Finite cut-off JT and Liouville quantum gravities

on the disk at one loop

Soumyadeep Chaudhuri and Frank Ferrari

Service de Physique Théorique et Mathématique
Université Libre de Bruxelles (ULB) and International Solvay Institutes
Campus de la Plaine, CP 231, B-1050 Bruxelles, Belgique

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Within the path integral formalism, we compute the disk partition functions of
two dimensional Liouville and JT quantum gravity theories coupled to a matter
CFT of central charge $c$, with cosmological constant $\Lambda$, in the limit
$c\rightarrow-\infty$, $|\Lambda|\rightarrow\infty$, for fixed $\Lambda/c$ and
fixed and finite disk boundary length $\ell$, to leading and first subleading
order in the $1/|c|$ expansion. In the case of Liouville theory, we find
perfect agreement with the asymptotic expansion of the known exact DOZZ
partition function. In the case of JT gravity, we obtain the first explicit
results for the partition functions at finite cut-off, in the three versions
(negative, zero and positive curvature) of the model. Our findings are in
agreement with predictions from the recent proposal for a microscopic
definition of JT gravity, including the $c\rightarrow-\infty$ expansion of the
Hausdorff dimension of the boundary. We also provide evidence for the
emergence of an effective Schwarzian description at length scales much greater
than the curvature length scale.

###### Contents

  1. 1 Introduction
  2. 2 Definition of the models and path integrals
    1. 2.1 Base manifold
    2. 2.2 Classical actions and counterterms
      1. 2.2.1 Classical actions
      2. 2.2.2 Counterterms
    3. 2.3 Path integrals
      1. 2.3.1 Formal definition
      2. 2.3.2 Conformal gauge
      3. 2.3.3 The Liouville action
      4. 2.3.4 Gauge fixing $\text{PSL}(2,\mathbb{R})$
      5. 2.3.5 The path integral for Liouville gravity
      6. 2.3.6 The path integral for JT gravity
    4. 2.4 One-loop path integrals
      1. 2.4.1 The case of Liouville gravity
      2. 2.4.2 The case of JT gravity
  3. 3 Leading order: classical solutions
    1. 3.1 Equations of motion
    2. 3.2 Solution for Liouville gravity
    3. 3.3 Solutions for JT gravity
  4. 4 One-loop order: quadratic fluctuations
    1. 4.1 The quadradic action $\widetilde{S}^{(2)}_{\text{eff}}$ for Liouville gravity
    2. 4.2 The quadradic action $\widetilde{S}^{(2)}_{\text{eff}}$ for JT gravity
      1. 4.2.1 Computation of $\Xi[\xi]$
      2. 4.2.2 Quadratic action in the flat case
      3. 4.2.3 Quadratic action in the curved cases
    3. 4.3 Stability analysis
  5. 5 One-loop order: path integrals
    1. 5.1 One-loop counterterms
    2. 5.2 Bulk determinants
      1. 5.2.1 Flat bulk determinants
      2. 5.2.2 Curved bulk determinants
    3. 5.3 Gauge fixing
    4. 5.4 Liouville gravity
      1. 5.4.1 Boundary path integral
      2. 5.4.2 Putting everything together
    5. 5.5 Flat space JT gravity
      1. 5.5.1 Boundary path integral
      2. 5.5.2 Putting everything together
    6. 5.6 Curved space JT gravity
      1. 5.6.1 Boundary path integral
      2. 5.6.2 Putting everything together
  6. 6  Discussion
    1. 6.1 Flat space limit
    2. 6.2 Critical exponents, improvement and all-loop predictions
    3. 6.3 Naive semiclassical limit and expected area
    4. 6.4 The Schwarzian limit
  7. A Boundary determinants

##  1 Introduction

Recently, a general framework has been proposed for the study of finite cut-
off JT quantum gravity from first principles [1, 2, 3], from both a
discretized, statistical physics point of view and a continuum, field
theoretic point of view. This framework provides a rigorous microscopic
definition of the model and thus allows, in principle, to study the full
quantum theory for any finite values of the parameters.

JT quantum gravity may be defined in negative, zero or positive curvature and
we shall deal with the three cases in parallel. We limit ourselves to
Euclidean signature and to the disk topology. We consider the case in which JT
gravity is coupled to an arbitrary matter CFT of central charge $c$. We focus
on the computation of the partition functions, denoted by $Z^{0}$, $Z^{-}$ and
$Z^{+}$ in the zero, negative and positive curvature theories, respectively,
using the path integral formalism.

According to [1], the boundary in JT gravity is a fractal curve and thus has
no standard geometrical length. The naive length is replaced by a renormalized
quantum length parameter $\beta$,111The parameter $\beta$ is a microscopic
parameter of the theories at finite cut-off and must not be confused with
another parameter, often denoted by $\beta$ as well in the literature, that
appears in the so-called Schwarzian limit of the negative curvature model. To
avoid confusion, we shall denote by $\beta_{\text{S}}$ the Schwarzian limit
parameter in the following. that has anomalous Hausdorff dimension
$d_{\text{H}}=1/\nu$, such that $\beta^{\nu}$ has the dimension of length. The
critical exponent $\nu$ is predicted to be [1]

$\nu=\frac{1}{2}\Biggl{[}1+\sqrt{\frac{c}{c-24}}\Biggr{]}\,.$ (1.1)

The parameter $\beta$ is very similar to the diffusion time $\beta$ for a
Brownian process. The critical exponent $\nu=1/2$ for pure, $c=0$, JT gravity,
actually coincides with the one for Brownian curves, even though the JT
gravity boundaries are not arbitrary random closed loops [1, 2]. The zero
curvature theory, with cosmological constant $\Lambda$, thus has a unique
dimensionless parameter $\Lambda\beta^{2\nu}$ on top of the central charge
$c$. The partition function at fixed boundary quantum length is predicted in
[1] to take the general form

$Z^{0}=\beta^{-1-2\nu\vartheta}g^{0}\bigl{(}\Lambda\beta^{2\nu},c\bigr{)}$
(1.2)

for some non-trivial function $g^{0}$ that has a smooth small $\Lambda$ limit
and a critical exponent

$\vartheta=2-\frac{c}{12}\,\cdotp$ (1.3)

In non-zero curvature, the curvature scale $L$ allows to build a second
dimensionless parameter $\beta/L^{1/\nu}$. The partition function then takes
the general form

$Z^{\pm}=\beta^{-1-2\nu\vartheta}g^{\pm}\bigl{(}\Lambda\beta^{2\nu},\beta/L^{1/\nu},c\bigr{)}$
(1.4)

for functions $g^{+}$ and $g^{-}$ that must be such that

$\lim_{L\rightarrow+\infty}g^{\pm}\bigl{(}\Lambda\beta^{2\nu},\beta/L^{1/\nu},c\bigr{)}=g^{0}\bigl{(}\Lambda\beta^{2\nu},c\bigr{)}\,.$
(1.5)

The vast majority of the published works on JT gravity focuses on the negative
curvature case in the so-called Schwarzian limit, see e.g. [4] and [2] for a
review. The limit is defined as

$\Lambda
L^{2}\rightarrow-\infty\,,\quad\frac{\ell_{\text{S}}}{L}\rightarrow\infty\,,\quad\frac{2\ell_{\text{S}}}{|\Lambda|L^{3}}=\beta_{\text{S}}\quad\text{fixed,}$
(1.6)

assuming that a long-distance effective smooth boundary length parameter
$\ell_{\text{S}}$ exists. In this limit, it is conjectured that an effective
description of the theory, in terms of the so-called reparameterization ansatz
and the Schwarzian action, becomes valid. A microscopic definition of the
model is then not needed to derive its leading order properties. This
effective description is similar to the IR Schwarzian description in the SYK
or tensor models [5]. It should emerge when $\ell_{\text{S}}\gg L$. The
microscopic properties, like the fractal structure of the boundary, become
relevant below the curvature length scale $L$. The limit
$\ell_{\text{S}}/L\rightarrow\infty$ is usually called the infinite cut-off
limit, consistently with the holographic interpretation. Note that, strictly
speaking, the smooth length parameter $\ell_{\text{S}}$ itself emerges only in
this limit and should be expressed in terms of the microscopic parameters
$\beta$, $L$ and $\Lambda$ [2].

One of the main motivation for the present work is to use some of the ideas
presented in [1, 2, 3], combined with an interesting semiclassical
approximation scheme explained below, to derive, from first principles and for
the first time, explicit results in negative curvature JT gravity away from
the Schwarzian limit. We shall also use the same set of ideas and tools to
study the positive and zero curvature models.

Instrumental in our analysis is an old idea of Zamolodchikov [6], that was
applied in the past to derive the first non-trivial quantum results for the
Liouville quantum gravity theory on the sphere. Zamolodchikov’s idea was
discussed further recently in the context of Liouville gravity [7, 8, 9] and
also to study generalisations of the Liouville model that occur when non-
conformal matter is coupled to gravity [10, 11]. In our context, it amounts to
studying the limit

$c\rightarrow-\infty\,,\quad|\Lambda|\rightarrow+\infty\,,\quad\frac{\Lambda}{|c|}=\frac{2\mu}{3}\quad\text{and}\quad\beta\quad\text{fixed.}$
(1.7)

In this limit, it turns out that the path integral is dominated by a smooth
classical geometry, at least for some range of the parameter $\mu$. We can
then, in principle, compute all the observables in the model in a systematic
loopwise $1/|c|$ expansion, using the saddle-point method. Consistent with
this idea, the exact result (1.1) admits an expansion

$\nu=\sum_{k\geq
0}\nu_{k}|c|^{-k}=1-\frac{6}{|c|}+\frac{108}{|c|^{2}}+\cdots\,,$ (1.8)

the term proportional to $|c|^{-k}$ being a $k$-loop contribution. Note that
the leading order value $\nu=1$ corresponds to a smooth boundary, consistently
with the fact that the limit $c\rightarrow-\infty$ is semiclassical. In this
limit we can thus identify $\beta$ with a genuine geometrical length at
leading order. For this reason, we shall use the notation

$\beta=\ell\,.$ (1.9)

The loop corrections to the exponent $\nu$ will give $\ell$ an anomalous
dimension. This anomalous dimension is itself the signature of the fractal
nature of the boundary in the microscopic description of the models. Our one-
loop computation below will confirm this expectation and we will find a
perfect match with the $-6/|c|$ term in (1.8), see in particular the
discussion in Sec. 6. More generally, the logarithm of the partition functions
are expected to have a large $|c|$ expansion of the form

$\ln Z^{\eta}=\sum_{k\geq 0}|c|^{1-k}f^{\eta}_{k}\,.$ (1.10)

The main new result of our paper is to compute the leading and first
subleading terms in this expansion, see Eqs. (3.26), (3.29), (3.36) and (3.37)
for the tree-level contributions and Eqs. (5.28), (5.34) and (5.35) for the
one-loop contributions, the dimensionless parameters $x$ and $y$ being defined
in (3.25). Combining the one-loop results with the exact statements (1.1) and
(1.2), one can even extract predictions at any loop order, see Sec. 6.

Interestingly, the usual Liouville quantum gravity theory on the disk can also
be studied in the limit (1.7). The path integral turns out to be dominated by
the same classical geometry relevant to the negative curvature JT model. As
for the JT models, the $1/|c|$ expansion can thus be studied systematically,
at least in principle, generalising the sphere computation of [6]. In
particular, one expects an expansion

$\ln Z_{\text{L}}=\sum_{k\geq 0}|c|^{1-k}f_{k}$ (1.11)

for the Liouville disk partition function $Z_{\text{L}}$, of the same form as
for the JT theory, Eq. (1.10).

Although our main interest is in JT, we have also studied the expansion (1.11)
in the Liouville theory, having in mind two main motivations. First, rigorous
path integral computations in Liouville theory are very few [7, 8, 9] and the
calculation of the one-loop contribution $f_{1}$ was never done before. This
calculation is indeed much more involved on the disk than on the sphere, both
for technical and conceptual reasons. But one must overcome essentially the
same difficulties to do the JT gravity calculations and thus one can get
$f_{1}$ for Liouville with very little additional effort if one has gotten
$f_{1}^{\pm}$ for JT. Actually, the calculations in Liouville and JT can be
naturally presented in parallel, and this is what we shall do below. Second,
an exact formula for the partition function of Liouville gravity on the disk
was proposed by FZZT in [12]. The ideas used by FZZT to obtain their results,
based on general assumptions in the operator formalism, generalising earlier
works by DOZZ [13] which apply to the sphere, are independent of the path
integral approach that we use in our work. Comparing their prediction with our
explicit tree-level and one-loop computations then allows to check our methods
in a very non-trivial way. Explicitly, in terms of the dimensionless
parameter333Liouville theory is defined only for positive or zero cosmological
constant.

$x=\frac{\ell\sqrt{\mu}}{2\pi}\,\raise 2.0pt\hbox{,}$ (1.12)

FZZT predicts

$Z_{\text{L}}=w\ell^{\gamma-3}x^{1-\gamma}K_{1-\gamma}\Biggl{[}\sqrt{\frac{\pi|c|}{6\sin\frac{\pi}{1-\gamma}}}\,x\Biggr{]}\,,$
(1.13)

where $K$ is the modified Bessel function of the second kind and $\gamma$ the
so-called string susceptibility exponent, given by the famous KPZ formula

$\gamma=\frac{1}{12}\Bigl{(}c-1-\sqrt{(1-c)(25-c)}\Bigr{)}\,.$ (1.14)

The constant $w$ is an overall $\ell$-independent and non-universal, scheme-
dependent, proportionality factor. We conventionally fix it by imposing that

$Z_{\text{L}}(\ell,0)=\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\gamma-3}\,,$ (1.15)

which yields

$w=\frac{8\pi^{3-\gamma}}{\Gamma(\gamma-1)}\biggl{(}\frac{6\sin\frac{\pi}{1-\gamma}}{\pi|c|}\biggr{)}^{\frac{\gamma-1}{2}}\,.$
(1.16)

In the limit (1.7), the exact result (1.13) yields, as expected, a
semiclassical expansion of the form (1.11), with explicit predictions for the
$k$-loop contributions. At tree-level and one-loop, one straightforwardly
gets, from the asymptotics of the Bessel function,

$\displaystyle f_{0}$
$\displaystyle=-\frac{1}{6}\ln\frac{\ell}{2\pi}+\frac{1}{6}\ln\frac{1+\sqrt{1+x^{2}}}{2}-\frac{1}{6}\bigl{(}\sqrt{1+x^{2}}-1\bigr{)}$
(1.17) $\displaystyle f_{1}$
$\displaystyle=-\frac{25}{6}\ln\frac{\ell}{2\pi}+\frac{13}{6}\ln\Bigl{[}\frac{1}{2}\bigl{(}1+\sqrt{1+x^{2}}\bigr{)}\Bigr{]}-\frac{13}{12}\bigl{(}\sqrt{1+x^{2}}-1\bigr{)}-\frac{1}{4}\ln\bigl{(}1+x^{2}\bigr{)}\,.$
(1.18)

Our direct semiclassical evaluation of the path integral below will reproduce
very precisely the above coefficients $f_{0}$ and $f_{1}$, see Sec. 3.2 and
5.4.

At the technical level, the main difficulty in performing the one-loop
calculations is that one needs to evaluate explicitly non-trivial bulk
determinants in a finite-size geometry, see Sec. 5.2 and reference [17]. But
the main subtlety is conceptual. After going to conformal gauge, the metric
degrees of freedom are encoded in the conformal factor, the logarithm of which
is called the Liouville field. The crucial point, emphasized in [1, 3], is
that _the bulk and the boundary Liouville fields must be treated as
independent integration variables in the path integral._ In other words, the
Liouville field has free boundary conditions. This point is equally
fundamental in the cases of the JT gravity and the Liouville theories.444A
discussion of the case of the Liouville theory on the disk in the grand
canonical ensemble, with boundary cosmological constant $\lambda$ and in the
limit $\lambda/|c|\rightarrow-\infty$, appears in [9] (the limit
$\lambda/|c|\rightarrow-\infty$ in the grand canonical ensemble is similar to
the limit $\ell\rightarrow\infty$ in the fixed $\ell$ ensemble on which we
focus in our work). As far as we understand, the crucial boundary degrees of
freedom were not taken into account in [9].

The plan of the paper is as follows. In Section 2, we present the models,
introducing the relevant classical geometries and actions and treating the
basics of the path integral formulation, including a discussion of conformal
gauge, the separation between the bulk and boundary degrees of freedom, the
role of the Liouville action and the gauge fixing of the
$\text{PSL}(2,\mathbb{R})$ group of disk automorphisms. We are also putting in
place some essential ingredients of the one-loop calculations. In Section 3,
we determine the classical geometries that dominate in the limit (1.7) by
solving the relevant field equations in Liouville gravity and in the various
versions of JT gravity. This yields the tree-level coefficients $f_{0}^{0}$,
$f_{0}^{\pm}$ and $f_{0}$ in the expansions (1.10) and (1.11). In Section 4,
we compute the quadratic fluctuations around the classical backgrounds. These
fluctuations determine the one-loop quantum corrections. We also discuss the
stability of the various backgrounds. In Section (5), we perform the relevant
Gaussian integrations. The bulk fluctuations produce a finite-size determinant
$\det_{\text{D}}(\Delta+M^{2})$, where $\Delta$ is the positive Laplacian with
Dirichlet boundary condition on the round disk of circumference $\ell$
embedded in hyperbolic space, the two-sphere or the Euclidean plane and
$M^{2}$ is fixed to the particular values $+2/L^{2}$, $-2/L^{2}$ or zero,
respectively. The determinants for arbitrary values of $M^{2}$ are studied in
[17]. Remarkably, they can be evaluated explicitly in terms of elementary
functions and the Euler $\Gamma$ function for the special mass parameters that
are relevant for our purposes. The boundary fluctuations yield a class of
functional determinants that we compute in Appendix A. Putting all together
(Faddeev-Popov factor for the $\text{PSL}(2,\mathbb{R})$ gauge fixing,
contributions of the gauge-fixed modes, on-shell Liouville action contribution
from reparameterization ghosts, bulk determinant and boundary determinant)
yields the one-loop coefficients $f_{1}^{0}$, $f_{1}^{\pm}$ and $f_{1}$.
Finally, in Section 6, we discuss our results in the light of the exact
predictions (1.1) and (1.2) and comment on the naive
$\Lambda\rightarrow-\infty$, fixed $\ell$, semiclassical limit of the models
and on the Schwarzian limit. In particular, we show explicitly that the usual
Schwarzian action is only an effective description valid on length scales much
larger than the curvature scale $L$, whereas the correct UV description is
governed by a completely different action that comes from the boundary
Hilbert-Liouville term, as discussed in [1]. We also find nice evidence that
the effective Schwarzian description emerges from the microscopic theory, if
one uses a renormalized version of the naive scaling used to define the
Schwarzian limit, consistently with the general ideas presented in [2].

##  2 Definition of the models and path integrals

The models we are studying are formally defined by giving a base manifold, a
classical action and a path integral measure. In subsections 2.1, 2.2, 2.3, we
follow the discussion in [1, 3], in which much more details can be found,
providing a synthetic and self-contained account of the points relevant to the
present paper. In subsection 2.4, we specialise to the one-loop approximation
we are interested in, reducing the problem to the computation of the classical
saddles and the evaluation of the Gaussian path integrals around the saddles,
which will be done explicitly in subsequent Sections.

