Semi-classical dilaton gravity and the very blunt defect expansion

Jorrit Kruthoff$\,{}^{1}$ and Adam Levine$\,{}^{2}$

1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540

2Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

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

We explore dilaton gravity with general dilaton potentials in the semi-
classical limit viewed both as a gas of blunt defects and also as a semi-
classical theory in its own right. We compare the exact defect gas picture
with that obtained by naively canonically quantizing the theory in geodesic
gauge. We find a subtlety in the canonical approach due to a non-perturbative
ambiguity in geodesic gauge. Unlike in JT gravity, this ambiguity arises
already at the disk level. This leads to a distinct mechanism from that in JT
gravity by which the semi-classical approximation breaks down at low
temperatures. Along the way, we propose that new, previously un-studied
saddles contribute to the density of states of dilaton gravity. This in
particular leads to a re-interpretation of the disk-level density of states in
JT gravity in terms of two saddles with fixed energy boundary conditions: the
disk, which caps off on the outer horizon, and another, sub-leading complex
saddle which caps off on the inner horizon. When the theory is studied using a
defect expansion, we show how the smooth classical geometries of dilaton
gravity arise from a dense gas of very blunt defects in the $G_{N}\to 0$
limit. The classical saddle points arise from a balance between the attractive
force on the defects toward negative dilaton and a statistical pressure from
the entropy of the configuration. We end with speculations on the nature of
the space-like singularity present inside black holes described by certain
dilaton potentials.

###### Contents

  1. 1 Introduction
    1. 1.1 Analyzing the exact answer
    2. 1.2 Canonical quantization and classical saddles
    3. 1.3 Dense gas of very blunt defects
  2. 2 Review of Dilaton Gravity and its Matrix Model Dual
    1. 2.1 Lorentzian signature
    2. 2.2 Matrix model dual
    3. 2.3 Re-summing the string equation
  3. 3 New saddles and the exact density of states
    1. 3.1 Saddle point analysis of the exact density of states
    2. 3.2 Ground state energy
    3. 3.3 Alternate dilaton contour and the semi-classical density of states
    4. 3.4 Bulk interpretation of the saddles
    5. 3.5 A (near) phase transition
    6. 3.6 The critical coupling
  4. 4 Canonical quantization in geodesic gauge
    1. 4.1 Length potential at large positive and negative length
    2. 4.2 Density of states and the WKB approximation
    3. 4.3 A subtlety with geodesic gauge in $W_{\lambda}(\Phi)$ gravity
  5. 5 Corrections to the length basis inner product
    1. 5.1 Trumpet geometries
    2. 5.2 Disk geometries ending on a closed geodesic and sewing the pacifier
    3. 5.3 The pacifier contribution to $\braket{\ell}{\ell^{\prime}}$
    4. 5.4 Pacifiers and the non-perturbative shift in the ground state energy
  6. 6 Classical dilaton gravity from a gas of very blunt defects
    1. 6.1 Classical solutions for the defect density at fixed boundary length and fixed defect number
    2. 6.2 Behavior of solutions at large $\kappa$
    3. 6.3 Finding the saddle point in $\kappa$
    4. 6.4 The Weil-Peterson volume for a large number of very blunt defects and a single closed geodesic
  7. 7 Discussion & Speculations
    1. 7.1 Firewalls from pacifiers?
    2. 7.2 Defect expansion in Lorentzian signature
    3. 7.3 A simpler theory close to the singularity
    4. 7.4 Future directions
  8. A Boundary particle fluctuations
    1. A.1 Fixed defect number fluctuations
  9. B Geodesics on the trumpet

##  1 Introduction

Simplified toy models of quantum gravity in two spacetime dimensions have
taught us much about the low-energy and long time structure of black holes [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11].111See [7] for a review of the progress.
Despite this progress, we still lack a complete microscopic understanding of
what happens to the notion of spacetime in the strong gravity regime deep
inside the black hole interior; in the context of AdS/CFT, a sharp signature
of the near-singularity region of the black hole interior in the boundary
theory remains obscure.

To move toward a better understanding of this deep interior regime from a
microscopic point of view, we will in this work turn again to simplified two-
dimensional models of quantum gravity. While Jackiw-Teitelboim (JT) gravity
[12, 13] has taught us much about the near-extremal limit of higher
dimensional black holes, in pure JT the curvature is constant everywhere in
the spacetime. We would like a better understanding of asymptotically AdS
black holes where the spacetime curvature in the interior becomes large and
where there are no subtleties due to the presence of Cauchy horizons, as
happens in classical solutions to JT gravity.

A simple way to modify the interior geometry of an asymptotically AdS2
spacetime away from rigid AdS2 is to consider dilaton gravity with a deformed
dilaton potential. Consider the spacetime action with bulk term222The more
physical way of thinking about $G_{N}$ is as being related to the renormalized
value of the dilaton at the boundary, $\Phi_{b}=\Phi_{r}/\varepsilon$ [14].
The classical $G_{N}\to 0$ limit corresponds to large renormalized values.
Viewed as a dimensional reduction, this means the transverse sphere has large
renormalized values.

$\displaystyle
S_{bulk}=S_{0}\chi(M)+\frac{1}{2G_{N}}\int_{M}d^{2}x\sqrt{g}\left(\Phi
R+U(\Phi)\right).$ (1.1)

The first term controls topological suppression and is proportional to the
Euler character of the spacetime. In this work, we will ignore higher topology
effects, effectively setting $S_{0}=\infty$. The dilaton equation of motion
imposes $R=-U^{\prime}(\Phi)$. In JT gravity, $U(\Phi)=2\Phi$ and the dilaton
becomes a Lagrange multiplier, enforcing $R=-2$ exactly. More generally, we
can deform $U(\Phi)$ at negative values of $\Phi$ while preserving the linear
behavior at large positive $\Phi$. This ensures that we still have an
asymptotically AdS boundary.

Certain classes of such theories were solved exactly at the disk topology
level and shown to be dual to matrix models in [15, 16, 17, 18]. In [15, 16],
these theories were solved when the deformation to $U$ was a general linear
combination of exponentials of the form

$\displaystyle
U(\Phi)-2\Phi=\sum_{\alpha_{i}}\lambda_{i}e^{-2\pi\frac{(1-\alpha_{i})}{G_{N}}\Phi},\
\ \pi<2\pi(1-\alpha_{i})<2\pi.$ (1.2)

The authors in [15, 16] proceeded by expanding the gravitational path integral
in the couplings $\lambda_{i}$ and viewing the exponential operators as
inserting defects in the spacetime. For $\alpha$ in the range above, these
defects have a conical deficit between $\pi$ and $2\pi$. We will refer to such
defects as _sharp_. In the work of [17, 18], this restriction on $\alpha$ was
extended to the range $0<2\pi(1-\alpha)<\pi$. Expanding in a gas of
exponential deformations with $\alpha$ in this range creates defects in the
spacetime that we will refer to as _blunt_. Various new subtleties arise in
this case due to contact term ambiguities in the definition of the exponential
of the dilaton operator. Such contact terms are related to the ability of two
blunt defects to merge without violating the upper bound in eq. (1.2). When
dilaton gravity is viewed as a limit of the (deformed) $(2,p)$ minimal string,
these contact terms are related to operator mixing of the vertex operators, as
analyzed in [19].

In this work, we would like to focus on theories dominated by a smooth
classical geometry, which arise in the classical $G_{N}\to 0$ limit. The
sharp-eyed reader may note the presence of a $G_{N}$ in the exponential in eq.
(1.2), which would seemingly not produce good classical geometries. This
$G_{N}$ is due to the overall normalization of the action we have picked in
eq. (1.1). A simple re-scaling of $\Phi\to\Phi G_{N}$ brings us back to the
more standard normalization used in [16, 15, 17, 18]. This is related to the
fact that sharp defects create large disturbances in the geometry and so do
not reproduce smooth saddle point geometries. Instead, to find smooth
geometries, one can work in the normalization of eq. (1.1) but focus on the
“very blunt” limit, where $G_{N}\to 0$ with $(1-\alpha)/G_{N}$ fixed. In this
limit, it is more convenient to work with a re-scaled
$(1-\alpha)\to(1-\alpha)G_{N}$. In terms of this re-scaled and shifted
$\alpha$, the action takes the form333The sum over $i$ in the action may be
replaced by an integral over $\alpha$ as long as the upper limit on $1-\alpha$
is less than $1/G_{N}$ and the results of [17, 18] will still hold.

$\displaystyle S_{bulk}=\frac{1}{2G_{N}}\int d^{2}x\sqrt{g}\left(\Phi
R+2\Phi+\sum_{i}\lambda_{i}e^{-2\pi(1-\alpha_{i})\Phi}\right).$ (1.3)

We can now simply apply the saddle-point approximation to this action in the
classical limit. In the interest of concreteness, we will often restrict our
attention to a single exponential deformation of strength $\alpha$, but we
will point out where our discussion is more general. In particular, for much
of this paper we will work with the specific action

$\displaystyle S_{bulk}=\frac{1}{2G_{N}}\int d^{2}x\sqrt{g}\left(\Phi
R+2\Phi+\lambda 2\pi(1-\alpha)e^{-2\pi(1-\alpha)\Phi}\right).$ (1.4)

As we will detail in Sec. 2, the classical solutions to eq. (1.4) for
$\lambda>0$ (but not too large as discussed in Sec. 2) look like black holes
with either three horizons or one horizon. When there are three horizons, the
Penrose diagram is an infinite grid of universes.444See [20] for a realization
of such diagrams in the context of higher dimensional dS. As far as the
authors know, such diagrams have not been realized by higher dimensional AdS
constructions. We will not let this fact bother us due to the existence of a
well-defined dual matrix model for our potentials. When there is one horizon,
the diagram looks like that of AdS-Schwarzschild in higher dimensions. As we
increase the energy, there is a critical energy, $E=E_{c}(\lambda,\alpha)$, at
which the diagram jumps from having three horizons to one. Part of the
motivation for this work was to find a signature of this transition in the
matrix model.

Given the set-up, we now turn to studying the theory given by eq. (1.4) (and
more generally (1.3)) in the $G_{N}\to 0$ limit. We proceed with three
complementary approaches which we now briefly outline.

###  1.1 Analyzing the exact answer

In the first part of the paper, we do a more detailed analysis of the exact
results of [17] in the very blunt limit than what has previously appeared in
the literature. As we will review in Sec. 3, the density of states for the
theory with action eq. (1.3) has a nice form derived in [17] which is

$\displaystyle\rho^{\rm exact}(E)=\frac{e^{S_{0}}}{2\pi G_{N}}\int_{E^{\rm
exact}_{0}}^{E}du\frac{1}{\sqrt{E-u}}\frac{d\mathcal{F}(u)}{du},$
$\displaystyle\mathcal{F}(E)=\int_{\mathcal{C}}\frac{d\Phi_{h}}{2\pi
i}\left(\Phi_{h}-\sqrt{W(\Phi_{h})-E}\right)e^{\frac{2\pi\Phi_{h}}{G_{N}}},$
(1.5)

where the contour $\mathcal{C}$ is an inverse Laplace transform contour and so
lies to the right of all non-analyticities. Here the function $W(\Phi_{h})$ is
the pre-potential of the theory in question so that

$\displaystyle W^{\prime}(\Phi)=U(\Phi).$ (1.6)

The ground state energy $E^{\rm exact}_{0}$ is determined by finding the
largest root of the equation

$\displaystyle\mathcal{F}(E^{\rm exact}_{0})=0.$ (1.7)

The equation $\mathcal{F}(u(x))=x$ is often referred to as the string equation
in the context of matrix models. Knowledge of this function completely
determines the disk-level density of states, and allows one to solve the
matrix model at all orders in the genus expansion.

In these equations the variable $\Phi_{h}$ is integrated over and so does not
have any a priori meaning. We will propose that it should be interpreted as
the value of the dilaton at the horizon. To get a more concrete understanding
of the density of states in the $G_{N}\to 0$ limit, we can perform the
$\Phi_{h}$ integral above by saddle point. We find a saddle point for every
root of the equation $W(\Phi_{h})=E$. Assuming that $\Phi_{h}$ is the dilaton
at the horizon, we will interpret these saddles geometrically by finding new
solutions to the gravity path integral with fixed energy boundary conditions.
Usually, the density of states is found by working at fixed boundary
length/inverse temperature, $\beta$, and then inverse Laplace transforming to
fixed energy. We will take a more direct approach where we work with fixed
energy boundary conditions. This amounts to fixing the condition on the
dilaton at the boundary

$\displaystyle\Phi_{b}K-\partial_{n}\Phi_{b}=E/2.$ (1.8)

Such boundary conditions were discussed in [21] in the context of JT gravity.

Figure 1: We illustrate what the other saddle point is that contributes to the
density of states in JT gravity. This amounts to a complex geometry that is
the AdS2 disk. As we will describe in Sec. 3, the radial coordinate, which can
be taken to be the dilaton, follows a complex contour that avoids the pole in
the metric at the outer horizon and ends on the negative root of the equation
$\Phi^{2}=E$. The metric in the range $-\sqrt{E}<\Phi<\sqrt{E}$ looks like
that of an anti-sphere, whose metric is Euclidean up to an overall minus sign.
This solution has constant curvature $R=-2$.

Using the boundary condition in eq. (1.8) allows us to reinterpret the disk
level density of states in JT gravity in terms of a contribution from two
saddles, one which gives an $e^{2\pi\sqrt{E}/G_{N}}$ contribution - the
standard disk saddle - and another which gives
$-e^{-2\pi\sqrt{E}/G_{N}}$.555As of now, we do not have a good bulk
interpretation of the minus sign for this saddle, except by doing the inverse
Laplace transform exactly. Arriving at the minus sign via saddle point methods
requires a more careful steepest descent analysis which we discuss in Sec. 3.
This second saddle is complex and can be described by the disk geometry but
with a complex contour for the radial coordinate in Schwarzschild gauge. This
contour goes into the complex plane and avoids the pole in the metric at the
origin of the disk, ending instead on the pole of the metric associated with
the inner horizon. Morally, this solution looks like a disk with an anti-
sphere connected onto the horizon. We illustrate this saddle point in Fig. 1.
This saddle can be associated with the presence of an inner horizon in the
classical Lorentzian spacetime.666As we will point out, this saddle violates
the Kontsevich-Segal (KS) criterion of [22, 23], although several necessary
violations of the KS criterion have now been discovered [24].

###  1.2 Canonical quantization and classical saddles

Next, we compare these results to those found by studying the theory defined
by eq. (1.3) semi-classically. In particular, we will avoid resorting to an
expansion in a gas of very blunt defects. We show how to re-derive some of the
results in [17]. In particular, we will show how to canonically quantize
general dilaton gravities in geodesic gauge, analogous to what was done in JT
[14, 5]. Just as in JT gravity, we find that the canonically quantized theory
is equivalent to a non-relativistic particle propagating in a potential. The
position of this particle corresponds to the renormalized length of the
Einstein-Rosen bridge connecting the two sides of the black hole.

Comparing with the exact density of states in eq. (1.1), we will find that
canonical quantization leads to a (naive) discrepancy with the exact result in
eq. (1.1). This discrepancy is most sharply seen by comparing the predicted
ground state energies for the matrix model from canonical quantization and
from eq. (1.1). We will see that the two answers differ by an amount which is
non-perturbatively small in $1/G_{N}$.

Figure 2: We give an example of what the pacifier spacetimes look like
qualitatively, illustrating that there can be multiple geodesics connecting
the same pair of boundary points. There exists a closed geodesic of length $b$
and a horizon at $\Phi=\Phi_{h}$.

We will resolve this discrepancy by pointing out that geodesic gauge is
generically ill-defined due to non-perturbative effects in $1/G_{N}$. These
effects are due to the ability of the spacetime to “nucleate” a closed
geodesic, which leads to multiple geodesics connecting the same pair of
boundary points. One should contrast this with JT gravity where geodesic gauge
breaks down only non-perturbatively in $S_{0}$ [25]. In JT gravity with a
single asymptotic boundary, a closed geodesic can appear only on higher genus
geometries, since there are no constant negative curvature Euclidean
geometries with a closed geodesic and the topology of the disk. The difference
in our work is that for theories with more general dilaton potentials, the
geometry can contain a closed geodesic _at the disk level_ since we no longer
have the constant negative curvature constraint. We will call such spacetimes
which have disk topology together with a closed geodesic _pacifier spacetimes_
, illustrated in Fig. 2.777Pacifier spacetimes share a qualitative resemblance
to the centaur geometries of [26], which connect an approximately constant
$R=+2$ sphere to an $R=-2$ exterior along a closed geodesic. We distinguish
the pacifier from the centaur because in general the pacifier may not have
everywhere positive curvature in the spacetime region bounded by the closed
geodesic.  As we will argue, the contribution of pacifier spacetimes to the
density of states is suppressed non-perturbatively in $1/G_{N}$ (but not
$S_{0}$) relative to the leading geometry but becomes enhanced at very low
energies. We will argue that this enhancement corresponds to a non-
perturbatively small shift in the ground state energy of the matrix model. We
find that the leading non-perturbative correction to $E_{0}$ due to the
inclusion of pacifier spacetimes agrees with that predicted by eq. (1.1), up
to one-loop factors since we have not done a thorough one-loop analysis of the
pacifier geometries.

