11institutetext: Yuki Sato 22institutetext: National Institute of Technology,
Tokuyama College, Gakuendai, Shunan, Yamaguchi 745-8585, Japan, 22email:
<EMAIL_ADDRESS>33institutetext: Nagoya University, Chikusaku, Nagoya
464-8602, Japan, 33email<EMAIL_ADDRESS>
# CDT and Hořava-Lifshitz QG
in Two Dimensions

Yuki Sato

###### Abstract

Two-dimensional causal dynamical triangulations ($2$d CDT) is a lattice model
of quantum geometry. In $2$d CDT, one can deal with the quantum effects
analytically and explore the physics through the continuum limit. The
continuum theory is known to be two-dimensional projectable Hořava-Lifshitz
quantum gravity ($2$d projectable HL QG). In this chapter, we wish to review
the very relation between $2$d CDT and $2$d projectable HL QG in detail.

Keywords

Lattice quantum gravity; Causal dynamical triangulations (CDT); Hořava-
Lifshitz quantum gravity; Two-dimensional models; Wormholes

Note This is a preprint of the chapter in the Handbook of Quantum Gravity. It
belongs to the section entitled Causal Dynamical Triangulations.

##  1 Introduction

Two-dimensional toy models of quantum gravity are a very useful playground for
understanding the quantum nature of geometry quantitatively. This is because
many of them are simple enough to be dealt with analytically and complex
enough to observe nontrivial quantum effects. Lattice regularizations in
particular are known to be quite powerful tools for investigating non-
perturbative quantum effects analytically. Among these, two-dimensional
Euclidean dynamical triangulations ($2$d EDT) Ambjorn:1985az ; Ambjorn:1985dn
; David:1984tx ; Billoire:1985ur ; Kazakov:1985ea ; Boulatov:1986jd  (see a
pedagogical textbook Ambjorn:1997di ) and two-dimensional causal dynamical
triangulations ($2$d CDT) Ambjorn:1998xu  (see a detailed review
Ambjorn:2022btk ) are good practical examples. The former and the latter,
respectively, are Euclidean and Lorentzian lattice models based on Regge’s
discretization of geometries Regge:1961px .

$2$d EDT discretizes Euclidean geometries by equilateral triangles and defines
a regularized quantum amplitude as a sum over distinct triangulated
geometries. Matrix models and combinatorics can be used for calculating such a
statistical sum analytically (whenever possible). By virtue of analytic
treatments, one can explicitly remove the regularization through the continuum
limit to calculate physical observables. What is remarkable is that exactly
the same value of observables can be reproduced from a genuine continuum field
theory called the Liouville quantum gravity Polyakov:1981rd ; Knizhnik:1988ak
; David:1988hj ; Distler:1988jt . This means that $2$d EDT serves as a well-
defined regularization of the Liouville quantum gravity.

$2$d CDT is a Lorentzian lattice model of quantum geometry which respects a
global time foliation and prohibits the creation of so-called baby universes.
One can calculate the sum over such Lorentzian triangulated geometries using
simple combinatorics, and take the continuum limit to remove the
regularization. All these processes can be performed analytically at least for
the plain model without coupling to a matter. It has been shown in Ref.
Ambjorn:2013joa  that the resulting continuum theory is known to be in the
same universality class of projectable Hořava-Lifshitz quantum gravity
(projectable HL QG) horava  in two dimensions, which is different from the
Liouville quantum gravity 111In fact, it has been shown that a direct lattice
discretization of $2$d projectable HL gravity, which has a lattice action
different from that of $2$d CDT, reproduces the same large-scale physics in
the continuum limit Glaser:2016smx ..

$2$d HL QG is a quantum field theory in two dimensions, which has a preferred
foliation structure. This model is invariant only under the subclass of
diffeomorphisms that respects the foliation, known as the foliation-preserving
diffeomorphisms222At the cost of full diffeomorphism invariance, HL QG has
been designed originally as a model of quantum gravity in higher dimensions
such that it has a good convergence at UV in keeping with unitarity, and would
approximately recover the diffeomorphism invariance at IR horava . $2$d
projectable HL QG is a certain version of HL QG where a part of the time-time
component of the metric called the lapse function is projectable, i.e. a
function only of time. In this chapter, we wish to explain in detail the
relation between projectable HL QG and CDT in two dimensions333The relation
between HL QG and CDT in four dimension has been pointed out first in Ref.
Horava:2009if  by looking at an observable called the spectral dimension, and
the resemblance of the CDT phase diagram to a Lifshitz phase diagram has been
shown in Ref. Ambjorn:2010hu ..

In fact one can generalize $2$d CDT in such a way that the creation and
annihilation of (a finite number of) baby universes, and the formation of
wormholes (handles) are allowed to occur in keeping with the foliation
structure. This model is called the generalized CDT (GCDT) introduced first as
a continuum theory Ambjorn:2007jm ; Ambjorn:2008ta  and later defined at the
discrete level Ambjorn:2008gk ; Ambjorn:2013csx . The full continuum
description of GCDT is given by the so-called string field theory for CDT
Ambjorn:2008ta  in which the string means the one-dimensional closed spatial
universe, and the baby universes and wormholes can be realized in terms of the
splitting and joining interactions of strings.

One of remarkable facts is that focusing on a certain amplitude, i.e. loop-to-
loop amplitude, one can read off a one-dimensional effective theory that takes
in all possible baby universe and wormhole contributions in an effective
manner Ambjorn:2009wi ; Ambjorn:2009fm : The $1$d effective theory is a one-
body quantum theory even though GCDT is a many-body theory that allows both
creation and annihilation of strings. It is known that one can correctly
reproduce the $1$d effective theory if quantizing the projectable HL gravity
with a certain bi-local interaction term Ambjorn:2021wou ; Ambjorn:2021ysb .
This topic will be treated in this chapter.

Furthermore, the $1$d effective theory that includes all contributions of baby
universes and wormholes is known to be reproduced if one assumes that the
cosmological constant of the continuum limit of $2$d CDT is not really a
constant but fluctuates in time Ambjorn:2021wdm . This idea leads to a certain
realization of Coleman’s mechanism about the cosmological constant
Coleman:1988tj  in the context of CDT Ambjorn:2021wdm , which will be also
explained in this chapter.

The rest of this chapter is organized as follows. In Sec. 2, a self-contained
introduction to $2$d CDT is presented. We show that taking the continuum limit
the physics of $2$d CDT can be described as a one-dimensional quantum system
with a Hamiltonian. $2$d projectable HL QG is explained in Sec. 3. Through the
path-integral quantization we read off the quantum Hamiltonian that is
equivalent to the one obtained in the continuum limit of $2$d CDT. Thereby one
can confirm that the continuum limit of $2$d CDT is $2$d projectable HL QG. In
Sec. 4, we introduce GCDT that is a generalization of $2$d CDT such that baby
universes and wormholes are introduced so as to be compatible with the
foliation, and determine the $1$d effective theory obtained through the sum
over all genera. In particular, we explain in detail that quantizing $2$d
projectable HL gravity with a bi-local interaction yields the $1$d effective
theory, and discuss Coleman’s mechanism in $2$d CDT. Sec. 5 is devoted to
summary.

##  2 $2$d causal dynamical triangulations

Two-dimensional causal dynamical triangulations ($2$d CDT) Ambjorn:1998xu  is
a lattice model of quantum geometries based on Regge’s discretization
Regge:1961px . In this section, we give an overview of $2$d CDT, and in
particular explain how to obtain the quantum Hamiltonian through the continuum
limit.

We start with a two-dimensional globally hyperbolic manifold equipped with a
global time foliation:

$\displaystyle\mathcal{M}=\bigcup_{t\in\mathbb{R}}\Sigma_{t}\ ,$ (1)

where each leaf $\Sigma_{t}$ is a one-dimensional Cauchy “surface” (line). One
approximates the manifold with a foliation in such a way that the continuous
label $t$ is discretized by integers, i.e. $t\in\mathbb{Z}$; each leaf (line)
is partitioned by vertices connected by isometric edges; vertices among
neighboring time steps are connected by isometric edges to form a
triangulation of strip (see Fig. 1).

Figure 1: A triangulation of a strip: Thick and thin lines are space-like and
time-like edges, respectively.

The edges at a given time step and those connecting vertices in different time
steps are, respectively, space-like and time-like edges since the squared edge
lengths of the space-like edge $a^{2}_{s}$ and the time-like edge $a^{2}_{t}$
are given by

$\displaystyle a^{2}_{s}=\varepsilon^{2}\ ,\ \ \
a^{2}_{t}=-\alpha\varepsilon^{2}\ ,$ (2)

where $\alpha$ is a positive number and $\varepsilon$ is a lattice spacing
that serves as a UV cutoff.