###  2.1 Base manifold

In this paper, the base manifold is the disk

$\mathscr{D}=\bigl{\\{}z\in\mathbb{C}\bigm{|}\lvert z\rvert<1\bigr{\\}}\,.$
(2.1)

The boundary is $\text{S}^{1}=\partial\mathscr{D}=\\{z\in\mathbb{C}\,|\,\lvert
z\rvert=1\\}$ and the closed disk is denoted by
$\bar{\mathscr{D}}=\mathscr{D}\cup\partial\mathscr{D}$. We often use polar
coordinates $(\rho,\theta)$ or Cartesian coordinates $(x^{a})$ defined by

$z=\rho e^{i\theta}=x^{1}+ix^{2}\,.$ (2.2)

The group of disk automorphisms, which are bijective holomorphic maps from
$\mathscr{D}$ to $\mathscr{D}$, is $\text{PSL}(2,\mathbb{R})$, acting as

$z^{\prime}=e^{i\alpha}\frac{z-z_{0}}{1-\bar{z}_{0}z}\,\raise
2.0pt\hbox{,}\quad|z_{0}|<1\,,\ (z,z^{\prime})\in\bar{\mathscr{D}}^{2}\,.$
(2.3)

Note that these transformations extend smoothly to $\bar{\mathscr{D}}$.

The canonical flat round metric of perimeter $2\pi$ on $\mathscr{D}$ is
denoted by

$\delta=|{\rm d}z|^{2}={\rm d}\rho^{2}+\rho^{2}{\rm d}\theta^{2}\,.$ (2.4)

The similar metric of perimeter $\ell$ is

$\delta_{\ell}=\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{2}\,\delta\,.$ (2.5)

The canonical negative curvature round metric of perimeter $\ell$ and Ricci
scalar $R=-2/L^{2}$ is

$\delta^{-}_{\ell}=\frac{4L^{2}r_{0}^{2}|{\rm
d}z|^{2}}{(1-r_{0}^{2}|z|^{2})^{2}}$ (2.6)

with

$\frac{\ell}{L}=\frac{4\pi r_{0}}{1-r_{0}^{2}}$ (2.7)

or equivalently

$r_{0}=\frac{2\pi
L}{\ell}\Biggl{[}-1+\sqrt{1+\frac{\ell^{2}}{4\pi^{2}L^{2}}}\Biggr{]}\,.$ (2.8)

Note that $0\leq r_{0}<1$. The area $A^{-}$ and extrinsic curvature of the
boundary $k^{-}$ are

$A^{-}=4\pi L^{2}\frac{r_{0}^{2}}{1-r_{0}^{2}}=L\ell r_{0}\,,\quad
k^{-}=\frac{2\pi}{\ell}\frac{1+r_{0}^{2}}{1-r_{0}^{2}}=\frac{2\pi}{\ell}\sqrt{1+\frac{\ell^{2}}{4\pi^{2}L^{2}}}\,.$
(2.9)

The metric (2.6) may be seen as arising from the natural embedding of the disk
into hyperbolic space.

The two canonical positive curvature round metrics of perimeter $\ell$ and
Ricci scalar $R=+2/L^{2}$ are of the form

$\delta^{+}_{\ell}=\frac{4L^{2}r_{0}^{2}|{\rm
d}z|^{2}}{(1+r_{0}^{2}|z|^{2})^{2}}$ (2.10)

with

$\frac{\ell}{L}=\frac{4\pi r_{0}}{1+r_{0}^{2}}\,\cdotp$ (2.11)

Note that $\ell$ is bounded above by $2\pi L$. The area $A^{+}$ and extrinsic
curvature of the boundary $k^{+}$ are

$A^{+}=4\pi L^{2}\frac{r_{0}^{2}}{1+r_{0}^{2}}=L\ell r_{0}\,,\quad
k^{+}=\frac{2\pi}{\ell}\frac{1-r_{0}^{2}}{1+r_{0}^{2}}\,\cdotp$ (2.12)

There are two possible solutions for $r_{0}$, which coincide if and only if
$\ell=2\pi L$,

$r_{0}^{>}=\frac{2\pi
L}{\ell}\Biggl{[}1+\sqrt{1-\frac{\ell^{2}}{4\pi^{2}L^{2}}}\Biggr{]}\,,\quad
r_{0}^{<}=\frac{2\pi
L}{\ell}\Biggl{[}1-\sqrt{1-\frac{\ell^{2}}{4\pi^{2}L^{2}}}\Biggr{]}\,,$ (2.13)

that we shall call the “large” and “small” disks. Note that
$r_{0}^{>}r_{0}^{<}=1$. The solution $r_{0}^{>}$ corresponds to the large
disk, having the greatest area. These metrics may be seen as arising from
embedding the disk into the round two-sphere of radius $L$. The embeddings may
be chosen in such a way that the two embedded disks share the same
$\text{S}^{1}$ boundary; the two metric correspond to “filling” one side of
the sphere or the other, see Fig. 1.

Figure 1: The two distinct positive curvature round disk metrics corresponding
to the same boundary length $\ell<2\pi L$, depicted as embeddings of the disk
into the round two-sphere. The “large” disk, with parameter $r_{0}^{>}$, is
shaded in light grey, the “small” disk, with parameter $r_{0}^{<}$, in dark
grey.

Finally, let us note that the Gauss-Bonnet formula on the disk

$\int_{\mathscr{D}}\\!{\rm
d}^{2}x\sqrt{g}\,R+2\oint_{\partial\mathscr{D}}\\!{\rm d}s\,k=4\pi$ (2.14)

has the following consequences, which are valid for any constant curvature
metric on $\mathscr{D}$ (and thus also for the canonical examples that we have
just mentioned). In zero curvature, the integral of the extrinsic curvature on
the boundary is fixed to $2\pi$,

$\oint_{\partial\mathscr{D}}k{\rm d}s=2\pi\,.$ (2.15)

In positive ($\eta=+1$) and negative ($\eta=-1$) curvature, this integral is
directly related to the area $A$ according to

$\oint_{\partial\mathscr{D}}k{\rm d}s=2\pi-\frac{\eta}{L^{2}}A\,.$ (2.16)

###  2.2 Classical actions and counterterms

####  2.2.1 Classical actions

##### Einstein-Hilbert term

The Gauss-Bonnet formula (2.14) implies that the Einstein-Hilbert term is
purely topological in two dimensions. It has no dynamical rôle. It can be
generated as a counterterm, see below.

##### The cosmological constant term

This is

$S_{\Lambda}=\frac{\Lambda}{16\pi}\int_{\mathscr{D}}{\rm
d}^{2}x\,\sqrt{g}=\frac{\Lambda}{16\pi}A[g]\,.$ (2.17)

In Liouville gravity, classical stability implies that $\Lambda$ must be
positive, because there exist metrics of fixed boundary length that have an
arbitrary large area. The condition $\Lambda\geq 0$ is also necessary and
sufficient for the stability of the quantum theory.555Classical stability and
quantum stability are not necessarily equivalent, because in the quantum
theory one must take into account the measure, i.e. the entropy of the
configurations one integrates over in the path integral. In JT gravity in
negative and zero curvature, classical stability is ensured for any
$\Lambda\in\mathbb{R}$ thanks to the isoperimetric inequalities, which put an
upper bound on the value of the area for a given boundary length,

$\displaystyle\frac{A}{2\pi L^{2}}$
$\displaystyle\leq-1+\sqrt{1+\Bigl{(}\frac{\ell}{2\pi
L}\Bigr{)}^{2}}\quad\text{if $R=-\frac{2}{L^{2}}$}\,\raise 2.0pt\hbox{,}$
(2.18) $\displaystyle A$ $\displaystyle\leq\frac{\ell^{2}}{4\pi}\quad\text{if
$R=0$}\,.$ (2.19)

In JT gravity in positive curvature, the theory is classically non-
perturbatively unstable if $\Lambda<0$, as explained in [14]. We shall discuss
perturbative stability in the context of the Zamolodchikov limit (1.7) in
Section 4.3.

##### The dilaton, or Lagrange multiplier, term

In JT gravity, the constraint of constant bulk curvature is imposed by a
Lagrange mutiplier term in the action. The Lagrange multiplier field is also
called in this context the “dilaton.” The associated term is traditionally
written as

$S_{\text{dil}}=-\frac{1}{16\pi}\Biggl{[}\int_{\mathscr{D}}\\!{\rm
d}^{2}x\sqrt{g}\,\Phi\Bigl{(}R-\frac{2\eta}{L^{2}}\Bigr{)}+2\oint_{\partial\mathscr{D}}\\!{\rm
d}s\,\Phi k\Biggr{]}\,.$ (2.20)

Note that the role of $\Phi$ is to impose $R=2\eta/L^{2}$ in the bulk, but the
boundary extrinsic curvature $k$ must be allowed to fluctuate. As a
consequence, $\Phi$ does not have boundary degrees of freedom and one imposes
Dirichlet boundary conditions

$\Phi_{|\partial\mathscr{D}}=\Phi_{\text{b}}=\text{constant.}$ (2.21)

The boundary term in (2.20) ensures that the variational principle is well-
defined with this boundary condition.666See e.g. [15] for a detailed
discussion of the boundary conditions and the variational principle in JT
gravity.

_Remark_ : the aforementioned distinction between bulk and boundary degrees of
freedom, which, following [1, 2, 3], will play a central rôle in our analysis,
may naively seem immaterial in the continuum formulation that we use in the
present paper. However, it becomes very clear in a regularized lattice version
of the models, as described in [2]. For instance, the bulk curvature $R$ is
entirely localized on bulk vertices, i.e. vertices that are in the interior of
the discretized disk, whereas the boundary vertices are associated with the
extrinsic curvature.

The dilaton term (2.20) can be conveniently rewritten as follows. Taking into
account that $R=2\eta/L^{2}$ is imposed in the bulk after integrating out
$\Phi$ and using (2.21) and the Gauss-Bonnet formula (2.14), the dilaton
action (2.20) can be replaced by an equivalent action

$S_{\text{dil}}^{\prime}=-\frac{1}{16\pi}\int_{\mathscr{D}}\\!{\rm
d}^{2}x\sqrt{g}\,\Phi\Bigl{(}R-\frac{2\eta}{L^{2}}\Bigr{)}+\frac{\eta\Phi_{\text{b}}}{8\pi
L^{2}}A-\frac{\Phi_{\text{b}}}{4}$ (2.22)

in the path integral. When $\eta\not=0$, we thus see that the cosmological
constant action (2.17) is redundant if the dilaton action is taken into
account, with the identification

$\Lambda=\frac{2\eta\Phi_{\text{b}}}{L^{2}}\,\cdotp$ (2.23)

##### The classical action for Liouville gravity coupled to a matter CFT

This is simply

$S^{\text{L}}_{\text{cl}}=S_{\Lambda}+S_{\text{CFT}}\,,$ (2.24)

where $S_{\text{CFT}}$ is the action for the matter CFT coupled to gravity.

##### The classical action for JT gravity coupled to a matter CFT

In non-zero curvature, one may use the action (2.20) plus the matter CFT
action. In zero curvature, one must then also add by hand the cosmological
constant term (2.17). Equivalently, and more conveniently, we can use the
action (2.22), supplemented with the cosmological constant term in zero
curvature. In all cases, and discarding the trivial constant
$-\Phi_{\text{b}}/4$ in (2.22),777This constant is not physical because it can
be absorbed in a counterterm, see below. we can work with

$S^{\text{JT}}_{\text{cl}}=-\frac{1}{16\pi}\int_{\mathscr{D}}\\!{\rm
d}^{2}x\sqrt{g}\,\Phi\Bigl{(}R-\frac{2\eta}{L^{2}}\Bigr{)}+S_{\Lambda}+S_{\text{CFT}}\,.$
(2.25)

####  2.2.2 Counterterms

As usual, the list of possible counterterms is established by looking at all
the possible local terms that respect diffeomorphism invariance and are
associated with a coupling of positive mass dimension.

In the Liouville gravity theory, the only possible counterterms are the bulk
terms $\int{\rm d}^{2}x\sqrt{g}R$, $\int{\rm d}^{2}x\sqrt{g}=A$ and the
boundary terms $\int{\rm d}s=\ell$ and $\int{\rm d}sk$. Thanks to the Gauss-
Bonnet formula, the term $\int{\rm d}^{2}x\sqrt{g}R$ can be replaced by a
constant and by the boundary term $\int{\rm d}sk$. We do not need to discuss
additional counterterms associated with the matter CFT; the treatment of the
path integral over the matter CFT, which is well-known, will be recalled
below. The counterterm action is thus of the form

$S^{\text{L}}_{\text{c.t.}}=\alpha_{1}+\alpha_{2}\ell+\alpha_{3}A+\alpha_{4}\int_{\partial\mathscr{D}}{\rm
d}s\,k\,,$ (2.26)

where the coefficients $\alpha_{j}$ may depend on the short-distance lattice
cut-off $\ell_{0}$, the cosmological constant $\Lambda$ and the CFT central
charge $c$.

Finite pieces in the counterterm coefficients $\alpha_{j}$ correspond to
undetermined parameters in the theory. The coefficient $\alpha_{1}$ is fixed
by imposing the normalization condition (1.15). The coefficient $\alpha_{2}$
yields an arbitrary boundary cosmological constant factor $e^{-\lambda\ell}$,
for some finite $\lambda$, in the partition function. The coefficient
$\alpha_{3}$ renormalises the bulk cosmological constant $\Lambda$. Finally,
at the one-loop order that we consider in the present paper, the effect of
$\alpha_{4}$ can be absorbed in $\alpha_{1}$ and $\alpha_{3}$, see below.

In JT gravity, all the counterterms of the Liouville theory are of course
allowed as well, but $\alpha_{4}$ can always be absorbed in $\alpha_{1}$ and
$\alpha_{3}$ thanks to the constant bulk curvature constraint. However, based
on power counting, more general counterterms of the form

$\int\\!{\rm
d}^{2}x\sqrt{g}\,g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi\,,\quad\int\\!{\rm
d}^{2}x\sqrt{g}\,V(\Phi)\,,\quad\int\\!{\rm d}^{2}x\sqrt{g}\,W(\Phi)R\,,$
(2.27)

for arbitrary polynomials $V$ and $W$, may also a priori be generated. If a
kinetic term for $\Phi$ or non-linear terms in $V$ and $W$ were generated by
quantum corrections, this would imply that $\Phi$ is no longer a Lagrange
multiplier field and thus that the constraint of constant bulk curvature
cannot be enforced at the quantum level. In other words, JT gravity, defined
as a theory of random constant curvature metrics, would not make sense quantum
mechanically. This, of course, goes against all evidence, in particular the
fact that a microscopic, UV-complete definition of the models exists [1, 2,
3]. The only new allowed counterterms are thus $\int{\rm d}^{2}x\sqrt{g}\Phi$
and $\int{\rm d}^{2}x\sqrt{g}\Phi R$. In the flat theory, the second term is
absent and the first forbidden, since it would modify the $R=0$ constraint to
$R\not=0$. In the curved theories, the two terms are equivalent since $R$ is
fixed. They correspond to a renormalization of $L$. The counterterm action in
the JT models will thus take the general form

$S^{\text{JT}}_{\text{c.t.}}=\alpha_{1}+\alpha_{2}\ell+\alpha_{3}A+|\eta|\alpha_{4}\int_{\mathscr{D}}{\rm
d}^{2}x\sqrt{g}\,\Phi\,.$ (2.28)

###  2.3 Path integrals

####  2.3.1 Formal definition

The partition functions we wish to compute have a formal path integral
representation of the form

$Z=\frac{1}{\operatorname{\text{Vol}}(\operatorname{Diff}(\mathscr{D}))}\int_{\mathscr{M}_{\ell}}Dg\,Z_{\text{CFT}}[g]\,e^{-S_{\Lambda}[g]}\,.$
(2.29)

The action $S_{\Lambda}$ is the cosmological constant term given in (2.17).
The CFT partition function $Z_{\text{CFT}}[g]$ is obtained after integrating
out the matter CFT degrees of freedom in a fixed background metric $g$. In the
case of the Liouville theory, the space $\mathscr{M}_{\ell}$ is the full space
of metrics on the disk $\mathscr{D}$, with a fixed boundary length $\ell$. In
the case of the JT models, it is the space of constant bulk curvature metrics,
$R=2\eta/L^{2}$, with a fixed boundary length $\ell$. The measure $Dg$ is a
formal ultralocal integration measure over these spaces. The prefactor
$1/\operatorname{\text{Vol}}(\operatorname{Diff}(\mathscr{D}))$ indicates
formally that the gauge redundancy associated with the group of disk
diffeomorphisms must be fixed.

It is important to note that, in the full quantum theory, the very notion of
geometric bulk area or boundary length must be revisited, since the typical
quantum geometry is not smooth. For instance, it is emphasized in [1, 2, 3]
that, in JT gravity, the boundary is a fractal, of non-trivial Hausdorff
dimension $d_{\text{H}}=1/\nu$ given by (1.1), and thus its geometric length
$\ell$ is ill-defined. The length parameter is replaced by a quantum length
parameter $\beta$ which is such that $\beta^{\nu}$ has the dimension of
length. As already discussed in the introduction, it does make sense to talk
about the smooth boundary length $\ell$ in the framework of the
$c\rightarrow-\infty$ expansion, since the limit is dominated by a smooth
geometry for which $\nu=1$, consistently with the expansion (1.8) at leading
order. One-loop corrections are expected to produce an anomalous dimension for
$\ell$ and this is what we shall find, see the discussion in Sec. 6.

####  2.3.2 Conformal gauge

To make (2.29) more precise, and as advocated in [1, 3] in the case of JT
gravity, we go to conformal gauge, in which the metric takes the form

$g=e^{2\Sigma}\delta\,,$ (2.30)

where $\delta$ is the round flat metric defined in Eq. (2.4). This can be done
without loss of generality, because any metric on the disk can be put in the
form (2.30) by acting with a disk diffeomorphism. We call
$\Sigma:\bar{\mathscr{D}}\rightarrow\mathbb{R}$ the “Liouville field” and we
note

$\sigma=\Sigma_{|\partial\mathscr{D}}$ (2.31)

its boundary value.

The crucial point, emphasized in [1, 3], is that the Liouville field is not
constrained by boundary conditions on $\partial\mathscr{D}$. One says that
$\Sigma$ has “free” boundary conditions.888This terminology might be a bit
misleading, since free boundary conditions really means no boundary condition
at all. Actually, in the case of JT gravity, the degrees of freedom are
entirely described by the boundary Liouville field $\sigma$ [1, 3] and
constraining this field in any way (Neumann, Dirichlet or any other way) would
be totally inconsistent.999In the case of the Liouville theory, for which both
the bulk and the boundary Liouville fields fluctuate, a subtle coincidence
makes the free and the Neumann boundary conditions equivalent, see e.g. the
discussion in [3]. But we insist that, from first principles, the correct
boundary conditions are the free boundary conditions, for both the usual
Liouville model and JT gravity. The bulk and boundary degrees of freedom in
the Liouville field must thus be carefully treated as independent variables.
Following [1, 3], we perform the explicit separation between the bulk and the
boundary by writing

$\Sigma=\Sigma_{\text{B}}+\Sigma_{\sigma}\,,$ (2.32)

where the bulk piece $\Sigma_{\text{B}}$ satisfies Dirichlet boundary
conditions,

$\Sigma_{\text{B}|\partial\mathscr{D}}=0\,,$ (2.33)

and the piece $\Sigma_{\sigma}$ coincides, by definition, with $\sigma$ on the
boundary, and is extended into the bulk by using a convenient prescription
that fixes the bulk values in terms of the boundary values,

$\Sigma_{\sigma|\partial\mathscr{D}}=\sigma\,,\quad\Sigma_{\sigma}(x)\
\text{depends only on the boundary $\sigma$ for all $x\in\mathscr{D}$}\,.$
(2.34)

At the conceptual level, the precise prescription for defining
$\Sigma_{\sigma}$ is, to a large extent irrelevant, but, in practice, it is
chosen to simplify the analysis of the particular problem at hand, see [1, 3]
and below.

