# Template

###### Abstract

We revisit the gravity path integral formalism of JT gravity. We explain how
to gauge fix the path integral in the presence of asymptotic boundaries and
conical defects, and resolve an ambiguity regarding the dilaton gravity
operator that creates a conical defect. Along the way we study JT gravity
coupled to matter on surfaces with defects of special opening angles,
obtaining expressions for partition and two-point functions of matter fields.
The two point function involves a summation over all geodesics on the surface,
including self-intersecting geodesics, which we formally manage to include.

Revisiting the second order formalism of JT gravity

Guanda Lin1, Mykhaylo Usatyuk1,2

1 Center for Theoretical Physics and Department of Physics, Berkeley, CA,
94720, USA
2 Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Determinants and correlators on hyperbolic surfaces
    1. 2.1 Building hyperbolic geometries
    2. 2.2 Two-point functions and determinants on quotient surfaces
    3. 2.3 JT examples
      1. 2.3.1 Double trumpet
      2. 2.3.2 Conical defect
      3. 2.3.3 Two conical defects
      4. 2.3.4 The handle disk
  3. 3 JT gravity path integral
    1. 3.1 Gauge fixing the path integral
      1. 3.1.1 Integrating over moduli space
    2. 3.2 Examples
      1. 3.2.1 Disk
      2. 3.2.2 Conical defect
      3. 3.2.3 Conical defect: $z$ coordinates
      4. 3.2.4 General surfaces
  4. 4 Discussion
  5. A Consistency of disk measure with general surface
  6. B Double trumpet gluing measure
  7. C The determinant calculation

##  1 Introduction

Jackiw-Teitelboim (JT) gravity is a simple two dimensional model that has been
central to many recent developments in quantum gravity[1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11]. In this paper we will be revisiting various unresolved aspects of
the second order formalism of the JT gravity path integral.

The most common approach to JT gravity is through the first order formalism
[7, 12], especially when dealing with the issue of gauge fixing the path
integral. There has been some progress on understanding JT gravity from the
second order formalism. For compact surfaces the path integral was properly
gauge fixed in [7], whereas for surfaces with asymptotically AdS boundaries
significant progress was made in [13, 14, 15, 16]. Furthermore, when
considering matter coupled to JT it is easiest to use gravitational variables.

In this paper we will consider JT gravity on surfaces with asymptotically AdS
boundaries with the addition of conical defects[17, 18, 19, 20]. We will
calculate determinants of Laplace operators, matter field correlators, and
gauge fix the gravity path integral on such surfaces from the perspective of
the second order formalism.

Determinants of Laplace operators have recently appeared in studies of JT
gravity coupled to matter where they correspond to partition functions of
matter fields on the geometry[13, 21, 22], and in the evaluation of Lorentzian
JT gravity amplitudes with topology changing wormholes [23, 24]. Similarly,
correlation functions of matter fields on wormhole geometries have recently
appeared in [25, 26, 27, 22, 21], where they probe the length of the wormhole.
Conical defect geometries have played an important role in the counting of
black hole microstates[10], and matter field correlators on defect geometries
were considered in [26, 28]. The procedure for gauge fixing the gravitational
path integral is well known from the string perturbation theory literature[29,
30, 31], and requires the evaluation of various determinants on hyperbolic
surfaces. This naturally connects the problem of gauge fixing the path
integral to understanding matter minimally coupled to JT gravity.

We will find that the second order formalism has certain advantages. A proper
gauge fixing will resolve an ambiguity regarding the correct definition of the
conical defect creation operator. Furthermore, it is straightforward to
incorporate matter, and we obtain closed-form expressions for determinants and
two-point functions of matter coupled to JT gravity. We now summarize our main
results.

### Summary of results

In section 2 we begin by explaining how to construct constant negative
curvature surfaces by taking a quotient of the upper half-plane by a group
$\Gamma\subset\text{PSL}(2,\mathbb{R})$, giving a surface
$\Sigma=\mathbb{H}/\Gamma$. Using the quotient construction a variety of
hyperbolic surfaces can be constructed including compact surfaces, surfaces
with asymptotic AdS boundaries, and surfaces that include conical defects with
special opening angles $\theta=\frac{2\pi}{n}$ with integer $n\geq 2$. We then
explain how to compute determinants of Laplace type operators on surfaces
obtained through the quotient method. The final result is the following.

Figure 1: The determinant and two point function are reproduced by a sum over
all geodesics on the surface $\Sigma$. On the left we have highlighted three
geodesics that contribute to the determinant on the surface with one handle
and one defect. On the right we have shown some geodesics that contribute to
the two-point function, with the geodesics ending on the operator insertions
at the boundary. There are infinitely many additional geodesics on these
surfaces that contribute to these quantities.

##### Determinants.

Let $\Sigma=\mathbb{H}/\Gamma$ be a hyperbolic surface, either compact or with
asymptotic AdS boundaries, with some number of conical defects. Define the
scalar Laplacian $\Delta=-g^{ab}\nabla_{a}\nabla_{b}$. The determinant of the
Laplace operator on the surface $\Sigma$ is given by

$\det\left({\Delta}+s(s-1)\right)_{\Sigma}=\underbrace{Z_{\text{hyp.}}(s)}_{\text{geodesics}}\underbrace{Z_{\text{ell.}}(s)}_{\text{defects}}\left(\text{const}\right),$
(1.1)

see section 2.2 and appendix C for additional technical details. The most
important contribution to the determinant is the Selberg zeta function111For
compact surfaces this form of the determinant is well known [32, 33], and
comes from the Selberg trace formula. For surfaces with asymptotically AdS
boundaries the techniques for calculating determinants are relatively recent
[34, 35, 36], and one of the new results is to incorporate conical defects in
the calculation, alongside evaluating the determinant for the vector Laplacian
$\det\left(\Delta_{1}+s(s-1)\right)_{\Sigma}$ in appendix C.

$Z_{\text{hyp.}}(s)=\prod_{\ell_{\gamma}\in\mathcal{L}_{\Sigma}}\prod_{m=0}^{\infty}\left(1-e^{-(s+m)\ell_{\gamma}}\right),$
(1.2)

which is defined to be a sum over all closed geodesics $\mathcal{L}_{\Sigma}$
on the surface $\Sigma$, with lengths given by $\ell_{\gamma}$. See figure 1.
The integer $m$ counts the winding of the geodesics. The additional term
$Z_{\text{ell.}}(s)$ comes from the presence of conical defects, and does not
have an obvious geometric interpretation. We also obtain a similar expression
for the determinant of the vector Laplacian
$\det\left(\Delta_{1}+s(s-1)\right)_{\Sigma}$, see appendix C.

##### Correlators.

As an intermediate step in the determinant calculation we obtain a closed form
expression for the two-point function of a scalar field
$\langle\phi(x)\phi(y)\rangle$ on an arbitrary surface $\Sigma$. When the
points $x,y$ are sent to asymptotic AdS boundaries the geodesic approximation
becomes exact, and the correlator reduces to a sum over all geodesics
connecting $x$ and $y$. Dressing the operator insertions to the boundary
Schwarzian, we find that the correlator for a field of mass $m^{2}=s(s-1)$ on
a surface $\Sigma=\mathbb{H}/\Gamma$ is given by

$G_{\Sigma}(\tau_{1},\tau_{2})=\sum_{\text{geodesics
}\gamma}e^{-s\ell_{\gamma}}~{}=\sum_{T\in\Gamma}\left(\frac{F^{\prime}(\tau_{1})(T\cdot
F(\tau_{2}))^{\prime}}{\left(F(\tau_{1})-T\cdot
F(\tau_{2})\right)^{2}}\right)^{s},$ (1.3)

where the sum over all geodesics connecting the points is given by the sum
over the group $\Gamma$ and the Schwarzian fluctuations are given by
$F(\tau)$, see equation (2.27).222In this formula we have the action $T\cdot
F=\frac{aF+b}{cF+d}$. The formula also applies in the case that the operators
are inserted on different asymptotic boundaries with independent Schwarzian
fluctuations $F_{i}(\tau)$. See section 2.3 for an example. This formula
includes a summation over all geodesics connecting the two operators including
self-intersecting geodesics, see figure 1.

In section 2.3 we work out a number of examples for both determinants and
correlators. We reproduce previous results for the defect and double trumpet
geometries, and obtain new expressions for the two defect and handle disk
geometries.

##### Gravity path integral.

In section 3 we turn to the problem of gauge fixing the gravity path integral
for JT gravity. Consider performing the JT gravity path integral on surfaces
$\Sigma$ of genus $g$ with asymptotically AdS boundaries of regularized
lengths $\vec{\beta}=\left(\beta_{1},\ldots,\beta_{n}\right)$, along with the
insertion of $k$ conical defect operators $\mathcal{V}_{\alpha}$ giving rise
to conical defects with opening angles $2\pi\left(1-\alpha_{i}\right)$
specified by $\vec{\alpha}=\left(\alpha_{1},\ldots,\alpha_{k}\right)$. We show
that the path integral reduces to the Weil-Petersson measure on the moduli
space $\mathcal{M}_{g,\vec{\alpha},\vec{\beta}}$ of associated hyperbolic
surfaces. More compactly,

$Z_{g,{\vec{\alpha}}}(\beta_{1},\ldots,\beta_{n})=\int\frac{\mathcal{D}g\mathcal{D}\Phi}{\text{V}(\text{Diff})}e^{-I_{\text{J}T}[g,\Phi]}\mathcal{V}_{\alpha_{1}}\ldots\mathcal{V}_{\alpha_{k}}=\int_{\mathcal{M}_{g,{\vec{\alpha},\vec{\beta}}}}d\left(\text{Weil-
Pet.}\right)e^{-I_{\text{bdy}}},$ (1.4)

where on the right-hand side the boundary action is for the Schwarzian on the
asymptotic AdS boundaries. In the above we have used determinants computed in
section 2.333We are only able to compute the necessary determinants for
certain values of $\alpha$, but we argue that the reduction to the Weil-
Petersson measure should occur for all $\alpha$. We are now left with the
problem of evaluating the integral on the right-hand side. It’s well known
from the string perturbation theory literature how to perform the integral
over moduli space [29, 31], formally we have

$\int_{\mathcal{M}}d(\text{Weil-Pet.})\hskip
2.84544pte^{-I_{\text{bdy}}}=\underbrace{\prod_{n}\int
dm_{n}d\overline{m}_{n}}_{\text{coordinates on
$\mathcal{M}$}}\underbrace{\frac{\det\langle\mu,\phi\rangle\det\langle\overline{\mu},\overline{\phi}\rangle}{\sqrt{\det\langle\phi,\phi\rangle\det\langle\overline{\phi},\overline{\phi}\rangle}}}_{\text{WP
measure}}\hskip 2.84544pte^{-I_{\text{bdy}}[m_{n},\overline{m}_{n}]}.$ (1.5)

The above equation can be understood as making a choice of coordinates $m_{n}$
for the space we are integrating over, and computing the Weil-Petersson
measure in that set of coordinates. The quantities appearing in the measure
are known as quadratic differentials $\phi$ and Beltrami differentials $\mu$,
and we explain the technical details of the measure in section 3.1.1.

Computing the measure for a particular choice of coordinates is challenging
for general surfaces, but in simple examples it is possible. In the case of
the disk with and without a conical defect the measure can be evaluated, and
in section 3.2 we perform the full integral over moduli space finding
agreement with the standard Schwarzian calculation.

##### Conical defect operator.

With the above we are ready to determine the dilaton gravity operator
$\mathcal{V}_{\alpha}$ that creates a conical defect when inserted into the
path integral (1.4). When a defect is added to the surface the real dimension
of the moduli space increases by two, which corresponds to the two directions
the defect can be moved on the surface. We consider a surface $\Sigma$ with a
conical defect, and choose two of the coordinates $m_{n}$ for the moduli space
to be the position $x$ of the defect. Using (1.5) we can immediately write
down the correct operator to be

$\mathcal{V}_{\alpha}=\underbrace{\int_{\Sigma}d^{2}x}_{\text{defect
position}}\underbrace{\frac{\langle\mu,\phi_{1}\rangle\langle\overline{\mu},\overline{\phi}_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle\langle\overline{\phi}_{1},\overline{\phi}_{1}\rangle}}}_{\text{measure
for $x$ coords}}e^{-2\pi\left(1-\alpha_{i}\right)\Phi(x)},$ (1.6)

in the above we have reintroduced the exponential of the dilaton since in the
measure $\eqref{eqn:intro_measure}$ we have already integrated out the
dilaton.444When combined with the JT action, integrating out the dilaton
creates a delta function source for the Ricci scalar at position $x$. For
additional details on the definition of this measure see section 3.1.1 around
equation (3.21).

This measure cannot be explicitly evaluated for a general surface $\Sigma$.
However, in section 3.2.4 we explain that in the special limit that $\alpha\to
1$, with opening angle $\theta=2\pi\alpha$, we can evaluate the measure on an
arbitrary surface $\Sigma$. We find that the conical defect operator takes the
form

$\text{General surface:}\qquad\lim_{\alpha\to
1}\mathcal{V}_{\alpha}=2\pi(1-\alpha)\int_{\Sigma}d^{2}x\sqrt{g(x)}e^{-2\pi(1-\alpha)\Phi(x)}+\mathcal{O}\left((1-\alpha)^{2}\right),$
(1.7)

up to corrections in the $\left(1-\alpha\right)$ expansion.555One interesting
aspect is that the metric $g(x)$ in (1.7) is for the surface $\Sigma$ with no
defect, which is important for our argument for the recursion relation for
Weil-Petersson volumes found by [20]. This is consistent with previous
expectations for the form of the operator in the blunt defect limit $\alpha\to
1$ [18, 17, 19, 20], and constitutes a gravity path integral argument for the
form of the operator. In the case of the disk we can do better, and in section
3.2.3 We evaluate the full measure and find

$\text{Disk:}\qquad\mathcal{V}_{\alpha}=\pi(1-\alpha^{2})\int
d^{2}x\sqrt{g(x)}e^{-2\pi(1-\alpha)\Phi(x)}.$ (1.8)

In appendix A we show how the calculation on the disk can be thought of as
performing a resummation of the $\left(1-\alpha\right)$ expansion appearing
for general surfaces (1.7). We thus conjecture that the general form of the
conical defect operator on an arbitrary surface should take the form

$\textbf{Conjecture for general surfaces:
}\qquad\mathcal{V}_{\alpha}=\pi(1-\alpha^{2})\int_{\Sigma}d^{2}x\sqrt{g(x)}e^{-2\pi(1-\alpha)\Phi(x)},$
(1.9)

after the resummation in (1.7) is performed. Our argument for the above form
is that the operator should be independent of the surface it is inserted on,
as is the case when these operators are defined as limits of minimal string
operators [19, 20], and so the answer we find on the disk should carry over to
other surfaces.

##### Applications to dilaton gravity.

There are two immediate applications of the conical defect operator. The first
is it clarifies the correct dilaton potential that corresponds to JT gravity
coupled to a gas of conical defects. In [18, 17, 19, 20] JT gravity coupled to
a gas of conical defects was defined by a summation over conical Weil-
Petersson volumes with a coupling $\lambda$ weighing each defect. The bulk
dilaton gravity action that corresponds to this theory is thus given by

$I[g,\Phi]=-\frac{1}{2}\int d^{2}x\sqrt{g}\left(\Phi R+2U(\Phi)\right),$
(1.10)

with the dilaton potential following from equation (1.9)

$U(\Phi)=\Phi+\pi\lambda(1-\alpha^{2})e^{-2\pi\left(1-\alpha\right)\Phi}.$
(1.11)

At each order in the $\lambda$ expansion the path integral will localize onto
singular hyperbolic surfaces and reproduce the appropriate Weil-Petersson
volume with the required coupling $\lambda^{k}$ for surfaces with $k$ defects.

A secondary application is that the conical defect operator can be used to
give a gravity path integral argument for a recursion relation of Weil-
Petersson volumes found by [20]. Namely, in the limit that one defect on a
surface becomes blunt the volume becomes related to the volume without the
defect through

$\frac{dV_{g,m,n+1}\left(\vec{\alpha}_{n+1},\vec{b}_{m}\right)}{d\alpha_{n+1}}\bigg{\rvert}_{\alpha_{n+1}=1}=4\pi^{2}\chi\left(\Sigma\right)V_{g,m,n}\left(\vec{\alpha}_{n},\vec{b}_{m}\right).$
(1.12)

In section 3.2.4 we give a gravity path integral argument for this recursion
relation using the conical defect operator.

##  2 Determinants and correlators on hyperbolic surfaces

###  2.1 Building hyperbolic geometries

We now explain how to build constant negative curvature geometries with
conical defects and asymptotic boundaries by taking quotients of the upper
half-plane (UHP) by an appropriate group. For additional details see [36]. The
quotient construction is useful for calculating determinants and correlation
functions on such surfaces since the method of images can be used. Consider
the upper half-plane $\mathbb{H}$ with the standard AdS2 metric

$ds^{2}=\frac{dzd\overline{z}}{\left(\operatorname{Im}z\right)^{2}}\,,$ (2.1)

The group of isometries is given by PSL$(2,\mathbb{R})$, and elements of the
group $T\in\text{PSL}(2,\mathbb{R})$ are either hyperbolic, elliptic, or
parabolic. Hyperbolic elements satisfy $\Tr T>2$, elliptic satisfy $\Tr T<2$,
while parabolic satisfy $\Tr T=2$. We will primarily be interested in
hyperbolic and elliptic elements, and the most general form of these elements
is given by

$\text{Hyperbolic:
}T_{\ell}=\Lambda^{-1}\left(\begin{array}[]{ll}e^{\ell/2}&0\\\
0&e^{-\ell/2}\end{array}\right)\Lambda,\qquad\text{Elliptic:
}T_{\theta}=\Lambda^{-1}\left(\begin{array}[]{ll}\cos(\frac{\theta}{2})&-\sin(\frac{\theta}{2})\\\
\sin(\frac{\theta}{2})&~{}~{}\cos(\frac{\theta}{2})\end{array}\right)\Lambda\,,$
(2.2)

where $\Lambda\in\text{PSL}(2,\mathbb{R})$. The action of $T$ on the upper
half-plane is defined by

$T\cdot z=\frac{az+b}{cz+d}\,.$ (2.3)

As an example, setting $\Lambda=1$ the hyperbolic element acts as
$T_{\ell}\cdot z=e^{\ell}z$. Elliptic elements leave a fixed point in the
interior which will become a bulk conical defect when we consider the quotient
geometry.666Parabolic elements leave a fixed point at the asymptotic boundary.
The fixed point corresponds to a conical singularity with deficit angle
$2\pi$, and is known as a cusp. We will not consider cusps because the
quotient method becomes much more complicated since the cusp lives on the
asymptotic boundary. However, all of our claims should obviously generalize to
surfaces that include cusps.

