Spacetime and Internal Symmetry from Split Bioctonions and the Two Extra SU(3)’s of E₈ × ωE₈

1. Introduction

Over the last few years we have proposed and developed an $E_8\times \omega E_8$ theory of unification which aims to unify the standard model with gravitation described by the general theory of relativity [1,2,3]. Here, $\omega$ is the split complex number. It is assumed that each of the two $E_8$ branches as $SU\left(3\right)\times E_6$ and each of the two resulting $E_6$ undergoes a trinification $E_6\to SU\left(3\right)\times SU\left(3\right)\times SU\left(3\right)$. The split complex number plays a crucial role, enabling the emergence of a Lorentzian signature for spacetime, and enabling the emergence of chiral fermions. Its origin in our theory can be traced to (left acting) octonionic chains made from the algebra $\mathbb{O}\oplus \omega \mathbb{O}$ of split bioctonions. Complex split bioctonions generate the Clifford algebra $Cl\left(7\right)\cong Cl\left(6\right)\oplus \omega Cl\left(6\right)$ which is used to obtain one generation of standard model chiral quarks and leptons [4].

The trinification provides the following interpretation of the branching of the two $E_8$, as discussed in detail in [3]:

$$\begin{array}{cc}\hfill E_{8L}& \to SU{\left(3\right)}_L^{geom}\times E_{6L}\stackrel{E_{6L}}{\to }SU{\left(3\right)}_c\times SU{\left(3\right)}_{F,L}\times SU{\left(3\right)}_L\stackrel{SU{\left(3\right)}_L}{\to }SU{\left(2\right)}_L\times U{\left(1\right)}_Y\hfill \\ \hfill E_{8R}& \to SU{\left(3\right)}_R^{geom}\times E_{6R}\stackrel{E_{6R}}{\to }SU{\left(3\right)}_{c^{\prime }}\times SU{\left(3\right)}_{F,R}\times SU{\left(3\right)}_R\stackrel{SU{\left(3\right)}_R}{\to }SU{\left(2\right)}_R\times U{\left(1\right)}_{Ydem}\hfill \end{array}$$

(1)

Of the three $SU\left(3\right)$s arising from the branching of $E_{6L}$, the $SU{\left(3\right)}_c$ implements the color gauge symmetry of QCD. Furthermore, $SU{\left(3\right)}_{F,L}$ is the non-gauged global flavor symmetry which is responsible for three left-handed fermion generations [3] described by the exceptional Jordan algebra $J_3\left({\mathbb{O}}_c\right)$. $SU{\left(3\right)}_L$ branches as $SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ giving rise to the electroweak sector, and the $SU{\left(2\right)}_L$ acts only on left-handed particles.

As regards the branching of the second $E_6$, which we label $E_{6R}$, the $SU{\left(3\right)}_{c^{\prime }}$ symmetry is preferably not gauged (so as to be consistent with phenomenology) - it is global and explicitly broken at the electroweak scale. Its role is to rotate the so-called Jordan frame which arises in the Peirce decomposition of the exceptional Jordan algebra matrices [3]. The $\omega SU{\left(3\right)}_{F,R}$ is the global flavor symmetry which describes three generations of right-handed fermions. The $SU{\left(3\right)}_R$ branches as $SU{\left(2\right)}_R\times U{\left(1\right)}_{Ydem}$. Here, the spontaneously broken $SU{\left(2\right)}_R$ gives rise to general relativity, and the unbroken $U{\left(1\right)}_{dem}$ arising from $U{\left(1\right)}_{Ydem}\to U{\left(1\right)}_{dem}$ is a new force, dubbed dark electromagnetism. It is sourced by the square-root of mass and can be made cosmologically and phenomenologically safe. The emergence of general relativity in this manner, and the preceding gravi-weak unification, was briefly discussed in [1] and is analysed in detail in a forthcoming publication [5].

