YITP-21-88

IPMU21-0054

JT Gravity Limit of Liouville CFT and Matrix Model

Kenta Suzukia and Tadashi Takayanagia,b,c

aCenter for Gravitational Physics,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
bInamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411 Japan
cKavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan

In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on
two-dimensional anti de-Sitter spaces and a semiclassical limit of $c<1$ two-
dimensional string theory. The world-sheet theory of the latter consists of a
space-like Liouville CFT coupled to a non-rational CFT defined by a time-like
Liouville CFT. We show that their actions, disk partition functions and
annulus amplitudes perfectly agree with each other, where the presence of
boundary terms plays a crucial role. We also reproduce the boundary Schwarzian
theory from the Liouville theory description. Then, we identify a matrix model
dual of our two-dimensional string theory with a specific time-dependent
background in $c=1$ matrix quantum mechanics. Finally, we also explain the
corresponding relation for the two-dimensional de-Sitter JT gravity.

###### Contents

  1. 1 Introduction
  2. 2 JT Gravity
    1. 2.1 Disk geometry
    2. 2.2 Euclidean wormhole geometry
    3. 2.3 One-loop partition function
  3. 3 Liouville CFT
    1. 3.1 Liouville action
    2. 3.2 JT gravity as a semiclassical limit of Liouville action
    3. 3.3 FZZT brane interpretation
    4. 3.4 Minisuperspace wavefunctions
    5. 3.5 Annulus amplitude
    6. 3.6 Boundary Schwarzian theory
  4. 4 $c=1$ Matrix Model Interpretation
    1. 4.1 $c=1$ Matrix model and deformation
    2. 4.2 Collective field description
  5. 5 Two-dimensional de Sitter Gravity
  6. 6 Conclusions
  7. A Boundary $\rho K^{(0)}$ contribution
  8. B Boundary conditions of classical Liouville theory
  9. C On-shell actions of Liouville theory
    1. C.1 Disk geometry
    2. C.2 Punctured disk geometry
  10. References

##  1 Introduction

Jackiw-Teitelboim (JT) gravity is a two-dimensional theory of quantum gravity
coupled to a real scalar field [1, 2]. This model played a crucial role for
our recent understanding of (nearly) AdS2/CFT1 correspondence [3, 4, 5, 6] and
it can be viewed as a leading low temperature universal sector of higher
dimensional near-extremal black holes [7, 8]. The model also brought us
improved insights for black hole information paradox via replica wormholes [9,
10, 11, 12]. JT gravity is especially remarkable when it is defined on a
Euclidean negatively curved backgrounds, including a sum over higher genus
topologies in the bulk. In this set-up, it was shown that its dual is given by
a double-scaled Hermitian matrix integral [13]. The discovery of this duality
led many authors further investigations, for example, generalization to other
matrix ensembles [14], correlation functions [15], non-perturbative effects
[16], and fixing matrix eigenvalues by eigenbranes [17], as well as attempts
of this type of averaged duality to higher dimensions (in particular
AdS3/CFT2) [18, 19, 20, 21].

In the work of [13], it was also suggested that JT gravity is a particular
semiclassical limit of the old minimal string theory [22, 23, 24, 25]. This
connection between JT gravity and the minimal string was further studied in
[26, 27, 28, 29, 30]. An important feature of this relation is that JT gravity
looks like the worldsheet theory of the minimal string, which consists of a
(space-like) Liouville CFT plus a minimal CFT (besides the ghost sector).
Hence, we might call this particular semiclassical limit of the minimal string
“JT string” [13] 111A similar idea for a worldline theory was developed in
[31]., but it is not completely clear how the degrees of freedom of JT gravity
arise from the minimal string. One useful way to describe a minimal CFT, which
is rational, is the well-known Coulomb gas formalism with a BRST truncation
[32] 222For a recent discussion on this Coulomb gas formalism, see [33].. In
this paper, we will initiate such investigation by replacing the minimal CFT
by a time-like Liouville CFT [34, 35, 36, 37, 38, 39, 40]. The time-like
Liouville theory is constructed by adding a Liouville potential on top of a
time-like linear-dilaton theory. As opposed to the mininal CFT, we do not
perform any truncation of the spectrum in the time-like Liouville CFT and thus
it is a non-rational CFT. Our worldsheet theory consisting of a time-like
Liouville theory (or a time-like linear-dilaton theory) together with the
usual space-like Liouville CFT. This is properly regarded as a time-dependent
background in two dimensional string theory, rather than the minimal string
333On the other hand, the target space of the minimal string was discussed in
[41, 42, 43]. and indeed this has a dual description by considering a time-
dependent background of the well-known $c=1$ matrix quantum mechanics [44, 22,
45] as found in [38]. With this knowledge as a hint, we will try to
investigate the target space picture of JT string.

A different connection between JT gravity and Liouville field theory was
pointed out in [46]. Also a relation between Schwarzian theory and Liouville
quantum mechanics was studied in [47, 48, 49] and another Liouville field
theory was obtained from the large $q$ limit of the SYK model [50, 51, 52].
However, the connection between these theories and our Liouville field theory
discussed in this paper is not clear.

The remainder of the paper is organized as follows. In section 2, we summarize
JT gravity computation of on-shell actions of the hyperbolic disk, as well as
of the “double-trumpet” (i.e. two-boundary Euclidean wormhole) geometries.
This section mostly follows the discussion of [5, 13], but using more
appropriate coordinates for later Liuoville theory discussion. The on-shell
action of the double-trumpet geometry is constructed by two copies of “single-
trumpet” geometries, and the latter is obtained by a simple analytical
continuation from a punctured disk geometry (i.e. hyperbolic disk with a
conical defect) discussion in [53, 54, 55, 56, 30].

In section 3.1, we define our double Liouville theory by a space-like and a
time-like Liouville theory, whose total central charge is 26 coincides with
that of world-sheet theory of two dimensional string theory. In section 3.2,
we demonstrate that the action of this double Liouville theory is reduced to
the JT gravity action in the semiclassical limit with appropriate field
redefinitions. This argument was originally presented in [28, 29]. In order to
match with JT gravity result with a Dirichlet boundary condition at asymptotic
AdS2, we study the FZZT brane [57, 58] boundary condition in our double
Liouville theory 444FZZT branes in JT gravity have been recently discussed in
[59].. With this boundary condition, we compute the classical on-shell action
of our double Liouville theory for the disk geometry, by direct evaluation of
the action with on-shell background solutions in section 3.3, and by
semiclassical saddle-point evaluation of the minisuperspace wave functions in
in section 3.4. In Liouville CFT literature, the on-shell action of the
double-trumpet geometry is called annulus amplitude. We compute the annulus
amplitude of our double Liouville theory by using the same methods as disk
geometry in section 3.5. Finally in section 3.6, we evaluate the boundary
action of our double Liouville theory, which leads to the Schwarzian theory as
in JT gravity.

In section 4.1, we present a matrix model description of the JT gravity limit
of the two-dimensional string theory whose matter sector is described by the
time-like Liouville CFT. We argue that this is given by the $c=1$ matrix model
in a time-dependent background and we determine the corresponding Fermi
surface of this background describing the eigenvalue fluctuations. Next, in
section 4.2, we study the collective field theory of this background. This
shows that the JT limit is described by a two-dimensional string theory with a
time-like Liouville wall.

In section 5, we describe the corresponding relation for the two-dimensional
de-Sitter JT gravity [60, 61]. This is simply archived by changing the sign of
the bulk cosmological constant of the Liouville CFT. Hence the dual matrix
quantum mechanics does not change from the one explained in section 4 and
indeed we are able to confirm this in our matrix quantum mechanics
description.

We give our conclusions in section 6 and the appendices contain some of
computational details and supplemental discussions.

##  2 JT Gravity

In this section, we study the partition function of JT gravity on a fixed
background. In particular, we are interested in the hyperbolic disk and
double-trumpet geometries.

We write the action of JT gravity on a two dimensional manifold $M$ as
follows:

$\displaystyle
I_{\textrm{JT}}\,=\,-2\int_{M}\sqrt{g}\,\Phi(R+2)\,-\,4\int_{\partial
M}\sqrt{\gamma}\,\Phi K\,+\,I_{\textrm{ct}}\,,$ (2.1)

where $\gamma$ and $K$ are the induced metric and trace of the extrinsic
curvature on the boundary $\partial M$. One can also add the Einstein-Hilbert
action to the above action, but it is just topological and gives a constant
contribution $-S_{0}\chi(M)$ to the on-shell action, where $S_{0}$ is a
coefficient of the Einstein-Hilbert action and $\chi(M)$ is the Euler
characteristic of the manifold $M$. In this case, the dialtion field must
satisfy $S_{0}\gg\Phi$ at any position. However, for a fixed background
geometry as we will consider in the rest of the paper, this topological
suppression factor does not play any important role. Therefore, we will omit
the Einstein-Hilbert action in the following.