The story we describe here is reminiscent of the picture found in [15] where
instantonic corrections to the 3d gravity path integral for near extremal
black holes lead to a non-perturbatively small shift in the ground state
energy. The instantons considered in [15] create defects in the spacetime, as
viewed from the dimensional reduction to 2d gravity. In Sec. 6, we will argue
that the pacifiers arise as a dense gas of these defects in the very blunt
limit. We describe this very blunt defect limit shortly, but we hope that the
methods of this portion of the paper may shed light on how to treat dilaton
gravities more generally when a simple defect gas picture is not present.

We also comment that already at the level of Lorentzian geometries, the theory
specified by the action in eq. (1.4) with $\lambda>0$ displays a breakdown of
geodesic gauge. We point out in Sec. 4 that the same theory defined with
$\lambda<0$ does not display such a breakdown in Lorentzian signature and yet
the ground state energy of the model as predicted by the exact answer in eq.
(1.1) disagrees with canonical quantization. This discrepancy can instead be
interpreted as a modification of the inner product used in the theory.

###  1.3 Dense gas of very blunt defects

Figure 3: This figure illustrates how a gas of very blunt defects can
approximate a smooth spacetime. On the left, we start with a few sharp
defects, with the arrows indictating that in the classical limit the defect
gas becomes simultaneously denser and blunter.

Finally, we return to the defect gas picture and re-analyze the problem in the
very blunt, classical limit. We find that in this limit the defect gas
expansion simplifies and is dominated by a very large number (scaling like
$1/G_{N}$) of very blunt defects. We show how such a large number of defects
builds up a smooth geometry by introducing a new, continuous degree of freedom
on the spacetime: the defect density. Given a spacetime with a large number,
$L$, of defects of strength $\alpha$, then the spacetime curvature can be
written as

$\displaystyle R=-2+2\pi(1-\alpha)LG_{N}\rho(x),\ \
\rho(x)\equiv\frac{1}{L}\sum_{i=1}^{L}\frac{\delta^{2}(x-x_{i})}{\sqrt{g}},$
(1.9)

where the $x_{i}$ are the positions of each defect. Scaling $L$ with
$1/G_{N}$, we arrive at an action for a continuous density of defects,
$\rho(x)$. As we will show, this action has saddles for the density which
approximate the semi-classical geometry. These saddles arise due to a balance
between the attraction of defects toward more negative values of the dilaton
and a statistical pressure from the entropy of the defect configuration. This
is how the very blunt limit reproduces the classical geometry. We illustrate
this effect in Fig. 3.

We attempt to compute the Weil-Peterson volumes of geometries with a single
geodesic boundary of length $b$ and many very blunt defects. We also show that
pacifier spacetimes arise due to geometries with a number of defects so large
that their “net” opening angle is negative. These geometries have a number of
defects $L$ with $2\pi(1-\alpha)G_{N}L>2\pi$ in terms of the re-scaled
$1-\alpha$. This matches the results of [15, 18, 16], where it was shown that
geometries with $LG_{N}>\frac{1}{1-\alpha}$ necessarily have a closed geodesic
separating the defects from the asymptotic boundary. Our pacifier spacetimes
are just the smooth, very blunt approximation to such geometries.

We now briefly summarize the layout of the paper. In Sec. 2, we review the
classical solutions to dilaton gravity, going to a gauge where the dilaton is
used as a radial coordinate on the geometry. We also review the exact matrix
model results. In Sec. 3, we analyze the exact density of states given by [17,
18] in the very blunt limit. We interpret the result in terms of new complex
saddles. In Sec. 4, we perform the naive canonical quantization of dilaton
gravity in geodesic gauge and find a discrepancy with the exact answer. In
Sec. 5, we resolve this ambiguity by identifying a non-perturbative breakdown
in geodesic gauge due to the appearance of pacifier spacetimes. We further
show that the inclusion of pacifier spacetimes corresponds to a non-
perturbatively small shift in the ground state energy of the model away from
that predicted by canonical quantization. In Sec. 6, we reformulate the very
blunt defect expansion in terms of a density of defects. We solve the defect
density theory classically and find agreement with the classical limit of
dilaton gravity. Finally, in Sec. 7, we end with some speculations about what
all of this might imply about the nature of the singularity inside the black
hole interior.

##  2 Review of Dilaton Gravity and its Matrix Model Dual

We now review the basics of classical dilaton gravity. The on-shell Euclidean
solutions to dilaton gravity with rotational symmetry are most easily written
down in the gauge in which $\Phi$ is the radial coordinate [27]. For a general
dilaton potential $U(\Phi)$ as in eq. (1.1), we introduce the dilaton
(pre)-potential $W$ defined by

$U(\Phi)=W^{\prime}(\Phi).$ (2.1)

In terms of $W$, one obtains the solution in Euclidean signature

$\mathrm{d}s^{2}=G(\Phi)\mathrm{d}\tau^{2}+\frac{\mathrm{d}\Phi^{2}}{G(\Phi)}$
(2.2)

with

$G(\Phi)=W(\Phi)-W(\Phi_{h}).$ (2.3)

Here $\Phi_{h}$ is the value of the dilaton at the horizon, where $G$
vanishes. For a potential $U$ that becomes linear at large, positive $\Phi$,
this geometry approaches AdS2 and does not differ much from the usual JT disk
solution. If we want to fix the temperature of the solution, we need to solve
the equation

$\beta=\frac{4\pi}{U(\Phi_{h})}.$ (2.4)

As discussed in the introduction, we will focus on the potential as given in
eq. (1.4) with pre-potential

$\displaystyle W(\Phi)=W_{\lambda}(\Phi)\equiv\Phi^{2}-\lambda
e^{-2\pi(1-\alpha)\Phi}.$ (2.5)

The exponential behaviour of the potential at large negative $\Phi$ leads to
an infinite number of (complex) solutions to eq. (2.4) for a given $\beta$.
There is also a real solution that is a smooth deformation of the usual JT
disk. This solution corresponds to when $\Phi_{h}$ has the largest real part
of all possible solutions to eq. (2.4).

The ADM energy of these solutions is given in terms of $\Phi_{h}$ by

$E_{\rm AdM}=\lim_{\Phi\to\infty}\left(\Phi^{2}-G(\Phi)\right)=W(\Phi_{h}).$
(2.6)

When $\Phi_{h}$ is real, then so is $E_{\rm AdM}$. When $\lambda$ is positive
and not too large, there is a solution with zero temperature or
$\beta\to\infty$. According to eq. (2.4), this solution occurs when
$U(\Phi_{h})=0$. Throughout this work we will refer to this solution as
occurring at the “the outermost mininum of the potential,” denoted by
$\Phi_{0}$. Furthermore, we will denote the AdM energy of this solution by
$E_{0}=W(\Phi_{0})$. In Sec. 4, we will rederive this ground state energy from
canonical quantization.

Figure 4: This figure illustrates the pre-potential $W_{\lambda}(\Phi)$ for
$\alpha=0$ and $\lambda=.001$. We label the energy of the zero temperature
classical solution, which is set by the value of $W_{\lambda}(\Phi)$ at its
outermost minimum, $E_{0}$. We also label the critical energy $E_{c}$.
Solutions with $E>E_{c}$ have one horizon while solutions with $E_{c}>E>E_{0}$
have three.

Note that the equation $U_{\lambda}(\Phi_{h})=0$ does not always have a real
solution for all $\lambda$ and $\alpha$. As we turn up $\lambda$, there is a
critical $\lambda=\lambda_{c}$ above which there no longer exists a real,
local minimum for $W_{\lambda}$. A brief computation shows that
$\lambda_{c}=(2\pi^{2}e(1-\alpha)^{2})^{-1}$ and
$\Phi^{c}_{h}=-(2\pi(1-\alpha))^{-1}$. The ground state energy of the model
with this critical $\lambda$ is
$E_{0}=-(4\pi^{2}(1-\alpha)^{2})^{-1}=-\Phi_{h}^{2}$. As we will see in the
next section, in the classical limit this critical $\lambda$ corresponds to
the coupling (up to exponentially small corrections) where the ground state
energy of the dual matrix model becomes complex.

In summary, for $\lambda<\lambda_{c}$ the smallest energy that is continuously
connected to solutions of infinite energy is obtained by solving for
$\Phi_{h}$ in the equation $\partial_{\Phi_{h}}E_{\rm ADM}=0$ and gives a non-
trivial function of $\lambda$. As we will see later on, this is actually a
crude estimate of the critical $\lambda$ and $E_{0}(\lambda)$ and the exact
version of the ground state energy $E_{0}$ as a function of $\lambda$ is
obtained in a more intricate manner, following the matrix model techniques
developed in [16, 15, 17]. Essentially we are asking here too much of the
semiclassical theory in order to give the ground state energy. As discussed in
the introduction, we will propose that other (complex) solutions play an
important role in determining the exact ground state energy of the model.

###  2.1 Lorentzian signature

The more exciting feature of these solutions is their Lorentzian continuation.
Asymptotically the geometry is still AdS2, but once we go inside the black
hole, the correction to the JT potential dominates. When
$\lambda<\lambda_{c}$, $W_{\lambda}(\Phi)$ is non-monotonic and one can have
three types of solutions. As we vary the AdM energy of the solution, we vary
between these three possible types of solutions, which have differing numbers
of horizons. There are two critical energies $E_{c}(\lambda)$ and
$E_{0}(\lambda)$ at which the number of horizons changes.

Figure 5: This figure illustrates the Penrose diagram for solutions with
prepotential $W_{\lambda}(\Phi)$ with $\lambda_{c}>\lambda>0$. The diagram on
the left shows the solution at low energies $E$ where $W_{\lambda}(\Phi)=E$
has three real roots. The diagram at the right where there is just one real
real root. The jagged lines denote regions where both the curvature blows up
and $\Phi\to-\infty$. The solid lines denote asymptotically AdS boundaries and
the dashes lines are horizons. Note that we have drawn the singularities as
horizontal but they will have an energy dependent shape.

  1. 1.

One horizon: The energy $E$ is chosen to obey $E>E_{c}$ or $E<E_{0}$ such that
the solution to $W_{\lambda}(\Phi)=E$ has a single real zero in the complex
dilaton plane. The Lorentzian solution has a spacelike singularity and the
Penrose diagram looks qualitatively similar to that of AdS Schwarzschild in
higher dimensions. We illustrate the Penrose diagram for this case on the
right of Fig. 5.

  2. 2.

Two horizons: The energy is such that there is one single zero and one double
zero. The double zero is an example of a degenerate zero, much like an
extremal horizon. This occurs when $E=E_{c}(\lambda)$ or when $E=E_{0}$, with
the double zero occuring at the outer-horizon in the latter case and at the
inner-horizon in the former case.

  3. 3.

Three horizons: The solution has energy $E_{0}<E<E_{c}$ such that there are
three real solutions to $W_{\lambda}(\Phi)=E$. In this case the solution is
exotic, as its Penrose diagram extends to an infinite grid in
$\mathbf{R}^{2}$, i.e. there are an infinite number of universes. This
situation can only happen when $W_{\lambda}(\Phi)$ is non-monotonic. We
illustrate the Penrose diagram for this case on the left of Fig. 5. We will
refer to the three real solutions to $W_{\lambda}(\Phi)=E$ as the inner,
middle and outer solution, where the inner solution has the most negative
dilaton value and the outer has the most positive.

Clearly there exists a transition where one goes between three and one horizon
with a degenerate intermediate two-horizon solution. Again, this transition
can only occur when $W_{\lambda}(\Phi)$ is non-monotonic and so
$\lambda<\lambda_{c}$.

One of the most interesting features of these solutions is that they have
spacelike singularities. In the three horizon case for $E_{0}<E<E_{c}$, the
spacelike singularities are hidden by a Cauchy horizon and so will evade any
simple probes from the boundary. The single horizon solutions, however, are
much like the black holes we know in higher dimensions. In particular, the
Ricci scalar $R\to+\infty$ when $\Phi\to-\infty$. Note that the curvature
becoming large and positive is exactly what happens to the sectional curvature
$R_{rtrt}$ in higher-dimensional black holes. A diffeomorphism invariant way
of characterizing the singularity is to fire in light rays from both
boundaries and ask at which boundary time $t_{*}$ do the two light rays meet
at the singularity [28, 29]. This is given by the integral

$t_{\star}=\int\mathrm{d}t={\rm
PV}\int_{-\infty}^{\infty}\frac{d\Phi}{G(\Phi)},$ (2.7)

which is finite due to to the exponential decay of the integrand close to the
singularity. Furthermore, calculating this integral for a fixed temperature
solution, reveals that $t_{\star}$ can be positive and negative. For
$t_{\star}<0$, the singularity is bent downwards, while for $t_{\star}>0$ it
bends up [28]. This also implies that one can have so-called “bouncing” space-
like geodesics in these geometries, just as in the higher dimensional case
studied in [28]. Indeed, these bouncing geodesics will be important for
understanding subtleties in the canonical approach discussed below. See Sec.
4.3 for more discussion.

Interestingly, as we will show in the next section, the exact matrix model
dual found via an expansion in a gas of defects seems to leave out solutions
with $E<E_{0}$, at least to leading order in $1/G_{N}$. In other words,
classically the ground state energy of the matrix model is $E=E_{0}$, up to
non-perturbatively small corrections.

###  2.2 Matrix model dual

As is well-known, ordinary JT gravity has a matrix model dual [4]. In [15, 16,
17, 18] this was extended to non-trivial dilaton gravities consisting of the
exponential form discussed above (with certain restrictions that we will come
to in a bit) by doing an expansion in $\lambda$ and resumming that expansion.
The thing one gains by doing an expansion in $\lambda$ is that each insertion
of an exponential in the dilaton changes the dilaton equations of motion to

$\displaystyle\sqrt{g}(R+2)=4\pi(1-\alpha)G_{N}\delta^{(2)}(x-x_{0}).$ (2.8)

The solution represents (to leading order in the genus) a disk with a deficit
angle of amount $2\pi(1-\alpha)G_{N}$ at $x=x_{0}$. In [15, 16], only sharp
defects were considered with $\frac{1}{G_{N}}>(1-\alpha)>\frac{1}{2G_{N}}$. In
[18, 17], the answer was extended to all $\alpha$.

At order $\lambda^{k}$ in the $\lambda$ expansion, one has $k$ such defects
and one needs to integrate over their positions, dividing out by
diffeomorphism redundancy. This amounts to integrating over the moduli space
of hyperbolic surfaces with $k$ conical defects. For
$k<\frac{1}{G_{N}(1-\alpha)}=k_{c}$, this integral was done in [18]. As
mentioned in the introduction, the proper treatment of this moduli space for
$\alpha$ in the blunt range requires dealing with contact term ambiguities in
the definition of the exponential operator. Geometrically, these ambiguities
arise due to the possible merging of two defects into a single defect of twice
deficit angle.

Figure 6: This figure illustrates the transition in the bulk geometry when the
defect number, $L$, is above the bound
$L>L_{c}=\lfloor\frac{1}{G_{N}(1-\alpha)}\rfloor$. We illustrate this
phenomenon for the choice $\frac{1}{2G_{N}}>1-\alpha>\frac{1}{3G_{N}}$ so that
the transition number is $L_{c}=2$. This transition number corresponds to when
the net deficit angle of all the defects is more than $2\pi$. In the minimal
string picture, where the defects correspond to insertions of the tachyon
operator on the worldsheet, this phenomenon corresponds to the fact that there
are a finite number of such primaries in the minimal model [17].

For $k>k_{c}$, the net deficit angle of all the defects together is greater
than $2\pi$. In this case, one can always separate the region with defects
from the asymptotic boundary using a closed geodesic [16, 18]. We illustrate
this in Fig. 6 for the case $\frac{1}{2}>(1-\alpha)G_{N}>\frac{1}{3}$ so that
a closed geodesic appears when the defect number $L\geq 3$. For defect number
above this bound, one needs to compute the volumes of moduli space with a
closed geodesic boundary and $k$ defects with deficit $2\pi(1-\alpha)G_{N}$.
For $\alpha$ in the sharp range, this can be accomplished in a straightforward
way from moduli space volumes for surfaces with $k$ geodesic boundaries via
analytic continuation to complex geodesic boundary lengths [16, 15]. More
generally one can use the recursion relations outlined in [18].

