Topological Spinors and Their Observation

The appearance of topological spinors may then finally be interrogated under observation byג

. Let x ∈ U ⊂ X be a point in the neighborhood of a local observation גU .

If ΨגU (x) ∈ /S(C) is a topological spinor, then under an observation by ג

∗ on X it will appear as

Space-Time Spinors

‘Internal’ Quantum Numbers (4.1)

so that an observer may be lead into error. While the topological spinor under observation is not generated by any algebraic auxiliary data unconnected to X, it is quite likely to appear as if it contains auxiliary internal quantum numbers if the observer is unaware of the Observerse structure involving Y , as the pull back fields via ג

∗ would have the false appearance of being native to X. A tell tale sign that one might be looking at such a unified structure with a single origin in X would be the presence of a power of 2 in the dimension count of the auxiliary quantum numbers for /Sx (Nג)x)).

Specifically, on an even dimensional manifold 26 of signature (i, j) = n we would expect to see internal quantum numbers of dimension

‘Internal’ QN

depending on whether the theory was non-chiral (Dirac) or either effectively or fundamentally chiral (Weyl). In particular, we might expect the later in regions U ⊂ Xi,j where gravity and curvature are weak and the former where they are strong and resistant to effective decoupling. And, to our way of thinking,

this, with a doublet of charged and uncharged leptons together with a tri-colored positively charged quark and its negatively charged weak isospin doublet partners along with the anti-particles of all the preceding, appears to be exactly what we see repeated over three apparent low energy families. Lastly, we note from personal communication that Frank Wilczek appears to have wondered about the spinorial coincidence (even in print), but did not find it compelling enough to pursue beyond noting its existence and the lack of incorporation within a physical framework. It is our hope that recognizing that

Figure 3: Wilczek On Internal Spinors.

the ‘10’ implicit within both the Georgi-Glashow and Pati-Salam theories may be tied to the 10 coupled equations of General Relativity may be considered compelling.

4.1 Maximal Compact and Complex Subgroup Reductions of Structure Group

Assume for the moment that we have a global observation:

or, in our case, simply that we have a given metric on X. The push-forward map induced on the tangent bundle T X given by

splits the tangent bundle on Y along ג)X) ⊂ Y into the image of Dג and its 10 dimensional complement V 10 along the fiber of metrics for every x ∈ X. This creates an identification with the chimeric bundle C = V10 ⊕ H∗ 4 as the vertical piece is already a subset of both T Y and C by construction, and Im(Dג= (

Having seen that the natural break down of our Chimeric Spin(7, 7) bundle under observation leads to a decomposition into tangent and normal components of dimensions 4 and 10 respectively, it is natural to ask what reductions of structure group are most natural to expect. Two immediately suggest themselves. Given any normal bundle that is even dimensional, there is a natural question as to whether it admits a complex or quaternionic structure. Here the dimension 10 being equal to 2 mod 4 suggests seeking a complex structure through reduction of structure group to U(3, 2) ⊂ Spin(6, 4).

Conversely, given the non-compact nature of Spin(6, 4) it is natural to wonder whether the structure group breaking to a maximal compact subgroup with better stability behavior is advantaged. Thus while non-compact groups are certainly considered from time to time, if the subgroup was broken to a maximal compact sub-group, we would be anthropically screened from seeing how nature accommodates non-compact symmetry and thus without guidance as to how to find a theory which extends to the general case.

To this end, we might also consider both reductions simultaneously and ask how a reduction to a maximal compact subgroup would appear if it were accompanied by a simultaneous reduction to accommodate a 5 complex dimensional normal bundle. Starting from Spin(1, 3) × Spin(6, 4)

where the Standard Model group is found within the intersection of the simultaneous reductions up to a reductive factor of U(1) if the special unitary group SU(3, 2) is not privileged over the full unitary group U(3, 2) to begin with.