The counter term is added to make the on-shell action finite and it reads [62]

$\displaystyle I_{\textrm{ct}}\,=\,4\int_{\partial M}\sqrt{\gamma}\Phi\,.$
(2.2)

###  2.1 Disk geometry

Let us first consider the disk topology. The variation of the dilation field
$\Phi$ in the bulk action of (2.1) simply sets $R=-2$, so the background
solution is (Euclidean) AdS2. Still there is a non-trivial equation for the
dilation field $\Phi$ coming from the variation of the metric. The the global
coordinates of the (Euclidean) AdS2 solution, which we are interested in, is
given by

$\displaystyle ds^{2}\,$
$\displaystyle=\,\frac{4}{(1-r^{2})^{2}}(dr^{2}+r^{2}d\theta^{2})\,,$
$\displaystyle\Phi\,$ $\displaystyle=\,A\cdot\frac{1+r^{2}}{1-r^{2}}\,,$ (2.3)

where $A$ is an integration constant of the solution. As in the usual angular
coordinates, $r$ is the radial coordinate $0<r<1$, where the boundary is
located at $r=1$, and $\theta$ is an angular coordinate with periodicyty
$2\pi$.

We regulate the UV divergence coming from the AdS asymptotic boundary $r\to 1$
by introducing a UV cutoff $\delta$ as [5]

$\displaystyle r\,=\,1-\delta\,,\qquad(0<\delta\ll 1)\,,$ (2.4)

which leads to the asymptotic behaviour of the background solutions

$\displaystyle ds^{2}\,$
$\displaystyle\simeq\,\frac{1}{\delta^{2}}(dr^{2}+r^{2}d\theta^{2})\,,$
$\displaystyle\Phi\,$ $\displaystyle\simeq\,\frac{A}{\delta}\,.$ (2.5)

Eventually we will be interested in the $\delta\to 0$ limit. We can choose the
boundary length to be $\beta$ by redefining the angular coordinate

$\displaystyle u\,=\,\frac{\beta}{2\pi}\,\theta\,,$ (2.6)

where $u$ can be understood as the time of the boundary theory. Then the
corresponding UV cut off $\epsilon$ is now defined as
$\epsilon=\frac{\beta}{2\pi}\,\delta$, such that

$\displaystyle ds^{2}\,\simeq\,\frac{1}{\epsilon^{2}}\,du^{2}\,,\qquad\
\Phi\,\simeq\,\frac{\bar{\Phi}}{\epsilon}\,,$ (2.7)

where we set the renormalized value of the dilaton:
$\bar{\Phi}=A\cdot\frac{\beta}{2\pi}$.

Since the background solution satisfies $R=-2$, the bulk action of (2.1)
vanishes on-shell. For the remaining boundary action, by substituting the
background solution, the on-shell action is evaluated as

$\displaystyle I^{\scriptsize\mbox{on-shell}}_{\textrm{JT}}\,$
$\displaystyle=\,-4A\int
d\theta\,\frac{1+r^{2}}{(1+r)^{2}}\bigg{|}_{r=1-\delta}$
$\displaystyle\simeq\,-4\pi A=-8\pi^{2}\frac{\bar{\Phi}}{\beta}\,,$ (2.8)

where for the second line, we neglected order $\delta$ contributions.

###  2.2 Euclidean wormhole geometry

Next we would like to study two-boundary Euclidean wormhole geometry (i.e.
“double-trumpet”). As shown in [13], such geometry can be constructed by
sewing two copies of “single-trumpet”. Also it is known that single-trumpet
geometry is related to the punctured disk (i.e. disk with a conical
singularity) by analytical continuation of its conical deficit angle [54, 55].
It seems more convenient to take this detour route via punctured disk (in
particular for comparison with the later Liouville theory discussion), where
we just need to insert a bulk “vertex operator”, instead of directly studying
the single-trumpet geometry.

Effectively, this means that we need to study a bulk one-point function of the
form $\big{\langle}e^{-4\pi b\alpha\Phi(0)}\big{\rangle}$, 555The minus sign
compared to the Liouville vertex operator comes from the overall sign
difference of the bulk action. where the operator is inserted at the center of
disk and the defect angle is $\pi b\alpha$. Now this modifies the JT bulk
action as

$\displaystyle
I_{\textrm{JT}}^{\textrm{bulk}}\,=\,-2\int_{M}\sqrt{g}(R+2)\Phi\,+\,4\pi
b\alpha\Phi(0)\,,$ (2.9)

while the boundary action and counterterm do not change. The corresponding
background solution is given by

$\displaystyle ds^{2}\,$
$\displaystyle=\,\frac{4\nu^{2}\,r^{2\nu-2}}{(1-r^{2\nu})^{2}}\,(dr^{2}+r^{2}d\theta^{2})\,,$
$\displaystyle\Phi\,$ $\displaystyle=\,A\cdot\frac{1+r^{2\nu}}{1-r^{2\nu}}\,,$
(2.10)

where we defined

$\displaystyle\nu\,=\,1-\frac{b\alpha}{2}\,.$ (2.11)

The on-shell action can be computed from the boundary action as before

$\displaystyle I^{\scriptsize\mbox{on-shell}}_{\textrm{JT}}\,$
$\displaystyle=\,-4\nu A\int
d\theta\frac{1+r^{2\nu}}{(1+r^{\nu})^{2}}\bigg{|}_{r=1-\delta}$
$\displaystyle\simeq\,-4\pi\nu^{2}A=-8\pi^{2}\nu^{2}\frac{\bar{\Phi}}{\beta}\,.$
(2.12)

This indeed agrees with the tree-level punctured disk partition function [53].
The single-trumpet partition function is obtained by

$\displaystyle 2\pi\nu\,\to\,i\tilde{\alpha}\,,$ (2.13)

where $\tilde{\alpha}$ is the geodesic length of the other end.

Finally the tree-level contribution to the double-trumpet partition function
is given by [13]

$\displaystyle\int_{0}^{\infty}d\tilde{\alpha}\,\tilde{\alpha}\,e^{-2\tilde{\alpha}^{2}\frac{\bar{\Phi}}{\beta_{1}}}e^{-2\tilde{\alpha}^{2}\frac{\bar{\Phi}}{\beta_{2}}}\,=\,\frac{\beta_{1}\beta_{2}}{4\bar{\Phi}(\beta_{1}+\beta_{2})}\,.$
(2.14)

###  2.3 One-loop partition function

Next, we would like to study one-loop contributions to the partition function.
Since the pure JT gravity is one-loop exact [63, 13], this completes the
partition function. To this end, we promote the bulk coordinates to functions
of the “boundary time” $u$ as $r\to r(u)$ and $\theta\to\theta(u)$ [5]. This
is simply a reparametrization between the bulk Euclidean time near the
boundary and the physical boundary time $u$. Now, because of the boundary
condition (2.7), the two functions are not independent, but

$\displaystyle\qquad g_{uu}\,=\,\frac{1}{\epsilon^{2}}\,=\,\frac{4(r^{\prime
2}+r^{2}\theta^{\prime 2})}{(1-r^{2})^{2}}\qquad\Rightarrow\qquad
r(u)\,=\,1-\epsilon\,\theta^{\prime}(u)\,+\,\mathcal{O}(\epsilon^{2})\,,$
(2.15)

where the prime denotes a derivative respect to the boundary time $u$. Using
this relation, we would like to evaluate the on-shell action (2.8) up to
$\mathcal{O}(\epsilon^{0})$ order. To do this, we need to be careful because
the boundary cutoff is now located at $r(u)=1-\epsilon\theta^{\prime}(u)$.
Detail evaluation of the extrinsic curvature on this cutoff can be found in
Appendix A of [8], and this leads to

$\displaystyle
K\,=\,1\,+\,\epsilon^{2}\,\textrm{Sch}\big{(}\tan(\tfrac{\theta}{2})\,,u\big{)}\,+\,\mathcal{O}(\epsilon^{3})\,,$
(2.16)

where the Schwarzian derivative is defined by

$\displaystyle\textrm{Sch}(F,x)\,=\,\frac{F^{\prime\prime\prime}(x)}{F^{\prime}(x)}\,-\,\frac{3}{2}\left(\frac{F^{\prime\prime}(x)}{F^{\prime}(x)}\right)^{2}\,.$
(2.17)

Therefore, we find the Schwarzian action

$\displaystyle
I^{\textrm{bdy}}_{\textrm{JT}}\,=\,-4\bar{\Phi}\int_{0}^{\beta}du\,\textrm{Sch}\big{(}\tan(\tfrac{\theta}{2})\,,u\big{)}\,.$
(2.18)

From this action, expanding around the saddle-point solution as
$\theta(u)=2\pi u/\beta+\epsilon(u)$, the quadratic action is obtained as

$\displaystyle I^{(2)}_{\textrm{JT}}\,$
$\displaystyle=\,\frac{\bar{\Phi}\beta^{2}}{2\pi^{2}}\int_{0}^{\beta}du\,\left[\epsilon^{\prime\prime}{}^{2}(u)\,-\,\left(\frac{2\pi}{\beta}\right)^{2}\epsilon^{\prime}{}^{2}(u)\right]$
$\displaystyle=\,\frac{8\pi^{2}\bar{\Phi}}{\beta}\sum_{m}\,m^{2}(m^{2}-1)\,\epsilon_{m}^{2}\,,$
(2.19)

where we used Fourier expansion $\epsilon(u)=\sum_{m}e^{2\pi
imu/\beta}\,\epsilon_{m}$. For the disk geometry, among the Fourier modes, we
have to exclude the zero modes corresponding $m=0,\pm 1$ [63]. Combining with
the appropriate measure factor [13], now the one-loop partition function is
obtained as

$\displaystyle Z_{\scriptsize\mbox{one-
loop}}\,=\,\left(\frac{\bar{\Phi}}{2\pi\beta}\right)^{\frac{3}{2}}\,.$ (2.20)

##  3 Liouville CFT

In this section, we define our double Liouville theory by a combination of a
space-like and a time-like Liouville theory and show that such theory agrees
with JT gravity in the semiclassical limit.

