Partition functions of $\mathcal{N}=(2,2)$ supersymmetric sigma models and special geometry on the moduli spaces of Calabi–Yau manifolds

We study a new example of a mirror relation between the exact partition functions of $\mathcal{N}{=}(2,2)$ supersymmetric gauged linear sigma models on the sphere $S^2$ and the special Kähler geometry on the moduli spaces of Calabi–Yau manifolds. Using exact calculations, we show this relation indeed holds for Calabi–Yau manifolds of the Berglund–Hubsch type with two moduli.