#
Monstrous M-Theory^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

**1**⊕

**196,883**, where

**1**is identified with the dilaton, and

**196,883**is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M${}^{2}$-theory, which includes Horowitz and Susskind’s bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten’s extremal Monster SCFT in terms of representations of $S{O}_{24}$, the massless little group in $25+1$; the purely bosonic nature of the involved $S{O}_{24}$-representations may be traced back to the unique feature of 24 dimensions, which allow for a non-trivial generalization of the triality holding in 8 dimensions. Last but not least, a certain subsector of M${}^{2}$-theory, when coupled to a Rarita–Schwinger massless field in $26+1$, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be $\mathcal{N}=1$ supergravity theory in $26+1$ space–time dimensions.

## 1. Introduction

**196,883**⊕

**1**, which first hinted at Monstrous Moonshine [4], entails the fact that the dilaton scalar field $\varphi $ in $25+1$ is a singlet of $\mathbb{M}$ itself. As such, the irreducibility under $\mathbb{M}$ is crucially related to dilatonic gravity in $25+1$ space–time dimensions. The existence of a “weak” form of the $S{O}_{8}$-triality for $S{O}_{24}$, which we will name $\mathit{\lambda}$-triality, gives rise to a p (⩾0)-parametrized tower of “weak” trialities involving p-form spinors in 24 dimensions, which we will regard as massless p-form spinor fields in $25+1$ space–time dimensions. Such “weak” trialities are instrumental to providing most of the Monstrous gravity theories with a fermionic (massless) spectrum such that the spectrum is still acted upon by the Monster $\mathbb{M}$.

**196,883**in relation to the total number of massless degrees of freedom of Monstrous gravities in $D=25+1$; as such, this also elucidates the definition of $\mathbb{M}$ as the automorphism group of the Griess algebra (the degree two piece of the Monster VOA), which was considered to be artificial in that it was thought to involve an algebra of two or more unrelated spaces [3,6,23].

## 2. Evidence for Monstrous M-Theory

#### 2.1. Bosonic M-Theory in $D=26+1$

**2024**) results. In the present work, we will consider the case in which the $D=25+1$ massless 1-form ($\mathbf{24}$) and 3-form (

**2024**) persist, and actually they give rise to the so-called $\mathit{\lambda}$-triality, which is the generalization of $S{O}_{8}$-triality up to $S{O}_{24}$ in a “weaker”, namely, reducible, way (“Weak” triality was suggested by Eric Weinstein in 2016, at the Advances in Quantum Gravity conference (San Francisco)), of the form

#### 2.2. Lifting the M2-Brane to $D=26+1$ and the Leech Lattice

#### 2.3. Superalgebras and Central Extensions

#### 2.3.1. From 10 + 1 …

#### 2.3.2. … to 26 + 1

**98,304**Rarita–Schwinger field, as we will see in Section 5.2.

#### 2.4. M-Branes, Horava–Witten and the Monster SCFT

**8192**spinor then factorizes as $(\mathbf{32},\mathbf{128})\oplus (\mathbf{32},{\mathbf{128}}^{\prime})$, thus isolating a hidden ${\mathbf{128}}^{(\prime )}$ spinor, which can be used to form ${\mathfrak{e}}_{8}={\mathfrak{so}}_{16}\oplus {\mathbf{128}}^{(\prime )}$. Intriguingly, this may suggest an origin for Horava–Witten theory [28,29], which requires an eleven-manifold ${M}^{11}$ with boundary, whose boundary points are the ${\mathbb{Z}}_{2}$ fixed points in ${M}^{11}$; in this theory, a M2-brane stretched between these fixed points yields the strongly coupled heterotic string [28,29]. On the other hand, in the presence of the broken

**8192**spinor, we see a possible reason for the ${E}_{8}$ symmetry that arises at the fixed points, as the hidden spinor fermions may contribute to anomalies induced via the orbifold of the M10-brane worldvolume.

