On Equivalences in Calabi–Yau Geometry from String Theory

We investigate several equivalence notions arising in the study of Calabi-Yau manifolds and their interactions with ideas from string theory. The focus is on bimeromorphic equivalence in complex geometry, Morita equivalence in noncommutative geometry, and twisted K-theory as a receptacle for D-brane charges in backgrounds with flux. Using tools from derived categories, Fukaya categories, and operator K-theory, we analyze how these equivalences appear across geometric, categorical, and physical frameworks. Particular attention is given to Fujiki class C manifolds, Hilbert C*-modules, and the role of homological mirror symmetry in relating these structures. Several examples and applications are discussed, illustrating how string-motivated constructions provide a unifying perspective on equivalence phenomena in Calabi-Yau geometry.