0.0}{0.55}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.25}]{Half_wormhole_fixed_energy.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{31.2982pt}{-18.77887pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.377pt}{-20.74458pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$M_{-\mathrm{i}E_{i}^{1/2}}(b)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{)}\quad,$ (3.20)

This has similar flavor to the eigenbranes picture of [22], with the important
distinction that here we have a semiclassical interpretation without
complicated sum over geometries. We have thus found that for each Hamiltonian
$\textsf{H}_{0}$ there is a correspondence

$\textsf{H}_{0}\quad\Longleftrightarrow\quad Z_{\text{brane}}(b)\,,$ (3.21)

which, in turn, can be understood as a linear combination of FZZT
wavefunctions.

###  3.4 Magnitude of UV corrections

We want to understand when the half-wormhole or brane one-point function in
(3.15) is a small correction to the leading black hole (as expected of UV
corrections). We can distinguish two situations, depending on whether
$\textsf{H}_{0}$ is a typical (realistic) draw of the original JT gravity
ensemble, or an atypical draw.

For typical draws we can accurately estimate the importance of the half-
wormhole or brane one-point function by computing ensemble averaged over
$\textsf{H}_{0}$, using the same measure as in the original JT gravity
ensemble. We denoted these averages by
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle}\dots\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rangle$,
to distinguish from the original matrix ensemble. Notice now that
$Z_{\text{brane}}(b)$ as given by (3.2) plays exactly the same role in the
ensemble average over $\textsf{H}_{0}$, as the role played by
${{\mathcal{O}}}_{\rm G}(b)$ in (2.11) in the original ensemble average. In
particular, ensemble averaging over $\textsf{H}_{0}$ one recovers the exact
analogue of (2.12)

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle}Z_{\text{brane}}(b){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rangle}=\sum_{g=1}^{\infty}e^{\textsf{S}_{0}(1-2g)}\,V_{g,1}(b)\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle}Z_{\text{brane}}(b_{1})\dots
Z_{\text{brane}}(b_{2}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rangle}_{\text{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}conn}}=\sum_{g=0}^{\infty}e^{\textsf{S}_{0}(2-2g-n)}\,V_{g,n}(b_{1}\dots
b_{n})\,.$ (3.22)

The last equation can graphically be represented as

$\leavevmode\hbox to78.68pt{\vbox
to48.37pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-48.16534pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.7}]{red_average_half_wormholes.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0);
{}{{}}{}{{}}{}\pgfsys@moveto{64.87299pt}{-42.67957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{54.59518pt}{-44.83234pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\text{conn}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad=\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.7}]{n-wormhole.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\quad\text{higher
genus}\,.$ (3.23)

Thus, even though the wormholes have canceled in the geometric expansion of
the partition function $Z(\beta)$, they are still encoded in the statistics of
the brane one-point function (or half-wormhole). From (3.23) we see that all
the moments of the one-point function are suppressed by powers of
$e^{-\textsf{S}_{0}}$ when we compute the partition function, so we conclude
that for any typical draws $\textsf{H}_{0}$ the UV corrections to the
partition function are indeed suppressed as compared to the black hole
geometry $Z(\beta)-Z_{\text{disk}}(\beta)\sim O(1)$, which is nontrivial since
both terms on the left are $\sim O(e^{\textsf{S}_{0}})$.

For atypical draws $\textsf{H}_{0}$ there is no such cancellation occurring,
and $Z(\beta)-Z_{\text{Disk}}(\beta)\sim O(e^{\textsf{S}_{0}})$. This implies
that the brane one-point function scales as $\sim O(e^{\textsf{S}_{0}})$, thus
the half-wormhole contribution is not parametrically smaller than the disk
contribution. This is no surprise, atypical draws are simply poorly described
by JT gravity, so for those matrices $\textsf{H}_{0}$, this construction makes
little sense from the get-go. In summary,

$\displaystyle\textsf{H}_{0}\text{ typical member of the JT ensemble}$
$\displaystyle\quad\Longleftrightarrow\quad\text{brane corrections suppressed
by $\sim O(e^{-\textsf{S}_{0}})$}$ $\displaystyle\textsf{H}_{0}\text{ atypical
member of the JT ensemble}$
$\displaystyle\quad\Longleftrightarrow\quad\text{brane corrections never
suppressed}\,.$

We note that for typical $\textsf{H}_{0}$ and self-averaging quantities, the
red average in (3.23) can essentially be dropped. The wormhole computation for
such observables approximates the answer in each member of the ensemble, which
itself is captured by only the black hole and the half-wormhole geometries
(3.15). As a concrete example, consider the spectral form factor [14]. At
exponentially late times $t>t_{\text{ramp}}$, the spectral form factor is not
self-averaging, but a smeared version where we time average over an interval
$\Delta T\gg 1$ _is_ self-averaging.181818This is because at times on the ramp
both the standard deviation of the spectral form, and the spectral form factor
itself, scale as $\sim t/\beta$. After time averaging, the standard deviation
scales as $\sim t/(\beta\Delta T)$, and the averaged signal as $\sim t/\beta$.
This implies the following approximate equality191919The figure above involves
an analytic continuation from Euclidean to Lorentzian geometry which here we
consider occurs along a geodesic in the geometry. For the contribution of the
half-wormhole we take this geodesic to be located along a geodesic that
separates the spacetime into a region that contains a brane and one that does
not. Thus, with our choice, the brane is located entirely in the Euclidean
preparation region of the bra or the ket states. This choice however does not
appear to be unique and it would be interesting to explore other possible
continuations to Lorentzian signature.

