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 that 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 h1,1(X)≤ 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 Z₂₂ symmetry where only one of the Z₂ factors was previously known. In addition, a new freely-acting Z₂ symmetry is constructed for a manifold with h1,1(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.