A duality conjecture and research program asserting deep correspondences between the symplectic geometry of a Calabi–Yau manifold and the complex algebraic geometry of its mirror, including equivalences between certain categories and equalities of enumerative invariants.

Principle

The organizing idea is that the derived Fukaya category of Lagrangian submanifolds and the derived category of coherent sheaves on the mirror Calabi–Yau are equivalent; symplectic counts of holomorphic disks/Gromov–Witten invariants correspond to complex periods and deformation data on the mirror.

Demonstration

In the classical example, predictions about the numbers of rational curves of given degree on a quintic threefold (symplectic side) are obtained by computing period integrals of the mirror family (complex side), producing explicit enumerative formulas later confirmed by direct algebraic geometry computations.

Misapplication

Treating mirror symmetry as a literal pointwise isomorphism of manifolds, assuming the homological statement holds without verifying technical hypotheses (e.g., monotonicity, transversality, or existence of required compactifications), or applying Calabi–Yau mirror arguments naively to unrelated varieties can lead to incorrect conclusions.

Consequence

When valid, mirror symmetry translates difficult enumerative or deformation problems from one geometric category to more tractable computations on the mirror, yields categorical equivalences that organize invariants, and suggests deep structural links between symplectic and algebraic geometry.

Reversal

The conceptual reversal exchanges the roles of complex and symplectic geometry: objects that are sheaves on one side correspond to Lagrangians on the other; failure of equivalence on one side illuminates obstructions (e.g., disk bubbling) on the other.

Boundary

Primarily formulated for Calabi–Yau manifolds and certain singular or log generalizations; extensions to Fano varieties or Landau–Ginzburg models require modified statements. It excludes arbitrary varieties without mirror constructions or cases lacking necessary analytic or categorical foundations.

Semantic Tension

There is tension between the physicists' enumerative predictions and the mathematicians' homological/categorical formulations: one is computational and conjectural, the other structural and categorical; also tension between formal equivalence of categories and concrete geometric or analytic existence of correspondences.

Synthesis

Mirror symmetry is the principle that a Calabi–Yau manifold and its mirror encode the same geometric information in dual languages—symplectic vs complex—often realized as an equivalence between the Fukaya category and the derived category of coherent sheaves, with enumerative invariants and deformation data matching across the equivalence.