One way to build a hyperbolic surface is to take a subgroup of the isometry
group $\Gamma\subset\text{PSL}(2,\mathbb{R})$ and take the quotient of the
upper half-plane by this subgroup $\Gamma\backslash\mathbb{H}$. That is, we
identify two points as equivalent if $z\cong T\cdot z$ for any $T\in\Gamma$.
To build a good hyperbolic surface we must restrict to Fuchsian groups
$\Gamma\subset\text{PSL}(2,\mathbb{R})$, which are discrete subgroups of
$\text{PSL}(2,\mathbb{R})$. This restricts the elliptic elements we are
allowed to consider to $T_{\theta}^{n}=\pm\operatorname{I}$, which only allows
opening angles $\theta=2\pi/n$ with integer $n\geq 2$.777Restricting to
Fuchsian groups is equivalent to imposing the condition that a sufficiently
small ball around the identity element, in the $SL(2,\mathbb{R})$ group
manifold, contains no other elements. Intuitively, when quotienting $z\cong
T\cdot z$ we do not want there to exist elements $T\in\Gamma$ arbitrarily
close to the identity, since then we will be identifying arbitrarily close
points $z,T\cdot z$, and the quotient surface will be degenerate. We will take
the Fuchsian group to have a finite number of generators $T_{i}$ given by some
set set of hyperbolic and elliptic elements (2.2).

We are interested in hyperbolic surfaces with both asymptotic boundaries and
conical defects. Locally, a conical defect at a point $x_{i}$ is characterized
by the fact that we can travel around the point by going through an angle
$\theta<2\pi$. For a conical defect of opening angle $\theta=2\pi\alpha$ this
translates to a condition on the scalar curvature given by

$\frac{1}{2}\sqrt{g}\left(R+2\right)=2\pi\left(1-\alpha\right)\delta^{2}(x-x_{i})\,,$
(2.4)

where we restrict to $\alpha\in(0,1)$. A natural question is whether there
exists a hyperbolic surface with $k$ defects with angles specified by
$\vec{\alpha}=\left(\alpha_{1},\ldots,\alpha_{k}\right)$. It turns out that
such a surface always exists provided that the specification of the defects
$\alpha_{i}$ does not violate the Gauss-Bonnet theorem [37, 38]

$2\pi\chi(\Sigma)=\frac{1}{2}\int_{\Sigma}\sqrt{g}R+\int_{\partial\Sigma}\sqrt{h}K\,,$
(2.5)

where $K$ is the extrinsic curvature on the boundary of the surface, $h$ is
the induced boundary metric, and $\chi(\Sigma)=2-2g-n$ where $n$ is the number
of boundaries and $g$ is the genus. Conical defects fall into two classes:
sharp defects with opening angle $\theta\leq\pi$, and blunt defects with
opening angle $\theta>\pi$ [19, 20]. This translates to the condition
$\alpha\leq\frac{1}{2}$ for sharp and $\alpha>\frac{1}{2}$ for blunt.888One
nice property of sharp defects is that for surfaces with an asymptotic
boundary and at least two sharp defects, the defects are separated from the
boundary by a closed geodesic homotopic to the boundary. This can be seen from
the Gauss-Bonnet theorem. The quotient construction only allows us to build
surfaces with sharp defects $\alpha\leq\frac{1}{2}$.

We now summarize existence theorems on the types of surfaces that can be built
using the quotient method. First, consider the moduli space $\mathcal{M}_{g}$
of compact surfaces of genus $g\geq 2$. For each compact hyperbolic surface
$\Sigma$ there exists a Fuchsian group $\Gamma$ such that
$\Sigma=\Gamma\backslash\mathbb{H}$, and such a representation of the surface
is known as a Fuchsian model. As an example, for $g=2$ the Fuchsian group is
generated by $4$ hyperbolic elements $T_{i}$ with a non-trivial
constraint999The constraint enforces that the closed loop generated by the
action of the following group element will be contractible on the surface. on
the generators

$\Gamma=\langle
T_{1},T_{2},T_{3},T_{4}~{}|~{}T_{4}^{-1}T_{3}^{-1}T_{4}T_{3}T_{2}^{-1}T_{1}^{-1}T_{2}T_{1}=1\rangle.$
(2.6)

In general, the Fuchsian group of a compact genus $g$ surface is generated by
$2g$ distinct hyperbolic elements satisfying non-trivial constraints.

Now consider the moduli space $\mathcal{M}_{g,\vec{\alpha}}$ of compact
surfaces of genus $g$ with $k$ conical defects with deficit angles specified
by $\vec{\alpha}=\left(\alpha_{1},\ldots,\alpha_{k}\right)$. When the opening
angles take special values $\theta_{i}=2\pi\alpha_{i}$ with
$\alpha_{i}=\frac{1}{n_{i}}$ and integer $n_{i}\geq 2$ then every surface can
be obtained by a quotient with a suitable Fuchsian group[39, 40]. The group
will contain $k$ elliptic elements that leave a fixed point
$z_{i}\in\mathbb{H}$ which becomes the location of the conical
defect.101010The order of the elliptic element determines the strength of the
opening angle to be $2\pi/n$. Surfaces with other deficit angles cannot be
constructed by the quotient method.

In JT gravity we typically consider non-compact surfaces with $n$ asymptotic
boundaries with regularized lengths
$\vec{\beta}=\left(\beta_{1},\ldots,\beta_{n}\right)$. All hyperbolic surfaces
with asymptotic boundaries, but without conical defects,
$\mathcal{M}_{g,\vec{\beta}}$ can be obtained by a quotient with a Schottky
group (see theorem 15.3 in [36]). We are not aware of a similar theorem in the
case that we include conical defects
$\mathcal{M}_{g,\vec{\alpha},\vec{\beta}}$, but we will assume that if we
limit the opening angles $\vec{\alpha}$ to the previously mentioned special
values that these surfaces can also be obtained through a quotient.

###  2.2 Two-point functions and determinants on quotient surfaces

The quotient construction makes it possible to calculate the determinant and
two-point function on the associated surface. In this subsection we explain
how to evaluate these quantities.

Let $\Sigma$ be a hyperbolic surface with metric $g$ obtained by quotienting
$\mathbb{H}$ with a Fuchsian group $\Gamma$. The scalar Laplacian is defined
to be $\Delta=-g^{ab}\nabla_{a}\nabla_{b}$. The two-point function on the
surface $\Sigma$, also known as the resolvent, is defined through the equation

$\left(\Delta+s(s-1)\right)R_{\Sigma}(s;z,w)=\frac{\delta^{2}(z-w)}{\sqrt{g}}\,,$
(2.7)

where $z,w$ are two points on the surface $\Sigma$, and $s\geq 1$ is related
to the mass of a scalar field through $m^{2}=s(s-1)$. For surfaces obtained
through the quotient method we can obtain a formula for the resolvent by using
the method of images. The key idea is that the resolvent on the upper half
plane $R_{\mathbb{H}}$, which is explicitly known, locally satisfies the
desired equation (2.7), but globally we must have that
$R_{\Sigma}(s;z,w)=R_{\Sigma}(s;T\cdot z,w)$ since $z\cong T\cdot z$ for all
$T\in\Gamma$. Performing a sum over the quotient group gives a function that
precisely satisfies all the properties that define the resolvent on $\Sigma$

$R_{\Sigma}(s;z,w)=\sum_{T\in\Gamma}R_{\mathbb{H}}(s;T\cdot z,w).$ (2.8)

Once the resolvent is known it is straightforward to calculate the
determinant. For smooth compact surfaces it is well known that the determinant
of the Laplacian reduces to a product over geodesics due to the Selberg trace
formula[32, 33]. However, for non-compact surfaces this method is not
applicable, and other techniques need to be used[34, 35, 36]. We will review
the technique for surfaces with asymptotic boundaries[36], and extend it to
include conical defects. The basic procedure of the calculation and key
results will be presented here, leaving technical details to appendix C.

To compute the determinant of an operator we must solve the eigenvalue problem
$\left(\Delta+s(s-1)\right)\phi_{n}=\lambda_{n}\phi_{n}$ and take the product
of all the eigenvalues. However, doing this directly is quite challenging. A
simpler technique is to define the determinant through the trace of the
resolvent. As an example, consider the case that the eigenvalues are discrete
with eigenfunctions $\phi_{n}$. The resolvent is given by

$R_{\Sigma}(s;z,w)=\sum_{n}\frac{\phi_{n}(z)\phi_{n}(w)}{\lambda_{n}+s(s-1)}\,,$
(2.9)

where the eigenfunctions satisfy
$\int_{\Sigma}d^{2}z\sqrt{g}\phi_{n}(z)\phi_{m}(z)=\delta_{nm}$. One can check
that the determinant defined as a product of eigenvalues is related to the
resolvent trace through

$\frac{1}{2s-1}\frac{d}{ds}\log(\det(\Delta+s(s-1)))={\rm
tr}R_{\Sigma}(s)\equiv\int_{\Sigma}d^{2}z\sqrt{g}R_{\Sigma}(s;z,z)\,.$ (2.10)

Conversely, if a closed form expression for the resolvent can be obtained then
the determinant can be obtained by inverting the above formula.

For simple surfaces the resolvent trace is easy to evaluate since the
resolvent is known on the UHP, and we can perform the sum over the Fuchsian
group defining the surface. Consider the double trumpet with geodesic throat
of size $\ell$. The Fuchsian group is generated by a single hyperbolic element
$\Gamma=\langle T_{\ell}\rangle$, producing the surface
$\Sigma=\mathbb{H}/\langle T_{\ell}\rangle$. The resolvent trace is obtained
after using equations (2.8) and (2.10)

$\displaystyle\sum_{\begin{subarray}{c}m\in\mathbb{Z}_{\neq
0}\end{subarray}}\int_{\mathbb{H}/\langle
T_{\ell}\rangle}d^{2}z\sqrt{g}R_{\mathbb{H}}\left(s;T_{\ell}^{m}\cdot
z,z\right)=\frac{2\ell}{2s-1}\sum_{m=1}^{\infty}\frac{e^{-sm\ell}}{1-e^{-m\ell}}\,.$
(2.11)

The integer $m$ counts the number of windings of the closed geodesic. The
integral can be implemented within a fundamental domain given by
$1<|z|<e^{\ell}$; as for details of integration, see appendix C.

For general surfaces the summation over the group (2.8) will reduce the
resolvent trace to simpler building blocks, such as the above calculation on
the double trumpet. Let us explain how this works. The sum over the group can
be broken up into a sum over conjugacy classes $[\gamma]$ of primitive
elements as
$\sum_{\Gamma}=\sum_{[\gamma]}\sum_{g\in\Gamma/C_{\langle\gamma\rangle}}$.111111A
primitive element $\gamma$ is not a power of any other element in the group.
The centralizer is defined as the set of elements that leave the group
generated by $\gamma$ to be invariant
$C_{\langle\gamma\rangle}=\\{g\langle\gamma\rangle
g^{-1}=\langle\gamma\rangle|g\in\Gamma\\}$. The set of left cosets
$\Gamma/C_{\langle\gamma\rangle}$ generates a list of distinct elements that
we should conjugate $\gamma$ by sum over the full conjugacy class. We must
take the centralizer of the group generated by $\gamma$,
$C_{\langle\gamma\rangle}$ because we want to break up the sum into conjugacy
classes of $[\gamma^{n}]$. It turns out that the centralizer for primitive
$\gamma$ is simply given by $C_{\langle\gamma\rangle}=\langle\gamma\rangle$,
which we use notationally throughout. To avoid over-counting, we must sum over
an element from each coset $g\in\Gamma/C_{\langle\gamma\rangle}$ and conjugate
$g\gamma g^{-1}$ to sum over the entire conjugacy class. The primitive
elements will either be hyperbolic or elliptic which geometrically corresponds
to either closed geodesics or conical defects respectively, and we must also
include a sum over conjugacy classes of multiples of the primitive elements
$\gamma^{n}$. The final result is that every group element can be reached by a
conjugation $g^{-1}\gamma^{n}g$ of some power of a primitive element $\gamma$.
Suppose we have a list of primitive elements denoted by $\Pi$, the summation
over the group is then given by121212When $\gamma$ is elliptic the summation
over powers $\gamma^{m}$ should be cutoff when we reach $\gamma^{m}=I$.

$R_{\Sigma}(s;z,w)=R_{\mathbb{H}}\left(s;z,w\right)+\sum_{\gamma\in\Pi}\sum_{g\in\Gamma/\langle\gamma\rangle}\sum_{m\neq
0}^{\infty}R_{\mathbb{H}}\left(s;g^{-1}\gamma^{m}g\cdot z,w\right)\,,$ (2.12)

where the first term is the identity contribution.

When calculating the trace of the resolvent (2.10) the sum over
$\Gamma/\langle T_{\ell}\rangle$ for hyperbolic elements $T_{\ell}$ will
transform the domain of integration from the surface $\Sigma$ to the
fundamental domain of a double trumpet with a closed geodesic $\ell$, of which
we already have the answer in (2.11)

$\displaystyle\sum_{\begin{subarray}{c}m\in\mathbb{Z}_{\neq
0}\end{subarray}}\sum_{g\in\Gamma/\langle
T_{\ell}\rangle}\int_{\mathbb{H}/\Gamma}d^{2}z\sqrt{g}R_{\mathbb{H}}\left(s;z,T^{m}_{\ell}g\cdot
z,g\cdot z\right)$
$\displaystyle=\sum_{\begin{subarray}{c}m\in\mathbb{Z}_{\neq
0}\end{subarray}}\int_{\mathbb{H}/\langle
T_{\ell}\rangle}d^{2}z\sqrt{g}R_{\mathbb{H}}\left(s;z,T_{\ell}^{m}\cdot
z\right)$
$\displaystyle=\frac{2\ell}{2s-1}\sum_{m=1}^{\infty}\frac{e^{-sm\ell}}{1-e^{-m\ell}}\,,$
(2.13)

where the RHS is already given in (2.11). This gives a contribution to the
determinant (2.10) for each closed geodesic $\ell$ on the surface $\Sigma$.
The same logic applies to elliptic elements $T_{\theta}$ in the sum (2.12)
except now the fundamental domain becomes a cone.

The full determinant calculation is quite involved and carried out in appendix
C. We quote the final result. Consider a hyperbolic surface $\Sigma$ with $k$
conical defects with opening angles $\theta_{i}=\frac{2\pi}{n_{i}}$ with
integer $n_{i}\geq 2$. The surface can be compact or with asymptotic AdS
boundaries. The determinant is given by

$\det\left({\Delta}+s(s-1)\right)=\underbrace{Z_{\text{hyp.}}(s)}_{\text{geodesics}}\underbrace{Z_{\text{ell.}}(s)}_{\text{defects}}\left(\text{const}\right)$
(2.14)

The most important contribution is given by $Z_{\text{hyp.}}$, which is the
Selberg zeta-function on $\Sigma$. It comes from the summation over hyperbolic
elements in (2.12). It is defined by a product over contributions from all
closed geodesics on the surface. Define the set of lengths $\ell_{\gamma}$ of
primitive closed geodesics to be $\mathcal{L}_{\Sigma}$,131313Excluding
exceptional cases, geodesics are endowed with an orientation. For each
geodesic $\gamma$ there is a mirror geodesic with opposite orientation and the
same length. Both a geodesic and it’s mirror need to be independently included
in the Selberg zeta-function. An example of a non-orientable geodesic is given
by the two defect surface considered below. and the Selberg zeta-function for
the surface $\Sigma$ is defined to be

$Z_{\text{hyp.}}(s)=\prod_{\ell_{\gamma}\in\mathcal{L}_{\Sigma}}\prod_{m=0}^{\infty}\left(1-e^{-(s+m)\ell_{\gamma}}\right),$
(2.15)

which can be derived from (2.11). In the above equation, the first product is
over all primitive geodesics, while the second is over all integer $m$
windings of the geodesic. Primitive geodesics include self-intersecting ones
as well as those that touch conical defects. We also mention that geodesics
with opposite orientations are counted as distinct in (2.15).

The second term comes from the summation over elliptic elements in (2.12).
Since elliptic elements give conical defects we label this the “defect”
contribution, and it is given by

$Z_{\text{ell.}}(s)=\prod_{i=1}^{k}\prod_{r=0}^{n_{i}-1}\Gamma\left(\frac{s+r}{n_{i}}\right)^{\frac{2r+1-n_{i}}{n_{i}}}.$
(2.16)

For each conical defect we get a highly non-analytic contribution in the
opening angles $\theta_{i}=\frac{2\pi}{n_{i}}$. Unlike the Selberg zeta-
function, the contribution due to elliptic elements does not seem to have an
obvious geometric interpretation. We emphasize that every term in equation
(2.14) depends on the choice of deficit angles in some way. Either implicitly
through the types of geodesics that are included in the Selberg zeta-function,
or explicitly as some function of the angles as above. The constant term in
(2.14) is due to the identity element contribution and given in appendix C.

