# Template

###### Abstract

We propose a definition for the Lorentzian Jackiw-Teitelboim (JT) gravity path
integral that includes Lorentzian topology changing configurations. The
construction is inspired by the bosonic string genus expansion on singular
Lorentzian worldsheets, with geometries known as lightcone diagrams playing a
prominent role. The Lorentzian path integral is defined through a suitable
analytic continuation of the Euclidean path integral, and includes metrics
that become degenerate at isolated points allowing for Lorentzian topology
changing transitions. We discuss the relation between Euclidean JT amplitudes
and the proposed Lorentzian amplitudes.

Comments on Lorentzian topology change in JT gravity

Mykhaylo Usatyuk

Center for Theoretical Physics and Department of Physics, University of
California, Berkeley,
CA 94720, USA

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Lightcone diagrams
    1. 2.1 Euclidean and Lorentzian lightcone diagrams
    2. 2.2 Negative curvature lightcone diagrams
  3. 3 JT gravity on lightcone diagrams
    1. 3.1 Lorentzian JT path integral
  4. 4 Discussion
  5. A Lorentzian pair of pants
  6. B Details on punctured Riemann surfaces and measures

##  1 Introduction

Euclidean wormholes have played an important role in recent developments in
quantum gravity. The study of two dimensional Jackiw-Teitelboim (JT) gravity
[1] has been central to these developments. Topology changing Euclidean
configurations have been found to be important in the calculation of a unitary
page curve [2, 3], the spectral form factor[4, 1, 5], and a variety of other
interesting effects, see [6, 7, 8] for a few selected examples.

However, so far a satisfactory Lorentzian explanation for Euclidean wormhole
calculations has been lacking. The standard procedure is to calculate a
Euclidean amplitude and then analytically continue it to obtain a Lorentzian
result. The role that Euclidean wormholes play in the Lorentzian theory is not
apparent in this analytic continuation. Similarly, in the standard canonical
quantization treatment of Lorentzian JT gravity[9] the topology of the
Lorentzian spacetime is fixed, and it’s unclear how the Euclidean path
integral formulation of the theory[1] can be related to the Lorentzian
formulation.

In this work we formulate a Lorentzian theory of JT gravity that includes
Lorentzian topology changing configurations. The theory is defined through a
special analytic continuation of the standard Euclidean path integral.
Euclidean wormholes are turned into topology changing Lorentzian
configurations with degenerate points in the metric, see figure 1.

Figure 1: Lorentzian topology changing transition where two spatial circles
evolves into three circles. The metric is Lorentzian everywhere except the
splitting points (blue circles) where it is degenerate. At the splitting
points the topology of spatial slices changes. The spacetime is an analytic
continuation of a genus two Euclidean geometry with five circular boundaries.

Our proposal is inspired by a formulation of bosonic string theory on
degenerate Lorentzian worldsheets known as the interacting string picture[10,
11]. In [12, 13] it was argued that the interacting string picture is
equivalent to the standard Euclidean path integral formulation to all orders
in the genus expansion. This is accomplished by gauge-fixing the Euclidean
path integral to lightcone diagrams[12], which are a special class of metrics
that can be analytically continued to Lorentzian signature where they become
the topology changing geometries of the interacting string picture.

To define the Lorentzian JT path integral we closely follow the construction
of the bosonic string genus expansion with singular Euclidean/Lorentzian
worldsheets [12, 13, 14]. We start with the standard Euclidean JT path
integral and with a suitable gauge choice and analytic continuation we end up
with a Lorentzian path integral over degenerate metrics. We now briefly
explain this construction, leaving technical details to the main sections.

### Summary of results

We begin with the two dimensional Euclidean gravity path integral with
specified boundary conditions. In this paper we consider boundary conditions
given by $n$ geodesic circles of given lengths $\vec{b}=(b_{1},\cdots,b_{n})$.
For simplicity we assume the bulk geometry has fixed genus $g$ and is fully
connected. The path integral is computed by

$Z=\int\frac{\mathcal{D}g_{\mu\nu}\mathcal{D}\Phi}{\text{Vol}}e^{-I_{{\mathrm{JT}}}[g,\Phi]}=\hskip
1.99168pt\raisebox{-51.21495pt}{\includegraphics[width=99.58464pt]{Section1/V_n2.pdf}}$
(1.1)

The integral over Euclidean metrics can be split into an integral over the
Weyl factor $\omega$ and over the moduli $\hat{g}$, where all metrics can be
represented as $g=e^{2\omega}\hat{g}$. The moduli space of $\hat{g}$ is the
moduli space of punctured Riemann surfaces $\mathcal{M}_{g,n}$ of genus $g$
with $n$ punctures. It was shown by Giddings and Wolpert [12] that this Moduli
space has a representative metric $\hat{g}$ that is flat with curvature
singularities at isolated points, this is known as a Euclidean lightcone
metric. Choosing the Euclidean lightcone metric as our representative metric
$\hat{g}$ gives us a path integral over singular Euclidean geometries

$Z=\int_{\rm moduli}{\mathrm{d}}({\rm
measure})\int\mathcal{D}\omega\mathcal{D}\Phi
e^{-I_{{\mathrm{JT}}}[e^{2\omega}\hat{g},\Phi]}=\raisebox{-51.21495pt}{\includegraphics[width=79.6678pt]{Section1/ELCSec1.pdf}}$
(1.2)

In the above figure, the circles (blue) corresponds to points where the metric
$\hat{g}$ becomes degenerate $\det\hat{g}=0$. It is at these points that the
topology of spatial slices changes. The integral over the moduli is partially
over all possible locations of the degenerate points. The Euclidean lightcone
diagram has a globally defined euclidean time $\tau$ in terms of which the
metric is flat everywhere except the degenerate points. The boundary
conditions are now specified at $\tau=\pm\infty$, and we must make a choice to
send boundary conditions either to the future or the past. In the above figure
the boundary of length $b_{1}$ has been sent to the past while the boundaries
of length $b_{2},b_{3}$ have been sent to the future.

To turn the above Euclidean path integral into a Lorentzian path integral, we
will analytically continue the Euclidean lightcone geometries. Since the time
$\tau$ is globally defined, we can analytically continue $\tau\to i\tau$ to
get a Lorentzian signature metric $\hat{g}$, known as a Lorentzian lightcone
diagram. We now have a path integral over degenerate Lorentzian metrics that
incorporate topology changing transitions

$Z_{\mathrm{L}}=\int_{\rm moduli}{\mathrm{d}}({\rm
measure})\int\mathcal{D}\omega\mathcal{D}\Phi
e^{iI_{{\mathrm{JT}}}[e^{2\omega}\hat{g},\Phi]}.$ (1.3)

Our _definition_ for the Lorentzian JT path integral will be the above
integral over Lorentzian lightcone diagrams. The role of the Weyl factor
$\omega$ will be to give the geometry constant negative curvature away from
the degenerate points.

In the above procedure we started with the usual Euclidean path integral and
through a gauge choice and a suitable analytic continuation we ended up with a
Lorentzian path integral. It might then be expected that the amplitudes
computed with this Lorentzian path integral should agree with the
corresponding Euclidean amplitudes. Indeed, this is the argument of D’Hoker
and Giddings[13] that the interacting string picture[11] is equal to the
Euclidean path integral to all orders in the genus expansion111Giddings and
D’Hoker [13] stopped short of analytically continuing the Euclidean lightcone
diagrams to Lorentzian signature. Thus they argued that the analytically
continued interacting string picture was equivalent to the usual Euclidean
path integral.. In the case of JT gravity there are some additional subtleties
that arise due to the degenerate points, and we return to this question in
section 3 and in the discussion. We now briefly summarize the rest of the
paper.

In Section 2 we review the basic aspects of Euclidean and Lorentzian lightcone
diagrams. We also review the work of Giddings and Wolpert [12] where it was
shown that Euclidean lightcone diagrams give a single cover of the moduli
space of punctured Riemann surfaces. Lastly, we discuss the problem of finding
a Weyl factor to turn a Euclidean/Lorentzian lightcone metric into a constant
negative curvature geometry with degenerate points.