**8192**spinor projects down to $\mathbf{4096}$. This is in agreement with the Monster SCFT [9], where the fixed points of the orbifold contain 4096 twisted states. This differs from the orbifold reduction of bosonic M-theory, where the fixed points have no extra degrees of freedom due to the absence of chiral bosons and fermions [19]. A $D=26+1$ M-theory with $\mathbf{24}\xb7\mathbf{4096}=\mathbf{98,304}$ Rarita–Schwinger field would have fermionic anomalies at each orbifold fixed point that must be canceled by vector multiplets as in the $D=10+1$ M-theory case [28,29]. One would expect a generalization of ${E}_{8}$ symmetry at each fixed point that contains at least $\mathbf{24}\xb7{\mathbf{2}}^{\mathbf{12}}=\mathbf{98,304}$ vector multiplets for RNS twisted sector states. The Griess algebra provides such minimal degrees of freedom, and thus could possibly be used to cancel anomalies at the fixed points. Another possibility is the Leech algebra, which we introduce in a later section.

**8192**spinor from $D=26+1$, which is expected from an orbifold reduction, analogous to the case of $D=10+1$ M-theory where the $\mathbf{32}$ spinor is projected to a $\mathbf{16}$ [28,29]. A ${S}^{1}/{\mathbb{Z}}_{2}$ orbifold reduction reduces the

**8192**spinor to $\mathbf{4096}$ spinor, where under $S{O}_{24}$ one has

**4096**=

**2048**⊕

**2048**${}^{\prime}$. The $\mathbf{2048}$ spinors can yield worldsheet fermions in $D=25+1$. Such $S{O}_{24}$ spinors are seen in Duncan’s SCFT with Conway group symmetry [24]. These spinors can be used to build RNS states in $D=25+1$, that generalize the gravitino and dilatino states of type IIA in $D=9+1$ from 128 to 98,304 degrees of freedom.

**98,304**degrees of freedom, and thus is a candidate for the origin of the dimension 2 Ramond fields in a SCFT. The remaining 98,280 states come from a discretized transverse space, where in the $Ad{S}_{4}\otimes {S}^{23}$ near-horizon geometry of the M2-brane in $D=26+1$ the 23-sphere is discretized by the 196,560 norm four Leech lattice vectors. This is consistent with the Conway group $C{o}_{0}$ being a maximal finite subgroup of the $\mathcal{R}$-symmetry group $S{O}_{24}$. The ${S}^{1}/{\mathbb{Z}}_{2}$ orbifold reduces the 196,560 vectors to 98,280, while also reducing $Ad{S}_{4}\otimes {S}^{23}$ to $Ad{S}_{3}\otimes {S}^{23}$, and breaking the discrete $\mathcal{R}$-symmetry group $C{o}_{0}$ down to the simple Conway group (It is interesting to observe that $\mathbf{98,280}$ is not the dimension of a unique irreducible representation of $C{o}_{1}$, but rather it can be decomposed as a sum of irreducible representation of $C{o}_{1}$ [30]. Remarkably, one finds a decomposition only in terms of irreducible representations of $C{o}_{0}$ which all survive (and stay irreducible) under the maximal reduction $C{o}_{0}\to C{o}_{1}$, namely $\mathbf{98,280}=\mathbf{80,730}\oplus \mathbf{17,250}\oplus \mathbf{299}\oplus \mathbf{1}$.) $C{o}_{1}\simeq C{o}_{0}/{\mathbb{Z}}_{2}$ [6], thus making contact with Witten’s holographic interpretation of the Monster [12] with $C{o}_{1}$ as a discrete R-symmetry.

**276**⊕

**2048**⊕

**2048**${}^{\prime}$, where worldsheet fermions are suggested. This implies a $D=25+1$ string theory with $\mathbf{2048}\oplus {\mathbf{2048}}^{\prime}$ worldsheet fermions that generalizes the $D=9+1$ superstring with $S{O}_{8}$ spinors. In the treatment given below, we propose a $D=26+1$ origin for such a string theory, supported by the fermionization of the Monster CFT [13], which suggests a $(2+1)$-dimensional fermionic gravitational Chern–Simons term that can live on the boundary of $Ad{S}_{4}$. Once again, given an ${S}^{1}/{\mathbb{Z}}_{2}$ orbifold reduction of $D=26+1$ M-theory with fermions, one does expect anomalies, and to cancel such anomalies may necessitate the use of the Leech lattice ${\Lambda}_{24}$ at each fixed hyperplane. The resulting $D=25+1$ closed string theory is then very similar to the Bimonster string theory introduced by Harvey et al. in [11].

