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

TL;DR

This paper develops an algebraic and analytic framework for a Spencer-type elliptic complex that reveals mirror duality and parameter invariance at the chain level, providing a unified foundation for topological invariants and localization phenomena.

Contribution

It introduces a novel algebraic realization of mirror duality in a Spencer-type complex, demonstrating invariance under parameter transformations and connecting to topological invariants.

Findings

Sign flips and rescaling correspond to conjugations of the differential.

Harmonic space dimensions are invariant under the mirror map.

The hypercohomology index depends only on characteristic classes, independent of parameters.

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.