F-theory on Genus-One Fibrations

TL;DR

Braun and Morrison extend F-theory to genus-one fibrations without requiring a section and show an F-theory limit exists as the fiber volume vanishes, with the $\tau$-function and $SL(2,\mathbb{Z})$ monodromy encoding IIB data. They link genus-one fibrations to their Jacobians, introduce the Tate-Shafarevich group as a global degree of freedom, and discuss how torsion fluxes yield well-defined M-theory limits and potential frozen singularities. The authors analyze six-dimensional anomaly cancellation, revealing localized neutral hypermultiplets at codimension-two fibers and novel monodromies that realize gauge structures beyond standard split/non-split cases in explicit toric models (e.g., $SU(5)\times U(1)$ and higher abelian factors). They furthermore generalize the Shioda–Tate–Wazir formula to genus-one fibrations with multi-sections, clarifying how Mordell–Weil data and TS twists determine gauge content and moduli, and they connect fiberwise M-theory/IIB duality to TS twists. Overall, the work broadens the landscape of F-theory compactifications and highlights new geometric and physical structures arising from genus-one fibrations without sections.

Abstract

We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various $SU(5)\times U(1)^n$ and $E_6$ models are presented as examples. To each genus-one fibration one can associate a $τ$-function on the base as well as an $SL(2,\mathbb{Z})$ representation which together define the IIB axio-dilaton and 7-brane content of the theory. The set of genus-one fibrations with the same $τ$-function and $SL(2,\mathbb{Z})$ representation, known as the Tate-Shafarevich group, supplies an important degree of freedom in the corresponding F-theory model which has not been studied carefully until now. Six-dimensional anomaly cancellation as well as Witten's zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimension-two fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of non-Abelian gauge groups.

Figures (9)

• Figure 1: The split $SU(5)$ top and associated extended Dynkin diagram, the $I_5$ Kodaira fiber.
• Figure 2: The $U(1)$ charge of a $\mathbf{\underline{5}}$ hypermultiplet localized at a codimension-two fiber where the $I_5$ discriminant (green) degenerates to $I_6$ (red). Pick the red node where the section generating the Mordell-Weil group intersects the fiber. The number next to the node is the ratio of the $U(1)$ charge relative to a $\mathbf{\underline{1}}$ hypermultiplet.
• Figure 3: The $SU(5)\times U(1)^3$ top.
• Figure 4: The $SU(5)$ top without section.
• Figure 5: Left: A localized hypermultiplet charged under a $U(1)$ from the difference of two sections. Right: A global monodromy preventing the $U(1)$ gauge charge, resulting in a localized uncharged hypermultiplet.