The focus of the present article is the two extra $SU\left(3\right)$s which we label $SU{\left(3\right)}_L^{geom}$ and $SU{\left(3\right)}_R^{geom}$, which arise in the decomposition of the two $E_8$, and which are apart from the six $SU\left(3\right)s$ arising in the trinification of the two $E_6$. As was already proposed by us in [1], the role of $SU{\left(3\right)}_L^{geom}$ and $\omega SU{\left(3\right)}_R^{geom}$ is to describe spacetime and internal symmetry space. The split complex number is crucial in enabling a Lorentzian signature to emerge from combination of two compact groups, as we rigorously demonstrate below. The split bioctonions also play an important role here: their split quaternionic subalgebra here gives rise to a 6D spacetime with signature (3,3), which upon electroweak symmetry breaking gives rise to two overlapping 4D spacetimes with signature (3,1) and (1,3) respectively. One of these is our familiar 4D spacetime, curved by gravitation. The other 4D spacetime has flipped signature and a (1,1) intersection with our spacetime - its directions are interpreted as being curved by the weak force [5]. The complementary remaining quaternionic subalgebras describe internal symmetry space for holding the unbroken symmetries $SU{\left(3\right)}_c$ and $SU{\left(3\right)}_{c^{\prime }}$. A decomposition of the adjoint $\mathbf{8}$ of $SU{\left(3\right)}_{L,R}^{geom}$ shows an elegant mapping to the octonions as described below.

Thus the goal of the present short note is to rigorously demonstrate how $SU{\left(3\right)}_L^{geom}$ and $\omega SU{\left(3\right)}_R^{geom}$ provide the scaffolding on which the fields contained in $E_{6L}\times E_{6R}$ live. This construction realises the all-encompassing role of $E_8\times \omega E_8$, making the unified symmetry a source of space-time as well as of matter. It is in this spirit that we call the primitive entities possessing $E_8\times \omega E_8$ symmetry ‘atoms of space-time-matter’ [2].

2. Roadmap

Our mass-ratio programme [3] places matter in $E_{6L}\times E_{6R}$ and uses the exceptional Jordan algebra $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ to derive charged-fermion square-root mass ratios and mixings. The present paper supplies the geometric scaffolding promised [see Sec. XII.H of [3]] in the uplift to $E_8\times \omega E_8$: the two extra $\mathrm{SU}\left(3\right)$’s are taken as geometric structure groups that yield precisely a $(3,3)$ base, two 4D spacetimes, and two real 4D internal fibres. The $E_6^{L,R}$ fields then live on this scaffold. A complementary 6D gravi–weak reduction shows how two 4D leaves arise dynamically from a 6D BF theory [5].

What is new here. 

(i) A detailed octonionic identification of the 4D fibres with the realification of $\mathbf{2}\oplus \overline{\mathbf{2}}$, using $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon$ and an intrinsic complex structure on $\mathbb{H}\epsilon$. (ii) A clean role for each extra $U\left(1\right)$ as the Spin^c line on the $\mathbb{C}{P}^2$ fibre. (iii) A unified presentation tying the split-bioctonion base, the $E_8$ branching, and the 6D gravi–weak localisation.

Plan. 

Section 3 and Section 4 build $M_6$ from split bioctonions and embed the two 4D spacetimes. Section 5 and Section 6 relate the $\mathrm{SU}{\left(3\right)}_{L,R}^{geom}$ branching to $\Im \mathbb{H}$ and $\mathbb{H}\epsilon$ and identify the 4D fibres with $T\mathbb{C}{P}^2$. Section 7 clarifies the two $U\left(1\right)$’s. Section 8 explains how the $E_{6L}\times E_{6R}$ fields live on the proposed scaffold, and Section 9 gives the big picture and interpretation.

3. Split-Bioctonionic Base $\mathbf{(}{\mathit{M}}_{\mathbf{6}}\mathbf{,}\mathit{g}\mathbf{)}$

Definition 1

(Split bioctonions $\mathbb{O}\oplus \omega \mathbb{O}$ [4] and metric). Let ${\mathbb{C}}_s$ denote split-complex numbers with generator ${\omega }^2=+1$ and let $\mathbb{O}$ be the octonions. Choose quaternionic subalgebras ${\mathbb{H}}_L\subset \mathbb{O}$ and ${\mathbb{H}}_R\subset \mathbb{O}$. Define

$$M_6:=\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R,g(x_L\oplus \omega x_R,y_L\oplus \omega y_R):=\langle x_L,y_L\rangle -\langle x_R,y_R\rangle ,$$

(2)

with $\langle ·,·\rangle$ the Euclidean inner product induced by the octonion norm.

