F-theory on all toric hypersurface fibrations and its Higgs branches

Abstract

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in P 2 , P 1 × P 1 and the recently studied P 2 (1, 1, 2), yield F-theory realizations of SUGRA theories with discrete gauge groups Z 3 , Z 2 and Z 4 . This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I 2 -fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP 1 has a U(1)-gauge field JHEP01 (2015)142 induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3.

Global Tensor‐Matter Transitions in F‐Theory

“…For a more detailed discussion we refer to [23,53] and Appendix A. In toric hypersurface constructions one embeds a torus fiber in one of the 16 toric ambient spaces given e.g.…”

“…In general, these equations simplify to a single polynomial equation depending on the sectionsŝ a [53],…”

“…V (I (4) ) p.48 in [53] (0, 0, 2) Table 2: Base-independent matter spectrum for the upper polytope in Figure 6. point of a face and a (4, 6, 12) singularity at d 6 = z 0 = 0 is created with multiplicity…”

Tall sections from non-minimal transformations

“…The height of the generating section S has the physical interpretation as the corresponding u(1) anomaly coefficient when B is a surface [6,23]. 5 The anomaly cancellation conditions then imply that…”

“…In [4], the authors study a family of F-theory vacua, first discovered in [5], whose gauge algebra is given by u(1) and has matter with charge three-times of the minimal charge.…”