###  2.3 JT examples

Let us briefly explain how these determinants and resolvents arise for JT
gravity coupled to matter. Consider a massive scalar field minimally coupled
to JT gravity. The path integral is given, up to boundary terms, by

$Z_{\text{JT +
matter}}=\int\mathcal{D}g\mathcal{D}\Phi\exp\left(\frac{1}{2}\int\sqrt{g}\phi\left(R+2\right)\right)\hskip
1.42271ptZ_{\text{matter}}[g],$ (2.17)

where the matter partition function is

$Z_{\text{matter}}[g]=\int\mathcal{D}\phi
e^{-S[\phi,g]}=\frac{1}{\sqrt{\det\left(\Delta+m^{2}\right)}},\qquad
S[\phi,g]=\frac{1}{2}\int_{\Sigma}\sqrt{g}\left(\phi\Delta\phi+m^{2}\phi^{2}\right).$
(2.18)

where we define the mass through $m^{2}=s(s-1)$, with $s\geq 1$ the scaling
dimension of the boundary operator dual to $\phi$. We see the matter partition
function reduces down to a determinant. After integrating out the dilaton we
localize onto the moduli space of constant negative curvature
geometries,141414By appropriately deforming the JT gravity action we can also
localize onto the moduli space of constant negative curvature surfaces with
conical points which we will describe in detail in section 3. and so is given
by the determinant calculated in section 2.2. Another interesting observable
when matter is present is the two-point function/resolvent
$\langle\phi(z)\phi(w)\rangle$ given by $R_{\Sigma}(s;z,w)$. On the upper half
plane the scalar Laplacian is given by

$ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}},\qquad\Delta=-y^{2}\left(\partial_{x}^{2}+\partial_{y}^{2}\right).$
(2.19)

Defining the complex coordinates $z=x+iy$, the resolvent is given by

$R_{\mathbb{H}}(s;z,w)=\frac{\Gamma(s)^{2}}{4\pi\Gamma(2s)}\sech^{2s}\left(\frac{\ell(z,w)}{2}\right)\hskip
1.42271pt{}_{2}F_{1}\left(s,s,2s;\sech^{2}\left(\frac{\ell(z,w)}{2}\right)\right).$
(2.20)

where $\ell(z,w)$ is the geodesic distance between $z$ and $w$. In JT gravity
we want to dress the two-point function to the boundary schwarzian. Since the
geodesic distance goes to infinity near the boundary the two-point function
simplifies to

$R_{\mathbb{H}}(z,w)\mathrel{\mathop{=}\limits_{\ell\to\infty}}\frac{4^{s}\Gamma(s)^{2}}{4\pi\Gamma(2s)}e^{-s\ell(z,w)}\left(1+\mathcal{O}\left(e^{-\ell}\right)\right)\,,$
(2.21)

and so the geodesic approximation becomes exact. On an arbitrary surface
$\Sigma$ the two-point function is obtained by summing over the group in
equation (2.8) so that we obtain

$\displaystyle R_{\Sigma}(s;z,w)$
$\displaystyle=\frac{\Gamma(s)^{2}}{4\pi\Gamma(2s)}\sum_{T\in\Gamma}\sech^{2s}\left(\frac{\ell(z,T\cdot
w)}{2}\right)\hskip
1.42271pt{}_{2}F_{1}\left(s,s,2s;\sech^{2}\left(\frac{\ell(z,T\cdot
w)}{2}\right)\right)$ (2.22)
$\displaystyle\hskip-6.0pt\mathrel{\mathop{=}\limits_{\ell\to\infty}}\frac{4^{s}\Gamma(s)^{2}}{4\pi\Gamma(2s)}\sum_{T\in\Gamma}e^{-s\ell(z,T\cdot
w)}\left(1+\mathcal{O}\left(e^{-\ell}\right)\right).$

This implements a summation over all possible geodesics connecting the points
$z,w$. To dress the operator insertions to the boundary fluctuations we
parameterize the boundary by the curve $(x,y)=(F(\tau),\epsilon
F^{\prime}(\tau))$ where $\tau$ is an affine time, $\epsilon$ is the boundary
cutoff, and we have enforced the condition that the induced metric on the
boundary is $\sqrt{h}=1/\epsilon$. We consider inserting operators at two
boundary points labelled by times $\tau_{1},\tau_{2}$. The complex coordinates
at these times are $z=F(\tau_{1})+i\epsilon F^{\prime}(\tau_{1})$ and
$w=F(\tau_{2})+i\epsilon F^{\prime}(\tau_{2})$. The geodesic distance between
these points is given by

$\ell(z,w)=\log\left(\frac{1}{\epsilon^{2}}\frac{\left(F(\tau_{1})-F(\tau_{2})\right)^{2}}{F^{\prime}(\tau_{1})F^{\prime}(\tau_{2})}+\mathcal{O}\left(\frac{1}{\epsilon}\right)\right)\,.$
(2.23)

We will define the two-point function dressed to the boundary fluctuations by
$G(\tau_{1},\tau_{2})$, and subtract off the usual divergences to obtain

$G(\tau_{1},\tau_{2})=\lim_{\epsilon\rightarrow
0}\frac{4\pi\Gamma(2s)}{4^{s}\Gamma(s)^{2}\epsilon^{2s}}R_{\mathbb{H}}(s;z,w)=\left(\frac{F^{\prime}(\tau_{1})F^{\prime}(\tau_{2})}{\left(F(\tau_{1})-F(\tau_{2})\right)^{2}}\right)^{s}\,.$
(2.24)

This is the standard result for the matter two-point function dressed to the
Schwarzian boundary at the level of the disk. In terms of the thermal circle
reparameterization $f(\tau)$ defined by
$F(\tau)\equiv\tan\frac{\pi}{\beta}f(\tau)$ the regularized resolvent is

$G(\tau_{1},\tau_{2})=\left(\frac{f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})}{\frac{\beta^{2}}{\pi^{2}}\sin^{2}\frac{\pi}{\beta}\left(f(\tau_{1})-f(\tau_{2})\right)}\right)^{s}\,.$
(2.25)

On a more complicated surface obtained from a quotient by $\Gamma$ there are
additional geodesics connecting the two boundary points. Since $w\cong T\cdot
w$ for $T\in\Gamma$ the geodesic connecting $z$ to $T\cdot w$ on the disk
becomes a distinct geodesic connecting $z$ to $w$ on the quotient geometry. We
can find the geodesic distance between the point $z$ and $T\cdot w$ to be

$\ell\left(z,T\cdot
w\right)=\log\left(\frac{1}{\epsilon^{2}}\frac{\left(F(\tau_{1})-T\cdot
F(\tau_{2})\right)^{2}}{F^{\prime}(\tau_{1})\left(T\cdot
F(\tau_{2})\right)^{\prime}}+\mathcal{O}\left(\frac{1}{\epsilon}\right)\right),\qquad
T\cdot F(\tau)=\frac{aF(\tau)+b}{cF(\tau)+d},$ (2.26)

The boundary dressed two-point function on a quotient surface can then be
immediately obtained from equation (2.22)

$G_{\Sigma}(\tau_{1},\tau_{2})=\lim_{\epsilon\rightarrow
0}\frac{4\pi\Gamma(2s)}{4^{s}\Gamma(s)^{2}\epsilon^{2s}}\sum_{T\in\Gamma}e^{-s\ell(z,T\cdot
w)}=\sum_{T\in\Gamma}\left(\frac{F^{\prime}(\tau_{1})(T\cdot
F(\tau_{2}))^{\prime}}{\left(F(\tau_{1})-T\cdot
F(\tau_{2})\right)^{2}}\right)^{s}.$ (2.27)

This formula includes all possible geodesics on the surface $\Sigma$
connecting the specified boundary points, and different group elements $T$
lead to geodesics with different windings/self-intersections on $\Sigma$. This
includes geodesics that self-intersect any number of times. However, we are
still left with the challenge of integrating over the Schwarzian and
performing the sum over the group elements. We now explain some simple
examples.

####  2.3.1 Double trumpet

The metric for the double trumpet can be written as

$ds^{2}=dr^{2}+\cosh^{2}\left(r\right)d\theta^{2},\qquad\theta\cong\theta+b.$
(2.28)

The two asymptotic boundaries are at $r\to\pm\infty$, and the only primitive
geodesic of length $b$ is at $r=0$. The double trumpet is obtained by taking
the quotient of the upper half-plane $\mathbb{H}/\Gamma$ by the group
$\Gamma=\langle T_{b}\rangle$ generated by the element

$T_{b}=\begin{pmatrix}\exp\left(\frac{b}{2}\right)&0\\\
0&\exp\left(-\frac{b}{2}\right)\end{pmatrix},$ (2.29)

which identify points in the UHP plane by $z\cong e^{b}z$. In the fundamental
domain the geodesic throat of length $b$ goes along the imaginary axis from
$z=i$ to $z=ie^{b}$. The group elements are hyperbolic and take the form
$T_{b}^{m}$ for $m\in\mathbb{Z}$, and correspond to the geodesics that winds
$m$ times around the throat. The resolvent is given by

$R_{\mathbb{H}/\langle
T_{b}\rangle}(s;z,w)=\sum_{m}R_{\mathbb{H}}(s;T_{b}\cdot z,w)\,.$ (2.30)

The resolvent trace and determinant have been discussed in section 2.2 and we
simply repeat the result

${\rm tr}R_{\mathbb{H}/\langle
T_{b}\rangle}=\frac{2b}{2s-1}\sum_{m=1}^{\infty}\frac{e^{-smb}}{1-e^{-mb}},\qquad\det\left({\Delta}+s(s-1)\right)_{\text{DT}}=\prod_{m=0}^{\infty}\left(1-e^{-\left(s+m\right)b}\right)^{2}\,.$
(2.31)

where the quantity in parenthesis is squared since there are two orientations
for the primitive geodesic, and the determinant should be understood to be up
to multiplicative constants.

##### One sided two-point function.

We first consider the correlator where we insert two matter operators
$\langle\phi_{L}(\tau_{1})\phi_{R}(\tau_{2})\rangle$ onto the same asymptotic
boundary and dress them to the boundary fluctuations. We take the operator to
be located at $z=F_{R}(\tau_{1})+i\epsilon F_{R}^{\prime}(\tau_{1})$ and
$w=F_{R}(\tau_{2})+i\epsilon F_{R}^{\prime}(\tau_{2})$. The quotient $z\cong
T\cdot z$ enforces a periodicity constraint on the boundary151515This is
different from the $\tanh$ one in the JT review, but equivalent The Schwarzian
derivative itself is $\operatorname{PSL}(2,\mathbb{R})$ invariant: if
$F=\frac{aG+b}{cG+d}$, then $\\{F,t\\}=\\{G,t\\}$.

$F_{R}(\tau+\beta)=T_{b}\cdot F_{R}(\tau),\qquad T_{b}^{m}\cdot
F(\tau)=e^{mb}F(\tau).$ (2.32)

We can introduce the parameterization
$F_{R}(\tau)=\exp\left(\frac{b}{\beta}f_{R}(\tau)\right)$ with
$f_{R}(\tau+\beta)=f_{R}(\tau)+\beta_{R}$ which respects this identification.
We can immediately apply (2.27) to find the two-point function

$G^{\text{RR}}_{\text{DT}}(\tau_{1},\tau_{2})=\sum_{m=-\infty}^{\infty}\left(\frac{F_{R}^{\prime}(\tau_{1})(T_{b}^{m}\cdot
F_{R}(\tau_{2}))^{\prime}}{(F_{R}(\tau_{1})-T_{b}^{m}\cdot
F_{R}(\tau_{2}))^{2}}\right)^{s}=\sum_{m=-\infty}^{\infty}\left(\frac{f_{R}^{\prime}(\tau_{1})f_{R}^{\prime}(\tau_{2})}{\frac{4\beta_{R}^{2}}{b^{2}}\sinh^{2}\left[\frac{b}{2\beta_{R}}\left(f_{R}(\tau_{1})-f_{R}(\tau_{2}+m\beta_{R})\right)\right]}\right)^{s}\,.$
(2.33)

We have used that the summation over the group $\Gamma$ is the same as a
summation over powers of $T_{b}^{m}$. The sum over $m$ is over windings of the
geodesic around the double trumpet as it connects the two points. Positive and
negative integers indicate which direction the geodesic winds around the
trumpet. The largest contribution is given by the shortest geodesic which
doesn’t wind and is given by $m=0$. This reproduces the one sided two-point
function on the double trumpet computed in [41].

Figure 2: Two-point function on the double trumpet. We have geodesics
connecting the two operator insertions that wind an integer $m$ times around
the trumpet.

##### Two sided two-point function.

The correlator with matter insertions on opposite boundaries
$\langle\phi_{L}(\tau_{1})\phi_{R}(\tau_{2})\rangle$ also immediately follows
from the previous result. The operators are now dressed to independent
boundary wiggles $z=F_{L}(\tau_{1})+i\epsilon F_{L}^{\prime}(\tau_{1})$ and
$w=F_{R}(\tau_{2})+i\epsilon F_{R}^{\prime}(\tau_{2})$. The quotient enforces
the periodicity constraint on both boundaries
$F_{L,R}(\tau+\beta_{L,R})=T_{b}\cdot F_{L,R}(\tau)$, and we can define
$F_{L,R}(\tau)=\mp\exp\left(\frac{b}{\beta_{L,R}}f_{L,R}(\tau)\right)$ which
satisfies the constraint as long as
$f_{L,R}(\tau+\beta_{L,R})=f_{L,R}(\tau)+\beta_{L,R}$ where the two boundaries
can have independent temperatures $\beta_{L,R}$. The relative minus sign is
because the left boundary is at $\real(z)<0$, so we must have that $F_{L}<0$
whereas $F_{R}>0$. Performing the sum over the group (2.27) as in the one
sided case we obtain

$G^{\text{LR}}_{\text{DT}}(\tau_{1},\tau_{2})=\sum_{m=-\infty}^{\infty}\left(\frac{F_{L}^{\prime}(\tau_{1})(T_{b}^{m}\cdot
F_{R}(\tau_{2}))^{\prime}}{(F_{L}(\tau_{1})-T_{b}^{m}\cdot
F_{R}(\tau_{2}))^{2}}\right)^{s}=\sum_{m=-\infty}^{\infty}\left(\frac{f_{L}^{\prime}(\tau_{1})f_{R}^{\prime}(\tau_{2})}{\frac{4\beta_{L}\beta_{R}}{b^{2}}\cosh^{2}\left[\frac{b}{2}\left(\frac{f_{L}(\tau_{1})}{\beta_{L}}-\frac{f_{R}(\tau_{2}+m\beta_{R})}{\beta_{R}}\right)\right]}\right)^{s}\,.$
(2.34)

For the two-sided correlator the geodesic can also wind $m$ times around the
trumpet. The dominant contribution is given by the geodesic with smallest
length, which is given by $m=0$ winding.

####  2.3.2 Conical defect

Figure 3: Two-point function on the conical defect geometry. The geodesics
connecting the two operators can wind up to $n-1$ times around the defect.

We now consider the geometry with a single conical defect of opening angle
$\theta=\frac{2\pi}{n}$ by taking the quotient by the group $\Gamma=\langle
T_{\theta}\rangle$. The generator is given by

$T_{\theta}=\left(\begin{array}[]{ll}\cos\frac{\pi}{n}&-\sin\frac{\pi}{n}\\\
\sin\frac{\pi}{n}&~{}~{}\cos\frac{\pi}{n}\end{array}\right)\,,$ (2.35)

and the conical defect is located at the fixed point $z=i$. This group has $n$
elements corresponding to powers of $T_{\theta}^{m}$ for $m=0,\ldots,n-1$. The
determinant on this geometry is not very interesting since the cone has no
closed geodesics. Instead we will consider the two-point function dressed to
the boundary schwarzian. As in the case of the disk, we insert the two
operators at $z=F(\tau_{1})+i\epsilon F^{\prime}(\tau_{1})$ and
$w=F(\tau_{2})+i\epsilon F^{\prime}(\tau_{2})$. The identification $z\cong
T_{\theta}\cdot z$ enforces the condition

$F(\tau+\beta)=T_{\theta}\cdot F(\tau).$ (2.36)

We can also use the parameterization
$F(\tau)=\tan\frac{\theta}{2\beta}f(\tau)$, with
$f(\tau+\beta)=f(\tau)+\beta$. Applying the two-point function formula (2.27)
we have

$G_{\text{Defect}}(\tau_{1},\tau_{2})=\sum_{m=0}^{n-1}\left(\frac{F^{\prime}(\tau_{1})(T_{\theta}^{m}\cdot
F(\tau_{2}))^{\prime}}{\left(F(\tau_{1})-T_{\theta}^{m}\cdot
F(\tau_{2})\right)^{2}}\right)^{s}=\sum_{m=0}^{n-1}\left(\frac{f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})}{\frac{4\beta^{2}}{\theta^{2}}\sin^{2}\frac{\theta}{2\beta}\left[f(\tau_{1}+m\beta)-f(\tau_{2})\right]}\right)^{s}\,,$
(2.37)

There are $n$ geodesics connecting the two operator insertions on the
boundary. These geodesics wind around the defect up to $n-1$ times which is
the summation over $m$. Note that for $n=1$ the opening angle becomes
$\theta=2\pi$ and we get the answer for the disk. The above is not the
complete answer, since we must still integrate over the boundary fluctuations.

If we consider the case of a massless field $s=1$ we can perform the summation
over winding geodesics to find

$G^{s=1}_{\text{Defect}}(\tau_{1},\tau_{2})=\frac{f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})}{\frac{\beta^{2}}{\pi^{2}}\sin^{2}\frac{\pi}{\beta}\left[f(\tau_{1})-f(\tau_{2})\right]},$
(2.38)

which is precisely the two-point function for a massless field on the disk
(2.25). This is the expected answer for a conformal scalar since the
correlator on the defect geometry is related to the correlator on the disk
through a conformal rescaling. In this case the rescaling is trivial at the
boundary so the answers agree. We see that including all of the winding
geodesics is crucial to recover the correct properties of the matter
fields.161616We still have to integrate over the schwarzian mode. On the
defect geometry the $n=2$ boundary modes are no longer zero modes and must be
included in the integral. Even though we have reduced the two point function
to the standard disk answer, we must still include these boundary modes when
computing the two-point function.

####  2.3.3 Two conical defects

To get a disk with two conical defects we need to generate a Fuchsian group
with two elliptic generators $\Gamma=\langle
T_{\theta_{1}},T_{\theta_{1}}\rangle$. We choose the generators to be

Figure 4: The disk with two defects can be represented in the UHP by the
pictured fundamental domain, with the two defects represented by “x”. The
single primitive closed geodesic is in blue, and travels between the defects.
We also have any number of windings of this geodesic.

