F-Theory Compactifications with Multiple U(1)-Factors: Addendum

TL;DR

This addendum generalizes six-dimensional F-theory compactifications with a rank-two Mordell–Weil group to arbitrary bases $B$, yielding $U(1)\times U(1)$ gauge symmetry with base-independent matter representations but base-dependent multiplicities. It then constructs a non-generic fibration realizing $SU(5)\times U(1)\times U(1)$ over any base and computes the full spectrum, including singlets and non-Abelian matter, with multiplicities expressed via base intersection data and a genus-dependent adjoint contribution. Across both setups, the authors verify complete anomaly cancellation by evaluating the six-dimensional anomaly equations using the Shioda map and base divisor data. The results provide a general framework for consistent 6D F-theory models with multiple abelian factors and non-Abelian sectors on broad classes of bases, facilitating model-building with flexible geometric settings.

Abstract

The purpose of this note is to extend the results obtained in [arXiv:1303.6970] in two ways. First, the six-dimensional F-theory compactifications with U(1) x U(1) gauge symmetry on elliptic Calabi-Yau threefolds, constructed as a hypersurface in $dP_2$ fibered over the base $B=\mathbb{P}^2$ [arXiv:1303.6970], are generalized to Calabi-Yau threefolds elliptically fibered over an arbitrary two-dimensional base B. While the representations of the matter hypermultiplets remain unchanged, their multiplicities are calculated for an arbitrary B. Second, for a specific non-generic subset of such Calabi-Yau threefolds we engineer SU(5) x U(1) x U(1) gauge symmetry. We summarize the hypermultiplet matter representations, which remain the same as for the choice of the base $B=\mathbb{P}^2$ [arXiv:1306.3987], and determine their multiplicities for an arbitrary B. We also verify that the obtained spectra cancel anomalies both for U(1) x U(1) and SU(5) x U(1) x U(1).

Figures (1)

• Figure 1: Left: $I_4$ singularity over a generic point of the divisor $z=0$, marked according to the interaction of the nodes with $\{S_P,S_Q,S_R\}$. Center: $A_5$ singularity with local SU(6) enhancement at $z=P_1=0$. Right: $D_5$ singularity with local SO(10) enhancement at $z=\beta_5=0$. The dashed lines indicate nodes of the Dynkin diagram of SU(5) that split at a codimension two locus under consideration.