# Symmetries, Information and Monster Groups before and after the Big Bang

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

The Mode is an enclosed, detectable manifestation of the Substance… The Substance is equipped with infinite attributes(Spinoza, Ethica, pars I)

## 1. Introduction

^{54}elements [1]. It is noteworthy that the Monster Module displays the highest known number of symmetries [2]. It has been recently proposed that the symmetries, widespread invariances occurring at every level of organization in our universe, may be regarded as the most general feature of physical systems, perhaps also more general than thermodynamic constraints [3,4]. Therefore, giving insights into the Monster symmetries would provide a very general approach to systems function, universe evolution and energetic dynamics. Here we show how a novel symmetry-based, topological approach sheds new light on the Monster’s features. We provide a foundation for the Monster’s physical counterparts, cast in a fashion that has the potential to be operationalized, which can be used for the assessment of our universe’s evolution and, in particular, pre-Big Bang scenarios.

## 2. Topological Tools

#### 2.1. The Standard Version of the Borsuk–Ulam Theorem (BUT)

**Borsuk–Ulam**

**Theorem.**

**Proof.**

**Energy-Borsuk–Ulam**

**Theorem.**

**Proof.**

#### 2.2. BUT Variants

^{n}can also stand for other types of numbers. The n value can be also cast as an integer, a rational or an irrational number. This allows us to use the n parameter as a versatile tool for the description of systems symmetries [3]. A BUT variant tells us that we can find a pair of opposite points an n-dimensional sphere, that display the same encoding not just on a R

^{n}manifold, but also on an n−1 sphere. A symmetry break occurs when the symmetry is present at one level of observation, but hidden at another level [4]. This means that symmetries can be found when evaluating the system in a proper dimension, while they disappear (are hidden or broken) when the same system is embedded in just one dimension lower.

^{n}manifold does not map just to a R

^{n}

^{−1}Euclidean space, but straight to a S

^{n−1}manifold. In other words, the Euclidean space is not mentioned in this formulation. Indeed, in many applications, e.g., in fractal systems, we do not need a Euclidean manifold at all. A manifold, in this case S

^{n}, may exist in and on itself, by an internal point of view, and does not need to be embedded in any dimensional space [13]. Therefore, we do not need an S

^{n}manifold curving into a dimensional space R

^{n}: we may think that the manifold just does exist by itself. An important consequence of this BUT version is that a n-sphere may map on itself. The mapping of two antipodal points to a single point in a dimension lower can be a projection internal to the same n-sphere.

^{n}and S

^{n−1}. An n-sphere S

^{n}is equipped with two antipodal points, standing for symmetries according to BUT.BUT is enriched by considering an n-sphere ${S}^{n}$ as a manifold, which is a Hausdorff space with a countable basis where each point has a neighbourhood that is homeomophic to some Euclidean space. Briefly, a space is Hausdorff, provided distinct points belong to disjoint neighbourhoods (distinct points live in separate houses [16]). A mapping $f:X\to Y$ is homeomorphic, provided f on X is 1-1, onto Y and has a continuous inverse. For the sake of intuition, we illustrate the notion of homeomorphic neighbourhoods in terms of planar homeomorphic energistic neighbourhoods. Let $p,x\in {S}^{n}$, radius $r>0$ and $\Vert {e}_{x}-{e}_{p}\Vert =\sqrt{{{e}_{x}}^{2}+{{e}_{p}}^{2}}$ (norm of energies associated with the points x, p) for an open neighbourhood $N\left(p,r\right)$ defined by

^{n−1}lies, a symmetry break/dimensionality reduction occurs, and a single point is achieved [11]. It is widely recognized that a decrease in symmetry goes together with a reduction in entropy and free-energy (in a closed system). This means that the single mapping function on S

^{n−1}displays energy parameters lower than the sum of the two corresponding antipodal functions on S

^{n}. Therefore, a decrease in dimensions gives rise to a decrease of energy and energy requirements. BUT no longer depends on thermodynamic parameters, but rather on topological features such as affine connections and homotopies. The energy-BUT concerns not just energy, but also information. Indeed, two antipodal points contain more information than their single projection in a lower dimension. Dropping down a dimension means each point in the lower dimensional space is simpler, because each point has one less coordinate. In sum, energy-BUT provides a way to evaluate the decrease of energy in topological, other than thermodynamic, terms.

