# An examination of the de Sitter space dual to
the Monster conformal field theory

Seyed Khaki111Academy of Fundamental Studies, Munich 82041, Germany
Email<EMAIL_ADDRESS>
###### Abstract

In 2007, Witten proposed that the Monster conformal field theory (CFT) is very
likely the dual CFT of 3D pure gravity in Anti-de Sitter space. Based on the
fact that the Hilbert space of quantum gravity in asymptotically de Sitter
space is finite, in this article, an attempt is made to study the de Sitter
space dual to the Monster CFT. As the Monster CFT is an orbifold theory, it
has two distinct sectors, an untwisted and a twisted one. It is observed that
the boson of the underlying scalar field in the ground state of the untwisted
sector is as heavy as the Higgs boson and the vacuum energy of the twisted
sector approximately coincides with the Cosmological constant. Although
further studies are required to shed more light on these observations, a
couple of proposals are suggested to the best of our knowledge aiming to
explain them.

##  1 Introduction

In 2007, Witten proposed that the Monster conformal field theory (CFT) is very
likely the dual CFT of 3D pure gravity in Anti-de Sitter (AdS) space [1].
Unfortunately, he did not examine the dual de Sitter (dS) space as he noted
"not knowing how to define any mathematically precise observables, we do not
know what to try to calculate". However, it is known that the Hilbert space of
quantum gravity in asymptotically dS space is finite [2][3][4](for a review
see [5]), which we found it as a reliable ground to calculate the vacuum
energies in the dual dS space. The plan of the article is the following. In
Section 1, we shortly introduce the orbifold theory with an emphasis on the
coexistence of two distinct untwisted and twisted sectors. Then, with a brief
introduction to the Monster CFT in Section 2, we calculate the vacuum energies
of the corresponding untwisted and twisted sectors and report that they may
address the Hierarchy and the Cosmological constant problems, respectively. In
Section 3, we try to illustrate these observations offering a couple of
proposals to the best of our knowledge, where we stress the roles of the
sporadic groups and the Leech lattice in mass generation and the spacetime
symmetries, respectively. In the last part, we assess aspects related to the
grand unified theory (GUT) including the calculation of the GUT scale for the
HE superstring theory.

###  1.1 Orbifold Theory

In mathematics, an orbifold generalizes the concept of the manifold to a
quotient space of a Euclidean space by a finite group. In string theory, the
orbifold CFTs are interesting as they provide rich spaces for string
compactification where strings can propagate smoothly in a consistent way even
though their classical geometry can be singular [6][7]. Starting from a
particular manifold, the orbifold can be regarded as gauging some discrete
worldsheet symmetries. Consider a unitary CFT with discrete symmetry group
$G$. One can mod out $G$ by projecting the Hilbert space to a subspace
invariant under the action of $G$ [8]:

$\mathcal{H}\rightarrow\mathcal{H}_{G}\>:=\>\frac{1}{|G|}\sum\limits_{g\in
G}g\>\mathcal{H}$ (1)

The partition function of this theory constructed by inserting operators of
this Hermitian projection in the trace is not modular invariant. To ensure
modular invariance, one should apply the boundary conditions for local
operators introducing a new space called twisted sector, where the fields
defined on the closed strings are now periodic up to an action from $G$. The
full modular invariant partition function then consists of the twisted and
untwisted sectors.

$Z=\>\frac{1}{|G|}\sum\limits_{g,h\in
G|[g,h]=1}\text{Tr}_{\mathcal{H}_{h}}\>g\>q^{(L_{0}-c/24)}\>\bar{q}^{(\bar{L}_{0}-\bar{c}/24)}$
(2)

where, in the h-twisted sector, we are restricted to symmetries $g$ that do
not change the sector ($gh=hg$) [8].

$\displaystyle Z=\>\frac{1}{|G|}\sum\limits_{gh=hg}\>\>\>\
g\>\>\leavevmode\hbox to29.25pt{\vbox
to29.25pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (3) $\displaystyle
h$

Spacetime points with the same G-orbit can be simply identified by the twisted
sector for a closed string and for an open string stretched between those
points. Notably, each conjugacy class of $G$ is associated with a distinct
superselection sector. While the class containing the identity yields the
untwisted sector, the other classes result in the twisted sector. One may also
view the new twisted states as strings stuck at fixed points. After all, the
twisted boundary conditions give rise to a new ground state and a distinct
collection of oscillators.

##  2 Observations

Before the employment of orbifolds in string theory, mathematicians were
studying asymmetric orbifold constructions to probe the properties of the
largest sporadic group, the Monster group $\mathbb{M}$ [9]. Construction of
the Monster CFT in 1984, which emerged independently in the context of string
compactifications, can be regarded as the earliest example of an orbifold CFT
[10]. Since then, many studies uncovered various properties of this theory by
addressing different aspects of it (for developments see [11], for a recent
review see [12]), and investigated its physical implications in the context of
the string theory. One of the milestones of these attempts is the Witten’s
proposal in [1], wherein he proposed that the dual CFT of 3D pure gravity with
negative Cosmological constant is very likely the Monster CFT. In this
section, we consider the 3D pure gravity with positive Cosmological constant
dual to the Monster CFT and calculate the corresponding vacuum energies of the
untwisted and twisted sectors.

