The Moduli Space of Holonomies and the Twisted Ihara Zeta Function

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 Consider a triple:

$$(\mathcal I, z, \Sigma)$$

where $\mathcal I$ is the canonical stratified surface defined in previous posts (smooth away from a singular set) embedded in $[-1,1]^3$, $z$ is the vertical projection (interpreted as time), and $\Gamma=\Sigma_0$ is the critical level set.

Define a movie that is given by:

$$\Sigma_z :=\begin{cases}\{v_1, v_2, v_3, v_4\} & \text{if } z = \pm 1, \\\\\coprod_{i=1}^4 S^1_i & \text{if } z \in (-1, 0) \cup (0, 1), \\\\\Gamma & \text{if } z = 0,\end{cases}$$

where each $S^1_i$ denotes a topological circle, and $\Gamma$ is a finite connected 4-regular multi-graph formed by the merging of four disjoint circles into two overlapping ovals, with their four intersection points defining the graphs vertices. 

For each $z \in (-1,0) \cup (0,1)$, the level set $\Sigma_z$ consists of four disjoint circles $S^1_1, \ldots, S^1_4$. Along each $S^1_i$, we define a real line field:

$$V_i \subset T\mathbb{R}^3|_{S^1_i},$$

consisting of unit-length vectors orthogonal to the slicing plane $\{z = \text{const}\}$ at each point. That is, if $T_x S^1_i \subset T_x \mathbb{R}^3$ is the tangent to the circle at $x$, then $V_i(x)$ is a 1-dimensional real subspace orthogonal to both $T_x S^1_i$ and to the horizontal plane.

As $z \to 0$, the four disjoint circles merge into the connected graph $\Gamma$. At this stage, the four real vertical bundles $\{V_i\}$ glue together to form a single real line bundle:

$$V_\Gamma \to \Gamma,$$

defined by identifying vertical directions at the junction points where circles coalesce. This gluing process descends the structure of $\bigoplus V_i$ into a globally defined vertical field over $\Gamma$.

Finally, we complexify the resulting real bundle to obtain a flat complex line bundle:

$$\mathcal{L}_\Gamma := V_\Gamma \otimes_\mathbb{R} \mathbb{C} \to \Gamma,$$

which serves as the geometric input for defining holonomy and zeta invariants on the graph.

This bundle defines a local system on $\Gamma$, represented by a unitary character:

$$\rho: \pi_1(\Gamma) \to \mathrm{U}(1).$$

The holonomy representation $\rho$ on $\Gamma$ is determined by the vertical bundles $V_i$ via the induced identifications through the collapse process:

$$\pi_1\left(\coprod S^1_i\right) \cong \mathbb{Z}^4 \longrightarrow \pi_1(\Gamma) \xrightarrow{\rho} \mathrm{U}(1).$$

Each loop in $\Gamma$ corresponds to a word in the fundamental group, and the value of $\rho$ on that loop is computed from the phase induced by the vertical directions traced through the merging process.

Proposition: Given vertical real line bundles $V_i$ over $S^1_i$ whose fibers are orthogonal to the slicing planes, the collapse of the disjoint circles to $\Gamma$ induces a gluing of $\{V_i\}$ into a single real bundle $V_\Gamma \to \Gamma$. Upon complexification, this defines a flat complex line bundle $\mathcal{L}_\Gamma \to \Gamma$, whose associated unitary representation $\rho: \pi_1(\Gamma) \to \mathrm{U}(1)$ encodes the combined holonomy data.

Define a twisted Ihara zeta function over $\Gamma$ denoted $\zeta_{\Gamma}(u,\rho)$:

$$\zeta_{\Gamma}(u, \rho) := \prod_{[P]} \left(1 - \rho(P) u^{\ell(P)}\right)^{-1},$$

where $[P]$ runs over all prime, tail-less, backtrackless cycles in $\Gamma$, and $\ell(P)$ is the length of the cycle, with $\rho(P)$ representing the holonomy around $P$.

Equivalently, the zeta function satisfies the determinant identity:

$$\zeta_\Gamma(u, \rho)^{-1} = \det(I - u A_\rho + u^2 Q),$$

where $A_\rho$ is the twisted adjacency matrix incorporating holonomies, and $Q$ is the degree matrix with $Q_{ii} = \deg(v_i) - 1$.

The holonomy representation $\rho$ belongs to the moduli space of unitary representations:

$$\mathrm{Hom}(\pi_1(\Gamma), \mathrm{U}(1)) \cong \mathrm{U}(1)^r,$$

where $r = b_1(\Gamma)$ is the first Betti number. This torus parametrizes the family of flat line bundles over $\Gamma$.

A finite group $\Delta \subset \mathrm{Aut}(\Gamma)$ acts on $\pi_1(\Gamma)$ by automorphisms and hence induces a pullback action on the moduli torus:

$$\rho^\delta(\gamma) := \rho(\delta^{-1} \cdot \gamma), \quad \delta \in \Delta.$$

This defines a group action:

$$\Delta \curvearrowright \mathrm{U}(1)^r,$$

with associated orbits $\mathcal{O}_\rho$ and the quotient moduli space:

$$\mathcal{M}_\Delta := \mathrm{U}(1)^r / \Delta.$$

The twisted Ihara zeta function defines a family of spectral invariants:

$$\mathcal{Z}_\Gamma : \mathrm{U}(1)^r \longrightarrow \mathbb{C}[[u]], \quad \rho \mapsto \zeta_\Gamma(u, \rho).$$

And this sheaf encodes how spectral data evolves over the moduli space of holonomies.

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