In Section 3 we fill in the technical details of the path integral over
lightcone diagrams. We explain how boundary conditions are implemented, and we
discuss the integration measure over the moduli space of lightcone diagrams.
The inclusion of degenerate points introduces certain ambiguities into the
path integral, and we discuss how these ambiguities modify the relation of the
Euclidean amplitudes to the Lorentzian amplitudes.

In the Appendices we construct the Lorentzian pair of pants with constant
negative curvature, and we include additional details on punctured Riemann
surfaces and the integration measure.

##  2 Lightcone diagrams

In this section we review aspects of Euclidean and Lorentzian lightcone
diagrams, their connection to punctured Riemann surfaces, and how to construct
constant negative curvature analogues of lightcone diagrams.

###  2.1 Euclidean and Lorentzian lightcone diagrams

A Euclidean/Lorentzian lightcone diagram is a two dimensional geometry with
degenerate metric $g$ built out of flat Euclidean/Lorentzian cylinders joined
together at singular points, see figure 2. The two basic properties of a
lightcone diagram are the number of asymptotic cylinders $n$ and the genus
$g$. The number of cylinders running off to infinity is given by $n\geq 2$
with a fixed number extending to past or future infinity222At least one
cylinder must run to the future and one to the past.. In the next section we
will see that in/out states are specified by introducing boundary conditions
on the asymptotic cylinders.

Figure 2: Euclidean/Lorentzian lightcone diagram of genus two with five
boundaries. The geometry is flat except at interactions points $\tau_{i}$ with
delta functions of curvature. A lightcone diagram is built by gluing pairs of
pants together. The interaction times are labelled by $\tau$, the twist angles
by $\theta$, and the free internal radius by $\rho$.

The genus $g$ and number of boundaries $n$ determine the number of times the
diagram splits apart and joins together in the interior of the geometry. At
the splitting and joining points the metric is degenerate, and there are
$2g-2+n$ such degenerate points. We will also call these interaction/singular
points, and denote where they occur by the subscript $z_{I}$. The curvature on
a Euclidean lightcone diagram is given by

$\frac{1}{2}\sqrt{g}R=-2\pi\sum_{I=1}^{2g-2+n}\delta^{2}(z-z_{I}),\qquad\qquad\det
g(z_{I})=0.$ (2.1)

Note that since the determinant of the metric is zero at the interaction
points the action on a lightcone diagram can be badly behaved at these points.
Even though the metric is singular, the singularity is sufficiently mild. The
Gauss-Bonnet theorem continues to hold

$\frac{1}{2}\int\sqrt{g}R=2\pi\chi,$ (2.2)

where $\chi=2-2g-n$. All the curvature required for Gauss-Bonnet to be
satisfied is localized at the interaction points. In Euclidean signature we
introduce coordinates $\tau,\sigma$ on the diagrams where $\tau$ is a time
coordinate and $\sigma$ is a periodic coordinate around the cylinder. We can
represent the diagrams by making identifications in the complex plane
$w=\tau+i\sigma$, see Fig. 3 for an example of how to build a pair of pants.
To travel around a singular point we must go through angle $4\pi$ in $w$
coordinates, so the singularity is a double cover of the Euclidean plane.

We can easily find the metric near the singular points using complex
coordinates

$ds^{2}=|z|^{2\alpha}dzd\bar{z}=e^{2\omega}dzd\bar{z},\qquad\qquad\frac{1}{2}\sqrt{g}R=-2\pi\alpha\delta^{(2)}(z).$
(2.3)

Figure 3: Euclidean Pair of pants constructed by identification in the complex
$w=\tau+i\sigma$ plane. The middle line (dashed) is infinitesimally split up
to the interaction time $\tau$ and identified with the top and bottom segments
as indicated. The singular point is indicated by the circle (blue). A closed
loop around the singular point goes through angle $4\pi$.

which is a cone of opening angle $2\pi(1+\alpha)$ with conformal mode
$\omega=\frac{1}{2}\log|z|^{2\alpha}$. The singularities of the Euclidean
lightcone diagram correspond to $\alpha=1$. We can see this by the coordinate
$w=z^{2}$, going through angle $2\pi$ in $z$ takes us through an angle of
$4\pi$ in $w$, which is a double cover of the Euclidean plane as required.

### Lorentzian lightcone diagrams

The above discussion was largely restricted to Euclidean lightcone diagrams.
However, by an analytic continuation of the time coordinate $\tau\to i\tau$ we
obtain a metric that is Lorentzian everywhere except at the degenerate points.
These geometries are Lorentzian lightcone diagrams, and the metrics on them
are known as almost Lorentzian metrics. The curvature is given by

$\frac{1}{2}\sqrt{-g}R=2\pi
i\sum_{I=1}^{2g-2+n}\delta^{2}(x-x_{I}),\qquad\qquad\det g(x_{I})=0.$ (2.4)

These geometries are constructed by gluing Lorentzian pants along geodesic
boundaries. The pair of pants cannot be given an everywhere Lorentzian metric,
but with the inclusion of the above singularity at the splitting point it
becomes possible to equip it with an almost Lorentzian metric [15, 16]. We can
explicitly write down a regularized complex metric near the splitting point of
the pants. In the conventions of [16] this metric is given by

$ds^{2}=\left(x^{2}+y^{2}+\gamma\right)\left(dx^{2}+dy^{2}\right)-(2\pm
i\epsilon)(xdx-ydy)^{2}.$ (2.5)

The splitting point is located at $x=y=0$, and we have introduced regulators
$\gamma,\epsilon>0$. The almost Lorentzian pants are obtained from the above
metric as we take the regulators to zero, with appropriate identifications of
the $x,y$ plane[15]. Analogous to the Euclidean case, the almost Lorentzian
pants are a double cover of Minkowski space. The $\gamma$ regulator
temporarily removes the degenerate point and the $\epsilon$ regulator deforms
the metric into what is known as an allowable complex metric333A complex
metric is defined as allowable if the following condition is satisfied[16,
17]. Consider a basis where the metric is diagonal
$g_{ij}=\lambda_{i}\delta_{ij}$, if
$\sum_{i}|\operatorname{Arg}\lambda_{i}|<\pi$ at every point on the manifold
then the metric is allowable. Allowable metrics satisfy certain nice
properties. For example, this condition ensures that the path integral of a
$p$-form gauge field converges on the background $g$. [16]. This is important
for us because allowable complex metrics satisfy an analytically continued
version of the Gauss-Bonnet theorem [15, 16].

Ultimately we will be integrating over all lightcone diagrams, and our
Euclidean action will contain a topological term suppressing each geometry by
it’s Euler characteristic $e^{S_{0}\chi}$, where $S_{0}$ is some large
constant. Upon analytic continuation of the time coordinate the topological
term will be proportional to

$\frac{i}{2}\int d^{2}x\sqrt{-g}R.$ (2.6)

For allowable complex metrics the above integral is equal to an analytically
continued version of the Gauss-Bonnet theorem [16]

$\frac{1}{2}\int d^{2}x\sqrt{-g}R=-2\pi i\chi.$ (2.7)

All metrics considered in this paper can be deformed to be allowable, and so
we maintain the same topological expansion in Lorentzian as in Euclidean
signature. We can directly see that the above formula holds for the Lorentzian
lightcone metric (2.4). An important property is that given an allowable
complex metric $g$, $e^{2\omega}g$ is also allowable if $\omega$ is real.
Since the Lorentzian pants are allowable so is any rescaling of the pants,
which will be useful when we need to consider the constant negative curvature
pants geometry.