**Region-Based Borsuk–Ulam**

**Theorem**

**(ReBUT).**

**Proof.**

^{n}, where each component of $\Phi \left(\text{str}A\right)$ is a feature value of $\text{str}A$.

#### 2.3. Quantum String Axioms

- Every string has an action.
- If $\text{str}A,\neg \text{str}A$ are antipodal, then $actio{n}_{strA}=actio{n}_{\neg strA}$.
- Separate strings with k features with the same description are antipodal.
- There is a set $\left\{\neg \text{str}A\right\}$ of antipodal strings for every string $\text{str}A$.

**Lemma.**[strBUT].

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

^{n}to S

^{n−1}, we need to work with lower dimensional spaces containing regions where each point in S

^{n−1}has one less coordinate than a point in S

^{n}.

^{d}and M

^{d−1}are topological spaces equipped with a strong descriptive proximity relation. Recall that in a topological space M, every subset in M and M itself are open sets. A set E in M is open, provided all points sufficiently near E belong to E [21]. The description-based functions in BUT are strongly proximally continuous and their domain can be mathematical, physical or biological features of world line shapes. Let A, B be subsets in the family of sets in M (denoted by ${2}^{M}$) and let $f:{2}^{M}\to {R}^{n},A\in {2}^{M},f(A)=$ a feature vector that describes A. That is, $f(A),f(B)$ are descriptions of A and B. Nonempty sets are strongly near, provided the sets of have elements in common. The function f is strongly proximally continuous, provided A strongly near B implies $f(A)$ is strongly near $f(B)$. This means that strongly near sets have nonempty intersection. From a BUT perspective, multiple sets of objects in M

^{d}are mapped to $f\left(A\cap B\right)$, which is a description of those objects common to A and B. In other words, the functions in BUT are set-based embedded in a strong proximity space. In particular, each set is a set of contiguous points in a path traced by a moving particle. The path is called a world line. Pairs of world lines have squiggly, twisted shapes opposite each other on the surface of a manifold. Unlike the antipodes in a conventional hypersphere assumed by the BUT, the antipodes are now sets of world lines that are discrete and extremely disconnected. Sets are extremely disconnected, provided the closure of every set is an open set [20], is in the discrete space and the intersection of the closure of the intersection of every pair of antipodes is empty. The shapes of the antipodes are separated and belong to a computational geometry. That is, the shapes of the antipodal world lines approximate the shapes in conventional homotopy theory [22]. The focus here is on the descriptions (sets of features) of world line shapes. Mappings onsets with matching description, or, in other words, mappings on descriptively strongly proximal sets, here means that such mappings preserve the nearness of pairs of sets. The assumption made here is that antipodal sets live in a descriptive Lodato proximity (DLP) space. Therefore, antipodal sets satisfy the requirements for a DLP [12]. Let $\delta $ be a DLP and write $A\delta B$ to denote the descriptive nearness of antipodes A and B, and let f be a DLP continuous function. This means $A\delta B\text{implies}f(A)\delta f(B)=f(A)\cap f(B)\ne \varnothing $.

**Example.**

#### 2.4. Generalized BUT (genBUT)

^{d}are mapped to a single set of objects in M

^{d−1}and vice versa. The sets of objects, which can be mathematical, physical or biological features, do not need to be antipodal and their mappings need not to be continuous. The term matching description means the sets of objects display common feature values or symmetries. M stands for a manifold with any kind of curvature, either concave, convex or flat. M

^{d−1}may also be a part of M

^{d}. The projection from S to R in not anymore required, just M is required. The notation d stands for a natural, or rational, or irrational number. This means that the need for spatial dimensions of the classical BUT is no longer required. The process is reversible, depending on energetic constraints. Note that a force, or a group, an operator, an energetic source, is required, in order to project from one dimension to another.

