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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Delta Groupoid Associated to an Enriched Graph

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 Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph embedded in $\mathbf R^3$. Define an enrichment of $\Gamma$ to be a triple:

$$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$


where, $\phi: E \to \mathcal{C}$ is a coloring or stratification of the edge set into a finite set of colors $\mathcal{C}$. $\mathcal{L}$ is a local system (i.e., a functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$). It is assumed a priori that connections are flat. $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation.


Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. The Delta groupoid $\mathcal{D}_\Gamma$ is defined as follows:


Objects are enrichments $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$. Morphisms are invertible maps


$$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$


that satisfy the following properties:


The underlying graph $\Gamma$ is fixed, thus the Ihara zeta function is preserved:

    $$Z_\Gamma(u) = Z_{\delta(\Gamma)}(u)$$

The coloring is transformed by a pullback or relabeling:

    $$\phi' = \delta^* \phi$$

The local system is transformed functorially:

    $$\mathcal{L}' = \delta^* \mathcal{L}, \quad \rho' = \rho \circ \delta_*$$

Composition is defined by:

$$(\delta_2 \circ \delta_1)(\mathcal{X}) = \delta_2(\delta_1(\mathcal{X}))$$

Each object has an identity morphism $\mathrm{id}_{\mathcal{X}}$. Every morphism $\delta \in \mathrm{Hom}(\mathcal{X}, \mathcal{Y})$ has an inverse $\delta^{-1} \in \mathrm{Hom}(\mathcal{Y}, \mathcal{X})$.


As can be seen above, the morphisms $\delta$ preserve $Z_{\Gamma}(u),$ but in general do not preserve $\mathcal L,\phi, \rho.$ In essence, $\delta$ "sees" both the invariance of $Z_{\Gamma}(u)$ and the variance of $\mathcal L,\phi,\rho$. This is because $Z_{\Gamma}$ is insensitive to $\delta$. $Z_{\Gamma}$ is a spectral object that detects cycles, not the higher enrichment data. Thus a given $\delta$ can transform an initialized bundle with flat connection into a bundle whose curvature is not zero.


Given two enrichments $\mathcal{X}, \mathcal{X}' \in \mathcal{E}_\Gamma$ such that $\delta(\mathcal{X}) = \mathcal{X}'$, under what conditions does

$$H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \mathcal{L}')$$

hold?


Let $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$, and suppose $\delta \in \mathcal{D}_\Gamma$ is a Delta morphism defined by a geometric transformation, such as a rotation about the normal to the $z = 0$ plane.


Assume further that:

- $\delta$ is a graph automorphism of $\Gamma \subset \mathbb{R}^3$, where $\Gamma$ is embedded in the $z = 0$ plane.

- $\delta$ also acts as an automorphism of the local system $\mathcal{L}$, in the sense that it maps fibers linearly and preserves the parallel transport structure.

- The connection on $\mathcal{L}$ is flat, and the orientation of vectors in $\mathcal{L}$ is compatible with the planar embedding.


Then $\delta$ preserves not only the Ihara zeta function $Z_\Gamma(u)$, but also the holonomy representation $\rho$ (up to isomorphism), and the twisted cohomology:

$$\rho \simeq \rho \circ \delta_*, \qquad H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \delta^* \mathcal{L}).$$


In particular, such a $\delta$ defines a morphism in a distinguished subgroupoid of $\mathcal{D}_\Gamma$ that preserves all enrichment data. 


However, in general, $\delta$ will only preserve $Z_{\Gamma},$ not $\phi,\mathcal L,\rho$.

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