###  2.1 Monster CFT

In the construction of the Monster CFT, the 26D target space of the bosonic
string is compactified to 2D [46], where the space of 24 transverse dimensions
is discretized by the Leech lattice, $\Lambda_{24}$, in a way that momenta of
the left- and right-moving are components of
$\Lambda_{24}^{L}\otimes\Lambda_{24}^{R}$. The CFT is then obtained by gauging
the $\mathbb{Z}_{2}$ symmetry and calculating the partition function of the
form [13][14]

$\displaystyle Z=\>\frac{1}{2}\left(1\>\leavevmode\hbox to29.25pt{\vbox
to29.25pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-0.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+g\>\leavevmode\hbox
to29.25pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to29.25pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+g\>\leavevmode\hbox
to29.25pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$ (4)
$\displaystyle 11\hskip 119.50148ptg\hskip 48.36958ptg$

which results in the modular invariant function $Z(\tau)=j(\tau)-744$ where
$j(\tau)$ is the celebrated $j$-function, $j(\tau)=e^{-2\pi
i\tau}+744+196884e^{2\pi i\tau}+21493760e^{4\pi i\tau}+...$ .

###  2.2 Untwisted Vaccum

Since $\mathbb{M}$ is the symmetry group of the theory, for each
$g\in\mathbb{M}$, one has an automorphism of the CFT,

$g:\mathcal{H}\rightarrow\mathcal{H}$ (5)

The ground state of the theory (the first term in the partition function),
which is a tachyonic scalar field, also respects $\mathbb{M}$ symmetry, i.e.,

$g\ket{\Omega}=\ket{\Omega}$ (6)

The ground state energy of a CFT with the central charge $c=24k$ is
$L_{0}=-c/24=-k$, which for the Monster CFT with $c=24$ becomes $L_{0}=-1$.
This negative energy is due to the assumption of an infinite number of
harmonic oscillators oscillating in 24 transverse dimensions, whose ground
state energies add up to $24\times(1+2+3+...)/2=24\times\zeta(-1)/2=-1$.
Particularly, the theory is a vertex operator algebra (VOA), which by
definition, is a graded infinite-dimensional vector space
$\mathcal{V}=\oplus^{\infty}_{n=0}\mathcal{V}_{n}$. A dual AdS space, which
accepts such an infinite-dimensional space, admits $L_{0}=-1$ too. However, it
is known that the Hilbert space of quantum gravity in asymptotically dS space
is finite [2][3][4](for a review see [5]). Therefore, a dual dS space of the
Monster CFT does not admit $L_{0}=-1$. It accepts only a finite number of
oscillators in its ground state which yields a finite and positive vacuum
energy.

As noted in (6), the ground state of the theory preserves $\mathbb{M}$
symmetry, i.e., for each $g\in\mathbb{M}$ we have a modular transformation
that leaves the ground state unchanged. In other words, the microstates of the
ground state are invariant under each modular transformation $g\in\mathbb{M}$.
Since there is $|\mathbb{M}|$ number of $g$, we have $|\mathbb{M}|$ modular
transformations in the ground state. When the number of modular elements is
finite, the field over which the transformation is defined must be finite.
Over an underlying finite field $\mathbb{F}_{q}$ ($\mathbb{Z}/q\mathbb{Z}$)
with $q$ microstates, the number of modular transformations is
$|PSL(2,q)|=q^{3}-q$ [15] (when the number of modular transformations is
infinite, they form the modular group $PSL(2,\mathbb{Z})$ which is defined
over the infinite field of $\mathbb{Z}$). The energy of each microstates
oscillating in 24-DOF is equal to the energy of 24 microstates oscillating in
1-DOF. That is, $q$ microstates oscillating in 24 DOF are equivalent (in terms
of energy) to $24q$ microstates oscillating in 1-DOF, which leads to
$|PSL(2,24q)|=(24q)^{3}-(24q)$ number of modular transformations. Thus, by
$\mathbb{F}_{q}$, the underlying scalar field in the ground state of the dual
dS space, and considering the central charge, we counted the modular
transformations that the Monster CFT suggests to be $|\mathbb{M}|\approx
8\times 10^{53}$. Thus,

$(24q)^{3}-(24q)\approx 8.08\times 10^{53}$ (7)

Solving for $q$ provides three roots with length $|q|\approx 3.88\times
10^{16}$. This is the string length ($\ell_{s}$) of the scalar (boson of)
$\mathbb{F}_{q}$ in the Planck units. The mass of the closed string is then

$m_{s}=\frac{2}{\ell_{s}}\approx\frac{2}{3.88\times 10^{16}}\approx 5.15\times
10^{-17}$ (8)

Accordingly, this is the dimensionless mass in the Planck units. Recovering
the dimension by the reduced Planck mass $M_{P}\approx 2.435\times
10^{18}\rm\>GeV$, yields $M_{s}\approx 5.15\times 10^{-17}\times M_{P}\approx
125.4\rm\>GeV$. This value is almost identical to the experimentally measured
mass of the Higgs boson ($125.1\>\rm GeV$) in the Large Hadron Collider (LHC)
in 2012 [16]. Taking into account an antiholomorphic counterpart, the ground
state energies add up to a total vacuum expectation value (VEV) of $\approx
250\rm\>GeV$.

Besides the string theoretic explanations for this observation, there is also
a simple interpretation without string terminology. The quantum mechanical
equivalent length of a particle with mass $M$ is the reduced Compton
wavelength $\lambda=\frac{\hbar}{Mc}$ which in the Planck units becomes
$\lambda=\frac{1}{m}$ where $m$ is now a dimensionless mass normalized by the
reduced Planck mass. How many Planck wavelengths can be embedded in the
Compton wavelength of the Higgs boson? It is $\lambda_{Higgs}\approx 10^{16}$
(which is the length of the open string). How many Planck cubes (cubes of
Planck length) can be embedded in a Higgs cube (cubes of $\lambda_{Higgs}$
length)? It is approximately $10^{48}$. How many Planck cubes can be embedded
in a 24-Higgs cube (cubes of length $24\lambda_{Higgs}$)? It is almost the
order of Monster group $\approx 8\times 10^{53}$!