The upshot is that at genus zero, these methods result in an exact expression
for the density of states,

$\rho_{g=0}(E)=\frac{e^{\textsf{S}_{0}}G_{N}}{2\pi}\int_{E^{\rm
exact}_{0}}^{E}\frac{du}{\sqrt{E-u}}\partial_{u}\mathcal{F}(u)$ (2.9)

with the ground state energy $E^{\rm exact}_{0}$ given by the largest real
zero of $\mathcal{F}$,

$\displaystyle\mathcal{F}(E_{0}^{\rm exact})=0.$ (2.10)

As mentioned in the introduction, in the orthogonal polynomial approach to the
matrix model, this function $\mathcal{F}$ defines the so-called _string
equation_ , $\mathcal{F}(u(x))=x$, which is used to systematically compute
higher-orders in the matrix integral. See [30] for a nice review of this
technology. The function $\mathcal{F}(u)$ depends on the dilaton potential
$W$; for the potential given by eq. (2.5), the function $\mathcal{F}$ is given
by the expression [18, 17]

$\mathcal{F}(u)=\sum_{L=0}^{\lfloor\frac{1}{(1-\alpha)G_{N}}\rfloor}\frac{\left(\lambda\pi(1-\alpha)\right)^{L}}{G_{N}L!}\left(\frac{2\pi(1-L(1-\alpha)G_{N})}{\sqrt{u}}\right)^{L-1}I_{L-1}\left(2\pi(1-L(1-\alpha)G_{N})\frac{\sqrt{u}}{G_{N}}\right),$
(2.11)

where $I_{L}$ is the modified Bessel function of the first kind. To get this
formula from [17], we have re-scaled all of our energy variables $E\to
E/G_{N}^{2}$ and $u\to u/G_{N}^{2}$ as well as the $(1-\alpha)$ variable. We
are also working with a different normalization for our coupling $\lambda$.

Note that the limit on $L$ is set by the floor of $\frac{1}{(1-\alpha)G_{N}}$.
From the minimal string perspective, where this formula was first derived, the
upper limit on the sum arises due to the fact that there are a finite number
of primaries in the minimal model CFT. From the geometric perspective,
however, the cutoff on the sum comes from the condition discussed above that
the net deficit angle of all of the defects needs to be less than $2\pi$ or
else the geometry nucleates a closed geodesic. It is interesting that the
presence of a closed geodesic in the geometry signals an overcompleteness from
the minimal string point of view; there are a finite number of tachyon vertex
operator primaries, and as we add more insertions eventually we stop
generating new states. In this work, we also find that the presence of a
closed geodesic signals linear dependencies between naively orthogonal states.

Note, however, that the truncation of $\mathcal{F}$ at order
$\lambda^{\lfloor\frac{1}{(1-\alpha)G_{N}}\rfloor}$ does not mean that all
physical observables, such as the density of states or partition function
truncates at a finite order in $\lambda$ as well. This is so because in these
observables $E^{\rm exact}_{0}$ also makes an appearance, and $E_{0}^{\rm
exact}$ does have an infinite expansion in $\lambda$, due to solving eq.
(2.10). In Sec. 5, we will argue that the corrections coming from $E^{\rm
exact}_{0}$ are actually very interesting geometrically. Furthermore, note
also that in the semiclassical limit, where $G_{N}\to 0$, the sum truncates at
a very large order and in fact the truncation errors are non-perturbative in
$1/G_{N}$. In equations we have

$\displaystyle\mathcal{F}^{\rm semi-classical}(u)\approx\mathcal{F}^{\rm
exact}(u)+\mathcal{O}\left(\lambda^{\frac{1}{G_{N}(1-\alpha)}}\right),$ (2.12)

where

$\displaystyle\mathcal{F}^{\rm semi-
classical}(u)\equiv\sum_{L=0}^{\infty}\frac{\left(\lambda\pi(1-\alpha)\right)^{L}}{G_{N}L!}\left(\frac{2\pi(1-L(1-\alpha)G_{N})}{\sqrt{u}}\right)^{L-1}I_{L-1}\left(2\pi(1-L(1-\alpha)G_{N})\frac{\sqrt{u}}{G_{N}}\right).$
(2.13)

Interestingly, the modified string equation defined by eq. (2.13) can arise
from using a different dilaton contour for the theory, as discussed in Sec.
3.3.

###  2.3 Re-summing the string equation

The authors in [17] showed that one can in fact nicely re-sum the exact
expression for $\mathcal{F}$ given in (2.11). They noted that the modified
Bessel functions in (2.11) have a simple form in terms of an inverse Laplace
transform over an auxiliary variable $y$. Re-writing (2.11) in this manner
allows one to simply re-sum the expansion in $\lambda$ to get

$\displaystyle\mathcal{F}^{\rm exact}(u)=\frac{1}{2\pi
iG_{N}^{2}}\int_{\mathcal{C}}dy\
\left(y-\sqrt{W(y)-u}\right)e^{2\pi\frac{y}{G_{N}}},$ (2.14)

where the contour $\mathcal{C}$ is an inverse Laplace transform contour and so
lies to the right of all non-analyticities in the integrand. The function
$W(y)$ is just the dilaton pre-potential.888Note that this formula has been
derived for general $W(y)$ such that $\delta W(y)\equiv W(y)-y^{2}$ has an
inverse Laplace transform, $\mathcal{L}^{-1}(W)(a)$, which has support only in
the range $a\in[0,2\pi/G_{N}]$. Plugging this formula for $\mathcal{F}$ into
the expression in eq. (2.9), we find

$\displaystyle\rho_{g=0}(E)$
$\displaystyle=\frac{e^{S_{0}}}{8\pi^{2}iG_{N}}\int_{E_{0}^{\rm
exact}}^{E}\frac{du}{\sqrt{E-u}}\int_{\mathcal{C}}dy\frac{e^{2\pi\frac{y}{G_{N}}}}{\sqrt{W(y)-u}}$
$\displaystyle=\frac{e^{S_{0}}}{2\pi G_{N}}\int_{\mathcal{C}}\frac{dy}{2\pi
i}e^{\frac{2\pi y}{G_{N}}}\tanh^{-1}\left(\sqrt{\frac{E-E_{0}^{\rm
exact}}{W(y)-E_{0}^{\rm exact}}}\right),$ (2.15)

where in the second line we did the $u$ integral. Again, although we are
focusing on a specific dilaton (pre-)potential, eq. (2.14) is expected to be
true for at least any dilaton potential whose deformation away from JT gravity
admits an expansion in defects.999We expect, however, that the expression in
(2.14) is quite general. It is tantalizing that the formulae in these
equations look strikingly similar to those in the discussion of the Wheeler de
Witt wavefunctions of dilaton gravity, as in [31, 32]. We suspect that using
these formulae together with the formalism of [33] could lead to a direct
derivation of the disk density of states. By a direct derivation, we mean one
that does not proceed by expanding in a gas of defects. Interpreting $y$ as
the dilaton value at the horizon, this formula can be read as imposing the
smoothness condition for the geometry at the horizon. Similar ideas for
computing the density of states in JT gravity were explored in [33]. There the
imposition of the smoothness condition was interpreted as the insertion of a
defect operator in the bulk. The insertion of $e^{2\pi y/G_{N}}$ can be
thought of as related to this defect operator. More generally, decomposing the
path integral into sums over fixed horizon area states was discussed in [34,
35, 36].

##  3 New saddles and the exact density of states

In the previous section we met an exact formula for the genus zero density of
states in our dilaton gravity theory. As discussed above, these functions
admit a semi-classical expansion in the $G_{N}\to 0$ limit, yielding the ideal
playground to see if we can reproduce them from the gravity theory. First we
will analyze the expression in (2.3) via a saddle point analysis. We will find
saddle points in the $y$ integral at each solution to $W(y)=E$. We will then
interpret these solutions in terms of novel dilaton gravity saddles with fixed
energy boundary conditions.

###  3.1 Saddle point analysis of the exact density of states

We start by analyzing the $y$ integral in the first line of eq. (2.3) at fixed
$u$ and then perform the $u$ integral. We want to compute the integral

$\displaystyle\mathcal{F}^{\prime}(u)=\frac{1}{2\pi
iG_{N}}\int_{\mathcal{C}}dy\frac{e^{2\pi\frac{y}{G_{N}}}}{\sqrt{W(y)-u}}.$
(3.1)

The $y$ integral has saddle points obeying the equation

$\displaystyle\frac{2\pi}{G_{N}}-\frac{U(y)}{2(W(y)-u)}=0\implies
u=W(y)+G_{N}\frac{U(y)}{4\pi}.$ (3.2)

Figure 7: This figure illustrates the analytic structure of the integrand in
eq. (3.1) in the complex $y$ plane. The contour of integration is pictured as
the vertical arrow on the right. Each branch cut due to the square root in
(3.1) is indicated by a black jagged line. The saddle points are indicated by
black dots and the branch points by small red dots. The contour lines are
lines of constant real part for the integral. The steepest descent contours
are then orthogonal to these lines and pass through each black dot. More
positive real part is in orange and more negative in blue.

In other words, near each solution to $u=W(y)$ there is a saddle point. For
our potential $W(y)=W_{\lambda}(y)$, we can even see easily which saddles
contribute for a given $u$. As discussed in Sec. 2, for $E_{c}>u>E_{0}$, there
are three real solutions to this equation and an infinite number of complex
solutions, due to the exponential behavior of the potential. For the integrand
in eq. (3.1), there are branch cuts emanating from each solution, with the two
largest real solutions connected by a branch cut between each other. We
illustrate this analytic structure in Fig. 7.

We can deform the contour to the steepest descent contours emanating from each
saddle point.101010Note that the factor of $i$ in the denominator of eq. (3.1)
is cancelled due to the fact that the steepest descent contours run parallel
to the imaginary axis near the saddle points. We see that each saddle
contributes, although the middle horizon (what would be the inner horizon in
JT gravity) must be treated differently since the saddle point sits on a
branch cut. Actually this saddle point contributes an amount which is
ambiguous because we are evaluating $\mathcal{F}^{\prime}$ on a Stokes line.
We can break this ambiguity by continuing $G_{N}$ to be slightly complex. This
saddle point then contributes something which is purely imaginary.

One might be concerned that the middle horizon then contributes a complex
number to the density of states. This is fixed due to the fact that to get the
density of states we have to integrate $\mathcal{F}^{\prime}(u)$ over $u$.
This integral _also_ localizes to a saddle in the $G_{N}\to 0$ limit near the
endpoint of the integral at $u=E$. For the term corresponding to the middle
horizon, this integral is localized at a saddle point at $u_{*}=E-cG_{N}$ with
$c>0$, but this turns out to be an unstable saddle point. Deforming the $u$
contour so that it picks up this saddle along the steepest descent contour
leads to another factor of $\pm i$ from rotating the contour. Again, we lie on
a Stokes line and so we need to continue $G_{N}$ slightly off the real axis to
determine the sign in front of this second factor of $i$. One can check that
the sign in front of these two $i$’s is directly correlated with the sign of
the small imaginary part given to $G_{N}$ in order to move off the Stoke’s
line. The upshot is that together these two factors of $i$ give an overall
minus sign. In the context of JT gravity, this minus sign is what gives the
$-e^{-2\pi\sqrt{E}/G_{N}}$ in the expression for the density of states,
$\rho_{JT}(E)\sim\sinh(2\pi\sqrt{E}/G_{N})$.

Instead of going into more detail about the steepest descent analysis we have
just mentioned, we will instead turn to a slightly simpler computable than the
density of states which illustrates the key points of interest for us.

###  3.2 Ground state energy

Since computing the full density of states is rather complicated, we will
instead focus for the rest of the paper on a simpler aspect of these formulae:
the ground state energy of the matrix model. As described above, the ground
state energy is fixed by the demand that the density of states has a square-
root edge characteristic of a matrix model. This statement leads to the
equation

$\displaystyle\mathcal{F}(E_{0}^{\rm exact})=0,$ (3.3)

where again the density of states is given in terms of $\mathcal{F}$ by eq.
(2.9).

Using a similar saddle point analysis to that described above, we can estimate
$E_{0}^{\rm exact}$ in the $G_{N}\to 0$ limit, using eq. (2.14). Before
picking up saddle points, however, we can first deform the contour so that it
wraps all the branch-cuts emanating from each branch point. Again, each
solution $y$ to $W(y)=u$ is a branch point in the complex $y$-plane.
Furthermore, assuming there exists an outermost minimum to $W(y)$, then at
small enough $u$ there are two branch points with the largest real part, as
illustrated in Fig. 7. These set the small $G_{N}$ behavior of $\mathcal{F}$
for generic $u$. Thus, when the contribution due to the branch cut connecting
these two points vanishes, then $\mathcal{F}$ vanishes up to corrections due
to all the other solutions with more negative real part. One can easily see
that the contribution from these two outermost solutions vanishes when the two
solutions merge. This is because each branch point corresponds to a simple
zero of $W(y)-u$ and so $\sqrt{W(y)-u}$ will in turn have a simple zero when
the two solutions coincide at $u=E_{0}$. This means we have

$\displaystyle\mathcal{F}(E_{0})=\mathcal{O}(e^{-1/G_{N}}),$ (3.4)

so that it vanishes up to corrections which are exponentially small in
$1/G_{N}$. To find the exact ground state energy, $E^{\rm exact}_{0}$, it is
then natural at small $G_{N}$ to expand around $E_{0}$. Doing so gives the
equation,

$\displaystyle E_{0}^{\rm exact}=E_{0}+\delta E_{0},\ \ \
\mathcal{F}(E_{0})|_{\mathcal{O}(e^{-1/G_{N}})}\approx-\mathcal{F}^{\prime}(E_{0})\delta
E_{0}.$ (3.5)

So far, what we have said about the ground state energy holds for general
$W(y)$ such that there exists an outermost minimum to the potential. For the
specific choice, $W(y)=W_{\lambda}(y)$ with $\lambda_{c}>\lambda>0$, then the
leading correction to $\mathcal{F}(E_{0})$ will come from the innermost real
solution to $W(y)=E_{0}$. We denote this solution by $y_{i}$. All the other
complex saddles have more negative real part than than this solution and so
are suppressed in the semi-classical limit.

To compute the contribution of this innermost real branch point at
$y=y_{i}(E_{0})$ to $\mathcal{F}(E_{0})$, we just compute the piece of the
deformed contour wrapping this branch cut. We then get the integral

$\displaystyle\mathcal{F}(E_{0})\approx\frac{1}{G_{N}\pi}\int_{-\infty}^{y_{i}(E_{0})}dy\left(\sqrt{E_{0}-W_{\lambda}(y)}\right)\,e^{\frac{2\pi
y}{G_{N}}}+\text{contributions from sub-leading branch points}.$ (3.6)

Written in this form, the integral has a saddle point along the defining
contour at the position $y_{*}$ defined by

$\displaystyle
W_{\lambda}(y_{*})=E_{0}-\frac{G_{N}W_{\lambda}^{\prime}(y_{*})}{4\pi}.$ (3.7)

To leading order in $G_{N}$, including the one-loop correction, we find

$\displaystyle\mathcal{F}(E_{0})\approx\frac{1}{2G_{N}}\left(\frac{G_{N}}{2\pi}\right)^{3/2}\sqrt{W_{\lambda}^{\prime}(y_{i})}e^{\frac{2\pi
y_{i}}{G_{N}}+\frac{\pi^{2}}{2}}.$ (3.8)

Furthermore, $\mathcal{F}^{\prime}(E_{0})$ can be calculated by noticing that
when the two outermost branch points collide, then they form a simple pole in
the expression for $\mathcal{F}^{\prime}$ in eq. (3.1). Picking up this pole
gives

$\displaystyle\mathcal{F}^{\prime}(E_{0})\approx\frac{\sqrt{2}}{G_{N}\sqrt{W_{\lambda}^{\prime\prime}(y_{o})}}e^{\frac{2\pi
y_{o}}{G_{N}}}+\text{ exponentially sub-leading}.$ (3.9)

This leads to

$\displaystyle\delta E_{0}\approx(\mathrm{one-loop})\times e^{\frac{2\pi\Delta
y}{G_{N}}}=-\frac{1}{4}\sqrt{W_{\lambda}^{\prime}(y_{i})W_{\lambda}^{\prime\prime}(y_{o})}\left(\frac{G_{N}}{2\pi}\right)^{3/2}e^{\frac{2\pi\Delta
y}{G_{N}}+\frac{\pi^{2}}{2}}$ (3.10)

where $\Delta y=y_{i}-y_{o}$ is the difference between the inner root and the
outer root of the equation $W_{\lambda}(y)=E_{0}$. Note that this difference
is negative so that the change in energy is indeed exponentially suppressed in
$1/G_{N}$.

