On genus one fibered Calabi-Yau threefolds with 5-sections

Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged linear sigma models (GLSMs) with non-Abelian gauge groups, where the dual geometries arise as phases of the same GLSM. Using the modular bootstrap approach of topological string theory, one can compute all-genus Gopakumar-Vafa invariants of these Calabi-Yaus. We observe that homological projective duality acts as an element of $\Gamma_0(5)$ on the topological string partition function and the partition functions of dual geometries transform into each other. Moreover, we study the geometries from an M-/F-theory perspective. We compute the F-theory spectrum and show how the genus one-fibered Calabi-Yaus are connected to certain Calabi-Yaus in toric varieties via a series of Higgs transitions. Based on the F-theory physics, we conjecture that dual geometries are elements of the same Tate-Shafarevich group. Our analysis also leads to a classification of 5-section geometries, as well as the construction of F-theory models with charge 5 hypermultiplets.