###  3.1 Liouville action

We consider the CFT consists of a space-like Liouville field $\phi$ and the
time-like one $\chi$, defined by the bulk action

$\displaystyle I^{\textrm{bulk}}_{\rm L}\,=\,\int
d^{2}x\sqrt{g}\,\Big{[}(\partial_{\mu}\phi)^{2}-(\partial_{\mu}\chi)^{2}+\mu
e^{2b\phi}-\mu e^{-2b\chi}\Big{]}\,,$ (3.1)

on a flat space R2

$\displaystyle ds^{2}\,=\,dr^{2}\,+\,r^{2}d\theta^{2}\,.$ (3.2)

Here we choose the renormalized bulk cosmological constant of the time-like
Liouville theory as $-\mu$ in order to agree with JT gravity [28, 29]. The
background charges are fixed as $Q=b+1/b$ for the space-like Liouville field
$\phi$ and $q=1/b-b$ for the time-like field $\chi$ with a parameter $0<b\leq
1$. Therefore, each theory has its central charge $c_{\phi}=1+6Q^{2}$ and
$c_{\chi}=1-6q^{2}$, so that the total central charge is
$c=c_{\phi}+c_{\chi}=26$, consistent with the anormaly cancellation condition
of the bosonic string theory.

The string coupling behaves as $g_{s}=e^{Q\phi+q\chi}$, and it is useful to
note that the conformal dimensions of bulk vertex operators read

$\displaystyle h\big{(}e^{2\alpha\phi}\big{)}\,=\,\alpha(Q-\alpha)\,,\qquad
h\big{(}e^{2\alpha\chi}\big{)}\,=\,\alpha(\alpha-q)\,.$ (3.3)

This shows that the bulk vertex operator of the form
$e^{Q\phi+q\chi+2iP\phi\pm 2iP\chi}$ is dimension one.

In the presence of boundary we need to add the boundary term

$\displaystyle I^{\textrm{bdy}}_{\rm L}\,=\,2\int
d\theta(Q\phi+q\chi)K^{(0)},$ (3.4)

where $K^{(0)}$ is the trace of the extrinsic curvature in the flat space at
$r=1$.

###  3.2 JT gravity as a semiclassical limit of Liouville action

In this subsection, we will explain the semiclassical limit of the double
Liouville theory, which gives the JT gravity result. (See also appendix F of
[29].)

We introduce field redefinition of the two Liouville fields by

$\displaystyle\phi\,=\,\rho/b-b\Phi\,,\qquad$
$\displaystyle\chi\,=\,-\,\rho/b-b\Phi\,,$
$\displaystyle\rho\,=\,\frac{b}{2}(\phi-\chi)\,,\qquad$
$\displaystyle\Phi\,=\,-\,\frac{1}{2b}(\phi+\chi)\,.$ (3.5)

This redefinition gives us 666 In order to derive this boundary action, we
first approximated $Q\approx 1/b$ and $q\approx 1/b$ in the boundary action,
then used (3.5). Instead if we use the original $Q$ and $q$ at the same time
as (3.5), this leads to an additional term containing $\rho K^{(0)}$. This
term does not change the finite contributions of the following discussion, but
leaves slight subtlety for the counter term. We will comment on this
contribution in appendix A.

$\displaystyle I^{\textrm{bulk}}_{\rm L}\,$ $\displaystyle=\,\int
d^{2}x\sqrt{g}\,\Big{[}-4(\partial\rho)\cdot(\partial\Phi)\,-\,2\mu
e^{2\rho}\sinh(2b^{2}\Phi)\Big{]}\,,$ (3.6) $\displaystyle
I^{\textrm{bdy}}_{\rm L}\,$ $\displaystyle=\,-4\int d\theta\,\Phi K^{(0)}\,.$
(3.7)

Now, we regard $\rho$ as the conformal factor of the JT gravity metric as

$\displaystyle ds_{\textrm{JT}}^{2}\,=\,e^{2\rho}(dr^{2}+r^{2}d\theta^{2})\,.$
(3.8)

After including this Weyl factor and writing in terms of the JT metric, we
find

$\displaystyle I^{\textrm{bulk}}_{\rm L}+I^{\textrm{bdy}}_{\rm L}\,=\,-2\int
d^{2}x\sqrt{g_{\textrm{JT}}}\,\Big{[}\Phi
R+\mu\sinh(2b^{2}\Phi)\Big{]}\,-\,4\int
d\theta\sqrt{\gamma_{\textrm{JT}}}\,\Phi K\,,$ (3.9)

where $R$ is the Ricci scalar for the JT gravity metric. In the semiclassical
($b\to 0$) limit with $\mu b^{2}=1$, we can see that the Liouville theory
defined by (3.1) and (3.4) gets equivalent to JT gravity. This type of
dilaton-gravity model was also studied in [64, 65, 66] as a Yang-Baxter
deformation of JT gravity, where parameter $b$ measures the deformation. It
would be interesting to understand this connection, but we leave this for a
future work.

The counter term (2.2) in JT gravity is expressed in terms of Liouville fields
as

$\displaystyle I_{ct}\,=\,4\int
d\theta\sqrt{\gamma}\left[-\frac{1}{2b}(\phi+\chi)e^{\frac{b}{2}(\phi-\chi)}\right]\,.$
(3.10)

In the semiclassical $b\to 0$ limit, we can employ the following decomposed
expression for the counter term:

$\displaystyle I_{ct}\,\simeq\,4\int
d\theta\sqrt{\gamma}\left[-\frac{1}{2b^{2}}(e^{b\phi}-e^{-b\chi})\right]\,.$
(3.11)

Now let us evaluate the on-shell action explicitly. It is straightforward to
confirm that in the $b\to 0$ limit, the solution to Liouville fields with the
Dirichlet boundary condition is written as 777 This solution corresponds to
(B.10) with $c^{2}=1-2b^{2}A$.

$\displaystyle\phi\,=\,\frac{1}{b}\log\left(\frac{2}{1-r^{2}}\right)\,-\,Ab\,\frac{1+r^{2}}{1-r^{2}}\,,$
(3.12)
$\displaystyle\chi\,=\,-\frac{1}{b}\log\left(\frac{2}{1-r^{2}}\right)\,-\,Ab\,\frac{1+r^{2}}{1-r^{2}}\,,$
(3.13)

which is equivalent to the AdS2 solution (2.3) i.e.

$\displaystyle e^{2\rho}\,=\,\frac{4}{(1-r^{2})^{2}}\,,\qquad\
\Phi\,=\,A\,\frac{1+r^{2}}{1-r^{2}}\,.$ (3.14)

Then we can compute each on-shell action as

$\displaystyle I^{\textrm{bulk}}_{\rm L}\,$ $\displaystyle=\,-\,8A\int
d\theta\,\frac{r^{2}(1+r^{2})}{(1-r^{2})^{2}}\,,$ (3.15) $\displaystyle
I^{\textrm{bdy}}_{\rm L}\,$ $\displaystyle=\,-4A\int
d\theta\,\frac{1+r^{2}}{1-r^{2}}\,,$ (3.16) $\displaystyle I_{\textrm{ct}}\,$
$\displaystyle=\,8A\int d\theta\,\frac{r(1+r^{2})}{(1-r^{2})^{2}}\,,$ (3.17)

and by summing them we obtain

$\displaystyle I^{\textrm{bulk}}_{\rm L}+I^{\textrm{bdy}}_{\rm
L}+I_{ct}\,=\,-4A\int d\theta\,\frac{1+r^{2}}{(1+r)^{2}}\,.$ (3.18)

This indeed agrees with the JT on-shell action (2.8).

###  3.3 FZZT brane interpretation

In the previous description (3.13), we put the UV cut off $r=1-\delta$ by
hand. Instead, it is also useful to regularize the solution itself with the
boundary condition $\phi_{0}=\phi(r=1)$ and $\chi_{0}=\chi(r=1)$. This is
expected to correspond to the semi-classical limit of FZZT-brane.888More
precisely this type of boundary condition is the Legendre transform of the
FZZT-brane boundary condition with a fixed boundary length. Original FZZT-
brane (at least classically) corresponds to a fixed energy boundary condition
of JT gravity [67]. We will summarize boundary conditions of classical
Liouville theory in appendix B.  We can find appropriate solutions by solving
the Liouville equation as follows:

$\displaystyle e^{2b\phi}\,$
$\displaystyle=\,\frac{p_{\phi}^{2}e^{2b\phi_{0}}}{(1-(1-p_{\phi})r^{2})^{2}}\,,$
$\displaystyle e^{-2b\chi}\,$
$\displaystyle=\,\frac{p_{\chi}^{2}e^{-2b\chi_{0}}}{(1-(1-p_{\chi})r^{2})^{2}}\,,$
(3.19)

where $p_{\phi}$ and $p_{\chi}$ satisfy $p_{\phi}^{2}e^{2b\phi_{0}}=4(1-p)$
and $q_{\chi}^{2}e^{-2b\chi_{0}}=4(1-q)$. Explicitly we find

$\displaystyle p_{\phi}\,$
$\displaystyle=\,-2e^{-2b\phi_{0}}+2e^{-b\phi_{0}}\sqrt{1+e^{-2b\phi_{0}}}\,\simeq\,2e^{-b\phi_{0}}-2e^{-2b\phi_{0}}\,>\,0\,,$
(3.20) $\displaystyle p_{\chi}\,$
$\displaystyle=\,-2e^{2b\chi_{0}}+2e^{b\chi_{0}}\sqrt{1+e^{2b\chi_{0}}}\,\simeq\,2e^{b\chi_{0}}-2e^{2b\chi_{0}}\,>\,0\,.$
(3.21)