Furthermore, We would like to point out that one can observe that
$|PSL(2,q)|=q^{3}-q$ appears (or is encoded) in the Virasoro algebra,
$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0}$, too. In
particular, over a field $k$ of characteristic $q$, the Witt algebra is
defined to be the Lie algebra of derivations of the ring $k[z]/z^{q}$, where
the Witt algebra is spanned by $L_{m}$ for $-1\leq m\leq q-2$. Hence, the
central extension term $m^{3}-m$ reflects the cubic form $q^{3}-q$ of the
number of modular transformations over a finite field.

###  2.3 Twisted Vacuum

As noted in the introduction, in orbifold theories like the Monster CFT, the
twisted sector introduces a new ground state and a distinct collection of
oscillators. The group of finite modular transformations $PSL(2,q)$ (a.k.a
$A_{n}(q)$) is a Chevalley group of Lie type which can be viewed as a Lie
group over $F_{q}$. If $F_{q}$ is a (unique up to isomorphism) finite field of
size $q$, there is a unique quadratic separable extension of $F_{q}$, such
that the extension field is a finite field of order $q^{2}$. Such a quadratic
extension field is involved in the twisted Chevalley group of Lie type,
$PSU(2,q)$ (a.k.a ${}^{2}A_{n}(q^{2})$) [15]. While the order of both twisted
and untwisted groups are $q^{3}-q$, the difference is that, in the twisted
group, two fields are involved; one of order $q$ and one quadratic of order
$q^{2}$. In the previous observation, we calculated the vacuum energy of the
fixed field in the untwisted sector. Now, we calculate the vacuum energy of
the extended field of order $q^{2}$ in the twisted sector. For the fixed
field, we calculated $\>|q|\approx 3.88\times 10^{16}$. Hence, for the
extension field, we have $q_{twist}=q^{2}=1.5\times 10^{33}$ and therefore
$m_{twist}=\frac{2}{1.5\times 10^{33}}\approx 1.33\times 10^{-33}$, which
yields $vev_{twist}=2m_{twist}=2.66\times 10^{-33}$. If we recover the
dimension, as we did above by the reduced Planck mass, we will obtain a
$VEV_{twist}$ of order $10^{-15}\>GeV$. Remarkably, the observational data
reports a vacuum energy density of $\rho_{vac}\approx 2.6\times
10^{-47}{GeV}^{4}$ [17] which is equivalent to $2.2\times 10^{-12}\>GeV$.

This observation suggests that calculating $VEV_{twist}$ is probably
estimating $\rho_{vac}$. Therefore, now, we know that we should not recover
the mass by the reduced Planck mass, $\sqrt{\frac{c\hbar}{8\pi G}}$, because
it is not purely quantum mechanical as it is ’reduced’ by the gravitational
factor $8\pi$ (coming from the Einstein field equations). Instead, to estimate
the VEV only from the quantum mechanics, we should recover the mass by the
Planck mass, $\sqrt{\frac{c\hbar}{G}}\approx 1.22\times 10^{19}\>GeV$.
Accordingly, $VEV_{twist}=vev_{twist}\times 1.22\times 10^{19}\approx
3.2\times 10^{-14}\>GeV$. Noting that the Cosmological constant problem is
known as the worst theoretical prediction in the history of physics [18](for a
review see [19]), this prediction is encouraging. Consequently, this
observation may explain the reason for the discrepancy between the
experimentally measured VEVs in particle physics and cosmology. Apparently, at
the subatomic level of particle physics, the VEV of the untwisted sector,
whereas in cosmology, the VEV of the twisted sector is measured.

##  3 Proposals

###  3.1 Sporadic Groups

We provided evidence that the Higgs field has probably the Monster group
symmetry in its ground state. Here, we explain why the symmetry of sporadic
groups is capable of mass generation and why they can be the only simple
groups that have this property.

In physics, and particularly in particle physics, the study of symmetry in the
context of group theory plays an essential role. One can argue that it is
natural to expect the elementary particles to be described by atoms of
symmetry, the simple groups. The simple groups are either finite or infinite.
Despite the fact that the absolute infinities (divergences) are not physical,
more focus in the physics community was on the infinite groups. One reason was
perhaps the mathematical difficulty of dealing with finite groups.
Nevertheless, due to the hard work of hundreds of mathematicians, the finite
simple groups are classified into a couple of general categories: infinite and
finite [20]. The finite category contains the sporadic groups (Figure 1) in
which the Monster group is the largest.