## 3. “Weak” Trialities in 24 Dimensions

#### 3.1. $\mathit{\lambda}$-Triality

#### 3.2. $\mathit{\psi}$-Triality

#### 3.3. Iso-Dimensionality among (Sets of) p-Forms: An Example

## 4. Monstrous Dilatonic Gravity in 25 + 1

**a**- They all contain gravity (in terms of one 26-bein, then yielding one metric tensor ${g}_{\mu \nu}$) and one dilaton scalar field $\varphi $; thus, the Lagrangian density of their gravito-dilatonic sector reads as follows (Throughout our analysis, we rely on the conventions and treatment given in Secs. 22 and 23 of [33]):$$\mathcal{L}={e}^{-2\varphi}\left(R-4{\partial}_{\mu}\varphi {\partial}^{\mu}\varphi \right).$$
**b**- The relations among all such theories are due to the $\mathit{\lambda}$-triality ${\tilde{\mathbb{T}}}_{\mathit{\lambda}}$ (34) and (35), the weak $\mathit{\psi}$-triality ${\tilde{\mathbb{T}}}_{\mathit{\psi}}$ (36) and (37), as well as the bosonic map $\mathcal{B}$ (38) and (39) of ${\mathfrak{so}}_{24}$ (real compact form of ${\mathfrak{d}}_{12}$), which is the Lie algebra of the massless little group.
**c**- By constraining the theories to contain only one graviton and only one dilaton, the total number of degrees of freedom of the massless spectrum must sum up to$$\begin{array}{ccc}& & 1+299+\mathrm{47,104}\xb7\left(\#\psi \right)+24\xb7\left(\#{\wedge}^{1}\right)+2048\xb7\left(\#\lambda \right)\hfill \\ & & +42,504\xb7\left(\#{\wedge}^{5}\right)+\mathrm{10,626}\xb7\left(\#{\wedge}^{4}\right)+2024\xb7\left(\#{\wedge}^{3}\right)+276\xb7\left(\#{\wedge}^{2}\right)\hfill \\ & =& \mathrm{196,884}.\hfill \end{array}$$

**196,883**is the dimension of its smallest non-trivial representation [1]. For this reason, the gravito-dilatonic theories under consideration will all be named Monstrous gravities. They will be characterized by the following split:

#### 4.1. Classification

- ${\mathbf{s}}_{1}$, a length-5 string, providing the number of independent “helicity”-h massless fields, with $h=2,\frac{3}{2},1,\frac{1}{2},0$, respectively denoted by g (graviton), $\psi $ (Rarita–Schwinger field), ${\wedge}^{1}$ (1-form potential), $\lambda $ [spinor field (The spinor field gets named gaugino (or dilatino) in the presence of supersymmetry.)], and $\varphi $ (dilaton); as pointed out above, we fix $\#g=\#\varphi =1$ throughout (additionally, note that any theory with $\#{\wedge}^{1}\u2a7e1$ is a Maxwell–Einstein–dilaton theory in $25+1$ space–time dimensions):$${\mathbf{s}}_{1}:=\left(\#g,\#\psi ,\#{\wedge}^{1},\#\lambda ,\#\varphi \right)=\left(1,\#\psi ,\#{\wedge}^{1},\#\lambda ,1\right);$$
- ${\mathbf{s}}_{2}$, a length-4 string, providing the number of independent p-form brane potentials, for the smallest values of p, namely for $p=5,4,3,2$,$${\mathbf{s}}_{2}:=\left(\#{\wedge}^{5},\#{\wedge}^{4},\#{\wedge}^{3},\#{\wedge}^{2}\right).$$

**a**–

**c**listed above, in five groups, labeled with Latin numbers: 0, 1, 2, 3, 4, specifying the number $\#\psi $ of $h=3/2$ RS fields. The $\mathit{\psi}$-triality ${\tilde{\mathbb{T}}}_{\mathit{\psi}}$ (36) and (37) of ${\mathfrak{so}}_{24}$ maps such five groups among themselves. Then, each of these groups will be split into four subgroups, labeled with Greek letters: $\alpha $, $\beta $, $\gamma $ and $\delta $, respectively characterized by the following values of $\#{\wedge}^{1}$ and $\#\lambda $:

**0**- Group 0 ($\psi $-less theories):$$\begin{array}{ccc}\hfill \underset{\mathbf{purely}\phantom{\rule{4.pt}{0ex}}\mathbf{bosonic}}{\underset{\left(B|F\right)=\left(196,884|0\right),\phantom{\rule{3.33333pt}{0ex}}\lambda \text{-}\mathrm{less}}{\alpha}}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,0,3,0,1\right)& \left(0,16,12,8\right)& \mathrm{bosonic},\phantom{\rule{4.pt}{0ex}}{\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,12,12,8\right)& \mathrm{bosonic}\\ iii& {}^{\u2033}& \left(2,8,12,8\right)& \mathrm{bosonic}\\ iv& {}^{\u2033}& \left(3,4,12,8\right)& \mathrm{bosonic}\\ v& {}^{\u2033}& \left(4,0,12,8\right)& \mathrm{bosonic},\phantom{\rule{4.pt}{0ex}}{\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{194,836}|2048\right)}{\beta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,0,2,1,1\right)& \left(0,16,11,8\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,12,11,8\right)& -\\ iii& {}^{\u2033}& \left(2,8,11,8\right)& -\\ iv& {}^{\u2033}& \left(3,4,11,8\right)& -\\ v& {}^{\u2033}& \left(4,0,11,8\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{192,788}|4096\right)}{\gamma}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,0,1,2,1\right)& \left(0,16,10,8\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,12,10,8\right)& -\\ iii& {}^{\u2033}& \left(2,8,10,8\right)& -\\ iv& {}^{\u2033}& \left(3,4,10,8\right)& -\\ v& {}^{\u2033}& \left(4,0,10,8\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{190,740}|6144\right),\phantom{\rule{3.33333pt}{0ex}}{\wedge}^{1}\text{-}\mathrm{less}}{\delta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,0,0,3,1\right)& \left(0,16,9,8\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,12,9,8\right)& -\\ iii& {}^{\u2033}& \left(2,8,9,8\right)& -\\ iv& {}^{\u2033}& \left(3,4,9,8\right)& -\\ v& {}^{\u2033}& \left(4,0,9,8\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$
**1**- Group 1 ($\#\psi =1$ theories):$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{149,780}|\mathrm{47,104}\right),\phantom{\rule{3.33333pt}{0ex}}\lambda \text{-}\mathrm{less}}{\alpha}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,1,3,0,1\right)& \left(0,12,10,6\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,8,10,6\right)& -\\ iii& {}^{\u2033}& \left(2,4,10,6\right)& -\\ iv& {}^{\u2033}& \left(3,0,10,6\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{147,732}|\mathrm{49,152}\right)}{\beta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,1,2,1,1\right)& \left(0,12,9,6\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,8,9,6\right)& -\\ iii& {}^{\u2033}& \left(2,4,9,6\right)& -\\ iv& {}^{\u2033}& \left(3,0,9,6\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{145,684}|\mathrm{51,200}\right)}{\gamma}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,1,1,2,1\right)& \left(0,12,8,6\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,8,8,6\right)& -\\ iii& {}^{\u2033}& \left(2,4,8,6\right)& -\\ iv& {}^{\u2033}& \left(3,0,8,6\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{143,636}|\mathrm{53,248}\right),\phantom{\rule{3.33333pt}{0ex}}{\wedge}^{1}\text{-}\mathrm{less}}{\delta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,1,0,3,1\right)& \left(0,12,7,6\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,8,7,6\right)& -\\ iii& {}^{\u2033}& \left(2,4,7,6\right)& -\\ iv& {}^{\u2033}& \left(3,0,7,6\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$
**2**- Group 2 ($\#\psi =2$ theories):$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{102,676}|\mathrm{94,208}\right),\phantom{\rule{3.33333pt}{0ex}}\lambda \text{-}\mathrm{less}}{\alpha}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,2,3,0,1\right)& \left(0,8,8,4\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,4,8,4\right)& -\\ iii& {}^{\u2033}& \left(2,0,8,4\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{100,628}|\mathrm{96,256}\right)}{\beta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,2,2,1,1\right)& \left(0,8,7,4\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,4,7,4\right)& -\\ iii& {}^{\u2033}& \left(2,0,7,4\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{98,580}|\mathrm{98,304}\right)}{\gamma}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,2,1,2,1\right)& \left(0,8,6,4\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,4,6,4\right)& -\\ iii& {}^{\u2033}& \left(2,0,6,4\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{96,532}|\mathrm{100,352}\right),\phantom{\rule{3.33333pt}{0ex}}{\wedge}^{1}\text{-}\mathrm{less}}{\delta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,2,0,3,1\right)& \left(0,8,5,4\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,4,5,4\right)& -\\ iii& {}^{\u2033}& \left(2,0,5,4\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$
**3**- Group 3 ($\#\psi =3$ theories):$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{55,572}|\mathrm{141,312}\right),\phantom{\rule{3.33333pt}{0ex}}\lambda \text{-}\mathrm{less}}{\alpha}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,3,3,0,1\right)& \left(0,4,6,2\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,0,6,2\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{53,524}|\mathrm{143,360}\right)}{\beta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,3,2,1,1\right)& \left(0,4,5,2\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,0,5,2\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{51,476}|\mathrm{145,408}\right)}{\gamma}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,3,1,2,1\right)& \left(0,4,4,2\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,0,4,2\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{49,428}|\mathrm{147,456}\right),\phantom{\rule{3.33333pt}{0ex}}{\wedge}^{1}\text{-}\mathrm{less}}{\delta}& :& \left[\begin{array}{cccc}& {\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ i& \left(1,3,0,3,1\right)& \left(0,4,3,2\right)& {\wedge}^{5}\text{-}\mathrm{less}\\ ii& {}^{\u2033}& \left(1,0,3,2\right)& {\wedge}^{4}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$
**4**- Group 4 ($\#\psi =4$ theories):$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(\mathrm{8,468}|\mathrm{188,416}\right),\phantom{\rule{3.33333pt}{0ex}}\lambda \text{-}\mathrm{less}}{\alpha}& :& \left[\begin{array}{ccc}{\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ \left(1,4,3,0,1\right)& \left(0,0,4,0\right)& {\wedge}^{5},{\wedge}^{4},{\wedge}^{2}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(6420|\mathrm{190,464}\right)}{\beta}& :& \left[\begin{array}{ccc}{\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ \left(1,4,2,1,1\right)& \left(0,0,3,0\right)& {\wedge}^{5},{\wedge}^{4},{\wedge}^{2}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(4372|\mathrm{192,512}\right)}{\gamma}& :& \left[\begin{array}{ccc}{\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ \left(1,4,1,2,1\right)& \left(0,0,2,0\right)& {\wedge}^{5},{\wedge}^{4},{\wedge}^{2}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill \underset{\left(B|F\right)=\left(2324|\mathrm{194,560}\right)}{\delta}& :& \left[\begin{array}{ccc}{\mathbf{s}}_{1}& {\mathbf{s}}_{2}& \mathbf{features}\\ \left(1,4,0,3,1\right)& \left(0,0,1,0\right)& {\wedge}^{5},{\wedge}^{4},{\wedge}^{2},{\wedge}^{1}\text{-}\mathrm{less}\end{array}\right]\hfill \end{array}$$