### Moduli space of lightcone diagrams

We now explain the moduli space of inequivalent Euclidean/Lorentzian lightcone
diagrams of genus $g$ with $n$ boundaries. The starting point is to specify
the geodesic radius $r_{i}$ of each of the $n$ boundaries and whether they run
off to past or future infinity. A positive radius indicates that the boundary
originates from past infinity, a negative radius indicates the boundary goes
to future infinity. We demand that the sum of the lengths vanishes
$\sum_{i}r_{i}=0$. So that the total length of incoming boundaries is equal to
the total length of outgoing boundaries. We denote the total length of
incoming/outgoing boundaries by $r_{\rm max}$.

A general diagram is constructed by gluing Euclidean/Lorentzian pairs of pants
together along geodesic boundaries444The pair of pants used in the
construction have boundaries with waist of length $b_{1}$ and legs of lengths
$b_{2},b_{3}$ with the constraint that $b_{1}=b_{2}+b_{3}$. There are other
Lorentzian pair of pants geometries without this constraint, but they are not
used in this construction.. The different ways to glue pants together gives
the moduli space of lightcone diagrams. From figure 2, the moduli space is
parameterized by the following coordinates

Interaction Times $\displaystyle\tau_{i}\in[0,\infty)\qquad\qquad~{}\hskip
1.70717pti=1,\ldots,2g+n-3$ Twist Angles
$\displaystyle\theta_{j}\in[0,2\pi)\qquad\qquad j=1,\ldots,3g+n-3$ Internal
Radius $\displaystyle\rho_{k}\in[0,r_{\rm max}]\qquad~{}~{}~{}\hskip
0.28453ptk=1,\ldots,g$

The moduli space is of real dimension $6g-6+2n$. The interaction times $\tau$
define the points at which the cylinders split apart at degenerate points. The
diagram can be translated in time so the first interaction time can always be
set at $\tau=0$. When gluing two cylinders together we can twist them by a
relative angle $\theta\in[0,2\pi)$, this gives the twist angles. Since the
pair of pants conserves total geodesic boundary length, the sum of radii of
the diagram at any intermediate time must be identical to the sum of beginning
or ending radii. We have a freedom in how the total geodesic length is
distributed at intermediate times between cylinders, which gives the last
parameter $\rho_{k}$ where we integrate over lengths of intermediate cylinders
subject to the constraint that the total length is fixed to $r_{\rm max}$.

The end result is that the integral over all lightcone diagrams is given by
integrating over interaction times $\tau$, twist angles $\theta$, and
intermediate radii $\rho$

$\int[d\tau][d\theta][\rho
d\rho]\equiv\frac{1}{S}\left(\prod_{i=1}^{2g+n-3}\int_{0}^{\infty}d\tau_{i}\right)\left(\prod_{j=1}^{3g+n-3}\int_{0}^{2\pi}d\theta_{j}\right)\left(\prod_{k=1}^{g}\int_{0}^{r_{\rm
max}}\rho_{k}d\rho_{k}\right).$ (2.8)

The above integration range actually overcounts identical lightcone diagrams,
and we must divide by a symmetry factor $S$ to integrate over one copy of the
moduli space[12]. This symmetry factor has never been explicitly computed, but
see [18] for a recent discussion.

### Punctured Riemann surfaces and lightcone diagrams

In this section we’d like to explain the connection between punctured Riemann
surfaces and lightcone diagrams. Giddings and Wolpert[12] showed that the
moduli space of Euclidean lightcone diagrams gives a single cover of the
moduli space of punctured Riemann surfaces $\mathcal{M}_{g,n}$. This is
accomplished by showing that every Riemann surface with $n\geq 2$ punctures
can be equipped with a unique Euclidean lightcone metric and vice versa. The
integral over the moduli space of punctured Riemann surfaces
$\mathcal{M}_{g,n}$ can therefore be represented as an integral over the
moduli space of Euclidean lightcone diagrams.

We now briefly explain how a given Riemann surface can be equipped with a
Euclidean lightcone metric. Consider a genus $g$ Riemann surface $\Sigma$ with
local coordinates $z$ and $n$ punctures located at points $z_{i}$. The goal
will be to find a unique meromorphic one-form $\omega=\omega(z)dz$ on $\Sigma$
with simple poles at the punctures $z_{i}$ with real residues $r_{i}$ such
that their sum is zero $\sum_{i=1}^{n}r_{i}=0$. The specific choice of the
$r_{i}$ does not matter. We also require that the integral of the form
$\omega$ is imaginary along any closed cycle555Without this condition there
are many one-forms $\omega$ with the appropriate simple poles, however they
fail to give a global lightcone diagram. One way to understand this condition
is that on a lightcone diagram a closed loop starts and ends at the same time
$\tau$, so around any closed loop $\Delta\tau+i\Delta\sigma=\oint\omega(z)dz$
must be purely imaginary. on the surface $\Sigma$

$\oint\omega(z)dz\in i\mathbb{R}.$ (2.9)

Figure 4: A genus three surface $\Sigma$ with two punctures. We can put a
metric on $\Sigma$ given by (2.10) to turn it into a Euclidean lightcone
diagram with two asymptotic cylinders.

Under these conditions, Giddings and Wolpert [12] showed that the one-form
$\omega$ exists, is unique, and that it defines a unique Euclidean lightcone
metric666The construction of $\omega$ involves the period matrix of the
Riemann surface, which is why each Riemann surface is mapped to a unique
lightcone diagram. on $\Sigma$ by

$ds^{2}=\omega\overline{\omega}=|\omega(z)|^{2}dzd\bar{z}=dwd\bar{w},$ (2.10)

see figure 4. In the last equality we have defined the lightcone coordinates
$w=\tau+i\sigma=\int_{z_{0}}^{z}\omega(z)dz$, where $z_{0}$ is an arbitrary
point on $\Sigma$. In these coordinates the metric is explicitly flat with the
only breakdown occurring near the poles or zeroes of the form. Near a puncture
$z\sim z_{i}$ the form behaves as $\omega=\frac{r_{i}}{z-z_{i}}dz+\ldots$, so
the metric will behave as

$ds^{2}=\frac{r^{2}_{i}}{|z-z_{i}|^{2}}dzd\bar{z}+\ldots$ (2.11)

The form will also have $2g-2+n$ zeros at certain points $z_{I}$ on $\Sigma$,
near such points the form behaves as $\omega=\left(z-z_{I}\right)dz+\ldots$,
and the metric will behave as

$ds^{2}=|z-z_{I}|^{2}dzd\bar{z}+\ldots$ (2.12)

Note that near $z_{I}$ the metric is that of the interaction point of a
Euclidean lightcone diagram in equation (2.3). Thus, the zeros of $\omega$
correspond to degenerate points where the topology of spatial slices of
$\Sigma$ change, while the simple poles are located at the punctures.

We now explain how to extract the moduli of the lightcone diagram
$(\tau,\theta,\rho)$ from the metric (2.10), see figure 5. One useful property
is that the time coordinate
$\tau=\operatorname{Re}\int_{z_{0}}^{z}\omega(z)dz$ is globally defined and
path independent, since it is defined as the integral of a globally defined
one-form777The integral along two paths can differ by at most an imaginary
constant if they enclose a puncture, but the time $\tau$ is the real part of
the integral so it does not matter.. The interaction time differences can be
determined by performing an integral between zeros of the form
$\tau_{I}-\tau_{J}=\operatorname{Re}\int_{z_{J}}^{z_{I}}\omega dz$. The radius
of an asymptotic cylinder is given by integrating around a puncture $\oint
dw=\oint_{z_{i}}\omega(z)dz=2\pi ir_{i}$. Near the punctures we have
$w=r_{i}\log(z-z_{i})$, and so the puncture is mapped to $\tau=\pm\infty$ if
the radius $r_{i}$ is negative/positive respectively.