###  3.3 Alternate dilaton contour and the semi-classical density of states

It is tempting to ask if the function $\mathcal{F}^{\rm semi-classical}$ in
(2.13), which ignores the upper limit on the sum in the expression for
$\mathcal{F}$ in eq. (2.11), has a representation analogous to eq. (2.14).

Figure 8: This figure illustrates the possible alternate contour for the $y$
variable that is used to compute the density of states in eq. (2.3). This
contour excludes contributions from saddle points other than those associated
with the two outermost horizons. This produces a result for the ground state
energy that is consistent with that predicted by canonical quantization.

For the dilaton potential $W_{\lambda}(y)$ with $0<\lambda<\lambda_{c}$,
consider an energy $u$ small enough such that there are three real solutions
to the equation $W_{\lambda}(y)=u$. Following the same steps presented in [17]
for re-summing the exact function $\mathcal{F}^{\rm exact}$, one finds that if
we modify $\mathcal{C}\to\mathcal{C}\cup\mathcal{C^{\prime}}$, where
$\mathcal{C}^{\prime}$ is a vertical contour that lies between the middle and
innermost solution but running in the opposite direction from $\mathcal{C}$,
then the $\lambda$ expansion does not truncate at a finite order. In other
words, we have

$\displaystyle\mathcal{F}^{\rm semi-class}(u)=\frac{1}{2\pi
iG_{N}^{2}}\int_{\mathcal{C}\cup\mathcal{C}^{\prime}}dy\
\left(y-\sqrt{W(y)-u}\right)e^{2\pi\frac{y}{G_{N}}}.$ (3.11)

We illustrate this other contour in Fig. 8. Note that this contour effectively
excludes any saddle point contributions from horizons other than the two
outermost horizons.

From this representation of $\mathcal{F}^{\rm semi-classical}$ it is clear
that the ground state energy $E_{0}$ as defined by $\mathcal{F}^{\rm semi-
classical}(E_{0})=0$ is just given by the value when the two outermost roots
of the equation $W(y)=E$ merge in the complex plane. This is because we can
deform the contours to pick up the branch cut connecting the outer two
horizons. This branch cut vanishes into a simple zero when the two roots
collide at $y_{0}$ given by $W(y_{0})=E_{0}$.

Unfortunately, this contour is only good at lower energies for $\lambda>0$.
For high enough energies, the inner contour $\mathcal{C}^{\prime}$ can be
pinched by the merging of the middle root and the inner-most root. These two
roots collide when the Penrose diagram truncates from an infinite number of
asymptotic regions to just one. This means that the formula for
$\mathcal{F}^{\rm semi-classical}$ encounters a non-analyticity at this
energy. It is tempting to interpret this non-analyticity in terms of a phase
transition. In Sec. 3.5, we will show that the exact answer as defined by
equations (2.3) does not see such a non-analyticity at this energy.

It is actually interesting to take this alternate contour seriously for a
moment and see what it predicts for the density of states by plugging
$\mathcal{F}^{\rm semi-classical}$ and $E_{0}$ into eq. (2.3) for
$\mathcal{F}^{\rm exact}$ and $E_{0}^{\rm exact}$, respectively. Examining the
second line of eq. (2.3), we can re-write this expression in a more
illuminating way as

$\displaystyle\rho_{g=0}(E)\sim\int_{\mathcal{C}\cup\mathcal{C}^{\prime}}\frac{dy}{2\pi
i}e^{\frac{2\pi
y}{G_{N}}}\left(\frac{1}{2}\log\left(W(y)-E\right)-\log\left(\sqrt{W(y)-E_{0}}+\sqrt{E-E_{0}}\right)\right).$
(3.12)

From the first term, we see that there are branch cuts emanating from all
solutions to $W(y)=E$ in the complex $y$ plane. From the second term, we see
that there appear to also be branch cuts emanating from solutions to
$W(y)-E_{0}$. In fact, since $E_{0}$ occurs at the outermost minimum, $y_{0}$,
of $W$, then the square root, $\sqrt{W(y)-E_{0}}$, in the second term of eq.
(3.12) is regular at $y_{0}$ since
$W(y)-E_{0}\approx\frac{W^{\prime\prime}(y_{0})}{2}(y-y_{0})^{2}$ for
$y\approx y_{0}$.

Since all the other solutions to either $W(y)=E_{0}$ or $W(y)=E$ are not
bounded by the alternate contour, we can then just deform
$\mathcal{C}\cup\mathcal{C}^{\prime}$ to pick up the branch cut in eq. (3.12)
running between the two outermost solutions to $W(y)-E$. We are then left with
the pleasingly simple answer

$\displaystyle\rho^{\rm semi-classical}(E)\sim
e^{\frac{2\pi}{G_{N}}W_{+}^{-1}(E)}-e^{\frac{2\pi}{G_{N}}W_{-}^{-1}(E)},$
(3.13)

where $W^{-1}_{\pm}(E)$ denotes the outermost $(+)$ and second outermost $(-)$
roots of the equation $W(y)=E$.

Since $E_{0}$ differs from $E_{0}^{\rm exact}$ by exponentially suppressed
contributions, eq. (3.13) actually agrees with the exact density of states up
to corrections that are exponentially small in $1/G_{N}$, at least in a
specific range of energies. In particular, for the potential $W_{\lambda}$
formula (3.13) is true only for energies not too large or not too small, since
these corrections can become unsuppressed in these limits. For energies $E$
too close to $E_{0}$, this formula breaks down due to edge effects near
$E_{0}^{\rm exact}$ in the formula eq. (2.3). These edge effects are the main
topic of this paper. For $E$ too large, this formula breaks down because the
middle real root of $W_{\lambda}(y)=E$ can merge with the innermost real root
as discussed above.

###  3.4 Bulk interpretation of the saddles

Note that the saddle points in the integral defining $\mathcal{F}$ obey the
same equations that defines the relationship between the dilaton at the
horizon and the boundary AdM energy in the classical solutions, as in eq.
(2.6). This suggests that we should interpret these solutions as associated to
bulk geometries with fixed energy boundary conditions.

Fixed energy boundary conditions in JT gravity are given by imposing fixed
$\partial_{n}\Phi$ and fixed $\Phi$ along the asymptotic boundary such that
$\partial_{n}\Phi-\Phi=E$.111111One may wonder how it is that doing the
inverse Laplace transform of fixed length boundary conditions leads to the
boundary conditions presented here. This point was nicely explained in [21].
Such boundary conditions allow for complex values of the dilaton at the
horizon. This is because the identification of $\tau\sim\tau+\beta$ is not
fixed to be real anymore. These solutions are then defined by analytic
continuations of the standard disk geometry into the complex dilaton plane, so
that the dilaton takes values along a complex contour instead of along the
real axis.

Note that we can discuss these complex gometries just in the context of pure
JT gravity with fixed energy boundaries. In JT gravity, to find the classical
solutions with these boundary conditions, we just have to solve the equation
$E=\Phi_{h}^{2}$. In that case, there is the standard disk saddle, with
$\Phi_{h}$ given by $\Phi_{h}=\sqrt{E}$. The contribution to the path integral
of this saddle gives a term like $e^{2\pi\sqrt{E}/G_{N}}$. There is, however,
another saddle whose dilaton at the horizon takes on the value of the other
root to this equation, $\Phi_{h}=-\sqrt{E}$.

To make sense of such a saddle, we can consider a complex geometry with metric
given by

$\displaystyle ds^{2}=(W(\Phi)-E)d\tau^{2}+\frac{d\Phi^{2}}{W(\Phi)-E}$ (3.14)

but where the dilaton follows the complex contour running from $+\infty$ down
to $-\sqrt{E}$, going around the other root at $+\sqrt{E}$ in the complex
plane. Of course, there is an ambiguity on how one actually chooses the
contour. As usual, all contours which are deformable to each other without
passing through non-analyticities in the action should be deemed equivalent.
This proposed saddle is illustrated in Fig. 1.

Note that upon continuing around the pole in the metric at $\Phi=+\sqrt{E}$,
the metric becomes that of the anti-sphere, i.e. the sphere with an overall
minus sign in front of its metric. Such a solution to JT gravity was also
considered in [37]. Note that this metric violates the Kontsevich-Segal
criterion for allowable metrics. Presumably, this saddle point, when its
fluctuations have been properly accounted for, reproduces the sub-leading term
in the $\sinh(2\pi\sqrt{E})$ density of states for JT gravity. We have not
done a direct bulk analysis of this minus sign, but it would be interesting to
understand how this works in more detail. One path forward would be to more
directly derive the formulae in eq. (2.14) and (2.3) from the bulk. We hope to
study this possibility in the future.

###  3.5 A (near) phase transition

As mentioned in Sec. 2, for $\lambda>0$ but not too large, there is a critical
energy $E_{c}(\lambda)$ above which the classical solutions go from an
infinite number of universes to a single universe. We can ask if the matrix
model’s thermodynamics see any signature of this transition. In particular, it
is tempting to look for a phase transition in the model.

As we crank up the energy, two innermost real roots of the equation
$W_{\lambda}(y)=E$ merge and then go off into the complex plane. They merge
precisely at critical energy. Since $\rho(E)\sim\mathcal{F}^{\prime}(E)$ semi-
classically, then we can just analyze $\mathcal{F}^{\prime}$ as a function of
energy. Just as discussed in the context of the ground state energy, when
these two roots of the equation $W_{\lambda}(y)=E$ merge, they form a simple
pole in (3.2). Picking up this pole gives a contribution of the form

$\displaystyle\mathcal{F}^{\prime}\supset\frac{\sqrt{2}e^{\frac{2\pi
y_{i}}{G_{N}}}}{G_{N}\sqrt{|W_{\lambda}^{\prime\prime}(y_{i})|}}.$ (3.15)

This contribution is finite and so there is no actual divergence in the
density of states. Interestingly, the one-loop factor is enhanced for this
saddle point by factors of $G_{N}$ because $W_{\lambda}^{\prime}(y_{i})=0$ at
this energy. One can understand this enhancement by examining the one-loop
fluctuations for the saddle points defined by eq. (3.2). This suggests that
the horizon dilaton fluctuations about these sub-leading saddles is becoming
strong at this point, since horizon area fluctuations appear to be diverging
like $1/G_{N}$ near this critical point.

Of course, this is not totally suprising: in the Penrose diagram, as we
approach this critical point, two horizons are merging and developing an
approximatedly dS2 region which lies behind the outermost horizon. It would be
interesting to understand if this mode could be related to the Schwarzian mode
near asymptotic future of the nearly de Sitter region as discussed in [38,
39]. Note that from the point of view of the full density of states of the
model, however, this is a non-perturbatively small effect which is swamped by
the leading classical answer. It would be interesting to find other parameter
regimes where this merger of the two inner branch points leads to a sharper
signal in boundary observables.

###  3.6 The critical coupling

In [40, 15, 16], the authors observed a special value of the coupling
$\lambda=\lambda_{c}^{MM}$ for the potential $W_{\lambda}(\Phi)$ such that the
exact string equation given in eq. (2.14) no longer has a real solution to
$\mathcal{F}^{\rm exact}(E_{0})=0$. The authors in [40] understood this as a
phase transition wherein the matrix model becomes non-perturbatively unstable
for $\lambda>\lambda_{c}^{MM}$. It is natural that this transition should be
associated to the critical coupling
$\lambda_{c}=\frac{1}{2\pi^{2}e(1-\alpha)^{2}}$ discussed in Sec. 2 at which
the dilaton potential $W_{\lambda}(\Phi)$ becomes monotonic. The reason is
that for $\lambda>\lambda_{c}$ the theory no longer has a classical solution
with $\beta=\infty$.

In the $G_{N}\to 0$ limit, it is easy to see that indeed the critical coupling
$\lambda_{c}^{MM}$ at which the matrix model becomes unstable is
(approximately) equal to the critical coupling $\lambda_{c}$ at which
$W_{\lambda}(\Phi)$ becomes monotonic. The reason can be seen easily from the
manipulations laid out in this section. In the $G_{N}\to 0$ limit, we have
shown that the ground state energy is (up to non-perturbative corrections)
associated with the value of $W_{\lambda}(\Phi)$ at the outermost minimum of
the potential. When this outermost minimum no longer exists, there is no
longer a way for $\mathcal{F}(E_{0})$ to (almost) vanish. The outermost
minimum vanishes from the real axis when the potential becomes monotonic and
so $\lambda_{c}\approx\lambda_{c}^{MM}$, up to non-perturbatively small
corrections.

##  4 Canonical quantization in geodesic gauge

At the disk level, JT gravity can be studied via the methods of canonical
quantization. In fact, dilaton gravity with more general dilaton potentials
can be quantized in this way as well. Canonical quantization of dilaton
gravity for general dilaton potentials was discussed in [31, 32, 41, 42]. To
proceed, we first write a general metric in AdM gauge as

$\displaystyle ds^{2}=-N^{2}dt^{2}+h(dx+N_{\perp}dt)^{2},$ (4.1)

where the time variable $t$ denotes a choice of Cauchy slice for the geometry
and $x$ denotes the spatial coordinate on the slice. The dynamical variables
on each slice can then be thought of as the dilaton and the scale factor
$h=e^{2\sigma}$. There is a large gauge redundancy in describing the state
using these variables, which we would like to fix. One natural choice is to
fix the extrinsic curvature of the slice to be zero. This is the so-called
geodesic gauge. Picking geodesic gauge has been discussed extensively in the
context of JT gravity, where it was used to solve for the disk-level
wavefunctions exactly [14, 5].

In dilaton gravity, the extrinsic curvature is canonically conjugate to the
dilaton, whereas the induced metric on a geodesic slice is conjugate to the
normal derivative of the dilaton [21]. By combining the Wheeler de-Witt and
momentum constraints, one can get a classical expression for the momentum
conjugate to the intrinsic metric which is

$\displaystyle
P_{\ell}=\frac{1}{G_{N}}\sqrt{\left(\partial_{u}\Phi\right)^{2}-W(\Phi)+E}$
(4.2)

where $E$ is the AdM mass of the solution.121212The factor of $1/G_{N}$ comes
from the re-scaling of $\Phi$ and the couplings required for the classical
limit. We denote the conjugate momentum by $P_{\ell}$ because it is conjugate
to the length of the geodesic slice, $\ell$, which is the only gauge invariant
quantity for a one-dimensional metric. Here $\partial_{u}$ means derivatives
with respect to a proper length variable along the slice. Using the ansatz
(2.2), one can show that this is precisely the equation for a spacelike
geodesic with momentum $P_{\ell}$, which is conserved due to the boost
symmetry of the spacetime.

We can solve this expression for the length in terms the momentum and energy,
to get an expression

$\displaystyle\ell=\lim_{\Phi_{b}\to\infty}2\left(\int_{W^{-1}(E-G_{N}P_{\ell}^{2})}^{\Phi_{b}}\frac{d\Phi}{\sqrt{W(\Phi)-E+G_{N}^{2}P_{\ell}^{2}}}-\log\Phi_{b}\right).$
(4.3)

Just as in JT gravity, the $\log\Phi_{b}$ term is a universal renormalization
of the length operator that makes it well-defined in the $\Phi_{b}\to\infty$
limit. Solving this for $V(\ell)\equiv E-G_{N}P_{\ell}^{2}$, we get an
expression for the energy in terms of the momentum and potential

$\displaystyle E=G_{N}^{2}P_{\ell}^{2}+V(\ell).$ (4.4)

In JT gravity, this equation becomes

$\displaystyle E=G_{N}^{2}P_{\ell}^{2}+4e^{-\ell},$ (4.5)

as found in [14]. For the models we consider here which are deformed away from
JT gravity by an exponential potential, the length potential is not
analytically solvable, except for in certain regimes. We plot the potential
analytically in Fig. 9 for $\lambda$ in the range $\lambda_{c}>\lambda>0$ and
for $\lambda=0$ (pure JT).

Figure 9: We plot the length potential for dilaton gravity with potential
$W_{\lambda}(\Phi)$. For $\lambda=0$, we just get the exponential potential of
[14]. For the $\lambda_{c}>\lambda>0$, we have a mildly modified potential
with a different asymptotic value at $\ell\to+\infty$. This corresponds to the
shift in the ground state energy of the model from $E_{0}=0$ to $E_{0}$, set
by the outermost minimum of $W_{\lambda}(\Phi)$. Note that the potentials
agree at $\ell\to-\infty$ since this probes the asymptotically AdS region.

