Calabi-Yau Genus-One Fibrations and Twisted Dimensional Reductions of F-theory

In this brief note we explore the space of genus one and elliptic fibrations within CY manifolds, their organizing principles, and how they relate to the set of all CY manifolds. We provide examples of genus one fibered manifolds that exhibit different Hodge numbers -- and physically lead to different gauge groups - than their Jacobian fibrations. We suggest a physical mechanism for understanding this difference in twisted circle reductions of 6-dimensional compactifications of F-theory.Comment: 12 pages, 1 figure. Appeared in the proceedings of the Nankai Symposium on Mathematical Dialogue