Mirror Symmetry of Spencer-Hodge Decompositions in Constrained Geometric Systems

2025/06/06 by Zheng, Dongzhe · 3 citations
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2506.05816

Abstract

This paper systematically investigates the interaction mechanism between metric structures and mirror transformations in Spencer complexes of compatible pairs. Our core contribution is the establishment of mirror symmetry for Spencer-Hodge decomposition theory, solving the key technical problem of analyzing the behavior of metric geometry under sign transformations. Through precise operator difference analysis, we prove that the perturbation R^k = -2(-1)^k ω⊗ δ^λ_\mathfrakg(s) induced by the mirror transformation (D,λ) ↦ (D,-λ) is a bounded compact operator, and apply Fredholm theory to establish the mirror invariance of harmonic space dimensions dim H^k_D,λ = dim H^k_D,-λ. We further prove the complete invariance of constraint strength metrics and curvature geometric metrics under mirror transformations, thus ensuring the spectral structure stability of Spencer-Hodge Laplacians. From a physical geometric perspective, our results reveal that sign transformations of constraint forces do not affect the essential topological structure of constraint systems, embodying deep symmetry principles in constraint geometry. This work connects Spencer metric theory with mirror symmetry theory, laying the foundation for further development of constraint geometric analysis and computational methods.

Citations

• Constructing Two Metrics for Spencer Cohomology: Hodge Decomposition of Constrained Bundles
• Dynamical Geometric Theory of Principal Bundle Constrained Systems: Strong Transversality Conditions and Variational Framework for Gauge Field Coupling

Cited by

• Exceptional Symmetry as a Source of Algebraic Cycles: Non-Constructive Methods for the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds
• Extension Research of Principal Bundle Constraint System Theory on Ricci-flat Kähler Manifolds
• Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry

Related

• Geometric Duality Between Constraints and Gauge Fields: Mirror Symmetry and Spencer Isomorphisms of Compatible Pairs on Principal Bundles
• Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry
• Constructing Two Metrics for Spencer Cohomology: Hodge Decomposition of Constrained Bundles
• The Rigidity of Constraint: A Spencer-Hodge Theoretic Approach to the Hodge Conjecture
• Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry