# Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries: Branched Coverings of 3- and 4-Manifolds

#### 2.1. As Warmup: Branched Coverings of 2-Manifolds

#### 2.2. Branched Coverings of 3-Manifolds

#### 2.3. Branched Covering of 4-Manifolds

#### 2.4. Branched Coverings of Knot Complements

## 3. Reconstructing a Spacetime: The K3 Surface and Particle Physics

- $\mathcal{M}$ is a smooth 4-manifold,
- any sequence of spacetime event has to converge to a spacetime event and
- any loop (time-like or not) must be contracted.

- $\mathcal{M}$ has to admit a smoothness structure with Ricci-flat metric representing the vacuum.

- there is a compact, contactable submanifold $A\subset \mathcal{M}$ (called Akbulut cork) so that cutting out Aand reglue it (by an involution) will produce a new smoothness structure,
- $\mathcal{M}$ splits topologically into$$|{E}_{8}\oplus {E}_{8}|\#\underset{3\left({S}^{2}\times {S}^{2}\right)}{\underbrace{\left({S}^{2}\times {S}^{2}\right)\#\left({S}^{2}\times {S}^{2}\right)\#\left({S}^{2}\times {S}^{2}\right)}}=2\left|{E}_{8}\right|\#3({S}^{2}\times {S}^{2})$$
- the 3-sphere ${S}^{3}$ is a submanifold of A.

## 4. From K3 Surfaces to Octonions, 3-Braids and Particles

#### 4.1. K3 Surfaces and Octonions

#### 4.2. From Immersed Surfaces in K3 Surfaces to Fermions and Knot Complements

#### 4.3. Fermions as Knot Complements

#### 4.4. Torus Bundle as Gauge Fields

- finite order (orders $2,3,4,6$): the tangent bundle is three-dimensional,
- Dehn-twist (left/right twist): the tangent bundle is a sum of a two-dimensional and a one-dimensional bundle,
- Anosov: the tangent bundle is a sum of three one-dimensional bundles.

- finite order: 2 isotopy classes (= no/even twist or odd twist),
- Dehn-twist: 2 isotopy classes (= left or right Dehn twists),
- Anosov: 8 isotopy classes (= all possible orientations of the three line bundles forming the tangent bundle).

- torus bundle with no/even twists: one isotopy class,
- torus bundle with twist (Dehn twist or odd finite twist): three isotopy classes,
- torus bundle with Anosov map: eight isotopy classes.

- torus bundle with no twists: one isotopy class with ${t}^{2}$,
- torus bundle with twist: three isotopy classes with ${t}_{1}^{2}+{t}_{2}^{2}+{t}_{3}^{2}$,
- torus bundle with Anosov map: eight isotopy classes with ${\sum}_{a=1}^{8}{t}_{a}^{2}$.

#### 4.5. Fermions, Bosons and 3-Braids

## 5. Electric Charge and Quasimodularity

#### 5.1. Electric Charge as Dehn Twist of the Boundary

#### 5.2. Electric Charge as a Frame of the Knot Complement

#### 5.3. The Charge Spectrum

#### 5.4. Vanishing of the Magnetic Charge and Quasimodularity

## 6. Drinfeld–Turaev Quantization and Quantum States

- The surface S (branching set of $Cob({Y}_{1},{Y}_{2})$) is inducing a representation ${\pi}_{1}\left(S\right)\to SL(2,\mathbb{C})$.
- The space of all representations $X(S,SL(2,\mathbb{C}))=Hom({\pi}_{1}\left(S\right),SL(2,\mathbb{C}))/SL(2,\mathbb{C})$ has a natural Poisson structure (induced by the bilinear on the group) and the Poisson algebra $(X(S,SL(2,\mathbb{C}),\left\{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right\})$ of complex functions over them is the algebra of observables.
- The skein algebra ${K}_{t}(S\times [0,1])$ is the quantization of the Poisson algebra $(X(S,SL(2,\mathbb{C})),\left\{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right\})$ with the deformation parameter $t=exp(h/4)$ (see also [69]) .

- The module ${K}_{-1}\left(M\right)$ for $t=-1$ is a commutative algebra.
- Let S be a surface. Then, ${K}_{t}(S\times [0,1])$ carries the structure of an algebra.

## 7. Fermions and Number Theory

## 8. The K3 Surface and the Number of Generations

## 9. Conclusions and Outlook

- We constructed a spacetime, the K3 surface and derive some numbers like the cosmological constant or some energy scales and neutrino masses agreeing with experimental data.
- We derived from a representation of K3 surfaces by branched covering a simple picture: fermions are hyperbolic knot complements, whereas bosons are link complements (torus bundles).
- We obtained the gauge group from this picture (at least in principle).
- We derived the correct charge spectrum and obtained one generation.
- We conjectured about the number of generations and global symmetry (the $PGL(3,4)$) to get the mixing between the generations.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Boundary of the branching set change—Left: Move 1 leading to the Trefoil knot, Right: Move 2 leading to the Hopf link.

**Figure 7.**Branching sets—

**Left**: 6-Plat for the knot cpomplement,

**Center**: a 3-Braid as 6-Braid as example,

**Right**: 6-Braid for the link complement

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Asselmeyer-Maluga, T.
Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics. *Symmetry* **2019**, *11*, 1298.
https://doi.org/10.3390/sym11101298

**AMA Style**

Asselmeyer-Maluga T.
Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics. *Symmetry*. 2019; 11(10):1298.
https://doi.org/10.3390/sym11101298

**Chicago/Turabian Style**

Asselmeyer-Maluga, Torsten.
2019. "Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics" *Symmetry* 11, no. 10: 1298.
https://doi.org/10.3390/sym11101298