Weierstrass models of elliptic toric $K3$ hypersurfaces and symplectic cuts

Pena²,

Oehlmann³

et al. 2015

140

570

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.

Section: Sectionmentioning

confidence: 99%

Section: Fibration By the Biquadric: X Fmentioning

confidence: 99%

“…, 16, in a lattice N = Z 2 . 16 These 16 reflexive polyhedra are given in a convenient presentation in figure 2 [17]. As indicated there, the polyhedra F i and F 17−i for i = 1, .…”

Section: Explicit Examples: Calabi-yau Hypersurfaces In 2d Toric Varimentioning

confidence: 99%

“…The multiplicities of the charged hyper multiplets are presented in table 17. These have been computed following section 2.3.…”

Section: Jhep01(2015)142mentioning

confidence: 99%

Section: Jhep01(2015)142mentioning

confidence: 99%

See 3 more Smart Citations

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

Pena²,

Oehlmann³

et al. 2015

140

570

Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1)×U(1)×U(1) gauge symmetry

Cvetič

Piragua

et al. 2014

106

We analyze general F-theory compactifications with U(1)×U(1)×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 CalabiYau 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 nefpartition 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 MordellWeil groups. We also compute explicitly the corresponding 14 multiplicities of massless hypermultiplets of a six-dimensional F-theory compactification for a general base B.

Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors

Cvetič

Grassi

et al. 2014

Self Cite

112

300

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP 2 . We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)×U(1) and SU(5)×U(1)×U(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B = P 3 . We determine the general G 4 -flux that belongs to H (2,2) V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G 4 -flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New ChernSimons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.