The gauge fixing (2.30) has the effect to break the diffeomorphism gauge
redundancy down to the $\text{PSL}(2,\mathbb{R})$ group of automorphisms of
the disk, Eq. (2.3), which corresponds to the three-dimensional subgroup of
$\operatorname{Diff}(\mathscr{D})$ that maps a metric in conformal gauge to
another metric in conformal gauge.101010Strictly speaking, there is an
additional $\mathbb{Z}_{2}$ redundancy $z\mapsto\bar{z}$ that remains unbroken
in conformal gauge and which produces an overall factor of 2 in the partition
function. But overall multiplication constants in $Z$ are unphysical, because
they can be absorbed in counterterms; thus, we do not indicate them
explicitly. In conformal gauge, the formal path integral (2.29) reduces to

$Z=\frac{1}{\operatorname{\text{Vol}}(\text{PSL}(2,\mathbb{R}))}\int_{\mathscr{L}_{\text{B}}\times\mathscr{L}_{\text{b}}}D\Sigma_{\text{B}}D\sigma\,\delta\Bigl{(}\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta-\ell\Bigr{)}\,J[g]Z_{\text{gh}}[g]Z_{\text{CFT}}[g]\,e^{-S_{\Lambda}[g]}\,.$
(2.35)

The factor $Z_{\text{gh}}[g]$ is the Faddeev-Popov determinant which, as is
well-known, corresponds to the partition function of a ghost CFT of central
charge $c_{\text{gh}}=-26$. The $\delta$-function enforces explicitly the
constraint of fixed boundary length

$\int_{\partial\mathscr{D}}{\rm d}s=\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta=\ell\,,$ (2.36)

since we work in this ensemble in the present paper.111111As explained in [3],
imposing the constraint of fixed length in this precise way and not, for
instance, using
$\smash{\delta\bigl{(}\frac{1}{\ell}\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta-1\bigr{)}}$, corresponds to counting the metrics with no marked point
on the boundary. The space
$\mathscr{L}_{\text{B}}\times\mathscr{L}_{\text{b}}$ over which we integrate
is the full space of Liouville fields on the closed disk $\bar{\mathscr{D}}$,
which decomposes into a bulk piece $\mathscr{L}_{\text{B}}$ and a boundary
piece $\mathscr{L}_{\text{b}}$, following (2.32). The measures
$D\Sigma_{\text{B}}$ and $D\sigma$ are formal reparameterization invariant and
background-independent ultralocal measures that we can identify with the
volume forms associated with the metrics

$\displaystyle||\delta\Sigma_{\text{B}}||^{2}$
$\displaystyle=\int_{\mathscr{D}}{\rm
d}^{2}x\sqrt{g}\,(\delta\Sigma_{\text{B}})^{2}=\int_{0}^{1}{\rm
d}\rho\,\rho\int_{0}^{2\pi}{\rm
d}\theta\,e^{2\Sigma}(\delta\Sigma_{\text{B}})^{2}$ (2.37)
$\displaystyle||\delta\sigma||^{2}$
$\displaystyle=\int_{\partial\mathscr{D}}{\rm
d}s\,(\delta\sigma)^{2}=\int_{0}^{2\pi}{\rm
d}\theta\,e^{\sigma}(\delta\sigma)^{2}$ (2.38)

on the spaces $\mathscr{L}_{\text{B}}$ and $\mathscr{L}_{\text{b}}$
respectively. The contribution

$J[g]=\delta\Bigl{(}R-\frac{2\eta}{L^{2}}\Bigr{)}$ (2.39)

is an additional piece that enters in the case of JT gravity, enforcing the
constraint of constant bulk curvature and coming from integrating out the
dilaton field $\Phi$.

####  2.3.3 The Liouville action

The Liouville action on the disk is defined by

$S_{\text{L}}[\Sigma]=\int_{\mathscr{D}}\partial_{a}\Sigma\partial_{a}\Sigma\,{\rm
d}^{2}x+2\int_{\partial\mathscr{D}}\sigma\,{\rm d}\theta\,.$ (2.40)

The metric dependence of the disk partition function $\mathscr{Z}[g]$ of an
arbitrary CFT of central charge $\mathscr{C}$ in the gauge (2.30) is given by

$\mathscr{Z}[g]=e^{\frac{\mathscr{C}}{24\pi}S_{\text{L}}[\Sigma]}\mathscr{Z}[\delta]\,.$
(2.41)

The combined ghost plus matter system entering in (2.35) has total central
charge $\mathscr{C}=c-26$. Discarding the overall constant
$Z_{\text{gh}}[\delta]Z_{\text{CFT}}[\delta]$, we can thus rewrite (2.35) as

$Z=\frac{1}{\operatorname{\text{Vol}}(\text{PSL}(2,\mathbb{R}))}\int_{\mathscr{L}_{\text{B}}\times\mathscr{L}_{\text{b}}}D\Sigma_{\text{B}}D\sigma\,\delta\Bigl{(}\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta-\ell\Bigr{)}\,J[g]\,e^{-S_{\Lambda}[\Sigma]-\frac{26-c}{24\pi}S_{\text{L}}[\Sigma]}\,.$
(2.42)

####  2.3.4 Gauge fixing $\text{PSL}(2,\mathbb{R})$

The $\text{PSL}(2,\mathbb{R})$ group of automorphisms of the disk being non-
compact, it is necessary to impose additional gauge conditions, that
supplement the conformal gauge condition (2.30), to make sense of (2.42). As
explained in detail in [3], a convenient set of gauge conditions fixing the
non-compact part of $\text{PSL}(2,\mathbb{R})$ is121212These gauge conditions
are actually globally defined, in particular, they don’t have Gribov
ambiguities.

$\int_{0}^{\frac{2\pi}{3}}e^{\sigma}{\rm
d}\theta=\int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}}e^{\sigma}{\rm
d}\theta=\frac{\ell}{3}\,\cdotp$ (2.43)

The associated Faddeev-Popov determinant is readily calculated to be
$\exp(\sigma(\frac{2\pi}{3})+\sigma(\frac{4\pi}{3}))$ [3]. Rearranging the
$\delta$-functions in a nice symmetric way, we obtain

$Z=\int_{\mathscr{L}_{\text{B}}\times\mathscr{L}_{\text{b}}}D\Sigma_{\text{B}}D\sigma\,\delta\Bigl{(}\int_{0}^{\frac{2\pi}{3}}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\delta\Bigl{(}\int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\delta\Bigl{(}\int_{\frac{4\pi}{3}}^{2\pi}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\\\
e^{\sigma(\frac{2\pi}{3})+\sigma(\frac{4\pi}{3})}J[g]\,e^{-S_{\Lambda}[\Sigma]-\frac{26-c}{24\pi}S_{\text{L}}[\Sigma]}\,.$
(2.44)

####  2.3.5 The path integral for Liouville gravity

The path integral formula for the Liouville partition function $Z_{\text{L}}$
is simply given by Eq. (2.44) with $J[g]=1$.

####  2.3.6 The path integral for JT gravity

In the case of JT gravity, the insertion of the functional $\delta$-function
(2.39) in the path integral enforces the constant bulk curvature constraint
which, in conformal gauge, reads

$R-\frac{2\eta}{L^{2}}=2e^{-2\Sigma}\Delta_{\delta}\Sigma-\frac{2\eta}{L^{2}}=0\,,$
(2.45)

where $\Delta_{\delta}=-\partial_{a}\partial_{a}$ is the positive Laplacian
for the metric $\delta=|dz|^{2}$. The Liouville field thus satisfies, in the
bulk, the so-called Liouville equation

$\Delta_{\delta}\Sigma=\frac{\eta}{L^{2}}e^{2\Sigma}\,.$ (2.46)

As reviewed in [1, 3], this equation implies that the bulk Liouville field
$\Sigma$ is uniquely expressed in terms of the boundary Liouville field
$\sigma$,

$\Sigma(x)=\Sigma[\sigma](x)\,.$ (2.47)

Note that in zero curvature, the Liouville equation is simply the Laplace
equation, which means that $\Sigma$ is a harmonic function. It is then well-
known that $\Sigma$ is uniquely specified by its boundary value $\sigma$, and
the relation (2.47) can be written down explicitly using the Poisson kernel.
This is explained and fully exploited in [1, 3]. In non-zero curvature, the
Liouville equation is non-linear and thus much more complicated than the
Laplace equation. However, the crucial fact that its solution is uniquely
specified in terms of the boundary data is still valid [16], even though a
closed-form formula expressing $\Sigma$ in terms of $\sigma$ no longer exists.
This fundamental result shows that, in JT gravity, the degrees of freedom are
entirely encoded in $\sigma$ alone, a fundamental feature already stressed
above.

This being understood, it is clear [1, 3] that the most natural choice in the
decomposition (2.32) in Liouville gravity is obtained by setting

$\Sigma_{\sigma}=\Sigma[\sigma]\,.$ (2.48)

The $\delta$-function in (2.39) then imposes $\Sigma_{\text{B}}=0$. The path
integral over $\Sigma_{\text{B}}$ is then evaluated straightforwardly by
expanding the argument of the $\delta$-function to linear order in
$\Sigma_{\text{B}}$,

$R-\frac{2\eta}{L^{2}}=2\Bigl{(}\Delta_{\sigma}-\frac{2\eta}{L^{2}}\Bigr{)}\Sigma_{\text{B}}+O(\Sigma_{\text{B}}^{2})\,,$
(2.49)

where $\Delta_{\sigma}$ is the positive Laplacian for the constant curvature
metric with boundary Liouville field $\sigma$. We thus get

$J[g]=\frac{\delta(\Sigma_{\text{B}})}{2|\det_{\text{D}}(\Delta_{\sigma}-\frac{2\eta}{L^{2}})|}$
(2.50)

where $\det_{\text{D}}(\Delta_{\sigma}-\frac{2\eta}{L^{2}})$ denotes the
functional determinant of $\Delta_{\sigma}-\frac{2\eta}{L^{2}}$ with Dirichlet
boundary conditions, consistently with (2.33). Integrating out
$\Sigma_{\text{B}}$ in (2.44) thus yields a simplified formula for the JT
gravity partition function in terms of a path integral over the boundary
Liouville field only,

$Z^{(\eta)}=\int_{\mathscr{L}_{\text{b}}}D\sigma\,\delta\Bigl{(}\int_{0}^{\frac{2\pi}{3}}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\delta\Bigl{(}\int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\delta\Bigl{(}\int_{\frac{4\pi}{3}}^{2\pi}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}\Bigr{)}\\\
e^{\sigma(\frac{2\pi}{3})+\sigma(\frac{4\pi}{3})}\,\frac{e^{-S_{\Lambda}[\Sigma_{\sigma}]-\frac{26-c}{24\pi}S_{\text{L}}[\Sigma_{\sigma}]}}{|\det_{\text{D}}(\Delta_{\sigma}-\frac{2\eta}{L^{2}})|}\,\cdotp$
(2.51)

###  2.4 One-loop path integrals

####  2.4.1 The case of Liouville gravity

We now focus on a one-loop calculation in the limit (1.7). In this limit, it
is natural to group together the terms proportional to $|c|$ in the action,
writing

$S_{\Lambda}+\frac{26-c}{24\pi}S_{\text{L}}=\frac{|c|}{24\pi}S_{\text{eff}}+\frac{13}{12\pi}S_{\text{L}}$
(2.52)

for an effective action

$S_{\text{eff}}[\Sigma]=S_{\text{L}}[\Sigma]+\mu\int_{\mathscr{D}}e^{2\Sigma}\,{\rm
d}^{2}x=\int_{\mathscr{D}}\bigl{(}\partial_{a}\Sigma\partial_{a}\Sigma+\mu
e^{2\Sigma}\bigr{)}\,{\rm d}^{2}x+2\int_{\partial\mathscr{D}}\sigma\,{\rm
d}\theta\,.$ (2.53)

Recall that $\mu$ was defined in (1.7). The path integral is dominated by a
saddle, computed in Section 3, that we note $\Sigma_{*}$. More generally, we
note with a $*$ any quantity evaluated on the saddle. We expand around the
saddle by writing

$\Sigma=\Sigma_{*}+\frac{\Xi}{\sqrt{|c|}}$ (2.54)

which, using the decomposition (2.32), is equivalent to expanding

$\Sigma_{\text{B}}=\Sigma_{\text{B}*}+\frac{\Xi_{\text{B}}}{\sqrt{|c|}}\,\raise
2.0pt\hbox{,}\quad\sigma=\sigma_{*}+\frac{\xi}{\sqrt{|c|}}\,\cdotp$ (2.55)

There are then two equivalent ways to deal with the $\delta$-functions in the
path integral (2.44). A first way amounts to using the integral representation
$\delta(x)=\frac{1}{2\pi}\int e^{i\lambda x}{\rm d}\lambda$ in terms of a
Lagrange multiplier $\lambda$. One introduces a Lagrange multiplier for each
$\delta$-function, adds the associated terms to the effective action (2.53)
and, together with the fields $\Sigma_{\text{B}}$ and $\sigma$, expands the
Lagrange multipliers around their saddle point values. The advantage of this
approach is that, by construction, the expansion of the effective action does
not contain linear terms. It can also be used systematically to any loop
order.

A second way, which is a bit more straightforward at one-loop, is to deal
directly with the $\delta$ functions, without introducing Lagrange
multipliers. The expansion of the effective action (2.53), to the order we
need for a one-loop calculation, has the form

$S_{\text{eff}}[\Sigma]=S_{\text{eff}*}+\frac{1}{|c|^{1/2}}S_{\text{eff}}^{(1)}[\xi]+\frac{1}{|c|}S_{\text{eff}}^{(2)}[\Xi_{\text{B}},\xi]+O\bigl{(}1/|c|^{3/2}\bigr{)}\,,$
(2.56)

where the linear term $S_{\text{eff}}^{(1)}$ turns out to depend on the
boundary field fluctuations $\xi$ only.131313This is easy to check, see
Section 5. The arguments of the $\delta$ functions are themselves expanded as

$\int_{\frac{2\pi k}{3}}^{\frac{2\pi(k+1)}{3}}e^{\sigma}{\rm
d}\theta-\frac{\ell}{3}=\frac{1}{\sqrt{|c|}}\Biggl{[}\int_{\frac{k\ell}{3}}^{\frac{(k+1)\ell}{3}}\xi\,ds_{*}+\frac{1}{2\sqrt{|c|}}\int_{\frac{k\ell}{3}}^{\frac{(k+1)\ell}{3}}\xi^{2}\,ds_{*}+O\bigl{(}1/|c|\bigr{)}\Biggr{]}\,.$
(2.57)

At one-loop, the $\delta$-function constraints thus ensure that we can make
the substitutions

$\int_{\frac{k\ell}{3}}^{\frac{(k+1)\ell}{3}}\xi\,ds_{*}\longmapsto-\frac{1}{2|c|^{1/2}}\int_{\frac{k\ell}{3}}^{\frac{(k+1)\ell}{3}}\xi^{2}\,ds_{*}$
(2.58)

in the effective action (2.56) without changing the value of the path
integral. This allows to kill all the linear terms in the expansion. At the
one-loop order, we are left with a purely quadratic action that we denote by
$\tilde{S}^{(2)}_{\text{eff}}$. The quadratic or higher order terms in the
$\delta$-functions can then be dropped.

Moreover, still at one-loop, the non-linear metrics (2.37), (2.38) can be
replaced by the usual linear metrics

$||\delta\Xi_{\text{B}}||_{*}^{2}=\frac{1}{|c|}\int_{\mathscr{D}}{\rm
d}^{2}x\sqrt{g_{*}}\,(\delta\Xi_{\text{B}})^{2}\,,\quad||\delta\xi||_{*}^{2}=\frac{1}{|c|}\int_{\partial\mathscr{D}}{\rm
d}s_{*}\,(\delta\xi)^{2}\,,$ (2.59)

which yield the measures $D_{*}\Xi_{\text{B}}$ and $D_{*}\xi$. Putting
everything together in (2.44), we finally get

$\ln Z_{\text{L}}=\ln Z_{\text{L},\,\text{tree}}+\ln
Z_{\text{L},\,\text{1-loop}}+O(1/|c|)$ (2.60)

with

$Z_{\text{L},\,\text{tree}}=e^{-\frac{|c|}{24\pi}S_{\text{eff}*}}$ (2.61)

and

$Z_{\text{L},\,\text{1-loop}}=e^{-\frac{13}{12\pi}S_{\text{L}*}+\sigma_{*}(\frac{2\pi}{3})+\sigma_{*}(\frac{4\pi}{3})}\\\
\int
D_{*}\Xi_{\text{B}}D_{*}\xi\,\delta\Bigl{(}\int_{0}^{\frac{\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{\ell}{3}}^{\frac{2\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{2\ell}{3}}^{\ell}\xi\,{\rm
d}s_{*}\Bigr{)}\,e^{-\frac{1}{24\pi}\tilde{S}^{(2)}_{\text{eff}}[\Xi_{\text{B}},\xi]}\,.$
(2.62)

####  2.4.2 The case of JT gravity

The same analysis can be repeated for JT gravity, starting from (2.51). The
main difference with the Liouville gravity case is that we only expand the
boundary field

$\sigma=\sigma_{*}+\frac{\xi}{\sqrt{|c|}}\,\raise 2.0pt\hbox{,}$ (2.63)

the bulk fluctuations being fixed in terms of $\xi$ as explained in subsection
(2.3.6). We obtain

$\ln Z^{(\eta)}=\ln Z^{(\eta)}_{\text{tree}}+\ln
Z^{(\eta)}_{\text{1-loop}}+O(1/|c|)$ (2.64)

with

$Z^{(\eta)}_{\text{tree}}=e^{-\frac{|c|}{24\pi}S_{\text{eff}*}}$ (2.65)

and

$Z^{(\eta)}_{\text{1-loop}}=\frac{e^{-\frac{13}{12\pi}S_{\text{L}*}+\sigma_{*}(\frac{2\pi}{3})+\sigma_{*}(\frac{4\pi}{3})}}{|\det_{\text{D}}(\Delta_{*}-\frac{2\eta}{L^{2}})|}\\\
\int D_{*}\xi\,\delta\Bigl{(}\int_{0}^{\frac{\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{\ell}{3}}^{\frac{2\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{2\ell}{3}}^{\ell}\xi\,{\rm
d}s_{*}\Bigr{)}e^{-\frac{1}{24\pi}\tilde{S}^{(2)}_{\text{eff}}[\xi]}\,.$
(2.66)

The similarity between (2.62) and (2.66) is startling, but there is also a
fundamental difference: in Liouville gravity, we have to integrate over both
bulk and boundary metric fluctuations, whereas in the case of JT only the
boundary degrees of freedom fluctuate.

We now turn to the explicit evaluation of the path integrals (2.62) and
(2.66). The saddle-point solutions and the associated tree-level contributions
are given in the next Section. The computation of the quadratic fluctuations
$\tilde{S}^{(2)}_{\text{eff}}$ is presented in Section 5. The calculation of
the associated Gaussian path integrals, which amounts to evaluating some
functional determinants, is detailed in Section 5.2, using in particular
results that are published separately [17].

