Mirror symmetry conjecture for dual reflexive Gorenstein cones

Let $\frac{}{}$ be a pair of $\bar d$-dimensional dual reflexive Gorenstein cones $\frac{}{}$ and $\frac{}{}$ of index $r$. The associated generalized Calabi–Yau manifolds arise as zeros of global sections of the corresponding line bundles on the toric varieties associated with the cones. Mirror symmetry conjecture for dual reflexive Gorenstein cones. Every pair of $\bar d$-dimensional dual reflexive Gorenstein cones $\frac{}{}$ and $\frac{}{}$ of index $r$ gives rise to an $N=2$ superconformal theory with central charge

$$c=3\bigl(\bar d-2(r-1)\bigr).$$

Moreover, the superpotentials of the corresponding Landau–Ginzburg theories define two families of generalized Calabi–Yau manifolds associated with $\frac{}{}$ and $\frac{}{}$, and these families are exchanged by the mirror involution. This is the paper’s main conjecture linking dual reflexive Gorenstein cones, $N=2$ superconformal theories, Landau–Ginzburg models, and mirror symmetry; the source gives no resolution status beyond its formulation as a conjecture.