$\displaystyle\frac{1}{\Delta T}\int_{t-\Delta T}^{t+\Delta
T}\mathrm{d}\tau\,Z(\beta-\mathrm{i}\tau)Z(\beta+\mathrm{i}\tau)$
$\displaystyle=\frac{1}{\Delta T}\int_{t-\Delta T}^{t+\Delta
T}\mathrm{d}\tau\,\,\bigg{|}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.3}{0.0}{0.0}{0.3}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{Lorentzian_cont.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (-1,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{|}^{2}\,\,$
(3.24) $\displaystyle{\approx}\,\,\begin{cases}\,\,\bigg{|}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.21}{0.0}{0.0}{0.21}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{Lorentzian_straightforward.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (-1,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\bigg{|}^{2}&\sim\frac{e^{\textsf{S}_{0}}}{t^{3}}e^{\frac{4\pi\beta}{t^{2}}}\,\text{
for }t<t_{\text{ramp}}\\\ \,\,\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.25}{0.0}{0.0}{0.25}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{averagedSFF.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (-1,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\sim\frac{t}{\beta}\,\hskip
28.45274pt\text{ for }t_{\text{ramp}}<t<t_{\text{plateau}}\end{cases}$

In going from the second to the third line, we used the fact that the time
averaged spectral form factor in quantum systems with (typical) Hamiltonians
$\textsf{H}_{0}$ is self-averaging, such that one can effectively apply
(3.23). So, at early times one sees that both the half-wormhole (in our model)
and geometric wormhole (in JT gravity) are subleading but, when
$t>t_{\text{ramp}}$, the half-wormhole contribution starts dominating in the
time-averaged spectral form factor, and crucially the result is well
approximated by the JT gravity wormhole. This is why simple gravity models,
with wormholes, but without branes, are oftentimes good approximations.

###  3.5 Additional comments

A few comments are in order.

  1. 1.

An important ingredient that has led to the simple expression for the brane
couplings (3.14) comes from allowing correlations between branes on closed
universes and branes on geometries that include an asymptotic boundary. One
could attempt to absorb the closed universes in dressed brane correlators
(black) that only attach to asymptotic universes, which would be non-Gaussian

(3.25)

However, some of the terms in this expansion are divergent, which suggests
that the more natural expansion is the original one, which does allow branes
connecting closed and asymptotic universes. We believe that the closed
universes are an important conceptual ingredient in this story. From the
perspective of dilaton gravity with nonlocal interactions (2.3) it also makes
most sense to think about things in terms of there being closed universes as
well.

  2. 2.

Tracing the presence of the brane correlators back to the non-local
deformations of the action, we see that the cancellation of all wormholes is
due to the perturbative expansion of these nonlocal deformations around the
leading black hole saddle. For example, the cancellation of the leading two-
boundary wormhole is obtained by expanding the non-local deformation around
the two copies of the black hole saddle-point (3.3). This re-emphasizes the
importance of tracking such non-local terms in the low-energy effective theory
of gravity. We will further discuss the possible meaning of such nonlocal
terms in the discussion in section 6.

  3. 3.

It is essential for the arguments of section 3.1 to work, that there is an
exponential of two-brane correlators in our theory which results in a sum over
all possible brane insertions. In these points, we substantially from [40],
and it is why we obtain factorization to all orders in the $e^{-S_{0}}$
expansion.

  4. 4.

The simplicity of the result is a consequence of the fact that the integration
measure and integration range when gluing trumpets to any surface $\Sigma$ in
(3.6), is independent of the topology and moduli of $\Sigma$. This is because
gluing a trumpet to $\Sigma$ changes the mapping class group just by
introducing an extra factor $\text{MCG}_{\Sigma}\times\mathbb{Z}$.

  5. 5.

The brane correlators are independent of the parameters that determine the
leading black hole geometry, indeed (3.14) is independent of
$e^{\textsf{S}_{0}}$ and $\beta$. This suggests a possible non-geometric
origin for the brane correlators, consistent with our intuition about the
stringy realization of these UV corrections in the section 1 and section 6.

  6. 6.

Factorization implies null-states in the baby-universe Hilbert space. One
might worry that the negativity of the connected brane correlator $Z_{\rm
brane}(b_{1},b_{2})$ might lead to a lack of reflection positivity in the
gravitational Hilbert space of the theory. To show that this is not the case
we will use the baby universe Hilbert space formalism of [20]. In this Hilbert
space, states are obtained from boundary creation operator, which we can take
to create asymptotic boundaries with boundary length $\beta$ and fixed
dilaton. The norm of any such state can be expressed as202020We thank D.
Stanford and Z. Yang for useful comments in this direction.

$\langle\Psi|\Psi\rangle=\alpha^{\dagger}\mathcal{Z}\alpha\,,\qquad\text{
where
}\qquad|\Psi\rangle=\sum_{i=1}^{n_{\text{bdies}}}\alpha_{i}\,|Z(\beta)^{i}\rangle\,,$
(3.26)

where $n_{\text{bdies}}$ is the maximum number of boundaries involved in the
state $|\Psi\rangle$. Factorization implies that the inner product matrix
$\mathcal{Z}_{ij}=Z(\beta)^{i+j}$ (whose dimension is $n_{\text{bdies}}\times
n_{\text{bdies}}$) has rank $1$ with its one non-zero eigenvalue being
positive and equal to $\sum_{i=1}^{n_{\text{bdies}}}Z(\beta)^{2i}$. This
implies that $\langle\Psi|\Psi\rangle\geq 0$ for all $\alpha$, and thus
reflection positivity is not violated in this baby universe Hilbert space.
Furthermore, the fact that $\mathcal{Z}$ is rank $1$ is a signature of the
existence of null-states which are associated to imposing factorization within
in our model. In the usual GNS construction these are then modded out and the
final Hilbert space is one-dimensional, consistent with being in a single
$\alpha$-state. The geodesic boundaries can be thought of as an additional set
of universe creating operators $\widehat{\mathcal{O}_{\rm G}}(b)$, with
correlations (3.2). These correlations are only accessible to observers with
access to different universes, and are inaccessible to observers within a
single universe. Observers within a single universe can only measures the
total correlation (which vanishes), see section 6.