$T_{\theta_{1}}=\begin{pmatrix}\cos\frac{\theta_{1}}{2}&e^{-\ell/2}\sin\frac{\theta_{1}}{2}\\\
e^{\ell/2}\sin\frac{\theta_{1}}{2}&\cos\frac{\theta_{1}}{2}\end{pmatrix}\,,\quad
T_{\theta_{2}}=\begin{pmatrix}\cos\frac{\theta_{2}}{2}&e^{\ell/2}\sin\frac{\theta_{2}}{2}\\\
e^{-\ell/2}\sin\frac{\theta_{2}}{2}&\cos\frac{\theta_{2}}{2}\end{pmatrix}\,.$
(2.39)

In the UHP the fixed point of $T_{\theta_{1}}$ is at $z=ie^{-\ell/2}$ while
the fixed point of $T_{\theta_{2}}$ is at $z=ie^{\ell/2}$, which are the
locations of the conical defects with opening angles $\theta_{1},\theta_{2}$
respectively. The simplest example, which captures many features of multi
defect surfaces, is to consider the case where both of the angles take the
value $\theta_{1,2}=\pi$. In this case the simplest group products take the
form

$T_{\theta_{1}}^{2}=T_{\theta_{2}}^{2}=\mathbb{I},\qquad T_{2\ell}\equiv
T_{\theta_{2}}T_{\theta_{1}}=\begin{pmatrix}e^{\ell}&0\\\
0&e^{-\ell}\end{pmatrix},$ (2.40)

where we see that a product of two elliptic elements becomes a hyperbolic
element $T_{2\ell}$. There are thus three primitive elements in the group: two
elliptic $T_{\theta_{1}},T_{\theta_{2}}$ and one hyperbolic $T_{2\ell}$. From
the general formula for the determinant (2.14) we immediately get171717The
careful reader may wonder why the Selberg zeta function does not get a square
like the double trumpet case. The reason is that the closed geodesic in this
case is not orientable. The closed geodesic with total length $2\ell$ is
actually obtained by gluing a copy of a geodesic segement with length $\ell$
back to itself in the opposite direction.

$\det(\Delta+s(s-1))=\underbrace{\prod_{m=0}^{\infty}\left(1-e^{-2\ell(s+m)}\right)}_{\text{closed
geodesic}}\underbrace{\frac{\Gamma\left(s+\frac{1}{2}\right)}{\Gamma\left(s\right)}.}_{\text{defect
contribution}}$ (2.41)

The Selberg zeta function contribution is given by the single geodesic
traversing between the two defects, which has length $2\ell$. This originates
from the single primitive hyperbolic element $T_{2\ell}$. We have also
included the contribution coming from the elliptic elements in the resolvent,
which correspond to the two defects. One interesting feature of this surface
is that the hyperbolic element is generated by products of elliptic elements,
and that the resulting closed geodesic touches the conical defects.

To calculate the two-point function (2.27) we must sum over the group. By
inspection we can find that all elements of the group take the form

$T_{2\ell}^{m}=\begin{pmatrix}e^{m\ell}&0\\\ 0&e^{-m\ell}\end{pmatrix}\,,\quad
T_{\theta_{1}}T_{2\ell}^{m}=\begin{pmatrix}0&e^{\left(m+\frac{1}{2}\right)\ell}\\\
e^{-\left(m+\frac{1}{2}\right)\ell}&0\end{pmatrix}\,,$ (2.42)

with $m$ an integer. From the fundamental domain of the schwarzian we see we
must identify $F(\tau+\beta)=e^{2\ell}F(\tau)$, which can be expanded in terms
of the identification $F(\tau)=\exp\left(\frac{2\ell}{\beta}f(\tau)\right)$.
We can immediately write the answer for the resolvent to be

$\displaystyle
G_{\text{Defects}}(\tau_{1},\tau_{2})=\sum_{m=-\infty}^{\infty}\left(\frac{F^{\prime}(\tau_{1})(T_{2\ell}^{m}\cdot
F(\tau_{2}))^{\prime}}{\left(F(\tau_{1})-T_{\theta}^{m}\cdot
F(\tau_{2})\right)^{2}}\right)^{s}+\left(\frac{F^{\prime}(\tau_{1})(T_{\theta_{1}}T_{2\ell}^{m}\cdot
F(\tau_{2}))^{\prime}}{\left(F(\tau_{1})-T_{\theta}^{m}\cdot
F(\tau_{2})\right)^{2}}\right)^{s}$
$\displaystyle=\sum_{m=-\infty}^{\infty}\left(\frac{f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})}{\frac{\beta^{2}}{b^{2}}\sinh^{2}\left[\frac{b}{\beta}\left(f(\tau_{1})-f(\tau_{2}+m\beta)\right)\right]}\right)^{s}+\left(\frac{f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})}{\frac{\beta^{2}}{b^{2}}\sinh^{2}\left[\frac{b}{\beta}\left(f(\tau_{1})+f(\tau_{2}+m\beta)-\frac{\beta}{2}\right)\right]}\right)^{s}\,.$
(2.43)

The first term corresponds to geodesics that wind around the throat, whereas
the second term are geodesics that wind around the defects.

The above example was for the special case where both defects have deficit
angle $\pi$, but the construction immediately extends to more general opening
angles smaller than $\pi$. We briefly explain the main differences.

  * •

To ensure the quotient surface has one asymptotic boundary and two defects,
the parameter $\ell$ must satisfy

$e^{\ell}>\frac{\sin\theta_{1}\left(1+\cos\theta_{2}\right)}{\sin\theta_{2}\left(1-\cos\theta_{1}\right)}\,.$
(2.44)

Geometrically, it means that two sharp defects can not be arbitrarily close to
each other.

  * •

There are infinitely many closed geodesics on the surface, generated by the
primitive “words” of $T_{\theta_{1,2}}$ such as
$T_{\theta_{1}}^{2},T_{\theta_{1}}T_{\theta_{2}},T_{\theta_{2}}^{2}T_{\theta_{1}}^{7},\ldots$
as long as the “word” is hyperbolic.

  * •

Such geodesics generically self-intersect. For example,
$T_{\theta_{1}}T_{\theta_{2}}^{-1}$ is a geodesic winding around both defects
once with a self-intersection.

####  2.3.4 The handle disk

We now consider the handle disk which has a non-trivial topology. To build the
surface we can cut the handle disk along two geodesics, and flatten it onto
the UHP with the identification in figure 5. Therefore, the Fuchsian group is
generated by two hyperbolic elements that identify the associated geodesics:
the first semicircle with the third and the second semicircle with the
fourth.181818Fuchsian groups generated by identifying pairs of semicircles are
called the Schottky groups. The generators will be denoted by $T_{1}$ and
$T_{2}$,191919The form of the hyperbolic generators is given as follows. A
hyperbolic element that identifies semi-circles centered along the real axis
at points $c_{1,2}$ with radii $r_{1,2}$ is
$T(c_{1},c_{2};r_{1},r_{2})=\left(r_{1}r_{2}(c_{12}-r_{12})(c_{12}+r_{12})\right)^{-\frac{1}{2}}\begin{pmatrix}-c_{2}c_{12}+r_{2}r_{12}&(c_{1}c_{2}+r_{1}r_{2})(c_{12}-r_{12})\\\
-c_{12}&c_{1}c_{12}-r_{1}r_{12}\end{pmatrix}$ where $c_{12}=|c_{1}-c_{2}|$ and
$r_{12}=|r_{1}-r_{2}|$. and the fundamental domain is to remove the four half
disks from $\mathbb{H}$.

Figure 5: The handle disk can be constructed by identifying the orange and
blue geodesic semi-circles in the UHP. The fundamental domain is the exterior
of all the semi-circles. The two identifications are the generators of the
Fuchsian group that generate the surface. The red and purple curves are closed
geodesics on the surface. In the right figure we omit the purple closed
geodesic because it is broken into four segments with the pictured fundamental
domain.

As in the case of the two-defect surface any hyperbolic word
$T_{1}^{k_{1}}T_{2}^{k_{2}}T_{1}^{k_{3}}\ldots\,$, on the condition that it is
primitive, corresponds to a primitive closed geodesic on the handle disk. A
generic hyperbolic word corresponds to an integer winding of a primitive
geodesic. As an example, the dark red geodesic in figure 5 is associated to
$T_{2}$ while the purple one is the closed geodesic separating the torus with
the geodesic boundary, generated by $T_{1}T_{2}T_{1}^{-1}T_{2}^{-1}$. More
complicated closed geodesics typically have self-intersections. Each closed
geodesics (with an associated hyperbolic word) contributes to the determinant,
and so it is difficult to write down a simple expression for the full
determinant.

For the two-point function we must consider boundary anchored geodesics such
as the orange and the blue geodesics in figure 5. It is convenient to use a
different fundamental domain for the handle disk where the asymptotic boundary
is in one connected segment of the UHP. To achieve this, one can first
diagonalize $T_{1}T_{2}T_{1}^{-1}T_{2}^{-1}$, which is associated to the
geodesic separating the asymptotic boundary from the handle, to be
$-\text{diag}(e^{\ell/2},e^{-\ell/2})$. This modifies the generators of the
group to be $\tilde{T}_{1},\tilde{T}_{2}$ with generated group
$\tilde{\Gamma}$.202020To get $\tilde{T}_{1,2}$ and $\tilde{\Gamma}$, one
should conjugate every element in $\Gamma$ with a proper SL(2,$\mathbb{R}$)
matrix $V$ so that $V\cdot T_{1}T_{2}T_{1}^{-1}T_{2}^{-1}\cdot
V^{-1}=-\text{diag}(e^{\ell/2},e^{-\ell/2})$. Then $\tilde{\Gamma}$ is
$V\cdot\Gamma\cdot V^{-1}$, and the generators are $\tilde{T}_{i}=V\cdot
T_{i}V^{-1}$ for which an explicit although complicated form can be obtained.
The resulting fundamental domain is pictured in figure 6. The fundamental
domain looks very similar to the double trumpet, and we can parameterize the
Schwarzian boundary by $z=F(\tau)+i\epsilon F^{\prime}(\tau)$ with
$F(\tau)=\exp\left(\frac{\ell}{\beta}f(\tau)\right)$. The two point function
is given by

$G(\tau_{1},\tau_{2})=\sum_{\gamma\in\tilde{\Gamma}}\left(\frac{F^{\prime}(\tau_{1})\left(\gamma\cdot
F(\tau_{2})\right)^{\prime}}{\left(F(\tau_{1})-\gamma\cdot
F(\tau_{2})\right)^{2}}\right)^{s}\,.$ (2.45)

Note that the parameterization is the same as the one-sided two-point function
on the double trumpet (2.33), but the handle disk case is much more
complicated due to its group $\tilde{\Gamma}$. There are distinct classes of
geodesics on the handle disk, and we now explain how the sum over geodesics
can be categorized into equivalence classes using large diffeomorphisms.

Figure 6: An alternative representation of the handle disk fundamental domain.
All the semi-circles are geodesics, and fundamental domain is the bounded
region inside all the semi-circles. The identifications are red/pink,
blue/light blue/gray, green/green. The purple geodesic winds around the
handle. In this representation the asymptotic boundary is in one connected
segment. The two green semi-circles are mapped to each other by the group
element $\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}.$

The moduli space of the handle disk is composed of schwarzian fluctuation and
the moduli of the torus with geodesic boundary denoted by $\mathcal{M}_{1,1}$.
The moduli space is obtained from Teichmuller space $\mathcal{T}_{1,1}$ after
modding out by the action of large diffs, also known as the mapping class
group (MCG). The MCG is generated by three elements $\\{\sigma,P,U\\}$ and has
a simple action on the generators of the Fuchsian group [42]

$\begin{array}[]{ll}\sigma(\tilde{T}_{1})=\tilde{T}_{1}^{-1},&P(\tilde{T}_{1})=\tilde{T}_{2},\\\
\sigma(\tilde{T}_{2})=\tilde{T}_{2},&P(\tilde{T}_{2})=\tilde{T}_{1},\end{array}\
\begin{aligned} &\tilde{U}(T_{1})=\tilde{T}_{1}\tilde{T}_{2},\\\
&U(\tilde{T}_{2})=\tilde{T}_{2}\,.\end{aligned}$ (2.46)

The MCG acts on a group element built from products of the generators by
acting on each generator independently. This action can be understood as a map
between cycles of the surface. Since each cycle has an associated closed
geodesic, we can think of this action as generating a map between closed
geodesics on the surface. When considering boundary anchored geodesics, this
becomes a map between boundary anchored geodesics. As an example, it can be
checked that the MCG preserves the group element
$\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}$ up to
conjugation,212121For example,
$\sigma\left(\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}\right)$
is $\tilde{T}_{1}^{-1}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2}^{-1}$, which is
conjugate to
$\left(\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}\right)^{-1}$
by $\tilde{T}_{1}$. so the geodesic splitting the handle from the asymptotic
boundary is fixed. We can consider a group element $\gamma\in\tilde{\Gamma}$
corresponding to a particular boundary anchored geodesic and act on it with
the MCG to generate the equivalence class $\\{\gamma\\}_{\rm MCG}$.222222The
geodesic corresponding to $\gamma$ is the geodesic connecting the boundary
points $F(\tau_{1})$ and $\gamma\cdot F(\tau_{2})$ if we unwrap the handle
disk on the UHP. As an example the element $\tilde{T}_{1}$ corresponds to the
purple boundary anchored geodesics that winds the handle in figure 6. We can
generate all other geodesics that wind the handle and do not self-intersect by
acting with the MCG

$\text{non self-intersecting geodesics: }\\{\tilde{T}_{1}\\}_{\rm
MCG}=\\{\tilde{T}_{1},\tilde{T}_{2},\tilde{T}_{1}\tilde{T}_{2},\tilde{T}_{1}^{2}\tilde{T}_{2},\tilde{T}_{1}^{3}\tilde{T}_{2},\tilde{T}_{1}^{2}\tilde{T}_{2}^{3},\ldots\\}\,.$
(2.47)

The summation over the elements $\\{\tilde{T}_{1}\\}_{\rm MCG}$ in the two
point function (2.33) implements a sum over all non-self intersecting
geodesics that wind the handle. These contributions were resumed in [25] and
proved to give the ramp behavior for the late time two-point function.

This method can also be used to partially classify self-intersecting geodesics
that wind around the handle. A complete closed form classification appears to
be highly non-trivial, but for one self-intersection we can find explicit
examples of simple equivalence classes. For boundary anchored once self-
intersecting geodesics that wind the handle, two equivalence classes are given
by starting with base elements $\tilde{T}_{1}^{2}\tilde{T}_{2}^{2}$,
$\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}$
and acting with the MCG

one self-intersection:
$\displaystyle\\{\tilde{T}_{1}^{2}\tilde{T}_{2}^{2}\\}_{\rm
MCG}=\\{\tilde{T}_{1}^{2}\tilde{T}_{2}^{2},\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2}^{-1},\tilde{T}_{1}^{3}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2},\ldots\\},$
(2.48)
$\displaystyle\\{\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1}\\}_{\rm
MCG}=\\{\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1},\tilde{T}_{1}\tilde{T}_{2}\tilde{T}_{1}^{3}\tilde{T}_{2}\tilde{T}_{1}^{-1}\tilde{T}_{2}^{-1},\ldots\\}\,.$

Note that the above is not a complete classification of all once self-
intersecting geodesics, but we believe the first equivalence class will always
contain the shortest self-intersecting geodesics on the surface.232323The
argument is that the length of self-intersecting geodesics increases with
additional windings, and the number of windings is controlled by the word-
length of the generator. We can write down other equivalence classes
systematically using the group theory description.242424Note that geodesics
with more self-intersections are generated by words with at least six
generators. The greater length of the words implies a greater length of the
corresponding geodesic. It would be interesting to perform the integral over
the boundary fluctuations in (2.45) for the self-intersecting geodesics
falling into the equivalence classes (2.48) to confirm that they do not
contribute to the late time two-point function [25].

##  3 JT gravity path integral

###  3.1 Gauge fixing the path integral

We now explain how to gauge fix the gravity path integral for JT gravity. For
the case of compact surfaces see [7], while for non-compact surfaces also see
[13, 14, 15, 16]. The integral is defined by

$Z=\int\frac{\mathcal{D}g\mathcal{D}\Phi}{\text{V}(\text{Diff})}e^{-I_{\text{J}T}[g,\Phi]},$
(3.1)

where we divide out by the volume of diffeomorphisms. The JT gravity action is
given by

$I_{\text{J}T}[g,\Phi]=-S_{0}\chi(\Sigma)-\bigg{[}\frac{1}{2}\int_{\Sigma}\sqrt{g}\Phi\left(R+2\right)+\underbrace{\int_{\partial\Sigma}\sqrt{h}\Phi\left(K-1\right)}_{-I_{\text{bdy}}}\bigg{]},$
(3.2)

where the first term is topological and $\chi(\Sigma)=2-2g-b$ for surfaces
with $g$ handles and $b$ boundaries. When we include asymptotic boundaries we
choose boundary conditions to fix the induced metric $\sqrt{h}=1/\epsilon$ and
the proper length of the boundary to be $\beta/\epsilon$, and we fix the
dilaton to asymptotically approach $\phi_{b}=\gamma/\epsilon$.

To integrate over metrics we must first specify a measure on the moduli space
of all metrics, that is we must specify a metric $\langle.\hskip
1.70709pt,.\rangle$ on the tangent space of moduli space. The tangent space is
naturally the space of metric deformations, and the standard metric for the
space of these deformations is the ultra-local measure[43]

$\langle\delta g_{ab},\delta
g_{cd}\rangle=\mathcal{N}\int_{\Sigma}\sqrt{g}g^{ac}g^{bd}\delta g_{ab}\delta
g_{cd},$ (3.3)

which can be extended to other tensor deformations in an obvious way. The
measure is defined up to a normalization constant $\mathcal{N}$ that we will
fix later. Metric deformation decompose into three orthogonal parts

$\delta g_{ab}=\underbrace{\omega g_{ab}}_{\text{weyl}}\hskip
2.84544pt\oplus\hskip
2.84544pt\underbrace{\operatorname{range}P_{1}}_{\text{small diff}}\hskip
2.84544pt\oplus\hskip
2.84544pt\underbrace{\operatorname{ker}P_{1}^{{\dagger}}}_{\text{moduli}},$
(3.4)

where $(P_{1}V)_{ab}=\nabla_{a}V_{b}+\nabla_{b}V_{a}-(\nabla_{c}V^{c})g_{ab}$,
and $(P_{1}^{\dagger}\delta g)_{a}=-2\nabla^{b}\delta g_{ab}$. A general
infinitesimal metric deformation is a combination of a Weyl transformation, a
small diffeomorphism, and a deformation of the moduli. Since the metric
deformations are orthogonal (3.4) the path integral measure breaks up into
integrals over these deformations.

