Seamed Riemann Surfaces

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 Consider a seamed orbifold Riemann surface constructed as follows. Start with the face poset of a three-dimensional cube. Let $\mathcal A_3$ denote the set of antipodal vertex pairs $$a=\{v,v^*\}.$$ To each such pair $a\in\mathcal A_3$, associate a spherical $2$-orbifold $$S^2(\alpha,\alpha)$$ whose two cone points are attached to the antipodal vertices $v$ and $v^*$. Equivalently, and more naturally from the complex-analytic point of view, we regard these spherical orbifolds as copies of $\mathbb{CP}^1$ and glue them along a common seam complex $\Gamma$. Thus the complex foam is $$\mathcal I_{\mathbb C}=\left(\bigsqcup_{a\in\mathcal A_3}^{4}\mathbb{CP}^1_a\right)\big/\!\sim_\Gamma .$$ The cubical scaffold should be understood primarily as an organizational device. Rather than constructing the foam by gluing disks one at a time, as in more traditional constructions of seamed Riemann surfaces or Klein foams, the cube records the incidence structure in advance: whenever two sheets o...
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Stratified Holonomy Dynamics of the Ihara Zeta function of $\Gamma$

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Let $\Gamma$ be a finite connected graph (e.g., the 1-skeleton of a stratified space or foliated complex), and let $\mathcal{R}$ denote the stratified space of holonomy representations:

$$\mathcal{R} = \bigsqcup_{G \subseteq GL_n(\mathbb{C})} \mathcal{R}_G,\quad \mathcal{R}_G := \left\{ \rho : \pi_1(\Gamma) \to G \right\}.$$


Each stratum $\mathcal{R}_G$ corresponds to a distinct choice of structure group $G$, such as $U(1)$, $SU(2)$, or  $GL_n(\mathbb{C})$.


Define a stratified holonomy evolution governed by a sequence of generalized symmetry transformations


$$\Psi_k : \mathcal{R}_{G_k} \longrightarrow \mathcal{R}_{G_{k+1}},$$


which may be continuous or discrete, invertible or not, and may preserve or enhance the structure group.


The evolution of representations is given by the discrete recurrence:


$$\rho_{k+1} = \Psi_k(\rho_k), \qquad \rho_k \in \mathcal{R}_{G_k},\; \rho_{k+1} \in \mathcal{R}_{G_{k+1}}.$$


This defines a dynamical system over the stratified moduli space $\mathcal{R}$, where transitions between strata correspond to changes in the structure group, such as:


$$U(1) \longrightarrow GL_2(\mathbb{C}) \longrightarrow GL_3(\mathbb{C}) \longrightarrow \cdots$$


or possibly to reductions:


$$GL_n(\mathbb{C}) \longrightarrow U(1).$$


Associated to each $\rho_k$, we define a (possibly twisted) Ihara-type zeta function:


$$\zeta_k(u) := \prod_{[p]} \det\left( I - \rho_k(p) u^{\ell(p)} \right)^{-1},$$


where the product is over equivalence classes of primitive closed paths $[p]$ in $\Gamma$, and $\ell(p)$ is the length of the path.


The evolution of zeta invariants under $\Psi_k$ is given by the pushforward:


$$\zeta_{k+1}(u) = \Psi_{k*}\left( \zeta_k(u) \right),$$


which reflects any change in the spectral type or determinant structure induced by the transformation of the holonomy representation.


This framework allows for discrete-time dynamics across the space of representations and zeta functions, capturing both internal deformations (when $G_{k+1} = G_k$) and structural enhancements (when $G_{k+1} \supsetneq G_k$) driven by the generalized symmetry $\Psi_k$.

Comments

  1. AnonymousMay 8, 2025 at 6:37 PM

    This is quite interesting. Looking forward to the next post

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    1. John ZimmermanMay 8, 2025 at 6:46 PM

      Thanks - my goal is to post 2-3 times per week :)

      Delete
  2. AnonymousMay 8, 2025 at 6:44 PM

    Is \Gamma the same as in previous posts? Is it arbitrary or fixed?

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    1. John ZimmermanMay 8, 2025 at 6:51 PM

      Yes good question. \Gamma is fixed to be the intersection locii of the canonical \mathcal I stratified complex (see previous posts for that), with x,y,z=0 coordinate planes. In some posts I constructed \mathcal I in X^3=[0,1]^3 but it can easily be rescaled to the more accessible [-1,1]^3=Y^3. In theory \Gamma and \mathcal I could generalize but I haven't thought about that much.

      Delete

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