  7. 7.

The $b$ integral in (3.20) has saddlepoints. The integrand of (3.20) is even,
so we can extend the integral to the entire real axis and deform the contour
through the saddlepoint at $b_{i}=2\beta\mathrm{i}E^{1/2}_{i}$ (3.16). Thus
for $E_{i}>0$ (which is the set of energies we focus on) this saddle is at
_imaginary_ $b$. This means we can actually view it (on-shell) as a defect
geometry with defect angle

$\alpha_{i}=2\pi+\mathrm{i}b_{i}=2\pi-2\beta E_{i}^{1/2}.$ (3.27)

It would be interesting to understand the precise meaning of the emergence of
defect geometries here and in particular its fate upon analytic continuation
to Lorentzian signature. In particular how we cut the Euclidean geometry and
glue it to a Lorentzian section.

All statements thus for hold to all orders in $e^{-\textsf{S}_{0}}$
perturbation theory. We now prove that actually (3.2) yields (3.3) even non-
perturbatively, the correlated branes truly fix the eigenvalues of the theory
to $E_{i}$.

##  4 Factorization and discreteness from matrix integral localization

Here we explain from the matrix integral description of two dimensional
dilaton gravity why introducing extra bilocal interactions in the spacetime
action is sufficient to obtain factorizing gravity systems, and how the brane
one point functions discretize the spectrum.

The partition functions of the relevant matrix ensembles are defined as [81,
19]

$\displaystyle\mathcal{Z}$
$\displaystyle=\prod_{i=1}^{L}\int_{\mathcal{{{\mathcal{C}}}}}d\lambda_{i}\exp\bigg(-L\sum_{i=1}^{L}V(\lambda_{i})\bigg{missing})\prod_{i<j}^{L}(\lambda_{i}-\lambda_{j})^{2}=\prod_{i=1}^{L}\int_{\mathcal{{{\mathcal{C}}}}}d\lambda_{i}\exp\bigg(-L\sum_{i=1}^{L}V(\lambda_{i})+\sum_{i\neq
j}^{L}\log(\lambda_{i}-\lambda_{j})\bigg{missing})\,,$ (4.1)

where the potential $V(E)$ determines which dilaton gravity we are studying.
Typical observables in this matrix integral are products of spectral densities

$\rho(E)=\sum_{i=1}^{L}\delta(E-\lambda_{i}).$ (4.2)

This corresponds in gravity with computing amplitudes with fixed energy
boundaries, which are inverse Laplace transforms of the typical fixed length
partition functions $Z(\beta)$. The equations of motion from varying the
action in (4.1) with respect to $\lambda_{i}$ relate the potential with the
large $L$ leading order spectrum $\rho_{0}(E)$ [82]

$\quad
LV(\lambda)=2\fint_{\Gamma}dE\,\log(\lambda-E)\,\rho_{0}(E)+\text{constant}\quad\Rightarrow\quad
LV^{\prime}(\lambda)=2\fint_{\Gamma}dE\,\frac{1}{\lambda-E}\,\rho_{0}(E)\,,$
(4.3)

where $\Gamma$ is the cut along which $\rho_{0}(E)$ has non-zero support. This
principal value integral can be inverted to determine $\rho_{0}(E)$ in terms
of the potential [83]. The constant is the Lagrange multiplier which is used
to fix the normalization of $\rho_{0}(E)$ in the collective field formulation;
its integral over the region $\Gamma$ is $L$. Subleading corrections and
higher point correlators of $\rho(E)$ can be calculated by expanding around
the saddle $\rho_{0}(E)$. The Vandermonde encodes non-trivial correlation
between different eigenvalues [14].

The gravitational dual of correlated eigenvalues are connected geometries
(i.e. wormholes) between different asymptotic regions. For example the leading
order variance of the spectral density is computed in gravity by computing the
wormhole between two different asymptotic regions [19]

$\rho(E_{1},E_{2})_{\text{conn}}=-\frac{1}{4\pi^{2}}\frac{E_{1}+E_{2}}{E_{1}^{1/2}E_{2}^{1/2}}\frac{1}{(E_{1}-E_{2})^{2}}+\text{subleading
higher genus corrections.}$ (4.4)

Converting this to fixed $\beta$, one simply sums over the wormhole geometry,
with any number of handles.

###  4.1 Localization concept

In section 3 we saw that one can obtain a factorizing gravitational theory by
including branes with specific correlation functions. In particular, we saw
that bilocal interactions between different branes can conspire to precisely
cancel the geometric wormhole correlation between different asymptotic
boundaries. It is surprising that bilocal interactions are sufficient to
attain factorization, since a priori there seems little gravitational reason
to expect that generic multi-local interactions would not feature.