We will first consider the case where we are integrating over all metrics on a
compact surface of genus $g\geq 2$. To perform the integral we gauge fix to
conformal gauge and write the metric as
$g=f^{*}\left(e^{2\omega}\hat{g}\right)$ with $f$ a small diffeomorphism,
$\omega$ a Weyl factor, and $\hat{g}$ a constant negative curvature metric
$\hat{R}=-2$. The measure works out to be[31, 29, 30]

$Z_{g}=\int_{\mathcal{M}_{g}}\underbrace{d\left(\text{Weil-Pet.}\right)\hskip
2.84544pt(\det\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}\mathcal{D}\omega}_{\mathcal{D}g/\text{Vol}}\mathcal{D}\Phi
e^{-I_{\text{J}T}[e^{2\omega}\hat{g},\Phi]}e^{-26S_{L}[\omega,\hat{g}]}.$
(3.5)

The integral is carried out over the moduli space of constant negative
curvature surfaces $\mathcal{M}_{g}$ of genus $g$.252525When integrating over
small diffeomorphisms we pick up a factor of the volume of small diffs. The
ratio of small diffs with all diffs gives the set of large diffs, which
reduces the integral
$\frac{\text{Vol$($Diff${}_{0})$}}{\text{Vol(Diff)}}\int_{\mathcal{T}_{g}}=\int_{\mathcal{M}_{g}}$
to the moduli space of $R=-2$ surfaces. Here $\mathcal{T}_{g}$ is the set of
metrics mod Weyl $[e^{2\omega}g]$ before quotienting by diffeomorphisms, also
known as Teichmuller space. In the above
$(\det\hat{P}_{1}^{\dagger}\hat{P}_{1})^{1/2}$ appears from the ghost path
integral when gauge fixing to conformal gauge, and the notation $\hat{P_{1}}$
implies the operator is defined with respect to $\hat{g}$. We have an integral
over the Weyl factor $\mathcal{D}\omega$, and the Liouville action $S_{L}$
appears due to the conformal anomaly arising from defining the integration
measures with respect to $\hat{g}$.262626Every integral in (3.5), such as
$\mathcal{D}_{\hat{g}}\Phi$, is defined with the appropriate ultra-local
measure (3.3) with metric $\hat{g}$. The Liouville action will give no
contribution after we perform the $\mathcal{D}\omega$ integral so we discard
it from now on. When restricted to constant negative curvature metrics
$\hat{g}$, the measure (3.3) is by definition the Weil-Petersson measure [44],
which is the origin of $d\left(\text{Weil-Pet.}\right)$.

We can write the part of the JT gravity action (3.2) that contains the dilaton
as

$I[e^{2\omega}\hat{g},\Phi]=-\frac{1}{2}\int_{\Sigma}\sqrt{\hat{g}}\Phi\left(\hat{R}-2\hat{\nabla}^{2}\omega+2e^{2\omega}\right).$
(3.6)

We can perform the integral over the dilaton by rotating the integration
contour to be along the imaginary axis, giving a delta function constraint
that localizes onto constant negative curvature geometries

$\int\mathcal{D}\omega\int_{i\mathbb{R}}\mathcal{D}\Phi
e^{-I_{\text{J}T}[e^{2\omega}\hat{g},\Phi]}=\int\mathcal{D}\omega\hskip
2.84544pt\delta\left(\hat{R}-2\hat{\nabla}^{2}\omega+2e^{2\omega}\right)=\frac{1}{\det(-2\hat{\nabla}^{2}+4)},$
(3.7)

where in the last equality we localize onto $\omega=0$. A straightforward
calculation shows that the gauge fixing differential operator takes the form
$\hat{P}_{1}^{\dagger}\hat{P}_{1}=2\left(-\hat{\nabla}_{1}^{2}+1\right)$ on a
surface with constant negative curvature, with $\hat{\nabla}^{2}_{1}$ the
vector laplacian. Putting everything together we find that the JT gravity path
integral (3.5) is

$\displaystyle Z_{g}$
$\displaystyle=(\text{const})^{\chi}\int_{\mathcal{M}_{g}}d\left(\text{Weil-
Pet.}\right)\frac{\det\left(-\hat{\nabla}_{1}^{2}+1\right)^{1/2}}{\det(-\hat{\nabla}^{2}+2)}$
(3.8)
$\displaystyle=(\text{const})^{\chi}\int_{\mathcal{M}_{g}}d\left(\text{Weil-
Pet.}\right).$ (3.9)

In the above we have discarded some multiplicative constants from the
determinants, since in zeta function regularization multiplying all the
eigenvalues by a constant shifts the determinant by
$(\text{const})^{\chi}$.272727All of these constants can be absorbed into a
redefinition of $S_{0}$ in the topological term of the action (3.2).
Furthermore we also used our result for the vector laplacian determinant

$\frac{\det\left(-\hat{\nabla}_{1}^{2}+1\right)^{1/2}}{\det(-\hat{\nabla}^{2}+2)}=2^{\frac{1}{2}\chi},$
(3.10)

given in equation (C.10). The above conclusion immediately generalizes beyond
compact surfaces.

Consider the moduli space of genus $g$ hyperbolic surface with $n$ asymptotic
boundaries of regularized lengths
$\vec{\beta}=\left(\beta_{i},\ldots,\beta_{n}\right)$ and $k$ conical defects
with opening angles specified by $2\pi\alpha_{i}$. We take
$\vec{\alpha}=\left(\alpha_{1},\ldots,\alpha_{k}\right)$ with
$\alpha_{i}=n_{i}^{-1}$ and integer $n_{i}\geq 2$. To localize onto geometries
with such singularities we must insert an operator $\mathcal{V}_{\alpha}$ into
the path integral (3.1). We explain the dilaton gravity form of this operator
slightly later, but for now in all integrals we assume we have inserted this
operator and integrated over the dilaton to localize onto the relevant
hyperbolic geometries.

When integrating over hyperbolic surfaces with arbitrary conical singularities
and asymptotic boundaries we again obtain the Weil-Petersson measure for the
associated moduli spaces. This is because the measure (3.3) for metrics with
conical singularities is again by definition the Weil-Petersson metric on such
spaces[45]. Carrying out the JT gravity path integral in the same way as for a
compact surface we arrive at

$\displaystyle Z_{g,{\vec{\alpha}}}(\beta_{1},\ldots,\beta_{n})$
$\displaystyle=(\text{const})^{\chi}\int_{\mathcal{M}_{g,{\vec{\alpha},\vec{\beta}}}}d\left(\text{Weil-
Pet.}\right)\frac{\det\left(-\hat{\nabla}_{1}^{2}+1\right)^{1/2}}{\det(-\hat{\nabla}^{2}+2)}e^{-I_{\text{bdy}}}$
$\displaystyle=(\text{const})^{\chi}\int_{\mathcal{M}_{g,{\vec{\alpha},\vec{\beta}}}}d(\text{boundary
wiggles})d\left(\text{bulk moduli}\right)e^{-I_{\text{bdy}}}.$ (3.11)

In the above we have split the integral over the moduli $d\left(\text{Weil-
Pet.}\right)$ into the boundary wiggles and the bulk moduli. The Weil-
Petersson measure (3.3) will induce a measure for the boundary wiggles, which
we will calculate in section 3.2.1 and see is given by the standard schwarzian
measure. While we are only able to compute the required determinants for
special defect angles, it seems likely that the cancellation between
determinants will remain the case for general conical defects.

####  3.1.1 Integrating over moduli space

We now explain how to formally carry out the integration over moduli space.
For reviews see [29, 31, 46, 47, 48]. From (3.4) the infinitesimal
deformations of the moduli correspond to variations of the metric $\delta
g\in\operatorname{Ker}P_{1}^{\dagger}$. In conformal gauge where the metric is
off-diagonal this translates to the condition $\overline{\partial}\delta
g_{zz}=\partial\delta g_{\bar{z}\bar{z}}=0.$ This implies that the moduli
deformations are holomorphic/anti-holomorphic deformations of the form

$\phi_{n}=\phi_{n}(z)dz^{2},\qquad\overline{\phi_{n}}=\overline{\phi_{n}}(\bar{z})d\bar{z}^{2}.$
(3.12)

These are known as quadratic differentials, and they provide a basis for
infinitesimal deformations of the moduli.

We briefly explain some properties of quadratic differentials. On a compact
genus $g\geq 2$ surface there is a basis of $3g-3$ globally defined pairs of
holomorphic and anti-holomorphic quadratic differentials giving a moduli space
of real dimension $6g-6$. When we consider hyperbolic surfaces with conical
defects, the quadratic differentials are allowed to have simple poles at the
location of the defect $\phi\sim z^{-1}dz^{2}$ [45].282828For examples of
quadratic differential on surfaces with defects see [49]. These deformations
correspond to moving the conical defect on the surface, and so the real
dimension of the moduli space with $n$ conical defects is $6g-6+2n$. When we
consider the inclusion of asymptotic boundaries the boundary fluctuations are
also part of the moduli, and there are infinitely many quadratic differentials
associated to turning on modes of the boundary fluctuations as we will see in
section 3.2.1.

An arbitrary infinitesimal moduli deformation, in conformal gauge, can be
written in terms of quadratic differentials as292929The normalization factor
of (3.14) is our convention. We will fix the normalization factor of the
metric perturbation inner product later, and are satisfied with the
proportionality here as a motivation.

$\delta g=\sum_{n}\delta
c_{n}\phi_{n}(z)dz^{2}+\delta\overline{c}_{n}\overline{\phi}_{n}(\bar{z})d\bar{z}^{2},$
(3.13)

where $\delta c_{n}$ are deformation parameters. Using the inner product for
metric deformations (3.3) we can define an inner product for quadratic
differentials, given by

$\langle\phi_{n},\phi_{m}\rangle\equiv\int_{\Sigma}d^{2}z\sqrt{g}g^{z\overline{z}}g^{z\overline{z}}\phi_{n}(z)\overline{\phi}_{m}(\bar{z}),$
(3.14)

where the second quantity is to be complex conjugated. It is standard to
ignore the normalization in (3.3), restoring it only when computing the full
path integral measure.

To define the measure on moduli space we will need one more quantity known as
the Beltrami differential $\mu$. To integrate over moduli space we must first
choose coordinates $m_{n},\bar{m}_{n}$ on the space by specifying a set of
metrics $g(z,\bar{z};m_{n},\bar{m}_{n})$ that give a slice through the space.
We must then compute the Jacobian for this set of coordinates. This can be
accomplished as follows. Consider a metric $g(m_{n},\bar{m}_{n})$ on a surface
which in some local patch takes form
$ds^{2}=e^{2\omega}|dz|^{2}=2g_{z\bar{z}}|dz|^{2}$ for some complex coordinate
$z$. As we change the moduli by moving to a nearby metric at $m_{n}+\delta
m_{n}$ the new metric can no longer be $\propto|dz|^{2}$ since it is not in
the same equivalence class as the original metric. However, there must be
complex coordinates $z^{\prime}=z+\delta m_{n}\,v(z,\bar{z})+\ldots$ where the
new metric takes the form $e^{2(\omega+\delta\omega)}|dz^{\prime}|^{2}$. We
thus find that the deformed metric in the old $z$ coordinates is

$ds^{\prime 2}=e^{2(\omega+\delta\omega)}\left(dzd\bar{z}+\delta m_{n}\hskip
1.9919pt\overline{\mu}_{n}dz^{2}+\delta\overline{m}_{n}\hskip
1.9919pt\mu_{n}d\bar{z}^{2}+\ldots\right),\qquad\delta(ds^{2})=\delta
m_{n}e^{2\omega}\overline{\mu}_{n}dz^{2}+\text{c.c.}$ (3.15)

where we have defined the Beltrami differentials $\mu=\overline{\partial}v$,
and $\overline{\mu}=\partial v$,303030The Beltrami differential are commonly
viewed as $(-1,1)$ forms $\mu=\mu_{\bar{z}}^{~{}z}dz^{-1}d\bar{z}$ which can
be integrated against $(2,0)$ forms $\phi_{zz}dz^{2}$, and it is convention to
define the metric deformation by beltrami differentials as
$\overline{\mu}dz^{2}$. which capture how the metric infinitesimally changes
as we change the $m_{n}$ coordinates. In the second equation we have written
the variation to linear order.313131All these statements are up to Weyl
rescalings since rescalings do not move the metric in moduli space. The
overlap between the quadratic differentials and the change in the metric due
to a deformation in $m_{n}$ coordinates can be obtained from the measure
(3.3), which is most commonly expressed as an overlap between the Beltrami
differentials $\mu$ and the quadratic differentials

$\langle\mu_{n},\phi_{m}\rangle\equiv 2\int_{\Sigma}d^{2}z\mu_{n}\hskip
1.42271pt\phi_{m},$ (3.16)

where the overlap is defined without a conjugation. Notice that deforming the
metric by a quadratic differential $\phi(z)dz^{2}$ (3.13) is equivalent to
deforming it by a beltrami differential
$\overline{\mu}=\frac{\phi}{2g_{z\bar{z}}}$ to linear order in the
deformation. To compute the correct measure we must project the above
deformation onto the space of genuine moduli deformations (as opposed to
diffeomorphisms and Weyl rescalings). Projecting onto the basis of quadratic
differentials we get

$\delta g=\delta
m_{n}\phi_{j}\langle\overline{\mu}_{n},\overline{\phi}_{i}\rangle\langle\phi_{j},\phi_{i}\rangle^{-1}+\delta\overline{m}_{n}\overline{\phi}_{j}\langle\mu_{n},\phi_{i}\rangle\langle\overline{\phi}_{j},\overline{\phi}_{i}\rangle^{-1},$
(3.17)

where the notation $\langle...\rangle^{-1}$ is an inverse matrix. Using (3.3)
we can compute the metric

$ds^{2}_{\text{Weil-
Pet.}}=2\mathcal{N}\langle\overline{\mu}_{n},\overline{\phi}_{j}\rangle\langle\mu_{m},\phi_{i}\rangle\langle\phi_{i},\phi_{j}\rangle^{-1}dm_{n}d\overline{m}_{m}.$
(3.18)

Taking the square root of the determinant and treating the metric as a product
of matrices, we find that in terms of $m_{n}$ coordinates the integral over
moduli space takes the form

$\int_{\mathcal{M}}d(\text{Weil-Pet.})\hskip
2.84544pte^{-I}=\prod_{n}\mathcal{N}\int
dm_{n}d\overline{m}_{n}\frac{\det\langle\mu,\phi\rangle\det\langle\overline{\mu},\overline{\phi}\rangle}{\sqrt{\det\langle\phi,\phi\rangle\det\langle\overline{\phi},\overline{\phi}\rangle}}\hskip
2.84544pte^{-I[m_{n},\overline{m}_{n}]}.$ (3.19)

We have also included a possibility of some action $I$ that depends on the
moduli. In JT gravity this action will be given by the boundary term in (3.2).
We have also reinstated the overall normalization constant $\mathcal{N}$
appearing in the measure (3.3). In [13], and in appendix B using quadratic
differentials, it can be shown that in the second order formalism the gluing
measure for geodesic boundaries is given by

$4\mathcal{N}\int bdb,$ (3.20)

and so to get standard form of the Weil-Petersson measure we must choose the
normalization $\mathcal{N}=\frac{1}{4}$. We keep the normalization constant
present throughout, inserting it’s numerical value when necessary.

#### Conical defect operator

From the expression for the Weil-Petersson measure we can extract the dilaton
gravity operator that creates a conical defect on the surface. When we have a
defect we have two additional moduli that move the defect. We choose the
moduli to be parameterized by the location $z_{i}$ of the defect, and we can
formally write the measure as

$\mathcal{N}\int_{\Sigma}d^{2}z_{i}\frac{\langle\mu,\phi_{1}\rangle\langle\overline{\mu},\overline{\phi}_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle\langle\overline{\phi}_{1},\overline{\phi}_{1}\rangle}},$
(3.21)

where $\mu$ is the beltrami differential that infinitesimally moves the defect
to $z_{i}+\delta z_{i}$, and $\phi_{1}$ is the quadratic differential with a
simple pole at $z_{i}$. From the JT gravity perspective this is the measure
after the dilaton has already been integrated out, and so restoring it we
arrive at the dilaton gravity operator that creates a conical defect

$\mathcal{V}_{\alpha}=\mathcal{N}\int_{\Sigma}d^{2}z_{i}\frac{\langle\mu,\phi_{1}\rangle\langle\overline{\mu},\overline{\phi}_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle\langle\overline{\phi}_{1},\overline{\phi}_{1}\rangle}}e^{-2\pi(1-\alpha)\Phi(z_{i})}.$
(3.22)

One complication for evaluating the above integral is that quadratic
differentials are not generally known, apart from their singular behavior near
the defect. However, for the hyperbolic disk everything can be worked out
explicitly, and in section 3.2.3 we evaluate the measure on the disk. In
section 3.2.4 we argue that for general surfaces in the blunt defect limit
$\alpha\to 1$ the measure can be worked out to leading order in the
$(1-\alpha)$ expansion by using the universal divergent behavior of the
quadratic differential.

###  3.2 Examples

####  3.2.1 Disk

In this section we explain how to apply the above formalism for integrating
over moduli space to reproduce disk partition function $Z(\beta)$ in JT
gravity[14, 15, 16]. The AdS2 disk metric in complex coordinates is given
by323232Our conventions for complex coordinates are $z=x+iy$ with $\int
d^{2}z=2\int d^{2}x$. The dirac delta function is defined by
$\overline{\partial}\partial\log|z|^{2}=2\pi\delta^{2}(z,\bar{z})$, with $\int
d^{2}z\delta^{2}(z,\bar{z})=1$. A Weyl rescaled metric
$g=e^{2\omega}dzd\bar{z}$ has curvature
$R=-8e^{-2\omega}\overline{\partial}\partial\omega$.