$2$d CDT deals with a set of restricted class of Lorentzian triangulations as
discussed above. In particular, we consider that the topology of the one-
dimensional universe (a graph consisting of vertices and edges at a given
time) is either $S^{1}$ or $[0,1]$, and the topology will not change during
(discrete) time propagation. Since the topology is fixed the curvature term
plays no role in two dimensions and only the discrete analogue of the
cosmological constant term, $\Lambda_{0}\int d^{2}x\sqrt{-g}$ is used as the
lattice action of $2$d CDT:

$\displaystyle
S_{T}[\lambda,\alpha]=-\frac{\lambda}{\varepsilon^{2}}\left(\frac{\sqrt{4\alpha+1}}{4}\varepsilon^{2}\
n(T)\right)\ ,$ (3)

where $\lambda/\varepsilon^{2}$ is the bare cosmological constant with the
dimensionless number $\lambda$, $n(T)$ the number of triangles in a
triangulation $T$, and the term inside the parentheses denotes the total area
of the triangulation. It is useful to rotate to the Euclidean signature which
can be performed by changing $\alpha\to-\alpha-i0$. Accordingly the lattice
action (3) changes as follows:

$\displaystyle iS_{T}[\lambda,\alpha]\to
iS_{T}[\lambda,-\alpha-i0]=-\lambda\frac{\sqrt{4\alpha-1}}{4}n(T)\equiv-\lambda
n(T)\ ,$ (4)

where $\alpha$ is chosen to be greater than $1/4$; otherwise, the triangle
inequalities will not be satisfied after the rotation. In any case we have
absorbed the parameter $\alpha$ by the redefinition of the dimensionless
cosmological constant $\lambda$.

The amplitude of the one-dimensional universe that starts with $\ell_{1}$
edges and ends up with $\ell_{2}$ edges after the discrete time step $t$ is
given by the sum over all allowed triangulations:

$\displaystyle
G^{(a)}_{\lambda}(\ell_{1},\ell_{2};t)=\sum_{T\in\mathcal{T}^{(a)}(\ell_{1},\ell_{2},t)}e^{-\lambda
n(T)}=\sum_{n}e^{-\lambda n}\mathcal{N}^{(a)}(\ell_{1},\ell_{2},n)\ ,$ (5)

where $\mathcal{T}^{(a)}$ is a set of triangulations whose topology is
$[0,1]\times[0,1]$ for $a=0$ and $S^{1}\times[0,1]$ for $a=1$, and

$\displaystyle\mathcal{N}^{(a)}(\ell_{1},\ell_{2},n)=\\#\left\\{T\in\mathcal{T}^{(a)}\
\bigl{|}\ n(T)=n\right\\}\ .$ (6)

When defining the amplitude (5), we do not allow the one-dimensional universe
to vanish during the discrete time propagation. For later convenience, we
introduce a marked amplitude:

$\displaystyle
G^{(-1)}_{\lambda}(\ell_{1},\ell_{2};t)=\ell_{1}G^{(1)}_{\lambda}(\ell_{1},\ell_{2};t)\
,$ (7)

where one of the edges in the initial one-dimensional universe is marked. This
is because there exist $\ell_{1}$ possible ways of marking the edges. The
three kinds of amplitude should satisfy the composition law:

$\displaystyle G^{(1)}_{\lambda}(\ell_{1},\ell_{2};t_{1}+t_{2})$
$\displaystyle=\sum^{\infty}_{\ell=1}G^{(1)}_{\lambda}(\ell_{1},\ell;t_{1})\
\ell\ G^{(1)}_{\lambda}(\ell,\ell_{2};t_{2})\ ,$ (8) $\displaystyle
G^{(0)}_{\lambda}(\ell_{1},\ell_{2};t_{1}+t_{2})$
$\displaystyle=\sum^{\infty}_{\ell=1}G^{(0)}_{\lambda}(\ell_{1},\ell;t_{1})\
G^{(0)}_{\lambda}(\ell,\ell_{2};t_{2})\ ,$ (9) $\displaystyle
G^{(-1)}_{\lambda}(\ell_{1},\ell_{2};t_{1}+t_{2})$
$\displaystyle=\sum^{\infty}_{\ell=1}G^{(-1)}_{\lambda}(\ell_{1},\ell;t_{1})\
G^{(-1)}_{\lambda}(\ell,\ell_{2};t_{2})\ ,$ (10)

where for $a=1$ one needs to multiply the amputated amplitudes by $\ell$ since
there exist $\ell$ possible ways of gluing to recover the whole amplitude.

It is convenient to introduce the generating function of the number of
triangulations. Using the notation

$\displaystyle g=e^{-\lambda}\ ,$ (11)

we define the generating function:

$\displaystyle\widetilde{G}^{(a)}(g,x,y;t)$
$\displaystyle=\sum^{\infty}_{\ell_{1}=1}\sum^{\infty}_{\ell_{2}=1}x^{\ell_{1}}y^{\ell_{2}}G^{(a)}_{\lambda}(\ell_{1},\ell_{2};t)$
$\displaystyle=\sum^{\infty}_{\ell_{1}=1}\sum^{\infty}_{\ell_{2}=1}\sum_{n}x^{\ell_{1}}y^{\ell_{2}}g^{n}\mathcal{N}^{(a)}(\ell_{1},\ell_{2},n)\
,$ (12)

where in the context of quantum gravity, $x$ and $y$ are related to the
boundary cosmological constants, $\lambda_{1}$ and $\lambda_{2}$, that control
the size of the boundaries:

$\displaystyle x=e^{-\lambda_{1}}\ ,\ \ \ y=e^{-\lambda_{2}}\ .$ (13)

One can reconstruct the amplitude from the generating function through the
following relation:

$\displaystyle
G^{(a)}_{\lambda}(\ell_{1},\ell_{2};t)=\oint_{\mathcal{C}_{1}}\frac{dx}{2\pi
ix^{\ell_{1}+1}}\oint_{\mathcal{C}_{2}}\frac{dy}{2\pi iy^{\ell_{2}+1}}\
\widetilde{G}^{(a)}(g,x,y;t)\ ,$ (14)

where the contour $\mathcal{C}_{1}$ ($\mathcal{C}_{2}$) is chosen to enclose
$x=0$ ($y=0$) and to ensure the convergence of $\widetilde{G}^{(a)}(g,x,y;t)$.
One can derive the relation (14) using the identity:

$\displaystyle\oint_{\mathcal{C}}\frac{dz}{2\pi iz^{n+1}}=\delta_{n,0}\ ,\ \ \
(n\in\mathbb{Z})\ ,$ (15)

where the contour $\mathcal{C}$ encloses $z=0$. We provide the composition law
for the generating function when $a=0,-1$ in preparation for later
calculations:

$\displaystyle\widetilde{G}^{(a)}(g,x,y;t_{1}+t_{2})=\oint_{\mathcal{C}}\frac{dz}{2\pi
iz}\widetilde{G}^{(a)}(g,x,z^{-1};t_{1})\ \widetilde{G}^{(a)}(g,z,y;t_{2})\ ,\
\ (a=0,-1)\ ,$ (16)

where the contour encloses $z=0$, and for fixed $g$, $x$ and $y$ lies inside
the radius of convergence for $\widetilde{G}^{(a)}(g,x,z^{-1};t_{1})$ as the
series in $1/z$ and for $\widetilde{G}^{(a)}(g,z,y;t_{2})$ as the series in
$z$, which is possible as we will see.

In what follows, we will discuss the one-step amplitude
$G^{(a)}_{\lambda}(\ell_{1},\ell_{2};1)$. This is because it becomes an
important object when computing the whole amplitude.

###  2.1 Counting triangulations

In this section we focus on the one-step amplitude
$G^{(a)}_{\lambda}(\ell_{1},\ell_{2};1)$ which is the sum over triangulations
of a strip as shown in Fig. 1:

$\displaystyle
G^{(a)}_{\lambda}(\ell_{1},\ell_{2};1)=e^{-\lambda(\ell_{1}+\ell_{2})}\mathcal{N}^{(a)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})\
,$ (17)

and count the number of triangulations
$\mathcal{N}^{(a)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})$. Based on simple
combinatorics, one can calculate the case of $a=1$ which has the
$S^{1}\times[0,1]$ topology:

$\displaystyle\mathcal{N}^{(1)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})=\frac{1}{\ell_{1}+\ell_{2}}\begin{pmatrix}\ell_{1}+\ell_{2}\\\
\ell_{1}\end{pmatrix}=\frac{(\ell_{1}+\ell_{2}-1)!}{\ell_{1}!\ell_{2}!}\ .$
(18)

Because of the property (7), one can easily compute the case of $a=-1$ that
the topology is $S^{1}\times[0,1]$ and one of the edges in the initial one-
dimensional universe is marked:

$\displaystyle\mathcal{N}^{(-1)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})=\ell_{1}\mathcal{N}^{(1)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})=\frac{\ell_{1}(\ell_{1}+\ell_{2}-1)!}{\ell_{1}!\ell_{2}!}\
.$ (19)

Concerning the case of $a=0$ whose topology is $[0,1]\times[0,1]$, there exist
several possibilities depending on the restriction on the leftmost and
rightmost triangles. If the rightmost triangle is the upward triangle
(downward triangle) and the leftmost triangles is the downward triangle
(upward triangle), then the counting of triangulations yields

$\displaystyle\mathcal{N}^{(0)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})=\begin{pmatrix}\ell_{1}+\ell_{2}-2\\\
\ell_{1}-1\end{pmatrix}=\frac{(\ell_{1}+\ell_{2}-2)!}{(\ell_{1}-1)!(\ell_{2}-1)!}\
.$ (20)

In the following, we will use eq. (20) in the case of $a=0$ for computational
simplicity444 If we do not impose any restriction on the leftmost and
rightmost triangles, the number of triangulations becomes
$\mathcal{N}^{(0)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})=(\ell_{1}+\ell_{2})!/(\ell_{1}!\ell_{2}!)$.
.