The infinite category contains 3 classes, the cyclic groups $\mathbb{Z}_{q}$,
the alternating groups $A_{n}$, and the groups of Lie type containing
$PSL(n+1,q)$, $O(2n+1,q)$, $PSp(2n,q)$, $O^{+}(2n,q)$, $PSU(n+1,q)$,
$O^{-}(2n,q)$, $F_{4}(q)$, $G_{2}(q)$, $E_{6}(q)$, $E_{7}(q)$, $E_{8}(q)$,
${}^{2}D_{4}(q)$, and ${}^{2}E_{6}(q)$. These groups are determined by their
input parameters $q$, $n$, or both. The adjective ’infinite’ refers to the
fact that there is an infinite number of these groups in each class. For
instance, if we select $n=2$ in $PSU(n,q)$, we get an infinite number of
groups $PSU(2,2),PSU(2,3),PSU(2,2^{2}),...$, with the order spectrum of
$6,24,120,...$ . However, on the other side, 26 sporadic groups in the finite
category are fully determined. Meaning that there is no input parameter to
determine. They are completely fixed by nature and their frozen order spectrum
is $|M_{11}|=7920,|M_{12}|=95040,...,\>|\mathbb{M}|\approx 8\times 10^{53}$.
These orders are the only constants within the simple groups’ orders. By
constant, we mean a fixed specified number (without any arbitrary input) that
when put on the RHS of (7), determines the field’s order on the LHS and,
therefore, a mass scale. In other words, the orders of sporadic groups are the
only orders that are not a function of $q$ and $n$, leading to a fixed number
of modular transformations and, therefore, a fixed mass. This property is
evident in the Monster CFT partition function. There are no massless DOFs in
the theory which hints that the theory is profoundly engaged with ’mass’ and
all its parameters are totally fixed.

In addition, noting the inverse relation of mass and length, the energy
spectrum of the infinite category is UV convergent but IR divergent. However,
only the spectrum of the sporadic groups is both UV and IR convergent. The UV
and IR convergences in the sporadic groups correspond to its natural cutoffs,
respectively the smallest and largest sporadic groups. By considering Monster
CFT we assessed the IR of the sporadic groups which led to the electroweak
scale of the Higgs mass. Now, let us probe the UV of the sporadic groups,
namely the Mathieu groups $|M_{12}|=95040$ and $|M_{11}|=7920$. The initial
moonshine development, the Monstrous moonshine (for a recent review of
moonshines see [12]), is recently extended by the link between the largest
Mathieu group $M_{24}$ representation and the elliptic genus of $K3$ surfaces
[21]. Although there are many aspects of the Mathieu moonshine that are
mysterious and yet unclear, it ultimately refers to the massive sector [22].
So, not wishing to look a gift horse in the mouth, noting the above
observations, let us apply our method to Mathieu groups irrespective of the
existence of VOAs/CFTs with the automorphism of such groups. We emphasize
that, here, our aim is merely to obtain an approximation of the mass scales
corresponding to these symmetries. In this way, for
$(24q)^{3}-(24q)\approx|M_{12}|$ and $(24q)^{3}-(24q)\approx|M_{11}|$, we
obtain $q\approx 1.9$ and $q\approx 0.8$, which yield the integer values of
$q=2$ and $q=1$, respectively. These values then give rise to the mass scales
of $M_{P}$ and $2M_{P}$ for closed string (with
$m_{s}=\frac{2}{l_{s}}=\frac{2}{q}$). Also, $q=1$ leads to $M_{P}$, for open
string with $m_{s}=\frac{1}{l_{s}}=\frac{1}{q}$.

Consequently, the [IR , UV] pair of the sporadic groups [$\mathbb{M}$ ,
$M_{11}$] coincide with the physical [IR , UV] pair of [Higgs , Planck]
scales. Based on this observation, we propose that the Higgs field was perhaps
following the symmetries of the sporadic groups in its evolution. Remarkably,
at the end of the spectrum, there are two phase transition points at the pair
$[q\approx|\mathbb{M}|^{1/3}$ , $q=1]$. Before discussing these points in the
next part, note that the intermediate scales are covered by intermediate
sporadic groups, where CFTs for 9 intermediate sporadic groups, for example,
are studied in [23].

####  3.1.1 Phase Transition

##### UV at $q=1$

Mathematically, based on the standard definition of the ’field’, 2 is the
smallest number of elements of a field. However, in mathematics, the field
with a single element [24], $\mathbb{F}_{1}$, is a known object that behaves
like a finite field with one element, if such a field could exist. It is worth
mentioning that it has connections to noncommutative geometry and the Riemann
hypothesis [25]. Physically, such a collapse of the field concept at $q=1$
matches with the current picture of modern physics where it is believed that
the typical concept of spacetime collapses above the Planck scale. Moreover,
at Planck scales, the Schwarzschild radius and the string length (Compton
wavelength) are of the same order. If the mass is squeezed smaller than its
Schwarzschild radius it becomes a black hole, a type of phase transition.

##### IR at $q\approx|\mathbb{M}|^{1/3}$

This phase transition (which gave rise to the electroweak scale of the Higgs
boson) is the electroweak spontaneous symmetry-breaking. It is then natural to
ask: How a phase transition can happen in a finite system? The well-known work
of Lee and Yang [26][27] nicely describes the mechanism of such a phase
transition. According to the Lee-Yang theory, in the infinite-size limit of a
finite-size system, when the complex zeros of the partition function become
numerous and dense along a certain arc, a phase transition can be triggered.
In our system, the partition function of the Monster CFT is the modular
invariant $J$-function, and such a certain arc is the circle of the discussed
finite field $\mathbb{F}_{q}$. In this way, it turns out that in the evolution
of the system, through numerous symmetry breaking, the number of zeros of the
$J$-function grows to $q\approx 3.88\times 10^{16}$ and then at this critical
value, a phase transition of the Lee-Young type occurs. Moreover, in the
original model assessed by Lee and Yang, a system containing $N$ Ising
$\frac{1}{2}$-spins $\sigma_{i}$ with the Hamiltonian
$H=\sum_{i<j}J_{ij}\sigma_{i}\sigma_{j}-\sum_{i}H_{i}\sigma_{i}$ is
considered, where, $J_{ij}$ is the coupling constants of pair ferromagnetic
interactions, and $H_{i}$ is the value of the magnetic field acting on the
spin $\sigma_{i}$. In a homogeneous magnetic field with $H_{i}=H$, the
partition function $Z_{N}(z_{1},z_{2},...,z_{N})$ is a polynomial of degree
$N$ in the fugacity $z=exp(-2H\beta)$. According to the Circle theorem, zeros
of $Z_{N}$ lie on the unit circle ($|z|=1$), for $N$ imaginary values of $H$
[28].

