From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917427548258304 |
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| author | Rahman, Abdul |
| author_facet | Rahman, Abdul |
| contents | In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_Σ:= (V_Σ,E_Σ,c_Σ)$, where $V_Σ$ is the finite node-indexed vertex set, $E_Σ$ is the nodewise coupling space, and $c_Σ$ 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 $S_Σ:= (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{Φ_k,Ψ_k\}_{k=1}^r,Sh(S_Σ))$, we define the extended vertex set $V_Σ^{\mathrm{ext}} := V_Σ\sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_Σ:= (V_Σ^{\mathrm{ext}},\rightsquigarrow_Σ)$, and its canonical binary decategorification $\mathcal I_Σ:= (V_Σ^{\mathrm{ext}},I_Σ)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_Σ:= (V_Σ,E_Σ,c_Σ,\mathcal F_Σ,I_Σ)$, where $\mathcal F_Σ:= \{(Φ_k,Ψ_k)\}_{k=1}^r$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly Rahman, Abdul Algebraic Geometry High Energy Physics - Theory In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_Σ:= (V_Σ,E_Σ,c_Σ)$, where $V_Σ$ is the finite node-indexed vertex set, $E_Σ$ is the nodewise coupling space, and $c_Σ$ 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 $S_Σ:= (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{Φ_k,Ψ_k\}_{k=1}^r,Sh(S_Σ))$, we define the extended vertex set $V_Σ^{\mathrm{ext}} := V_Σ\sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_Σ:= (V_Σ^{\mathrm{ext}},\rightsquigarrow_Σ)$, and its canonical binary decategorification $\mathcal I_Σ:= (V_Σ^{\mathrm{ext}},I_Σ)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_Σ:= (V_Σ,E_Σ,c_Σ,\mathcal F_Σ,I_Σ)$, where $\mathcal F_Σ:= \{(Φ_k,Ψ_k)\}_{k=1}^r$ 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. |
| title | From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2604.20110 |