The one-step generating functions can be derived inserting eqs. (18), (19) and
(20) into eq. (12):

$\displaystyle\widetilde{G}^{(1)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{\ell_{1}=1}\sum^{\infty}_{\ell_{2}=1}x^{\ell_{1}}y^{\ell_{2}}g^{\ell_{1}+\ell_{2}}\mathcal{N}^{(1)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})$
$\displaystyle=-\ln\left(\frac{1-gx-gy}{(1-gx)(1-gy)}\right)\ ;$ (21)
$\displaystyle\widetilde{G}^{(-1)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{\ell_{1}=1}\sum^{\infty}_{\ell_{2}=1}x^{\ell_{1}}y^{\ell_{2}}g^{\ell_{1}+\ell_{2}}\mathcal{N}^{(-1)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})$
$\displaystyle=\frac{g^{2}xy}{(1-gx)(1-gx-gy)}\ ;$ (22)
$\displaystyle\widetilde{G}^{(0)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{\ell_{1}=1}\sum^{\infty}_{\ell_{2}=1}x^{\ell_{1}}y^{\ell_{2}}g^{\ell_{1}+\ell_{2}}\mathcal{N}^{(0)}(\ell_{1},\ell_{2},n=\ell_{1}+\ell_{2})$
$\displaystyle=\frac{g^{2}xy}{1-gx-gy}\ .$ (23)

In fact, one can also obtain eq. (22) through
$\widetilde{G}^{(-1)}(g,x,y;1)=x\frac{\partial}{\partial
x}\widetilde{G}^{(1)}(g,x,y;1)$.

Alternatively, it is possible to compute the one-step generating functions
directly by simple combinatorics:

$\displaystyle\widetilde{G}^{(1)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{s=1}\frac{1}{s}\left(\sum^{\infty}_{k=1}(gx)^{k}\sum^{\infty}_{l=1}(gy)^{l}\right)^{s}=-\ln\left(\frac{1-gx-
gy}{(1-gx)(1-gy)}\right)\ ;$ (24) $\displaystyle\widetilde{G}^{(-1)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{k=0}\left(gx\sum^{\infty}_{l=0}(gy)^{l}\right)^{k}-\sum^{\infty}_{k=0}(gx)^{k}=\frac{g^{2}xy}{(1-gx)(1-gx-
gy)}\ ;$ (25) $\displaystyle\widetilde{G}^{(0)}(g,x,y;1)$
$\displaystyle=\sum^{\infty}_{s=1}\frac{1}{s}\left(\sum^{\infty}_{k=1}(gx)^{k}\sum^{\infty}_{l=1}(gy)^{l}\right)^{s}=\frac{g^{2}xy}{1-gx-
gy}\ .$ (26)

###  2.2 Continuum limit

All is now set for computing the amplitude in the continuum limit. In this
section, however, instead of directly computing the amplitude in the continuum
limit, we will derive the differential equation that the continuum amplitude
satisfies.

Before going into details any further, let us explain some basic facts of the
continuum limit. In order to remove the cutoff $\varepsilon$ through the
continuum limit, one has to tune the bare coupling constants $(g,x,y)$ to
their critical values $(g_{c},x_{c},y_{c})$. At the critical values, the
generating function hits the radii of convergence and therefore becomes non-
analytic. Approaching such a critical point, infinitely many triangles and
boundary edges become important in the summation of the generating function,
i.e., essentially, the average number of triangles and boundary edges become
infinity at the critical point. Having this in mind, one may intuitively
understand that the continuous surface would be obtained if
$(g,x,y)\to(g_{c},x_{c},y_{c})$ and $\varepsilon\to 0$ in a correlated manner.

Introducing $\lambda_{c}=-\ln[g_{c}]$, $\lambda_{1c}=-\ln[x_{c}]$ and
$\lambda_{2c}=-\ln[y_{c}]$, one can transmute the dimension of the lattice
spacing $\varepsilon$ into the dimension of the renormalized bulk and boundary
cosmological constants through the continuum limit:

$\displaystyle\Lambda=\lim_{\begin{subarray}{c}\lambda\to\lambda_{c}\\\
\varepsilon\to 0\end{subarray}}\frac{\lambda-\lambda_{c}}{\varepsilon^{2}}\ ,\
\ \ X=\lim_{\begin{subarray}{c}\lambda_{1}\to\lambda_{1c}\\\ \varepsilon\to
0\end{subarray}}\frac{\lambda_{1}-\lambda_{1c}}{\varepsilon}\ ,\ \ \
Y=\lim_{\begin{subarray}{c}\lambda_{2}\to\lambda_{2c}\\\ \varepsilon\to
0\end{subarray}}\frac{\lambda_{2}-\lambda_{2c}}{\varepsilon}\ ,$ (27)

where $\Lambda$ is the renormalized bulk cosmological constant, and $X$ and
$Y$ are the renormalized boundary cosmological constants. Therefore, the
divergent bare cosmological constants get additive renormalizations so as to
obtain the finite renormalized cosmological constants that set the scale at
IR.

In the following, we discuss the continuum limit in detail with respect to
each topology of spacetime.

####  $S^{1}\times[0,1]$ topology

We consider the case of $a=-1$, i.e. $S^{1}\times[0,1]$ topology with a marked
boundary. Using the composition law (16) and the one-step generating function
(25), one obtains

$\displaystyle\widetilde{G}^{(-1)}(g,x,y;t+1)$
$\displaystyle=\oint_{\mathcal{C}}\frac{dz}{2\pi
iz}\widetilde{G}^{(-1)}(g,x,z^{-1};1)\ \widetilde{G}^{(-1)}(g,z,y;t)$
$\displaystyle=\oint_{\mathcal{C}}\frac{dz}{2\pi
i}\frac{g^{2}x}{(1-gx)^{2}(z-g/(1-gx))}\frac{\widetilde{G}^{(-1)}(g,z,y;t)}{z}$
$\displaystyle=\frac{gx}{1-gx}\
\widetilde{G}^{(-1)}\left(g,\frac{g}{1-gx},y;t\right)\ .$ (28)

In the last equality, we have picked up a pole at $z=g/(1-gx)$, and there
exists no pole at $z=0$ since $\frac{\widetilde{G}^{(-1)}(g,z,y;t)}{z}$ is
regular. Through iterative use of eq. (28), one can analytically compute the
generating function, and extract the information of the critical point
Ambjorn:1998xu . However, we do not compute the generating function directly
to obtain the critical point. Instead, we follow the procedure shown in
Ambjorn:2022btk : One assumes the existence of the critical point, and
determines the value of the critical coupling constants from the consistency.

We assume the critical point characterized by the critical coupling constants
$(g_{c},x_{c},y_{c})$ and use the following parametrization:

$\displaystyle g=g_{c}e^{-\varepsilon^{2}\Lambda},\ \ \ x=x_{c}e^{-\varepsilon
X}\ ,\ \ \ y=y_{c}e^{-\varepsilon Y}\ .$ (29)

Assuming the scalings

$\displaystyle T=\varepsilon t\ ,\ \ \ L_{1}=\varepsilon\ell_{1}\ ,\ \ \
L_{2}=\varepsilon\ell_{2}\ ,$ (30)

we introduce the renormalized amplitude and the renormalized generating
function at the critical point by the multiplicative renormalizations:

$\displaystyle G^{(-1)}_{\Lambda}(L_{1},L_{2};T)$
$\displaystyle=\lim_{\varepsilon\to 0}C_{\varepsilon}\
G^{(-1)}_{\lambda}(\ell_{1},\ell_{2};t)\ ,$ (31)
$\displaystyle\widetilde{G}^{(-1)}_{\Lambda}(X,Y;T)$
$\displaystyle=\lim_{\varepsilon\to 0}\widetilde{C}_{\varepsilon}\
\widetilde{G}^{(-1)}(g,x,y;t)\ ,$ (32)

where $C_{\varepsilon}$ and $\widetilde{C}_{\varepsilon}$ are real functions
of $\varepsilon$ that will be fixed later. One can determine the critical
coupling constants for the consistency. Using the scaling behavior (32), eq.
(28) can yield the sensible continuum limit if the critical coupling constants
satisfy

$\displaystyle\frac{g_{c}x_{c}}{1-g_{c}x_{c}}=1\ ,\ \ \
\frac{g_{c}}{1-g_{c}x_{c}}=x_{c}\ \ \ \Rightarrow\ \ \ g_{c}=\frac{1}{2}\ ,\ \
\ x_{c}=1\ .$ (33)

Let us determine the function $C_{\varepsilon}$ in such a way that the
composition law (10) holds in the continuum limit as

$\displaystyle
G^{(-1)}_{\Lambda}(L_{1},L_{2};T_{1}+T_{2})=\int^{\infty}_{0}dL\
G^{(-1)}_{\Lambda}(L_{1},L;T_{1})G^{(-1)}_{\Lambda}(L,L_{2};T_{2})\ ,$ (34)

which is possible if $C_{\varepsilon}=\varepsilon^{-1}$. The function
$\widetilde{C}_{\varepsilon}$ can be determined in such way that eq. (12)
makes sense in the continuum limit, i.e.