Furthermore, let us explain such a phase transition from another perspective.
In the evolution of the system, through numerous symmetry breaking, the
entropy increases accordingly. Eventually, at the final state, where the
symmetries of the system are all broken, the second law of thermodynamics
forces the system to continue breaking the symmetries (increasing entropy).
However, the system is saturated and does not have more capacity. Therefore,
it must change its phase spontaneously in order to increase entropy.

In addition, this phase transition is not the first case related to the number
theory. A phase transition in a number-theoretical system is first reported in
the prominent work of Bost and Connes [29]. Particularly, the Bost-Connes
system, which is a quantum dynamical system originating from the statistical
theory of prime numbers, whose partition function is the Riemann zeta
function, exhibits a phase transition with spontaneous symmetry breaking. One
can naturally imagine such a phase transition by noting the well-known
connection between the j-function, which encodes information about the
elliptic curves, and the Riemann zeta function, which encodes information
regarding prime numbers (generally between modular forms and L-functions)
[30][31].

####  3.1.2 Neutrinos

The sporadic groups are divided into 2 separate families, namely the Monster
family and the Pariah family which are 6 groups (Lyons, O’Nan group, and
Rudvalis, and three Janko groups) that are not subquotients of the Monster
group (see Figure 1). Therefore, the model of sporadic groups predicts, at
least, two sources of mass. Since in the SM, it is expected to have another
source of mass except the Higgs boson for the Neutrinos, we propose that the
source of mass for the Neutrinos is the Pariah family. Although the Neutrino’s
masses are not in the spectrum of the Pariah family, the seesaw mechanism
[32], for instance, can explain their small masses. There are a couple of
supports for this proposal. First, the seesaw mechanism assumes the existence
of a very large mass scale comparable to the GUT scale which can be found
within the spectrum of the Pariah family (e.g., the scale of Rudvalis is
$\approx 2\times 10^{16}\>GeV$). Second, the UV of the Pariah family and the
Monster family are close to each other. Particularly, the UV of the Pariah
family corresponds to the smallest Janko group $|J_{1}|=175,560$ which is
close to $|M_{12}|=95,040$.

###  3.2 Leech Lattice

We provided evidence that the Monster CFT probably describes the 3D (2D space
+ 1D time) pure gravity in dS space which leads to the prediction of a flat
space plus time. The observational data reported from different sources (e.g.,
WMAP [33], BOOMERanG [34], and Planck [17]) confirm that the universe is
globally flat. Although this is a verification for our proposal, we locally
observe a 3D space in the presence of matter. Recall that the resulting 2D
space is produced by the compactification of 24D from the target space of 26
bosonic strings, where $26-24=2$. Consequently, there are 2 ways to obtain 3D
instead of 2D space. The first way is to start from the target space of 27D
instead of 26D (the existence of such a bosonic string theory is conjectured
in [35]). Because the target space is fixed, in this case, we should have
observed a 3D space globally, which is not correct. The second way is that the
24 transverse dimensions locally decrease to 23D in the presence of matter,
and conversely, by removing the matter, 23D transforms to the original 24D. In
this way, inserting matter results in a local 3D space, and removing it yields
a 2D space. The possibility of such a distortion depends on the nature of the
compactified 24 dimensions. As mentioned, the 24 internal dimensions are
discretized by the 24-dimensional Leech lattice $\Lambda_{24}$, which is an
even unimodular lattice with no roots. The local distortion of $\Lambda_{24}$
to a 23D lattice requires the existence of a 23D lattice with the following
properties.

##### 1.

It must be unimodular evidently. A simple interpretation is that the volume of
each lattice cell must be one. In string theory, in the context of
compactification, the ubiquitous unimodular property usually comes with the
even property as it is in $\Lambda_{24}$. This is not the case here because we
are not searching for a lattice for compactification. The compactification is
done in 24D with $\Lambda_{24}$, and the aim is now to find a similar neighbor
lattice in 23D which makes the distortion possible after perturbation by
matter.

##### 2.

It must have the same number of roots as $\Lambda_{24}$ (i.e., zero). There
are two reasons for this requirement. First, it is the no-roots property that
allows $\Lambda_{24}$ to have the densest sphere packing among all lattices in
24D and become universally optimal [36]. Implying that between all point
configurations of the same density, it provides the minimum possible Gaussian
energy [37]. The no-roots property makes $\Lambda_{24}$ unique such that
between 24 unimodular lattices in 24D, $\Lambda_{24}$ is the only one with
this property. It is, therefore, straightforward that these exceptional
features stem from the exceptional property of having no roots. The second
reason is that when we expect a distortion between two neighbor lattices, then
we should expect that they are similar in nature to make a smooth distortion
possible. If they have dissimilar numbers of roots, then every time a matter
is inserted and removed, the roots should be produced and removed which does
not make sense. Moreover, the removal of roots is what we expect in physics.
However, if the number of roots is dissimilar, then reciprocal distortion
implies the coexistence of removal and adding roots.

##### 3.

It must be an unstable excitation mode of $\Lambda_{24}$ because the local
distortions (local 3D) caused by a matter must be recovered (2D) after the
matter is removed, otherwise, we should not have observed a global flat
universe. In terms of symmetries, it means that it should be less symmetrical
than $\Lambda_{24}$ as distortion breaks symmetries and leads to an excited
state with a higher level of energy.

Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Lattices | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Dimension | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25
---|---|---|---|---|---|---|---|---|---
Lattices | 0 | 0 | 0 | 0 | 0 | 0 | $O_{23}$ | $\Lambda_{24}\>,\>O_{24}$ | 0
Table 1: Unimodular lattices with no roots

Among 117 unimodular lattices in 23D, only one lattice has no roots which is
the shorter Leech lattice $O_{23}$ (with minimal norm 3). Looking at Table 1
of unimodular lattices gives an insightful picture [38][39]. It seems that the
property of having no roots is extremely exceptional and meaningful (at least
in lower dimensions; since in higher dimensions they rapidly grow, e.g., in
32D there are around $10^{16}$ unimodular lattices without root). Under the
critical 26D, there are only three lattices with no roots, $\Lambda_{24}$ and
the odd Leech lattice $O_{24}$ in 24D, and $O_{23}$ in 23D! Remarkably,
$O_{23}$ is odd which makes sense regarding the 3rd property (if it was even
then it had all the required properties to be stable). On the other hand, the
symmetry group of $\Lambda_{24}$ and $O_{23}$ are the double covers of the
sporadic Conway groups $Co_{1}$ and $Co_{2}$, respectively. This explains the
close relationship between $\Lambda_{24}$ and $O_{23}$, since $Co_{2}$ is not
only a subquotient of $Co_{1}$ (see Figure 1) (in sporadic group wordings,
$Co_{2}$ is involved in $Co_{1}$) but also it is its largest maximal subgroup.
In other words, $O_{23}$ fits inside $\Lambda_{24}$ and a distortion between
them is possible.

A descriptive example of such a conversion is the Binary Golay code (BGC). One
way of constructing the Leech lattice is via the error-correcting BGC. There
are two closely related BGCs, namely the extended BGC, $G_{24}$, and the
perfect BGC, $G_{23}$, whose symmetries are the sporadic groups of Mathieu
$M_{24}$ and $M_{23}$, respectively. Similarly, $M_{23}$ is a subquotient of
$M_{24}$. $G_{24}$ can correct up to 3 errors in a codeword of 24 letters and
detect up to 7. $G_{23}$ has a length of 23 letters and can be obtained from
$G_{24}$ by deleting one letter. Conversely, $G_{24}$ can transform to
$G_{23}$ by adding a parity letter. Also, in Figure 1, one sees that $M_{24}$
is a subquotient of $Co_{1}$, and $M_{23}$ is a subquotient of $Co_{2}$,
suggesting that the relationship (transformation) between these Conway groups
is similar to the relationship (transformation) between these Mathieu groups.
On the other hand, just as the Monster VOA is associated with $\Lambda_{24}$,
a vertex operator superalgebra, called the shorter Moonshine module [40],
whose automorphism group is $2\times B$, is associated with $O_{23}$, where
$B$ is the Baby monster group (see the upper bold lines in Figure 1).

Figure 1: Diagram of sporadic groups. A line implies that the lower group is a
subquotient of the upper one. The colors indicate different generations,
namely the Mathieu groups (khaki), Leech lattice groups (grey), other
subgroups of the Monster (blue), and the Pariah groups (purple).

1 10000 1 10000 In sum, we propose that the fabric of spacetime is very likely
the Leech lattice which, in the presence of matter, is perturbed and locally
distorts to the shorter Leech lattice. Remarkably, the model explains why it
is not possible to have more than 3 and less than 2 dimensions of space. It is
because, as mentioned, there are no 22 and 25-dimensional lattices without
roots to make it possible for $\Lambda_{24}$ and $O_{23}$ to further transform
to other dimensions. We stress that, in the model, the compactification is
accomplished only in 24D and $\Lambda_{24}$ is the original candidate for the
fabric of spacetime. Hence, $O_{23}$ is only the consequence of the existence
of matter and it does not show up in a universe without matter.

####  3.2.1 Validation

Above, it is proposed that the presence of matter causes $\Lambda_{24}$ to
distort to $O_{23}$. What is the smallest mass that can do such a distortion?
If our proposal is correct, then the model has to predict the amount of mass
at which the gravitational effects of the quantum realm become important. This
mass scale is known to be the Planck scale, where the Compton wavelength (the
equivalent quantum length) of a matter is equal to the Schwarzschild radius
(the equivalent gravitational length). In string theory, it is the critical
scale of a fundamental string. In the model, deforming $\Lambda_{24}$ to
$O_{23}$ is equivalent to transforming from a more symmetrical state of
$2.Co_{1}$ to a less symmetrical state of $2.Co_{1}$, where
$|Co_{1}|=4157776806543360000$ and $|Co_{2}|=42305421312000$. In this process,
symmetries break $\frac{2|Co_{1}|}{2|Co_{2}|}=98280$ times, or in more
accurate terminology, symmetries become hidden. As before, this is the number
of modular transformations $|PSL(2,24q)|=(24q)^{3}-(24q)$ needed to break
$\Lambda_{24}$ to $O_{23}$. Therefore, we have $(24q)^{3}-(24q)=98280$ which
yields the length of $|q|\approx 1.92$ where the closest integer is $q=2$.
This length yields the closed string mass of $m_{s}=\frac{2}{2}=1$ which means
$M_{s}=M_{P}$.

