Multiple mirrors and the JKLMR conjecture

We address the problem of the fulfillment of the conjecture proposed by Jockers et al. (JKLMR conjecture) on the equality of the partition function of a supersymmetric gauged linear sigma model on the sphere $S^2$ and the exponential of the Kähler potential on the moduli space of Calabi–Yau manifolds. The problem is considered for a specific class of Calabi–Yau manifolds that does not belong to the Fermat type class. We show that the JKLMR conjecture holds when a Calabi–Yau manifold $X(1)$ of such type has a mirror partner $Y(1)$ in a weighted projective space that also admits a Calabi–Yau manifold of Fermat type $Y(2)$. Moreover, the mirror $X(2)$ for $Y(2)$ has the same special geometry on the moduli space of complex structures as the original $X(1)$.