Proposition 1

(Signature and isometries). $(M_6,g)$ has signature $(3,3)$. The action of unit quaternions by conjugation on each $\Im \mathbb{H}$ yields an isometry action of $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$, which sits inside the maximal compact $\mathrm{SO}\left(4\right)$ of $\mathrm{Spin}(3,3)\cong \mathrm{SL}(4,\mathbb{R})$.

4. Two Embedded 4D Spacetimes

Pick unit vectors $t_L\in \Im {\mathbb{H}}_L$ and $t_R\in \Im {\mathbb{H}}_R$. Define

$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(R\right)}& :=\Im {\mathbb{H}}_R\oplus \mathrm{span}\left\{t_L\right\}(\mathrm{signature}(3,1)),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathcal{M}}_4^{\left(L\right)}& :=\Im {\mathbb{H}}_L\oplus \mathrm{span}\left\{\omega t_R\right\}(\mathrm{signature}(1,3)).\hfill \end{array}$$

(4)

These two Lorentzian 4-planes intersect in a neutral $(1,1)$ 2-plane $\mathrm{span}\{t_L,\omega t_R\}$. This is the kinematic version of the two-leaf picture; a dynamical realisation from a 6D BF theory is given in [5].

5. The Two Extra $\mathbf{SU}\mathbf{\left(}\mathbf{3}\mathbf{\right)}$’s and Their Branching

On each side of $E_8\times \omega E_8$ one has the maximal chain

$$E_8\supset E_6\times \mathrm{SU}\left(3\right),\mathbf{248}=(\mathbf{78},\mathbf{1})\oplus (\mathbf{1},\mathbf{8})\oplus (\mathbf{27},\mathbf{3})\oplus (\overline{\mathbf{27}},\overline{\mathbf{3}}).$$

(5)

We use the two extra $\mathrm{SU}\left(3\right)$’s as geometry. Pick the standard embedding $\mathrm{SU}\left(2\right)\subset \mathrm{SU}\left(3\right)$ and take the complementary $U\left(1\right)$ generator proportional to $\mathrm{diag}(1,1,-2)$. With the convenient normalisation where the doublet has unit charge, the adjoint branches as

$$\mathbf{8}⟶{\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}\oplus {\mathbf{1}}_0.$$

(6)

The ${\mathbf{3}}_0$ from $SU{\left(3\right)}_L^{geom}$ will supply the three directions of $\Im {\mathbb{H}}_L$ and the ${\mathbf{3}}_0$ from $SU{\left(3\right)}_R^{geom}$ will supply the three directions of $\Im {\mathbb{H}}_R$; together forming $M_6$. The $\mathbf{2}\oplus \overline{\mathbf{2}}$ will furnish a real 4D internal fibre, one each from the two $SU{\left(3\right)}^{geom}$.

6. Octonionic Realisation: $\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\mathit{\epsilon }$ and the 4D Fibre

Fix a quaternionic subalgebra $\mathbb{H}=\langle 1,u,v,uv\rangle \subset \mathbb{O}$ and choose $\epsilon \in \Im \mathbb{O}$ orthogonal to $\mathbb{H}$ with ${\epsilon }^2=-1$. Then

$$\mathbb{O}=\mathbb{H}\oplus \mathbb{H}\epsilon ,\Im \mathbb{O}=\Im \mathbb{H}\oplus \mathbb{H}\epsilon .$$

(7)

Define on the real 4-space $\mathbb{H}\epsilon$ the complex structure J by left multiplication with a fixed unit $u\in \Im \mathbb{H}$:

$$J\left(a\epsilon \right):=\left(ua\right)\epsilon ,J^2=-{\mathrm{Id}}_{\mathbb{H}\epsilon }.$$

(8)

Let $\mathrm{SU}\left(2\right)$ act on $\mathbb{H}$ by unit-quaternion conjugation and trivially on $\epsilon$; let $U\left(1\right)$ act as phases $e^{\theta J}$ on $(\mathbb{H}\epsilon ,J)$. Then:

Proposition 2

(Identifications). (a) $\Im \mathbb{H}\cong \mathrm{Adj}\mathrm{SU}\left(2\right)\cong {\mathbf{3}}_0$ as real representations.

(b) 

$(\mathbb{H}\epsilon ,J)\cong {\mathbf{2}}_{+1}$ as a complex $\mathrm{SU}\left(2\right)\times U\left(1\right)$ representation. Forgetting the complex structure, the underlying real representation is

$$\mathbb{H}\epsilon \cong {\left({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}\right)}_{\mathbb{R}},$$

(9)

a real 4-vector space. This is the internal fibre $F_4$.