##  3 Leading order: classical solutions

###  3.1 Equations of motion

From the discussion of the previous section, we know that, at leading order in
the $c\rightarrow-\infty$ limit, the path integral is dominated by the minimum
of the effective action (2.53), taking into account the constraint of fixed
boundary length

$\int_{0}^{2\pi}e^{\sigma}{\rm d}\theta=\ell$ (3.1)

and, in the case of JT gravity, the additional constraint of constant bulk
curvature

$R=-2e^{-2\Sigma}\partial_{a}\partial_{a}\Sigma=\frac{2\eta}{L^{2}}\,\raise
2.0pt\hbox{,}\quad\text{$\eta=-1$, $1$ or $0$}\,.$ (3.2)

To impose these constraints at the classical level, it is convenient to
introduce Lagrange multipliers $\zeta$ and $\varphi$. We thus use the
classical action

$S=S_{\text{eff}}[\Sigma]+\zeta\Bigl{(}\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta-\ell\Bigr{)}+\vartheta\int_{\mathscr{D}}{\rm
d}^{2}x\,\varphi\Bigl{(}\partial_{a}\partial_{a}\Sigma+\frac{\eta}{L^{2}}e^{2\Sigma}\Bigr{)}\,,$
(3.3)

where $\vartheta=0$ in the case of the Liouville theory and $\vartheta=1$ in
the case of the JT theory. The Lagrange multiplier field $\varphi$ satisfies
Dirichlet boundary conditions

$\varphi_{|\partial\mathscr{D}}=0\,,$ (3.4)

since it imposes the constant curvature constraint in the bulk only. The
Liouville field $\Sigma$ has free boundary conditions, as already emphasized
several times above. The variation of (3.3) with respect to $\Sigma$ reads

$\delta S=\int_{\mathscr{D}}{\rm
d}^{2}x\,\biggl{[}-2\partial_{a}\partial_{a}\Sigma+2\Bigl{(}\mu+\frac{\vartheta\eta}{L^{2}}\varphi\Bigr{)}e^{2\Sigma}+\vartheta\partial_{a}\partial_{a}\varphi\biggr{]}\delta\Sigma\\\
+\delta\zeta\Bigl{(}\int_{0}^{2\pi}e^{\sigma}{\rm
d}\theta-\ell\Bigr{)}+\vartheta\int_{\mathscr{D}}{\rm
d}^{2}x\,\delta\varphi\Bigl{(}\partial_{a}\partial_{a}\Sigma+\frac{\eta}{L^{2}}e^{2\Sigma}\Bigr{)}\\\
+\int_{0}^{2\pi}{\rm
d}\theta\,\biggl{[}\Bigl{(}2\frac{\partial\Sigma}{\partial\rho}(\rho=1,\theta)-\vartheta\frac{\partial\varphi}{\partial\rho}(\rho=1,\theta)+\zeta
e^{\sigma}+2\Bigr{)}\delta\sigma+\vartheta\varphi\frac{\partial\delta\Sigma}{\partial\rho}(\rho=1,\theta)\biggr{]}\,.$
(3.5)

From this we derive the field equations for the saddle point solution, which
is denoted with a $*$ consistently with our previous conventions,

$\displaystyle\partial_{a}\partial_{a}\Sigma_{*}=\frac{1}{2}\vartheta\partial_{a}\partial_{a}\varphi_{*}+\Bigl{(}\mu+\frac{\vartheta\eta}{L^{2}}\varphi_{*}\Bigr{)}e^{2\Sigma_{*}}\,,$
(3.6)
$\displaystyle\frac{\partial\Sigma_{*}}{\partial\rho}(\rho=1,\theta)+1=\frac{1}{2}\vartheta\frac{\partial\varphi_{*}}{\partial\rho}(\rho=1,\theta)-\frac{1}{2}\zeta_{*}e^{\sigma_{*}}\,,$
(3.7)

together with the constraint (3.1) and, for JT, the additional constraint
(3.2). Note that Eq. (3.7) comes from the boundary term in (3.5), which does
not automatically vanish because the Liouville field has free boundary
conditions. The fact that boundary terms in the variation of the action yield
additional equations of motion may seem unfamiliar. Traditionally, one imposes
boundary conditions precisely in such a way that the boundary terms
identically vanish. However, this is a general feature of the variational
principle in conformal gauge and it is perfectly consistent. A detailed
discussion of this point is provided in [3].

The analysis of the equations of motion (3.6) and (3.7) are simplified by
rewritting them in a more geometric way. Note that the bulk curvature $R$ of
the metric $g=e^{2\Sigma}\delta$ is expressed in terms of $\Sigma$ as in Eq.
(3.2). Similarly, the extrinsic curvature of the disk boundary for the same
metric $g$ is given by

$k=e^{-\sigma}\Bigl{(}\frac{\partial\Sigma}{\partial\rho}(\rho=1,\theta)+1\Bigr{)}\,.$
(3.8)

Note also that the positive Laplacian in the metric $g$ is

$\Delta=-e^{-2\Sigma}\partial_{a}\partial_{a}$ (3.9)

and the normal derivative on the boundary

$\frac{\partial}{\partial n}=e^{-\sigma}\frac{\partial}{\partial\rho}\,\cdotp$
(3.10)

Equipped with these formulas, we find that (3.6) and (3.7) are equivalent to

$\displaystyle R_{*}$
$\displaystyle=-2\mu+\vartheta\Bigl{(}\Delta_{*}-\frac{2\eta}{L^{2}}\Bigr{)}\varphi_{*}\,,$
(3.11) $\displaystyle k_{*}$
$\displaystyle=-\frac{1}{2}\zeta_{*}+\frac{1}{2}\vartheta\frac{\partial\varphi_{*}}{\partial
n_{*}}\,\cdotp$ (3.12)

Finally, let us mention that the equations of motion in conformal gauge always
have a family of solutions related by the $\text{PSL}(2,\mathbb{R})$
automorphisms of the disk. This redundancy is fixed by further imposing the
gauge conditions (2.43).

Now let us turn to the analysis of the different cases we are interested in.

###  3.2 Solution for Liouville gravity

Recall that $\mu\geq 0$ in Liouville gravity. We choose $\mu>0$, since the
case $\mu=0$ can be obtained by taking the limit $\mu\rightarrow 0^{+}$. Eq.
(3.11), $R_{*}=-2\mu$, implies that the solution has constant negative
curvature. Locally, it is given by a patch of hyperbolic space $\text{H}^{2}$.
Eq. (3.12) implies that $k_{*}=-\frac{1}{2}\zeta_{*}$ is also a constant. The
disk boundary can thus be seen as a curve of constant extrinsic curvature in
$\text{H}^{2}$ whose interior must be the disk. Together with the conditions
(2.43), this identifies uniquely the metric to be $g=\delta_{\ell}^{-}$ given
in Eq. (2.6), with $L^{2}=1/\mu$. Note that one may also use a more pedestrian
approach to solve the equations of motion, starting from a radial ansatz
$\Sigma=\Sigma(\rho)$ and directly solving the resulting differential
equations. Explicitly, using the parameter $x$ defined in (1.12), we have

$\Sigma_{*}=\ln\frac{2r_{0}}{\sqrt{\mu}}-\ln\bigl{(}1-r_{0}^{2}\rho^{2}\bigr{)}$
(3.13)

for

$r_{0}=\frac{1}{x}\bigl{(}-1+\sqrt{1+x^{2}}\bigr{)}\,.$ (3.14)

It is then straightforward to evaluate, on the saddle point solution, the
area, the Liouville action,

$A_{*}=\frac{4\pi}{\mu}\frac{r_{0}^{2}}{1-r_{0}^{2}}=\frac{\ell
r_{0}}{\sqrt{\mu}}\,\raise 2.0pt\hbox{,}\quad
S_{\text{L}*}=\ell\sqrt{\mu}\,r_{0}+4\pi\ln\frac{2r_{0}}{\sqrt{\mu}}\,\raise
2.0pt\hbox{,}$ (3.15)

and the effective action

$S_{\text{eff}*}=S_{\text{L}*}+\mu
A_{*}=2\ell\sqrt{\mu}\,r_{0}+4\pi\ln\frac{2r_{0}}{\sqrt{\mu}}\\\
=4\pi\ln\frac{\ell}{2\pi}+4\pi\Bigl{[}\sqrt{1+x^{2}}-1-\ln\frac{1+\sqrt{1+x^{2}}}{2}\Bigr{]}\,.$
(3.16)

Using (2.61), this yields the tree-level partition function $\ln
Z_{\text{L},\,\text{tree}}(\ell,\mu)=-\frac{|c|}{24\pi}S_{\text{eff}*}=|c|f_{0}$,
the tree-level coefficient $f_{0}$ being precisely as in (1.17).

###  3.3 Solutions for JT gravity

The constraint (3.2) implies that, locally, the solution is given by a patch
of hyperbolic space, Euclidean space or the round two-sphere, if $\eta=-1$,
$\eta=0$ or $\eta=+1$, respectively. The equations of motion (3.11) and (3.12)
are then equivalent to

$\displaystyle\Bigl{(}\Delta_{*}-\frac{2\eta}{L^{2}}\Bigr{)}\varphi_{*}=2\mu+\frac{2\eta}{L^{2}}\,\raise
2.0pt\hbox{,}$ (3.17) $\displaystyle R_{*}=\frac{2\eta}{L^{2}}\,\raise
2.0pt\hbox{,}\quad
k_{*}=-\frac{1}{2}\zeta_{*}+\frac{1}{2}\frac{\partial\varphi_{*}}{\partial
n_{*}}\,\raise 2.0pt\hbox{,}$ (3.18)

together with the boundary length constraint (3.1).

Let us note that, if we had not included the Liouville action term in the
effective action (2.53), the equations of motion would take the simplified
form

$\displaystyle\Bigl{(}\Delta_{*}-\frac{2\eta}{L^{2}}\Bigr{)}\varphi_{*}=2\mu\,,$
(3.19) $\displaystyle R_{*}=\frac{2\eta}{L^{2}}\,\raise
2.0pt\hbox{,}\quad\frac{\partial\varphi_{*}}{\partial n_{*}}=\zeta_{*}\,.$
(3.20)

One can straightforwardly check that, as expected, these equations also follow
from varying the traditional dilaton action (2.20) in conformal gauge, with
the identification

$\Phi=\frac{|c|}{3}\bigl{(}\eta\mu L^{2}+\varphi\bigr{)}\,.$ (3.21)

The solutions to the JT classical equations of motion are well-known, even
though their analysis in conformal gauge, which is presented in [3], may be
unfamiliar. They imply that
$\smash{\frac{\epsilon^{\mu\nu}}{\sqrt{g_{*}}}\partial_{\nu}\varphi}$ is a
Killing vector of the metric.141414In conformal gauge, this is obtained by
combining Eq. (3.19) and the fact that $\varphi_{*}$ satisfies both Dirichlet,
Eq. (3.4), and Neumann, second equation in (3.20), boundary conditions, see
[3] for detailed explanations. The form of the Killing vectors in constant
curvature are of course well-known and, together with the Dirichlet boundary
condition (3.4) and the fact that the boundary must be topologically
$\text{S}^{1}$, this fixes the classical metrics to be $\delta_{\ell}^{-}$,
$\delta_{\ell}$ or one of the two $\delta_{\ell}^{+}$, as given by Eq. (2.6),
(2.5) and (2.10), when $\eta=-1$, $0$ or $+1$, respectively. In particular, in
positive curvature, a smooth classical solution exists if and only if
$\ell\leq 2\pi L$.

When the Liouville action is taken into account, the Neumann constraint
$\partial\varphi_{*}/\partial n_{*}=\text{constant}$ on the boundary is
replaced by the second equation in (3.18). This more general equation allows a
priori for a normal derivative varying along the boundary, and thus also a
varying extrinsic curvature. Nevertheless, it is clear that the classical
solutions for the metric in the JT theory will remain solutions of our more
general set of equations of motion. Indeed, if we _assume_ that
$\partial\varphi_{*}/\partial n_{*}$ is a constant, then clearly the two sets
of equations of motion (3.17), (3.18) and (3.19), (3.20) are equivalent, up to
trivial constant shifts in $\varphi_{*}$ and $\zeta_{*}$.

The fact that the equations (3.17) and (3.18) may have more general solutions,
breaking the $\text{U}(1)$ symmetry of the round metrics $\delta_{\ell}$ or
$\delta_{\ell}^{\pm}$, is an interesting possibility. In particular, one may
expect that such solutions become relevant in the positive curvature case when
$\ell>2\pi L$.151515The breaking of the $\text{U}(1)$ invariance in this
context is reminiscent of a Gross-Witten phase transition [3]. However, for
the purpose of the present work, we shall assume that the relevant solutions
are the symmetric ones. In particular, we limit ourselves to the case
$\ell\leq 2\pi L$ in the positive curvature theory. A discussion of the local
and global stability of the symmetric solutions will be presented below,
especially in Section 5.

To summarize, the saddle-point solutions and saddle point values of the action
are given as follows.

##### Negative curvature

The metric being $g_{*}=\delta_{\ell}^{-}$ as in Eq. (2.6), we get

$\displaystyle\Sigma_{*}^{-}$
$\displaystyle=\ln(2Lr_{0})-\ln\bigl{(}1-r_{0}^{2}\rho^{2}\bigr{)}$ (3.22)
$\displaystyle\varphi_{*}^{-}$ $\displaystyle=\frac{2\mu
L^{2}r_{0}^{2}}{1+r_{0}^{2}}\frac{1-\rho^{2}}{1-r_{0}^{2}\rho^{2}}\,\raise
2.0pt\hbox{,}$ (3.23)

with $r_{0}$ given in (2.8). This yields

$S_{\text{L}*}^{-}=\frac{\ell}{L}\,r_{0}+4\pi\ln(2Lr_{0})\,.$ (3.24)

It is convenient to introduce the dimensionless parameters

$x=\frac{\ell}{2\pi L}\,\raise 2.0pt\hbox{,}\quad y=\mu L^{2}\,.$ (3.25)

Using the formula (2.9) for the area, we get
$S_{\text{eff}*}^{-}=S_{\text{L}*}^{-}+\mu A_{*}^{-}$, which yields the tree-
level partition function $\ln
Z^{-}_{\text{tree}}=-\frac{|c|}{24\pi}S_{\text{eff}*}^{-}=|c|f^{-}_{0}$, with
the tree-level coefficient $f^{-}_{0}$ given by

$f_{0}^{-}=-\frac{1}{6}\ln\frac{\ell}{2\pi}+\frac{1}{6}\ln\frac{1+\sqrt{1+x^{2}}}{2}-\frac{1}{12}(1+y)\bigl{(}\sqrt{1+x^{2}}-1\bigr{)}\,.$
(3.26)

##### Zero curvature

The metric is the flat disk (2.5), which yields

$\displaystyle\Sigma_{*}^{0}=\ln\frac{\ell}{2\pi}\,\raise
2.0pt\hbox{,}\quad\varphi_{*}^{0}=\frac{\mu\ell^{2}}{8\pi^{2}}(1-\rho^{2})\,,$
(3.27) $\displaystyle S_{\text{L}*}^{0}=4\pi\ln\frac{\ell}{2\pi}\,\raise
2.0pt\hbox{,}\quad A_{*}^{0}=\frac{\ell^{2}}{4\pi}$ (3.28)

and thus $\ln
Z_{\text{tree}}^{0}=-\frac{|c|}{24\pi}S_{\text{eff}*}^{0}=-\frac{|c|}{24\pi}(S_{\text{L}*}^{0}+\mu
A_{*}^{0})=|c|f_{0}^{0}$ for a tree-level coefficient $f_{0}^{0}$ given by

$f_{0}^{0}=-\frac{1}{6}\ln\frac{\ell}{2\pi}-\frac{\mu\ell^{2}}{96\pi^{2}}\,\cdotp$
(3.29)

Figure 2: The critical line $y_{\text{c}}(x)$, defined by Eq. (3.35),
separating the region where the small disk dominates (above the curve) and the
region where large disk dominates (below the curve).

##### Positive curvature

We find two distinct solutions in this case, corresponding to the large and
small disks depicted in Fig. 1. The metric is of the form
$g_{*}=\delta_{\ell}^{+}$ as in Eq. (2.10), i.e.

$\displaystyle\Sigma_{*}^{+}$
$\displaystyle=\ln(2Lr_{0})-\ln\bigl{(}1+r_{0}^{2}\rho^{2}\bigr{)}$ (3.30)
$\displaystyle\varphi_{*}^{+}$ $\displaystyle=\frac{2\mu
L^{2}r_{0}^{2}}{1-r_{0}^{2}}\frac{1-\rho^{2}}{1+r_{0}^{2}\rho^{2}}\,\raise
2.0pt\hbox{,}$ (3.31)

with $r_{0}$ given by one of the two formulas in (2.13). This yields

$S_{\text{L}*}^{+}=-\frac{\ell}{L}\,r_{0}+4\pi\ln(2Lr_{0})\,.$ (3.32)

Using the formula (2.12) for the area, we get
$S_{\text{eff}*}^{+}=S_{\text{L}*}^{+}+\mu A_{*}^{+}$; in terms of the
parameters $x$ and $y$ defined in (3.25), this is

$\displaystyle S_{\text{eff}*}^{+>}=4\pi\ln
L+2\pi(y-1)\bigl{(}1+\sqrt{1-x^{2}}\bigr{)}-4\pi\ln\frac{1-\sqrt{1-x^{2}}}{2x}$
(3.33) $\displaystyle S_{\text{eff}*}^{+<}=4\pi\ln
L+2\pi(y-1)\bigl{(}1-\sqrt{1-x^{2}}\bigr{)}-4\pi\ln\frac{1+\sqrt{1-x^{2}}}{2x}$
(3.34)

for the large and small disk solutions, respectively.

The relevant solution is the one which has the smallest effective action. One
finds that the small disk dominates if and only if

$y>y_{\text{c}}(x)=1-\frac{1}{\sqrt{1-x^{2}}}\ln\frac{1+\sqrt{1-x^{2}}}{1-\sqrt{1-x^{2}}}\,\raise
2.0pt\hbox{,}$ (3.35)

the crossover between the small and the large disk occurring when the strict
inequality in the above equation is replaced by an equality. We have plotted
in Fig. 2 the critical line $y_{\text{c}}$ as a function of $x$. The
interpretation is quite simple. If the only term present in the effective
action were the area term, then the small disk would be favoured when $y>0$
and the large disk would be favoured when $y<0$. The plot shows that the
Liouville piece in the effective action tends to stabilise the small disk,
especially for small boundary length. Note that, if $y\geq
y_{\text{c}}(1)=-1$, then the small disk always dominates, for all allowed
boundary lengths. Stability issues are further discussed in the next section.