However from the matrix model side this has a simple explanation. Introducing
bilocal interactions in gravity, corresponds in the matrix integral with
introducing a double-trace deformation,

$\exp\bigg(\sum_{m,n=0}^{\infty}f_{2}^{mn}\,\Tr H^{m}\,\Tr
H^{n}\bigg{missing})=\exp\bigg(\sum_{i,j=1}^{L}f_{2}(\lambda_{i},\lambda_{j})\bigg{missing})\,.$
(4.5)

This modifies the repulsive force between eigenvalues. The basic intuition how
a discrete spectrum then arises is that we will tune the repulsive
interactions and the potential in such a way that the eigenvalues are frozen
at the locations of the eigenvalues of a matrix $\textsf{H}_{0}$, drawn from
the JT ensemble. Since the undeformed matrix integral itself is comprised of
just single-and double trace terms, it is not necessary to include higher-
trace deformations to achieve this.

More concretely, we will prove that the deformation corresponding to the brane
insertions discussed in section 3, result in a deformed matrix integral of the
form

$\displaystyle\mathcal{Z}(\textsf{H}_{0})=\lim_{q\to\infty}\prod_{i=1}^{L}\int_{{\mathcal{C}}}d\lambda_{i}\,\exp\bigg(q\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})\bigg{missing})\,,$
(4.6)

where $q$ is a parameter, larger than any other quantity in the matrix
integral, whose origin we explain shortly. Because $q\to\infty$ the integrals
over $\lambda_{i}$ localize onto the stationary points of the “action”
$I(\lambda_{i},E_{j})$, given in (4.30). We will prove that, with the brane
deformations of section 3, the stationary points are212121There can be other
saddles, but we will prove below that these are the dominant ones, to which
the integral localizes.

$\frac{\partial}{\partial\lambda_{i}}\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})=0\quad\Leftrightarrow\quad\lambda_{1}\dots\lambda_{L}=\text{permutations
of }E_{1}\dots E_{L}\,.$ (4.7)

Therefore, indeed, the set of random eigenvalues $\lambda_{i}$ are frozen to
the energy spectrum $E_{i}$ of $\textsf{H}_{0}$.222222There are different ways
of achieving this localization, here we use the single-and double-trace
deformations of section 3. This differs from the matrix integral action used
in [57], which involved a deformation that broke the $U(L)$ invariance. Thus,
for all purposes, the matrix integral (4.6) reduces for $q\to\infty$ to

$\mathcal{Z}(\textsf{H}_{0})=\prod_{i=1}^{L}\int_{{\mathcal{C}}}d\lambda_{i}\,\prod_{j=1}^{L}\sum_{i_{j}=1}^{L}\delta(\lambda_{i_{j}}-E_{j})\,.$
(4.8)

This same result was obtained with the eigenbrane picture [22, 18] and the
proposal in [57], but notably the gravitational description we presented in
section 3 is much simpler; we never lose control of gravity as the full theory
(1.3) has a good semiclassical interpretations.

We note that already in [57], there were signs that nonlocal interactions
should become important in a factorizing theory of gravity, as pointed out for
example in the discussion there. For large $1/\sigma^{2}$, which is analogous
to large $q$ here, multi-trace deformations in the Harish-Chandra integral
become important. We believe that a more careful analysis of the Harish-
Chandra integral in the double scaling limit, should reveal the same Gaussian
double-trace deformation as the one which we shall discuss in the next
section. It would be interesting to make that precise.

We now derive the form of $I$ that follows from inserting the correlated
branes of section 3, and prove that it indeed satisfies (4.7).

###  4.2 Argument

Based on the gravity setup of section 2, we are led to the following
deformation of the JT matrix integral

$\displaystyle\exp\bigg(\int_{0}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b)\,z_{\text{brane}}(b)+\frac{1}{2}\int_{0}^{\infty}\mathrm{d}b_{1}b_{1}\int_{0}^{\infty}\mathrm{d}b_{2}b_{2}\,{{\mathcal{O}}}_{\rm
G}(b_{1}){{\mathcal{O}}}_{\rm
G}(b_{2})\,z_{\text{brane}}(b_{1},b_{2})\bigg{missing})\,.$ (4.9)

Recall from section 2 that FZZT boundaries corresponds in the matrix integral
to insertions of

${{\mathcal{O}}}_{\text{FZZT}}(z)=\sum_{i=1}^{L}\log(\lambda_{i}+z^{2})-\frac{L}{2}V(-z^{2})\,,$
(4.10)

where $z^{2}=-E$. Notice also the potential term for FZZT branes. Furthermore,
as discussed in section 2, we can write a smeared FZZT brane as an operator
that creates geodesic boundaries in the spacetime232323To see that this is
indeed equivalent to (2.11), note the relation of the potential with the genus
zero spectrum (4.3).

$\displaystyle{{\mathcal{O}}}_{\rm G}(b)=-\frac{1}{2\pi
i}\int_{-i\infty}^{+i\infty}\mathrm{d}z\,e^{bz}\,{{\mathcal{O}}}_{\rm
FZZT}(z)=\sum_{i=1}^{L}\frac{2}{b}\,e^{-b\varepsilon}\cos(b\lambda_{i}^{1/2})-\frac{L}{2}W(b)\,,$
(4.11)

where we introduced the inverse Laplace transformed potential

$W(b)=\frac{1}{2\pi
i}\int_{-i\infty}^{+i\infty}\mathrm{d}z\,e^{bz}\,V(-z^{2}).$ (4.12)

Here, the regulator $\varepsilon$ appears because FZZT branes are actually
located an $\varepsilon$ left of the imaginary axis, resolvents have small
imaginary parts in their energy arguments [19]. This regulator was not
important in section 3, but here we should properly keep track of it.

