The geometric mirror pair conjecture

Let $(X,Y)$ be a geometric mirror pair, meaning that there is a parameter space $U$ and, for each $t\in U$, a correspondence $Z_t\subset X\times Y$, sections of the projections, Ricci-flat metrics on $X$ and $Y$ whose generic fibers are special Lagrangian $n$-tori, and canonically dual generic fibers. Let $\pi_X:U\to{\cal M}_{Ric}(X)$ and $\pi_Y:U\to{\cal M}_{Ric}(Y)$ be the natural maps to the moduli spaces of Ricci-flat metrics, and let the notation $\widetilde U_X$, $\widetilde U_Y$, ${\cal M}^{sc}_{CFT}$, ${\cal M}_A(\sigma)$ and $\overline{{\cal M}}_{cx}$ have the meanings specified in the source. Geometric mirror pair conjecture. The parameter space and data can be chosen so that $(X,Y)$ passes the topological mirror test; both $\pi_X$ and $\pi_Y$ lift to generically finite maps to the indicated semiclassical conformal-field-theory moduli spaces; and, when $h^{2,0}(X)=h^{2,0}(Y)=0$, the indicated partial compactifications and boundary points exist so that $(\widetilde\pi_X)_*(\widetilde\pi_Y)^*$ extends to $\mu^{-1}\times\mu'$, consisting of mirror maps in both directions. In particular, $(X,Y)$ passes the Hodge-theoretic mirror test. This conjecture links the geometric special-Lagrangian formulation of mirror symmetry with the topological and Hodge-theoretic tests, but the source supplies no resolution.