We are interested in the limit $b\to 0$ with

$\displaystyle\delta=e^{-\frac{b}{2}(\phi_{0}-\chi_{0})}\ll 1,$ (3.22)
$\displaystyle\eta\equiv b(\phi_{0}+\chi_{0})=-\frac{2b^{2}A}{\delta}\ll 1,$
(3.23)

while we require $\Phi_{0}=-\frac{1}{2b}(\phi_{0}+\chi_{0})\gg 1$. Then the
on-shell action is evaluated as

$\displaystyle I^{bulk}_{L}\simeq\int
d\theta\left[\eta\left(\frac{2}{b^{2}\delta}-\frac{2}{b^{2}}+\frac{\delta}{b^{2}}\right)\right]\simeq
A\int d\theta\left[-\frac{4}{\delta^{2}}+\frac{4}{\delta}-2\right],$ (3.24)
$\displaystyle I^{bdy}_{L}\simeq\frac{2}{b}\int
d\theta(\phi_{0}+\chi_{0})\simeq-\frac{4A}{\delta}\int d\theta.$ (3.25)

The counter term is evaluated as

$\displaystyle I_{ct}=4\int
d\theta\left[-\frac{1}{2b}(\phi+\chi)e^{\frac{b}{2}(\phi-\chi)}\right]\simeq\frac{4}{\delta}\int
d\theta\left(-\frac{1}{2b}(\phi_{0}+\chi_{0})\right)\simeq\frac{4A}{\delta^{2}}\int
d\theta.$ (3.26)

Therefore the total on-shell action becomes

$\displaystyle I^{bulk}_{L}+I^{bdy}_{L}+I_{ct}=-2A\int d\theta=-4\pi A,$
(3.27)

which reproduces the JT gravity result (2.8).

It is also useful to evaluate the contributions from $\phi$ and $\chi$ sector,
separately. First we note

$\displaystyle\phi_{0}=-\frac{1}{b}\log\delta+\frac{\eta}{2b},$ (3.28)
$\displaystyle\chi_{0}=\frac{1}{b}\log\delta+\frac{\eta}{2b}.$ (3.29)

Then, the evaluation of the on-shell action of each sector (detail evaluations
are presented in appendix C) is given by

$\displaystyle I^{\scriptsize\mbox{on-shell}}_{\phi}\,=\,\int
d\theta\left[-A+\frac{2}{b^{2}}(\log 2-1){-\frac{\delta}{b^{2}}}\right]\,,$
(3.30)

and

$\displaystyle I^{\scriptsize\mbox{on-shell}}_{\chi}\,=\,\int
d\theta\left[-A-\frac{2}{b^{2}}(\log 2-1)+\frac{\delta}{b^{2}}\right]\,.$
(3.31)

The total of two sectors agrees with the above result (3.27).

###  3.4 Minisuperspace wavefunctions

We can also derive the disk partition function from the minisuperspace wave
functions in the $b\to 0$ limit. We follow the minisuperspace analysis in
[57], where the Liouville action looks like $I^{FZZ}_{L}=\frac{1}{4\pi}\int
dx^{2}[(\partial_{a}\phi)^{2}+4\pi\mu^{FZZ}e^{2b\phi}]$ and it is
$\frac{1}{4\pi}$ times our original action. In our convention of
$\mu(=4\pi\mu^{FZZ})$, we set $\mu b^{2}=1$ as before. This leads to

$\displaystyle\kappa\,=\,\sqrt{\frac{\mu^{FZZ}}{\sin\pi
b^{2}}}\,\simeq\,\frac{1}{2\pi b^{2}}\,.$ (3.32)

In [57], the bulk one-point function with a fixed boundary length was obtained
as

$\displaystyle W^{(\phi)}_{\alpha}(l_{\phi})\,$
$\displaystyle\equiv\,\big{\langle}e^{2\alpha\phi}\big{\rangle}_{l_{\phi}}\,=\,\frac{2}{b}\left(\pi\mu\gamma(b^{2})\right)^{\frac{Q-2\alpha}{2b}}\,\frac{\Gamma(2\alpha
b-b^{2})}{\Gamma(1+\frac{1}{b^{2}}-\frac{2\alpha}{b})}\,K_{\frac{Q-2\alpha}{b}}(\kappa
l_{\phi})\,,$ (3.33) $\displaystyle W^{(\chi)}_{\alpha}(l_{\chi})\,$
$\displaystyle\equiv\,\big{\langle}e^{2\alpha\chi}\big{\rangle}_{l_{\chi}}\,=\,\frac{2i}{b}\left(-\pi\mu\gamma(-b^{2})\right)^{\frac{-q+2\alpha}{2b}}\,\frac{\Gamma(2\alpha
b+b^{2})}{\Gamma(1-\frac{1}{b^{2}}+\frac{2\alpha}{b})}\,K_{\frac{-q+2\alpha}{b}}(-\kappa
l_{\chi})\,,$ (3.34)

where $\gamma(x)=\Gamma(x)/\Gamma(1-x)$ and we defined the one-point function
of the time-like theory by simple analytical continuation

$\displaystyle\phi\,\to\,-i\chi\,\quad
b\,\to\,-ib\,,\quad\alpha\,\to\,i\alpha\,,\quad
Q\,\to\,iq\,,\quad\mu\,\to\,-\mu\,,$ (3.35)

from the result of the space-like theory.

Since the conformal dimension of the vertex operator $e^{b(\phi+\chi)}$ is
one, in the minisuperspace approximation for $b\to 0$ limit, we expect that
the disk amplitude with the Dirichlet boundary conditions $\phi=\phi_{0}$ and
$\chi=\chi_{0}$ reads

$\displaystyle\big{\langle}e^{b(\phi_{0}+\chi_{0})}\big{\rangle}_{\rm
disk}\,\sim\,K_{\frac{1}{b^{2}}}(\kappa l_{\phi})K_{\frac{1}{b^{2}}}(-\kappa
l_{\chi})\,,$ (3.36)

where we set

$\displaystyle l_{\phi}\,$ $\displaystyle\equiv\,2\pi
e^{b\phi_{0}}\,\simeq\,2\pi\cdot\delta^{-1}(1+\eta/2)\,,$ $\displaystyle
l_{\chi}\,$ $\displaystyle\equiv\,2\pi
e^{-b\chi_{0}}\,\simeq\,2\pi\cdot\delta^{-1}(1-\eta/2)\,.$ (3.37)

By using the approximation formula

$\displaystyle K_{\nu}(z)\,\simeq\,e^{\frac{\nu^{2}}{2z}-z}\,,$ (3.38)

for $\nu\gg 1$ and $z\gg\nu$, we can estimate the disk amplitude as

$\displaystyle\big{\langle}e^{b(\phi_{0}+\chi_{0})}\big{\rangle}_{\rm
disk}\,\sim\,e^{\frac{\pi}{b^{2}l_{\phi}}-\frac{\pi}{b^{2}l_{\chi}}}\,\simeq\,e^{A}\,.$
(3.39)

By multiplying the $4\pi$ factor to adjust to our original convention this
leads to

$\displaystyle\big{\langle}e^{b(\phi_{0}+\chi_{0})}\big{\rangle}_{\rm
disk}\,\sim\,e^{4\pi A}\,.$ (3.40)

This indeed agrees with the JT gravity calculation (2.8).

We can also consider each contribution separately. In order to agree with
(3.30) and (3.31), we need to include precise coefficients as in (3.33) and
(3.34). Since

$\displaystyle K_{\frac{1}{b^{2}}}(\kappa
l_{\phi})\,\sim\,e^{\frac{\pi}{b^{2}l_{\phi}}}\,\approx\,e^{\frac{\delta}{2b^{2}}\left(1-\frac{\eta}{2}\right)}\,,$
(3.41) $\displaystyle K_{\frac{1}{b^{2}}}(-\kappa
l_{\chi})\,\sim\,e^{-\frac{\pi}{b^{2}l_{\chi}}}\,\approx\,e^{-\frac{\delta}{2b^{2}}\left(1+\frac{\eta}{2}\right)}\,,$
(3.42)

we find the contribution from the spacelike and timelike sector as

$\displaystyle W^{(\phi)}_{\frac{b}{2}}(l_{\phi})\,$
$\displaystyle\simeq\,\exp\left[-\frac{1}{2}\left(\frac{\delta\eta}{2b^{2}}-\frac{\delta}{b^{2}}+\frac{2}{b^{2}}(\log
2-1)\right)\right]$ (3.43) $\displaystyle
W^{(\chi)}_{\frac{b}{2}}(l_{\phi})\,$
$\displaystyle\simeq\,\exp\left[-\frac{1}{2}\left(\frac{\delta\eta}{2b^{2}}+\frac{\delta}{b^{2}}-\frac{2}{b^{2}}(\log
2-1)\right)\right]$ (3.44)

respectively. These agree with (3.30) and (3.31) once we multiply the $4\pi$
factor as above.