Regardless, one could imagine quantizing this theory in the obvious way by
making $P,\ell$ into operators which obey the standard commutation relations.
This way of quantizing the theory ignores certain gauge ambiguities which will
be important for us.131313There are also ordering ambiguities in solving eq.
(4.3) for the length potential, which could in principle be important, but
following [31] we will ignore these entirely. What we will show in the
remainder of this section is that this naive quantization procedure gets the
correct answer for the ground state energy of the matrix model as well as the
low energy behavior of the density of states, up to corrections which are
exponentially small in $1/G_{N}$ (but enhanced at very low energies).

###  4.1 Length potential at large positive and negative length

While the length potential cannot be solved for analytically except for in a
very specific class of examples, it can be solved for in certain limits. At
large positive lengths, we can try to solve equation (4.3) for $V(\ell)$. The
only way to get a length which is arbitarily large is if we have tuned $E$ to
be near the outermost minimum of the potential $W(\Phi)$. Let the value that
$\Phi$ takes at this location be denoted by $\Phi_{0}$ so that

$\displaystyle W(\Phi_{0})=E_{0}.$ (4.6)

Focusing on the zero momentum geodesic for now, for such low energies we have

$\displaystyle\ell\approx\int_{W^{-1}(V(\ell))}\frac{d\Phi}{\sqrt{W^{\prime\prime}(\Phi-\Phi_{0})^{2}/2+(E_{0}-V(\ell))}}\approx-\frac{2\sqrt{2}}{\sqrt{W^{\prime\prime}(\Phi_{0})}}\left(\log\sqrt{\frac{V(\ell)-E_{0}}{\Lambda}}\right)$
(4.7)

up to a constant, $\Lambda$, which is independent of $V(\ell)$. This constant
can only be found numerically since it receives contributions from pieces of
the $\Phi$ integral away from the endpoint. Solving for the length potential
at large lengths then takes the form

$\displaystyle V(\ell)=E_{0}+\Lambda
e^{-\ell\sqrt{W^{\prime\prime}(\Phi_{0})/2}}\left(1+\mathcal{O}(1/\ell)\right).$
(4.8)

This means that a quantization of this model along the lines described above
leads to a prediction for the ground state energy of the dual matrix model as
occuring at $E_{0}$ defined to be the value of the dilaton (pre)-potential at
its outermost minimum. As discussed in Sec. 3, this answer almost matches with
the exact answer in the semi-classical $G_{N}\to 0$ limit. It is off from the
exact answer due to a non-perturbatively small shift in the ground state
energy of the model. In the next section, we will interpret this shift as
coming from a non-perturbatively small overlap between different states of
fixed geodesic length. In other words, at non-perturbative in $1/G_{N}$
orders, geodesic length states of differing lengths are in fact not linearly
independent, and so we should not expect canonical quantization in the
geodesic length gauge to get the correct answer unless this linear dependancy
is properly accounted for.

We can also solve for the potential at large negative $\ell$. Due to the large
shift by $\log\Phi_{b}$ in the renormalization of $\ell$ in (4.3), the way to
get large negative $\ell$ is when the turning point at the lower limit of the
integral goes to $+\infty$. This is when $V(\ell)$ is very large. In this
limit, the geodesic only probes the geometry at large $\Phi$ and so the
dilaton potential limits back to JT gravity. Thus, we see that the length
potential $V(\ell)$ has the behavior

$\displaystyle V(\ell)\sim 4e^{-\ell}\ \text{as }\ell\to-\infty,\ \text{and}\
V(\ell)\sim E_{0}+\Lambda e^{-\ell\sqrt{W^{\prime\prime}(\Phi_{0})/2}}\
\text{as}\ \ell\to+\infty.$ (4.9)

This agrees with the numerically generated answer shown in Fig. 9. Given this
answer for the predicted ground state of the model, in the specific example
where $W(\Phi)=W_{\lambda}(\Phi)$, we can ask what happens if we start
cranking up the coupling in the potential. As we discussed in Sec. 2, there is
a critical $\lambda=\lambda_{c}$ above which the potential becomes monotonic.
We can ask what happens to the length potential $V(\ell)$ at this critical
coupling. Indeed, we see that the quantum mechanics defined by the potential
$V(\ell)$ becomes ill-defined for very large $\ell$.141414One other way to
understand this is that for $\lambda>\lambda_{c}$ there are no longer
geodesics in the classical Lorentzian solutions that are anchored to the two
AdS boundaries for arbitrarily late times. As mentioned in Sec. 3.6, this
coupling $\lambda_{c}$ is associated with the critical coupling at which the
matrix model becomes non-perturbatively unstable. It is clear that much about
the exact matrix model’s disk-level density of states can be learned just from
analyzing the quantum mechanics defined by quantizing the theory in geodesic
gauge.

###  4.2 Density of states and the WKB approximation

Although the length potential defined by (4.3) is not analytically tractable
except for special dilaton potentials, in the semi-classical limit we can try
to solve Schroedinger’s equation via the WKB approximation. In that limit, the
wavefunctions in the length basis for energy eigenstates are given by linear
combinations of

$\displaystyle\psi_{E}(\ell)\supset\frac{1}{(E-V(\ell))^{1/4}}\exp\left(\frac{\pm
i}{G_{N}}\int^{\ell}_{V^{-1}(E)}d\ell^{\prime}\sqrt{E-V(\ell^{\prime})}\right)$
(4.10)

which is the answer in the classically allowable region. Using the results
above, we see that at very large $\ell$ and fixed $E$, we get that the
wavefunctions are

$\displaystyle\psi_{E}(\ell)\sim\frac{1}{(E-E_{0})^{1/4}}\exp\left(\pm
i\ell\frac{\sqrt{E-E_{0}}}{G_{N}}\right).$ (4.11)

At large $\ell$, we thus have that the general real solution takes the
form151515We will focus on real wavefunctions because this is what the
Euclidean path integral will necessarily output for the length wavefunction in
the energy eigenbasis, $\braket{\ell}{E}$.

$\displaystyle\psi_{E}(\ell)=\frac{1}{2(E-E_{0})^{1/4}}\left(C(E)\exp\left(i\ell\frac{\sqrt{E-E_{0}}}{G_{N}}\right)+C^{*}(E)\exp\left(-i\ell\frac{\sqrt{E-E_{0}}}{G_{N}}\right)\right).$
(4.12)

In fact, due to an argument adapted from [43], these wavefunctions are enough
to in principle determine the disk-level density of states, assuming that
fixed geodesic length states are delta-function orthogonal. The argument goes
as follows: let the density of states be given by $\rho_{0}(E)$. Then the
wavefunctions $\psi_{E}(\ell)$ should obey the orthogonality relations

$\displaystyle\int_{E_{0}}^{\infty}dE\rho_{0}(E)\psi^{*}_{E}(\ell)\psi_{E}(\ell^{\prime})=\delta(\ell-\ell^{\prime}),$
$\displaystyle\int_{-\infty}^{\infty}d\ell\psi_{E}(\ell)\psi_{E^{\prime}}(\ell)=\frac{\delta(E-E^{\prime})}{\rho_{0}(E)}.$
(4.13)

Given the above orthogonality relations, the fact that the wavefunctions decay
doubly exponentially in $\ell$ due to (4.9) tells us that the only way to get
a delta function in energy out of the integral over $\ell$ is from the large,
positive $\ell$ integral. This means that

$\displaystyle 1/\rho_{0}(E)=|C(E)|^{2}.$ (4.14)

Naively $C(E)$ is arbitrary. The trick, however, is then to view the disk-
level partition function as a return amplitude in Euclidean time, from the
$-\infty$ geodesic length state back to itself in $\beta$ time:

$\displaystyle Z(\beta)\sim\braket{-\infty}{e^{-\beta H}}{-\infty}=\int
dEe^{-\beta E}\rho_{0}(E)\psi_{E}(-\infty)\psi_{E}(-\infty).$ (4.15)

At large negative $\ell$, the wavefunctions decay doubly-exponentially in
$\ell$. Stripping off this decay and absorbing it into a re-definition of
$S_{0}$, we see that this formula produces the correct partition function
provided that $\psi_{E}(\ell)$ is independent of $E$ at large, negative
$\ell$. The demand that $C(E)$ be such that the transmission amplitude for
$\ell\to-\infty$ is independent of $E$ is in fact enough to fix the form of
$C(E)$, up to an energy independent constant.

We can then view solving for the density of states as a scattering problem,
where waves come in from large positive $\ell$ and scatter off the potential.
The above discussion says that there is a boundary condition at large negative
$\ell$ which demands that the transmission coefficient is independent of $E$.
Note that demanding this boundary condition amounts to doing the Euclidean
gravitational path integral at the disk level. In particular, this argument
should not be read as a new derivation of the density of states in gravity.
Rather, one should view this calculation as performing the Eucliean path
integral by slicing the geometry along geodesics.

In principle, one could try to solve for these transmission coefficients
$C(E)$ in the semi-classical limit to arrive at the density of states.
Instead, we will opt to determine their behavior at $E$ near the ground state,
$E_{0}$, as we will need this formula later. It would be interesting to use
WKB techniques of [29, 44] to try to compute $C(E)$ semi-classically for all
energies. At low energies, however, the problem simplifies. This is due to the
fact that at low energies the wavefunction really only feels the potential at
large lengths of order $\log 1/(E-E_{0})$. To understand this, we note that
the wavefunction rapidly decays for $\ell$ in the classically forbidden
region. Thus, the integral in eq. (4.2) over $\ell$ can be truncated to values
of $\ell$ that are only in the classically allowed region, up to corrections
that are non-perturbatively suppressed in $1/(E-E_{0})$. In this region, since
the lengths are long (greater than $\log 1/(E-E_{0})$, then we can use the
expression in eq. (4.9) for the potential.

The solution for these potentials is just the same as in the JT gravity case
and are given by Bessel functions, namely

$\displaystyle\psi_{E}(\ell)\sim
K_{\frac{2i\sqrt{(E-E_{0})}}{G_{N}}\sqrt{\frac{2}{W^{\prime\prime}(\Phi_{0})}}}\left(2\sqrt{\frac{2\Lambda}{W^{\prime\prime}(\Phi_{0})G_{N}}}e^{-\frac{\ell}{2}\sqrt{\frac{W^{\prime\prime}(\Phi_{0})}{2}}}\right),$
(4.16)

where again $\Lambda$ is a constant which needs to be determined numerically.
These wavefunctions are just the same as those of JT gravity but with a re-
scaled and shifted energy and so the density of states is just that of JT
gravity with a re-scaled and shifted energy

$\displaystyle\rho(E)\sim\sinh\left(2\pi\sqrt{\frac{2}{W^{\prime\prime}(\Phi_{0})}}\frac{\sqrt{E-E_{0}}}{G_{N}}\right)$
(4.17)

for $E$ close enough to $E_{0}$. Note that this formula just requires that we
are at low enough energies so that the second equation in eq. (4.9) holds.
Since the large $\ell$ approximation of $V(\ell)$ cares only about $\lambda$,
we have in particular not made the assumption that $E-E_{0}\sim G_{N}^{2}$.
Thus, the formula in eq. (4.17) should hold for a parametrically large range
of energies. Intuitively this formula should be obvious; we have zoomed in on
energies where the dilaton potential is well-approximated by a quadratic and
so we have reduced the theory back to JT gravity approximately.

Of course, since we are focusing on a specific region in energy space and
since our method can miss an overall, energy independent constant in the
density of states, the normalization in this formula may not be correct to
match onto higher energies. To find the correct normalization, we need to
solve for the density of states at higher energies and match the two
expressions in intermediate energies. At high enough energies, the semi-
classical answer comes purely from the leading disk saddle point. Its
contribution to the density of states gives

$\displaystyle\rho(E)\sim e^{2\pi W^{-1}(E)/G_{N}},$ (4.18)

where $W^{-1}(E)$ is the branch of $W^{-1}$ with the largest real part. At
lower energies this gives

$\displaystyle\rho(E)\sim\exp\left(\frac{2\pi}{G_{N}}\left(\Phi_{0}+\sqrt{\frac{2\left(E-E_{0}\right)}{W^{\prime\prime}(\Phi_{0})}}\right)\right),$
(4.19)

where again $\Phi_{0}$ is the value of the dilaton at the outermost minimum of
the potential $W(\Phi)$. We then expect the density of states goes like

$\displaystyle\rho(E)\sim
e^{\frac{2\pi}{G_{N}}\Phi_{0}}\sinh\left(2\pi\sqrt{\frac{2}{W^{\prime\prime}(\Phi_{0})}}\frac{\sqrt{E-E_{0}}}{G_{N}}\right)$
(4.20)

up to order one factors. Note that this formula agrees with the semi-classical
density of states in eq. (3.13) predicted by using the alternate contour of
Sec. 3.3. In eq. (4.20) we are effectively evaluating $\rho^{\rm semi-
classical}(E)$ in eq. (3.13) at energies close enough to $E_{0}$ so that
$W(y)$ can be approximated by a Gaussian around $y=y_{0}$. Apparently, slicing
the path integral in geodesic gauge leads to an answer that prefers the
alternate contour. As we will argue in Sec. 5, this is because the alternate
contour excludes the geometries which cause a breakdown in geodesic gauge.

###  4.3 A subtlety with geodesic gauge in $W_{\lambda}(\Phi)$ gravity

So far in this section, we have been agnostic about the form of $W(\Phi)$,
except that we have demanded that $W$ has an outermost minimum at $\Phi_{0}$,
where $W(\Phi_{0})=E_{0}$. If we restrict our attention to the potential
$W_{\lambda}(\Phi)$ for $\lambda_{c}>\lambda>0$, such that an outermost
minimum exists, there is a subtlety with geodesic gauge already at the level
of the Lorentzian classical solutions. As discussed in Sec. 2, for solutions
with AdM energy $E>E_{c}$ (or $E<E_{0}$), the Lorentzian universe truncates to
a single AdS boundary. We can solve for the geodesics anchored between two
boundary points in such spacetimes. Just as in higher dimensional AdS black
holes [28], one can show that there are bouncing geodesics for $E>E_{c}$. This
means that there can be more than one geodesic anchored to the same two
boundary points in the same solution. The length of a geodesic anchored to
fixed left and right boundary times then does not uniquely described the
state; geodesic gauge has broken down.

We will see in the next section that a similar phenomenon, albeit one that is
inherently Euclidean, can explain the discrepancy between the exact answer for
the ground state energy as given by eq. (2.10) and the answer predicted by a
canonical analysis. It would be interesting, however, to understand how to
incorporate this Lorentzian breakdown into the framework of [31] and to see if
this might lead to a Lorentzian derivation of the exact ground state energy.

In this work, we have been interested in classical black hole interiors that
look drastically different from those in pure JT gravity. For the purposes of
canonical quantization, however, it may be less confusing to consider the
theory with dilaton potential $W_{\lambda}(\Phi)$ where now $\lambda<0$. In
this case, $W_{\lambda}(\Phi)$ looks qualitatively just like the JT pre-
potential with a single unique minimum. In this case, geodesic gauge does not
breakdown for any energy and we can proceed just as in JT gravity. Still,
however, we find a discrepancy between the exact ground state energy from eq.
(2.10) and that predicted by canonical quantization in geodesic gauge. We will
come back to this case in Sec. 5 and Sec. 7. We will argue that indeed
geodesic gauge still breaks down in these theories but in a way that requires
the use of complex geometries. In this case, it is hard to imagine a way of
deriving the exact ground state energy for $\lambda<0$ by only considering
purely real, Lorentzian geometries. Instead, we can interpret this discrepancy
as a modification of the inner product of the theory as suggested in [25].

##  5 Corrections to the length basis inner product

In the previous section, we explained how a canonical analysis naively
disagrees with the $E_{0}^{\rm exact}$ predicted by eq. (2.11). It is
important to understand what we got wrong in applying the canonical framework.
What we will argue in this section is that we assumed that the geodesic length
basis is delta function normalized, which is a false assumption. This non-
trivial overlap comes from spacetime geometries which have the same topology
as the disk but also contain a closed geodesic loop in the spacetime.

Figure 10: Two types of geometries that are summed over in more general
theories of dilaton gravity. In JT gravity, however, only geometries like the
one on the left appear, which have higher genus than the disk and so are
topologically suppressed. In the theories considered in this work, one also
needs to include contributions from geometries like that on the right, which
appear at the disk level.

