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, $Ω^\bullet(X)\otimes \mathrm{Sym}^\bullet(G)$. The complex is governed by a total differential $D_{λ,ψ}$ depending on a section $ψ\inΓ(G)$ and a real parameter $λ$. 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 ($λ\mapsto -λ$ or $ψ\mapsto -ψ$) and rescaling ($λ\mapsto αλ$) of the deformation parameters correspond to simple conjugations of the differential $D_{λ,ψ}$ 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 $ψ\mapsto -ψ$. Algebraically, the Grothendieck--Riemann--Roch index formula for the complex's hypercohomology is shown to be manifestly independent of $(λ, ψ)$, 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.