###  3.5 Annulus amplitude

In this section, we would like to study annulus amplitude by semiclassical
approximation. To this end, we again take the route we used for JT gravity in
section 2.2. Namely, we study a bulk one-point function
$\big{\langle}e^{2\pi\alpha(\phi+\chi)}\big{\rangle}$ inserted at the center
of disk. The corresponding semiclassical solutions are [68]

$\displaystyle e^{2b\phi}\,$
$\displaystyle=\,\frac{p_{\phi}^{2}e^{2b\phi_{0}}\,r^{2\nu-2}}{(1-(1-p_{\phi})r^{2\nu})^{2}}\,,$
$\displaystyle e^{-2b\chi}\,$
$\displaystyle=\,\frac{p_{\chi}^{2}e^{-2b\chi_{0}}\,r^{2\nu-2}}{(1-(1-p_{\chi})r^{2\nu})^{2}}\,,$
(3.45)

where $\nu$ is defined in (2.11) and now

$\displaystyle p_{\phi}^{2}e^{2b\phi_{0}}\,$
$\displaystyle=\,4\nu^{2}(1-p_{\phi})\,,$ $\displaystyle
p_{\chi}^{2}e^{-2b\chi_{0}}\,$ $\displaystyle=\,4\nu^{2}(1-p_{\chi})\,.$
(3.46)

We use the same parametrization of $\delta$ and $\eta$ as before (3.23). We
present detail evaluation of the on-shell action in appendix C and simply
quote the final result here:

$\displaystyle I^{bulk}_{L}+I^{bdy}_{L}+I_{ct}\,=\,-2\nu^{2}A\int
d\theta\,=\,-4\pi\nu^{2}A\,.$ (3.47)

This agrees with the JT gravity result (2.12). Then, the tree-level annulus
amplitude is now given by

$\displaystyle\int_{0}^{\infty}d\tilde{\alpha}\,\tilde{\alpha}\,e^{-2\tilde{\alpha}^{2}\frac{\bar{\Phi}}{\beta_{1}}}e^{-2\tilde{\alpha}^{2}\frac{\bar{\Phi}}{\beta_{2}}}\,=\,\frac{\beta_{1}\beta_{2}}{4\bar{\Phi}(\beta_{1}+\beta_{2})}\,.$
(3.48)

We can also recover these results from the minisuperspace wavefunctions:

$\displaystyle
W_{P}(l)\,=\,\big{(}\pi\mu\gamma(b^{2})\big{)}^{-\frac{iP}{b}}\,\frac{\Gamma(1+2ibP)}{\Gamma(-\frac{2iP}{b})}\,K_{\frac{2iP}{b}}(\kappa
l)\,.$ (3.49)

For spacelike theory, we substitute $2iP=\nu(b-b^{-1})$ with $\kappa l=\kappa
l_{\phi}$, while for timelike theory we use $2iP=\nu(b+b^{-1})$ with $\kappa
l=-\kappa l_{\chi}$. This leads to the contribution from the spacelike and
timelike sector

$\displaystyle\exp\left[-\frac{1}{2b^{2}}\left(2\nu\log
2\nu-2\nu+\nu^{2}\delta-\frac{\nu^{2}\delta\eta}{2}\right)\right]\,,$ (3.50)
$\displaystyle\exp\left[-\frac{1}{2b^{2}}\left(-2\nu\log
2\nu+2\nu-\nu^{2}\delta-\frac{\nu^{2}\delta\eta}{2}\right)\right]\,,$ (3.51)

respectively.

###  3.6 Boundary Schwarzian theory

In this subsection, we study the boundary theory of our double Liouville
theory defined in section 3.1. For this discussion, it’s more convenient to
use complex coordinates: $z=re^{i\theta}$ and $\bar{z}=re^{-i\theta}$.
Therefore, now the metric is given by $ds^{2}=dzd\bar{z}$ and the bulk action
becomes

$\displaystyle I^{bulk}_{L}\,=\,\int
dzd\bar{z}\Big{[}4\partial\phi\bar{\partial}\phi\,-\,4\partial\chi\bar{\partial}\chi\,+\,\mu(e^{2b\phi}-e^{-2b\chi})\Big{]}\,.$
(3.52)

This action is invariant under the conformal transformation

$\displaystyle z\,\to f(z)\,,\qquad\bar{z}\,\to\,\bar{f}(\bar{z})\,,$ (3.53)

together with

$\displaystyle\phi(z,\bar{z})\,$
$\displaystyle\to\,\phi(f(z),\bar{f}(\bar{z}))\,-\,\frac{1}{2b}\,\log(\partial
f\bar{\partial}\bar{f})\,,$ (3.54) $\displaystyle\chi(z,\bar{z})\,$
$\displaystyle\to\,\chi(f(z),\bar{f}(\bar{z}))\,+\,\frac{1}{2b}\,\log(\partial
f\bar{\partial}\bar{f})\,.$ (3.55)

We note that the bulk action does not have independent transformation for each
Liouville theory, because this is a coordinate transformation (3.53).

At the boundary, the holomorphic and anti-holomorphic functions are related by
the condition $f(\bar{z})=\bar{f}(z)$. This means that at the boundary, we
have the diffeomorphism of the angular coordinate as $\theta\to f(\theta)$.
This symmetry is explicitly broken by the boundary term (3.4). Taking $b\to 0$
limit and combining with the counterterm (3.10), this boundary action is

$\displaystyle I_{L}^{\textrm{bdy}+\textrm{ct}}\,=\,\frac{2}{b}\int
d\theta\,(\phi+\chi)(K^{(0)}-1)\,.$ (3.56)

In order to evaluate this action, we can again go back to the polar
coordinates and the boundary condition gives $r(\theta)=1-\epsilon
f^{\prime}(\theta)$ as in (2.15). This leads to

$\displaystyle\phi+\chi\,=\,-\frac{2Ab}{\epsilon}\,+\,\mathcal{O}(\epsilon^{0})\,,$
(3.57) $\displaystyle
K^{(0)}\,=\,1\,+\,\left(\textrm{Sch}\big{(}\tan(\tfrac{f}{2})\,,\theta\big{)}\,+\,\frac{1}{2}\right)\epsilon\,+\,\mathcal{O}(\epsilon^{2})\,.$
(3.58)

Therefore, at the order $\epsilon^{0}$, we find the Schwarzain action up to a
constant as

$\displaystyle
I_{\textrm{Sch}}\,=\,-4A\int_{0}^{2\pi}d\theta\,\textrm{Sch}\big{(}\tan(\tfrac{f}{2})\,,\theta\big{)}\,.$
(3.59)

This agrees with (2.18). Since all results of JT gravity with a fixed
background topology is obtained by the Schwarzain theory, this confirms that
our double Liouville theory indeed gives JT gravity results in the
semiclassical limit ($b\to 0$).

##  4 $c=1$ Matrix Model Interpretation

The CFT defined by a space-like Liouville CFT coupled to a free scalar $c=1$
CFT, such that total central charge is $26$, defines a world-sheet theory of
two dimensional string theory, so called $c=1$ string theory. It is well-known
that this string theory is dual to the c=1 matrix model [44, 22, 45].

A $c<1$ extension of two dimensional string theory can be obtained by
considering a space-like Liouville CFT coupled to a time-like Liouville theory
as in (3.1). This string theory can be regarded as a non-critical string
theory with $c\leq 1$ non-rational matter. Its matrix model description was
given in [38] by deforming the $c=1$ matrix model state. As a special case,
this includes the familiar $c=1$ string theory. To relate this to the JT
gravity, we are interested in the limit $b\to 0$, which means that the matter
central charge is $c=-\infty$.

###  4.1 $c=1$ Matrix model and deformation

The $c=1$ matrix [69, 70, 71] is defined by the following matrix quantum
mechanics:

$\displaystyle S_{M}=\int
dt\,\textrm{Tr}\left[(D_{t}\Phi)^{2}+\Phi^{2}\right],$ (4.1)

where $\Phi(t)$ is a $N\times N$ hermitian matrix and
$D_{t}\bullet=\partial_{t}\bullet-i[A_{t},\bullet]$ is the covariant
derivative, so that the action is invariant under $U(N)$ symmetry. By using
this $U(N)$ gauge symmetry, we can diagonalize the matrix $\Phi$ into $N$
eigenvalues. They are identical to $N$ free fermions, each of which is
described by the Hamiltonian with the inverse harmonic potential
$H=p^{2}-x^{2}$.

In the $N\to\infty$ limit, the fermions fills the energy level up to the Fermi
surface

$\displaystyle H=p^{2}-x^{2}=-\mu,$ (4.2)

where we only fill the left-hand side of the potential $x<0$. This defines the
vacuum state of $c=1$ matrix model.

The $c\leq 1$ string theory is defined by the world-sheet theory

$\displaystyle I^{c<1}=\int
dx^{2}\left[(\partial_{a}\phi)^{2}-(\partial_{a}\chi)^{2}+\mu e^{2b\phi}+\nu
e^{-2b\chi}\right].$ (4.3)

This can be boosted into a deformation of $c=1$ string via the map:

$\displaystyle\tilde{\phi}\,$
$\displaystyle=\,\frac{Q}{2}\phi+\frac{q}{2}\chi,$
$\displaystyle\tilde{\chi}\,$
$\displaystyle=\,\frac{q}{2}\phi+\frac{Q}{2}\chi,$ (4.4)

which leads to

$\displaystyle I^{c<1}=\int
dx^{2}\left[(\partial_{a}\tilde{\phi})^{2}-(\partial_{a}\tilde{\chi})^{2}+\mu
e^{(b^{2}+1)\tilde{\phi}+(b^{2}-1)\chi}+\nu
e^{(1-b^{2})\tilde{\phi}-(1+b^{2})\tilde{\chi}}\right].$ (4.5)

This background (4.5) corresponds to a series of excited states in the $c=1$
matrix model given by the following Fermi surface [38] :

$\displaystyle(p-x)(-p-x)^{b^{2}}=\mu
e^{-(1-b^{2})t}+\nu(-p-x)^{2b^{2}}e^{-(1+b^{2})t}.$ (4.6)

In our case of the $b\to 0$ limit with $\mu=-\nu=1/b^{2}$, the Fermi surface
takes the following characteristic form:

$\displaystyle(p-x)e^{t}=-2\log\left[(-p-x)e^{-t}\right].$ (4.7)

This is plotted in Fig.1. At $t=-\infty$ we start with the Fermi surface
$p+x<0$ and eventually it approaches to the wedge $p+x<0$ and $p-x>0$ in the
late limit $t\to\infty$.