One important point is that, in equation (4.9), we should be inserting
_renormalized_ brane correlators

$\lim_{q\to\infty}\frac{1}{1+q}\,z_{\text{brane}}(b)=\sum_{j=1}^{L}\frac{2}{b}\cos(bE_{j}^{1/2})-\frac{L}{2}W(b)\,,\quad\lim_{q\to\infty}\frac{1}{q}\,z_{\text{brane}}(b_{1},b_{2})=-\frac{1}{b_{1}}\delta(b_{1}-b_{2})$
(4.13)

This expression for the propagator $z_{\text{brane}}(b_{1},b_{2})$ follows
from the geometric propagator $Z_{\text{brane}}(b_{1},b_{2})$ of section 3. To
understand how this renormalization comes by, notice that in the undeformed JT
matrix integral one finds

$\left\langle{{\mathcal{O}}}_{\rm G}(b_{1}){{\mathcal{O}}}_{\rm
G}(b_{2})\right\rangle=\frac{1}{b_{1}}\delta(b_{1}-b_{2})+O(e^{-\textsf{S}_{0}})\,.$
(4.14)

Geometrically this term represents the connected degenerate cylinder, which we
explicitly excluded from our geometric model. However, in the matrix integral
we have no choice, these amplitudes do exist. The full two-point function of
${{\mathcal{O}}}_{\rm G}(b)$ in the model with the double trace interaction is
then a Dyson series, because we can insert pairs of ${{\mathcal{O}}}_{\rm
G}(b)$ which are correlated via the brane two-point function
$z_{\text{brane}}(b_{1},b_{2})$. One should view
$z_{\text{brane}}(b_{1},b_{2})$ as a _bare_ brane propagator, and
$Z_{\text{brane}}(b_{1},b_{2})$ as the _dressed_ propagator, obtained by
resumming a Dyson series of $z_{\text{brane}}$ propagators connected through
degenerate cylinders

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.80}]{Prop_redefinition.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;.$ (4.15)

Note that no other closed universe geometries appear in (4.15); they are
already included in the topological expansion defined in section 3, which was
expressed in terms of blue correlators. In (4.15), all diagrams contribute at
the same order in $e^{\textsf{S}_{0}}$. This is similar to how in QFT one
would treat a mass term perturbatively by resumming diagrams. In formulas, the
above Dyson equation reads

$\displaystyle Z_{\text{brane}}(b_{1},b_{2})$
$\displaystyle=z_{\text{brane}}(b_{1},b_{2})+\int_{0}^{\infty}\mathrm{d}b_{3}b_{3}\,z_{\text{brane}}(b_{1},b_{3})z_{\text{brane}}(b_{3},b_{2})$
$\displaystyle\quad+\int_{0}^{\infty}\mathrm{d}b_{3}b_{3}\int_{0}^{\infty}\mathrm{d}b_{4}b_{4}\,z_{\text{brane}}(b_{1},b_{3})z_{\text{brane}}(b_{3},b_{4})z_{\text{brane}}(b_{4},b_{2})+\dots$
(4.16)

As $Z_{\text{brane}}(b_{1},b_{2})=-\delta(b_{1}-b_{2})/b_{1}$, the solution is
$z_{\text{brane}}(b_{1},b_{2})=-q\delta(b_{1}-b_{2})/b_{1}$, where $q$ must
satisfy

$-1=-q+q^{2}-q^{3}+\dots=-\frac{q}{1+q}.$ (4.17)

This has two solutions $q\to\pm\infty$, but since we want the matrix integral
including the deformation (4.9) to localize we choose $q\to+\infty$. One might
additionally be concerned that (4.17) is only convergent for $q\to+\infty$
upon analytic continuation in $q$. However, (4.17) is solely used to determine
which operator deformation in the matrix integral is equivalent to the brane
insertions discussed in section 3; in the resulting matrix integral, no such
convergence issues are encountered.

The renormalization of the bare propagator by a factor $q$ also affects the
first term in (4.9), i.e. the full one-point function of ${{\mathcal{O}}}_{\rm
G}(b)$ in the deformed matrix model. Comparing this again with our finding in
section 3, we see that including degenerate cylinders results in the series

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.6}{0.0}{0.0}{0.6}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.80}]{Oneptfunc_redefinition.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;.$ (4.18)

which, concretely, can be rewritten as

$\displaystyle Z_{\text{brane}}(b)$
$\displaystyle=z_{\text{brane}}(b)+\int_{0}^{\infty}\mathrm{d}b_{1}\,b_{1}z_{\text{brane}}(b,b_{1})z_{\text{brane}}(b_{1})$
$\displaystyle\quad+\int_{0}^{\infty}\mathrm{d}b_{1}\,b_{1}\mathrm{d}b_{2}\,b_{2}z_{\text{brane}}(b,b_{1})z_{\text{brane}}(b_{1},b_{2})z_{\text{brane}}(b_{2})+\dots\,.$
(4.19)

Plugging in the result for $z_{\text{brane}}(b_{1},b_{2})$ that we just
derived, this equation becomes

$Z_{\text{brane}}(b)=\lim_{q\to\infty}\frac{1}{1+q}\,z_{\text{brane}}(b)\,,$
(4.20)

and so we need $z_{\text{brane}}(b)=(q+1)Z_{\text{brane}}(b)$ modulo
subleading terms in $q$ which will not be of importance. We emphasize that
requiring consistency with results from section 3, which demanded
factorization and discreteness, forces $q\to\infty$. Finite $q$ does not
result in a factorizing or discrete system, not only because the matrix
integral does not localize, but this also simply would not match with our
geometric derivation. To summarize, the factors of $q$ come from taking into
account the degenerate cylinders.