Thus the octonionic split realises the branching (6) concretely: $\Im \mathbb{H}$ gives the 3’s for spacetime directions; $\mathbb{H}\epsilon$ gives the 4 internal directions.

6.1. Relation to $\mathbb{C}{P}^2$ and Kaluza–Klein Intuition

At a point $\left[n\right]\in \mathbb{C}{P}^2=\mathrm{SU}\left(3\right)/S(U\left(2\right)\times U\left(1\right))$ one has

$$T_{\left[n\right]}\mathbb{C}{P}^2\cong \mathrm{Hom}(\mathbb{C}n,{\mathbb{C}}^3/\mathbb{C}n)\cong {\mathbb{C}}^2\cong {\mathbf{2}}_{+1},$$

(10)

so the real tangent is 4-dimensional. The octonionic identification above gives a canonical isomorphism

$$F_4\simeq T\mathbb{C}{P}^2(\mathrm{real}\mathrm{rank}4\mathrm{at}\mathrm{each}\mathrm{point}).$$

(11)

This matches the minimal Kaluza–Klein choice for an $\mathrm{SU}\left(3\right)$ internal, explaining why four internal real directions are “right” for a QCD-like sector at the level of geometry.

7. The Two $\mathit{U}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$’s

The $U\left(1\right)$ in (6) is the isotropy phase in $S\left(U\right(2)\times U(1\left)\right)$. On $(\mathbb{H}\epsilon ,J)$ it acts by $e^{\theta J}$. For each extra $\mathrm{SU}\left(3\right)$ we therefore obtain a natural line bundle whose connection is the Spin^c twist needed on $\mathbb{C}{P}^2$ (which is non-spin). We do not identify the octonion real line $\mathbb{R}·1\subset \mathbb{O}$ with this $U\left(1\right)$; rather, the $U\left(1\right)$ from $SU{\left(3\right)}_{L,R}^{geom}\to SU\left(2\right)\times U\left(1\right)$ is a Lie-algebra direction acting on the ${\mathbb{C}}^2$ tangent. It is the Spin^c line on $T\mathbb{C}{P}^2$. In model-building one may consider mixing these geometric $U\left(1\right)$’s with abelian factors inside $E_6$, but the Spin^c role is canonical and model-independent.

8. How the ${\mathit{E}}_{\mathbf{6}}^{\mathit{L}}\mathbf{\times }{\mathit{E}}_{\mathbf{6}}^{\mathit{R}}$ Fields Sit on the Scaffold

We treat the two extra $\mathrm{SU}\left(3\right)$’s as structure only. Over $M_6$ take principal bundles for $E_6^L$ and $E_6^R$; matter sits in associated $\mathbf{27}$ vector bundles and gauge fields in adjoint bundles. Tangent/internal decomposition uses $T{M}_6=(\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R)$ and the two fibres $F_4^{L,R}\simeq {\mathbb{H}}_{L,R}{\epsilon }_{L,R}$ from above. The visible interactions come from $E_6^{L,R}$; the extra $\mathrm{SU}\left(3\right)$’s supply geometry and Spin^c lines on the internal fibres.

9. Big Picture and Interpretation

Three layers. 

• Geometric $\mathbf{SU}{\left(\mathbf{3}\right)}_{\mathit{L},\mathit{R}}^{\mathrm{geom}}$ from $E_8\supset E_6\times \mathrm{SU}\left(3\right)$ on each side: not gauged. Purpose: carve the base and fibre geometry. After $\mathrm{SU}\left(3\right)\to \mathrm{SU}\left(2\right)\times U\left(1\right)$,

  $$T{M}_6\cong (\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R),F_4^X\cong {({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1})}_{\mathbb{R}}\simeq T\mathbb{C}{P}^2,X=L,R,$$

