Codimension-3 singularities and Yukawa couplings in F-theory

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a splitÃ 4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) grand unified theories in F-theory. The desingularization is given by a small resolution of singularities. TheÃ 4 fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions of the elliptically fibered four-fold. These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the three-folds that defines the type IIB orientifold limit of the F-theory. The full resolution has interesting properties, specially for fibers in codimension-three: the rank of the singular e-print archive: http://lanl.arXiv.org/abs/1107.0733v2 MBOYO ESOLE AND SHING-TUNG YAUfiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagrams.

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“…It is conjectured that the I 5 fiber enhances to a I 6 fiber over Σ 5 and to a I * 1 fiber over Σ 10 (see for example [22,29]):…”

Section: Conjectured Fiber Geometrymentioning

confidence: 99%

“…The curve Σ 5 contains additional points Π 7 : w = P = R = 0 over which the singular fiber I 6 is conjectured to enhance further to a fiber of type I 7 [22,29]:…”

Section: Conjectured Fiber Geometrymentioning

confidence: 99%

Small resolutions of SU(5)-models in F-theory

Esole

Yau

2013

Advances in Theoretical and Mathematical Physics

125

311

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“…The exceptional sections are 16) and the intersection of the associated Cartan divisors is as follows…”

Section: Ementioning

confidence: 99%

The Tate form on steroids: resolution and higher codimension fibers

Lawrie

Schäfer‐Nameki

2013

J. High Energ. Phys.

147

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F-theory on singular elliptically fibered Calabi-Yau four-folds provides a setting to geometrically study four-dimensional N = 1 supersymmetric gauge theories, including matter and Yukawa couplings. The gauge degrees of freedom arise from the codimension 1 singular loci, the matter and Yukawa couplings are generated at enhanced singularities in higher codimension. We construct the resolution of the singular Tate form for an elliptic Calabi-Yau four-fold with an ADE type singularity in codimension 1 and study the structure of the fibers in codimension 2 and 3. We determine the fibers in higher codimension which in general are of Kodaira type along minimal singular loci, and are thus consistent with the low energy gauge-theoretic intuition. Furthermore, we provide a complementary description of the fibers in higher codimension, which will also be applicable to non-minimal singularities. The irreducible components in the fiber in codimension 2 correspond to weights of representations of the ADE gauge group. These can split further in codimension 3 in a way that is consistent with the generation of Yukawa couplings. Applying this reasoning, we then venture out to study non-minimal singularities, which occur for A type along codimension 3, and for D and E also in codimension 2. The fibers in this case are non-Kodaira, however some insight into these singularities can be gained by considering the splitting of fiber components along higher codimension, which are shown to be consistent with matter and Yukawa couplings for the corresponding gauge groups.

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“…Recently, F-theory has been used to obtain local models of Grand Unified Theories (GUTs), starting from the papers [30][31][32][33]. Global completions of GUT models have also been intensively studied [35][36][37][38][39][40][41][42][43], with special focus on SU(5) configurations.…”

Section: Absractmentioning

confidence: 99%

Tate form and weak coupling limits in F-theory

Esole

Savelli

2013

J. High Energ. Phys.

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AbsractWe consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several non-equivalent directions to flow from strong to weak coupling leading to different gauge groups.

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Codimension-3 singularities and Yukawa couplings in F-theory

Cited by 160 publications

References 69 publications

Small resolutions of SU(5)-models in F-theory

Small resolutions of SU(5)-models in F-theory

The Tate form on steroids: resolution and higher codimension fibers

Tate form and weak coupling limits in F-theory

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