## 3. Embedding the Monster Group in M^{d−1}

^{d}manifold, where d stands for their abstract dimension 196,884. Encompassing the two parameters in a M

^{d}manifold allows us to provide a topological commensurability between the Monster Module and the j-function. When we reduce the dimensions to S

^{196,883}, we achieve a single function, e.g., the Monster Lie group. It easy to see that if we map the two functions to a dimension lower, in this case M

^{196,883}, we achieve a single function that retains the features of both. This single function stands for the Monster Group, which is the automorphic Lie group acting on the Monster Module (Figure 3, upper part). In topological terms, as always, two functions on a S

^{n}sphere lead to a single function on a S

^{n−1}sphere.

## 4. Of Monsters and Universes

#### 4.1. Dimensions Reduction

^{d}structure encompassing our universe do not need to be necessarily of the huge order of 10

^{54}. Indeed, the Monster group includes several subgroups, classified into the sporadic groups (e.g., Mathieu groups, Leech lattice groups, and so on) [25]: this means that the universe might arise either from the Monster group, or one of its subgroups. In such a vein, one might think different possible physical scenarios:

- (a)
- The Monster group is progressively formed starting from its subgroups, with a gradual building from blocks.
- (b)
- The Monster group is the original structure giving rise to our universe.
- (c)
- The Monster group, the original structure, splits in its subgroups, then one of the subgroups gives rise to our universe.

#### 4.2. Topological Relationships between the Monster and String Theories

^{8}(C) and a lattice vertex operator algebra equipped with a rank 24 Leech lattice [23,26]. Several features of the Monster, either its Module, or its group and subgroups, have been associated with different physical theories. Some examples are depicted in Figure 3 (lower part). For example, links between Monstrous Moonshine and string theories have been proposed: the Monster might stand for the symmetry of a string theory for a Z

^{2}-orbifold of free bosons on a Leech lattice torus, in the context of a conformal field theory equipped with partition function j. Recent papers link other sporadic groups (e.g., Monster subgroups) with modular forms, suggesting a more central role for the Umbral Moonshine conjecture [27]. On the other hand, Witten proposed that pure gravity in AdS

_{3}(anti deSitter) space with maximally negative cosmological constant is AdS/CFT dual to a holomorphic CFT (conformal field theory), with the numbers of the Moonshine coming into play [28]. CFT/AdS is dual to string theories and is involved in many theoretical models: CFT, Chern-Simon-Matter, Super Jang–Mills, Superconformal algebras. The AdS/CFT correspondence means that conformal field theory is like a lower-dimensional hologram, which captures information about the higher-dimensional quantum gravity theory: this is one of the typical frameworks easily describable by a BUT topological apparatus.

#### 4.3. The Problem of Singularity

^{90}Planckian size, disconnected regions [31]. Currently, those regions make up our observable universe and resemble one another. The presence of the homogeneous Monster Module before the Big Bang explains, together with the inflationary period, why all the initial disconnected regions displayed the same features.

#### 4.4. The Monster and the Spacetime

^{196,884}manifold, lies in infinite dimensions and is atemporal, the lower level, embedded in an S

^{196,883}manifold, requires the introduction of the parameter time.

## 5. Quantifying Physical Monster’s Parameters

#### 5.1. Towards the Monster’s Enthalpy

_{0}, F

_{0}, T

_{0}, E

_{0}) and at the present time (H

_{1}, F

_{1}, T

_{1}, E

_{1}). The current level E

_{1}of entropy in the universe is estimated in 2.6 ± 0.3 × 10

^{122}K [33,34], while T

_{1}is neglectable. If the universe displays four dimension as currently believed, every dimension contains approximately an average entropy of: E

_{1}/4.