Figure 1: Profiles of of Fermi surfaces for $t=-4$ (blue), $t=-2$ (red) $t=0$
(green), $t=2$ (orange) and $t=4$ (purple). The Fermi seas extend in the left-
hand side of the Fermi surfaces.

###  4.2 Collective field description

Next we study the effective theory which captures excitations on our Fermi
surface (4.7) by employing the collective field theory [72, 73] (see also [74,
75]). We introduce the collective field by density of eigenvalues [76] as

$\displaystyle\varphi(t,x)\,\equiv\,\sum_{i=1}^{N}\delta(x-\lambda_{i}(t))\,,$
(4.8)

where $\lambda_{i}(t)$ is an eigenvalue of the matrix $\Phi(t)$. The
collective field after Fourier transforming from $x$ to $k$ corresponds to the
gauge invariant (Wilson) loop operators

$\displaystyle\varphi_{k}(t)\,=\,\int
dx\,e^{ikx}\varphi(t,x)\,=\,\sum_{i=1}^{N}e^{ik\lambda_{i}}\,=\,\textrm{Tr}\Big{(}e^{ik\Phi(t)}\Big{)}\,.$
(4.9)

In terms of this collective field, the matrix quantum mechanics is described
by

$\displaystyle S_{\rm coll}\,=\,\int dtdx\,\left(\frac{1}{\varphi}\left(\int
dx\,\partial_{t}\varphi\right)^{2}-\frac{\pi^{2}}{3}\varphi^{3}+(x^{2}-\mu)\varphi\right)\,.$
(4.10)

When the perturbation of the Fermi surface is infinitesimally small, this is
described by small fluctuations of the collective field around the background
solution:

$\displaystyle\varphi(t,x)\,=\,\varphi_{0}(t,x)\,+\,\frac{1}{\sqrt{\pi}}\,\partial_{x}\eta(t,x)\,,$
(4.11)

where

$\displaystyle\varphi_{0}(t,x)\,=\,\frac{1}{2\pi}\,(p_{+}-p_{-})\,.$ (4.12)

Then, the quadratic action of the fluctuation $\eta$ becomes that of a real
massless scalar with the kinetic term [72, 73]:

$\displaystyle I_{sc}=-\frac{1}{2}\int
dtdx\sqrt{g}g^{\mu\nu}\partial_{\mu}\eta\partial_{\nu}\eta,$ (4.13)

where the effective metric $g_{\mu\nu}$ is defined by

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{2p_{+}p_{-}}{p_{+}-p_{-}}dt^{2}-\frac{2(p_{+}+p_{-})}{p_{+}-p_{-}}dtdx+\frac{2}{p_{+}-p_{-}}dx^{2}.$
(4.14)

Before we study the collective field dynamics of our background, it is helpful
to quickly review the standard result for the usual $c=1$ vacuum, which is
described by the Fermi surface $p^{2}-x^{2}=-1$. In this case, we can express
the Fermi surface by introducing a parameter $\sigma$ or $\tilde{\sigma}$
which parameterizes the branch $p=p_{+}(x)$ or $p=p_{-}(x)$, respectively, as
follows:

$\displaystyle(-p_{+}-x)e^{-t}=e^{\sigma},\ \ \ (p_{+}-x)e^{t}=e^{-\sigma},$
$\displaystyle(-p_{-}-x)e^{-t}=e^{\tilde{\sigma}},\ \ \
(p_{-}-x)e^{t}=e^{-\tilde{\sigma}},$ (4.15)

where $\sigma=-\tilde{\sigma}-2t$ runs in the range

$\displaystyle-\infty<\sigma\leq\sigma_{0}(t),$
$\displaystyle\sigma_{0}(t)\leq\tilde{\sigma}<\infty,$ (4.16)

with $\sigma_{0}(t)=-t$.

Thus the metric (4.24) reads

$\displaystyle
ds^{2}=\frac{1}{2}(e^{-t-\sigma}-e^{t+\sigma})\left[-dt^{2}+d(t+\sigma)^{2}\right].$
(4.17)

If we introduce a new coordinate

$\displaystyle q=\sigma+t,$ (4.18)

then the effective action takes the standard form

$\displaystyle I_{sc}=-\frac{1}{2}\int
dt\int^{0}_{-\infty}dq\left[(\partial_{t}\eta)^{2}-(\partial_{q}\eta)^{2}\right].$
(4.19)

This shows the well-known fact that the target spacetime geometry of $c=1$
matrix model is a half space $q\leq 0$ with a Liouville wall at $q=0$, as
sketched in the left of Fig.2.

Figure 2: Structures of Spacetimes of the collective field theory. The left
and right picture describe the standard $c=1$ vacuum and our background,
described by (4.25), respectively.

Now let us consider the collective field theory for our background defined by
the time-dependent fermi surface (4.7). The fixed $x$ line intersects with the
Fermi surface at two points, written by $p=p_{+}(x)$ and $p=p_{-}(x)$ as
depicted in Fig.3. We introduce a new parameter $\sigma$ and $\tilde{\sigma}$
instead of $x$, which parameterize the branches $p=p_{+}(x)$ and $p=p_{-}(x)$,
respectively as:

$\displaystyle(-p_{+}-x)e^{-t}=e^{\sigma},\ \ \ (p_{+}-x)e^{t}=-2\sigma,$
$\displaystyle(-p_{-}-x)e^{-t}=e^{\tilde{\sigma}},\ \ \
(p_{-}-x)e^{t}=-2\tilde{\sigma}.$ (4.20)

The parameters $\sigma$ and $\tilde{\sigma}$ take the values in the range
(4.16), where we can explicitly find $\sigma_{0}(t)$, where the two branches
get degenerate, as

$\displaystyle\sigma_{0}(t)=-2t+\log 2.$ (4.21)

We can solve $x$ and $p_{\pm}$ as

$\displaystyle x=-\frac{1}{2}e^{t+\sigma}+\sigma
e^{-t}=-\frac{1}{2}e^{t+\tilde{\sigma}}+\tilde{\sigma}e^{-t},$ $\displaystyle
p_{+}=-\frac{1}{2}e^{t+\sigma}-\sigma e^{-t},$ $\displaystyle
p_{-}=-\frac{1}{2}e^{t+\tilde{\sigma}}-\tilde{\sigma}e^{-t}.$ (4.22)

It is also useful to note the relations

$\displaystyle p_{+}-p_{-}=2e^{-t}(\tilde{\sigma}-\sigma),\ \ \ \
p_{+}+p_{-}=-e^{t+\sigma}-2\tilde{\sigma}e^{-t}.$ (4.23)

Figure 3: Structure of Fermi surfaces paramerized by $\sigma$ and
$\tilde{\sigma}$. We also showed momentum functions $p_{\pm}(x)$.

In our back ground, the metric is explicitly found as

$\displaystyle
ds^{2}=2\left(-\frac{1}{2}e^{t+\sigma}+e^{-t}\right)dtd\sigma+\frac{e^{t}}{\tilde{\sigma}-\sigma}\left(-\frac{1}{2}e^{t+\sigma}+e^{-t}\right)^{2}d\sigma^{2}.$
(4.24)

However note that the metric is meaningful only up to Weyl rescaling due to
the conformal invariance of the massless scalar. This leads to the effective
action

$\displaystyle I_{sc}=\frac{1}{2}\int
dt\int^{\sigma_{0}}_{-\infty}d\sigma\left[\frac{1-\frac{1}{2}e^{2t+\sigma}}{\tilde{\sigma}-\sigma}(\partial_{t}\eta)^{2}-2(\partial_{t}\eta)(\partial_{\sigma}\eta)\right].$
(4.25)

If we take the asymptotic limit given by $\sigma\to-\infty$ and
$\tilde{\sigma}\to\infty$, this action is simply approximated by

$\displaystyle I_{sc}=\frac{1}{2}\int
dt\int^{\sigma_{0}(t)}_{-\infty}d\sigma\left[-2(\partial_{t}\eta)(\partial_{\sigma}\eta)\right],$
(4.26)

which suggests that $t$ and $\sigma$ plays a role of null coordinate. The
metric is approximated by

$\displaystyle ds^{2}\simeq 2e^{-t}dtd\sigma.$ (4.27)

On the other had, in the opposite limit $\sigma\simeq\sigma_{0}$, the action
is estimated as

$\displaystyle I_{sc}=\frac{1}{2}\int
dt\int^{\sigma_{0}(t)}_{-\infty}d\sigma\left[\frac{1}{2}(\partial_{t}\eta)^{2}-2(\partial_{t}\eta)(\partial_{\sigma}\eta)\right],$
(4.28)

which shows that $t$ is null and $\sigma$ is space-like. The metrix is
approximated by

$\displaystyle
ds^{2}\simeq(\sigma_{0}-\sigma)e^{-t}\left(2dtd\sigma+\frac{1}{2}d\sigma^{2}\right).$
(4.29)

From the above analysis shows that this target spacetime geometry is given by
a time-like Liouville wall, as sketched in the right of Fig.2. Therefore, we
expect that this is a regular time-dependent background in $c=1$ matrix.