$\displaystyle\widetilde{G}^{(-1)}_{\Lambda}(X,Y;T)=\int^{\infty}_{0}dL_{1}\int^{\infty}_{0}dL_{2}\
e^{-XL_{1}}e^{-YL_{2}}G^{(-1)}_{\Lambda}(L_{1},L_{2};T)\ ,$ (35)

which is possible if $\widetilde{C}_{\varepsilon}=\varepsilon$.

Now we wish to take the continuum limit of eq. (28). For notational
convenience, we redefine the renormalized coupling constants as follows:

$\displaystyle
g=\frac{1}{2}e^{-\varepsilon^{2}\Lambda}\equiv\frac{1}{2}\left(1-\frac{1}{2}\varepsilon^{2}\Lambda\right),\
\ \ x=e^{-\varepsilon X}\equiv 1-\varepsilon X\ ,\ \ \ y=e^{-\varepsilon
Y}\equiv 1-\varepsilon Y\ .$ (36)

Plugging eqs. (30), (36) into eq. (28), one obtains the differential equation:

$\displaystyle\frac{\partial}{\partial
T}\widetilde{G}^{(-1)}_{\Lambda}(X,Y;T)=-\frac{\partial}{\partial
X}\left[(X^{2}-\Lambda)\widetilde{G}^{(-1)}_{\Lambda}(X,Y;T)\right]\ .$ (37)

Doing a little math, one can also derive the continuum description of eq.
(14):

$\displaystyle
G^{(-1)}_{\Lambda}(L_{1},L_{2};T)=\int^{c+i\infty}_{c-i\infty}\frac{dX}{2\pi
i}\int^{c+i\infty}_{c-i\infty}\frac{dY}{2\pi i}\ e^{L_{1}X}e^{L_{2}Y}\
\widetilde{G}^{(-1)}_{\Lambda}(X,Y;T)\ ,$ (38)

where $c$ is a suitable real number. Using the inverse Laplace transform (38)
and eq. (37), one obtains the differential equation that the continuum
amplitude satisfies:

$\displaystyle\frac{\partial}{\partial
T}G^{(-1)}_{\Lambda}(L_{1},L_{2};T)=-\hat{H}^{(-1)}(L_{1})\
G^{(-1)}_{\Lambda}(L_{1},L_{2};T)\ ,$ (39)

where

$\displaystyle\hat{H}^{(-1)}(L)=-L\frac{\partial^{2}}{\partial L^{2}}+\Lambda
L\ .$ (40)

As a result, one can interpret the continuum limit of $2$d CDT as a quantum
system of the one-dimensional universe with length $L$ that propagates in time
$T$ following the quantum Hamiltonian (40). The quantum Hamiltonian (40) is
Hermitian with respect to the inner product:

$\displaystyle\int^{\infty}_{0}\frac{dL}{L}\phi^{*}(L)\
(\hat{H}^{(-1)}\psi)(L)=\int^{\infty}_{0}\frac{dL}{L}(\hat{H}^{(-1)}\phi)^{*}(L)\
\psi(L)\ .$ (41)

The differential equation for the un-marked amplitude can be easily read off
inserting the continuum limit of eq. (7)

$\displaystyle
G^{(-1)}_{\Lambda}(L_{1},L_{2};T)=L_{1}G^{(1)}_{\Lambda}(L_{1},L_{2};T)\ ,$
(42)

into eq. (39):

$\displaystyle\frac{\partial}{\partial
T}G^{(1)}_{\Lambda}(L_{1},L_{2};T)=-\hat{H}^{(1)}(L_{1})\
G^{(1)}_{\Lambda}(L_{1},L_{2};T)\ ,$ (43)

where

$\displaystyle\hat{H}^{(1)}(L)=-\frac{\partial^{2}}{\partial L^{2}}L+\Lambda
L\ .$ (44)

The quantum Hamiltonian (44) is Hermitian with respect to the inner product:

$\displaystyle\int^{\infty}_{0}LdL\phi^{*}(L)\
(\hat{H}^{(1)}\psi)(L)=\int^{\infty}_{0}LdL(\hat{H}^{(1)}\phi)^{*}(L)\
\psi(L)\ .$ (45)

####  $[0,1]\times[0,1]$ topology

Let us consider the case of $a=0$, i.e. $[0,1]\times[0,1]$ topology. We
basically follow the procedure shown in Sect. 2.2. Using the composition law
(16) and the one-step generating function (26), one obtains

$\displaystyle\widetilde{G}^{(0)}(g,x,y;t+1)$
$\displaystyle=\oint_{\mathcal{C}}\frac{dz}{2\pi
iz}\widetilde{G}^{(0)}(g,x,z^{-1};1)\ \widetilde{G}^{(0)}(g,z,y;t)$
$\displaystyle=\oint_{\mathcal{C}}\frac{dz}{2\pi
i}\frac{g^{2}x}{(1-gx)(z-g/(1-gx))}\frac{\widetilde{G}^{(0)}(g,z,y;t)}{z}$
$\displaystyle=gx\ \widetilde{G}^{(0)}\left(g,\frac{g}{1-gx},y;t\right)\ .$
(46)

From eq. (46), one may obtain a sensible continuum limit if the critical
coupling constants are the same as before, i.e.
$(g_{c},x_{c},y_{c})=(1/2,1,1)$, and if the multiplicative renormalization is
treated carefully:

$\displaystyle\widetilde{G}^{(0)}_{\Lambda}(X,Y;T)=\lim_{\varepsilon\to
0}\frac{\varepsilon}{2^{t}}\widetilde{G}^{(0)}(g,x,y;t)\ .$ (47)

In fact, this assumption yields the correct continuum limit. Plugging eqs.
(30), (36) into eq. (46) and using eq. (47), one obtains the differential
equation:

$\displaystyle\frac{\partial}{\partial
T}\widetilde{G}^{(0)}_{\Lambda}(X,Y;T)=-\left(X+(X^{2}-\Lambda)\frac{\partial}{\partial
X}\right)\widetilde{G}^{(0)}_{\Lambda}(X,Y;T)\ .$ (48)

Defining the continuum amplitude in such a way that the inverse Laplace
transform (38) holds in the case of $a=0$ as well, i.e.

$\displaystyle
G^{(0)}_{\Lambda}(L_{1},L_{2};T)=\int^{c+i\infty}_{c-i\infty}\frac{dX}{2\pi
i}\int^{c+i\infty}_{c-i\infty}\frac{dY}{2\pi i}\ e^{L_{1}X}e^{L_{2}Y}\
\widetilde{G}^{(0)}_{\Lambda}(X,Y;T)\ ,$ (49)

and using eq. (49), the differential equation (48) becomes

$\displaystyle\frac{\partial}{\partial
T}G^{(0)}_{\Lambda}(L_{1},L_{2};T)=-\hat{H}^{(0)}(L_{1})\
G^{(0)}_{\Lambda}(L_{1},L_{2};T)\ ,$ (50)

where $\hat{H}^{(0)}$ is the quantum Hamiltonian obtained in Refs.
DiFrancesco:2000nn ; Durhuus:2001sp :

$\displaystyle\hat{H}^{(0)}(L)=-\frac{\partial}{\partial
L}L\frac{\partial}{\partial L}+\Lambda L\ .$ (51)

The quantum Hamiltonian (51) is hermitian with respect to the inner product:

$\displaystyle\int^{\infty}_{0}dL\phi^{*}(L)\
(\hat{H}^{(0)}\psi)(L)=\int^{\infty}_{0}dL(\hat{H}^{(0)}\phi)^{*}(L)\ \psi(L)\
.$ (52)

###  2.3 Short summary of $2$d CDT

As discussed in Sect. 2.2, the continuum limit of $2$d CDT is described by the
quantum mechanics of a one-dimensional universe with length $L$ that
propagates in time $T$ based on the Hamiltonian $\hat{H}^{(a)}$:

$\displaystyle\hat{H}^{(-1)}=-L\frac{\partial^{2}}{\partial L^{2}}+\Lambda L\
,\ \ \ \hat{H}^{(1)}=-\frac{\partial^{2}}{\partial L^{2}}L+\Lambda L\ ,\ \ \
\hat{H}^{(0)}=-\frac{\partial}{\partial L}L\frac{\partial}{\partial L}+\Lambda
L\ ,$ (53)

where the label $a$ classifies the topology of the one-dimensional universe:
$S^{1}$ and $[0,1]$ for $a=1$ and $a=0$, respectively. When $a=-1$, the closed
one-dimensional universe is marked. Let us define the eigenstates of $L$ as
$|L\rangle_{a}$ that satisfy the completeness relation:

$\displaystyle\hat{1}=\int^{\infty}_{0}L^{a}dL\ |L\rangle_{a}{}_{a}\langle L|\
\ \ \Leftrightarrow\ \ \ {}_{a}\langle
L^{\prime}|L\rangle_{a}=\frac{1}{L^{a}}\delta(L-L^{\prime})\ .$ (54)

Note that $|L\rangle_{-1}=L|L\rangle_{1}$. One can then express the amplitudes
as matrix elements:

$\displaystyle G^{(1)}_{\Lambda}(L_{1},L_{2};T)$ $\displaystyle={}_{1}\langle
L_{2}|e^{-T\hat{H}^{(1)}}|L_{1}\rangle_{1}\ ,$ (55) $\displaystyle
G^{(-1)}_{\Lambda}(L_{1},L_{2};T)$ $\displaystyle={}_{1}\langle
L_{2}|e^{-T\hat{H}^{(-1)}}|L_{1}\rangle_{-1}\ ,$ (56) $\displaystyle
G^{(0)}_{\Lambda}(L_{1},L_{2};T)$ $\displaystyle={}_{0}\langle
L_{2}|e^{-T\hat{H}^{(0)}}|L_{1}\rangle_{0}\ ,$ (57)

Using eq. (54), one can show that the composition laws hold: For $a=-1,0$,

$\displaystyle G^{(a)}_{\Lambda}(L_{1},L_{2};T_{1}+T_{2})=\int^{\infty}_{0}dL\
G^{(a)}_{\Lambda}(L_{1},L;T_{1})G^{(a)}_{\Lambda}(L,L_{2};T_{2})\ ,$ (58)

and for $a=1$,

$\displaystyle G^{(1)}_{\Lambda}(L_{1},L_{2};T_{1}+T_{2})$
$\displaystyle=\int^{\infty}_{0}dL\
G^{(1)}_{\Lambda}(L_{1},L;T_{1})LG^{(1)}_{\Lambda}(L,L_{2};T_{2})\ .$ (59)

##  3 $2$d projectable Hořava-Lifshitz quantum gravity

We wish to introduce the classical field theory that reproduces the continuum
limit of $2$d CDT once it is quantized. The field theory is a certain version
of the two-dimensional Hořava-Lifshitz gravity ($2$d HL gravity).