_{1}, because a monotonical increase already occurred. This means that E

_{1}is, more or less, the maximum value of entropy achievable in the whole life of the universe, and also means that the free-energy F

_{1}is currently very low. Therefore:

_{1}= H

_{1}.

_{1}almost equals the total enthalpy H

_{1}of the universe. Vice versa, at the Big Bang, F

_{0}and T

_{0}were very high and E

_{0}close to zero. This means that, at the Big Bang:

_{0}= H

_{0}− T

_{0}.

_{M}is:

_{M}= E

_{1}/4 × 196,883.

_{M}stands roughly for the same value:

_{M}= E

_{1}/4 × 196,883

^{54}elements, and if our universe has four dimensions, we have 10

^{50}elements in our universe.

#### 5.2. Information

#### 5.3. Watching the Monster: Vertex Algebra

## 6. Questions and Conclusions

- (a)
- Where does the Monster take such a huge amount of enthalpy? It takes us in pre-, pre-Big Bang scenarios. This is the same problem with inflationist models, that do not explain where the energy of the required false vacuum comes from. A link between the Monster group and the false vacuum might be speculated.
- (b)
- What is the role of the j-function in the pre Big Bang period? Does it provide energy?
- (c)
- How does the Monstrous Moonshine look like? We could either imagine a timeless, immutable manifold where just the Monster Group movements take place, or as we did, a dynamical, time-dependent structure.
- (d)
- Does the curvature of the Monster Module change with the passage of time? This could be a very useful information, in order to elucidate the hypothesized step from an ancient anti DeSitter hyperbolic universe to the current, flat one.
- (e)
- Our universe might not arise directly from the Monster, but by one of its subgroups, e.g., the Th group (Figure 3), which is correlated with the successful superstring 10D theory. Is it possible to split the Leech lattice in which the Monster group is embedded, in order to achieve the lower dimensional E8 lattice where the Th group’s movements take place? It is central to remind that the step from an E8 lattice to the Leech lattice requires ×3 multiplication and peculiar rotations.
- (f)
- The topological step from the vertex operator algebra to the Lie Monster Group requires a continuous function. Are we in front of a “super” gauge field? In other words, is there a gauge field that causes the first projection depicted at the top of Figure 3? In a topological framework, the feature that links the symmetries at a higher level with the single point at a lower level is the continuous function. If we assess two antipodal points as symmetries, and the single point as symmetry breaks and local transformations, a gauge field could be required, in order to restore the (apparently hidden) symmetry.