We now prove that (4.9) indeed gives the localization described around (4.7).
The workhorse formula which we use to compute all integrals in (4.9) is

$-4\int_{\delta}^{\infty}\frac{\mathrm{d}b}{b}e^{-b\varepsilon}\cos(bE_{1}^{1/2})\cos(bE_{2}^{1/2})=P(E_{1},E_{2})+4\log\delta\,,$
(4.21)

Here $\delta$ regulates the small $b$ divergence of this integral, which will
drop out of all observables, and $P(E_{1},E_{2})$ is given by

$\displaystyle P(E_{1},E_{2})$
$\displaystyle=\log\Big(\left(E_{1}^{1/2}-E_{2}^{1/2}\right)^{2}+\varepsilon^{2}\Big{missing})+\log\Big(\left(E_{1}^{1/2}+E_{2}^{1/2}\right)^{2}+\varepsilon^{2}\Big{missing})$
(4.22) $\displaystyle=\left\\{\begin{array}[]{ll}2\log|E_{1}-E_{2}|&\text{ for
}E_{1}\neq E_{2}\\\ \log\varepsilon^{2}+\log(4|E_{1}|)&\text{ for
}E_{1}=E_{2}\end{array}\right.\,.$ (4.25)

The quadratic term in ${{\mathcal{O}}}_{\rm G}(b)$ in (4.9) can now be
immediately evaluated

$\displaystyle-\frac{q}{2}\int_{\delta}^{\infty}\mathrm{d}b\,b\,{{\mathcal{O}}}_{\rm
G}(b){{\mathcal{O}}}_{\rm G}(b)$
$\displaystyle=\frac{q}{2}\sum_{i,j=1}^{L}P(\lambda_{i},\lambda_{j})+qL\sum_{i=1}^{L}\int_{\delta}^{\infty}\mathrm{d}b\,W(b)\,e^{-b\varepsilon}\cos(b\lambda_{i}^{1/2})$
$\displaystyle\qquad-\frac{q}{2}\frac{L^{2}}{4}\int_{\delta}^{\infty}\mathrm{d}b\,b\,W(b)W(b)+2qL^{2}\log\delta$
$\displaystyle=\frac{q}{2}\sum_{i,j=1}^{L}P(\lambda_{i},\lambda_{j})-qL\sum_{i=1}^{L}V(\lambda_{i})+\text{constant.}$
(4.26)

The constant on the second line is independent of $\lambda_{i}$, and thus
cancel when normalizing our matrix integral. Subsequently we drop such terms.
On the last line we used the definition of the inverse Laplace transformed
potential $W(b)$ in (4.12), and did the integrals over $b$ and $z$ explicitly.
Similarly, the linear in ${{\mathcal{O}}}(b)$ part in (4.9) evaluates to

$\displaystyle\int_{0}^{\infty}dbb\,{{\mathcal{O}}}_{\rm
G}(b)z_{\text{brane}}(b)$
$\displaystyle=-(1+q)\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})+(1+q)L\sum_{i=1}^{L}V(\lambda_{i})\,.$
(4.27)

Combining the these two expressions, we find that (4.9) results in the
insertion of

$\displaystyle\exp\bigg(\frac{q}{2}\sum_{i,j=1}^{L}P(\lambda_{i},\lambda_{j})-(1+q)\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})+L\sum_{i=1}^{L}V(\lambda_{i})\bigg{missing})\,.$
(4.28)

The last term cancels the potential in the original matrix integral, so we end
up with the matrix integral

$\displaystyle\mathcal{Z}(\textsf{H}_{0})$
$\displaystyle=\lim_{q\to\infty}\prod_{i=1}^{L}\int_{{\mathcal{C}}}\mathrm{d}\lambda_{i}\,\exp\bigg(\sum_{i\neq
j}^{L}\log(\lambda_{i}-\lambda_{j})+\frac{q}{2}\sum_{i,j=1}^{L}P(\lambda_{i},\lambda_{j})-(q+1)\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})\bigg{missing})$
(4.29)
$\displaystyle=\lim_{q\to\infty}\prod_{i=1}^{L}\int_{{\mathcal{C}}}\mathrm{d}\lambda_{i}\,\exp\bigg(\frac{q+1}{2}\sum_{i\neq
j}^{L}\log(\lambda_{i}-\lambda_{j})+\frac{q}{2}\sum_{i=1}^{L}P(\lambda_{i},\lambda_{i})-(q+1)\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})\bigg{missing})\,.$

Here we have used (4.25) to identify the off-diagonal terms as changing the
power of the Vandermonde. The second and third term on the second line can be
viewed as the new potential of the matrix integral, which now depends on the
spectrum of $\textsf{H}_{0}$. The action in (4.6) for $q\to\infty$ becomes

$\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})=\frac{1}{2}\sum_{i,j=1}^{L}P(\lambda_{i},\lambda_{j})-\sum_{i,j=1}^{L}P(E_{i},\lambda_{j})+O(1/q)\,.$
(4.30)

The $O(1/q)$ terms just get evaluated on-shell in the $q\to\infty$ limit.

We can now check that the equations of motion (4.7) indeed have the solution
$\lambda_{1}\dots\lambda_{L}=$ permutations of $E_{1},\dots,E_{L}$, by
noticing that for $\varepsilon\to 0$ (one should only take $\varepsilon\to 0$
at the very end)

$\frac{\partial}{\partial\lambda_{i}}P(\lambda_{i},\lambda_{i})=\frac{1}{\lambda_{i}}\,,\quad\frac{\partial}{\partial\lambda_{i}}P(E_{j},\lambda_{i})=\frac{1}{2E_{j}}\text{
when }\lambda_{i}=E_{j}\,.$ (4.31)

Note that the solution is independent of the regulator $\varepsilon$ needed in
order to regulate the integral in (4.11). The on-shell action of this solution
is large and scales with both $-\log(\varepsilon)$ and $q$, making it the
dominant solution, and the variance is small, therefore the matrix integrals
localizes on these solutions for $q\to\infty$.