$ds^{2}=\frac{4dzd\overline{z}}{\left(1-|z|^{2}\right)^{2}}.$ (3.23)

This is related to the standard metric on the hyperbolic disk through the
coordinate change

$ds^{2}=dr^{2}+\sinh^{2}r\hskip 1.70709ptd\tau^{2},\qquad
z=e^{i\tau}\tanh\frac{r}{2}.$ (3.24)

As explained previously, deformations of the moduli correspond to holomorphic
quadratic differentials and their conjugates. On the disk it is simple to
write down a basis of holomorphic functions given by

$\phi_{n}=\sqrt{\frac{n^{3}-n}{2\pi}}z^{n-2}dz^{2},\qquad\overline{\phi}_{n}=\sqrt{\frac{n^{3}-n}{2\pi}}\overline{z}^{n-2}d\overline{z}^{2},\qquad\langle\phi_{n},\phi_{m}\rangle=\delta_{n,m},$
(3.25)

where for holomorphicity we restrict to integers $n\geq 2$. Infinitesimal
deformations of the disk are given by deforming by linear combinations of the
quadratic differentials

$ds^{2}=\frac{4dzd\overline{z}}{\left(1-|z|^{2}\right)^{2}}+c_{n}\sqrt{\frac{n^{3}-n}{2\pi}}z^{n-2}dz^{2}+\overline{c}_{n}\sqrt{\frac{n^{3}-n}{2\pi}}\overline{z}^{n-2}d\overline{z}^{2}.$
(3.26)

Note that the above metric is no longer AdS2. We will take our coordinates for
moduli space to be given by $c_{n},\overline{c}_{n}$, and so we must work out
the measure for these coordinates along with the boundary action of JT gravity
(3.2) in terms of $c_{n},\overline{c}_{n}$. The measure is straightforward
since for the Beltrami differentials in (3.19) we just use the dual of the
quadratic differentials
$\overline{\mu}_{n}=\frac{\phi_{n}}{2g_{z\overline{z}}}$. Working out the
action is slightly more complicated, which we now do.

The finite version of the metric for the above deformation was worked out by
[16], and is given by333333We mention that we still mostly work with
infinitesimal perturbations of the metric and the following finite metric
deformation is not unique.

$\displaystyle
ds^{2}=e^{C+\overline{C}}\frac{\left(1-|z|^{2}\right)^{2}}{\left(1-|e^{C}z|^{2}\right)^{2}}\left[\left(|1+z\partial
C|^{2}+|\overline{z}\overline{\partial}C|^{2}\right)\frac{4dzd\overline{z}}{\left(1-|z|^{2}\right)^{2}}{+}\left(1+z\partial
C\right)\varepsilon^{\prime\prime\prime}(z)dz^{2}+\text{c.c.}\right],$ (3.27)
$\displaystyle
C(z,\overline{z})={{-}}\frac{1}{2z}\left(\varepsilon(z)-z^{2}\overline{\varepsilon}(\overline{z})-z(1-z\overline{z})\overline{\varepsilon}^{\prime}(\overline{z})-\frac{1}{2}(1-z\overline{z})^{2}\overline{\varepsilon}^{\prime\prime}(\overline{z})\right).$

The functions $\varepsilon(z),\overline{\varepsilon}(\overline{z})$ are
holomorphic/anti-holomorphic, and the primes
$\varepsilon^{\prime}(z),\overline{\varepsilon}^{\prime}(\overline{z})$ denote
holomorphic/anti-holomorphic derivatives. The metric (3.27) is AdS2, which can
be seen by using coordinates $w=ze^{C}$ in terms of which it takes the
standard form (3.23). This coordinate change is a large diffeomorphism and
corresponds to a physically distinct configuration. The relation between
(3.26) and (3.27) comes from identifying

$\varepsilon(z)=\sum_{n\geq
2}\frac{1}{\sqrt{2\pi(n^{3}-n)}}\,c_{n}z^{n+1},\qquad\overline{\varepsilon}(\overline{z})=\sum_{n\geq
2}\frac{1}{\sqrt{2\pi(n^{3}-n)}}\,\overline{c}_{n}\overline{z}^{n+1}.$ (3.28)

Expanding the metric (3.27) to leading order in $\varepsilon$ we recover the
infinitesimal form (3.26) up to a Weyl factor of order $\varepsilon^{2}$.

We now explain how turning on these deformations corresponds to turning on
modes of the Schwarzian. In $z$ coordinates (3.27) the boundary cutoff is
located at fixed radial distance while in $w=ze^{C}$ coordinates the metric
takes the form

$ds^{2}=\frac{4dwd\overline{w}}{(1-|w|^{2})^{2}},$ (3.29)

and the cutoff surface fluctuates. Taking $z=e^{i\theta}$ we find that at the
boundary and linear order of $c_{n}$

$w=e^{if(\theta)},\qquad f(\theta)=\theta+\sum_{n\geq
2}\frac{1}{2i\sqrt{2\pi(n^{3}-n)}}\left(c_{n}e^{in\theta}-\overline{c}_{n}e^{-in\theta}\right).$
(3.30)

We would like to find the cutoff surface in terms of the $f(\theta)$
reparameterization. We can do this perturbatively in the cutoff $\delta$
finding $w\left(\theta\right)=e^{if(\theta)}\left(1-\delta
f^{\prime}(\theta)+\frac{\delta^{2}f^{\prime}(\theta)^{2}}{2}+\mathcal{O}(\delta^{3})\right)$.
Where we have found the radial distance in terms of $f(\theta)$ using that the
induced metric along the cutoff surface is fixed to be $\sqrt{h}=1/\delta$.
The extrinsic curvature can then be evaluated

$\displaystyle
K=1+\delta^{2}\left(\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3f^{\prime\prime
2}}{2f^{\prime 2}}+\frac{1}{2}f^{\prime
2}\right)+\mathcal{O}(\delta^{3})=1+\delta^{2}\operatorname{Sch}\left(\tan\frac{f(\theta)}{2},\theta\right),$
(3.31)

which is the standard Schwarzian answer. From (3.30) we see that the quadratic
differentials $\phi_{n},\overline{\phi}_{n}$ precisely correspond to turning
on the modes of the Schwarzian. The boundary action (3.2) can now be
evaluated. Reinstating the temperature $\beta$ and the value of the dilaton
$\phi=\gamma/\delta$ we find

$\displaystyle I_{\text{bdy}}$
$\displaystyle=-\frac{2\pi\gamma}{\beta}\int_{0}^{2\pi}d\theta\left(\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3f^{\prime\prime
2}}{2f^{\prime 2}}+\frac{1}{2}f^{\prime 2}\right),$ (3.32)
$\displaystyle=I_{0}+\frac{\pi\gamma}{2\beta}\sum_{n\geq 2}n\hskip
0.28436ptc_{n}\overline{c}_{n},\qquad I_{0}=-\frac{2\pi^{2}\gamma}{\beta}.$

Where in the second line we have used that the Schwarzian path-integral is
one-loop exact [50] so that it is sufficient to expand to quadratic order in
$c_{n},\overline{c}_{n}$ and carry out the integral[7, 51], and $I_{0}$ is the
classical contribution. We can now evaluate the path integral over the moduli
space using (3.19)

$\displaystyle Z_{\text{disk}}\left(\beta\right)$ $\displaystyle=\prod_{n\geq
2}\mathcal{N}\int
dc_{n}d\overline{c}_{n}\left|\frac{\det\langle\mu,\phi\rangle}{\sqrt{\det\langle\phi,\phi\rangle}}\right|^{2}e^{-I_{\text{bdy}}}=e^{\frac{2\pi^{2}\gamma}{\beta}}\prod_{n\geq
2}\mathcal{N}\int
dc_{n}d\overline{c}_{n}\exp\left(-\frac{\pi\gamma}{2\beta}\sum_{n\geq
2}n\hskip 0.28436ptc_{n}\overline{c}_{n}\right)$ (3.33)
$\displaystyle=e^{\frac{2\pi^{2}\gamma}{\beta}}\prod_{n\geq
2}\mathcal{N}\frac{4\beta}{\gamma
n}=\frac{1}{8}\sqrt{\frac{\gamma^{3}}{2\pi\mathcal{N}^{3}\beta^{3}}}\hskip
2.84544pte^{\frac{2\pi^{2}\gamma}{\beta}}.$

In the first line we used our normalization for the quadratic differentials
$\langle\phi_{n},\phi_{m}\rangle=\delta_{n,m}$ and that the dual beltrami
differentials $\overline{\mu}_{n}=\frac{\phi_{n}}{2g_{z\overline{z}}}$ satisfy
$\langle\mu_{n},\phi_{m}\rangle=\delta_{n,m}$. In the last line we used zeta
function regularization. This reproduces the JT gravity disk partition
function obtained from the first order formalism [7].

####  3.2.2 Conical defect

We now explain how the above calculation is modified in the case of a disk
with a single conical defect of opening angle $2\pi\alpha$. If the defect is
at the center of the disk then the metric and Ricci scalar are given by

$ds^{2}=\frac{4\alpha^{2}|z|^{2(\alpha-1)}}{(1-|z|^{2\alpha})^{2}}dzd\bar{z}\,,\qquad\frac{1}{2}\sqrt{g}\left(R+2\right)=2\pi(1-\alpha)\delta^{(2)}(z).$
(3.34)

The quadratic differentials are now given by

$\phi_{n}=\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544ptz^{n-2},\qquad\overline{\phi}_{n}=\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544pt\overline{z}^{n-2}d\overline{z}^{2},\qquad\langle\phi_{n},\phi_{m}\rangle=\delta_{n,m}.$
(3.35)

When a conical defect is present on the geometry the quadratic differentials
are allowed to have simple poles[45] at the location of the defect so we now
allow for $n\geq 1$ modes. This increases the real dimension of the moduli
space by two, and the new deformation should be interpreted as moving the
defect on the surface. The quadratic differentials infinitesimally deform the
metric by

$ds^{2}=\frac{4\alpha^{2}|z|^{2(\alpha-1)}}{(1-|z|^{2\alpha})^{2}}dzd\bar{z}+c_{n}\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544ptz^{n-2}dz^{2}+\overline{c}_{n}\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544pt\overline{z}^{n-2}d\overline{z}^{2}.$ (3.36)

Similar to the case of the disk, there is a vector field $\xi$ that
infinitesimally removes these deformations, and we can find it by solving for

$2\nabla_{z}\xi_{z}=-c_{n}\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544ptz^{n-2},\qquad
2\nabla_{\bar{z}}\xi_{\bar{z}}=-\overline{c}_{n}\sqrt{\frac{n^{3}-n\alpha^{2}}{2\pi}}\hskip
2.84544pt\overline{z}^{n-2},\qquad\nabla_{z}\xi_{\bar{z}}+\nabla_{\bar{z}}\xi_{z}=0.$
(3.37)

The vector field can be found and takes the form

$\displaystyle\xi^{\bar{z}}=\frac{\overline{c}_{n}\overline{z}^{n+1}}{2\sqrt{2\pi(n^{3}-n\alpha^{2})}}-\frac{c_{n}\overline{z}(z\overline{z})^{-\alpha}\left(2\alpha^{2}z^{n}(z\overline{z})^{\alpha}+n^{2}z^{n}\left(1-(z\overline{z})^{\alpha}\right)^{2}+n\alpha
z^{n}\left(1-(z\overline{z})^{2\alpha}\right)\right)}{4\alpha^{2}\sqrt{2\pi(n^{3}-n\alpha^{2}})},$
(3.38)
$\displaystyle\xi^{z}=\frac{c_{n}z^{n+1}}{2\sqrt{2\pi(n^{3}-n\alpha^{2})}}-\frac{c_{n}z(z\overline{z})^{-\alpha}\left(2\alpha^{2}\overline{z}^{n}(z\overline{z})^{\alpha}+n^{2}\overline{z}^{n}\left(1-(z\overline{z})^{\alpha}\right)^{2}+n\alpha\overline{z}^{n}\left(1-(z\overline{z})^{2\alpha}\right)\right)}{4\alpha^{2}\sqrt{2\pi(n^{3}-n\alpha^{2}})}.$

The finite form of the deformed metric can be recovered by choosing
coordinates $w=z\exp\left(\frac{1}{z}\xi^{z}\right)$ where in $w$ coordinates
we have the metric (3.34) (compare to (3.27)), and we implicitly sum over
$c_{n},\overline{c}_{n}$ in $\xi^{z}$. We can now follow the same procedure to
find the shape of the cutoff in $w$ coordinates as we followed for the disk.
Following the curve $z=e^{i\theta}$ we find in $w$ coordinates

$w=e^{if(\theta)},\qquad f(\theta)=\theta+\sum_{n\geq
1}\frac{1}{2i\sqrt{2\pi(n^{3}-n\alpha^{2})}}\left(c_{n}e^{in\theta}-\overline{c}_{n}e^{-in\theta}\right).$
(3.39)

Fixing the induced metric along the cutoff surface, we again find the same
functional form with a defect present
$w\left(\theta\right)=e^{if(\theta)}\left(1-\delta
f^{\prime}(\theta)+\frac{\delta^{2}f^{\prime}(\theta)^{2}}{2}+\mathcal{O}(\delta^{3})\right)$.
Calculating the extrinsic curvature along this curve we find

$\displaystyle
K=1+\delta^{2}\left(\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3f^{\prime\prime
2}}{2f^{\prime 2}}+\frac{\alpha^{2}}{2}f^{\prime
2}\right)+\mathcal{O}(\delta^{3})=1+\delta^{2}\operatorname{Sch}\left(\tan\frac{\alpha
f(\theta)}{2},\theta\right).$ (3.40)

Which is precisely the Schwarzian with a conical defect[52] which is again
one-loop exact. Restoring temperature and the dilaton as in the case of the
disk we find the action to quadratic order to be

$\displaystyle I_{\text{bdy}}$
$\displaystyle=-\frac{2\pi\gamma}{\beta}\int_{0}^{2\pi}d\theta\left(\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3f^{\prime\prime
2}}{2f^{\prime 2}}+\frac{\alpha^{2}}{2}f^{\prime 2}\right),$ (3.41)
$\displaystyle=I_{0}+\frac{\pi\gamma}{2\beta}\sum_{n\geq 1}n\hskip
0.28436ptc_{n}\overline{c}_{n},\qquad
I_{0}=-\frac{2\pi^{2}\gamma}{\beta}\alpha^{2}.$

Integrating over the moduli space with (3.19) we need to now include the $n=1$
mode compared with the disk calculation

$\displaystyle Z_{\text{defect}}\left(\beta,\alpha\right)$
$\displaystyle=\prod_{n\geq 1}\mathcal{N}\int
dc_{n}d\overline{c}_{n}\left|\frac{\det\langle\mu,\phi\rangle}{\sqrt{\det\langle\phi,\phi\rangle}}\right|^{2}e^{-I_{\text{b}dy}}=e^{\frac{2\pi^{2}\gamma}{\beta}\alpha^{2}}\prod_{n\geq
1}\mathcal{N}\int
dc_{n}d\overline{c}_{n}\exp\left(-\frac{\pi\gamma}{2\beta}\sum_{n\geq
1}n\hskip 0.28436ptc_{n}\overline{c}_{n}\right)$ (3.42)
$\displaystyle=e^{\frac{2\pi^{2}\gamma}{\beta}\alpha^{2}}\prod_{n\geq
1}\mathcal{N}\frac{4\beta}{\gamma
n}=\sqrt{\frac{\gamma}{8\pi\mathcal{N}\beta}}e^{\frac{2\pi^{2}\gamma}{\beta}\alpha^{2}},$

where again the measure simplifies due to our normalization (3.35) as in the
case of the disk (3.33). This again matches the calculation of the partition
function with a single defect [52, 7].

####  3.2.3 Conical defect: $z$ coordinates

In the previous section we integrated over moduli space using $c_{n}$
coordinates which correspond to deformations of the metric by quadratic
differentials. In this section we will carry out the previous calculation
using a coordinate choice that is more physically intuitive, namely the
position $z_{i}$ of the defect on the disk. This choice of coordinates is
applicable to the $n=1$ mode of the Schwarzian which can be thought of as
moving the defect on the disk. To integrate over the moduli space with $z_{i}$
coordinates we must again work out both the measure and the action in (3.19),
which we now do.

The metric with a conical defect at a general position $z_{i}$ can be obtained
by applying an $\text{SL}(2,\mathbb{R})$ transformation to the metric with a
defect at the center

$ds^{2}=\frac{4\alpha^{2}|w(z)|^{2(\alpha-1)}}{(1-|w(z)|^{2\alpha})^{2}}|w^{\prime}(z)|^{2}dzd\bar{z}\,,\qquad
w(z)=\frac{z-z_{i}}{1-z\overline{z}_{i}},$ (3.43)

where in $w$ coordinates the defect is at $w=0$, but is at $z=z_{i}$ in $z$
coordinates. To work out the measure in (3.19) we need to know the Beltrami
differential $\mu$ which contains information on how the metric
infinitesimally changes as we move the defect $z_{i}\to z_{i}+\delta z_{i}$.
As explained in section 3.1.1, this can be obtained by finding a set of
coordinates $z^{\prime}(z,\bar{z})$ where the deformed metric is flat
$\propto|dz^{\prime}|^{2}$ and the defect is mapped to the new point
$z^{\prime}(z_{i})=z_{i}+\delta z_{i}$ with the asymptotic boundary
undeformed.343434The coordinates $z^{\prime}$ should not affect the boundary
of the disk since such a deformation would both move the defect and turn on
some additional Schwarzian modes, and we want to isolate the deformation that
moves the defect. To extract the Beltrami we only need this coordinate chart
to linear order in $\delta z_{i}$, where it is given by

$z^{\prime}(z,\bar{z})=\frac{\delta
z_{i}\frac{1-z\bar{z}}{1-z_{i}\bar{z_{i}}}+z(1-z_{i}\bar{z})}{\delta
z_{i}\bar{z}_{i}\frac{1-z\bar{z}}{1-z_{i}\bar{z_{i}}}+(1-z_{i}\bar{z})}.$
(3.44) $|dz^{\prime}|^{2}\propto|dz+\delta z_{i}\hskip 1.13791pt\mu
d\bar{z}+\mathcal{O}((\delta
z_{i})^{2})|^{2},\qquad\mu=-\frac{(z-z_{i})(1-z\bar{z}_{i})}{(1-z_{i}\bar{z}_{i})(1-\bar{z}z_{i})^{2}}\,.$
(3.45)

Additionally, the quadratic differential that moves the defect when it is
located $z=z_{i}$ is given by353535It can be checked that
$\langle\mu,\phi_{n}\rangle=0$ for $n\geq 2$, implying that the only action of
$\mu$ is to move the defect.