In JT gravity, fixed geodesic length states have overlaps, denoted by
$\braket{\ell}{\ell^{\prime}}$, that are defined by the Euclidean
gravitational path integral with fixed geodesic length boundary conditions
[25]. The geometries that contribute to such a path integral for
$\ell\neq\ell^{\prime}$ are non-perturbatively small in $S_{0}$. The non-zero
overlap comes from spacetimes which contain a trumpet geometry. In JT gravity
though, the trumpet can only be glued to other trumpets or a pair of pants,
and so $\braket{\ell}{\ell^{\prime}}$ only receives contributions from
spacetimes with higher topology. What we will see in this section is that in
more general dilaton potentials the trumpet can instead be glued to a geometry
which caps off and so has the topology of the disk. We will refer to such
geometries with the topology of the disk but with a closed geodesic as
_pacifier spacetimes_ for their resemblance to a baby’s pacifier.

We begin by discussing the trumpet geometry with closed geodesic length $b$
and fixed energy boundary conditions in a general model with potential $W$
such that $W$ has an outermost minimum. We will then discuss geometries with
the topology of the disk that also end on a closed geodesic of length $b$ for
the specific potential $W_{\lambda}$ with $0<\lambda<\lambda_{c}$. We will
comment on these types of geometries for the case $\lambda<0$. Finally, we sew
these two geometries together to form a pacifier and show how such geometries
lead to a non-trivial inner-product between fixed length states.

As discussed in Sec. 2, the classical solutions of dilaton gravity are labeled
by the energy or size of the thermal circle at infinity. Given these
asymptotic parameters, the geometry is completely determined by the dilaton
pre-potential via equations (2.2) and (2.3). Let the outermost minimum of this
potential be at dilaton value $\Phi_{0}$ with associated energy
$E_{0}=W(\Phi_{0})$. One can consider classical solutions with asymptotic
energy $E<E_{0}$.161616In fact, just such solutions were considered in [45,
26].

We have shown in this paper, however, that such geometries in the $G_{N}\to 0$
limit have energy below the ground state energy of the dual matrix model,
since $E^{\rm exact}_{0}=E_{0}$ up to non-perturbatively small corrections.
Apparently, such classical geometries are excluded from the matrix model
spectrum. This does not mean that the matrix model does not know about these
geometries. Rather, they are present in the quantum effects of the model. As
we now show, these types of solutions can contribute to the inner product of
geodesic length states. An equivalent way of stating this issue is that for
general dilaton potentials geodesic gauge is non-perturbatively ill-defined
due to the possibility of nucleating a closed geodesic in the geometry. We
explain this in detail now.

###  5.1 Trumpet geometries

In our model, we would like to find the dominant contribution to the path
integral given two boundary conditions - one a geodesic circle of length $b$
and the other an asymptotic boundary of energy $E$. In fact, we will start
with the problem at fixed boundary length/temperature and then inverse Laplace
transform the answer back to fixed energy.

We now show that there is a fixed temperature saddle which ends on these
boundaries. Consider saddles with metric of the usual on-shell form

$\displaystyle ds^{2}=(W(\Phi)-E)d\tau^{2}+\frac{d\Phi^{2}}{W(\Phi)-E},$ (5.1)

but where $E<E_{0}$. In Euclidean signature, the geodesic equation is
equivalent to

$\displaystyle\left(\partial_{u}\Phi(u)\right)^{2}=W(\Phi(u))-E-P_{E}^{2},$
(5.2)

where $P_{E}$ is the Euclidean momentum of the geodesic and $u$ is the proper
length coordinate along the geodesic. We remind the reader that this equation
is just that of a classical particle with coordinate $\phi$ in a potential
$-W(\Phi)$ and with total energy $-E-P_{E}^{2}$.

Using this equation, we see that there is an unstable equilibrium point for
this particle at $\Phi_{0}$, where $W(\Phi_{0})=E_{0}$. This corresponds to a
geodesic that just sits at $\Phi_{0}$ and so is a closed geodesic. The
momentum of the geodesic that sits at this point is given by

$\displaystyle P^{2}_{E}=E_{0}-E>0.$ (5.3)

The length of this geodesic can then be related to $\beta$ and $E$ by the
equation

$\displaystyle b=\beta\sqrt{E_{0}-E}.$ (5.4)

We would now like to calculate the contribution of the trumpet to the gravity
path integral. The on-shell action of the solution in (5.1) with energy given
by (5.4) is quite simple. We have

$\displaystyle S$
$\displaystyle=\frac{1}{2G_{N}}\int_{M}d^{2}x\sqrt{g}\left(\Phi
R+U(\Phi)\right)+\frac{\Phi_{b}}{G_{N}}\int_{\partial M}\sqrt{h}K.$
$\displaystyle=\frac{\beta}{2G_{N}}\int_{\Phi_{0}}^{\Phi_{b}}d\Phi(-\Phi
W^{\prime\prime}(\Phi)+W^{\prime}(\Phi))+\frac{\Phi_{b}}{G_{N}}\int_{\partial
M}\sqrt{h}K.$ (5.5)

Integrating by parts in $\Phi$, using that $W(\Phi_{0})=E_{0}$ and the
expression for the on-shell value of $K$ in terms of $E$, we arrive at the
total on-shell action

$\displaystyle S=-\frac{\beta E_{0}}{G_{N}}+\frac{\beta
E}{2G_{N}}=-\frac{1}{2G_{N}}\left(\beta E_{0}+\frac{b^{2}}{\beta}\right).$
(5.6)

In the second equality, we use the expression $\eqref{eqn:trumpetenergy}$ for
$E$ as a function of $b,\beta$.

Since later we will be interested in low energies, it is important to also
include the one-loop contribution from the boundary mode, which should become
strongly coupled at low energies and temperatures. We do the one-loop
calculation in Appendix A, and, assuming that the measure for the boundary
mode is modified only up to $\beta$-independent constants from the JT gravity
answer, we find that the trumpet with geodesic boundary of length $b$ and
boundary of renormalized length $\beta$ contributes

$\displaystyle Z_{trumpet}(b,\beta)\sim\frac{1}{\beta^{1/2}}e^{-\frac{\beta
E_{0}}{2G_{N}}-\frac{b^{2}}{2\beta G_{N}}},$ (5.7)

up to $\beta$-independent order one factors. Again, the determination of the
order one factors would require a more careful analysis of the boundary
particle measure that we leave for future work.

Inverse Laplace transforming, we land on the JT gravity answer but with an
energy shifted by $E\to E-E_{0}$. This looks as if the ground state of the
model has shifted to that predicted by canonical quantization. This answer in
energy space then becomes

$\displaystyle
Z_{\mu,trumpet}(E,b)\sim\frac{1}{\sqrt{\pi(E-E_{0})}}\cos\left(\frac{b\sqrt{E-E_{0}}}{G_{N}}\right),$
(5.8)

up to energy-independent prefactors. As in JT gravity, this answer diverges at
the shifted ground state energy, $E=E_{0}$.

###  5.2 Disk geometries ending on a closed geodesic and sewing the pacifier

Figure 11: On the left, we illustrate a geometry with a closed geodesic of
length $b$ and the topology of a disk. On the right we point out how this cap
can be associated with a piece of a solution with energy $E<E_{0}$. We remind
the reader that the difference between the black horizontal line and the curve
of $W_{\lambda}(\Phi)$ is the metric component $g_{\tau\tau}$. The dilaton at
the horizon $\Phi_{h}(b)$ lies on the third real branch of the function
$W^{-1}$.

Given the answer for the trumpet with geodesic boundary of length $b$, we
would like to sew in a disk geometry with a closed geodesic of length $b$.
This is most explicit for the potential $W_{\lambda}(\Phi)$ with
$0<\lambda<\lambda_{c}$. Again we look for solutions of the form in eq. (5.1)
with $E<E_{0}$. In this energy range, the solution can cap off at the dilaton
value given by $W_{\lambda}(\Phi)=E$, where now $\Phi$ is on the innermost
real branch of $W_{\lambda}^{-1}(E)$. The equation relating $b$ to $\beta$ and
$E$ in (5.4) still holds. We get one more equation by demanding that the
geometry end at a smooth cap. Going through the usual steps, this equation
reads

$\displaystyle\beta=\frac{4\pi}{U(\Phi_{h})},$ (5.9)

where $\Phi_{h}(b)$ and $E(b)$ are related via the equation

$\displaystyle W_{\lambda}(\Phi_{h}(b))=E(b).$ (5.10)

These two equations together with equation (5.4) can be used to solve for
$\beta(b),E(b)$ and $\Phi_{h}(b)$ in terms of $b$.

We plug this solution into the action to solve for the on-shell action. In
this case, the only boundary term would come from the boundary at the closed
geodesic, but this vanishes because $K=0$ there. We are thus left with just
the bulk term which gives

$\displaystyle S_{disk}(b)$ $\displaystyle=\frac{1}{2G_{N}}\int
d^{2}x\sqrt{g}\left(-\Phi
W_{\lambda}^{\prime\prime}(\Phi)+W_{\lambda}^{\prime}(\Phi)\right)$
$\displaystyle=\frac{2\pi\Phi_{h}(b)}{G_{N}}-\frac{\beta(b)}{G_{N}}(W_{\lambda}(\Phi_{h}(b))-E_{0})$
$\displaystyle=\frac{1}{G_{N}}\left(2\pi\Phi_{h}(b)+b\sqrt{E_{0}-W_{\lambda}(\Phi_{h}(b))}\right)$
(5.11)

For the potential $W_{\lambda}(\Phi)$, analytic formulae for $\beta(b)$ and
$E(b)$ are not possible for general $b$. We can instead solve for them in
various limits. In particular, in the limit that $b\to 0$ then $E\to E_{0}$
but $\beta$ goes to a constant so that the second term in eq. (5.2) vanishes.
The constant that $\beta$ approaches is set by $U(\Phi)$ at the innermost
solution to the equation $W_{\lambda}(\Phi)=E_{0}$. As we increase $b$, $E(b)$
decreases and so $\Phi_{h}$ decreases.

In the opposite regime where $b$ becomes large, equation (5.4) tells us that
$E$ becomes more negative so that the equation $W_{\lambda}(\Phi_{h})=E$ leads
to $\Phi_{h}\approx-\frac{1}{2\pi(1-\alpha)}\log\frac{|E|}{\lambda}$, but then
equation (5.9) tells us that $\beta\sim 1/|E|)$. Plugging this back into
(5.4), we see that in fact $b\sim 1/\sqrt{|E|}$ at large $E$, which is in
contradiction with $b$ getting large. Therefore at large $b$ there seem to not
exist saddles of the type we found at small $b$. Since we will mostly be
interested in the small $b$ limit here, we just point out that we have not
analyzed all possible classical saddles which have the topology of the disk
and a closed geodesic of length $b$. Perhaps there are more exotic classical
saddles at large $b$ where the dilaton at the horizon is complex, akin to what
we discussed in Sec. 3. We leave further exploration of these geometries for
future work.

We now briefly comment on these geometries for $\lambda<0$. We can start again
with a solution with energy $E(b)<E_{0}$. In this case, there is no obvious
real geometry that ends on the geodesic of length $b$. That said, there are
infinitely many complex geometries that end on the complex roots of the
equation $W(\Phi(b))=E(b)$. Just as the disk-sphere saddle in JT gravity,
these can be understood by analytically continuing the dilaton contour. Using
this contour to evaluate the on-shell action in fact will give the same answer
as in eq. (5.2), but now with $\Phi_{h}(b)$ complex.

Finally, note that in this analysis we have not analyzed the one-loop
contributions to this disk geometry with a closed geodesic. Since we have not
analyzed these factors, we should not expect to match the one-loop
contribution to the exact answer for the shift in the ground state energy. We
leave a more careful one-loop analysis for future work, perhaps using the
recent techniques of [46].

Figure 12: The cap with a closed geodesic of length $b$ is sewn to the
exterior trumpet along the closed geodesic.

Nevertheless, having constructed the trumpet and the disk bounded by a closed
geodesic, we now sew these two spacetimes to form the pacifier, akin to what
is done in JT gravity. We glue the two spacetimes along the closed geodesic
and then integrate over its length, $b$. There is of course a question of the
measure for this integral. Since we expect there to again be a zero-mode
corresponding to twisting the disk-gometry relative to the trumpet, we expect
the measure to again be the same as in JT gravity.171717Note that the measure
for $b$ is an informed guess and we have not derived it. All in all we get
that these partially on-shell and partially off-shell geometries contribute

$\displaystyle\rho_{\rm
pacifier}(E)\sim\frac{1}{\sqrt{\pi(E-E_{0})}}\int_{0}^{\infty}dbb\
\cos\left(\frac{b\sqrt{E-E_{0}}}{G_{N}}\right)\times Z_{disk}(b),$ (5.12)

up to energy independent prefactors. Here, $Z_{disk}(b)$ is given by the
exponential of the action in (5.2).

###  5.3 The pacifier contribution to $\braket{\ell}{\ell^{\prime}}$

Given eq. (5.12) for the contribution to the density of states from pacifier-
type spacetimes, we can ask how the pacifier affects the inner product in the
geodesic length basis. First, we point out that the trumpet does in fact
contribute to the length inner product. This is because for any two points on
the asymptotic boundary of the trumpet, there are two geodesics connecting
those points. In order to show this, we do an analysis of the geodesics on the
trumpet in App. B.

However, supposing access to the length basis wavefunctions $\psi_{E}(\ell)$
defined in Sec. 4, we can easily get the length basis inner-product from
expression eq. (5.12). To see this, we can use the fact that the fixed energy
wavefunctions $\psi_{E}(\ell)$ discussed in Sec. 4 form a complete basis in
length so that

$\displaystyle\int_{-\infty}^{\infty}d\ell\psi_{E}^{*}(\ell)\psi_{E^{\prime}}(\ell)=\frac{\delta(E-E^{\prime})}{\rho_{0}(E)}$
(5.13)

where $\rho_{0}(E)$ is the density of states predicted by canonical
quantization. Using this relation, we see that

$\displaystyle\braket{\ell}{\ell^{\prime}}\supset\int_{E_{0}}^{\infty}dE\frac{1}{\sqrt{\pi(E-E_{0})}}\int_{0}^{\infty}dbb\
\cos\left(\frac{b\sqrt{E-E_{0}}}{G_{N}}\right)\times
Z_{disk}(b)\times\psi_{E}^{*}(\ell)\psi_{E}(\ell^{\prime}).$ (5.14)

In JT gravity, at the level of the disk topology, this inner product is
trivial,
$\braket{\ell}{\ell^{\prime}}=\delta(\ell-\ell^{\prime})+\mathcal{O}(e^{-S_{0}})$,
since the only purely AdS2 geometry with boundary conditions given by two
fixed length geodesics ending on the same asymptotic time is the trivial
geometry. In other words, at infinite $S_{0}$, the analog of $Z_{disk}(b)$ is
$0$.181818Or really, the analog of the wavefunction $Z_{disk}(b)$ averages to
zero in JT gravity, when JT gravity is viewed as an ensemble of dual quantum
theories. This fact that in a specific member of the JT ensemble,
$Z_{disk}(b)$ is a random-number perhaps suggests a connection between fixed
members of the ensemble and deformed dilaton gravity with random couplings, as
was suggested in [47, 48]. In fact, the deformations considered in [47, 48]
are of the exponential type considered here, but with complex $\alpha$, since
they correspond to opening up geodesic holes in the spacetime. If, however,
one allows surfaces of higher topology, this inner product can acquire off-
diagonal contributions. This fact was used to argue that at late times a black
hole in JT gravity may tunnel into a white hole [49]. See [25] for further
discussion of this point. This means that if one works at large $S_{0}$ in JT
gravity, the length basis is a “good” basis in the sense that length states of
different length $\ell$ are independent. We have seen here that with more
general dilaton potentials, the length basis is a bad basis already at the
disk level since $Z_{disk}(b)$ is non-zero.

###  5.4 Pacifiers and the non-perturbative shift in the ground state energy

We now show that the expression for $\rho_{\rm pacifier}(E)$ in eq. (5.12)
contributes an enhancement to the full density of states near $E\approx
E_{0}$. We interpret this as a non-perturbatively small shift in the ground
state energy of the model due to the pacifier spacetime. We compare with the
exact answer found Sec. 3.

It is clear that the expression in (5.12) diverges like $1/\sqrt{E-E_{0}}$ at
small $\delta E\equiv E-E_{0}$ so that

$\displaystyle\rho_{\rm
pacifier}(E)\approx\frac{c(\lambda,\alpha)}{\sqrt{E-E_{0}}}+\mathcal{O}((E-E_{0})^{0}).$
(5.15)

We would like to compute the coefficient $c(\lambda,\alpha)$. This requires us
to do the $b$ integral in expression (5.12) at small $\delta E$. Since we are
in the semi-classical, $G_{N}\to 0$ limit, we could can look for a saddle in
$b$.

