2025/06/07 by Zheng, Dongzhe · 2 citations
#14C30 #14F40 #32S35 #53C56 #58A20 #58J10 #FOS: Mathematics #General Mathematics (math.GM)
paper · doi:10.48550/arxiv.2506.06610
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)⊗ Sym^\bullet(G). The complex is governed by a total differential Dλ,ψ depending on a section ψ∈Γ(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 chain-level. We demonstrate that sign flips (λ↦ -λ or ψ↦ -ψ) and rescaling (λ↦ αλ) of the deformation parameters correspond to simple conjugations of the differential Dλ,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↦ -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.