The lengths of intermediate cylinders are given by integrating along cycles at
constant time $\tau$, which gives $\oint dw=\oint\omega(z)dz=2\pi i\rho_{i}$.
Integration over cycles with non-constant $\tau$ allows us to reconstruct the
twist angles $\theta$, see [12] for additional details. The above procedure
gives us the moduli $(\tau,\theta,\rho)$ which uniquely fixes a lightcone
diagram for a given Riemann surface $\Sigma$, but the reverse direction also
holds in mapping a lightcone diagram to a Riemann surface[12].

Figure 5: A torus with two punctures mapped to a lightcone diagram with two
interaction points and two asymptotic cylinders. The circles (blue) on the
torus denote where the one-form $\omega$ has zeros. The two cycles on the
torus are illustrated with one at constant time $\tau$ (orange). The loops
$\gamma_{i}$ enclose the punctures which run off to infinity in the lightcone
metric.

To summarize, given a punctured Riemann surface $\Sigma$ we can equip it with
a unique Euclidean lightcone metric. Integrating over the moduli space of
punctured Riemann surfaces $\mathcal{M}_{g,n}$ is thus equivalent to
integrating over the moduli space of Euclidean lightcone diagrams.

$\int_{\mathcal{M}_{g,n}}\ldots=\int[d\tau][d\theta][\rho d\rho]\ldots$ (2.13)

In the above we have left out the integration measure to which we return to in
section 3.

###  2.2 Negative curvature lightcone diagrams

We are ultimately interested in JT gravity, so we want to construct the
constant negative curvature analogues of lightcone diagrams. We would like to
start with a Euclidean/Lorentzian lightcone metric $\hat{g}$ and turn it into
a constant negative curvature geometry $g=e^{2\omega}\hat{g}$ with a Weyl
factor $\omega$, with suitable singularities at the degenerate points. The
reason for this is that when considering JT gravity we will find we have a
constraint $R=-2$ away from the degenerate points, but there is no condition
on the behavior of $\omega$ at such points. We find that the Weyl factor is
not unique, and we have a large class of geometries that differ by the
strength of the conical singularity at the degenerate points. The only
constraint is that the resulting geometries satisfy Gauss-Bonnet.

#### Negative curvature Euclidean lightcone diagrams

Since $\hat{g}$ already satisfies Gauss-Bonnet, we will choose Weyl factor to
weaken the delta functions at the splittings points, and redistribute the
curvature uniformly so that Gauss-Bonnet remains satisfied. Consider a Riemann
surface $\Sigma$ where we prescribe conical singularities of opening angles
$2\pi\alpha_{I}$ at points $z_{I}$, where the metric locally behaves as
$ds^{2}=|z|^{2(1-\alpha_{I})}dzd\bar{z}$ near $z_{I}$. Does there exist a
constant negative curvature metric $g$ on $\Sigma$ with the prescribed
singularity structure? The answer turns out to be yes, assuming the
specification of $\alpha_{I}$ does not violate Gauss-Bonnet

$\frac{1}{4\pi}\int_{\Sigma\backslash\\{z_{i}\\}}\sqrt{g}R=\chi(\Sigma)+\sum_{I}\left(1-\alpha_{I}\right)<0.$
(2.14)

Under the above condition, it was shown888For a review of various existence
and uniqueness theorems on metrics with conical singularities see [19].[20,
21] that the metric $g$ exists, is unique, and is in the conformal equivalence
class of metrics on $\Sigma$. We conclude that starting from a Euclidean
lightcone metric $\hat{g}$, we can find a unique $\omega$ such that
$g=e^{2\omega}\hat{g}$ has constant negative curvature and any desired
singularity structure consistent with Gauss-Bonnet

$\frac{1}{2}\sqrt{g}\left(R+2\right)=-2\pi\sum_{I=1}^{2g-2+n}(1-\alpha_{I})\delta^{2}(z-z_{I}).$
(2.15)

We have written the curvature in this way to emphasize that the effect of
$\alpha_{I}$ is to take curvature from the interaction points and redistribute
it uniformly across the rest of the surface $\Sigma\backslash\\{z_{i}\\}$.
Note that this amounts to studying JT gravity with conical excesses, which
appears to be the closest relative to Euclidean lightcone diagrams. However,
the utility of lightcone diagrams is they can be easily analytically continued
to singular Lorentzian geometries, which we now discuss.

#### Negative curvature Lorentzian lightcone diagrams

We would like to construct negative curvature Lorentzian lightcone diagrams in
a similar way by rescaling the metric $g=e^{2\omega}\hat{g}$, where
$g,\hat{g}$ are almost Lorentzian geometries. As far as we are aware, not much
is known about almost Lorentzian metrics and the existence of such Weyl
factors. However, we would like to argue that it is plausible that there
exists an appropriate $\omega$ such that the curvature of
$g=e^{2\omega}\hat{g}$ takes the form

$\frac{1}{2}\sqrt{-g}\left(R+2\right)=2\pi\sum_{I=1}^{2g-2+n}\left(i+\alpha_{I}\right)\delta^{2}(z-z_{I}),$
(2.16)

under the condition that $\alpha_{I}$ is chosen so that the complex Gauss-
Bonnet theorem is not violated, which requires that $\alpha_{I}>0$. The
purpose of the $\alpha_{I}$ can can be seen from the form of Gauss-Bonnet for
complex metrics[16, 15]

$\frac{1}{2}\int_{\Sigma}\sqrt{-g}R=-2\pi i\chi(\Sigma).$ (2.17)

The effect of the $\alpha_{I}$ is to cancel out the contribution of the bulk
volume

$\frac{1}{4\pi}\int_{\Sigma\backslash\\{z_{I}\\}}\sqrt{-g}R+\sum_{I=1}^{2g-2+n}\alpha_{I}=0,$
(2.18)

so that the remaining terms in (2.16) give the Euler characteristic. While we
cannot prove that such Weyl factor exists, we think it is reasonable that it
does for the following reasons. It’s existence would be the natural extension
of the Euclidean results of [20, 21] which gives the unique metric (2.15). In
both cases the role of the $\alpha_{I}$ is to cancel out the bulk volume term
to satisfy Gauss-Bonnet. Furthermore, since the Lorentzian lightcone metric
$\hat{g}$ is complex allowable, so is $e^{2\omega}\hat{g}$, which implies that
Gauss-Bonnet (2.17) holds for both metrics. If $e^{2\omega}\hat{g}$ is to have
constant negative curvature, it’s singular points must contribute in a way to
cancel out the bulk volume, otherwise Gauss-Bonnet would be violated. Finally,
in appendix A we construct the Lorentzian constant negative curvature pairs of
pants with the same singularity structure as advocated above.

To summarize, we believe it is reasonable that there exists a Weyl factor that
turns a Lorentzian lightcone diagram into a constant negative curvature
analogue given by (2.16), with the only constraint on $\alpha_{I}$ being that
Gauss-Bonnet (2.17) is satisfied. One important point is that while flat
lightcone diagrams are unique, the constant negative curvature analogues are
not. When integrating over the Weyl factor we must make a choice of which
geometries to include, which amounts to deciding which $\alpha_{I}$ to
include. We are not able to give a conclusive answer to this question, but we
return to it in the discussion.

##  3 JT gravity on lightcone diagrams

JT gravity is a two dimensional dilaton gravity theory with metric
$g_{\mu\nu}$ and dilaton $\Phi$. We will start with the Euclidean JT gravity
path integral with boundary conditions given by $n$ geodesic circles of given
lengths $\vec{b}=(b_{1},\cdots,b_{n})$. We assume the bulk geometry has genus
$g$ and is connected, with the extension to the full topological expansion
easily following. This amplitude can be computed by performing the path
integral over all compact geometries $\Sigma$ of genus $g$ with $n$ vertex
operators $V$ inserted999The other way to compute this amplitude is to perform
the path integral over surfaces with $n$ boundaries with appropriate boundary
conditions. However, the formalism of lightcone diagrams most easily applies
to the situation where boundary conditions are specified by vertex operator
insertions.