It is also worthwhile to further explore the significance of $98280$ hidden
symmetries. The minimal (norm four) vectors that span $\Lambda_{24}$ split
into 3 shapes of $(4^{2},0^{22}),\>(2^{8},0^{16})$, and $(\mp 3,\pm 1^{23})$
respectively with sizes $97152,\>1104,\>$ and $98304$, where, 97152 + 1104 +
98304 = 196560. First, note that $98280=\frac{196560}{2}$, i.e., the smallest
faithful permutation representation of $Co_{1}$ is on 98280 opposite pairs
$(x,-x)$ of norm 4 vectors. When exactly half of the minimal vectors become
hidden (break), it suggests the connection to the BPS-saturated states (where
the BPS bound is saturated when half of the supersymmetry generators are
unbroken) which we discuss in the next point. Second, the size of the last
shape, ($98304=2^{12}.24$) which originates from BGC (i.e., a 12-dimensional
subspace in a 24D space over $\mathbb{F}_{2}$) is $98280+24$ which means
$98280=24(2^{12}-1)$. Here there are two observations. On one hand, one can
see that in the Monster CFT, 196884 states with $h=2$ consist of 98280
opposite state pairs, plus $2^{12}.24$ states coming from the twisted sector,
plus $1+2+...+24$ states are of the form
$\alpha_{-1}^{i}\alpha_{-1}^{j}\ket{0}$. On the other hand, we noticed that
the reported spectrum of the Monstrous CHL models in [41] also contains two
kinds of irreducible representations, one short (BPS) 1-dimensional
representation and a long representation of dimension $2^{12}$ and the theory
has 24 spacetime supersymmetries. Before mentioning their work, recall that
the BPS states are those states that are annihilated by some nonzero odd
element in the superalgebra forming short representations of an extended
supersymmetry algebra [42]. It sounds interesting since we also have a
symmetry breaking from the even Leech lattice (with minimal norm 4) to the odd
shorter Leech lattice (with minimal norm 3).

Their work involves compactification (on the spatial circle) in the heterotic
Monster theory $\mathbb{M}\times Co_{0}$ from $(1+1)D$ to $(0+1)D$. The
heterotic Monster contains a supersymmetric variant of Monster CFT, which is
first remarked in [14] and then constructed in [43]. In particular, the left-
moving side has the same construction as the Monster CFT. The right-moving NS
sector is a $N=1$ superconformal field theory with $c=12$ whose symmetry group
is the familiar $2Co_{1}$, built in terms of the $E_{8}$ lattice. Although
there is not a direct geometric interpretation for the internal CFT, it can be
obtained as ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifold of the product
of the holomorphic Leech lattice CFT and the anti-holomorphic $E_{8}$ lattice
SCFT. The compactification of heterotic strings on torus $\mathbb{T}^{8}$
yields a $(1+1)D$ theory with $(8,8)$ spacetime supersymmetry. While the
${\mathbb{Z}}_{2}$ orbifold acting on the left-moving (bosonic) sector
respects them, the ${\mathbb{Z}}_{2}$ orbifold acting on the supersymmetric
right-moving side breaks half of them. Then, incorporating 16 extra
supersymmetries with the same spacetime chirality by the twisted sector,
yields $(0,24)$ supersymmetry of the resulting theory. The full partition
function of the Moster heterotic theory is
$Z_{NS}(q,\bar{q})=Z_{\mathbb{M}}(q).\bar{Z_{NS}}(\bar{q})$ and
$Z_{R}(q,\bar{q})=Z_{\mathbb{M}}(q).\bar{Z_{R}}(\bar{q})$ where,
$Z_{\mathbb{M}}(q)$ is the partition function of the Monster CFT,
$\bar{Z_{NS}}(\bar{q})=-\bar{Z_{R}}(\bar{q})=\frac{\Theta_{E_{8}}\theta_{2}^{4}}{2\eta^{12}}=8+2048q+49152q^{2}+...$,
and $\Theta_{E_{8}}$ is the theta function of the $E_{8}$ lattice. Note that
the partition function in the bosonic left-movers is a worldsheet object
whereas the supersymmetric right-movers belong to spacetime [41]. Thus,
distortion (symmetry breaking) from $\Lambda_{24}$ to $O_{23}$ seemingly
breaks the spacetime supersymmetry.

###  3.3 Parallel Chains

In attempts for the unification, it is understood that the notable gauge
theory candidates, that can accommodate the fermion families, are the GUT
groups of $SU(5)$, $SO(10)$, and $E_{6}$; respectively with a chiral matter
description of quark-leptons in the $\bar{5}+10$, 16, 27-dimensional
representation. Even though $SU(5)$ and $SO(10)$ nicely match with the fermion
quantum numbers, the gifts we received during the process of trying for
unification (e.g., predictions of proton decay and neutrino masses) justify
the desire for larger GUT groups like $E_{6}$. In this context, the
introduction of group embedding $E_{6}\rightarrow SO(10)\rightarrow
SU(5)\rightarrow SM$ was justified. The $E_{6}$ group, which is an exceptional
simple Lie group, also shows up in the embedding chain of the exceptional Lie
groups $E_{8}\rightarrow E_{7}\rightarrow E_{6}\rightarrow F_{4}\rightarrow
G_{2}$. In this embedding, notably, $E_{6}$ is the only exceptional group with
complex representations, and therefore, the only eligible one to contain
chiral fermions and be a candidate for GUT. In this regard, a natural question
was raised as to why nature should not allow the chain of exceptional groups
above $E_{6}$ [44]? What is exceptional with this exceptional group?

