# Publications associated with Particle Theory

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

Communications in Mathematical Physics (2017) 1-50

© 2017 The Author(s) 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 (Formula presented.) 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 (Formula presented.) symmetry where only one of the (Formula presented.) factors was previously known. In addition, a new freely-acting (Formula presented.) symmetry is constructed for a manifold with (Formula presented.). 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.

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