Discrete Symmetries of Calabi-Yau Hypersurfaces in Toric Four-Folds

We analyze freely-acting discrete symmetries of Calabi-Yau three-folds defined as hypersurfaces in ambient toric four-folds. An algorithm which allows the systematic classification of such symmetries which are linearly realised on the toric ambient space is devised. This algorithm is applied to all Calabi-Yau manifolds with $h^{1,1}(X)\leq 3$ obtained by triangulation from the Kreuzer-Skarke list, a list of some $350$ manifolds. All previously known freely-acting symmetries on these manifolds are correctly reproduced and we find five manifolds with freely-acting symmetries. These include a single new example, a manifold with a $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry where only one of the $\mathbb{Z}_2$ factors was previously known. In addition, a new freely-acting $\mathbb{Z}_2$ symmetry is constructed for a manifold with $h^{1,1}(X)=6$. While our results show that there are more freely-acting symmetries within the Kreuzer-Skarke set than previously known, it appears that such symmetries are relatively rare.