Treating each branch of the cosine in (5.12) separately and using the
expression for $S_{disk}(b)$ in (5.2), we find that the saddle point equation
for $b$ is

$\displaystyle 2\pi\Phi_{h}^{\prime}(b)\pm
i\sqrt{E-E_{0}}+\sqrt{E_{0}-W(\Phi_{h}(b))}-\frac{bU(\Phi_{h})\Phi_{h}^{\prime}(b)}{2\sqrt{E_{0}-W(\Phi_{h}(b))}}=0.$
(5.16)

Using the expression for $\beta$ given in (5.9) together with equation (5.4),
we get that the first and last terms cancel. So the saddle point equation for
$b$ just sets

$\displaystyle E=W(\Phi_{h}(b)).$ (5.17)

This is not surprising since integrating over $b$ should just enforce a jump
condition that $\Delta\partial_{n}\phi=0$ at the junction where the two
geometries were sewed. Such a jump condition indeed enforces that
$E=W(\Phi_{h}(b))$. After setting $b$ on-shell, we see that our geometry just
becomes that associated to the complex saddle of the type discussed in Sec.
3.4.

In the end, we have

$\displaystyle\rho_{\rm
pacifier}\sim\frac{1}{\sqrt{E-E_{0}}}\exp\left(\frac{2\pi
W^{-1}(E_{0})}{G_{N}}\right)\times\left(\text{one-loop}\right),$ (5.18)

where the one-loop factor comes from fluctuations about the pacifier geometry.
Combining this with the prediction from canonical quantization for the low-
energy density of states given in (4.20), we get that the non-perturbative
shift in the ground state energy takes the form

$\displaystyle\delta E\sim(\text{one-loop})\times
e^{2\pi\left(W^{-1}_{inner}(E_{0})-W^{-1}_{outer}(E_{0})\right)/G_{N}}$ (5.19)

where $W^{-1}_{inner/outer}(E_{0})$ is the inner/outer-most root of the
equation $W(\Phi)=E_{0}$. This agrees with the expression for the shift in the
ground state energy from evaluating the exact density of states in eq. (3.10),
up to one-loop factors. Again, we have not done a thorough analysis of the
one-loop factors for the pacifier geometries, but this is obviously an
important calculation for the future.

##  6 Classical dilaton gravity from a gas of very blunt defects

So far we have explored the classical solutions of dilaton gravity and their
contributions to the path integral in the $G_{N}\to 0$ limit. As explained in
Sec. 1 and outlined in [15, 16, 18], an alternative route can be taken for the
class of dilaton potentials we consider, which are deformed away from JT
gravity by an exponential in the dilaton - $e^{-2\pi(1-\alpha)\phi}$ (or
linear combinations thereof with different powers $\alpha$). This route
amounts to expanding perturbatively in these deformations and integrating
insertions of exponential operators over the spacetime.

More explicitly, for a dilaton potential of the form

$\displaystyle U(\Phi)=2\Phi+2\pi(1-\alpha)\lambda e^{-2\pi(1-\alpha)\Phi}$
(6.1)

one can expand the partition function at small $\lambda$. At order $L$ in the
$\lambda$ expansion, the gravity partition function receives a correction of
the form

$\displaystyle
Z_{grav}|_{\mathcal{O}(\lambda^{L})}\sim\frac{\left(2\pi(1-\alpha)\lambda\right)^{L}}{L!}\int
D\Phi Dg\
\int\left(\prod_{i=1}^{L}d^{2}x_{i}\right)\exp\left(-I_{JT}-\sum_{i=1}^{L}2\pi(1-\alpha)\Phi(x_{i})\right)$
(6.2)

with $I_{JT}$ given by

$\displaystyle
I_{JT}=-\frac{S_{0}}{2\pi}\left(\frac{1}{2}\int_{M}\sqrt{g}R+\int_{\partial
M}\sqrt{h}K\right)-\frac{1}{2}\int_{M}\sqrt{g}\Phi(R+2)-\int_{\partial
M}\sqrt{h}\Phi(K-1).$ (6.3)

Note that for the moment, we have a different normalization for the action
than what we discussed in eq. (1.1) in Sec. 1. The normalization in (6.3)
matches the convention of [17]. To compare with previous sections, we would
like to work in a saddle point approximation to the dilaton gravity path
integral. To do this, it is helpful to re-scale $\Phi\to\Phi/G_{N}$, with
$G_{N}$ some small, dimensionless number. We can think of $G_{N}$ as
controlling the size of the boundary value of the dilaton, which in JT gravity
controls for example the Schwarzian coupling. As described in the
introduction, to get classical dilaton gravity, we should then re-scale the
couplings

$\displaystyle\lambda 2\pi(1-\alpha)\to\frac{\lambda 2\pi(1-\alpha)}{G_{N}},$
$\displaystyle(1-\alpha)\to G_{N}(1-\alpha).$ (6.4)

In the $G_{N}\to 0$ limit, this corresponds to making the defect insertions
very blunt so that they do not perturb the geometry very much. We will see
that the saddle points of dilaton gravity are built up out of many (order
$1/G_{N}$) blunt defects. As we now show, the smooth geometries that dominate
the semi-classical dilaton gravity path integral can then be thought of as
made up of a dense gas of very blunt defects.

In this spirit, we introduce another field, which is the defect density field

$\displaystyle\rho_{0}(x)=\frac{1}{L}\sum_{i=1}^{L}\delta^{2}(x-x_{i})/\sqrt{g}.$
(6.5)

We would then like to focus on terms in the defect expansion in (6.2) with
fixed defect number $L$, where $L$ scales with $1/G_{N}$ as

$\displaystyle L\sim\kappa/G_{N}$ (6.6)

with $\kappa$ fixed. For such terms, we find that (6.2) becomes

$\displaystyle\frac{(\lambda 2\pi(1-\alpha))^{L}}{G_{N}^{L}L!}\int D\Phi
Dg\int\prod_{i}d^{2}x_{i}\sqrt{g_{i}}\exp\left(\frac{1}{2G_{N}}\int_{M}\sqrt{g}\left(\Phi
R+2\Phi-4\pi(1-\alpha)\kappa\rho_{0}(x)\Phi(x)\right)\right)$ (6.7)

where $\kappa\equiv LG_{N}$ is fixed as we take $G_{N}\to 0$ and $L\to\infty$.
For now, we are ignoring the Gibbons-Hawking boundary term in the action.

To proceed, we need to deal with the measure $\int\prod_{i}d^{2}x_{i}$. We
follow a standard procedure given in [50] and insert the identity by writing

$\displaystyle\int D\rho(x)\ \delta\left(\rho(x)-\rho_{0}(x)\right)=1$ (6.8)

and then exchanging the order of integration. Writing the delta function in
terms of an auxiliary density, $\hat{\rho}(x)$, we get

$\displaystyle\int
D\hat{\rho}(x)\int\prod_{i}\left(d^{2}x_{i}\sqrt{g_{i}}\right)\exp\left(i\int_{M}d^{2}x\sqrt{g}\
L\hat{\rho}(x)\left(\rho(x)-\frac{1}{L}\sum_{i}\frac{\delta^{2}(x-x_{i})}{\sqrt{g}}\right)\right)$
(6.9)

This becomes

$\displaystyle\int D\hat{\rho}(x)\exp\left(i\int_{M}d^{2}x\sqrt{g}\
L\hat{\rho}(x)\rho(x)\right)\left(\int d^{2}y\sqrt{g}e^{-i\int
d^{2}x\hat{\rho}(x)\delta^{2}(x-y)}\right)^{L}$ $\displaystyle=\int
D\hat{\rho}(x)\exp\left(i\int_{M}d^{2}x\sqrt{g}\
L\hat{\rho}(x)\rho(x)+L\log\left(\int d^{2}y\sqrt{g}\
e^{-i\hat{\rho}(y)}\right)\right)$ (6.10)

Evaluating this integral by saddle point to eliminate the $\hat{\rho}$
integral, we get that the saddle point equation is

$\displaystyle\rho(x)=\frac{e^{-i\hat{\rho}(x)}}{\int d^{2}y\sqrt{g}\
e^{-i\hat{\rho}(y)}}$ (6.11)

Plugging this solution back into the action we end up with the integral

$\displaystyle=\frac{(\lambda 2\pi(1-\alpha))^{L}}{G_{N}^{L}L!}\int D\Phi
DgD\rho_{=}\exp\left(\frac{1}{2G_{N}}\int_{M}\sqrt{g}\left(\Phi
R+2\Phi-4\pi(1-\alpha)\kappa\rho(x)\Phi(x)-2\kappa\rho(x)\log\rho(x)\right)\right)$
(6.12)

where by $D\rho_{=}$ we mean integrating over all unit normalized $\rho$,
$\int d^{2}x\sqrt{g}\rho(x)=1$.

The term in the action proportional to $\rho\log\rho$ came from the path
integral measure upon changing from an integral over $x_{i}$’s to an integral
over $\rho$. This term has a simple interpretation in terms of the statistical
entropy. It counts the number of microscopic configurations of defects for a
given density profile, $\rho$. This entropic term creates a statistical
pressure for the defects to not be on top of each other. On the other hand,
due to the coupling to the dilaton, there is a pressure to bring the defects
together, toward the minimum of the dilaton. As advertised, and shown
explicitly below, these two forces balance to give stable solutions.

To deal with the normalization condition on the $\rho$ field, we can insert a
Lagrange multiplier to enforce this condition, and so the path integral at
fixed defect number becomes

$\displaystyle=$ $\displaystyle\frac{(\lambda
2\pi(1-\alpha))^{L}}{G_{N}^{L}L!}\int D\Phi DgD\rho\,dz\times$
$\displaystyle\exp\left(\frac{1}{2G_{N}}\int_{M}\sqrt{g}\left(\Phi
R+2\Phi-4\pi(1-\alpha)\kappa\rho(x)\Phi(x)-2\kappa\rho(x)\log\rho(x)+2iz\kappa(\rho(x)-1)\right)\right).$
(6.13)

This is the JT path integral with a large number, $L$, of defects on the
background, ignoring higher order contributions from the measure. We now want
to find saddle points for this integral.

###  6.1 Classical solutions for the defect density at fixed boundary length
and fixed defect number

Varying the action in (6) with respect to the $\rho,z$ and $\Phi$ gives

$\displaystyle 2\pi(1-\alpha)\Phi+\log\rho+1+iz=0,$ $\displaystyle\int
d^{2}x\sqrt{g}\rho(x)=1,$ $\displaystyle R=-2+4\pi(1-\alpha)\rho(x).$ (6.14)

The solution to the first two of these equations is just

$\displaystyle\rho(x)=Ce^{-2\pi(1-\alpha)\Phi(x)}$ (6.15)

with $C$ determined by the normalization condition

$\displaystyle\int d^{2}x\sqrt{g}\rho(x)=\int
d^{2}x\sqrt{g}Ce^{-2\pi(1-\alpha)\Phi(x)}=1.$ (6.16)

Before we solve for the metric, we have already learned something interesting.
Classically the defects bunch up exponentially toward the region of more
negative dilaton.191919This exponential decay of the density is exactly
analogous to the Debye screening effect in charged metals at finite
temperature. We thank Steve Shenker for emphasizing this point to us.

To solve for the metric, we follow the steps laid out in Sec. 2 and [27] and
again look for solutions of the form

$\displaystyle ds^{2}=A(r)d\tau^{2}+\frac{dr^{2}}{G(r)}$ (6.17)

where we make the identification $\tau\to\tau+\beta$. Varying the action with
respect to $G$ and then using a gauge freedom to set $G=A$, we get

$\displaystyle\Phi^{\prime}(r)A^{\prime}(r)=2\Phi(r)-4\pi(1-\alpha)\kappa\rho(r)\Phi(r).$
(6.18)

Plugging in the ansatz in (6.17) and varying the action in (6) with respect to
$A$ further gives

$\displaystyle\Phi^{\prime\prime}=0.$ (6.19)

We see that just as before we can set $r=\Phi$.

Assuming that the horizon occurs at $\Phi=\Phi_{h}$ and using the form of the
metric in (6.17), the normalization condition in (6.16) implies

$\displaystyle C=\frac{2\pi(1-\alpha)e^{2\pi(1-\alpha)\Phi_{h}}}{\beta}.$
(6.20)

Plugging in for $\rho$ in (6.18) and integrating twice, the metric component
$A(\Phi)$ then takes the form

$\displaystyle A(\Phi)=W_{\lambda(\kappa)}(\Phi)-E.$ (6.21)

with $E$ an integration constant, which one can check is the AdM mass of the
solution. Here we have defined the fixed-$\kappa$ pre-potential as

$\displaystyle
W_{\lambda(\kappa)}(\Phi)=\Phi^{2}-\lambda(\kappa)e^{-2\pi(1-\alpha)\Phi},$
(6.22)

where we have defined an effective coupling at fixed $\kappa$ to be

$\displaystyle\lambda(\kappa,\beta)\equiv\frac{\kappa
C}{\pi(1-\alpha)}=\frac{2\kappa}{\beta}e^{2\pi(1-\alpha)\Phi_{h}(\kappa,\beta)}.$
(6.23)

One should think of eq. (6.22) as the effective dilaton pre-potential induced
by a $\kappa/G_{N}$ number of defects. Note that the effective coupling of the
theory depends on the asymptotic boundary length $\beta$. This dependence is a
result of the non-local condition on the theory that the defect density be
normalized to one, eq. (6.16).

Demanding that this metric is smooth at the horizon enforces that

$\displaystyle\beta=\frac{(2\pi-2\pi(1-\alpha)\kappa)}{\Phi_{h}}.$ (6.24)

A way to interpret this equation is that the dilaton at the horizon $\Phi_{h}$
is the generator of rotations and so $2\pi-\beta\Phi_{h}$ is the conical
deficit of the geometry.202020It may be confusing to the reader that we are
demanding that the horizon be smooth but yet are interpreting the solution in
terms of defects, which are not smooth. This is because we have taken the
scaling limit of large defect number and small $(1-\alpha)$, where the blunt
defects condense and approximate a smooth geometry. While in pure JT gravity
without defects the solution had a full angle of rotation by $2\pi$, now the
defects have reduced this angle by an amount $2\pi(1-\alpha)\kappa$ which in
the original, un-rescaled variables is just $2\pi(1-\alpha)L$. Furthermore,
note that the energy is related to the dilaton at the horizon, $\Phi_{h}$, by

$\displaystyle\Phi_{h}^{2}-\frac{2\kappa\Phi_{h}}{2\pi-2\pi(1-\alpha)\kappa}=\left(\frac{2\pi(1-(1-\alpha)\kappa)}{\beta}\right)^{2}-\frac{2\kappa}{\beta}=E.$
(6.25)

We would now like to plug everything in and compute the on-shell action. The
total action including boundary terms at fixed, renormalized boundary length
$\beta$ is then

$\displaystyle\frac{1}{2G_{N}}\left[\beta\int_{\Phi_{h}}^{\Phi_{b}}d\Phi\left(-\Phi
A^{\prime\prime}+2\Phi-4\pi(1-\alpha)\kappa\rho\Phi-2\kappa\rho\log\rho+2iz\kappa\rho\right)-2iz\kappa\right]+\frac{\Phi_{b}}{G_{N}}\int
du\sqrt{h}(K-1).$ (6.26)

One way to evaluate the on-shell action is to integrate by parts on $A$. The
boundary term from this integration by parts will kill the other boundary term
proportional to the extrinsic curvature. Using that
$A^{\prime}(\Phi_{h})\beta=4\pi$ as well as the equations of motion, we then
get that the total on-shell action is

$\displaystyle S_{on-
shell}(\kappa,\beta)=\frac{1}{G_{N}}\left((\pi-\pi(1-\alpha)\kappa)\Phi_{h}(\beta)+\kappa\log\frac{\beta}{2\pi(1-\alpha)}\right).$
(6.27)

Up to one-loop contributions to this answer, which we briefly discuss in App.
A, this exactly matches the answer found by [18] if we reintroduce $G_{N}$’s
by sending $(1-\alpha)\to(1-\alpha)/G_{N}$ and $\kappa=LG_{N}$. This formula
has an interpretation in terms of an “effective” defect with opening angle
$(1-(1-\alpha)\kappa)$. Indeed, [18] found the answer given by (6.27) by doing
the exact integral over the moduli space of hyperbolic disks with $L$ defects,
which they found localizes to the configuration where all $L$ defects sit on
top of each other. The picture in [18] then seems different from what we have
here since in our picture the defects are spread out over the geometry (albeit
with a density decaying exponentially in $\Phi$). We would like to understand
the origin of this difference more clearly.