##  5 Two-dimensional de Sitter Gravity

In this section, let us briefly consider the Liouville CFT and the matrix
quantum mechanics which correspond to the two-dimensional de Sitter JT gravity
[60, 61]. From the discussion of section 3.2, it looks like we just need to
change the sign of the bulk cosmological constant (i.e. $\mu\to-\mu$) in order
to get dS2 JT gravity in the semicalassical limit. Since the bulk cosmological
constant appears inside of the logarithm as in (B.10), the change of the sign
induces a pure imaginary factor $i$ inside of the log in (3.13), so that for
the background solutions we now have

$\displaystyle\phi\,$
$\displaystyle=\,\frac{1}{2b}\log\left(\frac{-4}{(1-r^{2})^{2}}\right)\,-\,Ab\,\frac{1+r^{2}}{1-r^{2}}\,,$
(5.1) $\displaystyle\chi\,$
$\displaystyle=\,-\frac{1}{2b}\log\left(\frac{-4}{(1-r^{2})^{2}}\right)\,-\,Ab\,\frac{1+r^{2}}{1-r^{2}}\,.$
(5.2)

Then for the metric, we find the “negative AdS2” as discussed in [60]

$\displaystyle e^{2\rho}\,=\,-\,\frac{4}{(1-r^{2})^{2}}\,,\qquad\
\Phi\,=\,A\,\frac{1+r^{2}}{1-r^{2}}\,.$ (5.3)

This overall minus sign gives a $-i$ factor for the tree-level on-shell action
as

$\displaystyle I^{bulk}_{L}+I^{bdy}_{L}+I_{ct}\,=\,4\pi iA\,.$ (5.4)

For the Schwarzian action discussed in section 3.6, again the overall minus
sign of the metric gives an overall $-i$ factor. Therefore, the wavefunction
of the two-dimensional de Sitter universe $\Psi_{+}$ is given from the AdS
partition function by [60]

$\displaystyle\Psi_{+}\,=\,Z_{{\rm AdS}_{2}}\big{(}A_{\rm AdS}\to-iA_{\rm
dS}\big{)}.$ (5.5)

In the random matrix viewpoint, the original disk partition function $Z_{{\rm
AdS}_{2}}$ in AdS2 is given by $\mbox{Tr}[e^{-L\hat{H}}]$, where $\hat{H}$ is
the Hamiltonian and $L$ is the boundary length. This is transformed into the
wave function $\Psi_{+}$ in dS2 by replacing $L$ with $-iL$ as
$\mbox{Tr}[e^{iL\hat{H}}]$ as argued in [60]. This implies that the matrix
quantum mechanics, we discussed in section 4.1, does not change, except that
we replace the boundary length parameter $L$ with $-iL$. Indeed, we can
confirm that the matrix model description is equivalent both for AdS2 and dS2.
This is because in terms of the Fermi surface of the matrix model (4.6), the
change of the sign of the bulk cosmological constant corresponds to the
exchange of $x$ and $p$. Therefore, the Fermi surface (see Fig. 1) now moves
into $p-x<0$ quadrant as $t$ grows. However, this does not mean any physical
change and indeed the both are related by a canonical transformation.

##  6 Conclusions

In this paper, we argued an equivalence between the JT gravity on an anti de-
Sitter space and the two dimensional string theory defined by a time-like
Liouville CFT coupled to the space-like Liouville one. We confirmed that their
actions, disk partition functions and annulus amplitudes perfectly agree with
each other, where the presence of boundary terms plays an important role. We
also reproduced the boundary Schwarzian theory from the Liouville theory
description. Our two dimensional string theory looks different from the
minimal string theory in that the latter assumes the truncation to a rational
CFT, while the former does not. Nevertheless, as we explicitly worked out in
this paper, our two dimensional string theory which is equivalent to the JT
gravity, has a non-perturbative description in terms of a time-dependent
background in the $c=1$ matrix quantum mechanics. It will be an important
future problem to prove the proposed correspondence between the JT gravity and
the matrix quantum mechanics by explicitly evaluating the disk amplitudes of
the latter. For this we need to precisely identify the form of loop operator
in the matrix quantum mechanics, which fits with our $c<1$ two dimensional
string theory. It will be also intriguing to explore what we can learn about
holography in de-Sitter space

## Acknowledgements

We are grateful to Douglas Stanford for comments on the draft and Tomonori
Ugajin for useful discussions. This work is supported by the Simons Foundation
through the “It from Qubit” collaboration. TT is also supported by Inamori
Research Institute for Science and World Premier International Research Center
Initiative (WPI Initiative) from the Japan Ministry of Education, Culture,
Sports, Science and Technology (MEXT), by JSPS Grant-in-Aid for Scientific
Research (A) No. 21H04469 and by JSPS Grant-in-Aid for Challenging Research
(Exploratory) 18K18766.

##  Appendix A Boundary $\rho K^{(0)}$ contribution

As we commented in the footnote 6, if we use (3.5) directly in (3.4), what we
obtain is

$\displaystyle I^{\textrm{bdy}}_{\rm L}\,=\,4\int
d\theta\,(\rho-\Phi)K^{(0)}\,.$ (A.1)

Therefore, we find an additional term containing $\rho K^{(0)}$ in the
boundary action. In this appendix we make some comments on the effects of this
additional term.

First let us consider the contribution of this term to the on-shell action

$\displaystyle I^{\textrm{bdy}}_{\rm L}\,\supset\,4\int d\theta\,\rho
K^{(0)}\,\simeq\,2b\int
d\theta\,(\phi_{0}-\chi_{0})\,\simeq\,-8\pi\log\delta\,.$ (A.2)

This is diverging, so it does not give any finite contribution to the on-shell
action, but we need an additional counterterm to eliminate this diverging
contribution. The additional counterterm can be taken as the following
expression

$\displaystyle I^{\textrm{ct}}_{\rm extra}\,=\,-4\int
d\theta\,\rho\,\simeq\,8\pi\log\delta\,,$ (A.3)

which indeed eliminates the diverging contribution coming from the additional
boundary term.

Next, we consider the derivation of the Schwarzian action discussed in section
3.6. If we include the additional boundary and counter terms, instead of
(3.56), we now obtain

$\displaystyle I_{\rm L}^{\textrm{bdy}+\textrm{ct}}\,=\,4\int
d\theta\,(\rho-\Phi)(K^{(0)}-1)\,.$ (A.4)

Using the expansion $r(\theta)=1-\epsilon f^{\prime}(\theta)$, we find

$\displaystyle\rho\,=\,\frac{b}{2}(\phi-\chi)\,=\,-\log\epsilon\,+\,\mathcal{O}(\epsilon)\,.$
(A.5)

Combining with (3.58), we can see that the additional boundary term does not
lead to any finite contribution in the $\epsilon\to 0$ limit. Therefore, even
if we include the additional boundary term, the effective action is still
given by the Schwarzian action (3.59).

It might be interesting to further clarify this discrepancy between the double
Liouville theory and JT gravity, but we leave this study to future work and we
will not further comment on this additional boundary term in this paper.

##  Appendix B Boundary conditions of classical Liouville theory

In this appendix, we study classical solutions of Liouville theory defined on
a disk with all possible boundary conditions. Since all discussion of this
appendix applies to the timelike Liouville theory in the same manner, here we
focus only on the spacelike Liouville theory:

$\displaystyle S\,=\,\int
d^{2}x\sqrt{g}\,\Big{[}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,+\,\mu\,e^{2b\phi}\Big{]}\,.$
(B.1)

We use the coordinates (3.2) and in the following we consider static
solutions. (Here by static, we mean $\partial_{\theta}\phi=0$.) Variation of
the action on this coordinates can be written explicitly as

$\displaystyle\delta
S\,=\,2\int_{r=1}d\theta\,(\partial_{r}\phi)\,\delta\phi\,+\,2\int
drd\theta\,r\,\left[-\frac{1}{r}\partial_{r}(r\partial_{r}\phi)\,+\,b\mu\,e^{2b\phi}\right]\delta\phi\,.$
(B.2)

In order for the boundary term to vanish, we can take either boundary
condition

$\displaystyle{\rm Dirichlet}:\qquad\phi(r=1)\,=\,\phi_{0}\quad(=\,{\rm
constant})$ (B.3) $\displaystyle{\rm
Neumann}:\qquad\partial_{r}\phi\big{|}_{r=1}\,=\,0\,.$ (B.4)

If we include an additional boundary action

$\displaystyle S_{\rm bdy}\,=\,\int_{r=1}d\theta\,4\pi\mu_{B}\,e^{b\phi}\,,$
(B.5)

we can have the FZZT brane (modified Neumann) boundary condition [57, 58]:

$\displaystyle{\rm
FZZT}:\qquad\partial_{r}\phi\,+\,2\pi\mu_{B}b\,e^{b\phi}\,=\,0\,,\qquad({\rm
at}\ r=1)$ (B.6)

The ZZ brane boundary condition [77] is a special case of the Dirichlet
boundary condition with $\phi_{0}=\infty$.

