Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) x U(1) x U(1) Gauge Symmetry

TL;DR

The authors construct global F-theory vacua with gauge symmetry $U(1)\times U(1)\times U(1)$ by building elliptically fibered Calabi–Yau manifolds whose Mordell–Weil group has rank three. The generic elliptic fiber is realized as a complete intersection in $\mathbb{P}^3$, resolved by embedding into $\mathrm{Bl}_3\,\mathbb{P}^3$, and the set of all such fibrations over a base $B$ is finite and classified by reflexive polytopes from nef-partitions. A thorough codimension-two singularity analysis yields 14 massless matter representations in 6D and 4D, including a novel tri-fundamental representation under all three $U(1)$ factors, with explicit six-dimensional multiplicities that satisfy anomaly cancellation. The results reveal universal structures of codimension-two singularities for higher rank Mordell–Weil groups and pave the way for further studies of non-Abelian enhancements and chiral 4D constructions in this rich Abelian sector of F-theory.

Abstract

We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abelian gauge symmetry by constructing the general elliptically fibered Calabi-Yau manifolds with a rank three Mordell-Weil group of rational sections. The general elliptic fiber is shown to be a complete intersection of two non-generic quadrics in P^3 and resolved elliptic fibrations are obtained by embedding the fiber as the generic Calabi-Yau complete intersection into Bl_3 P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitely many Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on these Calabi-Yau manifolds are shown to be labeled by integral points in reflexive polytopes constructed from the nef-partition of Bl_3 P^3. We determine all 14 massless matter representations to six and four dimensions by an explicit study of the codimension two singularities of the elliptic fibration. We obtain three matter representations charged under all three U(1)-factors, most notably a tri-fundamental representation. The existence of these representations, which are not present in generic perturbative Type II compactifications, signifies an intriguing universal structure of codimension two singularities of the elliptic fibrations with higher rank Mordell-Weil groups. We also compute explicitly the corresponding 14 multiplicities of massless hypermultiplets of a six-dimensional F-theory compactification for a general base B.

Figures (1)

• Figure 1: Toric fan of $\text{Bl}_3\mathbb{P}^3$ and the 2D projections to the three coordinate planes, each of which yielding the polytope of $dP_2$.