Sections, multisections, and U(1) fields in F-theory
David R. Morrison, Washington Taylor
TL;DR
This work provides a geometric framework linking genus-one F-theory fibrations without a global section to the moduli space of Weierstrass models via Jacobian fibrations ${\cal J}^k$, clarifying how abelian ${U(1)}$ gauge sectors arise from extra sections and how they relate to discrete symmetries through partial Higgsing. It shows that ${\cal S}_k \subseteq {\cal J}^k \subseteq {\cal W}$ and that, in 6D, every ${U(1)}$ can be obtained by Higgsing an ${SU(2)}$ with adjoint matter, with a concrete mechanism for enhancing to ${SU(2)}$ and generating a diagonal $\mathbb{Z}_2$ when appropriate. The paper provides explicit 6D examples on several bases (e.g., ${\mathbb{P}}^2$, ${\mathbb{F}}_n$) to illustrate how two-section/bisection geometries realize ${U(1)}$ and ${SU(2)}$ structures, while highlighting cases where $(4,6)$ singularities obstruct conventional unHiggsing. Extending to 4D, the authors discuss how flux can modify the picture, outlining the expected persistence of the core Higgsing–unHiggsing storyline in more general F-theory vacua.
Abstract
We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to abelian U(1) fields, arise as a subspace of the set of genus-one fibrations with multisections. Higgsing of certain matter multiplets charged under abelian gauge fields in the corresponding supergravity theories break the U(1) gauge symmetry to a discrete gauge symmetry group. We further show that in six dimensions every U(1) gauge symmetry arising in an F-theory model can be found by Higgsing an SU(2) gauge symmetry with adjoint matter, and that a similar structure holds for F-theory geometries giving 4D supergravity theories.