The starting point is the same class of manifold with a foliation (1) where
$\Sigma_{t}$ is a one-dimensional space labelled by $t$:

$\displaystyle\Sigma_{t}=\\{x^{\mu}\in\mathcal{M}\ |\ f(x^{\mu})=t\\}\ ,\ \ \
\text{with}\ \ \ \mu=0,1\ .$ (60)

Choosing that $f(x^{\mu})=x^{0}$, the time direction can be decomposed into
the two directions, i.e. the normal and the tangential to $\Sigma_{t}$:

$\displaystyle\left(\partial_{t}\right)^{\mu}=\frac{\partial x^{\mu}}{\partial
t}=Nn^{\mu}+N^{1}E^{\mu}_{1}\ ,$ (61)

where $n^{\mu}$ and $E^{\mu}_{1}$ are, respectively, the unit normal vector
and the tangent vector defined as

$\displaystyle n^{\mu}=\left(\frac{1}{N},-\frac{N^{1}}{N}\right)\ ,\ \ \
E^{\mu}_{1}=\delta^{\mu}_{1}\ .$ (62)

Here $N$ and $N^{1}$ are the Lapse function and the shift vector. Through the
use of eq. (61), one can parametrize the metric $g_{\mu\nu}$ on $\mathcal{M}$
as follows:

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-N^{2}dt^{2}+h_{11}\left(dx+N^{1}dt\right)\left(dx+N^{1}dt\right)\
,$ (63)

where $t=x^{0}$ and $x=x^{1}$; $h_{11}$ is the spatial metric on $\Sigma_{t}$
defined as $h_{11}=E^{\mu}_{1}E^{\nu}_{1}g_{\mu\nu}$.

$2$d HL gravity is a field theory that preserves the structure of the time
foliation, or in other words, it is invariant under the foliation-preserving
diffeomorphisms (FPD):

$\displaystyle t\to t+\xi^{0}(t)\ ,\ \ \ x\to x+\xi^{1}(t,x)\ .$ (64)

The fields transform under FPD as follows:

$\displaystyle\delta_{\xi}h_{11}$
$\displaystyle=\xi^{0}\partial_{0}h_{11}+\xi^{1}\partial_{1}h_{11}+2h_{11}\partial_{1}\xi^{1}\
,$ (65) $\displaystyle\delta_{\xi}N_{1}$
$\displaystyle=\xi^{\mu}\partial_{\mu}N_{1}+N_{1}\partial_{\mu}\xi^{\mu}+h_{11}\partial_{0}\xi^{1}\
,$ (66) $\displaystyle\delta_{\xi}N$
$\displaystyle=\xi^{\mu}\partial_{\mu}N+N\partial_{0}\xi^{0}\ .$ (67)

where $N_{1}=h_{11}N^{1}$. Here if a function is a constant on each foliation
$\Sigma_{t}$, such a function is called projectable. In fact, implementing FPD
the projectable Lapse function, i.e. $N=N(t)$, stays as a function only of
time. The HL gravity with the projectable Lapse function is dubbed the
projectable HL gravity. Since it is $2$d projectable HL gravity that
reproduces the continuum limit of $2$d CDT once it is quantized, from now we
focus on this special version of HL gravity.

The action of $2$d projectable HL gravity is given by

$\displaystyle I=\int dt\ \mathcal{L}=\frac{1}{\kappa}\int dtdx\
N(t)\sqrt{h(t,x)}\left((1-\eta)K^{2}(t,x)-2\widetilde{\Lambda}\right)\ ,$ (68)

where $\mathcal{L}$ is the Lagrangian; $\eta$, $\widetilde{\Lambda}$ and
$\kappa$ are a dimensionless parameter, the cosmological constant and the
(dimensionless) gravitational coupling constant, respectively; $h$ is the
determinant of the spatial metric $h_{11}$, i.e. $h=h_{11}$; $K$ is the trace
of the extrinsic curvature $K_{11}$ given by

$\displaystyle K_{11}=\frac{1}{2N}\left(\partial_{0}-2\nabla_{1}N_{1}\right)\
,\ \ \ \text{with}\ \ \
\nabla_{1}N_{1}=\partial_{1}N_{1}-\Gamma^{1}_{11}N_{1}\ .$ (69)

Here $\Gamma^{1}_{11}$ is the spatial Christoffel symbol:

$\displaystyle\Gamma^{1}_{11}=\frac{1}{2}h^{11}\partial_{1}h_{11}\ .$ (70)

One can in principle add higher spatial derivative terms to the action (68).
However, such terms are not necessary because the model is renormalizable in
two dimensions without introducing them, and therefore we omit such terms.

The continuum limit of $2$d CDT can be precisely obtained if quantizing $2$d
projectable HL gravity with the following identification of parameters:

$\displaystyle\Lambda=\frac{\widetilde{\Lambda}}{2(1-\eta)}\ ,\ \ \ \eta<1\ ,\
\ \ \kappa=4(1-\eta)\ ,$ (71)

where $\Lambda$ is the renormalized cosmological constant of CDT defined by
eq. (36).

###  3.1 Quantization

Let us overview the quantization of $2$d projectable HL gravity shown in Ref.
Ambjorn:2013joa  (see also Ref. Li:2014bla  for another article examining this
issue).

We introduce the conjugate momentum of $\sqrt{h}$ as $\Pi$, which satisfy the
Poisson bracket:

$\displaystyle\left\\{\sqrt{h(t,x)},\Pi(t,x^{\prime})\right\\}=\delta(x-x^{\prime})\
.$ (72)

Through the Legendre transformation of the Lagrangian (68), one obtains the
Hamiltonian of $2$d projectable HL gravity:

$\displaystyle H=\int
dx\left(\Pi(t,x)\partial_{t}\sqrt{h(t,x)}\right)-\mathcal{L}=N(t)\mathcal{C}(t)+\int
dx\ N_{1}(t,x)\mathcal{C}^{1}(t,x)\ .$ (73)

Since $2$d projectable HL gravity is a singular system due to the invariance
under FPD, there exist two kinds of constraint:

$\displaystyle\mathcal{C}^{1}(t,x)$
$\displaystyle=-\frac{\partial_{1}\Pi(t,x)}{\sqrt{h(t,x)}}\approx 0\ ,$ (74)
$\displaystyle\mathcal{C}(t)$ $\displaystyle=\int dx\
\left(\frac{\kappa}{4(1-\eta)}\Pi^{2}(t,x)\sqrt{h(t,x)}+\frac{2}{\kappa}\widetilde{\Lambda}\sqrt{h(t,x)}\right)\approx
0\ ,$ (75)

where $\mathcal{C}^{1}(t,x)\approx 0$ is the momentum constraint, and
$\mathcal{C}(t)\approx 0$ is the Hamiltonian constraint which is global
because of the projectable Lapse function555 The Hamiltonian and the momentum
constraints come from the consistency conditions that the primary constraints,
$\Pi_{N}\approx 0$ and $\Pi_{N_{1}}\approx 0$, should be preserved under the
time flow where $\Pi_{N}$ and $\Pi_{N_{1}}$ are the conjugate momenta of $N$
and $N_{1}$, respectively. .

The strategy is to solve the momentum constraint (74) at the level of
classical theory, i.e.

