Mirror Symmetry of Spencer-Hodge Decompositions in Constrained Geometric Systems

TL;DR

This paper establishes mirror symmetry in Spencer-Hodge decomposition theory, demonstrating invariance of harmonic spaces and metrics under sign transformations, thus revealing deep symmetry principles in constrained geometric systems.

Contribution

It introduces the first systematic analysis of mirror symmetry in Spencer complexes, proving invariance of harmonic dimensions and metrics under mirror transformations.

Findings

Mirror symmetry for Spencer-Hodge decompositions is established.

Harmonic space dimensions are invariant under sign changes.

Constraint and curvature metrics remain unchanged under mirror transformations.

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 $\mathcal{R}^k = -2(-1)^k \omega \otimes \delta^{\lambda}_{\mathfrak{g}}(s)$ induced by the mirror transformation $(D,\lambda) \mapsto (D,-\lambda)$ is a bounded compact operator, and apply Fredholm theory to establish the mirror invariance of harmonic space dimensions $\dim \mathcal{H}^k_{D,\lambda} = \dim \mathcal{H}^k_{D,-\lambda}$. 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.