F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections

TL;DR

The paper constructs F-theory vacua with a two-factor Abelian gauge sector by modeling the elliptic fiber with rank-two Mordell–Weil group as the Calabi–Yau onefold in $dP_2$, deriving explicit birational maps to Tate and Weierstrass forms and identifying the two rational sections that give $U(1)\times U(1)$. It develops a global classification of elliptic fibrations over bases, focusing on $B=\mathbb{P}^2$, and computes the full six-dimensional matter spectrum, including explicit multiplicities governed by intersections $n_2$ and $n_{12}$, with all results consistent with anomaly cancellation. The work provides three concrete toric Calabi–Yau models (two with $U(1)\times U(1)$ and one with $SU(5)\times U(1)^2$), and demonstrates a birational equivalence between different fiber realizations via extremal transitions, highlighting the necessity of non-holomorphic zero sections in higher-rank Abelian sectors. The methods generalize to higher-rank Mordell–Weil groups and offer a pathway toward systematic Abelian sector classifications in global F-theory compactifications, including potential extensions to Calabi–Yau fourfolds and $G_4$-flux analyses.

Abstract

We study F-theory compactifications with U(1)xU(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B=P^2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)xU(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P^2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)xU(1) and SU(5)xU(1)xU(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl_(1,0,0)P^2(1,2,3) and Bl_(0,1,0)P^2(1,1,2) employing birational maps and extremal transitions.

Figures (6)

• Figure 1: Fan of $\mathbb{P}^2$ on the left and its dual polytope on the right.
• Figure 2: Fan of $dP_2$ on the left and its dual polytope on the right.
• Figure 3: On the left it is shown a regular fiber with rational sections at generic points. In the center, a singular fiber is displayed with a section crossing the singularity where charged matter sits. On the right, the curve is shown after resolution. The isolated curve gives rise to an hypermultiplet charged under the corresponding Abelian gauge field.
• Figure 4: Fibers where two out of the three sections collide. The intersections occur when $s_7$ or/and $s_9$ vanish on the base $B$.
• Figure 5: How the fiber degenerates at the three loci (-1,1), (0,2) and (-1,-2) respectively.