$\displaystyle\mathcal{C}^{1}(t,x)=0\ \ \ \Rightarrow\ \ \ \Pi(t,x)=\Pi(t)\ ,$
(76)

meaning that the conjugate momentum becomes a function only of time. Applying
eq. (76), the Hamiltonian (73) reduces to the one for the one-dimensional
system:

$\displaystyle
H=N(t)\left(\frac{\kappa}{4(1-\eta)}\Pi^{2}(t)L(t)+\frac{2}{\kappa}\widetilde{\Lambda}L(t)\right)\
,\ \ \ \text{with}\ \ \ L(t)=\int dx\sqrt{h(t,x)}\ ,$ (77)

where $L(t)$ is the invariant length of the one-dimensional universe. Let us
discuss solutions to the Hamiltonian constraint. If
$(\eta-1)\widetilde{\Lambda}>0$, one has a solution:

$\displaystyle\Pi^{2}=\frac{8(\eta-1)}{\kappa^{2}}\ \widetilde{\Lambda}\ ,$
(78)

which means that the extrinsic curvature is a constant. On the other hand, if
$(\eta-1)\widetilde{\Lambda}<0$, the only solution is

$\displaystyle L=0\ .$ (79)

Hereafter we apply the parametrization (71): We choose
$(\eta-1)\widetilde{\Lambda}<0$, set the unimportant dimensionless
gravitational constant as $\kappa=4(1-\eta)$, and redefine the cosmological
constant as $\Lambda=\frac{\widetilde{\Lambda}}{2(1-\eta)}$. Since $\kappa>0$,
this means that $\eta<1$ which selects the correct sign of the kinetic term,
and the positive cosmological constant $\widetilde{\Lambda}>0$. The dynamics
of the classical $1$d system with the Hamiltonian (77) can be alternatively
described by the following action:

$\displaystyle S=\int^{1}_{0}dt\left(\frac{\dot{L}^{2}(t)}{4N(t)L(t)}-\Lambda
N(t)L(t)\right)\ ,$ (80)

where $\dot{L}(t):=\frac{d}{dt}L(t)$. We then introduce the proper time:

$\displaystyle\tau(s)=\int^{s}_{0}dt\ N(t)\ ,\ \ \ s\in[0,1]\ .$ (81)

Since the proper time (81) is invariant under the reparametrization of time,
$t\to t+\xi^{0}(t)$, if one fixes the Lapse function as $N(\tau)=1$, the
length of the one-dimensional universe $L(\tau)$ is also invariant under the
time redefinition. Therefore, it makes sense to discuss the amplitude such
that the one-dimensional universe with the length $L_{1}:=L(\tau=0)$
propagates in the proper time $\tau$, and ends up with the universe whose
length is given by $L_{2}:=L_{2}(\tau=T)$.

With this understanding, we consider such an amplitude based on the path
integral. For convenience, we rotate $t\to it$, which is possible thanks to
the foliation and introduce the Euclidean action:

$\displaystyle
S_{E}=\int^{1}_{0}dt\left(\frac{\dot{L}^{2}(t)}{4N(t)L(t)}+\Lambda
N(t)L(t)\right)\ .$ (82)

Using the Euclidean action (82), the amplitude becomes

$\displaystyle\mathcal{G}_{\Lambda}(L_{1},L_{2};T)=\int\frac{\mathcal{D}N(t)}{\text{Diff
[0,1]}}\int^{L(1)=L_{2}}_{L(0)=L_{1}}\mathcal{D}L(t)\ e^{-S_{E}[N(t),L(t)]}\
,$ (83)

where

$\displaystyle T:=\int^{1}_{0}dt\ N(t)\ .$ (84)

We fix the Lapse function as $N(\tau)=1$ introducing the corresponding
Faddeev-Popov (FP) determinant. Since the FP determinant only gives an overall
constant, we will omit it in the following. After the gauge fixing, the
amplitude (83) becomes

$\displaystyle\mathcal{G}_{\Lambda}(L_{1},L_{2};T)=\int^{L(T)=L_{2}}_{L(0)=L_{1}}\mathcal{D}L(\tau)\
\exp\left[-\int^{T}_{0}d\tau\left(\frac{\dot{L}^{2}(\tau)}{4L(\tau)}+\Lambda
L(\tau)\right)\right]\ ,$ (85)

where $\dot{L}(\tau):=\frac{d}{d\tau}L(\tau)$.

So far, we have not specified the integral measure $\mathcal{D}L(\tau)$. We
apply the three kinds of measure given by

$\displaystyle\mathcal{D}^{(a)}L(\tau)=\prod^{\tau=T}_{\tau=0}L^{a}(\tau)\
dL(\tau)\ ,\ \ \ (a=0,\pm 1)\ .$ (86)

Accordingly, we consider the three kinds of amplitude, i.e.
$\mathcal{G}^{(a)}_{\Lambda}(L_{1},L_{2};T)$, and rewrite them introducing the
quantum Hamiltonian $\hat{H}^{(a)}$:

$\displaystyle\mathcal{G}^{(a)}_{\Lambda}(L_{1},L_{2};T)={}_{a}\langle
L_{2}|e^{-T\hat{H}^{(a)}}|L_{1}\rangle_{a}\ ,$ (87)

where the eigenstates of $L$ satisfy the completeness relation:

$\displaystyle 1=\int^{\infty}_{0}L^{a}dL\ |L\rangle_{a}{}_{a}\langle L|\ \ \
\Leftrightarrow\ \ \ {}_{a}\langle
L^{\prime}|L\rangle_{a}=\frac{1}{L^{a}}\delta(L-L^{\prime})\ .$ (88)

In order to read off the quantum Hamiltonian $\hat{H}^{(a)}$, we discretize
the proper time interval in steps of $\varepsilon$, and calculate the one-step
matrix element $\mathcal{G}^{(a)}_{\Lambda}(L,L^{\prime};\varepsilon)$. The
normalization can be fixed so as to satisfy the following equation:

$\displaystyle\lim_{\varepsilon\to 0}\int^{\infty}_{0}L^{a}dL\
\mathcal{G}^{(a)}_{\Lambda}(L,L^{\prime};\varepsilon)=1\ ,$ (89)

which comes from the completeness relation (88). The result is

$\displaystyle\mathcal{G}^{(a)}_{\Lambda}(L,L^{\prime};\varepsilon)=\frac{(LL^{\prime})^{(1-a)/2}}{L^{\prime}\sqrt{4\pi\varepsilon
L^{\prime}}}e^{-\frac{(L-L^{\prime})^{2}}{4\varepsilon
L^{\prime}}-\Lambda\varepsilon L^{\prime}}\ .$ (90)

Integrating the one-step amplitude together with a function,
$\psi_{a}(L)={}_{a}\langle L|\psi\rangle$, for $\varepsilon\ll 1$, one can
read off the quantum Hamiltonian:

$\displaystyle\psi_{a}(L^{\prime};\varepsilon)$ $\displaystyle={}_{a}\langle
L^{\prime}|e^{-\varepsilon\hat{H}^{(a)}}|\psi\rangle$
$\displaystyle=\int^{\infty}_{0}L^{a}dL\ {}_{a}\langle
L^{\prime}|e^{-\varepsilon\hat{H}^{(a)}}|L\rangle_{a}{}_{a}\langle
L|\psi\rangle$
$\displaystyle\cong\psi_{a}(L^{\prime})-\varepsilon\hat{H}^{(a)}\psi_{a}(L^{\prime})+\mathcal{O}(\varepsilon^{3/2})\
.$ (91)

Using eq. (90), one obtains

$\displaystyle\hat{H}^{(-1)}(L)=-L\frac{d^{2}}{dL^{2}}+\Lambda L\ ,\ \ \
\hat{H}^{(0)}(L)=-\frac{d}{dL}L\frac{d}{dL}+\Lambda L\ ,\ \ \
\hat{H}^{(1)}(L)=-\frac{d^{2}}{dL^{2}}L+\Lambda L\ ,$ (92)

The quantum Hamiltonians (92) obtained by quantizing $2$d projectable HL
gravity are precisely equivalent to those obtained by the continuum limit of
$2$d CDT (see eqs. (40), (44) and (51)). The amplitudes are related as
follows:

$\displaystyle\mathcal{G}^{(-1)}_{\Lambda}(L,L^{\prime};T)=L^{\prime}G^{(-1)}_{\Lambda}(L,L^{\prime};T)\
,\ \ \
\mathcal{G}^{(a)}_{\Lambda}(L,L^{\prime};T)=G^{(a)}_{\Lambda}(L,L^{\prime};T)\
,\ \ \ (a=0,1)\ .$ (93)

Thereby, we understand that the classical field theory that reproduces the
continuum limit of $2$d CDT once it is quantized is indeed $2$d projectable HL
gravity. The projectable Lapse function allows us to introduce the
reparametrization-invariant proper time, and to reduce the $2$d field theory
to the $1$d system.

##  4 Sum over all wormholes and baby universes

In the CDT model, the spatial topology change is not allowed to occur by
definition. One can generalize the $2$d CDT model in such a way that spatial
topology changes do occur in keeping with the foliation structure, and the
universality class is the same as that of $2$d CDT. Such a model is called
generalized CDT (GCDT). GCDT can be constructed as both discretized and
continuum models. Here of course the continuum model can be obtained by the
continuum limit of the discretized model, but one can directly construct the
continuum GCDT model promoting the one-dimensional quantum-mechanical system
discussed in Sec. 2 to a $2$d field theory that includes the splitting and
joining interactions of the one-dimensional spatial universe. Such a field
theory is dubbed the string field theory for CDT, in which the string means
the one-dimensional universe Ambjorn:2008ta . In this section, we introduce
the string field theory for CDT, and briefly explain the fact that one can
take the sum over all wormholes (i.e. handles) and baby universes
Ambjorn:2009wi ; Ambjorn:2009fm . Here the baby universe is a portion of
geometry that is pinched off from the “parent universe” and vanishes into the
vacuum. We also introduce an effective one-body theory that reproduces the
many-body effects coming from the splitting and joining interactions. We then
discuss those effects in the context of HL gravity Ambjorn:2021wou . In the
end, we show that a sort of Coleman’s mechanism works when taking into account
all contributions of wormholes and baby universes non-perturbatively
Ambjorn:2021wdm .

