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)andle;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 and#8484;2×and#8484;2 symmetry where only one of the and#8484;2 factors was previously known. In addition, a new freely-acting and#8484;2 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.