When the small disk dominates, i.e. when $y>y_{\text{c}}$, the tree-level
partition function is given by $\ln
Z^{+}_{\text{tree}}=-\frac{|c|}{24\pi}S_{\text{eff}*}^{+<}=|c|f^{+}_{0}$, with
a tree-level coefficient $f^{+}_{0}$ given by

$f_{0}^{+}=-\frac{1}{6}\ln\frac{\ell}{2\pi}+\frac{1}{6}\ln\frac{1+\sqrt{1-x^{2}}}{2}-\frac{1}{12}(y-1)\bigl{(}1-\sqrt{1-x^{2}}\bigr{)}\,,\quad
y>y_{\text{c}}\,.$ (3.36)

Similarly, when the large disk dominates, i.e. when $y<y_{\text{c}}$, the same
tree level coefficient is given by

$f_{0}^{+}=-\frac{1}{6}\ln\frac{\ell}{2\pi}+\frac{1}{6}\ln\frac{1-\sqrt{1-x^{2}}}{2}-\frac{1}{12}(y-1)\bigl{(}1+\sqrt{1-x^{2}}\bigr{)}\,,\quad
y<y_{\text{c}}\,.$ (3.37)

##  4 One-loop order: quadratic fluctuations

###  4.1 The quadradic action $\widetilde{S}^{(2)}_{\text{eff}}$ for Liouville
gravity

We expand the effective action (2.53) to quadratic order, using (2.54), the
field equation (3.6) and the explicit solution (3.13). This yields, in the
notations of Eq. (2.56), after using an integration by part,

$\displaystyle S_{\text{eff}}^{(1)}$
$\displaystyle=\frac{2(1+r_{0}^{2})}{1-r_{0}^{2}}\int_{\partial\mathscr{D}}\xi\,{\rm
d}\theta\,,$ (4.1) $\displaystyle S_{\text{eff}}^{(2)}$
$\displaystyle=\int_{\mathscr{D}}\Big{(}\partial_{a}\Xi\partial_{a}\Xi+2\mu
e^{2\Sigma_{*}}\Xi^{2}\bigr{)}\,{\rm d}^{2}x\ .$ (4.2)

By performing the substitution indicated in (2.58), we get the purely
quadratic action

${\tilde{S}}_{\text{eff}}^{(2)}=\int_{\mathscr{D}}\Big{(}\partial_{a}\Xi\partial_{a}\Xi+2\mu
e^{2\Sigma_{*}}\Xi^{2}\bigr{)}\,{\rm
d}^{2}x-\frac{1+r_{0}^{2}}{1-r_{0}^{2}}\int_{\partial\mathscr{D}}\xi^{2}\,{\rm
d}\theta\,.$ (4.3)

We now carry out the explicit splitting between the bulk and boundary degrees
of freedom, in accordance with Eq. (2.32). We choose to define
$\Sigma_{\sigma}$ by the conditions

$\partial_{a}\partial_{a}\Sigma_{\sigma}=\mu
e^{2\Sigma_{\sigma}}\,,\quad\Sigma_{\sigma|\partial\mathscr{D}}=\sigma\,.$
(4.4)

As required, this prescription fixes uniquely $\Sigma_{\sigma}$ in terms of
the boundary field $\sigma$. The advantage of this particular choice is that
the bulk and the boundary degrees of freedom decouple in the action (4.3).
Indeed, expanding $\sigma$ as in (2.55) yields a corresponding expansion

$\Sigma_{\sigma}=\Sigma_{\sigma*}+\frac{\Xi_{\sigma}}{\sqrt{|c|}}$ (4.5)

for $\Sigma_{\sigma}$. At leading order at large $|c|$, which is all what we
need for our one-loop calculation, the conditions (4.4) are then equivalent to

$\bigl{(}\Delta_{*}+2\mu\bigr{)}\Xi_{\sigma}=\bigl{(}-e^{-2\Sigma_{*}}\partial_{a}\partial_{a}+2\mu\bigr{)}\Xi_{\sigma}=0\,,\quad\Xi_{\sigma|\partial\mathscr{D}}=\xi\,.$
(4.6)

Plugging $\Xi=\Xi_{\text{B}}+\Xi_{\sigma}$ in (4.3), and using (4.6), one then
finds that the terms proportional to $\Xi_{\text{B}}\Xi_{\sigma}$ exactly
cancel. We are left with

${\tilde{S}}_{\text{eff}}^{(2)}[\Xi_{\text{B}},\xi]={\tilde{S}}_{\text{eff,B}}^{(2)}[\Xi_{\text{B}}]+{\tilde{S}}_{\text{eff,b}}^{(2)}[\xi]\,,$
(4.7)

for purely bulk and boundary quadratic actions given by

$\displaystyle{\tilde{S}}_{\text{eff,B}}^{(2)}[\Xi_{\text{B}}]$
$\displaystyle=\int_{\mathscr{D}}{\rm
d}^{2}x\sqrt{g_{*}}\,\Xi_{B}\bigl{(}\Delta_{*}+2\mu\bigr{)}\Xi_{B}\,,$ (4.8)
$\displaystyle{\tilde{S}}_{\text{eff,b}}^{(2)}[\xi]$
$\displaystyle=\int_{\partial\mathscr{D}}\xi(\partial_{\rho}\Xi_{\sigma})_{|\partial\mathscr{D}}-\frac{1+r_{0}^{2}}{1-r_{0}^{2}}\int_{\partial\mathscr{D}}\xi^{2}\,{\rm
d}\theta\,.$ (4.9)

Note that, by definition, and consistently with (2.33), $\Xi_{\text{B}}$
satisfies Dirichlet boundary conditions,

$\Xi_{\text{B}|\partial\mathscr{D}}=0\,.$ (4.10)

The boundary action can be evaluated explicitly as follows. We expand in
Fourier modes

$\xi(\theta)=\sum_{n\in\mathbb{Z}}\xi_{n}e^{in\theta}\,,\quad\Xi_{\sigma}(\rho,\theta)=\sum_{n\in\mathbb{Z}}\Xi_{\sigma,n}(\rho)e^{in\theta}\,.$
(4.11)

Since the fields are real, we have

$(\xi_{n})^{*}=\xi_{-n}\,,\quad(\Xi_{\sigma,n})^{*}=\Xi_{\sigma,-n}\,.$ (4.12)

The conditions (4.6) yields

$\biggl{[}-\frac{(1-r_{0}^{2}\rho^{2})^{2}}{4r_{0}^{2}}\Bigl{(}\frac{{\rm
d}^{2}}{{\rm d}\rho^{2}}+\frac{1}{\rho}\frac{{\rm d}}{{\rm
d}\rho}-\frac{n^{2}}{\rho^{2}}\Bigr{)}+2\biggr{]}\Xi_{\sigma,n}(\rho)=0\,,\quad\Xi_{\sigma,n}(1)=\xi_{n}\,.$
(4.13)

This can be solved straightforwardly, imposing in particular the regularity of
the solution at the origin $\rho=0$,

$\Xi_{\sigma,n}(\rho)=\frac{1-r_{0}^{2}}{1+r_{0}^{2}+|n|(1-r_{0}^{2})}\frac{1+r_{0}^{2}\rho^{2}+|n|(1-r_{0}^{2}\rho^{2})}{1-r_{0}^{2}\rho^{2}}\rho^{|n|}\xi_{n}.$
(4.14)

Substituting in Eq. (4.9), we get

${\tilde{S}}_{\text{eff,b}}^{(2)}[\xi]=-2\pi\frac{1-r_{0}^{2}}{1+r_{0}^{2}}|\xi_{0}|^{2}+4\pi\sum_{n\geq
2}\frac{(n^{2}-1)(1-r_{0}^{2})}{1+r_{0}^{2}+n(1-r_{0}^{2})}|\xi_{n}|^{2}\,.$
(4.15)

###  4.2 The quadradic action $\widetilde{S}^{(2)}_{\text{eff}}$ for JT
gravity

As explained in Sec. 2.3.6, in the case of JT gravity, the bulk field
$\Sigma_{\text{B}}$ is set to zero and the Liouville field is entirely
determined in terms of its boundary value, $\Sigma=\Sigma_{\sigma}$, by the
condition (2.46). To compute the quadratic action, we thus proceed in two
steps. We start by expanding as usual $\Sigma=\Sigma_{*}+|c|^{-1/2}\Xi$ and
$\sigma=\sigma_{*}+|c|^{-1/2}\xi$, see Eq. (2.54) and (2.55), we express $\Xi$
in terms of $\xi$ up to terms of order $|c|^{-1/2}$, which will be needed,

$\Xi=\Xi[\xi]=\Xi^{(0)}[\xi]+\frac{\Xi^{(1)}[\xi]}{\sqrt{|c|}}+O\bigl{(}|c|^{-1}\bigr{)}\,.$
(4.16)

The expansion (2.56) of the effective action (2.53) then reads

$\displaystyle S_{\text{eff}}^{(1)}$
$\displaystyle=2\int_{\mathscr{D}}\bigl{(}\partial_{a}\Sigma_{*}\partial_{a}\Xi^{(0)}+\mu
e^{2\Sigma_{*}}\Xi^{(0)}\bigr{)}\,{\rm
d}^{2}x+2\int_{\partial\mathscr{D}}\xi\,{\rm d}\theta\,$ (4.17) $\displaystyle
S_{\text{eff}}^{(2)}$
$\displaystyle=2\int_{\mathscr{D}}\bigl{(}\partial_{a}\Sigma_{*}\partial_{a}\Xi^{(1)}+\mu
e^{2\Sigma_{*}}\Xi^{(1)}\bigr{)}\,{\rm
d}^{2}x+\int_{\mathscr{D}}\bigl{(}\partial_{a}\Xi^{(0)}\partial_{a}\Xi^{(0)}+2\mu
e^{2\Sigma_{*}}(\Xi^{(0)})^{2}\bigr{)}\,{\rm d}^{2}x$ (4.18)

and can be explicitly evaluated.

####  4.2.1 Computation of $\Xi[\xi]$

Expanding Eq. (2.46) to the desired order in the $1/|c|^{1/2}$ expansion
yields the following differential equations for $\Xi^{(0)}$ and $\Xi^{(1)}$,

$\displaystyle\Bigl{(}\Delta_{*}-\frac{2\eta}{L^{2}}\Bigr{)}\Xi^{(0)}=0\,,$
(4.19)
$\displaystyle\Bigl{(}\Delta_{*}-\frac{2\eta}{L^{2}}\Bigr{)}\Xi^{(1)}=\frac{2\eta}{L^{2}}\bigl{(}\Xi^{(0)}\bigr{)}^{2}\,.$
(4.20)

As usual, $\smash{\Delta_{*}=e^{-2\Sigma_{*}}\partial_{a}\partial_{a}}$ refers
to the positive Laplacian on the classical background geometries around which
we expand. The relevant classical solutions for $\Sigma_{*}$ were given in
Sec. 3.3. The equations (4.19) and (4.20) must be solved with the boundary
conditions

$\Xi^{(0)}_{|\partial\mathscr{D}}=\xi\,,\quad\Xi^{(1)}_{|\partial\mathscr{D}}=0$
(4.21)

and imposing regularity at $\rho=0$.

The first equation (4.19) is similar to (4.6), which was solved in (4.14).
Expanding in Fourier modes

$\xi(\theta)=\sum_{n\in\mathbb{Z}}\xi_{n}e^{in\theta}\,,\quad\Xi^{(0)}(\rho,\theta)=\sum_{n\in\mathbb{Z}}\Xi^{(0)}_{n}(\rho)e^{in\theta}\,,$
(4.22)

we find

$\Xi^{(0)}_{n}(\rho)=\frac{1+\eta r_{0}^{2}}{1-\eta r_{0}^{2}+|n|(1+\eta
r_{0}^{2})}\frac{1-\eta r_{0}^{2}\rho^{2}+|n|(1+\eta
r_{0}^{2}\rho^{2})}{1+\eta r_{0}^{2}\rho^{2}}\rho^{|n|}\xi_{n}\,.$ (4.23)

Similarly, expanding

$\Xi^{(1)}(\rho,\theta)=\sum_{n\in\mathbb{Z}}\Xi^{(1)}_{n}(\rho)e^{in\theta}\,,$
(4.24)

one finds from (4.20) that the Fourier modes $\Xi^{(1)}_{n}$ satisfy an
ordinary linear second order differential equation that may be solved by
elementary methods. In the case of the zero mode $\Xi^{(1)}_{0}$, which is, as
we shall see below, the only function we need to evaluate explicitly, the
differential equation reads

$\biggl{[}-\frac{(1+\eta r_{0}^{2}\rho^{2})^{2}}{4r_{0}^{2}}\Bigl{(}\frac{{\rm
d}^{2}}{{\rm d}\rho^{2}}+\frac{1}{\rho}\frac{{\rm d}}{{\rm
d}\rho}\Bigr{)}-2\eta\biggr{]}\Xi^{(1)}_{0}(\rho)=2\eta\sum_{n\in\mathbb{Z}}\bigl{(}\Xi^{(0)}_{n}(\rho)\bigr{)}^{2}\,.$
(4.25)

Its solution, smooth at the origin and satisfying the boundary condition
(4.21), $\Xi^{(1)}_{0}(1)=0$, is given by

$\Xi^{(1)}_{0}(\rho)=-\frac{2\eta r_{0}^{2}(1+\eta r_{0}^{2})}{1+\eta
r_{0}^{2}\rho^{2}}\\\ \sum_{n\in\mathbb{Z}}\biggl{[}\frac{1+\eta
r_{0}^{2}}{1+\eta r_{0}^{2}\rho^{2}}\,\rho^{2|n|+2}-\frac{1-\eta
r_{0}^{2}\rho^{2}}{1-\eta
r_{0}^{2}}\biggr{]}\frac{|\xi_{n}|^{2}}{\bigl{[}1-\eta r_{0}^{2}+|n|(1+\eta
r_{0}^{2})\bigr{]}^{2}}\,\cdotp$ (4.26)

For future reference, let us note that this yields

${\frac{{\rm d}\Xi^{(1)}_{0}}{{\rm d}\rho}}\bigl{(}\rho=1\bigr{)}=-\frac{4\eta
r_{0}^{2}}{1-\eta^{2}r_{0}^{4}}\sum_{n\in\mathbb{Z}}\frac{1+\eta^{2}r_{0}^{4}+|n|(1-\eta^{2}r_{0}^{4})}{\bigl{[}1-\eta
r_{0}^{2}+|n|(1+\eta r_{0}^{2})\bigr{]}^{2}}\,|\xi_{n}|^{2}\,.$ (4.27)

For $\eta=0$, the right-hand side of the above equation vanishes and for
$\eta=\pm 1$ one has $\eta^{2}=1$.

####  4.2.2 Quadratic action in the flat case

When $\eta=0$, $\Sigma_{*}=\ln\frac{\ell}{2\pi}=\text{constant}$,
$\Xi^{(0)}_{n}=\rho^{|n|}\xi_{n}$ and $\Xi^{(1)}=0$. The derivation of the
quadratic action is then particularly simple. From (4.17) and (4.18) we
immediately get

$\displaystyle S_{\text{eff}}^{(1)}$
$\displaystyle=2\Bigl{(}1+\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}\int_{0}^{2\pi}\xi\,{\rm
d}\theta\,,$ (4.28) $\displaystyle S_{\text{eff}}^{(2)}$
$\displaystyle=2\pi\sum_{n\in\mathbb{Z}}\biggl{(}|n|+\frac{\mu\ell^{2}}{4\pi^{2}}\frac{1}{|n|+1}\biggr{)}|\xi_{n}|^{2}\,.$
(4.29)

Using the rule (2.58) then yields

${\tilde{S}}_{\text{eff}}^{(2)}=-2\pi\Bigl{(}1-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}|\xi_{0}|^{2}+4\pi\sum_{n\geq
2}\frac{n-1}{n+1}\Bigl{(}n+1-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}|\xi_{n}|^{2}\,.$
(4.30)

####  4.2.3 Quadratic action in the curved cases

Integrating by part and using $\partial_{a}\partial_{a}\Sigma_{*}=-\eta
e^{2\Sigma_{*}}/L^{2}$, we first rewrite (4.17) as

$S_{\text{eff}}^{(1)}=2\Bigl{(}\mu+\frac{\eta}{L^{2}}\Bigr{)}\int_{\mathscr{D}}e^{2\Sigma_{*}}\Xi^{(0)}{\rm
d}^{2}x+2\int_{0}^{2\pi}\Bigl{(}1+\frac{{\rm d}\Sigma_{*}}{{\rm
d}\rho}(\rho=1)\Bigr{)}\xi\,{\rm d}\theta\,.$ (4.31)

Using the classical solutions (3.22), (3.30) and the solution (4.23) for
$\Xi^{(0)}$ then yields

$S_{\text{eff}}^{(1)}=\frac{2(r_{0}^{4}+2\mu
L^{2}r_{0}^{2}+1)}{1-r_{0}^{4}}\int_{0}^{2\pi}\xi\,{\rm d}\theta\,.$ (4.32)

Similarly, integrating by part, using in particular
$\Xi^{(1)}_{|\partial\mathscr{D}}=0$ and the equations of motion (4.19),
(4.20), allows to rewrite (4.18) as a purely boundary term,

$S_{\text{eff}}^{(2)}=\int_{0}^{2\pi}\biggl{[}\xi\frac{\partial\Xi^{(0)}}{\partial\rho}-\bigl{(}1+\eta\mu
L^{2}\bigr{)}\frac{\partial\Xi^{(1)}}{\partial\rho}\biggr{]}\,{\rm
d}\theta\,.$ (4.33)

By using (4.23) and (4.27), this is straightforwardly evaluated. Using (4.32)
and the substitution (2.58) then yields

${\tilde{S}}_{\text{eff}}^{(2)}=-2\pi\frac{(1+\eta r_{0}^{2})(1-2(\mu
L^{2}+2\eta)r_{0}^{2}+r_{0}^{4})}{(1-\eta r_{0}^{2})^{3}}\,|\xi_{0}|^{2}\\\
+4\pi\frac{1+\eta r_{0}^{2}}{1-\eta r_{0}^{2}}\sum_{n\geq
2}\frac{(1-r_{0}^{4})n-2(\mu L^{2}+2\eta)r_{0}^{2}+1+r_{0}^{4}}{\bigl{[}1-\eta
r_{0}^{2}+n(1+\eta r_{0}^{2})\bigr{]}^{2}}\,(n^{2}-1)|\xi_{n}|^{2}\,.$ (4.34)

Note that one gets, from either (2.8) or (2.13) (for $r_{0}=r_{0}^{<}$), that
$r_{0}\sim\frac{\ell}{4\pi L}$ in the flat space $L\rightarrow\infty$, fixed
$\ell$, limit. Then, as expected, we find the flat space result (4.30) as the
limit of (4.34), starting either from $\eta=-1$ or from $\eta=+1$ for the
small disk solution.

###  4.3 Stability analysis

Let us analyse the local stability of the classical geometries around which we
expand. Clearly, local stability is a necessary condition for the validity of
our analysis. It is equivalent to the convergence of the Gaussian integrals
over the quadratic fluctuations that we perform in the next Section to derive
the one-loop partition functions.

A first remark is in order. By inspecting the form of the quadratic
fluctuations, Eq. (4.15), (4.30) or (4.34), one notices that the boundary
modes $\xi_{\pm 1}$ are absent and that the sign in front of the zero mode
contribution $|\xi_{0}|^{2}$ may be negative. These flat directions and/or
zero mode instabilities are unphysical and harmless, because they are due to
the $\text{PSL}(2,\mathbb{R})$ gauge redundancy of conformal gauge on the
disk. It is indeed straightforward to check (see the next Section) that the
$\delta$-functions in the path integrals (2.62) or (2.66) set $\xi_{0}=0$ and
determine $\xi_{\pm 1}$ in terms of the other modes.

##### Liouville theory

In the Liouville theory, we have limited our analysis above to the case of a
non-negative cosmological constant, because the theory is classically unstable
for $\mu<0$ and the known solution, Eq. (1.13), makes sense only for $\mu\geq
0$, suggesting that the full quantum theory exists only for $\mu\geq
0$.161616The fact that the theory is ill-defined for $\mu<0$ can also be
understood by looking at the moments $\langle A^{k}\rangle$ of the area in the
$\mu=0$ theory. An obvious necessary condition for the existence of the theory
at $\mu<0$ is that these moments must be finite for all $k$. However, they are
predicted to diverge for sufficiently large values of $k$, both in the matrix
model approach and by the exact formula (1.13).

Nevertheless, let us briefly discuss the case $\mu<0$ as well. When $\ell\leq
2\pi/\sqrt{-\mu}$, one finds two classical solutions, a small and a large
disk, exactly as in the positive curvature JT theory with the identification
$L=1/\sqrt{-\mu}$. A comparison of the values of the classical action for
these two solutions shows that the small disk always dominates over the large
one. The computation of the quadratic fluctuations around these saddles,
following exactly the same strategy as in Sec. 4.1, then shows that the
dominant saddle, i.e. the small disk, is locally stable.171717One finds that
the large disk is locally unstable, due to a negative eigenvalue of the
operator $\Delta_{*}+2\mu$ appearing in the bulk action (4.8) for $\mu<0$
[17].