  with $\Im \mathbb{H}$ realizing ${\mathbf{3}}_0$ and $\mathbb{H}\epsilon$ realizing ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}$. The two $U\left(1\right)$’s act as Spin^c line connections on the $\mathbb{C}{P}^2$ fibres.
• Gauge $\mathbf{SU}\left(\mathbf{3}\right)$’s inside each $E_6$ (trinification): these are dynamical. On the left: $E_6^L\to \mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(3\right)}_L\times \mathrm{SU}{\left(3\right)}_{F,L}$ with $\mathrm{SU}{\left(3\right)}_L\to \mathrm{SU}{\left(2\right)}_L\times U{\left(1\right)}_Y$. On the right: $E_6^R\to \mathrm{SU}{\left(3\right)}_{c^{\prime }}\times \mathrm{SU}{\left(3\right)}_R\times \mathrm{SU}{\left(3\right)}_{F,R}$ with $\mathrm{SU}{\left(3\right)}_R\to \mathrm{SU}{\left(2\right)}_R\times U{\left(1\right)}_{\mathrm{Ydem}}$.
• Localization and Lorentz breaking in 6D: two Higgs order parameters define localized 4D leaves ${\Sigma }_R,{\Sigma }_L\subset M_6$ via a covariant two-form density ${\rho }_X$. Normal 2-frames $U_X$ implement $SO(3,3)\to SO(3,1)$ on ${\Sigma }_R$ and $SO(3,3)\to SO(1,3)$ on ${\Sigma }_L$, eating 9 Lorentz coset modes per leaf and leaving the tangent spin connections massless [5].

Where Do the Unbroken Gauge Groups Live? 

Because every sector action is wedged with ${\rho }_X$, dynamics is localized on the leaves. Hence unbroken $\mathrm{SU}{\left(3\right)}_c$ (and $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ if retained) are 4D gauge symmetries on the relevant $\Sigma$’s. The 6D base $M_6$ persists as the ambient bundle base, but low-energy fields do not propagate in the bulk.

What are the fibres relative to spacetime? 

$F_4^{L,R}$ are rank-4 internal vector bundles over $M_6$, canonically $F_4^X\simeq {\mathbb{H}}_X{\epsilon }_X\cong {({\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1})}_{\mathbb{R}}\simeq T\mathbb{C}{P}^2$. They are not extra spacetime directions. Restriction to a leaf gives internal fibres $F_4^X{|}_{{\Sigma }_X}$ on which internal interactions act. The two 4D spacetimes come from the six tangent directions $(\Im {\mathbb{H}}_L)\oplus (\Im {\mathbb{H}}_R)$ plus one opposite-side normal on each leaf.

Two Consistent Options for $\mathbf{SU}{\left(\mathbf{3}\right)}_{{\mathit{c}}^{\prime }}$.

• Decoupled/hidden: break or confine $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ above the localization scale; only visible $\mathrm{SU}{\left(3\right)}_c$ remains on ${\Sigma }_R$.
• Gauged on a leaf: keep $\mathrm{SU}{\left(3\right)}_{c^{\prime }}$ dynamical on one leaf (typically ${\Sigma }_R$ or ${\Sigma }_L$). Portal terms can live on $X_3={\Sigma }_R\cap {\Sigma }_L$ under the BF matching conditions.

Dictionary (one line). 

$$\begin{array}{|c|}\hline \mathrm{structural}\mathrm{SU}{\left(3\right)}_{L,R}^{\mathrm{geom}}\Rightarrow (M_6,F_4^{L,R})\mathrm{vs}\mathrm{gauge}\mathrm{SU}\left(3\right)\subset E_6^{L,R}\Rightarrow 4\mathrm{D}\mathrm{forces}\mathrm{on}{\Sigma }_{L,R}.\\ \hline\end{array}$$

10. Summary

• The $(3,3)$ base $M_6$ is $\Im {\mathbb{H}}_L\oplus \omega \Im {\mathbb{H}}_R$; the two 4D spacetimes are 4-planes obtained by adding a single normal from the opposite side.
• The extra $\mathrm{SU}\left(3\right)$’s branch as $\mathbf{8}\to {\mathbf{3}}_0\oplus {\mathbf{2}}_{+1}\oplus {\overline{\mathbf{2}}}_{-1}\oplus {\mathbf{1}}_0$.
• $\Im \mathbb{H}$ realises ${\mathbf{3}}_0$; $\mathbb{H}\epsilon$ realises ${(\mathbf{2}\oplus \overline{\mathbf{2}})}_{\mathbb{R}}$ and is the 4D fibre $F_4\simeq T\mathbb{C}{P}^2$.
• Each geometric $U\left(1\right)$ is the Spin^c line on $\mathbb{C}{P}^2$; we do not confuse it with the octonion real line.