$Z=\int\frac{\mathcal{D}g_{\mu\nu}\mathcal{D}\Phi}{\text{Vol}}e^{-I_{{\mathrm{JT}}}[g,\Phi]}V_{1}\ldots
V_{n},$ (3.1)

where Vol is the volume of the diffeomorphism group. The Euclidean JT action
on a compact surface $\Sigma$ is given by [22, 23, 24]

$I_{\mathrm{JT}}[g,\Phi]=-\frac{S_{0}}{4\pi}\int_{\Sigma}\sqrt{g}R-\frac{1}{2}\int_{\Sigma}\sqrt{g}\Phi(R+2).$
(3.2)

The first term is purely topological. Taking into account the vertex
operators, the total topological contribution will be given by $e^{S_{0}\chi}$
where $\chi=2-2g-n$. We ignore this term from now on, although at certain
points various constants will be absorbed into the definition of $S_{0}$. The
vertex operator that inserts a geodesic boundary of length $b$ is given by
[25]

$V=e^{-S_{0}}\int_{\Sigma}d^{2}z\sqrt{g}e^{-2\pi\Phi}\cos\left(b\Phi\right).$
(3.3)

The path integral over all two dimensional Euclidean metrics can be decomposed
into an integral over the moduli space $\mathcal{M}_{g}$ of genus $g$ Riemann
surfaces, and over the Weyl factor $\omega$ of the metric [26, 27]

$\int\frac{\mathcal{D}g_{\mu\nu}\mathcal{D}\Phi}{\text{Vol}}=\int_{\mathcal{M}_{g}}{\mathrm{d}}\mu\int\mathcal{D}\omega\mathcal{D}\Phi
e^{-25S_{L}[\omega,\hat{g}]}.$ (3.4)

The above path integral is performed over metrics $g=e^{2\omega}\hat{g}$,
where $\hat{g}$ is a choice of representative metric101010The standard choice
is to pick $\hat{g}$ such that it has constant negative curvature for $g\geq
2$. for each point in the moduli space $\mathcal{M}_{g}$. In the above
integral ${\mathrm{d}}\mu$ is the standard integration measure on the moduli
space $\mathcal{M}_{g}$ [26, 27], since it is quite complicated we write it
schematically and go into additional detail in appendix B. The Liouville
action $S_{L}$ will turn out to be proportional to the Euler characteristic
after we integrate out the dilaton, and we absorb it into the definition of
the topological term. This integral treats the moduli of the Riemann surface
on a different footing from the positions of the vertex operators. Giddings
and D’Hoker [13] gave an argument that the integral over vertex operator
positions $\int d^{2}z_{i}\sqrt{g}$ can be absorbed into the integral over
$\mathcal{M}_{g}$ to give an integral over the moduli space of punctured
Riemann surfaces $\mathcal{M}_{g,n}$ of genus $g$ with $n$ punctures[13]

$\int_{\mathcal{M}_{g}}{\mathrm{d}}\mu\prod_{i=1}^{n}\left(e^{-S_{0}}\int
d^{2}z_{i}\sqrt{g}e^{-2\pi\Phi}\cos\left(b_{i}\Phi\right)\right)=\int_{\mathcal{M}_{g,n}}{\mathrm{d}}\mu\prod_{i=1}^{n}W_{i}.$
(3.5)

The measure ${\mathrm{d}}\mu$ on the right is now for the moduli space of
punctured Riemann surfaces $\mathcal{M}_{g,n}$, and various factors have been
absorbed into the definition of the wavefunctions $W_{i}$. The integral over
$\mathcal{M}_{g,n}$ automatically takes into account the integral over all
distinct points $z_{i}$ at which the wavefunctions can be inserted, and it
could have been our starting point for the Euclidean path integral111111The
two are equal up to overall normalization factors that can be absorbed, see
[13, 27]. (3.1). On the right hand side we have defined the wavefunctions

$W_{i}=e^{-2\pi\Phi(z_{i})}\cos\left(b_{i}\Phi(z_{i})\right),$ (3.6)

located at the punctures at points $z_{i}$ on the surface. The role that the
wavefunctions $W_{i}$ play is that they impose boundary conditions at the
punctures in the following way. Consider performing the path integral over the
dilaton near a particular puncture $z_{i}$ with wavefunction $W_{i}$. When we
integrate over the dilaton $\Phi$ near the puncture we get the following
constraint on the behavior of the metric

$\frac{1}{2}\sqrt{g}(R+2)=(2\pi\pm ib_{i})\delta^{2}(z-z_{i}).$ (3.7)

Each wavefunction comes with two branches denoted by $\pm ib$, which arise
from the expansion of $\cos(b\Phi)$, and so the constraint is a linear
combination of the two branches121212It might at first be surprising that a
geodesic boundary is given by an imaginary conical defect since a geodesic
boundary does not give a delta function of curvature. This was explained in
Appendix A of [25]. A geodesic boundary is given by the end of the trumpet
geometry, after reaching the geodesic on the trumpet the radial coordinate can
be analytically continued into the imaginary axis to reach an imaginary
defect. We should thus imagine that the geometry is continued into the complex
radial direction near the insertion points.. Since each wavefunction comes
with two branches, performing the dilaton integral over the entire surface
gives a sum over all possible branches of the constraints.

The end result is we have re-written the path integral over two dimensional
Euclidean metrics with vertex operator insertions as a path integral over the
moduli space of punctured Riemann surfaces with wavefunctions inserted at the
punctures

$Z=\int_{\mathcal{M}_{g,n}}{\mathrm{d}}\mu\int\mathcal{D}\omega\mathcal{D}\Phi
e^{-I_{\mathrm{JT}}[e^{2\omega}\hat{g},\Phi]}W_{1}\ldots W_{n}.$ (3.8)

In the above integral ${\mathrm{d}}\mu$ is the standard measure on the moduli
space $\mathcal{M}_{g,n}$ [13, 27]. The path integral measure for
$\mathcal{M}_{g,n}$ is quite complicated and is written out in appendix B
alongside additional details on the moduli space.

### Lightcone diagrams

To perform the above integral we must choose a representative metric $\hat{g}$
for each point in $\mathcal{M}_{g,n}$, and then perform the integral over the
Dilaton $\Phi$ and Weyl factor $\omega$. At this point we must make a choice
to separate the $n$ wavefunctions $W_{i}$ into those that run to the Euclidean
past or future131313In bosonic string theory we would send the vertex
operators defining the in state to the Euclidean past, while vertex operators
that define the out state would be sent to the future.. We must have at least
one wavefunction in the future and one in the past for the Euclidean lightcone
metric $\hat{g}$ to be a good gauge choice. Using the results summarized in
section 2, we choose the representative metric $\hat{g}$ to be given by the
unique Euclidean lightcone diagram for each point in $\mathcal{M}_{g,n}$,
which we write below for convenience

$\frac{1}{2}\sqrt{\hat{g}}\hat{R}=-2\pi\sum_{I=1}^{2g-2+n}\delta^{2}(z-z_{I}),\qquad\qquad\det\hat{g}(z_{I})=0.$
(3.9)

As explained in section 2, the Euclidean lightcone metric $\hat{g}$ has a
globally defined Euclidean time $\tau$. In terms of the $\tau$ coordinate on
$\Sigma$, the wavefunctions $W_{i}$ we choose to define the in state are
located at $\tau=-\infty$, while those that define the out state are at
$\tau=\infty$. In lightcone diagram gauge the measure ${\mathrm{d}}\mu$ takes
a particularly simple form[13, 27]

$\int_{\mathcal{M}_{g,n}}{\mathrm{d}}\mu=\int[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}},$
(3.10)

up to a term proportional to the Euler characteristic that we absorbed into
$S_{0}$. The prime on the determinant indicates that we exclude zero modes. We
include more details on the derivation of the above measure in appendix B. The
determinant of the Laplacian is defined to be with respect to the Euclidean
lightcone metric $\hat{g}$ on $\Sigma$, and we discuss the determinants in
more detail slightly later. We end up with the following path integral

$Z=\int_{W_{i}}[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}}\int\mathcal{D}\omega\mathcal{D}\Phi
e^{-I_{\mathrm{JT}}[e^{2\omega}\hat{g},\Phi]}.$ (3.11)

We have denoted the integral over lightcone diagrams as $\int_{W_{i}}$ to
emphasize that the wavefunctions $W_{i}$ impose boundary conditions at the
ends of the cylinders at $\tau=\pm\infty$. Note that so far all we have done
is rewritten (3.1) in a particular gauge choice. This is essentially the
argument for the equivalence of the Polyakov formulation of bosonic string
theory and the interacting string picture [10, 11] on Euclidean lightcone
diagrams [13, 28]. The only new ingredients we have encountered in considering
this formalism for JT gravity is that we have to integrate over the Weyl
factor, and there is no canonical choice for separating the boundaries into
those that are sent to the past or future.