We thus have an explicit example of a case for which the dominant saddle is
locally stable, allowing to define a consistent semiclassical large $|c|$
expansion, but which is globally unstable and do not correspond to a well-
defined theory at the non-perturbative level.

When $\mu>0$, the bulk action (4.8) is positive-definite thanks to the
Breitenlohner-Freedman bound and the boundary action (4.15) is also manifestly
positive-definite (after setting $\xi_{0}=0$ as explained above). These ensure
that the saddle point in this case is locally stable. As we have already shown
in section 3, the computation of the classical action at the saddle point
correctly reproduces the leading term in the expansion of the FZZT result at
the $c\rightarrow-\infty$ limit. This means that the saddle that we are
considering is indeed the global minimum of the quantum effective action in
the aforementioned regime.

##### Flat space JT gravity

For flat-space JT gravity, Eq. (4.30) shows that our classical solution is
stable if and only if

$\mu\ell^{2}<24\pi^{2}\,.$ (4.35)

This instablility is easily interpreted: when $\mu$ is too large and positive,
the cosmological constant term in the action favours geometries with a
branched-polymer structure, that have very small area, over the smooth
inflated classical solution. Note that the full quantum theory is expected to
be perfectly well-defined for arbitrary positive $\mu$. It is just no longer
dominated by a smooth classical geometry. When $\mu<0$, the saddle is always
stable. This is not surprising, since we are in a regime where the geometry is
semi-classical, and the isoperimetric inequality ensures the classical
stability of the model. Note that the existence of the model for any negative
$\mu$ in the full quantum theory is a much more subtle issue, which remains to
be settled but which can be tackled, in principle, with the methods presented
in [1]; see also the discussion in [2].

##### Negative curvature JT

The situation for negative curvature JT is similar to the flat case. The
action (4.34) is positive-definite when $\xi_{0}=0$ if and only if
$3-r_{0}^{4}-2(\mu L^{2}-2)r_{0}^{2}>0$ which, using the parameters $x$ and
$y$ defined in (3.25), is equivalent to

$y<3+\frac{2}{x^{2}}\bigl{(}1+2\sqrt{1+x^{2}}\bigr{)}\,.$ (4.36)

This inequality gives the domain of stability of the saddle point, see Fig. 3.
For any given $\ell$, there is a strictly positive upper bound for the
cosmological constant. The existence of this upper bound, as well as the
stability for all negative $\mu$, can be interpreted exactly as in the flat
case. The existence of the finite $c$ quantum theory for any negative $\mu$ is
strongly believed to be valid but, strictly speaking, and as in the flat case,
this remains to be proven [2].

Figure 3: Domain of stability of the saddle point in negative curvature.

##### Positive curvature JT

The case of positive curvature JT is the most intricate. The stability
conditions, derived from (4.34), are $3-r_{0}^{4}-2(\mu L^{2}+2)r_{0}^{2}>0$
for the small disk ($r_{0}<1$) and $3-r_{0}^{4}-2(\mu L^{2}+2)r_{0}^{2}<0$ for
the large disk ($r_{0}>1$). In terms of the variables $x$ and $y$, this is
equivalent to

$\begin{split}&y<y_{\text{s}}(x)=-3+\frac{2}{x^{2}}\bigl{(}1+2\sqrt{1-x^{2}}\bigr{)}\quad\text{(small
disk stability),}\\\
&y>y_{\text{l}}(x)=-3+\frac{2}{x^{2}}\bigl{(}1-2\sqrt{1-x^{2}}\bigr{)}\quad\text{(large
disk stability).}\end{split}$ (4.37)

The domains of local stability are depicted in Fig. 4, together with the
critical line $y_{\text{c}}(x)$ separating the regions where the small disk
dominates (above $y_{\text{c}}$) and the large disk dominates (below
$y_{\text{c}}$). For any fixed allowed value of $x$, we find different
regimes:

i) If $y>y_{\text{s}}(x)$, the large disk is the only locally stable solution,
but we know that it is actually metastable, since the small disk has a lower
action. The small disk itself is locally unstable. In this region of large
positive cosmological constant, we expect a branched polymer phase and there
is no dominating smooth classical geometry.

ii) If $y_{\text{l}}(x)<y<y_{\text{s}}(s)$, both the small and the large disks
are locally stable, but the large disk is metastable for
$y_{\text{c}}(x)<y<y_{\text{s}}(x)$ and the small disk is metastable for
$y_{\text{l}}(x)<y<y_{\text{c}}(x)$.

iii) If $y<y_{\text{l}}(x)$, the small disk is the only locally stable
solution, but we know that it is metastable, since the large disk has a lower
action. The large disk itself is locally unstable. The instability for
$y<y_{\text{l}}(x)$ is specific to the positive curvature theory. This is
consistent with the results in [14], where it is explained that positive
curvature JT gravity is actually classically unstable for any negative value
of the cosmological constant.

So we see that our semiclassical, $c\rightarrow-\infty$ limit, analysis in
positive curvature makes sense only in an interval
$y\in]y_{\text{l}}(x),y_{\text{s}}(x)[$ of values for the cosmological
constant. It is tempting to conjecture that the full, finite $c$, quantum
theory will exist only for sufficiently large cosmological constant. Below a
certain critical value $\mu_{\text{c}}\leq 0$, the classical instability of
the theory probably gives rise to the inconsistency of the full quantum theory
[2].

Figure 4: Domains of stability of the saddle points in positive curvature. The
critical line, in black, separates the regions where the small disk dominates
(above) and the large disk dominates (below). The lightest shade of gray
indicates the domain (upper-right region) in which only the large disk is
locally stable. The slightly darker shade of gray indicates the domain (lower-
right region) in which only the small disk is locally stable. The darkest
shade of gray indicates the domain in which both disks are locally stable. See
the main text for a detailed discussion

##  5 One-loop order: path integrals

###  5.1 One-loop counterterms

The general form of the counterterms were discussed in Sec. 2.2.2. At one-
loop, their contribution to the partition function is straightforwardly
obtained by evaluating each term on the relevant classical backgrounds. The
finite contributions that come from counterterms are fundamental ambiguities
in the logarithm of the partition functions, that can be eliminated by
redefining the parameters. It is thus useful to write down their explicit
form. In the results presented in the next subsections, we shall often simply
write “$+\ \text{counterterms}$” to indicate that these terms might be added,
with undetermined coefficients that can depend on the local couplings but not
on $\ell$ and thus not on $r_{0}$ or equivalently on $x$.

In the Liouville theory, the counterterm action (2.26), written in terms of
the parameter $r_{0}$ or equivalently in terms of the parameter $x$, yields

$S^{\text{L}}_{\text{c.t.,
1-loop}}=c_{1}+c_{2}\frac{r_{0}}{1-r_{0}^{2}}+\frac{c_{3}}{1-r_{0}^{2}}=\tilde{c}_{1}+\tilde{c}_{2}x+\tilde{c}_{3}\sqrt{1+x^{2}}$
(5.1)

for arbitrary numerical constants $c_{i}$s and $\tilde{c}_{i}$s.

In flat JT gravity, the counterterms are of the same kind, and evaluating on
the classical background yields

$S^{\text{JT},0}_{\text{c.t., 1-loop}}=c_{1}+c_{2}\ell+c_{3}\ell^{2}$ (5.2)

for numerical constants $c_{i}$s.

In curved JT gravity, there are similar counterterms, but also a new
contribution that comes from the last term in Eq. (2.28). This term can be
evaluated by using (3.21) and the classical solutions (3.23) and (3.31),

$\int_{\mathscr{D}}{\rm d}^{2}x\sqrt{g_{*}}\,\varphi_{*}^{\eta}=4\pi\mu
L^{4}\frac{r_{0}^{4}}{1-r_{0}^{4}}\,\cdotp$ (5.3)

Overall, we get

$S^{\text{JT},\eta}_{\text{c.t., 1-loop}}=c_{1}+c_{2}\frac{r_{0}}{1+\eta
r_{0}^{2}}+\frac{c_{3}}{1-r_{0}^{2}}+\frac{c_{4}}{1+r_{0}^{2}}=\tilde{c}_{1}+\tilde{c}_{2}x+\tilde{c}_{3}\sqrt{1-\eta
x^{2}}+\frac{\tilde{c}_{4}}{\sqrt{1-\eta x^{2}}}$ (5.4)

for dimensionless constants $c_{i}$ or $\tilde{c}_{i}$ that may depend on $y$.

###  5.2 Bulk determinants

In Liouville gravity, integrating out the bulk fluctuations $\Xi_{\text{B}}$
in the one-loop path integral (2.62), by using the form (4.8) of the action
for the quadratic fluctuations, produces a bulk contribution to the partition
function of the form

$Z_{\text{L, 1-loop,
Bulk}}=\frac{1}{\sqrt{\det_{\text{D}}(\Delta_{*}+2\mu)}}\,\raise
2.0pt\hbox{,}$ (5.5)

where $\Delta_{*}$ is the positive Laplacian on the negative curvature disk
$R=-2\mu$ of fixed boundary length $\ell$ and the determinant is computed with
Dirichlet boundary conditions.181818This is because the bulk field
$\Xi_{\text{B}}$ satisfies Dirichlet boundary conditions, see Eq. (4.10).
Similarly, the JT gravity one-loop path integral (2.66) has a factor

$Z_{\text{1-loop,
Bulk}}^{(\eta)}=\frac{1}{|\det_{\text{D}}(\Delta_{*}-\frac{2\eta}{L^{2}})|}\,\raise
2.0pt\hbox{,}$ (5.6)

where $\Delta_{*}$ is the positive Laplacian on the constant curvature disk
$R=2\eta/L^{2}$ of fixed boundary length $\ell$ and the determinant is also
computed with Dirichlet boundary conditions. Note that the determinant needed
for the Liouville theory is the same as the one needed for negative curvature
JT, with the identification $\mu=1/L^{2}$.

We are thus facing a non-trivial problem: evaluating the determinants
$\det_{\text{D}}(\Delta_{*}-\frac{2\eta}{L^{2}})$ on _finite size_ constant
curvature disks. As far as we know, this calculation has never been done
before. Since the result is in itself new and interesting (in non-zero
curvature), and the developments required to derive the result are rather
lengthy and of a different spirit from the ideas of the present paper, we have
decided to present the derivation in a separate paper [17]. In [17], we
actually discuss more general determinants, of the form
$\det_{\text{D}}(\Delta_{*}+M^{2})$, for arbitrary mass parameter $M$. The
result of the calculations, made within the $\zeta$-function regularization
scheme, are as follows.

####  5.2.1 Flat bulk determinants

For

$\Delta_{*}=-\Bigl{(}\frac{2\pi}{\ell}\Bigr{)}^{2}\partial_{a}\partial_{a}$
(5.7)

the positive Laplacian on a flat disk of circumference $\ell$, we have

$\ln{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\bigl{(}\Delta_{*}+M^{2}\bigr{)}\biggr{]}=\frac{1}{3}\ln
2-\frac{1}{2}\ln(2\pi)-\frac{5}{12}-2\zeta_{\text{R}}^{\prime}(1)\\\
+\frac{1}{2}\bigl{(}\gamma-1-\ln
2\bigr{)}\Bigl{(}\frac{M\ell}{2\pi}\Bigr{)}^{2}+\biggl{[}-\frac{1}{3}+\frac{(M\ell)^{2}}{8\pi^{2}}\biggr{]}\ln\frac{\ell}{\ell_{\text{IR}}}\\\
+\sum_{n\in\mathbb{Z}}\ln\biggl{[}2^{|n|}|n|!\Bigl{(}\frac{2\pi}{M\ell}\Bigr{)}^{|n|}I_{|n|}\Bigl{(}\frac{M\ell}{2\pi}\Bigr{)}\exp\Bigl{(}-\frac{M^{2}\ell^{2}}{16\pi^{2}(|n|+1)}\Bigr{)}\biggr{]}$
(5.8)

where $\zeta_{\text{R}}$ is the Riemann $\zeta$-function, $\gamma$ is Euler’s
constant and $I_{n}$ the Bessel function of the second kind. We have
introduced an arbitrary infrared renormalization scale $\ell_{\text{IR}}$ to
make the determinants dimensionless.

For our purposes, we need the case $M=0$, which yields

$\ln{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\Delta_{*}\biggr{]}=-\frac{1}{3}\ln\ell+\text{counterterms}.$
(5.9)

Of course, this very simple result at $M=0$ can be derived straightforwardly
from the conformal anomaly, since $\det_{\text{D}}\Delta_{*}$ is the partition
function of a $c=-2$ conformal field theory. We have included the full formula
(5.8) for completeness but also to compare with the negative and positive
curvature cases below, for which the conformal anomaly does not provide the
result we need.

####  5.2.2 Curved bulk determinants

For

$\Delta_{*}^{\eta}=-\frac{(1+\eta
r_{0}^{2}\rho^{2})^{2}}{4L^{2}r_{0}^{2}}\partial_{a}\partial_{a}$ (5.10)

the positive Laplacian on a constant curvature disk, $R=2\eta/L^{2}$, of
circumference $\ell$ (see Eq. (2.8) and Eq. (2.13) for the relation between
the parameters $r_{0}$ and $\ell$), we have

$\ln{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\bigl{(}\Delta_{*}^{\eta}+M^{2}\bigr{)}\biggr{]}=-\frac{1}{2}\ln(2\pi)-\frac{5}{12}-2\zeta_{\text{R}}^{\prime}(1)\\\
+\frac{\eta A^{\eta}}{3\pi
L^{2}}+\frac{\gamma-1}{2\pi}A^{\eta}M^{2}+\biggl{[}-\frac{1}{3}+\frac{A^{\eta}M^{2}}{2\pi}\biggr{]}\ln\frac{2\pi
Lr_{0}}{\ell_{\text{IR}}}\\\
+\sum_{n\in\mathbb{Z}}\ln\Biggl{[}F\Bigl{(}\frac{1}{2}+\frac{1}{2}\sqrt{1-4\eta(ML)^{2}},\frac{1}{2}-\frac{1}{2}\sqrt{1-4\eta(ML)^{2}},|n|+1,\frac{\eta
A^{\eta}}{4\pi L^{2}}\Bigr{)}\\\
\exp\biggl{(}-\frac{A^{\eta}M^{2}}{4\pi(|n|+1)}F\Bigl{(}1,1,|n|+2,\frac{\eta
A^{\eta}}{4\pi L^{2}}\Bigr{)}\biggr{)}\Biggr{]}\,,$ (5.11)

where $F$ is the usual hypergeometric function and $A^{\eta}$ is the area,
given in terms of $r_{0}$ in Eq. (2.9) and (2.12). Note that it is
straightforward to check that the flat space limit of (5.11),
$L\rightarrow\infty$ at fixed $\ell$, reproduces the flat space result (5.8).
In particular, the hypergeometric functions yield the correct Bessel
functions, as can be checked easily by analysing the limit of the coefficients
in the defining series representations.

For our purposes, we need the special case $M^{2}=-2\eta/L^{2}$. The
hypergeometric function then drastically simplifies to

$F\Bigl{(}\frac{1}{2}+\frac{1}{2}\sqrt{1-4\eta(ML)^{2}},\frac{1}{2}-\frac{1}{2}\sqrt{1-4\eta(ML)^{2}},|n|+1,z\Bigr{)}\\\
=F\bigl{(}2,-1,|n|+1,z\bigr{)}=1-\frac{2z}{|n|+1}$ (5.12)

and the infinite sum in (5.11) can be explicitly evaluated [17],

$\sum_{n\in\mathbb{Z}}\ln\Biggl{[}\biggl{(}1-\frac{\eta A^{\eta}}{2\pi
L^{2}}\frac{1}{|n|+1}\biggr{)}\exp\biggl{(}\frac{\eta A^{\eta}}{2\pi
L^{2}}\frac{1}{|n|+1}F\Bigl{(}1,1,|n|+2,\frac{\eta A^{\eta}}{4\pi
L^{2}}\Bigr{)}\biggr{)}\Biggr{]}\\\ =\frac{\gamma\eta A^{\eta}}{\pi
L^{2}}+\ln\frac{1-\eta r_{0}^{2}}{1+\eta r_{0}^{2}}-\frac{1-\eta
r_{0}^{2}}{1+\eta r_{0}^{2}}\ln\bigl{(}1+\eta
r_{0}^{2}\bigr{)}^{2}-2\ln\Gamma\Bigl{(}\frac{2}{1+\eta r_{0}^{2}}\Bigr{)}\,.$
(5.13)

In the above equation, $\Gamma$ denotes Euler’s Gamma function. This yields

$\ln{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\bigl{(}\Delta_{*}^{\eta}-2\eta/L^{2}\bigr{)}\biggr{]}=-\Bigl{(}\frac{1}{3}+\frac{\eta
A^{\eta}}{\pi L^{2}}\Bigr{)}\ln r_{0}\\\ +\ln\frac{1-\eta r_{0}^{2}}{1+\eta
r_{0}^{2}}-\frac{1-\eta r_{0}^{2}}{1+\eta r_{0}^{2}}\ln\bigl{(}1+\eta
r_{0}^{2}\bigr{)}^{2}-2\ln\Gamma\Bigl{(}\frac{2}{1+\eta
r_{0}^{2}}\Bigr{)}+\text{counterterms.}$ (5.14)

For future reference, it is useful to express this result in terms of the
variable $x$, defined by Eq. (1.12) in Liouville or by Eq. (3.25) in JT. For
Liouville or for negative JT gravity, we get the same expression as a function
of $x$,191919For future convenience, we add constants and discard terms
proportional to $\sqrt{1+x^{2}}$, which are counterterms.

$\ln{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\bigl{(}\Delta_{*}^{-}+2/L^{2}\bigr{)}\biggr{]}=\Bigl{(}2\sqrt{1+x^{2}}-\frac{7}{3}\Bigr{)}\ln\frac{\ell}{2\pi}+\frac{7}{3}\ln\Bigl{[}\frac{1}{2}\bigl{(}1+\sqrt{1+x^{2}}\bigr{)}\Bigr{]}\\\
+\frac{1}{2}\ln(1+x^{2})-2\ln\Gamma\bigl{(}1+\sqrt{1+x^{2}}\bigr{)}+\text{counterterms.}$
(5.15)

For positive curvature JT, with $\varepsilon=+1$ for the large disk and
$\varepsilon=-1$ for the small disk,

$\ln\biggl{|}{\det}_{\text{D}}\biggl{[}\Bigl{(}\frac{\ell_{\text{IR}}}{2\pi}\Bigr{)}^{2}\bigl{(}\Delta_{*}^{+}-2/L^{2}\bigr{)}\biggr{]}\biggr{|}=-\Bigl{(}\frac{7}{3}+2\varepsilon\sqrt{1-x^{2}}\Bigr{)}\ln\frac{\ell}{2\pi}+\frac{7}{3}\ln\Bigl{[}\frac{1}{2}\bigl{(}1-\varepsilon\sqrt{1-x^{2}}\bigr{)}\Bigr{]}\\\
+\frac{1}{2}\ln(1-x^{2})-2\ln\Gamma\bigl{(}1-\varepsilon\sqrt{1-x^{2}}\bigr{)}+\text{counterterms.}$
(5.16)

Note that we have been careful in this case to indicate that we consider the
absolute value of the determinant. Indeed, for the large disk solution,
$r_{0}>1$, the operator $\Delta_{*}^{+}-2/L^{2}$ has one negative eigenvalue
[17], which is reflected in the negative sign of the argument of the logarithm
of the first term on the second line of Eq. (5.14).

