A Four-Sheeted Orbifold Foam as a Planck-Cell Carrier

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 One possible way to think about Planck-scale structure is not to begin with a smooth spacetime manifold, but with a finite geometric carrier on which quantum data can be placed. In this note I describe a speculative mathematical object of this kind: a four-sheeted football-orbifold foam suspended over the face poset of a cube. The proposal is not that ordinary spacetime literally contains small Euclidean cubes. Rather, the cube poset should be understood as a finite incidence scaffold: it records which zero, one, two, and three-dimensional strata are allowed to meet, while the actual geometric realization is an orbifold/Klein foam built over that combinatorial skeleton.  Let $P_{\Box}$ denote the face poset of a cube. The eight vertices of the cube serve as incidence markers for the eight cone points of the foam. Since a football orbifold has two cone points, one can naturally attach four football sheets by pairing the eight vertices into four antipodal pairs. Thus we begin w...
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The $\Gamma$-sets and $\zeta$-space.

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Assume $X$ is a smooth convex open manifold and $\partial X$ is a convex regular polytope.

Def: An *$\mathcal F$-completion,* or *foliational completion* of a *block*, $\mathcal B:=X\cup\partial X$ over its vertex set $V$, is $\mathscr CX_V:=\cup_{v_{ij}\in V} \mathcal F_{v_{ij}}$ where $i,j$ index the vertices. 

Example: Set $X=(0,1)^n$. Take a smooth foliation $\mathcal F$ for every pair of points $v_i,v_j\in \partial X=[0,1]^n-(0,1)^n$ satisfying $\mathrm{sup~dist}_n(v_i,v_j)=\sqrt{r}$. Let the leaves in each $\mathcal F$ be mutually diffeomorphic to the class $M=(0,\sqrt{n})\times S^{n-2}$ accumulating to $v_i,v_j$. We see that $r=\lbrace 1,\sqrt{n} \rbrace.$ Allowing only $r=\sqrt{n}$, we can look at sections of $\mathscr CX_V$:

Def: A $\Gamma$-set on $\mathscr CX_V$ is the union of all leaf intersections of the foliations i.e. 

$$\Gamma_{n}:= \lbrace \mathcal (\mathcal F_1 \cap \mathcal F_2)\cup (\mathcal F_2 \cap \mathcal F_3) \cup (\mathcal F_3 \cap \mathcal F_4) \cup \cdot\cdot\cdot \cup (\mathcal F_{n-1} \cap \mathcal F_{n})  \rbrace.$$

Then it follows trivially that for any given foliation, $\Gamma_1=X^1=(0,1)$ and $\Gamma_2=X^2=(0,1)^2$. 

>What is $\Gamma_3$ equal to?

It seems that $\Gamma_3$ is equal to three mutually orthogonal planes but I want some reassurance that this is true. I don't know the structure of the $\Gamma$-sets for $n>3$.

The following is meant to provide context as to why I am defining these concepts above in the first place:

Personally I am interested in the $\Gamma$-sets because if you define vector fields on elements of the $M$-class, you can add the vectors along sections of the $\Gamma$-sets resulting in vector bundles that alternate orientation when traveling along a path on the $\Gamma$-set. Thus the $\Gamma$-sets tell you about the geometry.

Another reason I'm interested in the construction in general is because one inherits a notion of decay with $\mathscr CX_V$. Specifically, the $M$-class rapidly decays to zero when traveling in the direction prescribed by the vector fields placed on the $M$-class. This can be formalized analogous to rapid decay of elements of the Schwartz space $\mathcal S(\Bbb R^n)$, which are functions, not surfaces. However it is not difficult to adapt this notion.

With this rapid decay we have a concept of a Schwartz space which plays nice with zeta functions because it is known that we can build zeta functions from elements in the Schwartz space.

   

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