**4**(66) and (69)) contain bosonic string theory, whose (massless, closed string) field content is $\#g=\#\varphi =\#{\wedge}^{2}=1$ (see e.g., [19]), as a subsector.

## 5. Monstrous M-Theory in 26 + 1

**42,504**+ 4 ×

**10,626**+ 6 ×

**2024**) + 4 ×

**276**+

**24**to the norm four (i.e., minimal) Leech vectors modulo ${\mathbb{Z}}_{2}$, and hence 196,884 = 1 + 299 + 98,280 + 98,304 which corresponds to the Griess algebra, namely to the sum of the two smallest representations of $\mathbb{M}$, namely the trivial (singlet)

**1**and the smallest non-trivial one

**196,883**. Such a theory will be henceforth named Monstrous M-theory, or simply M${}^{2}$-theory. Note that the disentangling of the 196,884 degrees of freedom into

**196,883**⊕

**1**occurs only when reducing the theory to $25+1$, in which case the dilaton $\varphi $ is identified with the singlet of $\mathbb{M}$: in other words, the (observation which firstly hinted the) Monstrous Moonshine [4] is crucially related to the ${S}^{1}$ compactification of M${}^{2}$-theory down to $25+1$ space–time dimensions.

#### 5.1. Lagrangian(s) for Bosonic Monstrous M-Theory

#### 5.2. $B=F$ in $26+1$

#### 5.2.1. $\mathcal{N}=1$ Supergravity in $26+1$?

**Conjecture**

## 6. Cohomological Construction of Lattices: From ${\mathfrak{e}}_{8}$ to the Leech Lattice

- M-theory in $10+1$ space–time dimensions, with $S{O}_{9}$ massless little group and massless spectrum given by $\mathbf{128}$ (gravitino $\mathit{\psi}$) =$\mathbf{84}$ (3-form potential ${\wedge}^{3}$) ⊕$\mathbf{44}$ (graviton $g\simeq {S}_{2}^{0}$); this corresponds to $D0$-branes (supergravitons) in BFSS M(atrix) model, carrying $256=128\left(B\right)+128\left(F\right)$ KK states [38].
- The would-be $\mathcal{N}=1$ supergravity in $26+1$ space–time dimensions, with $S{O}_{25}$ massless little group and massless spectrum given by
**98,304**(would-be gravitino $\mathit{\psi}$) $=3\times \mathbf{2300}\oplus 3\times \mathbf{12,650}\oplus \mathbf{53,130}$ (set of massless p-forms which is the “($26+1$)-dimensional analogue” of the 3-form in $10+1$) ⊕$\mathbf{324}$ (graviton $g\simeq {S}_{2}^{0}$); this would correspond to $D0$-branes (i.e., the would-be “supergravitons”) in the would-be BFSS-like M(atrix) model, carrying $\mathrm{196,608}=\mathrm{98,304}\left(B\right)+\mathrm{98,304}\left(F\right)$ KK states.

#### 6.1. $26+1\u27f610+1$ through Vinberg’s T-Algebras

#### 6.2. $10+1\u27f63+1$ through ${S}^{7}$ Fiber

## 7. Further Evidence for M${}^{\mathbf{2}}$-Theory: Monster SCFT and Massless p-Forms in $\mathbf{25}+\mathbf{1}$

## 8. Final Remarks

**Monstrous M-theory, Monstrous dilatonic gravities and Monstrous Moonshine**

**1**and the non-trivial one

**196,883**. A subsector of M${}^{2}$-theory yields Horowitz and Susskind’s bosonic M-theory [19]. Crucially, the disentangling of the 196,884 degrees of freedom into

**196,883**⊕

**1**occurs only when reducing M${}^{2}$-theory down to $25+1$, obtaining the massless spectrum (71), in which the dilaton $\varphi $ is identified with the singlet of $\mathbb{M}$: in other words, the (initial observation giving rise to) Monstrous Moonshine [4] is crucially related to the KK compactification of M${}^{2}$-theory down to a certain Monstrous dilatonic gravity (namely, the theory $0.\alpha .iii$ within the classification carried out in Section 4.1) in $25+1$ space–time dimensions.

**196,883**of $\mathbb{M}$, and thus they may provide an explanation of the (initial observation giving rise to) Monstrous Moonshine in terms of (higher-dimensional, gravitational) field theory.

**Black hole entropy in 2 + 1**

**Local SUSY in 26 + 1?**

**Leech lattice and Griess algebra**

**Developments**

- It would be interesting to explore the implications of the characterization of the $\mathbb{M}$ as acting on the whole massless spectrum of M${}^{2}$-theory in $26+1$ space–time dimensions.
- One could further study the maps discussed in Section 3; as pointed out above, no other Dynkin diagram (besides ${\mathfrak{d}}_{4}$) has an automorphism group of order greater than 2, and thus, such maps cannot be realized as an automorphism of ${\mathfrak{d}}_{12}$, nor they can be traced back to some structural symmetry of the Dynkin diagram of ${\mathfrak{d}}_{12}$ itself.
- Additionally, one could study the Lagrangian structure of M${}^{2}$-theory, as well as of its Scherk–Schwarz reduction to $25+1$.
- Further evidence may be gained by investigating whether the dimensions of representations of finite groups, such as the Baby Monster group $\mathbb{BM}$, the Conway group $C{o}_{0}$ and the simple Conway group $C{o}_{1}\simeq C{o}_{0}/{\mathbb{Z}}_{2}$, can all be rather simply interpreted as sums of dimensions of representation of $S{O}_{24}$ or $S{O}_{25}$ itself, and study the decomposition of the (smallest) coefficients of the partition functions of the SCFT derived from the Monster SCFT.
- Further study may concern the double copy structure of Monster dilatonic gravities in $25+1$, as well as of M${}^{2}$-theory, and its possibly supersymmetric subsector, in $26+1$.
- The investigation on the existence of local SUSY in $26+1$, and the determination of the corresponding Lagrangian and SUSY transformations is of the utmost relevance, of course.
- Last but not least, it would be interesting to study the massive spectrum of (massive variants of) Monstrous gravities and of M${}^{2}$-theory.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Chern–Simons Lagrangian Terms for Monstrous M-Theory

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Marrani, A.; Rios, M.; Chester, D.
Monstrous M-Theory. *Symmetry* **2023**, *15*, 490.
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