###  5.3 Gauge fixing

Let us now treat the gauge-fixing $\delta$-functions that appear in the one-
loop path integrals for both Liouville, Eq. (2.62), and JT, Eq. (2.66). The
analysis is simplified a little by rewriting the product of $\delta$-functions
as

$\delta\Bigl{(}\int_{0}^{\frac{\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{\ell}{3}}^{\frac{2\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{2\ell}{3}}^{\ell}\xi\,{\rm
d}s_{*}\Bigr{)}=\delta\Bigl{(}\int_{0}^{\ell}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{0}^{\frac{\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\delta\Bigl{(}\int_{\frac{\ell}{3}}^{\frac{2\ell}{3}}\xi\,{\rm
d}s_{*}\Bigr{)}\,.$ (5.17)

Using the usual mode expansion $\xi=\sum_{n\in\mathbb{Z}}\xi_{n}e^{in\theta}$,
the first $\delta$-function yields

$\delta\Bigl{(}\int_{0}^{\ell}\xi\,{\rm d}s_{*}\Bigr{)}=\delta\bigl{(}2\pi
e^{\sigma_{*}}\xi_{0}\bigr{)}=\frac{e^{-\sigma_{*}}}{2\pi}\delta(\xi_{0})$
(5.18)

and thus sets $\xi_{0}=0$ while producing a factor $e^{-\sigma_{*}}$ (we
discard as usual the overall numerical constant, which is a counterterm). The
other two $\delta$-functions produce in the same way two factors of
$e^{-\sigma_{*}}$ and allow to express $\xi_{\pm 1}$ in terms of the $\xi_{n}$
for $|n|\geq 2$. But the relevant quadratic actions governing the fluctuations
around the saddle, given in Eq. (4.15) for Liouville and in Eq. (4.30) and
(4.34) for JT, do not depend on $\xi_{1}$ or $\xi_{-1}$. These variables can
thus be trivially integrated out.

To get all the factors correctly, we must also take into account the form of
the formal path integral measure over $\xi$. This measure is derived from the
metric indicated in Eq. (2.59). It reads, modulo global unphysical numerical
factors,202020The ambiguities coming from arbitrary numerical factors
multiplying the metrics from which the path integral measures are derived turn
out, as expected, to be counterterms. This is discussed below.

$D_{*}\xi=\prod_{n\in\mathbb{Z}}\bigl{(}e^{\frac{\sigma_{*}}{2}}{\rm
d}\xi_{n}\bigr{)}\,.$ (5.19)

The integration over the modes $\xi_{0}$, $\xi_{1}$ and $\xi_{-1}$ thus
produce an overall factor

$e^{-3\sigma_{*}}\times
e^{\frac{3\sigma_{*}}{2}}=e^{-\frac{3\sigma_{*}}{2}}\,,$ (5.20)

with three factors of $e^{-\sigma_{*}}$ coming from the $\delta$-functions and
three factors of $e^{\frac{\sigma_{*}}{2}}$ coming from the measure.

###  5.4 Liouville gravity

####  5.4.1 Boundary path integral

The Gaussian integral over the boundary degrees of freedom in Liouville theory
can now be performed straightforwardly. From the form of the measure (5.19)
and of the boundary action (4.15), taking into account the factor (5.20), we
get the one-loop boundary contribution to the partition function,

$Z_{\text{L, 1-loop, bd.}}=e^{-\frac{3\sigma_{*}}{2}}\prod_{n\geq
2}\biggl{[}e^{\sigma_{*}}\frac{1+r_{0}^{2}+n(1-r_{0}^{2})}{(1-r_{0}^{2})(n^{2}-1)}\biggr{]}\,.$
(5.21)

The infinite product can be regularized and computed using standard
$\zeta$-function techniques. This is explained in detail in the Appendix.
Using the definition (A.1) and the result (A.13), we get

$Z_{\text{L, 1-loop,
bd.}}=e^{-\frac{3\sigma_{*}}{2}}D^{-1}\Bigl{(}1,3;2+\frac{1+r_{0}^{2}}{1-r_{0}^{2}};e^{-\sigma_{*}}\Bigr{)}=\sqrt{\frac{2}{\pi}}\,\frac{(e^{\sigma_{*}})^{-3+\frac{1+r_{0}^{2}}{1-r_{0}^{2}}}}{\Gamma\Bigl{(}2+\frac{1+r_{0}^{2}}{1-r_{0}^{2}}\Bigr{)}}\,\cdotp$
(5.22)

In terms of the variable $x=\frac{\ell\sqrt{\mu}}{2\pi}$, this reads

$\ln Z_{\text{L, 1-loop,
bd.}}=\Bigl{(}-3+\sqrt{1+x^{2}}\Bigr{)}\ln\frac{\ell}{2\pi}-\ln\Gamma\bigl{(}2+\sqrt{1+x^{2}}\bigr{)}+\text{counterterms.}$
(5.23)

_Remark:_ one may wonder what happens if the bulk and boundary metrics in
field space, Eq. (2.59), from which the formal path integral measures are
defined, are multiplied by arbitrary numerical constant $A_{\text{B}}$ and
$A_{\text{b}}$. Since these numerical constants are not fixed by physics, one
expects on general gounds that $A_{\text{B}}$ and $A_{\text{b}}$ can be
cancelled by counterterms. For the bulk piece, the constant can clearly be
absorbed in the IR cut-off $\ell_{\text{IR}}$ introduced in (5.11). This cut-
off appears in a contribution to the logarithm of the determinant which is
proportional to the area and is thus a counterterm (this also explains why we
have not kept it explicitly in Eq. (5.14)). For the boundary piece, the
constant $A_{\text{b}}$ can always be absorbed in the factor $z$ entering in
the definition of the determinant given in the Appendix, Eq. (A.7). It thus
yields a contribution proportional to $\zeta(0)$, given by Eq. (A.11). In the
present case, this is proportional to the area plus a constant and is thus, as
required, a counterterm. Let us note as well that overall numerical factors in
front of the actions, like the $\frac{1}{24\pi}$ factor sitting in front of
(2.62) and (2.66), can be absorbed by rescaling the integration variables.
This is equivalent to multiplying the metric in field space by a constant and
hence can be absorbed in counterterms. This is the reason why we usually
discard such terms.

####  5.4.2 Putting everything together

From (2.62), the full one-loop partition function is the sum

$\ln Z_{\text{L, 1-loop}}=-\frac{13}{12\pi}S_{\text{L}*}+2\sigma_{*}+\ln
Z_{\text{L, 1-loop, Bulk}}+\ln Z_{\text{L, 1-loop, bd.}}\,.$ (5.24)

Using (3.24), the first piece yields a contribution

$-\frac{13}{12\pi}S_{\text{L}*}+2\sigma_{*}=-\frac{7}{3}\ln\frac{\ell}{2\pi}+\frac{13}{3}\ln\Bigl{[}\frac{1}{2}\bigl{(}1+\sqrt{1+x^{2}}\bigr{)}\Bigr{]}+\text{counterterms}.$
(5.25)

The bulk contribution is, from (5.5), $-\frac{1}{2}$ times the logarithm of
the determinant given in Eq. (5.15), for $L^{2}=1/\mu$, and the boundary
contribution is given in Eq. (5.23). Adding up, recalling the identity
$\Gamma(1+u)=u\Gamma(u)$, yields a one-loop term $\ln Z_{\text{L,
1-loop}}=f_{1}$ that precisely match the result given in Eq. (1.18), which was
derived from the exact FZZT formula (1.13).212121Note that
$\sqrt{1+x^{2}}-1=\frac{\mu A^{-}}{2\pi}$ is an area counterterm and therefore
the term proportional to it in Eq. (1.18) can be discarded. As far as we know,
this provides the first non-trivial path integral test of FZZT. Maybe more
importantly, it gives us confidence in the robustness of our methods, which we
are now going to apply to JT gravity.

###  5.5 Flat space JT gravity

####  5.5.1 Boundary path integral

Similarly to the case of Liouville gravity, but now using the quadratic action
(4.30), we get the boundary partition function for flat JT gravity,

$\begin{split}Z^{0}_{\text{1-loop,
bd.}}&=e^{-\frac{3\sigma_{*}}{2}}\prod_{n\geq
2}\biggl{[}e^{\sigma_{*}}\frac{n+1}{(n-1)(n+1-\frac{\mu\ell^{2}}{8\pi^{2}})}\biggr{]}\\\
&=e^{-\frac{3\sigma_{*}}{2}}D^{-1}\bigl{(}1,3-\frac{\mu\ell^{2}}{8\pi^{2}};3;e^{-\sigma_{*}}\bigr{)}=\frac{1}{2\sqrt{2\pi}}(e^{\sigma_{*}})^{-2+\frac{\mu\ell^{2}}{8\pi^{2}}}\,\Gamma\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}\,.\end{split}$
(5.26)

_Remark:_ let us check, as we have done at the end of Sec. (5.4.1) for the
case of Liouville, that the arbitrary numerical constant $A_{\text{B}}$ and
$A_{\text{b}}$ can be absorbed in counterterms. It works for the bulk
constant, for exactly the same reason as in the case of Liouville. For the
boundary constant, the $\zeta$-function (A.11) being such that
$\zeta(0)=-\frac{1}{2}+\frac{\mu\ell^{2}}{8\pi^{2}}=-\frac{1}{2}+\frac{\mu
A^{0}}{2\pi}$ is a counterterm, it works as well.

####  5.5.2 Putting everything together

From (2.66), the logarithm of the full one-loop partition function is the sum
of $\ln Z^{0}_{\text{1-loop, bd.}}$ and of the additional contribution

$\begin{split}-\frac{13}{12\pi}S_{\text{L}*}^{0}+2\sigma_{*}-\ln{\det}_{D}\Delta_{*}&=\Bigl{(}-\frac{13}{3}+2+\frac{1}{3}\Bigr{)}\ln\frac{\ell}{2\pi}+\text{counterterms}\\\
&=-2\ln\frac{\ell}{2\pi}+\text{counterterms}\end{split}$ (5.27)

that we have evaluated using (3.28) and (5.9). This yields

$\ln
Z^{0}_{\text{1-loop}}=f_{1}^{0}=-\Bigl{(}4-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}\ln\frac{\ell}{2\pi}+\ln\Gamma\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}+\text{counterterms.}$
(5.28)

Note that the result makes sense as long as $\mu\ell^{2}<24\pi^{2}$,
consistently with the stability analysis in Sec. 4.3, Eq. (4.35).

###  5.6 Curved space JT gravity

####  5.6.1 Boundary path integral

The quadratic action (4.34) yields

$\begin{split}Z^{\eta}_{\text{1-loop,
bd.}}&=e^{-\frac{3\sigma_{*}}{2}}\prod_{n\geq
2}\biggl{[}e^{\sigma_{*}}\frac{1-\eta r_{0}^{2}}{1+\eta
r_{0}^{2}}\frac{\bigl{[}1-\eta r_{0}^{2}+n(1+\eta
r_{0}^{2})\bigr{]}^{2}}{\bigl{[}(1-r_{0}^{4})n-2(\mu
L^{2}+2\eta)r_{0}^{2}+1+r_{0}^{4}\bigr{]}(n^{2}-1)}\biggr{]}\\\
&=e^{-\frac{3\sigma_{*}}{2}}D^{-1}\Bigl{(}\frac{3-2(\mu
L^{2}+2\eta)r_{0}^{2}-r_{0}^{4}}{1-r_{0}^{4}},3,1;\frac{3+\eta
r_{0}^{2}}{1+\eta r_{0}^{2}},\frac{3+\eta r_{0}^{2}}{1+\eta
r_{0}^{2}};e^{-\sigma_{*}}\Bigr{)}\\\
&=\sqrt{\frac{2}{\pi}}\,(e^{2\sigma_{*}})^{-2+\frac{1+\mu
L^{2}r_{0}^{2}}{1-r_{0}^{4}}}\,\Gamma\Bigl{(}\frac{3-2(\mu
L^{2}+2\eta)r_{0}^{2}-r_{0}^{4}}{1-r_{0}^{4}}\Bigr{)}\biggl{[}\Gamma\Bigl{(}\frac{3+\eta
r_{0}^{2}}{1+\eta r_{0}^{2}}\Bigr{)}\biggr{]}^{-2}\,\cdotp\end{split}$ (5.29)

In terms of the usual variables $x=\frac{\ell}{2\pi L}$ and $y=\mu L^{2}$,
this reads

$\ln Z^{-}_{\text{1-loop,
bd.}}=\Bigl{(}-3+\sqrt{1+x^{2}}+\frac{(y-1)x^{2}}{2\sqrt{1+x^{2}}}\Bigr{)}\ln\frac{\ell}{2\pi}\\\
+\ln\Gamma\Bigl{(}2+\sqrt{1+x^{2}}-\frac{(y-1)x^{2}}{2\sqrt{1+x^{2}}}\Bigr{)}-2\ln\Gamma\bigl{(}2+\sqrt{1+x^{2}}\bigr{)}+\text{counterterms}$
(5.30)

in negative curvature, and

$\ln Z^{+}_{\text{1-loop,
bd.}}=\Bigl{(}-3-\varepsilon\sqrt{1-x^{2}}-\varepsilon\frac{(y+1)x^{2}}{2\sqrt{1-x^{2}}}\Bigr{)}\ln\frac{\ell}{2\pi}\\\
+\ln\Gamma\Bigl{(}2-\varepsilon\sqrt{1-x^{2}}+\varepsilon\frac{(y+1)x^{2}}{2\sqrt{1-x^{2}}}\Bigr{)}-2\ln\Gamma\bigl{(}2-\varepsilon\sqrt{1-x^{2}}\bigr{)}+\text{counterterms}$
(5.31)

in positive curvature, where $\varepsilon=+1$ and $\varepsilon=-1$ correspond
to the large disk and the small disk solutions respectively.

_Remark:_ let us check again that the numerical constants $A_{\text{B}}$ and
$A_{\text{b}}$ that may multiply the flat space metrics can be absorbed in
counterterms. It works for $A_{\text{B}}$ exactly as in the case of Liouville
or flat space JT. As for $A_{\text{b}}$, it will work if and only if
$\zeta(0)$, given by (A.11), is a counterterm. In the present case, we find

$\zeta(0)=-\frac{5}{2}+\frac{2}{1-r_{0}^{4}}+2\mu
L^{2}\frac{r_{0}^{2}}{1-r_{0}^{4}}\,\cdotp$ (5.32)

This is indeed a counterterm of the form given in Eq. (5.4).

####  5.6.2 Putting everything together

As in the flat case, using (2.66), the logarithm of the full one-loop
partition function is the sum of $\ln Z^{\eta}_{\text{1-loop, bd.}}$, of
$-\frac{13}{12\pi}S_{\text{L}*}+2\sigma_{*}$, which is given by (5.25) in
negative curvature and by

$-\frac{13}{12\pi}S_{\text{L}*}^{+}+2\sigma_{*}=-\frac{7}{3}\ln\frac{\ell}{2\pi}+\frac{13}{3}\ln\Bigl{[}\frac{1}{2}\bigl{(}1-\varepsilon\sqrt{1-x^{2}}\bigr{)}\Bigr{]}+\text{counterterms}$
(5.33)

in positive curvature, and of
$-\ln|\det_{\text{D}}(\Delta_{*}^{\eta}-2\eta/L^{2})|$ given by Eq. (5.15) and
(5.16). We thus obtain the one-loop partition functions, $\ln
Z_{\text{1-loop}}^{\eta}=f_{1}^{\eta}$,

$\displaystyle\begin{split}f_{1}^{-}&=-\Bigl{(}3+\sqrt{1+x^{2}}-\frac{(y-1)x^{2}}{2\sqrt{1+x^{2}}}\Bigr{)}\ln\frac{\ell}{2\pi}-\frac{1}{2}\ln(1+x^{2})\\\
&\hskip
85.35826pt+\ln\Gamma\Bigl{(}2+\sqrt{1+x^{2}}-\frac{(y-1)x^{2}}{2\sqrt{1+x^{2}}}\Bigr{)}+\text{counterterms,}\end{split}$
(5.34)
$\displaystyle\begin{split}f_{1}^{+}&=-\Bigl{(}3-\varepsilon\sqrt{1-x^{2}}+\varepsilon\frac{(y+1)x^{2}}{2\sqrt{1-x^{2}}}\Bigr{)}\ln\frac{\ell}{2\pi}-\frac{1}{2}\ln(1-x^{2})\\\
&\hskip
85.35826pt+\ln\Gamma\Bigl{(}2-\varepsilon\sqrt{1-x^{2}}+\varepsilon\frac{(y+1)x^{2}}{2\sqrt{1-x^{2}}}\Bigr{)}+\text{counterterms.}\end{split}$
(5.35)

As usual, $\varepsilon=+1$ for the large disk and $\varepsilon=-1$ for the
small disk.

##  6  Discussion

###  6.1 Flat space limit

A simple consistency check amounts to taking the flat space limit

$L\rightarrow\infty\,,\quad\text{$\ell$ and $\mu$ fixed}$ (6.1)

of our JT gravity results in negative curvature or in positive curvature when
the small disk dominates. In both cases, the tree level contributions, Eqs.
(3.26) and (3.36), and the one-loop contributions, Eqs. (5.34) and (5.35),
have the correct limits yielding the flat space results (3.29) and (5.28).

###  6.2 Critical exponents, improvement and all-loop predictions

From (3.29) and (5.28), the flat space partition function at one loop reads

$\ln
Z^{0}=-\frac{|c|\mu\ell^{2}}{96\pi^{2}}-\Bigl{(}\frac{|c|}{6}+4-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}\ln\frac{\ell}{2\pi}+\ln\Gamma\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}+O\bigl{(}|c|^{-1}\bigr{)}+\text{c.t.,}$
(6.2)

where c.t. stands for possible arbitrary counterterms.

Let us first make an elementary remark to clarify a potential puzzle. The
$\ln\frac{\ell}{2\pi}$ term seems to be inconsistent with dimensional
analysis, since $\ell$ is dimensionful. This is a familiar feature of one-loop
calculations. One can replace $\ln\frac{\ell}{2\pi}$ by
$\ln\frac{\ell}{\ell_{0}}$ for an arbitrary length scale $\ell_{0}$ by adding
counterterms, a constant and an area counterterm in the present case. In the
$\mu=0$ theory, if one uses a cut-off regularization, the only length scale
that can appear before renormalization is the UV cut-off scale, and thus
$\ell_{0}$ may be naturally identified with the UV cut-off in this case. This
cut-off can be absorbed in counterterms and replaced by an arbitrary
renormalization length scale.

Let us now use Eq. (6.2) for $\mu=0$. It implies that
$Z^{0}\propto\ell^{-\frac{|c|}{6}-4+O(|c|^{-1})}$. This result can be compared
with the exact prediction (1.2). Using (1.1) and (1.3), we find, when
$c\rightarrow-\infty$,

$2\nu\vartheta+1=\frac{|c|}{6}+4+O\bigl{(}|c|^{-1}\bigr{)}\,.$ (6.3)

Recalling that $\beta=\ell$ in our case, Eq. (1.9), we find a perfect match.

When $\mu\not=0$, Eq. (6.2) can be rewritten as

$\ln
Z^{0}=-\frac{|c|\mu}{24}\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{2\nu}-\Bigl{(}\frac{|c|}{6}+4\Bigr{)}\ln\frac{\ell}{2\pi}+\ln\Gamma\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}+O\bigl{(}|c|^{-1}\bigr{)}+\text{c.t.}$
(6.4)

for $\smash{\nu=1-\frac{6}{|c|}+O(|c|^{-2})}$. We thus observe that the
$\smash{\frac{\mu\ell^{2}}{8\pi^{2}}\ln\frac{\ell}{2\pi}}$ correction is
absorbed in an anomalous dimension for $\ell$, with an exponent matching the
critical exponent $\nu$ given in Eq. (1.8), at least up to the one-loop order.
Combining with the $\ell$-dependence at $\mu=0$ discussed in the previous
paragraph, we see that our one-loop calculation precisely reproduces the
prediction of the exact formulas for both the exponents $\nu$ and $\vartheta$.
Let us also note, for later reference, that the one loop anomalous dimension
term in the partition function entirely comes from the boundary determinant
contribution, Eq. (5.26).