###  6.2 Behavior of solutions at large $\kappa$

Figure 13: We plot the effective dilaton potential $W_{\lambda(\kappa)}(\Phi)$
for increasing values of $\kappa$ as one moves clockwise from the upper left.
There is a critical $\kappa$ where the potential becomes non-monotonic. At
this point, the energy of the solution $W_{\lambda(\kappa)}(\Phi_{h})$ is
below the outermost minimum of the potential and so the geometry develops a
throat. At larger $\kappa$, the energy goes above the outermost minimum and
the throat pinches off.

As discussed in Sec. 2, something special happens when the number of defects
is so large that their net angle deficit is bigger than $2\pi$. Indeed, this
number of defects is precisely the order at which the string equation
truncates in the $\lambda$ expansion, as shown in eq. (2.11). Intuitively,
this is because the defects have soaked up all the opening angle in the
geometry. In response, a closed geodesic nucleates in the geometry akin to
what happens in the pacifier spacetimes of Sec. 5. We will show that indeed
one can think of the pacifier as also arising from a dense gas of defects with
total defect number larger than $\kappa_{c}$. To understand this, we would
like to see how our density picture sees the development of a throat for
increasing defect number.

In our scaling limit, the net deficit angle for the defects becomes greater
than $2\pi$ when $2\pi(1-\alpha)LG_{N}>2\pi$ or in terms of the re-scaled
defect number, $\kappa$, we have

$\displaystyle\kappa>\kappa_{c}\equiv 1/(1-\alpha).$ (6.28)

In our representation in terms of a smooth density of defects, near
$\kappa=\kappa_{c}$ the spacetime geometry changes dramatically. To see this,
we can plot the effective dilaton pre-potential, $W_{\lambda(\kappa)}(\Phi)$,
which appears in the metric of our fixed $\kappa$ saddle, which we reproduce
here:

$\displaystyle
W_{\lambda(\kappa)}(\Phi)=\Phi^{2}-\frac{2\kappa}{\beta}e^{2\pi(1-\alpha)(\Phi_{h}(\beta)-\Phi)},$
$\displaystyle\Phi_{h}(\beta)=\frac{2\pi(1-(1-\alpha)\kappa)}{\beta}.$ (6.29)

According to these formulae, we can think of dialing $\kappa$ as dialing the
coupling in the dilaton potential $W_{\lambda}(\Phi)$ discussed in Sec. 2
where here the effective coupling is again

$\displaystyle\lambda(\kappa)=\frac{2\kappa}{\beta}e^{2\pi(1-\alpha)\frac{2\pi(1-(1-\alpha)\kappa)}{\beta}}.$
(6.30)

In Sec. 2, we found that there was a critical coupling
$\lambda_{c}=\frac{1}{e2\pi^{2}(1-\alpha)^{2}}$ above which the pre-potential
$W_{\lambda}(\Phi)$ becomes monotonic. From this, we have the following
characterization of the fixed-$\kappa$ pre-potential:

$\displaystyle W_{\lambda(\kappa)}(\Phi)\ \text{is monotonic if
}\lambda(\kappa)>\lambda_{c},$ $\displaystyle W_{\lambda(\kappa)}(\Phi)\
\text{is non-monotonic if }\lambda(\kappa)<\lambda_{c}.$ (6.31)

As we increase $\kappa$ from near zero, one can show that $\lambda(\kappa)$
begins increasing and eventually passes $\lambda_{c}$. At very large $\kappa$,
however, the exponential in eq. (6.30) begins to take over and
$\lambda(\kappa)$ rapidly goes back below $\lambda_{c}$. Note that all the
while the dilaton at the horizon, $\Phi_{h}$, decreases according to eq.
(6.22). As one can check, once $\lambda(\kappa)$ drops below $\lambda_{c}$
from above, $\Phi_{h}$ is in fact less than the dilaton value,
$\Phi_{outermin}$, at the outer-most minimum of $W_{\lambda(\kappa)}$. In
fact, when $\lambda(\kappa)$ reaches $\lambda_{c}$ for the second time, one
can show that the AdM energy of the solution, $E(\kappa)$, is below
$W_{\lambda(\kappa)}(\Phi_{outermin})$. This means the geometry has a throat.
As $\kappa$ increases further, the energy of the solution increases until
$\kappa$ reaches $\kappa_{\text{pinch}}$ where the throat degenerates and the
geometry pinches off. The geometry can be continued in $\kappa$ beyond this
point but it becomes complex. We are not sure how seriously to take the saddle
when the throat has degenerated. We suspect that by this point the saddle has
left the integration contour. The evolution of $W_{\lambda(\kappa)}(\Phi)$ as
$\kappa$ is increased is illustrated in Fig. 13.

In short, as we crank up the defect number fixing the boundary temperature,
the density of defects builds up near the horizon, creating a throat that
eventually pinches off. As discussed in the next subsection, we expect that
this pinching off effectively truncates the integral over $\kappa$ in eq.
(6.3) rendering the full answer finite, although this deserves further study.

###  6.3 Finding the saddle point in $\kappa$

Having computed the contribution to the fixed boundary length path integral at
fixed defect number, $\kappa$, we would like to see how the sum over defect
number reproduces the classical answer. As we will now show, the sum over
defect number has a saddle point in defect number, $\kappa_{*}$, which depends
upon the dilaton coupling $\lambda$ in such a way that the on-shell effective
pre-potential, defined in eq. (6.22), is equal to the actual pre-potential of
the theory at fixed $\lambda$:

$\displaystyle W_{\lambda(\kappa_{*})}(\Phi)=W_{\lambda}(\Phi),\ \
\lambda(\kappa_{*})=\lambda.$ (6.32)

To see this, we go back to the full sum over number of defects which defines
the full gravitational path integral which is

$\displaystyle=$
$\displaystyle\sum_{L}\frac{(\lambda\pi(1-\alpha))^{L}}{G_{N}^{L}L!}\int D\Phi
DgD\rho\,dz\times$
$\displaystyle\exp\left(\frac{1}{2G_{N}}\int_{M}\sqrt{g}\left(\Phi
R+2\Phi-4\pi(1-\alpha)\kappa\rho(x)\Phi(x)-2\kappa\rho(x)\log\rho(x)+2iz\kappa(\rho(x)-1)\right)\right).$
(6.33)

If we focus on terms in the sum where $L$ scales with $G_{N}$ like
$L=\kappa/G_{N}$, then we can replace the sum with an integral over $\kappa$.
At small $G_{N}$, the large $L$ contribution to the disk partition function
then becomes

$\displaystyle Z_{grav}$ $\displaystyle\sim\frac{1}{G_{N}}\int^{??}_{\sim
G_{N}}d\kappa\int D\Phi
DgD\rho\,dz\times\exp\left(\frac{\kappa}{G_{N}}\log\left(\frac{\lambda\pi(1-\alpha)}{\kappa}\right)+\frac{\kappa}{G_{N}}\right)\times$
$\displaystyle\exp\left(\frac{1}{2G_{N}}\int_{M}\sqrt{g}\left[\Phi
R+2\Phi-4\pi(1-\alpha)\kappa\rho(x)\Phi(x)-2\kappa\rho(x)\log\rho(x)+2iz\kappa(\rho(x)-1)\right]\right).$
(6.34)

where we used Stirling’s approximation on the $L!$ in (6.3). We have left the
upper limit on $\kappa$ ambiguous for now but come back to this point
momentarily. The lower limit is set by the value at which the density
approximation becomes untrustworthy which is $\kappa$ of order $G_{N}$.

Differentiating the action with respect to $\kappa$, we see that we should
supplement the equations in eq. (6.1) with the equation

$\displaystyle\log\left(\frac{\lambda\pi(1-\alpha)}{\kappa}\right)-\int
d^{2}x\sqrt{g}\left(2\pi(1-\alpha)\rho\Phi+\rho\log\rho\right)+iz\left(\int
d^{2}x\sqrt{g}\rho-1\right)=0.$ (6.35)

Using the three equations in eq. (6.1) and again defining the effective
coupling constant $\lambda(\kappa)\equiv\frac{C\kappa}{\pi(1-\alpha)}$ where
$\rho(x)=Ce^{-2\pi(1-\alpha)\Phi(x)}$, we see that eq. (6.35) implies

$\displaystyle\lambda(\kappa_{*})=\lambda,$ (6.36)

as we wanted to show. This confirms that we should think of these smooth on-
shell geometries as a dense collection of defects with total defect number set
by eq. (6.36).

We can be a bit more explicit for $\kappa_{*}$ by using the equation for
$\lambda(\kappa)$ in (6.30). Further use of eq. (6.24) allows us to solve for
$\kappa_{*}$ in terms of $\lambda,\beta$ as

$\displaystyle\kappa_{*}=\frac{-\beta\mathcal{W}\left(-2\pi^{2}\lambda(1-\alpha)^{2}e^{-4\pi^{2}(1-\alpha)/\beta}\right)}{4\pi^{2}(1-\alpha)^{2}},$
(6.37)

where $\mathcal{W}$ is the Lambert W-function, defined by the equation
$\mathcal{W}(x)e^{\mathcal{W}(x)}=x$.

Interestingly, we see that for high temperatures where
$1/\beta\gtrsim\log(1/G_{N})$ then $\kappa\lesssim G_{N}$. In this regime, it
seems that we leave the large $L=\kappa/G_{N}$ approximation. This happens
because at such high temperatures the Euclidean disk geometry is so close to
the regular JT gravity solution that you need only a few defects to account
for the difference. For these temperatures and energies, the difference
between the JT metric at the horizon and the exact metric is polynomially
suppressed in $G_{N}$. Note, that by the discussion in Sec. 2, we expect these
geometries to have a space-like singularity in the interior. Apparently then
just a few exterior defects can modify the black hole interior drastically.

One final point is that the saddle point equation for $\kappa_{*}$ in (6.37)
in fact has one other real solution in $\kappa$, since the Lambert W-function
can take multiple real values on its different branches. This can be seen
clearly in Fig. 14 for particular values of $\lambda,\beta$ and $\alpha$.
Geometrically, these solutions with larger $\kappa$ will be the solutions
which we discussed in the previous subsection, Sec. 6.2, which have $\Phi_{h}$
on the third branch of the inverse of the effective dilaton potential,
$W_{\lambda(\kappa)}^{-1}$. One can check numerically however that this second
saddle point $\kappa_{*}^{(2)}$ always appears to occur above the critical
number, $\kappa_{\mathrm{pinch}}$, where the throat in the geometry pinches
off as discussed in the previous subsection. This suggests to us that this
other, sub-leading saddle point in the $\kappa$ integral will not contribute
to the actual answer. Furthermore, we can see from Fig. 14 that the $\kappa$
integral in Eq. (6.3) does not converge unless there is an upper limit on the
integral. Again, we suspect that this cutoff is provided by the effects
discussed in the previous subsection. This point clearly deserves further
study.

Figure 14: This figure shows a plot of the integrand in eq. (6.3) for
$\lambda\approx.01,\alpha=0$ and $\beta\approx 60$. This shows that the
integrand has two saddle points at small $G_{N}$: one at small $\kappa$ which
is stable and another, unstable saddle point at larger $\kappa$. The integral
of this function is not convergent on the full real line and so must get
regulated. We speculate on how this happens in the main text.

###  6.4 The Weil-Peterson volume for a large number of very blunt defects and
a single closed geodesic

Using these ideas, it is tempting to try to compute the Weil-Peterson volumes
for a geometry with a large number of blunt defects and a single geodesic
boundary.212121The setup gets more complicated for higher numbers of geodesic
boundaries but we suspect something akin to these techniques might generalize.
To understand this, we will look for saddle points with fixed defect number
$\kappa$ on a geometry which is bounded by a geodesic of length $b$. We will
see that this is only possible if $\kappa>\kappa_{c}$, as demanded by a simple
application of the Gauss-Bonnet theorem.

To find solutions with a closed geodesic, we take inspiration from the
discussion in Sec. 5.2 and we look for solutions with the AdM energy below the
outermost minimum of $W_{\lambda(\kappa)}$. As discussed in Sec. 5.2, the
closed geodesic will then sit at constant $\Phi=\Phi_{*}$, where
$W_{\lambda(\kappa)}$ has this outermost minimum at $\Phi_{*}$. The equations
of motion just as before give the metric component in terms of the dilaton

$\displaystyle
A(\Phi)=\Phi^{2}-\frac{2\kappa}{2\pi(1-\alpha)}Ce^{-2\pi(1-\alpha)\Phi}-E(b,\kappa)$
(6.38)

where the time coordinate has periodicity $\tau\to\tau+\beta$, which will be
determined by the equations below. Before doing that, we need to determine $C$
from the normalization condition

$\displaystyle\beta\int_{\Phi_{h}}^{\Phi_{*}}d\Phi\rho(\Phi)=1\implies
C=\frac{2\pi(1-\alpha)}{\beta(e^{-2\pi(1-\alpha)\Phi_{h}}-e^{-2\pi(1-\alpha)\Phi_{*}})},$
(6.39)

where $\Phi_{h}$ is the value of the dilaton at the cap so that

$\displaystyle
E(b,\kappa)=W_{\lambda(\kappa)}(\Phi_{h})=\Phi_{h}^{2}-\frac{2\kappa}{\beta(1-e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})})}.$
(6.40)

Smoothness at the cap demands

$\displaystyle\beta\Phi_{h}+\frac{2\pi(1-\alpha)\kappa}{1-e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})}}=2\pi.$
(6.41)

We have two further conditions from the fact that $\Phi_{*}$ occurs at the
minimum of the dilaton potential and another coming from demanding that the
proper length of this geodesic is $b$. These two equations read

$\displaystyle 2\Phi_{*}+\frac{4\pi(1-\alpha)\kappa
e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})}}{\beta(1-e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})})}=0$
(6.42)

and

$\displaystyle\Phi_{*}^{2}-\frac{2\kappa
e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})}}{\beta(1-e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})})}-E(b,\kappa)=\frac{b^{2}}{\beta^{2}}.$
(6.43)

Note that if we take eq. (6.42), mulitply through by $\beta$ and subtract it
off from eq. (6.41), we get the constraint

$\displaystyle\beta(\Phi_{h}-\Phi_{*})=2\pi-2\pi(1-\alpha)\kappa.$ (6.44)

From this we wee that if $\Phi_{*}>\Phi_{h}$ then $\kappa>\kappa_{c}$ as
expected. Equation (6.44) follows simply from Guass-Bonnet applied to these
geometries. Note that eq.’s (6.40), (6.41), (6.41) and (6.42) comprise four
equations for the four unknowns $\Phi_{h},\Phi_{*},E,\beta$ which in principle
allow us to solve for these parameters in terms of the fixed quantities
$\kappa$ and $b$. Unfortunately, solving these equations analytically is not
possible except in certain limits.

One interesting effect that one can see numerically is that for fixed
$\kappa>1/(1-\alpha)$, as we crank up $b$, there are not solutions with
arbitrarily large $b$. Note that this is precisely what happened when we
analyzed $Z_{disk}(b)$ in Sec. 5. The reason is that as we crank up $b$, the
value of the dilaton at the horizon increases even though the energy of the
solution, $E(b,\kappa)$, is decreasing. This leads to an increase in the
effective coupling constant in $\lambda(\kappa)$. At a critical value of $b$,
the effective coupling constant $\lambda(\kappa,b)$ hits the critical coupling
constant $\lambda_{c}$ discussed above and the metric in eq. (6.38) becomes
monotonic as a function of $\Phi$. This means it no longer contains a real
closed geodesic.

To compute the WP volume for a hyperbolic geometry with $L=\kappa/G_{N}$
number of defects and a single geodesic boundary of length $b$, we then just
need to compute the Shannon entropy of this solution since there are no
boundary terms present in the problem. The density is

$\displaystyle\rho(\Phi)=Ce^{-2\pi(1-\alpha)\Phi}$ (6.45)

where $C$ is given above. Introducing the notation

$\displaystyle r=e^{2\pi(1-\alpha)(\Phi_{h}-\Phi_{*})}$ (6.46)

we have

$\displaystyle
C=\frac{2\pi(1-\alpha)}{\beta}\frac{e^{2\pi(1-\alpha)\Phi_{h}}}{1-r}.$ (6.47)

Note that by construction $r<1$ so $C>0$ as it should be. Then the Shannon
term becomes

$\displaystyle-\int d^{2}x\sqrt{g}\rho\log\rho=-\log
C+2\pi(1-\alpha)C\beta\int_{\Phi_{h}}^{\Phi_{*}}d\Phi
e^{-2\pi(1-\alpha)\Phi}\Phi$ $\displaystyle=-\log
C+\frac{1-r+2\pi(1-\alpha)\Phi_{h}-r2\pi(1-\alpha)\Phi_{*}}{1-r}$
$\displaystyle=\log\left(\frac{\beta(r)(1-r)}{2\pi(1-\alpha)}\right)+\frac{r}{1-r}\log
r+1.$ (6.48)