Generalizing elliptic-curve mirror symmetry to higher-dimensional Calabi–Yau manifolds

Extend the explicit mirror map identification (ρ ↔ τ) and associated categorical equivalence established for the elliptic curve to higher-dimensional Calabi–Yau manifolds, addressing the dependence of Fukaya category compositions on the Kähler form and determining the necessary features of the unique Calabi–Yau metric to enable such generalizations.

Background

The paper explains that for elliptic curves (dimension one) the Homological Mirror Symmetry correspondence can be formulated without full A∞ data, with the mirror map relating the complex parameter τ to the complexified Kähler parameter ρ.

It then states that pushing this to higher-dimensional Calabi–Yau manifolds remains open, in part because compositions in the Fukaya category depend sensitively on the Kähler form arising from the unique Ricci-flat Calabi–Yau metric, whose explicit form is not known in general.