$\phi_{1}=\frac{1}{w}(\partial
w)^{2}dz^{2}=\frac{(1-z_{i}\bar{z}_{i})^{2}}{(z-z_{i})(1-z\bar{z}_{i})^{3}}dz^{2},\qquad\langle\phi_{1},\phi_{1}\rangle=\frac{2\pi}{1-\alpha^{2}}.$
(3.46)

Which is precisely $\phi_{1}=\frac{1}{w}dw^{2}$ in $w$ coordinates, implying
the norm relation noted above. Using the Beltrami and quadratic differential
we can compute the overlap

$\langle\mu,\phi_{1}\rangle=2\int
dzd\bar{z}\frac{1-|z_{i}|^{2}}{(1-z\bar{z}_{i})^{2}(1-z_{i}\bar{z})^{2}}=\frac{4\pi}{1-|z_{i}|^{2}}\,.$
(3.47)

We can now immediately compute the Weil-Petersson measure (3.19) for the
$z_{i}$ coordinates that move the defect

$\mathcal{N}\int
d^{2}z_{i}\left|\frac{\langle\mu,\phi_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle}}\right|^{2}=8\pi\mathcal{N}\int
d^{2}z_{i}\frac{1-\alpha^{2}}{(1-|z_{i}|^{2})^{2}}=4\pi\mathcal{N}(1-\alpha^{2})\int
d^{2}z_{i}\sqrt{g}\,,$ (3.48)

where in the last equality we have noticed the measure takes the form of the
metric on the disk without a defect. We are now left to compute the action in
the $z_{i}$ coordinates. We follow the same procedure as in the previous
section, if we take the cutoff to be at $z=e^{i\theta}$ then in
$w=e^{if(\theta)}$ coordinates (3.43) we have

$w(e^{i\theta})=e^{i\theta}\frac{1-z_{i}e^{-i\theta}}{1-\bar{z}_{i}e^{i\theta}}\
\Rightarrow\
f(\theta)=\theta+i^{-1}\log\left(\frac{1-z_{i}e^{-i\theta}}{1-\bar{z}_{i}e^{i\theta}}\right)\,.$
(3.49)

Note that this reparameterization only takes into account the $n=1$ mode of
the Schwarzian. We previously worked out the action for $f(\theta)$ in (3.41),
giving

$\displaystyle I^{n=1}_{\text{bdy}}$
$\displaystyle=-\frac{2\pi\gamma}{\beta}\int_{0}^{2\pi}d\theta\left(\frac{f^{\prime\prime\prime}}{f^{\prime}}-\frac{3f^{\prime\prime
2}}{2f^{\prime 2}}+\frac{\alpha^{2}}{2}f^{\prime 2}\right),$
$\displaystyle=-\frac{2\pi\gamma}{\beta}\int_{0}^{2\pi}d\theta\frac{1}{2}\left[\alpha^{2}\frac{(1-|z_{i}|^{2})^{2}}{|1-z_{i}e^{-i\theta}|^{4}}-\frac{(z_{i}e^{-i\theta}+\bar{z}_{i}e^{i\theta}-2)(z_{i}e^{-i\theta}+\bar{z}_{i}e^{i\theta}-2|z_{i}|^{2})}{|1-z_{i}e^{-i\theta}|^{4}}\right],$
$\displaystyle=-\frac{2\pi^{2}\gamma}{\beta}\left(1-(1-\alpha^{2})\frac{1+|z_{i}|^{2}}{1-|z_{i}|^{2}}\right).$
(3.50)

We can now write the full answer using (3.19), (3.48), and (3.2.3). Including
all the $n\geq 2$ modes of the Schwarzian we get

$Z_{\text{defect}}(\beta,\alpha)=4\pi\mathcal{N}\int
d^{2}z_{i}\frac{\left(1-\alpha^{2}\right)}{\left(1-|z_{i}|^{2}\right)^{2}}e^{-I^{n=1}_{\text{bdy}}}\times\prod_{n\geq
2}\mathcal{N}\frac{4\beta}{\gamma
n}=e^{\frac{2\pi^{2}\gamma}{\beta}\alpha^{2}}\prod_{n\geq
1}\mathcal{N}\frac{4\beta}{\gamma n},$ (3.51)

which is the expected answer (3.42). From the above we see that when
integrating over the position of the defect we pick up a measure factor
proportional to $(1-\alpha^{2})$ along with an action proportional to the
same. In the limit $\alpha\to 1$ the defect becomes very blunt and one expects
that it does not backreact strongly on the geometry, so that semiclassical
methods can be applied. However, we see that the path integral measure is also
becoming important in this limit, and must be taken into account.

We can rewrite the integral over $z_{i}$ in a way that connects it to the
semiclassical calculation of a single defect on the disk. The contribution of
the $n=1$ mode of the Schwarzian is

$Z^{n=1}_{\text{defect}}(\beta,\alpha)=4\pi\mathcal{N}(1-\alpha^{2})\int
d^{2}z_{i}\sqrt{g(z_{i})}e^{-\pi(1-\alpha^{2})\phi_{\text{cl.}}(z_{i})},\qquad\phi_{\text{cl.}}(z_{i})=\frac{2\pi\gamma}{\beta}\frac{1+|z_{i}|^{2}}{1-|z_{i}|^{2}},$
(3.52)

where in the above $\sqrt{g}$ is the metric on the hyperbolic disk without a
defect and $\phi_{\text{cl.}}$ is the classical solution for the dilaton on
the disk. It has previously been observed that a semiclassical calculation
[17, 18] of the disk with a single defect seems to agree with the full
Schwarzian calculation if an appropriate measure factor is inserted for the
integral over the dilaton. The semiclassical calculation is given by

$\lim_{\alpha\to 1}Z_{\text{defect}}^{\text{semiclassical}}\approx
2\pi\left(1-\alpha\right)\int
d^{2}z\sqrt{g}e^{-2\pi\left(1-\alpha\right)\phi_{\text{cl.}}(z)},$ (3.53)

where we approximate the integral over the dilaton by it’s classical value,
and a measure factor of $2\pi\left(1-\alpha\right)$ is inserted by hand to get
agreement with the Schwarzian. Comparing with (3.52) we see why this semi-
classical calculation works, the integral over the position cancels out the
measure in both calculations giving the same answer.

####  3.2.4 General surfaces

In this section we will give a general argument that the correct form of the
path integral measure for a conical defect in the limit that the angle becomes
blunt is given by

$\lim_{\alpha\to 1}\mathcal{N}\int
d^{2}z\left|\frac{\langle\mu,\phi_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle}}\right|^{2}=2\pi(1-\alpha)\int
d^{2}z\sqrt{g(z)}+\mathcal{O}\left(\left(1-\alpha\right)^{2}\right),$ (3.54)

where we have inserted $\mathcal{N}=\frac{1}{4}$ to normalize the operator
with respect to the standard gluing measure $\int bdb$. Consider a surface
$\Sigma$ with at least one conical defect363636For simplicity, we do not
consider geodesic boundary temporarily. located at $z=z_{i}$. The quadratic
differential that moves this defect has a simple pole at $z_{i}$

$\phi_{1}=\frac{1}{z-z_{i}}dz^{2}+\ldots\,,$ (3.55)

where the subleading terms are holomorphic on $\Sigma$.373737We can also
include a residue at the pole, as we did for the disk in (3.46) and (3.47),
but it will cancel out of the path integral measure. In principle we need to
know the quadratic differential on the entire surface to calculate the
measure, but in the blunt defect limit $\alpha\to 1$ we will argue that the
singular behavior of the quadratic differential dominates.

To calculate the inner products we also need the metric on the surface
$ds^{2}=e^{2\omega}dzd\bar{z}$ where we are in local coordinates $z$ near the
defect. Demanding that the opening angle is $2\pi\alpha$ at $z_{i}$ the metric
is given by solving Liouville’s equation383838We are solving for the
constraint
$\frac{1}{2}\sqrt{g}\left(R+2\right)=2\pi(1-\alpha)\delta^{2}(z-z_{i})$ and we
have used that $R=-2e^{-2\omega}\hat{\nabla}^{2}\omega+e^{-2\omega}\hat{R}$
for a Weyl transformed metric $g=e^{2\omega}\hat{g}$. with a source

$-2\overline{\partial}\partial{\omega}+\frac{1}{2}e^{2\omega}=2\pi(1-\alpha)\delta^{2}(z-z_{i})\,.$
(3.56)

Solving for the metric perturbatively in $(1-\alpha)$ we find

$\omega=\omega_{0}-\frac{1}{2}(1-\alpha)\big{[}\log|z-z_{i}|^{2}+r\big{]}+\mathcal{O}\left(\left(1-\alpha\right)^{2}\right)\,,$
(3.57)

where $\omega_{0}$ is the Weyl factor for the surface $\Sigma$ without the
defect. The linear order term in $(1-\alpha)$ is split up into a logarithm
which gives the delta function singularity, along with a holomorphic function
$r$ that satisfies $\overline{\partial}\partial
r=e^{2\omega_{0}}\log|z-z_{i}|$.

One final part we need for the calculation of the measure is the beltrami
differential $\mu$ that corresponds to infinitesimally moving the defect from
$z_{i}\to z_{i}+\delta z_{i}$. As explained in the case of the disk, the
beltrami differential can be extracted from a coordinate chart
$z^{\prime}=z+\delta z_{i}\hskip 1.42271ptv(z,\bar{z})+\mathcal{O}((\delta
z_{i})^{2})$ such that $z^{\prime}(z_{i})=z_{i}+\delta z_{i}$, which implies
that $v(z_{i},\bar{z}_{i})=1$. We only need this coordinate chart to linear
order in $\delta z_{i}$. The beltrami differential is then be given by
$\mu=\overline{\partial}v(z,\bar{z})$. We will also demand that
$v|_{\text{bdy}}=0$ which is the condition that we are infinitesimally moving
the defect but not changing any asymptotically AdS boundaries, which will
allow us to integrate by parts.

We can now calculate all the components of the path integral measure for
moving the defect position in $z_{i}$ coordinates:

$\langle\mu,\phi_{1}\rangle=\int_{\Sigma}d^{2}z\hskip
1.42271pt\overline{\partial}v\left(\frac{1}{z-z_{i}}+\text{holo.}\right)=2\pi,$
(3.58)

where we have integrated by parts and assumed all boundary terms
vanish,393939This can be compared to the calculation performed on the disk
where we obtain the same answer by performing the full integral (3.47). used
that $\phi_{1}$ is holomorphic away from the defect, and that
$v(z_{i},\bar{z}_{i})=1$.404040In our conventions
$\overline{\partial}z^{-1}=2\pi\delta^{2}(z)$ and $\int
d^{2}z\delta^{2}(z)=1$. Computing the inner product of the quadratic
differential we find

$\displaystyle\langle\phi_{1},\phi_{1}\rangle$
$\displaystyle=2\int_{\Sigma}d^{2}ze^{-2\omega}\left|\frac{1}{z-z_{i}}+\ldots\right|^{2}=2\int_{\Sigma}d^{2}ze^{-2\omega_{0}}\frac{|z-z_{i}|^{2(1-\alpha)}}{|z-z_{i}|^{2}}+\text{less
singular},$ (3.59) $\displaystyle\mathrel{\mathop{=}\limits_{\alpha\to
1}}-\frac{4\pi
e^{-2\omega_{0}(z_{i})}}{(1-\alpha)}+\mathcal{O}(1)=-\frac{2\pi}{(1-\alpha)\sqrt{g(z_{i})}}+\mathcal{O}(1).$

where in the last line we take $\alpha\to 1$ and localize onto the most
singular part of the integral near $z=z_{i}$,414141The integral must be
performed with a radial cutoff around $z_{i}$, after which the leading order
answer in $(1-\alpha)$ is cutoff independent. and in the second equation we
notice that we have picked up a factor of the metric $\sqrt{g}$ with the
defect removed from the surface, evaluated at $z_{i}$. Putting everything
together, we find that the Weil-Petersson measure (3.19) for the integral over
the defect position in the blunt angle limit is given by

$\lim_{\alpha\to 1}\mathcal{N}\int
d^{2}z_{i}\left|\frac{\langle\mu,\phi_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle}}\right|^{2}=2\pi\left(1-\alpha\right)\int
d^{2}z_{i}\sqrt{g(z_{i})}+\mathcal{O}\left((1-\alpha)^{2}\right).$ (3.60)

Note that on the right $g(z_{i})$ is the metric on the surface without the
conical defect. From the above we immediately have that the dilaton-gravity
operator that inserts a conical defect in the blunt defect limit is given by

$\lim_{\alpha\to 1}\mathcal{V}_{\alpha}=2\pi(1-\alpha)\int
d^{2}z_{i}\sqrt{g(z_{i})}e^{-2\pi(1-\alpha)\Phi(z_{i})}+\mathcal{O}\left((1-\alpha)^{2}\right).$
(3.61)

#### Recursion Relation For WP Volumes

This result also allows us to give a gravitational path integral argument for
a recursion relation of Weil-Petersson volumes with blunt defects derived in
[20]. Consider the Weil-Petersson volume of surfaces $\Sigma$ of genus $g$
with $m$ geodesic boundaries of lengths $\vec{b}_{m}=(b_{1},\ldots,b_{m})$ and
$n+1$ conical defects of opening angles $2\pi\alpha_{i}$ with
$\vec{\alpha}_{n+1}=(\alpha_{1},\ldots,\alpha_{n},\alpha_{n+1})$. In the limit
where one of the defects becomes blunt $\alpha_{n+1}\to 1$ [20] proved the
following relation

$\frac{dV_{g,m,n+1}\left(\vec{\alpha}_{n+1},\vec{b}_{m}\right)}{d\alpha_{n+1}}\bigg{\rvert}_{\alpha_{n+1}=1}=-2\pi|\Sigma|V_{g,m,n}\left(\vec{\alpha}_{n},\vec{b}_{m}\right),$
(3.62)

where $|\Sigma|$ is the hyperbolic area of the surface with $n$ defects
satisfying

$|{\Sigma}|=-2\pi\left(2-2g-m-\sum_{i=1}^{n}(1-\alpha_{i})\right)=-\frac{1}{2}\int_{\Sigma/\\{x_{i}\\}}\sqrt{g}R.$
(3.63)

We can prove this formula be decomposing the volume into an integral over a
coordinate parameterizing the position of the $\alpha_{n+1}$ defect, and all
the other moduli of the surface. Working near the blunt defect limit the
integral over the defect takes the simplified form (3.60)

$V_{g,m,n+1}\left(\vec{\alpha}_{n+1},\vec{b}_{m}\right)=\int
d\left(\text{other
moduli}\right)\times\left(2\pi(1-\alpha_{n+1})\int_{\Sigma/\\{z_{j}\\}}d^{2}z\sqrt{g}+\mathcal{O}\left((1-\alpha_{n+1})^{2}\right)\right),$
(3.64)

where the integral over the “other moduli” take the form of the measure (3.19)
with various determinants of beltrami and quadratic differentials. The measure
for the other moduli implicitly depends on $\alpha_{n+1}$ through the
appearance of the defect metric in the inner products defining the measure.
However, as we take $\alpha_{n+1}\to 1$ this dependence goes away since the
surface no longer has a defect. Therefore, in the limit that the defect
vanishes the integral over the other moduli becomes the Weil-Petersson volume
of the surface without the $\alpha_{n+1}$ defect. Another important point as
explained around (3.60) is that the metric $\sqrt{g}$ appearing in the above
is the metric for the surface without the $\alpha_{n+1}$ defect. We have also
excluded the integral over the points $z_{j}$ where the other
$\vec{\alpha}_{n}$ defects are located, as these configurations are at the
boundary of moduli space.424242This exclusion is necessary to reproduce the
relation (3.62) since the Euler characteristic excludes these points. Taking a
derivative we immediately find

$\frac{dV_{g,m,n+1}\left(\vec{\alpha}_{n+1},\vec{b}_{m}\right)}{d\alpha_{n+1}}\bigg{\rvert}_{\alpha_{n+1}=1}=\left(-2\pi\int_{\Sigma/\\{z_{j}\\}}d^{2}z\sqrt{g}\right)\times
V_{g,m,n}\left(\vec{\alpha}_{n},\vec{b}_{m}\right),$ (3.65)

which is the desired recursion relation (3.62). Note that this argument also
goes through if we replace the measure for the defect with
$\pi\left(1-\alpha^{2}\right)\int d^{2}z\sqrt{g}$, as we found for the disk.

From the above argument it might be suspected that the volumes also satisfy
$\lim_{\alpha_{n}\to
1}V_{g,n,m}\left(\vec{\alpha}_{n},\vec{b}_{m}\right)\stackrel{{\scriptstyle?}}{{=}}0$.
In [20] this was shown to be true when there are no geodesic boundaries, but
is false when such boundaries are present. We do not have a gravity path
integral argument for this. One possibility is that with geodesic boundaries
there are additional boundary terms that enter into the measure through
(3.58), where we assumed that all boundary terms vanished.

##  4 Discussion

In this paper we studied various unresolved aspects of the gravitational path
integral of JT gravity. We carried out the gauge fixing of the path integral
in second order formalism for general hyperbolic surfaces with asymptotic
boundaries and conical singularities. The second order formalism allowed us to
clarify the procedure for calculating the proper measure for the conical
defect operator, and resolved the question of which dilaton gravity potential
should be used for JT gravity coupled to a gas of conical defects [17, 18, 19,
20]. This also allowed us to give a gravity path integral argument for certain
recursion relations of Weil-Petersson volumes derived using algebraic geometry
techniques [20]. An open problem is to carry out the full computation of the
measure for the conical defect operator to all orders in the $(1-\alpha)$
expansion on a general surface to prove the conjectured form given in equation
1.9, which we were only able to fully compute on the disk topology.434343As
explained in the introduction, our reasoning for extending the normalization
of the operator found on the disk to arbitrary surfaces is that the operator
should be surface independent.