Actually, using the fact that the function $g^{0}$ in the exact scaling
formula (1.2) for the partition function can only depend on the dimensionless
parameter $\mu(\frac{\ell}{2\pi})^{2\nu}$, we can trust the “improved” tree-
level term $\smash{-\frac{|c|\mu}{24}(\frac{\ell}{2\pi})^{2\nu}}$ beyond one
loop. We claim, following a standard lore in quantum field theory, that it
will generate correctly the leading logarithmic contributions in
$(\ln\ell)^{k}$ at any $k^{\text{th}}$ loop order. Similarly, the one-loop
term involving the $\Gamma$ function can also be improved, following the same
logic, to make it consistent with the exact scaling form (1.2), by replacing
$\frac{\mu\ell^{2}}{4\pi^{2}}$ by $\mu(\frac{\ell}{2\pi})^{2\nu}$ in its
argument. This yields non-trivial predictions for the subleading logarithms,
at any loop order greater than or equal to two. For instance, we get terms

$\Biggl{[}-\frac{3\mu\ell^{2}}{4\pi^{2}}\bigl{(}\ln\frac{\ell}{2\pi}\bigr{)}^{2}+\frac{3\mu\ell^{2}}{2\pi^{2}}\biggl{[}\Psi\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}-\frac{3}{4}\biggr{]}\ln\frac{\ell}{2\pi}\Biggr{]}\frac{1}{|c|}$
(6.5)

at two loops for $\ln Z^{0}$, the leading logarithm coming from the improved
tree-level contribution and the subleading logarithm coming from both the
improved tree-level and one-loop contributions, with
$\Psi=\Gamma^{\prime}/\Gamma$ the Euler $\Psi$ function. Overall, our improved
one-loop result for the partition function in flat space reads

$Z^{0}=\ell^{-1-2\nu\vartheta}\Gamma\Bigl{(}3-\frac{1}{2}\mu\bigl{(}\frac{\ell}{2\pi}\bigr{)}^{2\nu}\Bigr{)}e^{-\frac{|c|\mu}{24}(\frac{\ell}{2\pi})^{2\nu}}\quad\text{(one-
loop improved).}$ (6.6)

Of course, we do not claim that the above formula is “exact” in any sense. In
particular, it displays the instability for $\mu(\frac{\ell}{2\pi})^{2\nu}\geq
6$, which is inherent to the $c\rightarrow-\infty$ expansion, but which is not
present in the exact finite $c$ theory, where a crossover is expected to occur
from a smooth inflated geometry to a branched polymer phase when $\mu$ is very
large and positive.

The same logic can be applied in non-zero curvature, with the additional
subtlety that we now have a natural IR scale $L$ given by the curvature of
space. We focus on the case of negative curvature, for conciseness, since the
case of positive curvature can be discussed exactly along the same lines.
Improving the tree-level term (3.26) by replacing $x=\ell/(2\pi L)$ by the
renormalized parameter

$\hat{x}=\frac{1}{L}\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\nu}$ (6.7)

to take into account the length anomalous dimension, and expanding at large
$|c|$, produces the one-loop logarithmic contribution

$-\Bigl{(}1-\sqrt{1+x^{2}}-\frac{(y-1)x^{2}}{2\sqrt{1+x^{2}}}\Bigr{)}\ln\frac{\ell}{2\pi}\,\cdotp$
(6.8)

Note that, as usual, replacing $\ln\frac{\ell}{2\pi}$ by
$\ln\frac{\ell}{\ell_{0}}$ for an arbitrary scale $\ell_{0}$ in this
expression is immaterial, because the modification can be absorbed in
counterterms, whose general forms are given in Eq. (5.4). As in zero
curvature, this anomalous dimension contribution is generated by the boundary
determinant, Eq. (5.30). There is an additional logarithmic piece

$2\Bigl{(}1-\sqrt{1+x^{2}}\Bigr{)}\ln\frac{\ell}{2\pi}$ (6.9)

in the full one-loop result, Eq. (5.34), whose origin can be traced back to
the contribution of the bulk determinant (5.15). Such $\mu$-independent
logarithmic terms can exist in non-zero curvature because there is a scale $L$
in the problem which allows to interpret them as contributions, proportional
to $\ln\frac{\ell}{L}$, to the function $g^{-}(0,\beta/L^{1/\nu},c)$ defined
in (1.4). Overall, we can thus write an improved version of $\ln Z^{-}$ as

$\ln
Z^{-}=-\Bigl{(}\frac{|c|}{6}+4\Bigr{)}\ln\frac{\ell}{2\pi}+\frac{|c|}{6}\ln\frac{1+\sqrt{1+\hat{x}^{2}}}{2}-\frac{|c|}{12}(1+y)\bigl{(}\sqrt{1+\hat{x}^{2}}-1\bigr{)}\\\
+2\Bigl{(}1-\sqrt{1+\hat{x}^{2}}\Bigr{)}\ln\hat{x}-\frac{1}{2}\ln(1+\hat{x}^{2})+\ln\Gamma\Bigl{(}2+\sqrt{1+\hat{x}^{2}}-\frac{(y-1)\hat{x}^{2}}{2\sqrt{1+\hat{x}^{2}}}\Bigr{)}\\\
\text{(one-loop improved).}$ (6.10)

The first term, proportional to $\ln\frac{\ell}{2\pi}$, gives the overall
power of $\ell$ in the partition function, with the same critical exponent as
in the flat case, see Eq. (1.2), (1.4) and (6.3), whereas the other terms
contribute to $\ln g^{-}$, where $g^{-}$ is the dimensionless function defined
in (1.4).

###  6.3 Naive semiclassical limit and expected area

At the classical level, and after integrating out the dilaton field $\Phi$
that imposes the constraint of constant curvature, the action for JT gravity
reduces to the cosmological constant term given in Eq. (2.17). The limit
$\Lambda\rightarrow-\infty$ at fixed boundary length $\ell$ may therefore seem
to correspond, at least superficially, to a semi-classical limit, with an
expansion parameter $1/|\Lambda|$ playing the role of $\hbar$. At leading
order, the configurations that dominate must maximise the area for a given
$\ell$. By the standard isoperimetric inequalities, for negative and zero
curvature, this is achieved precisely by round disks embedded in hyperbolic
space or in Euclidean space, which are the saddle point solutions that we have
found in Section 3.3, Eqs. (3.22) and (3.27). In positive curvature, the area
is not bounded above, even at fixed $\ell$ [14]. For this reason, we restrict
the discussion to the cases of zero and negative curvature in this subsection.

We are going to show that our results strongly suggest that the limit
$\Lambda\rightarrow-\infty$ at fixed boundary length $\ell$ is actually _not_
a semi-classical limit in JT gravity. This confirms explicitly the general
arguments and physical picture presented in [2].

In the framework of the present paper, we take the $c\rightarrow-\infty$ limit
first. The naive semi-classical limit (not to be confused with the
$c\rightarrow-\infty$ limit!) is then

$\mu\rightarrow-\infty\,,\quad\ell\quad\text{fixed.}$ (6.11)

One could study directly the partition functions in this limit, but to better
understand what is going on it is more physical and instructive to look at the
expectation value of the area

$\langle A\rangle^{\eta}=-\frac{24\pi}{|c|}\frac{\partial\ln
Z^{\eta}}{\partial\mu}=-24\pi\biggl{(}\frac{\partial
f_{0}^{\eta}}{\partial\mu}+\frac{1}{|c|}\frac{\partial
f_{1}^{\eta}}{\partial\mu}+O\bigl{(}|c|^{-2}\bigr{)}\biggr{)}\,.$ (6.12)

Let us first focus on the theory in flat space. From (3.29) and (5.28), we get

$\langle
A\rangle^{0}=\frac{\ell^{2}}{4\pi}-\frac{3\ell^{2}}{\pi|c|}\ln\frac{\ell}{2\pi}+\frac{3\ell^{2}}{\pi|c|}\Psi\Bigl{(}3-\frac{\mu\ell^{2}}{8\pi^{2}}\Bigr{)}+O\bigl{(}|c|^{-2}\bigr{)}\,.$
(6.13)

In the limit (6.11), this yields

$\langle
A\rangle^{0}=\frac{\ell^{2}}{4\pi}-\frac{3\ell^{2}}{\pi|c|}\ln\frac{\ell}{2\pi}+\frac{3\ell^{2}}{\pi|c|}\ln\frac{|\mu|\ell^{2}}{8\pi^{2}}+O\bigl{(}|c|^{-2}\bigr{)}+O\bigl{(}|\mu|^{-1}\bigr{)}\,.$
(6.14)

The first logarithm on the right-hand side of this equation is interpreted, as
in the previous subsection, as coming from the anomalous dimension of $\ell$.
It is thus absorbed by replacing $(\frac{\ell}{2\pi})^{2}$ by
$(\frac{\ell}{2\pi})^{2\nu}$ in the leading term
$\ell^{2}/(4\pi)=\pi(\frac{\ell}{2\pi})^{2}$. The second logarithm, on the
other hand, points to a divergence of the area when $|\mu|\rightarrow\infty$.
Of course, strictly speaking, our one-loop calculation can be trusted only if
the one-loop correction is much smaller than the leading correction. But the
leading logarithms can be resummed, along the lines of the previous
subsection, to yield an improved area expectation value that we may write in
the following form,

$\langle
A\rangle^{0}\underset{\mu\rightarrow-\infty}{\sim}\pi\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{2\nu}\biggl{[}\frac{|\mu|}{2}\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{2\nu}\biggr{]}^{2\xi}\,,$
(6.15)

for a new critical exponent $\xi$ that we predict to be such that

$\xi=\frac{6}{|c|}+O\bigl{(}|c|^{-2}\bigr{)}\,.$ (6.16)

One may notice that $\xi=1-\nu$ at this order, but we have no rationale to
believe that such a simple relation between $\nu$ and $\xi$ has to be valid to
all orders in the $1/|c|$ expansion. Actually, one may conjecture the relation
(6.24) between $\xi$ and $\nu$, see [2] and the discussion below.

The divergence of the expected area in the limit (6.11), predicted by Eq.
(6.15), immediately implies that this limit is not dominated by a given smooth
background. Thus, we do not have a semiclassical expansion in the limit (6.11)
in JT gravity [2]. A crucial physical feature at the origin of this effect is
the fact that the boundary is a fractal in the finite $|c|$ theory. In
particular, its geometrical length is ill-defined (infinite) and thus the
isoperimetric inequality does not prevent the divergence of the enclosed area,
even at fixed finite quantum length $\ell$.

We can repeat the above analysis in the negative curvature model. Using (3.26)
and (5.34), we find the large $|c|$ expansion

$\langle A\rangle^{-}=2\pi L^{2}\bigl{(}\sqrt{1+{x^{2}}}-1\bigr{)}\\\
-\frac{3\ell^{2}}{\pi|c|}\frac{1}{\sqrt{1+x^{2}}}\biggl{[}\ln\frac{\ell}{2\pi}-\Psi\Bigl{(}2+\sqrt{1+x^{2}}+\frac{(1+|\mu|L^{2})x^{2}}{\sqrt{1+x^{2}}}\Bigr{)}\biggr{]}+O\bigl{(}|c|^{-2}\bigr{)}\,,$
(6.17)

generalising (6.13). Taking $\mu$ to be large and negative, without making any
asumption on $x$ or on its renormalized version $\hat{x}$ defined in Eq.
(6.7), we can find an improved form of the expectation value,

$\langle A\rangle^{-}=2\pi
L^{2}\Biggl{[}\biggl{(}1+\hat{x}^{2}\Bigl{(}\frac{|\mu|(\frac{\ell}{2\pi})^{2\nu}}{\sqrt{1+\hat{x}^{2}}}\Bigr{)}^{2\xi}\biggr{)}^{1/2}-1\Biggr{]}\quad\text{for
$y=\mu L^{2}\ll-1$,}$ (6.18)

generalising (6.15). This formula shows that the model can be in two
qualitatively different regimes when $\mu L^{2}$ is large and negative. If
$\hat{x}\ll 1$, the right-hand side of Eq. (6.18) goes to the flat space
regime described by (6.15). On the other hand, if

$\hat{x}\biggl{(}|\mu|\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{2\nu}\biggr{)}^{\xi}\gg
1\,,$ (6.19)

one gets

$\langle A\rangle^{-}\sim 2\pi
L\Big{(}\frac{\ell}{2\pi}\Big{)}^{\nu}\biggl{(}\frac{|\mu|(\frac{\ell}{2\pi})^{2\nu}}{\sqrt{1+\hat{x}^{2}}}\biggr{)}^{\xi}\,.$
(6.20)

As in the flat space case, the expected area thus diverges when
$\mu\rightarrow-\infty$, showing that the limit is not semi-classical in the
negative curvature theory either. But the asymptotic regime is of a very
different nature than in flat space, because the typical geometries are very
large compared to the curvature length scale. If one assumes that $\hat{x}\gg
1$ on top of (6.19), the formula further simplifies to

$\langle A\rangle^{-}\sim 2\pi
L\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\nu}\biggl{(}|\mu|\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\nu}L\biggr{)}^{\xi}\,.$
(6.21)

The regime in which (6.21) is valid is conjectured to match with the regime
where an effective Schwarzian description of the model is possible [2]. In
this regime, the curvature length scale $L$ actually plays the role of an
ultraviolet cut-off!

The asymptotic formula (6.21) allows to identify the macroscopic smooth length
parameter $\ell_{\text{S}}$ [2], that is traditionally used in the Schwarzian
description of JT gravity, see Eq. (1.6) and below. The idea [2] is that for a
large, near-hyperbolic effective smooth geometry, one has

$\langle A\rangle^{-}\sim L\ell_{\text{S}}\,.$ (6.22)

Comparing with (6.21) yields

$\ell_{\text{S}}=2\pi\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\nu}\biggl{(}|\mu|\Bigl{(}\frac{\ell}{2\pi}\Bigr{)}^{\nu}L\biggr{)}^{\xi}\,.$
(6.23)

It is interesting to note that the overall power of $\ell$ in the right-hand
side of this equation is one, at the order we have computed, since
$(1+\xi)\nu=1+O(|c|^{2})$. It is natural to conjecture [2] that this property
will remain valid at finite $c$ and thus that

$\xi=\frac{1}{\nu}-1\,.$ (6.24)

A simple argument in favour of this identity is that boundary cosmological
constant counterterms must be proportional to the quantum length parameter
$\ell$ in the microscopic theory [2]. Consistency with the treatment of such
counterterms in the effective description then implies that $\ell_{\text{S}}$
must be proportional to $\ell$. More evidence for this relation is given in
[2].

Let us note that the above discussion follows closely and is perfectly
consistent with the general discussion in [2], which does not rely on the
$c\rightarrow-\infty$ limit. To close this paper, we are now going to discuss
further the Schwarzian limit of the negative curvature partition function in
our framework and find nice confirmations of the interpretations we have just
given.

###  6.4 The Schwarzian limit

The Schwarzian limit, or near-hyperbolic limit, or near-AdS limit, is a
particular asymptotic regime of the negative curvature theory defined by
(1.6). In the limit, the disk geometry is usually assumed to be close to the
geometry of the full hyperbolic space, with a smooth and gently wiggling
boundary embedded in hyperbolic space, described by the so-called
reparameterization ansatz and governed by the Schwarzian action. As emphasized
in [1, 2], and as should be clear from the discussion of the present paper,
the Schwarzian description is nothing more than an _effective_ description of
JT gravity, valid on length scales much greater than $L$. In particular, the
parameter $\ell_{\text{S}}$ appearing in the very definition (1.6) of the
limit, which is usually presented as being the length of the disk boundary, is
actually an effective parameter itself. We have outlined, in the previous
subsection, how the Schwarzian “macroscopic” regime can emerge, in the
framework of the $c\rightarrow-\infty$ limit, with an effective length
parameter given by (6.23); see also the detailed discussion in [2].

Our goal in this last subsection is to work out some more details on the
limit, emphasizing the crucial difference between the microscopic description
of the theory and the effective, long-wavelength description associated with
the Schwarzian, and giving more evidence for the emergence of this
description.

Let us first note that, taking into account the rescaling of the cosmological
constant, Eq. (1.7), the Schwarzian limit corresponds in our case to

$y=\mu
L^{2}\rightarrow-\infty\,,\quad\ell_{\text{S}}\rightarrow\infty\,,\quad\frac{3\ell_{\text{S}}}{|\mu|L^{3}}=b_{\text{S}}=\text{constant,}$
(6.25)

the fixed parameter $b_{\text{S}}$ being related to the usual parameter
$\beta_{\text{S}}$ by

$b_{\text{S}}=|c|\beta_{\text{S}}\,.$ (6.26)

Let us emphasize that the scaling (6.25) used to define the Schwarzian limit
is a renormalized version of the naive scaling that would involve $\ell$
instead of $\ell_{\text{S}}$. Using this renormalized version is natural from
the discussion of the previous subsection and from the general discussion in
[2], but the statement that this is the correct way to define the limit is a
non-trivial claim. The naive “bare” parameter, defined by

$\tilde{b}_{\text{S}}=\frac{3\ell}{|\mu|L^{3}}\,\raise 2.0pt\hbox{,}$ (6.27)

is related to the “renormalized” parameter $b_{\text{S}}$ by

$b_{\text{S}}=\tilde{b}_{\text{S}}(|\mu|L)^{\xi}=\tilde{b}_{\text{S}}\Bigl{(}1+\frac{6}{|c|}\ln\bigl{(}L^{-1}|y|\bigr{)}+O\bigl{(}|c|^{-2}\bigr{)}\Bigr{)}\,.$
(6.28)

This is found by using (6.23), (1.8) and (6.16). Clearly, keeping
$\tilde{b}_{\text{S}}$ or $b_{\text{S}}$ fixed in the limit are two
inequivalent prescriptions. We are going to check explicitly below that it is
indeed the renormalized scaling, for which $b_{\text{S}}$ is kept fixed (and
thus $\tilde{b}_{\text{S}}\rightarrow 0$), that yields a well-defined limit.
For future reference, let us express the Schwarzian scaling in terms of our
usual variables $x$ and $y$,

$x\rightarrow\infty\,,\quad y\rightarrow-\infty\,,\quad
b_{\text{S}}=\tilde{b}_{\text{S}}\bigl{(}L^{-1}|y|\bigr{)}^{\xi}=\frac{6\pi
x}{|y|}\bigl{(}L^{-1}|y|\bigr{)}^{\xi}\quad\text{fixed.}$ (6.29)

##### Leading order

The leading order partition function (3.26) reduces to

$f_{0}^{-}=-\frac{1}{12}(1-|y|)(x-1)-\frac{1}{6}\ln(2L)+\frac{|y|}{24x}+o(1)\,,$
(6.30)

where $o(1)$ represents terms that go to zero in the Schwarzian limit. The
$x$-independent (or, equivalently, $\ell$-independent) terms, and the terms
proportional to $x$ (or equivalently, proportional to $\ell$ or to
$\ell_{\text{S}}$), are counterterms, interpreted as renormalizing the zero-
temperature entropy and ground state energy of the holographic dual,
respectively (we are using this language here, because it is traditional in
the Schwarzian limit). We shall no longer indicate these terms explicitly. The
physically relevant piece is the term $\frac{|y|}{24x}$. At leading order in
the $c\rightarrow-\infty$ expansion, the background is smooth and classical,
$\ell_{\text{S}}$ and $\ell$ are identified and there is no distinction
between the naive and the renormalized scaling. This term then yields the
familiar