From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly

TL;DR

This paper constructs an interaction and incidence layer from finite-node conifold degenerations, linking algebraic data, schober packages, and quiver theory to support advanced geometric and physical structures.

Contribution

It introduces a functorial incidence package and assembles a finite quiver-theoretic structure from schober data, establishing invariance and compatibility with existing geometric frameworks.

Findings

The constructed package is canonically determined by the schober datum.

The package is invariant under equivalence of schober realizations.

It supports the development of graded interaction, stability, and wall-crossing theories.

Abstract

In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma)$, where $V_\Sigma$ is the finite node-indexed vertex set, $E_\Sigma$ is the nodewise coupling space, and $c_\Sigma$ is the coefficient vector of the corrected global extension class. The purpose of the present paper is to construct the corresponding interaction and incidence layer. Starting from the finite-node schober package $S_\Sigma := (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{\Phi_k,\Psi_k\}_{k=1}^r,Sh(S_\Sigma))$, we define the extended vertex set $V_\Sigma^{\mathrm{ext}} := V_\Sigma \sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_\Sigma := (V_\Sigma^{\mathrm{ext}},\rightsquigarrow_\Sigma)$, and its canonical binary decategorification $\mathcal I_\Sigma := (V_\Sigma^{\mathrm{ext}},I_\Sigma)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma,\mathcal F_\Sigma,I_\Sigma)$, where $\mathcal F_\Sigma := \{(\Phi_k,\Psi_k)\}_{k=1}^r$ is the functorial coupling datum. We prove that this package is canonically determined by the finite-node schober datum, compatible with the corrected perverse extension and its mixed-Hodge-module refinement, and invariant under equivalence of finite-node schober realizations. This yields the interaction and incidence layer required for later graded interaction, stability, BPS, and wall-crossing structures.