We introduce an operator that creates a marked closed string (i.e. a marked
closed one-dimensional universe) with length $L$, $\Psi^{\dagger}(L)$, and an
operator that annihilates a length-$L$ marked closed string, $\Psi(L)$. These
operators satisfy the following commutators:

$\displaystyle[\Psi(L),\Psi^{\dagger}(L^{\prime})]=L\delta(L-L^{\prime})\ ,\ \
\ [\Psi(L),\Psi(L^{\prime})]=[\Psi^{\dagger}(L),\Psi^{\dagger}(L^{\prime})]=0\
.$ (94)

The vacuum state $|\text{vac}\rangle$ is defined by the equation:
$\Psi(L)|\text{vac}\rangle=0$. The CDT amplitude (57) can be expressed by
sandwiching the one-body Hamiltonian:

$\displaystyle\mathcal{G}^{(-1)}_{\Lambda}(L_{1},L_{2};T)=L_{2}G^{(-1)}_{\Lambda}(L_{1},L_{2};T)=\langle\text{vac}|\Psi(L_{2})\
e^{-T\mathcal{H}^{(-1)_{\text{free}}}}\
\Psi^{\dagger}(L_{1})|\text{vac}\rangle\ ,$ (95)

where

$\displaystyle\mathcal{H}^{(-1)}_{\text{free}}=\int^{\infty}_{0}\frac{dL}{L}\Psi^{\dagger}(L)\left(-L\frac{\partial^{2}}{\partial
L^{2}}+\Lambda L\right)\Psi(L)\ .$ (96)

Hereafter we omit the superscript $(-1)$ for avoiding notational complexity.
Adding splitting and joining interactions into the free Hamiltonian (96), one
obtains the full Hamiltonian of the string field theory for CDT:

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{\text{free}}-g_{s}\int^{\infty}_{0}dL_{1}\int^{\infty}_{0}dL_{2}\Psi^{\dagger}(L_{1})\Psi^{\dagger}(L_{2})\Psi(L_{1}+L_{2})$
$\displaystyle\ \ \ -\alpha
g_{s}\int^{\infty}_{0}dL_{1}\int^{\infty}_{0}dL_{2}\Psi^{\dagger}(L_{1}+L_{2})\Psi(L_{1})\Psi(L_{2})$
$\displaystyle\ \ \ -\int^{\infty}_{0}\frac{dL}{L}\delta(L)\Psi(L)\ ,$ (97)

where the second, third and fourth terms respectively mean the splitting
interaction with the string coupling constant $g_{s}$, the joining interaction
with the coupling constant $\alpha g_{s}$, and the term associated with a
string vanishing into the vacuum. Here the parameter $\alpha$ is introduced
for counting the number of handles (i.e. wormholes). One can in principle
calculate the amplitude for the process such that $m$ closed strings propagate
in time and end up with $n$ closed strings:

$\displaystyle
A(L_{1},\cdots,L_{m};L^{\prime}_{1},\cdots,L^{\prime}_{n};T)=\langle\text{vac}|\Psi(L^{\prime}_{1})\cdots\Psi(L^{\prime}_{n})e^{-T\mathcal{H}}\Psi^{\dagger}(L_{1})\cdots\Psi^{\dagger}(L_{m})|\text{vac}\rangle\
,$ (98)

###  4.1 Effective theory

Let us consider the full propagator $A(L_{1};L_{2};T)$ that includes the sum
over all genera and baby universes. We can set $\alpha=1$ without loss of
generality since the parameter $\alpha$ plays a supplementary role and we are
interested in taking the sum over all genus contributions. Somewhat
miraculously, the full propagator defined in the many-body system with the
Hamiltonian (97) can be effectively described by the one-body system
Ambjorn:2009wi ; Ambjorn:2009fm :

$\displaystyle A(L_{1};L_{2};T)=\langle
L_{2}|e^{-T\hat{H}_{\text{eff}}(L_{1})}|L_{1}\rangle\ ,$ (99)

where $\hat{H}$ is the effective Hamiltonian given by

$\displaystyle\hat{H}_{\text{eff}}(L)=-L\frac{d^{2}}{dL^{2}}+\Lambda
L-g_{s}L^{2}\ .$ (100)

This is possible because there exists a bijection called Ambjørn-Budd
bijection Ambjorn:2013csx  such that one can map each geometry generated in
GCDT to a branched polymer with loops at the discrete level. The last term in
the Hamiltonian (100), $-g_{s}L^{2}$, expresses all the effects originated
with the baby universes and the wormholes. Note that the Hamiltonian (100) is
not bounded from below because of the last term, but in fact this system is
known to be “classical incomplete,” which means that the Hamiltonian has
discrete energy spectra, and a set of square integrable eigenfunctions (see
e.g. Ref. Ambjorn:1992ve  for a pedagogical explanation about the classical
incomplete systems). A similar deformation has been observed in the $c=1$ non-
critical string theory Moore:1991sf ; Betzios:2020nry .

The full propagator (100) can be also described in terms of the path-integral:

$\displaystyle
A(L_{1};L_{2};T)=\int^{L(T)=L_{2}}_{L(0)=L_{1}}\mathcal{D}L(\tau)\
\exp\left[-\int^{T}_{0}d\tau\left(\frac{\dot{L}^{2}(\tau)}{4L(\tau)}+\Lambda
L(\tau)-g_{s}L^{2}(\tau)\right)\right]\ ,$ (101)

where the integral measure is given by

$\displaystyle\mathcal{D}L(\tau)=\prod^{\tau=T}_{\tau=0}L^{-1}(\tau)dL(\tau)\
.$ (102)

In order for the functional integral (101) to be well defined, one need to
choose the boundary conditions on $L(\tau)$ at infinity such that the kinetic
term counteracts the unboundedness of the potential. If one generalizes the
integral measure (102) as

$\displaystyle\mathcal{D}^{(a)}L(\tau)=\prod^{\tau=T}_{\tau=0}L^{a}(\tau)dL(\tau)\
,\ \ \ (a=0,\pm 1)\ ,$ (103)

one can recover all possible orderings of the effective Hamiltonian (100)
following the procedure explained in Sec. 3.1.

Interestingly, one can reproduce the full propagator (101) if one considers
that the cosmological constant $\Lambda$ in eq. (85) is not a constant but
fluctuates independently in time around $\Lambda$, following the Gaussian
distributions with a standard deviation $\sigma=2\sqrt{g_{s}}$:

$\displaystyle A(L_{1};L_{2};T)=\int\mathcal{D}\nu(\tau)\
e^{-\frac{1}{4g_{s}}\int^{T}_{0}d\tau\
\nu^{2}(\tau)}\mathcal{G}_{\Lambda+\nu}(L_{1},L_{2};T)\ ,$ (104)

where

$\displaystyle\mathcal{G}_{\Lambda+\nu}(L_{1},L_{2};T):=\int^{L(T)=L_{2}}_{L(0)=L_{1}}\mathcal{D}L(\tau)\
\exp\left[-\int^{T}_{0}d\tau\left(\frac{\dot{L}^{2}(\tau)}{4L(\tau)}+(\Lambda+\nu(\tau))L(\tau)\right)\right]\
.$ (105)

Therefore, all the contributions coming from the sum over all wormholes and
baby universes can be fully taken in if the cosmological “constant” in (the
continuum limit of) $2$d CDT or projectable HL quantum gravity where no
wormholes and baby universes exist is not really a constant but fluctuates in
time. This would lead to a realization of Coleman’s mechanism that will be
discussed in Sec. 4.3.

In the next section, we will show that the full propagator can be also
obtained if quantizing $2$d projectable HL gravity with an effective wormhole
interaction term.

###  4.2 Wormhole interaction in $2$d projectable HL gravity

Let us consider the $2$d projectable HL gravity with a space-like wormhole
interaction given by the following action:

$\displaystyle I_{\text{w}}$ $\displaystyle=\frac{1}{\kappa}\int dtdx\
N(t)\sqrt{h(t,x)}\left((1-\eta)K^{2}(t,x)-2\widetilde{\Lambda}\right)$
$\displaystyle\ \ \ +\beta\int dtN(t)\int
dx_{1}dx_{2}\sqrt{h(t,x_{1})}\sqrt{h(t,x_{2})}\ ,$ (106)

where $\beta$ is a dimension-full coupling constant. The last bi-local term
can be interpreted as an effective interaction term for a space-like wormhole
connecting two distant regions at a given $t$. This bi-local term is allowed
to be included since it is invariant under FPD.