To compute observables such as $Z(\beta_{1}\dots\beta_{n})$ in this deformed
matrix integral, we have to normalize the answer by the matrix integral
partition function (4.6), to find

$\displaystyle Z(\beta_{1}\dots\beta_{n})$
$\displaystyle=\frac{1}{\mathcal{Z}(\textsf{H}_{0})}\lim_{q\to\infty}\prod_{i=1}^{L}\int_{{\mathcal{C}}}d\lambda_{i}\,\exp\bigg(q\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})\bigg{missing})\,\sum_{i_{1}=1}^{L}e^{-\beta_{1}\lambda_{i_{1}}}\dots\sum_{i_{n}=1}^{L}e^{-\beta_{n}\lambda_{i_{n}}}$
$\displaystyle=\sum_{i_{1}=1}^{L}e^{-\beta_{1}E_{i_{1}}}\dots\sum_{i_{n}=1}^{L}e^{-\beta_{n}E_{i_{n}}}=\Tr
e^{-\beta_{1}\textsf{H}_{0}}\dots\Tr e^{-\beta_{n}\textsf{H}_{0}}\,.$ (4.32)

We emphasize that the $q$-dependent one-loop determinants, the $L!$ factor
coming from the total number of solutions, and constants such as that in (4.2)
drop out between the numerator and the denominator. Thus, our matrix integral
computation confirms that, even non-perturbatively, our model (that involves
correlated spacetime branes) has resolved the factorization puzzle and yields
a discrete spectrum.

###  4.3 Additional comments

Several comments are in place:

  1. 1.

Factorization and discreteness are intimately linked.  It is interesting to
see what becomes of the theory with only the bilocal deformation, so with
$z_{\text{brane}}(b)=0$ and $Z_{\text{brane}}(b)=0$. According to the
discussion of section 3, this should be sufficient to obtain a factorizing
theory of gravity. For this we consider the matrix integral with only the
deformation (4.2). One sees that the localization argument still applies, but
now the saddle point equations feature the original potential $V(E)$

$\frac{\partial}{\partial\lambda_{i}}\sum_{i,j=1}^{L}I(\lambda_{i},E_{j})=0\quad\Leftrightarrow\quad\frac{1}{\lambda_{i}}+\sum_{j\neq
i}\frac{1}{\lambda_{j}-\lambda_{i}}=LV^{\prime}(\lambda_{i})\,.$ (4.33)

These are the equations of motion for some $(\upalpha,\upbeta)$ ensemble with
$\upalpha=1+2\upbeta$ with $\upbeta\to\infty$ [21, 84]. In the limit
$q\to\infty$ and in particular $q\gg L$ the usual techniques for solving
matrix models break down. In particular, one can no longer make the
approximation that the spectrum is continuous, and the loop equations collapse
in this regime to the above discrete set of equations.242424In the standard
large $L$ saddle-point, the equations of motion are imposed for all $\lambda$
in some to-be-determined spectral cut. One can interpret the discrete set here
as the matrix integral having $L$ cuts around the eigenvalues, with width
going to zero for $q\to\infty$. This behavior of nearly-factorized matrix
integrals was also observed using different techniques in [57]. The limit
$q\to\infty$ _freezes_ the eigenvalues to one classical solution of (4.33) (up
to permutations). Even though we fixed $Z_{\text{brane}}(b)=0$, the
eigenvalues are fixed non-perturbatively, away from the continuous leading
density of states $\rho_{0}$, to the solutions of the saddle-point equation
(4.33) (once again, well-approximated by $\rho_{0}$). A similar localization
in the context of the $\upbeta\to\infty$ limit of the $\upbeta$ ensembles
(with a Gaussian potential) was discussed in [85], where the matrix integral
localized on the zeros of the $L$-th Hermite polynomial.252525The matrix model
action in that case looks like $\upbeta\sum_{i\neq
j}\log(\lambda_{i}-\lambda_{j})-\upbeta L\sum_{i}\lambda_{i}$ and so at large
$\upbeta$ it localizes on the solutions of $\sum_{i\neq
j}(\lambda_{i}-\lambda_{j})^{-1}=2L\lambda_{j}$, which are the equations for
the zeros of the Hermite polynomial $H_{L}(\sqrt{L}x)$. The solutions for
large $\upbeta$ in our case $(\upalpha,\upbeta)=(1+2\upbeta,\upbeta)$ would
then again be the roots of the relevant orthogonal polynomial. In summary,
turning on only the bilocal deformation (which is sufficient to obtain
factorization), already discretizes the spectrum non-perturbatively. The
parameter $q$ did not make any immediate appearance in section 3, but here we
see that treating the limit $q\to\infty$ non-perturbatively, gives an intimate
relation between factorization and discreteness.

  2. 2.

Large deformations can lead to small changes. Consider $\textsf{H}_{0}$ to be
a typical matrix in the JT matrix integral ensemble. While the deformations
performed in the matrix integral are very large (since we were forced to take
the $q\to\infty$ limit), typical observables in the matrix integral are only
affected at subleading order in $e^{-\textsf{S}_{0}}$. This is so because the
leading order density of states $\rho_{0}(E)$ remains unaffected after turning
on the deformation. This is equivalent to our observation in section 3.4 that
the half-wormhole correction is subleading to the black hole solution.