###  3.1 Lorentzian JT path integral

Since Euclidean lightcone diagrams come with a global notion of Euclidean time
$\tau$, we can analytically continue $\tau\to i\tau$ to get Lorentzian
lightcone diagrams with metric

$\frac{1}{2}\sqrt{-\hat{g}}\hat{R}=2\pi
i\sum_{I=1}^{2g-2+n}\delta^{2}(z-z_{I}),\qquad\qquad\det\hat{g}(z_{I})=0.$
(3.12)

Our _definition_ for the Lorentzian JT path integral will be given by the
above analytic continuation applied to (3.11), which gives us

$Z_{\text{L}}=\int_{W_{i}}[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}}\int\mathcal{D}\omega\mathcal{D}\Phi
e^{iI_{\mathrm{JT}}[e^{2\omega}\hat{g},\Phi]}.$ (3.13)

The Lorentzian JT action is now given by

$I_{\mathrm{JT}}[g,\Phi]=\frac{S_{0}}{4\pi}\int_{\Sigma}\sqrt{-g}R+\frac{1}{2}\int_{\Sigma}\sqrt{-g}\Phi(R+2).$
(3.14)

where we use the notation $g=e^{2\omega}\hat{g}$. We will ignore the first
term which is topological by complex Gauss-Bonnet (2.17) and focus on the
second term. On a lightcone diagram we use (3.12) to find that the second term
is given by

$\frac{1}{2}\int_{\Sigma}\sqrt{-g}\Phi(R+2)=2\pi
i\sum_{I=1}^{2g-2+n}\Phi(z_{I})+\frac{1}{2}\int_{\Sigma\backslash\\{z_{I}\\}}\sqrt{-g}\Phi\left(R+2\right).$
(3.15)

We see that the JT action on a lightcone diagram picks up point terms at the
degenerate points where the geometry undergoes a topology changing transition.
The usual JT gravity constraint of constant negative curvature $R=-2$ no
longer holds at such points. If the constraint did hold on all of $\Sigma$ the
path integral would be zero since a suitable Lorentzian topology changing
metric does not exist. There is one issue, if we integrate over the dilaton
$\Phi(z_{I})$ at the interaction point the path integral will diverge.
However, since there are $2g-2+n$ interaction points we can absorb this
infinite divergence into the definition of $S_{0}$ in the topological term.

We can now perform the path integral over the dilaton $\Phi$. Since we are in
Lorentzian signature, we choose our contour for $\Phi$ to be along the real
axis. Rewriting the action in terms of the Weyl factor we find

$\int\mathcal{D}\Phi\exp\left(i\int_{\Sigma\backslash\\{z_{I}\\}}\sqrt{-\hat{g}}\Phi\left(-\hat{\nabla}^{2}\omega+e^{2\omega}\right)\right)=\delta\left(-\hat{\nabla}^{2}\omega+e^{2\omega}\right)=\frac{\delta\left(\omega-\omega_{0}\right)}{\det\left(-\hat{\nabla}^{2}+2e^{2\omega_{0}}\right)}.$
(3.16)

Here $\omega_{0}$ is a configuration that satisfies the delta function
constraint. This constraint enforces that the metric $g=e^{2\omega}\hat{g}$
has constant negative curvature $R=-2$ away from the interaction points. The
end result is the following path integral

$Z_{\text{L}}=\int_{W_{i}}[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}}\int\mathcal{D}\omega\frac{\delta\left(\omega-\omega_{0}\right)}{\det\left(-\hat{\nabla}^{2}+2e^{2\omega_{0}}\right)}.$
(3.17)

The reason we have not carried out the integral over $\omega$ is the
following. As explained in section 2.2, there exist multiple Weyl factors that
satisfy the constraint (3.16) while having different behavior at the
degenerate points given by (2.16), which we restate here for convenience

$\frac{1}{2}\sqrt{-g}\left(R+2\right)=2\pi\sum_{I=1}^{2g-2+n}\left(i+\alpha_{I}\right)\delta^{2}(z-z_{I}).$
(3.18)

We must choose our contour of integration for $\omega$ to decide which
configurations should be included. A configuration is specified by picking out
a preferred choice of $\alpha_{I}$. There appear to be two natural choices. We
can choose a certain $\alpha_{I}$ as special and the Weyl factor $\omega_{0}$
can be forced to localize onto such a geometry for all lightcone diagrams. The
other obvious option is to integrate over all possible $\omega$ (i.e. all
$\alpha_{I}$). The benefit of choosing a particular $\alpha_{I}$ as special
would be that we would get the closest analogue to Euclidean JT gravity where
only one $\omega$ contributes for each point in moduli space
$\mathcal{M}_{g,n}$. For simplicity we will assume that the Weyl factor
localizes to a single configuration, after which we are left with an integral
over the moduli space $\mathcal{M}_{g,n}$ with given measure

$Z_{\rm L}=\int_{W_{i}}[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}}\frac{1}{\det\left(-\hat{\nabla}^{2}+2e^{2\omega_{0}}\right)}.$
(3.19)

The integration range is defined in (2.8), and the wavefunctions $W_{i}$
implement boundary conditions as discussed around (3.7). The end result is
that we must choose a set of boundary conditions and then evaluate the above
integral.

We now discuss the definition of the functional determinants appearing in the
above integral. Even though the integral is over Lorentzian lightcone
geometries, we will choose the functional determinants to be analytically
continued and defined on the corresponding Euclidean lightcone geometries to
make them well defined. The Laplacian $\hat{\nabla}^{2}$ is defined with
respect to the degenerate lightcone metric (2.1). The corresponding
determinant $\operatorname{det}(-\hat{\nabla}^{2})$ has been studied by [29,
14, 30, 31], and in principle it is known in terms of basic objects on the
underlying punctured Riemann surface. However, the second determinant is more
complicated. Using the conformal anomaly we can relate it to
$\operatorname{det}(-\nabla^{2}+2)$ where $\nabla^{2}$ is now defined with
respect to a constant negative curvature metric $g=e^{2\omega}\hat{g}$ with
conical singularities at the splitting points. There has been some recent
progress on computing closely related determinants141414In [32] various
determinants were studied on punctured Riemann surfaces with conical
singularities of opening angles $\frac{2\pi}{n+1}$ with $n\in\mathbb{Z}^{+}$.
However, in our case we require conical excesses instead of defects so the
results do not appear immediately applicable. [32], but as of now determinants
on singular Riemann surfaces are not fully understood. Regardless, evaluating
the above determinants will introduce some dependence on the moduli into the
above integral.