Following essentially the same procedure explained in Sec. 3.1, let us
quantize the system defined by the action (106). Introducing the conjugate
momentum of the density $\sqrt{h}$ as $\Pi$, we introduce the Poisson bracket
(72). Implementing the Legendre transform, one obtains the corresponding
Hamiltonian:

$\displaystyle H_{\text{w}}=N(t)\ \mathcal{C}_{\text{w}}(t)+\int dxN_{1}(t,x)\
\mathcal{C}^{1}_{\text{w}}(t,x)\ ,$ (107)

where

$\displaystyle\mathcal{C}^{1}_{\text{w}}(t,x)$
$\displaystyle=-\frac{\partial_{1}\Pi(t,x)}{\sqrt{h(t,x)}}\approx 0\ ,$ (108)
$\displaystyle\mathcal{C}_{\text{w}}(t)$ $\displaystyle=\int dx\
\biggl{(}\frac{\kappa}{4(1-\eta)}\Pi^{2}(t,x)\sqrt{h(t,x)}+\frac{2}{\kappa}\widetilde{\Lambda}\sqrt{h(t,x)}$
$\displaystyle\ \ \ -\beta\sqrt{h(t,x)}\int
dx_{2}\sqrt{h(t,x_{2})}\biggl{)}\approx 0\ .$ (109)

The constraints (108) and (109) are the momentum constraint and the
Hamiltonian constraint, respectively. Solving the momentum constraint (108) at
the classical level as before, the Hamiltonian (107) reduces to the following
one-dimensional one:

$\displaystyle
H_{\text{w}}=N(t)\left(\frac{\kappa}{4(1-\eta)}\Pi^{2}(t)L(t)+\frac{2}{\kappa}\widetilde{\Lambda}L(t)-\beta
L^{2}(t)\right)\ ,$ (110)

where $L(t):=\int dx\sqrt{h(t,x)}$. The Hamiltonian (110) is subject to the
Hamiltonian constraint:

$\displaystyle
L(t)\left(\frac{\kappa}{4(1-\eta)}\Pi^{2}(t)+\frac{2}{\kappa}\widetilde{\Lambda}-\beta
L(t)\right)\approx 0\ .$ (111)

Here we choose the CDT parametrization (71). A solution to the Hamiltonian
constraint (111) is

$\displaystyle\Pi^{2}=-\Lambda+\beta L\geq 0\ ,\ \ \ \text{for}\ \ \
\sqrt{\Lambda}L\geq 1/\xi\ ,$ (112)

where $\xi$ is a dimensionless parameter given by $\xi=\beta/\Lambda^{3/2}$.
For $\sqrt{\Lambda}L<1/\xi$, the only allowed solution is $L=0$.

When quantizing the system, if we follow the same procedure described in Sec.
3.1, and set $\beta=g_{s}$, one can reproduce the path-integral of the full
propagator (101). Remember the boundary condition for the path-integral (101),
i.e. the kinetic term should counteract the unboundedness of the potential
term at $L=\infty$. This balance between the kinetic and potential terms is
precisely what is reflected in the classical Hamiltonian constraint (112).

###  4.3 Coleman’s mechanism

In this section, we discuss a sort of Coleman’s mechanism in the context of
two-dimensional gravity based on CDT briefly.

Let us define the two kinds of Wheeler-deWitt equation:

$\displaystyle\hat{H}W_{0}(L)=0\ ,\ \ \ \hat{H}_{\text{eff}}W(L)=0\ ,$ (113)

where $\hat{H}:=\hat{H}^{(-1)}$ introduced in eq. (53). The solutions to the
Wheeler-deWitt equations are the Hartle-Hawking wave functions given by

$\displaystyle W_{0}(L)=e^{-\sqrt{\Lambda}L}\ ,\ \ \
W(L)=\frac{\text{Bi}(\xi^{-2/3}-\xi^{1/3}\sqrt{\Lambda}L)}{\text{Bi}(\xi^{-2/3})}+c\
\text{Ai}(\xi^{-2/3}-\xi^{1/3}\sqrt{\Lambda}L)\ ,$ (114)

where Ai and Bi are the standard Airy functions, $\xi$ is a dimensionless
string coupling constant measured by the cosmological constant, i.e.,
$\xi:=g_{s}/\Lambda^{3/2}$; and $c$ is an undetermined dimensionless constant.
The Hartle-Hawking wave function $W_{0}(L)$ is the one for (the continuum
theory of) $2$d CDT, i.e. neither baby universe nor wormhole contributions are
included. On the other hand, $W(L)$ is the Hartle-Hawking wave function
including all possible contributions of baby universes and wormholes non-
perturbatively.

We wish to explore the behavior of the non-perturbative Hartle-Hawking wave
function $W(L)$ (see Fig. 2).

Figure 2: A plot of the Hartle-Hawking wave functions, $W_{0}$ (dashed line)
and $W$ (solid line), for $\xi=1/3$ and $c=0$: The horizontal axis is
$\sqrt{\Lambda}L$ and the vertical axis is either $W_{0}$ or $W$.

For $\sqrt{\Lambda}L\ll 1/\xi$, one obtains the asymptotic expansion:

$\displaystyle W(L)\sim e^{-\sqrt{\Lambda}L}=W_{0}(L)\ .$ (115)

Therefore, when the size of the one-dimensional universe is small enough, the
physics is very close to the one without baby universes and wormholes, and it
is essentially governed by the cosmological constant. The wave function in
this region decreases exponentially, and this behavior does not change at any
finite order of perturbation.

However, once the size of the universe is large enough, i.e.
$\sqrt{\Lambda}L>1/\xi$, the wave function starts oscillating, and the
behavior is governed by the string coupling constant instead of the
cosmological constant:

$\displaystyle W(L)\sim 1/(g^{1/3}_{s}L)^{1/4}\ .$ (116)

This drastic change happens due to the infinitely many wormholes and baby
universes. The similar behavior has been observed in the context of non-
critical string theory Moore:1991sf ; Betzios:2020nry .

From the discussion above, we observe that a sort of Coleman’s mechanism
works: For a large universe, the cosmological constant is not important enough
to govern the physics666 The Hartle-Hawking wave function $W(L)$ is not
normalizable, but similar arguments are valid on the normalizable energy
eigenstates Ambjorn:2021wdm . .

##  5 Summary

We have reviewed the relation between two-dimensional causal dynamical
triangulations ($2$d CDT) and two-dimensional projectable Hořava-Lifshitz
quantum gravity ($2$d projectable HL QG).

In the first part, it has been shown that the physics described by the
continuum limit of $2$d CDT coincides with the one obtained quantizing $2$d
projectable HL gravity. This is confirmed because the quantum Hamiltonians of
both models are exactly the same. The system is expressed in terms of quantum
mechanics of a $1$d extended object, i.e. a $1$d universe.

It would be too hasty to consider that this scenario also holds for the higher
dimensional cases. In fact, it has been shown that in $2+1$ dimensions
numerical studies of the so-called locally causal dynamical triangulations
that relax the proper time foliation of CDT and require the local causality
reproduce an intriguing specialty of CDT, an emergence of the de-Sitter-like
geometry Jordan:2013awa  (see e.g. Ref. Loll:2019rdj  for the higher-
dimensional CDT). On the other hand, a Landau theory approach suggests a
relation between CDT and theories invariant under the foliation-preserving
diffeomorphisms in $2+1$ dimensions Benedetti:2014dra ; Benedetti:2016rwo
(see also the dedicated chapter of the Handbook of Quantum Gravity
Benedetti:2022ots ). This issue should be investigated further.

In the second part, we have introduced the generalized CDT (GCDT) that permits
baby universes and wormholes to form in keeping with the foliation, and in
particular the construction based on the string field theory for CDT has been
explained. Here the string means the $1$d universe, and the string field
theory is constructed in such a way that the free part reproduces the CDT
amplitudes, and the splitting and joining interactions of string are
introduced to create baby universes and wormholes.

Focusing on the loop-to-loop amplitude, we have introduced an effective $1$d
theory that includes all the contributions coming from the sum over all
possible baby universes and wormholes. From the point of view of HL gravity,
the effective theory can be precisely reproduced if introducing a bi-local
interaction term into the action of $2$d projectable HL gravity and if
quantizing the system. In addition, the effective theory can be also obtained
considering that the cosmological constant of $2$d CDT is not a constant but
it fluctuates in time. This leads to Coleman’s mechanism in $2$d CDT such that
for a large universe the cosmological constant is not important enough to
govern the physics.

Although we have not discussed issues of the coupling to matter, $2$d CDT
coupled to Yang-Mills theory has been solved analytically in Ref.
Ambjorn:2013rma , and it has been shown that the quantum Hamiltonian obtained
in Ref. Ambjorn:2013rma  can be reproduced quantizing $2$d projectable HL
gravity coupled to Yang-Mills theory Ipsen:2015ckl . In fact, we know very
little about the analytical treatment of the coupling to matter compared to
the situation of $2$d dynamical triangulations and the Liouville quantum
gravity. This direction needs to be explored in the future.

What is remarkable is that following the standard Wilsonian renormalization
group, one can take the continuum limit of the lattice model, $2$d CDT, and
find the continuum quantum field theory, $2$d projectable HL QG, which is in
the same universality class of $2$d CDT. A missing piece is the continuum
quantum field theory of GCDT that is described by metric components and allows
us to compute all the amplitudes defined by the string field theory for CDT,
although we have the effective field theory, the $2$d projectable HL gravity
with a bi-local interaction, that reproduces the restricted class of GCDT
amplitudes once it is quantized. We wish to unveil the underlying continuum
quantum field theory of GCDT, through which we can understand something
inherently interesting about quantum geometries for sure.

###### Acknowledgements.

YS would like to thank Jan Ambjørn, Lisa Glaser, Yuki Hiraga, Yoshiyasu Ito
and Yoshiyuki Watabiki, for wonderful collaborations on the topics related to
this chapter. Great ideas appeared in this chapter attribute to them. However,
if conceptual mistakes exist, YS is responsible for them.

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