When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion

TL;DR

The work develops a systematic framework to realize non-simply-connected gauge groups in F-theory by tuning Mordell–Weil torsion, focusing on $\oldsymbol{\mathbb{Z}_2}$ torsion in rank-one MW models and connecting to genus-one fibrations via bisections and Jacobians. It provides an explicit generic Weierstrass realization whose Calabi–Yau compactification yields $G = \frac{SU(2) \times SU(4)}{\mathbb{Z}_2} \times SU(2)$, and a specialized toric resolution demonstrating a torsional homology relation that enforces $SU(2)/\mathbb{Z}_2$ for one factor. The analysis extends to a Jacobian fibration $J(Y_b)$ with a $\,\mathbb{Z}_2$ torsional section, producing a $\frac{SU(2)}{\mathbb{Z}_2} \times \mathbb{Z}_2$ theory and revealing subtle issues in interpreting F-theory on genus-one fibrations. Collectively, the results illuminate how Mordell–Weil torsion reshapes global gauge structures, provide anomaly-consistent spectra, and raise questions about dualities and multi-section formulations in F-theory.

Abstract

We explore novel gauge enhancements from abelian to non-simply-connected gauge groups in F-theory. To this end we consider complex structure deformations of elliptic fibrations with a Mordell--Weil group of rank one and identify the conditions under which the generating section becomes torsional. For the specific case of Z2 torsion we construct the generic solution to these conditions and show that the associated F-theory compactification exhibits the global gauge group [SU(2) x SU(4)]/Z2 x SU(2). The subsolution with gauge group SU(2)/Z2 x SU(2), for which we provide a global resolution, is related by a further complex structure deformation to a genus-one fibration with a bisection whose Jacobian has a Z2 torsional section. While an analysis of the spectrum on the Jacobian fibration reveals an SU(2)/Z2 x Z2 gauge theory, reproducing this result from the bisection geometry raises some conceptual puzzles about F-theory on genus-one fibrations.

Figures (4)

• Figure 1: On the left: The toric polygon, referred to as $F_6$ in Klevers:2014bqa, of the fiber ambient space ${\rm Bl}_1 {\mathbb{P}}_{112}$ of the blown-up $U(1)$ model \ref{['eq:MP-fibration']}. On the right: The dual polygon, giving rise to the resolved hypersurface equation \ref{['eq:MP_hypersurface_s_resolved']}. As pointed out in Morrison:2012ei, by assuming a unit coefficient in front of the term $s\,w^2$, the two red monomials can be absorbed by a shift of $w$ by a multiple of $u$.
• Figure 2: On the left: The toric polygon, called $F_8$ in Klevers:2014bqa, of the $\gamma$-blow-up of ${\rm Bl}_1 {\mathbb{P}}_{112}$. On the right: The dual polygon, giving rise to the hypersurface equation for $\hat{Y}_{F_8}$. The $\gamma$-blow-up removes the vertex of the dual polygon corresponding to the $c_3$-term of the $U(1)$ model \ref{['eq:MP_hypersurface_s_resolved']}. As discussed in Klevers:2014bqa, this elliptic fibration has an $I_2$ locus above $b=0$. Were it not for a unit coefficient in front of $\gamma^2\,s\,w^2$ (which allows us to absorb the red terms), there would be another $I_2$ locus present. The non-toric tuning $c_1 \rightarrow 0$ enhances the $U(1)$ "torsionally".
• Figure 3: The structure of the codimension two singular fiber over the locus $b = c_2 = 0$, demonstrating the intersection pattern of the curves in (\ref{['eq:restricted_example_codim_2_locus_splitting']}); the numbers on each node indicate the multiplicity. It is a non-Kodaira singular fiber which is a contraction of the $I_0^*$ Kodaira fiber where one multiplicity one node is removed.
• Figure 4: Geometric construction of the Mordell--Weil group law. Each dashed line marks three points on the elliptic curve (solid curve) that add up to zero under the group law. The rational points $A,B,C$ satisfy $A \boxplus B = C$.