  3. 3.

Notice that (4.8) is consistent with the fact that observables in the gravity
theory dual to the fixed quantum mechanics, reproduce the observables in the
original dilaton gravity upon ensemble averaging again over the spectrum
$E_{i}$ of the fixed Hamiltonian $\textsf{H}_{0}$

$\displaystyle\frac{1}{\mathcal{Z}}\prod_{i=1}^{L}\int_{\mathcal{{{\mathcal{C}}}}}dE_{i}$
$\displaystyle\exp\bigg(-L\sum_{i=1}^{L}V(E_{i})\bigg{missing})\prod_{i<j}^{L}(E_{i}-E_{j})^{2}$
$\displaystyle\frac{L!}{\mathcal{Z}(\textsf{H}_{0})}\prod_{i=1}^{L}\int_{{\mathcal{C}}}d\lambda_{i}\,\prod_{j=1}^{L}\delta(\lambda_{j}-E_{j})\,{{\mathcal{O}}}(H)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle}{{{\mathcal{O}}}(\textsf{H}_{0})}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rangle}=\left\langle{{\mathcal{O}}}(H)\right\rangle\,.$
(4.34)

This is trivial and boring here, but has interesting gravity consequences,
namely that, as mentioned before in section 3.4, more complicated geometries
like wormholes are effectively encoded in the statistical properties of the
brane one-point functions, the coupling constants in the spacetime action.
This is the whole reason why the original dilaton gravity theory which
includes wormhole serves an extremely good approximation to the gravity dual
of one quantum system drawn from the ensemble.

##  5 Including bulk matter

So far, we have focused on effective field theories that did not include any
matter fields in the $2$d bulk. The inclusion of such fields introduces two
complications. First, we don’t know an exact matrix model that is dual to JT
gravity plus matter fields. Second, there are divergences due to Casimir
energies associated to spacetime wormholes with narrow necks. In this section
we will address the latter problem, and show that even with matter one can
obtain a discrete and factorizing theory.

###  5.1 Factorization

In our model with additional (correlated) boundaries, the inclusion of matter
requires us to put certain boundary conditions for the matter fields on those
additional boundaries (on the asymptotic boundaries we use the usual Dirichlet
boundary condition). At this point we will be agnostic about what boundary
conditions we impose at the branes, and label some basis of matter boundary
conditions by $\mathcal{B}$.

From the discussion in section 3 it is then clear that we can essentially run
the argument again with matter fields present, one finds that the only non-
zero brane correlator is (because, by definition, the set of all boundary
conditions $\mathcal{B}$ form an orthonormal basis)

$\text{Factorization}\quad\Leftrightarrow\quad Z_{\text{brane
matter}}((b_{1},\mathcal{B}_{1}),(b_{2},\mathcal{B}_{2}))=-\frac{1}{b_{1}}\delta(b_{1}-b_{2})\,\delta(\mathcal{B}_{1}-\mathcal{B}_{2})\,.$
(5.1)

Following the same arguments that resulted in (3.15), the total partition
function reduces to

$Z(\beta)=Z_{\text{disk}}(\beta)+\sum_{\mathcal{B}}\int_{0}^{\infty}\mathrm{d}b\,b\,Z_{\text{trumpet
matter}}(\beta,b,\mathcal{B})\,Z_{\text{brane matter}}(b,\mathcal{B})\,.$
(5.2)

Here the trumpet partition function is that of matter-coupled JT gravity.
Knowing this trumpet partition function, and the eigenvalues $E_{i}$ of the
Hamiltonian $\textsf{H}_{0}$ of the putative exact quantum mechanical dual,
one could in principle determine $Z_{\text{brane matter}}(b,\mathcal{B})$
explicitly, which will again depend on $E_{i}$. Ensemble averaging over
$\textsf{H}_{0}$ with the appropriate measure (which as mentioned above, for
matter coupled JT gravity, is not currently known) gives back the genus
expansion of wormholes plus matter.

This immediately raises the question what happens in our model to the
$e^{a/b}$ (with a constant $a>0$) divergence from the Casimir energy of the
matter fields, which features in the trumpet partition function
$Z_{\text{trumpet matter}}(\beta,b,\mathcal{B})$. In a full UV complete model
of JT gravity with matter, these divergence should not be there, some
(unknown) matrix model description should presumably resolve it. From our
perspective, we are imposing that (5.2) is finite (with some given discrete
spectrum). This means that this would-be divergence will by construction be
cancelled by a factor $e^{-a/b}$ in $Z_{\text{brane matter}}(b,\mathcal{B})$.
This is independent of the precise set of typical eigenvalues that one chooses
for $\textsf{H}_{0}$ and so upon averaging this divergence will remain absent,
just as one would expect from a UV complete description of JT plus matter.
Additionally, we expect that when averaging over $\textsf{H}_{0}$ the
contribution of geometries which have all moduli sizes sufficiently large is
however well approximated by simply coupling JT gravity to matter fields. It
would be interesting to make this precise, should a dual of JT plus matter
ever become available.

###  5.2 Probe matter

To be more concrete, let us consider probe matter, so we ignore any
backreaction on the matter fields but we entirely include the backreaction of
such probes on the metric. Consider a free bulk matter field. In the boundary
this corresponds to some operator $\mathcal{O}$ with dimension $\Delta$. Since
this is probe matter we can use the techniques developed in [52, 16, 86, 87,
88, 89, 90, 91] and we can use the fact that the free propagator on a
hyperbolic disk is $e^{-\Delta\ell}$ with $\ell$ the (regularized) geodesic