The equation of motion can be read off from the second term of (B.2) as

$\displaystyle-\frac{1}{r}\partial_{r}(r\partial_{r}\phi)\,+\,b\mu\,e^{2b\phi}\,=\,0\,,$
(B.7)

and the most general solution is given by

$\displaystyle e^{2b\phi(r)}\,=\,\frac{1}{\mu
b^{2}}\frac{c_{1}^{2}}{r^{2}\sinh[c_{1}(c_{2}-\log r)]^{2}}\,,$ (B.8)

where $c_{1}$ and $c_{2}$ are integration constants. We first require a
regularity condition at the center of the disk ($r\to 0$). Since

$\displaystyle e^{2b\phi(0)}\,\sim\,\frac{1}{\mu
b^{2}}\frac{4c_{1}^{2}}{r^{2-2c_{1}}}\,,$ (B.9)

the regularity condition fixes $c_{1}=1$. Now the solution (B.8) becomes

$\displaystyle e^{2b\phi(r)}\,=\,\frac{1}{\mu
b^{2}}\frac{4c^{2}}{(1-c^{2}r^{2})^{2}}\,,$ (B.10)

where we set $c=e^{-c_{2}}$.

The Dirichlet boundary condition requires

$\displaystyle e^{2b\phi(1)}\,=\,\frac{1}{\mu
b^{2}}\frac{4c^{2}}{(1-c^{2})^{2}}\,=\,e^{2b\phi_{0}}\,.$ (B.11)

Therefore, if we set $c^{2}=1-p$ (with $\mu b^{2}=1$), we find the solution
(3.19). The ZZ brane boundary condition is a special case of this result with
$c=1$, which gives

$\displaystyle e^{2b\phi(r)}\,=\,\frac{1}{\mu
b^{2}}\frac{4}{(1-r^{2})^{2}}\,.$ (B.12)

The FZZT brane boundary condition fixes the remaining constant as

$\displaystyle c\,=\,-\frac{2\pi\mu_{B}b^{2}}{\sqrt{\mu b^{2}}}\,.$ (B.13)

Therefore, the standard Neumann boundary condition ($\mu_{B}=0$) leads to the
trivial solution ($c=0$).

##  Appendix C On-shell actions of Liouville theory

In this appendix, we present some detail of the on-shell actions for the disk
geometry discussed in section 3.3 and for the punctured disk geometry
discussed in section 3.5.

###  C.1 Disk geometry

The on-shell action for $\phi$ reads

$\displaystyle I^{bulk}_{\phi}=\int
d\theta\left[\left(\frac{2}{b^{2}\delta}+\frac{2}{b^{2}}(\log
2-1)+\frac{2}{b^{2}}\log\delta-\frac{\delta}{b^{2}}\right)+\eta\left(\frac{1}{b^{2}\delta}-\frac{1}{b^{2}}+\frac{\delta}{2b^{2}}\right)\right],$
$\displaystyle I^{bdy}_{\phi}=\frac{2}{b}\int\phi_{0}d\theta=\int
d\theta\left[-\frac{2}{b^{2}}\log\delta+\frac{\eta}{b^{2}}\right],$
$\displaystyle I^{ct}_{\phi}=-\frac{2}{b^{2}}\int d\theta
e^{b\phi_{0}}=-\frac{2}{b^{2}}\int
d\theta\left[\frac{1}{\delta}+\frac{\eta}{2\delta}\right],$ (C.1)

which give the total action

$\displaystyle I^{bulk}_{\phi}+I^{bdy}_{\phi}+I^{ct}_{\phi}$ $\displaystyle=$
$\displaystyle\int d\theta\left[\frac{\delta}{2b^{2}}\eta+\frac{2}{b^{2}}(\log
2-1)-\frac{\delta}{b^{2}}\right]$ (C.2) $\displaystyle=$ $\displaystyle\int
d\theta\left[-A+\frac{2}{b^{2}}(\log 2-1){-\frac{\delta}{b^{2}}}\right].$

On the other hand, for the field $\chi$ we find

$\displaystyle I^{bulk}_{\chi}=\int
d\theta\left[\left(-\frac{2}{b^{2}\delta}-\frac{2}{b^{2}}(\log
2-1)-\frac{2}{b^{2}}\log\delta+\frac{\delta}{b^{2}}\right)+\eta\left(\frac{1}{b^{2}\delta}-\frac{1}{b^{2}}+\frac{\delta}{2b^{2}}\right)\right],$
$\displaystyle I^{bdy}_{\chi}=\frac{2}{b}\int\chi_{0}d\theta=\int
d\theta\left[\frac{2}{b^{2}}\log\delta+\frac{\eta}{b^{2}}\right],$
$\displaystyle I^{ct}_{\chi}=-\frac{2}{b^{2}}\int d\theta
e^{-b\chi_{0}}=-\frac{2}{b^{2}}\int
d\theta\left[-\frac{1}{\delta}+\frac{\eta}{2\delta}\right],$ (C.3)

which give the total action

$\displaystyle I^{bulk}_{\chi}+I^{bdy}_{\chi}+I^{ct}_{\chi}$ $\displaystyle=$
$\displaystyle\int d\theta\left[\frac{\delta}{2b^{2}}\eta-\frac{2}{b^{2}}(\log
2-1)+\frac{\delta}{b^{2}}\right]$ (C.4) $\displaystyle=$ $\displaystyle\int
d\theta\left[-A-\frac{2}{b^{2}}(\log 2-1)+\frac{\delta}{b^{2}}\right].$

###  C.2 Punctured disk geometry

As we consider bulk one-point function, we include the contribution from the
vertex operator into the bulk actions as

$\displaystyle I^{bulk}_{\phi}\,$ $\displaystyle=\,\int
d^{2}x\sqrt{g}\,\Big{[}(\partial_{a}\phi)^{2}+\mu
e^{2b\phi}\Big{]}\,-\,\alpha\int d\theta\,\phi(\epsilon)\,,$ $\displaystyle
I^{bulk}_{\chi}\,$ $\displaystyle=\,\int
d^{2}x\sqrt{g}\,\Big{[}-(\partial_{a}\chi)^{2}-\mu
e^{-2b\chi}\Big{]}\,-\,\alpha\int d\theta\chi(\epsilon)\,.$ (C.5)

Since the bulk fields $\phi$, $\chi$ are diverging at $r=0$, we regularize
this by integrating $\epsilon\leq r\leq 1$ and inserting the vertex operators
at $r=\epsilon$. Then, we simply eliminate all diverging contribution in
$\epsilon\to 0$ limit. Now the on-shell action for $\phi$ reads

$\displaystyle I^{bulk}_{\phi}\,=\,\frac{1}{b^{2}}\int
d\theta\left[\left(\frac{2}{\delta}-\nu(2+\nu\delta)+2\nu\log
2\nu+2\log\delta\right)\,+\,\eta\left(\frac{1}{\delta}-1+\frac{\nu^{2}\delta}{2}\right)\right]\,,$
$\displaystyle
I^{bdy}_{\phi}\,=\,\frac{2}{b}\int\phi_{0}d\theta\,=\,\frac{1}{b^{2}}\int
d\theta\Big{[}-2\log\delta+\eta\Big{]}\,,$ $\displaystyle
I^{ct}_{\phi}\,=\,-\,\frac{2}{b^{2}}\int d\theta
e^{b\phi_{0}}\,=\,-\frac{1}{b^{2}}\int
d\theta\left[\frac{2}{\delta}+\frac{\eta}{\delta}\right],$ (C.6)

which give the total action

$\displaystyle I^{bulk}_{\phi}+I^{bdy}_{\phi}+I^{ct}_{\phi}\,$
$\displaystyle=\,\frac{1}{b^{2}}\int
d\theta\left[\frac{\nu^{2}\delta\eta}{2}+2\nu\log
2\nu-\nu(2+\nu\delta)\right]\,.$ (C.7)

On the other hand, for the field $\chi$ we find

$\displaystyle I^{bulk}_{\chi}\,=\,\frac{1}{b^{2}}\int
d\theta\left[\left(-\frac{2}{\delta}+\nu(2+\nu\delta)-2\nu\log
2\nu-2\log\delta\right)\,+\,\eta\left(\frac{1}{\delta}-1+\frac{\nu^{2}\delta}{2}\right)\right]\,,$
$\displaystyle
I^{bdy}_{\chi}\,=\,\frac{2}{b}\int\chi_{0}d\theta\,=\,\frac{1}{b^{2}}\int
d\theta\Big{[}2\log\delta+\eta\Big{]}\,,$ $\displaystyle
I^{ct}_{\chi}\,=\,-\frac{2}{b^{2}}\int d\theta
e^{-b\chi_{0}}\,=\,-\frac{1}{b^{2}}\int
d\theta\left[-\frac{2}{\delta}+\frac{\eta}{\delta}\right],$ (C.8)

which give the total action

$\displaystyle I^{bulk}_{\chi}+I^{bdy}_{\chi}+I^{ct}_{\chi}\,$
$\displaystyle=\,\frac{1}{b^{2}}\int
d\theta\left[\frac{\nu^{2}\delta\eta}{2}-2\nu\log
2\nu+\nu(2+\nu\delta)\right]\,.$ (C.9)

Therefore, the total of the spacelike and timelike contributions is

$\displaystyle I_{\phi}\,+\,I_{\chi}\,=\,-2\nu^{2}A\int
d\theta\,=\,-4\pi\nu^{2}A\,.$ (C.10)

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