### Relation to Euclidean amplitudes

We would like to discuss the relation of our proposed Lorentzian path integral
(3.19) to the standard Euclidean amplitudes. We first briefly review why the
Euclidean JT path integral gives Weil-Petersson volumes, originally explained
in [1]. For simplicity, consider a compact surface of genus $g$, the Euclidean
path integral over all smooth metrics on this surface reduces to [1]

$Z=\int_{\mathcal{M}_{g}}{\mathrm{d}}{\rm(WP)}(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}\int\mathcal{D}\omega\frac{\delta(\omega)}{\operatorname{det}(-\hat{\nabla}^{2}+2)},$
(3.20)

where we have gauge fixed to a smooth metric $\hat{g}$ of constant negative
curvature, and as a result the Weyl factor localizes to $\omega=0$. We have
written the integral in this way to compare to (3.17). The measure
${\mathrm{d}}{\rm(WP)}$ is known as the Weil-Petersson measure. In the above
we have two determinants:
$(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}$ which originates
from gauge fixing and is theory independent, and
$\operatorname{det}(-\hat{\nabla}^{2}+2)$ which is special to JT gravity and
arises from the integral over the dilaton $\Phi$. In [1] it was pointed out
that the ratio of these determinants is unity, up to a factor that can be
absorbed into the coupling constant $S_{0}$. We are left with an integral over
the Weil-Petersson measure, which gives us the Weil-Petersson volume

$Z=\int_{\mathcal{M}_{g}}{\mathrm{d}}{\rm(WP)}=V_{g}.$ (3.21)

We can similarly ask whether our Lorentzian path integral (3.19) localizes to
the Weil-Petersson measure. Possible issues might arise both from the theory
independent measure (3.10), and from the theory dependent contribution (3.16).
However, choosing lightcone diagrams as the representative metric for the
moduli space $\mathcal{M}_{g,n}$ is simply a gauge choice, so the theory
independent measure (3.10) cannot depend on this choice. Indeed, it was
pointed out in [13] that (3.10) secretly contains the Weil-Petersson measure
${\mathrm{d}}{\rm(WP)}$ in lightcone coordinates, and that the determinant of
the Laplacian $\operatorname{det}(-\hat{\nabla}^{2})$ roughly comes from the
punctured Riemann surface analogue of the gauge fixing determinant
$(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}$.

The only possible issue with recovering the Euclidean amplitudes can therefore
be from the determinant arising from the integral over the Weyl factor in
(3.16). However, this is precisely where we ran into an ambiguity since
$\omega$ is not constrained to behave in any particular way at the degenerate
points. Therefore the determinant in (3.16) should somehow depend on the
choice of singularity structure at the degenerate points, and it’s unclear if
we get an exact cancellation as in (3.20) for Euclidean JT. Without a better
understanding of determinants on singular surfaces it is difficult to say
anything more concrete. However, since the only deviation comes from a finite
set of points we believe it’s quite likely that the resulting amplitudes
computed by (3.19) share similar features with Euclidean JT gravity, we return
to this point in the discussion.

##  4 Discussion

In this work we have proposed a definition for the Lorentzian JT gravity path
integral that includes topology changing configurations. This is accomplished
by integrating over metrics which are Lorentzian signature everywhere except
at special points where the metric becomes degenerate, and where the spatial
topology changing transitions occur.

Our proposal is inspired by a formulation of bosonic string theory on singular
Lorentzian worldsheets [10, 11, 12, 13, 14], and we followed similar logic to
define the Lorentzian JT path integral. Using lightcone diagrams we
analytically continued the Euclidean path integral to define a Lorentzian path
integral over degenerate Lorentzian metrics. This analytic continuation gives
a Lorentzian interpretation for the Euclidean path integral genus expansion.
We end with a few comments and potential future directions.

### Relation to Euclidean JT and degenerate points

We found that the most serious ambiguity in the definition of our Lorentzian
theory is how to properly treat the degenerate points. There appears to be no
standard prescription for dealing with such points in Lorentzian signature,
although see [33] for a recent discussion. In Euclidean signature the
degenerate points can always be removed with a singular Weyl factor to give a
smooth Euclidean metric. However, in Lorentzian signature this cannot be done
since the degenerate points are crucial for the existence of an almost
Lorentzian metric with spatial topology change. This gives us some freedom to
choose the behavior of the metric at such points151515This ambiguity does not
appear when considering the bosonic string on lightcone diagrams[13] since we
don’t integrate over the Weyl factor. (2.16), and this choice propagates into
the final path integral in (3.19).

As discussed in section 3.1, to understand the relation between Lorentzian and
Euclidean amplitudes we must understand how the ambiguity at the singular
points modifies the path integral measure. Indeed, in both the Lorentzian and
Euclidean cases the domain of integration is given by the moduli space of
punctured Riemann surfaces $\mathcal{M}_{g,n}$, so the only difference can
arise from the respective measures on this space. Since the determinants on
singular Riemann surfaces appearing in (3.19) are not fully understood it is
difficult to give a conclusive answer. However, it is likely that this
ambiguity modifies the path integral measure, and so we don’t expect the
Lorentzian amplitudes to exactly match the Euclidean amplitudes.

Regardless, we expect the Lorentzian amplitudes to have qualitatively similar
behavior to Euclidean JT gravity amplitudes for reasons of random matrix
universality. In [5] it was shown that wormhole contributions in a wide range
of dilaton-gravity models, with integration measures deviating from pure JT
gravity, have universal behavior independent of the specific details of the
theory. Indeed, we do not expect the specific details of the integration
measure to seriously modify universal behavior, such as the existence of a
page curve[2, 3] or the spectral form factor [4, 1, 5].

### Towards non-perturbative Lorentzian physics

Non-perturbative Euclidean wormholes have played an important role in a wide
variety of recent calculations, see [2, 3, 4, 1, 5, 6, 7, 8] for some selected
examples. It would be interesting to understand the Lorentzian origin of these
calculations using the Lorentzian path integral developed in this work. As an
example, it would be interesting to understand the Replica wormhole
computations [2, 3] or the swap entropy [34] using topology changing
Lorentzian wormholes, see also [35, 36]. Additionally, it would be interesting
to understand how the inclusion of topology changing Lorentzian metrics
modifies the structure of the JT gravity Hilbert space[9], and whether there
is a canonical quantization interpretation for Lorentzian topology change.

To address the above questions it will be necessary to introduce
asymptotically AdS boundaries, which amounts to studying lightcone diagrams
with boundaries. These diagrams were studied in the context of open
strings[10, 11], but the proof of Giddings and Wolpert [12] was never extended
to them. Thus we cannot claim that these diagrams give a single cover of the
desired moduli space, but it seems quite likely that they do. Regardless, we
can directly define the Lorentzian JT theory with asymptotic AdS boundaries to
live on such lightcone diagrams, but for now we leave an examination of the
above questions to future work.

### Simpler representation of moduli space

One of the features of lightcone diagrams is that they provide a particularly
simple representation for the integral over the moduli space of punctured
Riemann surfaces $\mathcal{M}_{g,n}$. The standard procedure is to introduce
Fenchel-Nielsen coordinates on the moduli space in terms of which the the
domain of integration is incredibly complicated, see [1] for a discussion.
However, lightcone diagrams have a very simple region of integration (2.8)

$\int_{\mathcal{M}_{g,n}}\ldots=\frac{1}{S}\prod_{i,j,k}\int_{0}^{\infty}d\tau_{i}\int_{0}^{2\pi}d\theta_{j}\int_{0}^{r_{\rm
max}}\rho_{k}d\rho_{k}\ldots$ (4.1)

while the integration measure can be quite complicated, see (3.19). It would
be interesting to understand the integration measure in (3.19) in terms of the
lightcone coordinates on the moduli space. It might be possible that in
corners of moduli space the measure takes a simplified form in terms of these
coordinates. More generally, it would be interesting to understand if the
ratio of determinants in (3.19) simplifies as in the case of Euclidean JT
gravity.

### Acknowledgments

We thank Andreas Blommaert, Ven Chandrasekaran, Steve Giddings, Bob Knighton,
Adam Levine, and Henry Maxfield for useful discussions. We would especially
like to thank Don Marolf for numerous discussions on related topics. MU is
supported in part by the NSF Graduate Research Fellowship Program under grant
DGE1752814, by the Berkeley Center for Theoretical Physics, by the DOE under
award DE-SC0019380 and under the contract DE-AC02-05CH11231, by NSF grant
PHY1820912, by the Heising-Simons Foundation, the Simons Foundation, and
National Science Foundation Grant No. NSF PHY-1748958.

