SU(<em>n</em>) Holonomy Deviations in Calabi–Yau Manifolds (CY1–CY4)

We study holonomy-induced deviations arising from Levi-Civita parallel transport on Calabi-Yau manifolds of complex dimensions one through four. Using the Ricci-flat Kähler structure and the associated SU(n) holonomy reduction, we develop a unified framework for deviation operators that applies uniformly across dimensions. General expressions are formulated in terms of path-ordered transport, curvature endomorphisms, and non-Abelian Stokes techniques, clarifying how nontrivial holonomy effects persist despite vanishing Ricci curvature. A dimension-by-dimension analysis is presented, covering elliptic curves, K3 and abelian surfaces, Calabi-Yau threefolds, and Calabi-Yau fourfolds. We identify which holonomy contributions are suppressed by type constraints in Kähler geometry, which arise only at higher order, and how these features depend on the complex dimension. The paper is intended both as a reference for explicit holonomy and deviation computations and as a bridge to applications involving geometric phases and compactification effects.