The desire to continue the chain of exceptional groups above $E_{6}$ was
mainly due to the emergence of the heterotic $E_{8}\times E_{8}$ (HE)
superstring theory [45][46]. In the development of the string theory, the HE
superstring is a well-marked candidate. Therefore, let us calculate its
corresponding energy scale in a similar manner. Although the simple
exceptional group $E_{8}$ is an infinite group and, therefore, leads to a zero
mass, we exceptionally consider its Weyl group which is a finite group and
describes the symmetries of the weight lattice of $E_{8}$. As demonstrated in
[47], in the case of finite-dimensional representations, there exists a
canonical action of the Weyl groups on observables such that many physical
relations are a consequence of the Weyl group action rather than the action of
the original group of root systems on observables. Particularly,
$Weyl(E_{8})=2.O_{8}^{+}(2).2$, i.e., a stem extension by $\mathbb{Z}_{2}$ of
an extension of $\mathbb{Z}_{2}$ by the orthogonal group $O_{8}^{+}(2)$ [48].
The target space of the HE superstring is again the 26 bosonic strings where
compactification of 16D results in a 10D spacetime. Thus, we have
$|PSL(2,16q)|=(16q)^{3}-(16q)=|Weyl(E_{8})|=696729600$. Solving for $q$,
yields $\>|q|\approx 55.4$. The mass of the NS sector ground state is
$m_{s}=\frac{\sqrt{2}}{\ell s}$. Therefore, $M_{s}\approx 6\times
10^{16}\rm\>GeV$. This value is in agreement with the approximated GUT scale
of $10^{16}\>\>GeV$ with supersymmetry [49].

As mentioned, the heterotic version of the Monster CFT, $\mathbb{M}\times
Co_{0}$, is also constructed in [43]. Putting the symmetry groups of the left
and right moving sides in a bracket as a pair, [$\mathbb{M}$ , $Co_{1}$]
($Co_{0}$ is the double cover of $Co_{1}$), we found it worthwhile to find
other similar pairs (bold lines in Figure 1). In this manner, one level lower,
we find the pair [$B$ , $Co_{2}$] noting the discussed relationship between
$Co_{1}$ and $Co_{2}$. As noted, a vertex operator superalgebra, called the
shorter Moonshine module, whose symmetry group is $2\times B$, where $B$ is
the Baby Monster group, is also constructed in [40]. According to McKay’s
observation, the exceptional Lie groups $E_{8}$, $\>E_{7}$, and $\>E_{6}$ are
in correspondence respectively with $\mathbb{M}$, $B$, and $Fi_{24}$
[50][51][52] regarding the assignment of their extended Dynkin diagram to
certain conjugacy classes in the sporadic groups. So, the next pair should
start with $Fi_{24}$, and according to the subquotient relationships, it
becomes [$Fi_{24}$ , $M_{24}$]. The next pairs are then easy to find,
[$Fi_{23}$ , $M_{23}$] and [$Fi_{22}$ , $M_{22}$]. These pairs along the
abovementioned group embedding, considering McKay’s correspondence are
collected in Table 2 Note that the table is divided into two different
sections regarding the nature of the left element of the pairs (LEPs) and the
right element of the pairs (REPs). First, the REPs in the upper section are
the familiar Conway groups (from the 2nd generation of the sporadic groups).
The REPs in the lower section are the Mathieu groups (from 1st generation of
the sporadic groups). Second, the LEPs of the lower part are the Fischer
groups, where $Fi_{24}$ is not a subquotient of $B$ and is not involved in it.
Therefore, Table 2 may be elaborating and providing a hint for the above
question about the $E_{6}$ distinction. That is, not only $B$ does not contain
$F_{24}$ (which cuts the chain) but also the REPs of the upper section are
reserved for the spacetime symmetry of the Leech and shorter Leech lattices.

$E_{8}$ | [$\mathbb{M}$ , $Co_{1}$]
---|---
$E_{7}$ | [$B$ , $Co_{2}$]
$E_{6}$ | [$Fi_{24}$ , $M_{24}$]
$SO(10)$ | [$Fi_{23}$ , $M_{23}$]
$SU(5)$ | [$Fi_{22}$ , $M_{22}$]
Table 2: Parallel Chains

Finally, there is another rationale pointing at $E_{6}$. If we search for the
largest maximal subgroup of $\mathbb{M}$, we find $B$ which is also the
centralizer of an involution. Next, if we search for the largest maximal
subgroup of $B$, we do not find a sporadic group! It is the twisted Chevalley
group ${}^{2}E_{6}(2)$ (over $\mathbb{F}_{2}$) [20] which is also the the
centralizer of an involution. This latter embedding of ${}^{2}E_{6}(2)$ in $B$
is "a match made in Heaven" according to Stroth [53]. Also, its order can be
calculated by inserting $q=2$ in
$|^{2}E_{6}(q)|=|q^{36}(q^{12}-1)(q^{9}+1)(q^{8}-1)(q^{6}-1)(q^{5}+1)(q^{2}-1)|$
which approximately is $2^{78}\approx 10^{23}$. Over, e.g.
$q={|\mathbb{M}|}^{1/3}$, it has then around $10^{1326}$ points which seem
enough to reflect a favorable smooth manifold. For comparison, there are about
$10^{80}$ atoms in the universe. As a result, we stress the importance of the
embedding chain of $\mathbb{M}\rightarrow B\rightarrow\>^{2}E_{6}(2)$. In sum,
in light of the presented observations, we found these pieces of evidence
satisfactory to come to the conclusion that a fundamental theory of nature is
possibly discrete and finite.

## Acknowledgment

I especially thank my family for their support and appreciate the discussion
with Masud Naseri.

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