##  Appendix A Lorentzian pair of pants

In this section we explain how to construct the constant negative curvature
Lorentzian pair of pants161616We thank Don Marolf for suggesting this
construction.. This can be accomplished with an identification similar to the
one used for the flat pants in figure 3. We start with the Lorentzian $R=-2$
cylinder which is the analytic continuation of the double trumpet

$ds^{2}=\frac{-d\tau^{2}+d\sigma^{2}}{\cosh^{2}\tau},\qquad\tau\in\left(-\infty,\infty\right),\qquad\sigma\in[0,b_{1}].$
(A.1)

From the identification $\sigma\sim\sigma+b_{1}$ we have a spatial geodesic of
length $b_{1}$ at $\tau=0$. We now pick two points $x,x^{\prime}$ at constant
$\tau$ but at different values of $\sigma$, and we connect these points with
two geodesics traversing both sides of the cylinder, see figure 6. The
geodesics will have lengths $b_{2},b_{3}$ such that $b_{1}\geq b_{2}+b_{3}$.
Solving for the geodesic lengths in terms of $x,x^{\prime}$, it’s possible to
show that we can find $x,x^{\prime}$ to give any geodesic lengths
$b_{2},b_{3}$ satisfying this constraint. We now identify the points $x\sim
x^{\prime}$ and discard the portion of the geometry above the geodesics to get
a Lorentzian pair of pants with a singular point at $x\sim x^{\prime}$.

Figure 6: Building the Lorentzian constant negative curvature pair of pants
with geodesic boundaries by identifying points $x,x^{\prime}$. The dashed
lines are geodesics that form a figure eight with a singular point between
them. The geometry on the right is smooth everywhere except at the point
$x\sim x^{\prime}$.

When the geodesics meet at $x,x^{\prime}$ there is a jump in the tangent
vector, so there is a delta function in the Extrinsic curvature of the
boundary curve at those points. This is taken into account in the Gauss-Bonnet
theorem through the jump angles $\alpha_{i}$, of which there are two in the
above construction

$\frac{1}{2}\int\sqrt{g}R+\sum_{i}\alpha_{i}=2\pi\chi.$ (A.2)

We’ll absorb the jump angles into a delta function contribution to the Ricci
scalar at the point $x\sim x^{\prime}$. The above is the Euclidean version of
the Gauss-Bonnet theorem, the analytically continued version for almost
Lorentzian metrics is given by (2.7) [15, 16]

$\frac{1}{2}\int\sqrt{-g}R=-2\pi i\chi.$ (A.3)

and we see that the effect of the jump angles is to cancel out the bulk volume
contribution as explained in section 2.2. The scalar curvature of the pair of
pants is then given by

$\frac{1}{2}\sqrt{-g}\left(R+2\right)=\left(2\pi
i+\alpha\right)\delta^{2}(x).$ (A.4)

where we have relabeled the sum of jump angles as just $\alpha$. One way to
see that the imaginary delta function contribution appears after the
identification of $x\sim x^{\prime}$ is to note that without the Weyl factor
in (A.1) this is the construction of the flat Lorentzian pants, where this
contribution can be explicitly evaluated[15]. We can also attach pant legs to
the boundaries $b_{2}$ or $b_{3}$ as follows. Consider a Lorentzian cylinder
with two geodesic boundaries where one boundary has a corner so that the
geometry exists. We can glue the boundary with the corner to the pants at
$b_{2}$ or $b_{3}$ making sure the corner coincides with the point $x\sim
x^{\prime}$. This will give additional contributions to the $\alpha$ term, and
gives Lorentzian pants with any desired geodesic boundary lengths.

##  Appendix B Details on punctured Riemann surfaces and measures

In this section we include some additional details on punctured Riemann
surfaces and on the integration measure. A punctured surface $\Sigma$ is
defined by taking a compact surface $\overline{\Sigma}$ of genus $g$ and
removing $n$ distinct points
$\Sigma=\overline{\Sigma}\backslash\\{z_{1},\ldots,z_{n}\\}$, after which
$\Sigma$ is no longer compact and its Euler characteristic is given by
$\chi(\Sigma)=2-2g-n$. When performing an integral over a punctured surface
the punctures do not contribute to the integral. The moduli space of punctured
Riemann surfaces can be constructed by considering equivalence classes of
singular Euclidean metrics on $\Sigma$ [14]. We consider metrics that behave
as

$ds^{2}=\frac{c_{i}}{|z-z_{i}|^{2}}dzd\bar{z}+\ldots$ (B.1)

near the punctures at $z_{i}$, where $c_{i}$ is some constant. We consider all
such metrics related by smooth Weyl transformations to be in the same
conformal equivalence class. The set of all such equivalence classes gives us
the moduli space of punctured Riemann surfaces $\mathcal{M}_{g,n}$. In the
literature it is also common to include additional singular metrics in the
conformal equivalence class[12, 27, 29], such as the lightcone diagram metrics
described in section 2. These metrics are related to other metrics in the
equivalence class through Weyl transformations that are singular at isolated
points, see [11] for an explicit example at genus zero.

The path integral over metrics on a genus $g$ surface with $n$ punctures was
analyzed in [13] and is given by

$\int\frac{\mathcal{D}g}{\text{Vol}}=\int_{\mathcal{M}_{g,n}}[dm]\frac{\operatorname{det}\left\langle\mu_{\alpha},\phi_{\beta}\right\rangle}{\operatorname{det}\left(\phi_{\alpha},\phi_{\beta}\right)^{\frac{1}{2}}}(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}\int\mathcal{D}\omega
e^{-26S_{L}[\omega,\hat{g}]}.$ (B.2)

Where on the left side Vol is the volume of the diffeomorphism group of the
punctured surface, and we have gauge fixed to a metric $\hat{g}$ on the right.
Compare this to equation (3.8) in the main text171717Note that in (3.4) we are
integrating over $\mathcal{M}_{g}$ instead of punctured surfaces
$\mathcal{M}_{g,n}$. The integration measure slightly differs between the two
cases, see section 3 of [13].. In the above integral $m$ is a coordinate on
the moduli space and
$(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}$ comes from gauge
fixing to the representative metric $\hat{g}$, and in total the above is the
integration measure for the moduli space $\mathcal{M}_{g,n}$. For a detailed
explanation of the various determinants in the above measure see [26, 27, 13].
In the main text we introduced the shorthand notation ${\mathrm{d}}\mu$ for
the integration measure for simplicity, we now define it in terms of the above
objects for easier comparison to the literature[13, 26, 27]

$\int_{\mathcal{M}_{g,n}}{\mathrm{d}}\mu\equiv\int_{\mathcal{M}_{g,n}}[dm]\frac{\operatorname{det}\left\langle\mu_{\alpha},\phi_{\beta}\right\rangle}{\operatorname{det}\left(\phi_{\alpha},\phi_{\beta}\right)^{\frac{1}{2}}}(\operatorname{det}\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}.$
(B.3)

Gauge fixing to the lightcone metric $\hat{g}$, the various determinants in
the above integral were worked out in [13] (see equations (2.22) and (4.7-4.8)
in [13], see also section V.G in [27]). The product of determinants
significantly simplifies, and we find the final result

$\int\frac{\mathcal{D}g}{\text{Vol}}=\int[d\tau][d\theta][\rho
d\rho]\frac{2\pi\operatorname{det}^{\prime}(-\hat{\nabla}^{2})}{\int_{\Sigma}d^{2}z\sqrt{\hat{g}}}\int\mathcal{D}\omega
e^{-26S_{L}[\omega,\hat{g}]}.$ (B.4)

Compare this to equations (3.10) and (3.11). The prime on the determinant
indicates that we remove zero modes.

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