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**11 - 18**of**18**### and its Higgs Branches

, 2014

"... 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 fibrati ..."

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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 corre-spond 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 sec-tion, whose fibers are given by the toric hypersurfaces in P2, P1 × P1 and the recently studied P2(1, 1, 2), yield F-theory realizations of SUGRA theories with discrete gauge groups Z3, Z2 and Z4. 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 I2-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 dP1 has a U(1)-gauge field 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.

### Prepared for submission to JHEP MIT-CTP-4535 Calabi-Yau threefolds with large h2,1

"... Abstract: We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section (“EFS”) and have a large Hodge number h2,1. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topolo-gies connected throug ..."

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Abstract: We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section (“EFS”) and have a large Hodge number h2,1. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topolo-gies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h2,1 ≥ 350 by tuning co-efficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with h2,1 ≥ 350, as well as three appar-ently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach. ar X iv

### A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations.

"... We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q7(L,S) model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantiti ..."

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We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q7(L,S) model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations on the curve can be done using the chord-tangent group law. We will use this model to determine the spectrum of singular fibers of an elliptic fibration of rank one and compute a generating function for its Euler characteristic. With a view toward string theory, we determine a semi-stable degeneration which is understood as a weak coupling limit in F-theory. We show that it satisfies a non-trivial topological relation at the level of homological Chern classes. This identity ensures that the D3 charge in F-theory is the same as the one in the weak coupling limit. Email: esole at math.harvard.edu, jkang at physics.harvard.edu, yau at math.harvard.edu ar X iv

### Prepared for submission to JHEP UCSB Math 2014-12, MIT-CTP-4540 Sections, multisections, and U(1) fields in F-theory

"... Abstract: We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to abelian U(1) fields, arise as a subspace of the set of g ..."

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Abstract: We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to abelian U(1) fields, arise as a subspace of the set of genus-one fibrations with multisections. Higgsing of certain matter multiplets charged under abelian gauge fields in the corresponding supergravity theories break the U(1) gauge symmetry to a discrete gauge symmetry group. We further show that in six dimensions every U(1) gauge symmetry arising in an F-theory model can be found by Higgsing an SU(2) gauge symmetry with adjoint matter, and that a similar structure holds for F-theory geometries giving 4D supergravity theories.ar X iv

### Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory

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### Towards the Standard Model in F-theory

"... This article explores possible embeddings of the Standard Model gauge group and its mat-ter representations into F-theory. To this end we construct elliptic fibrations with gauge group SU(3) × SU(2) × U(1) × U(1) as suitable restrictions of a Bl2P2-fibration with rank-two Mordell-Weil group. We anal ..."

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This article explores possible embeddings of the Standard Model gauge group and its mat-ter representations into F-theory. To this end we construct elliptic fibrations with gauge group SU(3) × SU(2) × U(1) × U(1) as suitable restrictions of a Bl2P2-fibration with rank-two Mordell-Weil group. We analyse the five inequivalent toric enhancements to gauge group SU(3) × SU(2) along two independent divisors W3 and W2 in the base. For each of the resulting smooth fibrations, the representation spectrum generically consists of a bifundamental (3,2), three types of (1,2) representations and five types of (3,1) representations (plus conjugates), in addition to charged singlet states. The precise spectrum of zero-modes in these representations depends on the 3-form background. We analyse the geometrically realised Yukawa couplings among all these states and find complete agreement with field theoretic expectations based on their U(1) charges. We classify possible identifications of the found representations with the Standard Model field content exten-ded by right-handed neutrinos and extra singlets. The linear combination of the two abelian gauge group factors orthogonal to hypercharge acts as a selection rule which, depending on the specific model, can forbid dangerous dimension-four and-five proton decay operators. ar X iv

### Calabi-Yau threefolds over semi-toric base surfaces

"... 6D F-theory models and elliptically fibered ..."