  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.