## Conflicts of Interest

## References

- Conway, J.H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups; Conway, J.H., Ed.; Clarendon Press: Oxford, MS, USA, 1985. [Google Scholar]
- Du Sautoy, M. Finding Moonshine: A Mathematician’s Journey through Symmetry; Fourth Estate: London, UK, 2008. [Google Scholar]
- Tozzi, A.; Peters, J.F. A Topological Approach Unveils System Invariances and Broken Symmetries in the Brain. J. Neurosci. Res.
**2016**, 94, 351–365. [Google Scholar] [CrossRef] [PubMed] - Roldán, E.; Martínez, I.A.; Parrondo, J.M.R.; Petrov, D. Universal features in the energetics of symmetry breaking. Nat. Phys.
**2014**, 10, 457–461. [Google Scholar] [CrossRef] - Borsuk, M. Drei Sätze über die n-dimensionale euklidische Sphäre. Fundam. Math.
**1933**, 1, 177–190. (In German) [Google Scholar] - Crabb, M.C.; Jawaworski, J. Aspects of the Borsuk–Ulam theorem. J. Fixed Point Theory Appl.
**2013**, 13, 459–488. [Google Scholar] [CrossRef] - Bredon, G.E. Topology and Geometry (Graduate Texts in Mathematics); Springer: New York, NY, USA, 1993; p. 557. [Google Scholar]
- Beyer, W.A.; Zardecki, A. The early history of the ham sandwich theorem. Am. Math. Mon.
**2004**, 111, 58–61. [Google Scholar] [CrossRef] - Matoušek, J. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Weisstein, E.W. Antipodal Points. 2016. Available online: http://mathworld.wolfram.com/AntipodalPoints.html (accessed on 19 December 2016).
- Tozzi, A.; Peters, J.F. Towards a Fourth Spatial Dimension of Brain Activity. Cogn. Neurodyn.
**2016**, 10, 189–199. [Google Scholar] [CrossRef] [PubMed] - Peters, J.F. Computational Proximity: Excursions in the Topology of Digital Images; Intelligent Systems Reference Library, Ed.; Springer: Berlin, Germany, 2016. [Google Scholar]
- Weeks, J.R. The Shape of Space, 2nd ed.; Marcel Dekker, Inc.: New York, NY, USA; Basel, Switzerland, 2002. [Google Scholar]
- Watanabe, T.; Masuda, N.; Megumi, F.; Kanai, R.; Rees, G. Energy landscape and dynamics of brain activity during human bistable perception. Nat. Commun.
**2014**. [Google Scholar] [CrossRef] [PubMed] - Sengupta, B.; Tozzi, A.; Cooray, G.K.; Douglas, P.K.; Friston, K.J. Towards a Neuronal Gauge Theory. PLoS Biol.
**2016**, 14, e1002400. [Google Scholar] [CrossRef] [PubMed] - Naimpally, S.A.; Peters, J.F. Topology with Applications: Topological Spaces via Near and Far; World Scientific: Singapore, 2013. [Google Scholar]
- Weyl, H. Space-Time-Matter; Dutton Publishing: Boston, MA, USA, 1922. [Google Scholar]
- Olive, D.I.; Landsberg, P.T. Introduction to string theory: Its structure and its uses. Philos. Trans. R. Soc. Lond. A Math. Phys. Sci.
**1989**, 329, 319–328. [Google Scholar] [CrossRef] - Petty, C.M. Equivalent sets in Minkowsky spaces. Proc. Am. Math. Soc.
**1971**, 29, 369–374. [Google Scholar] [CrossRef] - Dochviri, I.; Peters, J.F. Topological sorting of finitely near sets. Math. Comp. Sci.
**2016**, 10, 273–277. [Google Scholar] [CrossRef] - Bourbaki, N. Elements of Mathematics. General Topology 1; Éditions Hermann: Paris, France, 1966; Chapters 1–4. [Google Scholar]
- Borsuk, M. Fundamental retracts and extensions of fundamental sequences. Fundam. Math.
**1969**, 64, 55–85. [Google Scholar] - Frenkel, I.; Lepowsky, J.; Meurman, A. A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Natl. Acad. Sci. USA
**1984**, 81, 3256–3260. [Google Scholar] [CrossRef] [PubMed] - Duncan, J.F.R.; Griffin, M.J.; Ono, K. Moonshine. Res. Math. Sci.
**2015**, 2. [Google Scholar] [CrossRef] - Gannon, T. Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Borcherds, R. Monstrous Moonshine and Monstrous Lie Superalgebras. Invent. Math.
**1992**, 109, 405–444. [Google Scholar] [CrossRef] - Eguchi, T.; Ooguri, H.; Tachikawa, Y. Notes on the K3 Surface and the Mathieu group M
_{24}. Exp. Math.**2011**, 20, 91–96. [Google Scholar] [CrossRef] - Witten, E. Three-dimensional gravity revisited. 2007; arXiv:0706.3359. [Google Scholar]
- Chow, T.L. Gravity, Black Holes, and the Very Early Universe. An Introduction to General Relativity and Cosmology; Springer: New York, NY, USA, 2008; p. 280. [Google Scholar]
- Collins, G.P. The shapes of space. Sci Am.
**2004**, 291, 94–103. [Google Scholar] [CrossRef] [PubMed] - Veneziano, G. A Simple/Short Introduction to Pre-Big-Bang Physics/Cosmology. 1998; arXiv:hep-th/9802057v2. [Google Scholar]
- Fixsen, D.J. The Temperature of the Cosmic Microwave Background. Astrophys. J.
**2009**, 707, 916–920. [Google Scholar] [CrossRef] - Egan, C.A.; Lineweaver, C.H. A Larger Estimate of the Entropy of the Universe. 2010; arXiv:0909.3983v3. [Google Scholar]
- Frampton, P.; Hsu, S.D.H.; Kephart, T.W.; Reeb, D. What is the entropy of the universe? 2008; arXiv:0801.1847. [Google Scholar]
- Ekstrand, J. Lambda: A Mathematica package for operator product expansions in vertex algebras. Comput. Phys. Commun.
**2011**, 182, 409–418. [Google Scholar] [CrossRef] - Weisstein, E.W. Klein’s Absolute Invariant. 2016. Available online: http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html (accessed on 19 December 2016).
- Frenkel, I.B.; Lepowsky, J.; Meurman, A. Vertex Operator Algebras and the Monster; Academic Press: Cambridge, MA, USA, 1988; Volume 134. [Google Scholar]
- Sprott, J.C. A dynamical system with a strange attractor and invariant tori. Phys. Lett.A
**2014**, 378, 1361–1363. [Google Scholar] [CrossRef]