Along the way we computed determinants of Laplace operators on hyperbolic
surfaces with asymptotic boundaries. These determinants are straightforwardly
related to partition functions of matter fields minimally coupled to JT
gravity. It would be interesting to better understand matter fields coupled to
JT with a gas of conical defects, where the bulk geometry would not be pure
AdS2. Tangentially, we computed the determinant of the vector Laplacian
$\det(\Delta_{1}+s(s-1))$ on a cone geometry. This can be related to the bulk
entanglement entropy of a gauge field on one side of the TFD. Since gauge
fields in two dimensions have no propagating degrees of freedom a non-trivial
entanglement entropy should arise from edge modes, and it would be interesting
to understand this better in AdS2.444444We thank Sean Colin-Ellerin for
discussion on this point.

We obtained an exact expression for the two-point function of matter fields on
an arbitrary surface $\Sigma$ obtainable through the quotient method. The
correlator takes into account all geodesics on the surface including self-
intersecting geodesics, but does not include an integration over the
Schwarizan mode. In [53] diagrammatic rules were derived for correlation
functions, including the appearance of $6j$ symbols when geodesics intersect,
but this has yet to be derived from the second order perspective. It would be
interesting to integrate over the boundary fluctuations and get a closed form
expression for the full correlator reproducing the expected $6j$ symbols. This
would also allow one to incorporate the contribution of self-intersecting
geodesics to the late time two-point function calculation on the handle-disk,
where it was argued in [25] that such contributions should decay with time.

### Acknowledgments

We thank David Borthwick, Sean Colin-Ellerin, Luca Iliesiu, Geoff Penington,
and Joaquin Turiaci for discussion. MU is supported in part by the NSF
Graduate Research Fellowship Program under grant DGE1752814, by the National
Science Foundation under Grants No. NSF PHY-1748958 and PHY-2309135, 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 Consistency of disk measure with general surface

In this section we explain how the exact calculation for a disk with a single
defect is consistent with the measure derived for a general surface. Recall
that we found the measure for the defect to be

$\mathcal{N}\int
d^{2}z_{i}\left|\frac{\langle\mu,\phi_{1}\rangle}{\sqrt{\langle\phi_{1},\phi_{1}\rangle}}\right|^{2}=\int
d^{2}z_{i}~{}\times~{}\begin{cases}\pi\left(1-\alpha^{2}\right)\sqrt{g(z_{i})},&\text{disk,}\\\
2\pi\left(1-\alpha\right)\sqrt{g(z_{i})}+\mathcal{O}\left(\left(1-\alpha\right)^{2}\right),&\text{general
surface}.\end{cases}$

The point is that since the disk calculation is exact we implicitly re-summed
the $\left(1-\alpha\right)$ corrections to the measure. We now make this
explicit in the case that the defect is at the center. The tower of
$\left(1-\alpha\right)$ corrections comes from the inner product for the
quadratic differential

$\langle\phi_{1},\phi_{1}\rangle_{\text{disk}}=\frac{4\pi}{1-\alpha^{2}}\frac{1}{\sqrt{g(0)}},\qquad\langle\phi_{1},\phi_{1}\rangle_{\text{general}}\mathrel{\mathop{=}\limits_{\alpha\to
1}}\frac{2\pi}{\left(1-\alpha\right)\sqrt{g(0)}}+\mathcal{O}\left(1\right).$
(A.1)

We can express the metric for the disk with a conical defect in the same form
we used for a general surface

$ds^{2}=e^{2\omega}dzd\bar{z},\qquad\omega=\omega_{0}+\log\left(|z|^{\alpha-1}\right)-\log\left(\frac{1-|z|^{2\alpha}}{\alpha^{2}(1-|z|^{2})}\right),\qquad\omega_{0}=\log\left(\frac{2}{1-|z|^{2}}\right),$
(A.2)

where $\omega_{0}$ is the Weyl factor for a disk without a defect. We can now
calculate the inner product on the disk, doing a series expansion in
$(1-\alpha)$ for the last term in the Weyl factor

$\displaystyle\langle\phi_{1},\phi_{1}\rangle_{\text{disk}}$
$\displaystyle=\int
d^{2}z\left(\frac{e^{2\omega_{0}}}{2}\right)^{-1}\frac{|z|^{2(1-\alpha)}}{|z|^{2}}\times\underbrace{\frac{(1-|z|^{2})^{2}}{\alpha^{2}(1-|z|^{2\alpha})^{2}}}_{\text{expand}},$
(A.3)
$\displaystyle=\frac{2\pi}{(1-\alpha)\sqrt{g(0)}}+\frac{\pi}{\sqrt{g(0)}}+\mathcal{O}\left(1-\alpha\right),$
(A.4) $\displaystyle\stackrel{{\scriptstyle\tiny\text{re-
sum}}}{{=}}\frac{4\pi}{(1-\alpha^{2})\sqrt{g(0)}}.$ (A.5)

Which is in agreement with the expression for general surfaces in the
$\left(1-\alpha\right)$ expansion, and the re-summation of the series
reproduces the full disk answer (A.1).

##  Appendix B Double trumpet gluing measure

The Weil-Petersson measure for gluing two geodesic boundaries is famously
given by $\int dbd\tau$ [54, 55], where $b\in[0,\infty)$ is the length of the
geodesic boundary and $\tau\in[0,b]$ is a relative twist between the two
boundaries being glued. In this appendix we will review how this measure can
be derived using Beltrami differentials.

##### The Beltrami differential.

We will derive the gluing measure for the double trumpet geometry which can be
represented by the quotient of the UHP with metric
$ds^{2}=\frac{dzd\overline{z}}{\left(\operatorname{Im}z\right)^{2}}$, where
the geometry is represented by $\mathbb{H}/\langle T_{b}\rangle$ where
$T_{b}\cdot z=e^{b}z$. The fundamental domain $\mathcal{F}$ is the region
between the two semicircles $r=1$ and $r=e^{b}$. The main idea is to
explicitly find the two quasiconformal deformations that infinitesimally
deform the geodesic length $b$ and twist $\tau$, and calculate the measure
from equation (3.19). The transformation that infinitesimally deforms the
length $b\to b+\epsilon$ is given by the map

$z\rightarrow f_{b}(z,\bar{z})=z(z\bar{z})^{\frac{\epsilon}{2b}}\,.$ (B.1)

This map satisfies $\left|f_{b}({|z|{=}e^{mb}})\right|=e^{m(b+\epsilon)}$,
meaning that semicircles with radius $e^{mb}$ are mapped to semicircles with
radius $e^{m(b+\epsilon)}$. Thus this transformation increases the geodesic
throat size of the double trumpet. For the twist the appropriate deformation
is given by

$z\rightarrow
f_{\tau}(z,\bar{z})=ze^{\epsilon\Phi(\theta)}\,,\qquad\Phi(0)=0,~{}~{}\Phi(\pi)=1\,.$
(B.2)

In the above $\Phi(\theta)$ is an arbitrary smooth function of the angle
$z=re^{i\theta}$. The properties of $\Phi$ ensure that the left boundary of
the double trumpet is smoothly infinitesimally twisted relative to the right
boundary. For example, a point on the left boundary $z=-1$ is sent to
$z=-(1+\epsilon)$. These maps are known as quasiconformal
transformations.454545Quasiconformal transformations satisfy
$|\bar{\partial}f(z,\bar{z})|<\partial f(z,\bar{z})$. A conformal
transformation can be viewed as a special class of quasiconformal
transformation with $\bar{\partial}f=0$. Geometrically, a conformal map
preserves angles and maps small circles to circles; while a quasiconformal map
does not preserve angles and maps small circles to ovals. Roughly speaking,
the eccentricity of the ovals are given by the ratio $\bar{\partial}f/\partial
f$. The transformation is known as an infinitesimaly quasiconformal
transformation if this ratio can be made infinitesimal. Such transformations
infinitesimally deform the metric in moduli space, since the new metric is no
longer Weyl equivalent to the original metric.

The Beltrami differential is then defined to be
$\mu=\frac{\bar{\partial}f}{\partial f}$.464646Note that the definition of
$\mu$ here is the complex conjugate of the definition from most of the math
literature. Taking $\epsilon$ to be infinitesimal we can define an
infinitesimal beltrami $\hat{\mu}=\lim_{\epsilon\rightarrow 0}\mu/\epsilon$
which through an abuse of notation is the definition of Beltrami differentials
used in the main text. As explained in the main text, these infinitesimal
beltrami differentials capture the infinitesimal change in the metric as we
deform the moduli. We can directly calculate $\hat{\mu}$ from (B.1) and (B.2)
and find

$\hat{\mu}_{\tau}=\frac{iz}{2\bar{z}}\Phi^{\prime}(\theta)dz^{-1}d\bar{z}\,,\quad\hat{\mu}_{b}=\frac{z}{2b\bar{z}}dz^{-1}d\bar{z}\,,$
(B.3)

where we have emphasized that these differentials should be thought of as
$(-1,1)$ tensors. It can however be the case that a Beltrami differential does
not correspond to a pure deformation of the moduli, for example it can deform
the metric by a Weyl transformation. It is thus customary to act on Beltrami
differentials by a projection operator that restricts them to pure moduli
deformations. This does not affect the calculation of the path integral
measure, but ensures that the differentials live in the tangent space to
moduli space. The projection is orginally given by [56]

$P[\hat{\mu}](z)\equiv\frac{6}{\pi}(\operatorname{Im}z)^{2}\int_{\mathbb{H}}d^{2}\xi\frac{\overline{\hat{\mu}}(\xi,\bar{\xi})}{({\xi}-\bar{z})^{4}}\,.$
(B.4)

This gives us the projected differentials474747Note that $P[\hat{\mu}_{\tau}]$
is independent of $\Phi^{\prime}(\theta)$, which shows that the apparent
entire function’s worth of degrees of freedom in $\Phi$ do not correspond to
genuine moduli deformations.

$P[\hat{\mu}_{b}]=\frac{(\text{Im}z)^{2}}{{z}^{2}b}dz^{-1}d\bar{z}\,,\quad
P[\hat{\mu}_{\tau}]=\frac{i(\text{Im}z)^{2}}{\pi{z}^{2}}dz^{-1}d\bar{z}.$
(B.5)

One remarkable property of this projection is
$\langle\hat{\mu},\phi\rangle=\langle P[\hat{\mu}],\phi\rangle$ for any
quadratic differential $\phi$, and since we are primarily interested in inner
products we use the same notation $\hat{\mu}$ to represent the Beltrami and
its projection below. This identity is the statement that the operator
projects out the subset of deformations in $\hat{\mu}$ that do not deform the
moduli.

##### Quadratic differential.

To compute the metric we also need the quadratic differentials, which are
thought of as forming the cotangent space to moduli space. These differentials
are quite challenging to compute and were found by Wolpert [55], we quote the
final result484848Note that our convention here is a bit different from
Wolpert’s convention. The differences are (1) our definition of the
$\langle\mu,\phi\rangle$ inner product is twice as Wolpert’s and (2) the
integral measure on Riemann surface $d^{z}$ is 2 times the Euclidean measure
$dxdy$ used by Wolpert.

$\phi_{b}=\frac{1}{\pi
z^{2}}dz^{2}\,,\qquad\phi_{\tau}=\frac{i}{bz^{2}}dz^{2}.$ (B.6)

They have the following inner products with the beltrami differentials
$\mu_{b},\mu_{\tau}$494949We slightly change the definition of the inner
product by taking the real part, since both $db$ and $d\tau$ are real
variables.

$\langle\mu_{b},\phi_{b}\rangle\equiv
2\text{Re}\int_{\mathcal{F}}\mu_{b}\bar{\phi}_{b}=\frac{2}{\pi
b}\int_{\mathcal{F}}\frac{(\text{Im}\xi)^{2}}{\xi^{2}\bar{\xi}^{2}}d^{2}\xi=2\,,\qquad\langle\mu_{\tau},\phi_{\tau}\rangle=2,$
(B.7)

where we have used the inner product in (3.16). It is straight forward to
check that the off-diagonal inner products vanish
$\langle\mu_{\tau},\phi_{b}\rangle=0$ because of the factor $i$ in the
$\mu_{\tau}$ expression.

##### The Weil-Petersson metric.

Given these results the Weil-Petersson metric for the moduli space is
straightforward to obtain. From the discussion in Section 3, we need to
evaluate several determinants. It is straight forward to calculate

$\det\langle\mu,\phi\rangle=\det\begin{pmatrix}\langle\mu_{b},\phi_{b}\rangle&\langle\mu_{b},\phi_{\tau}\rangle\\\
\langle\mu_{\tau},\phi_{b}\rangle&\langle\mu_{\tau},\phi_{\tau}\rangle\end{pmatrix}=\det\begin{pmatrix}2&0\\\
0&2\end{pmatrix}=4\,.$ (B.8)
$\det\langle\phi,\phi\rangle=\det\begin{pmatrix}\langle\phi_{b},\phi_{b}\rangle&\langle\phi_{b},\phi_{\tau}\rangle\\\
\langle\phi_{\tau},\phi_{b}\rangle&\langle\phi_{\tau},\phi_{\tau}\rangle\end{pmatrix}=\det\begin{pmatrix}2b&0\\\
0&2b^{-1}\end{pmatrix}=4\,.$ (B.9)

The measure is given by equation (3.19)

$d\mu_{\text{DT}}=\mathcal{N}\frac{\det\langle\mu,\phi\rangle\det\langle\bar{\mu},\bar{\phi}\rangle}{\det\langle\phi,\phi\rangle}db\,d\tau=4\mathcal{N}db\,d\tau\,,$
(B.10)

where the integral ranges over $\tau\in[0,b]$ and $b\in[0,\infty)$. To get the
standard form of the gluing measure $\int bdb$ we must choose the
normalization $\mathcal{N}=1/4$ for the measure (3.3) as claimed in the main
text.

##  Appendix C The determinant calculation

### Notation, conventions, and summary of results

We first list the notation and main results of the determinant calculation. We
denote the surface obtained by a quotient of the Fuchsian group $\Gamma$ as
$\Sigma=\mathbb{H}/\Gamma$. The fundamental domain is denoted by
$\mathcal{F}_{\Sigma}$ which is thought of as a subregion of $\mathbb{H}$. We
will sometimes not distinguish between $\mathcal{F}_{\Sigma}$ and $\Sigma$ for
simplicity. We define the Fuchsian group action to be $T(Z)$ where $T$ is a
Fuchsian group element and $Z$ will either be a subregion or a point in
$\mathbb{H}$.

An important geometrical invariant of the surface $\Sigma$ is the hyperbolic
area

$|\Sigma|=-\left(\frac{1}{2}\int_{\Sigma/\\{z_{i}\\}}d^{2}z\sqrt{g}R+\int_{\partial\Sigma}\sqrt{h}K\right)=2\pi\bigg{(}2g+n-2+\sum_{j=1}^{k}\left(1-n_{j}^{-1}\right)\bigg{)},$
(C.1)

where $g$ is the genus, $n$ is the number of boundaries (including cusps and
asymptotic boundaries), and $k$ is the number of defects located at positions
$z_{i}$ with deficit angles $\frac{2\pi}{n_{j}}$. For simplicity we exclude
surfaces with geodesic boundaries and cusps ($2\pi$ deficit angles) from the
determinant calculation. The area element of $\mathbb{H}$ is
$d^{2}z\sqrt{g}=y^{-2}dxdy$, and the $u$ variable for hyperbolic distance

$u=\cosh^{2}\left(\frac{\ell(z,z^{\prime})}{2}\right)=\frac{1}{2}\left(1+\cosh(\ell(z,z^{\prime}))\right)=\frac{(x-x^{\prime})^{2}+(y+y^{\prime})^{2}}{4yy^{\prime}}\,,$
(C.2)

where our complex coordinates are given by $z=x+iy$.

We will be interested in the scalar and vector Laplacians. The scalar
Laplacian is defined as $-g^{ab}\nabla_{a}\nabla_{b}$ acting on scalar
functions. Similarly, the vector Laplacian is
$\nabla_{1}^{2}=-g^{ab}\nabla_{a}\nabla_{b}$ acting on two component vectors
or covectors, and was introduced in the main text through
$\nabla_{1}^{2}=\frac{1}{2}P_{1}^{\dagger}P_{1}-1$. It will be more convenient
to conjugate the vector Laplacian and consider
$y^{-2}\nabla_{1}^{2}y^{2}+s(s-1)-1=\text{diag}\left(D_{-1}+s(s-1),D_{1}+s(s-1)\right)$
where we define the operators

$D_{\pm 1}=-\left(\partial_{x}\pm
i\partial_{y}\right)y^{2}\left(\partial_{x}\mp i\partial_{y}\right),$ (C.3)

where the conjugated operators acts on vectors with top component
$v^{x}+iv^{y}$ and bottom component $v^{x}-iv^{y}$. Conjugated operators have
the same spectrum, and so we can equivalently compute the determinant of the
above operators. The spectra of $D_{\pm 1}$ are identical, and so we find the
result that $\det(\Delta_{1}+1)=\det\left(D_{1}+2\right)^{2}$. The above
immediately implies that
$\det(\Delta_{1}+s(s-1)-1)=\det\left(D_{1}+s(s-1)\right)^{2}$. We will abuse
notation in this appendix and refer to $D_{\pm 1}$ as the vector Laplacian.

In the calculation of the determinant we will need the resolvent of the
associated operators, also known as the propagator, denoted by
$R^{0}_{\Sigma}$ and $R^{1}_{\Sigma}$, and their associated traces over the
surface ${\mathrm{t}r}R_{\Sigma}=\int_{\Sigma}d^{2}z\sqrt{g}R_{\Sigma}(z,z)$.
Note that the operator, the resolvent, and the trace are defined for a
specific surface $\Sigma$. The trace is an integral over the fundamental
domain $\mathcal{F}_{\Sigma}\subset\mathbb{H}$.

The main result for the determinants are given as follows 505050One can also
allow for $n_{c}$ cusps on the surface, which give a parabolic contribution
$Z_{\rm par.}=\left(\Gamma(s-1/2)(2s-1)2^{s-1}\right)^{n_{c}}$. With the
inclusion of cusps, the scalar determinant would include $Z_{\rm par.}$,
namely $\det\left(\ldots\right)\to\det\left(\ldots\right)\times Z_{\rm par.}$
[36]. We believe a similar formula holds for the vector determinant.

$\det\left({\Delta}_{0}+s(s-1)\right)={Z_{\text{hyp.}}(s)}{Z_{\text{ell.}}(s)}G_{\infty}(s)^{-\frac{|\Sigma|}{2\pi}}e^{-Bs(s-1)+D}\,.$
(C.4)