Mirror Duality in a Spencer-Type Complex: Analytic and Riemann-Roch Perspectives

Abstract

We introduce and analyze a Spencer-type elliptic complex on the space of differential forms valued in symmetric powers of an adjoint bundle, $\Omega^\bullet(X)\otimes \mathrm{Sym}^\bullet(G)$. The complex is governed by a total differential $D_{\lambda,\psi}$ depending on a section $\psi\in\Gamma(G)$ and a real parameter $\lambda$. The central result of this paper is an algebraic realization of mirror-type duality and parameter robustness at the \emph{chain-level}. We demonstrate that sign flips ($\lambda\mapsto -\lambda$ or $\psi\mapsto -\psi$) and rescaling ($\lambda\mapsto \alpha\lambda$) of the deformation parameters correspond to simple conjugations of the differential $D_{\lambda,\psi}$ by elementary zero-order automorphisms. This provides a unified, conceptual foundation for the invariance of topological invariants that is often established via case-by-case analytic methods. Analytically, this framework implies the invariance of harmonic space dimensions under the mirror map $\psi\mapsto -\psi$. Algebraically, the Grothendieck--Riemann--Roch index formula for the complex's hypercohomology is shown to be manifestly independent of $(\lambda, \psi)$, determined solely by the characteristic classes of a universal virtual bundle. The theory is fully compatible with equivariant localization and is verified with concrete applications on Calabi--Yau backgrounds, including K3 surfaces and elliptic curves. This framework thus offers a rigorous, chain-level explanation for the parameter robustness intrinsic to Witten-type deformations and localization phenomena, grounding them in a fundamental algebraic conjugation principle.