**Figure 1.**Torus Antipodal Strings. World lines with matching description preserve the nearness of pairs of sets. See text for further details.

**Figure 2.**(

**a**) 3D plot of $qr{e}^{iq\varphi}$ with nome q; (

**b**) 2D plot of real part of an elliptic module function $q={e}^{i\pi \tau}$. Symmetries in (

**a**) display a 3D plot and in (

**b**) a 2D plot of an elliptic module function. The j-function is an elliptic module function such as an elliptic theta function. A sample elliptic theta plot is shown in this Figure. Such functions are expressed in terms of a nome $q={e}^{i\pi \tau}.$ Then, for a complex number z, Jacobi theta functions are defined in terms the nome q, e.g., ${\vartheta}_{1}\left(z,q\right)\equiv {\displaystyle \sum _{n=-\infty}^{\infty}{{\displaystyle \left(-1\right)}}^{n-\frac{1}{2}}}{q}^{{\left(\frac{n+1}{2}\right)}^{2}}{e}^{\left(2n+1\right)iz}$.

**Figure 3.**Progressive loss of dimensions in sporadic groups can be encompassed in a Borsuk–Ulam Theorem framework. Note also the loss of symmetries from the highest dimension levels to the lowest ones. The Figure also illustrates how every sporadic group might display a theoretical physical counterpart.

**Figure 4.**Defining a vertex algebra on a torus helps us to visualize otherwise abstract structures. Starting from a vertex operator algebra (a very small portion is illustrated in (

**a**)) we made use of the attractors and the corresponding ordinary differential equations described by [38]. In (

**b**), the j-function on the attractor torus displays one coordinate initialized with a j-function value.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tozzi, A.; Peters, J.F.
Symmetries, Information and Monster Groups before and after the Big Bang. *Information* **2016**, *7*, 73.
https://doi.org/10.3390/info7040073

**AMA Style**

Tozzi A, Peters JF.
Symmetries, Information and Monster Groups before and after the Big Bang. *Information*. 2016; 7(4):73.
https://doi.org/10.3390/info7040073

**Chicago/Turabian Style**

Tozzi, Arturo, and James F. Peters.
2016. "Symmetries, Information and Monster Groups before and after the Big Bang" *Information* 7, no. 4: 73.
https://doi.org/10.3390/info7040073